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## Material Information- Title:
- A Coupled Modeling System to Predict Morphology Changes and a Comparison of Pressure Gradient Forces to Shear Stresses in the Nearshore
- Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Bathymetry ( jstor )
Modeling ( jstor ) Pressure gradients ( jstor ) Sediment transport ( jstor ) Sediments ( jstor ) Shear stress ( jstor ) Shorelines ( jstor ) Stress waves ( jstor ) Velocity ( jstor ) Waves ( jstor )
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- University of Florida
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- University of Florida
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- 7/30/2007
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A COUPLED MODELING SYSTEM TO PREDICT MORPHOLOGY CHANGES AND A COMPARISON OF PRESSURE GRADIENT FORCES TO SHEAR STRESSES IN THE NEARSHORE By WILLIAM L. REILLY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2005 Copyright 2005 By William L. Reilly This work is dedicated to my mother, father, and sister. ACKNOWLEDGMENTS I wish to thank my friends and family for their continued support and encouragement. I would not be where I am today if it was not for the guidance, direction, and advice of my wonderful family. It is my close friends and everyone that have meant something to me along the way that makes me who I am today. Dr. Donald Slinn, my graduate advisor, must be acknowledged for his continued support, not only academically but also in his efforts of helping me with my future endeavors. I also thank Dr. Robert Thieke and Dr. Ashish Mehta for taking the time to serve on my committee. I must also thank the other professors at the University of Florida in the Civil and Coastal Engineering Department for their dedication and effort of sharing their knowledge to the next generation of coastal engineers. Lastly, I would like to acknowledge a few other people who have made this project possible. Appreciation is given to Bill Birkemeier for his help in my growth of an aspiring engineer as well as the Field Research Facility for providing the data for this project. Finally, I would like to thank Nathaniel Plant as well as the other members of the Naval Research Lab for their continued support on this project. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF FIGURE S ......... ....................... ............. ........... vii ABSTRACT .............. ................. .......... .............. ix CHAPTER 1 OVERVIEW .................................. .. .................................. .............. . 1.1 P reface ................................................................ . 1 1.2 O organization ................................................................. 2 2 A COUPLED MODELING SYSTEM TO PREDICT MORPHOLOGY CHANGES IN THE NEARSHORE ........................................ ........................ 4 2.1 Introduction ............... ......... ......... ................... ............ ......... 4 2.2 O observations ................................................................ 6 2 .2 .1 B athy m etric D ata ..............................................................................6 2.2.2 M odel G rid ..............................................................7 2 .2 .3 W av e D ata ....................................................... 9 2 .3 M odel D description ............................................................10 2 .3 .1 W av e M o d el ......................................................................................... 12 2.3.2 H ydrodynam ic M odel .................. .................. ............... .... ............ ..... 13 2.3.3 Sediment Transport M odel ................... .......................... ....... .... ....18 2.4 Results.............................................. ......................... 22 2.4.1 O october 27th Sim ulation............................................................. 23 2.4.2 N ovem ber 6th Sim ulation.................................................. ............... 32 2.4.3 Com prison ................................................ ............... ............... 36 2 .5 D isc u ssio n ....................................................................................................... 3 7 2.6 Conclusion ..................................................................... ........ 40 2 .6 .1 F u tu re W o rk .......................................................................................... 4 0 2.6.2 Closing R em arks ................................................. ........ 41 3 PRESSURE GRADIENT FORCES AND SHEAR STRESSES ON SAND GRAINS UNDER SHOALING WAVES .................................... ...............42 3.1 Introduction ................................................... ............... .. .......42 3.2 Stress Form ulation .................. ............................ .. ...................... .. 43 3 .2 .1 S h ear Stress ............................................................4 3 3.2.2 Pressure G radient Stress ...................................... ...................... ........... 45 3 .2 .3 P h ase L ag .........................................................4 7 3.2.4 N onlinear W ave ............................................................ ...... ..... .... 48 3.3 V arying Param eters and Lim its ........................................ ......................... 50 3.4 R esults...................................................... 53 3 .4 .1 L in ear R esu lts ............................................................. ........ .... .... .. 5 3 3.4.2 Nonlinear Results .................................... .......... ..............58 3.5 Conclusion .................................................................... ......... 62 APPENDIX DERIVATION OF THE RESULTANT FORCE ON A SPHERE ...........65 L IST O F R E F E R E N C E S ...................... .. ............. .. ....................................................67 B IO G R A PH IC A L SK E TCH ...................................................................... ..................72 LIST OF FIGURES Figure page 2-1. Initial bathymetry for Oct. 27, 1999 and Nov 6, 1999....................... ...............9 2-2. Schematic of the coupled modeling system. .................................... ...............10 2-3. Fall Velocity as a function of offshore distance.................................................20 2-4. W ave characteristics for October 27, 1999. ............. ................................................22 2-5. Wave characteristics for November 6, 1999. ..........................................................23 2-6. Time series of the cross shore and alongshore velocities for October 27th .............24 2-7. Time averaged alongshore stream function for October 27th..................................25 2-8. Bathymetric change from hour 25 to hour 35. ................................. ...............26 2-9. Wave focusing around bathymetry for October 27th........................................27 2-10. Difference in bathymetry from original for Oct 27th..............................................29 2-11. Difference in bathymetry for October 27th with the shoreline held static ..............30 2-12. Difference in bathymetry for October 27th after 130 hours with the shoreline held static........................................................................................ 3 1 2-13. Time series of the cross shore and alongshore velocities for November 6th............32 2-14. Snapshot of the vorticity fields .................................................... ............... 33 2-15. Time averaged alongshore stream function for November 6th...............................34 2-16. 3-Dimensional bathymetry for November 6th. ....................................................35 2-17. Difference in bathymetry for November 6th with the shoreline held static .............36 3-1. Schem atic of a sand grain ...................... .... .................. ................. ............... 45 3-2. Phase lag of stresses. ............................................. .. .... ......... .. ...... 47 3-3. N onlinear w ave. ......................... ...................... .. .. ............. ........ 49 3-4. N onlinear w ave addition. ............................................... .............................. 50 3-5. Linear: T = 7sec, d = 0.3m m H = m ......................................... ............... 53 3-6. Linear: T= 7sec, d = 0.3m m H = m ......................................... ............... 54 3-7. Linear: T = 7sec, d = 0.5m m H = m ......................................... ............... 55 3-8. Linear: = 2.5sec, d = 0.3m m H = m ....................................... ............... 56 3-9. Linear: T= 2.5sec, d= 0.5mm H = m ..................................... ............... 57 3-10. Nonlinear: = 7sec, d= 0.3mm, H = m .......................................................58 3-11. N onlinear: T= 7sec, d= 0.3m m H = m ................................... .................59 3-12. Nonlinear: T= 7sec, d= 0.5mm H = m ................................... .................60 3-13. Nonlinear: T= 2.5sec, d = 0.3mm, H = m ................................. ............... 60 3-14. Nonlinear: T= 2.5sec, d = 0.5mm, H = m ................................. ............... 61 viii Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science A COUPLED MODELING SYSTEM TO PREDICT MORPHOLOGY CHANGES AND A COMPARISON OF PRESSURE GRADIENT FORCES TO SHEAR STRESSES IN THE NEARSHORE By William L. Reilly August 2005 Chair: Donald Slinn Major Department: Civil and Coastal Engineering Two separate, but related, topics are investigated in this study. The goal of the first half of the study is to simulate beach morphology on time scales of hours to days. Our approach is to develop finite difference solutions from a coupled modeling system consisting of nearshore wave, circulation, and sediment flux models. We initialize the model with bathymetry from a dense data set north of the pier at the Field Research Facility (FRF) in Duck, NC. The offshore wave height and direction are taken from the 8-meter bipod at the FRF and input to the wave-model, SWAN (Spectral WAve Nearshore). The resulting calculated wave induced force per unit surface area (gradient of the radiation stress) output from SWAN is used to drive the currents in our circulation model. Our hydrodynamic model is then integrated forward in time solving the 2- dimensional unsteady Navier Stokes Equations. The divergence of the time averaged sediment flux is calculated after one hour of simulation. The sediment flux model is based on the energetic approach of Bagnold and includes approximations for both bed- load and suspended load. The results of bathymetric change vary for different wave conditions. Typical results indicate that for wave heights on the order of one meter, shoreline advancement and sandbar evolution is observed on the order of tens of centimeters. While the magnitudes of the resulting bathymetric changes seem to be smaller than expected, the general shape and direction of transport appear to be reasonable. The second half of this study takes a systematic look at the ratio of horizontal forces on the seabed from shear stresses compared to the forces exerted by the pressure gradients of passing waves. This study was completed to investigate the importance of these pressure gradients to sediment transport. We develop analytic solutions, for linear and weakly nonlinear waves, to predict the forces felt by individual sand grains by passing waves. A range of wave frequencies and amplitudes, water depths, and grain sizes are varied to calculate the two horizontal forces. We demonstrate that the pressure gradient, for certain sediment sizes and wave regimes, can be sufficient to induce bed motion. A principal consequence of our findings is that near shore sediment transport parameterizations should not be based upon empirical relationships developed from steady open channel or even oscillatory flow experiments if they are not produced by surface gravity waves. This work will also help in the parameterization of sediment flux in the direction of wave advance due to asymmetric and skewed nonlinear wave shapes typical of shoaled and breaking waves. CHAPTER 1 OVERVIEW 1.1 Preface Accurate predictions of nearshore bathymetric change are challenging at all relevant scales. The difficulty lies in that the relevant scales span a very broad range. One could look at a very small scale process such as grain to grain interaction (O millimeters) or how a sandbar moves from a pre-storm position to a post-storm position (O meters) or even the scope of alongshore littoral cells (O kilometers). The largest spatial scales are of particular importance because they contain the majority of the spatial and temporal variability of nearshore bathymetric change [Lippmann and Holman, 1990; Plant et al., 1999]. In terms of temporal scales, similar breadth is encountered. One could look at very short time scales such as turbulent dissipation (O seconds) or tidal influence on sediment transport (O hours) or even chronic erosion (O years). To complicate things further, sediment suspension observed in the surf zone is also spatially and temporally intermittent on time periods of waves, storms, seasons or climate variability. Large changes in concentration occur over times shorter than a wave period [e.g., Downing et al., 1981] and at spatial scales shorter than a wavelength. The difficulty in modeling and prediction turns out to be acute at large scales, since the evolution at this scale requires the integration over all smaller scales [Roelvink and Broker, 1993]. The dominant causes of these small scale fluctuations in suspended sediment concentration are unclear and may include instabilities of the bottom boundary layer [Conley and Inman, 1994; Foster et al., 1995], vortex shedding from megaripples, or coherent turbulent flow structures [Hay and Bowen, 1994]. From a modeling perspective, these small scale processes are difficult to capture. They are driven by various forces including wave, current, and gravity driven flow. There is also significant experimental evidence that flow acceleration, which serves as a proxy for the horizontal pressure gradient in a coastal bottom boundary layer, has an effect on sediment transport [e.g., Hanes and Huntley, 1986; King, 1990; Gallagher et al., 1998; Butt and Russell, 1999; Drake and Calantoni, 2001; Elgar et al., 2001; Puleo et al., 2003]. 1.2 Organization The unifying theme of this project is its association with sediment transport. The first half of the study will attempt to predict the morphology of a real beach in the nearshore with a coupled modeling system. One part of the coupled modeling system is the calculation of the sediment transport. There are several mechanisms, many not yet well understood, that contribute to the movement of sediment in the nearshore. Due to this fact, multiple empirically based parameterizations are incorporated into the sediment transport formulation. The second half of the study will investigate one of these mechanisms that may be vital to accurately predicting sediment transport in the nearshore. Here we will explore the role that pressure gradient forces have on sand grains under shoaling waves compared to that of shear stresses. Chapter 2 will cover beach evolution on intermediate temporal and spatial scales. Section 2.1 presents an introduction and background on sediment transport models over the past few decades. Section 2.2 describes where the data has come from as well as how the model grid is set up. The individual parts of the modeling system are described in section 2.3 and how they are coupled together. Results are presented in section 2.4. Finally, a brief discussion and conclusion can be found in sections 2.5 and 2.6, respectively. The second half of the study on contributions of pressure gradients to sediment transport is presented in chapter 3. A brief background and introduction is given in section 3.1. Section 3.2 takes a closer look at the two stresses. It is here that we examine how at certain wave phases the pressure and shear forces work together to mobilize the bed and explore their interaction under a surface gravity wave. Section 3.3 describes our approach and clarifies the formulation. Results for linear and weakly nonlinear waves are presented in section 3.4. Finally, the paper will close with the conclusion in section 3.5. CHAPTER 2 A COUPLED MODELING SYSTEM TO PREDICT MORPHOLOGY CHANGES IN THE NEARSHORE 2.1 Introduction Process-oriented energetics-based total load sediment transport models [Bagnold, 1963] relate sediment transport to the near-bottom flow field and have often been used to predict beach evolution. Bagnold's [1963] arguments are physically reasonable and they seem to capture some of the essential aspects of nearshore sediment transport. The problem is that Bagnold's model yield specific, and thus rigid, parameterizations of some variable small-scale processes. Bagnold sought to parameterize interactions associated with gravitational and near-bed turbulent forces that drive transport under uni-directional river flows. However, several responses of the flow, such as the velocity profiles, the instantaneous bed shear stress, the sediment flux, and the total amount of the mobilized sediment cannot be fully parameterized by a quasi-steady free-stream velocity [Hsu and Hanes, 2004]. The formulas are based on riverine flow that is obviously not the condition that characterizes the nearshore environment. Since then other researchers have developed similar time-averaged versions of the energetic-based beach profile evolution models [Bowen, 1980; Bailard 1981; Stive 1986; Roelvink and Stive, 1989] using improved parameterization to represent net effects of the small-scale processes more accurately. Several recent studies have tried to compare observed bathymetric changes to predicted cross-shore profile changes from related models [Thornton et al., 1996; Gallagher et al., 1998]. These studies used measured near-bed velocities to drive the sediment transport model. Both studies concluded that the transport model predicted patterns of offshore transport that were accurate during undertow dominated conditions, where sediment transport was dominated by strong, seaward directed, near-bed, cross-shore mean flow. However, the slow onshore migration observed during low-energy wave conditions was not predicted well. Detailed hydrodynamic information is hardly ever available in any nearshore environment unless an intensive field study is being conducted. In contrast, wave data is readily available in many coastal environments. Instead of measuring the hydrodynamics from observations, this study will compute the hydrodynamics from measured boundary condition data. The sediment transport is then estimated from the computed hydrodynamics. Plant et al. [2004] found significant predictive skill in a similar approach by tuning several free parameters in the sediment transport model. Plant's study was able to find significant predictive skill for conditions dominated by onshore and offshore transport. However, in order to obtain this skill, it was necessary to allow the model parameters to vary with changing wave conditions. A forward stepping model is sensitive to model parameters, and these parameters should be independent [Plant et al., 2004]. In the present study, the sediment transport parameterization is fixed with one set of model coefficients. There are many different sediment transport mechanisms, all of which are not included in this model. Therefore the model is limited in that respect. Our model has representation for the depth averaged mean currents, and the combined wave and current boundary layer transports. The model does not yet explicitly include effects contributed by undertow, wave skewness and asymmetry, breaking waves and turbulence, and surface wave induced pressure gradients. The goal of this work has been to develop a rational framework for a beach morphology model coupled to a wave and mean current model. The precision of knowledge is still lacking pertaining to these unaddressed mechanisms. It is left to the community and future work to fine tune the relative contributions that may be attributable to these neglected mechanisms. Furthermore, current computer technology requires many approximations in order to simulate for multiple days. To run such extensive simulations you need to run with a 2-D circulation model and a phase averaged wave model. Therefore the undertow, wave skewness, and the turbulence intensities are all parameters that would have to be estimated by some yet-to-be-evaluated methods. As a result, these features were reasonably placed beyond the scope of this stage of the project. 2.2 Observations Observations used in this study were obtained from the Army Corps of Engineers' Field Research Facility (FRF), located near the town of Duck, North Carolina on a barrier island exposed to the Atlantic Ocean [Birkemeier and Holland, 2001]. We utilized data from the SHOWEX (SHOaling Waves EXperiment) field experiment conducted in the fall of 1999. This data set was chosen for its spatially broad and temporally dense set of bathymetry that extends well beyond the FRF property limits. 2.2.1 Bathymetric Data Bathymetry was collected by the CRAB (Coastal Research Amphibious Buggy). The CRAB is a 10 meter tall amphibious vehicle, capable of performing surveys to a depth of 8 meters when incident wave heights are less than about 2 meters [Birkemeier and Mason, 1984]. During the SHOWEX experiment, the spatial sampling pattern consisted of shore normal transects, with along-track sample spacing of less than one meter. The transects typically spanned a 500 meter wide portion of the nearshore and some transects occasionally covered more than 1200 meters in the cross-shore direction. The alongshore spacing of the survey transects were about 45 meters. The extent of the transects in the alongshore, over 2000 meters, is what made this data set most attractive. The two major bathymetric data sets that will be employed in this study are ones that were collected between Oct-26-1999 and Oct-28-1999 and between Nov-05-1999 and Nov-07-1999. 2.2.2 Model Grid Most previous cross-shore profile evolution models [Bowen, 1980; Bailard, 1981; Stive, 1986; Roelvink and Stive, 1989; Gallagher et al., 1998; Plant et al., 2001] assume alongshore uniformity. This study utilizes the extended data set and grids the entire domain. A domain just north of the pier was chosen to eliminate irregular isobaths around the pier. Furthermore, our wave data is collected just north of the pier as well, so minimal sheltering effects will occur. A constant grid spacing of 5 x 5 meters was chosen with 100 columns and 200 rows. The columns are oriented parallel to the shoreline and the rows perpendicular to the shoreline. Hence, the model domain represents an area of 1000 meters in the alongshore direction and 500 meters in the cross- shore direction. The data were interpolated using the scale controlled methods described by Plant et al. [2002], enforcing smoothness constraints in both the cross-shore and alongshore. A 2-D Hanning filter with an interpolation smoothing scale of 10 meters in the cross-shore and 120 meters in the alongshore was used to generate interpolated profiles. An important note must be made about the grid rotation. For best model results the domain should be most closely rotated so that the shoreline and/or the sandbar is/are aligned parallel to the onshore and offshore boundaries. The FRF has tried to do this with their local coordinate system which deviates from the standard latitude/longitude coordinate system. This rotation is approximately 18 degrees counterclockwise from true north. Although this rotation is a best fit to align the entire shoreline parallel with the coordinate system, a new rotation is calculated with the subset grid used in this study. To find the appropriate second rotation, cross-shore trends were calculated and then aligned. The resulting rotation for this data set is an additional slight counterclockwise tilt. With the alongshore data nearly uniform, a cross-shore trend can be calculated and the data can be filtered. Filtering is done with a spline curve. The filtering will effectively force the unrealistic data, with respect to the cross-shore trend, to have a weighting of zero. It will also force the first derivative to zero at the alongshore boundary. The hydrodynamic model has a few domain requirements. The first requirement is that the gradients be constant on all boundaries. The second requirement of the hydrodynamic model is that there be periodic lateral boundaries. The boundary conditions are enforced with a b-spline curve, which is more sophisticated than a spline curve. The smoothing scale used here is 40 meters in the cross-shore and 150 meters in the alongshore direction. Lastly the hydrodynamics model does not allow bathymetry that is not submerged. Because of this, some interfering must be done in the area of the shoreline. Prediction of the region near the shoreline is not a specific objective although its inclusion is required for boundary conditions on both the sediment transport and the hydrodynamics. This is accomplished by modifying the bathymetry for depths shallower than 40 centimeters and then easing it to a constant depth of 5 centimeters at the shoreline. Experience on this project indicates that this does not significantly influence the solutions obtained for the beach morphology in deeper water. To address the same constraint of having a submerged bathymetry, the tidal variation is not incorporated into this study but is maintained at mean sea level. Bathy at the beginning of hour 0 Bathy at the beginning of hour 0 for 27 Oct 1999 (m) for 06 Nov 1999 (m) 200 20 1- 1 -1 S3 3 o00 1':'' 1 CC 4 4 -6 -6 20 40 60 80 100 20 40 60 80 100 crossshore grid x (5m) crossshore grid x (5m) Figure 2-1. Initial bathymetry for Oct. 27, 1999 and Nov 6, 1999. The two initial bathymetries, which were surveyed in the same approximate area, have some significant differences. The survey on October 27th illustrates a more intricate sandbar system with large amplitude perturbations. This results in a more complex nearshore flow field. In contrast, the survey taken on November 6th exhibits a more linear and shore parallel bar structure. 