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R.A. Bagnold (1980) ACKNOWLEDGMENTS I would like to extend my gratitude first to my fellow officemates for their support, encouragement, assistance, and friendship. Robert, Jenn, Allison, Ty, and Jessica, thanks for alv, serving as a sounding board for my ideas, as well as my complaints. Second, I am grateful for the constructive criticism of my supervisory committee members, those that I have often sought out for advice, and particularly for the guidance and encouragement of my advisor, Don Slinn. Mutlu Sumer is deserving of special acknowledgment for the boundary l~ r data used in the modeldata comparison. Last, I owe a great debt of gratitude to my family, especially my wife, Shannon, as none of this would have been possible without their love and constant support. TABLE OF CONTENTS ACKNOW LEDGMENTS ................................. 4 LIST OF TABLES ....................... ............. 8 LIST OF FIGURES .................................... 9 LIST OF SYMBOLS ....................... ............ 14 A BSTRA CT . . . . . . . . . . 19 CHAPTER 1 INTRODUCTION ...................... .......... 20 1.1 Background ...................... ........... 20 1.2 Motivation ...................... ........... 23 1.3 Approach ....................... ............ 26 1.4 O utline. . . . . . . ..... . 28 2 SEDIMENT TRANSPORT PROCESSES ........ .......... .... 30 2.1 Introduction ...................... ............ 30 2.2 Governing Hydrodynamics ................... ..... 30 2.2.1 W aves ...................... .......... 30 2.2.2 Currents ...................... .......... 33 2.2.3 Waves and Currents ................... ..... 34 2.2.4 Tides and Tidal Currents .................. ... 35 2.3 Transport .. .. .. ... .. .. .. .. ... .. .. .. .. ...... 36 2.3.1 M odes . . . . . . . .... 36 2.3.1.1 Bedload . . . . . . .. .. 37 2.3.1.2 Suspended Load .................. .. 40 2.3.2 Regim es . . . . . . . .... 43 2.4 Turbulence .. .. .. ... .. .. .. .. .. .. .. .. .. ...... 46 2.4.1 Dynam ics .................. ............. 47 2.4.2 Closure ................... . . ... 48 2.4.2.1 RANS M odels .................. .... 48 2.4.2.2 Large Eddy Simulations .................. 49 2.4.2.3 Direct Numerical Simulations ............... 49 2.5 M odels . . . . . . . . .. .. 50 2.5.1 Types. .................. ............... 50 2.5.1.1 TimeAveraged .................. ... 50 2.5.1.2 QuasiSteady ............... .... .51 2.5.1.3 SemiUnsteady ............... .. .. 52 2.5.1.4 Unsteady ............... ...... .. 53 2.5.2 Integrated Approaches ............... ... .54 2.5.3 Shortcom ings ... .. .. .. .. ... .. .. .. ...... 56 3 BEDFORM DYNAMICS ............ 3.1 Introduction .............................. 59 3.2 Bedform Types .................. .............. .. 59 3.2.1 Rolling Grain Ripples .................. ..... .. 61 3.2.2 Vortex Ripples .................. . .... 63 3.2.3 ('!C i 'teristics . . . . .. . .. 66 3.3 Experiments .................. ................ .. 69 3.3.1 Laboratory .................. ............. .. 69 3.3.1.1 Oscillating Tray .................. .... .. 70 3.3.1.2 Oscillating Water Tunnel .................. .. 70 3.3.1.3 Flume .................. .......... .. 71 3.3.2 Field . . . . . . .. . ... 73 3.4 Ripple Predictors .................. ............. .. 74 3.4.1 Clifton (1976) .................. ........... .. 74 3.4.2 Nielsen (1981) .................. ........... .. 74 3.4.3 Grant and Madsen (1982) ................ ... .. 76 3.4.4 Wiberg and Harris (1994) .................. .... .. 77 3.4.5 Mogridge et al. (1994) .................. ..... .. 78 3.4.6 Khelifa and Ouellet (2000) ................ .... .. 79 3.4.7 Faraci and Foti (2002) ............ . ..... 80 3.4.8 W illiams et al. (2005) .................. ..... .. 81 3.4.9 Soulsby and Whitehouse (2006) ............. .. .. 81 3.5 Models ................. ................ ...... 82 4 MODEL DESCRIPTION ................... . 88 4.1 Introduction .................. . . .... 88 4.2 OneDimensional Linear Model .................. .... .. 88 4.2.1 Hydrodynamics .................. .......... .. 89 4.2.1.1 Governing Equations ................... ... .. 89 4.2.1.2 Num erics .................. ...... .. .. 92 4.2.2 Sediment Transport ................... . ... 93 4.2.3 Morphology .............. . . . .... 93 4.3 Two and ThreeDimensional Nonlinear Models ...... . . 93 4.3.1 Hydrodynamics .................. .......... .. 95 4.3.1.1 Governing Equations ............. .. .. .. 95 4.3.1.2 Turbulence Closure ................ .. 98 4.3.1.3 Numerics ... ............ ....... ..104 4.3.1.4 Boundary Conditions ............... 109 4.3.2 Sediment Transport .................. ........ .. 111 4.3.2.1 Bedload .................. ....... ..1 111 4.3.2.2 Suspended Load ................ .... .. 117 4.3.3 Morphology ............... . . ..... 123 4.3.3.1 FiniteDifference Methods ................. 125 4.3.3.2 Filtering Techniques ............... . 131 5 MODEL EXPERIMENTS .................. 5.1 Linear Model Experiments ........................... 146 5.2 Nonlinear Model Experiments .................. .... 147 5.2.1 Phase I: Model Validation ................. .... 147 5.2.2 Phase II: Model Capabilities ................. .... 149 5.2.3 Phase III: Sediment Transport Processes . . 150 6 RESULTS ................... ............ ...... 159 6.1 Linear Model Results .................. ........... 159 6.2 Nonlinear Model Results .................. ......... 160 6.2.1 Phase I: Model Validation ................. .... 160 6.2.1.1 Hydrodynamic Validation ................. .. 161 6.2.1.2 Computational Grid Tests ..... . . ..... 163 6.2.1.3 Model Domain Width Tests ................ 165 6.2.1.4 Sediment Transport Submodel Tests . . 170 6.2.1.5 Morphology Tests ................ .... .. 173 6.2.2 Phase II: Model Capabilities ................... .... .. 183 6.2.2.1 Steady Flow .................. .. ..... 184 6.2.2.2 Subcritical Flow ................. .... 185 6.2.2.3 Sheetflow Regime ................ .. 185 6.2.2.4 Ripple Growth, Equilibrium, and Decay . .... 187 6.2.2.5 Sediment Size Tests ................. . .. 193 6.2.3 Phase III: Sediment Transport Processes . . 197 6.2.3.1 Total Load Growth ............. .... . 198 6.2.3.2 Total Load Decay ................ .... .. 203 6.2.3.3 Bedload Only .................. ..... 205 6.2.3.4 Suspended Load Only ............... .. 206 7 CONCLUSIONS ................... . . ..... 274 7.1 LiveBed Model Evaluation .................. ..... .. 274 7.2 Sediment Transport and Bedform Dynamics ................ ..278 7.3 Future Applications .................. .......... .. 281 APPENDIX A EXPERIMENT MATRIX .................. ......... ..283 B FLUX TENDENCY PHASE DIAGRAM ................. .. ..285 REFERENCES .................. ................ .. .. 289 BIOGRAPHICAL SKETCH .................. ............. ..304 LIST OF TABLES Tabl 21 22 31 32 33 34 51 52 53 54 55 56 57 58 61 62 63 A Power law formulations for bedload transport . ..... Empirical pickup functions for suspended load transport . .. List of common values for orbital ripple length.. . . Summary of field and laboratory ripple experiments . . Parameters related to bedform dynamics. . ..... Ripple predictor key ...... . . Mesh parameters for grid comparisons..... . ..... Mesh parameters for domain width comparisons . . Abbreviation key for transport equation tests . ..... Morphology test simulation matrix....... . ..... Ripple predictor citation key for the morphology test simulations . Expected morphology results based on ripple predictors . . Phase II experiment matrix ................. . ..... Simulation parameters for the Phase III experiments . . Ripple predictor RMS error for morphology tests under labscale flows Ripple predictor RMS error for morphology tests under fieldscale flows Ripple predictor model performance index. . ..... Complete experimental matrix....... . ...... e Page . 58 . 58 . 85 . 86 . 87 . .. 87 . 155 . 155 . 155 . 156 . 156 . 157 . 158 . 158 . 272 . 272 . 273 . 284 LIST OF FIGURES Figure Page 11 Pictures of sand ripples in lab and field settings ................. .. 29 21 Velocity and stress time series for linear and nonlinear waves. . ... 57 22 Distribution of fluid and grain shear stress. .................. .... 57 31 Schematic of vortex shedding over a rippled bed ................. ..84 32 Effects of bedforms on boundary 1, r thickness ................. ..85 41 Schematic of onedimensional bedload model ........ . 135 42 Schematic of the threedimensional modeling domain ............. 135 43 Model control volumes with variable locations .............. 136 44 Schematic of variable mesh scaling ................ ...... 136 45 Mesh clustering and variable scaling ................ ..... 137 46 A comparison of bed shear stress formulations .... . ... 138 47 Estimation of critical Shields' curve by Brownlie (1981) . . ..... 139 48 Estimation of critical Shields' curve by van Rijn (1993) ........... .139 49 Particle forces acting on longitudinal and transverse slopes . . ... 140 410 Gravitational forces on a sphere .................. ........ .. 141 411 Behavior of bedload transport equations .................. ..... 141 412 Behavior of sediment pickup functions .................. ...... 142 413 Modification of relative settling velocity by concentration . . ..... 142 414 Sediment control volume and transport schematic ................ ..143 415 Amplification factors for the twodimensional lowpass morphology filter . 144 416 Amplification factors for the hybrid morphology filter .............. ..145 51 Horizontal and vertical grid spacing for grid comparisons ............ ..152 52 Model domains used in the domain width comparison tests . . .... 153 53 Schematic of the model domain used for the Phase III simulations . ... 154 61 Timestacks of bed elevation for the 1DH linear model experiment . ... 208 62 Time evolution of ripple height, wavelength, and steepness for the 1DH linear model experiment . . . . . .. . 209 3 Modeldata hydrodynamic comparison... . ...... 4 Modeldata comparison of boundary 1,r thickness . . 5 Assessment of hydrodynamic model error . ..... 6 Computational budget for grid resolution tests.. . . 7 Velocity standard deviations for grid comparisons . . 8 Cumulative average turbulent kinetic energy (TKE) for grid comparisons 9 Phase and volumeaveraged TKE for grid comparisons . . 10 Phase and yaveraged uvelocity profiles for grid comparisons . . 11 Phase and yaveraged vvelocity profiles for grid comparisons . . 12 Phase and yaveraged wvelocity profiles for grid comparisons . . 13 Computational budget for threedimensional grid tests . ..... 614 Cumulative average turbulent kinetic energy (TKE) for threedimensional grid com parsons . . . . . . . . . . 615 Phase and volumeaveraged TKE for threedimensional grid comparisons . 616 Phase and volumeaveraged velocity components for grid comparisons ..... 617 Average Fourier transforms of vvelocity for grid comparisons .. ........ 618 Spatiallyaveraged standard deviation of bed elevation in the third dimension . 619 Comparisons of equilibrium ripple height for grid comparisons .. ........ 620 Average Fourier transforms of bed elevation for grid comparisons ... 621 Effects of sediment transport submodels on equilibrium ripple characteristics . 622 Conservation of sediment mass ............. 623 Bedform statistics in labscale flows ............. 624 Fourier analysis timestack of bedform statistics for HL1v1 .. ......... 625 Ripple predictor comparisons for labscale morphology tests .. ......... 626 Bedform statistics in fieldscale flows .. .................... 627 Fourier analysis timestack of bedform statistics for MF2v2 and HF1v1 .. . 210 . 211 . 212 . 213 . 214 . 215 . 215 . 216 . 217 . 218 . 219 628 Ripple predictor comparisons for fieldscale morphology tests . . ... 232 629 Comparison of model results to the ripple predictor equations of Faraci and Foti (2002) . . . . . . .. . ... ... 233 630 Comparison of model results to the ripple predictor equations of Nielsen (1981), Khelifa and Ouellet (2000), and Williams et al. (2005) ........... .234 631 Comparison of model results to the ripple predictor equations of Mogridge et al. (1994) . . . . . . .. . ... ... 235 632 Bed elevation timestack for steady flow test ................ 236 633 Steady flow bed shear stress and sediment transport ... . . 237 634 Bed elevation timestack for subcritical flow test ................. ..238 635 Average bed shear stress and bedload transport in subcritical flow . ... 238 636 Bed elevation timestack for sheetflow sediment transport ............ ..239 637 Timeseries of bedform statistics in the sheetflow regime ............ ..239 638 Energy density timestack for the sheetflow test .................. ..240 639 Isocontours of sediment concentration in the sheetflow regime . ... 240 640 Average sediment concentration profile in the sheetflow regime . ... 241 641 Bed elevation timestacks for ripple growth, equilibrium, and decay . ... 242 642 Timestack of ripple spectra during growth from flat bed ............ ..243 643 Ripple statistics for growth, equilibrium, and decay ............... ..244 644 Time evolution of ripple heights during growth, equilibrium, and decay . 245 645 Bed elevation timestacks for ripple coarsening and bifurcation . ... 246 646 Timestacks of ripple spectra during ripple wavelength saturation . ... 247 647 Ripple statistics during height and wavelength evolution ............ ..248 648 Effects of hindered settling on model predictions of ripple height and length 249 649 Modification of average concentration profile due to hindered settling . 250 650 Timeseries of bedform statistics for fine gravel in subcritical flow . ... 250 651 Phaseaveraged stress and transport for fine gravel in subcritical flow . 251 652 Timestacks of bed elevation for fine gravel in a strong flow . .... 251 653 Bedform statistics for fine gravel in a strong flow ................ ..252 654 Phaseaveraged stress and transport for fine gravel in a strong flow ...... ..252 655 Timestacks of bed elevation for medium silt in a weak flow . ..... 253 656 Time evolution of ripple height and length for medium silt in a weak flow . 253 657 Phaseaveraged stress, transport, and sediment concentration for medium silt in a weak flow ............... ................ .. 254 658 Timestacks of bed elevation for medium silt in a strong, highlyconcentrated flow 255 659 Time evolution of bedform height and length for medium silt in a strong flow 255 660 Phaseaveraged stress, transport, and sediment concentration for medium silt in a strong, highlyconcentrated flow .................. ...... 256 661 Suspended sediment phase diagram .................. ..... 257 662 Time evolution of ripple height, wavelength, and steepness during ripple growth and equilibration (TL2v2) .................. ........... .. 258 663 Timestacks of bed elevation during ripple growth (TL2v2) . ..... 259 664 Cumulative averaged and corrected transport fluxes for TL2v2 . ... 259 665 Phase dependence of bedload, entrainment, and deposition during ripple growth and equilibrium .................. ............... .. .. 260 666 Phase dependence of corrected bedload and suspended load fluxes during ripple growth and equilibrium .................. ............. .. 261 667 Transport flux tendency phase diagrams for ripple growth and equilibrium 262 668 Time evolution of ripple characteristics during ripple decay (TL2v2d) . 263 669 Cumulative averaged and corrected transport fluxes for TL2v2d . ... 264 670 Phaseaveraged and corrected transport fluxes during ripple decay . ... 264 671 Phase dependence of transport constituents during ripple decay for TL2v2d. 265 672 Transport flux phase diagram for rapid ripple decay ............... ..266 673 Time evolution of ripple characteristics through bedload transport only . 267 674 Phase dependence of bedload transport during ripple growth for BL2v2 . 268 675 Time evolution of ripple characteristics through suspended load transport only 269 676 Phase dependence of vertical sediment fluxes during ripple decay for SL2v2 270 677 Time evolution of scaled ripple heights for Phase III simulations . ... 271 B1 Derivation of the flux tendency phase diagram of bedform growth, equilibrium, and decay . . . . . . . .. .. . . 288 LIST OF SYMBOLS Greek symbols a longitudinal bedslope. /3 transverse bedslope. X wave period parameter. ft time step. atf morphology hybrid filtering time step. 6xi x grid spacing. 6yj y grid spacing. 6Zk z grid spacing. 6 boundary lv. r thickness. A filter width. 6d displacement boundary l1.r thickness. Ad relative orbital excursion. 6ij Kronecker delta. rl free surface displacement. rlr ripple height. 7Y smoothing coefficient. K von Kdrmdn constant. Amix mixed l. r thickness. ,r ripple wavelength. p dynamic viscosity. v kinematic viscosity. Vt eddy viscosity. w angular frequency. Sediment angle of repose. 4b nondimensional bedload discharge. P p Ps Pmix as JA (r7 T, iri 0 ,Ow 02.5 Ocr General ((90)) nondimensional entrainment rate. velocity potential. mobility number. density of water. density of sediment. density of mixed IlV 1 r. Schmidt number. ripple height standard deviation. ripple length standard deviation. ripple steepness. bed shear stress. gravitational particle shear stress. maximum bed shear stress. residualstress tensor. anisotropic residualstress tensor. nondimensional (Shields) stress. variable finitedifference coefficient. nondimensional current (Shields) stress. nondimensional wave (Shields) stress. grain roughness Shields parameter. critical Shields parameter. critical Shields parameter for a horizontal bed. wavecurrent angle. primitive variable at time level n. operators filtered variable. phaseaveraged quantity. (c) timeaveraged quantity. P spatiallyaveraged quantity. cp! residual variable. Roman symbols fs Smagorinsky lengthscale. Zb lowpass filtered bed elevation. VQb signcorrected bedload transport flux. VQb signcorrected suspended load transport flux. C filtered volume concentration. Cb filtered volume concentration above bed. Sij filtered rate of strain. S characteristic filtered rate of strain. sR grain Reynolds number. Rp particle Reynolds number. R, wave orbital Reynolds number. Zb hybrid filtered bed elevation. k wavenumber vector. Q+ rightbiased sediment transport flux. Q leftbiased sediment transport flux. A orbital semiexcursion length. A+ van Driest constant. C volume concentration. Cf coefficient of friction. Cp bedform propagation phase speed. Cs Smagorinsky coefficient. Cf CourantFreidrichsLevy number. C,ix mixed I,r concentration. d grain diameter. d, dimensionless grain size. d5o median grain diameter. E(t) entrainment rate of bed material. f coefficient of friction. F(, ) Flux evaluated at time level n. fb coefficient of resisting friction. F, downslope component of particle weight. fw wave friction factor. F. buovint weight of particle. F, body force. f2.5 grain roughness friction factor. fmix coefficient of applied friction. g gravitational acceleration. h water depth. H wave height. k turbulent kinetic energy. k, residual kinetic energy. k, Nikuradse roughness length. k, wave number. kx wavenumber in xdirection. ky wavenumber in ydirection. n normal direction. np porosity of bed material. p pressure. Prt turbulent Prandtl number. Qb volumetric bedload transport rate. Qs volumetric suspended load transport rate. Qt total volumetric sediment transport rate. s specific gravity of sediment. S, nondimensional sediment parameter. T wave period. t time. TKE turbulent kinetic energy. U, friction velocity. Us fluid velocity at edge of boundary l1_r. Uoo freestream velocity. ub nearbed fluid velocity. Uc current velocity. ui velocity component. Umax maximum velocity amplitude. Umix mixed lv.r velocity. Us sediment velocity. Ws hindered settling velocity. W, constant settling velocity. xi spatial direction component. z+ normalized wall unit. Zb vertical location of bed. Zo reference height. Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SMALLSCALE SEDIMENT TRANSPORT PROCESSES AND BEDFORM DYNAMICS By Bret Maxwell Webb M1 v 2008 C('! Ii: Donald N. Slinn Major Department: Coastal and Oceanographic Engineering The generation of smallscale sedimentary structures in the coastal environment is a complex process that occurs over a wide separation of scales in both time and space. These bedforms are ubiquitous features of the nearshore region, and yet specific information regarding their behavior and characteristics is still lacking. Specifically, it is unclear whether the bedloaddominated processes of the linear regime are as equally responsible for the generation of bedforms in the nonlinear regime, where flow separation, and subsequent vortex formation, tend to govern the dynamics of the bottom boundary l~ r. While a simple onedimensional model is derived and used to explain incipient bedform growth in the linear regime, such an approach is not wellsuited at addressing the complexities of the wave bottom boundary l1.r. Utilizing a new threedimensional phaseresolving livebed model, we simulate the dynamics of bedforms, such as sand ripples, in the nonlinear regime. Through fortythree independent simulations, the model has been found to reproduce oscillatory boundary l r flow, as well as provide accurate predictions of ripple geometry in both lab and fieldscale flows. Model results confirm that in the linear regime, bedform growth is promoted purely through bedload sediment transport, but inertial properties of the sediment are equally as important. In the nonlinear regime, bedform growth is also dominated by bedload transport; however, the entrainment and deposition of bed material p1l i an important role in maintaining ripple equilibrium, whereas it is mostly responsible for ripple decay. CHAPTER 1 INTRODUCTION 1.1 Background Bedforms, such as sand ripples, are ubiquitous features of the coastal zone. And although they have been the subject of numerous investigationstheoretical, experimental, and numerical alikedating back to the late nineteenth century (Hunt, 1882; Darwin, 1883), definitive information about their dynamics in the coastal environment remains limited. This is not to ,i that theory and understanding have not been markedly advanced over the last century, but it underpins the necessity for continued research on bedform dynamics at scales both large and small. Morphological bedform features occurring in nature may range in scales from a few centimeters in height and tens of centimeters in length for wavegenerated sand ripples, to larger sandwave features like dunes and megaripples that can have heights of a few meters and lengths on the order of tens of meters. Regardless of their size, bedforms plih an important role in both the energy and sediment budgets of the nearshore environment. Bedforms have been found to strongly affect: energy dissipation in the wave bottom boundary 1lV r; sediment transport characteristics; and acoustic properties of the seafloor. While momentum transfer above plane beds is due mostly to turbulent diffusion, rippled beds induce flow separation resulting in organized vortices, or coherent motions, that dominate momentum transfer in the wave bottom boundary 1lr (WBBL) (\1 !,1 i 2004). These coherent motions are effective in dissipating wave energy in a nearbed boundary 1i, 2006). Above this nearbed 1l, r, coherent motions break down and are replaced by random turbulence (Davies and Villaret, 1997). Tunstall (1973) and Tunstall and Inman (1975) estimate that as much as 10' of wave energy may be dissipated through the process of vortex formation. Wave energy is also effectively dissipated through friction as bedforms affect the hydraulic roughness of the bed for waves and currents (Soulsby and Whitehouse, 2006). Bedform height controls the effective bottom roughness (Faraci and Foti, 2002), which is of practical engineering importance. Parametrization of hydraulic roughness (Grant and Madsen, 1982) are commonly applied to engineering wave and sediment transport models in order to account for the additional energy dissipation that takes place in the WBBL. Such practice permits the effects of bedforms on currents and waves to be treated in a bulk manner, rather than having to account for them explicitly in the modeling approach. The parametrization and application of roughness to models accounts for the thickening of the WBBL in the presence of bedforms (Wiberg and Harris, 1994), whereby energy is dissipated through form di I. skin friction, and turbulence damping due to an intense 1],.r of nearbed sediment transport (Grant and Madsen, 1982). The amount of dissipation due to bedforms is not trivial; field observations by Ardhuin et al. (2002) indicate that the form drag over large ripples is effective in wave attenuation across the continental shelf. Accurate representations of equivalent bottom roughness, then, are of particular importance for understanding the energy budget of the nearshore environment. However, prior characterizations of roughness and their subsequent implementation in nearshore models often neglect changes in bedform size, shape, and orientation, and do not account for phase lag effects induced by flow separation in the boundary 1l,.r. Bedforms are manifestations of differential sediment transport near the seabed, and can often be found in unique patterns along the seafloor as seen in Figures 1 (a)1l(d). Once large enough to induce flow separation, they have a profound impact on subsequent sediment transport. Enhanced sediment suspension over rippled beds not only increases potential sediment transport in the bottom boundary 1I,.r, but affects the background flow as well. Stratification of sediment in the water column results in a modification of the velocity profile (Coleman, 1981), turbulence damping (\McLean, 1991), and an increase in boundary roughness (Parker and Coleman, 1985). Therefore, sediment suspension over rippled beds serves as an additional mechanism for the dissipation of energy in the bottom boundary lvr. According to Nakato et al. (1977), suspension processes over rippled beds are dominated by the formation, ejection, and motion of eddies. Organized vortices that form in the leeside of ripples entrain sediment from troughs or valleys in each successive halfcycle of wave forcing. Immediately following flow reversal, the sedimentladen vortices are ejected into the main flow above the bedforms where the sediment is subsequently advected by the local fluid velocity field. Laboratory experiments by (van der Werf et al., 2005) ,t'. 1 that such behavior results in three distinct peaks in the nearbed concentration as a function of wave phase. Compared to suspension events over a flat bed, field data ti'. 1 that sedimentladen vortices in the wake of bedforms alters sediment suspension (Gallagher et al., 1998), and may also enhance the phase lag between suspension and transport (Inman and Bowen, 1963). Furthermore, van der Werf and Ribberink (2004) propose that rippleinduced phase differences between peak suspended sediment concentrations and peak fluid velocities result in net sediment transport rates directed offshore under surface gravity waves. Such phase differences, however, are not accounted for in common phaseaveraging, coupled hydrodynamic and sediment transport models. This may it. 1 why some crossshore sediment transport models fail when oscillatory currents are larger than steady currents (Gallagher et al., 1998). In recent years, the Navy has become increasingly interested in the acoustic properties of the seafloor, specifically with respect to the ability of sonar devices to detect both emergent and submerged munitions in the battle space environment (BSE). Bedforms affect the acoustic response for sonar applications, either enhancing or inhibiting the penetration of the sonar signal into the bed (Soulsby and Whitehouse, 2006). From data collected during the SAX99 field experiment, Piper et al. (2002) show that both the ripple height and wavelength strongly affect the level of sonar penetration for subcritical grazing angles. Additionally, the orientation of the ripples relative to the incident sonar field is of importance. The results of Piper et al. (2002) indicate that subsurface imaging is enhanced when the sonar propagation direction is perpendicular to the mean direction of ripple crests, whereby the amount of acoustic energy scattered by the bedforms into the sediment is increased. 1.2 Motivation In the linear regime of bedform growthwhere rolling grain ripples persistsediment transport takes place purely through bedload processes. The bedload transport mode consists of particles rolling, sliding, and saltating in small leaps on the order of a few grain diameters above the bed. The linear regime is dominated by friction and inertial lags between fluid forcing and particle transport. The absence of suspended sediment transport in the linear regimeand at subcritical threshold valueshave lead many to postulate that bedforms are generated purely through bedload transport. It is unclear, however, that this statement holds true in the nonlinear regime where coherent motions in the boundary 1i. r promote suspended load transport and induce phase differences between fluid forcing and sediment entrainment. The nonlinear regime contains both bedload and suspended load transport modes, but their relative contributions to ripple growth, equilibration, and decay are unknown. Additionally, the contributions of bedload and suspended load transport modes may vary as a function of ripple position (profile and planform) and fluid forcing. The quantitative roles of constructive (fluid forcing) and destructive (gravitational) forces in generating sedimentary structures are not immediately evident. It would be beneficial, therefore, to investigate the contributions of various transport modes to bedform dynamics, and to better understand the roles of constructive and destructive forces during ripple, growth, equilibration, and decay. Field and laboratory measurement techniques are incapable of gathering in situ data that clearly distinguish one transport mode from another. Such methods often require the assumption of a threshold based on volumetric concentration to separate the transport modes after the data have been collected. Additionally, measurement techniques for bedload and suspended load transport are commonly invasive, thereby altering the hydrodynamic and sediment transport fields that they are attempting to quantify. Numerical simulations may provide useful insight into sediment transport processes in smallscale bedform dynamics where physical experimentation is currently not possible. Through the University of Florida, we are participating in the Office of N i,. Research's (ONR) intensive study of Sand Ripples on the Inner Shelf (FY0408), a project involving fifteen principal investigators from across the nation (Ripples DRI). This interdisciplinary project combines field observations with laboratory experiments and numerical simulations from the biological, geological, and oceanographic sciences. The primary goals of the project, as identified by ONR, are to 1. measure and model morphology; 2. investigate morphological response to forcing perturbations; 3. measure and model rates of biodegradation; 4. measure and model the effects of grainsize distribution; and 5. understand the role in generating sedimentary structures. Our participation in the project primarily involves the first two goals outlined above, as well as the last. In order to address these objectives, we seek to develop a physicsbased model that couples hydrodynamics and morphology by updating the bed level at every, or nearly every, hydrodynamic timestep. The review of noncohesive sediment transport processes and bedform dynamics that follows in C'! lpters 2 and 3 should provide adequate guidance for determining a suitable model framework. Aside from the goals of the ONR Ripples DRI, the primary objective of this research is to investigate sediment transport characteristics during ripple growth, equilibration, and decay. In particular, we hope to provide answers to the following questions: 1. What are the relative contributions of bedload and suspended load to the generation or obliteration of sedimentary structures? 2. Are there specific locations in the ripple profile/planform where one transport mode dominates over the other? 3. Are there specific timesas a function of phasewhen one mode dominates over the other? 4. What are the dominant roles of: bedload, suspended load, and gravitational forces? To address our objectives, we propose the development of an entirely new modeling system capable of simulating phaseresolving smallscale sediment transport and morphology. Capabilities of the modeling system will be assessed by evaluating the hydrodynamics through modeldata comparisons, and by also performing simulations of bedform equilibration under a variety of scenarios (e.g. growth, equilibration, decay, coarsening, bifurcation, steady flows, short and longperiod flows, highlyconcentrated flows, and extreme sediment sizes). In order to answer the questions posed above, simulations of bedform growth, equilibration, and decay will be performed with bedload and suspended load; bedload only; and suspended load only. The design of the modeling system assumes that bedload transport is predicted using Einstein's dimensionless bedload flux and common powerlaw formulations, and that suspended load originates through an entrainment of sediment from the bed using semiempirical pickup functions available in existing literature. Thus, our system allows us to alternately turn bedload and suspended load on or off by simply setting bedload transport rates, or entrainment/deposition, to zero, respectively. We further assume that the bedload regime is comprised of two types of transport: one due to fluid forcing; and another due to gravitational forces. Doing so permits us to determine the roles of constructive and destructive forces independently, thereby allowing us to better understand their roles in generating sedimentary structures. The development of this modeling system is a secondary objective of this dissertation, and is necessary to address our scientific questions posed above. Specific details about the system are outlined below, as well as in the following chapters. 1.3 Approach Modeling sediment transport continues to be a challenge for fluid dynamicists. Indeed, much of the uncertainty in fluidsediment models continues to lie mainly with the particulate phase, whereas the fluid hydrodynamics are comparatively well understood. Sediment transport models fall mainly into two broad categories: 1. time (or phase) averaged; and 2. unsteady (or phaseresolving). Variations on phaseaveraged and phaseresolving models include quasisteady and semiunsteady models, respectively. Each of these models has its own benefits and deficiencies, making some more useful in certain situations than others. Briefly, time or phaseaveraged sediment transport models effectively integrate out intrawave processes by considering waveaveraged values of fluid velocity, sediment concentration, and sediment transport (Bijker, 1971). So called quasisteady models assume that the transport rate is proportional to the instantaneous nearbed flow field raised to some power (Bailard, 1981), but only provide an average transport rate over the wave period. Semiunsteady models incorporate additional complexity into the formulation of transport rates by considering phase lags either through parametrizations (Nielsen, 1988; DohmenJanssen, 1999) or by breaking a full wave period into wave halfcycles (Di i.iiiii and Watanabe, 1992). While computationally intensive, the most robust approach is to describe the sediment transport and hydrodynamics as coupled, timedependent processes. Very few of these models exist outside of the computational research arena, as their practicality for engineering applications is still somewhat limited due to their complexity [see Drake and Calantoni (2001); Gessler et al. (1999); and Lesser et al. (2004)]. To address a number of relevant engineering problems involving sediment transport processesfrom structureinduced scour to coastal erosionit would be advantageous to have a flexible, robust model capable of resolving timedependent bed morphology under various forcing conditions. Such a modeling system could also satisfy some of the primary objectives of the ONR Ripples DRI project outlined above. Here we seek to develop and evaluate the skill of a coupled fluidsediment numerical model that resolves instantaneous hydrodynamics, sediment transport, and timedependent bed morphology. It is not evident that such a model exists at the level of detail proposed. The desired phaseresolving model would be capable of both two and threedimensional simulations of (1) steady flow, (2) oscillatory flow, (3) combined oscillatory and steady flow, and (4) surface gravity waves. While the time integration of the governing equations will be limited to durations of 0(100 s), resolution of physical lengths will range from 0(104 m) to 0(1 m). Such detailed treatment of hydrodynamics near solid boundaries should permit the resolution of highintensity turbulent fluctuations that initiate sediment transport. Integration of the bed level through time will be governed by the sediment continuity (Exner) equation, where gradients in the bedload flux and the competition between entrainment and deposition of suspended solids provide an estimation of the instantaneous bed level. Sediment transport in the model will be estimated using bulk, empirical formulations available in existing literature. Natural feedback between morphology and the hydrodynamic flow field represents the oneway fluidsediment coupling in the model. Twoway coupling of the hydrodynamics and local suspended sediment concentration may be considered in the future. 1.4 Outline What follows is a general overview of knowledge pertaining to noncohesive sediment transport and bedform dynamics, cast in a manner that, hopefully, underscores the need for a hydrodynamic livebed model having the general characteristics outlined previously. A brief summary of noncohesive sediment transport processes is given in ('! Ilpter 2, with a discussion on the governing hydrodynamics or forcing, modes of transport, the role of turbulence, and a description of accepted models. C'!i lpter 3 provides information about general bedform properties and dynamics, with particular attention given to their classification and characteristics, mechanisms for growth, a summary of relevant laboratory and field experiments, semiempirical solutions for their geometrical properties, and an overview of models ranging from simple to complex. A description of newly developed models is provided in ('!i lpter 4 and outlines governing equations, as well as methodologies for their implementation in the modeling system. The model experiments are outlined in ('! Ilpter 5, and those simulation results are presented and discussed in ('!i lpter 6. Concluding statements are provided in Chapter 7, along with some words on potential future applications of the newlycreated modeling systems. (b) USA Wave Basin (c) Yucatan Peninsula, Mexico Figure 1 1. (d) Yucatan Peninsula, Mexico Pictures of sand ripples in lab [(a) and (b)] and field [(c) and (d)] settings. The lab pictures were taken in the University of South Alabama Wave Basin after draining, while the field pictures were taken in a depth of approximately 5 m of water off the East coast of the Yucatan Peninsula in Mexico. (a) USA Wave Basin CHAPTER 2 SEDIMENT TRANSPORT PROCESSES 2.1 Introduction Sediment transport processes in the coastal environment are highly complex. Transportation is both initiated and sustained by hydrodynamic forcing on the seabed and water column. While the knowledge of sediment transport processes has been significantly advanced in idealized, laboratory environments, detailed information about such processes remains ambiguous under the stochastic forcing that is present in nature. The following sections outline the governing hydrodynamics that drive sediment transport in the coastal environment, provide a description of common transport modes and regimes, discuss the mutual relationships between transport and turbulence, and deliver an overview of various sediment transport models. 2.2 Governing Hydrodynamics 2.2.1 Waves Surface gravity waves represent the most dominant mobilizing force for sediment in coastal areas (van der Werf, 2004). In some cases the orbital velocities under these waves can be quite large near the seabed, especially when waves steepen as they enter shallower water. Furthermore, realistic surface gravity waves are not linear, typically having steeper peaks and flatter troughs. A result of this nonlinear profile is an inherent .ivmmetry in the orbital velocity field with shortduration, highintensity velocities directed shoreward under wave crests and longerduration, lowerintensity velocities directed seaward under the broad, flat troughs (see Figure 2la). This inequality in the orbital velocity time series becomes very important for the determination of net sediment transport rates [see Ribberink and AlSalem (1990); Gallagher et al. (1998); and Elgar et al. (2001)]. As discussed earlier, the wave bottom boundary licr (WBBL) is an important mechanism for energy dissipation and influences the magnitude and direction of sediment transport in the nearshore. This boundary lIr develops near the seabed in response to friction between it and the fluid. A general expression for boundary li r thickness is given by 6 oc xVIT (21) where v is the kinematic viscosity and T is the wave period. The thickness of the boundary l~.r is also affected by the roughness of the boundary. Wave boundary l.. ir over smooth boundaries may be of 0(1 mm) while the thickness over rough boundaries, like a real seabed, increases to 0(1 cm). Due to inertial effects in the boundary 1lvr (slower moving fluid), the nearbed oscillatory flow is often out of phase with the freestream forcing. This results in a nearbed phaselead of approximately 450 for laminar flow and 15 for turbulent boundary lV. i with respect to the freestream forcing. The phase lead of nearbed oscillatory flow has profound significance for the estimation of sediment transport. Common sediment transport formulations incorporate some threshold for incipient motion, such as Shields' parameter (Shields, 1936), that requires the bed shear stress to exceed some critical value. A general expression of the bed shear stress is Tb =pCf U U (22) where p is the fluid density, Cf is a coefficient of friction (drag), and U is representative of the freestream forcing. Figure 2lb shows a representation of bed shear stress under linear and nonlinear waves using Equation 22 with p = 1025 kg/m3 and the canonical value of 0.0025 for Cf (Rivikre et al., 2004). This type of formulation for bed shear stress is typical of very general models and does not include any information about phase differences between flow at the bed and flow outside of the boundary lI r. It is evident, then, that application of Equation 22 for unsteady flow in the boundary l1ir will incorrectly predict the phase of sediment mobilization with respect to the applied forcing. Under purely horizontal, regular oscillatory flow this is of less significance as the net sediment transport from cycletocycle will be zero. The phase difference for mobilization and entrainment under nonlinear surface gravity waves, however, is significant as it can dictate the direction of net transport [see Elgar et al. (2001) and van der Werf and Ribberink (2004)]. An alternative to the general bed shear stress formula given by Equation 22, commonly employ, l in phaseaveraged or quasisteady transport models, is to consider the maximum bed shear stress for an individual wave based on a wave friction factor (f,). Such a method was proposed by Jonsson (1966): Tb,max = Pf.., (23) where Ub is the nearbed orbital velocity. The wave friction factor of Swart (1974) is applicable for fullydeveloped, rough turbulent flow and is given by f, exp 5.213( 04) 5.997 (24) where k, is the Nikuradse (1933) roughness lengthscale and A is the orbital semiexcursion length (A = UbU). Jonsson's (1966) formulation of the maximum bed shear stress is more complete than the formulation shown in Equation 22, as it accounts for the roughness of the bottom and uses the nearbed orbital velocity as opposed to the freestream forcing, but represents a bulk approximation of the stress during a wave period instead of treating it as a timedependent quantity. A large portion of the total bed stress is due to wave pressure variations over the seabed. According to McLean (1991), this part of the stress is not effective in mobilizing sediment because the lengthscale of the pressure variation is so much larger than the particle diameter. Foda (2003) claims, however, that outside of the surf zone the wave pressure pl .1 i an important role in sediment transport processes: lowenergy waves (mild pressure gradients) tend to drive sediment offshore, while highenergy waves (steep pressure gradients) result in onshore transport. These two conflicting theories may be harmonized by considering the context under which the statements were made. In an attempt to determine the importance of wave pressure on bedload transport, Foda (2003) treats the seabed as a viscoelastic fluid, not as individual particles; therefore, the lengthscales of the pressure variation and an "effectiu, length of the active viscoelastic fluid may not be greatly different. N. i!1i.re waves, due in large part to their .iviiili. l iry, produce secondary flows outside of the WBBL that plan an important role in suspended sediment transport. Considering the velocity time series for a nonlinear (Stokes) wave in Figure 2la, one can see that velocity under the wave crest is larger than that under the trough. This inequality results in a net displacement of a parcel of fluid, or suspended sediment, in the direction of wave propagation, often referred to as Stokes drift (van der Werf, 2004). In the nearshore region, this net displacement of shorewarddirected fluid is balanced, over long periods of time, by a seawarddirected return flow near the bed often called undertow. Elgar et al. (2001) .. 1 that offshore sandbar migration is linked to crossshore gradients of undertow, as these nearbed currents are often strong enough to mobilize sediment. Furthermore, Gallagher et al. (1998) believe that the intensification of undertow near a sandbar leads to crossshore gradients in the suspended sediment flux that further promote offshore bar migration. In addition to the crossshore directed secondary flows, surface gravity waves also induce boundary l1.r, or steady, streaming. Vertical velocities generated in the WBBL result in a diffusive vertical flux of momentum directed away from the horizontal boundary liv.r. Although weak compared to oscillatory flow outside of the boundary lV. r, steady streaming results in a nonzero timeaveraged bed shear stress and has a significant effect on suspended sediment transport (\ 1,iii, 2004). The steaminginduced flux away from the boundary 1.r represents an additional mechanism for the entrainment of sediment into the outer flow, where it can then be advected by the local flow field. 2.2.2 Currents Steady currents, such as the flow found in rivers and hydraulic channels, are effective in transporting sediment, once mobilized, through advection. Relative to the wave boundary 1I. r, the current boundary 1i.r is often one or two orders of magnitude larger as it develops over comparatively long durations of time. A number of scientists have considered sediment transport in currents: A t. l rPeter and Miiller (1948), Eintstein (1950), Smith and McLean (1977), Bagnold (1980), and van Rijn (1984). Much of the research indicates that the transporting capability of a steady flow is proportional to the stream power, or velocity, raised to some power [Bagnold (1980); van Rijn (1993)] such that q = au where q is the volumetric transport, and a and b are empirically determined constants (Long et al., 2007). 2.2.3 Waves and Currents The nearshore environment is often characterized as stochastic, having multiple realizations of wave frequency, direction, and height, as well as current magnitude and direction. Often times nearshore waves travel over shearing currentsa common example being a shoaling wave propagating over an alongshore current. The combined effects of wave and currentinduced velocities, however, cannot be found by a simple linear superposition of the constituents (van der Werf, 2004). Rather, the presence of a current modifies the waveinduced flow field in a nonlinear manner by alternatively adding and subtracting from the orbital velocities during different phases of the wave. The wave celerity and length are also affected by the current, where celerity increases (decreases) as wavelength decreases (increases) in response to an opposing (following) current. This effect is demonstrated for collinear waves and currents by considering the modified dispersion relationship (Dean and Dalrymple, 1991): L kw Uk, + gk tanh kh (25) where w is the angular wave frequency, Uc is the current velocity, k" is the wave number (k, = 27/L), g is gravitational acceleration, and h is water depth. It is also known that the waveinduced orbital velocities reduce the nearbed current magnitude (van der Werf, 2004). The nonlinear interactions between the current and waveinduced boundary 1v, i s have a significant effect on sediment transport. The stirring effect of waves coupled with a steady current has been found to increase the total sediment transport significantly (Camenen and Larson, 2005). Wave orbital velocity amplitude (A), current velocity (U,), and the angle of incidence between the wave and current (c,,) dictate the resultant behavior of nearbed fluid motion. While a current flowing perpendicular to the direction of wave propagation does not affect wave celerity or length, it does alter the net sediment transport. In their laboratory experiments, Lacy et al. (2006) find that for ,, = 900 the maximum transport is symmetric about the current direction resulting in wavephaseaveraged sediment transport in the downstream direction. As the wavecurrent angle decreases from 0,, = 900, the maximum instantaneous bed shear stress increases (similarly for O8, increasing from 900 to 1800). Lacy et al. (2006) found that the effect of the current was to increase the nearbed velocity amplitude, thereby increasing the relative importance of suspended sediment to bedload transport. Increased entrainment of mobilized sediment into the current boundary 1v.,r was also observed. 2.2.4 Tides and Tidal Currents In general, tides do not have a substantial impact on sediment transport in the nearshore region. Tidalinduced velocities are typically orders of magnitude smaller than the instantaneous, orbital velocities produced under a surface gravity wave having a frequency much higher than the tidal frequency. Therefore, tidalinduced velocities are often not large enough to initiate motion at the seabed, nor do they p1 i, a significant role in the advection of sediment to different locations. Of course there may be special circumstances, such as a very shallow inlet or estuary, where tidal currents are sufficiently strong to mobilize sediment in large quantities (i\ I!! r and Komar, 1980a). Tidal currents generated by shoaling internal tidalwaves can increase the shear stress sufficiently to develop nepheloid l1.1 r? or regions on high turbidity near the seabed. These tidal currents are influential in suspending nearbottom sediments in the absence of windgenerated waves and currents on the continental shelf and slope (Cacchione et al., 1994). Once mobilized, the suspended sediment may then be transported across the continental shelf by internal waves and tidal currents. Although they found stormgenerated waves to be the dominant forcing mechanism of transport on the continental shelf, Puig et al. (2001) i r. 1. 1 nearinertial internal waves as a mechanism for the maintenance of a nepheloid l1i.r and suspended sediment during milder wave climates. 2.3 Transport Incipient sediment motion has been studied extensively in the laboratory since the pioneering work of Shields (1936). The basic concept of incipient motion, offered by Shields (1936), is that the sediment mobilizing forces of fluid lift and drag must exceed the stabilizing force of gravity acting on the particle. The ratio of mobilizing to stabilizing forces is referred to as the Shields parameter and is given by 0 b (26) (s 1)G where Tb is the bed shear stress, s is the specific gravity of the sediment relative to water, g is gravity, and d is the particle diameter. The dimensionless Shields stress accounts for skin friction, but not form drag which is the second component of total drag experienced by the bed material. However, skin friction is the more effective component of drag on mobilizing sediment (McLean, 1991). 2.3.1 Modes Classical modes of sediment transport are divided into three categories: 1. wash load, 2. bedload, and 3. suspended load. The wash load is characterized by very fine sediment particles, transported by the fluid, that are not representative of the bed material (Fredsoe and Deigaard, 1992). Estimation of the wash load is difficult since it requires knowledge of sediment characteristics from the point of origin... a location possibly far removed from the area of interest. It is commonplace to discard wash load in the computation of the total sediment load, which is then comprised of the bedload and suspended load material. 2.3.1.1 Bedload Bedload is the part of the total sediment load that is in almost continuous contact with the bed (Fredsoe and Deigaard, 1992). Under nonbreaking waves, DohmenJanssen and Hanes (2002) find that bedload transport accounts for nearly 9i '. of the total sediment load. The bedload material tends to move along just above the static bed in a thickness a few tens of grain diameters at most (Bagnold, 1980). The movement of bedload particles is characterized by sliding, rolling, and/or saltation. Sliding occurs in more loosely packed beds and for higher mobilizing stresses, where the particles slide across one another in sheets. Bedload particles may also roll over top of their neighbors if the moment of lift and drag forces are just large enough to counteract the moment of stabilizing force of gravity acting on the particle (Luque and van Beek, 1976). Saltation occurs when a particle is liberated from the bed material by an intense stress and, once ejected into the flow, follows a more or less ballistic trajectory just a few grain diameters above the bed (Bagnold, 1973). Pioneering work in theoretical and empirical bedload transport was put forth by Eintstein (1950). Einstein's (1950) empirical bedload function was the first of such formulations to account for the randomness of flow and subsequent transport by equating the number of particles deposited (eroded) per unit area to the number of particles in motion (on the bed), and on the probability that the force balance on the particle is such that it will be deposited (eroded). A central concept of the Einstein bedload function is that bedload transport is proportional to the flow rate raised to some power. In contrast, the experiments of Luque and van Beek (1976) show that the average length of individual particle saltations is a constant, implying that the probability of deposition is independent of the flow rate. Such a proportionality, however, has also been ii::. 1. by empirical relationships "\!. I. rPeter and Miiller (1948); Bagnold (1980); Ribberink and AlSalem (1990); Nielsen (1992); Ribberink (1998); Camenen and Larson (2005)], and by analytical expressions derived from physical concepts as well [Bagnold (1966); Bowen (1980); Bailard (1981); Kovacs and Parker (1994)]. Eintstein (1950) postulated that a functional relationship existed between the bedload transport rate and the particle buov, i, v, such that a nondimensional bedload discharge may defined as Qb b Qb (27) V(s 1)gd3 where Qb is the volumetric rate of bedload transport per unit width, s is the specific gravity of the particle, g is gravitational acceleration, and d is the particle diameter. A number of physical laboratory experiments have been conducted to determine the functional relationship between fluid forcing and resulting transport, represented in Equation 27 by Kb. These functional relationships are termed I .. 1 1. since the transport rate is proportional to some measure of the fluid forcing (velocity or stress) raised to a power. A number of power laws, in various forms, have been iri I. 1 for transport by currents [e.g. A1t. rPeter and Miller (1948); Nielsen (1992); Ribberink (1998); Camenen and Larson (2005)] and by waves [e.g. Madsen and Grant (1976); Bailard and Inman (1981); Dib l i and Watanabe (1992); Soulsby (1997); Ribberink (1998); Camenen and Larson (2005)]. Most of these relationships are based on an exceedance of the critical Shields stress and take the form: b = a(O Oc)b (28) where a is some constant of proportionality, b is an empiricallyderived exponent providing the best fit of Equation 28 to a set of data, and 0 and Oc, are the instantaneous and critical Shields parameters, respectively. Various examples of bedload power laws are give in Table 21. Shields' description of incipient sediment motion is based on the principle that there is absolutely no motion when the applied stress is below the critical threshold. Therefore, when 0 < Ocr in Equation 28 the dimensionless bedload discharge 1b is actually equal to zero. Camenen and Larson (2005) provide expressions for 4b that have the general form: b = Cod exp ( e ) (29) where c, d, and e are empiricallyderived constants. Equation 29 ,i. 1 that transport begins as soon as a stress is applied to the sediment (when 0 > 0), contradicting Shields' concept of a critical threshold for sediment motion. Initial treatment of bedload transport treated moving particles in a bulk fashion, with the mobilizing stress being transmitted by the fluid to the bulk. By the definition of bedload given above, however, it would seem that some stress is transmitted between particles since they are in almost constant contact with one another. Bagnold (1954) addressed this subject by considering the intergranular stresses transmitted between particles under fluid shear and found a substantial radial dispersive pressure exerted between the particles. This dispersive pressure is distributed in such a manner that the moving grains are in equilibrium with their gravitational force. As shear stress is applied to the particle matrix, dilation occurs and the dispersive pressure decreases as the particles move further from each other. The total shear stress within the dispersed particle matrix is then a combination of intergranular fluid and particle stresses such that 7 = f + 7 (210) where Tf is the stress transmitted by the fluid within the pores, and 7T is the tangential dispersive stress that represents momentum exchanged due to particleparticle interactions (Fredsoe and Deigaard, 1992). The particle collisions that contribute to 7 subtract momentum from moving sediment particles that must be replaced by the fluid forcing. This momentum subtraction results in an apparent roughness greater than would be expected for a static bed (McLean, 1991). Through a series of laboratory experiments of particle transport in a closed rectangular flow channel, Luque and van Beek (1976) found that bedload reduces the maximum fluid shear stress at the bed level by exerting an average reaction force on the surrounding fluid. According to Equation 210, a reduction in fluid stress ii.;. I that the intergranular stresses must increase to maintain the total stress. At high bedload transport rates, then, a in ii y of the total stress is exerted by particle collisions while the fluid stresses remain small (Luque and van Beek, 1976). A schematic of the distribution of fluid and granular stresses for an idealized open channel flow is shown in Figure 22. 2.3.1.2 Suspended Load Besides bedload, the other constituent of the total sediment load is the suspended load. Suspended load may be defined as material advected by the fluid, maintained in suspension by turbulence, and having very little contact with the bed. In this respect the suspended load is transported by mechanisms similar to those responsible for the transport of wash load, but its origins differ. While wash load consists of particles not characteristic of bed material, suspended load sediment is entrained directly from the bed material. Hence, some knowledge of the bed material may be utilized in the characterization of suspended sediment properties. Suspended load p1l 'i, an important role in both sediment transport and hydrodynamic processes. Once entrained from the bed material, these sediments can be advected locally by nearbed wave orbital motion, and regionally by persistent crossshore and alongshore currents. As most natural sediment is multimodalhaving a naturally occurring distribution of grain sizesthe entrainment of smaller particles near the surface can leave behind a l1v.r of larger particles. Bagnold (1980) sl::. 1 that the presence of a larger grain size reduces the transport rate through a phenomenon called armoring. Total load transport rates may also be reduced through the effects of the suspended load on the nearbed flow field. McLean (1991) selI: 1. I that density stratification by suspended sediment damps turbulence, thereby limiting the ability of flow to transport mass and momentum vertically. As a result, the upward dispersion of suspended sediment maintained, against gravity, by random eddy currents is reduced (Bagnold, 1973), the ability of the flow to keep sediment in suspension is impaired, and the entrainment capacity of the flow decreases (M\cLean, 1991). Suspension and density stratification may also modify the velocity profile, which indirectly alters the boundary roughness through a subsequent change in bed morphology [Coleman (1981); Parker and Coleman (1985)]. Although suspended sediment may suppress turbulence, Nakato et al. (1977) found that both the mean and fluctuating vertical fluid velocity were amplified by suspension. Through a series of laboratory experiments, Coleman (1981) demonstrated the ability of suspended sediment to reduce turbulence, as well as modify the shape of the velocity profile. The distribution of velocity throughout the water column is of significance for net sand transport, as the suspended particles are advected by the fluid velocity. This is especially true for transport over a rippled bed where coherent motions, like organized vortices, dictate entrainment, suspension, and advection of significant quantities of sediment (van der Werf et al., 2006). Luque and van Beek (1976) found that the average transport velocity of suspended particles just above the bed was approximately equal to the turbulent fluid velocity minus a constant, which is speculated to be an inertial effect. Knowledge of the velocity profile, then, could provide some indication of suspended load concentrations. Such information is especially useful in the nearbed region, which p1 i,, an important role in the determination of net sand transport owing to the high concentrations of sediment found there (van der Werf et al., 2006). A common approach for estimating the nearbed velocity distribution, within the boundary l1 r, utilizes some functional form of the classical PrandtlKdrmin logarithmic law of the wall (von Kirmnn, 1930), or log law, given by ln + B (211) Ut K Z0 where u is fluid velocity, u, is friction velocity, K is the von Krrmnn constant, z is the vertical distance from the boundary (bed), and zo is a reference height. While various estimates of K and B appear in literature pertaining to boundary 1fr flows, Pope (2000) reports that generally all are within five percent of S= 0.41, B 5.2. (212) In the determination of velocity profiles containing suspended sediment, some have ii_, t, ,.1 that K decreases with increasing suspended sediment concentration [Raudkivi (1967); Graf (1971); Yalin (1977)]. A reduction of K in Equation 211 "t'., 1 that the ratio of the fluid velocity to friction velocity increases. The experiments of Coleman (1981), however, demonstrate that K is independent of sediment concentration. Values for the reference height in Equation 211 (zo) have commonly been related to the sediment grain diameter (Grant and Madsen, 1982). An estimate of Zo for intense sediment transport over flat beds in oscillatory flows, determined from the lab data of Carstens et al. (1969), is approximately 7 8 grain diameters. This concept of an equivalent sand grain roughness height (Ks) evolved from the work of Nikuradse (1933). Entrainment of bed particles into the flow may be computed in a manner similar to that used for bedload discharge (Equation 27). Following Eintstein (1950), a dimensionless entrainment parameter %p is given by E S (213) p S l),.J,, where E is the volumetric rate of entrainment of bed material, ps is the sediment density, s is the sediment specific gravity, g is gravitational acceleration, and d50 is a median grain diameter. Typically, one uses an available expression for 4~ and then computes the rate of entrainment E, or pickup rate. This methodology has been used for modeling suspended load transport in particle trajectory models (Hansen et al., 1991) as well as in more sophisticated hydrodynamic models Zedler and Street (2001). A number of relationships for 4~ have been empirically derived from laboratory experiments, and a few common relationships are given in Table 22. 