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Effects of Three-Dimensional Forcing on Alongshore Currents: A Comparative Study


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EFFECTS OF THREE-DIMENSIONAL FORCING ON ALONGSHORE CURRENTS: A COMPARATIVE STUDY By KRISTEN D. SPLINTER 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 2004

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Copyright 2004 by Kristen D. Splinter

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This thesis is dedicated to my parents, for their support and guidance throughout my education.

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ACKNOWLEDGMENTS I would like to thank my parents for their continual support throughout my education; pushing me to excel, and letting me find my way. I thank my sister, Karen, and my parents, for always being there when I needed someone to talk to. I would like to extend my appreciation to my advisor, Dr. Donald N. Slinn, for his patience and guidance throughout the process. I thank Drs. Robert Thieke (University of Florida), and Todd Holland and Joe Calantoni (Navel Research Laboratory, Stennis Space Center, MS) for giving me the opportunity to help with field research. It brought a new level of understanding to my work. I also thank Dr. Kraig Winters (Scripps Institute of Oceanography, San Diego, CA) for the use of his computer programs and for his continued support throughout. I would also like to thank Drs. Robert Dean and Robert Thieke for serving on my supervisory committee. I thank all of the students and faculty who have helped me along the way; Brian Barr, Jodi Eshleman, Jamie MacMahan, Jon Miller, Robert Weaver, and Bret Webb. Their guidance and friendship throughout this process, along with their patience with my computer problems and various dilemmas have been greatly appreciated. Finally, I thank all of my housemates and friends in Gainesville who made me feel welcome and part of a family so far away from home. iv

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS ............................................................................................iv LIST OF TABLES .......................................................................................................vii LIST OF FIGURES ....................................................................................................viii ABSTRACT .................................................................................................................xii CHAPTER 1 INTRODUCTION...................................................................................................1 Methods of Cross-Shore Mixing..............................................................................2 Shear Instabilities...............................................................................................2 Cross-Shore Circulation: Mass Flux and Undertow..........................................5 Surface Rollers...................................................................................................8 Alongshore Pressure Gradients..........................................................................9 Vertical Distribution of Wave Stresses............................................................10 Chapter Contents....................................................................................................17 2 MODEL COMPARISONS....................................................................................19 Two-Dimensional Alongshore Current Models for Barred Beaches.....................19 Quasi-3D Models: SHORECIRC..........................................................................23 Summary................................................................................................................23 3 PROBLEM AND MODEL SETUP......................................................................27 Model Domain and Solution Method....................................................................27 Boundary Conditions.......................................................................................28 Initial Conditions.............................................................................................29 Governing Equations.............................................................................................29 Forcing...................................................................................................................32 Two-Dimensional Sub-model..........................................................................32 Three-Dimensional Forcing Model..................................................................33 Scope of the Model................................................................................................39 v

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4 SIMULATIONS....................................................................................................42 The Effect of Grid Resolution................................................................................49 Alongshore-Averaged Alongshore Currents....................................................50 Alongshore and Depth-Averaged Alongshore Currents..................................51 Alongshore-Averaged Cross-shore Currents...................................................52 Vorticity...........................................................................................................55 Conclusions of Grid Testing............................................................................56 The Effect of Vertical Distribution........................................................................56 Cross-shore Flow.............................................................................................57 Alongshore Currents........................................................................................60 Vorticity...........................................................................................................70 Effect of Domain Length.......................................................................................76 5 SUMMARY AND DISCUSSION.........................................................................81 Findings and Results..............................................................................................81 Future Work...........................................................................................................84 6 LIST OF REFERENCES.......................................................................................86 7 BIOGRAPHICAL SKETCH.................................................................................90 vi

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LIST OF TABLES Table page 1 Comparison of 2D models........................................................................................25 2 Quasi-3D model summary......................................................................................26 3 Summary of simulations..........................................................................................42 vii