2.2.3 Wave Data The wave data was collected using the 8-meter array at the Field Research Facility. The FRF array consists of 15 pressure gauges (collectively referred to as gauge 3111) mounted approximately 0.5 meters off the bottom. It is located in the vicinity of the 8- meter isobath about 900 meters offshore and to the north of the research pier. Voltage analogs of pressure signals are hard-wired through 10-Hz, fourth-order, Butterworth filters (primarily to eliminate 60-Hz noise) and indicates an accuracy of the pressure equivalent of 0.006 meters of water for wave-induced fluctuations [Birkemeir and Holland, 2001]. These gages supplied estimates of the RMS (root mean squared) wave height, period, and dominant direction at three hour intervals. The dominant wave direction was subsequently rotated to fit our domain orientation. After temporal interpolation to hourly intervals matching the modeling systems computational intervals between bathymetric changes, the wave data supplied by the 8-meter array were applied to the time-varying boundary conditions. 2.3 Model Description With the initial measured bathymetry and the boundary conditions measured offshore, the wave field and circulation are computed. The computed hydrodynamics then drive a sediment transport formulation. The divergence of the transport is used to predict bathymetric changes. These bathymetric changes are then inserted back to the hydrodynamic model at the subsequent interval with new forcing from the wave model. A resulting coupled modeling system for predicting bathymetric evolution has been developed. Figure 2-2. Schematic of the coupled modeling system. Figure 2-2. Schematic of the coupled modeling system. The assumption is made that the seabed evolves slowly, such that the feedback of the changing bathymetry and the sediment transport to the hydrodynamic model are only updated on a bathymetric timescale. In other words, the time step taken by the hydrodynamic model (0.075 seconds) is extremely short compared to the morphologic timescale. Therefore, the slowly evolving bathymetry is updated only after intervals of approximately one hour when the hydrodynamics and sediment transport have been integrated forward using the previously computed bathymetry. This eliminates the need for computationally costly time dependant bathymetric updates and new estimates of the wave field at frequent intervals. This assumption is widely accepted and has been used by others experimenting with coupled modeling systems. Gallagher et al. [1998] and Thornton et al. [1996] use a coupled time interval of three hours while Plant et al. [2004] decided on a one hour interval. A time step interval of one hour was chosen for this model, after testing the sensitivity of the results to different time intervals. Different model durations were investigated briefly. Using measured wave characteristics has limited our runtime scope so that the model would run for a period that would incorporate significant bathymetric changes. For example, to see the response due to storm conditions, the model must simulate through a time period of intense wave conditions. Similarly, if the bathymetric alterations due to a sustained calm wave climate were of interest, the model needs to run throughout many days of calm wave conditions and the adjusted profile needs to be evaluated before any storm conditions occur in the data. As a result, model simulations become very dependent on the measured data. This is complicated due to the fact that the bathymetric data sets are surveyed over multiple days. To minimize error caused by evolving bathymetry during the surveys, the bathymetric observations were centered at a specific date within the survey and simulations were run from this initial time. This appears to be a valid assumption because surveys can not take place during intense wave conditions and therefore any bathymetric alterations would be minimal from start to finish of the entire survey. Typical model durations investigated here are approximately two days. Cases were chosen where significant bathymetric changes were large enough to be distinguished from measurement errors. The duration of our simulations were sometimes constrained by numerical instabilities caused by growing shoreline anomalies in the updated bathymetries (section 2.4.1). Others have found a similar range of most advantageous simulated durations. Holland et al. [2000] found that a 5-day interval was close to the optimum prediction interval. 2.3.1 Wave Model SWAN (Simulating WAves Nearshore), a third-generation stand-alone (phase- averaged) wave model was used to simulate waves over the bathymetry [Booij, 1999]. The offshore RMS wave height was converted to significant wave height and the wave data was submitted to the wave model. For time efficiency, the SWAN model grid spacing in the alongshore direction was increased from 5 meters to 25 meters. The 5 meter cross-shore spacing was preserved. To avoid a shadow zone of waves coming in at an angle, the SWAN input domain must be extended by a factor of three, doubling it in each alongshore direction. This is done by taking the alongshore boundary cross sections (this is also the mean trend cross section) and extending them out to a distance equal to the length of the original domain. The resulting domain is now three times the length of the original. Many parameters must be determined in order to initialize the wave model. A Gaussian-shaped frequency spectrum was chosen with a spectral width of 0.01 Hz and a directional spreading of five degrees. A constant depth induced wave breaking parameter was also decided on. The proportionality coefficient of the rate of dissipation is 1.0 and the ratio of maximum individual wave height over depth is 0.73. SWAN has the ability to output a number of different hydrodynamic properties. The one of interest to the hydrodynamics model is the gradient of the radiation stress. For small amplitude waves in irrotational flows, Longuet-Higgins and Stuart [1960] showed that the forcing due to waves is related to the wave radiation stress tensor S. The gradient of the wave radiation stress tensor is conveniently outputted by SWAN as the wave induced force per unit surface area: F- c and F = (2.1). a 9y a y The wave radiation stress tensor is defined as S = pg [n cos2 0 + n -1/2]Eduad S =S,= pgjnsinOcosOEdadO (2.2), S, = pgl[nsin2 0 + n 1/2]EdadO where n is the ratio of group velocity over phase velocity. The subscripts refer to the direction in which the forces act, where x points offshore and y points in the longshore direction. The gradient of radiation stress must be interpolated back to the original domain before being utilized by the hydrodynamic model. 2.3.2 Hydrodynamic Model Nearshore circulation can be modeled using the mass and momentum conservation equations that have been integrated over the incident wave timescale and depth. This model uses a simple forward stepping scheme in time with an interval of 0.075 seconds and the computational grid is set up identically to the bathymetric grid with 5 meter grid spacing. The model uses a third order Adams-Bashforth scheme [Heath, 2002] to calculate the time derivatives and a fourth order compact scheme to calculate spatial derivatives. For an initial condition, the fluid is started from rest (u(x, y, 0) = v(x, y, 0) = r(x, y, 0) = 0). A periodicity condition is imposed in the longshore direction. Free-slip or symmetric boundary conditions (/ = u = u = 0) are applied at both the shoreline and offshore boundaries using fourth order accurate ghost points. The depth averaged approach is used and assumed to be a reasonable approximation because of the large discrepancy of vertical to horizontal length scales. The vertical depth is under one percent of the horizontal extent of the domain. Time dependant movement of the free surface is included and a fourth order compact filter was added to the flow field in both directions. Following Ozkan-Haller and Kirby [1999], our computational fluid dynamics (CFD) model solves the two-dimensional, unsteady, Navier-Stokes equations for an incompressible, homogeneous fluid with variable water depth. + -(ud)+- (vd)= 0 at Ox ay au au au y. +u +v-= -g +x bx (2.3) at ax ay ax av av av ay -+u--+V--= -g- +r -'r Ot ax y y Fy Where u and v are the depth-averaged mean current velocities in the x and y directions, respectively. Here, 7 is the phase-averaged water surface elevation above the still water level, h is the water depth with respect to the still water level, and d = h + 7 is the total water depth. The hydrodynamic model will simulate unsteady alongshore currents in the surf zone driven by the gradients in radiation stress caused by obliquely incident breaking waves. The incident wave forcing effects are parameterized by TFx and TFy, and are expressed using the radiation stress formulation by Longuet-Higgins and Stewart [1964]. These terms reduce to F F rFx Fr' F (2.4) p d pd for straight and parallel contours. Bottom friction is taken into account through the nonlinear damping terms bx =f by f (2.5), d d where ( ) represents a phase-averaged quantity. The total instantaneous velocity vector, U, includes the cross-shore (u), longshore (v), mean, and oscillatory components of the velocity. U =u+v (2.6) U (u+u)+(v+v) Where u and v are the x and y mean velocity components, respectively and u and v are the x and y oscillatory components of velocity, respectively. The oscillatory component of the velocity can be represented as u= u cos0cos(o)t) S(2.7), v = uo sin 0 cos(oit) where wo is the radian frequency and uo is the orbital velocity in shallow water. Hr 1 u = (2.8), 2 sinh(kh) where c = 2ir/T and Tis the peak wave period. The local wave height and wave number are represented by H and k, respectively. The time average product of the instantaneous velocity magnitude ( U ) and the instantaneous alongshore or cross-shore velocity (u or v respectively), found in equation 2.5, are an important component of the circulation model. Fedderson et al. [2000] explains that direct estimation of (I u) requires a more detailed specification of the velocity field than is usually available, so the term I U nu) is often linearly parameterized even though linear parameterizations in the mean current frequently are inaccurate because the underlying assumptions (e.g., weak-currents) are violated. Except for the weakest flows, ( U u) depends strongly on the mean current and the total velocity variance oscillatoryy components of velocity) [Fedderson et al., 2000]. Mean and oscillatory velocities components are the critical constituents to calculating ( | u) and subsequently calculating the flow field well. To solve the difficult formulation of U Iu ), we use the nonlinear integral parameterization method. To do this, one must integrate over the wave period at every time step. U u) =1 i [Uo cos2 (cot)+2uo cos(c)t)(u cos0+vsin0)+u +V ]12[U+U0 cos 0cos(ctt)]dt S(2.9) | U| v: = -r[Uo cos (=at)+2uo cos(cmt)(ucos 0 + vsin0)+ + V ]'2[V+Uo cos Ocos(mct)]dt TT Following McIlwain and Slinn [2004], these integrals are evaluated numerically using Simpson's rule [Hornbeck, 1975] with 16 intervals. The above integrals require significant computational effort to evaluate and, in our case, will increase the computational time by a factor of three. If one is not willing to sacrifice this computational time, then the less accurate linear parameterization method should be used. The nonlinear damping terms z-b and z- are the free parameters in the model and truly are a source of uncertainty. The friction coefficient parameter, cf will effectively control the nature of the resulting motion. Depending on the values of the friction coefficient, fully developed fluctuations can behave in a variety of ways ranging from equilibrated, small-amplitude fluctuations to energetic, random fluctuations involving strong vorticies [Slinn et al., 1998]. Ozkan-Haller and Kirby [1999] and Slinn et al. [1998], both found that a stronger mean current, more energetic fluctuations in the velocities, faster propagation speeds, and more energetic vortex structures result as the friction coefficient cf is decreased. The flow field exhibits shear instabilities of the longshore current due to the reduction of this term and results in unsteady longshore progressive vortices. These shear instabilities are found to induce significant horizontal mixing in the surf zone and affect the cross-shore distribution of the mean longshore current [Ozkan-Haller and Kirby, 1999]. McIlwain and Slinn [2004] establish for the nonlinear integral parameterization method used here, the best agreement with observed data came from a cf value of 0.003. The effect of lateral mixing due to turbulence and the dispersive three-dimensional effect of the vertical fluctuations in the current velocities [Svendsen and Putrevu, 1994] have been neglected and left out of the shallow water equations. Ozkan-Haller and Kirby [1999] found, for reasonable mixing coefficients, that the mixing induced by the instabilities in the flow dominates over mixing due to eddy viscosity terms, which include the effects of turbulence and depth variation in the current velocities. They also found that the presence of the shear instabilities and the associated momentum mixing actually tends to suppress momentum mixing due to the eddy viscosity terms. 2.3.3 Sediment Transport Model For steady, two-dimensional unidirectional stream flow, Bagnold [1963] utilized an energetics-based sediment transport model assuming that the sediment is transported in two distinctly different modes. Sediment transport as bedload occurs via bed shear stress from the fluid flow plus the downslope contribution of gravity, while sediment transport as suspended load occurs via turbulent diffusion by the stream fluid. The total immersed weight sediment transport rate, i, can be represented as [Bagnold, 1966] i = ( +i = ( + -)o (2.10), S tan tanf (W/u)- tan/ where subscripts b and s refer to bedload and suspended load, respectively. The parameter / is the local bed slope and q is the angle of repose which is taken to be 28 [Julien, 1998]. The percent of power used for bedload and suspended load is represented with the efficiency factors es and s,. Following Thornton et al. [1996], b = 0.135 and E, = 0.015. The rate of energy production of the stream a), is equal to the product of the time-averaged bottom stress r and the mean free stream velocity, S= Tu (2.11). The shear stress at the bed can be represented by S= pWCfu u (2.12) Inserting equation 2.12 into equation 2.11 yields o =pCf u (2.13) where p, is the density of water. Mean and fluctuating velocity components contribute 3 to the nonlinear term just as described in section 2.3.2. The steady-flow transport equation is then extended to include oscillatory flows as well as steady flows [Bailard, 1981; Bailard and Inman, 1981]. The contribution of the longshore bottom stress is also included. The resulting time averaged immersed weight suspended sediment transport rate, Q, is S2U) tan Wx3)) 3 tani )) (2.14) t tjh12ev utan ly a K 13)) rr(t13 V)a t aev- any Q, tan K+ Kpw( | | W 1 5 with the units kg/s3 and ( ) represents a time average over many wave periods. The coefficients Kb and K [Gallagher et al., 1998] are __ Eb Pw Es K = w b and K= c- (2.15), Ps Pw tan 0 P p W where p, is the density of the sediment. Kb is dimensionless while Ks has the dimensions of s/m The coefficient of friction cf is the same as referred to earlier in section 2.3.2. Just as in the hydrodynamics model (section 2.3.2), the product (I U u) is solved using the nonlinear integral parameterization method integrating numerically with Simpson's rule. The sediment fall velocity is represented by W. The effect of grain size is handled only explicitly via the fall velocity. The offshore sands are often finer than the sand in the nearshore region and must be represented that way. Plant et al. [2004] found that the highest prediction skill was achieved by using a temporally constant but cross-shore variable distribution of sediment fall velocities. Sediment Fall Velocity 0.045 0.04 0.035 0.03 E S0.025 I W = 3.6213xu' 0.02 "L 0.015 0.01 0.005 0 0 100 200 300 400 500 600 Distance Offshore (m) Figure 2-3. Fall Velocity as a function of offshore distance.1 The bed evolution depends on the divergence of the time averaged transport rate, Q. Mass conservation in both the cross-shore and alongshore directions yields dh 1 dQ dQ -( xd + d ) (2.16). dt Agp, dx dy 1 Sediment fall velocity as a function of cross-shore location (following Sleath 1984) from data collected in 1984-1985 at the FRF [Stauble 1992]. A power series trendline is used to fit the data. Assuming the density of sediment packing is constant, A is the packing factor and taken to be 0.7 [Thornton et al., 1996]. The divergence of the time averaged transport rate is calculated at the end of each coupled time interval. The bathymetry can now be updated for the subsequent time interval. hn =hold + dh (2.17) The constraint of having a submerged bathymetry is still enforced after the bathymetry is updated. Therefore, to conserve mass, any bathymetry that emerges beyond the 5 centimeter depth contour will be scoured back to 5 centimeters and the remaining bathymetry at that node will be added to the closest cross-shore node in the offshore direction. This seems to only happen at the shoreline and may even represent shoreline accretion. But as mentioned before, prediction of the region near the shoreline is not a specific objective of this study. Extra filtering was incorporated to the sediment flux as well as the divergence of the sediment flux at the end of each coupled time interval to reduce any irregular isobaths primarily at the shoreline. With a fixed set of parameters in the hydrodynamic and sediment transport model, the calculations can become less stable when there is an oddity in the bathymetry especially near a boundary like the shoreline. Therefore this extra filtering was incorporated with a fourth order compact filter in both directions. While this technique was introduced for model stability issues, we note that this could even represent horizontal smoothing attributable to turbulent diffusion at the bed. Filtering may help our model stay numerically stable but it will consequently decrease the net localized accretion or erosion of small scale features. 2.4 Results Two model simulations were run in this study. One from the bathymetric data gathered in the region of October 27th, 1999 (hereinafter OCT27) and another from the bathymetric data surveyed around the date of November 6th, 1999 (hereinafter NOV6). The two simulations were started from approximately the same location area with each of their respected initial bathymetries (Section 2.2.2). They were stepped forward in time with measured wave data. The wave data varies quite dramatically between the two simulations. OCT27 (Fig. 2-4) represents a calmer wave climate period. While the first few hours are a bit stormy, it then calms down with wave heights hovering around 1 meter as a long South-East swell enters the nearshore. Wave Charade risl iE lor 1c999c127 1.5 -1- 3_= II AP 0 1 i 21 c 0 10 20 30 40 50 & 0 0 10 20 30 40 50 a l0 I I I aD 0 10 20 0 40 50 lime (hr) Figure 2-4. Wave characteristics for October 27, 1999. Wave C~ha rdaerel i io r 1999nov6 .A ic II 0.5 0 5 10 15 20 25 30 35 10 0 5 10 15 20 25 30 35 "r S100 d l y 0 5 10 15 20 25 30 35 lime (hr) Figure 2-5. Wave characteristics for November 6, 1999. Conditions were quite different for the NOV6 (Fig. 2-5) simulation. The first few hours of the simulation consist of waves with a long period approaching from the south- east but around hour 5 the wave climate dramatically strengthens. Waves begin to approach from the north-east with wave heights exceeding 1.5 meters and wave periods around 6 seconds. The performance of the model for the two simulations is more interesting because the initial bathymetries and the wave climates significantly differ. 2.4.1 October 27th Simulation The intricate initial bathymetry (Sec. 2.2) causes the flow field as well as the sediment transport to be quite complex for this simulation. The flow field and the transport also have a strong correlation with each other. When the waves are entering the nearshore from a large enough angle away from shore normal (+/- 10 degrees), a recognizable alongshore current can be identified. This is evident from time series of the alongshore and cross shore velocities at a single point in the flow field. v Velocity Vs. Time @ x=100m, y=500m for 1999oct27 0.5 0 - -0.5 -1 I L L L L 0 10 20 30 40 50 60 u Velocity Vs. Time @ x=100m, y=500m for 1999oct27 0.5 - EJ 0 =- > -0.5 I 0 10 20 30 40 50 60 time(hr) Figure 2-6. Time series of the cross shore and alongshore velocities for October 27th. The point chosen to record this velocity time series is located 100 meters offshore and 500 meters from the bottom of the grid. This point was selected because it is located close enough to the shore to track evidence of an alongshore current. Notice the cross shore velocity (u) time series will oscillate around 0 m/s and gives an indication of how unsteady the flow is. On the other hand, the alongshore velocity (v) time series indicates the alongshore direction of the flow. The correlation of wave direction and alongshore velocity direction can be seen as well as the unsteadiness of the cross shore velocity during the larger wave periods. The complex bathymetry causes the flow to exhibit a meandering around the perturbations in the sandbar similar to that reported by Slinn et al. [2000]. This is evident by looking at a stream function field of the alongshore velocity overlaying the existing bathymetry. Alongshore stream function between hour 25 & hour 35 200 10 10 20 30 40 50 60 70 80 90 100 1 -- S- 1 r / 4 A.) -5 -6 10 20 30 40 50 60 70 80 90 100 crossshore grid x (Sm) Figure 2-7. Time averaged alongshore stream function for October 27th. The stream function of the time averaged flow was calculated for a ten hour interval within the simulation. It is averaged from the beginning of hour 25 to the end of hour 34. During this time the waves are approaching from a south-east direction. Therefore the stream lines run from the bottom of the grid to the top. Note the more narrow spacing between the stream lines, denoting a stronger flow, in the nearshore. As you can see, the current senses the bottom and seems to curve around the high points. The flow field will continue to bend around these perturbations as long as the waves approach the shore with the same angle and height. The sediment transport is also correlated with the meandering of the alongshore current. Difference in bathy from hour 25 to hour 35 (m) 200 II 180 0.15 0.1 140 0 120 0.05 0.05 o100 - 80 o 80 60 -0.05 40- -0.1 20 - 10 20 30 40 50 60 70 80 90 100 crossshore grid x (5m) Figure 2-8. Bathymetric change from hour 25 to hour 35. Figure 2-8 shows the change in bathymetry recorded for the same simulated time span as the alongshore stream function above (Fig. 2-7). Looking at the bathymetry change away from the shoreline in the sandbar region (around 100m from the left edge of the grid), one can detect a sort of meandering of the sediment transport as well. A correlation is evident when compared to the stream function (Fig. 2-7). When the time averaged alongshore current is headed offshore, it is accompanied by an offshore transport. Conversely, when the stream turns back shoreward, an onshore sediment transport can be seen. Also note that the onshore transport regions are of stronger magnitude. This is likely related to the contribution from the wave velocities to the total velocity vector. When the principle wave axis and the mean current vector are aligned, the nonlinear product of the total velocity squared or cubed is much larger then when the two velocity components are not collinear. Figure 2-9 shows that local wave heights are also larger in these same regions. Bathy @ beginning of hour 6 (m) Hs @ beginning of hour 6 (m) 200 200 1.2 180 180 1 160 160 140 J 0.8 120 120 It is also apparent that the perturbations in the bathymetry are causing wave 0.4 focusing. This is a significant effect and often observed in nature but it also causes 0.2 -5h -6 20 40 60 80 100 20 40 60 80 100 crossshore grid x (5m) crossshore grid x (5m) Figure 2-9. Wave focusing around bathymetry for October 27 It is also apparent that the perturbations in the bathymetry are causing wave focusing. This is a significant effect and often observed in nature but it also causes additional complications for this study. The focusing creates large pockets of energy at the shoreline. Consequently, immense scouring and erosion can be found at these points. These shoreline perturbations can tend to grow and subsequently result in the flow becoming numerically unstable. Various methods to alleviate these difficulties were implemented with varying degrees of success. As stated above, prediction of the region near the shoreline is not our main specific objective although its inclusion is necessary for providing boundary conditions for both the transport divergence and the hydrodynamic formulations. Our main focus is on bar migration. Because of problems near the shoreline, however, our model simulation runs are limited to the duration until the hydrodynamics become numerically unstable around the odd and jagged shoreline bathymetry. When a fixed set of hydrodynamic and transport parameters are implemented, it is difficult for the model to cope with either variations in the wave climate or, in our case, peculiarities of the formulations in very shallow water near the boundary conditions. Shoreline stability and more accurate shoreline transformation is a future goal of this continuing study. A number of potentially fruitful avenues are open to further investigation. One that will be explored below is simply to make the shoreline static and let the remaining bathymetry evolve. The net bathymetry change exhibits a similar pattern of sandbar meandering and jagged cuspate features near the shoreline (Figure 2-10). These concentrated sharp shoreline perturbations are found directly behind the sandbar crests and are a direct effect of wave focusing. The magnitude of the sediment transport is relatively small; only about 20 cm of vertical change over 55 hours in the sandbar region. This may be realistic due to the calm wave climate but in our judgment more likely to be an underestimate. The next bathymetric survey occurred approximately 200 hours later and we were unable to integrate the modeling system forward to that duration because of the shoreline stability. This made it impossible to make direct comparisons of the model hindcasts with survey data. Difference from Original Bathy (m) after 55 hours for 1999oct27 2001 ;1 -^-,-------------------.. . ..--------0.