2.3.2 Regimes Sediment transport is often divided into the following three regimes (van der Werf, 2004): 1. bedload, 2. ripple, and 3. sheetflow. Clear definitions of these regimes in the literature, and the constitutive relationships that make them unique, are tenuous at best. The bedload regime is characterized by bedload transport of sand particles in a l r not more than a few grain diameters thick (Bagnold, 1980), and occurring over an otherwise horizontal bed. According to Bagnold (1946) this transport persists from the onset of motion to a value about twice the critical Shields parameter (Ocr < 0 < 20cr). For reference, a practical range of Ocr for the coastal environment varies between 0.03 and 0.06 (van der Werf, 2004). Although the distinction was not made at the time, early experiments on bedform development by Darwin (1883) and Ayrton (1904) provided qualitative information on the characteristics of incipient motion and transport in the bedload regime. Darwin (1883) ~i. 1. .1 that particles transported in the bedload regime, by oscillatory flow, would ., i'regate in regions on increased friction. Similarly, the oscillating tank experiments of Ayrton (1904) demonstrated that particles transported on a flat bed would congregate in a specific location, that being the location of a standing wave node where the vertical (horizontal) velocity is at a maximum (minimum). The oscillating tray experiments of Bagnold (1946) further reinforce the bedload regime concept, where sand particles transported in a thin, concentrated 1. r over a flat bed would .i: regate in areas of higher friction, and lower tangential forcing. Once a sufficient number of particles have gathered to induce flow separation, the transport behavior is modified due to the adverse pressure gradient formed in the lee (sheltered side) of the perturbation. This behavior was consistent in the experiments of Darwin (1883), Ayrton (1904), and Bagnold (1946), even though all three utilized different testing devices. According to Bagnold (1946), the growth of substantial vortices in the lee of a bedform occurs when the Shields parameter exceeds twice the critical value, or 0 > 20cr. As opposed to the turbulent diffusion of momentum that takes place above a horizontal bed, van der Werf et al. (2006) ir:. i that momentum transfer and sediment transport above rippled beds are dominated by organized vortex shedding in a nearbed lir approximately twice the ripple height. Coherent motions break down and are replaced by random turbulence above this lIvr (Davies and Villaret, 1997). Transport modes in the ripple regime include both bedload and suspended load. The particular mode of transport is determined by the ratio of particle settling velocity to friction velocity: wo (214) t, where w,8 is the particle settling (fall) velocity, and u, is the friction velocity (u, b/p). The principle mode of sediment transport shifts from suspended load to bedload when Equation 214 is equal to about 0.07 Nielsen (1979). The shedding of vortices from ripple crests in oscillatory flow, as well as their subsequent ejection into the outer flow, dictates the phase and quantity of sediment entrainment and deposition in the ripple regime. Huang and Dong (2002) ir. 1. 1 that rippled beds induce a nearbed current directed opposite from the wave propagation direction. Furthermore, van der Werf and Ribberink (2004) found that rippleinduced phase differences between peak suspended sediment concentrations and peak fluid velocities lead to net transport rates directed seaward under surface gravity waves. This ' . I that the timing, or phase, of sediment entrainment in the ripple regime is of particular importance for determining transport direction under .ivmmetric flow conditions. The sheetflow regime is characterized by a relatively thin ( 10 100d) 1iv.r of particles transported in a highly concentrated suspension. This regime persists for values of the Shields parameter much greater than the critical threshold (0 w 0.8 1.0) (van der Werf, 2004). For very large values of the maximum Shields parameter (w 10 times as large), sheetflow washes out ripples and planes off the bed (Li and Amos, 1999). Such conditions exist under surfzone waves as transient pressure gradients result in fleeting episodes of intense sheetflow transport (Drake and Calantoni, 2001). According to Dong and Zhang (1999), the sheetflow regime is of particular importance due to the high concentrations and large transport rates found within. Dominant forcing mechanisms within the sheetflow 1.r include intergranular and turbulent stresses, as well as the interaction forces of fluid lift, drag, and inertia between the fluid and sediment particles (Dong and Z!i ii: 1999). In the lower part of the sheetflow Il.1r, sediment concentrations are very high and the stresses generated by particle collisions significantly affect the velocity of fluid and sediment (Dong and Zhang, 1999). Through a series of laboratory experiments on sheetflow transport, Ribberink et al. (1994) ,.. 1 that a three i.i r system exists with an active pickup 1.v.r near the bed, a sheetflow liVr, and a suspension 1iv.r in the outer flow. Ribberink et al. (1994) found that a majority of the horizontal fluxes were concentrated in the sheetflow and pickup lv. i and an increasing phase lag between fluid velocity and sediment concentration with increasing elevation in the suspension lcr. Almost no phase lag between fluid velocity and concentration was found to exist in the sheetflow and pickup lI.i where net sediment transport rates were proportional to the thirdorder velocity moment ((u3)) and the concentration profile was predicted well by a power law formulation. 2.4 Turbulence In the rippled bed regime, nearbed momentum transfer is dominated by vortex shedding rather than random turbulence in oscillatory forcing conditions [Sleath (1991); Nielsen (1992); Malarkey and Davies (2004)]. Flow separation at ripple crests represents a significant mechanism for the production of turbulence in the boundary 1.Tr during the phase of maximum flow (Barr et al., 2004). Turbulence p1l ,i, an important role in the hydrodynamics of the bottom boundary l~.r and affects the suspension and transport of sediment. Turbulent eddies maintain an upward dispersion of suspended sediment against the counteracting force of gravity (Bagnold, 1973), and turbulent stresses serve as additional forcing in the sheetflow regime (Dong and Zhang, 1999). The subsequent stratification of suspended sediment, in turn, affects the background flow by modifying the velocity profile (Coleman, 1981), damping turbulence (M\cLean, 1991), and increasing friction (Parker and Coleman, 1985). Modeling the effects of the particle phase on fluid turbulence is difficult. Most numerical models cannot accurately simulate sedimentfluid interactions and coherent motions in the turbulent boundary l1vr (Zedler and Street, 2001). Closure of the nonlinear momentum equations, as it pertains to turbulence, has been approached a number of viv ranging from the more simple onedimensional eddy viscosity approach (Davies and Thorne, 2002) to the more complex large eddy simulation (Wang and Squires, 1996). One option for bypassing the explicit treatment of turbulence closure through one and twoequation models, or eddy viscosity models, is to perform direct numerical simulations of the particlefluid interactions [Elghobashi and Truesdell (1992); Drake and Calantoni (2001)]. Direct numerical simulations of high Reynolds numbers flows, however, are still computationally prohibitive for large spatial scales and long durations. 2.4.1 Dynamics Turbulence and sediment transport are interconnected in the sense that they affect one another. Perhaps the first to incorporate the role of turbulence in sediment transport formulations, Eintstein (1950) considered the probability of particle deposition and erosion based on the randomness of nearbed fluid velocity. Bagnold (1954) further considered the role of turbulence in the fluidparticle matrix through laboratory experiments on the dispersion of spheres in a shearing flow. In the clear fluid, Bagnold (1954) found the in i i ily of bed shear stress to be due almost wholly to turbulence, whereas increasing concentrations of sediment suppressed turbulence. These concepts were confirmed through additional experiments by Luque and van Beek (1976). Bagnold (1973) later ii. 1. . that the dissipation of turbulence in fluidparticle flows occurred through the development and maintenance of a mean velocity equal in magnitude, and opposite in direction, to the particle settling velocity. The role of turbulence in concentrated particle flow, then, is to maintain the sediment in suspension: work performed on the particles represents the energy dissipated by the turbulence (Parker and Coleman, 1985). Subsequently, the transport rates of solids in suspension is limited by the rate of generation of turbulent kinetic energy (Bagnold, 1973). In the rippled bed and sheetflow transport regimes, intense episodes of suspension can lead to density stratification. Stratification serves as an additional mechanism for the suppression of turbulence by limiting the ability of the flow to transport momentum and particles vertically (\McLean, 1991). This often results in higher suspended sediment concentrations in the nearbed region (Ribberink et al., 1994), which may limit the entrainment capacity of the flow (McLean, 1991). As turbulence is damped, drag decreases as does the relative velocity maintaining particles in suspension, thereby leading to particle settling (McLean, 1991). 2.4.2 Closure Turbulence closure typically refers to the solution of an additional equation that provides an estimate of turbulent stress. Various closure methods have been presented in the literature, where most seek to solve for the turbulent Reynolds stresses (uu'j) in the Reynoldsaveraged momentum equation(s). The most basic, and widely used, concept was first introduced by Boussinesq who postulated that turbulent stresses were proportional to mean turbulent velocity gradients such that u v( ui 2 k (215) where vt is a turbulent eddy viscosity, and k is the turbulent kinetic energy which is equal to onehalf the trace of the turbulent Reynoldsstress tensor. Equation 215 is often referred to as the generalized eddy viscosity concept (van der Werf, 2004). There are three basic types of eddy viscosity models: 1. zeroequation models; 2. oneequation models; and 3. twoequation models. Zeroequation models include both timeinvariant and timedependent formulations, mixing length models, and von Kdrmdn momentum integral methods (van der Werf, 2004). Oneequation models seek to solve for the turbulent kinetic energy k, such as in Davies and Li (1997). Popular twoequation closure models include k c and k U, where e and w are turbulent dissipation rates [see Wilcox (1998); Andersen (1999); Andersen and Freds0e (1999)]. 2.4.2.1 RANS Models ReynoldsAveraged N .',1. iStokes (RANS) models solve the Reynoldsaveraged momentum equations, where dependent variables (like velocity and pressure) are decomposed into mean and fluctuating components. Twoequation closure models, such as k e and k w models, are often employ, ,1 in the RANS approach. With respect to their twodimensional sediment transport model, Andersen and Fredsoe (1999) found the k u performed well in estimating turbulence above ripples in unsteady flow. C'!i ig and Scotti (2004) compared simulations of turbulent flow over stationary ripples using the k u closure model of Wilcox (1998) and large eddy simulations, and found deficiencies in the RANS approach. RANS models require averaging processes that can bias the highly unsteady, timedependent flow that occurs in an oscillatory boundary l.r above a rippled bed. 2.4.2.2 Large Eddy Simulations While RANS models seek to separate dependent quantities into mean and fluctuating components, the basic concept of large eddy simulation is that variables are decomposed into resolved (large) and filtered (small) motions. The LES approach assumes that the computational mesh is sufficiently fine enough to resolve the larger turbulent motions, and yet acknowledges the fact that smaller eddies will be filtered by the grid near physical boundaries, unless the resolution is near the viscous lengthscale (Pope, 2000). In high Reynolds numbers flows, the viscous lengthscale is often prohibitively small to resolve in the computational mesh so wall models are typically used to compensate for the lack of resolution there. Particularly relevant large eddy simulations of turbulent flow, with respect to sediment transport and bedforms, have been successfully performed by Wang and Squires (1996), Calhoun and Street (2001), Zedler and Street (2001), Barr et al. (2004), C'!I ,ig and Scotti (2004) and others. 2.4.2.3 Direct Numerical Simulations It is theoretically possible to bypass the turbulence closure problem altogether by performing direct numerical simulations (DNS) of turbulent flow. The N i.',1 Stokes equations fully describe all quantities of the timedependent flow field, in all three dimensions, if the computational mesh is capable of resolving all scales of motion. Until recently, the application of such models was limited to low or moderate Reynolds number flows over smooth boundaries (van der Werf, 2004). Technological advancements in microprocessors have made it viable to perform DNS simulations at high Reynolds numbers over complex physical boundaries. Recent examples of threedimensional direct numerical simulations of moderate and high Reynolds number flows over stationary ripples are found in Scandura et al. (2000) and Barr et al. (2004). Detailed threedimensional DNS modeling of fluidparticle, and particleparticle, interactions has also been performed by Elghobashi and Truesdell (1992) and Drake and Calantoni (2001). 2.5 Models A number of sediment transport models exist in published literature, ranging from simple empirical models to more complex threedimensional unsteady models. These models fall into one of four categories based on their treatment of transport and hydrodynamics: timeaveraged, quasisteady, semiunsteady, and fully unsteady. Each have benefits and limitations, some of which will be discussed in the following sections. Additionally, most coupled models of fluid hydrodynamics and sediment transport require parametrization of either the fluid or particle phase, but some exceptions do exist. The following sections provide a brief overview of typical model types, their assumptions, and also their shortcomings. 2.5.1 Types The most commonly used sediment transport models may be classified as either timeaveraged, quasisteady, semiunsteady, or fully unsteady. 2.5.1.1 TimeAveraged While simple, timeaveraged models are not necessarily considered robust. These models often rely on flow statistics that have been averaged over durations much longer than would be considered relevant in the coastal environment, such as an individual wave period. Timeaveraged suspended load transport may be computed using a C u integral approach: Q, = C(z) u(z)dz (216) Jo 0h where the dependent variables of sediment concentration (C) and fluid velocity (u) are averaged over one or many wave periods (van der Werf, 2004). A popular timeaveraged model for currentrelated suspended load and bedload transport is given by Bijker (1971). With respect to applications in the coastal environment, the 1i i ir, disadvantage of any timeaveraged sediment transport model is that the waverelated oscillatoryy) component of transport is integrated out of the solution. 2.5.1.2 QuasiSteady Quasisteady models seek to account for both the wave and currentrelated sediment transport components by equating the instantaneous transport to the instantaneous nearbed flow field raised to some power. Implementation of quasisteady transport models, however, are not meant to provide intrawave statistics of sediment transport. Rather, the quasisteady transport is computed as Q J1 u3(t)dt (217) T Jo where Q8 is the waveaveraged sediment transport rate, T is the wave period, and u(t) is an expression of the wave and current velocity field. Considering the example given by van der Werf (2004), the velocity timeseries (u(t)) for a secondorder Stokes wave superimposed on a steady current is given by u(t) = u + u1 cos(wt) + u2 cos(2wt) (218) where u is the current velocity and i1 and u2 are the wave velocity amplitudes. Substituting Equation 218 into Equation 217, and integrating over the wave period, gives the waveaveraged transport rate as a function of both wave and current velocities: 3 2 3 2 3 2, 2 Q, oc 3 1 u+2 4 + tu2). (219) An example of a widelyused quasisteady transport model is given by the total load formulation of Bailard (1981), which is an extension of the earlier energetic model of Bagnold (1966). Useful quasisteady and unsteady bedload formulations are also provided by Ribberink (1998). Due to the timedependent process of vortex shedding, however, quasisteady models are not sufficiently robust to estimate the suspended sediment transport above rippled beds (van der Werf et al., 2006). 2.5.1.3 SemiUnsteady An implicit assumption in timeaveraged and quasisteady transport models is that sediment transport is ahv, in phase with wave forcing. This is not ahv i the case, and inertial effects of both the boundary 1 , r flow and the weight of the sediment itself result in phase differences between applied forcing and subsequent transport [Parker (1975); Luque and van Beek (1976)]. Semiunsteady models attempt to account for these phase differences either through parametrizations, or by considering transport over two successive wave halfcycles. Nielsen (1988) developed a simple 5i ,1> and dump" model of waverelated sediment transport by assuming that transport over ripples occurs through two distinct mechanisms: first, sediment is transported along the ripple face (stoss) and carried over the crest while a lee vortex entrains sediment from the trough and second, liberated sand is lifted by the vortex and subsequently advected by the main flow. Since this process happens twice during each wave cycle, Nielsen (1988) accounts for the forward and backward transport separately through distinct entrainment coefficients based on the instantaneous velocity amplitudes. In this manner, the ,i i1> and dump" model is capable of accounting for wave .,vmmetry. Nielsen (1988) reports that the model provides reasonable estimates of transport for both fine and coarse sands, resolves the phase of maximum transport events, and is extremely practical as it requires only an estimate of a nearbed reference concentration. The concept of dividing transport into successive wave halfcycles to account for phase lags was popularized by the model of Dil iii and Watanabe (1992). Instead of computing an average transport rate over the entire wave period, the halfcycle model solves for the net transport rate over the entire wave period. This is done by taking the linear sum of the transport rates in the positive (first half) and negative (second half) phases of a wave. The halfcycle model consists of two elements of transport for each halfcycle: sediment entrained and transported within the same halfcycle, and sediment entrained during the previous halfcycle and transported during the following halfcycle (van der Werf, 2004). The halfcycle model is considered to be semi, il/, ,.i/; because while it is unsteady over the entire wave period, it is steady for each halfcycle. 2.5.1.4 Unsteady Unsteady models provide estimates of timedependent sediment transport, typically through integration with an unsteady hydrodynamics model. Timedependent values of fluid velocity and sediment concentration may be substituted into a C u integral approach, similar to Equation 216, in order to model the unsteady suspended sediment transport rate: oh Qs(t)= C(z,t) u(z,t)dz. (220) Most quasisteady bedload transport models [e.g. At. i, rPeter and Miller (1948); Ribberink (1998)] may also be adapted for unsteady use by considering a timedependent Shields parameter 0(t), based on the instantaneous bed shear stress, in the formulation (\! ,, en and Grant, 1976). For example, the popular A\t, i, rPeter and Miller (1948) bedload transport equation may be cast in an unsteady form: b(t = 8(0(t) 8,)3/2 (221) where )b(t) is the timedependent, dimensionless bedload function of Eintstein (1950). Various references to the 1 iv, rPeter and Miiller (1948) bedload equation have been made in the literature and some have ir I 1..1 that the coefficient (8) and exponent (3/2) vary somewhat depending on flow and sediment characteristics. Although the functional form was derived from laboratory data on steady flow, Madsen (1991) i r.. Is that this bedload formula performs well for coastal applications, too. Depending on the modeling approach, these unsteady approximations of suspended load and bedload transport rates may either be averaged over individual wave periods to provide estimates of net transport (van der Werf, 2004), or integrated over fractional time steps to provide discrete representations of instantaneous transport. 2.5.2 Integrated Approaches For unsteady models in particular, there are a variety of integrated fluidsediment transport models described in existing literature. The coupling of hydrodynamics and sediment transport often requires either special treatment of the fluid or parametrizations of the sediment phase. Some models solve the detailed hydrodynamics explicitly, often through DNS (Elghobashi and Truesdell, 1992) or using LES turbulence closure (Wang and Squires, 1996), and then incorporate oneway coupling by modeling the particle momentum equations (\! i::;y and Riley, 1983). Others choose to model the particulate phase by solving an advectiondiffusion equation for the suspended sediment concentration (Zedler and Street, 2001), along with a sediment pickup function (van Rijn, 1984) and appropriate boundary conditions (Nielsen et al., 1978). A slightly more detailed approach for simulating fluidparticle flow involves modeling the water and sediment as a continuum, such as in Phillips et al. (1992). An advantage of the continuum approach is that twoway coupling of mass and momentum is implicit in the formulation. The continuum model has three major components: mixture momentum equations, an advectiondiffusion equation for the mixture concentration, and often times a sophisticated diffusivity model. The application of continuum models to large coastal applications remains a relatively new area of research. Explicit treatment of threedimensional hydrodynamics, especially for DNS and LES models, can be timeconsuming. Unsteady hydrodynamic models, however, are necessary for accurately resolving the timedependent coherent motions that develop in the boundary liv. r above a rippled bed. Since the vortex motions determine a iii i ilry of entrainment and transport in the rippled bed regime (Sleath, 1991; Nielsen, 1992), it is possible to parametrize the bulk hydrodynamics by considering the transport of vorticity in the flow field rather than the momentum. Such an approach is described in Malarkey and Davies (2002), where a discrete vortex model (DVM) is used to simulate the timedependent vorticity field above a rippled bed in oscillatory flow. Malarkey and Davies (2002) present results of simulations using a simple inviscid DVM, and a slightly more advanced "cloudincell" (CIC) model that considers a distribution of vorticity point values within the overall vortex. These models are capable of estimating reasonable values of vorticity, but often fail to accurately capture the phase of vortex formation and ejection. Compared to RANS models, the discrete vortex models are bettersuited for resolving sharp gradients in the vorticity field since they do not suffer from the numerical diffusion related to advection (i\ I, Il:;.y and Davies, 2004). An alternative method for simulating coherent motions in the boundary 1,v.r was presented by Davies and Villaret (1997), who sel 1. that vortex shedding could be modeled as a "coi,. I I ,. stress represented by a timedependent, heightinvariant eddy viscosity. The convectivee eddy vi ... ii relates the convective shear stress to the velocity gradient. Examples of onedimensional convective eddy viscosity models are given by Davies and Thorne (2002) and Malarkey and Davies (2004). Model simulations by Malarkey and Davies (2004) sirl. 1 that the convective eddy viscosity approach is valid in a thickness of about one or two ripple heights above the mean bed where the coherent motions exist in a convective 1V.,r. More advanced modeling techniques exist for the simulation of multiphase flows in coastal applications. For example, Calantoni (2002) presents a discrete particle model for bedload transport in the surf zone. The discrete particle model simulates the dynamics of the fluid flow and accounts for the kinematics of each particle on an individual basis. Therefore, terms that are typically parametrized in other models, such as granular stresses and momentum transfer through elastic collisions, are accounted for explicitly through first principles. Solving momentum equations for the fluid, as well as each particle, is computationally expensive and applications of discrete particle models are currently reserved for research problems. An alternative approach is to model all of the sediment particles as a separate phase, much as the fluid phase is modeled as a homogeneous medium. Following this methodology, Dong and Zhang (1999) developed a twophase model of fluid and sediment flow in oscillatory sheetflow. The model solves the continuity and linearized momentum equations for fluid and sediment. Turbulent and intergranular stresses are incorporated in the solution algorithm, thereby providing a complete description of the interaction forces between the fluid and sediment phases. 2.5.3 Shortcomings Analytical models of sediment transport are not readily available. Most sediment transport relations may be characterized as either empirical or semiempirical at best. Indeed, much of the literature published on sediment transport has focused on providing parametrizations of transport based on laboratory, and sometimes field, experiments. This tends to be acceptable when developing a model for a specific application to a specific set of criteria (e.g. forcing, sediment size and gradation, transport mode and regime), but makes it difficult to develop universal sediment transport models. Regardless of the model framework chosen, the limiting factor on accuracy and predictability seems to lie in the treatment of sediment transport and not necessarily the hydrodynamics. (a) 1 05 1 (b) 3 2 _ 1 Linear . Stokes 1 2 3 4 5 Time (s) Figure 21. Comparison of (a) velocity and (b) shear stress time series for linear and nonlinear waves with period T = 6 s in water depth h = 1 m. Figure 22. Approximate distribution of fluid and grain shear stress in steady open channel flow. Adapted from Fredsoe and Deigaard (1992). Table 21. Common power law formulations for bedload sediment transport by currents and waves. Shields' parameter based on currents is Oc and for waves is O0. Reference Dimensionless Bedload Discharge Currents 4b AI, iv rPeter and Muller (1948) 8(0c Ocr)1.5 12v/O(Oc Ocr) 11(Oc _Ocr)1.65 Nielsen (1992) Ribberink (1998) Camenen and Larson (2005) Waves 12015 exp(4.50cr/Oc) )1/2 Madsen and Grant (1976) Soulsby (1997) Waves Bailard and Inman (1981) CbfwU(t)3/[(S 1)22 tan d5o] 11(18,(t)l OCr)1.65 l (t)/l W(t)] Ribberink (1998) Waves Dil, i i; I and Watanabe (1992) Camenen and Larson (2005) 0.001W~For5(F/ F)/ (8s 1),~.l a Ocwon + cw,offOw,m exp(bOcr/0w) Table 22. Empirical entrainment functions for suspended load sediment transport. Reference Dimensionless PickUp Rate, 4, FernandezLuque (1974) 0.02(0 Ocr)1.5 N .1;: 1i. a and Tsujimoto (1980) 0.02(1 0.035/0)3o van Rijn (1984) S(t) 12.5W, 0 3 ( 1),/.,, 5.1(0w Ocr)1.5 0.00033[(s 1),l',,/'v2]]0.1/(/ cr )1.5 CHAPTER 3 BEDFORM DYNAMICS 3.1 Introduction Differential sediment transport on the seabed creates morphological features that range in size from smallscale sand ripples to much larger, shoreparallel sandbars. Regardless of their size, the resulting morphological structure plh,v an important role in both sediment transport and hydrodynamic processes. Shoreparallel sandbars have been shown to intensify the crossshore currents that promote offshore bar migration through gradients in the crossshore sediment transport (Gallagher et al., 1998). Smallscale bedforms, such as sand ripples, determine the generation of turbulence and entrainment of sand near the bed (Hanes et al., 2001) largely through flow separation that leads to an organized pattern of timedependent vortex shedding in each halfcycle of a wave. These vortices often determine the timing of sediment entrainment and deposition (Nielsen, 1988), the amount of sediment carried in suspension (van der Werf et al., 2005), and are responsible for the dissipation of wave energy in the boundary 1. r (Tunstall, 1973; Tunstall and Inman, 1975). The presence of smallscale bedforms has also been found to cause natural sorting of sediment, resulting in a redistribution of fine and coarse material on the seabed (Faraci and Foti, 2002). The focus of this chapter is on smallscale bedforms, such as sand ripples, so further discussion of larger features like sandbars will be limited. The following sections provide a brief overview of bedform types, their classification and characteristics, mechanisms for their growth, their affect on the bottom boundary I .;r, and methods for predicting their length and height. Additional information on both simple and complex morphological models is also provided. 3.2 Bedform Types Smallscale bedforms may be broadly categorized by the dominant flow conditions under which they are formed: currentgenerated bedforms or wavegenerated bedforms. Early accounts of wavegenerated ripples in the sea date back to the observations of Hunt (1882), who found that "ripple n i Il: were formed on a sandy seabed by a slight oscillation of the water. Laboratory experiments by Darwin (1883), and later by Ayrton (1904), provided mostly qualitative data about the formation of ripples in oscillating flow. Wavegenerated bedforms are further distinguished by Bagnold (1946) as either rolling grain or vortex ripples. A distinction is made between these two types of wavegenerated ripples not only because they grow through different processes, but also because their characteristic length, height, and steepness scale differently. Briefly, rolling grain ripples grow through an .,.::regation of sand particles in areas of high friction (Darwin, 1883) but do not typically scale with waverelated quantities. Vortex ripples form once a perturbation on the bed is large enough to induce flow separation (Bagnold, 1946), such that an adverse pressure gradient is formed in the lee of the ripple crest (Ayrton, 1904). This process may occur either through the growth of rolling grain ripples or due to existing perturbations on the seafloor. Specific details about the mechanisms for growth and evolution are given in the following sections. Currentgenerated bedforms include those formed in rivers and open channel hydraulic flows. The similarity between these two environments is that the predominant forcing is, more or less, onedimensional steady flow. While not ahv, truly I. ily," the term is used here to convey an absence of a persistent oscillatory, or repeating, flow. Shallow tidallydominated inlets and estuaries may even be a special case of I. ,Ily" flow if the ambient oscillatory motion is weak compared to the tidalinduced currents. Currentgenerated ripples initially form in much the same way(s) as wavegenerated ripples, but once the bedform is large enough to induce flow separation the dynamics are different. Since the steady flow has a constant phase, a recirculation zone is only formed on one sidethe lee sideof the bedform. The dynamics of the bedform are controlled by a balance between sand transported along the stoss, sand transported over the crest onto the lee side, gravitational forces pulling sediment down the lee side, and the transport of sand up the lee side from the attached vortex. This commonly results in a bedform with an .,ivii,,. I ic profile where the stoss side has a milder slope than that of the lee side (Darwin, 1883). It is common for the lee side of an .,ivii,,. I ii. ripple to have a slope very close to the natural angle of repose. Once in equilibrium with the surrounding flow, steady currents are capable of maintaining the ripple through a dynamic equilibrium but are often displaced downstream due to higher velocities at the crest than at the trough (Darwin, 1883). 3.2.1 Rolling Grain Ripples Bagnold (1946) was the first to classify the two types of wavegenerated ripples as either rolling grain or vortex, but a physical distinction between them was made earlier by Darwin (1883). The laboratory experiments of Darwin (1883) demonstrated that oscillatory motion first created what he called transient ripples as a result of particle ..iregation in areas of high friction. The process of ..:'regation, he noted, increases the friction further and traps even more particles in that area. Others have characterized the process similarly, i .. 1i i; that each grain creates a region of weaker flow in its lee ultimately causing grains to group into transverse zones that form ripples with larger shadow zones (Bagnold, 1946; Blondeaux, 1990; Vittori and Blondeaux, 1990; Andersen, 1999). The particle model of Andersen (2001) demonstrates that rolling grain ripples form and coarsen due to gradients in the transport velocity from one side of the .r. 'egation to the other. Transient ripples were found to have a wavelength approximately onehalf of the final ripple wavelength in the experiments of Darwin (1883), which seems to be a consistent observation in other laboratory experiments (Sleath, 1976). Rolling grain ripples continue to grow in height through ... regation and trapping until the lee slopes are steep enough to allow a vortex to form in the sheltered space behind the crest (Bagnold, 1946), eventually forming vortex ripples (Scherer et al., 1999; Faraci and Foti, 2001). During the initial growth process, the steepness (height/length) of rolling grain ripples has been found to be consistently near 0.1 by Sleath (1976) and others. Darwin (1883) qualitatively Ii... I. 1 that the initial growth process only occurred between an upper and lower limit of fluid velocity, and that the limits were a function of average sediment properties like size and density. These upper and lower limits were found by Bagnold (1946) to be 20cr and 0r,, respectively. The stability of rolling grain ripples formed during laboratory experiments is unclear. Due to their low relief and transient nature, rolling grain ripples are rarely observed in the field. Rolling grain ripples were found to be stable in the experiments of Bagnold (1946) and Sleath (1976), but Miller and Komar (1980b) i,i. 1 that these two instances may be a result of the testing apparatus used during the experiments (oscillating tray). Experiments in rotating annular cells (Scherer et al., 1999; Stegner and Wesfreid, 1999) and oscillating water tunnels ii.. 1 that rolling grain ripples are transient, unstable features (\ I!1. r and Komar, 1980b). The prevailing theory, now, is that rolling grain ripples are transient bedforms that initiate the growth of larger, vortex ripples through an inverse cascade mechanism: the dynamical evolution involves coarsening from an initially short wavelength (rolling grain ripples) to a longer wavelength (vortex ripples), with saturation at a final equilibrium condition (Stegner and Wesfreid, 1999; Andersen et al., 2002). While Darwin (1883) and Bagnold (1946) provided qualitative descriptions of the growth of rolling grain ripples from an initially flat bed, Kennedy (1963, 1969) demonstrated that if a lag is assumed to exist between velocity and transport, an instability exists at the sandwater interface (Coleman and Melville, 1994). This lag is often attributed to inertial effects of the sediment (Parker, 1975), and has been identified as the ripple "i,'i.'i:; by Smith (1970). Initiation of growth through this ripple instability, however, should not be confused with the stability of rolling grain ripples described above; the former refers to a mechanism that initiates growth while the latter conveys an ability of the bedform to achieve an equilibrium condition with respect to its flow conditions. If a spatial lag is assumed to exist between velocity and transport, Parker (1975) finds, through a linear stability analysis of derived relations for flow and transport, an inherent instability of flat beds which leads to the formation of river antidunes. However, if velocity (shear) and transport are assumed to be in phase with one another, flat sand beds are unconditionally stable (Parker, 1975). Under pure bedload transport, a flat bed is also unstable at all wavelengths to small perturbations in topography (Smith, 1970). This sir.. I that one of two necessary conditions must be met for the initiation of ripple growth from a flat bed; either forcing and transport must be decoupled, or some initial perturbation in bed topography must be present. Hayakawa (1970) was able to develop a theory of dune formation in open channel flows by applying the ripple instability concept, and others have had similar success (Coleman and Melville, 1994). Some predictive models of rolling grain ripple geometry, based on the ripple instability concept, have also been developed. Sleath (1976) created a stability analysis model capable of predicting the wavelength of rolling grain ripples. More recently, Foti and Blondeaux (1995a) developed a predictive theory of ripple formation under waves that agrees well with experimental data. Assuming a fully turbulent boundary lv.vr, Foti and Blondeaux (1995a) obtained closed form solutions of turbulent oscillatory flow over a wavy surface through a linear stability analysis of the governing equations. Detailed particle models of rolling grain ripples also exist (Andersen, 2001). 3.2.2 Vortex Ripples Bagnold (1946) ir.. 1. I that wavegenerated ripples fall into one of two categories: rolling grain or vortex. The deciding factor, Bagnold determined, was based on the range of applied stress at the bed. Once the Shields parameter increases beyond 20cr, the lee slopes of the ripple grow to a height large enough that a vortex is formed in the lee (Bagnold, 1946). The process that Bagnold (1946) observed in the laboratory involved a transition first from a flat bed to one with rolling grain ripples; and second, from rolling grain ripples to vortex ripples. It is also possible for vortex ripples to form without rolling grain ripples if there is some sort of natural perturbation on the bed large enough to induce flow separation (Ayrton, 1904). So while rolling grain ripples are a sufficient mechanism for the growth of vortex ripples, the induced flow separation and subsequent vortex formation are necessary conditions. The ability of vortex ripples to grow from natural perturbations may also explain why rolling grain ripples are rarely observed in the field (Mi!!,. r and Komar, 1980b), whereas vortex ripples are often detected (M!1!!, r and Komar, 1980a; Hanes et al., 2001; Ardhuin et al., 2002). Vortex ripples pliv an integral role in both sediment transport and energy dissipation, as discussed previously. Central to the effects of vortex ripples on boundary lv. r processes is the formation and ejection of vortices from the ripple profile into the main flow during each successive halfcycle of oscillatory forcing. Vortices ejected over the ripple crest can entrain significant amounts of sediment within the boundary 1~. r that is subsequently advected by the outer flow (Gallagher et al., 1998; van der Werf et al., 2005). This is thought to increase the phase lag between suspension and transport relative to transport over a flat bed (Inman and Bowen, 1963), leading to timedependent suspended sediment transport processes that cannot be modeled in a phaseaveraged or quasisteady manner (van der Werf et al., 2006). The formation of vortices in the lee of bedforms results from flow separation at the crest, similar in many vv to flow separation around bluff bodies, cylinders, and the like. In her laboratory experiments, Ayrton (1904) identified two conditions necessary for vortex formation in the lee of ripples: 1. a reduction of pressure in the lee; and 2. an upwarddirected resultant gravity pressure along the lee ridge. The first condition results from flow separating at the crest, creating a sheltered region in the lee of the ripple. The second condition is due to an adverse pressure gradient that develops due to phase differences between the boundary li.r flow and the freestream forcingan inertial property of boundary l~vr flows. Adverse pressure gradients may also be attributed to surface gravity waves, but are not present in steady flows over flat beds. Ayrton (1904) speculated that vortex formation could not occur in steady currents and that ripples could not be created or maintained by such flow conditions. Recent laboratory experiments of bedform growth from a flat bed in steady currents have proved this to be false (\I! ii. 1990; Coleman and Melville, 1994, 1996). Under oscillatory forcing, vortices are formed and ejected in alternating halfcycles. A schematic of vortex formation and ejection as a function of fluid phase and forcing is shown in Figure 31. Vortices form at the beginning of each halfcycle (Figures 3la, 31c, 3le), growing in strength and size to maximum values at a phase of approximately 90 degrees (Earnshaw and Greated, 1998). During phases of maximum forcing, strong flow separation and vortex shedding near the ripple crest serve as mechanisms for the production of turbulence in the boundary 1, r (Barr et al., 2004). After contracting in size slightly due to a reduction in the applied forcing, the ejected vortices in Figures 3lc and 3le continue to increase in size as they move vertically into areas of decreasing pressure (Ayrton, 1904). Neglecting small perturbations due to turbulence, these processes are more or less symmetric from one halfcycle to the next. Under real waves, however, vortex formation and ejection is .,iviii,. iir i owing to Stokes' law (Earnshaw and Greated, 1998). This can have a significant effect on the net sediment transport in the crossshore as suspended load may be advected shoreward by the Eulerian flow. Flow separation and vortex development are imperative for the growth and stability of vortex ripples under a variety of forcing conditions (Nielsen, 1981). During the growth process, the nearbed hydrodynamics result in net sediment transport toward the ripple crest (Andersen and Freds0e, 1999). The impinging flow accelerates as it travels along the upwind side of the ripple (the stoss) creating strong shear stresses at the bed that drive sediment transport from the trough toward the crest. In the lee of the ripple, the counterrotating vortex induces a bed shear stress that is directed up the leeside of the ripple resulting in transport toward the crest. This is the primary growth mechanism for vortex ripples in nearequilibrium conditions. Experimental data of bedforms in a wave flume i. i; that the alternating vortices support the ripple in each successive halfcycle when 0 < 0.2 Faraci and Foti (2002). For 0 > 0.2, the stress at the crest is so large that the induced erosion cannot be overcome by the stabilizing effect of the vortices. 3.2.3 Characteristics The growth and evolution of vortex ripples from a flat bed may occur through one of two processes; one involves a transient state prior to equilibration and the other i.i, 1 relatively constant growth from a flat bed. For weak flows and under regular waves, vortex ripples form through coarsening of transient rolling grain ripples (Stegner and Wesfreid, 1999; Faraci and Foti, 2002). The coarsening process is characterized by an increase in wavelength, resulting in fewer ripples, with saturation to an equilibrium condition (Andersen et al., 2002). The coarsening process does not occur for bedforms in stronger flows or under irregular wave conditions where they reach equilibrium directly through constant growth (Stegner and Wesfreid, 1999; Faraci and Foti, 2002). An overwhelming amount of physical data demonstrates the ability of vortex ripples to transform from a smaller to a larger wavelength, but rarely does this happen in reverse. Vortex ripples in the field and lab have been found to exhibit strong hysteresis (Traykovski et al., 1999; Stegner and Wesfreid, 1999) whereby changes in stable ripple patterns from larger to shorter wavelengths are not observed. This i... I that larger ripples must be completely destroyed by an increase in the forcing conditions before a smaller stable ripple pattern grows in response to new forcing. Rolling grain ripples and vortex ripples each have unique characteristics due in large part to the forcing conditions that create them. The rolling grain ripples observed and measured in a number of lab experiments (Bagnold, 1946; Sleath, 1976; Faraci and Foti, 2002) consistently have a steepness (T,]/Ar) of 0.1 regardless of the type of flow and testing apparatus, where rl, and A, are the ripple height and wavelength, respectively. Vortex ripples tend to have a maximum steepness almost twice that of rolling grain ripples at approximately 0.18 (Nielsen, 1981; Faraci and Foti, 2002). For 0 < 0.2, Faraci and Foti (2002) find that the ripple steepness remains constant at its maximum value that corresponds well with the value given by 0.32 tan Q. The wavelength of vortex ripples commonly increase along with the heightkeeping a more or less constant steepnessuntil an equilibrium is reached. This equilibrium is a function of flow and sediment characteristics (Lofquist, 1978) and the final steepness tends to maximize resistance to the local flow conditions (Davies, 1980). Doucette and O'Donoghue (2006) identify three distinct processes through which the ripple wavelength evolves: slide, merge, and split. The slide process identifies a gradual increase (decrease) of wavelength as .il.i i :ent ripples move away from (toward) each other. A merge occurs when two larger flanking ripples increase in height and move toward each other, d. 1 vii._; the smaller ripple in between. A split refers to a situation when one large ripple decreases in size through the crest splitting into what appear to be two or more ripples of shorter wavelength. When bedload transport dominates over suspended load, vortex ripples grow through the processes of merge and slide and decay through the slide and split mechanisms (Doucette and O'Donoghue, 2006). The characteristics of vortex ripples have been found in rn inr: cases to scale with properties of the fluid forcing. Clifton (1976) first sl, 1. 1 that a particular ripple regime existed whereby the ripple wavelengths were constant multiples of the nearbed orbital diameter 2A: A Ua (31) where Umax is the maximum nearbed velocity amplitude and w is the angular frequency of the forcing. Here, A represents the orbital semiexcursion length which is onehalf the orbital diameter (2A). A number of relationships between ripple wavelength A, and orbital semiexcursion A have been presented in the literature, and range from Ar 0.78A for suspended dominated ripples (Andersen and Fredsoe, 1999) to the upper estimate of A, w 1.33A of Nielsen (1992) for bedload dominated conditions. Some common wavelength scaling relationships for orbital ripples are given in Table 31. An alternative theory of bedform scaling is offered by Wiberg and Harris (1994) who ir,.. 1 that for fieldscale flows, vortex ripples scale best with the grain size. This is not ahv, true, however, as Ardhuin et al. (2002) found that waveformed vortex ripple wavelengths on the continental shelf were proportional to the nearbed orbital excursions, not the grain size. Data from lab and field experiments do consistently show discrepancies between vortex ripple scaling [e.g. Miller and Komar (1980a); Nielsen (1981); Faraci and Foti (2002)] and many attribute these differences to the disparity between flow conditions. Whereas the in ii i ly of laboratory experiments are confined to regular oscillatory forcing with short periods, conditions in the field are represented by a broad spectrum of wave heights, frequencies, directions, and other flow properties. Attempts to unify appropriate vortex ripple scales for field and lab data have received increasing attention recently [e.g. Miller and Komar (1980a); O'Donoghue and Clubb (2001); Williams et al. (2005); Lacy et al. (2006)] and will be discussed further in the following sections. Wiberg and Harris (1994) expand on the earlier concepts of Clifton (1976) who identified three categories of symmetric ripples that differ in their characteristics: 1. orbital ripples; 2. anorbital ripples; and 3. suborbital ripples. Where orbital ripples scale with the orbital fluid motion and are commonly found in lab experiments, Wiberg and Harris (1994) ,.::. 1 that anorbital ripples scale with sediment properties and are indicative of vortex ripples found in the field. Suborbital ripples are classified as transitional bedforms lying somewhere between orbital and anorbital ripples. Wiberg and Harris (1994) state that anorbital and orbital ripple height T,] scale with the oscillatory boundary lv. r thickness 6 such that orbital ripple heights are approximately 26 while anorbital ripple heights are about 6/4. This is demonstrated in Figure 32 and shows that while anorbital ripples are completely immersed within the boundary lr, orbital ripples tend to protrude from it. The significance of this is made evident through the choice of roughness parametrizations in coastal hydrodynamic models, where one is forced to choose appropriate values of boundary roughness based on the interaction of bedforms with the boundary 1v'. 3.3 Experiments A number of experiments on bedform dynamics have been performed in an effort to provide better understanding of ripple growth and evolution, as well as sediment transport characteristics in the rippled bed regime. These experiments range from small to largescale laboratory experiments utilizing a number of different testing methods, to field experiments and observations of bedform dynamics under the stochastic conditions found in nature. The experimental data have been used to develop new empirically derived ripple predictors (discussed in the next section) and have lead to the conclusion that discrepancies exist between bedforms produced by lab and fieldscale flows. Additionally, some have found that the type of laboratory apparatus used in experiments may affect bedform characteristics and behavior. A brief overview of experiments on bedform dynamics is provided in the following sections and a summary of historical experiments and data sets is provided in Table 32, along with citation keys used in the following sections. 3.3.1 Laboratory Miller and Komar (1980b) provide a helpful review of commonlyused experimental techniques used for simulating flows in the laboratory. Although some experimental data from experiments in rotating annular cells exist in published bedform literature (Scherer et al., 1999; Stegner and Wesfreid, 1999), by far the most widely used experimental setups include: oscillating tray; oscillating water tunnel; and flume. Each of the devices listed, however, have unique characteristics that have some consequence on the characteristics and dynamics of bedforms. 3.3.1.1 Oscillating Tray The oscillating tray device has been used to study hydrodynamics in the vicinity of stationary (simulated) ripples (Earnshaw and Greated, 1998) and also bedform evolution in a sand bed (Bagnold, 1946; Manohar, 1955; Sleath, 1976, 1985; Scherer et al., 1999; Lacy et al., 2006). In these experiments, a tray of sand or stationary bedforms is oscillated at a particular frequency through a tank of still water thereby simulating the oscillatory motion of water waves. Lacy et al. (2006) used a modified setup to simulate combined wave and current forcing by passing a steady current over the tray as it oscillated in a flume at various angles to the steady flow. The flow induced by the oscillating tray apparatus, however, is not a direct analog of oscillatory wave motion over a static bed. Steady streaming effects found in oscillatory boundary lr flows are not simulated by the apparatus, thereby excluding the additional vertical flux of momentum directed away from the bed. The motion of the tray has also been found to induce a mean flow directed opposite to the direction of travel (Scherer et al., 1999). Observed bedform characteristics in oscillating tray experiments also appear to be unique, where Scherer et al. (1999) note that the apparatus imparts an additional inertial force to the bedforms. Furthermore, rolling grain ripples have only been found to be stable in oscillating tray experiments where they are transient in oscillating water tunnel and flume experiments, and rarely observed in the field (1i,!! r and Komar, 1980b). 3.3.1.2 Oscillating Water Tunnel Among others, the oscillating water tunnel has been used to study bedform characteristics by Foti and Blondeaux (1995a), V, o'i' v et al. (1999), O'Donoghue and Clubb (2001), and Doucette and O'Donoghue (2006). Similar to the oscillating tray apparatus, oscillating water tunnels are designed to simulate the oscillatory motion of waves. Most oscillating water tunnels consist of a long closedconduit horizontal test section and pistons located in vertical risers at each end that are used to drive the oscillatory flow. The resulting oscillatory flow is purely horizontal and mostly onedimensional. Like the oscillating tray apparatus, the horizontal flow is of constant phase and lacks any vertical component due to boundary l1.r streaming. In some cases it is possible to simulate a collinear steady current superimposed on the oscillatory motion. A distinct advantage of the oscillating water tunnel is that the flow characteristics can be precisely prescribed and controlled. While most flume and oscillating tray experiments are limited to labscale flows of shortperiod waves, low to moderate current velocities, and low to moderate mobility numbers, the oscillating water tunnel has been used to simulate wave conditions more commonly found in nature. O'Donoghue and Clubb (2001) and Doucette and O'Donoghue (2006) utilized the water tunnel apparatus to simulate longperiod waves and to analyze the differences between symmetric and .,ii, i,,. 1 ic forcing on bedform characteristics. While no significant difference in ripple height and wavelength was found for symmetric and .ivmmetric flows, O'Donoghue and Clubb (2001) did find that previously sl.. 1. ,1 methods for predicting bedform geometry were insufficient, especially for threedimensional ripples. 3.3.1.3 Flume Flumes have been used extensively to study bedform dynamics under steady currents (Yalin, 1985; Mantz, 1990; Coleman and Melville, 1994, 1996), waves (1M'1!. r and Komar, 1980b; Marsh et al., 1999; Faraci and Foti, 2002), and combined wave and current forcing (Khelifa and Ouellet, 2000; Lacy et al., 2006). An advantage of flume experiments over oscillating tray and oscillating water tunnel experiments is that the true oscillatory nearbed motion is simulated by the surface gravity waves. However, most laboratory flumes are only capable of producing shortperiod, unidirectional waves not truly representative of field conditions. Certain exceptions do exist, such as the large wave flumes at Oregon State and Delft University of Technology, and experiments in these facilities help bridge the information gap between labscale and fieldscale bedform characteristics (Mi !!. r and Komar, 1980b). As mentioned earlier, the oscillatory motion produced by oscillating tray and oscillating water tunnel experiments is not a true representation of waveinduced motion in the boundary 1.vr. Progressive waves induce a steady streaming in the boundary l vr that, while weak compared to the oscillatory flow, has a significant effect on sediment transport (\1 iii, 2004). Boundary livr streaming is characterized by a vertical diffusive flux of momentum directed away from the horizontal boundary 1 Vr. DohmenJanssen and Hanes (2002) ir:. 1 that this streaming results in net transport rates under waves that are 2.5 times larger than those in uniform horizontal oscillatory flow. Furthermore, the wave bottom boundary 1vir contains components of velocity in both horizontal and vertical directions. These vertical velocities, which are not exactly out of phase with the horizontal velocities in the bottom boundary 1;Vr, lead to a nonzero timeaveraged bed shear stress (\I i ii, 2004). Field conditions are rarely classified by one type and size of wave. Rather, field data are characterized by a broad spectrum of wave frequencies, directions, heights, and current velocities (\i!!,. r and Komar, 1980a). Even the few large wave flumes capable of producing longperiod, irregular waves are unable to simulate multidirectional waves or waves superimposed on a steady current. Therefore, methods for predicting ripple geometry that have been based on laboratory data tend to fail when applied to field data. In an effort to overcome at least one of these shortcomings, Khelifa and Ouellet (2000) performed laboratory experiments on bedform characteristics in combined wave and current flows. They accomplished this by using two intersecting flumes at Laval University and were able to simulate progressive waves combined with steady currents at 600 and 900 degree angles to one another. The resulting data were used to formulate new empirically derived expressions for bedform height and length in combined flows, but unfortunately their utility is somewhat limited by the short (< 2 s) wave periods generated in the flume. More recently, Lacy et al. (2006) attempted to overcome this scale discrepancy by simulating fieldscale wave periods (> 8 s) and current velocities with an oscillating tray in a current flume. The fullyinstrumented tray was filled with sand and then oscillated at various anglesboth oblique and orthogonalto a steady current generated in the flume. The data from their experiments are awaiting publication. 3.3.2 Field Observations of ripple characteristics under field conditions have been found to differ substantially (about 3i ii) from those found in laboratory experiments (Faraci and Foti, 2002). In the laboratory, Marsh et al. (1999) ,i. 1 that bedform length scales rapidly under the monochromatic forcing typical of most wave flumes. They speculate, however, that ripple wavelength must be difficult to change in the field since the bedforms are subjected to a broad spectrum of wave frequencies. Often times observed ripple characteristics are not in direct equilibrium with the local forcing and experience significant hysteresis (Traykovski et al., 1999). More specifically, bedforms in the field may be a result of an antecedent wave climate that remain static as local forcing conditions subside below the critical threshold required for sediment mobilization. This has made it difficult to formulate unified expressions for bedform geometry under fieldscale conditions. Additional complications arise when trying to associate ripple processes with a single, statistical representation of a forcing spectrum composed of many wave frequencies and directions. Miller and Komar (1980a) ir.., I 1 that significant wave parameters computed from field spectra should be used in the calculation of relevant bedform parameters such as maximum orbital velocity amplitude, orbital excursion length, mobility number, and Shields' parameter. Nevertheless, it remains difficult to formulate new expressions based on field data and the predictive capability of methods derived from fieldscale lab experiments is still poor. Observations and measurements of bedform characteristics in the field, however, continue to supplement existing theory and provide more detailed information about their geometry and behavior under natural forcing conditions (Li and Amos, 1999; Traykovski et al., 1999; Hanes et al., 2001; Ardhuin et al., 2002). 3.4 Ripple Predictors The following section presents some commonly used methodologies from existing literature for predicting ripple height ,]r and wavelength A,. These ripple predictors are based on extensive sets of data from lab and field experiments (see Table 32), and in many cases are based on nondimensional groups that relate sediment characteristics with flow properties. A number of these dimensionless parameters used to classify or characterize bedforms are listed in Table 33. 3.4.1 Clifton (1976) The semiquantitative model of Clifton (1976) (C76) is based on the nearbed orbital velocity Ub and nearbed orbital velocity .ivimmetry Aub: Ub = Au (32a) H2 Aub 14.8 sinh4 kh (32b) LT where H is wave height, L is wavelength, k is the wavenumber, and h is water depth. These parameters are then used to identify one of four bed states: (1) no sediment movement; (2) symmetric ripples; (3) .,vmmetric ripples; and (4) sheetflow. The symmetric ripple bed state is further divided by Clifton (1976) into orbital, anorbital, and suborbital ripples as discussed previously. 3.4.2 Nielsen (1981) Nielsen (1981) (N81) offers semiempirical formulations for the size and shape of vortex ripples derived from an analysis of various sets of lab and field data. A strong dependence on Shields' parameter 0 was found for ripple steepness while the ripple wavelength A, data collapsed best when plotted against the mobility number T (Brebner, 1980), which is a ratio between sediment destabilizing and stabilizing forces. The formulations offered by Nielsen (1981) are particularly useful since they differentiate between lab and field conditions. When compared to the BASEX data of Vincent and Osborne (1993), Marsh et al. (1999) find that the two sets of equations serve as bounding equations for the data. For relatively weak conditions (T < 20), the ripple wavelength and steepness are given by A, a 1.3A (33a) (r,/A,)max 0.32 tan Q. (33b) For conditions outside of this range, or for general use, Nielsen (1981) provides the following relationships for regular laboratory waves; A,/A = 2.2 0.3450.34 (2 < T < 230), (34a) qr/Ar 0.182 0.240, (34b) 0.275 0.022V for ( < 156 r ,/A = (34c) 0 for T > 156 and for irregular field waves; /A exp 693 0.371n8 (35a) 1000 + 0.75 In7 qj or/Ar = 0.342 0.34 5, (35b) r,/A = 2141.85 (4 > 10). (35c) Equations 34c and 35c correspond specifically to quartz sand with s = 2.65. Parameters such as A, T, and 02.5 used in the ripple predictors for irregular waves (Equations 35a 35c) should be based on significant wave parameters (Nielsen, 1981). In the preceding expressions, the grain roughness Shields parameter 02.5 is computed as 1/2 f2.5p(A)2 1 02.5 2p(.5 (36a) p(s 1)gd 2 where the grain roughness friction factor f2.5 is based on the formulation of Swart (1974) (Equation 24) with a roughness (k,) of 2.5d50, S 2 5d 0.194  f2.5 exp 5.213 2.5 5.977 (36b) 3.4.3 Grant and Madsen (1982) Based on the lab data of C69, the formulations of Grant and Madsen (1982) (GM82) for bedform characteristics are based on the skin friction component of boundary shear stress. Grant and Madsen (1982) identify two specific ranges for bedform behavior; an equilibrium range where ripple steepness remains constant and the wavelength changes with the shear stress; and a breakoff range at higher shear stress where ripple steepness decreases. For the equilibrium range (0/Ocr) < (O/Ocr)B, the ripple height and steepness are given by Equations 37a and 37b, respectively. Ripple characteristics in the breakoff range (0/Oc) > (0/Ocr)B are computed using Equations 38a and 38b. The breakoff range is determined as (0/0cr)B 1.8S.6, where S, is a dimensionless sediment parameter defined in Table 33. lr 0.22A(0/0,)0.16 (3 7a) T,//A, = 0.16(0/0r)0.04 (37b) rr = 0.48ASo0(8/o0) 15 (38a) r/Ar = 0.28S0.6(0/0)1.0 (38b) Unfortunately, accurate values of the boundary shear stress are not commonly known a prior, nor are they characterized by one specific value. These two shortcomings make the ripple predictor of Grant and Madsen (1982) a cumbersome choice for both estimating and comparing ripple characteristics. 