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LIST OF FIGURES Figure page 1 Longuet-Higgins and Stewarts description of the vertical distribution of the total radiation stresses.........................................................................................12 2 Dean and Dallys description of the components of shear stress distribution..........12 3 Svendsens description of the components of shear stress distribution....................13 4 Deigaard and Fredsoes description of the components of the shear stress distribution..........................................................................................................14 5 Sobey and Thiekes description of apparent radiation stress (s xx (z)) distribution..........................................................................................................15 6 Rivero and Arcillas description of the shear stress distribution for sloping topography..........................................................................................................15 7 Mellors description of the distribution of radiation stresses....................................17 8 Comparison of alongshore current profiles from 2D circulation for varying values of C f over time..........................................................................................21 9 Vorticity contours from 2D model over time...........................................................22 10 3D outlay of physical grid used in the short domain simulations.............................29 11 Cross-shore distribution of nondimensionalized wave radiation stress forcing coupled to topography............................................................................33 12 Case 8: Sample vertical distribution of nondimensionalized forcing......................35 13 Case 9: Sample vertical distribution of nondimensionalized forcing......................36 14 Case 10: Sample vertical distribution of nondimensionalized forcing....................36 15 Case 11: Sample vertical distribution of nondimensionalized forcing....................37 16 Case 12: Sample vertical distribution of nondimensionalized forcing....................38 17 Case 13: Sample vertical distribution of nondimensionalized forcing....................39 viii

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18 Time series for u-velocity.........................................................................................44 19 Time series for w-velocity........................................................................................44 20 Time series for v-velocity.........................................................................................45 21 Case 1 and 2: Vertical distribution of forcing used in the simulations. Case 1 (A) and 2 (B)....................................................................................................46 22 Comparison of depth and alongshore averaged currents for the initial simulations..........................................................................................................47 23 A typical vertical profile of cross-shore velocity vectors produced by depth linear forcing.......................................................................................................48 24 Surface vorticity contours for depth-uniform forcing as a function of time.............48 25 Surface vorticity contours for depth-dependent forcing as a function of time.........49 26 Cases 3-7: Contours of the alongshore-averaged alongshore velocity (v) for variable grid sizes in the x-z plane......................................................................50 27 Depth and alongshore-averaged alongshore currents at t = 5 min............................51 28 Depth and alongshore-averaged alongshore currents at t = 25 min..........................52 29 Case 3: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors....................................53 30 Case 4: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors....................................53 31 Case 5: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors....................................54 32 Case 6: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors....................................54 33 Case 7: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors....................................55 34 Comparison of the depth-averaged term as a function of cross-shore location................................................................................................................55 35 Cases 8-10: Comparison of alongshore and time-averaged u-velocity as a function of depth and location in the cross-shore domain..................................58 36 Cases 11-13: Comparison of alongshore and time-averaged u-velocity as a function of depth and location in the cross-shore domain..................................58 ix

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37 Vector plot of alongshore-averaged u-w velocities for Case 7.................................59 39 Alongshore-averaged u-velocity contours for Case 7, t = 10.86 min.......................60 40 Case 8: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period....................................................61 41 Case 9: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period....................................................62 42 Case 10: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period....................................................62 43 Case 11: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period....................................................63 44 Case 12: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period....................................................63 45 Cross-shore position as a function of time of the peak alongshore current for the 5 vertical distsributions tested.......................................................................64 46 Peak velocity of the alongshore current as a function of time.................................64 47 Case 8: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v).........................................................................................65 48. Case 9: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v).........................................................................................66 49 Case 10: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v).........................................................................................67 50 Case 11: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v).........................................................................................68 51 Case 12: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v).........................................................................................69 52 Comparison of surface vorticity component ( z ) and alongshore surface current (v) contours in the cross-shore (x) and alongshore (y) domain..............70 53 Sample comparison of spatially averaged, original vorticity, and time-averaged contours at vertical grid point nz-1.............................................72 54 Depth-averaged U(xo,yo,t)-velocity energy density spectrum for Case 8................74 55 Depth-averaged V(xo,yo,t)-velocity energy density spectrum for Case 8................74 x