-60.8_ 200, 1 n I 1 0.8 0.6 S0.4 crossshore grid x (5m) Figure 2-10. Difference in bathymetry from original for Oct 27th. -0.4 E -0.2 As mentioned above, a simulation was run holding the shoreline in place and letting the rest of the bathymetry evolve. This allows us to focus in on the transport magnitudes specifically in the sandbar region. But note, variation of the shoreline that may indeed happen and the consequential effects to the rest of the bathymetry are not included. A comparison of the two different approaches for similar time intervals reveal quite similar results but with the magnitude of the static shoreline approach slightly higher due to less filtering needed. Figure 2-11 shows the magnitudes of transport in the bar region for the same time interval as Figure 2-10 but with a static shoreline region. In the regions where sediment is moving onshore, some shoreward advancement of the sandbars can be seen, but only minimally. Difference from Original Bathy (m) after 55 hours for 1999oct27 200----- 180 0.2 160 - 0.15 140 810 - 100 -0 0 2 80 -0.05 40- 20 -0.1 10 20 30 40 50 60 70 80 90 100 crossshore grid x (5m) Figure 2-11. Difference in bathymetry for October 27th with the shoreline held static. The sediment transport appears to be building the bar more than moving it shoreward. This is even evident where there is a trough in the bar and net offshore transport is detected. The net sediment accretion is enclosed within sediment depletion on both the shoreward and offshore edges. While these magnitudes seem smaller than expected, the transport shape and directions appear reasonable and consistent with the model formulation. Holding the shoreline static also allows us to run our simulation further in time. A simulation of 130 hours was completed. Similar wave conditions were observed for the remaining 75 hours; relatively small waves (less than a meter) coming from the south east. Figure 2-12 shows the same general transport shape with larger magnitudes. Difference from Original Bathy (m) after 130 hours for 1999oct27 200 1 180 0.8 160 140 0.6 E 120 S0.4 100 10 20 30 40 50 60 70 80 90 100 crossshore grid x (5m) Figure 2-12. Difference in bathymetry for October 27th after 130 hours with the shoreline held static. The 130 hour simulation (just over 5 days) eventually incurred similar stability problems as the variable shoreline simulations. Instead of sharp cuspate features appearing and growing at the shoreline, the sandbar ultimately became jagged and caused the model to break down numerically. Therefore the model was never able to run a full 10 day simulation. Even after the 5 day simulation it is apparent that the sandbar evolution is not desirable. One would expect the sandbar perturbations introduced in October 27th survey to smooth out to the linear bar exhibited by the November 6th survey (Figure 2-1). This is obviously not the case from Figure 2-12. Here again, the bar seems to be building rather than smoothing out. 2.4.2 November 6th Simulation The initial bathymetry for NOV6 (Sec. 2.2) demonstrates a much more linear sandbar than OCT27's bathymetry. Furthermore, the wave climate and circulation is much more intense (Fig. 2-5). This is evident by the time series of the alongshore and cross shore velocities at the same point in the flow field. v Velocity Vs. Time @ x=100m, y=500m for 1999nov06 0.5 0 0.5 -"'A I -1.5 I I 0 5 10 15 20 25 30 35 0.4 i 0.2 0.2 0.4 LI I L 0 5 10 15 20 25 30 35 time(hr) Figure 2-13. Time series of the cross shore and alongshore velocities for November 6th The magnitude of the alongshore velocity is clearly more powerful and more unsteady in the cross shore direction than for the OCT27 simulation (Fig. 2-6). It is easy to see the unsteadiness when the vorticity fields are compared. The stream function is calculated during a 10 hour period of 1.5 meter waves coming from the north-east (so a particle would follow the stream line from the top of the grid to the bottom). Snapshot of Vorticity Field Snapshot of Vorticity Field for November 6, 1999 (s1) for October 27, 1999 (s1 200 -r 200 0.03 0.03 180 0 180 0.02 0.02 160 -160 - 0.01 0.01 140 140 0 120 120 S- 0.01 r 00.01 0) S100 100 S- -0.02 -C'.02 080 80 S-0.03 r 0.03 60 60 0.04 -0.04 40 40 20 20 U -0.06 0.06 20 40 60 80 100 20 40 60 80 100 crossshore grid x (5m) crossshore grid x (5m) Figure 2-14. Snapshot of the vorticity fields. The proximity of stream lines within the sandbar region show the relatively much stronger alongshore current with respect to the OCT27 simulation (Figure 2-7). Although there seems to be quite a bit of vortex shedding, the alongshore mean current flows in a very linear fashion parallel to the sandbar. Wave focusing and shoreline concentration, as stated above, appear to be a problem in this simulation as well. While the initial sandbar is nearly linear and parallel to the shoreline, it does have a distinct trough and high point located about 200 meters from the bottom of the grid. This may be easier to see in a 3-dimensional bathymetry as shown in Figure 2-16. Similar to the first simulation, the waves focus here, as well as at the northern portion of the grid, and subsequently the shoreline perturbations eventually cause the circulation model to become numerically unstable near the shoreline after 31 hours. Alongshore stream function between hour 18 & hour 28 200 180 i 160 II 140 120 2 100 | II Figure 2-15. Time averaged alongshore stream function for November 6th 0 80 40 -5 10 20 30 40 50 60 70 80 90 100 crossshore grid x (5m) Figure 2-15 Time averaged alongshore stream function for November 6" A total net sediment transport for the simulation after 31 hours of wave action was completed but, due to the magnitude of the shoreline perturbations, it is difficult to analyze the sandbar region specifically. Here again, another simulation was rn with the shoreline held static. This also allows the simulation to run for a full two days. In Figure 2-17 the sandbar region is looked at more closely. Wave height and wave direction stay consistent for the rest of the simulation; approximately one meter wave height and approach angle nearly shore normal. It is noted here again that holding the shoreline static did not significantly affect the magnitude or shape of the total net sediment transport. Figure 2-16. 3-Dimensional bathymetry for November 6th Similar to the NOV6 simulation, the magnitude of the sediment transport is also quite small; only about 35 cm of vertical change in the sandbar region. This is more likely to be an underestimate due to the fact that there is such an intense wave climate. The sediment transport in the sandbar region is nearly all moving onshore. The peak of the positive sediment transport is almost directly on top of the peak of the original sandbar and the depletion appears to be on the seaward slope of the bar. Again, the sediment transport seems to be building the bar more so than moving it. These magnitudes seem smaller than expected and the direction of transport seems suspect and will be discussed below, but the shape and areas of transport appear reasonable. Difference from Original Bathy (m) after 48 hours for 1999nov06 0.4 180 - 160 -0.3 140 0.2 S120 S1008 0 -I0 0 60 -0.1 40 20- -0.2 10 20 30 40 50 60 70 80 90 100 crossshore grid x (5m) Figure 2-17. Difference in bathymetry for November 6th with the shoreline held static. 2.4.3 Comparison OCT27's simulation as well as NOV6's simulation, to our judgment, produced less than expected sediment transport. The magnitude of transport is more acceptable for OCT27's case, because the wave climate is so mild, but seems more unrealistic for the case of NOV6's more energetic wave climate. It is hard to know what is sensible when comparable bathymetric surveys are spaced so far apart and the model is limited in time. The direction of transport correlates quite well with the stream line velocities of OCT27. With a straight and parallel initial sandbar, one would expect the shape of transport that was simulated for NOV6 but in the opposite direction. Sand bars on a natural beach typically move slowly shoreward when wave energy is low and move more rapidly offshore when waves are energetic and the wave driven circulation is strong [Winant et al., 1975; Aubrey, 1979; Jaffe et al., 1984; Wright and Short, 1984; Lippmann and Holman, 1990]. With NOV6's vigorous wave climate, one would expect a net offshore sediment transport. As noted above, the net shoreward transport can be explained by the wave and current velocities co-aligning and reasonably dominating. Furthermore, for both simulations, the peaks of the sandbars don't seem to be moving very much but rather predominantly building and steepening the bar. 2.5 Discussion Our model, which accounts for mean and oscillatory flow is based on the assumption that sediment suspension occurs instantaneously in response to fluid forcing. However, sediment suspension at one phase of the wave can be transported during a subsequent phase before settling to the bed [Hanes and Huntly, 1986]. Sediment transport where oscillatory flow is dominant, may also depend on fluid accelerations such as those caused by pressure gradients of the surface waves [Hallermeir, 1982; Hanes and Huntley, 1986; Gallagher, 1998] that are not accounted for in this model. With the fluid accelerations neglected, the alongshore current becomes an important mobilization force. Some have neglected this force [Bailard, 1981] but Thornton et al. [1996] found the best agreement between observed and predicted profile changes when the steady alongshore current contributed significantly to stirring of sediment that was subsequently transported offshore by the mean cross-shore flow. Improved predictions of profile changes using the energetic transport model are reported to result from including (in the wave and wave-driven circulation models) undertow [Stive and Battjes, 1984], wave asymmetry [Nairn and Southgate, 1993], breaking-induced turbulence [Roelvink and Stive, 1989], and infragravity waves [Sato and Mitsunobu, 1991]. Recently published extensions of the Bailard [1981] model [Hoefel and Elgar, 2003] demonstrated improved prediction skill compared to Gallagher, [1998], although the skill for the onshore bar migration sequence appears to be poorer than the results presented by Plant et al. [2004]. The extended models included a third- order statistic (i.e. skewness) of the acceleration computed from observed flow velocities. The timing of strong accelerations relative to onshore flow is hypothesized to produce net onshore sediment transport [Elgar et al., 2001]. Also added to their model were two additional free parameters, which will always increase the model hindcast skill [Davis, 1976]. A general inspection of parameterized nearshore process models by Plant et al. [2004] revealed that unresolved processes contribute to model errors. This suggests that additional information is necessary to adjust model parameters in order to minimize these errors. There is still not enough known about some of the processes that contribute to sediment transport. Dean and Dalrymple [2002] show the importance of the influence of wave-breaking-induced turbulence as a mobilizing agent within the surf zone. The dynamics of breaking waves on sand bars are understood only qualitatively. Gallagher [1998] suggests that vertical shear in mean cross-shore current may be significant. These various transport mechanisms are difficult to parameterize when little is known about them. To further complicate matters, their significance seems to vary with altering wave conditions. The relative importance of competing transport mechanisms is shown by Plant et al. [2001] to depend strongly on the relative wave height (defined as the ratio of the RMS wave height to the local depth). The shape and direction of the sediment transport for the OCT27 simulation looks as expected, except for the fact that the oddities in the bar are not really smoothing out but rather growing. This is an example of positive feedback. This is also the case for the shoreline perturbations which continue to grow and eventually make the model numerically unstable. In most cases in nature, there is negative feedback. When something gets out of equilibrium, nature usually slows it down and will try to bring it back. A good example of this is the air pressure of the earth. When there is a gradient of air pressure in the atmosphere, air in places of higher air pressure will shift air toward areas of lower pressure in the form of wind. Our model seems to have a positive feedback that must be addressed in future work. When a perturbation in the bathymetry is introduced, the model appears to magnify the disturbance rather than smooth it out as nature might under certain conditions. Not enough is known about all the mechanisms under shoaling waves to accurately predict or parameterize how this might be incorporated. Of course coastal zones also exhibit cases of positive feedback, in the form of erosional hotspots and non-uniform accretion to form beach cusps. Thus some of the model response, especially near the shoreline, may be representative of natural phenomenon. An important thing to remember is that even an ideal sediment transport model can be inaccurate. First, performance of a sediment transport model is difficult to evaluate in studies where the predicted transport also depends on the accuracy of the hydrodynamic model, wave model, and underlying bathymetry. Secondly, errors in initial conditions, such as bathymetric sampling errors, could lead to errors in model predictions. For example, ripples that might significantly affect both hydrodynamics and sediment transport are not typically resolved by most surveying practices [Plant et al., 2002]. Accurate results may develop for flow over a smooth bed but may be inaccurate for flow over ripples and megaripples. 2.6 Conclusion 2.6.1 Future Work The framework of a coupled beach morphology modeling system has been developed. It was tested for two data sets and environmental conditions at Duck, NC. The magnitude of sediment transport appears to be less than expected. Future work will include a more detailed study of the different components that contribute to sediment transport. From there, different coefficients can be assigned to various physically based terms accordingly to best fits to observations. Future work also includes modifications to attempt to numerically stabilize the shoreline region. When this is accomplished, model simulations should be able to run long enough in time to arrive at a second completed survey and model results can be compared to field data. The direction of transport of the NOV6 simulation is a call for concern. The model predicted sediment transport onshore when offshore transport was expected. This can also be addressed in future work when additional mechanisms are added to the transport equation, such as undertow, skewness of waves, pressure gradient forces, and scour caused by breaking waves. The hydrodynamics of the model are reasonable but include necessary engineering approximations for today's computational platforms. The alongshore stream function follows the bathymetry quite well with stronger flows developing in appropriate areas. The issues of shoreline retreat and advance and tidal variations in the water level eventually need to be addressed. The shoreline is an import boundary condition and should be as representative as possible of natural conditions if accurate predictions are to be accomplished throughout the flow field. 2.6.2 Closing Remarks The major achievement of this project has been to develop a rational framework for a beach morphology model coupled to a wave and mean current model and have it run continuously forced by measured data. With many of the correct physics represented in the model, the results are somewhat realistic but still considerably doubtful. This is considered acceptable because there is still much work to be done by the community to approximately represent more complex transport mechanisms. It is a vast problem and small steps are necessary. There is still value in analyzing the results of the coupled modeling system we have implemented. First, because it is a step forward from where the community was. Second, because the formulas we used for sediment transport are widely accepted and used. And finally, because it allows a more simplified interpretation of the response of this still complex physical and modeling system. With reasonable results and the correct fundamental physics in place, it is quite encouraging and a good foundation for future development. CHAPTER 3 PRESSURE GRADIENT FORCES AND SHEAR STRESSES ON SAND GRAINS UNDER SHOALING WAVES 3.1 Introduction There is significant experimental evidence that flow acceleration, which serves as a proxy for the horizontal pressure gradient in a coastal bottom boundary layer, has an effect on sediment transport. This evidence originates from U-tube experiments [e.g., King, 1990], field measurements in the surf zone [e.g., Hanes and Huntley, 1986; Gallagher et al., 1998; Elgar et al., 2001] and in the swash [e.g., Butt and Russell, 1999; Puleo et al., 2003], and three-dimensional discrete particle computer simulations [e.g., Drake and Calantoni, 2001]. They have found significant correlations under certain relevant conditions between phases of flow acceleration caused by pressure gradient forces from the surface gravity waves and sediment suspension and net cross-shore transport. Hsu and Hanes [2004] demonstrate that responses of sheet flow, such as the velocity profiles, the instantaneous bed shear stress, the sediment flux, and the total amount of the mobilized sediment, cannot be fully parameterized by a quasi-steady free- stream velocity and may be correlated with the magnitude of the local horizontal pressure gradient. Moreover, their numerical experiments indicate that catastrophic internal bed failure is a direct consequence of large horizontal pressure gradients. These recent papers investigate the link between fluid acceleration and sediment transport. This chapter will explore the degree to which the pressure gradient contributes compared to the shear stress on sediment mobilization under surface gravity waves in the surf zone. Generally there are two types of bed stresses that have the ability to mobilize sediment under surface gravity waves. One that is considered by all, is the shear stress exerted by the orbital velocities under a propagating wave. This stress results in a force acting tangentially to a surface such as a real flow over a sea bed. The other, maybe less recognized to have significance, is the horizontal pressure gradient stress. This stress is the consequence of the difference in hydrostatic pressure from one side of a particle to the other. When large enough, the difference in pressure can induce a sediment particle to become unstable and be mobilized. In many typical coastal regimes, the ratio of the pressure gradient stress to the shear stress is 1/10 or even 1/50. For this reason, it has been thought sufficient to neglect the contribution of the pressure gradient stress in formulating the total stress in the surf zone. The primary aim of this study is to demonstrate that there is a regime in which it is inaccurate to neglect the pressure gradient stress. This will be done by examining such parameters as wave period, grain size, wave height, and water depth. A secondary aim is to obtain an understanding of the degree to which the pressure gradient stress contributes to bed mobilization to be used in future parameterizations of sediment transport models. 3.2 Stress Formulation 3.2.1 Shear Stress The stress component from forces applied parallel to a surface is the shear stress. In our case, it is the force exerted by flowing water over a sea bed. We will be examining the shear stress resulting from this force. The stress is simply the force divided by the cross sectional area on which the force is applied. There is not a direct division of the cross sectional area in the shear stress, r, equation because the grain size is embedded within the empirical Darcy-Weisbach friction factor S= 8pfu u (3.1) We are interested in the bottom shear stress, so the horizontal velocity term, u, is the horizontal near bottom fluid velocity. This variable will be examined more in section 3 below. The density of sea water is represented by p which has a value of 1024 kg/m3. This equation has been developed through dimensional analysis and experimental data have been used to develop values of the friction factor,f For wave motion, the bottom friction is a nonlinear function and due to the absolute value sign becomes somewhat complicated to work with directly. In our model, the Stanton Diagram for friction factor under waves as a function of the relative roughness, r, was used to acquire the friction factor [Kamphuis, 1975]. Rough turbulent flow is assumed to obtain friction factors off of the Stanton diagram. A polynomial fit was then created to easily acquire friction factors for given parameters. f = 0.62r0 38 (3.2) It is important to note the ambiguity of the friction factor. The friction factors which are represented on the Stanton diagram are ones developed from measurements of bottom shear stress retarding the motion of a fluid in a unidirectional pipe flow. This leaves a bit of uncertainty in the level of approximation of this equation considering we are investigating sediment mobilization under surface gravity waves. We know that situations are seldom idealized. Therefore, possibly an empirical calculation, even if it is one derived from somewhat different circumstances, is a better representation of this physical occurrence. The evaluation of the shear stress equation is beyond the scope of this paper but the uncertainty and level of approximation inherent from representing it under ocean waves is not to be overlooked. 3.2.2 Pressure Gradient Stress Under a surface gravity wave there is a variable water level within each wave. The difference in hydrostatic pressure from the differing water levels will exert a net horizontal force on a sand grain. Here we represent a sand grain as a sphere. q (x) --Z=-- ---- - ZO - ZR Figure 3-1. Schematic of a sand grain. As everyone knows, sand grains are not perfect spheres, so a modest level of approximation is introduced at this point. The horizontal pressure gradient force is computed by integrating the normal force over the sphere surface. At each point on the surface of the sphere there is a force per unit area, P, on the solid acting normal to the surface. Multiplying this local force per unit area by the surface area on which it acts and integrating over the surface of the sphere will produce the resultant force. = 2;T 0;= F= f (-PrR sin insn )R2sin OdOdO (3.3) A detailed derivation of the resultant force can be found in the appendix. This force is analogous to a component of the inertial force calculation in the Morrison [1950] Equation. Ft = pV (l+C,) (3.4) dx This equation is made up of two components pdu du F- = p +CpV (3.5). dx dx The first part of this equation, or the "1" in equation 3.4, is the force that is corresponding to the pressure gradient force explained above. This pressure gradient force is then divided by the cross sectional area on which the force is acting. The final stress can now be represented by P, = pgk d (3.6), 3 ax 2 where the pressure stress is denoted as P,, g is the acceleration due to gravity, r//8x is the local gradient of the wave in the horizontal direction, and kp is the pressure response factor. k cosh(k(h + z)) (3.7) S cosh(kh) An important point to note is that these two equations, whose magnitudes will be compared to each other, have a different designation for the grain size. The shear stress calculation uses d90 (diameter of the sand that 90% is finer) while the pressure stress calculation usesdso. To account for this, the distribution of sand is assumed to obey a log normal probability distribution. That is, if normal probability paper is used for the cumulative percentage passing and the phi scale is used for the sand size, a straight line will result. Again, there is always uncertainty with a completely idealized equation. There are no empirical parameters in this equation and we know that no situation is ever completely idealized. Yet again, one must be cautious with the level of confidence put into an equation of this nature. 3.2.3 Phase Lag The two stresses in question vary along time and space intervals as a surface gravity wave passes over a location. Hence, it is equivalent to examine a point in space over a wave period or different spatial locations at an instant in time. It was decided to hold the time variable constant and vary the space variable to investigate the distribution of the stresses. It was found that the peak stresses are separated by a phase lag of 90 degrees. 0.5- Pressure Gradiant Stress 0.4 Shear Stress -- equilibrium 0.3 0.2 (- 0. 1 10.1 -0.2 0.3 0.4 05 x Figure 3-2. Phase lag of stresses. Notice at some phases of the wave, the shear stress and pressure stress work together in a constructive manner and at other phases they work against each other in a destructive manner. Because the two phases lag by exactly 90 degrees, when one of the stresses is at its maximum, the other is zero. It is not the individual stresses that will be compared to threshold values of mobilization but rather the total of the two which is the best indicator of bed mobilization. That is, the maximum of the total stress under the wave is the value that will be used to compare to the thresholds of mobilization to indicate whether a sand particle will be suspended under a wave for particular wave characteristics. A comparison of the two stress magnitudes at this certain phase, where the maximum total stress is found, will not be made but rather the maximum of each of the stresses under the wave will be compared. This is the only fair comparison because as alluded to before, there are parts under the wave which are completely dominated by the shear stress and others that are completely dominated by pressure stress. Comparing the two stresses at the point of maximum total stress is not as reasonable because the ratio would be swayed depending on where on the phase the total maximum stress was found. The reason to calculate the maximum total stress is to compare it to the threshold of mobilization to see if a particular sediment size will be suspended under certain conditions. 