3.4.4 Wiberg and Harris (1994) When plotted against the relative orbital diameter 2A/d, Wiberg and Harris (1994) (WH94) find that nondimensional ripple properties from field and lab data fall into unique groups. They propose that a lack of substantial overlap in forcing conditions between lab (regular waves, shortperiod flows) and field (irregular waves, longperiod flows) data makes it difficult to effectively classify ripples or to characterize their height and length. Indeed, there are few existing ripple predictors that perform equally as well at predicting lab and fieldscale ripples. Through an analysis of the laboratory data of C69, KF65, MK72, D74, and the field data of 157 and D74, Wiberg and Harris (1994) find that anorbital ripple wavelength Xno found in field data is independent of the nearbed orbital excursions. Instead, they find that ,no scales proportionally with the grain diameter d ranging between 400d < Ano < 600d, similar to Clifton and Dingier (1984). The methodology of Wiberg and Harris (1994) involves a sequence of steps that first assumes that the ripples are classified as anorbital ripples, where the wavelength and steepness are given by Equations 39a and 39b, respectively. Of course, the expression for the steepness has the added disadvantage of requiring an iterative process to solve for /ano given Aano and A. ano = 535d (39a) (/A)ano exp 0.095 In + 0.442 In  2.28 (39b) / ano ) Tano Once the anorbital ripple height has been calculated using Equation 39b, the following ranges are used to determine whether the ripple should be classified as orbital (Equation 310a), anorbital (Equation 310b), or suborbital (Equation 310c): 2A/ lno < 20 orbital ripples (310a) 2A/riano > 100 anorbital ripples (3O10b) 20 < 2A/riano < 100 suborbital ripples (310c) If the computed value of 2A/,ano does not fall in the anorbital range, then the ripple properties must then be computed using either orbital (Equations 3 1a and 31 b) or suborbital (Equations 312a and 312b) formulations. Wiberg and Harris (1994) ,i.. 1 that the wavelengths of suborbital ripples fall between those of orbital and anorbital ripples; therefore, Equation 312a represents a weighted geometric average of the orbital and anorbital ripple wavelengths given by Equations 3 1a and 39a, respectively. The suborbital ripple height is determined iteratively using Equation 312b, which is simply a modification of Equation 39b previously defined. Aorb = 1.24A (3 1a) (q/A),,b = 0.17 (311b) Asub = exp L 1n2 lnlOO ) (ln orb In Aano) + In Aano (312a) /)sub exp 0.095 In + 0.442 In 2.28 (312b) L \ sub rlsub 3.4.5 Mogridge et al. (1994) Mogridge et al. (1994) (\!')4) conducted a substantial analysis of existing lab (B46, M55, YR62, KF65, HW67, C69, MK72, S76, W93) and field (157, DI76, K88) data, along with a reduction of methodologies and data presented in Miller and Komar (1980a), Nielsen (1981), and Vongvisessoini 1i (1984). The data are evaluated in terms of the relative nearbed orbital diameter 2A/d and a wave period parameter X (see Table 33), and a dependence between ripple properties and the wave period parameter is found. For very small values of x (< 0.15 x 106)representative of field dataMogridge et al. (1994) find that the ripple wavelength depends only on the grain size and scales according to A = 1394d. Most of the laboratory data fall in the range X > 0.15 x 106 and expressions for maximum ripple wavelength (Equation 313a) and height (Equation 313b) are found through curve fitting. loglo(A/d) = 13.373 13.7720.02054 (3 13a) logjo(rl/d) = 8.542 10.8220.03967 (3 13b) These equations have been found to perform well at predicting bedform geometry in fieldscale oscillatory flows (O'Donoghue and Clubb, 2001), yet it should be noted that Equation 313b is based solely on laboratory data. 3.4.6 Khelifa and Ouellet (2000) Khelifa and Ouellet (2000) (KO00) performed laboratory experiments of bedforms in combined wave and current flows using an intersecting wave basin. The experiments consisted of shortperiod waves (T < 2.0s) of low velocity amplitude (Umax < 30cm/s), and relatively weak currents (Uc < 30cm/s). The new bedform data was added to existing data sets from wave and combined wavecurrent experiments, and Khelifa and Ouellet (2000) derived new empirical formulations for ripple wavelength (Equation 314a) and height (Equation 314b) in combined wavecurrent flows. The new formulations were compared to existing ripple predictors for wave forcing (1\ i!! r and Komar, 1980b; Nielsen, 1981; Mogridge et al., 1994; Tanaka and Shuto, 1984) and for combined wavecurrent flows (Tanaka and Shuto, 1984; Tanaka and Dang, 1996). A,,/A,, = 1.9 + 0.08 1n2(1 + ,) 0.74 ln(1 + T,,) (314a) /wc/A,, = 0.32 + 0.0171n2(1 + c) 0.142 ln(1 + ,,) (314b) Contrary to behavior in purely oscillatory motion, ripple height and wavelength were found to increase along with a modified wavecurrent mobility number for the combined wavecurrent flows. Khelifa and Ouellet (2000) attribute this incongruity to a currentdominated effect, but note that the expressions given by Equations 314a and 314b reproduced the observed increase. The parameters A,, and w,,c used above are modified versions of the orbital excursion amplitude and mobility number, respectively, that account for the combined wave and current effects by utilizing a wavecurrent velocity amplitude U,, [see Khelifa and Ouellet (2000)]. 3.4.7 Faraci and Foti (2002) Faraci and Foti (2002) (FF02) collected bedform data from laboratory experiments of noncohesive, quartz sand under regular and irregular waves at low mobility numbers. Discrepancies in ripple characteristics from the two forcing types were not evident, but the bedforms were observed to reach equilibrium through differing mechanisms. Under regular waves, Faraci and Foti (2002) find that the equilibrium ripples grow from a flat bed through a transitional state where rTl/A, A 0.10 that t'. I the presence of rolling grain ripples. The data for irregular waves ii:.: i that the ripples reached their equilibrium height and wavelength through constant growth. Bedform data were evaluated in terms of specific gravity s, the grain Reynolds number Rg, the orbital Reynolds number R,, and the mobility number T. In the equilibrium range, Faraci and Foti (2002) find that ripple wavelength and height are given by Equation 315a and Equation 315b, respectively, and that steepness follows the empirical relationship of Nielsen (1981) (Equation 315c). 0.08291 A/A, so09 (315a) QRg Tyr/A 1 02 I exp(0.0076R5 + 0.1681) (315b) rT,/A, 0.32 tan (315c) 3.4.8 Williams et al. (2005) The ripple predictor of Williams et al. (2005) (W05) is empirically derived from sets of field and lab data on bedform growth. They distinguish between two types of commonly observed ripples; small wavegenerated ripples (SWRs) that may be classified as orbital, anorbital, or suborbital having heights of centimeters and lengths of tens of centimeters; and large wavegenerated ripples (LWRs) that have long wavelengths (O(m)) and comparatively low amplitudes (O(cm)). Williams et al. (2005) note that the empirically derived formulations for SWRs prevalent in the literature perform badly at predicting the characteristics of LWRs. Through an analysis of many data sets, Williams et al. (2005) provide formulations for SWR and LWR characteristics based on the mobility number T. The ripple height and wavelength of LWRs are best predicted by Equation 316a and Equation 316b, respectively. 1LWR/A exp [ 0.2043 ln()2 + 1.2791n(0) 4.808] (316a) ALWR/A exp 0.20071n(0)2 + 1.4671n(Q) 1.718] (316b) For SWRs, the ripple height and wavelength are given by Equation 317a and Equation 317b, respectively. swR/A = exp 0.0282 n(Q)2 1.418 In() + 1.249] (317a) ASWR/A exp [0.0542 In(0)2 1.3071n(T ) + 2.843] (317b) 3.4.9 Soulsby and Whitehouse (2006) By reevaluating field and laboratory data of bedform characteristics Soulsby and Whitehouse (2006) (SW06) provide empirically derived formulations for ripple wavelength and height. They found that, when combined, the lab and field data collapsed best when plotted against a relative orbital excursion length Ad = A/d. The nondimensional ripple wavelength and height are given by Equation 318a and Equation 318b, respectively. rA/A= [L + 1.87 x 10"oAd(1 exp (2.0 x o4Ad)15}) (3 18a) 7qr/Ar = 0.15 I exp { (5000/Ad)3}] (318b) 3.5 Models While simple analytical models of bedform characteristics existsuch as the ripple predictors given in the previous sectionthere are few detailed models of hydrodynamics and sediment transport that seek to simulate unsteady bedform characteristics and timedependent morphology. This is due in large part to the complexity of accurately describing the timedependent, nonlinear processes that occur above vortex ripples. Where a linear stability win 1 ,i; was enough to effectively describe bedform growth in the linear regime, such an approach is not applicable for modeling the evolution of vortex ripples (Nielsen, 1981). Indeed, a linchpin of the stability approach of Foti and Blondeaux (1995a) requires that the flow remain in the linear regime thereby precluding flow separation and vortex shedding. Some empirical and semiempirical models of bedforms have been developed through laboratory experiments. Baas (1994) created an empirical model to simulate the development and equilibrium characteristics of smallscale bedforms in fine sand. The model was based on steady flow experiments in a flume, and demonstrated that the time required for equilibrium ripples to develop from a flat sand bed is related to an inverse power of the flow velocity. The semiempirical models of Andersen and Fredsoe (1999), Andersen et al. (2001), and Andersen et al. (2002) incorporate additional complexity through additional treatment of the hydrodynamics, providing more detailed information about the nonlinear evolution of vortex ripples. None of these models, however, seek to characterize the hydrodynamic, sediment transport, and morphological processes in great detail. A number of lab and field experiments, combined with advancements in computational technology, have improved basic understanding of coupled hydrodynamic and sediment transport processes. This has made it possible for scientists to develop more sophisticated morphological models. While scarce, some detailed models of the timedependent coupling between hydrodynamics and morphology exist (Gessler et al., 1999; Lesser et al., 2004). Gessler et al. (1999) developed a sediment transport and morphology component for the threedimensional hydrodynamic model CH3D, allowing for simulations of timedependent bed morphology. The model does not specifically address the generation or dynamics of bedforms but shows some skill at predicting erosion and deposition in river bends. While the NavierStokes equations are used in the horizontal plane, the model invokes the Boussinesq equation in the vertical that assumes a hydrostatic pressure field. The Boussinesq approximation is not expected to hold true above a rippled bed where coherent motions create a spatially dependent pressure field. Gessler et al. (1999) do, however, incorporate a number of useful components into CH3DSED such as the ability to model a number of different sediment sizes, sediment exchange between bedload and suspended load regimes, and turbulent damping by suspended particles. A morphological component integrated with the DELFT3D model was created by Lesser et al. (2004), resulting in a fully coupled threedimensional model of hydrodynamics and morphology. The hydrodynamic model solves the threedimensional shallow water equations, incorporates a number of turbulence closure schemes, and uses an advectiondiffusion equation to model the suspended load. Lesser et al. (2004) apply the modeling system to a number of coastal problems but tend to concentrate on largescale morphology. (a) phase = 60 deg. (b) phase = 120 deg. (c) phase = 240 deg. (d) phase = 300 deg. (e) phase = 400 deg. Figure 31. An illustration of vortex shedding over a rippled bed as a function of fluid phase. Vortex sizes, locations, and fluid phases are approximate and not to scale. The magnitude of large arrows indicate forcing strength and direction. As the flow accelerates from left to right, (a) a vortex forms in the lee of each ripple and (b) increases in size with increasing fluid velocity. The original vortices are ejected over the ripple crest upon flow reversal. New vortices form (c) in the lee of each ripple and (d) grow as the ejected vortices are advected by the main flow. The process repeats itself (e) during the next halfcycle. Adapted from Earnshaw and Greated (1998). Orbital Ripples Anorbital Ripples Figure 32. Effects of bedform type on boundary li. r thickness. Table 31. List of common orbital ripple length predictors based on the orbital semiexcursion. A differentiation is made for some of the values with SL referring to suspended load dominated bedforms and BL denoting bedload dominated bedforms. Reference A, Andersen and Fredsoe (1999) (SL) 0.78A Andersen and Fredsoe (1999) (BL) 1.25A Andersen et al. (2001) 1.28A Mogridge and Kamphuis (1972) 1.3A Miller and Komar (1980b) 1.3A Nielsen (1981) 1.3A Nielsen (1992) 1.33A Wiberg and Harris (1994) 1.24A Table 32. A summary of lab and field experiments, and ripple data sets. keys F and L refer to field and lab experiments, respectively. Reference Experiment Bagnold (1946) L Manohar (1955) L Inman (1957) F Yalin and Russell (1962) L Kennedy and Falcon (1965) L Horikawa and Watanabe (1967) L Carstens et al. (1969) L Mogridge and Kamphuis (1972) L Dingier (1974) F Tunstall and Inman (1975) L Dingier and Inman (1976) F Sleath (1976) L Lofquist (1978) L Nielsen (1979) L Miller and Komar (1980b) L Miller and Komar (1980a) F Boyd et al. (1988) F K.'iii, (1988) F Southard et al. (1990) L Osborne and Vincent (1993) F Vincent and Osborne (1993) F Willis et al. (1993) L Coleman and Melville (1994) L Hay and Wilson (1994) F Wheatcroft (1994) F Thornton et al. (1998) F Li and Amos (1999) F Marsh et al. (1999) L Traykovski et al. (1999) F Khelifa and Ouellet (2000) L Hanes et al. (2001) F O'Donoghue and Clubb (2001) L Ardhuin et al. (2002) F Faraci and Foti (2002) L Doucette and O'Donoghue (2006) L Lacy et al. (2006) L The experiment Citation Key B46 M55 157 YR62 KF65 HW67 C69 MK72 D74 TI75 DI76 S76 L78 N79 B88 K88 S90 OV93 V093 W93 C'\1)4 HW94 W94 T98 LA99 M99 T99 KO00 HO1 OC01 A02 FF02 D006 L06 Table 33. Common dimensionless parameters and variables used in the classification of bedforms and computation of bedform characteristics. Name Symbol Equation Orbital SemiExcursion A Umax /W A/d Relative Orbital Excursion Displacement Thickness Orbital Reynolds Number Grain Reynolds Number Sediment Parameter Mobility Number Wave Period Parameter 0.08A Aud/v (d/4v)[(s 1)gd]0.5 (Aw)2/[(s 1)gd] d/[(s )gT2] Table 34. A list of common ripple predictors, the values they provide, and classifications of their supporting data sets as either laboratory L, field F. Reference Nielsen (1981) Grant and Madsen (1982) Wiberg and Harris (1994) Mogridge et al. (1994) Khelifa and Ouellet (2000) Faraci and Foti (2002) Williams et al. (2005) Soulsby and Whitehouse (2006) Value Computed Data Sets 'Tr, Ar T'r, Ar Tlorb,ano,sub, Aorb,ano,sub rImax, Amax 7wc, Ae TrR, Ar tILWR,SWR, ALWR,SWR Trj, Ar L, F L L, F L, F L L L, F L, F CHAPTER 4 MODEL DESCRIPTION 4.1 Introduction This chapter provides a description of newlydeveloped models for simulating unsteady fluid hydrodynamics, noncohesive sediment transport, and timedependent morphology in linear and nonlinear forcing regimes. The models range from a simple onedimensional bedload model based on a simplified momentum equation, to more complex two and threedimensional models that solve the unsteady NavierStokes equations for fluid flow with a set of equations for turbulence closure, and simulate both suspended load and bedload sediment transport. Each of the new models may be classified as unsteady, or phaseresolving, and do not employ process averaging or parametrizations of flow or transport characteristics inherent in phaseaveraged, quasisteady, and semiunsteady models. Sediment transport is treated in a bulk fashion in the models: no attempt is made at solving independent sets of kinematic equations for each sand particle, although such models do exist (Calantoni, 2002). The following models are particularly wellsuited to handle a variety of forcing conditions, but each was developed with a specific application in mind; the onedimensional model is used to simulate dynamics in the linear regime; and the two and threedimensional models are applied to problems in the nonlinear regime where flow separation, coherent motions, and turbulence p1 iv significant roles in boundary 1. r sediment transport processes. It should be noted that the twodimensional model is simply a unique application of the full, threedimensional model, so they will be discussed simultaneously. 4.2 OneDimensional Linear Model The growth of bedforms from an initially flat bed occurs through two distinct regimes; a linear regime where the freestream forcing and applied bed stress are in phase with one another; and a nonlinear regime where phase lags exist between stress in the boundary 1v.r and fluid forcing in the outer flow. In the linear regime, Bagnold (1946) observed that rolling grain ripples formed through an .. . regation of sediment particles in areas of increased friction. Such a process is challenging to simulate with numerical models that do not incorporate parametrizations of spatiallydependent friction as there is no inherent mechanism to initiate ... regation. Linear stability models have proven to be useful tools in simulating transport and bedform growth in the linear regime, but often incorporate parametrizations specific to a set of fluid or sediment parameters (Parker, 1975; Sleath, 1976; Coleman and Melville, 1994; Foti and Blondeaux, 1995a). The following sections introduce the methodology used to formulate a new onedimensional model of transport and bedform dynamics in the linear regime, and the constitutive relationships for fluid flow and sediment transport are provided. A schematic of the model is provided in Figure 41, and will serve as a useful reference for explanation of methodologies and assumptions used to derive the governing equations employ, 1 in the new model. The objectives of the onedimensional model are to identify the instability of the flat bed leading to bedform growth in the linear regime, and to simulate the timedependent morphology resulting from gradients in the bedload transport. 4.2.1 Hydrodynamics 4.2.1.1 Governing Equations The onedimensional horizontal (1DH) model is based on a simplified version of the onedimensional momentum equation for fluid flow in the xdirection (Equation 41). A righthanded coordinate system is used where x and z are the horizontal and vertical directions as shown if Figure 41. Although the zdirection is shown in Figure 41, the computational domain is confined solely to the horizontal xdirection. Therefore, terms of v and w and derivatives in y and z implicit in the full onedimensional momentum equation are excluded from the simplified 1DH momentum equation: u+ au + i P X 0 (41) at Ox p Ox where u is the xdirected fluid velocity, p is pressure, p is fluid density, and 'T is a horizontal shear stress. For the purpose of simulating the single 1~,r mixture of sediment and water in the linear regime, the pressure term in Equation 41 is assumed to be hydrostatic where p = pg(qr z) (Dean and Dalrymple, 1991). Substituting the hydrostatic assumption for the pressure variable in the pressure gradient term of Equation 41 yields: Sa g9 (42) p x Qx where = (zb + Amix) is the displacement height of the surface mixture, and Zb is the height of the bed level. The thickness of the mixed lIr, Amix, is a free parameter that depends on three parameters: the number of grainthicknesses assumed in the 1I,r (N), the grain diameter (d), and the concentration in the mixed 1. r (Cmix). The relationship between the mixed l1~r thickness and the three parameters is given by Amix = Nd( Cmix). (43) The concentration of the mixed 1 i.r, like the l V.r thickness, is a free parameter. Bearing in mind that the maximum packing concentration of spherical particles is nearly 0.60, the concentration in the mixed l1,r (Cmix) will be somewhat less due to the presence of fluid in the 1l r. A typical assumption for model applications is that the mixed 1lr contains 5(0' sediment by volume, or Cix = 0.30. The choice of the mixed 1 ,,r concentration subsequently has an effect on the density of the mixed 1~,r: Pmix = PsCmix + P(1 Cmix), (44) where ps is the sediment density. Present simulations with the 1DH model assume that the sediment is comprised of quartz material with a specific gravity of s = 2.65. The horizontal stress gradient in Equation 41 is replaced by a dimensionally equivalent expression representing the balance of stresses on the mixed lir shown if Figure 41: 1 aTr 1 07 ,( rs (45) p OX PmixAmix ( where Tb is the bed shear stress resisting motion of the mixed 1li.r, and 'T is the surface stress applied to the mixed l?vr from the freestream forcing. The parametrizations of these two stress terms, Tb and r,, are given by Equations 47 and 48, respectively. Substituting the previous simplifications into the original onedimensional momentum equation (Equation 41), and denoting the velocity u in the mixture l,vr as Umix, the resulting 1DH general mixed1 iv.r momentum equation for the linear regime is given by auuix 9um ar 1 / + Umix +g + Ts + F, = 0 (46a) dt OX OX PmixAmix F, = Uuw cos(wt) (46b) where F, is a body force, U, is the freestream forcing velocity, w is the angular wave frequency, and t is time. The body force Fx represents the oscillatory nearbed forcing induced by a horizontal pressure gradient, and is applied uniformly to the fluid. Parametrizations for the stresses applied to (7) and resisting (Tr) the mixed l,vr are required to close the 1DH mixed livr momentum equation (Equation 46a). The resisting stress of the bed acting on the mixed 1lvr is given by b = Pmix fbUmix Umix (4 7) where fb is a coefficient of friction between the bed and the mixed l1. r. The constant stress applied to the mixed li r from the simulated freestream forcing is given by Ts= Pf. ,", "f (48) where fmix is a constant representing friction between the fluid and mixed l1' rs, and uf is the fluid velocity above the mixed li r. At any time t the fluid velocity Uf is found by integrating the body force F, (Equation 46b) with respect to time such that f(t) = E(t) dt= JU,w cos(wt) dt (49) 4.2.1.2 Numerics The 1DH mixedli, "r momentum equation (Equation 46a) is integrated forward in time using an explicit thirdorder AdamsBashforth timemarching scheme: n+1 =n+ [+j23F() 16F( 1) + 5F(92)] (410) where jo" may be taken as the mixture velocity umz% at time level n and the terms F(p"), F(p"1), and F(pn2) correspond to the evaluation of fluxes from Equation 46a at the n, n1, and n2 time levels, respectively. In general, a flux term F(p" from Equation 46a is evaluated as Fu%") u'.  n Ts n x (411) mx OX g ax Pmixnmix b while the fluxes at time levels n 1 and n 2 are evaluated in a similar fashion. Since the thirdorder AdamsBashforth scheme (Equation 410) requires information at two previous time levels (n 1 and n 2), a firstorder upwinding time integration (Equation 412) is used for the first two time steps. Sn+l = p + 1tF(pn) (412) The spatial discretization of the 1DH mixedl ivr momentum equation (Equation 46a) is achieved using secondorder central differences: O i+l 26 (413) Oxj 26x where 90 may be taken as any primitive variable of the 1DH mixed1 vivr momentum equation evaluated at grid location i and time level n, and 6x is the grid spacing. The horizontal grid spacing in the xdirection is assumed to be constant, where periodic boundary conditions are employ, 1 4.2.2 Sediment Transport The volumetric rate of bedload sediment transport (per unit width) in the 1DH model is assumed to be a function of the mixedliv r velocity, thickness, and concentration such that Qb UmixrmixCmix (414) where the mixed1 i r thickness Amix and concentration Cmix are assumed to remain constant. The magnitude of bedload transport predicted by Equation 414 will vary depending on values chosen for the number of particle thicknesses (N) and concentration (Cmix) in the mixedliv r through Equation 43. 4.2.3 Morphology The timedependent bed morphology is predicted by the sediment continuity, or Exner, equation: a zb aQb ( ) + 0 (415) at ax where Zb is the vertical location of the bed, Qb is the bedload transport rate, and n, is the porosity of the bed material. Equation 415 is integrated forward in time using the thirdorder AdamsBashforth scheme (Equation 410), and discretized spatially using secondorder central differences (Equation 413). 4.3 Two and ThreeDimensional Nonlinear Models In comparison to the onedimensional linear model presented previously, the two and threedimensional nonlinear models are quite complex. The complexity is justifiable since in the nonlinear regime of bedform growth it is no longer sufficient to assume that the applied stress at the bed is uniform in space, and in phase with the fluid forcing. Rather, phase differences in the nonlinear regime are manifest through inertial effects in the boundary 1, r where applied forcing on the sediment tends to lead the freestream oscillatory forcing outside the boundary 1 .r. Early investigations of bedform growth by Ayrton (1904) and Bagnold (1946) gave particular attention to this matter, noting the regime change from the frictionallydominated case of rolling grain ripples to the orbital, or vortex, ripples that form and grow due to flow separation. A unique, coupled modeling system has been developed to capture all of the salient mechanisms governing the behavior of orbital and anorbital ripples in the nonlinear regime. A schematic of the threedimensional modeling domain is provided in Figure 42. The modeling system employs a threedimensional, N ,i,. rStokes solver for the fluid hydrodynamics and incorporates a Large Eddy Simulation to model the subgrid scale (SGS) turbulent stresses. Semiempirical formulations, available in existing literature, govern the bedload and suspended load noncohesive sediment transport. Coupling between the fluid and sediment dynamics is achieved by solving a discretized version of the sediment continuity equation, which provides timedependent information about the spatial location of the bed (morphology). Unlike the discrete particle model of Drake and Calantoni (2001), or the twophase model of Dong and Zhang (1999), sediment is not inherent to the current modeling scheme. Instead, the sediment should be considered as a distinct second phase acting only as a passive scalar with its own inertial properties. The model does not yet consider the twoway coupling between fluid and sediment interactions that may lead to significant alterations of the hydrodynamic field through turbulence damping at high concentrations (McLean, 1991; Ribberink et al., 1994). Specific details about the hydrodynamics, turbulence closure, sediment transport, and morphology techniques employ, .1 in the new, nonlinear modeling system are provided in the following sections. 4.3.1 Hydrodynamics 4.3.1.1 Governing Equations In a righthanded coordinate system, where the x, y, and zdirections (xi, x2, x3) are associated with the u, v, and wvelocity components (ul, u2, u3), fluid hydrodynamics in the model are governed by the continuity (Equation 416a) and NavierStokes (Equation 416b) equations. u = 0 (416a) axi 9Oui 9Oui 9 fi,6ijp 9OuiA S+ U + + 6ijF = 0 (416b) at 9xj 9xj p 9xj ) F,= [U,,w cos(wt), 0, g] (416c) where ui is the fluid velocity, p is pressure, v is kinematic viscosity, 6ij is the Kronecker delta, and Fi is an appropriate body force applied to the fluid. As shown in Equation 416c, a constant horizontal pressure gradient is applied in the xdirection, and gravitational acceleration (g) acts in the zdirection. The horizontal pressure gradient drives an oscillatory flow having the form U(x, t) U sin(t) (417) where U, is the prescribed freestream forcing and w is the angular frequency of oscillation. The applied forcing given in Equation 416c is derived by simply taking the timederivative of the velocity timeseries given by Equation 417. The timedependent, finitedifference modeling system employs an Eulerian reference frame, such that the fluid motions must be tracked as they progress through the computational mesh, or grid. Primary dependent variables in the model consist of the three components of velocity (u, v, w), and the total fluid pressure (p). The velocity and pressure variables are arranged on a ,.: r'ed, Cartesian grid, such that numerical representations of pressure gradients are coincident with the locations of the facecentered, velocity variables in the momentum equation (Equation 416b). The 1 I,.: red arrangement ensures numerical fidelity, and reduces the computational overhead associated with variable memory storage. A representation of the i .1:: red grid is provided in Figures 43(a) and 43(b). Other features of the computational grid include optional variable mesh clustering and scaling, which can be utilized to enhance numerical resolution in desired regions while permitting a more coarse resolution for farfield solutions. With the current memory capacity of most computational platforms exceeding onegigabyte (1GB), the additional storage requirement of three vectors containing the variable grid spacing values is of minor importance; however, the variable mesh scaling does add complexity to the derivation of appropriate finitedifference schemes as 6x, 6y, and 6z are no longer constants. Nevertheless, the benefits of increased resolution in numerically intensive areas such as boundary 1iv.r and regions of organized, coherent motions far outweigh the negatives. The variable mesh scaling is set using a simple geometric function: 6xi = A + Bxi + Cxf + Dx4 (418) where the coefficients A, B, C, and D are a function of the number of grid points, as well as the minimum specified grid I' ii; in each submesh. A schematic of the variable mesh scaling is shown by Figures 44(a) and 44(b). In order to simplify numerical algorithms used in the model, the variable mesh scaling is considered to be isotropic in the two orthogonal directions: for instance, the grid spacing in the xdirection will be a geometric function of the grid location i only, such that 6x(i) is constant in the y and zdirections. It is also possible to define submeshes in each of the three dimensions that permit clustering of grid points in multiple areas where higher resolution is needed (i.e., a boundary 1ivr and an interface between two fluids). Such an example is shown in Figures 45(a)45(d) where the x and ydirections use one clustered mesh, and the zdirection includes two clustered submeshes. In Figure 45(a), the grid volume ratio is defined as 6v Y (4 19a) 6max where JS^z = 6x(i) Sy(j) 6z(k) (419b) and 6max = max{6x(i)}max{y(j)} max{6z(k)}. (419c) For simulations incorporating free or rigid boundaries, care must be taken to avoid smoothing of flow properties across the boundary. Finitedifference formulations often require the averaging of flow properties on the computational grid to ensure the colocation of variable quantities. This type of averaging is common in the evaluation of convective flux terms in the momentum equation (Equation 416b), but contributes to "Eulerian smoothiin if flow properties are averaged across a discontinuous boundary such as a free surface (Nichols et al., 1980). The fractional Volume of Fluid (VOF) method of Hirt and Nichols (1981) is used to avoid the numerical inaccuracies associated with averaging flow properties across free boundaries. The volume of fluid is a time and spatiallydependent quantity denoted in the model as F(x, y, z, t). A step function representation of F is used, where F = 1 in a cell whose volume is completely filled with fluid, and F = 0 elsewhere. When averaged over a control volume, the fractional volume of fluid may fall within the range given by Equation 420. If Equation 420 is true for a particular control volume, then that control volume is assumed to contain an interface between the fluid corresponding to F = 1 and the alternate media represented by values of F = 0. The alternate media may either be void space, whose density and viscosity are undefined, or an alternate fluid with its own unique density. In such a manner, free surface or twophase flows may be simulated. 0 < F(x,y,z,t) < 1 (420) The fractional volume of fluid for any particular cell is advected through the computational mesh corresponding to the local velocity field. At this particular time, there is no diffusion of the VOF scalar. The governing equation for F(x, y, z, t) is OF OF F+ F 0 (421) In the VOF method, boundary normals are identified by evaluating the location of highest spatial gradients of F throughout the computational mesh. The value of F for a cell containing an interface may be used in conjunction with the boundary normal direction to reconstruct the slope of the interface in that cell. Since F is a step function, caution must be exercised when computing the spatial derivatives of Equation 421 to avoid unwanted smearing of the fractional volume between zero and one. A special flux approximation for F, the DonorAcceptor method, is used to retain the discontinuous nature of the free surface where necessary (Johnson, 1970). The DonorAcceptor method considers maximum and minimum values of F available for fluxing between control volumes based on the local advecting velocity field, and the fractional value of F contained in neighboring cells. In this way, no control volume can ever donate more mass to its neighbor than it has to give, and no volume can ever accept more mass from its downstream neighbor than it can contain within its own control volume. One limitation of the current VOF scheme is that it is only accurate to O[(6t)] in time and O[(6x)] in space. 4.3.1.2 Turbulence Closure The foundation of turbulence closure starts with a wellchosen decomposition of the instantaneous velocity field into mean (average) and fluctuating components. For instance, consider the standard Reynolds decomposition of a velocity component into mean and fluctuating components: U(x, t) = U(x, t) + u(x, t). (422) As demonstrated by Reynolds (1894), substituting the decomposition (Equation 422) into the N ,li, rStokes equation(s) (Equation 416b) provides a mean momentum equation. The resulting set of mean momentum equations is commonly referred to as the ReynoldsAveraged NavierStokes (RANS) equations and form the basis of twoequation turbulence models such as the commonplace k c and k w formulations. An alternative to the Reynoldsstress approach for modeling turbulence, largeeddy simulation (LES), has gained popularity in the disciplines of atmospheric and oceanographic sciences in recent years. In level of complexity and computational cost, LES lies somewhere between Reynoldsstress models and direct numerical simulations (DNS). However, in highly unsteady threedimensional flows, LES is found to be superior to Reynoldsaveraged models in predicting turbulent flow over bluff bodies (Pope, 2000). By comparing the estimates of RANS and LES model results to DNS simulations of turbulent flow over ripples, ('!i ing and Scotti (2004) demonstrate that a twoequation (k w) RANS model underestimates both the magnitude of Reynolds stress, as well as the oscillatory amplitude of vertical velocity. Largeeddy simulation has been applied to a number of turbulent flows over simple and complex geometries with promising results [e.g. Zedler and Street (2001); Calhoun and Street (2001); Mahesh et al. (2004); Tseng and Ferziger (2004)]. The central concept of LES is that largerscale turbulent motions are well resolved in the computational mesh, and therefore directly represented, while the smallscale motions are modeled. There are four conceptual steps involved in largeeddy simulation: 1. filtering of dependent variables, 2. derivation of the filtered governing equations, 3. modeling the subgrid scale residualstress tensor (closure), and 4. solve the filtered momentum equations from step 2. The filtered velocity field is given by the convolution U(x) G(r)U(x r)dr (423) where G(r) is a homogeneous filter. Common filter types include box, Gaussian, sharp spectral, Cauchy, and Pao as given by Pope (2000). Regardless of the filter chosen, their effect on the velocity field is to remove small wavenumber fluctuations while maintaining the overall trends of the velocity field. Prior to general filtering concepts, Deardorff (1970) I... 1. 1 that mean quantities could be represented by volumeaveraging their values over a cell in a rectangular grid. That is, for a cell centered at x, a mean quantity is given by _1 r3+h 3/2 xa2+h2/2 xi+hi/2 U(x, t) U(x, t)dxidx2dx3 (424) 1Jh2h3 J3h3/2 Ja2h2/2 Jxihi/2 where hi, h2, andh3 are filter widths of a threedimensional, inhomogeneous grid. The equivalent homogeneous filter, for use in Equation 423, is 3 G(r) J H A() r() (425) i= 1 where A(i) is representative of the filter size. Applying this volumeaveraging filter to the dependent variables on a I:.. red, rectangular grid results in a filtertogrid ration of two. According to C'! i.  and Moin (2003), a filtertogrid ratio of at least four is desired for a secondorder finitedifference scheme. A filtertogrid ratio of four may be obtained by passing an unfiltered variable through a succession of secondorder filters: U(x, t) = U(x, t) + 7 (U(x 6x, t) 2U(x, t) + U(x + 6x, t)) (426a) U(x, t) = U(x, t) + U(0x 6x, t) 2U(x, t) + U(x + 6x, t)) (426b) where 6x is the grid spacing and 7, is a smoothing coefficient, typically taken equal to 0.25. Current model simulations employ the more simple volumeaveraging approach given by Equation 424, as it is much easier to apply to complex geometries near the bed. While the approach of Reynolds (1894) involves the decomposition of dependent variables into average and fluctuating quantities, LES employs a decomposition that separates variables into filtered (resolved) and residual (modeled) components. Consider the following decompositions for the velocity and pressure fields: ui = ui + u (427a) p =p + p' (427b) where ui and p are the filtered velocity and pressure and u' and p' are the residual components of velocity and pressure, respectively. Derivation of the filtered momentum equations is achieved by substituting the velocity (Equation 427a) and pressure (Equation 427b) decompositions into the conservation of mass and momentum equations given by Equation 416a and Equation 416b, respectively. After simplification, the filtered continuity and momentum equations are au= 0 (4 28a) axi Oui Oui a ( ijp Oui R\  + U + v + ) + VjF = 0 (428b) at Ujaxj a xx where P^ is the residualstress tensor, which is equivalent to the Reynoldsstress tensor. More specifically, Tr, represents the difference between the filtered product (T)uj) and the product of the filtered velocities (uiuj), which arises through decomposition of the velocity field in the N ',i. rStokes equation. In order to close the system of filtered equations (Equations 428a and 428b), it is necessary to introduce a third equation that relates the residualstress tensor (r,) to the filtered velocity field. Turbulence closure is provided by a linear eddyviscosity model proposed by Smagorinsky (1963). The Smagorinsky (1963) model assumes that the anisotropic residualstress tensor is proportional to the filtered rate of strain (Sij) through the following equation: ,j = 2vt Sj (429) where vt is the socalled eddy viscosity and the filtered rate of strain is 42 xj 30x( Si i(aita i)+ (430) The relationship between the anisotropic and isotropic residualstress tensors is given by Equation 431a, where k, (Equation 431b) is the residual kinetic energy. Sr[T + 2 k6 (431a) k, t \ (431b) The Smagorinsky (1963) model defines the eddy viscosity as Vt f2 S (CsA)2 (432) where is is the Smagorinsky lengthscale, Cs is the Smagorinsky coefficient, A is representative of the filter width, and S is the characteristic filtered rate of strain, S 2S JS (433) In the current modeling approach, the Smagorinsky lengthscale is defined as Cs 0.17 (Pope, 2000) and the filter width is taken as A = ( ,5.Iz)1/3. However, using a constant value of is in the viscous wall region is not necessarily correct as it leads to a nonzero residual viscosity at the wall (Pope, 2000); therefore, the definition of is requires additional treatment in the nearwall region, especially for turbulent shear flows. According to Pope (2000), special consideration must be given to the viscous wall region in LES turbulence closure models, and two methods are identified. Largeeddy simulation with nearwall resolution (LESNWR) is an appropriate choice when at least ',I '. of the total energy is resolved by the filter and grid. If the filter and grid are only capable of resolving 'II'. of the energy outside of the boundary lvr (the nearwall eddy motions are not resolved), then largeeddy simulation with nearwall modeling (LESNWM) should be considered. Even though the numerical model considered here employs variable mesh i ,ii. it is computationally prohibitive to resolve eddy motions at the viscous lengthscale as the number of grid points necessary to do so is large in the highReynolds number flows considered here. Therefore, LESNWM is implemented in the turbulence closure scheme by applying a van Driest damping function to fs similar to Moin and Kim (1982): is = CsA [1 exp (z+/A+)] (434a) z+= z (434b) where z+ is the normalized distance from the wall, A+ = 26 is the van Driest constant (Pope, 2000), and u, = I/bp is the friction velocity. Use of the van Driest damping function is a rather simple wall model which allows the eddy viscosity to go to zero at the bed. While more sophisticated wall models exist, such as those proposed by Piomelli and Balaras (2002), the method of Moin and Kim (1982) provides the flexibility necessary to adapt to the changing bed level during simulations. Moreover, the linear eddyviscosity concept employ, 1 in the Smagorinsky (1963) model itself has come under some scrutiny for application to highReynolds number flows over complex geometries (Ferziger, 1996b). One of the limitations of the Smagorinsky (1963) model is that it does not include backscatter: the eddy viscosity t is alv,i greater than or equal to zero. Conceptually, a negative eddy viscosity implies an inverse energy cascade, or one where energy is transferred from small to largescale motions. Dynamic subgridscale eddyviscosity models, such as the one proposed by Germano et al. (1991), are more robust and allow for negative values of eddy viscosity. However, such an approach requires additional filtering of the eddy viscosity field (Zikanov et al., 2002) to prevent numerical instability when the eddy viscosity is negative over large spatial extents (Ferziger, 1996b). 4.3.1.3 Numerics Governing equations in the hydrodynamic modeling system are discretized on the orthogonal Cartesian grid and solved using a variety of finitedifference techniques in space and time. Discretization of the first derivatives implicit in the advective accelerations of the NavierStokes equation (Equation 416b) are complicated by the nonconstant grid spacing. Since two .,.i ,i:ent control volumes may have different widths or heights, it is not possible to solve the momentum equation in conservative form where u O(UU) (435) Oxj Oxj For the computation of advective fluxes on computational grids with nonuniform spacing, Nichols et al. (1980) report no reduction in accuracy if the nonconservative notation of Equation 416b is employ, 1l Furthermore, Nichols et al. (1980) demonstrate that the advective fluxes may be discretized using a thetaweighted variable differencing scheme such that 0 = 1 corresponds to pure onesided donor cell differencing and 0 = 0 represents a standard secondorder central difference. Values of 0 between zero and one may also be used to construct a mixed finitedifference scheme incorporating the benefits of improved accuracy inherent in the secondorder solution, while retaining some diffusive properties of the firstorder scheme. An example of the thetaweighted discretization for an advective flux term at the grid location (i + ~ j, k) is given by Equations 436a and 436b. aui+ U+ [i+l (u'i . ae J +2' UtL^ } + :1 1 '2 {i (Ui ) U6+Ui+13U 2 9i 6xH [ 6xi 2 2 2xi+l 2 (u U U e J + 6x + +l + 2 (6xi+l 6x ) (436b) U +1 With the exception of the special thetaweighted discretization of the advective fluxes, the remainder of first and second spatial derivatives are solved using modified, secondorder two and threepoint finitedifference stencils. The finitedifference schemes are derived through application of a Taylor series expansion about a particular grid location, such as (i + ,j, k), while accounting for the nonuniform grid spacing. The resulting secondorder twopoint stencil is just a special case of Equation 436a with 0 0 and without the leading velocity variable ui+i. An example of the secondorder threepoint stencil used for second derivatives for a point at (i + j, k) is given by Equation 437. 02U,, r u. i Ui ui, 3 =2 2 2 + + iix+ (437) X2 6xi(6xi + 6xi+) 6xi+(6xi6+ The hydrodynamic modeling system utilizes two different types of explicit timemarching, or time advancement, finitedifference schemes; the firstorder Euler upwinding scheme and the thirdorder AdamsBashforth scheme. As given in Duran (1999), the general formulation of the Euler and AdamsBashforth schemes are, respectively, given by Equations 438a and 438b. n+l =/ + 6tF(pn) (438a) n+ n + 6 [+ 23F(qn) 16F(p ) + 5F(9s2)] (4 38b) where po is a generic substitute for any primitive variable, the superscripts n denote the time level, and F(yp) represents the evaluation of flux terms at the indicated time level. For the case of the momentum equation (Equation 4 16b), the flux F(p") would represent the evaluation of advective fluxes, viscous fluxes, and the pressure gradient term evaluated at time level n. As evident in Equation 438b, the thirdorder AdamsBashforth scheme requires information at four different time levels, two of which are evaluations of fluxes in previous time steps. Since in many cases the flow conditions are not known a priori, a new simulation starting at t = 0 requires an explicit timemarching scheme that only considers fluxes evaluated at the current time level. In the current modeling system, Equation 438a is used for the first two time steps, and those fluxes are stored for use in Equation 438b once the third time step has been reached. Caution should be exercised when using Equation 438a with secondorder spatial differencing. Such a marriage is often referred to as the forwardtime centeredspace scheme, or FTCS, and it has strict stability requirements. For convection dominated problems, the FTCS scheme is unconditionally unstable (Fletcher, 2000). Typical applications of the N ',i. iStokes equation (Equation 416b), such as those considered in the current model, are more representative of transport equations with a balance between advection (convection) and diffusion. Fletcher (2000) states that the FTCS scheme is conditionally stable when applied to advectiondiffusion transport equations, so long as Cf< (439) where Cfl = u6t/6x is typically called the Courant number of the CFL (CourantFreidrichsLevy) criterion. When starting simulations from equilibrium conditions, typical u velocities are negligibly small, and in the current modeling system the time step 6t is also very small. In general, the thirdorder AdamsBashforth scheme (Equation 438b) has a CFL limit similar to Equation 439, but the utility of the scheme is its reduction in amplification and phase errors. The amplification and phase errors associated with the firstorder Euler scheme (Equation 438a) are of 0[(6t)2], while the errors of the thirdorder AdamsBashforth scheme (Equation 438b) are of 0[(6t)4]. In our explicit timemarching model, the hydrodynamic time steps are much less than unity; therefore, the errors associated with the thirdorder AdamsBashforth scheme are often orders of magnitude smaller than those of the firstorder Euler scheme. In all varieties of numerical modeling, the appropriate choice of a time step is often precursory. The chosen time step may often be too large or too small, depending on the current flow conditions and the resolution of the computational grid. It is often convenient, then, to utilize a dynamic time step that adjusts accordingly to the instantaneous flow conditions and grid resolution. This type of adjustable time step is utilized in the current modeling system to optimize the time efficiency of the code. The instantaneous model time step is selected by taking the minimum of time steps computed using the CFL limit, as well as a viscous diffusivee) stability criterion, given by the formulations of Equations 440a and 440b, respectively. A moderate amount of conservatism is applied to the CFL limit in Equation 440a, such that the instantaneous time step will alv produce a Cfl < 0.5, keeping well within the stability requirements and reducing potential phase and amplification errors associated with the magnitude of 6t. 1 6xi 6yj 6zk 6t < min i j' I (440a) 2 Ui,j,k I I' I 1 6X26y26z2 v6t < (4 40b) 2 62 + 6y2 + z2 The explicit timemarching formulations given by Equations 438a and 438b are strictly valid for a constant time step at. No formal changes are required when using a dynamic time step in Equation 438a, as there is only a single time step being evaluated between time levels n and n + 1. The derivation of a thirdorder AdamsBashforth scheme with a nonconstant time step requires the application of a Taylor series expansion about time levels n 2, n 1, n, and n + 1, and a number of substitutions. The result of such a procedure produces roughly eight handwritten pages of numerical derivation, and the final result is given by Equation 441a, in its most compact form possible. at[ F(cp") SF(t )t2 ) 6(&[) t2 + 3(t)(t_)+ + 2( + (2) 3(t)( ) + 2(6t)2 (4 (6t2)[62 U1t1 6t_ = t t"1 6t2 t"2 (4 41c) The basic solution algorithm of the hydrodynamic code can be summarized in four procedures: 1. Compute an initial first guess for the velocity field u4 using the discretized momentum equation (Equation 416b) and variables at time level n. 2. Apply appropriate boundary conditions to the velocity field. 3. Iteratively solve for the pressure variable at the newest time level p"+l until the continuity equation (Equation 416a) has been satisfied within some userdefined threshold (typically e < 1 x 103). 4. Update the volume of fluid scalar F(x, y, z, t) to advect any material interfaces through the computational mesh. Direct solutions of the pressure field are not considered in the current hydrodynamic model due to the additional complexity of multiple free boundaries. Instead, a twostep projection method similar to that of Chorin (1968) is employ, 1 to iteratively adjust cell pressures and velocities until continuity is obeyed in each control volume. The first step of the projection involves computing the initial velocity field estimate u4 at an intermediate time level between n and n + 1, by solving the discretized form of the NavierStokes equation (Equation 416b). If the divergence of the velocity field (the lefthand side of Equation 416a) is greater than zero, the cell pressure is reduced to encourage inflow to the control volume. If the divergence of the velocity field is less than zero, the cell pressure is increased to discourage inflow. These incremental cell pressures are used in the second step of the projection to update the velocity estimates and pressure field. The general formulations for the second step of the projection are given by Equations 442a 442c (Nichols et al., 1980). A successive overrelaxation (SOR) method is used to enhance the (441b) convergence of the iterative solution, where a typical value of 2 = 1.8 is applied to the incremental pressure adjustments shown in Equations 442b and 442c. Pi,j,k + Pj,k + 6p (442a) li+'Jk i k + 2t 6 (442b) u = j ut 6ot (4 42c) 4.3.1.4 Boundary Conditions The imposition of suitable boundary conditions for numerical models is alv7x challenging. To be as efficient as possible, computational grids are often selected to be no larger than what is necessary to resolve the desired phenomenon. If the size of the domain is not sufficiently large such that the simulated processes are far removed from the numerical boundaries, inaccuracies in the modeled processes will arise. Various combinations of six different types of boundary conditions are employ, l1 in the current hydrodynamic model. Possible options include freeslip and noslip walls, periodic boundaries, inflowoutflow boundaries, timedependent radiative boundaries, and sponge I ir . The free and noslip walls are often termed mixed boundary conditions because they incorporate both Neumann and Dirichlet specifications. Periodic boundary conditions are useful in modeling monochromatic processes where one dominant oscillatory frequency is readily identifiable: they are intended to mimic an infinitely long domain in the direction of periodicity. Inflowoutflow boundaries employ Dirichlettype boundary conditions where the flow or pressure is specified either upstream or downstream to prohibit reflection from the boundary. The timedependent radiative boundary condition may be utilized to simulate a surface gravity wave for free surface simulations. When the timedependent boundary condition for wave forcing is used in free surface simulations, it is often advantageous to employ a sponge l1 vr at the opposite end of the modeling domain. Such a lvr minimizes reflections from the far boundary, and prevents the need for an outflow boundary condition. As given in Boccotti (2000), the expressions for the free surface and threedimensional velocity potential are given by Equations 443a and 443b, respectively. H q(x, y, t) = cos(kmx + ky ut) (443a) H g cosh k(h + z) S(x, y, z, t) =  sin(kxx + kyy ut) (443b) 2 u cosh kh Here, 0, is the velocity potential, H is the wave height, h is the water depth, kx and ky are the wavenumbers in the x and ydirections, respectively, and k is the wavenumber magnitude such that k = / + k$. Assuming that the model describes irrotational waves at the radiative boundary, a velocity potential for the fluid exists and the total fluid velocity may be taken as i = Vo,. The three components of velocity used to force the timedependent wave signal may then be found using the following expressions: 0 0 '*.. '.*.. x' ay dz These velocities constitute the Dirichlet boundary conditions applied along a vertical wall in the modeling domain, such as at (1,j, k) for all j and k indices. Current applications of the hydrodynamic modeling system to the simulation of bedforms employ a freeslip boundary on the top (lid) of the domain, a noslip condition along the bed (sandwater interface), and periodic boundary conditions along the vertical walls in the x and ydirections. A summary of the Neumann and Dirichlet boundary conditions used for the top and bottom of the modeling domain are provided below. ap 0 aap 0 az az au 0 u 0 Top : Bottom : = 0 v 0 w 0 w 0 An example of the periodic, Dirichlet boundary condition used on the vertical domain walls in the xdirection is provided below. The periodic boundary conditions result in an overlapping of six computational grid points in order to define three common, periodic points. The same methodology may be applied to the v and wvelocity components, as well as the pressure variable p. A similar procedure is used to define the periodic boundary conditions in the ydirection. Ul,j,k Uimax2,j,k Sides : Uimaxl,j,k 1 Uimnax,j,k ,k 4.3.2 Sediment Transport 4.3.2.1 Bedload The transport of noncohesive sediment in the bedload regime occurs through a destabilization of sediment particles due to the applied fluid stress. The total stress within the dispersed particle matrix, however, is a function of both the applied fluid stress and a resisting stress due to particleparticle interactions (Bagnold, 1954). Luque and van Beek (1976) found that the particleparticle stresses dominated the total stress only for very large bedload transport rates; therefore, only the applied fluid stress will be considered in the current bedload formulations. Most modern sediment transport models assume that bed particles remain static until the ratio of destabilizing forces to stabilizing forces reaches some critical threshold. This nondimensional shear stress considers the ratio of applied fluid stress to the buo'in_y of a sediment particle resting in a fluid, and is often called the Shields stress after the experiments of Shields (1936). The instantaneous Shields stress is given by Equation 444, where rb(t) is the instantaneous applied fluid stress, s is the specific gravity of the sediment, g is gravitational acceleration, and d is the particle diameter. While Shields' stress is typically taken as a constant for use in steady flows, Madsen and Grant (1976) found it suitable to apply it discretely in time by using the instantaneous bed shear stress. 0(t) = () (444) (s l)gd It is often common in both coastal and riverine applications to use a standard, quadratic drag law (Equation 445) to estimate the bed shear stress. In Equation 445, f is an empirical drag coefficient (f f 2.5 x 103), and Us is the fluid velocity at the edge of the boundary 1liv r. Tb pf U0 Us (445) Common bed shear stress parametrizations, however, do not account for phase differences that are often present in oscillatory boundary 1. rs. Inertial effects in the boundary li. r cause the fluid velocity at the bed to lead the freestream forcing, resulting in a phase lag outside of the boundary 1.r with respect to the applied forcing on the bedload sediment. In such cases, the bed shear stress parametrization given by Equation 445 leads to errors in both magnitude and direction (see Figure 46). If instead we consider the Newtonian definition of fluid stress, where stress is proportional to velocity shear, it is possible to define the instantaneous bed shear stress as au Tb T/an (446) where p is the fluid dynamic viscosity, u is the fluid velocity near the bed, and n is the direction normal to the bed. Using this formulation, however, results in bed shear stress values that are not in agreement with the order of magnitude of stress predicted by Equation 445. To bring these formulations into agreement, the turbulent eddy viscosity predicted by the Smagorinsky (1963) method is added to the absolute viscosity, resulting in a more accurate representation of the total viscosity near the bed (Equation 447). The relationship between Equations 445 447 is demonstrated in Figure 46 as a function of fluid phase over a simulated ripple. b ( + PVt) (4 47) Through a series of laboratory experiments Shields (1936) determined the critical threshold for incipient sediment motion on a horizontal bed. In terms of the nondimensional Shields stress (Equation 444, this critical threshold is a function of the applied forcing, as well as fluid and sediment properties. Values of the critical Shields stress may be read from Shields' curve (Shields, 1936), but are also commonly found in textbooks (Julien, 1998). Shields' curve may be reproduced using the empirical approximation of Brownlie (1981) for steady, open channel flow: 0* = 0.22 R 0.6 + 0.06 exp(17.77R0.6) (448a) u,d Rip (4 11) where 0* is the critical Shields stress for a horizontal bed, and Rp is a particle Reynolds number based on the friction velocity u,. The predicted curve of Equation 448a is plotted in Figure 47 as a function of the particle Reynolds number. For a given value of Rp, sediment motion will occur for values of 0r above Shields' curve in Figure 47, while no motion is predicted for values falling below the curve. There is some speculation that Shields' method of extrapolation to the point of zero transport may overestimate the critical stress as both grain stress and form drag due to bedforms are inherent in the methodology (Chiew and Parker, 1194). This may partially explain the scattering of data often seen when plotted against the original Shields curve (Brownlie, 1981; Chiew and Parker, 1194). Theoretical derivations of Shields' curve have also been ii:. i .1 by Ikeda (1982) and Wiberg and Smith (1985), but only considered a mostly horizontal bed. For unsteady flow applications it is more convenient to use an empirically derived expression for the critical Shields stress based on unchanging properties such as grain size. The empirical expression of van Rijn (1993) is given by Equations 449a and 449b, where d, is a dimensionless grain size parameter. Values of 0* predicted by Equation 449a are plotted as a function of d, and d50 in Figure 48. 0.24d*1 1 < d, < 4 0.14d 0.64 4 < d* < 10 r = 0.04d;01 10 < d < 20 (449a) 0.01:'./"" 20 < d, < 150 0.055 d, > 150 d d50 2l) 3 (449b) To this point, we have only considered incipient motion on a horizontal bed. Chiew and Parker (1194) evaluated Shields' (1936) curve for incipient motion through laboratory experiments and found that the critical Shields stress was overestimated (underestimated) for downward (upward) slopes. This ii:: I that the slope of the bed also pl . a role in determining the critical threshold of motion. Modification of the critical Shields parameter for transverse and longitudinal bedslopes is achieved using Sw r A tan2 ] tan1 Ocr O cos [ cosa 1 tan (450) Stan2 tan1 where f is the angle of transverse slope, a is the angle of longitudinal slope, and Q is the angle of sediment repose (Calantoni, 2002). Figures 49(a) and 49(b) demonstrate the modification of the critical Shields parameter due to bedslope, as well as the linear superposition of fluid and gravitational forcing. In Figure 49(b), for example, Equation 450 acts to increase (decrease) the critical stress when the fluid forcing and gravitational component of the sediment oppose (align with) one another. The bedslope modification for the critical Shields parameter Ocr given by Equation 450 applies only to bedslope angles less than the angle of sediment repose, or a, 3 < Q. Therefore, simply increasing or decreasing the critical threshold will not prevent bedforms from becoming too steep, nor will it simulate gravityinduced transport. Watanabe (1988) ,.. i .,1 modifying the local transport rates using 8zb Qb= QbO O, I (451a) Ozb Qby Qbyo lQbol aO (451b) where E (F 2) is a diffusivity parameter and Qb.o and Qbyo are the x and y components of bedload transport on a flat bed, respectively. The effect of Equations. 451a and 451b is to enhance downslope transport while diminishing upslope transport. Unfortunately, the choice of an appropriate value for the diffusivity parameter E is somewhat vague as it is often used as a tuning parameter (Watanabe, 1988; Johnson and Zyserman, 2002). A poor choice of E leads to excessive smoothing (diffusion) of the transport and morphology when it is too large, and is not effective at preventing unrealistically steep bedforms when chosen too small. An alternative method to simulating gravitational transport is to derive a particle shear stress based on the buovint weight of the spherical particle submerged in the fluid, and the local bedslope angle. A schematic of the particle and resultant static forces is shown in Figure 410 where a and f are the longitudinal and transverse bedslopes, respectively. The particle shear stress is derived by considering the buovint weight of the sphere where F = 3 (ps p)g (452a) F = Fsin a (452b) Fgy = F, sin f and the downslope gravitational components Fx and Fgy act over an area roughly equal to the crosssectional area of a sphere such that 2 mT = ,gd(ps p) sin a for a > Q (452d) and 2 7~= 3gd(p, p)sin 3 for 3 > (452e) When a > (> Q), the particle shear stress of Equation 452d (Equation 452e) is added to the bed shear stress Tb (rT) to simulate gravityinduced transport. Current model simulations ,.., 1 that the addition of the gravitational stress component is sufficient to prevent bedforms from growing steeper than the angle of repose. The nondimensional bedload function of Eintstein (1950) (Equation 27) describes the relationship between the volumetric bedload transport rate and the particle buov, i v. By measuring the rate of bedload transport in a laboratory, it is then possible to derive a relationship for f)b from Equation 27 that equates the amount of transport to the level of applied forcing. A number of functional relationships for )b have been empirically derived for currents (\ I. v rPeter and Miiller, 1948; Nielsen, 1992; Ribberink, 1998; Camenen and Larson, 2005) and parametrizations have also been developed for waves (\! k, .en and Grant, 1976; Bailard and Inman, 1981; D1i.b iiii and Watanabe, 1992; Soulsby, 1997; Ribberink, 1998; Camenen and Larson, 2005). Most of these functional equations (see Table 21) follow a 1"'". i 1 . formulation as given by Equation 28, where the exceedance of the critical Shields stress is raised to a specified power, and then multiplied by a constant of proportionality. Examples of commonly used power laws for bedload transport by currents (Equation 453) and by waves (Equation 454) are found in At, v, rPeter and Miller (1948) and Ribberink (1998), respectively. b((t) 8 ( (t) 0 (453) 10(01l (452c) 4eb(t) t)t 1.65 (454) More recently, Camenen and Larson (2005) . 