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56 Time sample of velocity fluctuations (black) and the low-pass filtered (red) depth-averaged u-velocity at 0.03 Hz taken from energy spectrum data...........75 57 Time sample of raw data (black) and the low-pass filtered (red) depth-averaged v-velocity at 0.046 Hz...............................................................75 58 Influence of domain length on alongshore currents for depth-uniform forcing: Alongshore and depth-averaged alongshore current profiles as a function of cross-shore position and time...........................................................77 59 Influence of domain length on alongshore currents for Case 10 forcing: Alongshore and depth-averaged alongshore current profiles as a function of cross-shore position and time.........................................................................78 60 Influence of domain length on alongshore currents for Case 11 forcing: Alongshore and depth-averaged alongshore current profiles as a function of cross-shore position and time.........................................................................78 61 Case 12: Contours of the depth-averaged vorticity as a function of time.................79 62 Case 15: Contours of the depth-averaged vorticity as a function of time................79 63 Case 16: Contours of the depth-averaged vorticity as a function of time................80 xi

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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 EFFECTS OF THREE-DIMENSIONAL FORCING ON ALONGSHORE CURRENTS: A COMPARATIVE STUDY By Kristen D. Splinter August 2004 Chair: Donald N. Slinn Major Department: Civil and Coastal Engineering Our study introduced a new 3D, time-dependent, non-hydrostatic, Large Eddy Simulation (LES) nearshore circulation model. It was wave-phase-averaged, in a curvilinear, bottom conforming, -coordinate system; with a rigid lid that was capable of examining depth-dependent, low-frequency, and nearshore current response to breaking waves in the surf zone. A principal advantage of our model was that it included dynamics of the undertow and vertical mixing that led to enhanced cross-shore mixing. Uncertainties associated with modeling bottom friction were reduced because bottom stress was modeled with the no-slip condition by using a high-resolution vertically clustered grid with O (1-10 cm) scales near the seabed. The LES turbulence closure scheme produced reasonable time-averaged alongshore and cross-shore vertical velocity profiles. The model was formulated for alongshore-uniform bathymetry, and included a shore-parallel sand bar. Forcing conditions for the alongshore currents in the model were coupled to beach topography and specified surface wave conditions, using the xii

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Thornton-Guza (1983,1986) wave breaking sub-model. The model included wave set-up through an induced cross-shore pressure gradient. A variety of vertical distributions of the wave-induced radiation stresses were tested. Results indicated that vertical profiles for undertow, mean alongshore current profiles, and dynamics of alongshore-current shear instabilities depended on the particular approximation chosen for the forcing. In contrast to simulations using a depth-averaged 2D model, the initial instabilities developed on the vertical shear in the water column and produced relatively short wavelength disturbances that caused the flow to transition to turbulence before organizing into larger scale flow features. The strength, stability, and location of the alongshore current were dependent on the vertical distribution of forcing and domain length. Depth-varying forcing suppressed the development of the large-scale shear instabilities and influenced the time-dependent effects of the unsteady flows. Short domains [O (20 m)] were dominated by the effects of the vertical distribution of forcing that produced stronger mean alongshore currents with peaks located shoreward of the nearshore bar. Long domains [O (200 m)], however, included the effects of shear waves, and produced mean alongshore currents with broader cross-shore profiles, and less distinct peaks located further in the bar trough. Including the effects of anisotropic mixing, vertically dependent forcing, and large-scale shear instabilities produced mean alongshore currents that were of reasonable magnitude centered in the bar trough of a barred beach system. xiii

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CHAPTER 1 INTRODUCTION The circulation patterns of the surfzone are important because they affect the alongshore and cross-shore movement of sediment, and the strength of the overall currents. Two distinct types of idealized nearshore circulation models are commonly used to investigate aspects of surfzone dynamics. Two-dimensional (2D) cross-shore (x-z) circulation models are often used to study sediment transport and undertow. These generally assume alongshore uniformity, and are sensitive to the vertical distribution of wave shear stresses. The second class of models simulate 2D alongshore (x-y) currents. These generally consider variations of the current and topography in both the alongshore and cross-shore directions, but are approximated by a depth-averaged current. Some alongshore current models have proven useful for predicting the current structure on plane beaches for certain wave conditions, but are less useful for a barred beach system. According to wave radiation stress theory ( Longuet-Higgins, 1970 ), alongshore currents are produced near the location of wave breaking. In a barred beach system, this commonly occurred near the bar crest, and at the shoreline. Some field observations, however, showed that a mean current maximum developed in the trough region, a location where reduced breaking occurred ( Church et al., 1992 ). Our purpose was to gain a better understanding of the physics that influenced alongshore currents. Most contemporary nearshore circulation numerical models ( Allen et al., 1996 ; Slinn et al., 1998 2000 ; zkan-Haller and Kirby, 1999 ; Van Dongeren and Svendsen, 2000 ) used various forms of the 2D depth-integrated flow approximation. Although 1