3.2.4 Nonlinear Wave Everyone knows that ocean waves are not perfectly linear in nature. Many different types of nonlinearities exist. Especially as ocean waves approach breaking, a strong nonlinearity in the leading edge can be observed as a steep face. One can imagine that if a wave were to steepen, the steep leading edge of the wave would experience a far greater horizontal pressure gradient than the linear case. 1.5 Nonlinear Wave 0.5 -- Linear Wave -- equilibrium -0 -0.5 -1 0 3.14 6.28 X Figure 3-3. Nonlinear wave. To analyze the contribution of the nonlinear wave it can first be broken down into many constituent waves by Fourier decomposition. A single nonlinear wave can be represented by a set of many smaller linear waves, all of them superimposed. Once broken down, the contribution by each smaller individual set of waves can be analyzed and tallied. To find the phase position of the maximum total stress under a wave, only the first mode (lx) of the decomposition was varied. The nonlinear signal is not a complicated wave pattern, therefore most of the energy is found in the lowest frequency waves of the breakdown. Due to the fact that the nonlinear wave is of similar amplitude and frequency to the linear wave, the contributions of the multi-x waves drop dramatically after the lx wave. In other words, only the lowest frequency of the stresses' decomposition is Nonlinear Equation Figure 3-4. Nonlinear wave addition. varied to find the maximum total stress phase position. The phase position is now used in the superposition of the remaining multi frequency waves. Just as in the linear case, at some phases of the wave, one of the stresses will dominate the other stress. Although this time the maximum pressure gradient stress will be larger due to the steeper gradient and thus increase the ratio of pressure gradient stress to shear stress. 3.3 Varying Parameters and Limits A closer look at the two stress equations reveals that they both decay exponentially with increasing kh. This kh parameter is the measure of wave type. Larger kh values indicate a more deep water wave (kh > ;r) and smaller kh values signify a more shallow water wave (kh < ;r/10). Within the pressure gradient stress equation (eq. 3.4), the pressure response factor, kp, includes the exponential function, hyperbolic cosine (or cosh). ex+e cosh(x) = (3.8) 2 The exponential function is located in the denominator which will cause the magnitude of the pressure gradient stress to decay at a rate of e-kh k = osh(kh) for the sea bed. (3.9) Likewise, within the shear stress equation (eq. 3.1) the horizontal velocity, u, also includes the exponential function in the denominator. S gHk cos(kx cosh(kh) fortheseabed. (3.10) The horizontal velocity is squared in the shear stress equation (eq. 3.1) while kp is not raised to a power in the pressure gradient stress equation (eq. 3.4). Therefore, the magnitude of the shear stress will decay at a rate of e 2kh while the pressure gradient stress will decay at a rate of e kh. At low values ofkh, the shear stress is usually much more than the pressure gradient stress. With increasing kh values, both the stresses will decrease exponentially but the shear stress will decrease much faster. The magnitude of the pressure gradient stress will eventually overtake the magnitude of the shear stress. But will it be too much like a deep water wave when the pressure gradient stress has any sort of contribution? We are seeking to identify a realistic regime where the pressure gradient stress plays a substantial role in mobilizing sediment but is not in too deep of water that the combined stresses are too weak to initiate sediment movement. This limits our analysis domain to inside the critical limit of mobilization. Our investigation is only shoreward of this point which is a function of grain size. These critical limits were first explored by Shields [1936]. He determined the threshold condition by measuring sediment transport for different values of total stress at least twice as large as the critical value and then extrapolated to the point of vanishing sediment transport. For turbulent flows over rough boundaries, the critical stress becomes linearly proportional to the sediment size. Based on a comparison of data from the Highway Research Board [1970], a relationship between critical stress and median grain size on a flat horizontal surface was formulated for granular material in table form [Julien, 1998]. A polynomial fit was then created to acquire a critical stress for a given sediment size. rnccal =0.11+135.37d50 +225189.58d2 -91766304.35d3 (3.11), where d5s is in meters and r is measured in pascals. The shoreward extent of our analysis domain will be taken at the breaker limit. Our analysis is based on linear theory which loses validity once a wave breaks. During breaking and after, other wave induced bed stresses become dominant like breaking- induced turbulence [Roelvink and Stive, 1989]. Therefore our analysis domain is limited to seaward of the break point. An approximate depth limited breaking boundary can be obtained from McCowan [1894], who determined that waves break when their height becomes equal to a fraction of the water depth. Hb =0.78hb (3.12) The subscript b denotes the value at breaking. This is a crude approximation and there has been more complex ways to acquire breaking parameters but this one will be used for simplicity. Steepness limited breaking must also be taken into account. Miche [1944] developed a simple equation for wave breaking in any water depth. SHmax 1 (3.13) L 7 This indicates that the maximum wave height, H, is limited to one-seventh of the wave length, L. Also a crude approximation, but simply stated within our analysis. 3.4 Results 3.4.1 Linear Results As stated above, shear stress dominates pressure stress in many typical wave conditions. An easy way to show this is by varying the water depth and seeing the resulting stresses for typical wave characteristics. 10' 01 .. i Pressure Gradiant Stress S- Shear Stress Critical Stress Limit (no movement seaward) S- Depth Limited Breaking Shoreward a.10, 1021 25 50 75 h(m) Figure 3-5. Linear: T= 7sec, d= 0.3mm, H = Im In Figure 3.5, the x-axis is the water depth measured in meters and the y-axis is the resulting stresses measured in pascals on a log scale. Here, the shear dominates well below the critical limit of mobilization. The magnitude of the pressure force does eventually draw level and surpass the shear stress but this happens in water depths much too deep for the combination of the two stresses to initiate sediment mobilization. This graph is a bit misleading because the intersections of either of the two stresses and the critical limit have no real significance. It is the addition of the two stresses which needs to be compared to the critical limit. It is not even the total of the two maximum stresses but rather the maximum of the total stress calculation under a wave which is being compared against the critical limit. A better way to represent the stresses is to look at the ratio of the two stresses in question versus the kh parameter in which they vary so greatly. / 0.9 0.8 0.7 0.6 0.5 0.4 Ratio of Ps/tau S Depth Limited Breaking Shoreward 0.3 Critical Stress Limit (no movement seaward) Ps and tau of Equal Importance 0.2 Steepness Limited Breaking Shoreward 0.1 0 2 4 6 8 kh Figure 3-6. Linear: T= 7sec, d= 0.3mm, H= Im The x-axis is the non-dimensional wave type, kh, and the y-axis is the ratio of pressure gradient stress to shear stress. This way the graph is non-dimensional and the crossing of the ratio line and the critical limit has significance. This intersection will be the ratio of the maximum stresses just as an incoming wave initiates sediment motion. The greatest ratio will always occur at the offshore boundary when the critical stress is just met and the sediment is first mobilized. With these wave conditions the pressure gradient stress to shear stress ratio only reaches 0.14, or the magnitude of the pressure gradient stress is only 14% of that of the shear stress. With these wave conditions it seems reasonable to simply neglect the pressure gradient stress and still not sacrifice much accuracy. Varying the dependant parameters, one can examine the relative importance of the pressure gradient stress to shear stress ratio. The wave height and period will remain the same and the diameter of the grains will be increased from 0.3 mm to 0.5 mm. / 0.9 0.8 0.7 0.6 0.5 S Ratio of Ps/tau 0.4 Depth Limited Breaking Shoreward Critical Stress Limit (no movement seaward) 0.3 Ps and tau of Equal Importance Steepness Limited Breaking Shoreward 0.2 0.1 O0 2 4 6 8 kh Figure 3-7. Linear: T= 7sec, d= 0.5mm, H= Im Here, the maximum ratio of pressure gradient stress to shear stress only increases slightly. The ratio only reaches a value around 0.18, still not very significant to the total stress contribution. An increase in the contribution of the pressure gradient stress implies that the greater surface area of a larger grain aids the pressure gradient stress more so than the shear stress, but only slightly. The variation of the wave period will now be analyzed. The grain size diameter is returned to a value of 0.3 mm and the wave period is decreased from 7 seconds to 2.5 seconds. Decreasing the wave period and keeping the same wave height will effectively make the wave shorter and steeper. This will cause a bigger disparity of hydrostatic pressure from one side of a sand grain to the other. ,/ 0.9 - 0.8- 0.7- = 0.6- 0.57 0.4 - 0.3 - Ratio of Psltau 0.2 Depth Limited Breaking Shoreward -- Critical Stress Limit (no movement seaward) Ps and tau of Equal Importance 1- -- Steepness Limited Breaking Shoreward 0 2 4 6 8 kh Figure 3-8. Linear: = 2.5sec, d= 0.3mm, H= Im In this example, the ratio of pressure gradient stress to shear stress has increased noticeably by making the wave period shorter. The ratio at the point of mobilization has increased to over 0.3. With such a short wave, one must be conscious of the wave being too steep and breaking. Here, the graph also indicates that the wave will break due to depth limited breaking before it reaches its steepness limitation. The wave height was also varied, but little change in maximum ratio was observed. The only noticeable outcome was a change in the break point. The increased wave height may mobilize a greater domain of sediment but our domain of applicability decreased because the breaker limitation and the maximum ratio remain similar. With the wave period shortened and the grain size increased, the pressure stress will find its most favorable contribution. In the next case, the wave period is shortened again to 2.5 seconds and the grain size diameter is increased to 0.5 mm. With these wave characteristics, the maximum pressure stress to shear stress ratio has now reached 0.42. ,/ 0.9 - 0.8- 0.7- = 0.6- C1 0.5 - 0.4 - 03- Ratio of P/ tau _- Depth Limited Breaking Shoreward Critical Stress Limit (no movement seaward) 0.2 Ps and tau of Equal Importance -- Steepness Limited Breaking Shoreward 0.1 - 0 2 4 6 8 kh Figure 3-9. Linear: T= 2.5sec, d= 0.5mm, H= Im Note where the sediment is mobilized along the kh axis. At the point of mobilization the kh value is 3.2 and at the point of breaking the kh value is 1.5. Here our analysis domain falls within the intermediate wave type category. 3.4.2 Nonlinear Results The nonlinear results will be analyzed in a similar fashion to the linear case. Similar results are observed but swayed more toward the contributions from the pressure gradient. Figure 3-10 shows the dimensional graph varying the water depth and viewing the resulting stresses. 0 i' 10 -- Pressure Gradiant Stress Shear Stress Critical Stress Limit (no movement seaward) S Depth Limited Breaking Shoreward 102 103 25 50 75 h(m) Figure 3-10. Nonlinear: = 7sec, d= 0.3mm, H= Im This again is for our proto-typical wave conditions. With the same axis lengths and scale as the linear graph, it is apparent that the shear stress and pressure gradient stress crossing point occurs in much shallower water and closer to the critical limit. What also should be understood from this graph is that there is a regime from about 10 meters water depth into the break limit where the pressure stress alone is enough to mobilize the sediment. The nonlinear case is also converted to a non-dimensional graph in Figure 3-11. The stresses are represented as the ratio of maximum pressure to shear stress and the kh parameter is varied. 0.9 - 0.8 - 0.7 3 0.6 0.5 0.4 Ratio of Ps/tau SDepth Limited Breaking Shoreward 0.3 Critical Stress Limit (no movement seaward) Ps and tau of Equal Importance 0.2 / Steepness Limited Breaking Shoreward 0.1 / 0 0 I I I I 21 I I I I I I I 0O 2 4 kh Figure 3-11. Nonlinear: T= 7sec, d= 0.3mm, H= Im The ratio at the point of mobilization has increased to over 0.5. This is considerably larger compared to the linear case, 0.14, with the same wave characteristics. The same dependant variables will be varied with the nonlinear wave at the same degree to see its effect on the stresses compared to that of the linear case. First the grain size diameter will be increased to 0.5 mm. The increase in grain size diameter has a similar effect as the linear case. The change in grain size results in a minimal change of the maximum ratio. The pressure gradient stress to shear stress ratio has now reached a value of 0.7. Figure 3-12 shows that the maximum of the pressure gradient stress and the shear stress are now nearly the same magnitude at the point of mobilization. 0.9 0.8 0.7 0.6 0.5 Ratio f Ps/tau 0.4 Depth Limited Bi Critical Stress Li 0.3 / Ps and tau of Eq 0.- Steepness Limit 0.2 0.1 0 2 kh Figure 3-12. Nonlinear: T= 7sec, d breaking Shoreward mit (no movement seaward) ual Importance ed Breaking Shoreward 4 0.5mm, H= lm - Ratio of Ps/tau Depth Limited Breaking Shoreward - Critical Stress Limit (no movement seaward) Ps and tau of Equal Importance - Steepness Limited Breaking Shoreward U 0 2 4 kh Figure 3-13. Nonlinear: T= 2.5sec, d= 0.3mm, H= Im The grain size is returned to 0.3 mm and the wave period is now decreased to 2.5 seconds in figure 3-13. With the ratio of the stresses reaching and surpassing 1.0 and the sediment still being mobilized, the magnitude of the pressure gradient stress is now more than the shear stress just as the sediment is mobilized. At the point of mobilization the pressure gradient stress is about 20% greater than the shear stress. The wave height has little effect as mentioned earlier in the linear case. The two dependent variables, wave period and grain size, make the largest contributions to maximizing the contribution of the pressure gradient stress. Similar to the linear case, the grain size and wave period are changed to maximize the contribution by the pressure gradient stress. 1.8 1.6 - 1.4 1.2 0- 0.8 0.6 Ratio of Ps tau Depth Limited Breaking Shoreward Critical Stress Limit (no movement seaward) 0.4 Ps and tau of Equal Importance -- Steepness Limited Breaking Shoreward 0.2 O0 ----I-I I I I-- 0 2 4 kh Figure 3-14. Nonlinear: T= 2.5sec, d= 0.5mm, H= Im The ratio has now reached 1.6 at the point of mobilization. The magnitude of the maximum pressure gradient stress is 60% greater than that of the shear stress at the point of mobilization. The kh axis indicates again that our analysis takes place within the intermediate wave type range. 3.5 Conclusion A simple analytical model based on empirical formulas was used to examine the relative contributions of pressure and shear stresses to bed mobilization. First, the parameters that were varied will be discussed. It became evident that an increase in grain size diameter contributed to the pressure gradient stress contribution. Clearly a larger surface area for the horizontal pressure gradient force to act upon is more favorable then for the shear force to act upon. A decrease in wave period also produced a favorable contribution for the pressure gradient stress. A decreased period will result in a shorter and steeper wave. The steeper gradient will result in a greater horizontal pressure gradient and a noticeably greater pressure gradient stress contribution. Both changes in these parameters are in favor of the pressure gradient stress but to different degrees. The dependent variable comparison gives the impression that the ratio of the two stresses is more sensitive to the change in wave period rather than grain size. As mentioned before, the wave height appeared to have minimal effect on the ratio. While the break point would move, the maximum ratio of pressure gradient stress to shear stress stayed approximately the same. The analysis gives us some distinct regimes where the pressure gradient stress holds considerable influence on the total stress. Short period waves, 2 5 seconds, will result in an influential contribution by the pressure gradient stress. For a 2.5 second wave and 0.3 mm grain size, the magnitude of the maximum pressure gradient stress reaches over 30% of that of the shear stress for the linear wave case and is actually 20% more than the maximum shear stress for the nonlinear wave case. The solutions also show that it was not uncommon for the pressure gradient stress alone to be sufficient to induce particle mobilization. This suggests that for short period waves, the pressure gradient stress can be quite important. Increased grain size also resulted in an enhanced contribution by the pressure gradient stress, but not to the extent of the wave period fluctuations. Larger grain sizes, 0.3- 1.0 mm, appear to favor the pressure gradient contribution. These characteristics were developed for our analysis in intermediate water depths, 1-15 meters, and resulted in intermediate wave types. Altering both the wave period and the grain size to realistic ocean quantities, a 2.5 second wave and a mean sand grain diameter of 0.5 mm, one can detect a significant contribution from the pressure gradient stress. The linear case suggests that the magnitude of the maximum pressure gradient stress is over 40% to that of the maximum shear stress while the nonlinear case indicates that the pressure gradient stress is actually 60% more than that of the shear stress when the sediment is first mobilized. One certainly can not have much confidence in setting a threshold for the total stress when possibly half the magnitude of the stress is neglected if pressure gradient contributions are neglected. The analysis has also shown where the pressure gradient stress makes its greatest contribution. The maximum contribution made by the pressure gradient stress was always located at the critical point of mobilization or more easily stated, just as an incoming wave begins to move sediment around. This is also the place within our analysis domain where a wave exhibits its most linear form. As a wave approaches breaking, it will take on more of a nonlinear form similar to the one analyzed in this study. A breaking wave may even approach a vertical wall of water just prior to breaking which would increase the pressure gradient stress significantly. This suggests that the pressure gradient stress could become important along our entire analysis domain, from the critical stress limit all the way to the point of breaking. Consequently, for certain wave and grain characteristics, it would be precarious to not include the pressure gradient stress in the total stress calculation and expect high levels of precision. Since the pressure gradient stress caused by surface gravity waves show some relative contribution to the total stress at the sea-bed, one must rethink the way that near shore sediment transport parameterizations are found. These parameterizations should not be reasonably based upon empirical relationships developed from open channel or even oscillatory flow experiments if they are not produced by surface gravity waves. By including pressure gradient stresses it will also help in the parameterization of sediment flux in the direction of wave advance due to asymmetric and skewed nonlinear wave shapes typical of shoaled and breaking waves. APPENDIX DERIVATION OF THE RESULTANT FORCE ON A SPHERE a,7 a(x) a0= ------------------------------ z=0 - Zo -z ZR y 7(x) = ax z, =RcosO -=2;T 0e=; F= (-P rR sin s sin )R sin OdOdO =o0 0=o P r=R = Pt + Pgz + pg l(x) rR + PgzR P r=R = Pa + pgz + pgaR sin 0 sin + pgR cos 0 F = {-[Patm + pgzo + pgaR sin 8 sin # + pgR cos 0]sin 6 sin )}R2 sin OdO6d 0 0 27;T 7 27; 7; F = f -PtR2 sin2 0sinpdOdo+ f f-pgzR2 sin2 0sin OdOd 0 0 0 0 27T 7- 27f T- + J -pgaR3 sin3 0 sin2 4dOd+ f f-pgR3 cos0 sin 2 0sin OdOd 00 00 Sf 3 30. O sin 2o F,= -pgaR in 4-- 0o (2 4 o= dO =f0 0 ) 3 1 2 F = -7TpgaR3.- cos O(sin2 0+2) 0 3 0 47rpgaR3 3 7TR2 pgaR 3 -;ipgaR3 sin3 0dO Fx4= 4-7pgaR3 3 4pgkp do 3 Ox 2 LIST OF REFERENCES Aubrey, D.G., Seasonal patterns of onshore/offshore sediment movement, J. 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Sallenger, The role of suspended sediment in shore-normal beach profile changes, in Proc. 19th Int. CoastalEng. Conf, pp 1983- 1996, Am. Soc. of Civ. Eng., New York, 1984. Julien, P.Y., Erosion and Sedimentation, Cambridge University Press, New York, 1998. Kamphuis, J.W., Friction factor under oscillatory waves, J. Waterways, Harbors Coastal Eng. Div., ASCE, Vol. 101, 135-144, 1975. King, D.B., Studies in oscillatory flow bed load sediment transport, Ph.D. thesis, University of California, San Diego, 1990. Lippmann, T.C. and R.A. Holman, The spatial and temporal variability of sand bar morphology, J Geophys. Res., 95, 11575-11590, 1990. Longuet-Higgins, M.S. and R.W. Stewart, The changes in the form of short gravity waves on steady non-uniform currents, J Geophys. Res., 8, 565-583, 1960. Longuet-Higgins, M.S. and R.W. Stewart, Radiation stress in water waves, a physical discussion with application, Deep Sea Res., 11, 529-563, 1964. McCowan, J., On the Highest Wave of Permanent Type, Philos. Mag. J. Sci., Vol. 38, 1894. McIlwain, S. and D.N. Slinn, Modeling alongshore currents over barred beaches, submitted to the Journal of Waterway, Port, Coastal, and Ocean Engineering, 2004. Miche, M., Movements ondulatoires des mers en profondeur constant ou decroissante, Annales des Ponts et Chaussees, 25-78, 131-164, 270-292, 369-406, 1944. Morrison, J.R., M.P. O'Brien, J.W. Johnson, and S.A. Schaaf, The force exerted by surface waves on piles, Petrol. Trans., AIME, Vol. 189, 1950. Nairn, R.B. and H.N. Southgate, Deterministic profile modeling of nearshore processes. Part 2. Sediment transport and beach profile development, Coastal Eng., 19, 57-96, 1993. Ozkan-Haller, H.T. and J.T Kirby, Nonlinear evolution of shear instabilities of the longshore current: A comparison of observations and computations, J. Geophys. Res., 104, C11, 25953-25984, 1999. Plant, N.G., R.A. Holman, M.H. Freilich, and W.A. Birkemeier, A simple model for interannual sandbar behavior, J. Geophys. Res., 104, 15, 755-15, 776, 1999. Plant, N.G., B.G. Ruessink, and K.M. Wijnberg, Morphologic properties derived from a simple cross-shore sediment transport model, J. Geophys. Res., 106, 945-958, 2001. Plant, N.G., K.T. Holland, and J.A. Puleo, Analysis of the scale of errors in nearshore bathymetric data, Mar. Geol., 191, 71-86, 2002. Plant, N.G., K.T. Holland, J.A. Puleo, and E.L. Gallagher, Prediction skill of nearshore profile evolution models, J. Geophys. Res., 109, C01006, 2004. Puleo, J.A., K.T. Holland, N.G. Plant, D.N. Slinn, and D.M. Hanes, Fluid acceleration effects on suspended transport in the swash zone, J. Geophys. Res., 108(C11), 3350, doi: 10.1029/2003JC001943, 2003. Roelvink, J.A., and M.J.F. Stive. Bar generating cross shore flow mechanisms on a beach, J. Geophys. Res., 94, 4785-4800, 1989. Roelvink, J.A. and I. Broker, Cross-shore profile models, CoastalEng., 21, 163-191, 1993. Sato, S. and N. Mitsunobu, A numerical model of beach profile change due to random waves, in Proc. Coastal Sediments '91, pp 674-687, Am. Soc. of Civ. Eng., New York, 1991. Shields, A., Anwendung der aehnlichkeitsmechanik und der turbulenz forschung auf die geschiebebewegung, Mitteilungen der Preussische Versuchanstalt fur Wasserbau undSchiffbau, Berlin, 1936. Sleath, J.F.A., Sea Bed Mechanics, pp 335, John Wiley, New York, 1984. Slinn, D.N., J.S. Allen, P.A. Newberger, and R.A. Holman, Nonlinear shear instabilities of alongshore currents over barred beaches, J Geophys. Res., 103, 18, 357-18, 379, 1998. Slinn, D.N., J.S. Allen, and R.A. Holman, Alongshore currents over variable beach topography, J. Geophys. Res., 105, C7, 16,971-16,998, 2000. Stauble, D.K., Long-term profile and sediment morphodynamics: Field Research Facility case history, Tech. Rep. CERC-92-7, U.S. Army Corp of Eng., 1992. Stive M.J.F., A model for cross-shore sediment transport, in Proc. 20th Int. Coastal Eng. Conf., pp 1550-1564, Am. Soc. of Civ. Eng., Taipei, Taiwan, 1986. Stive M.J.F. and J.A. Battjes, A model for offshore sediment transport, in Proc. 19th Int. CoastalEng. Conf, pp 1420-1436, Am. Soc. of Civ. Eng., New York, 1984. Svendsen, I.A. and U. Putrevu, Nearshore mixing and dispersion, Proc. R. Soc. London, Ser. A, 445, 561-576, 1994. Thornton, E.B., R.T. Humiston, and W.A. Birkemeier, Bar-trough generation on a natural beach, J. Geophys. Res., 101, 12097-12110, 1996. 71 Winant, C.D., D.L. Inman, and C.E. Nordstrom, Description of seasonal beach changes using empirical eigenfunctions, J Geophys. Res., 80, 1979-1986, 1975. Wright, L.D. and A.D. Short, Morphodynamic variability of surf zones and beaches: a synthesis, Mar. Geol., 26, 93-118, 1984. BIOGRAPHICAL SKETCH I was born and in a small coastal town located on the New Jersey Shore called Point Pleasant. I spent my first 6 years in a small house in Toms River, NJ, only to move to a more rural area located just south. It is here in Lanoka Harbor that I spent the majority of my adolescence. In this house I lived with my two loving parents and my older sister, Maggie. I was also blessed with the great experience of living with my grandparents too. Growing up in this small pristine town was very enjoyable. I spent the majority of my time outside playing sports, surfing, and riding my bike with my friends. As I grew older, I became more involved in organized sports. By high school I participated in three varsity sports (football, indoor track, and baseball) and was the captain of two by senior year. I was also very active in many clubs and groups as well as playing in the marching band for one year. After graduating high school in 1999 I decided to attend North Carolina State University. It was not an easy decision, because going to a large university meant that it would be difficult to play any varsity sport. I got involved in different ways by joining a social fraternity (Sigma Chi) as well as an honors fraternity (Chi Epsilon). Being the intramural sports chairman and participating in philanthropy events around the community were a few ways I spent my free time in college. I was even able to play a varsity sport when I walked onto the track team for one year. Of course the academic demands of civil engineering were first and foremost. I feel I was provided with an excellent base of skills in civil engineering at NC State to excel in any sub discipline chosen. It was my junior year, after I heard a presentation by one of our professors, that I knew coastal engineering was the field for me. It combined something I enjoyed studying, engineering, as well as something I loved, the beach. I was fortunate enough to obtain an internship in Duck, NC at the USACE Field Research Facility for the summer after my junior and senior years of undergraduate. Here I got a taste of the coastal engineering field by participating in many field experiments and leading a public tour. After graduating from NC State University in May of 2003, it seemed like an easy decision to continue my academic experience in graduate school studying coastal engineering at the University of Florida. The decision was made easier after being offered a graduate research assistant position studying under Dr. Donald Slinn. I have demonstrated growth and development in my short time at U.F. by speaking and presenting at two major conferences and expect to have two papers published from my thesis work. While working under Dr. Slinn I also managed to have a successful internship at the Naval Research Laboratory and will be graduating with honors this summer with my Master of Science degree. This August I begin my practical coastal engineering experience when I start a full time job with Olsen & Associates, a small coastal engineering firm located in Jacksonville, FL. I am eager to apply my knowledge in a practical atmosphere and am also excited about my short move to the Jacksonville area. |