1I,1 that most power laws underestimate bedload transport for Shields stress values just slightly larger than the critical threshold; and of course they failed to predict any motion for Shields stress values below the critical threshold. Camenen and Larson (2005) offered an alternative equation (Equation 455) for bedload transport based on the concept of continuing transport for all values of Shields stress greater than zero. +bt(t) 21.t b(t)= 12 0 (t) i. exp(4.50cr/10(t)) (455) The behavior of Equations 453 455, as well as the power law of Nielsen (1992) (Table 21), is demonstrated in Figure 411 for a range of Shields stress values typically found in the flatbed and rippledbed transport regimes. The inset panel in Figure 411 shows that Equation 455 predicts larger values of Nb below and slightly above the specified critical Shields stress of Ocr = 0.04, but predicts smaller values of )b as Shields' stress increases, when compared to the power law formulations. The choice of which bedload transport equation to use is somewhat arbitrary, but Madsen (1991) advocates strongly for use of the M iv, rPeter and Miller (1948) power law in coastal applications. Substituting Equation 453 for Kb(t) in Equation 27 yields a direct solution for the volumetric bedload transport rate: 0(t) l5 Qb(t) 8 0 (1 I (t)l 15 ( 1)gd3 (4 56) where the term 0(t)/l0(t) is used to assign direction to the bedload transport rate in keeping with the sign of the bed shear stress Tb. 4.3.2.2 Suspended Load Parametrizations for the volumetric suspended sediment transport rate exist (McLean, 1991; van der Werf et al., 2006), but their implementation in a timedependent, threedimensional model would be unnecessarily ad hoc. Often times these parametrizations require a priori knowledge of either the flow field or the sediment transport conditions, such as a reference concentration above the bed. Others have ii:. I. 1 adapting power laws similar to those used for bedload transport by increasing the critical threshold level and the exponent (Yalin, 1977; Hogg et al., 1997). An alternative approach for computing the suspended load transport is the Cuintegral approach discussed in Nielsen (1992). In this approach, the instantaneous volumetric suspended load transport rate is defined as fD Qs Jt) Cz,t )z (4 57) where Zb is the vertical location of the bed, D is the depth, C is the filtered volumetric concentration, and us, is the sediment velocity. The suspended load transport rate Qs, given in Equation 457 is computed for the horizontal x and ydirections by substituting the appropriate component of the sediment velocity u,, given by Equation 459. The Cuintegral approach requires an estimation of the local volumetric concentration field, which may be determined from either a particle trajectory model (Nielsen, 1992), or by modeling an advectiondiffusion equation (Equation 458) for the scalar quantity (Zedler and Street, 2001). While both approaches incorporate some order of empiricism into the modeling approachmainly through the selection of a pickup functionthe particle trajectory approach also requires the solution of a momentum equation for the sediment phase. The trajectory method introduces further uncertainty through parametrizations of drag and inertial forces; therefore, the particle trajectory model is not used to simulate suspended load. Instead, the volumetric concentration field is modeled using a + a (u w)C V + ai 0 (458) at 9xj as ) 9xjI where C is the filtered volume concentration, vt is the turbulent eddy viscosity predicted by the Smagorinsky (1963) model, w, is the hindered settling velocity (Equation 461a), and as is the Schmidt number. The Schmidt number as represents the balance between momentum and mass diffusivity (viscosity and molecular diffusion), and is often used as a scaling parameter in boundary l1vr models. We assume here that as = 1 for our computations, as in Zedler and Street (2001). In the model, it is assumed that the sediment velocity is defined by Equation 459. ui [u, v, (w w,)] (4 59) The settling velocity of a single particle in quiescent fluid is a function of sediment characteristics and fluid properties. A general expression for this constant settling velocity (wso), as proposed by van Rijn (1984), is 10vlj' 0.01(s 1)gd3 1 ws d =O 1 + 2O.(s ) 1 (460) where d is the mean grain diameter, and s is the specific gravity of sediment (taken here as s = 2.65). However, the settling velocity is affected by the local concentration of particles in the fluid; regions of high concentration have a smaller settling velocity relative to that of a single particle, as the fluid drag on the concentrated region is increased. Richardson and Zaki (1954) provide a parametrization for the hindered settling velocity based on the local concentration and sediment characteristics: w, w,(1 C)q (461a) 4.35 P0.03 0.2 < Rp < 1.0 q = 4.45 R 010 1.0 < Rp < 500 (461b) 2.39 500 < ~R dw(4 RP so (4 61c) V where C is the filtered volume concentration and Rp is the settling particle Reynolds number. A graphical plot showing the reduction of settling velocity for increasing concentrations is shown in Figure 413 for three different values of the settling particle Reynolds number. For volumetric concentrations less than about 0.5' ,. Figure 413 demonstrates that there is very little reduction of the settling velocity wo. The effects of hindered settling, as predicted by Equation 461a, are negligible for very dilute flows, but are significant for highly concentrated flows (Figure 413). The affect of hindered settling velocity on sediment transport and morphology is discussed in C'! lpter 6, and is found to have a modest effect due to the relatively low suspended sediment concentrations modeled in most simulations (106 < C < 103). Speculation exists regarding the applicability of Equation 458 to simulating suspended sediment transport, mostly due to the absence of particlefluid or particleparticle interactions (Zedler and Street, 2001). In the advectiondiffusion equation, we assume that the volumetric concentration is a conservative scalar deviating from the behavior of a fluid particle only by the particle settling velocity. In this manner, only the fluidparticle interaction is modeled, and other interactions such as turbulence damping from particlefluid interactions, and energy dissipation through particleparticle interactions, are neglected. When working with sand particles whose density is much greater than that of fluid, Elghobashi (1994) report that particlefluid interactions are important when 106 < C < 103, and that particleparticle interactions become significant when C > 103. However, Villaret and Davies (1995) ,i. 1 that Equation 458 is appropriate for concentrations less than approximately C < 0.1 and has been successfully used for concentrations up to C a 1 Current simulations of the model system have persistent concentration values in the range 106 < C < 103 where Elghobashi (1994) states that particlefluid interactions should be considered; therefore, it may be necessary to consider the particlefluid turbulence damping in future versions of the modeling system. There are inherent difficulties in modeling the concentration field with the advectiondiffusion equation (Equation 458). Similar to the volume of fluid function F, the concentration values vary through the range 0 < C < 1. Without the inclusion of fluxlimiting schemes, traditional finitedifference formulations tend to result in values of concentration less than zero and greater than one. These inaccuracies result in nonconservative values of the concentration scalar, sI.. . ii; that sediment is either gained or lost in the computational grid. To overcome these difficulties, we adopt a twostep process for solving Equation 458. First, the DonorAcceptor method of Johnson (1970) is used to estimate the proper amount of concentration (mass) to flux from one cell to its neighbor based on the advecting velocity and the amount of mass available to either donate or accept. The concentration values are updated in time to an intermediate value between time level n and time level n + 1. The second step of the solution sweeps through the concentration field and computes the diffusive fluxes using the threepoint stencil for a secondorder central difference given by Equation 437. Once the diffusive fluxes have been calculated, a final firstorder time advancement from the intermediate time level to n + 1 is performed. This twostep process is given by Equations 462a and 462b in abbreviated form. Although the second step is accurate to O[(6x)2], the leading order of error for the entire process is similar to the VOF method with 0[(6t), (6x)]. C* = C" 6t(uj wo)VC (462a) C"' = C* + 6t V + vt VC (462b) The boundary conditions on the concentration field are specified as freeslip along the top of the rigid lid domain, and periodic in the x and ydirections (along the vertical walls of the model domain). As written in Equation 458, the advectiondiffusion equation does nothing more than transport sediment from one place to another based on the local hydrodynamics. Hence, a special boundary condition must be used along the bed to entrain the sediment into the flow (Nielsen et al., 1978). The Neumann boundary conditions for the top and bottom of the modeling domain are given by Top : C 0 Bottom : ac E(t) 5z an v where E(t) is a timedependent sediment pickup function. Similar to the nondimensional bedload function of Eintstein (1950), a nondimensional entrainment function may be expresses as E 4 (463) Ps (s 1)gd which relates the volumetric rate of bed material exchange to the buoi i, i of the sediment particle. Various forms of 4, have been ii .. I1. in the literature. Similar to the power law bedload relations for I)b, the nondimensional pickup functions often are formulated in terms of some exceedance of a critical Shields stress raised to an empirically determined exponent. Three examples of nondimensional entrainment functions are found in FernandezLuque (1974), N I, iI: i. and Tsujimoto (1980), and van Rijn (1984), and are given by Equations 464a 464c. Ip = 0.02(10 0cr)15 (464a) 4, 0.020( 0.0) (464b) (i 0.00033 [(s )gd 0"1 0 0, (464c) The behavior of Equations 464a 464c with respect to 0 are demonstrated in Figure 412. Although Equations 464a 464c were empirically determined for steady flow, Nielsen (1992) ' 1. .1 that the entrainment function of van Rijn (1984) could be adapted to unsteady flow by considering the timedependent Shields parameter 0(t). The adapted form of the entrainment function given by Nielsen (1992) is E(t) = 0.00033 S )0. dj ( 0 W ) 1.5 (465) 00.2 C T when 0(t) > Oc0, and E(t) = 0 if the timedependent Shields parameter (Equation 444) is below the critical value. The pickup rate E(t) has units of volume rate of exchange of bed material per unit area per unit time. Also, when substituted for 4p in Equation 463, Equation 464c is equivalent to Equation 465 with the exception of the timedependent Shields parameter. 4.3.3 Morphology The twoway coupling between fluid hydrodynamics and sediment transport is produced through differential changes in 11llr:metry with respect to time. These timedependent, morphological changes produce positive feedback loops between the hydrodynamics and resulting sediment transport. In this case, fluid forcing applied to the sediment produces differential transport in the horizontal (bedload) and vertical (deposition and entrainment) directions leading to changes in bed elevation, which further act to alter the nearbed hydrodynamics. In the most simple form, the time rate of change of bed elevation is predicted by the sediment continuity, or Exner, equation: ( n) + VQt 0 (466) where n, is the porosity of the bed material, zb is the vertical location of the bed, and Qt is the total sediment transport (Qt = Qb + Q), or the sum of the bedload and suspended load components. In Equation 466, V is the gradient operator acting on Qt in the x and ydirections such that VQ (Qt + Qt Q Ox a y differentiates the total sediment transport Qt with respect to x and y, individually. The simple sediment continuity equation (Equation 466) has been applied to a number of general riverine and coastal morphology problems (Foti and Blondeaux, 1995b; Gessler et al., 1999; Parker et al., 2000; Johnson and Zyserman, 2002; Long et al., 2007). In the current modeling system, the suspended load sediment transport rate is predicted by the Cuintegral approach given by Equation 457. Instead of lumping the bedload Qb and suspended load Q8 transport rates together as Qt in Equation 466, it is advantageous to separate them into horizontal and vertical flux components for application of the control volume approach employ, ,1 in the modeling system. If the divergence of suspended load transport is instead replaced by the balance between the rates of entrainment and deposition, the resulting twodimensional sediment continuity equation may be written as aZb (1 np) + VQb + E woCb 0 (467) where E is the entrainment rate, wo is the constant settling velocity, and Cb is the average volume concentration immediately above the bed (Nielsen, 1992). When taken as a whole, the term (E W8oCb) in Equation 467 is equivalent to the divergence of the suspended load transport rate VQs, as it represents the differential flux of sediment through the top of the control volume. The physical interpretation of the differential vertical flux term is that when the rate of entrainment is greater than the rate of deposition (E WsoCb) > 0, the time rate of change of zb is negative, corresponding to a decrease in bed elevation. Conversely, when the rate of deposition is larger than the rate of entrainment (E WsoCb) < 0, the time rate of change of Zb is positive, corresponding to an increase in bed elevation. A schematic of a twodimensional (x z) sediment control volume is provided in Figure 414, along with a representation of the terminology used in Equation 467. Numerous variations of the Exner equation (Equation 467) are found in existing literature. For instance, Parker et al. (2000) include the effects of sediment mixtures, and Gessler et al. (1999) incorporate multiple sediment size classes and simulate the exchange of particles between the bedload and suspended load lV. r. Such intricacies are reserved for employment in this modeling system at a later date since merely solving Equation 467 numerically presents special challenges. Details regarding the numerical solution of Equation 467 and filtering requirements follow. 4.3.3.1 FiniteDifference Methods The Exner equation (Equation 467) may be generalized as a one or twodimensional advection/convection equation. Such equations are difficult to solve numerically, as there is no physical mechanism (diffusion) for balancing the advection component. Additionally, the simulation of bedforms require special "shock capturing techniques so as not to artificially smooth steepsided morphology. The following finitedifference formulations were tested and found to be insufficient for solving Equation 467: Forward Time, Backward Space (FTBS); Forward Time, Centered Space (FTCS); and FTCSLaxWendroff (FTCSLW). The FTBS upwinding scheme is first order accurate in time and space, and has a complex amplification factor resulting in strong dissipation of zb (Fletcher, 2000). The dissipation of the upwinding scheme is sufficiently large to control the advection, but artificially damps perturbations in the bed level. When applied to the simple Exner equation (Equation 466), the FTBS upwinding scheme results in the following discretization for the xdirection: zbi Zb 26x(1t np) 1 ) + Q )] (4 68) where 6x and Rt are taken as constants for the sake of brevity, and Cp, is a bedform propagation phase speed given by (Long et al., 2007) as aQt C,(zb) < ax (469) The FTBS upwind scheme is conditionally stable for Cp, < 1. Estimation of Cp has undesirable effects when applied to morphology with little to no change in slope ( = 0), as the value of Cp, approaches infinity (Long et al., 2007); therefore, discretization requiring the estimation of Cp should not be used when simulating smallscale bedforms or mostly flat beds. The FTCS scheme is unconditionally unstable for all applications to Equation 466, but is conditionally stable for advectiondiffusion transport equations, as discussed earlier. The leading order truncation error of the FTCS scheme for the Exner equation is second order in space and first order time O[(6x2, 6t)], and when applied to the advection equation results in an imaginary amplification factor (Fletcher, 2000). According to Long et al. (2007), a typical method for counteracting the instability of the FTCS scheme is to add artificial dissipation, or viscous effects, to the finite difference formulation. The Lax and Wendroff (1960) scheme addresses this issue by incorporating diffusion of the bed elevation (zb). As presented in Johnson and Zyserman (2002) and also in Long et al. (2007), the discretization of Equation 466 for the FTCSLaxWendroff scheme yields: +l at 6 t2C zbn = z1 (Q Q ) + ( 1}t1 2 2z + z b ) (4 70) bi bi 26x(1 np) "+I ti1 26X2 bi where again, 6x and 6t are assumed to be constant in space and time, respectively. The addition of the diffusive term in Equation 470 produces a scheme that is second order in both space and time O[(6x2, 6t2)], and is conditionally stable for Cp, t < 1. As noted by Long et al. (2007), an overestimation of C,, since it is taken to the second power in front of the diffusive term in Equation 470, produces excessive smoothing of the morphology; however, if Cpi is underestimated the diffusive term is no longer large enough to balance the advection, and Equation 470 reduces to a standard FTCS scheme. Currently, the twodimensional Exner equation (Equation 467) is solved using an EulerWENO (Weighted Essentially NonOscillatory) scheme after Long et al. (2007). The "Euler" part refers to the first order time marching scheme, while \V\NO" describes the discretization of the horizontal sediment transport fluxes. WENO methods are based on the ENO (Essentially NonOscillatory) method of Harten (1983), which employs a smoothed stencil to evaluate sediment fluxes over a number of .,Ii i,:ent grid points. By taking a weighted average of the ENO stencils, Long et al. (2007) demonstrate the ability to avoid oscillations, and excessive smoothing, near sharp discontinuities in the bed elevation. The WENO scheme has been successfully applied to crossshore morphology models by Long et al. (2007), but there are no references to its utility in smallscale modeling. For a simple, onedimensional Exner equation (Equation 466) where Q = Qt for simplification, the flux of sediment is approximated by the WENO method as dQ Qi+ Qi1/2 2 (471) dx 6x where Qi 1 and Qj_1 are approximations of the sediment transport rate at grid points S+ and i respectively. Each approximation of the sediment transport rate may be further divided into left and rightbiased fluxes such that Q+ = Q + Q+, (472a) 2 2 +2 and Qi Q + (472b) 2 ? 2 2 where Q and Q+ correspond to bedform propagation in the negative and positive xdirections, respectively. For brevity, the remainder of the formulations will only focus on the approximation of the transport component Qi+ given by Equation 472a, but a similar methodology may be applied to Qi_ by shifting indices back one full step. Moreover, the following formulations are easily applied for transport in two directions (x, y) by replacing all i indices with the corresponding j index. The left and rightbiased fluxes in Equation 472a each have their own unique fivepoint finitedifference stencil and weighting coefficients. The leftbiased flux is given by = 70 + 72Q + 3+ for C +> 0 (473a) Q_= 0 for Cp 1<0 1 7 11 S 3 Qi2 Qi1 + Qi 1 5 1 Q Qi1 + Qi + Qi+ 2 6 6 3 (473b) (473c) (473d) 1 5 1 Q3 = 1 Q + Qi+ tQi+2 (4 73e) are three finitedifference stencils used to compute the value of Qi+. The weighting coefficients 71, 72, and 73 in Equation 473a, as determined by Jiang and Shu (1996), produce a fifthorder accurate estimation of the transport component at i + I and are computed as follows: 71 = 1 I + 2 + 3 72 +2 73 3 7 + 2 + 3 (474a) (474b) (474c) 0.1 1 0. (4 75a) (S1 + C)2 0.6 S (475b) (S2& )2 0.3 1 0. (475c) (S3+ )2 with c is taken as a very small number (t 1020) and the smoothness measurements Si, S2, and S3 are computed as: 13 1 S1 = (vi 2v2 + V3)2 + (1 4V2 + 33)2 S2 (V= 2 2, + 4)2 + (V2 V4)2 12 4 and where (4 76a) (4 76b) 13 S3 = NV3 12 1 2v4 + .)2 + (3V3 4 1 = Qi2 v2 Qi1 V3 Qi V4 = Qi+ ' = Qi+2. Following a similar procedure, the rightbiased flux Q+ of Equation 472a is defined '+2 2 1Ii^ 1 )2 3 1Qi+ _, + 3Qi+l for Cp 0 for C, >0 +2 1 5 1 1 5 1 Qi + Qi 6Qi+l2 +2 3 6 6 11 Q3 Q B2 6 (478c) (478d) (4 78e) 7 1 7Qi+2 + Qi+3 6 3 are three finitedifference stencils used to compute the value of Qi+ The weighting coefficients 71, 72, and 73 in Equation 478a are computed as follows: 1 + 2 + 3 1I + 2 + 63 (479a) (479b) with 4v4 + .)2 (4 76c) (4 77a) (477b) (477c) (477d) (4 77e) and where 2i (4 78a) (478b) 73 3 (479c) where S 0.3 0 .= (480a) (S1+e)2 0.6 (2 (2 (480b) (S2 +)2 0.1 (1 (480c) (S3 +)2 where S1, S2, and S3 are computed as 13 1 S1 = (v2 2 + U4)2 + (2 4V3 + 3U4)2 (4 81a) 12 4 13 1 2 = ( 24 +' )2 + ( .)2 (4 81b) 12 4 13 1 3 (4 2, .. + + (4 '4 .. + 2 (4 81c) 12 4 where ',. =Qi+3 is added to the previous definitions given by Equations 477a 477e. Unlike the FTBS (Equation 468) and FTCSLaxWendroff (Equation 470) schemes that required an estimation of the bedform propagation phase speed Cp, only the sign, or direction of bedform propagation, is needed in the WENO scheme (see Equations 473a 473b and Equations 478a 478b). One simple method of computing the propagation phase speed is given by the simple upwinding scheme of Roe (1987) where CP 1 Q I Q for z~ bi+ z. (482) 2 (1 np) zbi+ bi When Cp > 0 the bedform propagates from left to right, or in the positive xdirection. 2+ Alternatively, the bedform propagation direction is from right to left when Cp,+ < 0. The WENO scheme, however, only uses the sign of Cp, to select the left or rightbiased fluxes, so a much less restrictive method is given by (Qi+l Q)(bi ) (483a) + I(Qi+l Qi)(zbi+l bi)l which produces values of Cp { 1,0, 1} corresponding to bedform propagation in the {x, +x, +x} directions. A similar methodology is applied to the ydirection for twodimensional transport calculations where r+ (QJ+I Qi)(zbj+ Zbj) (483b) C (483b) I+ (Qj+l Qj)(zbj+, bj ) produces values of C, = {1, 0, 1} corresponding to bedform propagation in the {y, +y,+y} directions. Combining the WENO discretization of the horizontal sediment transport flux terms with the firstorder Euler timemarching scheme, and applying them to our twodimensional Exner equation (Equation 467) results in the following finitedifference formulation: n+1, ,) [(Q/+. 1 . Qb. i, + Qb.. Qb ,+ jw (484) (1r, n,) [K 2 2 ) K 2 + WCbij (484) ( np) _6x 6y which is firstorder accurate in time and fifthorder accurate in space O[(6x5,6t)]. 4.3.3.2 Filtering Techniques The fifthorder WENO finitedifference scheme presented in the previous section is often not .. ressive enough to damp highfrequency oscillations of the bed level by merely smoothing the transport gradients. Johnson and Zyserman (2002) demonstrate that highfrequency spatial perturbations in simulated morphology are produced through a dependence of the bed celerity with bed elevation. As a consequence, these highfrequency signals can bread even higher harmonics until the entire bed elevation signal is truncated by numerical inaccuracies. It is often necessary, therefore, to apply some artificial smoothing to the morphology in order to damp the high wavenumber errors while allowing the low wavenumber signal to dominate. This type of smoothing may be accomplished by invoking a lowpass filter of the type given by Equation 426a for a onedimensional field. For morphology that varies in both the x and ydirections, however, the twodimensional filter of Johnson and Zyserman (2002) is emplovc., 5ij 45i,j +'s(QPi+l,j 2i,j + il,j) + 's(Vi,j+l 2pi,j + 4ijl) + 1 + 2 s(i+1,j+l i+lj1 + 9i l1 ilj+l) (485) where p is any primitive variable, ( is the lowpass filtered result, and 7s is a smoothing coefficient. Unlike the LES filtering process where Equation 426a is applied independently in the x, y, and zdirections, smoothing the morphology requires use of the twodimensional filter as it includes crossslope components represented by the last group of terms on the righthand side of Equation 485. Johnson and Zyserman (2002) show that, when performed, a linear stability analysis of Equation 485 results in the following amplification factor: Gi p 1 47t + 27,(cos (x + cos (y + sin ( sin (y) (486a) ( 2 cos (486b) a S2 sin Ob ( (4 86c) a where Ob is the direction of bedform propagation, and a is the number of grid points per wavelength. The 2D lowpass filter given by Equation 485 is stable (unstable) when 7s < 0.25 (> 0.25): values of 7y > 0.25 result in amplification factors GIp greater than unity. The amplification factors of the 2D lowpass filter are shown in Figures 415(a) and 415(b) for smoothing coefficient values of s = 0.125 and 7, = 0.25, respectively. While application of Equation 485 to the timedependent morphology successfully damps the highfrequency disturbances, it also tends to damp lowfrequency (or largescale) features over a given amount of time, many of which may be physically realistic. This behavior is demonstrated in Figures 415(a) and 415(b), where even for , = 0.125 up to 5'. of the signal is damped for bed features resolved by ten or more grid points (Ar > 10 6x). When applied on morphological timescales, subsequent damping of 5'. during each application of Equation 485 may completely obscure physically realistic bedforms, especially for the smallscale features considered in this study. The shortcomings of the singleapplication 2D lowpass filter (Equation 485) may be overcome by considering the hybrid filtering algorithm i., 1. I by Jensen et al. (1999). As outlined by Jensen et al. (1999), the filtering algorithm consists of the following foursteps: 1. Filtering Zb to obtain Zb; 2. Computing 6zb Zb b; 3. Filtering 6zb to obtain 6zb; and 4. Updating the new bed elevation using Zb =b + 6b. Jensen et al. (1999) report that this hybrid filtering algorithm effectively damps signals on the order of the grid size O[(6x, 6y)]. The ii.: 1.1 hybrid filtering algorithm is applied to new estimates of the bed elevation z+1, which are provided by the 2D Exner equation (Equation 467), in the following manner: S+l '+1 (z+1 2+ + z ) + 7 1 +l 2 "+1 + ,+ + 2n+1 z z" zb ) (4 87a) Sbi+ j ,j +1 i I "+ 1 'nr+l I Z+l + +, (4 87a) + 28b +1,^ _i+ijji + 'l iij+i n+l = n+l_ n+1 (487b) Sn+1 +1 l (Z7s(z +l 26n+1 + zn+l) + +7s(6n+l1 267n+1 + 6n+l )+ 1 0+1 b  + 1(6z n+1 n +l= n+l + 6 (487d) where ?n+l is the final smoothed estimate of the bed elevation at the newest time level. In the current modeling scheme, one must be cautious as to how often the hybrid filtering algorithm given by Equations 487a 487d are applied to the morphology as it is still possible to overfilter the bed elevation if used too frequently (i.e., at every hydrodynamic timestep). An optimum frequency interval (6tf) for the hybrid smoothing applied to the current modeling system corresponds to a range of roughly 0.05 s < 6tf < 0.1 s. Applying the 2D lowpass filter (Equation 485) in the manner ii.. I. 1 by Jensen et al. (1999) (Equations 487a 487d), Johnson and Zyserman (2002) find the amplification factor of the total algorithm to be Gf = G1_p(2 GIp) (488) where Glp is the amplification factor of the singleapplication 2D lowpass filter given by Equation 486a. Amplification factors computed with Equation 488 for values of s = 0.125 and 7s = 0.25 are plotted as a function of wavelength resolution (number of grid points per unit disturbance wavelength) in Figures 416(a) and 416(b), respectively. When compared to the behavior of the singleapplication lowpass filter (Equation 485) for = 0.125, Figure 416(a) demonstrates that the hybrid filtering algorithm damps < 1 of the morphology signal for features resolved by six grid points (6 6x) or more, while leaving features resolved by ten grid points or more unbiased. Although the behavior of the hybrid filtering algorithm for >7 = 0.25 completely damps the 2 6x wave, as shown in Figure 416(b), it continues to have a diminutive effect on features resolved at the 10 6x wavelength. Model simulations of smallscale bedforms in the current study have confirmed the assertion of Johnson and Zyserman (2002) that using s = 0.125 sufficiently damps highfrequency oscillations in the bed elevation. Figure 41. A schematic of the onedimensional linear model of bedload transport. z L 0 Figure 42. A representation of a generic threedimensional model domain with domain length labels Lx, L,, and L, the bed elevation zb, and velocity vectors. The blue color represents fluid and the brown color represents sediment. z 1 pl U i+ ,j,k (k ) x(i) z l ) (a) x z plane 5(k) *  i y(j ) *tV,j +J, YJb ) Y( + 1) (b) y z plane Figure 43. Model control volumes showing the location of primitive model variables and definition of indices for a I.: red grid in the x z and y z planes. Z z II 6xi ti1' J SZk (a) x z plane (b) y z plane Figure 44. A representation of variable mesh scaling in the x z and y z planes. For each plane, the horizontal grid spacing is uniform in the orthogonal direction. 136 z ,X w,j,c+ 7 1 kh + 2 i i  z(k)   (k Fi,3,k 0 mink ^JL 2 Pi,3, '( ) 2             _/i (a) Grid volume ratio y(cm) (c) Grid spacing in y 3xyz 3. max 6 1 09 08 07 ' 06 05 C 04  02 0 01 x(cm) (b) Grid spacing in x 50 40 30 20 10 0 02 04 06 08 1 12 14 16 18 Grid Spacing, 6z (cm) (d) Grid spacing in z representation of (a) variable mesh clustering and mesh scaling in the (b) , (c) y, and (d) zdirections. 18 16 S14 1 12 S108 U , 06 06 04 02 Figure 45. A X _ 5  I 1/10 z_ _ gp 5  1   x (cm) 150 o 3U/10 15 531/10 5 x (cm) 10 S15 S10 7 10 10 0 E 5  1, 15 5 (cm) 0 10 0 x (c 10 x(cm) 15 0 10 /10 S5   1o x (cm) 8 6 /10 10 /10 5 0 /10 S5 X(cm) 10 10 8 10 5 ...5r....7777757 '7 1 E 5 1  Predictions of bed shear stress with three different formulations. The red, green, and blue lines correspond to the stress formulations give in Equations 445, 446, and 447, respectively. The solid black line denotes the location of the bed, in the horizontal, relative to the stress predictions. Figure 46. 0, 10' 102 10 Figure 47. 10' 102 103 9 (U Ud/v) Estimation of Shields' curve for incipient sediment motion based on the equation of Brownlie (1981). 0 0.2 d50 (cm) dso (cm) 0 0.2 d ( ) 0.4 0. 0 50 100 15 Figure 48. Critical Shields stress curve based on the dimensionless grain size method of van Rijn (1993). Flow Direction W Wsin a a 31 W sin,,3 > Flow Direction W ia (a) Transverse slope (b) Longitudinal slope Figure 49. Particle forces acting on longitudinal and transverse slopes. F9 Figure 410. A schematic of gravitational forces acting on a sphere resting on a sloping bed with angle a or 3 to the horizontal. 0.12 1 1 MPM48 0.1 N92  R98 / ...... CL05 / / 0.0 / / / / 0.04 / / 0.02 / / o I, 0 0.02 0.04 0.06 0.08 0.1 00 Figure 411. Behavior of bedload transport equations at low Shields numbers. The critical threshold for this example is taken arbitrarily as cr = 0.04. Legend names correspond to the transport relations of M1, i, rPeter and Miller (1948) (\!PM48), Nielsen (1992) (N92), Ribberink (1998) (R98), and Camenen and Larson (2005) (CL05). The limits of the inset panel are represented by the dark black box on the larger figure. 0.002 0.0015 0.001 0.0005 0 FL74 NT80 VR84 004 0045 005 . 0.04 0.06 0.08 C 0 Figure 412. Behavior of sediment pickup functions at low Shields numbers. The critical threshold for this example is taken arbitrarily as r, = 0.04. Legend names correspond to the transport relations of FernandezLuque (1974) (FL74), N ,; .1: .vt and Tsujimoto (1980) (NT80), and van Rijn (1984) (VR84). The limits of the inset panel are represented by the dark black box on the larger figure. 0.8 =9p, 0.5 0.6 9I = 40 p \9= 9 7000 0.4 0.2 0 0.2 0.4 0.6 0.8 1 C (cm3 / cm3) Figure 413. The modification of the relative settling velocity as a function of concentration using the equations of Richardson and Zaki (1954). 0 0001 6E05k 4E05 2E05 (903 0030 I I I I * i. WsoCb E C(x,y,z,t) Qb Qb + dQ4 dx dx Figure 414. A twodimensional sediment control volume showing relevant transport terminology and a schematic of sediment distribution. In the model, the true control volume extends into the ydirection, too. 0.8 0.6 0.4 0.2  e=o  Ob = 22.5 ........... = 45 I I I I I I I I 2 4 6 8 10 12 14 16 18 Grid Points/Wavelength (a) 7, 0.125 6 8 10 12 14 Grid Points/Wavelength (b) y, 0.25 Figure 415. Amplification factors for the 2D lowpass morphology filter (after Johnson and Zyserman (2002)). 2 4 6 8 10 12 14 Grid Points/Wavelength 16 18 20 (a) 7 = 0.125 Grid Points/Wavelength (b) y, 0.25 Figure 416. Amplification factors for the fourstep hybrid morphology filter of Jensen et al. (1999) (after Johnson and Zyserman (2002)). 0.8 0.6 0.4 0.2 /i Ob = 0  eb = 22.5 ........... b = 45 CHAPTER 5 MODEL EXPERIMENTS A total of forty simulations are performed to evaluate the capabilities of our 1DH linear, and 3D nonlinear, phaseresolving livebed modeling systems. An additional four simulations are conducted to address our specific scientific questions regarding the roles of bedload and suspended load sediment transport, as a function of both space and time, in generating and d, lr ,ii ;. bedforms. Additional details regarding simulation parameters for all fortyfour experiments can be found in Table A1 of Appendix A. Information regarding the experiments follows, and is first segregated by model type, and then into three distinct phases for the nonlinear model experiments. 5.1 Linear Model Experiments A proofofconcept simulation is proposed with the 1DH model of bedform growth in the linear regime. The simulation will be used to illustrate the validity of our derived model dynamics, and similarly prove that bedforms in the linear regime may be formed completely through bedload sediment transport, as si l. 1. I in the literature. Other than the basic domain and flow parameters, there are two free parameters that must be chosen: N and Cmix. For the simulation here, we assign a mixedli ir thickness of N = 2 graindiameters, and a mixed1li vr concentration of Cix = 0.3, which is onehalf of the commonly accepted closepacking concentration of 0.6. A 300 pm grain diameter is selected for this experiment. In reality, the region of active bedload sediment transport may be as much as 10 20 graindiameters in thickness (Bagnold, 1980), but the model results seem relatively insensitive to this parameter. For the coefficient of resisting friction in Equation 47, the canonical value of fb = 2.5 x 103 is selected. The periodic length of the xdomain is set at Lx = 60 cm, with a horizontal grid spacing of 6x = 2.0 cm. The freestream velocity and period are chosen as U, = 30 cm/s and T = 4 s, respectively. 5.2 Nonlinear Model Experiments 5.2.1 Phase I: Model Validation Evaluation of the hydrodynamic modeling system is carried out through modeldata comparisons using the oscillatory boundary l1.r data of Jensen et al. (1989). Specifically, simulations of Test 10 from Jensen et al. (1989) are performed with the following flow characteristics: Uo = 200 cm/s, and T = 9.72 s. The flow parameters tested produce an orbital Reynolds number of R = 6 x 106. In the horizontal periodic x y plane, the model domain has lengths Lx = Ly = 3.2 cm, whereas the entire 30 cm water column is modeled in the vertical. Model predictions of the horizontally and phaseaveraged (10 wave periods) streamwise boundary l1.r velocity, as well as the boundary l1.r thickness, are compared to the Test 10 experimental data reported in Jensen et al. (1989), and obtained from Mutlu Sumer. Beyond the modeldata comparisons of boundary lv.r flow, four simulations with domain sizes, flow parameters, and boundary conditions similar to those that will be used extensively in the following investigation are performed to determine appropriate grid metrics. Additionally, these tests are performed to ensure convergence of normalized flow metrics for a variety of horizontal and vertical grid resolutions. The variable mesh resolution for the computational grid tests is shown in Figures 51(a)51(d), and the grid parameters are provided in Table 51. These resolution tests are performed over a single, stationary, sinusoidal ripple having ,r = 1.0 cm and A = 13.6 cm, in a flow with Uo = 40 cm/s and T = 2 s. Prior to performing the majority of model simulations, seven simulations with increasingly large domain widths in the spanwise (y) direction are performed with the coupled, livebed model. These domain width tests are conducted to determine the effect of the third dimension on both hydrodynamics and predicted morphology, the results of which will be used to determine appropriate domain sizes for the Phase III simulations. Model domain parameters for the seven domain width tests are given in Table 52, and the undistorted model domains are shown graphically in Figures 52(a)52(f). Simulations are performed with a freestream velocity of U" = 40 cm/s, wave period of T = 2 s, and grain size of d = 0.3 mm, and the bottom boundary is initialized with a 2D sinusoidal ripple having r] = 1.0 cm and A, = 13.6 cm. The domain width tests are run for a total of 40 wave periods, or t = 80 s. These flow parameters will be used extensively throughout the following investigation, and correspond to what we are deeming our baseline experimental parameters. In order to evaluate the sensitivity of morphology predictions to the choice of sediment transport submodels, four simulations are proposed. The sediment transport submodel tests employ the bedload transport equations of AT. , rPeter and Miller (1948) (Equation 453) and Camenen and Larson (2005) (Equation 455), and the pickup functions of van Rijn (1984) (Equation 464c) and FernandezLuque (1974) (Equation 464a), in the varying combinations listed in Table 53. These particular transport submodel combinations are chosen for their uniqueness, or popularity, in available literature; for example, the powerlaw bedload formulations are conceptually the same, differing only through the choice of a constant and an exponent; however, the formulation of Camenen and Larson (2005) is unique in the literature, as it is one of very few that do not incorporate the commonly accepted critical threshold concept. For the sake of brevity, a quasithreedimensional domain is used, having only two physical, nonperiodic grid points in the spanwise direction. The domain and initial morphology are similar to that shown in Figure 52(a). The model simulations are initialized with a single, sinusoidal ripple having rr = 1.0 cm and A, = 13.6 cm. Flow parameters correspond to the baseline case where U, = 40 cm/s and T = 2 s, yielding a mobility number = 32.95. The final suite of simulations performed in Phase I are aimed at assessing the ability of the 3D livebed modeling system at predicting morphology over a range of flow conditions. With respect to the mobility number T, a nondimensional parameter commonly used in semiempirical ripple predictors for determining equilibrium height and length, the experiments range by one order of magnitude from low (T w 10) to high (T w 100). A total of nine morphology test simulations are conducted utilizing the quasithreedimensional configuration, where only two nonperiodic points are used in the spanwise direction. The simulations are performed with 200 pm, 300 pm, and 400 pm grain diameters, flow velocities ranging from 30 cm/s < U o < 80 cm/s, for labscale (T = 2 s) and fieldscale (T = 8 s) flows. Where possible, the morphology is initialized with two parabolicshaped ripples having T1r = re/2 and Ar = A,, where r1e and A, correspond to the equilibrium ripple height and length, respectively, computed using the semiempirical ripple predictor of Nielsen (1981) for both fieldscale (regular) and labscale (irregular) flows. Forcing and sediment characteristics of the nine morphology test simulations are given in Table 54. The expected ripple heights and lengths, as predicted by the equations of Nielsen (1981), Grant and Madsen (1982), Mogridge et al. (1994), Wiberg and Harris (1994), Khelifa and Ouellet (2000), Faraci and Foti (2002), Williams et al. (2005), and Soulsby and Whitehouse (2006), are provided in Table 56. 5.2.2 Phase II: Model Capabilities A total of fourteen simulations are performed in Phase II to determine additional model capabilities, as well as evaluate the sensitivity of the coupled modeling system to initial morphology, periodic domain length, and sediment size. Simulation parameters, such as model domain lengths, sediment size, and flow characteristics, are provided in Table 57. These test cases are also performed with the quasithreedimensional model configuration, which only employs two nonperiodic grid points in the spanwise direction. We feel that the limited domain width will be mostly insignificant for the purposes of exploring the model sensitivity and parameter space, as mostly qualitative evaluations of system capabilities will be pursued. In general, the model experiments conducted in Phase II are illustrative of typical results. One simulation of bedform propagation under a steady current is performed, as well as an experiment yielding subcritical flow conditions to ensure that the morphology and transport submodels do not produce spurious results. A high mobility number simulation (T = 206) of morphology and transport in the sheetflow regime is also proposed, and will assess the capability of the suspended load transport submodel at predicting typical concentrations in sheetflow lI. i, as well as the morphological response to the highlyconcentrated and energetic flow conditions. Half of the model experiments conducted in Phase III are aimed at evaluating the sensitivity of morphology submodel predictions to initial conditions, periodic domain lengths, and the effects of hindered settling; these simulations address ripple growth from 1/2 the expected height, as well as growth from a nearly flat bed; morphological response under equilibrium conditions; ripple decay from initialized morphology that is out of equilibrium with the flow conditions; as well as ripple wavelength evolution through the processes of ripple coarsening (slide) and bifurcation (split). While the majority of simulations conducted with the livebed modeling system use fine to medium cohesionless sand, four experiments are performed with extreme grain sizes ranging from silt, d = 25 pm, to gravel, d = 6000 pm. Although most of the semiempirical ripple predictors discussed previously are not wellsuited for predicting morphology under such extreme conditions, the predicted bed states are compared to a recent bed state phase diagram provided by Kleinhans (2005) for morphology under waveinduced flows. A total of four experiments are conducted with these extreme sediment sizes, with one simulation expected to develop a consistent morphology, and one to remain below the critical threshold of motion, for both grain sizes. 5.2.3 Phase III: Sediment Transport Processes The third and final phase of model experiments are formulated, and conducted, to determine the relative contributions of bedload and suspended load sediment transport, as a function of both space and time, during ripple growth, equilibrium, and decay. Four simulations are proposed to address these specific scientific objectives: one employing total load sediment transport for ripple growth; another simulating ripple decay under total sediment load transport; a third incorporating only bedload sediment transport; and the fourth employing only suspended load transport in the bedlevel updating scheme. Using the results of many of the simulations performed in Phases I and II, specifically those pertaining to acceptable grid resolution (GRID1) and model domain width (Ly > 2.0 cm), we select the most desirable parameters for computational mesh and domain size. The four Phase III production simulations are all performed with the full threedimensional livebed modeling system to ensure accurate representation of turbulent structures that form in the spanwise direction. A schematic of the model domain showing the initial morphologytwo parabolicshaped ripples having r1, = 1.0 cm and A, = 13.6 cm is provided in Figure 53. Model domain lengths, as well as flow and sediment characteristics, of the final simulations are provided in Table 58. The flow parameters for the Phase III simulations correspond to our baseline test case where U" = 40 cm/s and T = 2 s. These four simulations are initially run for 3 wave periods to allow equilibration of model hydrodynamics prior to engaging the transport and morphology submodels, after which time they are run out for an additional 72 wave periods for a total simulation time of t = 150 s. S4 2 0 10: 10 8 6 N 4 2 0 10: Figure 51. 10' Grid Spacing (cm) (a) GRID1 10 Grid Spacing (cm) (c) GRID3 S4 2 0  10 10: 10r 8 6 S 4 2 0 10 10: 10 Grid Spacing (cm) (b) GRID2 10 Grid Spacing (cm) (d) GRID4 The horizontal (6x, by) and vertical (6z) grid spacing for (a) GRID1, (b) GRID2, (c) GRID3, and (d) GRID4 as a function of the vertical zdimension. The values of grid spacing 6x, by, and 6z are plotted on a logarithmic scale. k 2 0 2 4 1 o ( 01 (a) WIDE3D_5 r j 12 1 4" 68 0 0 (c) WIDE3D_2) (e) WIDE3D_40 (b) WIDE3D_10 (d) WIDE3D_30 10 0 8 1 IzIR (f) WIDE3D_50 Figure 52. The model domains for the domain width comparison tests. The domain for WIDE3D_2 was omitted since it is only two grid points wide. Color contours represent the volumetric sediment concentration C. 153 10 N 5 0  1 019 6 788 4 6 " klb,4 z 20 C 0 95 09 15 0 85 08 0 75 07 N 10 065 06 "C 055 05 0 45 04 0 35 03 0 25 02 25 015 15 005 Figure 5 3. A schematic of the model domain and initial morphology used in the Phase III simulations: TL2v2, BL2v2, and SL2v2. The morphology is initialized with two parabolicshaped ripples having = 1.0 cm and A, = 13.6 cm. 154 Table 51. Numerical mesh parameters for the grid comparison tests. Grid L,, L,, Lz 6x, 6y, 6z nx, ny, nz Tot tal Number of Points cm 13.6, 0.4, 11.0 13.6, 0.8, 11.0 13.6, 0.2, 11.0 13.6, 0.2, 11.0 cm 0.2, 0.2, 0.05 0.4, 0.4, 0.1 0.1, 0.1, 0.025 0.1, 0.1, 0.05 71, 5, 92 37, 5, 47 139, 5, 182 139, 5, 92 Table 52. Numerical mesh parameters for the domain width comparison tests. Lx, Ly,, Lz cm 13.6, 0.4, 11.0 13.6, 1.0, 11.0 13.6, 2.0, 11.0 13.6, 4.0, 11.0 13.6, 6.0, 11.0 13.6, 8.0, 11.0 13.6, 10.0, 11.0 Table 53. Abbreviations used in Reference M. i,. rPeter and Muller (1948) Camenen and Larson (2005) FernandezLuque (1974) van Rijn (1984) 6x, 6y, 6z cm 0.2, 0.2, 0.05 0.2, 0.2, 0.05 0.2, 0.2, 0.05 0.2, 0.2, 0.05 0.2, 0.2, 0.05 0.2, 0.2, 0.05 0.2, 0.2, 0.05 nx, ny, nz 71, 5, 92 71, 8, 92 71, 13, 92 71, 23, 92 71, 33, 92 71, 43, 92 71, 53, 92 the sediment transport equation tests. Equation Transport Mode 4 53 Bedload 455 Bedload 464a Suspension 464c Suspension Total Number of Points 32660 52256 84916 150236 215556 280876 346196 Abbreviation MPM CL FL VR GRID1 GRID2 GRID3 GRID4 32660 8695 126490 63940 Grid WIDE3D_ WIDE3D_ WIDE3D_ WIDE3D_ WIDE3D_ WIDE3D_ WIDE3D_ Table 54. Case Name LL1v3 LF1v1 ML1v2 ML2v2 ML3v1 MF1v2 MF2v2 HL1v1 HF1vI The morphology test simulation matrix showing alphanumeric case names, fluid forcing and sediment parameters, and values of nondimensional parameters commonly used in semiempirical ripple predictors. Uo0 (cm/s) 30 30 30 50 40 30 50 80 60 A (cm) 9. 38. 9. 15. f 12. 38. 63. 25. 76. d Ipm 400 400 200 300 300 200 300 300 200 13. 13. 27. 51. 33. 27. 51. 132. 111. X (x106) 6. 18 0. 386 3. 089 4. 63 4. 63 0. 193 0. 290 4. 63 0. 193 (x104) 2. 86 11. 5 2. 86 7. 96 5. 09 11. 5 31. 8 20. 4 45. 8 Table 55. Citation keys of ripple predictor equations used in the morphology test simulations. Reference Nielsen (1981) Grant and Madsen (1982) Mogridge et al. (1994) Wiberg and Harris (1994) Khelifa and Ouellet (2000) Faraci and Foti (2002) Williams et al. (2005) Soulsby and Whitehouse (2006) Equations Equations Equations Equations Equations Equations Equations Equations Equations 3a35c 7a38b 13a313b 9a312b 14a314b 15a315c 16a317b 18a318b Citation Key N81 GM82 M94 WH94 KO00 FF02 W05 SW06 _^ 03 c 0 Cl 0 C O  "o t oc 0 0 0 0 0 0 ; V ^ , ** oi TO O ( t CC CtL 'I t L' ' C o0 CD coc O ^ s~ 3 o~ooo ooo C.) c> CI I03 7t 0* 0 cc I 03 I  ^E~)r n ~~ ioLoo oo 0; a D 1 0c 3 0^ 0 r Cl^ ClOO^ ci o 03 oc oo6 So  '. 1 t 0 B 030 030303 00  S s~ ^oc^ o oc Lr!o c 0; 0 S o , 03 0l 0  0 Cl CO Cl 03 Cl 0 3 Cl 0n bic CTJCD o ii3~o dC~ jC j h V. . C'A OC C' C. C I h V 0 0  0 ~ ~ o cCl C lC l O V h 0 CI 0 c CO Vc CO 1i CC V "3 ,_,0 0 1 0 10 0 4 oi Cl ho i Cco 0 CIA st' ' y. CI bC 0&O0hO&O&LC~ OCl 0S 0 H_ O C0 V CO1 C1 C t5 0S 0 C C<^1 ( 00 30 H t C1^t 5 w~~. E01'0 O L c ?^^_C ^ ^0 ^^ ; 0 0 S C ^ o o CM^ o o Z < ^ 0   Table 57. A list of experiments and relevant simulation parameters for simulations performed in Phase II. Case Name STEADY SUBCR SHEET GROW1 GROW2 EQUILIB DECAY HS1v2 SLIDE SPLIT GRAVLOW GRAVHI SILTLOW SILTHI Lx (cm) 27. 2 27. 2 13. 6 13. 6 13. 6 13. 6 13. 6 27. 2 27. 2 27. 2 30. 0 30. 0 30. 0 30. 0 Ly (cm) 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 Lz (cm) 12. 0 11. 0 11. 0 11. 0 11. 0 11. 0 11. 0 11. 5 11. 5 11. 5 11. 5 11. 5 11. 5 11. 5 nx ny nz U. (cm/s) 136 2 110 30 136 2 90 5 68 2 90 100 68 2 90 40 68 2 90 40 68 2 90 40 68 2 90 40 136 2 100 40 136 2 100 40 136 2 100 40 150 2 90 30 150 2 90 80 150 2 90 10 150 2 90 40 Table 58. A full list of experiments and relevant simulation parameters for the Phase III Case Name TL2v2 TL2v2d BL2v2 SL2v2 model experiments of sediment transport processes and bedform dynamics. Lx Ly Lz nx ny nz U. T R d (cm) (cm) (cm) (cm/s) (s) (x104) (pm) 27. 2 3. 2 21. 5 136 16 110 40 2. 0 5. 0929 300 27. 2 3. 2 21. 5 136 16 110 40 2. 0 5. 0929 300 27. 2 3. 2 21. 5 136 16 110 40 2. 0 5. 0929 300 27. 2 3. 2 21. 5 136 16 110 40 2. 0 5. 0929 300 (x104) 0. 00 0. 07958 31. 83 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 2. '.*,, 20. 372 0. 3183 5. 0929 d (3m) 300 300 300 300 300 300 300 300 300 300 6000 6000 25 25 CHAPTER 6 RESULTS Simulation results for the linear and nonlinear phaseresolving livebed models of hydrodynamics and sediment transport are provided in the following sections. A total of fortyfour (44) simulations, outlined in Chapter 5 and listed in Table A 1, were performed, and provide insight about model skill and capabilities. The model results reinforce two important concepts: first, bedform growth in the linear regime is a process dominated by sediment inertia, and second, growth in the nonlinear regime is dominated by the formation of vortices through flow separation at the crest of bedforms. 6.1 Linear Model Results The results of the 1DH conceptual model experiments indicate that in order to create bedforms in the linear regime, it is necessary to incorporate a spatial lag between the mixture velocity and transport in Equation 414. Without such a lag, the 1DH model predicts a flattening of the initialized morphology, with no recovery or generation of bedforms. This behavior has both physical and numerical significance, and is discussed in terms of linear stability in Parker (1975). The linear stability analysis of Parker (1975) demonstrates that flat sand beds are unconditionally stable in a linear regime where velocity and transport are assumed to be in phase with one another. Physically, this spatial lag may be interpreted as an inertial effect of the sediment, whereby the weight of the sediment causes transport to lag behind the applied forcing. The spatial lag 6s is implemented in the 1DH transport equation (Equation 414) according to Equation 61. For the conceptual 1DH experiment discussed here, the lag was found to be 6s = 6 cm, which yields a lag distancewavelength ratio, ks/27 =0.1, similar to the results of Parker (1975). Qb(X) = Umix(X 6s)AmixCmix (61) Timestacks of bed elevation from the 1DH linear model experiment are plotted in Figure 61. After a flattening of the initial morphology, having an amplitude equal to three grain diameters and a wavelength equal to the periodic domain, the bedform instability begins to develop, and grows to a point where the linear model approximations are no longer valid. The time evolution of bedform height and length is demonstrated in Figure 6 2(a), and shows the rapid scaling of wavelength coincident with bedform development at approximately t = 19 s. Rolling grain ripples, characteristic of the linear regime, typically have steepness values of r = 0.1 (Sleath, 1976). A timeseries of bedform steepness (or = qr/Ar) is given in Figure 62(b), and shows an increase in steepness to about onehalf the expected value for rolling grain ripples before nonlinearity in the mixed velocity causes the model to fail. More specifically, this model failure is a result of a nonlinearity between the mixed velocity directed up the bedform slopes and a competing gravitational stress directed in the opposite sense. The results of the experiment using our conceptual 1DH bedload transport model, however, successfully point out that in the linear regime, where applied stress and transport are in phase with one another, inertial characteristics of the sediment are responsible for bedform development. 6.2 Nonlinear Model Results Results of the nonlinear model simulations are divided into three phases corresponding to: model validation of hydrodynamics and morphology; an exploration of model capabilities, sensitivity, and parameter space for sediment transport and morphology; and information about the roles of bedload and suspended load transport in generating sedimentary structures. 6.2.1 Phase I: Model Validation A total of twentyfive (25) simulations were performed to evaluate the predictive skill of the hydrodynamic and livebed submodels, as well as their sensitivity to numerical resolution and domain size. The results of these simulations are provided in the following sections. 6.2.1.1 Hydrodynamic Validation Assessment of the LES hydrodynamic model was accomplished by comparing model predictions of the phaseaveraged streamwise velocity to the oscillatory boundary l1 v. data of Jensen et al. (1989) (Test 10). The tests simulate an oscillatory boundary l1'vr over a smooth boundary having U, = 200 cm/s and T = 9.72 s, and an orbital Reynolds number of R, = 6 x 106. The results of the hydrodynamic validation are shown in Figures 63(a)63(1). Streamwise velocity profiles predicted by the hydrodynamic model were phaseaveraged over nine wave periods, with the first cycle being ignored due to model spinup .,viiii:, I ry. Experimental data for the second half of the phaseaveraged wave cycle were not available, but the results are simply a mirror image of the first halfcycle. Laboratory velocity profile data were only collected over the bottom 15 cm of the water column, so only that portion of the model profile is shown; however, the model domain extended to L, = 30 cm to simulated the full water column. For additional clarity, only every other data point is plotted in Figures 63(a)63(1). Shortcomings in the hydrodynamic model are evident in the comparison of predicted and observed velocity profile data shown by Figures 63(a)63(1). Although the shape of the predicted velocity profile tends to agree with the results, the model velocity tends to lag slightly behind the laboratory data outside of the boundary 1l .r. On further inspection, the lag appears to be somewhat consistent having a phase lag of approximately 3, or about t = 0.08 s when considering the simulated wave period of T = 9.72 s. This disagreement may be the result of systemic timekeeping errors, either in the laboratory data, or manifest through a summation of residual computational errors in the model. Differences are also seen in the modeled and observed boundary 11rvr thickness, as demonstrated in Figure 64. With exception to a few phases, the LES model consistently overestimates the thickness of the boundary 1v r (6) by approximately 2 5 mm. The marked overestimation of 6 at t = 107r/12 is a result of the predicted flow beginning to reverse phase ahead of the observed data, whereby the boundary l?v1r is released prior to reforming near the wall in the succeeding phase interval of t = ll1r/12. This premature flow reversal, as seen in Figures 63(j) and 63(k), is an inertial property of the boundary 1l .r, where the slower flow (thicker boundary 1i~vr) predicted by the model possesses less momentum than the faster flow (thinner boundary 1.~r ) observed in the experiment. The overestimation of the boundary 1 vr thickness, and the resulting differences between modeled and observed data in boundary 1v.r phase, are most likely due to an overestimation of the turbulent eddy viscosity Vt. Such overestimations are characteristic of the Smagorinsky (1963) closure method in high Reynolds number flows, but have been shown to perform well in limited cases (Ferziger, 1996a; Meneveau, 1994). Additional inaccuracies in the boundary l vr predictions may be the result of the nearwall model, which uses the van Driest damping function of Moin and Kim (1982) to ensure that Vt goes to zero at the wall. Ferziger (1996b) argues that the application of van Driest damping as a nearwall model is difficult to justify in LES models, and more sophisticated models are offered by Piomelli and Balaras (2002). An evaluation of average model error, with respect to the experimental data, is shown in Figures 6 5(a) and 6 5(b). Model predictions of velocity were obtained at twice the resolution of the available lab data in the lower 15 cm of the water column; therefore, for the purpose of determining model error through direct comparison, it was necessary to interpolate the model predictions to the distribution of experimental data over the vertical. The interpolation was performed using an inverse distance weighted (IDW) procedure, but no attempt was made to ensure conservation over the profile. The phase and horizontallyaveraged absolute percent difference between model predictions and experimental data is plotted in Figure 6 5(a) as a function of vertical location and wave phase. As noted earlier, the poorest agreement in Figure 6 5(a) tends to be during periods of flow reversal, and within the boundary 11~vr. This is more clearly demonstrated by the verticallyaveraged values of absolute difference plotted in Figure 65(b). With the exception of the large differences found at times of flow reversal, the model predictions are generally within i l'. of the expected values. The relatively poor performance of the LES model, especially within the boundary l~v.r, does have implications for the performance of the sediment transport and morphology submodels. An overestimation of the boundary 1l r thickness, due mostly to a larger vt, yields lower vertical velocity gradients, and, therefore, lower values of bed shear stress (Equation 446). This difficulty may be overcome by accounting for the effects of the eddy viscosity in the shear stress definition, as previously discussed in C'! lpter 4, and demonstrated in Equation 447. A thicker boundary l1, r may also lead to higher concentrations of suspended sediment over a greater distance above the bed. The phase lead in the boundary l~ r (ahead of the data), and corresponding lag in the outer flow, may also affect the timing of entrainment and deposition of suspended sediments, although we would expect the presence of vortices over rippled beds to p1 i, a much more dominant role in this regard. It should be noted that while the LES model yielded, in some cases significant, differences between expected and observed results, the 1D! iiP ily of simulations used for assessing characteristics of sediment transport and morphology were performed at Reynolds numbers two orders of magnitude smaller than that used in the modeldata comparison. 6.2.1.2 Computational Grid Tests A number of quasithreedimensional (having only a few grid points in the y or spanwise direction) simulations were performed in order to determine the effects of grid resolution on normalized flow metrics, as well as computational expense. Four grids with varying degrees of resolution (see Table 51) were tested for an oscillatory flow with U, = 40 cm/s and T = 2 s over a fixed, sinusoidal bottom boundary. The overall computational expense, with respect to computational resolution, is demonstrated in Figure 66, and the relationship between the two is found to follow a logarithmic relationship. The total number of grid points within the tested domains ranged from 8695 at the lowest resolution, to 126490 for the highest resolution, and required 1.2 and 401.5 hours of wallclock time to simulate 30 s of model time, respectively. The most timeconsuming component of the modeling system is by far the iterative pressure solver, requiring increasingly larger number of iterations as the number of grid points increases. As exhibited by the overall computational performance plotted in Figure 66, a roughly tenfold increase in the number of grid points results in a fourhundredfold increase in required wallclock time. While the computational expense, with respect to time, is sensitive to the number of grid points used, certain normalized flow metrics should not vary significantly with numerical resolution. The timedependent, volumeaveraged u, v, and wvelocity standard deviations are plotted in Figures 67(a)67(c). The standard deviations were filtered (50 passes with 7s = 0.25) using a diffusive smoothing filter (Equation 426a) to enhance the lowfrequency detail of the signals. Convergence of the average streamwise velocity standard deviation signal is observed in Figure 67(a) for all four grids. Similar convergence is found in the vertical velocity standard deviation signal for GRID1, GRID3, and GRID4 in Figure 67(c), with the lowest resolution grid (GRID2) predicting slightly lower values. The spanwise velocity standard deviation signals shown in Figure 67(b) are nearly dubious since the domains only have two nonperiodic points in the ydirection; however, GRID2 demonstrates an inability of the numerics to reach a steady state in the spanwise direction within the simulated model time. The turbulent kinetic energy [k = (u'2 + v'2 + w'2)/2] is also used as a normalized metric for comparing the effects of grid resolution on model predictions. Figure 68 shows the cumulative volumeaveraged turbulent kinetic energy (TKE) as a function of time for the grid comparison tests. Similar to the velocity standard deviations, the TKE signals were filtered (50 passes with 7s = 0.25) using a diffusive smoothing filter (Equation 426a) to enhance the signaltonoise ratio. The maximum absolute difference between the four signals in Figure 68 is approximately 10C' and is reduced by a factor of onehalf when the lower resolution GRID2 is removed. The phase and volumeaveraged TKE (((TKE))) is plotted in Figure 69 as a function of forcing phase (wt), and demonstrates similarly good agreement between GRID1, GRID3, and GRID4, indicating numerical convergence of the grid parametrics. Phase and spanwiseaveraged velocity profiles for the grid comparison tests are plotted as a function of ripple location in Figures 610(a) 610(f), Figures 611(a) 611(f), and Figures 612(a) 612(f), for the ((u)), ((v)), and ((w)) velocity components, respectively. Since the applied horizontal pressure gradient produces a symmetric oscillatory flow, only the first onehalf of the forcing cycle is shown in each of the figures. Although the average velocity profiles are not necessarily considered a normalized metric, there appears to be generally good agreement between the predictions of the four grids tested. The most significant differences in profile shape and magnitude are found for the ((w)) velocity component plotted in Figures 612(a) 612(f), and may be attributed to the difference in resolution of the sinusoidal bottom boundary between the four grids. While the velocity profile and normalized metric comparisons tended to yield favorable results for GRID1, GRID3, and GRID4, the former (GRID1) was selected for use in the remainder of the simulations as it provided the most advantageous balance of resolution and expense. 