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2 valid, because the water was shallow, O (1 m), compared to its horizontal dimensions, O (100 m), this approximation may have obscured several important three-dimensional (3D) features. Two such features were: A depth-dependent vertical distribution of the cross-shore radiation stress could have produced shoreward momentum flux in the upper portion of the water column, and a resulting return flow in the lower portion of the water column, known as undertow. Under some conditions, breaking waves could have effectively mixed the momentum across the water depth at relatively high frequency. Together these processes might have led to enhanced cross-shore diffusion rates that could have decreased the mean cross-shore shear of the alongshore current, and shifted the peak current toward the bar trough. Ongoing research to explain the shoreward shift of the alongshore current on a barred beach has focused on four main topics: shear waves, cross-shore circulation, wave rollers, and alongshore pressure gradients. These topics and aspects of the vertical distribution of the wave radiation stresses are reviewed next. Methods of Cross-Shore Mixing Shear Instabilities Strong alongshore currents on the order of 1 m/s have been generated in the surfzone by the alongshore momentum flux caused by obliquely incident surface gravity waves. The transfer of momentum from the wave field to the mean current was characterized by the cross-shore gradient of the wave radiation stress (e.g., dS xy /dx) ( Longuet-Higgins, 1970 ). Greenwood and Sherman (1986) discussed lateral mixing of alongshore current profiles over barred beaches, and suggested that for a plane beach solution, the presence of bars would enhance lateral mixing by increasing the cross-shore velocity gradients. Observations at Duck, North Carolina by Oltman-Shay et al. (1989)

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3 showed the presence of alongshore propagating velocity perturbations associated with the presence of shear in the alongshore currents. zkan-Haller and Kirby (1999) and Slinn et al. (1998, 2000) showed that the nonlinear evolution of shear instabilities provided significant horizontal mixing of alongshore momentum that affected the cross-shore distribution of the alongshore current. The stability and structure of shear instabilities depended on several factors. These included magnitude and cross-shore profile of the alongshore current, beach topography, presence of nonzero gradients of the horizontal Reynolds stresses ( Dodd and Thornton, 1990 ), and bed shear stress ( Dodd et al., 2000 ). Shear instabilities were distinguished from gravity waves because their periods [O (10 2 -10 3 s)], were too long for their associated wave lengths [O (10 2 m)] ( Oltman-Shay et al., 1989 ) based on the dispersion relationship. (E.g., a 100 m long wave in 5 m of water should have had a period of approximately 14.5 s). Shear waves were most pronounced in areas where strong alongshore O (1 m/s) currents were present, and had total RMS shear wave fluctuations varying between 10 and 40% of the maximum alongshore current ( Noyes et al., 2004 ). They were generated as the alongshore current grew in strength and became unstable, which caused the alongshore current to meander in a sinusoidal pattern, and first appeared as an alongshore progressive wave feature ( Bowen and Holman, 1989 ). They were often strongest, and most prevalent on barred beach profiles where wave breaking over the bar generated a strong shear in the alongshore current, and enhanced the growth rate of the shear waves ( Dodd et al., 2000 ). Field observations by Noyes et al. (2004) also suggested that these instabilities were generated in the region where the greatest shear occurred, just seaward of the location of the maximum alongshore velocity. The instabilities were characterized by an