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PAGE 1 A COUPLED MODELING SYSTEM TO PREDICT MORPHOLOGY CHANGES AND A COMPARISON OF PRESSURE GRADIENT FORCES TO SHEAR STRESSES IN THE NEARSHORE By WILLIAM L. REILLY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2005 PAGE 2 Copyright 2005 By William L. Reilly PAGE 3 This work is dedicated to my mother, father, and sister. PAGE 4 iv ACKNOWLEDGMENTS I wish to thank my friends and fa mily for their continued support and encouragement. I would not be where I am today if it was not for the guidance, direction, and advice of my wonderful family. It is my close friends and ever yone that have meant something to me along the way th at makes me who I am today. Dr. Donald Slinn, my graduate advisor, must be acknowledged for his continued support, not only academically but also in hi s efforts of helping me with my future endeavors. I also thank Dr. Robert Thieke and Dr. Ashish Mehta for taking the time to serve on my committee. I must also thank th e other professors at the University of Florida in the Civil and Coastal Engineering De partment for their dedication and effort of sharing their knowledge to the next ge neration of coastal engineers. Lastly, I would like to acknowledge a few ot her people who have made this project possible. Appreciation is given to Bill Birk emeier for his help in my growth of an aspiring engineer as well as the Field Resear ch Facility for providing the data for this project. Finally, I would like to thank Nathaniel Plant as well as the other members of the Naval Research Lab for their co ntinued support on this project. PAGE 5 v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF FIGURES..........................................................................................................vii ABSTRACT....................................................................................................................... ix CHAPTER 1 OVERVIEW.................................................................................................................1 1.1 Preface.................................................................................................................... 1 1.2 Organization...........................................................................................................2 2 A COUPLED MODELING SYSTEM TO PREDICT MORPHOLOGY CHANGES IN THE NEARSHORE............................................................................4 2.1 Introduction.............................................................................................................4 2.2 Observations...........................................................................................................6 2.2.1 Bathymetric Data..........................................................................................6 2.2.2 Model Grid...................................................................................................7 2.2.3 Wave Data....................................................................................................9 2.3 Model Description................................................................................................10 2.3.1 Wave Model...............................................................................................12 2.3.2 Hydrodynamic Model.................................................................................13 2.3.3 Sediment Transport Model.........................................................................18 2.4 Results...................................................................................................................2 2 2.4.1 October 27th Simulation..............................................................................23 2.4.2 November 6th Simulation............................................................................32 2.4.3 Comparison.................................................................................................36 2.5 Discussion.............................................................................................................37 2.6 Conclusion............................................................................................................40 2.6.1 Future Work................................................................................................40 2.6.2 Closing Remarks........................................................................................41 3 PRESSURE GRADIENT FORCES AND SHEAR STRESSES ON SAND GRAINS UNDER SHOALING WAVES..................................................................42 3.1 Introduction...........................................................................................................42 PAGE 6 vi 3.2 Stress Formulation................................................................................................43 3.2.1 Shear Stress................................................................................................43 3.2.2 Pressure Gradient Stress.............................................................................45 3.2.3 Phase Lag....................................................................................................47 3.2.4 Nonlinear Wave..........................................................................................48 3.3 Varying Parameters and Limits............................................................................50 3.4 Results...................................................................................................................5 3 3.4.1 Linear Results.............................................................................................53 3.4.2 Nonlinear Results.......................................................................................58 3.5 Conclusion............................................................................................................62 APPENDIX DERIVATION OF THE RESULTANT FORCE ON A SPHERE...........65 LIST OF REFERENCES...................................................................................................67 BIOGRAPHICAL SKETCH.............................................................................................72 PAGE 7 vii LIST OF FIGURES Figure page 2-1. Initial bathymetry for Oct. 27, 1999 and Nov 6, 1999...............................................9 2-2. Schematic of the coupled modeling system.............................................................10 2-3. Fall Velocity as a function of offshore distance.......................................................20 2-4. Wave characteristics for October 27, 1999..............................................................22 2-5. Wave characteristics for November 6, 1999............................................................23 2-6. Time series of the cross shore and alongshore velocities for October 27th..............24 2-7. Time averaged alongshore stream function for October 27th...................................25 2-8. Bathymetric change from hour 25 to hour 35..........................................................26 2-9. Wave focusing around bathymetry for October 27th................................................27 2-10. Difference in bathymetry from original for Oct 27th................................................29 2-11. Difference in bathymetry for October 27th with the shoreline held static................30 2-12. Difference in bathymetry for October 27th after 130 hours with the shoreline held static..................................................................................................................31 2-13. Time series of the cross shore and alongshore velocities for November 6th............32 2-14. Snapshot of the vorticity fields.................................................................................33 2-15. Time averaged alongshore stream function for November 6th.................................34 2-16. 3-Dimensional bathymetry for November 6th..........................................................35 2-17. Difference in bathymetry for November 6th with the shoreline held static..............36 3-1. Schematic of a sand grain.........................................................................................45 3-2. Phase lag of stresses.................................................................................................47 PAGE 8 viii 3-3. Nonlinear wave........................................................................................................49 3-4. Nonlinear wave addition..........................................................................................50 3-5. Linear: T = 7sec, d = 0.3mm, H = 1m....................................................................53 3-6. Linear: T = 7sec, d = 0.3mm, H = 1m....................................................................54 3-7. Linear: T = 7sec, d = 0.5mm, H = 1m....................................................................55 3-8. Linear: = 2.5sec, d = 0.3mm, H = 1m....................................................................56 3-9. Linear: T = 2.5sec, d = 0.5mm, H = 1m.................................................................57 3-10. Nonlinear: = 7sec, d = 0.3mm, H = 1m..................................................................58 3-11. Nonlinear: T = 7sec, d = 0.3mm, H = 1m..............................................................59 3-12. Nonlinear: T = 7sec, d = 0.5mm, H = 1m..............................................................60 3-13. Nonlinear: T = 2.5sec, d = 0.3mm, H = 1m............................................................60 3-14. Nonlinear: T = 2.5sec, d = 0.5mm, H = 1m............................................................61 PAGE 9 ix Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science A COUPLED MODELING SYSTEM TO PREDICT MORPHOLOGY CHANGES AND A COMPARISON OF PRESSURE GRADIENT FORCES TO SHEAR STRESSES IN THE NEARSHORE By William L. Reilly August 2005 Chair: Donald Slinn Major Department: Civil and Coastal Engineering Two separate, but related, topics are investigated in this study. The goal of the first half of the study is to simu late beach morphology on time scales of hours to days. Our approach is to develop finite difference solutions from a coupled modeling system consisting of nearshore wave, circulation, a nd sediment flux models We initialize the model with bathymetry from a dense data se t north of the pier at the Field Research Facility (FRF) in Duck, NC. The offshore wa ve height and directi on are taken from the 8-meter bipod at the FRF and input to the wave-model, SWAN (Spectral WAve Nearshore). The resulting cal culated wave induced force pe r unit surface area (gradient of the radiation stress) output from SWAN is used to drive the curren ts in our circulation model. Our hydrodynamic model is then integrated forward in time solving the 2dimensional unsteady Navier Stokes Equations The divergence of the time averaged sediment flux is calculated after one hour of simulation. The sediment flux model is PAGE 10 x based on the energetics approach of Bagnol d and includes approximations for both bedload and suspended load. The results of bathymetric change vary for different wave conditions. Typical results indicate that for wave heights on th e order of one meter, shoreline advancement and sandbar evolution is observed on the orde r of tens of centimeters. While the magnitudes of the resulting bathymetric change s seem to be smaller than expected, the general shape and direction of tran sport appear to be reasonable. The second half of this study takes a syst ematic look at the ratio of horizontal forces on the seabed from shear stresses compared to the forces exerted by the pressure gradients of passing waves. This study was completed to investigate the importance of these pressure gradients to sediment transport. We deve lop analytic solutions, for linear and weakly nonlinear waves, to predict th e forces felt by individual sand grains by passing waves. A range of wa ve frequencies and amplitude s, water depths, and grain sizes are varied to calculate the two horizontal forces. We demonstrate that the pressure gradie nt, for certain sediment sizes and wave regimes, can be sufficient to induce bed moti on. A principal conse quence of our findings is that near shore sediment transport parameterizations should not be based upon empirical relationships devel oped from steady open channel or even oscillatory flow experiments if they are not produced by surface gravity waves. This work will also help in the parameterization of sediment flux in the direction of wave advance due to asymmetric and skewed nonlinear wave shap es typical of shoale d and breaking waves. PAGE 11 1 CHAPTER 1 OVERVIEW 1.1 Preface Accurate predictions of nearshore bat hymetric change are challenging at all relevant scales. The difficulty lies in that th e relevant scales span a very broad range. One could look at a very small scale proce ss such as grain to grain interaction ( O millimeters) or how a sandbar moves from a pre-storm position to a post-storm position ( O meters) or even the scope of alongshore littoral cells ( O kilometers). The largest spatial scales are of particular importance becau se they contain the majority of the spatial and temporal variability of nearshore bathymetric change [ Lippmann and Holman, 1990 ; Plant et al., 1999 ]. In terms of temporal scales, similar breadth is encountered. One could look at very short time scal es such as turbulent dissipation ( O seconds) or tidal influence on sediment transport ( O hours) or even chronic erosion ( O years). To complicate things further, sediment suspension ob served in the surf zone is also spatially and temporally intermittent on time periods of waves, storms, seasons or climate variability. Large changes in concentration occur over times shorter than a wave period [e.g., Downing et al., 1981 ] and at spatial scales shor ter than a wavelength. The difficulty in modeling and pred iction turns out to be acute at large scales, since the evolution at this scale requires the integration over all smaller scales [ Roelvink and Broker, 1993 ]. The dominant causes of these small scale fluctuations in suspended sediment concentration are unclear and may include instabilities of the bottom boundary layer PAGE 12 2 [ Conley and Inman, 1994 ; Foster et al., 1995 ], vortex shedding from megaripples, or coherent turbulent flow structures [ Hay and Bowen, 1994 ]. From a modeling perspective, these small scale processes ar e difficult to capture. They are driven by various forces including wave, cu rrent, and gravity driven flow. There is also significant experimental evidence that flow acceleration, which serves as a proxy for the horizontal pressure gradient in a coastal bottom boundary layer, has an effect on sediment transport [e.g., Hanes and Huntley, 1986 ; King, 1990 ; Gallagher et al., 1998 ; Butt and Russell, 1999 ; Drake and Calantoni, 2001 ; Elgar et al., 2001 ; Puleo et al., 2003 ]. 1.2 Organization The unifying theme of this project is its as sociation with sediment transport. The first half of the study will attempt to pr edict the morphology of a real beach in the nearshore with a coupled mode ling system. One part of th e coupled modeling system is the calculation of the sediment transport. There are several mechanisms, many not yet well understood, that contribute to the movement of sediment in the nearshore. Due to this fact, multiple empirically based parameteri zations are incorporated into the sediment transport formulation. The second half of the study will investigate one of these mechanisms that may be vital to accurate ly predicting sediment transport in the nearshore. Here we will e xplore the role that pressure gradient forces have on sand grains under shoaling waves compared to that of shear stresses. Chapter 2 will cover beach evolution on inte rmediate temporal and spatial scales. Section 2.1 presents an introduction and bac kground on sediment transport models over the past few decades. Section 2.2 describes wh ere the data has come from as well as how the model grid is set up. The individual pa rts of the modeling system are described in section 2.3 and how they are coupled together Results are presented in section 2.4. PAGE 13 3 Finally, a brief discussion and conclu sion can be found in sections 2.5 and 2.6, respectively. The second half of the study on contributions of pressure gradients to sediment transport is presented in chapter 3. A brief background and introduction is given in section 3.1. Section 3.2 takes a clos er look at the two stresses. It is here that we examine how at certain wave phases the pressure and shear forces work together to mobilize the bed and explore their interact ion under a surface gravity wave. Section 3.3 describes our approach and clarifies the formulation. Resu lts for linear and weakly nonlinear waves are presented in section 3.4. Finall y, the paper will close with the conclusion in section 3.5. PAGE 14 4 CHAPTER 2 A COUPLED MODELING SYSTEM TO PREDICT MORPHOLOGY CHANGES IN THE NEARSHORE 2.1 Introduction Process-oriented energetics-based tota l load sediment transport models [ Bagnold, 1963 ] relate sediment transport to the near-botto m flow field and have often been used to predict beach evolution. BagnoldÂ’s [1963] arguments are physically reasonable and they seem to capture some of the essential aspects of nearshore sediment transport. The problem is that BagnoldÂ’s model yield specific, and thus rigid, parameterizations of some variable small-scale processes. Bagnold sought to paramete rize interactions associated with gravitational and near-bed turbulent forces that driv e transport under uni-directional river flows. However, several responses of the flow, such as the velocity profiles, the instantaneous bed shear stress, the sediment flux, and the total amount of the mobilized sediment cannot be fully parameterized by a quasi-steady free-stream velocity [ Hsu and Hanes, 2004 ]. The formulas are based on riveri ne flow that is obviously not the condition that characterizes the nearshore environment. Since then other researchers have develope d similar time-averaged versions of the energetic-based beach prof ile evolution models [ Bowen, 1980 ; Bailard 1981 ; Stive 1986 ; Roelvink and Stive, 1989 ] using improved parameterization to represent net effects of the small-scale processes more accurately. Seve ral recent studies have tried to compare observed bathymetric changes to predicted cross-shore profile changes from related models [ Thornton et al., 1996 ; Gallagher et al., 1998 ]. These studies used measured PAGE 15 5 near-bed velocities to drive the sediment tr ansport model. Both studies concluded that the transport model predicted patterns of of fshore transport that were accurate during undertow dominated conditions, where sedi ment transport was dominated by strong, seaward directed, near-bed, cross-shore mean flow. However, the slow onshore migration observed during low-energy wave conditions was not predicted well. Detailed hydrodynamic information is hard ly ever available in any nearshore environment unless an intensive field study is being conducted. In contrast, wave data is readily available in many coastal environmen ts. Instead of measuring the hydrodynamics from observations, this study will compute the hydrodynamics from measured boundary condition data. The sediment transport is then estimated from the computed hydrodynamics. Plant et al. [2004] found significant predicti ve skill in a similar approach by tuning several free parameters in the sediment transport model. PlantÂ’s study was able to find significant predictive skill for conditions dominated by onshore and offshore transport. However, in order to obtain this skill, it was necessary to allow the model parameters to vary with changing wave conditions. A forward stepping model is sensitive to model parameters, and these parameters should be independent [ Plant et al., 2004 ]. In the present study, the sediment tran sport parameterization is fixed with one set of model coefficients. There are many different sediment trans port mechanisms, all of which are not included in this model. Therefore the model is limited in that respect. Our model has representation for the depth av eraged mean currents, and th e combined wave and current boundary layer transports. The model does not yet explicitly include effects contributed by undertow, wave skewness and asymmetr y, breaking waves and turbulence, and PAGE 16 6 surface wave induced pressure gradients. The goal of this work has been to develop a rational framework for a beach morphology mode l coupled to a wave and mean current model. The precision of knowledge is sti ll lacking pertaining to these unaddressed mechanisms. It is left to the community a nd future work to fine tune the relative contributions that may be attributable to these neglected mechanisms. Furthermore, current computer technology requires many approximations in order to simulate for multiple days. To run such extensive simulations you need to run with a 2-D circulation model and a phase averaged wa ve model. Therefor e the undertow, wave skewness, and the turbulence intensities are all parameters that would have to be estimated by some yet-to-beevaluated methods. As a re sult, these features were reasonably placed beyond the scope of this stage of the project. 2.2 Observations Observations used in this study were obtai ned from the Army Corps of EngineersÂ’ Field Research Facility (FRF), located near the town of Duck, Nort h Carolina on a barrier island exposed to the Atlantic Ocean [ Birkemeier and Holland, 2001 ]. We utilized data from the SHOWEX (SHOaling Waves EXperi ment) field experiment conducted in the fall of 1999. This data set was chosen for its spatially broad and temporally dense set of bathymetry that extends well beyond the FRF property limits. 2.2.1 Bathymetric Data Bathymetry was collected by the CRAB (C oastal Research Amphibious Buggy). The CRAB is a 10 meter tall amphibious vehi cle, capable of performing surveys to a depth of 8 meters when incident wave heights are less than about 2 meters [ Birkemeier and Mason, 1984 ]. During the SHOWEX experiment, the spatial sampling pattern consisted of shore normal transects, with along-track sample spacing of less than one PAGE 17 7 meter. The transects typically spanned a 500 meter wide portion of the nearshore and some transects occasionally covered more than 1200 meters in the cr oss-shore direction. The alongshore spacing of the survey transect s were about 45 meters. The extent of the transects in the alongshore, over 2000 meters, is what made this data set most attractive. The two major bathymetric data sets that will be employed in this study are ones that were collected between Oct-26-1999 a nd Oct-28-1999 and between Nov-05-1999 and Nov-07-1999. 2.2.2 Model Grid Most previous cross-shore profile evolution models [ Bowen, 1980 ; Bailard, 1981 ; Stive, 1986 ; Roelvink and Stive, 1989 ; Gallagher et al., 1998 ; Plant et al., 2001 ] assume alongshore uniformity. This study utilizes th e extended data set and grids the entire domain. A domain just north of the pier was chosen to eliminate irregular isobaths around the pier. Furthermore, our wave data is collected just nor th of the pier as well, so minimal sheltering effects will occur. A constant grid spacing of 5 x 5 meters was chosen with 100 columns and 200 rows. Th e columns are oriented parallel to the shoreline and the rows perpendicular to th e shoreline. Hence, the model domain represents an area of 1000 mete rs in the alongshor e direction and 500 meters in the crossshore direction. The data were interpolated using the scale controlled methods described by Plant et al. [2002] enforcing smoothness constraint s in both the cross-shore and alongshore. A 2-D Hanning filte r with an interpolation smoo thing scale of 10 meters in the cross-shore and 120 meters in the alongshore was used to generate interpolated profiles. An important note must be made about the grid rotation. For best model results the domain should be most closely rotated so th at the shoreline and/ or the sandbar is/are PAGE 18 8 aligned parallel to the onshore and offshor e boundaries. The FRF has tried to do this with their local coordinate system which de viates from the standa rd latitude/longitude coordinate system. This rotation is approxi mately 18 degrees counterclockwise from true north. Although this rotation is a best fit to align the entire shoreline para llel with the coordinate system, a new rotation is calculated w ith the subset grid used in this study. To find the appropriate second rotati on, cross-shore trends were ca lculated and then aligned. The resulting rotation for this data set is an additional slight counterclockwise tilt. With the alongshore data nearly uniform, a cross-shore trend can be calculated and the data can be filtered. Filtering is done with a spline curve. The filtering will effectively force the unrealistic data, with re spect to the cross-shore trend, to have a weighting of zero. It will also force the first derivative to ze ro at the alongshore boundary. The hydrodynamic model has a few domain requirements. The first requirement is that the gradients be consta nt on all boundaries. The second requirement of the hydrodynamic model is that there be periodic lateral boundaries. The boundary conditions are enforced with a b-spline curve, which is more sophisticated than a spline curve. The smoothing scale used here is 40 meters in the cross-shore and 150 meters in the alongshore direction. La stly the hydrodynamics model does not allow bathymetry that is not submerged. Because of this, some interfering must be done in the area of the shoreline. Prediction of the region near the shoreline is not a specific objective although its inclusion is required fo r boundary conditions on both the sediment transport and the hydrodynamics. This is accomplished by modifyi ng the bathymetry for depths shallower than 40 centimeters and then easing it to a constant depth of 5 centimeters at the shoreline. Experience on this project indicat es that this does not significantly influence PAGE 19 9 the solutions obtained for the beach morphology in deeper water. To address the same constraint of having a submerged bathymetry, th e tidal variation is not incorporated into this study but is maintained at mean sea level. Figure 2-1. Initial bathymetry for Oct. 27, 1999 and Nov 6, 1999. The two initial bathymetries, which were surveyed in the same approximate area, have some significant differences. The survey on October 27th illustrates a more intricate sandbar system with large amplitude perturbations. This results in a more complex nearshore flow field. In contrast, the survey taken on November 6th exhibits a more linear and shore parallel bar structure. 