6.2.1.3 Model Domain Width Tests Simulations with increasingly larger spanwise (y) domains were performed in order to evaluate the effects of domain width, and therefore threedimensionality, on both normalized flow metrics and the equilibrium bed morphology. These tests are useful for determining the extent of the third dimension required for resolving pertinent details in flow and morphology, as it is computationally beneficial to use the smallest domain possible. The simulations were performed with a twodimensional (x z) oscillatory flow having U, = 40 cm/s and T = 2 s, over a solitary sinusoidal ripple initialized at lr = 1.2 cm for a total duration of 40 wave periods, or t = 80 s. Mesh parameters for the seven threedimensional (3D) tests are provided in Table 52, and range from domains having two to fifty grid points in the ydirection. The overall computational budget for the seven domains tested is plotted in Figure 613, and shows a logarithmic relationship between the number of grid points and wallclock time similar to that found in the resolution tests (Figure 66). A tenfold increase in the number of grid points used in the threedimensional tests results in simulation times that are nearly fourtimes longer. Unfortunately, simulations were run on different platforms with some of the larger tests executed on more recent CPU models; therefore, a direct comparison or analysis between the simulations is not appropriate, but is nevertheless insightful. It should also be noted here that the 3D tests utilized the sediment transport and morphology submodels to update the bed elevation at every timestep, thereby requiring an increasing number of iterations by the pressure solver to ensure flow continuity within the fluid domain. A more extensive evaluation of the sediment transport and morphology submodels is provided in the following sections. Similar to the numerical resolution tests discussed previously, normalized metrics of flow and morphology are used to compare the effects of the domain width on model predictions. The filtered (Equation 426a) cumulative volumeaveraged TKE (TKE) is plotted in Figure 614 as a function of time for the seven 3D grid comparisons. The cumulative TKE for all seven grids is found to grow immediately during model spinup, and then achieves an approximately average value of TKE = 200 cm2/s2. As opposed to the fixed bed resolution tests, there is more high frequency oscillation in the cumulative signal due to the change in morphology at every hydrodynamic timestep; however, they do tend to attain a mean value over the duration of the simulation. Figure 615 shows the phase and volumeaveraged TKE (((TKE))) as a function of forcing phase (wt) for the seven 3D domains. Phaseaveraged quantities are computed using ensemble averages over 39 wave cycles, with the first cycle being excluded due to model spinup .,iiliii I ry. With the exception of the 40 (WIDE3D_40) and 50point domains (WIDE3D_50), the phase and volumeaveraged TKE appears to be similar between the grids. The slightly lower values of average TKE in the two largest domains, as indicated by Figures 614 and 615, may indicate higher levels of either physical or numerical dissipation in the grid. The general effect of the third dimension (y) should be to increase dissipation due to the formation of turbulence. Without a sufficient width in the spanwise direction, rotation generated in the x z plane is prohibited from breaking down into turbulent structures, thereby limiting the dissipation of energy through turbulence production. In theory, one would seek to increase the spanwise dimension until full development of the spanwise velocity is achieved. The phase and volumeaveraged u, v, and w velocity components are plotted in Figures 616(a)616(c) as a function of forcing phase for the threedimensional grid comparisons. The oscillatory uvelocity signal is clearly defined in Figure 616(a), and no discernible differences are evident between the seven different grids. The width of the third dimension (Ly) does appear to impact the development and behavior of the spanwise (v) velocity, with the three largest domains reaching general agreement. Interestingly, the four largest domains ranging in width from 4 cm to 10 cm show a mean spanwise velocity oriented in the positive ydirection; however, even the largest values (0.3 cm/s) are two orders of magnitude smaller than the streamwise forcing velocity. The magnitudes of the phase and volumeaveraged vertical (w) velocity are fleetingly small, having values less than 1 x 103 cm/s, and appear to have a zero mean. This I... 1 that the development of vortices on either side of the ripple crest are wellbalanced during ripple development. A Fourier ain i1, i using a Fast Fourier Transform (FFT) package, was performed to determine the dominant energycontaining lengthscales for flow in the spanwise direction of the seven 3D domain tests. Figures 617(a) and 617(b) show the time and spatiallyaveraged direct Fourier transforms of the v velocity as a function of wavenumber (ky). Applying direct Fourier transforms to the v velocity provides information regarding the relationship between energy density and dominant lengthscales in the velocity field. As shown in Figure 617(a), the largest values of energy density for each domain correspond to the lowest wavenumbers (k = 1/Ly). The lowest wavenumber for each domain corresponds to the full periodic length of that domain. If the peak energy density were to occur at a higher wavenumber, then the domains would contain significant amounts of energy at smaller integral lengthscales, or wavenumbers, in the ydirection. Furthermore, the spectral density plots in Figure 617(a) tend to saturate for the four largest domains having widths Ly > 2.0 cm. Figure 617(b) indicates that the largest velocity amplitudes also occur at the lowest wavenumbers, but are still one order of magnitude smaller than the streamwise forcing amplitude (U, = 40 cm/s). The absence of significant amounts of energy at integral lengthscales smaller than the periodic length of the ydirection for the threedimensional domains, as well as the small values of velocity amplitude in the ydirection, reinforce the twodimensional nature of the applied oscillatory forcing. In addition to the effects of the domain width on hydrodynamics, particularly as they relate to the generation of turbulence, it is beneficial to evaluate those effects on equilibrium ripple morphology. Even for the strictly twodimensional horizontal pressure gradient applied to generate the oscillatory boundary 1, r forcing, some modification of the flow, and therefore morphology, is anticipated in the spanwise dimension. The standard deviation of the bed elevation a,, in the spanwise dimension, averaged over the xdirection, is plotted as a function of time in Figure 618. The standard deviation of the bed elevation in the third (y) dimension provides an indication of ripple variability, about the mean, in the third dimension for the different threedimensional grids. The smallest domain, containing only two physical grid points (WIDE3D_2), has essentially zero variability in the ydirection as shown in Figure 618. The lack of variability in the third dimension for WIDE3D_2 indicates that it is behaving more twodimensionally, with little to no transport in the ydirection. For domains having greater than five physical grid points in the third dimension, Figure 618 demonstrates that variability of the ripple profile in the ydirection saturates at about a, = 0.75 cm, and remains relatively constant after the first few wave periods. Therefore, the width of the spanwise dimension does appear to influence properties of the morphology in the third dimension, but the variability for domains larger than or equal to Ly = 2.0 cm appears to be relatively similar, and remains constant in time. The most significant effect of the ydomain width (Ly) on the morphology would manifest itself through discrepancies in the equilibrium ripple characteristics. Equilibrium ripple heights for the seven domains are shown in Figure 619, and have been averaged over thirty wave periods (20 s < t < 80 s). The standard deviation of ,]r about the timeaveraged value (Tr,) is also given in Figure 619 for each domain test. There appears to be good agreement for lr not only amongst the seven domains, but also to the equilibrium ripple height (Tr = 1.89 cm) computed using the ripple predictor of Nielsen (1981) for regular flows. There is approximately a twelvepercent difference between the highest and lowest predictions of l, for the seven domains, with the smallest ]r = 1.80 cm occurring for the quasithreedimensional domain (WIDE3D_2). For domains having widths greater than or equal to Ly = 2.0 cm, there is less than I!'. difference in the average equilibrium ripple heights. Since all seven domain tests were run with one full period of a ripple, the equilibrium ripple length remains locked at A = 13.6 cm for all simulations. Fourier analysis was also used to evaluate the impact of the domain width on the development of significant bed features in the third dimension. Figures 620(a) and 620(b) show the time and spatiallyaveraged direct Fourier transforms of the bed elevation zb in the ydirection. The transforms were first applied to the morphology in the ydirection, averaged spatially in the x and zdirections above the ripple crests, and then timeaveraged over the last thirty wave periods of the simulations. Similar to the results of the Fourier analysis applied to the v velocity field, the maximum spectral density of zb occurs at the lowest wavenumber (k, = 1/L,) for each domain, corresponding to the full periodic length of each domain in the ydirection (Figure 620(a)). The energy contained in perturbations of the ripple profile in the ydirection, even at the lowest wavenumbers (largest lengthscales), is weak with most peaks less than 0.1 cm3. The maximum perturbation amplitudes, shown in Figure 620(b), are also small and tend to be less than 0.1 cm with peaks occurring at the lowest wavenumber. Once again, there appears to be saturation in both the spectral density and perturbation amplitude at the lowest wavenumbers for the four largest threedimensional domains (Ly > 2.0 cm), with very little energy contained in the higher wavenumbers. Taking into account the impact of the third dimension on both hydrodynamics and morphology, it appears advantageous to perform the final production simulations with domain widths of Ly > 2.0 cm. This length is in direct proportion to the expected equilibrium ripple height, as predicted by Nielsen (1981), sl., I ii.; a dependence of domain width on ripple height. Similar observations were made by Barr et al. (2004) for oscillatory flow over fixed, rippled boundaries. 6.2.1.4 Sediment Transport Submodel Tests Four combinations of various bedload transport and sediment pickup equations, found in available literature, were used to determine the sensitivity of morphology to transport submodels. As outlined in Table 53, the following combinations of transport submodels are shown in Figures 621(a) and 621(b): MPMVR, AT i rPeter and Miller (1948) and van Rijn (1984); MPMFL, At, i* rPeter and Miller (1948) and FernandezLuque (1974); CLVR, Camenen and Larson (2005) and van Rijn (1984); CLFL, Camenen and Larson (2005) and FernandezLuque (1974). The instantaneous ripple height, shown in Figure 621(a), was calculated using Equation 62, where a ,(t) is the standard deviation of the bed elevation zb in the streamwise direction at each time t. The ripple height formulation given in Equation 62 was I  I. '1 by Hanes et al. (2001), and has more recently been used in laboratory analysis of transient ripples by Doucette and O'Donoghue (2006). In this current study, the method of Hanes et al. (2001) for calculating ripple heights is favored over the more traditional method of identifying maximums and minimums in the bed elevation signaltypically through application of a zerocrossing techniquefor two reasons: first, the zerocrossing techniques are not applicable when simulating only one ripple wavelength; and second, Equation 62 is much less sensitive to the rapid changes in bed elevation experienced during the simulations as it relies on average statistics of the morphology instead of singlepoint measurements. r (t) 2 V2a(t) (62) Timeaveraged statistics of the ripple heights for each combination of sediment transport submodels are shown in the inset plot of Figure 621(a), where the error bars represent the standard deviation of the instantaneous ripple height about the mean (lr). The timeaveraged ripple heights (Qlr) were determined by averaging ripple height statistics over the last twenty wave periods of the simulation, The data indicate that there is some dependence of the predicted morphology on the choice of sediment transport submodels, with a difference of roughly ten percent between the highest and lowest average ripple height predictions, and a factor of two difference between the ripple height standard deviations. The maximum departure from the expected ripple height (rr = 1.89 cm) is just over seven percent, and corresponds to the sediment transport submodel (I PMFL) employing the bedload transport equation (Equation 453) of At. i, rPeter and Miller (1948) and sediment pickup function (Equation 464a) of FernandezLuque (1974). The MPMVR and CLFL transport submodels yielded the minimum departure from the expected ripple height at +1.5'. and 1.5'. respectively; however, the CLFL submodel yielded the largest standard deviation of all the transport submodels at a = 0.11 cm, while the MPMVR submodel produced the smallest at a = 0.057 cm. The data in Figure 621(a) may be used to assess the predictive capabilities of submodel combinations, as well as their constituent components. For example, the MPMFL and CLFL submodels both employ the sediment pickup function of FernandezLuque (1974), which yielded the largest standard deviations in the bed elevation timeseries. In contrast, the MPMVR and CLVR submodels produced the smallest standard deviations and ripple height predictions both within 2.,' of the expected value. These results sl., I the the sediment pickup function of van Rijn (1984) may perform better at predicting suspension events, with respect to timedependent bed morphology, to that of FernandezLuque (1974). Isolating the performance of the bedload transport equations from the data in Figure 621(a) is more difficult, but the combination (\PIMVR) of the AM. i' rPeter and Miller (1948) bedload transport equation and sediment pickup function of van Rijn (1984) appears to produce the best overall results. This conclusion agrees well with those of Madsen (1991) and Garcia and Parker (1991), who recommended using the A1. i*, rPeter and Miiller (1948) bedload transport equation and van Rijn (1984) pickup function, respectively. The ripple steepness, defined as r/A,, for the sediment transport submodel comparison tests is shown in Figure 621(b). Since the ripple wavelength A, for these simulations was fixed at one periodic length in the streamwise direction (A, = 13.6 cm), the ripple steepness timeseries mimics that of the ripple height shown in Figure 621(a). Classical orbital ripple predictors si. 1 that there is only one value of ripple steepness, or ratio of ripple height to wavelength, for a particular combination of fluid forcing and sediment characteristics [e.g. Nielsen (1981)]; therefore, the ripple steepness should remain constant during ripple growth as the height and wavelength grow large in proportion to one another. This constant ripple steepness was observed in the laboratory experiments of O'Donoghue and Clubb (2001), but is not reflected in the results of Figure 621(b) since Ar remains fixed at the periodic length of the streamwise domain. Instead, the ripple steepness grows quickly along with the ripple height until the morphology is in equilibrium with the hydrodynamics, after which time it remains relatively constant at roughly r/Ar = 0.14. It is encouraging that the ripple steepness remains both relatively constant and below the maximum (gravitational) limit as indicated in Figure 621(b), as it sirl 1 that the morphology reaches a dynamic equilibrium with the fluid forcing characteristics instead of gravitational limits imposed by the sediment's avalanche angle, or static angle of sediment repose. The maximum steepness limit shown in Figure 621(b) corresponds to 0.32 tan 4, a practical limit of steepness for vortex ripples indicated by laboratory data (Nielsen, 1992; Faraci and Foti, 2002). In the transport submodel tests, Q = 300 is the static angle of sediment repose for the cohesionless sediment grain size (ds0 0.3 mm) assumed in the simulations (Julien, 1998). Anecdotal evidence of stable rolling grain ripples has appeared in the literature, but typically only for laboratory experiments utilizing oscillating trays (Bagnold, 1946) or oscillating annular cells (Stegner and Wesfreid, 1999). Analytical models of rolling grain ripples, such as those of Sleath (1976), Andersen (2001), and Andersen et al. (2001), have also i. 1. the presence of stable rolling grain ripples, typically having a characteristic steepness of rr/Ar = 0.10 similar to that iti. 1i 1 by early experiments of Bagnold (1946). Furthermore, the experiments of Bagnold (1946) ,;;i 1. .1 that ripple development involved a primary formation of stable rolling grain ripples, with a transition to vortex ripples at higher Shields' stresses. Laboratory experiments in oscillating water tunnels, however, ,'1.: 1 that rolling grain ripples are transient at best; and Miller and Komar (1980b) state that they are rarely observed in the field. The ripple steepness results in Figure 621(b) show no clear evidence of a transient state at the rolling grain ripple steepness of 0.10, and instead ii.:: 1 that there is continuous transformation of the steepness as the ripple grows in height. 6.2.1.5 Morphology Tests A total of nine simulations were conducted over a range of fluid forcing and sediment characteristics to: 1. Assess the predictive capabilities of the sediment transport and morphology submodels; and 2. Evaluate the performance of the coupled modeling system over a range of mobility numbers (T) (Brebner, 1980). The mobility number T is a metric that describes the ratio of mobilizing (destabilizing) to gravitational (stabilizing) forces for a sand bed, and is commonly used in semiempirical ripple predictors (Nielsen, 1981; Khelifa and Ouellet, 2000; Faraci and Foti, 2002; Williams et al., 2005) to determine equilibrium ripple height and length. As there are many references to the role of the wave periodand hence the nearbed wave orbital amplitudeon determining ripple characteristics (\il r and Komar, 1980a; Mogridge et al., 1994; Wiberg and Harris, 1994), simulations were conducted at what are considered labscale flows having wave periods of T = 2 s, and fieldscale flows having periods of T = 8 s. The simulation parameters for the morphology tests, along with the expected ripple heights and lengths determined by the ripple predictors discussed in ('!i Ilter 3, are provided in Tables 54 and 56, respectively. Prior to discussing any results from the morphology test cases, it is imperative that we demonstrate conservation of mass within the coupled modeling system. Conservation of fluid mass in the hydrodynamic model is ensured through the SOR iterative pressure solver, which iterates until continuity is satisfied within IV ui < 1 x 103. As outlined in C'! lpter 4, the morphology submodel utilizes gradients in bedload transport flux, as well as the entrainment and deposition modeled through the sediment concentration field, to update the bed elevation at every hydrodynamic timestep. There are then three possible areas where conservation errors may be introduced: (1) the finitedifference formulations used to estimate the bedload transport fluxes, (2) incorrect implementation of the sediment pickup function as a bottom boundary condition on the scalar concentration field, and (3) numerical errors associated with the firstorder donoracceptor method used for tracking the sediment concentration field. Figure 622 demonstrates the conservative nature of the sediment transport and morphology component of the modeling system, and is representative of typical model performance. At every time step (6t = 0.0001 s), the percent change in sediment volume, from the initial conditions, was computed and recorded; the resulting timeseries, after filtering at 1 Hz to remove noise, is shown in Figure 622. The signal mean, regardless of the filtering process, remains well below 0.1.  ., I ii; that conservation of sediment mass is obtained. In general, most of the available literature on bedforms tends to agree that orbital, vortex ripples scale in some proportion to the nearbed wave orbital semiamplitude A = Umax/. A selection of such relationships between ripple wavelength A, and A are provided in Table 31, and show general agreement for bedloaddominated transport regimes. Commonly, ripple predictor equations for height and length are nondimensionalized by the semiamplitude A and cast in terms of the mobility number T. The most commonly observed behavior of orbital ripples in fine to coarse sand, in terms of T, is that for an increase in destabilizing forces (larger T), the ripple height tends to decrease while the ripple length increases. The laboratory data of YR62, KF65, C69, N79 (refer to Table 32 for citation keys) tend to confirm this behavior for T < 150, above which r1,  0 and A,  oc, and the bed becomes mostly flat. In order to determine the skill of the coupled modeling system at predicting ripple geometry in labscale flows, a total of five simulations were performed having mobility numbers ranging by an order of magnitude from 13 to 130. Using the classification system of Wiberg and Harris (1994), the bedforms in the five labscale flow morphology tests fall into the orbital range; therefore, the bedforms are expected to scale with hydrodynamic properties within the wave bottom boundary livr (WBBL). Each simulation was allowed to run for three wave periods (t = 6 s) before the morphology subroutine was engaged to allow the hydrodynamic .,iiii.ii I ry associated with model spinup to subside, Simulations were initialized with two parabolicshaped ripples having heights approximately onehalf of the expected equilibrium and wavelengths corresponding to the expected value As a baseline predictor for all simulations, the regular and irregular ripple predictors of Nielsen (1981) were used to determine the expected geometry of labscale and fieldscale bedforms, respectively. Timeseries of bedform statistics from the labscale morphology tests are show in Figures 623(a)623(e). The bed elevation signal was recorded at a frequency of 10 Hz during model simulations, and then postprocessed to determine both timedependent and timeaveraged ripple height and length. The timedependent ripple length in Figures 623(a)623(d) were computed using a zero downcrossing technique after averaging the signal in the ydirection. We found the zero downcrossing technique to routinely overestimate the ripple height by a factor of 1.25, so the statistical method of Hanes et al. (2001) (Equation 62) was used to determine the instantaneous ripple height. The method of Hanes et al. (2001) is also particularly helpful when analyzing a signal that contains only one bedform, as there are not successive downcrossings in such a signal. For the more intense flow of simulation HL1v1, a direct Fourier transform was applied to the spanwiseaveraged bed elevation to determine the timedependent ripple height and wavelength (Figure 624). The ripple height and length were determined from the Fourier analysis by first selecting the wavelength A, with the largest energy density, and then by multiplying the real Fourier coefficient, which describes the signal amplitude at the selected wavelength, by two. Often, if the morphology is highly irregular it is impossible to use the zero downcrossing technique to select wavelengths once the mean of the bed elevation signal is removed, as it cannot identify successive crossings of zero. The ripple heights computed using the Fourier analysis generally agree well with the equation of Hanes et al. (2001), but the analysis is particularly helpful at selecting the energycontaining wavelengths in the elevation signal. Average ripple heights and lengths, along with the associated signal standard deviations, are denoted by the symbols and error bars to the right of the timeseries in Figures 623(a)623(e). The timeaveraged ripple characteristics are obtained by averaging the quantities over the last twenty waveperiods (t = 20 s) of the signals. The timeaveraged ripple heights ((Tlr)) and lengths ((A,)) obtained from the labscale morphology simulation are compared to several semiempirical ripple predictors in Figures 625(a) and 625(b), respectively. The citation keys listed in the legends of Figures 625(a) and 625(b) correspond to the ripple predictors defined in Table 55, and discussed in C'i ipter 3. The expected ripple heights and lengths for the lab and fieldscale flow simulations, determined using the ripple predictors, are provided in Table 56. A onetoone comparison of modeled abscissaa) and expected ordinatee) ripple height r1, is shown in Figure 625(a), where the dark solid line ii. ;perfect agreement, and the dashed () and dotted (...) lines represent deviations of 10'. and :IG'. from the expected values, respectively. The symbols are colored according to their numeric value of the mobility number T, and correspond to the included color scale. There appears to be much better agreement in the low to medium mobility numbers in Figure 625(a), whereas only one comparison of ripple height (N81) falls within the :l' I. lines for the high mobility number test. The model predictions of equilibrium ripple height appear to agree best with the N81 and SW06 ripple predictors, with all of the comparisons falling within the 3:l'. range. With the exception of the high mobility number case, all of the N81 comparisons fall within 1(0' of the expected value. As demonstrated in Figure 625(a), the model tends to follow the expected trend of ripple height variation (decreasing) with mobility number (increasing). A similar observation is made regarding the trend of ripple length (increasing) with mobility number (increasing) when considering the onetoone comparison of modeled abscissaa) and expected ordinatee) ripple lengths in Figure 625(b). Admittedly, the comparison of predicted and expected ripple wavelengths is less definitive, since the periodic length of the streamwise domain is thought to p1 i, a role in wavelength selection. Since the simulations were initialized at the expected wavelength, as determined by N81, the low to medium mobility number cases show perfect agreement when model results are compared to the N81 values in Figure 625(b). Comparisons using SW06 and M94 all fall within :I '. of the expected values, while nearly all of the comparisons using FF02 fall within 10''. of the expected values. The rootmeansquare error (RMSE) between modeled and expected values for ripple height and length were computed for each ripple predictor used in the onetoone comparisons of Figures 625(a) and 625(b), and are provided in Table 61. The RMSE for ripple height and length were computed using Equations 63a and 63b, respectively, where N is the number of tests, rli and Ai are the average ripple height and length obtained by the model, and qi and Ai are the expected ripple height and length, respectively. A total of eight ripple predictors were used in direct comparisons to model results, for five different labscale flows, resulting in a total of 40 data points for use in modelpredictor evaluation. Each ripple predictor was assigned a score of 1 through 8 based on the computed value of TrRMSE and ARMSE, with 1 representing the modelpredictor comparison with the lowest RMS error, and 8 signifying the largest RMS error between model and expected results. The individual scores for TIRMSE and ARMSE were then combined, yielding an overall score for each modelpredictor comparison. The eight ripple predictors listed in Table 61 are ranked from lowest to highest score, representing the smallest and largest combined RMSE, respectively. The analysis of RMSE ,i.. I that while the model results for r, agree best with those of Nielsen (1981), the average ripple wavelengths A tend to be closer to the predictor of Faraci and Foti (2002). This is an interesting result since morphology simulations were initialized using expected ripple wavelengths predicted by Nielsen (1981); therefore, some modification of the ripple wavelength is found to occur during the morphology simulations. Modification of the ripple wavelength during the simulations is encouraging, as it i ... 1 that the periodic streamwise domain is not prohibiting changes in the ripple geometry. T1RMSE N i)2 (63a) (1 i=l ) ARMSE = A i)2 (63b) i= There is much debate in the literature on bedforms over what parameters ripples measured in field conditions actually scale best with. One reason for the debate stems from the vast discrepancies found in collected ripple data from the field. Traykovski et al. (1999) attributes some of these differences to hysteresis, whereby ripples in the field often do not relax from larger to smaller wavelengths. Furthermore, Miller and Komar (1980a) ,i. 1 that the spread of field data on ripple geometry may suffer from data collected during times when bedforms are not in true equilibrium with the observed hydrodynamic conditions. This lack of harmony in the expected ripple geometry for fieldscale flows has created an opportunity for a number of investigators to relate expected characteristics to a variety of different parameters. While Wiberg and Harris (1994) argue that ripples commonly found in the field can be characterized as anorbital, scaling with sediment size as opposed to hydrodynamic properties. Conversely, Nielsen (1981) and Mogridge et al. (1994) sl. 1 a relationship between expected ripple geometry and hydrodynamic properties, but do not agree on the nondimensional scaling groups used, with the former using the mobility number T and the latter using the wave period parameter X. According to the classification system of Wiberg and Harris (1994), the expected bedforms in the fieldscale morphology tests are expected to be suborbital, with the exception of the high mobility number case which is anticipated to fall in the anorbital range. Suborbital ripples represent a distinct subset of data containing ripples that do not tend to scale well with either hydrodynamic or sediment properties. A total of four morphology simulations were conducted under fieldscale, or longer wave period, flows where T = 8 s. Similar to the labscale simulations, the bedforms were initialized at onehalf the expected height, and at the expected equilibrium wavelength, using the predictor of Nielsen (1981) for irregular, or fieldtype flows. Model simulations were run for two wave periods (t = 16 s) prior to engaging the morphology submodel. Figures 626(a)626(d) show the timeseries of bedform statistics for the labscale flow simulations. The symbols and error bars to the left of the timeseries correspond to the timeaveraged ripple heights and lengths, and their respective standard deviations, respectively. The timeaveraged ripple height (Trr) and length (Ar) were computed by averaging the instantaneous results over the last five wave periods of the simulation corresponding to a time of t = 40 s. Again, the bed elevation was recorded at a frequency of 10 Hz throughout the simulation, and then postprocessed to determine the average height and wavelength of the bedforms. A zero downcrossing technique was used to compute the average bedform wavelength, and Equation 62 was used to compute the average ripple height as a function of time, in Figures 626(a) and 626(b). Fourier a' 1 i was used to determine the ripple height and wavelength for the more energetic simulations MF_2v2 and HF_1v1, and the timestacks are provided in Figures 627(a) and 626(d). Direct comparisons of model results to expected ripple height and length for the fieldscale flow tests are shown in Figures 628(a) and 628(b), respectively. Expected ripple heights and lengths for the fieldscale morphology tests are also provided in Table 56 for each of the eight ripple predictors used in the direct comparisons. With respect to the labscale flows, the direct comparisons of model results to predicted ripple height and length show a much larger spread in the expected results. The solid dark line in the onetoone plots shown in Figures 628(a) and 628(b) represents perfect agreement between model predictions and expected results, while the dashed () and dotted ( ..) lines represent deviations of 10'.' and nl' from perfect agreement. For the onetoone comparison of average ripple heights (9Tr) plotted in Figure 628(a), only the W05 and FF02 ripple predictors consistently fall within n:' of perfect agreement. The symbols in Figures 628(a) and 628(b) are colored according to the corresponding value of T for that test case. Similar to the behavior found in the labscale flows, the model consistently predicts a reduction in r1, for an increase in the mobility number, which may also be taken as a proxy for the strength of the flow. The onetoone comparisons of modeled and expected ripple wavelengths are demonstrated in Figure 628(b), and reveal a similarly large amount of disagreement between the eight ripple predictors used. The N81, W05, and WH94 predictors appear to perform similarly with regard to predicting equilibrium ripple wavelength, with most of the direct comparisons falling within ~3' of perfect agreement. The RMS errors between model predictions and expected ripple height and wavelength for the fieldscale flow tests were computed using Equations 63a and 63b. Each ripple predictor was assigned a score of 1 through 8 based on the computed RMS error for both ripple height and wavelength, with 1 corresponding to the smallest error, and 8 representing the largest error. The scores for T/RMSE and ARMSE were then added, yielding a total combined score for each ripple predictor. These scores are listed in Table 62, and r..;. i that the model predictions of equilibrium ripple geometry in fieldscale flows agree best with the ripple predictors of Williams et al. (2005), Faraci and Foti (2002), and Nielsen (1981). While exhibiting a larger degree of spread between the predictions of ripple geometry in fieldscale flows, the RMS errors listed in Table 62 are consistently of the same order of magnitude as those for the labscale flows provided in Table 61. The similarity of these errors . ; that the model is equally adept at predicting ripple geometry in both lab and fieldscale flow regimes. The combined scores for ripple predictors in labscale (Table 61) and fieldscale (Table 62) are further combined to produce a model performance index (\! PI) score in Table 63. The MPI scores in Table 63 are reflective of the capability of the modeling system to predict morphology over a large range of flow conditions. The rankings show that for all of the morphology simulations, the best agreement between model predictions and expected ripple height and length is obtained when using the ripple predictors of Faraci and Foti (2002), Nielsen (1981), and Williams et al. (2005). The predictors of Mogridge et al. (1994) and Khelifa and Ouellet (2000) complete the top five, with the equations of Wiberg and Harris (1994), Soulsby and Whitehouse (2006), and Grant and Madsen (1982) following in that order. Further analysis of model predictions from the morphology tests is performed with respect to the five ripple predictors having the lowest MPI scores in Table 63 (FF02, N81, W05, M94, and KO00). The analysis is conducted by plotting the model predictions against the respective ripple predictor equations given in ('!i Ilter 3. The ripple predictor equations (Equations 315b and 315a) of Faraci and Foti (2002) are plotted in Figures 629(a) and 629(b) as a function of nondimensional ripple height and length, respectively. Equation 315b is a function of both mobility number (') and orbital Reynolds number ( f), so the results are plotted as a series of curves in Figure 629(a). The family of curves for R, correspond to the median values of the ranges indicated in the figure legend, where open and filled symbols signify labscale and fieldscale flow simulations, respectively. The model predictions appear to agree well with the family of curves, and capture the general trends of bedform behavior where ripple height decreases as the mobility number and orbital Reynolds numbers increase. The nondimensional ripple wavelength, computed using Equation 315a, is plotted in Figure 629(b) as a function of the orbital Reynolds number R, for a family of curves corresponding to the grain Reynolds numbers ?Rg shown in the figure legend. Again, the open and closed symbols in Figure 629(b) denote the labscale and fieldscale flow simulations, respectively, and the symbol shapes correspond to the range of grain Reynolds number provided in the legend. The model predictions of ripple wavelength, when nondimensionalized by the wave orbital semiamplitude, agree well with the family of curves given by Faraci and Foti (2002). Figure 629(b) demonstrates that the model predictions are following two expected relationships, where the wavelength increases with increasing values of Rg, and decreases with larger values of R,. Model predictions of equilibrium ripple height and length, for all nine morphology simulations, are plotted in Figures 630(a) and 6 30(b), respectively, along with the ripple predictors of Nielsen (1981), Khelifa and Ouellet (2000), and Williams et al. (2005). The expressions of N81, KO00, and W05 are functions of T exclusively, so each curve in Figures 630(a) and 630(b) represents the ripple predictor corresponding to the key given in the figure legend. Furthermore, the open and closed symbols represent the labscale and fieldscale morphology simulations, respectively. The references to N81r and N81i in the figures correspond to the ripple predictors of Nielsen (1981) for regular (lab) and irregular (field) flows. The model predictions of nondimensional ripple height for labscale flows appear to agree very well with the rippled predictor of Nielsen (1981) for regular flows. For the fieldscale results, better agreement between model predictions and expected results are found for the Williams et al. (2005) ripple predictor, specifically for the short wave ripple (SWR) equations. Similarly, the model predictions of ripple wavelength in Figure 630(b), when nondimensionalized by the wave orbital semiamplitude A, for labscale flows agree best with the curve of Nielsen (1981) for regular flows, while the fieldscale morphology results are better predicted by the equations of Williams et al. (2005) for short wave ripples (SWR). Figures 631(a) and 631(b) demonstrate the agreement between model predictions of ripple height and length, respectively, and the ripple predictors of Mogridge et al. (1994) for lab and fieldscale morphology results. The nondimensional ripple characteristics in Figures 631(a) and 631(b) are plotted as a function of the wave period parameter X. Although the formulation of X is similar to that of T, the wave period parameter depends explicitly on the wave period squared, which is found in the denominator of the nondimensional relationship. Therefore, larger wave periods yield lower values of X while shorter, labscale flows yield values of X that can be one or two orders of magnitude larger. The equations of Mogridge et al. (1994) provide estimates of the maximum expected ripple height and length as a function of X, with shorter period flows (larger X values) having smaller ripple heights and lengths. Figures 631(a) and 631(b) demonstrate fair agreement between model predictions and expected values of ripple height and length, respectively, particularly for the labscale flows (open symbols). 6.2.2 Phase II: Model Capabilities Fourteen (14) simulations were performed to evaluate the skill and capability of the coupled modeling system at predicting morphology over a range of flow and sediment parameters, as well as its sensitivity to initial conditions. The results of these simulations are provided in the sections that follow. 6.2.2.1 Steady Flow A timestack of the spanwiseaveraged bed elevation is shown in Figure 632. The bed state diagrams of Southard and Boguchwal (1990) and Kleinhans (2005) SI:. 1 that ripples should form for the given flow (U, = 30 cm/s) and sediment (d = 0.3 mm) characteristics, while stronger forcing would tend to produce twodimensional dunes. If the applied bed stress in the steady flow case was overestimated, owing in part to the larger eddy viscosity predicted by the LES submodel, then the formation of the dune in the model simulation would be in agreement with the bed state diagrams of Southard and Boguchwal (1990) and Kleinhans (2005). Although ripples did not form, the translation of the hump is similar to the results of numerical simulations performed by Long et al. (2007), which may be used to verify that the EulerWENO morphology subroutine is working properly. As seen in Figure 632, the morphology reaches a steady state shortly after t = 100 s. This steady state is the result of a thickening, or diffusion, of the boundary 11 r to the point where the applied bed shear stress drops below the critical threshold for motion, and sediment transport goes to zero, as shown in Figure 633(a). As the boundary 1iv r diffuses vertically through the water column, the velocity shear near the bed tends to decrease. This reduction in velocity shear is typically balanced by an increase in the turbulent eddy viscosity in order to maintain a relatively constant bed shear stress (Equation 447). Timeseries of spatiallyaveraged nearbed velocity shear and eddy viscosity are plotted in Figure 633(b) and demonstrate such behavior; however, the increase in eddy viscosity does not appear to be substantial enough to overcome the reduction of velocity shear in maintaining a higher bed shear stress. 6.2.2.2 Subcritical Flow The flow and sediment characteristics of the subcritical flow simulation are such that no sediment motion is anticipated. As found in Julien (1998), the threshold for sediment motion of a sediment with a particle size of d = 0.3 mm corresponds to a critical shear stress of Tr, = 0.21 Pa, or 2.1 dyne/cm2. The phase and spatiallyaveraged bed shear stress and bedload transport rate are plotted in Figure 635 as a function of fluid phase. The yellow region in Figure 635 denotes the range of fluid phase over which the average bed shear stress falls below the threshold for motion. For the subcritical flow simulation, the average stress clearly stays well below the critical stress, resulting in zero bedload transport. However, the timestack of bed elevations plotted in Figure 634 i.;. I that there is a subtle change from the initial to final ripple shapes. The smoothing of the bed profile evident in Figure 634 is a result of the mildly diffusive morphology filter used to prevent spurious oscillations in the bed elevation signal. Although the filter tends to smooth the ripple profile over time, it does not appear to have an effect on the ripple height, as demonstrated by the color contours in Figure 634. 6.2.2.3 Sheetflow Regime As discussed in C'! Ilpter 2, the sheetflow regime of suspended sediment transport is characterized by persistent, high volumes of sediment contained in a nearbed 1iv.r approximately 10 to 100 graindiameters thick. The suspended sediment transport regime shifts from the vortexdominated rippled bed regime to the sheardominated sheetflow regime for values of 0 w 0.8 1.0 (van der Werf, 2004). For values of 0 > 100oc, Li and Amos (1999) find that intense sheetflow transport produces a flat bed. Dingier (1974) ~i I i that, in terms of the mobility number, the planar bed state is achieved for T > 240. Although the suspended sediment transport submodel is not anticipated to perform well at such high sediment concentrations (see C'! Ilpter 4), a simulation was performed to evaluate model behavior in the sheetflow regime. A timestack of the bed elevation made from successive vectors of Zb(x,t), recorded at a frequency of 10 Hz, is shown in Figure 636. The morphology was initialized with a parabolicshaped ripple having lr = 2.0 cm and A, = 13.6 cm. The bed elevation timestack in Figure 636 clearly demonstrates the effects of the intense sheetflow transport on the morphology signal. As expected, no persistent bed state is reached during 40 wave periods of the simulation. The mobility number corresponding to the flow and sediment characteristics of the sheetflow simulation (U, = 100 cm/s, T = 2 s, and d = 0.3 mm) yield a value of T = 206, which is less than the flat bed limit (T > 240) of Dingier (1974) mentioned previously. Time and horizontallyaveraged values of the Shields parameter are found to be (0) = 0.825, which is within the range of values anticipated in the sheetflow regime (van der Werf, 2004). Timeseries of ripple height and length are provided in Figure 637, and the symbols and error bars at the end of the timeseries denote average values and standard deviations computed over the last twenty wave periods of the simulation. The timeseries of ripple characteristics was constructed from samplings of zb(t) taken at a frequency of 10 Hz, and then an !.i. .1 using direct Fourier transforms to determine the instantaneous ripple heights and lengths containing the largest energy density. A plot of spectral energy density as a function of both ripple length A, and time, is given in Figure 638, and demonstrates a lack of coherency in the bed state during the intense sheetflow transport. The advectiondiffusion equation (Equation 458) used to model the volumetric concentration field in the coupled modeling system is not expected to produce reliable answers for concentrations C > 1 (Villaret and Davies, 1995). Additionally, the modeling system does not account for particleparticle interactions that are ii:: I '1 to pl i, an important role for concentrations C > 103 (Elghobashi, 1994). The time and volumeaveraged suspended sediment concentration during the sheetflow simulation was computed as (C) = 2.15 x 102, or (C) a 60 g/l. Isocontours of the timeaveraged volumetric concentration field (C(x, y, z)) are shown in Figure 639, and are used to determine the average nearbed concentrations predicted by the suspended sediment submodel. Values of (C(x, y, z)) isocontours correspond to the provided legend, and ii.:: i average concentrations as high as 265 g/1 within 1 cm of the average bed elevation ((C(x, y, z)) = 0.6). The time and horizontallyaveraged concentration profile (C(z)) is also provided in Figure 640 and shows similar behavior due to the horizontal stratification of suspended sediment above the mostly flat bed. Ribberink and AlSalem (1995) report suspended concentrations as high as 500 g/1 in the upper sheetflow 1lv.r, where peaks in suspended sediment concentration are approximately in phase with the freestream velocity. This agreement in phase is atypical of suspended sediment behavior in the rippledbed regime, where peak concentrations tend to lag behind the oscillatory forcing outside of the boundary 1 .Ir. In volumetric units, sheetflow concentrations of 500 g/1 equate to C w 1'. for quartz sand having a specific gravity of S = 2.65. 6.2.2.4 Ripple Growth, Equilibrium, and Decay A total of seven unique simulations were performed in order to evaluate the sensitivity of ripple height and wavelength to: Initial morphology characteristics; Periodic domain length; and Hindered settling. These experiments were conducted using the baseline simulation flow and sediment characteristics of U, = 40 cm/s, T = 2 s, and d = 300 pm. Additional simulation parameters for GROW1, GROW2, EQUILIB, DECAY, SLIDE, SPLIT, and HS1v2, such as domain lengths and grid resolution, are listed in Table A1. The purpose of these simulations is to determine how different aspects of the coupled modeling system affect the expected morphology. Eight predictions (N81, GM82, M94, WH94, KO00, FF02, W05, and SW06) of equilibrium ripple height re1 and wavelength A, for our baseline test are given in Table 56, and correspond to the simulation case name ML3v1. Where applicable, the ripple predictor of Nielsen (1981) (N81) was used to determine expected ripple heights rT1 and lengths A, a prior. The first set of four experiments, namely GROW1, GROW2, EQUILIB, and DECAY, were performed to evaluate the sensitivity of ripple height and length to the initial morphology used in the model simulations. Timestacks of the spanwiseaveraged bed elevation zb, recorded at a frequency of 10 Hz during the simulations, are plotted in Figures 641(a)641(d) for the four simulations. The morphology for GROW1 was initialized at e/2 and A, (based on N81) using a parabolic ripple shape. In GROW2, a small Gaussian hump 5 graindiameters in height (1.5 mm) and approximately 1 cm in length was used to break the model symmetry experienced when using a perfectly smooth bed. The initial morphology used in the EQUILIB simulation corresponds to the expected ripple height and length, r and A,, as given by Nielsen (1981). In DECAY, the ripple wavelength was initialized at A, while the initial ripple height was set at 150'. of the expected value re. The primary mechanism for growth from e/2, as demonstrated in Figure 641(a), appears to be through a deepening of the ripple trough, and a subsequent increase in the height of the ripple crest. This is similar to the expected behavior of ripple growth in the orbital range, which was discussed in C'! lpter 3, where shear at the bed is directed up the lee and stoss ripple slopes during both phases of flow due to flow separation in the lee of the ripple crest. The crestdirected stress results in net transport of sediment from the troughs to the crests during ripple growth, until it reaches an equilibrium with entrainment of sediment from the crest and deposition in the trough. The behavior exhibited by GROW2 (Figure 641(b)), where the simulation was started from a mostly flat bed, is more indicative of growth in the linear regime, where groups of sand grains tend to cluster together to form rolling grain ripples until the crests are high enough to induce flow separation and vortex formation, finally saturating at the equilibrium wavelength. A timestack of energy density is plotted in Figure 642 as a function of A, and time for the GROW2 simulation, and demonstrates that there is no coherency between energy and A, until saturation is reached at approximately t = 60 s, or after 30 wave cycles. Once flow separation begins, the time evolution of the ripple is similar to that of GROW1 where sediment is taken from the trough and moved toward the crest, thereby increasing the height. As anticipated, the ripple geometry remains relatively constant throughout the EQUILIB simulation shown in Figure 641(c), where the morphology was initialized at r]e and A,. When initialized at 150'. of the expected height, the results of the DECAY simulation shown in Figure 641(d) demonstrate a rapid decrease in height due to gravitational bedload transport, after which the morphology remains relatively constant. The time evolution of ripple height and wavelength for the GROW1, GROW2, EQUILIB, and DECAY simulations are shown in Figures 643(a)643(d), respectively. These time series were reconstructed from the bed elevation Zb(X, y, t) sampled at 10 Hz, averaged in the spanwise direction, and then analyzed using Equation 62 and a zero downcrossing technique to determine the ripple height qr,(t) and wavelength A,(t) statistics, respectively, as a function of simulation time. Whereas growth from r1e/2 to a relatively constant value occurred in about 10 wave cycles in Figure 643(a), growth from a flat bed took nearly 50 wave cycles to reach a constant value (Figure 643(b). After an initial rapid reduction in height due to gravitation transport, continued ripple decay shown in Figure 643(d) occurred over approximately 15 wave cycles until a value consistent with the final average was obtained. The time evolution of ripple height for all four simulations is plotted in nondimensional form in Figure 644. Here, the timedependent ripple height rl,(t) is scaled by the expected value r1e as predicted by the ripple predictor of Nielsen (1981), while the simulation time is scaled by the total duration of each experiment, tf, provided in the figure. Figure 644 demonstrates that equilibration of ripple height from an initially flat bed took approximately 150'. longer than required when starting from onehalf re. With exception of the flat bed case, GROW1, EQUILIB, and DECAY all reach and/or remain at the expected ripple height 10' . The preceding figures of ripple height evolution demonstrate that the final ripple height predicted by the coupled modeling system is irrespective of the initial morphology for a given flow condition (U, and T) and sediment characteristics (d). It would also be beneficial to ensure that the choice of a periodic domain in the streamwise direction is not forcing the selection of a particular wavelength, like L,, for the ripple geometry. Two simulations, SLIDE and SPLIT, were performed to determine the effects of periodic domain length L, on Ar, and demonstrate ripple coarsening and bifurcation. Timestacks of spanwiseaveraged bed elevation zb(t) from the SLIDE and SPLIT simulations are plotted in Figures 645(a) and 645(b), respectively. In SLIDE, the morphology was initialized with e/2 and A,/2, while the periodic domain was set at 2 A,. The process of ripple coarsening, whereby ripples with wavelengths out of equilibrium with the given flow condition increase in length until saturation at the expected wavelength A, is achieved. This coarsening process is clearly evident in the timestacks of Figure 645(a), as well as in the timestacks of ripple spectra plotted in Figure 646(a). The ripple spectra demonstrate the energy density as a function of ripple wavelength A,, and show that the initial energy confined to the smaller ripples having A, = 6.8 cm decays over the first 14 s until saturation at A, = 13.6 cm, after which time the equilibrium ripples continue to grow in height. The immediate shift from one wavelength to another is characteristic of the laboratory data reported by Doucette and O'Donoghue (2006), who classify such behavior as ripple slide (see C'!i pter 3). A timeseries of ripple statistics for the SLIDE simulation, shown in Figure 647(a), demonstrates that the saturation of A, at t = 14 s is accompanied by a rapid increase in ripple height until a dynamic equilibrium is attained. The symbol and error bars to the right of the timeseries represent the timeaveraged ripple height ((Tr) = 1.98 cm) and standard deviation (" .), respectively, computed using the last 20 wave periods of the simulation. According to the ripple predictor of Nielsen (1981) for regular waves, the SLIDE simulation over predicts the expected ripple height of r = 1.89 cm by roughly 5'. while underestimating the expected ripple length of A = 13.6 cm by nearly the same amount. The inverse of the coarsening experiment is represented by the SPLIT simulation, whereby one ripple initialized with a Gaussian shape, and having a continuous length equal to L,, splits to form two ripples having A, = L,/2. Timestacks of spanwiseaveraged bed elevation zb, sampled from the simulation at a frequency of 10 Hz, are plotted in Figure 645(b) and show the progression from initial to final morphology. The time evolution of ripple height is more clearly shown by the timeseries plotted in Figure 647(b), where the ripple height ,] corresponds to the left ordinate. These timeseries were produced by analyzing the bed elevation field zb(x, y, t) at a frequency of 10 Hz, making use of Equation 62 and a zero downcrossing technique to determine the spatially averaged ripple height and length, respectively. The timeaveraged ripple height ((rr) = 2.39 cm) and standard deviation (5'.), computed over the last 20 wave periods of the simulation, are given by the square symbol and error bars. With respect to the equilibrium ripple height predicted by Nielsen (1981), the SPLIT simulation over predicts the average ripple height by 12.' ; when combined with the rather large value of r = 5' over the last twenty wave periods, it is possible that the morphology had not yet reached an equilibrium with the flow conditions. As is often the case with incoherent bed elevation signals, the zero downcrossing technique failed to consistently identify an appropriate ripple wavelength as demonstrated in the timeseries of Figure 647(b); therefore, Fourier analysis was employ, 1 to identify ripple wavelengths containing the most energy in the signal zb(x, t). Timestacks of ripple spectra are provided in Figure 646(b); initially, the energy in the signal is contained at A, = Lx, but shifts to Ar = Lx/2 after approximately 7 wave periods. According to Doucette and O'Donoghue (2006), this type of wavelength evolution is most similar to the process of ripple split, where one ripple with a large wavelength decreases through a bifurcation into two or more ripples having shorter wavelengths. As demonstrated in Figure 413, the modification of the constant settling velocity due to hindered settling, in suspensions having C < 5 x 103, is minimal. A simulation (HS1v2) having the flow and sediment characteristics of our baseline test case (U, = 40 cm/s and d = 0.3 mm) was performed and incorporated the hindered settling velocity in the advectiondiffusion equation (Equation 458) for sediment concentration. While time and volumeaveraged values of sediment concentration are small for the baseline simulations, (C) < 103, instantaneous values can often be two or three times as large; therefore, it would be beneficial to evaluate modifications of stratification and ripple morphology from hindered settling. The time and horizontallyaveraged concentration profiles for simulations HS1v2 and ML3v1 (baseline) are plotted in Figure 649. The reduced gradients of OC/az in HS1v2 demonstrate an increase in vertical stratification within the boundary 1 ,.r due to hindered settling, as anticipated (Baldock et al., 2004). Time and volumeaveraged concentrations ((C)) for HS1v2 and ML3v1 were found to be 5.7 x 103 and 3.3 x 103, respectively. For the typical values of (C) observed in the baseline simulations, modification of the constant settling velocity due to hindered settling is only 0."' Timeseries of ripple height and length for the comparative simulations are given in Figures 648(a) and 648(b), respectively. For the simulations considered here, there does not appear to be a significant effect on the ripple characteristics. The timeaveraged ripple height (over 20 wave periods) (plr) for the two simulations differ by only 2.5'. while the average wavelengths (A,) differ by 2.>'. One noticeable difference between the time evolution of ripple height in Figure 648(a) is the rate of growth, which appears to be higher for the simulation with hindered settling. This increase in rate may be a result of suppressed entrainment of sediment from the bed due to the lower vertical gradients in the nearbed concentration field. Recently, Baldock et al. (2004) published modifications to the exponent used in the hindered setting velocity formulation of Richardson and Zaki (1954). These modifications were a result of settling velocity experiments using real sand, and demonstrated an increase in typical exponents by a factor of approximately 1.5. For the flow and sediment characteristics used in the baseline simulations, estimates of w8 by Richardson and Zaki (1954) and Baldock et al. (2004) differ by only 0.'"., therefore, employing the more recent results of Baldock et al. (2004) for real sand would not have a significant effect on the stratification or the morphology. 6.2.2.5 Sediment Size Tests With respect to the coupled modeling system, the previous simulations have addressed model sensitivity to flow characteristics (mobility number, '), transport regime, initial morphology bias, and periodic domain length. These simulations, however, have only explored the parameter space of grain size in a very narrow range: 0.2 < d < 0.4 mm. It is anticipated that the grain size will have an effect on morphology and sediment transport predictions through the Shields parameter (Equation 444), the fall velocity (Equation 460), and the rate of entrainment of bed material (Equation 465). Four simulations were conducted to test model sensitivity to large and small sediment sizes, each for weak and strong flows. The bed state diagram of Kleinhans (2005) was used to determine extreme combinations of flow and sediment characteristics that would yield bedforms in one simulation and not the other, for each sediment size. The large sediment size tested, d = 6 mm, is classified as pebble on the Wentworth scale, and fine gravel in the Unified Soil Classification (USC) system (Dean and Dalrymple, 2004). The smaller sediment size, d = 0.025 mm, corresponds to silt in both the Wentworth and USC systems, or a medium silt in Julien (1998). There are some computational reasons why these two sediment sizes represent extremes for the current modeling system: first, it doesn't make physical sense to have a sediment diameter that is smaller than even your highest numerical resolution through the bed, and second, using a grain size as large as d = 6 mm would typically warrant a parametrization of bottom roughness to account for the additional friction at the bed. The fine gravel, having d = 6 mm, was tested in a weak flow having the characteristics U, = 30 cm/s and T = 2 s, yielding a mobility number of = 0.93 (GRAVLOW). Due to the dependence of T on the grain size, the same flow characteristics yielded a value of T = 28 for a fine sand having d = 0.2 mm in the morphology test case ML1v2 in which ripples formed. According to the bed state diagram of Kleinhans (2005), no motion is anticipated for the combination of large grain size and comparatively weak fluid forcing used in the GRAVLOW simulation. Indeed, the ratio of destabilizing to stabilizing forces represented by the Shields parameter 0 is difficult to elevate above the critical threshold of Ocr = 0.055 due to the weight of the particle. The phase and horizontallyaveraged bed shear stress and sediment transport rates are plotted in Figure 651, and demonstrate that the the shear stress applied to the bed is never large enough to surpass the critical threshold for motion, which is represented by the shaded yellow area in the figure. The simulation is initialized with a flat bed, and the timeseries of bedform height and length shown in Figure 650 indicate that no morphology develops during the simulation. The second simulation performed for the fine gravel employ .1 stronger fluid forcing with U, = 80 cm/s and T = 2 s yielding a mobility number of T = 6.59 for the GRAVHI test. Comparatively, the HL1v1 test case used the same fluid forcing, but yielded a mobility number of T = 132 due to the medium sand size of 0.3 mm. The bed state diagram of Kleinhans (2005), for oscillatory flows, ,ii. I that the flow parameters of the GRAVHI simulation are such that gravel ripples may be expected. Timestacks of bed elevation (Figure 652), and timeseries of ripple height evolution (Figure 653), demonstrate that some morphology develops from the initially flat bed; however, the computed average bedform height in Figure 653 is less than 1 mm, and our grain size is d = 6 mm. Hence, it is difficult to ?v whether these results are actually meaningful. It is encouraging, however, that the model predicts both bedload and suspended sediment transport during periods of strong flow, as demonstrated in Figure 654, as transport would be necessary to generate the gravel ripples. Simulations of the siltsized sediment with d = 0.025 mm in weak and strong flows are represented by the simulations SILTLOW and SILTHI, respectively. In SILTLOW, the fluid forcing characteristics of U, = 10 cm/s and T = 2 s combine with the small sediment size to yield a mobility number T = 24.7. Timestacks of the spanwiseaveraged bed elevation zb, sampled at a frequency of 10 Hz during the simulation, are plotted in Figure 655 and demonstrate that the bed remained flat through approximately 30 wave cycles, after which time a lowcrested feature began to form. The time evolution of bedform height and length, determined through Fourier analysis, is shown in Figure 656 and reveals three wavelengths in the morphology: A, = 4.7 cm, A, = 7.6 cm, and the largest, A, = 30 cm. The average height of the feature having the wavelength A, = 30 cm is rr = 0.03 cm, which is on the order of ten grain sizes. It is unclear whether this feature has physical relevance, or if it is a result of positive feedback between the suspended load transport, shown in Figure 657(a), and bedlevel updating submodel. The bed state diagram of Kleinhans (2005) predicts no formation of persistent morphology for the combination of weak flow and small sediment used in the SILTLOW simulation. Figures 657(a) and 657(b) show the phaseaveraged relationships between bed shear stress and sediment transport; and bed shear stress and sediment concentration, respectively. Average values of bed shear stress predicted by the model eclipse the critical threshold required for sediment motion, represented by the shaded yellow areas in these figures (Julien, 1998), but only yield significant values of suspended load transport and not bedload transport. The phase and volumeaveraged suspended sediment concentration, plotted in Figure 657(b), ,.; ; an average value of (C) a 4 x 103 during the simulation. Fluid forcing characteristics for the simulation of silt in strong flows were chosen such that we would expect highlyconcentrated flows, as well as the development of morphology. When combined with the small sediment size d = 0.025 mm, the forcing characteristics U, = 40 cm/s and T = 2 s yield a very large mobility number: T = 395. The bed state diagram of Kleinhans (2005) predicts the development of hummocks or other complicated, threedimensional bedforms for the parameters used in the SILTHI simulation. Timestacks of bed elevation for the SILTHI simulation are shown in Figure 658 and demonstrate a very complicated time evolution of bedform heights and lengths through the processes of slide, split, and merge (Doucette and O'Donoghue, 2006). Timeseries of spatiallyaveraged ripple height and length are plotted in Figure 659, along with timeaveraged values and standard deviations obtained over the last 20 wave periods of the simulation. The predicted average height (Tr) = 2.074 cm and length (A,) = 18.07 cm of the final morphology agree reasonably well with the predicted values of Grant and Madsen (1982) (e = 2.17 cm, Ae 14.5 cm) and Soulsby and Whitehouse (2006) (re = 1.80 cm, Ae 12.0 cm). The phase and spatiallyaveraged bed shear stress and sediment transport rates are plotted in Figure 660(a), and sel 1 a dominance of suspended load transport over bedload transport for the SILTHI simulation. With respect to the phaseaveraged suspended load transport ((Q,)) for the SILTLOW experiment (Figure 657(a)), the phase lag of ((Q,)) for the SILTHI test is enhanced by approximately 200. These phase lags are characteristic of the upper sheetflow li.r (Ribberink et al., 1994), but are typically not found within the active pickup l. Ii~ near the bed. The phase lag between average shear stress and average sediment concentration is approximately 450, as shown in Figure 660(b). Time and volumeaveraging the sediment concentration field during the SILTHI simulation yields a value of (C) = 0.025, or (C) = 66.25 g/1. According to Elghobashi (1994), such large values are indicative of a range where particleparticle interactions become important; therefore, it is unclear whether the current suspended sediment submodel (Equation 458) is applicable here, although Villaret and Davies (1995) cite successful applications of similar models for concentrations as high as C 1 A phase diagram of model applicability, with respect to the suspended sediment transport submodel, is provided in Figure 661. The phase diagram shows the predicted time and volumeaveraged suspended sediment concentrations from 15 different model simulations conducted over large ranges of fluid forcing and sediment size. The color of each data point in Figure 661 represents the predicted value of (C) for the specific combination of orbital Reynolds number R, and sediment size. Orbital Reynolds numbers are plotted on a logarithmic (base ten) scale in Figure 661, while the sediment sizes correspond to the Phi scale of Krumbein (1936), where grain size is distributed from large to small progressing from left to right along the abscissa. For reference, typical results of a baseline simulation, such as ML3v1, are indicated on the phase diagram. 