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4 exchange of energy from the mean current to a perturbation velocity field. Cross-shore mixing of alongshore momentum through the Reynolds stresses usually occurred when the cross-shore gradient of the depth-averaged alongshore velocity went through an inflection point (Dodd and Thornton (1990 ) and Rayleigh (1880) ). Research to explain the field observations of shear instabilities has been based on linear theory, weakly nonlinear theory, and nonlinear analysis of shear wave growth. Bowen and Holman (1989) used linear stability analysis of depth-integrated currents to show that these disturbances could have been produced by instabilities developed by the cross-shore shear of the alongshore current. Allen et al. (1996) used a nonlinear model to show that the finite amplitude behavior of unstable flows on plane beaches could be described by the ratio of frictional to advective terms (Q). As the friction factor was decreased, the flows changed from weakly unstable, to chaotic. They also examined the effect of different alongshore domain lengths, and concluded that shear wave instabilities initially grew at the wave length corresponding to the fastest-growing linearly unstable mode ( 2/ok ). As domain length increased, however, the initial disturbances of the flow merged, and their length scales evolved into longer wave length propagating disturbances when Q was moderate to large ( Allen et al., 1996 ). Slinn et al. (1998) extended the work of Allen et al. (1996) by applying coupled forcing to a barred beach profile. They found that short wave-length disturbances continued to dominate the velocity spectrum even for large values of Q. They presented four different flow regimes that were attained by altering the bottom friction. As bottom friction was decreased, instabilities of the current changed from stable shear waves, to fluctuating vortex patches, to shedding vortex pairs, and finally to turbulent shear flows.

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5 Finite amplitude structures generated were independent of domain length for a barred beach system, and remained at length scales predicted by linear theory: on the order of 100 to 200 m. The strength of these fluctuations increased as bottom friction decreased. This agreed well with Dodd (1994 ) and Falques and Iranzo (1994 ) who found that increasing the bed shear stress and eddy viscosity reduced the current velocity and dampened the shear instabilities. Once the alongshore instabilities grew to finite amplitude, they continued to evolve and shed vortices from the mean current. Bowen and Holman (1989 ) suggested that this might be an important feature of cross-shore mixing in this environment, because shear instabilities might provide rates of turbulent diffusion up to ten times greater than those produced by breaking gravity waves. Cross-Shore Circulation: Mass Flux and Undertow Cross-shore circulation was usually excluded from 2D depth-averaged alongshore current models because of the approximation that alongshore and cross-shore currents were approximately vertically uniform. Although this approximation was often sufficient for large-scale nearshore models, cross-shore circulation existed and could potentially have influenced the alongshore current. Instead, mass flux and undertow were studied in 2D cross-shore circulation models. Cross-shore circulation was driven by the onshore component of the wave radiation stress gradient, dS xx /dx. The undertow was largely a result of the local variation between the depth-dependent radiation stress and the depth-uniform horizontal pressure gradient due to set-up. These were in equilibrium with each other over the water depth ( Svendsen, 1984a) but unbalanced throughout the water column. As the waves traveled through the surfzone and breaking occurred, their wave heights decreased, and resulted in a decreased shoreward mass flux. This produced a

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6 downwards directed flow ( Svendsen, 1984a ) from the free surface, and also encouraged cross-shore circulation. The mass flux above the trough level from the shoaling waves pushed the alongshore current near the surface shoreward, while the alongshore current near the bottom was pulled seaward by the increased return flow in the undertow. Svendsen and Lorenz (1989) noted that in the surfzone, the wave-averaged cross-shore current always had a shoreward velocity component near the free surface, and an off-shore directed component near the bottom. Results from cross-shore circulation models were sensitive to boundary conditions and integral constraints. Cross-shore models tended to invoke: Depth-integrated mass balance (net zero mass flux). A no-slip condition on the bottom boundary ( Svendsen and Hansen, 1988 ). The choice of the turbulent eddy-viscosity term in these models also contributed to the predicted undertow profile. Haines and Sallenger (1994) found that the eddy-viscosity varied significantly over the cross-shore in the surfzone, and also varied in the vertical near the top and bottom boundaries. Faria et al. (2000) disagreed, and found that a depth-dependent eddy viscosity did not reduce the differences between model results and field data. Using a constant eddy-viscosity inside and outside the boundary layer, with the no-slip bottom boundary condition, however, yielded onshore velocities in the bottom boundary layer. This result, however, contradicted field data from Hansen and Svendsen (1984) and Haines and Sallenger (1994) who observed a strong undertow jet over the bar during their field experiments. Two modeling approaches have been developed to address this problem. Stive and Wind (1996) suggested replacing the no-slip condition at the bottom with a specified shear stress at the wave trough level.