2.2.3 Wave Data The wave data was collected using the 8-mete r array at the Field Research Facility. The FRF array consists of 15 pressure gauge s (collectively referred to as gauge 3111) PAGE 20 10 mounted approximately 0.5 meters off the bottom. It is located in th e vicinity of the 8meter isobath about 900 meters offshore and to the north of the rese arch pier. Voltage analogs of pressure signals are hard-wired through 10-Hz, fourth -order, Butterworth filters (primarily to eliminate 60-Hz noise) and indicates an accuracy of the pressure equivalent of 0.006 meters of water for wave-induced fluctuations [ Birkemeir and Holland, 2001 ]. These gages supplied estimates of the RMS (root mean squared) wave height, period, and dominant di rection at three hour inte rvals. The dominant wave direction was subsequently rotated to f it our domain orientation. After temporal interpolation to hourly intervals matching th e modeling systems computational intervals between bathymetric changes, the wave data supplied by the 8-meter array were applied to the time-varying boundary conditions. 2.3 Model Description With the initial measured bathymetry and the boundary conditions measured offshore, the wave field a nd circulation are computed. The computed hydrodynamics then drive a sediment transport formulation. The divergence of the transport is used to predict bathymetric cha nges. These bathymetric changes are then inserted back to the hydrodynamic model at the subsequent interval with new forcing from the wave model. A resulting coupled modeling system for pr edicting bathymetric evolution has been developed. Figure 2-2. Schematic of th e coupled modeling system. PAGE 21 11 The assumption is made that the seabed e volves slowly, such that the feedback of the changing bathymetry and the sediment transport to the hydrodynamic model are only updated on a bathymetric timescale. In ot her words, the time step taken by the hydrodynamic model (0.075 seconds) is extremel y short compared to the morphologic timescale. Therefore, the slowly evolving ba thymetry is updated only after intervals of approximately one hour when the hydrodynamics and sediment transport have been integrated forward using the pr eviously computed bathymetry. This eliminates the need for computationally costly time dependant ba thymetric updates and new estimates of the wave field at frequent interv als. This assumption is wide ly accepted and has been used by others experimenting with coupled modeling systems. Gallagher et al. [1998] and Thornton et al. [1996] use a coupled time interval of three hours while Plant et al. [2004] decided on a one hour interval. A time step interval of one hour was chosen for this model, after testing the sensitivity of th e results to different time intervals. Different model durations were investig ated briefly. Using measured wave characteristics has limited our runtime scope so that the model would run for a period that would incorporate significant bathymetric change s. For example, to see the response due to storm conditions, the model must simula te through a time period of intense wave conditions. Similarly, if the bathymetric alte rations due to a sustained calm wave climate were of interest, the model needs to run throughout many days of calm wave conditions and the adjusted profile needs to be evalua ted before any storm conditions occur in the data. As a result, model simulations become very dependent on the measured data. This is complicated due to the fact that the bathymetric data sets are surveyed over multiple days. To minimize error caused by evolvi ng bathymetry during the surveys, the PAGE 22 12 bathymetric observations were centered at a specific date within the survey and simulations were run from this initial time. This appears to be a valid assumption because surveys can not take place during intense wave conditions and therefore any bathymetric alterations would be minimal from start to finish of the entire survey. Typical model durations investigated here are approximately two days. Cases were chosen where significant bathymetric change s were large enough to be distinguished from measurement errors. The duration of our simulations were sometimes constrained by numerical instabilities caused by growing shoreline anomalies in the updated bathymetries (section 2.4.1). Others have found a similar range of most advantageous simulated durations. Holland et al. [2000] found that a 5-day inte rval was close to the optimum prediction interval. 2.3.1 Wave Model SWAN (Simulating WAves Nearshore), a third-generation st and-alone (phaseaveraged) wave model was used to s imulate waves over the bathymetry [ Booij, 1999 ]. The offshore RMS wave height was converted to significant wave height and the wave data was submitted to the wave model. For time efficiency, the SWAN model grid spacing in the alongshore direc tion was increased from 5 meters to 25 meters. The 5 meter cross-shore spacing was preserved. To avoid a shadow zone of waves coming in at an angle, the SWAN input domain must be ex tended by a factor of three, doubling it in each alongshore direction. This is done by taking the alongshore boundary cross sections (this is also the mean trend cross section) and extending them out to a distance equal to the length of the original domain. The resu lting domain is now three times the length of the original. PAGE 23 13 Many parameters must be determined in order to initialize the wave model. A Gaussian-shaped frequency spectrum was chosen with a spectral width of 0.01 Hz and a directional spreading of five degrees. A constant depth i nduced wave breaking parameter was also decided on. The propor tionality coefficient of the rate of dissipation is 1.0 and the ratio of maximum individual wa ve height over depth is 0.73. SWAN has the ability to out put a number of different hydrodynamic properties. The one of interest to the hydrodynamics model is the gradient of the radiation stress. For small amplitude waves in irrotational flows, Longuet-Higgins and Stuart [1960] showed that the forcing due to waves is re lated to the wave radiation stress tensor S. The gradient of the wave radiation stress tensor is conveniently outputted by SWAN as the wave induced force per unit surface area: andxyyxyy xx xySSS S FF x yxy (2.1). The wave radiation stress tensor is defined as 2 2[cos12] sincos [sin12]xx xyyx yySgnnEdd SSgnEdd SgnnEdd (2.2), where n is the ratio of group velocity over phase velocity. The subscripts refer to the direction in which the forces act, where x points offshore and y points in the longshore direction. The gradient of ra diation stress must be inter polated back to the original domain before being utilized by the hydrodynamic model. 2.3.2 Hydrodynamic Model Nearshore circulation can be modeled us ing the mass and mome ntum conservation equations that have been integrated over the incident wave timescale and depth. This PAGE 24 14 model uses a simple forward stepping scheme in time with an interval of 0.075 seconds and the computational grid is set up identically to the bathymetric grid with 5 meter grid spacing. The model uses a third order Adams-Bashforth scheme [ Heath, 2002 ] to calculate the time derivatives and a fourth order compact scheme to calculate spatial derivatives. For an initial condition, the fluid is started from rest ((,,0)(,,0)(,,0)0) uxyvxyxy A periodicity condition is imposed in the longshore direction. Free-slip or symmetric boundary conditions (0) v u xx are applied at both the shoreline and offs hore boundaries using fourth order accurate ghost points. The depth averaged approach is used and assumed to be a reasonable approximation because of the large discrepancy of vertical to horizontal length scales. The vertical depth is under one percent of the horizontal extent of the domain. Time dependant movement of the free surface is incl uded and a fourth order compact filter was added to the flow field in both directions. Following Ozkan-Haller and Kirby [1999] our computational fluid dynamics (CFD) model solves the two-dimensional, unsteady, Navier-Stokes equations for an incompressible, homogeneous fluid with variable water depth. ()()0Fxbx Fybyudvd txy uuu uvg txyx vvv uvg txyy (2.3) Where u and v are the depth-averaged mean current velocities in the x and y directions, respectively. Here, is the phase-averaged water surface elevation above the PAGE 25 15 still water level, h is the water depth with respect to the still water level, and dh is the total water depth. The hydrodynamic model will simulate uns teady alongshore currents in the surf zone driven by the gradients in radiation stress caused by obliquely incident breaking waves. The incident wave forcing effects are parameterized by Fx and Fy and are expressed using the radia tion stress formulation by Longuet-Higgins and Stewart [1964] These terms reduce to y x FxFy wwF F dd (2.4) for straight and parallel contours. Bottom friction is taken into account through the nonlinear damping terms bxfbyfUuUv cc dd (2.5), where represents a phase-averaged quantity. The total instantaneous velocity vector, U includes the cross-shore () u longshore () v mean, and oscillatory components of the velocity. ()() Uuv Uuuvv (2.6) Where uand vare the x and y mean velocity components, respectively and uand v are the x and y oscillatory compone nts of velocity, respectively. The oscillatory component of the velocity can be represented as coscos() sincos()o ouut vut (2.7), PAGE 26 16 where is the radian frequency and ou is the orbital velocity in shallow water. 1 2sinh()oH u kh (2.8), where 2 T and T is the peak wave period. The local wave height and wave number are represented by H and k respectively. The time average product of the in stantaneous velocity magnitude ( U ) and the instantaneous alongshore or cross-shore velocity ( u or v respectively), found in equation 2.5, are an important component of the circulation model. Fedderson et al. [2000] explains that direct estimation of || Uu requires a more detailed specification of the velocity field than is usually available, so the term || Uu is often linearly parameterized even though linear parameterizat ions in the mean current frequently are inaccurate because the underlying assumptions (e.g., weak-currents) are violated. Except for the weakest flows, || Uu depends strongly on the mean current and the total velocity variance (oscillatory components of velocity) [ Fedderson et al., 2000 ]. Mean and oscillatory velocities components are the critical constituen ts to calculating || Uu and subsequently calculating the flow field well. To solve the difficult formulation of || Uu we use the nonlinear integral parameterization method. To do this, one mu st integrate over the wave period at every time step. 22 2212 22 22121 ||[cos()2cos()(cossin)][coscos()] 1 ||[cos()2cos()(cossin)][coscos()]ooo T ooo TUuututuvuvuutdt T Uvututuvuvvutdt T (2.9) PAGE 27 17 Following McIlwain and Slinn [2004] these integrals are evaluated numerically using SimpsonÂ’s rule [ Hornbeck, 1975 ] with 16 intervals. The above integrals require significant computational effort to evaluate and, in our case, will increase the computational time by a factor of three. If one is not willing to sacrifice this computational time, then the less accurate lin ear parameterization method should be used. The nonlinear damping terms bx and by are the free parameters in the model and truly are a source of unc ertainty. The friction coefficient parameter, fc will effectively control the nature of the resulting motion. Depending on the values of the friction coefficient, fully developed fluctuations can behave in a variety of ways ranging from equilibrated, small-amplitude fluctuations to energetic, random fl uctuations involving strong vorticies [ Slinn et al., 1998 ]. Ozkan-Haller and Kirby [1999] and Slinn et al. [1998] both found that a stronger mean current, more energetic fl uctuations in the velocities, faster propagation speeds, and more energetic vortex structures result as the friction coefficient f c is decreased. The flow field exhibits shear instabilities of the longshore current due to the re duction of this term and results in unsteady longshore progressive vortices. These sh ear instabilities are found to induce significant horizontal mixing in the surf zone and affect the cr oss-shore distribution of the mean longshore current [ Ozkan-Haller and Kirby, 1999 ]. McIlwain and Slinn [2004] establish for the nonlinear integral parameterization method used here, the best agreement with observed data came from a fc value of 0.003. The effect of lateral mixing due to turbulence and the di spersive three-dimensional effect of the vertical fluctuati ons in the current velocities [ Svendsen and Putrevu, 1994 ] have been neglected and left out of the shallow water equations. Ozkan-Haller and Kirby PAGE 28 18 [1999] found, for reasonable mixing coeffi cients, that the mixing induced by the instabilities in the flow domi nates over mixing due to eddy viscosity terms, which include the effects of turbulence and de pth variation in the current velocities. They also found that the presence of the shear instabilities and the associated momentum mixing actually tends to suppress momentum mixing due to the eddy viscosity terms. 2.3.3 Sediment Transport Model For steady, two-dimensional unidirectional stream flow, Bagnold [1963] utilized an energetics-based sediment transport model assu ming that the sediment is transported in two distinctly different modes. Sediment transport as bedl oad occurs via bed shear stress from the fluid flow plus the downslope cont ribution of gravity, while sediment transport as suspended load occurs via turbulent diffusi on by the stream fluid. The total immersed weight sediment transport rate, i, can be represented as [ Bagnold, 1966 ] () tantan ()tanbs bsiii Wu (2.10), where subscripts b and s refer to bedload and suspended load, respectively. The parameter is the local bed slope and is the angle of repose which is taken to be 28 [ Julien, 1998 ]. The percent of power used for bedload and suspended load is represented with the efficiency factors b and s Following Thornton et al. [1996] 0.135b and 0.015s The rate of energy pr oduction of the stream is equal to the product of the time-averaged bottom stress and the mean free stream velocity, u (2.11). The shear stress at the bed can be represented by wfCuu (2.12) PAGE 29 19 Inserting equation 2.12 into equation 2.11 yields 3wfCu (2.13) where w is the density of water. Mean and fl uctuating velocity co mponents contribute to the nonlinear term 3u, just as described in section 2.3.2. The steady-flow transport equation is then extended to include oscillatory flows as well as steady flows [ Bailard, 1981 ; Bailard and Inman, 1981 ]. The contribution of the longshore bottom stress is also included. Th e resulting time averaged immersed weight suspended sediment transport rate, Q is 2335 2335tan (||||) (||tan||) tan tan (||||) (||tan||) tanxs xbwswx y s ybwswyQKUuUKUuU W QKUvUKUvU W (2.14) with the units 3kgs and represents a time average over many wave periods. The coefficients bK and s K [ Gallagher et al., 1998 ] are and tanwbws bfsf swswKcKc W (2.15), where s is the density of the sediment. bK is dimensionless while s K has the dimensions of sm. The coefficient of friction f c is the same as referred to earlier in section 2.3.2. Just as in the hydrodyn amics model (section 2.3.2), the product || Uu is solved using the nonlinear integral paramete rization method integra ting numerically with SimpsonÂ’s rule. PAGE 30 20 The sediment fall velocity is represented by W. The effect of grain size is handled only explicitly via the fall velocity. The o ffshore sands are often finer than the sand in the nearshore region and must be represented that way. Plant et al. [2004] found that the highest prediction skill was achieved by usi ng a temporally constant but cross-shore variable distribution of sediment fall velocities. Sediment Fall VelocityW = 3.6213x-0.9210 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0100200300400500600 Distance Offshore (m)Fall Velocity (m/s) Figure 2-3. Fall Velocity as a function of offshore distance.1 The bed evolution depends on the divergence of the time averaged transport rate, Q. Mass conservation in both the crossshore and alongshore directions yields 1 ()y x sdQ dQ dh dtgdxdy (2.16). 1 Sediment fall velocity as a function of cross-shore location (following Sleath 1984 ) from data collected in 1984-1985 at the FRF [ Stauble 1992 ]. A power series trendline is used to fit the data. PAGE 31 21 Assuming the density of sedi ment packing is constant, is the packing factor and taken to be 0.7 [ Thornton et al., 1996 ]. The divergence of the tim e averaged transport rate is calculated at the end of each coupled time interval. The bathymetry can now be updated for the subsequent time interval. newoldhhdh (2.17) The constraint of having a submerged ba thymetry is still enforced after the bathymetry is updated. Therefore, to c onserve mass, any bathymetry that emerges beyond the 5 centimeter depth contour will be scoured back to 5 centimeters and the remaining bathymetry at that node will be added to the closest cross-shore node in the offshore direction. This seems to only happe n at the shoreline a nd may even represent shoreline accretion. But as mentioned before prediction of the region near the shoreline is not a specific objective of this study. Extra filtering was incorporated to the se diment flux as well as the divergence of the sediment flux at the end of each coupled ti me interval to reduce any irregular isobaths primarily at the shoreline. With a fixe d set of parameters in the hydrodynamic and sediment transport model, the calculations can become less stable when there is an oddity in the bathymetry especially near a boundary like the shoreline. Therefore this extra filtering was incorporated with a fourth order compact filter in both directions. While this technique was introduced for model stab ility issues, we note th at this could even represent horizontal smoothing attributable to turbulent diffusion at the bed. Filtering may help our model stay numerically stable but it will consequently decrease the net localized accretion or erosion of small scale features. PAGE 32 22 2.4 Results Two model simulations were run in this study. One from the bathymetric data gathered in the region of October 27th, 1999 (hereinafter OCT27) and another from the bathymetric data surveyed around the date of November 6th, 1999 (hereinafter NOV6). The two simulations were started from approximately the same location area with each of their respected initial bathymet ries (Section 2.2.2). They we re stepped forward in time with measured wave data. The wave data varies quite dramatically between the two simulations. OCT27 (Fig. 2-4) represents a calmer wa ve climate period. While the first few hours are a bit stormy, it then calms down w ith wave heights hovering around 1 meter as a long South-East swell enters the nearshore. Figure 2-4. Wave characte ristics for October 27, 1999. PAGE 33 23 Figure 2-5. Wave characteri stics for November 6, 1999. Conditions were quite differe nt for the NOV6 (Fig. 2-5) simulation. The first few hours of the simulation consist of waves w ith a long period approaching from the southeast but around hour 5 the wave climate dram atically strengthens. Waves begin to approach from the north-east with wave he ights exceeding 1.5 meters and wave periods around 6 seconds. The performance of the model for the two simulations is more interesting because the initial bathymetries and the wave climates significantly differ. 2.4.1 October 27th Simulation The intricate initial bathymetry (Sec. 2. 2) causes the flow field as well as the sediment transport to be quite complex for this simulation. The flow field and the transport also have a strong correlation with each other. PAGE 34 24 When the waves are entering the nearshor e from a large enough angle away from shore normal (+/10 degrees), a recognizable al ongshore current can be identified. This is evident from time series of the alongshore a nd cross shore velocities at a single point in the flow field. Figure 2-6. Time series of the cross shore and alongshore velocities for October 27th. The point chosen to record this velocity time series is locat ed 100 meters offshore and 500 meters from the bottom of the grid. Th is point was selected because it is located close enough to the shore to track evidence of an alongshore current. Notice the cross shore velocity (u) time series will oscillate around 0 m/s and gives an indication of how unsteady the flow is. On the other hand, the alongshore velocity (v) time series indicates the alongshore direction of th e flow. The correlation of wave direction and alongshore PAGE 35 25 velocity direction can be seen as well as the unsteadiness of the cross shore velocity during the larger wave periods. The complex bathymetry causes the fl ow to exhibit a meandering around the perturbations in the sandbar similar to that reported by Slinn et al. [2000] This is evident by looking at a stream function field of the alongshore velo city overlaying the existing bathymetry. Figure 2-7. Time averaged alongsho re stream function for October 27th. The stream function of the time averaged fl ow was calculated for a ten hour interval within the simulation. It is averaged from the beginning of hour 25 to the end of hour 34. During this time the waves are approaching fr om a south-east direction. Therefore the stream lines run from the bottom of the grid to the top. Note the more narrow spacing between the stream lines, denoting a stronger fl ow, in the nearshore. As you can see, the PAGE 36 26 current senses the bottom and seems to curve around the high points. The flow field will continue to bend around these pe rturbations as long as the wa ves approach the shore with the same angle and height. The sediment transport is also correlate d with the meanderi ng of the alongshore current. Figure 2-8. Bathymetric cha nge from hour 25 to hour 35. Figure 2-8 shows the change in bathymetry recorded for the same simulated time span as the alongshore stream function above (Fig. 2-7) Looking at the bathymetry change away from the shoreline in the sandbar region (around 10 0m from the left edge of the grid), one can detect a sort of meandering of the sediment transport as well. A correlation is evident when compared to the stream function (Fig 2-7). When the time averaged alongshore current is headed offshore, it is accompanied by an offshore transport. Conversely, when the stream turns back shoreward, an onshore sediment transport can be seen. Also note PAGE 37 27 that the onshore transport regions are of stronger magnitude. Th is is likely related to the contribution from the wave velocities to the total velocity vector. When the principle wave axis and the mean current vector are aligned, the nonlinear product of the total velocity squared or cubed is much larger th en when the two velocity components are not collinear. Figure 2-9 shows that local wave he ights are also larger in these same regions. Figure 2-9. Wave focusing ar ound bathymetry for October 27th. It is also apparent that the perturba tions in the bathymetry are causing wave focusing. This is a significant effect a nd often observed in nature but it also causes additional complications for this study. The focusing creates large pockets of energy at the shoreline. Consequently, immense scouring and erosion can be found at these points. These shoreline perturbations can tend to gr ow and subsequently result in the flow becoming numerically unstable. Various met hods to alleviate these difficulties were PAGE 38 28 implemented with varying degrees of success. As stated above, prediction of the region near the shoreline is not our main specific obj ective although its inclusion is necessary for providing boundary conditions for both th e transport divergence and the hydrodynamic formulations. Our main focus is on bar migration. Because of problems near the shoreline, however, our model simulation runs are limited to the duration until the hydrodynamics become numerically unstabl e around the odd and jagged shoreline bathymetry. When a fixed set of hydrodynamic and transport parameters are implemented, it is difficult for the model to cope with either variations in the wave climate or, in our case, peculiarities of the formulations in very shallow water near the boundary conditions. Shoreline stability and mo re accurate shoreline transformation is a future goal of this continuing study. A number of potentially fruitful avenues are open to further investigation. One that will be explored below is simply to make the shoreline static and let the remaining bathymetry evolve. The net bathymetry change exhibits a similar pattern of sandbar meandering and jagged cuspate features near the shoreline (Figure 2-10). These concentrated sharp shoreline perturbations are found directly behind the sandbar crests and are a direct effect of wave focusing. The magnitude of the sediment transport is relatively small; only about 20 cm of vertical change over 55 hours in the sandbar region. This may be realistic due to the calm wave climate but in our judgm ent more likely to be an underestimate. The next bathymetric survey occurred appr oximately 200 hours later and we were unable to integrate the modeling system forward to that duration because of the shoreline stability. This made it impossible to make direct comparisons of the model hindcasts with survey data. PAGE 39 29 Figure 2-10. Difference in bathym etry from original for Oct 27th. As mentioned above, a simulation was r un holding the shoreline in place and letting the rest of the bathymetry evolve. This allows us to focus in on the transport magnitudes specifically in the sandbar region. But