6.2.3 Phase III: Sediment Transport Processes In order to address our specific scientific questions posed in ('! Ilpter 1, four independent simulations were performed to evaluate the roles of bedload and suspended load sediment transport during ripple growth, equilibrium, and decay. The simulation results also provide previously undocumented information regarding the roles of bedload and suspended load transport as a function of the ripple profile, as well as phase of fluid forcing. These comparative simulations were conducted by first obtaining the morphological response to bedload and suspended load for our baseline test case having U, = 40 cm/s, T =2 s, and d = 0.3 mm (TL2v2). The morphology in TL2v2 was initialized with r, = 1.0 cm and A, = 13.6 cm, and model hydrodynamics were simulated for three wave periods (t = 6 s) prior to engaging the sediment transport and morphology submodels. While the TL2v2 simulation simulates ripple growth in response to the baseline simulation conditions, an additional experiment (TL2v2d) was performed to evaluate the behavior of ripple decay under total load conditions. In TL2v2d, the morphology was initialized with Tr, = 3.0 cm and A, = 13.6 cm. Two additional simulations were then performed incorporating only bedload (BL2v2) or suspended load (SL2v2) transport modes. As a result of the computational grid and model domain width test simulations performed in Phase I, these final production runs were performed using the resolution of GRID1, and conducted with the threedimensional model having domain widths of 3.2 cm. The resolution throughout the ripple profile (6x = 6y = 0.2 cm, 6z = 0.05 cm) is such that each control volume would contain approximately 85 spherical particles having d = 0.3 mm and a porosity np = 0.4. These final production simulations, then, are the equivalent of modeling 9.2 x 106 sand particles. 6.2.3.1 Total Load Growth The total load simulation TL2v2 was initialized with two parabolicshaped ripples having 1, = 1.0 cm and A, = 13.6 cm. The time evolution of ripple height, shown in Figure 662(a), indicates that after an initially rapid increase in height during the first 20 wave periods (O9]r/t = 3.5 x 102 cm/s), the rate of growth slows considerably (Or]r/t = 4.7 x 104 cm/s) until t U 110 s when a stable ripple height is reached. A timeaverage of the ripple height taken over the last 20 wave periods of the simulation yields a value of (Tr) = 2.43 cm having a standard deviation of 0.;:' The average ripple height predicted by the TL2v2 simulation is within 5'. of the prediction of Nielsen (1981) using Equation 33b. As demonstrated in Figure 662(b), there is very little modification of the ripple wavelength, from the initialized length, during the course of the simulation. The timeaverage ripple wavelength, taken over the last 20 wave periods, was found to be (Ar) = 13.68 cm and had a standard deviation of 0.7'. Time evolution of the ripple steepness, computed as ,/Ar,, is provided in Figure 662(c), and generally follows the evolution of height since the length remains nearly constant. The predicted shape of the bedforms, clearly demonstrated by the bed elevation timestacks in Figure 663, is mostly triangular with slightly smoothed crests. The smoothing at the crests is most likely an artifact of the bed level filtering process, the effects of which were demonstrated by the SUBCRIT simulation. However, the mostly triangular shape of the ripple flanks is characteristic of shapes often observed in the field (e.g. Figures 1 l(c) and 1 l(d)) and also found in the laboratory experiments of Stegner and Wesfreid (1999). For their lowerfrequency experiments (w = 0.5 Hz), which corresponds to the angular frequency used in the baseline simulations, Stegner and Wesfreid (1999) observed triangularshaped ripples with peaked crests and linear flanks while the higherfrequency tests ( = 1 Hz) produced a more sinusoidal ripple having smooth crests and troughs. The TL2v2 simulation results ii:: 1 an initially linear growth rate for the morphology, as demonstrated in Figure 662(a), followed by a gradual approach to the final value of rr = 2.43 cm. The red boxes in Figure 662(a) highlight two specific areas of interest: first, the period of growth that occurs during the first 20 wave cycles, and second, a period of equilibrium during the last 20 wave cycles of the simulation. For the period of growth, the rate of increase is found to be r9r/0t = 3.5 x 102 cm/s, and is followed by a marked decrease over the next 33 wave periods to 9Or/at = 4.7 x 104 cm/s. These same areas of growth and equilibration are identified in Figure 664, where the cumulative spatiallyaveraged and corrected transport fluxes are plotted as a function of time. We call these fluxes "corrected," because we have altered the sign of the transport fluxes to produce a more intuitive convention where positive (+) fluxes correspond to bedform growth, and negative () fluxes represent bedform decay (Penko, 2007). The alteration of sign must be performed due to fluxes either above or below the bed elevation zb = 0 representing different growth tendencies. The cumulative, spatiallyaveraged and corrected bedload (Qt) and suspended load (Qf) fluxes are computed using Equations 64a and 64b, respectively. In Figure 664, the corrected fluxes sl: 1 that the bedload fluxes are responsible for the i_, .ii i ly of the ripple growth throughout the simulation, and, therefore, represents a constructive force with respect to ripple growth. Although the corrected suspended flux is found to be negative initially, thereby acting to decrease the ripple height (a destructive force), it too becomes increasingly positive during the simulation. Once in the equilibrium phase of ripple evolution, corresponding to the rightmost box in Figure 664, we see that the cumulative average bedload and suspended load fluxes are of the same magnitude. This sl: 1 a balance between bedload and suspended load during times of constant ripple geometry. nx,ny Vb (6 4a) nx, n n S(wC E)t i, 1,1 VQ (64b) nx ny The qualitative relationship between bedload transport, entrainment, and deposition to the phase of hydrodynamic forcing appear to vary little during times of ripple growth and equilibration. The phaseaveraged bed shear stress, gravitational stress, bedload transport rate, and vertical sediment fluxes are provided in Figures 665(a) and 665(b) during ripple growth, and Figures 665(c) and 665(d) during ripple equilibrium. Although the bedload transport rate and vertical sediment fluxes decrease in magnitude from growth to equilibration, their behavior with respect to the phase of applied shear remains similar. The bedload transport rate simply mimics the phasedependence of the applied shear, with both leading the freestream forcing by approximately 600. Phase leads such as these are common in both laminar and turbulent oscillatory boundary 1~.v. r, but this value is higher than values typically found in the literature (Nielsen and Callaghan, 2003). It is unclear whether this more pronounced phase lead has physical significance, owing to the evolution of the bedforms, or is an artifact of deficiencies in the LES closure model discussed previously. The vertical sediment fluxes ,II 1 a slightly different behavior, with deposition 1 S:ii_; slightly behind periods of maximum flow, and entrainment leading the maximum flow (Figures 665(b) and 665(d)). The model clearly demonstrates the inverse relationship between the entrainment and deposition of solids at the bed; when entrainment is large, deposition is small; and when deposition is large, entrainment is low. While Figures 665(a)665(d) do show the phasedependence of bedload transport and vertical sediment fluxes, they do not provide information as to whether those sediment transport processes are producing net growth or decay. If we instead take the horizontallyaveraged and corrected bedload and suspended load fluxes given by Equations 64a and 64b, and then phaseaverage them over 20 wave periods during times of growth and equilibrium, we clearly see the effects of fluid forcing on building and decaying fluxes in Figures 666(a) and 666(b). During ripple growth, the phaseaveraged and corrected fluxes indicate that bedload and suspended load fluxes are of the same magnitude, and behave inversely; when building (positive) bedload fluxes are low (1 ,.iir; behind maximum freestream forcing), the growth of the bedform is dominated by vertical fluxes; when building (positive) bedload fluxes are large (1 ..i i; slightly behind flow reversals), the bedforms experience decay through vertical sediment fluxes. When the curves in Figure 666(a) are integrated and the areas summed, the result is a net positive in sediment fluxes corresponding to ripple growth; furthermore, we find that i.' of the transport fluxes associated with growth may be attributed to bedload processes, while the remaining ;:".. is due to the vertical sediment fluxes of entrainment and deposition. These results are, of course, consistent with our observations of ripple growth through the time evolution of ripple height shown in Figure 662(a). The phasedependence of building and decaying sediment fluxes is similarly intriguing during ripple equilibrium, as plotted in Figure 666(b), , . Iii':; an unexpected compensation between the two transport modes to maintain equilibrium. More specifically, when the vertical sediment fluxes are building (decaying) the ripple following the maximum freestream fc.n ivr. the bedload fluxes are compensating by decaying (building). Integrating the two curves in Figure 666(b), we find the net action of bedload and vertical sediment fluxes to be nearly zero, which is consistent with our observations of ripple equilibrium. Additionally, comparing their integrated magnitudes si .. i that only :'. of the transport fluxes are due to bedload, while i', of the fluxes may be attributed to suspension and deposition events. Thus far, we have only discussed the temporal and phase dependence of the sediment transport fluxes with respect to ripple growth and equilibrium. Using the sign convention for building (+) and decaying () fluxes as described earlier, but dropping the horizontal averaging, we define the timeaveraged and corrected bedload and suspended load transport fluxes in Equations 65a and 65b, respectively. Such a convention is useful for determining the distribution and tendency of transport fluxes throughout the ripple profile. Timeaveraging over 20 wave periods during ripple growth and equilibrium, these corrected fluxes are plotted in Figures 667(a) and 667(b), respectively. Each symbol represents a point on the horizontal bed plane, and their colors correspond to the timeaveraged bed elevation (zb) during the respective period. Plotting the bedload fluxes against the suspended load fluxes in this manner yields quadrants of growth and decay for each transport constituent, as well as a line of equilibrium. A more thorough explanation of the transport flux tendency phase diagram is provided in Appendix B. During times of ripple growth, the data in Figure 667(a) indicate a bedloaddominated (closer to the horizontal axis) growth at the ripple crest (warmer colors), and a suspensiondominated (closer to the vertical axis) growth in the ripple troughs (cooler colors). This flux tendency phase diagram confirms the observations of ripple growth from the timeseries of ripple heights (Figure 662(a)), as the concentration of data points is highest above the equilibrium line (the solid diagonal line), in the growth tendency region. The locus of transport flux data during ripple equilibrium is found to fall almost entirely along the equilibrium line in Figure 667(b), which also agrees with the observation of ripple equilibrium during the last 20 wave periods of the TL2v2 simulation. The distribution of the data points in Figure 667(b) ir.. I that in order to maintain equilibrium, the ripple crest experiences an almost perfect balance between bedload flux growth and vertical flux decay; the opposite is observed in the troughs, where suspended fluxes promote growth and balance the bedload fluxes that cause decay. nt fX nxly . Q 1 ij 1,1 (6 5a) (V b,) fllk^l l (65a) SZ (wsCb E) (VQ) 1 ij 11 (65b) 6.2.3.2 Total Load Decay The results of the total load simulation TL2v2 provide information about the spatial and phase dependence of the various transport components with respect to ripple growth and equilibrium. A similar test was performed to evaluate the phase and spatial dependence of the transport components, as well as the bedform behavior, during times of decay (TL2v2d). This was achieved by repeating the simulation parameters of TL2v2, with the exception of initializing the morphology with a ripple height that was threetimes larger, or rl, = 3.0 cm. The initial ripple wavelength was maintained at A, = 13.6 cm, thus creating a bedform that would be out of equilibrium in both height and steepness (or = 0.22 cm/cm). Initially, it was unclear whether the ripple wavelength would also have to adjust to compensate for the higher than normal steepness; maximum steepness values are found in vanishing flow conditions and have typical values of cr = 0.32 tan Q or about a, = 0.18 for a sediment having an angle of repose of = 300 (Nielsen, 1992). As demonstrated by the time evolution of ripple wavelength shown in Figure 668(b), A, tended to remain constant throughout the simulation. In order to then reach the anticipated equilibrium steepness ,r = 0.18, plotted in Figure 668(c), the ripple height d.' i, d rapidly over the first 10 wave periods as shown in Figure 668(a). Following the initial rapid decay in height, the ripple experienced a much more gradual decay over the next 40 wave periods until it finally reached an equilibrium value near t = 106 s, remaining constant for the remainder of the simulation. The timeaveraged ripple height and standard deviation, taken over the last 20 wave periods of the simulation, are plotted at the end of the timeseries in Figure 668(a). Applying the segregation of signs to the transport fluxes to signify their tendency to promote bedform growth (+) or decay (), and then using Equations 64a and 64b, we find that the initial rapid decay of ripple height is due to the vertical transport fluxes of entrainment and deposition. These results are plotted in Figure 669, and indicate that the bedload transport flux is perfectly balanced with equal magnitudes promoting both growth and decay, while the suspended load fluxes are clearly negative, indicating their tendency to promote ripple height decay. By extracting the first 20 s of data shown in Figure 669 and phaseaveraging over those 10 wave periods corresponding to the time of rapid ripple height decay, we are able to plot the phaseaveraged and corrected transport fluxes, as shown in Figure 670. During rapid decay, the phaseaveraged bedload transport fluxes are perfectly balanced in magnitude between growth and decay (((VQ )) = 0), while the vertical sediment fluxes are strongly negative throughout most of the wave cycle ((((wsCb E) ) < 0). When integrated over the wave cycle, the phase and spatiallyaveraged and corrected vertical sediment flux is negative, verifying its tendency to promote ripple decay. As a function of wave phase, the vertical flux term tends to promote weak building during flow deceleration, and stronger decay during flow acceleration. The phase dependence of the bedload, entrainment, and deposition transport components is demonstrated in Figures 671(a) and 671(b), and shows a behavior similar to that observed during the simulation of growth and equilibrium. The bedload transport tends to mimic the phase behavior of the motivating bed stress; peaks in the entrainment signal lead the maximum freestream forcing; and deposition events lag slightly behind the maximum flow. One clear difference between the growth and decay simulations, however, is also evident in Figures 671(a) and 671(b); the magnitude of the gravitational stresses is much larger in the decaying phase, ii:: 1ii:; a strong gravitational component in the bedload sediment transport signal. Similar to the simulation of bedform growth and equilibrium discussed earlier, the transport fluxes also exhibit a spatial dependence with respect to their location along the ripple profile. The spatial dependence of the bedload and suspended load transport fluxes is demonstrated in the flux tendency phase diagram shown in Figure 672. Here, the data points represent the timeaveraged and corrected flux tendencies computed using Equations 65a and 65b and are colored according to their vertical position along the timeaveraged ripple profile ((zb)); warmer colors represent the timeaveraged ripple crest location, while cooler colors indicate the average ripple trough elevation. Using the flux tendency phase diagram maps provided in Appendix B, we can infer from Figure 672 that most of the ripple decay is experienced through suspension events at the crest, and deposition in the trough. The upper ripple flanks experience a weak tendency to build through bedload transport, while the lower flanks experience weak building through deposition events. 6.2.3.3 Bedload Only Having already determined the behavior and tendencies of bedload transport fluxes to promote ripple growth in the total sediment load simulations, performing a simulation incorporating only the bedload transport term in our bedlevel updating scheme (Equation 467) is done for the sake of completeness. In the BL2v2 simulation, the vertical sediment fluxes corresponding to sediment entrainment (E) and deposition (w8Cb) are set to zero in the transport and morphology submodels. If our assessment of bedload flux tendency in the total load simulations holds true, then the modeling system should still predict bedform growth and equilibration at or near the expected conditions. The time evolution of ripple height, length, and steepness is provided in Figures 673(a), 673(b), and 673(c), respectively, and confirms our previous evaluation. Figure 673(a) Ii. I that while the ripple height does grow to within 12''. of the value observed in the total load simulation TL2v2, the growth period occurs over approximately 30 wave periods instead of 20 as before. The slower observed growth rate (Olr/a/t = 2.1 x 102 cm/s) is due to the absence of suspended load transport in the BL2v2 simulation, which, according to Figure 664, also contributes to bedform growth under certain conditions. The phaseaveraged shear stress, gravitational stress, and bedload transport rates during ripple growth and equilibration in the BL2v2 simulation are provided in Figures 674(a) and 674(b), respectively. There are no substantial differences in the phase dependence of bedload transport during ripple growth or equilibrium, nor does the behavior differ significantly from the total load simulation results provided in Figures 665(a) and 665(c). However, Figures 674(a) and 674(b) do ii. 1 an enhanced phase lead of the bed shear stress and bedload transport, ahead of the maximum freestream velocity, as ripple height increases. This result is consistent with the enhanced phase lead found for the total load simulation (TL2v2). 6.2.3.4 Suspended Load Only Simulations of suspended load only were achieved by setting the bedload transport rate to zero in the bedlevel updating scheme given by Equation 467. Such an experiment holds little physical relevance, as it is unlikely to ever occur in nature; however, the results are illustrative. Previously, we observed that the suspended load transport fluxes contributed to positive bedform growth under certain circumstances during the total load simulation TL2v2 (see Figures 664 and 666(a)). If this were generally true, then would a simulation incorporating only the entrainment and deposition of bed material in the morphology submodel yield bedform growth? The answer, interestingly, is no. The time evolution of ripple height, length, and steepness for the SL2v2 simulation is shown in Figures 675(a), 675(b), and 675(c), respectively. After an initially rapid decay in ripple height over the first 10 wave periods, Figure 675(a) demonstrates a more gradual decay during the subsequent 10 wave periods, followed then by a generally static time where the height does not vary more than :'. After 72 wave periods, the final ripple height and length are (Tr) = 8.5 x 102 cm and (A) = 14.4 cm, respectively; the timeaveraged height over the last 20 wave periods of the simulation is on the order of 3 graindiameters, ~i. 1 iir; that the bed is mostly flat. The phase dependence of suspension and deposition events for the rapidly decaying ripple height observed in simulation SL2v2 shows significantly different behavior from that observed in previous simulations. Phaseaveraged bed shear stress, entrainment, and deposition are plotted in Figure 676, and are i. I i,. of turbulent flow and transport over a rough bed, where both lead the freestream forcing by approximately 18. In the total load and bedload simulations, periods of entrainment tended to lag slightly behind the maximum freestream forcing, while depositional events occurred during times of flow reversal. The depositional behavior, with respect to wave phase, is unchanged for the simulation incorporating only suspended load transport; however, peaks in the entrainment signal tend to closely follow the phase signal of the applied fluid forcing and bed stress. This result is encouraging, as it agrees well with the observations of Ribberink et al. (1994) who found almost no phase difference between flow and suspension in the sheetflow and pickup l V.ir for transport in turbulent flows over rough beds. The results of the total load, total load decaying, bedload, and suspended load simulations provide interesting detail about the phase and spatial tendencies of bedload and suspended load transport fluxes to promote ripple growth, equilibration, and decay. While bedload transport is completely capable of generating and sustaining orbital ripples in the simulated flow regime (moderate mobility number, T = 32.95), suspended load transport, in the absence of bedload sediment transport, tends to promote ripple decay through strong suspension events at the ripple crests, subsequently leading to deposition in the troughs. The time evolution of ripple heights for all four simulations are plotted in Figure 677. Here, ripple heights are scaled by the final equilibrium height of the total load simulation ((Tr) = 2.43 cm), and time is nondimensionalized by the wave period. Z, (cm) k 03 x 024 0 18 012 0 06 0 0 06 0 12 0 18 S0 24 0 32 15 10 0 10 5 20 30 40 xCn 50 60 0 Figure 61. The timestacks of bed elevation for the 1DH linear model experiment. 100 80 E 60 < a 40 20 20 Time (s) (a) Height and Length 4 8 12 16 20 Time (s) (b) Steepness Figure 62. Timeseries of ripple (a) height (left ordinate) and length (right ordinate), and (b) steepness from the 1DH linear model experiment. S0.06 C CL ) 0.04 0. 0. 0.02 0 0 0.2 ,r 0o C.) Cor E 2 M ^ a I I I I I I I I In o ICn o (wU3) z C, I I I I In o In o (wo) z So I0 o (wo) z 2, Co) E  C.) 4 I3: 2  vv E i 4 V v~ u ~~  t SCo1 13 , ^^  D O c" U 0 S o in oi (uwo) z c c 00 c ) I) C (uwo) z In o n o In o In (wU3) Z (wUo3) Z 0 O v . C) 'S 0 O 0 z m  0b a r 2d O 0I 1m r E .2 b '5;D Phase (radians) Figure 64. A comparison of the modeled ( 1vr thickness (6). ) and observed (o) phaseaveraged boundary Difference (%) 0 15 30 45 60 75 90 105 120 135 150 On 0.257c 0.57 0.757c Phase (radians) (a) PhaseAveraged Difference 0.257c 0.57 0.757c Phase (radians) (b) Phase and VerticallyAveraged Difference Figure 65. The (a) phaseaveraged and (b) phase and verticallyaveraged absolute percent difference between model and experimental data. 150 S125 C 2 100 5 Z 75 0 so a 50 : 25 10' I 103 104 105 106 Number of Grid Points Figure 66. The computational budget for the grid comparison tests (GRID1, GRID2, GRID3, GRID4). The equation and statistics of the leastsquares fit to the four data points are provided in the figure. S10   8Y 6 S GRID1 4  GRID2 4  GRID3 >  GRID4 > 2 I I I I I I I I I I 0 4 8 12 16 20 24 28 Time (s) (a) (u, 04 S02 0 4 8 12 16 20 24 28 Time (s) (b) a, 6 S4 0 4 8 12 16 20 24 28 Time (s) (c) u, Figure 67. The (a) u and (b) wvelocity standard deviations of volumeaveraged velocities as a function of time for GRID1 ( ), GRID2 (), GRID3 ( D and GRID4 (). 214 80 I^ E S60 I " 40 E C. Figure 68. Time (s) The cumulative, volumeaveraged turbulent kinetic energy (TKE) as a function of time for GRID1 ( ), GRID2 (), GRID3 (), and GRID4 (). 100 80  60 E S40 20 0 Figure 69. 17C Phase (radians) The phase and volumeaveraged turbulent kinetic energy (((TKE))) as a function of forcing phase for GRID1 ( ), GRID2 (), GRID3 (), and GRID4 (). 0 O 4 0 biCk oc E " u Q~ HCCl 0 cn 13  b bci z 0 ^ E s . X9 ~ 0 a oc (w3) z 2 0 &^ E^C 0 Hm ii 11 (w3o)z b* O 0 2m c,^ 0 ^ " o = r c ^ ^ ^  1 , o s i ^^ ^ i< ^ G m~ \N ^' ^ \ ^ 2  ' (Ur3) Z m bO __________ ^' i ~i  i   J:  ^ ^ _______________ d00 a I Cb ~^ fi n E o" 8 I ^, r^ CC \ ^ ^ ^ a ^_^ 3 S ri  r r c ' '  216 (w3) z (wU3)z 0 O cb 0 E   == ^I I ^ '* 0 a0 (w13)Z z ,Q , ' or m =0 ? ^ m E  ^ r 00^ cb O k 0 :i O 0  n0 E ! m o 0'' ^ ' (w3)z b S^ 217 o ^ ^  ' , QU /\ C ^ g S^   ^ ^ S 3 fOs cba ~ ~~ ~ ~ \i : ^ I \ ^, r m \ ~ r ^ ^^ = a_ ^00 (luo)d z~ i~ o bf i 217O (w3) z (wo3)z o 0 oo 430 0 0 4a 06 m 0 H I oc' C (w23) z I 00 0   ^ ^ ^  ^ ^ ^ 2 ____ ^ T 00 *^ a \ ^ ^ ^ * ^_  ^ ^ ^ ^ ^ ^ 1 ^ o ^ I ^ x co ^ 3 I \ ^ ^' ^ ^ ^ (iuo)z i kj 218 (w3) z (wU3)z 103 E 104 105 106 Number of Grid Points Figure 613. The computational budget for the threedimensional grid comparison tests (WIDE3D_2, WIDE3D_5, WIDE3D_10, WIDE3D_20, WIDE3D_30, WIDE3D_40, and WIDE3D_50). The equation and statistics of the leastsquares fit to the seven data points are provided in the figure. 102 SI WIDE3D_2 S10  WIDE3D_5 E S WIDE3D_10 0 WIDE3D 20 WIDE3D_30 WIDE3D 40 WIDE3D_50 100 I I 0 20 40 60 80 Time (s) Figure 614. The cumulative, volumeaveraged turbulent kinetic energy (TKE) as a function of time for domains having increasingly large widths in the third (y) dimension. 103 E 102 ) 101 Phase (radians) Figure 615. The phase and volumeaveraged turbulent kinetic energy (((TKE))) as a function of forcing phase for domains having increasingly larger widths in the third (y) dimension. 60 I I 40 20 \ 0 WIDE3D_2 20  WIDE3D_5  WIDE3D_10  WIDE3D_20 40  WIDE3D 30  WIDE3D 40 WIDE3D 50 60 0 5x 1M 1 5n' 2x Phase (radians) (a) uvelocity 1 i i 05  05 05  005 0 5 17 1 51 2. Phase (radians) (b) vvelocity 001 0 005  0 005  00 0 5x 17 1 5x 2x Phase (radians) (c) wvelocity Figure 616. The phase and volumeaveraged (a) ((u)), (b) ((v)), and (c) ((w)) velocity components for seven different threedimensional grids. 221 102 S101 E 100 a 10  a S102 : c) 10  104 10  102 ky (cm1) (a) Spectral Density S100 0 >. 10 E o 2 103 102 101 100 ky (cm) (b) Velocity Amplitude Figure 617. The time and spatiallyaveraged (x and z) (a) spectral density, and (b) amplitude of the vvelocity as a function of wavenumber (ky) for threedimensional grid comparisons. WIDE3D_2 WIDE3D 5 WIDE3D_10 WIDE3D_20 WIDE3D_30 WIDE3D_40 WIDE3D_50 WIDE3D_2  WIDE3D_5  WIDE3D_10  WIDE3D_20  WIDE3D_30  WIDE3D_40 WIDE3D 50 10 101 S e 10.2 ._ O m 104 WIDE3D 2 C o WIDE3D5 . WIDE3D_10 U) 10  WIDE3D_20  WIDE3D_30 106  WIDE3D_40 WIDE3D_50 10 0 20 40 60 80 Time (s) Figure 618. The xaveraged standard deviation of the bed elevation (ab) in the ydirection as a function of time for threedimensional grid comparisons. 2.5 0.5 2 4 6 k (cm) 8 10 Figure 619. Timeaveraged equilibrium ripple heights ((tr,)) as a function of domain width (Ly) in the third dimension. The error bars correspond to the standard deviation of the ripple height r1, about (Tr,). The dashed line represents the equilibrium ripple height as predicted by the equations of Nielsen (1981). WIDE3D_2 A WIDE3D_5 y WIDE3D_10 I WIDE3D_20 4 WIDE3D_30 S WIDE3D_40 WIDE3D_50  Nielsen(1981) I I I I I l 10  102 E U 104 C) S10o4 ) 10 , a 108 107 108 102 10  101 10 2 10 104 102 ky (cm1) (a) Spectral Density 101 100 ky (cm) (b) Perturbation Amplitude Figure 620. The time and spatiallyaveraged (x and z) (a) spectral density, and (b) amplitude of bed elevation perturbations in the ydirection as a function of wavenumber (ky) for threedimensional grid comparisons. WIDE3D_2 WIDE3D_5 WIDE3D_10 WIDE3D_20 WIDE3D_30 WIDE3D_40 WIDE3D 50 WIDE3D 2 WIDE3D 5 WIDE3D_10 WIDE3D_20 WIDE3D_30 WIDE3D_40 WIDE3D_50 2.5 2 E S1.5 1 0.5 0. Time (s) (a) Ripple Height Maximum Rolling Grain Ripples 20 40 60 8 Time (s) (b) Ripple Steepness Figure 621. The equilibrium ripple (a) height, and (b) steepness as a function of time for various combinations of bedload transport equations and sediment pickup functions (see Table 53 for abbreviation keys). The average ripple heights (rTr) and corresponding standard deviations are shown in the inset of panel (a), and represent statistics averaged over 20 wave periods (40 s < t < 80 s). 100 E 10  a) 102 C 0) 103 20 40 60 80 100 120 140 160 Time (s) Figure 622. The percent change in sediment volume as a function of time during a morphology test (LL1v3). The signal has been filtered at 1 Hz to remove some of the noise. 25 2 8 15 05 0 20 40 60 80 100 120 140 160 Time (s) (a) LL1v3 Time (s)  Height   Length L'0 .. L .. .. . . " 20 40 60 80 100 120 140 Time (s) (b) ML1v2 0 20 40 Time (s) (c) ML2v2 60 80 (d) ML3vl Time (s) (e) HLlvl Figure 623. Timeseries of bedform height and length for morphology tests in labscale flows (T = 2 s). The average height ((Tlr)) and length ((A,)) are given by the symbols at the end of the timeseries, and the error bars denote the standard deviation about the mean. Average bedform statistics are computed using the last twenty waveperiods in the signal. 227 .  Height * Length 15 4  10 ri (cm) 0 010203040506070809 1 ...... 1 E 05 25 0 0 5 Figure 624. A timestack of bedform statistics using direct Fourier analysis for simulation HL1v1, where the ripple heights are inferred from the real Fourier coefficients. 2 3 4 Modeled Ripple Height (cm) 130 118 106 94 82 70 58 46 34 0 22 10 (a) Ripple Heights 40 I /30 130 1 18 / / 106 C X 94 S 82 S 70 20 / 58 S 46 C, N81 34 S// WH94 221 A GM82 10 v M94 X1 x 4 KOOO L 0* FF02 S. W05 0 x SW06 0 I I 0 10 20 30 40 Modeled Ripple Length (cm) (b) Ripple Lengths Figure 625. Onetoone comparisons of modeled and expected ripple (a) heights and (b) wavelengths for the labscale morphology tests. The dark solid line represents perfect agreement, while the dashed () and dotted ( ... ) lines represent deviations of 1(0' and :i ', respectively. The coloring of the symbols indicates the numeric value of the mobility number T for that case.   Height 0 Length 40 80 120 160 200 240 280 320 Time (s) (a) LFlv1 SI I I 'I I I I 'I a Height  Length ~""ir 0 20 40 60 80 100 120 140 160 Time (s) (c) MF2v2  Height  Length 20 40 60 80 100 120 140 160 Time (s) (b) MF1v2 (d) HFlvi Figure 626. Timeseries of bedform height and length for morphology tests in fieldscale flows (T = 8 s). The average height ((Q,)) and length ((A,)) are given by the symbols at the end of the timeseries, and the error bars denote the standard deviation about the mean. Average bedform statistics are computed using the last twenty waveperiods in the signal. 230 Im ri (cm) 0 010203040506070809 1 160 ;1, .......ilE~ir.... 5 20 0 l~ (a) MF2v2 rnr (cm) 0 002 004 006 008 01 012 014 0 16018 02 r~j0 6 , 0 4 (b) HFlv1 Figure 627. A timestack of bedform statistics using direct Fourier analysis for simulations (a) MF2v2 and (b) HF1v1, where the ripple heights are inferred from the real Fourier coefficients. 14 E 6 01 2  0 U U : 00 110 100 90 80 70 60 50 40 30 S20 10 (a) Ripple Heights W 110 100 90 80 70 60 50 40 30 E 20 10 Modeled Ripple Length (cm) (b) Ripple Lengths Figure 628. Onetoone comparisons of modeled and expected ripple (a) heights and (b) wavelengths for the fieldscale morphology tests. The dark solid line represents perfect agreement, while the dashed () and dotted ( ... ) lines represent deviations of 1(0'. and 3:Ii '. respectively. The coloring of the symbols indicates the numeric value of the mobility number I for that case. 2 4 Modeled Ripple Height (cm) 100 S80 E o 60 J 40 S 20 . 20 40r w 20 Re = 25x103 Re =65x103 R =150 Re = 150x10/ "*'. \3 Re= 400x103/ E Re= (2030)x103 A Re= (5080)x103 v Re= (100200)x103 0 Re = (300500)x103 " 101 102 a 103 ci C (. 0 z 101 101 C 10 100 C 10 J 0 ' 10 C a) E a, c z 105 Orbital Reynolds Number, Re, (b) Nondimensional Ripple Length Figure 629. A comparison of model results to the ripple predictor equations of Faraci and Foti (2002) for nondimensional ripple (a) height and (b) length. The open and filled symbols denote the labscale and fieldscale morphology results, respectively. I1 ii 101 102 Mobility Number, y (a) Nondimensional Ripple Height 1021 104 Re, = 60  Re, =120 .................. Re = 240 a 30 < Re < 90 S 90 *\ I Mobility Number, y (a) Nondimensional Ripple Height 101 '" .. 102 I I I 101 102 Mobility Number, y (b) Nondimensional Ripple Length Figure 630. A comparison of model results to the ripple predictor equations of Nielsen (1981), Khelifa and Ouellet (2000), and Williams et al. (2005) for nondimensional ripple (a) height and (b) length. The open and filled symbols denote the labscale and fieldscale morphology results, respectively. 103 .S 1r l 102 Q. E 0. E E' 101 xI1 100 1c 104 104 101 CJ 0o E 101 1( 101 100 101 10 Wave Period Parameter, xx 106 (a) Nondimensional Ripple Height (b) Nondimensional Ripple Length Figure 631. A comparison of model results to the ripple predictor equations of Mogridge et al. (1994) for nondimensional ripple (a) height and (b) length. The open and filled symbols denote the labscale and fieldscale morphology results, respectively. z Zb (cm) 04 X 0 32 024 0 16 008 008 200 0 16 0 24 0 32 160 0 4 140 120 60 40 15 20 20 x (Cm) 30 Figure 632. A timestack of bed elevation for the steady flow simulation. o102 10. E 105 10 1 0 1I 106 , 108 1 101I Time (s) (a) Stress and Transport 140 0.04 u120 laz E 120 C vn V, N0 0.03 3 100 80 80 0.02 C 40 0.01 > I 4 20 0 20 40 60 80 100 120 140 160 180 20 Time (s) (b) li,, ,i and Eddy Viscosity Figure 633. Timeseries of spatiallyaveraged (a) bed shear stress and bedload transport, and (b) velocity shear and eddy viscosity for the steady flow simulation. Zb (cm) 055 0 495 044 0 385 033 0 275 022 0165 011 0 055 0 (cm) 10 Figure 634. A timestack of yaveraged bed elevation profiles for the subcritical flow simulation. 4 0: 7C 17l Phase (radians) 1.57 0.1 0.05 E 0.0 E 0.05 Figure 635. The phase and spatiallyaveraged bed shear stress and bedload transport as a function of phase. The yellow area shows the positive and negative limits for the threshold of motion. H b I I I I I I I I I I I I I I I Z Zb (cm) 2 18 16 1 1 * Figure 636. A timestack of yaveraged bed elevation profiles during sheetflow sediment transport. 6 5 4 E C 3 2 1  18 16 14 12 10 8 4 20 40 60 80 Time (s) Figure 637. A timeseries of average ripple height (r,) and length (A,) for the sheetflow simulation. The symbols .,.1i ,:ent to the timeseries denote the ripple height and length averaged over twenty wave periods, and the error bars represent the signal standard deviation from the mean. z _I E (cm C. E I, le Figure 638. A timestack of the energy density contained in the morphology profile for the sheetflow simulation. S(C)=0.001 S(C)=0.1 S(C = 0.5 Figure 639. Timeaveraged isocontours of sediment concentration in the sheetflow regime. 4x 3 2  N Sc ) (cmi/cm3) 0  0 0.2 0.4 0.6 0.8 1 ( ) (cm3/cm3) Figure 640. The time and horizontallyaveraged sediment concentration profile in the sheetflow regime. The inset plot shows the full extent of the profile to the top of the modeling domain. z, (cm) I2 "icm) (a) Growth from r,/2 z, (cm) 4 (c) Equilibrium Conditions (ne, A,) (b) Growth from flat bed z (cm) 2 7 24 ( CM) (d) Decay from 3r,/2 Figure 641. Timestacks of the spanwise averaged bed elevation zb for livebed simulations initialized with (a) r7 /2 and Ae, (b) a small Gaussian hump five graindiameters tall (1.5 mm), (c) r, and A,, and (d) 3,e/2 and Ae. z, (cm) 160 0 10 X (Crn 150 z Ytx E (cm' ) I: 1 5 J,\e e 05 0 Figure 642. Timestacks of spectral energy density as a function of ripple wavelength and time for the GROW2 simulation. 5  2 10 0 0 1 5 5 Height Length S20 40 60 80 Time (s) (a) Growth from ,e/2 4 .20 5 ~Height Length 3 15 5  E 2 10 U 5  1 5 5  0 20 40 Time (s) 60 80 (c) Equilibrium Conditions (ne, Ae) Time (s) (b) Growth from flat bed 0 20 40 Time (s) 60 80 (d) Decay from 3 e/2 Figure 643. Timeseries of ripple height T, and length A, for livebed simulations initialized with (a) re/2 and Ae, (b) a small Gaussian hump five graindiameters tall (1.5 mm), (c) qe and A,, and (d) 3 q,/2 and A,. The timedependent average ripple height (rl) corresponds to the left ordinate, while the average ripple wavelength (A,) corresponds to the right ordinate. The symbols and error bars denote the averaged and standard deviations of ripple height ((fq)) and length ((A,)) computed using the last twenty wave periods of the simulation. 244   Height 0 Length  Growth t,=80: Decay  Equilibrium S1.5 t,=160s: FlatBed .S I C. 0. 0 0 0.2 0.4 0.6 0.8 Nondimensional Time, t t, Figure 644. The time evolution of ripple height, scaled by the expected height from Nielsen (1981) rT, = 1.89 cm, versus the nondimensional simulation time. The simulation time is scaled by the total time duration of the simulations, tf, which are given in the figure. x (cm) (a) Ripple Coarsening Zb (cm) 2 17 14 I I *7 x (cm) (b) Ripple Bifurcation Figure 645. Timestacks of the spanwise averaged bed elevation zb for livebed simulations initialized with (a) qe/2 and A,/2, and (b) qe/2 and 2 A, demonstrating ripple evolution through slide and split, respectively . 1 IpP 8 1 0l 0 (a) Ripple Coarsening E (cm)' E III , .0 i. _l.I. 21 0 (b) Ripple Bifurcation Figure 646. The timestacks of spectral density as a function of ripple wavelength and time during wavelength saturation through (a) coarsening (slide), and (b) bifurcation (split). 15 10 _ 5 0 Time (s) (a) Ripple Coarsening 40 35 30 25 E 20 ) 15 10 5 0 Time (s) (b) Ripple Bifurcation Figure 647. Timeseries of ripple height tr, and length A, for livebed simulations initialized with (a) q7,/2 and A,/2, and (b) q7,/2 and 2 A,. The timedependent average ripple height (r,) corresponds to the left ordinate, while the average ripple wavelength (A,) corresponds to the right ordinate. The symbols and error bars denote the averaged and standard deviations of ripple height ((rf)) and length ((A,)) computed using the last twenty wave periods of the simulation. 0.5  20 40 60 8( Nondimensional Time, t / T (a) Scaled Ripple Heights 20 40 60 Nondimensional Time, t/ T (b) Scaled Ripple Lengths Figure 648. Timeseries of (a) ripple height, and (b) ripple length scaled by their expected values (rle = 1.89 cm, A, = 13.6 cm) from Nielsen (1981). Simulation time t is scaled by the wave period T = 2 s, resulting in a measure of the number of wave cycles completed. HS1v2 ML3v1 ........... ....... ._". .**.*.. .** * I 0 *O. >D o0icP z rcc ox 0 S HS1v2 S ML3v 1 Cc ) (cfm'cm) 0 1 0 0.2 0.4 0.6 0.8 1 ( c ) (cm/cm3) Figure 649. The time and horizontallyaveraged concentration profiles (C(z)) from simulations HS1v2 and ML3v1. The inset figure shows the full vertical extent of the profiles, and the dashed black line () shows the upper extent of the profiles plotted in the larger figure. 4 3.5 3 2.5 ^ 2 F 1.5 1 0.5 0 0.5 I I I 1 I I I I 20 40 60 80 100 Time (s) 120 140 160 30 25 20 15 E 10 5 0 Figure 650. A timeseries of bedform height and length for the GRAVLOW simulation. 250  Height 0 Length  Phase (radians) Figure 651. The phase and horizontallyaveraged bed shear stress (('b)) (left ordinate) and bedload ((Qb)) and suspended load ((Q,)) sediment transport (right ordinate). The yellow shaded areas indicate regions of subcritical flow as predicted by Julien (1998). z Zb (cm) 02 016 S012 008 004 0  004  008 0 12 01C x (cm) Figure 652. Timestacks of the spanwiseaveraged bed elevation zb for the GRAVHI simulation. U 1  Qb 0 I 0 0 0  5 1 .57 U7C 3.5  Height . Length 3 2.5 2 1.5 1 0.5 o 35 30 25 20 E 20 15 < 10 5 0 0 20 40 60 80 100 120 140 160 Time (s) Figure 653. A timeseries of bedform height (left ordinate) and length (right ordinate) for the GRAVHI simulation. The symbols and error bars to the right of each signal represent the timeaverage value and standard deviation computed over the last 20 wave periods of the simulation. 200 .15 150 / b 0 Q 10 100  c1 5 E E 50 U 4 C C .................................... ... i.1... .. ... E A 0.57 A SPho (r 5 100  10 150  200 215 0U 0.5c 1c 1.5c C Phase (radians) Figure 654. The phase and horizontallyaveraged bed shear stress ((Tb)) (left ordinate) and bedload ((Qb)) and suspended load ((Q,)) sediment transport (right ordinate) for the GRAVHI simulations. The yellow shaded areas indicate regions of subcritical flow as predicted by Julien (1998). Zb (cm) 003 0 024 0018 S0012 0006 0  0 006 0012 0018 0 024 003 Figure 655. Timestacks of spanwiseaveraged bed elevation zb for the SILTLOW simulation. I. (cm) 0uu S0008 0 006 0 004 0 002 Figure 656. Timestacks of bedform height T1, and length A, computed using direct Fourier transforms on the spanwiseaveraged bed elevation signal Zb(, t) sampled at 10 Hz. z I., Phase (radians) (a) Stress and Sediment Transport 1 a Phase (radians) 0.005 (b) Stress and Sediment Concentration Figure 657. The phase and horizontallyaveraged (a) bed stress ((rb)) (left ordinate) and bedload ((Qb)) and suspended load ((Q,)) sediment transport (right ordinate), and (b) bed stress ((rb)) (left ordinate) and sediment concentration ((C)) for the SILTLOW simulation. The yellow shaded areas indicate regions of subcritical flow as predicted by Julien (1998). Zb (cm) *2 Z X~  2 30 X(cm)1~~~i Figure 658. The timestacks of spanwiseaveraged bed elevation zb sampled at 10 Hz during the SILTHI simulation. 4 3.5 3 2.5 E 2 1.5 1 0.5 40 35 30 25 20 U 15 10 5 0 Time (s) Figure 659. A timeseries of bedform height r, (left ordinate) and length A, (right ordinate) for the I i:. o highlyconcentrated SILTHI simulation. E 0( I" a 5 10 10 E 0 5 10 Phase (radians) (a) Stress and Sediment Transport chase (radians) Phase (radians) (b) Stress and Sediment Concentration Figure 660. The phase and horizontallyaveraged (a) bed stress ((Tb)) (left ordinate) and bedload ((Qb)) and suspended load ((Q,)) sediment transport (right ordinate), and (b) bed stress ((rb)) (left ordinate) and sediment concentration ((C)) for the SILTHI simulation. The yellow shaded areas indicate regions of subcritical flow as predicted by Julien (1998). 6 (C) S1 OE02 4 0E03 1 6E03 ) Baseline 6 3E04 Baseline 2 5E04 t 2 5E04 S 1 OE04 4 ParticleFluid 4 0E05 0 1 6E05 0 6 3E06 E 2 5E06 1 OE06 2 FluidParticle 4 2 0 2 4 6 Diameter, Phi Units Figure 661. A phase diagram for sediment suspension showing the three ranges of interactions as i. 1. .1 by Elghobashi (1994): fluidparticle interactions, C < 106; particlefluid interactions, 106 < C < 103; and particleparticle interactions, C > 10 The data points are taken from a selection of simulations having ranges of orbital Reynolds number and sediment diameter of 700 < sR, < 460, 000 and 0.025 < d < 6 mm, respectively. U 3 2. 2 Equilibrium *H. Growth 0. 0 20 40 60 80 100 120 140 160 Time (s) (a) Ripple Height U 1 15 10 t a. 5   O I I I I I I, I ,I, I ,I, I I 0 20 40 60 80 100 120 140 160 Time (s) (b) Ripple Length E 0 0.2 I ~ I I I I I I I , 0.1  0 20 40 60 80 100 120 140 160 Time (s) (c) Ripple Steepness Figure 662. Timeseries of ripple (a) height, (b) length, and (c) steepness for the total load simulation, TL2v2. The shaded yellow areas correspond to upper and lower estimates or ripple geometry using Grant and Madsen (1982) and Nielsen (1981), respectively. The red boxes in (a) represent the 20waveperiods corresponding to the growth and equilibrium phases of height evolution. The symbols and error bars represent the timeaveraged ripple (a) height ((rlr)) and standard deviation (o,,), and (b) length ((A,)) and standard deviation (a\) computed using the last 20 wave periods of the simulation. 258 Z X Zb (cm) I 2 17 14 140 40 20 x (cm) ' Figure 663. The timestacks of spanwiseaveraged bed elevation zb during ripple growth and equilibrium for the total load simulation TL2v2. 20 40 60 80 Time (s) 100 120 140 160 Figure 664. The cumulative, spatiallyaveraged and corrected bedload and suspended load transport fluxes as a function of simulation time for TL2v2. 2 E S1.5 X U I 0.5 0 0 5 0.5 E 1 I I I I I I I I S Suspended Flux Bedload Flux , I I I I I I I , (a) Ripple Growth 1P r Phase (radians) (c) Ripple Equilibrium Phase (radians) (b) Ripple Growth Phase (radians) (d) Ripple Equilibrium Figure 665. The phase dependence of (a) bedload sediment transport (Qb), and (b) entrainment (E) and deposition (wsCb) during ripple growth, and equilibrium in (c) and (d) for the TL2v2 simulation. 005 0 W o 0o E 0 05 _= I E 8  6 E 2 S 4 6 8 10 1 5. 2. x LL S0.01 0 0.01 U 0 S0.02 o 0.02 Phase (radians) (a) Ripple Growth  VQb  w. E OTC 0.57c 17 1.5(7 2 Phase (radians) (b) Ripple Equilibrium Figure 666. The phase and horizontallyaveraged, corrected bedload (VQ ) and suspended load (VQ ) transport fluxes during ripple (a) growth and (b) equilibrium for the TL2v2 simulation. 0 U_ U3 x LU 02 U_ ci Q (zb) (cm) 15 13 1 1 09 07 05 03 01 01 03 05  Bedload Flux < + Bedload Flux (a) Ripple Growth S002 (cm) 0.02 / 0 (/ 17 + 14 11 S 0* 05 S ' 04 X .* 04 LLV 7 0.02 0. ).04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 Bedload Flux I + Bedload Flux (b) Ripple Equilibrium Figure 667. The timeaveraged and corrected bedload ((VQ~ )) and suspended load ((VQ )) transport fluxes during ripple (a) growth and (b) equilibrium for the TL2v2 simulation. The colored symbols represent points in the horizontal x y bed plane, and are contoured with respect to their location in the ripple profile with the warmer colors representing the ripple crest, and cooler colors denoting the ripple troughs. The dashed () lines signify the onetoone agreement in each quadrant. 3 ._ 2 SRapidGradual Equilibrium a 1 O I I I I I I I I I 0 20 40 60 80 100 120 140 160 Time (s) (a) Ripple Height U 15 10 ,. 5 O I I I I I ,I I I I 0 20 40 60 80 100 120 140 160 Time (s) (b) Ripple Length I I I I ~ I I I I I I S0.2 0.1  0.1 0. SI I I I I I I , 0 20 40 60 80 100 120 140 160 Time (s) (c) Ripple Steepness Figure 668. Timeseries of ripple (a) height, (b) length, and (c) steepness for the decaying total load simulation, TL2v2d. The shaded yellow areas correspond to upper and lower estimates or ripple geometry using Grant and Madsen (1982) and Nielsen (1981), respectively. The red boxes in (a) represent the rapid, gradual, and equilibrium phases of height evolution. The symbols and error bars represent the timeaveraged ripple (a) height ((ir,)) and standard deviation (o,,), and (b) length ((A,)) and standard deviation (ax) computed using the last 20 wave periods of the simulation. I I I I I I I 20 40 60 80 Time (s) 100 120 140 160 Figure 669. The cumulative, spatiallyaveraged and corrected bedload and suspended load transport fluxes as a function of simulation time during ripple decay (TL2v2d). 0.02 E x Uc i0 x L '0 0D D  0.01 0.01 0.02 ' 0 7n l1 Phase (radians) Figure 670. The phaseaveraged and corrected bedload and vertical sediment fluxes during rapid ripple decay in TL2v2d.  Suspended Flux Bedload Flux I I I I I I I I I  VQb w c E I II I I Phase (radians) E 0.05 0 Q. o 0 0.05  m 71 (a) Stress and Bedload Transport Phase (radians) 0.05 E x LL C 0 a) E 7 o 0.05 '= (b) Stress and Vertical Sediment Fluxes Figure 671. The phaseaveraged (a) bed shear stress, gravitational stress, and bedload transport rate, and (b) bed shear stress, gravitational stress, and vertical sediment fluxes for the TL2v2d simulation. a) / (Zb) (cm) 0.01 / ( M) 0/) t / 18 12 0 1 2 * / 14 J *08 S* 106 S* 04 LL lLl 02 [i ./ 0 2 s*/ e e 0.01 / 0. 02 0.02 0.01 0 0.01 0.02 Bedload Flux <= + Bedload Flux Figure 672. The timeaveraged and corrected bedload ((VQf)) and suspended load ((VQ')) transport fluxes during rapid ripple decay for the TL2v2d simulation. The colored symbols represent points in the horizontal x y bed plane, and are contoured with respect to their location in the ripple profile with the warmer colors representing the ripple crest, and cooler colors denoting the ripple troughs. The dashed () lines signify the onetoone agreement in each quadrant. 3 S~Equilibrium *u 2 2 ii  0. H. 1 Growth 0 I I I ,I ,I, I ,II I 0 20 40 60 80 100 120 140 160 Time (s) (a) Ripple Height 15 10 0 a. 5   O I I I IIII I I 0 20 40 60 80 100 120 140 160 Time (s) (b) Ripple Length A 0.3 2  E S0.2 0. .2  0 I I I I I I I I I , 0 20 40 60 80 100 120 140 160 Time (s) (c) Ripple Steepness Figure 673. Timeseries of ripple (a) height, (b) length, and (c) steepness produced solely through bedload transport. The shaded yellow areas correspond to upper and lower estimates or ripple geometry using Grant and Madsen (1982) and Nielsen (1981), respectively. The red boxes in (a) represent the growth and equilibrium phases of height evolution. The symbols and error bars represent the timeaveraged ripple (a) height ((Tr,)) and standard deviation (o,,), and (b) length ((A,)) and standard deviation (a\) computed using the last 20 wave periods of the simulation. 267 6 I. I 2I. I 6 005 6 005 0 4   0 ? 4 ^ O 4 4  0 0 0 05 0 0 5 6  8  10 015 1 115 2 U100K 05. 1I 150 2 0 Phase (radians) Phase (radians) (a) Ripple Growth (b) Ripple Equilibrium Figure 674. The phase dependence of bedload sediment transport (Qb) during ripple (a) growth, and (b) equilibrium for the BL2v2 simulation. 268 S0.8 Decay F S0.6 S0.4 S0.2  0 20 40 60 80 100 120 140 160 Time (s) (a) Ripple Height 40 I I I I I 30 .2  S20 1 L 10 0 20 40 60 80 100 120 140 160 Time (s) (b) Ripple Length E e0.15  0.1 O 0.05 0 20 40 60 80 100 120 140 160 Time (s) (c) Ripple Steepness Figure 675. Timeseries of ripple (a) height, (b) length, and (c) steepness produced solely through suspended load transport (SL2v2). The red box in (a) represents the time of rapid height decay. The symbols and error bars represent the timeaveraged ripple (a) height ((Tr,)) and standard deviation (o,,), and (b) length ((A,)) and standard deviation (a\) computed using the last 20 wave periods of the simulation. 15 0.15 S10 0.1 E o X S5 0.05 . S  0 0 5 0.05 U C 10 b 0.1 . E 15 c 0.15 Ws Cb 20 0.2 020 0.57c 17 1.57c 27C Phase (radians) Figure 676. The phase dependence of entrainment (E) and deposition (w2Cb) fluxes for rapid ripple decay during the suspensiononly simulation SL2v2. C 1 S05  TL2v2  TL2v2d BL2v2 O ___SL2v2 U) 0 20 40 60 Nondimensional Time, t/ T Figure 677. The evolution of ripple height for the total load (TL2v2), decaying total load (TL2v2d), bedload only (BL2v2), and suspended load only (SL2v2) simulations as a function of the number of wave cycles (t/T). The ripple heights are nondimensionalized by the timeaveraged ripple height ((7,)) of the TL2v2 simulation. Table 61. Ripple Pre Faraci and Mogridge e Nielsen (19 Soulsby an Khelifa and Williams et Grant and Wiberg an The RMS error between model and expected ripple height and length for morphology tests under labscale flows, with associated numerical scores and combined rankings. Numerical scores are assigned values ranging from 1 to 8 corresponding to the lowest and highest RMS errors, respectively. doctor r]RMSE Score ARMSE Score Combined Score (cm) (cm) Foti (2002) 0. 652 2 2. 25 1 3 t al. (1994) 1. 052 3 2. 77 2 5 81) 0. 114 1 4. 22 4 5 d Whitehouse (2006) 1. 57 6 3. 53 3 9 i Ouellet (2000) 1. 19 4 10. 2 8 12 Sal. (2005) 1. 30 5 10. 2 7 12 Madsen (1982) 2. 23 7 7. 055 6 13 i Harris (1994) 2. 38 8 6. 42 5 13 Table 62. The RMS error between model and expected ripple height and length for morphology tests under fieldscale flows, with associated numerical scores and combined rankings. Numerical scores are assigned values ranging from 1 to 8 corresponding to the lowest and highest RMS errors, respectively. Ripple Predictor r1RMSE Score ARMSE Score Combined Score (cm) (cm) Williams et al. (2005) 0. 435 1 5. 47 1 2 Faraci and Foti (2002) 0. 914 2 7. 66 4 6 Nielsen (1981) 1. 56 4 6. 42 2 6 Khelifa and Ouellet (2000) 1. 17 3 8. 76 5 8 Wiberg and Harris (1994) 2. 84 5 7. 34 3 8 Mogridge et al. (1994) 9. 69 7 30. 8 6 13 Soulsbv and Whitehouse (2006) 7. 40 6 37. 5 7 13 Grant and Madsen (1982) 10. 8 8 55. 9 8 16 Table 63. The model performance index (l\!PI) associated with the ripple predictors used to assess morphology test results. The MPI score is found by summing the combined scores from the labscale and fieldscale flow rankings in Tables 61 and 62, respectively. MPI Score Ripple Predictor Faraci and Foti (2002) Nielsen (1981) Williams et al. (2005) Mogridge et al. (1994) Khelifa and Ouellet (2000) Wiberg and Harris (1994) Soulsby and Whitehouse (2006) Grant and Madsen (1982) CHAPTER 7 CONCLUSIONS Through fortyfour simulations, we have not only demonstrated the capabilities of new phaseresolving livebed modeling systems in the linear and nonlinear regime, but have enhanced our understanding of smallscale sediment transport processes, and their role in generating bedforms. In some cases, we have reinforced widelyaccepted theories about the roles of sediment transport modes during ripple growth, while in other areas we have advanced knowledge on the subject of ripple equilibrium and decay. The following sections provide a brief summation of pertinent model results as they pertain to the validity and predictive capabilities of our newly developed livebed models at simulating morphology under a wide variety of flow regimes. Additionally, conclusions are drawn from the results of Phase III simulations, and shed light on the roles of bedload and suspended load transport during ripple growth, equilibrium, and decay in the nonlinear regime as a function of both ripple location and wave phase. 7.1 LiveBed Model Evaluation The simple onedimensionalhorizontal (1DH) model of bedload transport in the linear regime, derived from first principles, reinforces a concept widely held in literature on bedform growth for the previous century; incipient bedform growth from a flat sand bed may occur entirely through bedload sediment transport. An interesting result of our conceptual 1DH model simulation was discovered, and appears to emphasize the role of particle inertia in promoting an initial instability of the flat bed. A central theme, or assumption, of the simple 1DH model is that the motivating stress applied to the bed by the mixed 1lvr is constant across the bed, and in phase (space) with the sediment transport. The latter part of this assumption was found to be incorrect; in fact, a spatial lag between applied stress and particle transport is necessary for the linear instability of the sandwater interface to develop, as found in the analysis of Parker (1975). This phase lag is explained as an inertial effect of the sediment, whereby its weight creates a tendency for the particle motion to lag slightly behind the mobilizing forces. Without such a lag, the sand bed remains unconditionally stable at all wavelengths, and bedload transport within the mixed 1. r tends to promote a stable, flat bed. While our simple 1DH model is of little use for practical applications, it is nonetheless illustrative of the dominant sediment transport and particle processes that lead to bedform growth in the linear regime. The generation of rolling grain ripples on a flat bed may serve as a useful physical analog to the characteristics of the 1DH model. Through modeldata comparisons of phase and horizontallyaveraged velocity profiles in oscillatory boundary 1 r flow (Jensen et al., 1989), we demonstrate the ability of the LES hydrodynamic model to reproduce realistic flow characteristics in the boundary 1 , r. Although the model predictions of boundary 1 r flow exhibited a consistent lag of 30 with respect to the experimental data, and tended to overestimate the boundary 1 ,r thickness by 2 5 mm, simulation results are generally within Si"' of the observed values. The overestimation of boundary 1l.r thickness tends to promote early flow reversal near the bed, leading to undesirable phase discrepancies in the modeldata comparisons, particularly near flow reversal. These shortcomings may be harmonized by employing a more advanced closure model, as the Smagorinsky (1963) method is known to perform poorly in high Reynolds number flows, tending to overestimate the turbulent eddy viscosity; model predictions in the boundary l1..r, particularly near the wall, may further benefit from a more sophisticated nearwall model. Generally speaking, though, the LES model numerics tended to show positive performance in the numerical convergence tests, demonstrating an independence of normalized flow metrics to the choice of grid resolution. Aside from evaluating the hydrodynamic component of the coupled modeling system, the predictive capability of the sediment transport and morphology submodels was also investigated through the simulation of nine distinct livebed model experiments of morphology in a wide range of flow conditions. The distribution of flow and sediment characteristics used in these simulations yielded mobility numbers that ranged by one order of magnitude. Due to the overwhelming number of references to discrepancies in the characteristics of morphology created under labscale and fieldscale flow conditions, flow periods were chosen to mimic oscillatory flows likely to be found in both settings. In the absence of a unified set of laboratory and field data regarding bedform characteristics, model predictions of ripple height and length were compared to expected values computed using eight different semiempirical ripple predictors prevalent in existing literature on the subject [i.e. Nielsen (1981); Grant and Madsen (1982); Mogridge et al. (1994); Wiberg and Harris (1994); Khelifa and Ouellet (2000); Faraci and Foti (2002); Williams et al. (2005); Soulsby and Whitehouse (2006)]. As the 1 i in i ly of these expressions were developed using multiple sets of data on bedform and flow characteristics, the end result is equivalent to comparing our model predictions of morphology to over 30 different sets of laboratory and field data (see Table 32). Model predictions of ripple height and length tend to agree well with the ripple predictors of Nielsen (1981) for both field and labscale flows, similar to the observations of Marsh et al. (1999); however, the best overall fit to all of the morphology test predictions by the new livebed model was found using the ripple predictor of Faraci and Foti (2002). Owing to the large degree of scatter in the expected values of ripple height and length computed using the ripple predictors (see Table 56) for the fieldscale flow simulations, it is unclear whether the model is particularly adept at predicting morphology under such conditions. A lack of harmony in the analysis of field ripple data, therefore, precludes us from drawing any definite conclusions about the applicability of the livebed modeling system to fieldscale flows; however, the model predictions of ripple height and length for labscale flows, especially those at low to moderate mobility numbers, are encouraging. Numerous other simulations were performed in Phase II, the results of which highlight additional capabilities of the livebed modeling system, as well as address sensitivity issues regarding initial conditions, periodic domain length, and the effects of hindered settling on morphology. While it would seem that the methodology chosen for modeling the suspended load transport through an advectiondiffusion equation would not lend itself well in simulating suspension and transport in the sheetflow regime, model simulations of highlyconcentrated flow at high mobility numbers i ... I reasonable performance. Without a detailed modeldata comparison to sheetflow experiments, the only conclusion that may be drawn from the sheetflow simulation is that the suspended load transport submodel predicts concentrations in the lower and upper sheetflow lV. i typical of those found in laboratory experiments (Ribberink and AlSalem, 1995). Furthermore, the morphology model correctly predicted a flattening of the initial bedform, and the lack of a coherent signal in the bed elevation for such energetic flow and transport conditions. Simulations of our baseline flow and sediment parameters were conducted to evaluate the sensitivity of the morphology submodel to initial conditions. One experiment simulated the growth of a solitary ripple from onehalf its expected equilibrium height, while another simulated growth from a nearly flat bed (a small Gaussian hump having a height equivalent to 5 graindiameters). Still others simulated equilibrium ripple conditions, as well as decay from morphology initialized with a ripple height that was out of equilibrium with the flow conditions. An additional experiment of growth from onehalf the expected height was performed, and incorporated the effects of hindered settling (Richardson and Zaki, 1954) in the suspended sediment transport submodel. The final predictions of ripple height for all five cases varied by less than 1('. ,ii. li:.; that the initialized morphology p1. i' only a minor role in the final model predictions. Although the in, i.i iy of simulations were performed with morphology initialized at the equilibrium or expected ripple wavelength, in order to satisfy the periodic streamwise boundary conditions, two model experiments were conducted to determine the domain effect on wavelength selection. These simulations successfully reproduced both the processes of ripple coarsening and bifurcation, referred to in the literature as ripple slide and split (Doucette and O'Donoghue, 2006), whereby ripple evolution occurs through a change in wavelength first, and height second. Whereas the coarsening process involves the saturation of ripple wavelength at a value larger than what it started with, ripple bifurcation involves a splitting of the initial bedform into two distinct ripples having wavelengths smaller than the initial morphology. Both experiments yielded ripple heights and lengths at or very near the expected values. 7.2 Sediment Transport and Bedform Dynamics The final set of four experiments conducted for the Phase III experiments were formulated with our specific scientific objectives, outlined in C'! lpter 1, in mind. Each simulation was conducted using the full threedimensional livebed modeling system, where information regarding acceptable grid resolutions and spanwise domain widths were obtained through the Phase I model experiments. In order to determine the relative contributions of bedload and suspended load transport during ripple growth, equilibrium, and decay, and also to determine their dependence on both wave phase and location throughout the ripple profile, two total load transport simulations were conducted, each focusing on either ripple growth and equilibration, or decay and equilibration. Taking advantage of our unique model capabilities, two additional simulations were performed incorporating either bedload transport only, or suspended load transport only, in order to determine their respective roles in bedform dynamics in the absence of the other. Utilizing the total load simulation results of bedform growth, equilibrium, and decay, along with the newly developed flux tendency phase diagrams discussed in Chapter 6, and more thoroughly in Appendix B, we present previously unpublished information regarding the phase dependence of transport fluxes with respect to bedform dynamics. With respect to wave phase, the dominant transport fluxes leading to ripple growth occur during times of maximum acceleration and deceleration of the freestream flow. During maximum acceleration, the model predictions ,ii. i bedloaddominated growth, with decay occurring through suspension events. The opposite behavior is observed during times of maximum flow deceleration, with depositiondominated growth and a comparatively weak decay through bedload transport. These results also appear to hold true during times of ripple equilibrium, yet the suspension and bedload fluxes are balanced in a manner that yields no net growth or decay. The simulations of ripple decay under the influence of total sediment transport load ii.:: 1 drastically different behavior; here, the bedload transport fluxes contain equal magnitudes tending to promote both decay and growth, resulting in no net contribution throughout the entire wave phase. Conversely, the vertical sediment fluxes exhibit a dependence on wave phase similar to that observed during growth and equilibration, although the signal is displaced almost entirely to the decaying () tendency. In summary, the model simulations r. I that during ripple growth, i.'. (35'.) of the total fluxes may be attributed to bedload (suspended load) sediment transport; and during ripple equilibrium, I.'