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7 Svendsen et al. (1987) proceeded with the no-slip condition, but applied two eddy viscosity terms: with a smaller one in the boundary layer than in the interior of the fluid. Several lab experiments have been conducted to study the undertow profile. Matsunaga et al. ( 198 8, 1994 ) found instabilities in the off-shore directed flow seaward of the breaker line. These instabilities, which they termed a vortex train, consisted of a solitary layer of large eddies rotating about a horizontal, alongshore axis. Larger scale lab experiments conducted by Li and Dalrymple (1998) showed two layers of oppositely rotating vortex trains. These vortex trains caused vertical mixing of the cross-shore current and could potentially affect the alongshore current as well. Faria et al. (2000) presented model results for the vertical structure of the undertow. The undertow was not uniform with depth, over the cross-shore, or steady in time. Undertow flows were approximately parabolic in the vertical, and varied in magnitude with water depth. In the inner trough zone, they found the return flow was weak with very little vertical structure. Maximum offshore flows occurred at the shoreward face, and on top of the bar, which coincided with areas of intense wave breaking. Seaward of the bar, they found that the undertow was nearly depth uniform. The variation of the undertow across the surfzone could induce cross-shore mixing of the alongshore current. The changing strength of the undertow across a barred beach profile was explained by channel flow theory and mass conservation. As the cross-sectional area decreased in shallow water, the flow velocity must have increased to conserve volumetric flow rate. The same concept held on the offshore side of the bar where undertow currents decreased. Putrevu and Svendsen (1992) suggested that the vertical profile of the undertow could have caused considerable shearing of the alongshore current in the

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8 cross-shore direction. Including effects of variable cross-shore circulation could influence the position of the alongshore current. Surface Rollers The standard wave radiation stress theory assumed an instantaneous transfer of momentum from breaking waves to the mean water column. The wave roller was introduced to represent the intermediate process that gradually released energy and momentum from the breaking wave bore into the water column over some distance. The wave roller effectively redistributed the incident wave stress over the surfzone ( Lippmann et al., 1995 ) and was most noticeable on a barred beach ( Svendsen, 1984b ), where multiple breakpoints were common. The transfer of momentum was described as a two-step process. The momentum from a breaking wave was supplied to a turbulent roller that in turn transferred momentum through a surface shear stress ( Deigaard, 1993 ) to the water column at a finite rate. The roller was an approximation to the front face of a breaking wave, where turbulent fluid was conveyed at the wave celerity ( Lippmann et al., 1995 ). In their model, wave asymmetry, and phase speed determined the spatial distribution and intensity of the roller energy dissipation. Waves with smaller asymmetry, or increased local water depth, resulted in a shoreward shift of the break point, and an increased distance that the roller traveled. Together these effects shifted the current towards the trough region. The spatial lag of momentum input into the mean currents by the roller was an efficient means for offsetting the current maximum from over the bar into the trough. Although rollers were able to shift the momentum input shoreward of the bar, their effect alone did not accurately predict the cross-shore distribution of alongshore currents

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9 ( Lippmann et al., 1995 ). Reniers and Battjes (1997) presented laboratory results of alongshore currents over barred beaches that showed: The roller contribution helped to describe the cross-shore distribution of the observed shear stress. Lateral mixing alone had no significant influence on the location of the maximum velocity. The concept that emerged was that the velocity profile was shifted shoreward by the roller, and spread horizontally by the effect of lateral mixing. Alongshore Pressure Gradients In the same paper, Reniers and Battjes (1997) also suggested that alongshore pressure gradients were important for predicting alongshore current location. Laboratory experiments of wave driven currents over a barred beach in the absence of alongshore pressure gradients produced alongshore currents near the regions of breaking; over the bar, and at the shoreline. They hypothesized that alongshore currents located in the trough were explained by alongshore pressure gradients in the field due to uneven topography in the alongshore direction. Putrevu et al. (1995) conducted model studies that examined the importance of including the effects of alongshore non-uniformities on alongshore currents. Variable beach topography in the alongshore direction affected the breaker height and location of wave breaking in the cross-shore. Including the forcing from alongshore pressure gradients had a significant impact on the alongshore current location. They found that this component could be almost equal in magnitude to the forcing due to the alongshore component of the radiation stress term in some cases. The effect of the alongshore