note, variation of the shoreline that may indeed happen and the consequential effect s to the rest of the bathymetry are not included. A comparison of the two different approaches for similar time intervals reveal quite similar results but with the magnitude of the static shoreline approach slightly higher due to less filtering n eeded. Figure 2-11 shows the ma gnitudes of transport in the bar region for the same time interval as Figur e 2-10 but with a sta tic shoreline region. In the regions where sediment is moving onshore, some shoreward advancement of the sandbars can be seen, but only minimally. PAGE 40 30 Figure 2-11. Difference in bathymetry for October 27th with the shoreline held static. The sediment transport appear s to be building the bar more than moving it shoreward. This is even evident where there is a tr ough in the bar and net offshore transport is detected. The net sediment accretion is en closed within sediment depletion on both the shoreward and offshore edges. While these magnitudes seem smaller than expected, the transport shape and directions appear r easonable and consiste nt with the model formulation. Holding the shoreline static also allows us to run our simulation further in time. A simulation of 130 hours was completed. Similar wave conditions were observed for the remaining 75 hours; relatively small waves (less than a meter) coming from the south east. Figure 2-12 shows the same general transport shape with larger magnitudes. PAGE 41 31 Figure 2-12. Difference in bathymetry for October 27th after 130 hours with the shoreline held static. The 130 hour simulation (just over 5 days) eventually incurred similar stability problems as the variable shoreline simulations. Inst ead of sharp cuspate features appearing and growing at the shoreline, the sandbar ultimately became jagged and caused the model to break down numerically. Therefore the m odel was never able to run a full 10 day simulation. Even after the 5 day simulation it is apparent that the sandbar evolution is not desirable. One would expect the sandba r perturbations introduced in October 27th survey to smooth out to the linear ba r exhibited by the November 6th survey (Figure 2-1). This is obviously not the case from Figure 2-12. Here again, the bar seems to be building rather than smoothing out. PAGE 42 32 2.4.2 November 6th Simulation The initial bathymetry for NOV6 (Sec. 2.2) demonstrates a much more linear sandbar than OCT27Â’s bathymetry. Furthermor e, the wave climate and circulation is much more intense (Fig. 2-5). This is ev ident by the time series of the alongshore and cross shore velocities at the sa me point in the flow field. Figure 2-13. Time series of the cross s hore and alongshore veloci ties for November 6th. The magnitude of the alongshore velocity is clearly more powerful and more unsteady in the cross shore direction than for the OCT27 si mulation (Fig. 2-6). It is easy to see the unsteadiness when the vorticity fields are compared. The stream function is calculated dur ing a 10 hour period of 1.5 meter waves coming from the north-east (so a particle would follow the stream line from the top of the grid to the bottom). PAGE 43 33 Figure 2-14. Snapshot of the vorticity fields. The proximity of stream lines within th e sandbar region show the relatively much stronger alongshore current with respect to the OCT27 simulation (Figure 2-7). Although there seems to be quite a bit of vortex she dding, the alongshore mean current flows in a very linear fashion parallel to the sandbar. Wave focusing and shoreline concentration, as stated above, appear to be a problem in this simulation as well. Wh ile the initial sandbar is nearly linear and parallel to the shoreline, it does have a dis tinct trough and high point located about 200 meters from the bottom of the grid. This may be easier to se e in a 3-dimensional bathymetry as shown in Figure 2-16. Similar to the first simulation, the waves focus here, as well as at the northern portion of the grid, and subsequen tly the shoreline perturbations eventually PAGE 44 34 cause the circulation model to become numeri cally unstable near the shoreline after 31 hours. Figure 2-15. Time averaged alongsho re stream function for November 6th. A total net sediment transport for the si mulation after 31 hours of wave action was completed but, due to the magnitude of the shoreline perturbations, it is difficult to analyze the sandbar region specifically. Here again, another simulation was run with the shoreline held static. This also allows the si mulation to run for a full two days. In Figure 2-17 the sandbar region is looked at more closely. Wave hei ght and wave direction stay consistent for the rest of the simulation; approximately one me ter wave height and approach angle nearly shore normal. It is noted here again that holding the shoreline static did not significantly affect the magn itude or shape of th e total net sediment transport. PAGE 45 35 Figure 2-16. 3-Dimensional bathymetry for November 6th. Similar to the NOV6 simulation, the magnit ude of the sediment transport is also quite small; only about 35 cm of vertical ch ange in the sandbar region. This is more likely to be an underestimate due to the fact th at there is such an intense wave climate. The sediment transport in the sandbar region is nearly all moving ons hore. The peak of the positive sediment transport is almost di rectly on top of the peak of the original sandbar and the depletion appears to be on th e seaward slope of the bar. Again, the sediment transport seems to be building the bar more so than moving it. These magnitudes seem smaller than expected and th e direction of transport seems suspect and will be discussed below, but the shape a nd areas of transport appear reasonable. PAGE 46 36 Figure 2-17. Difference in bathymetry for November 6th with the shoreline held static. 2.4.3 Comparison OCT27Â’s simulation as well as NOV6Â’s simu lation, to our judgment, produced less than expected sediment transport. The ma gnitude of transport is more acceptable for OCT27Â’s case, because the wave climate is so mild, but seems more unrealistic for the case of NOV6Â’s more energetic wave climate. It is hard to know what is sensible when comparable bathymetric surveys are spaced so far apart and the model is limited in time. The direction of transport correl ates quite well with the stream line velocities of OCT27. With a straight and parallel initial sandbar, one would expect the shape of transport that was simulated for NOV6 but in the opposite direction. Sand bars on a natural beach typically move slowly shoreward when wa ve energy is low and move more rapidly offshore when waves are energetic and th e wave driven circulation is strong [ Winant et PAGE 47 37 al., 1975 ; Aubrey, 1979 ; Jaffe et al., 1984 ; Wright and Short, 1984 ; Lippmann and Holman, 1990 ]. With NOV6Â’s vigorous wave clim ate, one would expect a net offshore sediment transport. As noted above, the ne t shoreward transport can be explained by the wave and current velocities co-aligning and reasonably dominating. Furthermore, for both simulations, the peaks of the sandbars donÂ’ t seem to be moving very much but rather predominantly building a nd steepening the bar. 2.5 Discussion Our model, which accounts for mean and oscillatory flow is based on the assumption that sediment suspension occurs inst antaneously in response to fluid forcing. However, sediment suspension at one phase of the wave can be transported during a subsequent phase before settling to the bed [ Hanes and Huntly, 1986 ]. Sediment transport where oscillatory flow is dominant, may also depend on fluid accelerations such as those caused by pressure gradients of the surface waves [ Hallermeir, 1982 ; Hanes and Huntley, 1986 ; Gallagher, 1998 ] that are not accounted for in this model. With the fluid accelerations neglected, the al ongshore current becomes an important mobilization force. Some have neglected this force [ Bailard, 1981 ] but Thornton et al. [1996] found the best agreement between observed and predicted pr ofile changes when the steady alongshore current contributed significantly to stirring of sediment that was subsequently transported offshore by the mean cross-shore flow. Improved predictions of profile changes using the energetics transport model are reported to result from including (in the wa ve and wave-driven circulation models) undertow [ Stive and Battjes, 1984 ], wave asymmetry [ Nairn and Southgate, 1993 ], breaking-induced turbulence [ Roelvink and Stive, 1989 ], and infragravity waves [ Sato and Mitsunobu, 1991 ]. Recently published extensions of the Bailard [1981] model PAGE 48 38 [ Hoefel and Elgar, 2003 ] demonstrated improved prediction skill compared to Gallagher, [1998] although the skill for the onshore bar mi gration sequence appears to be poorer than the results presented by Plant et al. [2004] The extended models included a thirdorder statistic (i.e. skewness) of the accelerat ion computed from obser ved flow velocities. The timing of strong accelerations relative to onshore flow is hypothesized to produce net onshore sediment transport [ Elgar et al., 2001 ]. Also added to their model were two additional free parameters, which will always increase the model hindcast skill [ Davis, 1976 ]. A general inspection of parameteri zed nearshore process models by Plant et al. [2004] revealed that unresolved processes contribu te to model errors. This suggests that additional information is necessary to adjust model parameters in order to minimize these errors. There is still not enough known about some of the processes that contribute to sediment transport. Dean and Dalrymple [2002] show the importance of the influence of wave-breaking-induced turbulence as a mobilizing agent within the surf zone. The dynamics of breaking waves on sand bars are understood only qualitatively. Gallagher [1998] suggests that vertical shear in mean cross-shore current may be significant. These various transport mechanisms are difficult to parameterize when little is known about them. To further complicate matters, their si gnificance seems to vary with altering wave conditions. The relative importance of comp eting transport mechanisms is shown by Plant et al. [2001] to depend strongly on the relative wave height (defined as the ratio of the RMS wave height to the local depth). The shape and direction of the sediment transport for the OCT27 simulation looks as expected, except for the fact that the oddi ties in the bar are not really smoothing out PAGE 49 39 but rather growing. This is an example of pos itive feedback. This is also the case for the shoreline perturbations which continue to grow and eventually make the model numerically unstable. In most cases in na ture, there is negative feedback. When something gets out of equilibrium, nature us ually slows it down and will try to bring it back. A good example of this is the air pressure of the earth. When th ere is a gradient of air pressure in the atmosphere, air in places of higher air pressure will shift air toward areas of lower pressure in the form of wind. Our model seems to have a positive feedback that must be addressed in future work. When a perturbation in the bathymetry is introduced, the model appears to magnify th e disturbance rather than smooth it out as nature might under certain conditions. No t enough is known about all the mechanisms under shoaling waves to accurately predic t or parameterize how this might be incorporated. Of course coastal zones also exhibit cases of positive feedback, in the form of erosional hotspots and non-uni form accretion to form beach cusps. Thus some of the model response, especially near the shorel ine, may be representative of natural phenomenon. An important thing to remember is that ev en an ideal sediment transport model can be inaccurate. First, performance of a sediment transport model is difficult to evaluate in studies where the predicted transport also depends on the accuracy of the hydrodynamic model, wave model, and underlying bathymetry Secondly, errors in initial conditions, such as bathymetric sampling errors, could le ad to errors in model predictions. For example, ripples that might significantly affect both hydrodynamics and sediment transport are not typically resolved by most surveying practices [ Plant et al., 2002 ]. PAGE 50 40 Accurate results may develop for flow over a smooth bed but may be inaccurate for flow over ripples and megaripples. 2.6 Conclusion 2.6.1 Future Work The framework of a coupled beach morphology modeling system has been developed. It was tested fo r two data sets and environmen tal conditions at Duck, NC. The magnitude of sediment transport appears to be less than expected. Future work will include a more detailed study of the different components th at contribute to sediment transport. From there, different coefficients can be assigned to va rious physically based terms accordingly to best fits to observations. Future work also includes modifications to attempt to numerically stabili ze the shoreline region. When this is accomplished, model simulations should be able to run long enough in time to arrive at a second completed survey and model results can be compared to field data. The direction of transport of the NOV6 simu lation is a call for concern. The model predicted sediment transport onshore when o ffshore transport was expected. This can also be addressed in future work when addi tional mechanisms are added to the transport equation, such as undertow, skewness of wave s, pressure gradient forces, and scour caused by breaking waves. The hydrodynamics of the model are reasonab le but include necessary engineering approximations for todayÂ’s computational pl atforms. The alongshore stream function follows the bathymetry quite well with str onger flows developing in appropriate areas. The issues of shoreline retreat and advan ce and tidal variations in the water level eventually need to be addressed. The shoreline is an import boundary condition and PAGE 51 41 should be as representative as possible of na tural conditions if accurate predictions are to be accomplished throughout the flow field. 2.6.2 Closing Remarks The major achievement of this project has been to develop a rational framework for a beach morphology model coupled to a wave and mean current model and have it run continuously forced by measured data. With many of the correct physics represented in the model, the results are somewhat realisti c but still considerably doubtful. This is considered acceptable because th ere is still much work to be done by the community to approximately represent more complex transpor t mechanisms. It is a vast problem and small steps are necessary. There is still value in analyzing the results of the coupled modeling system we have implemented. First, because it is a step forward from where the community was. Second, because the formulas we used for sediment transport are widely accepted and used. And finally, because it allows a more simplified interpretation of the response of this still complex physical and modeling system With reasonable results and the correct fundamental physics in place, it is quite encouraging and a good foundation for future development. PAGE 52 42 CHAPTER 3 PRESSURE GRADIENT FORCES AND SHEAR STRESSES ON SAND GRAINS UNDER SHOALING WAVES 3.1 Introduction There is significant experimental evidence that flow acceleration, which serves as a proxy for the horizontal pressure gradient in a coastal bottom boundary layer, has an effect on sediment transport. This eviden ce originates from U-tube experiments [e.g., King, 1990 ], field measurements in the surf zone [e.g., Hanes and Huntley, 1986 ; Gallagher et al., 1998 ; Elgar et al., 2001 ] and in the swash [e.g., Butt and Russell, 1999 ; Puleo et al., 2003 ], and three-dimensional discrete pa rticle computer simulations [e.g., Drake and Calantoni, 2001 ]. They have found significan t correlations under certain relevant conditions between phases of flow acceleration caused by pressure gradient forces from the surface gravity waves and sediment suspension and net cross-shore transport. Hsu and Hanes [2004] demonstrate that responses of sheet flow, such as the velocity profiles, the instantaneous bed shear stress, the sediment flux, and the total amount of the mobilized sediment, cannot be fully parameterized by a quasi-steady freestream velocity and may be correlated with the magnitude of the local horizontal pressure gradient. Moreover, their numerical experime nts indicate that cat astrophic internal bed failure is a direct consequence of large horizon tal pressure gradients. These recent papers investigate the link between fluid acceleration and sediment transport. This chapter will explore the degree to which the pressure gradie nt contributes compared to the shear stress on sediment mobilization under surface gravity waves in the surf zone. PAGE 53 43 Generally there are two types of bed stre sses that have the ability to mobilize sediment under surface gravity waves. One that is considered by all, is the shear stress exerted by the orbital velocities under a propaga ting wave. This stress results in a force acting tangentially to a surface such as a real flow over a sea bed. The other, maybe less recognized to have significance, is the horizontal pressure grad ient stress. This stress is the consequence of the difference in hydrostatic pressure from one side of a particle to the other. When large enough, the difference in pressure can induce a sediment particle to become unstable and be mobilized. In many typical coastal regimes, the ratio of the pressure gradient stress to the shear stress is 1/10 or even 1/50. For this reason, it has been thought su fficient to neglect the contribution of the pressure grad ient stress in formulating the total stress in the surf zone. The primary aim of this study is to demonstrat e that there is a regime in which it is inaccurate to neglect the pressure gradient stress. This will be done by examining such parameters as wave period, grain size, wave height, and water depth. A secondary aim is to obtain an understanding of the degree to wh ich the pressure gradient stress contributes to bed mobilization to be used in future pa rameterizations of sediment transport models. 3.2 Stress Formulation 3.2.1 Shear Stress The stress component from forces applied pa rallel to a surface is the shear stress. In our case, it is the force exerted by flowi ng water over a sea bed. We will be examining the shear stress resulting from this force. The stress is simply the force divided by the cross sectional area on which the force is applied. There is not a direct division of the cross sectional area in the shear stress, equation because the grain size is embedded within the empirical Darcy-Weisbach friction factor f PAGE 54 44 1 8 f uu (3.1) We are interested in the bottom shear st ress, so the horizontal velocity term, u is the horizontal near bottom fluid velocity. This variable will be examined more in section 3 below. The density of sea water is represented by which has a value of 1024 3kgm This equation has been developed through dimensional analysis and experimental data have been used to develop values of the friction factor, f For wave motion, the bottom friction is a nonlinear function and due to the absolute value sign becomes somewhat complicated to work with directly In our model, the Stanton Diagram for friction factor under waves as a f unction of the relative roughness, r was used to acquire the friction factor [ Kamphuis, 1975 ]. Rough turbulent flow is assumed to obtain friction factors off of the Stanton diagram. A polynom ial fit was then created to easily acquire friction factors for given parameters. 0.380.62 f r (3.2) It is important to note the ambiguity of the friction factor. The friction factors which are represented on the Stanton diagra m are ones developed from measurements of bottom shear stress retarding the motion of a fl uid in a unidirectional pipe flow. This leaves a bit of uncertainty in the level of approximation of this e quation considering we are investigating sediment mobilization unde r surface gravity waves. We know that situations are seldom idealized. Therefore, possibly an empirical calculation, even if it is one derived from somewhat different circumst ances, is a better representation of this physical occurrence. The evaluation of the sh ear stress equation is beyond the scope of this paper but the uncertainty and level of approximation inherent from representing it under ocean waves is not to be overlooked. PAGE 55 45 3.2.2 Pressure Gradient Stress Under a surface gravity wave there is a vari able water level within each wave. The difference in hydrostatic pressure from th e differing water levels will exert a net horizontal force on a sand grain. Here we represent a sand grain as a sphere. Figure 3-1. Schematic of a sand grain. As everyone knows, sand grains are not pe rfect spheres, so a modest level of approximation is introduced at this point. The horizontal pressure gradient force is computed by integrating the normal force over the sphere surface. At each point on the surface of the sphere there is a force per unit area, P on the solid acting normal to the surface. Multiplying this local force per unit area by the surface area on which it acts and integrating over the surface of the sphe re will produce the resultant force. 2 2 00(sinsin)sinx rRFPRdd (3.3) A detailed derivation of the resultant fo rce can be found in the appendix. This force is analogous to a component of the inertial force calculation in the Morrison [1950] Equation. (1)Imdu FC dx (3.4) z0 (x) zR z x y Z=0 -------z PAGE 56 46 This equation is made up of two components Imdudu FC dxdx (3.5). The first part of this equation, or the Â“1Â” in equation 3.4, is the for ce that is corresponding to the pressure gradient force explained above. This pressure gradient force is then divided by the cross s ectional area on which the force is acting. The final stress can now be represented by 504 32spd Pgk x (3.6), where the pressure stress is denoted as s P g is the acceleration due to gravity, x is the local gradient of the wave in the horizontal direction, and pk is the pressure response factor. cosh(()) cosh()pkhz k kh (3.7) An important point to note is that th ese two equations, whose magnitudes will be compared to each other, have a different desi gnation for the grain size. The shear stress calculation uses 90d (diameter of the sand that 90% is finer) while the pressure stress calculation uses50d To account for this, the distribution of sand is assumed to obey a log normal probability distribution. That is, if normal probability paper is used for the cumulative percentage passing and the phi scale is used for the sand size, a straight line will result. Again, there is always uncertainty with a completely idealized equation. There are no empirical parameters in this equation and we know that no situation is ever completely PAGE 57 47 idealized. Yet again, one must be cautious with the level of confidence put into an equation of this nature. 3.2.3 Phase Lag The two stresses in question vary alo ng time and space intervals as a surface gravity wave passes over a location. Hence, it is equivalent to examine a point in space over a wave period or different sp atial locations at an instant in time. It was decided to hold the time variable constant and vary the space variable to investigate the distribution of the stresses. It was found that the peak stresses are separated by a phase lag of 90 degrees. Figure 3-2. Phase lag of stresses. Notice at some phases of the wave, the shear stress and pressure stress work together in a constructive manner and at other phases they work against each other in a destructive manner. Because the two phases lag by exactly 90 degrees, when one of the stresses is at its maximum, the other is zero. PAGE 58 48 It is not the individual stresses that will be compared to threshold values of mobilization but rather the total of the two which is the best indicator of bed mobilization. That is, the maximum of the to tal stress under the wave is the value that will be used to compare to the thresholds of mobilization to indicate whether a sand particle will be suspended under a wave fo r particular wave ch aracteristics. A comparison of the two stress ma gnitudes at this certain phase, where the maximum total stress is found, will not be made but rather the maximum of each of the stresses under the wave will be compared. This is the only fair comparison because as alluded to before, there are parts under the wave which are co mpletely dominated by the shear stress and others that are completely do minated by pressure stress. Comparing the two stresses at the point of maximum total stress is not as re asonable because the ratio would be swayed depending on where on the phase the total maximum stress was found. The reason to calculate the maximum total stress is to compar e it to the threshold of mobilization to see if a particular sediment size will be suspended under certain conditions. 3.2.4 Nonlinear Wave Everyone knows that ocean waves are not perfectly linear in nature. Many different types of nonlinearities exist. Esp ecially as ocean waves approach breaking, a strong nonlinearity in the leading edge can be ob served as a steep face. One can imagine that if a wave were to steepen, the steep l eading edge of the wave would experience a far greater horizontal pressure gradient than the linear case. PAGE 59 49 Figure 3-3. Nonlinear wave. To analyze the contribution of the nonlinea r wave it can first be broken down into many constituent waves by Fourier decomposition. A single nonlinear wave can be represented by a set of many smaller linear waves, all of them superimposed. Once broken down, the contribution by each smaller individual set of waves can be analyzed and tallied. To find the phase position of the maximum total stress under a wave, only the first mode (1 x ) of the decomposition was varied. Th e nonlinear signal is not a complicated wave pattern, therefore most of the energy is found in the lowest fr equency waves of the breakdown. Due to the fact that the nonlinear wave is of similar amplitude and frequency to the linear wave, the contributions of the multix waves drop dramatically after the 1 x wave. In other words, only the lowest frequency of the stressesÂ’ decomposition is PAGE 60 50 Figure 3-4. Nonlinear wave addition. varied to find the maximum total stress phase position. The phase position is now used in the superposition of the remaining multi frequency waves. Just as in the linear case, at some phases of the wave, one of the stresses will dominate the other stress. Although this ti me the maximum pressure gradient stress will be larger due to the steeper gr adient and thus increase the rati o of pressure gradient stress to shear stress. 