. (: '.) of the total fluxes are related to suspended load (bedload) sediment transport; and during rapid ripple decay, 10' of the net sediment fluxes are due to suspension and deposition events, because the bedload transport fluxes are equally balanced between promoting growth and decay. In order to explain the spatial dependence of the transport fluxes, as well as their tendency to either promote ripple growth or decay, we make use of the newly developed flux tendency phase diagram. For the purposes of this discussion, the spatial dependence is related to the ripple profile through three distinct regions: 1. the ripple crest; 2. the ripple flanks; and 3. the ripple troughs. The simulation results of the total load simulations ii. I that during times of growth, the ripple experiences bedloaddominated growth at the crest and upper flanks, while suspensiondominated growth occurs in the ripple troughs; a balance of fluxes is found to exist through the lower and middle ripple flanks. During times of ripple equilibrium, we find the locus of data points falling along the equilibrium (solid diagonal line) line in the flux tendency phase diagram, where ripple crests experience a balance between bedload growth and suspension decay, while ripple troughs tend to exhibit decay through bedload fluxes and growth through suspension, causing them to scour deeper. Not surprisingly, a i: i i i ly of data points are found in the lower left triangle of the flux tendency phase diagram during ripple decay. The highest density of data points tends to fall around the vertical axis, indicating very strong decaying fluxes produced through suspension at the ripple crests, and deposition in the ripple troughs. A comparatively weak growth tendency due to depositional fluxes is observed along the ripple flanks. These data reinforce the common interpretation of processes leading to bedform growth, whereby vortices in the lee of ripple crests tend to liberate particles from the ripple troughs, which subsequently experience crestdirected movement through bedload sediment transport. An unexpected result, however, is the lack of significant gravitationallyinduced bedload transport during ripple equilibrium. Although the phaseaveraged gravitational stresses where considerably larger for the total load decaying simulation, their role in substantial ripple decay must be minor. This lack of a significant gravitational component to the bed stress and transport is an encouraging result of the growth and equilibrium simulations, and '. ; that the livebed modeling system predicts morphology in equilibrium with the hydrodynamic flow conditions rather than steepnesslimited bedforms. The bedload and suspended load simulations verify the information extracted from the flux tendency phase diagrams, where we find that bedload is the dominant transport mechanism promoting bedform growth while the entrainment and deposition of bed material tend to promote decay. These simulations are unique as they are not something that could be easily achieved in a laboratory experiment; it would be unlikely that such experiments could be derived in the nonlinear regime where one could alternately turn bedload and suspended load on or off. One attainable analog would be to perform experiments with extremes in the grain size spectrum, whereby gravel and cobbles would move only through bedload transport, and fine silts and (1 lii.. 1 particles would mostly be transported through suspension. Using such extremes, however, presents additional difficulties due to their inertial properties and tendencies to create bedforms other than orbital ripples. In retrospect, the independent simulations using only bedload sediment transport or suspended load transport were frivolous, owing to the utility of the flux tendency phase diagrams derived from model simulations of total sediment transport load during periods of ripple growth, equilibrium, and decay. However, aspects of the simulations that could not have been directly inferred from the phase diagrams is the remarkable tendency of bedload transport to produce orbitaltype ripples in the nonlinear regime, and that of suspension events to rapidly decay bedforms in the absence of bedload sediment transport. The bedload simulations also demonstrated that when acting without suspended load sediment transport, the ripple growth rate is much slower when compared to that observed in the total load simulations, with the time to reach equilibrium increasing by 150'. In this regard, the additional bedload and suspended load transport simulations were particularly illustrative. 7.3 Future Applications An extensive list of "todo" items, with respect to the current modeling system, has been compiled over the course of this investigation. It would appear that the hydrodynamic modeling system could benefit from a more robust closure scheme, potentially a dynamic closure method that simulates backscatter of energy. Also, it would be advantageous to introduce spectral methods for the solution of the pressure field, thereby promoting future optimization of the entire livebed modeling system through domain decomposition and parallelization. With respect to the transport and morphology submodels, the EulerWENO scheme should be adapted to permit variable mesh scaling in the horizontal plane if simulations of largescale morphology are to be performed in the future. Additionally, an alternate method of simulating the suspended sediment concentration field may be considered for simulating more typical coastaltype flows having suspended sediment concentrations, by mass, on the order of 100 g/1. Of course, the 3D livebed modeling system could benefit from additional validation and calibration to recent laboratory data, such as those collected during the CROSSTEX experiments at the O.H. Hinsdale Wave Research Laboratory at Oregon State University, or the USGS oscillating tray experiments under combined wavecurrent flow reported in Lacy et al. (2006). APPENDIX A EXPERIMENT MATRIX The complete simulation matrix for test cases discussed in the text is given in Table Al. The experimental matrix lists the case name, domain sizes and resolutions, relevant flow characteristics, and sediment sizes. Table A1. A full list of experiments and relevant simulation parameters for test cases discussed in C'! plters 5 Case Name 1DH JSF4v8 GRID1 GRID2 GRID3 GRID4 WIDE3D_2 WIDE3D_5 WIDE3D_10 WIDE3D_20 WIDE3D_30 WIDE3D_40 WIDE3D_50 MPMVR MPMFL CLVR CLFL LL1v3 LFlvO ML1v2 ML2v2 ML3v1 MF1v2 MF2v2 HL1v1 HFlv1 STEADY SUBCR SHEET GROW GROW2 EQUILIB DECAY HSlv2 SLIDE SPLIT GRAVLOW GRAVHI SILTLOW SILTHI TL2v2 TL2v2d BL2v2 SL2v2 Lx (cm) 60.0 3.2 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 25.6 34.8 21.6 28.0 13.6 25.2 22.8 29.6 24.4 27.2 27.2 13.6 13.6 13.6 13.6 13.6 27.2 27.2 27.2 30.0 30.0 30.0 30.0 27.2 27.2 27.2 27.2 Ly (cm) 0.0 3.2 0.4 0.8 0.2 0.2 0.4 1.0 2.0 4.0 6.0 8.0 10.0 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 3.2 3.2 3.2 3.2 Lz (cm) 0.0 30.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.5 32.5 11.5 11.5 11.0 11.5 11.5 11.5 11.5 12.0 11.0 11.0 11.0 11.0 11.0 11.0 11.5 11.5 11.5 11.5 11.5 11.5 11.5 21.5 21.5 21.5 21.5 and 6. nx 30 32 68 34 136 136 68 68 68 68 68 68 68 68 68 68 68 128 174 108 140 68 126 114 148 122 136 136 68 68 68 68 68 136 136 136 150 150 150 150 136 136 136 136 ny nz Uo (cm/s) N/A N/A 30 32 100 200 3 90 40 3 45 40 3 180 40 3 90 40 2 90 40 5 90 40 10 90 40 20 90 40 30 90 40 40 90 40 50 90 40 2 90 40 2 90 40 2 90 40 2 90 40 2 100 30 2 160 30 2 100 30 2 100 50 2 90 40 2 100 30 2 100 50 2 100 80 2 100 60 2 110 30 2 90 5 2 90 100 2 90 40 2 90 40 2 90 40 2 90 40 2 100 40 2 100 40 2 100 40 2 90 30 2 90 80 2 90 10 2 90 40 16 110 40 16 110 40 16 110 40 16 110 40 T (s) 4.0 9.72 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 8.0 2.0 2.0 2.0 8.0 8.0 2.0 8.0 0.00 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 (x104) 5. 730 618.8 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 2. .i ,', 11.46 2. .i ,', 7.958 5.0929 11.46 31.83 20.37 45.84 0.00 0.07958 31.83 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 2. .i ,', 20.372 0.3183 5. 0929 5. 0929 5. 0929 5. 0929 5. 0929 d (pm) 300 N/A N/A N/A N/A N/A 300 300 300 300 300 300 300 300 300 300 300 400 400 200 300 300 200 300 300 200 300 300 300 300 300 300 300 300 300 300 6000 6000 25 25 300 300 300 300 APPENDIX B FLUX TENDENCY PHASE DIAGRAM An explanation regarding the use of the phase diagram, as well as a description of its derivation, is provided. The flux tendency phase diagram is a useful tool for determining the relationship between sediment transport fluxes and their tendency to promote either growth, equilibrium, or decay. While this investigation made particular use of the phase diagram while explaining the spatial dependence of transport fluxes throughout the ripple profile, the same methodology could be applied to determining phase dependence, as well, by coloring the symbols with respect to wave phase. Observations made regarding the behavior of transport fluxes during the Phase III simulations fully support the flux tendency phase diagram concept, as well as the inferences that can be drawn from it regarding the predilection of bedload and suspended load transport flux components to either promote growth or decay. The flux tendency diagram is elegant in its simplicity, and quite powerful in its utility. In order to utilize the flux tendency phase diagram shown in Figures Bl(a)Bl(d), it is first necessary to segregate transport flux components by their tendency to either promote bedform growth or decay. This process is similar to that used in Penko (2007), and is found to be quite illustrative. Transport fluxes producing bedform growth are assigned a positive (+) value, while those tending to yield bedform decay are given a negative () value. These assignments are given in Equations Bla and Blb. Special care must be taken to ensure sign fidelity when segregating transport fluxes above and below Zb = 0. Once the flux components have been successfully sorted by sign, which indicates its tendency for growth or decay, the data can then be plotted as scatter in the flux tendency phase diagram. 7Q+ VQb,s Qb,s 01b41 = vQbs for > 0 (Bla) VQ = VQb, for < 0 (Bb) The concept of the flux tendency phase diagram began as a simple onetoone plot of bedload transport flux versus suspended load fluxes with the raw, absolute flux data plotted as scatter. Utilizing the segregation principle outlined in Equations Bla and Blb required further division of the simple onetoone plot, as it initially only applied to the positivepositive quadrant since only the absolute flux magnitudes were being plotted. This primary division is shown in Figure Bl(a), and represents the first of four steps in the derivation of the flux tendency phase diagram. Here, the phase diagram is clearly divided into four distinct quadrants, labeled I IV using a convention similar to that found in Euclidean geometry. In Figure Bl(a), we see that the overall plot is bisected by the bedload and suspended load transport flux axes, indicating an origin in the center of the diagram. In this manner, quadrants to the right of the vertical bisector (I and II) are ii 1 . of bedload flux growth, while those to the left (III and IV) represent regions of bedform flux decay. Alternatively, quadrants above the horizontal bisector (I and IV) indicate bedform growth through suspended fluxes, with those below the horizontal bisector (II and III) denoting suspended flux decay. Now, each quadrant can be interpreted as its own distinct onetoone comparison of VQt and VQ', as demonstrated in Figure Bl(b), with the lines of perfect agreement represented by the dashed () lines. Data falling along the lines of perfect agreement indicate an equality in bedload and suspended load flux magnitudes for that quadrant. Points falling closer to the horizontal bisectors signify bedloaddominated tendencies, while those falling closer to the vertical bisector represent suspension or depositiondominated tendencies. In an attempt to generalize the flux tendency diagram further, the diagram described by Figure Bl(b) is simplified to define regions of growth (+) and decay () in Figure Bl(c). Moreover, the tendency of the onetoone lines in each quadrant is rationalized and labeled in Figure Bl(c), where the solid diagonal line represents a line of equilibrium; in quadrant II, bedload flux growth is counteracted by suspended flux decay along the onetoone line; the opposite is true in quadrant IV. In quadrant I, the line of perfect agreement produces growth regardless of the flux mechanism, while quadrant III represents total decay through contributions from both flux mechanisms. Regions of growth (+) and decay () are combined in Figure Bl(d), yielding the final flux tendency phase diagram. Here, the phase diagram is subdivided into only two distinct regions for simplicity: growth and decay. Each region may be further subdivided by the respective action on either side of the onetoone line in order to generalize the dominant flux behavior in that region of the phase diagram. Data points falling along the equilibrium line are indicative of morphology that is experiencing dynamic equilibrium through an equality in magnitudes of both bedload and suspended load transport fluxes. Simulation results of bedform growth, equilibrium, and decay confirm the utility of the flux tendency phase diagram.  Bedload Flux < Figure B1. >+ Bedload Flux (a) Ordered Quadrants (c) Reduction Bedload Flux + Bedload Flux (b) Dominant Fluxes and Tendency Bedload Flux =! i=> + Bedload Flux (b) Dominant Fluxes and Tendency 0  Bedload Flux =z> + Bedload Flux (d) Simplified Map An explanation of the transport flux tendency phase diagram of bedform growth, equilibrium, and decay. The quadrants are (a) ordered one through four using a common convention, (b) labeled corresponding to the dominant flux and its tendency, (c) divided in half by the onetoone (perfect agreement) line and labeled according to the growth (+) or decay () tendency, and (d) simplified to two triangular regions associated with either bedform growth (upper right green triangle) or decay (lower left red triangle). The solid diagonal line () represents ripple equilibrium, while the dashed line () and dasheddotted line ( ) represent balanced growth and decay, respectively. IV I III II REFERENCES Andersen, K., 1999. Ripples beneath surface waves and topics in shell models of turbulence. Ph.D. thesis, University of Copenhagen. 2.4.2, 3.2.1 Andersen, K., Abel, M., Krug, J., Ellegaard, C., S0ndergaard, L., Udesen, J., 2002. Pattern dynamics of vortex ripples in sand: nonlinear modeling and experimental validation. Physical Review Letters 88 (23). 3.2.1, 3.2.3, 3.5 Andersen, K. H., 2001. 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DCW Ind., La Canada, California. 2.4.2, 2.4.2.1 Williams, J., Bell, P., Thorne, P., 2005. Unifying large and small wavegenerated ripples. Journal of Geophysical Research 110, 118. (document), 3.2.3, 3.4.8, 34, 5.2.1, 55, 6.2.1.5, 6.2.1.5, 630, 61, 62, 63, 7.1 Willis, D., Davies, M., Mogridge, G., 1993. Laboratory observations of bedforms under directional irregular waves. Canadian Journal of Civil Engineering 20 (4), 550563. 32 Yalin, M., 1977. Mechanics of Sediment Transport. Pergamon Press, Inc., New York, New York. 2.3.1.2, 4.3.2.2 Yalin, M., 1985. On the determination of ripple geometry. Journal of Hydraulic Engineering 111 (8), 11481155. 3.3.1.3 Yalin, M., Russell, R., 1962. Similarity in sediment transport due to waves. In: Proceedings of the 8th International Conference on Coastal Engineering, Mexico City, Mexico. pp. 151167. 32 Zedler, E. A., Street, R. L., 2001. Largeeddy simulation of sediment transport: currents over ripples. Journal of Hydraulic Engineering 127 (6), 444452. 2.3.1.2, 2.4, 2.4.2.2, 2.5.2, 4.3.1.2, 4.3.2.2, 4.3.2.2, 4.3.2.2 Zikanov, O., Slinn, D., Dhanak, M., 2002. Turbulent convection driven by surface cooling in shallow water. Journal of Fluid Mechanics 464, 81111. 4.3.1.2 BIOGRAPHICAL SKETCH Bret Maxwell Webb was born in Fort Myers, Florida, on the 8th of May, 1979. The son of Ross and Stephanie Webb, and the younger of two children, Bret was raised in the City of Palms by his parents, grand parents, and extended family. Spending most of his time on the waters of San Carlos Bay and Pine Island Sound either sailing or fishing for Redfish, Bret did manage to schedule in some academics, graduating from Bishop Verot Catholic High School in 1997. After receiving a B.S. in Civil Engineering from the University of Florida in 2001, Bret pursued a graduate degree in Coastal and Oceanographic Engineering, obtaining a Master of Science degree in 2004, also from the University of Florida. Upon completion of the master's program, Bret was awarded an Alumni Fellowship from the Department of Civil and Coastal Engineering at the University of Florida, providing him funding to seek a doctorate in his chosen field. Bret has been studying sand ripples for the past four years, specifically developing numerical models to explain their behavior and characteristics. In August 2007, Bret was offered a position in academia at the University of South Alabama in the Department of Civil Engineering. Since that time, he has served as an instructor, teaching undergraduate courses in the civil engineering curriculum. Upon successful fulfillment of the requirements necessary to obtain his Doctor of Philosophy, Bret will return to the University of South Alabama to accept a tenure track faculty position. Bret currently resides in Mobile, Alabama, with his wife of five years, Shannon Leigh. PAGE 1 SMALLSCALESEDIMENTTRANSPORTPROCESSESANDBEDFORMDYNAMICSByBRETMAXWELLWEBBADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2008 1 PAGE 2 Copyright2008byBretMaxwellWebb 2 PAGE 3 Itisunlikelythatthenaturalprocessofsedimenttransportbyowingwaterwillbeunderstoodinprecisedynamicaltermsintheforeseeablefuture." R.A.Bagnold 3 PAGE 4 ACKNOWLEDGMENTSIwouldliketoextendmygratitudersttomyfellowocematesfortheirsupport,encouragement,assistance,andfriendship.Robert,Jenn,Allison,Ty,andJessica,thanksforalwaysservingasasoundingboardformyideas,aswellasmycomplaints.Second,Iamgratefulfortheconstructivecriticismofmysupervisorycommitteemembers,thosethatIhaveoftensoughtoutforadvice,andparticularlyfortheguidanceandencouragementofmyadvisor,DonSlinn.MutluSumerisdeservingofspecialacknowledgmentfortheboundarylayerdatausedinthemodeldatacomparison.Last,Ioweagreatdebtofgratitudetomyfamily,especiallymywife,Shannon,asnoneofthiswouldhavebeenpossiblewithouttheirloveandconstantsupport. 4 PAGE 5 TABLEOFCONTENTS ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 LISTOFSYMBOLS .................................... 14 ABSTRACT ........................................ 19 CHAPTER 1INTRODUCTION .................................. 20 1.1Background ................................... 20 1.2Motivation .................................... 23 1.3Approach .................................... 26 1.4Outline ...................................... 28 2SEDIMENTTRANSPORTPROCESSES ..................... 30 2.1Introduction ................................... 30 2.2GoverningHydrodynamics ........................... 30 2.2.1Waves .................................. 30 2.2.2Currents ................................. 33 2.2.3WavesandCurrents ........................... 34 2.2.4TidesandTidalCurrents ........................ 35 2.3Transport .................................... 36 2.3.1Modes .................................. 36 2.3.1.1Bedload ............................ 37 2.3.1.2SuspendedLoad ....................... 40 2.3.2Regimes ................................. 43 2.4Turbulence ................................... 46 2.4.1Dynamics ................................ 47 2.4.2Closure .................................. 48 2.4.2.1RANSModels ......................... 48 2.4.2.2LargeEddySimulations ................... 49 2.4.2.3DirectNumericalSimulations ................ 49 2.5Models ...................................... 50 2.5.1Types ................................... 50 2.5.1.1TimeAveraged ........................ 50 2.5.1.2QuasiSteady ......................... 51 2.5.1.3SemiUnsteady ........................ 52 2.5.1.4Unsteady ........................... 53 2.5.2IntegratedApproaches ......................... 54 2.5.3Shortcomings .............................. 56 5 PAGE 6 3BEDFORMDYNAMICS ............................... 59 3.1Introduction ................................... 59 3.2BedformTypes ................................. 59 3.2.1RollingGrainRipples .......................... 61 3.2.2VortexRipples .............................. 63 3.2.3Characteristics .............................. 66 3.3Experiments ................................... 69 3.3.1Laboratory ................................ 69 3.3.1.1OscillatingTray ........................ 70 3.3.1.2OscillatingWaterTunnel ................... 70 3.3.1.3Flume ............................. 71 3.3.2Field ................................... 73 3.4RipplePredictors ................................ 74 3.4.1Clifton1976 .............................. 74 3.4.2Nielsen81 .............................. 74 3.4.3GrantandMadsen82 ....................... 76 3.4.4WibergandHarris94 ........................ 77 3.4.5Mogridgeetal.994 ......................... 78 3.4.6KhelifaandOuellet000 ....................... 79 3.4.7FaraciandFoti02 ......................... 80 3.4.8Williamsetal.2005 .......................... 81 3.4.9SoulsbyandWhitehouse06 .................... 81 3.5Models ...................................... 82 4MODELDESCRIPTION .............................. 88 4.1Introduction ................................... 88 4.2OneDimensionalLinearModel ........................ 88 4.2.1Hydrodynamics ............................. 89 4.2.1.1GoverningEquations ..................... 89 4.2.1.2Numerics ........................... 92 4.2.2SedimentTransport ........................... 93 4.2.3Morphology ............................... 93 4.3TwoandThreeDimensionalNonlinearModels ............... 93 4.3.1Hydrodynamics ............................. 95 4.3.1.1GoverningEquations ..................... 95 4.3.1.2TurbulenceClosure ...................... 98 4.3.1.3Numerics ........................... 104 4.3.1.4BoundaryConditions ..................... 109 4.3.2SedimentTransport ........................... 111 4.3.2.1Bedload ............................ 111 4.3.2.2SuspendedLoad ....................... 117 4.3.3Morphology ............................... 123 4.3.3.1FiniteDierenceMethods .................. 125 4.3.3.2FilteringTechniques ..................... 131 6 PAGE 7 5MODELEXPERIMENTS .............................. 146 5.1LinearModelExperiments ........................... 146 5.2NonlinearModelExperiments ......................... 147 5.2.1PhaseI:ModelValidation ....................... 147 5.2.2PhaseII:ModelCapabilities ...................... 149 5.2.3PhaseIII:SedimentTransportProcesses ............... 150 6RESULTS ....................................... 159 6.1LinearModelResults .............................. 159 6.2NonlinearModelResults ............................ 160 6.2.1PhaseI:ModelValidation ....................... 160 6.2.1.1HydrodynamicValidation .................. 161 6.2.1.2ComputationalGridTests .................. 163 6.2.1.3ModelDomainWidthTests ................. 165 6.2.1.4SedimentTransportSubmodelTests ............ 170 6.2.1.5MorphologyTests ....................... 173 6.2.2PhaseII:ModelCapabilities ...................... 183 6.2.2.1SteadyFlow .......................... 184 6.2.2.2SubcriticalFlow ....................... 185 6.2.2.3SheetowRegime ....................... 185 6.2.2.4RippleGrowth,Equilibrium,andDecay .......... 187 6.2.2.5SedimentSizeTests ...................... 193 6.2.3PhaseIII:SedimentTransportProcesses ............... 197 6.2.3.1TotalLoadGrowth ...................... 198 6.2.3.2TotalLoadDecay ....................... 203 6.2.3.3BedloadOnly ......................... 205 6.2.3.4SuspendedLoadOnly .................... 206 7CONCLUSIONS ................................... 274 7.1LiveBedModelEvaluation .......................... 274 7.2SedimentTransportandBedformDynamics ................. 278 7.3FutureApplications ............................... 281 APPENDIX AEXPERIMENTMATRIX .............................. 283 BFLUXTENDENCYPHASEDIAGRAM ...................... 285 REFERENCES ....................................... 289 BIOGRAPHICALSKETCH ................................ 304 7 PAGE 8 LISTOFTABLES Table Page 2{1Powerlawformulationsforbedloadtransport ................... 58 2{2Empiricalpickupfunctionsforsuspendedloadtransport ............. 58 3{1Listofcommonvaluesfororbitalripplelength ................... 85 3{2Summaryofeldandlaboratoryrippleexperiments ................ 86 3{3Parametersrelatedtobedformdynamics ...................... 87 3{4Ripplepredictorkey ................................. 87 5{1Meshparametersforgridcomparisons ....................... 155 5{2Meshparametersfordomainwidthcomparisons .................. 155 5{3Abbreviationkeyfortransportequationtests ................... 155 5{4Morphologytestsimulationmatrix ......................... 156 5{5Ripplepredictorcitationkeyforthemorphologytestsimulations ........ 156 5{6Expectedmorphologyresultsbasedonripplepredictors .............. 157 5{7PhaseIIexperimentmatrix ............................. 158 5{8SimulationparametersforthePhaseIIIexperiments ............... 158 6{1RipplepredictorRMSerrorformorphologytestsunderlabscaleows ..... 272 6{2RipplepredictorRMSerrorformorphologytestsundereldscaleows ..... 272 6{3Ripplepredictormodelperformanceindex ..................... 273 A{1Completeexperimentalmatrix ............................ 284 8 PAGE 9 LISTOFFIGURES Figure Page 1{1Picturesofsandripplesinlabandeldsettings .................. 29 2{1Velocityandstresstimeseriesforlinearandnonlinearwaves. .......... 57 2{2Distributionofuidandgrainshearstress. ..................... 57 3{1Schematicofvortexsheddingoverarippledbed .................. 84 3{2Eectsofbedformsonboundarylayerthickness .................. 85 4{1Schematicofonedimensionalbedloadmodel .................... 135 4{2Schematicofthethreedimensionalmodelingdomain ............... 135 4{3Modelcontrolvolumeswithvariablelocations ................... 136 4{4Schematicofvariablemeshscaling ......................... 136 4{5Meshclusteringandvariablescaling ........................ 137 4{6Acomparisonofbedshearstressformulations ................... 138 4{7EstimationofcriticalShields'curveby Brownlie 1981 .............. 139 4{8EstimationofcriticalShields'curveby vanRijn 1993 .............. 139 4{9Particleforcesactingonlongitudinalandtransverseslopes ............ 140 4{10Gravitationalforcesonasphere ........................... 141 4{11Behaviorofbedloadtransportequations ...................... 141 4{12Behaviorofsedimentpickupfunctions ....................... 142 4{13Modicationofrelativesettlingvelocitybyconcentration ............. 142 4{14Sedimentcontrolvolumeandtransportschematic ................. 143 4{15Amplicationfactorsforthetwodimensionallowpassmorphologylter .... 144 4{16Amplicationfactorsforthehybridmorphologylter ............... 145 5{1Horizontalandverticalgridspacingforgridcomparisons ............. 152 5{2Modeldomainsusedinthedomainwidthcomparisontests ............ 153 5{3SchematicofthemodeldomainusedforthePhaseIIIsimulations ........ 154 6{1Timestacksofbedelevationforthe1DHlinearmodelexperiment ........ 208 9 PAGE 10 6{2Timeevolutionofrippleheight,wavelength,andsteepnessforthe1DHlinearmodelexperiment ................................... 209 6{3Modeldatahydrodynamiccomparison ....................... 210 6{4Modeldatacomparisonofboundarylayerthickness ................ 211 6{5Assessmentofhydrodynamicmodelerror ...................... 212 6{6Computationalbudgetforgridresolutiontests ................... 213 6{7Velocitystandarddeviationsforgridcomparisons ................. 214 6{8CumulativeaverageturbulentkineticenergyTKEforgridcomparisons .... 215 6{9PhaseandvolumeaveragedTKEforgridcomparisons .............. 215 6{10Phaseandy)]TJ/F15 11.9552 Tf 9.2985 0 Td[(averagedu)]TJ/F15 11.9552 Tf 9.2985 0 Td[(velocityprolesforgridcomparisons ......... 216 6{11Phaseandy)]TJ/F15 11.9552 Tf 9.2985 0 Td[(averagedv)]TJ/F15 11.9552 Tf 9.2985 0 Td[(velocityprolesforgridcomparisons ......... 217 6{12Phaseandy)]TJ/F15 11.9552 Tf 9.2985 0 Td[(averagedw)]TJ/F15 11.9552 Tf 9.2985 0 Td[(velocityprolesforgridcomparisons ......... 218 6{13Computationalbudgetforthreedimensionalgridtests .............. 219 6{14CumulativeaverageturbulentkineticenergyTKEforthreedimensionalgridcomparisons ...................................... 220 6{15PhaseandvolumeaveragedTKEforthreedimensionalgridcomparisons .... 220 6{16Phaseandvolumeaveragedvelocitycomponentsforgridcomparisons ..... 221 6{17AverageFouriertransformsofv)]TJ/F15 11.9552 Tf 9.2985 0 Td[(velocityforgridcomparisons .......... 222 6{18Spatiallyaveragedstandarddeviationofbedelevationinthethirddimension .. 223 6{19Comparisonsofequilibriumrippleheightforgridcomparisons .......... 223 6{20AverageFouriertransformsofbedelevationforgridcomparisons ........ 224 6{21Eectsofsedimenttransportsubmodelsonequilibriumripplecharacteristics .. 225 6{22Conservationofsedimentmass ........................... 226 6{23Bedformstatisticsinlabscaleows ......................... 227 6{24FourieranalysistimestackofbedformstatisticsforHL1v1 ............ 228 6{25Ripplepredictorcomparisonsforlabscalemorphologytests ........... 229 6{26Bedformstatisticsineldscaleows ........................ 230 6{27FourieranalysistimestackofbedformstatisticsforMF2v2andHF1v1 ..... 231 10 PAGE 11 6{28Ripplepredictorcomparisonsforeldscalemorphologytests ........... 232 6{29Comparisonofmodelresultstotheripplepredictorequationsof FaraciandFoti 2002 ......................................... 233 6{30Comparisonofmodelresultstotheripplepredictorequationsof Nielsen 1981 KhelifaandOuellet 2000 ,and Williamsetal. 2005 .............. 234 6{31Comparisonofmodelresultstotheripplepredictorequationsof Mogridgeetal. 1994 ......................................... 235 6{32Bedelevationtimestackforsteadyowtest .................... 236 6{33Steadyowbedshearstressandsedimenttransport ................ 237 6{34Bedelevationtimestackforsubcriticalowtest .................. 238 6{35Averagebedshearstressandbedloadtransportinsubcriticalow ........ 238 6{36Bedelevationtimestackforsheetowsedimenttransport ............. 239 6{37Timeseriesofbedformstatisticsinthesheetowregime ............. 239 6{38Energydensitytimestackforthesheetowtest ................... 240 6{39Isocontoursofsedimentconcentrationinthesheetowregime .......... 240 6{40Averagesedimentconcentrationproleinthesheetowregime .......... 241 6{41Bedelevationtimestacksforripplegrowth,equilibrium,anddecay ........ 242 6{42Timestackofripplespectraduringgrowthfromatbed ............. 243 6{43Ripplestatisticsforgrowth,equilibrium,anddecay ................ 244 6{44Timeevolutionofrippleheightsduringgrowth,equilibrium,anddecay ..... 245 6{45Bedelevationtimestacksforripplecoarseningandbifurcation .......... 246 6{46Timestacksofripplespectraduringripplewavelengthsaturation ......... 247 6{47Ripplestatisticsduringheightandwavelengthevolution ............. 248 6{48Eectsofhinderedsettlingonmodelpredictionsofrippleheightandlength .. 249 6{49Modicationofaverageconcentrationproleduetohinderedsettling ...... 250 6{50Timeseriesofbedformstatisticsfornegravelinsubcriticalow ........ 250 6{51Phaseaveragedstressandtransportfornegravelinsubcriticalow ...... 251 6{52Timestacksofbedelevationfornegravelinastrongow ............ 251 11 PAGE 12 6{53Bedformstatisticsfornegravelinastrongow ................. 252 6{54Phaseaveragedstressandtransportfornegravelinastrongow ....... 252 6{55Timestacksofbedelevationformediumsiltinaweakow ............ 253 6{56Timeevolutionofrippleheightandlengthformediumsiltinaweakow .... 253 6{57Phaseaveragedstress,transport,andsedimentconcentrationformediumsiltinaweakow ...................................... 254 6{58Timestacksofbedelevationformediumsiltinastrong,highlyconcentratedow 255 6{59Timeevolutionofbedformheightandlengthformediumsiltinastrongow .. 255 6{60Phaseaveragedstress,transport,andsedimentconcentrationformediumsiltinastrong,highlyconcentratedow .......................... 256 6{61Suspendedsedimentphasediagram ......................... 257 6{62Timeevolutionofrippleheight,wavelength,andsteepnessduringripplegrowthandequilibrationTL2v2 .............................. 258 6{63TimestacksofbedelevationduringripplegrowthTL2v2 ............ 259 6{64CumulativeaveragedandcorrectedtransportuxesforTL2v2 .......... 259 6{65Phasedependenceofbedload,entrainment,anddepositionduringripplegrowthandequilibrium .................................... 260 6{66Phasedependenceofcorrectedbedloadandsuspendedloaduxesduringripplegrowthandequilibrium ................................ 261 6{67Transportuxtendencyphasediagramsforripplegrowthandequilibrium ... 262 6{68TimeevolutionofripplecharacteristicsduringrippledecayTL2v2d ...... 263 6{69CumulativeaveragedandcorrectedtransportuxesforTL2v2d ......... 264 6{70Phaseaveragedandcorrectedtransportuxesduringrippledecay ........ 264 6{71PhasedependenceoftransportconstituentsduringrippledecayforTL2v2d ... 265 6{72Transportuxphasediagramforrapidrippledecay ................ 266 6{73Timeevolutionofripplecharacteristicsthroughbedloadtransportonly ..... 267 6{74PhasedependenceofbedloadtransportduringripplegrowthforBL2v2 ..... 268 6{75Timeevolutionofripplecharacteristicsthroughsuspendedloadtransportonly 269 6{76PhasedependenceofverticalsedimentuxesduringrippledecayforSL2v2 ... 270 12 PAGE 13 6{77TimeevolutionofscaledrippleheightsforPhaseIIIsimulations ......... 271 B{1Derivationoftheuxtendencyphasediagramofbedformgrowth,equilibrium,anddecay ....................................... 288 13 PAGE 14 LISTOFSYMBOLSGreeksymbols longitudinalbedslope. transversebedslope. waveperiodparameter. t timestep. tf morphologyhybridlteringtimestep. xi xgridspacing. yj ygridspacing. zk zgridspacing. boundarylayerthickness. lterwidth. d displacementboundarylayerthickness. d relativeorbitalexcursion. ij Kroneckerdelta. freesurfacedisplacement. r rippleheight. s smoothingcoecient. vonKarmanconstant. mix mixedlayerthickness. r ripplewavelength. dynamicviscosity. kinematicviscosity. t eddyviscosity. angularfrequency. sedimentangleofrepose. b nondimensionalbedloaddischarge. 14 PAGE 15 p nondimensionalentrainmentrate. v velocitypotential. mobilitynumber. densityofwater. s densityofsediment. mix densityofmixedlayer. S Schmidtnumber. rippleheightstandarddeviation. ripplelengthstandarddeviation. r ripplesteepness. b bedshearstress. g gravitationalparticleshearstress. b;max maximumbedshearstress. Rij residualstresstensor. rij anisotropicresidualstresstensor. nondimensionalShieldsstress. variablenitedierencecoecient. c nondimensionalcurrentShieldsstress. w nondimensionalwaveShieldsstress. 2:5 grainroughnessShieldsparameter. cr criticalShieldsparameter. cr criticalShieldsparameterforahorizontalbed. cw wavecurrentangle. 'n primitivevariableattimeleveln.Genearaloperators lteredvariable. hh'ii phaseaveragedquantity. 15 PAGE 16 h'i timeaveragedquantity. spatiallyaveragedquantity. '0 residualvariable.Romansymbols `S Smagorinskylengthscale. ^zb lowpasslteredbedelevation. rQb signcorrectedbedloadtransportux. rQs signcorrectedsuspendedloadtransportux. C lteredvolumeconcentration. Cb lteredvolumeconcentrationabovebed. Sij lteredrateofstrain. S characteristiclteredrateofstrain. PAGE 17 d graindiameter. d dimensionlessgrainsize. d50 mediangraindiameter. Et entrainmentrateofbedmaterial. f coecientoffriction. Fn Fluxevaluatedattimeleveln. fb coecientofresistingfriction. Fg downslopecomponentofparticleweight. fw wavefrictionfactor. Fw buoyantweightofparticle. Fx bodyforce. f2:5 grainroughnessfrictionfactor. fmix coecientofappliedfriction. g gravitationalacceleration. h waterdepth. H waveheight. k turbulentkineticenergy. kr residualkineticenergy. ks Nikuradseroughnesslength. kw wavenumber. kx wavenumberinx)]TJ/F15 11.9552 Tf 9.2985 0 Td[(direction. ky wavenumberiny)]TJ/F15 11.9552 Tf 9.2985 0 Td[(direction. n normaldirection. np porosityofbedmaterial. p pressure. Prt turbulentPrandtlnumber. Qb volumetricbedloadtransportrate. 17 PAGE 18 Qs volumetricsuspendedloadtransportrate. Qt totalvolumetricsedimenttransportrate. s specicgravityofsediment. S nondimensionalsedimentparameter. T waveperiod. t time. TKE turbulentkineticenergy. u frictionvelocity. U uidvelocityatedgeofboundarylayer. U1 freestreamvelocity. ub nearbeduidvelocity. Uc currentvelocity. ui velocitycomponent. Umax maximumvelocityamplitude. umix mixedlayervelocity. us sedimentvelocity. ws hinderedsettlingvelocity. wso constantsettlingvelocity. xi spatialdirectioncomponent. z+ normalizedwallunit. zb verticallocationofbed. zo referenceheight. 18 PAGE 19 AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySMALLSCALESEDIMENTTRANSPORTPROCESSESANDBEDFORMDYNAMICSByBretMaxwellWebbMay2008Chair:DonaldN.SlinnMajorDepartment:CoastalandOceanographicEngineeringThegenerationofsmallscalesedimentarystructuresinthecoastalenvironmentisacomplexprocessthatoccursoverawideseparationofscalesinbothtimeandspace.Thesebedformsareubiquitousfeaturesofthenearshoreregion,andyetspecicinformationregardingtheirbehaviorandcharacteristicsisstilllacking.Specically,itisunclearwhetherthebedloaddominatedprocessesofthelinearregimeareasequallyresponsibleforthegenerationofbedformsinthenonlinearregime,whereowseparation,andsubsequentvortexformation,tendtogovernthedynamicsofthebottomboundarylayer.Whileasimpleonedimensionalmodelisderivedandusedtoexplainincipientbedformgrowthinthelinearregime,suchanapproachisnotwellsuitedataddressingthecomplexitiesofthewavebottomboundarylayer.Utilizinganewthreedimensionalphaseresolvinglivebedmodel,wesimulatethedynamicsofbedforms,suchassandripples,inthenonlinearregime.Throughfortythreeindependentsimulations,themodelhasbeenfoundtoreproduceoscillatoryboundarylayerow,aswellasprovideaccuratepredictionsofripplegeometryinbothlabandeldscaleows.Modelresultsconrmthatinthelinearregime,bedformgrowthispromotedpurelythroughbedloadsedimenttransport,butinertialpropertiesofthesedimentareequallyasimportant.Inthenonlinearregime,bedformgrowthisalsodominatedbybedloadtransport;however,theentrainmentanddepositionofbedmaterialplaysanimportantroleinmaintainingrippleequilibrium,whereasitismostlyresponsibleforrippledecay. 19 PAGE 20 CHAPTER1INTRODUCTION1.1BackgroundBedforms,suchassandripples,areubiquitousfeaturesofthecoastalzone.Andalthoughtheyhavebeenthesubjectofnumerousinvestigationstheoretical,experimental,andnumericalalikedatingbacktothelatenineteenthcentury Hunt 1882 ; Darwin 1883 ,denitiveinformationabouttheirdynamicsinthecoastalenvironmentremainslimited.Thisisnottosaythattheoryandunderstandinghavenotbeenmarkedlyadvancedoverthelastcentury,butitunderpinsthenecessityforcontinuedresearchonbedformdynamicsatscalesbothlargeandsmall.Morphologicalbedformfeaturesoccurringinnaturemayrangeinscalesfromafewcentimetersinheightandtensofcentimetersinlengthforwavegeneratedsandripples,tolargersandwavefeatureslikedunesandmegaripplesthatcanhaveheightsofafewmetersandlengthsontheorderoftensofmeters.Regardlessoftheirsize,bedformsplayanimportantroleinboththeenergyandsedimentbudgetsofthenearshoreenvironment.Bedformshavebeenfoundtostronglyaect: energydissipationinthewavebottomboundarylayer; sedimenttransportcharacteristics;and acousticpropertiesoftheseaoor.Whilemomentumtransferaboveplanebedsisduemostlytoturbulentdiusion,rippledbedsinduceowseparationresultinginorganizedvortices,orcoherentmotions,thatdominatemomentumtransferinthewavebottomboundarylayerWBBL Marin 2004 .Thesecoherentmotionsareeectiveindissipatingwaveenergyinanearbedboundarylayerthatisapproximatelytwotimestherippleheight vanderWerfetal. 2006 .Abovethisnearbedlayer,coherentmotionsbreakdownandarereplacedbyrandomturbulence DaviesandVillaret 1997 Tunstall 1973 and TunstallandInman 20 PAGE 21 1975 estimatethatasmuchas10%ofwaveenergymaybedissipatedthroughtheprocessofvortexformation.Waveenergyisalsoeectivelydissipatedthroughfrictionasbedformsaectthehydraulicroughnessofthebedforwavesandcurrents SoulsbyandWhitehouse 2006 .Bedformheightcontrolstheeectivebottomroughness FaraciandFoti 2002 ,whichisofpracticalengineeringimportance.Parametrizationofhydraulicroughness GrantandMadsen 1982 arecommonlyappliedtoengineeringwaveandsedimenttransportmodelsinordertoaccountfortheadditionalenergydissipationthattakesplaceintheWBBL.Suchpracticepermitstheeectsofbedformsoncurrentsandwavestobetreatedinabulkmanner,ratherthanhavingtoaccountforthemexplicitlyinthemodelingapproach.TheparametrizationandapplicationofroughnesstomodelsaccountsforthethickeningoftheWBBLinthepresenceofbedforms WibergandHarris 1994 ,wherebyenergyisdissipatedthroughformdrag,skinfriction,andturbulencedampingduetoanintenselayerofnearbedsedimenttransport GrantandMadsen 1982 .Theamountofdissipationduetobedformsisnottrivial;eldobservationsby Ardhuinetal. 2002 indicatethattheformdragoverlargeripplesiseectiveinwaveattenuationacrossthecontinentalshelf.Accuraterepresentationsofequivalentbottomroughness,then,areofparticularimportanceforunderstandingtheenergybudgetofthenearshoreenvironment.However,priorcharacterizationsofroughnessandtheirsubsequentimplementationinnearshoremodelsoftenneglectchangesinbedformsize,shape,andorientation,anddonotaccountforphaselageectsinducedbyowseparationintheboundarylayer.Bedformsaremanifestationsofdierentialsedimenttransportneartheseabed,andcanoftenbefoundinuniquepatternsalongtheseaoorasseeninFigures 1{1a { 1{1d .Oncelargeenoughtoinduceowseparation,theyhaveaprofoundimpactonsubsequentsedimenttransport.Enhancedsedimentsuspensionoverrippledbedsnotonlyincreasespotentialsedimenttransportinthebottomboundarylayer,butaectsthebackgroundowaswell.Straticationofsedimentinthewatercolumnresultsinamodicationofthe 21 PAGE 22 velocityprole Coleman 1981 ,turbulencedamping McLean 1991 ,andanincreaseinboundaryroughness ParkerandColeman 1985 .Therefore,sedimentsuspensionoverrippledbedsservesasanadditionalmechanismforthedissipationofenergyinthebottomboundarylayer.Accordingto Nakatoetal. 1977 ,suspensionprocessesoverrippledbedsaredominatedbytheformation,ejection,andmotionofeddies.Organizedvorticesthatformintheleesideofripplesentrainsedimentfromtroughsorvalleysineachsuccessivehalfcycleofwaveforcing.Immediatelyfollowingowreversal,thesedimentladenvorticesareejectedintothemainowabovethebedformswherethesedimentissubsequentlyadvectedbythelocaluidvelocityeld.Laboratoryexperimentsby vanderWerfetal. 2005 suggestthatsuchbehaviorresultsinthreedistinctpeaksinthenearbedconcentrationasafunctionofwavephase.Comparedtosuspensioneventsoveraatbed,elddatasuggestthatsedimentladenvorticesinthewakeofbedformsalterssedimentsuspension Gallagheretal. 1998 ,andmayalsoenhancethephaselagbetweensuspensionandtransport InmanandBowen 1963 .Furthermore, vanderWerfandRibberink 2004 proposethatrippleinducedphasedierencesbetweenpeaksuspendedsedimentconcentrationsandpeakuidvelocitiesresultinnetsedimenttransportratesdirectedoshoreundersurfacegravitywaves.Suchphasedierences,however,arenotaccountedforincommonphaseaveraging,coupledhydrodynamicandsedimenttransportmodels.Thismaysuggestwhysomecrossshoresedimenttransportmodelsfailwhenoscillatorycurrentsarelargerthansteadycurrents Gallagheretal. 1998 .Inrecentyears,theNavyhasbecomeincreasinglyinterestedintheacousticpropertiesoftheseaoor,specicallywithrespecttotheabilityofsonardevicestodetectbothemergentandsubmergedmunitionsinthebattlespaceenvironmentBSE.Bedformsaecttheacousticresponseforsonarapplications,eitherenhancingorinhibitingthepenetrationofthesonarsignalintothebed SoulsbyandWhitehouse 2006 .FromdatacollectedduringtheSAX99eldexperiment, Piperetal. 2002 showthatboththe 22 PAGE 23 rippleheightandwavelengthstronglyaectthelevelofsonarpenetrationforsubcriticalgrazingangles.Additionally,theorientationoftheripplesrelativetotheincidentsonareldisofimportance.Theresultsof Piperetal. 2002 indicatethatsubsurfaceimagingisenhancedwhenthesonarpropagationdirectionisperpendiculartothemeandirectionofripplecrests,wherebytheamountofacousticenergyscatteredbythebedformsintothesedimentisincreased.1.2MotivationInthelinearregimeofbedformgrowthwhererollinggrainripplespersistsedimenttransporttakesplacepurelythroughbedloadprocesses.Thebedloadtransportmodeconsistsofparticlesrolling,sliding,andsaltatinginsmallleapsontheorderofafewgraindiametersabovethebed.Thelinearregimeisdominatedbyfrictionandinertiallagsbetweenuidforcingandparticletransport.Theabsenceofsuspendedsedimenttransportinthelinearregimeandatsubcriticalthresholdvalueshaveleadmanytopostulatethatbedformsaregeneratedpurelythroughbedloadtransport.Itisunclear,however,thatthisstatementholdstrueinthenonlinearregimewherecoherentmotionsintheboundarylayerpromotesuspendedloadtransportandinducephasedierencesbetweenuidforcingandsedimententrainment.Thenonlinearregimecontainsbothbedloadandsuspendedloadtransportmodes,buttheirrelativecontributionstoripplegrowth,equilibration,anddecayareunknown.Additionally,thecontributionsofbedloadandsuspendedloadtransportmodesmayvaryasafunctionofripplepositionproleandplanformanduidforcing.Thequantitativerolesofconstructiveuidforcinganddestructivegravitationalforcesingeneratingsedimentarystructuresarenotimmediatelyevident.Itwouldbebenecial,therefore,toinvestigatethecontributionsofvarioustransportmodestobedformdynamics,andtobetterunderstandtherolesofconstructiveanddestructiveforcesduringripple,growth,equilibration,anddecay. 23 PAGE 24 Fieldandlaboratorymeasurementtechniquesareincapableofgatheringinsitudatathatclearlydistinguishonetransportmodefromanother.Suchmethodsoftenrequiretheassumptionofathresholdbasedonvolumetricconcentrationtoseparatethetransportmodesafterthedatahavebeencollected.Additionally,measurementtechniquesforbedloadandsuspendedloadtransportarecommonlyinvasive,therebyalteringthehydrodynamicandsedimenttransporteldsthattheyareattemptingtoquantify.Numericalsimulationsmayprovideusefulinsightintosedimenttransportprocessesinsmallscalebedformdynamicswherephysicalexperimentationiscurrentlynotpossible.ThroughtheUniversityofFlorida,weareparticipatingintheOceofNavalResearch'sONRintensivestudyofSandRipplesontheInnerShelfFY0408,aprojectinvolvingfteenprincipalinvestigatorsfromacrossthenationRipplesDRI.Thisinterdisciplinaryprojectcombineseldobservationswithlaboratoryexperimentsandnumericalsimulationsfromthebiological,geological,andoceanographicsciences.Theprimarygoalsoftheproject,asidentiedbyONR,areto 1. measureandmodelmorphology; 2. investigatemorphologicalresponsetoforcingperturbations; 3. measureandmodelratesofbiodegradation; 4. measureandmodeltheeectsofgrainsizedistribution;and 5. understandtheroleingeneratingsedimentarystructures.Ourparticipationintheprojectprimarilyinvolvesthersttwogoalsoutlinedabove,aswellasthelast.Inordertoaddresstheseobjectives,weseektodevelopaphysicsbasedmodelthatcoupleshydrodynamicsandmorphologybyupdatingthebedlevelatevery,ornearlyevery,hydrodynamictimestep.ThereviewofnoncohesivesedimenttransportprocessesandbedformdynamicsthatfollowsinChapters 2 and 3 shouldprovideadequateguidancefordeterminingasuitablemodelframework. 24 PAGE 25 AsidefromthegoalsoftheONRRipplesDRI,theprimaryobjectiveofthisresearchistoinvestigatesedimenttransportcharacteristicsduringripplegrowth,equilibration,anddecay.Inparticular,wehopetoprovideanswerstothefollowingquestions: 1. Whataretherelativecontributionsofbedloadandsuspendedloadtothegenerationorobliterationofsedimentarystructures? 2. Aretherespeciclocationsintherippleprole/planformwhereonetransportmodedominatesovertheother? 3. Aretherespecictimesasafunctionofphasewhenonemodedominatesovertheother? 4. Whatarethedominantrolesof:bedload,suspendedload,andgravitationalforces?Toaddressourobjectives,weproposethedevelopmentofanentirelynewmodelingsystemcapableofsimulatingphaseresolvingsmallscalesedimenttransportandmorphology.Capabilitiesofthemodelingsystemwillbeassessedbyevaluatingthehydrodynamicsthroughmodeldatacomparisons,andbyalsoperformingsimulationsofbedformequilibrationunderavarietyofscenariose.g.growth,equilibration,decay,coarsening,bifurcation,steadyows,shortandlongperiodows,highlyconcentratedows,andextremesedimentsizes.Inordertoanswerthequestionsposedabove,simulationsofbedformgrowth,equilibration,anddecaywillbeperformedwith bedloadandsuspendedload; bedloadonly;and suspendedloadonly.ThedesignofthemodelingsystemassumesthatbedloadtransportispredictedusingEinstein'sdimensionlessbedloaduxandcommonpowerlawformulations,andthatsuspendedloadoriginatesthroughanentrainmentofsedimentfromthebedusingsemiempiricalpickupfunctionsavailableinexistingliterature.Thus,oursystemallowsustoalternatelyturnbedloadandsuspendedloadonorobysimplysettingbedloadtransportrates,orentrainment/deposition,tozero,respectively.Wefurtherassume 25 PAGE 26 thatthebedloadregimeiscomprisedoftwotypesoftransport:oneduetouidforcing;andanotherduetogravitationalforces.Doingsopermitsustodeterminetherolesofconstructiveanddestructiveforcesindependently,therebyallowingustobetterunderstandtheirrolesingeneratingsedimentarystructures.Thedevelopmentofthismodelingsystemisasecondaryobjectiveofthisdissertation,andisnecessarytoaddressourscienticquestionsposedabove.Specicdetailsaboutthesystemareoutlinedbelow,aswellasinthefollowingchapters.1.3ApproachModelingsedimenttransportcontinuestobeachallengeforuiddynamicists.Indeed,muchoftheuncertaintyinuidsedimentmodelscontinuestoliemainlywiththeparticulatephase,whereastheuidhydrodynamicsarecomparativelywellunderstood.Sedimenttransportmodelsfallmainlyintotwobroadcategories: 1. timeorphaseaveraged;and 2. unsteadyorphaseresolving.Variationsonphaseaveragedandphaseresolvingmodelsincludequasisteadyandsemiunsteadymodels,respectively.Eachofthesemodelshasitsownbenetsanddeciencies,makingsomemoreusefulincertainsituationsthanothers.Briey,timeorphaseaveragedsedimenttransportmodelseectivelyintegrateoutintrawaveprocessesbyconsideringwaveaveragedvaluesofuidvelocity,sedimentconcentration,andsedimenttransport Bijker 1971 .Socalledquasisteadymodelsassumethatthetransportrateisproportionaltotheinstantaneousnearbedoweldraisedtosomepower Bailard 1981 ,butonlyprovideanaveragetransportrateoverthewaveperiod.Semiunsteadymodelsincorporateadditionalcomplexityintotheformulationoftransportratesbyconsideringphaselagseitherthroughparametrizations Nielsen 1988 ; DohmenJanssen 1999 orbybreakingafullwaveperiodintowavehalfcycles DibajniaandWatanabe 1992 .Whilecomputationallyintensive,themostrobustapproachistodescribethesedimenttransportandhydrodynamicsascoupled,timedependentprocesses.Veryfewofthesemodels 26 PAGE 27 existoutsideofthecomputationalresearcharena,astheirpracticalityforengineeringapplicationsisstillsomewhatlimitedduetotheircomplexity[see DrakeandCalantoni 2001 ; Gessleretal. 1999 ;and Lesseretal. 2004 ].Toaddressanumberofrelevantengineeringproblemsinvolvingsedimenttransportprocessesfromstructureinducedscourtocoastalerosionitwouldbeadvantageoustohaveaexible,robustmodelcapableofresolvingtimedependentbedmorphologyundervariousforcingconditions.SuchamodelingsystemcouldalsosatisfysomeoftheprimaryobjectivesoftheONRRipplesDRIprojectoutlinedabove.Hereweseektodevelopandevaluatetheskillofacoupleduidsedimentnumericalmodelthatresolvesinstantaneoushydrodynamics,sedimenttransport,andtimedependentbedmorphology.Itisnotevidentthatsuchamodelexistsatthelevelofdetailproposed.Thedesiredphaseresolvingmodelwouldbecapableofbothtwoandthreedimensionalsimulationsof1steadyow,oscillatoryow,combinedoscillatoryandsteadyow,andsurfacegravitywaves.WhilethetimeintegrationofthegoverningequationswillbelimitedtodurationsofO00s,resolutionofphysicallengthswillrangefromO0)]TJ/F23 7.9701 Tf 6.5865 0 Td[(4mtoOm.Suchdetailedtreatmentofhydrodynamicsnearsolidboundariesshouldpermittheresolutionofhighintensityturbulentuctuationsthatinitiatesedimenttransport.IntegrationofthebedlevelthroughtimewillbegovernedbythesedimentcontinuityExnerequation,wheregradientsinthebedloaduxandthecompetitionbetweenentrainmentanddepositionofsuspendedsolidsprovideanestimationoftheinstantaneousbedlevel.Sedimenttransportinthemodelwillbeestimatedusingbulk,empiricalformulationsavailableinexistingliterature.Naturalfeedbackbetweenmorphologyandthehydrodynamicoweldrepresentstheonewayuidsedimentcouplinginthemodel.Twowaycouplingofthehydrodynamicsandlocalsuspendedsedimentconcentrationmaybeconsideredinthefuture. 27 PAGE 28 1.4OutlineWhatfollowsisageneraloverviewofknowledgepertainingtononcohesivesedimenttransportandbedformdynamics,castinamannerthat,hopefully,underscorestheneedforahydrodynamiclivebedmodelhavingthegeneralcharacteristicsoutlinedpreviously.AbriefsummaryofnoncohesivesedimenttransportprocessesisgiveninChapter 2 ,withadiscussiononthegoverninghydrodynamicsorforcing,modesoftransport,theroleofturbulence,andadescriptionofacceptedmodels.Chapter 3 providesinformationaboutgeneralbedformpropertiesanddynamics,withparticularattentiongiventotheirclassicationandcharacteristics,mechanismsforgrowth,asummaryofrelevantlaboratoryandeldexperiments,semiempiricalsolutionsfortheirgeometricalproperties,andanoverviewofmodelsrangingfromsimpletocomplex.AdescriptionofnewlydevelopedmodelsisprovidedinChapter 4 andoutlinesgoverningequations,aswellasmethodologiesfortheirimplementationinthemodelingsystem.ThemodelexperimentsareoutlinedinChapter 5 ,andthosesimulationresultsarepresentedanddiscussedinChapter 6 .ConcludingstatementsareprovidedinChapter 7 ,alongwithsomewordsonpotentialfutureapplicationsofthenewlycreatedmodelingsystems. 28 PAGE 29 aUSAWaveBasin bUSAWaveBasin cYucatanPeninsula,Mexico dYucatanPeninsula,MexicoFigure1{1.Picturesofsandripplesinlab[aandb]andeld[candd]settings.ThelabpicturesweretakenintheUniversityofSouthAlabamaWaveBasinafterdraining,whiletheeldpicturesweretakeninadepthofapproximately5mofwaterotheEastcoastoftheYucatanPeninsulainMexico. 29 PAGE 30 CHAPTER2SEDIMENTTRANSPORTPROCESSES2.1IntroductionSedimenttransportprocessesinthecoastalenvironmentarehighlycomplex.Transportationisbothinitiatedandsustainedbyhydrodynamicforcingontheseabedandwatercolumn.Whiletheknowledgeofsedimenttransportprocesseshasbeensignicantlyadvancedinidealized,laboratoryenvironments,detailedinformationaboutsuchprocessesremainsambiguousunderthestochasticforcingthatispresentinnature.Thefollowingsectionsoutlinethegoverninghydrodynamicsthatdrivesedimenttransportinthecoastalenvironment,provideadescriptionofcommontransportmodesandregimes,discussthemutualrelationshipsbetweentransportandturbulence,anddeliveranoverviewofvarioussedimenttransportmodels.2.2GoverningHydrodynamics2.2.1WavesSurfacegravitywavesrepresentthemostdominantmobilizingforceforsedimentincoastalareas vanderWerf 2004 .Insomecasestheorbitalvelocitiesunderthesewavescanbequitelargeneartheseabed,especiallywhenwavessteepenastheyentershallowerwater.Furthermore,realisticsurfacegravitywavesarenotlinear,typicallyhavingsteeperpeaksandattertroughs.Aresultofthisnonlinearproleisaninherentasymmetryintheorbitalvelocityeldwithshortduration,highintensityvelocitiesdirectedshorewardunderwavecrestsandlongerduration,lowerintensityvelocitiesdirectedseawardunderthebroad,attroughsseeFigure 2{1 a.Thisinequalityintheorbitalvelocitytimeseriesbecomesveryimportantforthedeterminationofnetsedimenttransportrates[see RibberinkandAlSalem 1990 ; Gallagheretal. 1998 ;and Elgaretal. 2001 ].Asdiscussedearlier,thewavebottomboundarylayerWBBLisanimportantmechanismforenergydissipationandinuencesthemagnitudeanddirectionofsedimenttransportinthenearshore.Thisboundarylayerdevelopsneartheseabedinresponse 30 PAGE 31 tofrictionbetweenitandtheuid.Ageneralexpressionforboundarylayerthicknessisgivenby/p T{1whereisthekinematicviscosityandTisthewaveperiod.Thethicknessoftheboundarylayerisalsoaectedbytheroughnessoftheboundary.WaveboundarylayersoversmoothboundariesmaybeofOmmwhilethethicknessoverroughboundaries,likearealseabed,increasestoOcm.Duetoinertialeectsintheboundarylayerslowermovinguid,thenearbedoscillatoryowisoftenoutofphasewiththefreestreamforcing.Thisresultsinanearbedphaseleadofapproximately45forlaminarowand15forturbulentboundarylayers,withrespecttothefreestreamforcing.Thephaseleadofnearbedoscillatoryowhasprofoundsignicancefortheestimationofsedimenttransport.Commonsedimenttransportformulationsincorporatesomethresholdforincipientmotion,suchasShields'parameter Shields 1936 ,thatrequiresthebedshearstresstoexceedsomecriticalvalue.Ageneralexpressionofthebedshearstressisb=CfUjUj{2whereistheuiddensity,Cfisacoecientoffrictiondrag,andUisrepresentativeofthefreestreamforcing.Figure 2{1 bshowsarepresentationofbedshearstressunderlinearandnonlinearwavesusingEquation 2{2 with=1025kg/m3andthecanonicalvalueof0.0025forCf Riviereetal. 2004 .Thistypeofformulationforbedshearstressistypicalofverygeneralmodelsanddoesnotincludeanyinformationaboutphasedierencesbetweenowatthebedandowoutsideoftheboundarylayer.Itisevident,then,thatapplicationofEquation 2{2 forunsteadyowintheboundarylayerwillincorrectlypredictthephaseofsedimentmobilizationwithrespecttotheappliedforcing.Underpurelyhorizontal,regularoscillatoryowthisisoflesssignicanceasthenetsedimenttransportfromcycletocyclewillbezero.Thephasedierenceformobilizationandentrainmentundernonlinearsurfacegravitywaves,however,issignicantasitcandictate 31 PAGE 32 thedirectionofnettransport[see Elgaretal. 2001 and vanderWerfandRibberink 2004 ].AnalternativetothegeneralbedshearstressformulagivenbyEquation 2{2 ,commonlyemployedinphaseaveragedorquasisteadytransportmodels,istoconsiderthemaximumbedshearstressforanindividualwavebasedonawavefrictionfactorfw.Suchamethodwasproposedby Jonsson 1966 :b;max=1 2fwu2b{3whereubisthenearbedorbitalvelocity.Thewavefrictionfactorof Swart 1974 isapplicableforfullydeveloped,roughturbulentowandisgivenbyfw=exp"5:213ks A0:194)]TJ/F15 11.9552 Tf 11.9552 0 Td[(5:997#{4whereksisthe Nikuradse 1933 roughnesslengthscaleandAistheorbitalsemiexcursionlengthA=ub!.Jonsson's966formulationofthemaximumbedshearstressismorecompletethantheformulationshowninEquation 2{2 ,asitaccountsfortheroughnessofthebottomandusesthenearbedorbitalvelocityasopposedtothefreestreamforcing,butrepresentsabulkapproximationofthestressduringawaveperiodinsteadoftreatingitasatimedependentquantity.Alargeportionofthetotalbedstressisduetowavepressurevariationsovertheseabed.Accordingto McLean 1991 ,thispartofthestressisnoteectiveinmobilizingsedimentbecausethelengthscaleofthepressurevariationissomuchlargerthantheparticlediameter. Foda 2003 claims,however,thatoutsideofthesurfzonethewavepressureplaysanimportantroleinsedimenttransportprocesses:lowenergywavesmildpressuregradientstendtodrivesedimentoshore,whilehighenergywavessteeppressuregradientsresultinonshoretransport.Thesetwoconictingtheoriesmaybeharmonizedbyconsideringthecontextunderwhichthestatementsweremade.Inanattempttodeterminetheimportanceofwavepressureonbedloadtransport, Foda 32 PAGE 33 2003 treatstheseabedasaviscoelasticuid,notasindividualparticles;therefore,thelengthscalesofthepressurevariationandaneective"lengthoftheactiveviscoelasticuidmaynotbegreatlydierent.Nearshorewaves,dueinlargeparttotheirasymmetry,producesecondaryowsoutsideoftheWBBLthatplananimportantroleinsuspendedsedimenttransport.ConsideringthevelocitytimeseriesforanonlinearStokeswaveinFigure 2{1 a,onecanseethatvelocityunderthewavecrestislargerthanthatunderthetrough.Thisinequalityresultsinanetdisplacementofaparcelofuid,orsuspendedsediment,inthedirectionofwavepropagation,oftenreferredtoasStokesdrift vanderWerf 2004 .Inthenearshoreregion,thisnetdisplacementofshorewarddirecteduidisbalanced,overlongperiodsoftime,byaseawarddirectedreturnownearthebedoftencalledundertow. Elgaretal. 2001 suggestthatoshoresandbarmigrationislinkedtocrossshoregradientsofundertow,asthesenearbedcurrentsareoftenstrongenoughtomobilizesediment.Furthermore, Gallagheretal. 1998 believethattheintensicationofundertownearasandbarleadstocrossshoregradientsinthesuspendedsedimentuxthatfurtherpromoteoshorebarmigration.Inadditiontothecrossshoredirectedsecondaryows,surfacegravitywavesalsoinduceboundarylayer,orsteady,streaming.VerticalvelocitiesgeneratedintheWBBLresultinadiusiveverticaluxofmomentumdirectedawayfromthehorizontalboundarylayer.Althoughweakcomparedtooscillatoryowoutsideoftheboundarylayer,steadystreamingresultsinanonzerotimeaveragedbedshearstressandhasasignicanteectonsuspendedsedimenttransport Marin 2004 .Thesteaminginduceduxawayfromtheboundarylayerrepresentsanadditionalmechanismfortheentrainmentofsedimentintotheouterow,whereitcanthenbeadvectedbythelocaloweld.2.2.2CurrentsSteadycurrents,suchastheowfoundinriversandhydraulicchannels,areeectiveintransportingsediment,oncemobilized,throughadvection.Relativetothewave 33 PAGE 34 boundarylayer,thecurrentboundarylayerisoftenoneortwoordersofmagnitudelargerasitdevelopsovercomparativelylongdurationsoftime.Anumberofscientistshaveconsideredsedimenttransportincurrents: MeyerPeterandMuller 1948 Eintstein 1950 SmithandMcLean 1977 Bagnold 1980 ,and vanRijn 1984 .Muchoftheresearchindicatesthatthetransportingcapabilityofasteadyowisproportionaltothestreampower,orvelocity,raisedtosomepower[ Bagnold 1980 ; vanRijn 1993 ]suchthatq=aubwhereqisthevolumetrictransport,andaandbareempiricallydeterminedconstants Longetal. 2007 .2.2.3WavesandCurrentsThenearshoreenvironmentisoftencharacterizedasstochastic,havingmultiplerealizationsofwavefrequency,direction,andheight,aswellascurrentmagnitudeanddirection.Oftentimesnearshorewavestravelovershearingcurrentsacommonexamplebeingashoalingwavepropagatingoveranalongshorecurrent.Thecombinedeectsofwaveandcurrentinducedvelocities,however,cannotbefoundbyasimplelinearsuperpositionoftheconstituents vanderWerf 2004 .Rather,thepresenceofacurrentmodiesthewaveinducedoweldinanonlinearmannerbyalternativelyaddingandsubtractingfromtheorbitalvelocitiesduringdierentphasesofthewave.Thewavecelerityandlengtharealsoaectedbythecurrent,wherecelerityincreasesdecreasesaswavelengthdecreasesincreasesinresponsetoanopposingfollowingcurrent.Thiseectisdemonstratedforcollinearwavesandcurrentsbyconsideringthemodieddispersionrelationship DeanandDalrymple 1991 :!=Uckw+p gktanhkh{5where!istheangularwavefrequency,Ucisthecurrentvelocity,kwisthewavenumberkw=2=L,gisgravitationalacceleration,andhiswaterdepth.Itisalsoknownthat 34 PAGE 35 thewaveinducedorbitalvelocitiesreducethenearbedcurrentmagnitude vanderWerf 2004 .Thenonlinearinteractionsbetweenthecurrentandwaveinducedboundarylayershaveasignicanteectonsedimenttransport.Thestirringeectofwavescoupledwithasteadycurrenthasbeenfoundtoincreasethetotalsedimenttransportsignicantly CamenenandLarson 2005 .WaveorbitalvelocityamplitudeA,currentvelocityUc,andtheangleofincidencebetweenthewaveandcurrentcwdictatetheresultantbehaviorofnearbeduidmotion.Whileacurrentowingperpendiculartothedirectionofwavepropagationdoesnotaectwavecelerityorlength,itdoesalterthenetsedimenttransport.Intheirlaboratoryexperiments, Lacyetal. 2006 ndthatforcw=90themaximumtransportissymmetricaboutthecurrentdirectionresultinginwavephaseaveragedsedimenttransportinthedownstreamdirection.Asthewavecurrentangledecreasesfromcw=90,themaximuminstantaneousbedshearstressincreasessimilarlyforcwincreasingfrom90to180. Lacyetal. 2006 foundthattheeectofthecurrentwastoincreasethenearbedvelocityamplitude,therebyincreasingtherelativeimportanceofsuspendedsedimenttobedloadtransport.Increasedentrainmentofmobilizedsedimentintothecurrentboundarylayerwasalsoobserved.2.2.4TidesandTidalCurrentsIngeneral,tidesdonothaveasubstantialimpactonsedimenttransportinthenearshoreregion.Tidalinducedvelocitiesaretypicallyordersofmagnitudesmallerthantheinstantaneous,orbitalvelocitiesproducedunderasurfacegravitywavehavingafrequencymuchhigherthanthetidalfrequency.Therefore,tidalinducedvelocitiesareoftennotlargeenoughtoinitiatemotionattheseabed,nordotheyplayasignicantroleintheadvectionofsedimenttodierentlocations.Ofcoursetheremaybespecialcircumstances,suchasaveryshallowinletorestuary,wheretidalcurrentsaresucientlystrongtomobilizesedimentinlargequantities MillerandKomar 1980a 35 PAGE 36 Tidalcurrentsgeneratedbyshoalinginternaltidalwavescanincreasetheshearstresssucientlytodevelopnepheloidlayers,orregionsonhighturbidityneartheseabed.Thesetidalcurrentsareinuentialinsuspendingnearbottomsedimentsintheabsenceofwindgeneratedwavesandcurrentsonthecontinentalshelfandslope Cacchioneetal. 1994 .Oncemobilized,thesuspendedsedimentmaythenbetransportedacrossthecontinentalshelfbyinternalwavesandtidalcurrents.Althoughtheyfoundstormgeneratedwavestobethedominantforcingmechanismoftransportonthecontinentalshelf, Puigetal. 