3.3 Varying Parameters and Limits A closer look at the two stress equations reveals that they both decay exponentially with increasing kh This kh parameter is the measure of wave type. Larger kh values indicate a more deep water wave ( kh > ) and smaller kh values signify a more shallow =+ + Nonlinear Equation PAGE 61 51 water wave ( kh < 10 ). Within the pressure gradient stress equation (eq. 3.4), the pressure response factor, pk includes the exponential func tion, hyperbolic cosine (or cosh). cosh() 2xxee x (3.8) The exponential function is located in the de nominator which will cause the magnitude of the pressure gradient stre ss to decay at a rate of khe 1 for the sea bed. cosh()pk kh (3.9) Likewise, within the shear stress equati on (eq. 3.1) the hor izontal velocity, u also includes the exponential func tion in the denominator. cos( ) for the sea bed. 2cosh() gHkkxt u kh (3.10) The horizontal velocity is squared in th e shear stress equation (eq. 3.1) while pk is not raised to a power in the pressure gradie nt stress equation (eq. 3.4). Therefore, the magnitude of the shear stress will decay at a rate of 2 khe while the pressure gradient stress will decay at a rate of khe At low values of kh the shear stress is usually much more than the pressure gradient stress. With increasing kh values, both the stresses will decrease exponentially but the shear stress w ill decrease much faster. The magnitude of the pressure gradient stress will eventually overtake the magnitude of the shear stress. But will it be too much like a deep water wave when the pressure gradient stress has any sort of contribution? We are seeking to identify a realistic regime where the pressure gradient stress plays a substantial role in m obilizing sediment but is not in too deep of water that the combined stresses are t oo weak to initiate sediment movement. PAGE 62 52 This limits our analysis domain to inside the critical limit of mobilization. Our investigation is only shoreward of this point which is a f unction of grain size. These critical limits were first explored by Shields [1936] He determined the threshold condition by measuring sediment transport for different values of total stress at least twice as large as the critical value and then extrapolated to the point of vanishing sediment transport. For turbulent flow s over rough boundaries, the critical stress becomes linearly proportional to the sediment size. Based on a comparison of data from the Highway Research Board [1970] a relationship between cri tical stress and median grain size on a flat horizontal surface was formul ated for granular material in table form [ Julien, 1998 ]. A polynomial fit was then created to acquire a critical stress for a given sediment size. 23 5050500.11135.37225189.5891766304.35Criticalddd (3.11), where 50d is in meters and is measured in pascals. The shoreward extent of our analysis doma in will be taken at the breaker limit. Our analysis is based on linea r theory which loses validity once a wave breaks. During breaking and after, other wa ve induced bed stresses beco me dominant like breakinginduced turbulence [ Roelvink and Stive, 1989 ]. Therefore our analysis domain is limited to seaward of the break point. An approxi mate depth limited breaking boundary can be obtained from McCowan [1894] who determined that waves break when their height becomes equal to a fraction of the water depth. 0.78bbHh (3.12) PAGE 63 53 The subscript b denotes the value at breaking. This is a crude approximation and there has been more complex ways to acquire breaking parameters but this one will be used for simplicity. Steepness limited breaking must also be taken into account. Miche [1944] developed a simple equation for wa ve breaking in any water depth. max1 7H L (3.13) This indicates that the maximum wave height, H, is limited to one-seventh of the wave length, L. Also a crude approximation, but si mply stated within our analysis. 3.4 Results 3.4.1 Linear Results As stated above, shear stress dominates pressure stress in many typical wave conditions. An easy way to show this is by varying the water depth and seeing the resulting stresses for typical wave characteristics. Figure 3-5. Linear: T = 7sec, d = 0.3mm, H = 1m PAGE 64 54 In Figure 3.5, the x-axis is the water depth measured in meters and the y-axis is the resulting stresses measured in pascals on a l og scale. Here, the shear dominates well below the critical limit of mobilization. The magnitude of the pressure force does ev entually draw level and surpass the shear stress but this happens in water depths mu ch too deep for the combination of the two stresses to initiate sediment mobilization. This graph is a bit misleading because the in tersections of either of the two stresses and the critical limit have no real significance. It is the addition of the two stresses which needs to be compared to the critical limit. It is not even the total of the two maximum stresses but rather the maximum of the tota l stress calculation under a wave which is being compared against the critical limit. A better way to represent the stresses is to look at the ratio of the two stresses in question versus the kh parameter in which they vary so greatly. Figure 3-6. Linear: T = 7sec, d = 0.3mm, H = 1m PAGE 65 55 The x-axis is the non-dimensional wave type, kh, and the y-axis is the ratio of pressure gradient stress to shear stress. This way the graph is non-dimensional and the crossing of the ratio line and the critical limit has significance. This intersection will be the ratio of the maximum stresses just as an incoming wave initiates sediment motion. The greatest ratio will always occur at the offshore boundary when th e critical stress is just met and the sediment is first mobilized With these wave conditions the pressure gradient stress to shear stress ratio only reac hes 0.14, or the magnitude of the pressure gradient stress is only 14% of that of the shear stress. With these wave conditions it seems reasonable to simply neglect the pressu re gradient stress and still not sacrifice much accuracy. Varying the dependant parameters, one can examine the relative importance of the pressure gradient stress to shear stress ratio. The wave height and period will remain the same and the diameter of the grains w ill be increased from 0.3 mm to 0.5 mm. Figure 3-7. Linear: T = 7sec, d = 0.5mm, H = 1m PAGE 66 56 Here, the maximum ratio of pressure gradie nt stress to shear stress only increases slightly. The ratio only reaches a value around 0. 18, still not very sign ificant to the total stress contribution. An increase in the contri bution of the pressure gradient stress implies that the greater surface area of a larger grain aids the pressure gradient stress more so than the shear stress, but only slightly. The variation of the wave period will now be analyzed. The grain size diameter is returned to a value of 0.3 mm and the wave period is decreased from 7 seconds to 2.5 seconds. Decreasing the wave period and keep ing the same wave height will effectively make the wave shorter and steeper. This will cause a bigger disp arity of hydrostatic pressure from one side of a sand grain to the other. Figure 3-8. Linear: = 2.5sec, d = 0.3mm, H = 1m In this example, the ratio of pressure gr adient stress to shea r stress has increased noticeably by making the wave period shorter. The ratio at the point of mobilization has increased to over 0.3. With such a short wa ve, one must be conscious of the wave being PAGE 67 57 too steep and breaking. Here, the graph also indicates that the wave will break due to depth limited breaking before it reaches its steepness limitation. The wave height was also varied, but little change in maximum ratio was observed. The only noticeable outcome was a change in th e break point. The increased wave height may mobilize a greater domain of sediment but our domain of applicability decreased because the breaker limitation and the maximum ratio remain similar. With the wave period shortened and the grain size increased, the pressure stress will find its most favorable contribution. In the next case, the wave period is shortened again to 2.5 seconds and the grain size diameter is increased to 0.5 mm. With these wave characteristics, the maximum pressure stress to shear stress ratio has now reached 0.42. Figure 3-9. Linear: T = 2.5sec, d = 0.5mm, H = 1m Note where the sediment is mobilized along the kh axis. At the point of mobilization the kh value is 3.2 and at the poin t of breaking the kh value is 1.5. Here our analysis domain falls within the intermediate wave type category. PAGE 68 58 3.4.2 Nonlinear Results The nonlinear results will be analyzed in a similar fashion to the linear case. Similar results are observed but swayed more toward the contributions from the pressure gradient. Figure 3-10 shows the dimensional graph varying the water depth and viewing the resulting stresses. Figure 3-10. Nonlinear: = 7sec, d = 0.3mm, H = 1m This again is for our proto-typical wave c onditions. With the same axis lengths and scale as the linear graph, it is apparent that the shear stress and pressure gradient stress crossing point occurs in much shallower water a nd closer to the critical limit. What also should be understood from this graph is that there is a regime from about 10 meters water depth into the break limit where the pressu re stress alone is enough to mobilize the sediment. PAGE 69 59 The nonlinear case is also converted to a non-dimensional graph in Figure 3-11. The stresses are represented as the ratio of maximum pressure to shear stress and the kh parameter is varied. Figure 3-11. Nonlinear: T = 7sec, d = 0.3mm, H = 1m The ratio at the point of mobilization has increased to over 0.5. This is considerably larger compared to the linear case, 0.14, with the same wave characteristics. The same dependant variables will be vari ed with the nonlinear wave at the same degree to see its effect on the stresses compared to that of the linear case. First the grain size diameter will be increased to 0.5 mm. The increase in grain size diameter has a similar effect as the linear case. The change in grain size results in a minimal change of the maximum ratio. The pressure gradient st ress to shear stress ratio has now reached a value of 0.7. Figure 3-12 shows that the ma ximum of the pressure gradient stress and the shear stress are now nearly the same ma gnitude at the poin t of mobilization. PAGE 70 60 Figure 3-12. Nonlinear: T = 7sec, d = 0.5mm, H = 1m Figure 3-13. Nonlinear: T = 2.5sec, d = 0.3mm, H = 1m The grain size is returned to 0.3 mm and the wave period is now decreased to 2.5 seconds in figure 3-13. With the ratio of the stresses reaching and surpassing 1.0 and the sediment still being mobilized, the magnitude of the pressure gradient stress is now more PAGE 71 61 than the shear stress just as the sediment is mobilized. At the poi nt of mobilization the pressure gradient stress is about 20% greater than the shear stress. The wave height has little effect as men tioned earlier in the linear case. The two dependent variables, wave pe riod and grain size, make th e largest contributions to maximizing the contribution of the pressure gradient stress. Similar to the linear case, the grain size and wave period are changed to maximize the contribution by the pressure gradient stress. Figure 3-14. Nonlinear: T = 2.5sec, d = 0.5mm, H = 1m The ratio has now reached 1.6 at the point of mobilization. The magnitude of the maximum pressure gradient stress is 60% greater than that of the shear stress at the point of mobilization. The kh axis indicates again that our analysis takes place within the intermediate wave type range. PAGE 72 62 3.5 Conclusion A simple analytical model based on empiri cal formulas was used to examine the relative contributions of pressure and shear stresses to bed mobilization. First, the parameters that were varied will be discussed. It became evident that an increase in grain size diameter contributed to the pressure grad ient stress contribution. Clearly a larger surface area for the horizontal pressure gradient force to act upon is more favorable then for the shear force to act upon. A decrease in wave period also produced a favorable contribution for the pressure gradient stress. A decreased period will result in a shorter and steeper wave. The steeper gradient wi ll result in a greater horizontal pressure gradient and a noticeably greater pressure gradient stress contribution. Both changes in these parameters are in favor of the pressure gradient stress but to di fferent degrees. The dependent variable comparison gives the impres sion that the ratio of the two stresses is more sensitive to the change in wave period ra ther than grain size. As mentioned before, the wave height appeared to have minimal effect on the ratio. While the break point would move, the maximum ratio of pressure gradient stress to shear stress stayed approximately the same. The analysis gives us some distinct re gimes where the pressure gradient stress holds considerable influence on the total stre ss. Short period waves, 2 5 seconds, will result in an influential contri bution by the pressure gradient stress. For a 2.5 second wave and 0.3 mm grain size, the magnitude of the maximum pressure gradient stress reaches over 30% of that of the shear stress for the linear wave case and is actually 20% more than the maximum shear stress for the nonlinear wave case. The solutions also show that it was not uncommon for the pressure gradient stress alone to be sufficient to induce particle mobilization. This suggests that fo r short period waves, the pressure gradient PAGE 73 63 stress can be quite important. Increased grain size also resulted in an enhanced contribution by the pressure gr adient stress, but not to th e extent of the wave period fluctuations. Larger grain si zes, 0.31.0 mm, appear to favor the pressure gradient contribution. These characteristics were developed for our analysis in intermediated water depths, 1-15 meters, and result ed in intermediate wave types. Altering both the wave period and the grain size to realistic ocean quantities, a 2.5 second wave and a mean sand grain diameter of 0.5 mm, one can detect a significant contribution from the pressure gradient st ress. The linear case suggests that the magnitude of the maximum pressure gradient stress is over 40% to that of the maximum shear stress while the nonlinea r case indicates that the pressu re gradient stress is actually 60% more than that of the shear stress when the sediment is first mobilized. One certainly can not have much confidence in se tting a threshold for the total stress when possibly half the magnitude of the stress is ne glected if pressure gr adient contributions are neglected. The analysis has also shown where the pressure gradient stress makes its greatest contribution. The maximum contribution ma de by the pressure gradient stress was always located at the critical point of mob ilization or more easily stated, just as an incoming wave begins to move sediment ar ound. This is also the place within our analysis domain where a wave exhibits its most linear form. As a wave approaches breaking, it will take on more of a nonlinear form similar to the one analyzed in this study. A breaking wave may even approach a ve rtical wall of water just prior to breaking which would increase the pressure gradient stress significantly. This suggests that the pressure gradient stress could become impor tant along our entire analysis domain, from PAGE 74 64 the critical stress limit all the way to the poi nt of breaking. Consequently, for certain wave and grain characteristics, it would be precarious to not include the pressure gradient stress in the total stress calculation and expect high levels of precision. Since the pressure gradient stress caused by surface gravity waves show some relative contribution to the total stress at the sea-bed, one mu st rethink the way that near shore sediment transport para meterizations are found. Thes e parameterizations should not be reasonably based upon empirical rela tionships developed from open channel or even oscillatory flow experiments if they ar e not produced by surface gravity waves. By including pressure gradient stresses it will al so help in the parameterization of sediment flux in the direction of wave advance due to asymmetric and skewed nonlinear wave shapes typical of shoaled and breaking waves. PAGE 75 65 APPENDIX DERIVATION OF THE RESULT ANT FORCE ON A SPHERE () cosR x ax zR 2 2 00(sinsin)sinx rRFPRdd 0 0() sinsincosatmR rRrR atm rRPPgzgxgz PPgzgaRgR 2 2 0 00 22 2222 0 0000 22 33232 0000{[sinsincos]sinsin}sin sinsinsinsin sinsincossinsinxatm xatmFPgzgaRgRRdd FPRddgzRdd gaRddgRdd z0 (x) zR z x y a x Z=0 -------z PAGE 76 66 2 3333 00 0 323 0 0sin2 sinsin 24 14 cos(sin2) 33x x xFgaRdgaRd FgaRFgaR 3 50 24 44 3 332x sp pgaR Fd gaRPgk ARx PAGE 77 67 LIST OF REFERENCES Aubrey, D.G., Seasonal patterns of onshore/offshore sediment movement, J. Geophys. Res., 84 6347-6354, 1979. Bagnold, R.A., Mechanics of marine sedimentation, in The Sea vol. 3, The Earth Beneath the Sea edited by M.N. Hill, pp 507-528, Wiley-Interscience, New York, 1963. Bagnold, R.A., An approach to the sediment transport problem from general physics, Prof. Paper 422-I U.S. Geol. Surv., 1966. Bailard, J.A., An energetics total load se diment transport model for a plane sloping beach, J. Geophys. 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Foster, D.L., R.A. Holman, and R.A. Beach, Sediment suspension events and shear instabilities in the bo ttom boundary layer, in Coastal Dynamics Â’95 pp 712-726, Am. Soc. Of Civ. Eng., New York, 1995. Gallagher, E.L., Observations of sand bar evolution on a natural beach, J. Geophys. Res., 103 3203-3215, 1998. Hanes, D.M. and D.A. Huntley, Continuous measurements of suspended sand concentration in a wave domi nated nearshore environment, Cont. Shelf Res., 6 585-596, 1986. Hay, A.E. and A.J. Bowen, Coherence scales of wave-induced suspended sand concentration fluctuations, J. Geophys. Res., 99 12, 749-12,765, 1994. Heath, M.T., Scientific Computing: An Introductory Survey, Second Edition, pp 407-410, McGraw-Hill, New York, 2002. Highway Research Board, Tentative desi gn procedure for riprap-lined channels, National Academy of Science, National C ooperative Highway Research Program Report no. 108, 1970. Hoefel, F. and S. Elgar, Wave-induced se diment transport and sandbar migration, Science 299, 1885-1887, 2003. Holland, K.T., C.L. Vincent, and R.A. Holman Statistical characte rization of nearshore morphodynamic behavior, in Coastal Sediments Â’99 edited by N.C. Kraus, pp 2176-2189, Am. Soc. Of Civ. Eng., Reston, Virginia, 2000. Hornbeck, R.W., Numerical Methods Prentice-Hall, New Jersey, 1975. Hsu, T.-J. and D.M. Hanes, Effects of wave shape on sheet flow sediment transport, J. Geophys. Res., 109, C05025, doi:10.1029/2003JC002075, 2004. PAGE 79 69 Jaffe, B.E., R.W. Sternburg, and A.H. Sallenge r, The role of suspended sediment in shore-normal beach profile changes, in Proc. 19th Int. Coastal Eng. Conf ., pp 19831996, Am. Soc. of Civ. Eng., New York, 1984. Julien, P.Y., Erosion and Sedimentation Cambridge University Press, New York, 1998. Kamphuis, J.W., Friction fact or under oscillatory waves, J. Waterways, Harbors Coastal Eng. Div., ASCE Vol. 101, 135-144, 1975. King, D.B., Studies in oscill atory flow bed load sediment transport, Ph.D. thesis, University of California, San Diego, 1990. Lippmann, T.C. and R.A. Holman, The spat ial and temporal variability of sand bar morphology, J. Geophys. Res., 95 11575-11590, 1990. Longuet-Higgins, M.S. and R.W. Stewart, Th e changes in the form of short gravity waves on steady non-uniform currents, J. Geophys. Res., 8 565-583, 1960. Longuet-Higgins, M.S. and R.W. Stewart, Ra diation stress in wate r waves, a physical discussion with application, Deep Sea Res., 11 529-563, 1964. McCowan, J., On the Highest Wave of Permanent Type, Philos. Mag. J. Sci. Vol. 38, 1894. McIlwain, S. and D.N. Slinn, Modeling alongshore currents over barred beaches, submitted to the Journal of Waterway, Port, C oastal, and Ocean Engineering 2004. Miche, M., Movements ondulatoires des mers en profondeur constante ou decroissante, Annales des Ponts et Chaussees 25-78, 131-164, 270-292, 369-406, 1944. Morrison, J.R., M.P. OÂ’Brien, J.W. Johnson, and S.A. Schaaf, The force exerted by surface waves on piles, Petrol. Trans. AIME Vol. 189, 1950. Nairn, R.B. and H.N. Southgate, Deterministic profile modeling of nearshore processes. Part 2. Sediment transport and beach profile development, Coastal Eng., 19 57-96, 1993. Ozkan-Haller, H.T. and J.T Kirby, Nonlinea r evolution of shear instabilities of the longshore current: A comparison of observations and computations, J. Geophys. Res., 104 C11, 25953-25984, 1999. Plant, N.G., R.A. Holman, M.H. Freilich, and W.A. Birkemeier, A simple model for interannual sandbar behavior, J. Geophys. Res., 104 15, 755-15, 776, 1999. Plant, N.G., B.G. Ruessink, and K.M. Wij nberg, Morphologic proper ties derived from a simple cross-shore sediment transport model, J. Geophys. Res., 106 945-958, 2001. PAGE 80 70 Plant, N.G., K.T. Holland, and J.A. Puleo, An alysis of the scale of errors in nearshore bathymetric data, Mar. Geol., 191 71-86, 2002. Plant, N.G., K.T. Holland, J.A. Puleo, and E.L. Gallagher, Prediction skill of nearshore profile evolution models, J. Geophys. Res., 109 C01006, 2004. Puleo, J.A., K.T. Holland, N.G. Plant, D.N. Slinn, and D.M. Hanes, Fluid acceleration effects on suspended trans port in the swash zone, J. Geophys. Res., 108 (C11), 3350, doi:10.1029/2003JC001943, 2003. Roelvink, J.A., and M.J.F. Stive. Bar ge nerating cross shore flow mechanisms on a beach, J. Geophys. Res., 94 4785-4800, 1989. Roelvink, J.A. and I. Broker, Cross-shore profile models, Coastal Eng., 21 163-191, 1993. Sato, S. and N. Mitsunobu, A numerical model of beach profile change due to random waves, in Proc. Coastal Sediments Â’91 pp 674-687, Am. Soc. of Civ. Eng., New York, 1991. Shields, A., Anwendung der aehnlichkeitsmech anik und der turbulenz forschung auf die geschiebebewegung, Mitteilungen der Preussische Versuchanstalt fur Wasserbau und Schiffbau, Berlin, 1936. Sleath, J.F.A., Sea Bed Mechanics pp 335, John Wiley, New York, 1984. Slinn, D.N., J.S. Allen, P.A. Newberger, a nd R.A. Holman, Nonlinear shear instabilities of alongshore currents over barred beaches, J. Geophys. Res., 103 18, 357-18, 379, 1998. Slinn, D.N., J.S. Allen, and R.A. Holm an, Alongshore currents over variable beach topography, J. Geophys. Res., 105 C7, 16,971-16,998, 2000. Stauble, D.K., Long-term profile and sedime nt morphodynamics: Field Research Facility case history, Tech. Rep. CERC-92-7 U.S. Army Corp of Eng., 1992. Stive M.J.F., A model for crossshore sediment transport, in Proc. 20th Int. Coastal Eng. Conf ., pp 1550-1564, Am. Soc. of Civ. Eng., Taipei, Taiwan, 1986. Stive M.J.F. and J.A. Battjes, A model for offshore sediment transport, in Proc. 19th Int. Coastal Eng. Conf ., pp 1420-1436, Am. Soc. of Civ. Eng., New York, 1984. Svendsen, I.A. and U. Putrevu, Nearshore mixing and dispersion, Proc. R. Soc. London, Ser. A, 445 561-576, 1994. Thornton, E.B., R.T. Humiston, and W.A. Birkemeier, Bar-trough generation on a natural beach, J. Geophys. Res., 101 12097-12110, 1996. PAGE 81 71 Winant, C.D., D.L. Inman, and C.E. Nordst rom, Description of seasonal beach changes using empirical eigenfunctions, J. Geophys. Res., 80 1979-1986, 1975. Wright, L.D. and A.D. Short, Morphodynamic variability of surf zones and beaches: a synthesis, Mar. Geol., 26 93-118, 1984. PAGE 82 72 BIOGRAPHICAL SKETCH I was born and in a small coastal town located on the New Jersey Shore called Point Pleasant. I spent my fi rst 6 years in a small house in Toms River, NJ, only to move to a more rural area located just south. It is here in Lanoka Harbor that I spent the majority of my adolescence. In this hous e I lived with my two loving parents and my older sister, Maggie. I was also blessed w ith the great experience of living with my grandparents too. Growing up in this small pristine town was very enjoyable. I spent the majority of my time outside playing sports, surfing, and ri ding my bike with my friends. As I grew older, I became more involved in organized sp orts. By high school I participated in three varsity sports (football, indoor track, and baseball) and was the captain of two by senior year. I was also very active in many clubs a nd groups as well as playing in the marching band for one year. After graduating high school in 1999 I decided to attend North Carolina State University. It was not an easy decision, b ecause going to a large uni versity meant that it would be difficult to play any varsity sport. I got involved in different ways by joining a social fraternity (Sigma Chi) as well as an honors fraternity (Chi Epsilon). Being the intramural sports chairman and partic ipating in philanthropy events around the community were a few ways I spent my free time in college. I was even able to play a varsity sport when I walked ont o the track team for one year. PAGE 83 73 Of course the academic demands of civil engineering were first and foremost. I feel I was provided with an excel lent base of skills in civ il engineering at NC State to excel in any sub discipline chos en. It was my junior year, after I heard a presentation by one of our professors, that I knew coastal engineering was the field for me. It combined something I enjoyed studying, engineering, as well as something I loved, the beach. I was fortunate enough to obtain an internship in Duck, NC at the USACE Field Research Facility for the summer after my junior and se nior years of undergra duate. Here I got a taste of the coastal engineer ing field by participating in many field experiments and leading a public tour. After graduating from NC State University in May of 2003, it seemed like an easy decision to continue my academic experien ce in graduate school studying coastal engineering at the University of Florida. The decision was made easier after being offered a graduate research assistant positi on studying under Dr. Donald Slinn. I have demonstrated growth and development in my short time at U.F. by speaking and presenting at two major conferences and exp ect to have two papers published from my thesis work. While working under Dr. Sli nn I also managed to have a successful internship at the Naval Research Labora tory and will be graduating with honors this summer with my Master of Science degree. This August I begin my practical coastal e ngineering experience when I start a full time job with Olsen & Associates, a sma ll coastal engineering firm located in Jacksonville, FL. I am eager to apply my knowledge in a practical atmosphere and am also excited about my short move to the Jacksonville area. |