2001 suggestednearinertialinternalwavesasamechanismforthemaintenanceofanepheloidlayerandsuspendedsedimentduringmilderwaveclimates.2.3TransportIncipientsedimentmotionhasbeenstudiedextensivelyinthelaboratorysincethepioneeringworkof Shields 1936 .Thebasicconceptofincipientmotion,oeredby Shields 1936 ,isthatthesedimentmobilizingforcesofuidliftanddragmustexceedthestabilizingforceofgravityactingontheparticle.TheratioofmobilizingtostabilizingforcesisreferredtoastheShieldsparameterandisgivenby=b s)]TJ/F15 11.9552 Tf 11.9551 0 Td[(1G{6wherebisthebedshearstress,sisthespecicgravityofthesedimentrelativetowater,gisgravity,anddistheparticlediameter.ThedimensionlessShieldsstressaccountsforskinfriction,butnotformdragwhichisthesecondcomponentoftotaldragexperiencedbythebedmaterial.However,skinfrictionisthemoreeectivecomponentofdragonmobilizingsediment McLean 1991 .2.3.1ModesClassicalmodesofsedimenttransportaredividedintothreecategories: 1. washload, 2. bedload,and 36 PAGE 37 3. suspendedload.Thewashloadischaracterizedbyverynesedimentparticles,transportedbytheuid,thatarenotrepresentativeofthebedmaterial FredseandDeigaard 1992 .Estimationofthewashloadisdicultsinceitrequiresknowledgeofsedimentcharacteristicsfromthepointoforigin...alocationpossiblyfarremovedfromtheareaofinterest.Itiscommonplacetodiscardwashloadinthecomputationofthetotalsedimentload,whichisthencomprisedofthebedloadandsuspendedloadmaterial.2.3.1.1BedloadBedloadisthepartofthetotalsedimentloadthatisinalmostcontinuouscontactwiththebed FredseandDeigaard 1992 .Undernonbreakingwaves, DohmenJanssenandHanes 2002 ndthatbedloadtransportaccountsfornearly90%ofthetotalsedimentload.Thebedloadmaterialtendstomovealongjustabovethestaticbedinathicknessafewtensofgraindiametersatmost Bagnold 1980 .Themovementofbedloadparticlesischaracterizedbysliding,rolling,and/orsaltation.Slidingoccursinmorelooselypackedbedsandforhighermobilizingstresses,wheretheparticlesslideacrossoneanotherinsheets.Bedloadparticlesmayalsorollovertopoftheirneighborsifthemomentofliftanddragforcesarejustlargeenoughtocounteractthemomentofstabilizingforceofgravityactingontheparticle LuqueandvanBeek 1976 .Saltationoccurswhenaparticleisliberatedfromthebedmaterialbyanintensestressand,onceejectedintotheow,followsamoreorlessballistictrajectoryjustafewgraindiametersabovethebed Bagnold 1973 .Pioneeringworkintheoreticalandempiricalbedloadtransportwasputforthby Eintstein 1950 .Einstein's950empiricalbedloadfunctionwastherstofsuchformulationstoaccountfortherandomnessofowandsubsequenttransportbyequatingthenumberofparticlesdepositederodedperunitareatothenumberofparticlesinmotiononthebed,andontheprobabilitythattheforcebalanceontheparticleissuchthatitwillbedepositederoded.AcentralconceptoftheEinsteinbedloadfunctionis 37 PAGE 38 thatbedloadtransportisproportionaltotheowrateraisedtosomepower.Incontrast,theexperimentsof LuqueandvanBeek 1976 showthattheaveragelengthofindividualparticlesaltationsisaconstant,implyingthattheprobabilityofdepositionisindependentoftheowrate.Suchaproportionality,however,hasalsobeensuggestedbyempiricalrelationships[ MeyerPeterandMuller 1948 ; Bagnold 1980 ; RibberinkandAlSalem 1990 ; Nielsen 1992 ; Ribberink 1998 ; CamenenandLarson 2005 ],andbyanalyticalexpressionsderivedfromphysicalconceptsaswell[ Bagnold 1966 ; Bowen 1980 ; Bailard 1981 ; KovacsandParker 1994 ]. Eintstein 1950 postulatedthatafunctionalrelationshipexistedbetweenthebedloadtransportrateandtheparticlebuoyancy,suchthatanondimensionalbedloaddischargemaydenedasb=Qb p s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd3{7whereQbisthevolumetricrateofbedloadtransportperunitwidth,sisthespecicgravityoftheparticle,gisgravitationalacceleration,anddistheparticlediameter.Anumberofphysicallaboratoryexperimentshavebeenconductedtodeterminethefunctionalrelationshipbetweenuidforcingandresultingtransport,representedinEquation 2{7 byb.Thesefunctionalrelationshipsaretermedpowerlaws"sincethetransportrateisproportionaltosomemeasureoftheuidforcingvelocityorstressraisedtoapower.Anumberofpowerlaws,invariousforms,havebeensuggestedfortransportbycurrents[e.g. MeyerPeterandMuller 1948 ; Nielsen 1992 ; Ribberink 1998 ; CamenenandLarson 2005 ]andbywaves[e.g. MadsenandGrant 1976 ; BailardandInman 1981 ; DibajniaandWatanabe 1992 ; Soulsby 1997 ; Ribberink 1998 ; CamenenandLarson 2005 ].MostoftheserelationshipsarebasedonanexceedanceofthecriticalShieldsstressandtaketheform:b=a)]TJ/F24 11.9552 Tf 11.9552 0 Td[(crb{8 38 PAGE 39 whereaissomeconstantofproportionality,bisanempiricallyderivedexponentprovidingthebesttofEquation 2{8 toasetofdata,andandcraretheinstantaneousandcriticalShieldsparameters,respectively.VariousexamplesofbedloadpowerlawsaregiveinTable 2{1 .Shields'descriptionofincipientsedimentmotionisbasedontheprinciplethatthereisabsolutelynomotionwhentheappliedstressisbelowthecriticalthreshold.Therefore,when PAGE 40 wherefisthestresstransmittedbytheuidwithinthepores,andgisthetangentialdispersivestressthatrepresentsmomentumexchangedduetoparticleparticleinteractions FredseandDeigaard 1992 .Theparticlecollisionsthatcontributetogsubtractmomentumfrommovingsedimentparticlesthatmustbereplacedbytheuidforcing.Thismomentumsubtractionresultsinanapparentroughnessgreaterthanwouldbeexpectedforastaticbed McLean 1991 .Throughaseriesoflaboratoryexperimentsofparticletransportinaclosedrectangularowchannel, LuqueandvanBeek 1976 foundthatbedloadreducesthemaximumuidshearstressatthebedlevelbyexertinganaveragereactionforceonthesurroundinguid.AccordingtoEquation 2{10 ,areductioninuidstresssuggeststhattheintergranularstressesmustincreasetomaintainthetotalstress.Athighbedloadtransportrates,then,amajorityofthetotalstressisexertedbyparticlecollisionswhiletheuidstressesremainsmall LuqueandvanBeek 1976 .AschematicofthedistributionofuidandgranularstressesforanidealizedopenchannelowisshowninFigure 2{2 .2.3.1.2SuspendedLoadBesidesbedload,theotherconstituentofthetotalsedimentloadisthesuspendedload.Suspendedloadmaybedenedasmaterialadvectedbytheuid,maintainedinsuspensionbyturbulence,andhavingverylittlecontactwiththebed.Inthisrespectthesuspendedloadistransportedbymechanismssimilartothoseresponsibleforthetransportofwashload,butitsoriginsdier.Whilewashloadconsistsofparticlesnotcharacteristicofbedmaterial,suspendedloadsedimentisentraineddirectlyfromthebedmaterial.Hence,someknowledgeofthebedmaterialmaybeutilizedinthecharacterizationofsuspendedsedimentproperties.Suspendedloadplaysanimportantroleinbothsedimenttransportandhydrodynamicprocesses.Onceentrainedfromthebedmaterial,thesesedimentscanbeadvectedlocallybynearbedwaveorbitalmotion,andregionallybypersistentcrossshoreand 40 PAGE 41 alongshorecurrents.Asmostnaturalsedimentismultimodalhavinganaturallyoccurringdistributionofgrainsizestheentrainmentofsmallerparticlesnearthesurfacecanleavebehindalayeroflargerparticles. Bagnold 1980 suggeststhatthepresenceofalargergrainsizereducesthetransportratethroughaphenomenoncalledarmoring.Totalloadtransportratesmayalsobereducedthroughtheeectsofthesuspendedloadonthenearbedoweld. McLean 1991 suggestedthatdensitystraticationbysuspendedsedimentdampsturbulence,therebylimitingtheabilityofowtotransportmassandmomentumvertically.Asaresult,theupwarddispersionofsuspendedsedimentmaintained,againstgravity,byrandomeddycurrentsisreduced Bagnold 1973 ,theabilityoftheowtokeepsedimentinsuspensionisimpaired,andtheentrainmentcapacityoftheowdecreases McLean 1991 .Suspensionanddensitystraticationmayalsomodifythevelocityprole,whichindirectlyalterstheboundaryroughnessthroughasubsequentchangeinbedmorphology[ Coleman 1981 ; ParkerandColeman 1985 ].Althoughsuspendedsedimentmaysuppressturbulence, Nakatoetal. 1977 foundthatboththemeananductuatingverticaluidvelocitywereampliedbysuspension.Throughaseriesoflaboratoryexperiments, Coleman 1981 demonstratedtheabilityofsuspendedsedimenttoreduceturbulence,aswellasmodifytheshapeofthevelocityprole.Thedistributionofvelocitythroughoutthewatercolumnisofsignicancefornetsandtransport,asthesuspendedparticlesareadvectedbytheuidvelocity.Thisisespeciallytruefortransportoverarippledbedwherecoherentmotions,likeorganizedvortices,dictateentrainment,suspension,andadvectionofsignicantquantitiesofsediment vanderWerfetal. 2006 LuqueandvanBeek 1976 foundthattheaveragetransportvelocityofsuspendedparticlesjustabovethebedwasapproximatelyequaltotheturbulentuidvelocityminusaconstant,whichisspeculatedtobeaninertialeect.Knowledgeofthevelocityprole,then,couldprovidesomeindicationofsuspendedloadconcentrations.Suchinformationisespeciallyusefulinthenearbedregion,whichplaysanimportantroleinthedeterminationofnetsandtransportowingtothehigh 41 PAGE 42 concentrationsofsedimentfoundthere vanderWerfetal. 2006 .Acommonapproachforestimatingthenearbedvelocitydistribution,withintheboundarylayer,utilizessomefunctionalformoftheclassicalPrandtlKarmanlogarithmiclawofthewall vonKarman 1930 ,orloglaw,givenbyu u=1 lnz zo+B{11whereuisuidvelocity,uisfrictionvelocity,isthevonKarmanconstant,zistheverticaldistancefromtheboundarybed,andzoisareferenceheight.WhilevariousestimatesofandBappearinliteraturepertainingtoboundarylayerows, Pope 2000 reportsthatgenerallyallarewithinvepercentof=0:41;B=5:2:{12Inthedeterminationofvelocityprolescontainingsuspendedsediment,somehavesuggestedthatdecreaseswithincreasingsuspendedsedimentconcentration[ Raudkivi 1967 ; Graf 1971 ; Yalin 1977 ].AreductionofinEquation 2{11 suggeststhattheratiooftheuidvelocitytofrictionvelocityincreases.Theexperimentsof Coleman 1981 ,however,demonstratethatisindependentofsedimentconcentration.ValuesforthereferenceheightinEquation 2{11 zohavecommonlybeenrelatedtothesedimentgraindiameter GrantandMadsen 1982 .Anestimateofzoforintensesedimenttransportoveratbedsinoscillatoryows,determinedfromthelabdataof Carstensetal. 1969 ,isapproximately7)]TJ/F15 11.9552 Tf 12.2887 0 Td[(8graindiameters.Thisconceptofanequivalentsandgrainroughnessheightsevolvedfromtheworkof Nikuradse 1933 .EntrainmentofbedparticlesintotheowmaybecomputedinamannersimilartothatusedforbedloaddischargeEquation 2{7 .Following Eintstein 1950 ,adimensionlessentrainmentparameterpisgivenbyp=E sp s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd50{13 42 PAGE 43 whereEisthevolumetricrateofentrainmentofbedmaterial,sisthesedimentdensity,sisthesedimentspecicgravity,gisgravitationalacceleration,andd50isamediangraindiameter.Typically,oneusesanavailableexpressionforpandthencomputestherateofentrainmentE,orpickuprate.Thismethodologyhasbeenusedformodelingsuspendedloadtransportinparticletrajectorymodels Hansenetal. 1991 aswellasinmoresophisticatedhydrodynamicmodels ZedlerandStreet 2001 .Anumberofrelationshipsforphavebeenempiricallyderivedfromlaboratoryexperiments,andafewcommonrelationshipsaregiveninTable 2{2 .2.3.2RegimesSedimenttransportisoftendividedintothefollowingthreeregimes vanderWerf 2004 : 1. bedload, 2. ripple,and 3. sheetow.Cleardenitionsoftheseregimesintheliterature,andtheconstitutiverelationshipsthatmakethemunique,aretenuousatbest.Thebedloadregimeischaracterizedbybedloadtransportofsandparticlesinalayernotmorethanafewgraindiametersthick Bagnold 1980 ,andoccurringoveranotherwisehorizontalbed.Accordingto Bagnold 1946 thistransportpersistsfromtheonsetofmotiontoavalueabouttwicethecriticalShieldsparametercr<<2cr.Forreference,apracticalrangeofcrforthecoastalenvironmentvariesbetween0:03and0:06 vanderWerf 2004 .Althoughthedistinctionwasnotmadeatthetime,earlyexperimentsonbedformdevelopmentby Darwin 1883 and Ayrton 1904 providedqualitativeinformationonthecharacteristicsofincipientmotionandtransportinthebedloadregime. Darwin 1883 suggestedthatparticlestransportedinthebedloadregime,byoscillatoryow,wouldaggregateinregionsonincreasedfriction.Similarly,theoscillatingtankexperimentsof Ayrton 1904 demonstratedthatparticlestransportedonaatbedwouldcongregate 43 PAGE 44 inaspeciclocation,thatbeingthelocationofastandingwavenodewheretheverticalhorizontalvelocityisatamaximumminimum.Theoscillatingtrayexperimentsof Bagnold 1946 furtherreinforcethebedloadregimeconcept,wheresandparticlestransportedinathin,concentratedlayeroveraatbedwouldaggregateinareasofhigherfriction,andlowertangentialforcing.Onceasucientnumberofparticleshavegatheredtoinduceowseparation,thetransportbehaviorismodiedduetotheadversepressuregradientformedintheleeshelteredsideoftheperturbation.Thisbehaviorwasconsistentintheexperimentsof Darwin 1883 Ayrton 1904 ,and Bagnold 1946 ,eventhoughallthreeutilizeddierenttestingdevices.Accordingto Bagnold 1946 ,thegrowthofsubstantialvorticesintheleeofabedformoccurswhentheShieldsparameterexceedstwicethecriticalvalue,or>2cr.Asopposedtotheturbulentdiusionofmomentumthattakesplaceaboveahorizontalbed, vanderWerfetal. 2006 suggestthatmomentumtransferandsedimenttransportaboverippledbedsaredominatedbyorganizedvortexsheddinginanearbedlayerapproximatelytwicetherippleheight.Coherentmotionsbreakdownandarereplacedbyrandomturbulenceabovethislayer DaviesandVillaret 1997 .Transportmodesintherippleregimeincludebothbedloadandsuspendedload.Theparticularmodeoftransportisdeterminedbytheratioofparticlesettlingvelocitytofrictionvelocity:wso u{14wherewsoistheparticlesettlingfallvelocity,anduisthefrictionvelocityu=p b=.TheprinciplemodeofsedimenttransportshiftsfromsuspendedloadtobedloadwhenEquation 2{14 isequaltoabout0.07 Nielsen 1979 .Thesheddingofvorticesfromripplecrestsinoscillatoryow,aswellastheirsubsequentejectionintotheouterow,dictatesthephaseandquantityofsedimententrainmentanddepositionintherippleregime. HuangandDong 2002 suggestedthatrippledbedsinduceanearbedcurrentdirectedoppositefromthewavepropagationdirection.Furthermore, vanderWerfand 44 PAGE 45 Ribberink 2004 foundthatrippleinducedphasedierencesbetweenpeaksuspendedsedimentconcentrationsandpeakuidvelocitiesleadtonettransportratesdirectedseawardundersurfacegravitywaves.Thissuggeststhatthetiming,orphase,ofsedimententrainmentintherippleregimeisofparticularimportancefordeterminingtransportdirectionunderasymmetricowconditions.Thesheetowregimeischaracterizedbyarelativelythin10)]TJ/F15 11.9552 Tf 12.7985 0 Td[(100dlayerofparticlestransportedinahighlyconcentratedsuspension.ThisregimepersistsforvaluesoftheShieldsparametermuchgreaterthanthecriticalthreshold0:8)]TJ/F15 11.9552 Tf 12.1308 0 Td[(1:0 vanderWerf 2004 .ForverylargevaluesofthemaximumShieldsparameter10timesaslarge,sheetowwashesoutripplesandplanesothebed LiandAmos 1999 .Suchconditionsexistundersurfzonewavesastransientpressuregradientsresultineetingepisodesofintensesheetowtransport DrakeandCalantoni 2001 .Accordingto DongandZhang 1999 ,thesheetowregimeisofparticularimportanceduetothehighconcentrationsandlargetransportratesfoundwithin.Dominantforcingmechanismswithinthesheetowlayerincludeintergranularandturbulentstresses,aswellastheinteractionforcesofuidlift,drag,andinertiabetweentheuidandsedimentparticles DongandZhang 1999 .Inthelowerpartofthesheetowlayer,sedimentconcentrationsareveryhighandthestressesgeneratedbyparticlecollisionssignicantlyaectthevelocityofuidandsediment DongandZhang 1999 .Throughaseriesoflaboratoryexperimentsonsheetowtransport, Ribberinketal. 1994 suggestthatathreelayersystemexistswithanactivepickuplayernearthebed,asheetowlayer,andasuspensionlayerintheouterow. Ribberinketal. 1994 foundthatamajorityofthehorizontaluxeswereconcentratedinthesheetowandpickuplayers,andanincreasingphaselagbetweenuidvelocityandsedimentconcentrationwithincreasingelevationinthesuspensionlayer.Almostnophaselagbetweenuidvelocityandconcentrationwasfoundtoexistinthesheetowandpickuplayers,wherenetsedimenttransportrateswereproportionaltothe 45 PAGE 46 thirdordervelocitymomenthu3iandtheconcentrationprolewaspredictedwellbyapowerlawformulation.2.4TurbulenceIntherippledbedregime,nearbedmomentumtransferisdominatedbyvortexsheddingratherthanrandomturbulenceinoscillatoryforcingconditions[ Sleath 1991 ; Nielsen 1992 ; MalarkeyandDavies 2004 ].Flowseparationatripplecrestsrepresentsasignicantmechanismfortheproductionofturbulenceintheboundarylayerduringthephaseofmaximumow Barretal. 2004 .Turbulenceplaysanimportantroleinthehydrodynamicsofthebottomboundarylayerandaectsthesuspensionandtransportofsediment.Turbulenteddiesmaintainanupwarddispersionofsuspendedsedimentagainstthecounteractingforceofgravity Bagnold 1973 ,andturbulentstressesserveasadditionalforcinginthesheetowregime DongandZhang 1999 .Thesubsequentstraticationofsuspendedsediment,inturn,aectsthebackgroundowbymodifyingthevelocityprole Coleman 1981 ,dampingturbulence McLean 1991 ,andincreasingfriction ParkerandColeman 1985 .Modelingtheeectsoftheparticlephaseonuidturbulenceisdicult.Mostnumericalmodelscannotaccuratelysimulatesedimentuidinteractionsandcoherentmotionsintheturbulentboundarylayer ZedlerandStreet 2001 .Closureofthenonlinearmomentumequations,asitpertainstoturbulence,hasbeenapproachedanumberofwaysrangingfromthemoresimpleonedimensionaleddyviscosityapproach DaviesandThorne 2002 tothemorecomplexlargeeddysimulation WangandSquires 1996 .Oneoptionforbypassingtheexplicittreatmentofturbulenceclosurethroughoneandtwoequationmodels,oreddyviscositymodels,istoperformdirectnumericalsimulationsoftheparticleuidinteractions[ ElghobashiandTruesdell 1992 ; DrakeandCalantoni 2001 ].DirectnumericalsimulationsofhighReynoldsnumbersows,however,arestillcomputationallyprohibitiveforlargespatialscalesandlongdurations. 46 PAGE 47 2.4.1DynamicsTurbulenceandsedimenttransportareinterconnectedinthesensethattheyaectoneanother.Perhapsthersttoincorporatetheroleofturbulenceinsedimenttransportformulations, Eintstein 1950 consideredtheprobabilityofparticledepositionanderosionbasedontherandomnessofnearbeduidvelocity. Bagnold 1954 furtherconsideredtheroleofturbulenceintheuidparticlematrixthroughlaboratoryexperimentsonthedispersionofspheresinashearingow.Intheclearuid, Bagnold 1954 foundthemajorityofbedshearstresstobeduealmostwhollytoturbulence,whereasincreasingconcentrationsofsedimentsuppressedturbulence.Theseconceptswereconrmedthroughadditionalexperimentsby LuqueandvanBeek 1976 Bagnold 1973 latersuggestedthatthedissipationofturbulenceinuidparticleowsoccurredthroughthedevelopmentandmaintenanceofameanvelocityequalinmagnitude,andoppositeindirection,totheparticlesettlingvelocity.Theroleofturbulenceinconcentratedparticleow,then,istomaintainthesedimentinsuspension:workperformedontheparticlesrepresentstheenergydissipatedbytheturbulence ParkerandColeman 1985 .Subsequently,thetransportratesofsolidsinsuspensionislimitedbytherateofgenerationofturbulentkineticenergy Bagnold 1973 .Intherippledbedandsheetowtransportregimes,intenseepisodesofsuspensioncanleadtodensitystratication.Straticationservesasanadditionalmechanismforthesuppressionofturbulencebylimitingtheabilityoftheowtotransportmomentumandparticlesvertically McLean 1991 .Thisoftenresultsinhighersuspendedsedimentconcentrationsinthenearbedregion Ribberinketal. 1994 ,whichmaylimittheentrainmentcapacityoftheow McLean 1991 .Asturbulenceisdamped,dragdecreasesasdoestherelativevelocitymaintainingparticlesinsuspension,therebyleadingtoparticlesettling McLean 1991 47 PAGE 48 2.4.2ClosureTurbulenceclosuretypicallyreferstothesolutionofanadditionalequationthatprovidesanestimateofturbulentstress.Variousclosuremethodshavebeenpresentedintheliterature,wheremostseektosolvefortheturbulentReynoldsstresses u0iu0jintheReynoldsaveragedmomentumequations.Themostbasic,andwidelyused,conceptwasrstintroducedbyBoussinesqwhopostulatedthatturbulentstresseswereproportionaltomeanturbulentvelocitygradientssuchthat)]TJET1 0 0 1 238.9002 538.2004 cmq[]0 d0 J0.4782 w0 0.2391 m21.0883 0.2391 lSQ1 0 0 1 238.9002 538.2004 cmBT/F24 11.9552 Tf 238.9002 528.2201 Td[(u0iu0j=t@ui @xj+@uj @xi)]TJ/F15 11.9552 Tf 13.1506 8.0877 Td[(2 3k{15wheretisaturbulenteddyviscosity,andkistheturbulentkineticenergywhichisequaltoonehalfthetraceoftheturbulentReynoldsstresstensor.Equation 2{15 isoftenreferredtoasthegeneralizededdyviscosityconcept vanderWerf 2004 .Therearethreebasictypesofeddyviscositymodels: 1. zeroequationmodels; 2. oneequationmodels;and 3. twoequationmodels.Zeroequationmodelsincludebothtimeinvariantandtimedependentformulations,mixinglengthmodels,andvonKarmanmomentumintegralmethods vanderWerf 2004 .Oneequationmodelsseektosolvefortheturbulentkineticenergyk,suchasin DaviesandLi 1997 .Populartwoequationclosuremodelsincludek)]TJ/F24 11.9552 Tf 11.9619 0 Td[(andk)]TJ/F24 11.9552 Tf 11.9619 0 Td[(!,whereand!areturbulentdissipationrates[see Wilcox 1998 ; Andersen 1999 ; AndersenandFredse 1999 ].2.4.2.1RANSModelsReynoldsAveragedNavierStokesRANSmodelssolvetheReynoldsaveragedmomentumequations,wheredependentvariableslikevelocityandpressurearedecomposedintomeananductuatingcomponents.Twoequationclosuremodels,suchask)]TJ/F24 11.9552 Tf 12.1095 0 Td[(andk)]TJ/F24 11.9552 Tf 12.1095 0 Td[(!models,areoftenemployedintheRANSapproach.Withrespect 48 PAGE 49 totheirtwodimensionalsedimenttransportmodel, AndersenandFredse 1999 foundthek)]TJ/F24 11.9552 Tf 12.2059 0 Td[(!performedwellinestimatingturbulenceaboveripplesinunsteadyow. ChangandScotti 2004 comparedsimulationsofturbulentowoverstationaryripplesusingthek)]TJ/F24 11.9552 Tf 11.986 0 Td[(!closuremodelof Wilcox 1998 andlargeeddysimulations,andfounddecienciesintheRANSapproach.RANSmodelsrequireaveragingprocessesthatcanbiasthehighlyunsteady,timedependentowthatoccursinanoscillatoryboundarylayerabovearippledbed.2.4.2.2LargeEddySimulationsWhileRANSmodelsseektoseparatedependentquantitiesintomeananductuatingcomponents,thebasicconceptoflargeeddysimulationisthatvariablesaredecomposedintoresolvedlargeandlteredsmallmotions.TheLESapproachassumesthatthecomputationalmeshissucientlyneenoughtoresolvethelargerturbulentmotions,andyetacknowledgesthefactthatsmallereddieswillbelteredbythegridnearphysicalboundaries,unlesstheresolutionisneartheviscouslengthscale Pope 2000 .InhighReynoldsnumbersows,theviscouslengthscaleisoftenprohibitivelysmalltoresolveinthecomputationalmeshsowallmodelsaretypicallyusedtocompensateforthelackofresolutionthere.Particularlyrelevantlargeeddysimulationsofturbulentow,withrespecttosedimenttransportandbedforms,havebeensuccessfullyperformedby WangandSquires 1996 CalhounandStreet 2001 ZedlerandStreet 2001 Barretal. 2004 ChangandScotti 2004 andothers.2.4.2.3DirectNumericalSimulationsItistheoreticallypossibletobypasstheturbulenceclosureproblemaltogetherbyperformingdirectnumericalsimulationsDNSofturbulentow.TheNavierStokesequationsfullydescribeallquantitiesofthetimedependentoweld,inallthreedimensions,ifthecomputationalmeshiscapableofresolvingallscalesofmotion.Untilrecently,theapplicationofsuchmodelswaslimitedtolowormoderateReynoldsnumberowsoversmoothboundaries vanderWerf 2004 .Technologicaladvancementsin 49 PAGE 50 microprocessorshavemadeitviabletoperformDNSsimulationsathighReynoldsnumbersovercomplexphysicalboundaries.RecentexamplesofthreedimensionaldirectnumericalsimulationsofmoderateandhighReynoldsnumberowsoverstationaryripplesarefoundin Scanduraetal. 2000 and Barretal. 2004 .DetailedthreedimensionalDNSmodelingofuidparticle,andparticleparticle,interactionshasalsobeenperformedby ElghobashiandTruesdell 1992 and DrakeandCalantoni 2001 .2.5ModelsAnumberofsedimenttransportmodelsexistinpublishedliterature,rangingfromsimpleempiricalmodelstomorecomplexthreedimensionalunsteadymodels.Thesemodelsfallintooneoffourcategoriesbasedontheirtreatmentoftransportandhydrodynamics:timeaveraged,quasisteady,semiunsteady,andfullyunsteady.Eachhavebenetsandlimitations,someofwhichwillbediscussedinthefollowingsections.Additionally,mostcoupledmodelsofuidhydrodynamicsandsedimenttransportrequireparametrizationofeithertheuidorparticlephase,butsomeexceptionsdoexist.Thefollowingsectionsprovideabriefoverviewoftypicalmodeltypes,theirassumptions,andalsotheirshortcomings.2.5.1TypesThemostcommonlyusedsedimenttransportmodelsmaybeclassiedaseithertimeaveraged,quasisteady,semiunsteady,orfullyunsteady.2.5.1.1TimeAveragedWhilesimple,timeaveragedmodelsarenotnecessarilyconsideredrobust.Thesemodelsoftenrelyonowstatisticsthathavebeenaveragedoverdurationsmuchlongerthanwouldbeconsideredrelevantinthecoastalenvironment,suchasanindividualwaveperiod.TimeaveragedsuspendedloadtransportmaybecomputedusingaC)]TJ/F24 11.9552 Tf 12.089 0 Td[(uintegralapproach: Qs=Zh0 Czuzdz{16 50 PAGE 51 wherethedependentvariablesofsedimentconcentration Canduidvelocityuareaveragedoveroneormanywaveperiods vanderWerf 2004 .Apopulartimeaveragedmodelforcurrentrelatedsuspendedloadandbedloadtransportisgivenby Bijker 1971 .Withrespecttoapplicationsinthecoastalenvironment,themajordisadvantageofanytimeaveragedsedimenttransportmodelisthatthewaverelatedoscillatorycomponentoftransportisintegratedoutofthesolution.2.5.1.2QuasiSteadyQuasisteadymodelsseektoaccountforboththewaveandcurrentrelatedsedimenttransportcomponentsbyequatingtheinstantaneoustransporttotheinstantaneousnearbedoweldraisedtosomepower.Implementationofquasisteadytransportmodels,however,arenotmeanttoprovideintrawavestatisticsofsedimenttransport.Rather,thequasisteadytransportiscomputedas Qs/1 TZT0u3tdt{17where Qsisthewaveaveragedsedimenttransportrate,Tisthewaveperiod,andutisanexpressionofthewaveandcurrentvelocityeld.Consideringtheexamplegivenby vanderWerf 2004 ,thevelocitytimeseriesutforasecondorderStokeswavesuperimposedonasteadycurrentisgivenbyut=u+^u1cos!t+^u2cos!t{18whereuisthecurrentvelocityand^u1and^u2arethewavevelocityamplitudes.SubstitutingEquation 2{18 intoEquation 2{17 ,andintegratingoverthewaveperiod,givesthewaveaveragedtransportrateasafunctionofbothwaveandcurrentvelocities: Qs/u3+3 2u^u21+3 2u^u22+3 4^u21^u2:{19Anexampleofawidelyusedquasisteadytransportmodelisgivenbythetotalloadformulationof Bailard 1981 ,whichisanextensionoftheearlierenergeticsmodelof 51 PAGE 52 Bagnold 1966 .Usefulquasisteadyandunsteadybedloadformulationsarealsoprovidedby Ribberink 1998 .Duetothetimedependentprocessofvortexshedding,however,quasisteadymodelsarenotsucientlyrobusttoestimatethesuspendedsedimenttransportaboverippledbeds vanderWerfetal. 2006 .2.5.1.3SemiUnsteadyAnimplicitassumptionintimeaveragedandquasisteadytransportmodelsisthatsedimenttransportisalwaysinphasewithwaveforcing.Thisisnotalwaysthecase,andinertialeectsofboththeboundarylayerowandtheweightofthesedimentitselfresultinphasedierencesbetweenappliedforcingandsubsequenttransport[ Parker 1975 ; LuqueandvanBeek 1976 ].Semiunsteadymodelsattempttoaccountforthesephasedierenceseitherthroughparametrizations,orbyconsideringtransportovertwosuccessivewavehalfcycles. Nielsen 1988 developedasimplegrabanddump"modelofwaverelatedsedimenttransportbyassumingthattransportoverripplesoccursthroughtwodistinctmechanisms:rst,sedimentistransportedalongtheripplefacestossandcarriedoverthecrestwhilealeevortexentrainssedimentfromthetroughandsecond,liberatedsandisliftedbythevortexandsubsequentlyadvectedbythemainow.Sincethisprocesshappenstwiceduringeachwavecycle, Nielsen 1988 accountsfortheforwardandbackwardtransportseparatelythroughdistinctentrainmentcoecientsbasedontheinstantaneousvelocityamplitudes.Inthismanner,thegrabanddump"modeliscapableofaccountingforwaveasymmetry. Nielsen 1988 reportsthatthemodelprovidesreasonableestimatesoftransportforbothneandcoarsesands,resolvesthephaseofmaximumtransportevents,andisextremelypracticalasitrequiresonlyanestimateofanearbedreferenceconcentration.Theconceptofdividingtransportintosuccessivewavehalfcyclestoaccountforphaselagswaspopularizedbythemodelof DibajniaandWatanabe 1992 .Insteadofcomputinganaveragetransportrateovertheentirewaveperiod,thehalfcyclemodel 52 PAGE 53 solvesforthenettransportrateovertheentirewaveperiod.Thisisdonebytakingthelinearsumofthetransportratesinthepositiversthalfandnegativesecondhalfphasesofawave.Thehalfcyclemodelconsistsoftwoelementsoftransportforeachhalfcycle:sedimententrainedandtransportedwithinthesamehalfcycle,andsedimententrainedduringtheprevioushalfcycleandtransportedduringthefollowinghalfcycle vanderWerf 2004 .Thehalfcyclemodelisconsideredtobesemiunsteady,becausewhileitisunsteadyovertheentirewaveperiod,itissteadyforeachhalfcycle.2.5.1.4UnsteadyUnsteadymodelsprovideestimatesoftimedependentsedimenttransport,typicallythroughintegrationwithanunsteadyhydrodynamicsmodel.TimedependentvaluesofuidvelocityandsedimentconcentrationmaybesubstitutedintoaC)]TJ/F24 11.9552 Tf 12.7541 0 Td[(uintegralapproach,similartoEquation 2{16 ,inordertomodeltheunsteadysuspendedsedimenttransportrate:Qst=Zh0Cz;tuz;tdz:{20Mostquasisteadybedloadtransportmodels[e.g. MeyerPeterandMuller 1948 ; Ribberink 1998 ]mayalsobeadaptedforunsteadyusebyconsideringatimedependentShieldsparametert,basedontheinstantaneousbedshearstress,intheformulation MadsenandGrant 1976 .Forexample,thepopular MeyerPeterandMuller 1948 bedloadtransportequationmaybecastinanunsteadyform:bt=8t)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr3=2{21wherebtisthetimedependent,dimensionlessbedloadfunctionof Eintstein 1950 .Variousreferencestothe MeyerPeterandMuller 1948 bedloadequationhavebeenmadeintheliteratureandsomehavesuggestedthatthecoecientandexponent=2varysomewhatdependingonowandsedimentcharacteristics.Althoughthefunctionalformwasderivedfromlaboratorydataonsteadyow, Madsen 1991 suggeststhatthisbedloadformulaperformswellforcoastalapplications,too.Dependingonthemodeling 53 PAGE 54 approach,theseunsteadyapproximationsofsuspendedloadandbedloadtransportratesmayeitherbeaveragedoverindividualwaveperiodstoprovideestimatesofnettransport vanderWerf 2004 ,orintegratedoverfractionaltimestepstoprovidediscreterepresentationsofinstantaneoustransport.2.5.2IntegratedApproachesForunsteadymodelsinparticular,thereareavarietyofintegrateduidsedimenttransportmodelsdescribedinexistingliterature.Thecouplingofhydrodynamicsandsedimenttransportoftenrequireseitherspecialtreatmentoftheuidorparametrizationsofthesedimentphase.Somemodelssolvethedetailedhydrodynamicsexplicitly,oftenthroughDNS ElghobashiandTruesdell 1992 orusingLESturbulenceclosure WangandSquires 1996 ,andthenincorporateonewaycouplingbymodelingtheparticlemomentumequations MaxeyandRiley 1983 .Otherschoosetomodeltheparticulatephasebysolvinganadvectiondiusionequationforthesuspendedsedimentconcentration ZedlerandStreet 2001 ,alongwithasedimentpickupfunction vanRijn 1984 andappropriateboundaryconditions Nielsenetal. 1978 .Aslightlymoredetailedapproachforsimulatinguidparticleowinvolvesmodelingthewaterandsedimentasacontinuum,suchasin Phillipsetal. 1992 .Anadvantageofthecontinuumapproachisthattwowaycouplingofmassandmomentumisimplicitintheformulation.Thecontinuummodelhasthreemajorcomponents:mixturemomentumequations,anadvectiondiusionequationforthemixtureconcentration,andoftentimesasophisticateddiusivitymodel.Theapplicationofcontinuummodelstolargecoastalapplicationsremainsarelativelynewareaofresearch.Explicittreatmentofthreedimensionalhydrodynamics,especiallyforDNSandLESmodels,canbetimeconsuming.Unsteadyhydrodynamicmodels,however,arenecessaryforaccuratelyresolvingthetimedependentcoherentmotionsthatdevelopintheboundarylayerabovearippledbed.Sincethevortexmotionsdetermineamajorityofentrainmentandtransportintherippledbedregime Sleath 1991 ; Nielsen 1992 54 PAGE 55 itispossibletoparametrizethebulkhydrodynamicsbyconsideringthetransportofvorticityintheoweldratherthanthemomentum.Suchanapproachisdescribedin MalarkeyandDavies 2002 ,whereadiscretevortexmodelDVMisusedtosimulatethetimedependentvorticityeldabovearippledbedinoscillatoryow. MalarkeyandDavies 2002 presentresultsofsimulationsusingasimpleinviscidDVM,andaslightlymoreadvancedcloudincell"CICmodelthatconsidersadistributionofvorticitypointvalueswithintheoverallvortex.Thesemodelsarecapableofestimatingreasonablevaluesofvorticity,butoftenfailtoaccuratelycapturethephaseofvortexformationandejection.ComparedtoRANSmodels,thediscretevortexmodelsarebettersuitedforresolvingsharpgradientsinthevorticityeldsincetheydonotsuerfromthenumericaldiusionrelatedtoadvection MalarkeyandDavies 2004 .Analternativemethodforsimulatingcoherentmotionsintheboundarylayerwaspresentedby DaviesandVillaret 1997 ,whosuggestedthatvortexsheddingcouldbemodeledasaconvective"stressrepresentedbyatimedependent,heightinvarianteddyviscosity.Theconvectiveeddyviscosity"relatestheconvectiveshearstresstothevelocitygradient.Examplesofonedimensionalconvectiveeddyviscositymodelsaregivenby DaviesandThorne 2002 and MalarkeyandDavies 2004 .Modelsimulationsby MalarkeyandDavies 2004 suggestthattheconvectiveeddyviscosityapproachisvalidinathicknessofaboutoneortworippleheightsabovethemeanbedwherethecoherentmotionsexistinaconvectivelayer.Moreadvancedmodelingtechniquesexistforthesimulationofmultiphaseowsincoastalapplications.Forexample, Calantoni 2002 presentsadiscreteparticlemodelforbedloadtransportinthesurfzone.Thediscreteparticlemodelsimulatesthedynamicsoftheuidowandaccountsforthekinematicsofeachparticleonanindividualbasis.Therefore,termsthataretypicallyparametrizedinothermodels,suchasgranularstressesandmomentumtransferthroughelasticcollisions,areaccountedforexplicitlythroughrstprinciples.Solvingmomentumequationsfortheuid,aswellaseach 55 PAGE 56 particle,iscomputationallyexpensiveandapplicationsofdiscreteparticlemodelsarecurrentlyreservedforresearchproblems.Analternativeapproachistomodelallofthesedimentparticlesasaseparatephase,muchastheuidphaseismodeledasahomogeneousmedium.Followingthismethodology, DongandZhang 1999 developedatwophasemodelofuidandsedimentowinoscillatorysheetow.Themodelsolvesthecontinuityandlinearizedmomentumequationsforuidandsediment.Turbulentandintergranularstressesareincorporatedinthesolutionalgorithm,therebyprovidingacompletedescriptionoftheinteractionforcesbetweentheuidandsedimentphases.2.5.3ShortcomingsAnalyticalmodelsofsedimenttransportarenotreadilyavailable.Mostsedimenttransportrelationsmaybecharacterizedaseitherempiricalorsemiempiricalatbest.Indeed,muchoftheliteraturepublishedonsedimenttransporthasfocusedonprovidingparametrizationsoftransportbasedonlaboratory,andsometimeseld,experiments.Thistendstobeacceptablewhendevelopingamodelforaspecicapplicationtoaspecicsetofcriteriae.g.forcing,sedimentsizeandgradation,transportmodeandregime,butmakesitdiculttodevelopuniversalsedimenttransportmodels.Regardlessofthemodelframeworkchosen,thelimitingfactoronaccuracyandpredictabilityseemstolieinthetreatmentofsedimenttransportandnotnecessarilythehydrodynamics. 56 PAGE 57 Figure2{1.ComparisonofavelocityandbshearstresstimeseriesforlinearandnonlinearwaveswithperiodT=6sinwaterdepthh=1m. Figure2{2.Approximatedistributionofuidandgrainshearstressinsteadyopenchannelow.Adaptedfrom FredseandDeigaard 1992 57 PAGE 58 Table2{1.Commonpowerlawformulationsforbedloadsedimenttransportbycurrentsandwaves.Shields'parameterbasedoncurrentsiscandforwavesisw. ReferenceDimensionlessBedloadDischarge Currentsb MeyerPeterandMuller 1948 8c)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr1:5 Nielsen 1992 12p cc)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr Ribberink 1998 11c)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr1:65 CamenenandLarson 2005 121:5cexp)]TJ/F15 11.9552 Tf 9.2985 0 Td[(4:5cr=cWaves1=2 MadsenandGrant 1976 12:5Ws3w=p s)]TJ/F15 11.9552 Tf 11.9551 0 Td[(1gd50 Soulsby 1997 5:1w)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr1:5Wavest BailardandInman 1981 bfwuwt3=[s)]TJ/F15 11.9552 Tf 11.9551 0 Td[(12g2tand50] Ribberink 1998 11jwtj)]TJ/F24 11.9552 Tf 17.9327 0 Td[(cr1:65wt=jwtjWaves DibajniaandWatanabe 1992 0:001Ws)]TJ/F23 7.9701 Tf 7.3141 4.3385 Td[(0:55)]TJ/F24 11.9552 Tf 11.8665 0 Td[(=j)]TJ/F27 11.9552 Tf 7.3142 0 Td[(j=p s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd50 CamenenandLarson 2005 ap cw;on+cw;offw;mexp)]TJ/F24 11.9552 Tf 9.2985 0 Td[(bcr=w Table2{2.Empiricalentrainmentfunctionsforsuspendedloadsedimenttransport. ReferenceDimensionlessPickUpRate,p FernandezLuque 1974 0:02)]TJ/F24 11.9552 Tf 11.9552 0 Td[(cr1:5 NagakawaandTsujimoto 1980 0:02)]TJ/F15 11.9552 Tf 11.9552 0 Td[(0:035=3 vanRijn 1984 0:00033[s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd50=2]0:1=cr)]TJ/F15 11.9552 Tf 11.9552 0 Td[(11:5 58 PAGE 59 CHAPTER3BEDFORMDYNAMICS3.1IntroductionDierentialsedimenttransportontheseabedcreatesmorphologicalfeaturesthatrangeinsizefromsmallscalesandripplestomuchlarger,shoreparallelsandbars.Regardlessoftheirsize,theresultingmorphologicalstructureplaysanimportantroleinbothsedimenttransportandhydrodynamicprocesses.Shoreparallelsandbarshavebeenshowntointensifythecrossshorecurrentsthatpromoteoshorebarmigrationthroughgradientsinthecrossshoresedimenttransport Gallagheretal. 1998 .Smallscalebedforms,suchassandripples,determinethegenerationofturbulenceandentrainmentofsandnearthebed Hanesetal. 2001 largelythroughowseparationthatleadstoanorganizedpatternoftimedependentvortexsheddingineachhalfcycleofawave.Thesevorticesoftendeterminethetimingofsedimententrainmentanddeposition Nielsen 1988 ,theamountofsedimentcarriedinsuspension vanderWerfetal. 2005 ,andareresponsibleforthedissipationofwaveenergyintheboundarylayer Tunstall 1973 ; TunstallandInman 1975 .Thepresenceofsmallscalebedformshasalsobeenfoundtocausenaturalsortingofsediment,resultinginaredistributionofneandcoarsematerialontheseabed FaraciandFoti 2002 .Thefocusofthischapterisonsmallscalebedforms,suchassandripples,sofurtherdiscussionoflargerfeatureslikesandbarswillbelimited.Thefollowingsectionsprovideabriefoverviewofbedformtypes,theirclassicationandcharacteristics,mechanismsfortheirgrowth,theiraectonthebottomboundarylayer,andmethodsforpredictingtheirlengthandheight.Additionalinformationonbothsimpleandcomplexmorphologicalmodelsisalsoprovided.3.2BedformTypesSmallscalebedformsmaybebroadlycategorizedbythedominantowconditionsunderwhichtheyareformed:currentgeneratedbedformsorwavegeneratedbedforms. 59 PAGE 60 Earlyaccountsofwavegeneratedripplesintheseadatebacktotheobservationsof Hunt 1882 ,whofoundthatripplemarks"wereformedonasandyseabedbyaslightoscillationofthewater.Laboratoryexperimentsby Darwin 1883 ,andlaterby Ayrton 1904 ,providedmostlyqualitativedataabouttheformationofripplesinoscillatingow.Wavegeneratedbedformsarefurtherdistinguishedby Bagnold 1946 aseitherrollinggrainorvortexripples.Adistinctionismadebetweenthesetwotypesofwavegeneratedripplesnotonlybecausetheygrowthroughdierentprocesses,butalsobecausetheircharacteristiclength,height,andsteepnessscaledierently.Briey,rollinggrainripplesgrowthroughanaggregationofsandparticlesinareasofhighfriction Darwin 1883 butdonottypicallyscalewithwaverelatedquantities.Vortexripplesformonceaperturbationonthebedislargeenoughtoinduceowseparation Bagnold 1946 ,suchthatanadversepressuregradientisformedintheleeoftheripplecrest Ayrton 1904 .Thisprocessmayoccureitherthroughthegrowthofrollinggrainripplesorduetoexistingperturbationsontheseaoor.Specicdetailsaboutthemechanismsforgrowthandevolutionaregiveninthefollowingsections.Currentgeneratedbedformsincludethoseformedinriversandopenchannelhydraulicows.Thesimilaritybetweenthesetwoenvironmentsisthatthepredominantforcingis,moreorless,onedimensionalsteadyow.Whilenotalwaystrulysteady,"thetermisusedheretoconveyanabsenceofapersistentoscillatory,orrepeating,ow.Shallowtidallydominatedinletsandestuariesmayevenbeaspecialcaseofsteady"owiftheambientoscillatorymotionisweakcomparedtothetidalinducedcurrents.Currentgeneratedripplesinitiallyforminmuchthesamewaysaswavegeneratedripples,butoncethebedformislargeenoughtoinduceowseparationthedynamicsaredierent.Sincethesteadyowhasaconstantphase,arecirculationzoneisonlyformedononesidetheleesideofthebedform.Thedynamicsofthebedformarecontrolledbyabalancebetweensandtransportedalongthestoss,sandtransportedoverthecrestontotheleeside,gravitationalforcespullingsedimentdowntheleeside,andthetransport 60 PAGE 61 ofsanduptheleesidefromtheattachedvortex.Thiscommonlyresultsinabedformwithanasymmetricprolewherethestosssidehasamilderslopethanthatoftheleeside Darwin 1883 .Itiscommonfortheleesideofanasymmetricrippletohaveaslopeveryclosetothenaturalangleofrepose.Onceinequilibriumwiththesurroundingow,steadycurrentsarecapableofmaintainingtheripplethroughadynamicequilibriumbutareoftendisplaceddownstreamduetohighervelocitiesatthecrestthanatthetrough Darwin 1883 .3.2.1RollingGrainRipples Bagnold 1946 wasthersttoclassifythetwotypesofwavegeneratedripplesaseitherrollinggrainorvortex,butaphysicaldistinctionbetweenthemwasmadeearlierby Darwin 1883 .Thelaboratoryexperimentsof Darwin 1883 demonstratedthatoscillatorymotionrstcreatedwhathecalledtransientripplesasaresultofparticleaggregationinareasofhighfriction.Theprocessofaggregation,henoted,increasesthefrictionfurtherandtrapsevenmoreparticlesinthatarea.Othershavecharacterizedtheprocesssimilarly,suggestingthateachgraincreatesaregionofweakerowinitsleeultimatelycausinggrainstogroupintotransversezonesthatformrippleswithlargershadowzones Bagnold 1946 ; Blondeaux 1990 ; VittoriandBlondeaux 1990 ; Andersen 1999 .Theparticlemodelof Andersen 2001 demonstratesthatrollinggrainripplesformandcoarsenduetogradientsinthetransportvelocityfromonesideoftheaggregationtotheother.Transientrippleswerefoundtohaveawavelengthapproximatelyonehalfofthenalripplewavelengthintheexperimentsof Darwin 1883 ,whichseemstobeaconsistentobservationinotherlaboratoryexperiments Sleath 1976 .Rollinggrainripplescontinuetogrowinheightthroughaggregationandtrappinguntiltheleeslopesaresteepenoughtoallowavortextoformintheshelteredspacebehindthecrest Bagnold 1946 ,eventuallyformingvortexripples Schereretal. 1999 ; FaraciandFoti 2001 .Duringtheinitialgrowthprocess,thesteepnessheight/lengthofrollinggrainrippleshasbeen 61 PAGE 62 foundtobeconsistentlynear0.1by Sleath 1976 andothers. Darwin 1883 qualitativelysuggestedthattheinitialgrowthprocessonlyoccurredbetweenanupperandlowerlimitofuidvelocity,andthatthelimitswereafunctionofaveragesedimentpropertieslikesizeanddensity.Theseupperandlowerlimitswerefoundby Bagnold 1946 tobe2crandcr,respectively.Thestabilityofrollinggrainripplesformedduringlaboratoryexperimentsisunclear.Duetotheirlowreliefandtransientnature,rollinggrainripplesarerarelyobservedintheeld.Rollinggrainrippleswerefoundtobestableintheexperimentsof Bagnold 1946 and Sleath 1976 ,but MillerandKomar 1980b suggestthatthesetwoinstancesmaybearesultofthetestingapparatususedduringtheexperimentsoscillatingtray.Experimentsinrotatingannularcells Schereretal. 1999 ; StegnerandWesfreid 1999 andoscillatingwatertunnelssuggestthatrollinggrainripplesaretransient,unstablefeatures MillerandKomar 1980b .Theprevailingtheory,now,isthatrollinggrainripplesaretransientbedformsthatinitiatethegrowthoflarger,vortexripplesthroughaninversecascademechanism:thedynamicalevolutioninvolvescoarseningfromaninitiallyshortwavelengthrollinggrainripplestoalongerwavelengthvortexripples,withsaturationatanalequilibriumcondition StegnerandWesfreid 1999 ; Andersenetal. 2002 .While Darwin 1883 and Bagnold 1946 providedqualitativedescriptionsofthegrowthofrollinggrainripplesfromaninitiallyatbed, Kennedy 1963 1969 demonstratedthatifalagisassumedtoexistbetweenvelocityandtransport,aninstabilityexistsatthesandwaterinterface ColemanandMelville 1994 .Thislagisoftenattributedtoinertialeectsofthesediment Parker 1975 ,andhasbeenidentiedastherippleinstabilityby Smith 1970 .Initiationofgrowththroughthisrippleinstability,however,shouldnotbeconfusedwiththestabilityofrollinggrainripplesdescribedabove;theformerreferstoamechanismthatinitiatesgrowthwhilethelatter 62 PAGE 63 conveysanabilityofthebedformtoachieveanequilibriumconditionwithrespecttoitsowconditions.Ifaspatiallagisassumedtoexistbetweenvelocityandtransport, Parker 1975 nds,throughalinearstabilityanalysisofderivedrelationsforowandtransport,aninherentinstabilityofatbedswhichleadstotheformationofriverantidunes.However,ifvelocityshearandtransportareassumedtobeinphasewithoneanother,atsandbedsareunconditionallystable Parker 1975 .Underpurebedloadtransport,aatbedisalsounstableatallwavelengthstosmallperturbationsintopography Smith 1970 .Thissuggeststhatoneoftwonecessaryconditionsmustbemetfortheinitiationofripplegrowthfromaatbed;eitherforcingandtransportmustbedecoupled,orsomeinitialperturbationinbedtopographymustbepresent. Hayakawa 1970 wasabletodevelopatheoryofduneformationinopenchannelowsbyapplyingtherippleinstabilityconcept,andothershavehadsimilarsuccess ColemanandMelville 1994 .Somepredictivemodelsofrollinggrainripplegeometry,basedontherippleinstabilityconcept,havealsobeendeveloped. Sleath 1976 createdastabilityanalysismodelcapableofpredictingthewavelengthofrollinggrainripples.Morerecently, FotiandBlondeaux 1995a developedapredictivetheoryofrippleformationunderwavesthatagreeswellwithexperimentaldata.Assumingafullyturbulentboundarylayer, FotiandBlondeaux 1995a obtainedclosedformsolutionsofturbulentoscillatoryowoverawavysurfacethroughalinearstabilityanalysisofthegoverningequations.Detailedparticlemodelsofrollinggrainripplesalsoexist Andersen 2001 .3.2.2VortexRipples Bagnold 1946 suggestedthatwavegeneratedripplesfallintooneoftwocategories:rollinggrainorvortex.Thedecidingfactor,Bagnolddetermined,wasbasedontherangeofappliedstressatthebed.OncetheShieldsparameterincreasesbeyond2cr,theleeslopesoftheripplegrowtoaheightlargeenoughthatavortexisformedinthelee Bagnold 1946 .Theprocessthat Bagnold 1946 observedinthelaboratoryinvolveda 63 PAGE 64 transitionrstfromaatbedtoonewithrollinggrainripples;andsecond,fromrollinggrainripplestovortexripples.Itisalsopossibleforvortexripplestoformwithoutrollinggrainripplesifthereissomesortofnaturalperturbationonthebedlargeenoughtoinduceowseparation Ayrton 1904 .Sowhilerollinggrainripplesareasucientmechanismforthegrowthofvortexripples,theinducedowseparationandsubsequentvortexformationarenecessaryconditions.Theabilityofvortexripplestogrowfromnaturalperturbationsmayalsoexplainwhyrollinggrainripplesarerarelyobservedintheeld MillerandKomar 1980b ,whereasvortexripplesareoftendetected MillerandKomar 1980a ; Hanesetal. 2001 ; Ardhuinetal. 2002 .Vortexripplesplayanintegralroleinbothsedimenttransportandenergydissipation,asdiscussedpreviously.Centraltotheeectsofvortexripplesonboundarylayerprocessesistheformationandejectionofvorticesfromtherippleproleintothemainowduringeachsuccessivehalfcycleofoscillatoryforcing.Vorticesejectedovertheripplecrestcanentrainsignicantamountsofsedimentwithintheboundarylayerthatissubsequentlyadvectedbytheouterow Gallagheretal. 1998 ; vanderWerfetal. 2005 .Thisisthoughttoincreasethephaselagbetweensuspensionandtransportrelativetotransportoveraatbed InmanandBowen 1963 ,leadingtotimedependentsuspendedsedimenttransportprocessesthatcannotbemodeledinaphaseaveragedorquasisteadymanner vanderWerfetal. 2006 .Theformationofvorticesintheleeofbedformsresultsfromowseparationatthecrest,similarinmanywaystoowseparationaroundblubodies,cylinders,andthelike.Inherlaboratoryexperiments, Ayrton 1904 identiedtwoconditionsnecessaryforvortexformationintheleeofripples: 1. areductionofpressureinthelee;and 2. anupwarddirectedresultantgravitypressurealongtheleeridge.Therstconditionresultsfromowseparatingatthecrest,creatingashelteredregionintheleeoftheripple.Thesecondconditionisduetoanadversepressuregradientthat 64 PAGE 65 developsduetophasedierencesbetweentheboundarylayerowandthefreestreamforcinganinertialpropertyofboundarylayerows.Adversepressuregradientsmayalsobeattributedtosurfacegravitywaves,butarenotpresentinsteadyowsoveratbeds. Ayrton 1904 speculatedthatvortexformationcouldnotoccurinsteadycurrentsandthatripplescouldnotbecreatedormaintainedbysuchowconditions.Recentlaboratoryexperimentsofbedformgrowthfromaatbedinsteadycurrentshaveprovedthistobefalse Mantz 1990 ; ColemanandMelville 1994 1996 .Underoscillatoryforcing,vorticesareformedandejectedinalternatinghalfcycles.AschematicofvortexformationandejectionasafunctionofuidphaseandforcingisshowninFigure 3{1 .VorticesformatthebeginningofeachhalfcycleFigures 3{1 a, 3{1 c, 3{1 e,growinginstrengthandsizetomaximumvaluesataphaseofapproximately90degrees EarnshawandGreated 1998 .Duringphasesofmaximumforcing,strongowseparationandvortexsheddingneartheripplecrestserveasmechanismsfortheproductionofturbulenceintheboundarylayer Barretal. 2004 .Aftercontractinginsizeslightlyduetoareductionintheappliedforcing,theejectedvorticesinFigures 3{1 cand 3{1 econtinuetoincreaseinsizeastheymoveverticallyintoareasofdecreasingpressure Ayrton 1904 .Neglectingsmallperturbationsduetoturbulence,theseprocessesaremoreorlesssymmetricfromonehalfcycletothenext.Underrealwaves,however,vortexformationandejectionisasymmetricowingtoStokes'law EarnshawandGreated 1998 .ThiscanhaveasignicanteectonthenetsedimenttransportinthecrossshoreassuspendedloadmaybeadvectedshorewardbytheEulerianow.Flowseparationandvortexdevelopmentareimperativeforthegrowthandstabilityofvortexripplesunderavarietyofforcingconditions Nielsen 1981 .Duringthegrowthprocess,thenearbedhydrodynamicsresultinnetsedimenttransporttowardtheripplecrest AndersenandFredse 1999 .Theimpingingowacceleratesasittravelsalongtheupwindsideoftheripplethestosscreatingstrongshearstressesatthebedthatdrivesedimenttransportfromthetroughtowardthecrest.Intheleeoftheripple,the 65 PAGE 66 counterrotatingvortexinducesabedshearstressthatisdirecteduptheleesideoftherippleresultingintransporttowardthecrest.Thisistheprimarygrowthmechanismforvortexripplesinnearequilibriumconditions.Experimentaldataofbedformsinawaveumesuggeststhatthealternatingvorticessupporttherippleineachsuccessivehalfcyclewhen<0:2 FaraciandFoti 2002 .For>0:2,thestressatthecrestissolargethattheinducederosioncannotbeovercomebythestabilizingeectofthevortices.3.2.3CharacteristicsThegrowthandevolutionofvortexripplesfromaatbedmayoccurthroughoneoftwoprocesses;oneinvolvesatransientstatepriortoequilibrationandtheothersuggestsrelativelyconstantgrowthfromaatbed.Forweakowsandunderregularwaves,vortexripplesformthroughcoarseningoftransientrollinggrainripples StegnerandWesfreid 1999 ; FaraciandFoti 2002 .Thecoarseningprocessischaracterizedbyanincreaseinwavelength,resultinginfewerripples,withsaturationtoanequilibriumcondition Andersenetal. 2002 .Thecoarseningprocessdoesnotoccurforbedformsinstrongerowsorunderirregularwaveconditionswheretheyreachequilibriumdirectlythroughconstantgrowth StegnerandWesfreid 1999 ; FaraciandFoti 2002 .Anoverwhelmingamountofphysicaldatademonstratestheabilityofvortexripplestotransformfromasmallertoalargerwavelength,butrarelydoesthishappeninreverse.Vortexripplesintheeldandlabhavebeenfoundtoexhibitstronghysteresis Traykovskietal. 1999 ; StegnerandWesfreid 1999 wherebychangesinstableripplepatternsfromlargertoshorterwavelengthsarenotobserved.Thissuggeststhatlargerripplesmustbecompletelydestroyedbyanincreaseintheforcingconditionsbeforeasmallerstableripplepatterngrowsinresponsetonewforcing.Rollinggrainripplesandvortexrippleseachhaveuniquecharacteristicsdueinlargeparttotheforcingconditionsthatcreatethem.Therollinggrainripplesobservedandmeasuredinanumberoflabexperiments Bagnold 1946 ; Sleath 1976 ; FaraciandFoti 2002 consistentlyhaveasteepnessr=rof0.1regardlessofthetypeof 66 PAGE 67 owandtestingapparatus,whererandraretherippleheightandwavelength,respectively.Vortexripplestendtohaveamaximumsteepnessalmosttwicethatofrollinggrainripplesatapproximately0:18 Nielsen 1981 ; FaraciandFoti 2002 .For<0:2, FaraciandFoti 2002 ndthattheripplesteepnessremainsconstantatitsmaximumvaluethatcorrespondswellwiththevaluegivenby0:32tan.Thewavelengthofvortexripplescommonlyincreasealongwiththeheightkeepingamoreorlessconstantsteepnessuntilanequilibriumisreached.Thisequilibriumisafunctionofowandsedimentcharacteristics Lofquist 1978 andthenalsteepnesstendstomaximizeresistancetothelocalowconditions Davies 1980 DoucetteandO'Donoghue 2006 identifythreedistinctprocessesthroughwhichtheripplewavelengthevolves:slide,merge,andsplit.Theslideprocessidentiesagradualincreasedecreaseofwavelengthasadjacentripplesmoveawayfromtowardeachother.Amergeoccurswhentwolargerankingripplesincreaseinheightandmovetowardeachother,destroyingthesmallerrippleinbetween.Asplitreferstoasituationwhenonelargerippledecreasesinsizethroughthecrestsplittingintowhatappeartobetwoormoreripplesofshorterwavelength.Whenbedloadtransportdominatesoversuspendedload,vortexripplesgrowthroughtheprocessesofmergeandslideanddecaythroughtheslideandsplitmechanisms DoucetteandO'Donoghue 2006 .Thecharacteristicsofvortexrippleshavebeenfoundinmanycasestoscalewithpropertiesoftheuidforcing. Clifton 1976 rstsuggestedthataparticularrippleregimeexistedwherebytheripplewavelengthswereconstantmultiplesofthenearbedorbitaldiameter2A:A=Umax !{1whereUmaxisthemaximumnearbedvelocityamplitudeand!istheangularfrequencyoftheforcing.Here,ArepresentstheorbitalsemiexcursionlengthwhichisonehalftheorbitaldiameterA.AnumberofrelationshipsbetweenripplewavelengthrandorbitalsemiexcursionAhavebeenpresentedintheliterature,andrangefromr0:78A 67 PAGE 68 forsuspendeddominatedripples AndersenandFredse 1999 totheupperestimateofr1:33Aof Nielsen 1992 forbedloaddominatedconditions.SomecommonwavelengthscalingrelationshipsfororbitalripplesaregiveninTable 3{1 .Analternativetheoryofbedformscalingisoeredby WibergandHarris 1994 whosuggestthatforeldscaleows,vortexripplesscalebestwiththegrainsize.Thisisnotalwaystrue,however,as Ardhuinetal. 2002 foundthatwaveformedvortexripplewavelengthsonthecontinentalshelfwereproportionaltothenearbedorbitalexcursions,notthegrainsize.Datafromlabandeldexperimentsdoconsistentlyshowdiscrepanciesbetweenvortexripplescaling[e.g. MillerandKomar 1980a ; Nielsen 1981 ; FaraciandFoti 2002 ]andmanyattributethesedierencestothedisparitybetweenowconditions.Whereasthemajorityoflaboratoryexperimentsareconnedtoregularoscillatoryforcingwithshortperiods,conditionsintheeldarerepresentedbyabroadspectrumofwaveheights,frequencies,directions,andotherowproperties.Attemptstounifyappropriatevortexripplescalesforeldandlabdatahavereceivedincreasingattentionrecently[e.g. MillerandKomar 1980a ; O'DonoghueandClubb 2001 ; Williamsetal. 2005 ; Lacyetal. 2006 ]andwillbediscussedfurtherinthefollowingsections. WibergandHarris 1994 expandontheearlierconceptsof Clifton 1976 whoidentiedthreecategoriesofsymmetricripplesthatdierintheircharacteristics: 1. orbitalripples; 2. anorbitalripples;and 3. suborbitalripples.Whereorbitalripplesscalewiththeorbitaluidmotionandarecommonlyfoundinlabexperiments, WibergandHarris 1994 suggestthatanorbitalripplesscalewithsedimentpropertiesandareindicativeofvortexripplesfoundintheeld.Suborbitalripplesareclassiedastransitionalbedformslyingsomewherebetweenorbitalandanorbitalripples. WibergandHarris 1994 statethatanorbitalandorbitalrippleheightrscalewiththeoscillatoryboundarylayerthicknesssuchthatorbitalrippleheightsareapproximately 68 PAGE 69 2whileanorbitalrippleheightsareabout=4.ThisisdemonstratedinFigure 3{2 andshowsthatwhileanorbitalripplesarecompletelyimmersedwithintheboundarylayer,orbitalripplestendtoprotrudefromit.Thesignicanceofthisismadeevidentthroughthechoiceofroughnessparametrizationsincoastalhydrodynamicmodels,whereoneisforcedtochooseappropriatevaluesofboundaryroughnessbasedontheinteractionofbedformswiththeboundarylayer.3.3ExperimentsAnumberofexperimentsonbedformdynamicshavebeenperformedinaneorttoprovidebetterunderstandingofripplegrowthandevolution,aswellassedimenttransportcharacteristicsintherippledbedregime.Theseexperimentsrangefromsmalltolargescalelaboratoryexperimentsutilizinganumberofdierenttestingmethods,toeldexperimentsandobservationsofbedformdynamicsunderthestochasticconditionsfoundinnature.Theexperimentaldatahavebeenusedtodevelopnewempiricallyderivedripplepredictorsdiscussedinthenextsectionandhaveleadtotheconclusionthatdiscrepanciesexistbetweenbedformsproducedbylabandeldscaleows.Additionally,somehavefoundthatthetypeoflaboratoryapparatususedinexperimentsmayaectbedformcharacteristicsandbehavior.AbriefoverviewofexperimentsonbedformdynamicsisprovidedinthefollowingsectionsandasummaryofhistoricalexperimentsanddatasetsisprovidedinTable 3{2 ,alongwithcitationkeysusedinthefollowingsections.3.3.1Laboratory MillerandKomar 1980b provideahelpfulreviewofcommonlyusedexperimentaltechniquesusedforsimulatingowsinthelaboratory.Althoughsomeexperimentaldatafromexperimentsinrotatingannularcellsexistinpublishedbedformliterature Schereretal. 1999 ; StegnerandWesfreid 1999 ,byfarthemostwidelyusedexperimentalsetupsinclude: oscillatingtray; 69 PAGE 70 oscillatingwatertunnel;and ume.Eachofthedeviceslisted,however,haveuniquecharacteristicsthathavesomeconsequenceonthecharacteristicsanddynamicsofbedforms.3.3.1.1OscillatingTrayTheoscillatingtraydevicehasbeenusedtostudyhydrodynamicsinthevicinityofstationarysimulatedripples EarnshawandGreated 1998 andalsobedformevolutioninasandbed Bagnold 1946 ; Manohar 1955 ; Sleath 1976 1985 ; Schereretal. 1999 ; Lacyetal. 2006 .Intheseexperiments,atrayofsandorstationarybedformsisoscillatedataparticularfrequencythroughatankofstillwatertherebysimulatingtheoscillatorymotionofwaterwaves. Lacyetal. 2006 usedamodiedsetuptosimulatecombinedwaveandcurrentforcingbypassingasteadycurrentoverthetrayasitoscillatedinaumeatvariousanglestothesteadyow.Theowinducedbytheoscillatingtrayapparatus,however,isnotadirectanalogofoscillatorywavemotionoverastaticbed.Steadystreamingeectsfoundinoscillatoryboundarylayerowsarenotsimulatedbytheapparatus,therebyexcludingtheadditionalverticaluxofmomentumdirectedawayfromthebed.Themotionofthetrayhasalsobeenfoundtoinduceameanowdirectedoppositetothedirectionoftravel Schereretal. 1999 .Observedbedformcharacteristicsinoscillatingtrayexperimentsalsoappeartobeunique,where Schereretal. 1999 notethattheapparatusimpartsanadditionalinertialforcetothebedforms.Furthermore,rollinggrainrippleshaveonlybeenfoundtobestableinoscillatingtrayexperimentswheretheyaretransientinoscillatingwatertunnelandumeexperiments,andrarelyobservedintheeld MillerandKomar 1980b .3.3.1.2OscillatingWaterTunnelAmongothers,theoscillatingwatertunnelhasbeenusedtostudybedformcharacteristicsby FotiandBlondeaux 1995a Voropayevetal. 1999 O'DonoghueandClubb 2001 ,and DoucetteandO'Donoghue 2006 .Similartotheoscillating 70 PAGE 71 trayapparatus,oscillatingwatertunnelsaredesignedtosimulatetheoscillatorymotionofwaves.Mostoscillatingwatertunnelsconsistofalongclosedconduithorizontaltestsectionandpistonslocatedinverticalrisersateachendthatareusedtodrivetheoscillatoryow.Theresultingoscillatoryowispurelyhorizontalandmostlyonedimensional.Liketheoscillatingtrayapparatus,thehorizontalowisofconstantphaseandlacksanyverticalcomponentduetoboundarylayerstreaming.Insomecasesitispossibletosimulateacollinearsteadycurrentsuperimposedontheoscillatorymotion.Adistinctadvantageoftheoscillatingwatertunnelisthattheowcharacteristicscanbepreciselyprescribedandcontrolled.Whilemostumeandoscillatingtrayexperimentsarelimitedtolabscaleowsofshortperiodwaves,lowtomoderatecurrentvelocities,andlowtomoderatemobilitynumbers,theoscillatingwatertunnelhasbeenusedtosimulatewaveconditionsmorecommonlyfoundinnature. O'DonoghueandClubb 2001 and DoucetteandO'Donoghue 2006 utilizedthewatertunnelapparatustosimulatelongperiodwavesandtoanalyzethedierencesbetweensymmetricandasymmetricforcingonbedformcharacteristics.Whilenosignicantdierenceinrippleheightandwavelengthwasfoundforsymmetricandasymmetricows, O'DonoghueandClubb 2001 didndthatpreviouslysuggestedmethodsforpredictingbedformgeometrywereinsucient,especiallyforthreedimensionalripples.3.3.1.3FlumeFlumeshavebeenusedextensivelytostudybedformdynamicsundersteadycurrents Yalin 1985 ; Mantz 1990 ; ColemanandMelville 1994 1996 ,waves MillerandKomar 1980b ; Marshetal. 1999 ; FaraciandFoti 2002 ,andcombinedwaveandcurrentforcing KhelifaandOuellet 2000 ; Lacyetal. 2006 .Anadvantageofumeexperimentsoveroscillatingtrayandoscillatingwatertunnelexperimentsisthatthetrueoscillatorynearbedmotionissimulatedbythesurfacegravitywaves.However,mostlaboratoryumesareonlycapableofproducingshortperiod,unidirectionalwavesnottrulyrepresentativeofeldconditions.Certainexceptionsdoexist,suchasthelargewave 71 PAGE 72 umesatOregonStateandDelftUniversityofTechnology,andexperimentsinthesefacilitieshelpbridgetheinformationgapbetweenlabscaleandeldscalebedformcharacteristics MillerandKomar 1980b .Asmentionedearlier,theoscillatorymotionproducedbyoscillatingtrayandoscillatingwatertunnelexperimentsisnotatruerepresentationofwaveinducedmotionintheboundarylayer.Progressivewavesinduceasteadystreamingintheboundarylayerthat,whileweakcomparedtotheoscillatoryow,hasasignicanteectonsedimenttransport Marin 2004 .Boundarylayerstreamingischaracterizedbyaverticaldiusiveuxofmomentumdirectedawayfromthehorizontalboundarylayer. DohmenJanssenandHanes 2002 suggestthatthisstreamingresultsinnettransportratesunderwavesthatare2:5timeslargerthanthoseinuniformhorizontaloscillatoryow.Furthermore,thewavebottomboundarylayercontainscomponentsofvelocityinbothhorizontalandverticaldirections.Theseverticalvelocities,whicharenotexactlyoutofphasewiththehorizontalvelocitiesinthebottomboundarylayer,leadtoanonzerotimeaveragedbedshearstress Marin 2004 .Fieldconditionsarerarelyclassiedbyonetypeandsizeofwave.Rather,elddataarecharacterizedbyabroadspectrumofwavefrequencies,directions,heights,andcurrentvelocities MillerandKomar 1980a .Eventhefewlargewaveumescapableofproducinglongperiod,irregularwavesareunabletosimulatemultidirectionalwavesorwavessuperimposedonasteadycurrent.Therefore,methodsforpredictingripplegeometrythathavebeenbasedonlaboratorydatatendtofailwhenappliedtoelddata.Inaneorttoovercomeatleastoneoftheseshortcomings, KhelifaandOuellet 2000 performedlaboratoryexperimentsonbedformcharacteristicsincombinedwaveandcurrentows.TheyaccomplishedthisbyusingtwointersectingumesatLavalUniversityandwereabletosimulateprogressivewavescombinedwithsteadycurrentsat60and90degreeanglestooneanother.Theresultingdatawereusedtoformulatenewempiricallyderivedexpressionsforbedformheightandlengthincombinedows, 72 PAGE 73 butunfortunatelytheirutilityissomewhatlimitedbytheshort<2swaveperiodsgeneratedintheume.Morerecently, Lacyetal. 2006 attemptedtoovercomethisscalediscrepancybysimulatingeldscalewaveperiods>8sandcurrentvelocitieswithanoscillatingtrayinacurrentume.Thefullyinstrumentedtraywaslledwithsandandthenoscillatedatvariousanglesbothobliqueandorthogonaltoasteadycurrentgeneratedintheume.Thedatafromtheirexperimentsareawaitingpublication.3.3.2FieldObservationsofripplecharacteristicsundereldconditionshavebeenfoundtodiersubstantiallyabout300%fromthosefoundinlaboratoryexperiments FaraciandFoti 2002 .Inthelaboratory, Marshetal. 1999 suggestthatbedformlengthscalesrapidlyunderthemonochromaticforcingtypicalofmostwaveumes.Theyspeculate,however,thatripplewavelengthmustbediculttochangeintheeldsincethebedformsaresubjectedtoabroadspectrumofwavefrequencies.Oftentimesobservedripplecharacteristicsarenotindirectequilibriumwiththelocalforcingandexperiencesignicanthysteresis Traykovskietal. 1999 .Morespecically,bedformsintheeldmaybearesultofanantecedentwaveclimatethatremainstaticaslocalforcingconditionssubsidebelowthecriticalthresholdrequiredforsedimentmobilization.Thishasmadeitdiculttoformulateuniedexpressionsforbedformgeometryundereldscaleconditions.Additionalcomplicationsarisewhentryingtoassociaterippleprocesseswithasingle,statisticalrepresentationofaforcingspectrumcomposedofmanywavefrequenciesanddirections. MillerandKomar 1980a suggestedthatsignicantwaveparameterscomputedfromeldspectrashouldbeusedinthecalculationofrelevantbedformparameterssuchasmaximumorbitalvelocityamplitude,orbitalexcursionlength,mobilitynumber,andShields'parameter.Nevertheless,itremainsdiculttoformulatenewexpressionsbasedonelddataandthepredictivecapabilityofmethodsderivedfromeldscalelabexperimentsisstillpoor.Observationsandmeasurementsofbedformcharacteristicsintheeld,however,continuetosupplementexistingtheoryandprovide 73 PAGE 74 moredetailedinformationabouttheirgeometryandbehaviorundernaturalforcingconditions LiandAmos 1999 ; Traykovskietal. 1999 ; Hanesetal. 2001 ; Ardhuinetal. 2002 .3.4RipplePredictorsThefollowingsectionpresentssomecommonlyusedmethodologiesfromexistingliteratureforpredictingrippleheightrandwavelengthr.TheseripplepredictorsarebasedonextensivesetsofdatafromlabandeldexperimentsseeTable 3{2 ,andinmanycasesarebasedonnondimensionalgroupsthatrelatesedimentcharacteristicswithowproperties.AnumberofthesedimensionlessparametersusedtoclassifyorcharacterizebedformsarelistedinTable 3{3 .3.4.1Clifton976Thesemiquantitativemodelof Clifton 1976 C76isbasedonthenearbedorbitalvelocityubandnearbedorbitalvelocityasymmetryub:ub=A!{2aub=14:8H2 LTsinh4kh{2bwhereHiswaveheight,Liswavelength,kisthewavenumber,andhiswaterdepth.Theseparametersarethenusedtoidentifyoneoffourbedstates:nosedimentmovement;symmetricripples;asymmetricripples;andsheetow.Thesymmetricripplebedstateisfurtherdividedby Clifton 1976 intoorbital,anorbital,andsuborbitalripplesasdiscussedpreviously.3.4.2Nielsen1981 Nielsen 1981 N81oerssemiempiricalformulationsforthesizeandshapeofvortexripplesderivedfromananalysisofvarioussetsoflabandelddata.AstrongdependenceonShields'parameterwasfoundforripplesteepnesswhiletheripplewavelengthrdatacollapsedbestwhenplottedagainstthemobilitynumber Brebner 1980 ,whichisaratiobetweensedimentdestabilizingandstabilizingforces.The 74 PAGE 75 formulationsoeredby Nielsen 1981 areparticularlyusefulsincetheydierentiatebetweenlabandeldconditions.WhencomparedtotheBASEXdataof VincentandOsborne 1993 Marshetal. 1999 ndthatthetwosetsofequationsserveasboundingequationsforthedata.Forrelativelyweakconditions<20,theripplewavelengthandsteepnessaregivenbyr1:3A{3ar=rmax0:32tan:{3bForconditionsoutsideofthisrange,orforgeneraluse, Nielsen 1981 providesthefollowingrelationshipsforregularlaboratorywaves;r=A=2:2)]TJ/F15 11.9552 Tf 11.9552 0 Td[(0:3450:34<<230;{4ar=r=0:182)]TJ/F15 11.9552 Tf 11.9551 0 Td[(0:241:52:5;{4br=A=8><>:0:275)]TJ/F15 11.9552 Tf 11.9551 0 Td[(0:022p for<1560for>156{4candforirregulareldwaves;r=A=exp693)]TJ/F15 11.9552 Tf 11.9552 0 Td[(0:37ln8 1000+0:75ln7!;{5ar=r=0:342)]TJ/F15 11.9552 Tf 11.9552 0 Td[(0:344p 2:5;{5br=A=21)]TJ/F23 7.9701 Tf 6.5865 0 Td[(1:85>10:{5cEquations 3{4c and 3{5c correspondspecicallytoquartzsandwiths=2:65.ParameterssuchasA,,and2:5usedintheripplepredictorsforirregularwavesEquations 3{5a { 3{5c shouldbebasedonsignicantwaveparameters Nielsen 1981 75 PAGE 76 Intheprecedingexpressions,thegrainroughnessShieldsparameter2:5iscomputedas2:5=1=2f2:5A!2 s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd=1 2f2:5{6awherethegrainroughnessfrictionfactorf2:5isbasedontheformulationof Swart 1974 Equation 2{4 witharoughnessksof2:5d50,f2:5=exp"5:2132:5d50 A!0:194)]TJ/F15 11.9552 Tf 11.9551 0 Td[(5:977#:{6b3.4.3GrantandMadsen1982BasedonthelabdataofC69,theformulationsof GrantandMadsen 1982 GM82forbedformcharacteristicsarebasedontheskinfrictioncomponentofboundaryshearstress. GrantandMadsen 1982 identifytwospecicrangesforbedformbehavior;anequilibriumrangewhereripplesteepnessremainsconstantandthewavelengthchangeswiththeshearstress;andabreakorangeathighershearstresswhereripplesteepnessdecreases.Fortheequilibriumrange=cr<=crB,therippleheightandsteepnessaregivenbyEquations 3{7a and 3{7b ,respectively.Ripplecharacteristicsinthebreakorange=cr>=crBarecomputedusingEquations 3{8a and 3{8b .Thebreakorangeisdeterminedas=crB=1:8S0:6,whereSisadimensionlesssedimentparameterdenedinTable 3{3 .r=0:22A=cr)]TJ/F23 7.9701 Tf 6.5865 0 Td[(0:16{7ar=r=0:16=cr)]TJ/F23 7.9701 Tf 6.5865 0 Td[(0:04{7br=0:48AS0:8=cr)]TJ/F23 7.9701 Tf 6.5865 0 Td[(1:5{8ar=r=0:28S0:6=cr)]TJ/F23 7.9701 Tf 6.5865 0 Td[(1:0{8bUnfortunately,accuratevaluesoftheboundaryshearstressarenotcommonlyknownapriori,noraretheycharacterizedbyonespecicvalue.Thesetwoshortcomingsmake 76 PAGE 77 theripplepredictorof GrantandMadsen 1982 acumbersomechoiceforbothestimatingandcomparingripplecharacteristics.3.4.4WibergandHarris1994Whenplottedagainsttherelativeorbitaldiameter2A=d, WibergandHarris 1994 WH94ndthatnondimensionalripplepropertiesfromeldandlabdatafallintouniquegroups.Theyproposethatalackofsubstantialoverlapinforcingconditionsbetweenlabregularwaves,shortperiodowsandeldirregularwaves,longperiodowsdatamakesitdiculttoeectivelyclassifyripplesortocharacterizetheirheightandlength.Indeed,therearefewexistingripplepredictorsthatperformequallyaswellatpredictinglabandeldscaleripples.ThroughananalysisofthelaboratorydataofC69,KF65,MK72,D74,andtheelddataofI57andD74, WibergandHarris 1994 ndthatanorbitalripplewavelengthanofoundinelddataisindependentofthenearbedorbitalexcursions.Instead,theyndthatanoscalesproportionallywiththegraindiameterdrangingbetween400d PAGE 78 Equation 3{10a ,anorbitalEquation 3{10b ,orsuborbitalEquation 3{10c :2A=ano<20orbitalripples {10a 2A=ano>100anorbitalripples {10b 20<2A=ano<100suborbitalripples {10c Ifthecomputedvalueof2A=anodoesnotfallintheanorbitalrange,thentheripplepropertiesmustthenbecomputedusingeitherorbitalEquations 3{11a and 3{11b orsuborbitalEquations 3{12a and 3{12b formulations. WibergandHarris 1994 suggestthatthewavelengthsofsuborbita 