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10 pressure gradient was dependent on the angle of wave incidence: it increased as waves approached closer to shore normal. Slinn et al. (2000) coupled topography and radiation stress gradients over alongshore variable topography in numerical model experiments to test this hypothesis. Their results showed that currents tended to follow contours of constant depth. Variable points of wave breaking caused momentum to be inputted at different cross-shore locations depending on topography. The alongshore pressure gradients and the radiation stress terms counteracted each other to some degree. They concluded that alongshore pressure gradients alone were insufficient to produce peak currents in the trough. They concluded, however, that topographic variability in the alongshore had a significant influence on nearshore circulation. Vertical Distribution of Wave Stresses In depth-integrated, 2D, circulation models on a uniform coast with straight and parallel contours, alongshore currents were forced by the cross-shore gradients of the depth integrated wave radiation stresses, dS xy /dx. With the advancement to quasi-3D and 3D modeling techniques, the distribution of the wave shear stresses within the water column became important. Earlier work describing the vertical distribution of wave stresses was simplified to examine shore normal waves, where dS xy /dx = 0. The only dynamic wave force present in these models was dS xx /dx, which drove set-up and cross-shore circulation. Longuet-Higgins and Stewart (1964) defined the relationship between the depth-dependent radiations stress, s xx (z) and the depth-integrated value that was used in

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11 2D circulation models as xxxxS= s()h z dz The wave radiation stress, S xx was divided into three parts, so. The breakdown of each component was: (1)(2)(3)xxxxxxxxSSSS 0(1)22xxhhSudzu dz (1) 000(2)200()()()xxhhhSppdzppdzwdz (2) (3)2012xxSpdzg (3) In shallow water, 0(1)(2)2xxxxhSSud z because u 2 >> w 2 This term was due to the momentum of the wave orbital motion, and was equal to the total energy density of the waves, E. It was distributed evenly over the water column and accounted for two-thirds of the total radiation stress. was the pressure term due to the waves, and was equal to E. It accounted for one-third of the total force, and was centered about the mean water level (MWL) (Figure 1). S (3)xxS xy lacked the pressure terms due to the waves, and was given by the relationship xyhSuv dz Deigaard and Fredsoe (1989) compared their findings to the works of Dally and Dean (1984) and Svendsen (1984a) for the vertical variation of the wave induced shear stress that included components of radiation stress, wave roller effects, and set-up. Their analyses were based on linear shallow water wave theory over a horizontal bottom. Dally and Dean (1984) modeled the vertical distribution of radiation stress following the approach of Longuet-Higgins and Stewart (1964) The resulting shear stress distribution was split up into the three components: wave motion, surface roller, and set-up. Dally

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12 and Dean (1984) did not include the effect of a surface roller. They assumed the set-up balanced the radiation stress gradient, and the result was a linear distribution of the shear stress across the water column, and a depth-uniform distribution of the radiation stress below the trough. The shear stress had a maximum at the MWL and was zero at the bed ( Figure 2 ). A B Figure 1. Longuet-Higgins and Stewarts description of the vertical distribution of the total radiation stresses. A) s xx (z). B) s xy (z) y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1B y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1A y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1C y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 D Figure 2. Dean and Dallys description of the components of shear stress distribution. y/D is the normalized water depth. 21* = -/16dHgdx A) Wave motion. B) Surface roller. C) Setup. D) Resulting shear stress distribution. Svendsens (1984a) results were similar to Dally and Dean (1984) He used the same linear shallow water wave theory, but included the effect of a surface roller. The

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13 surface roller added an additional component to the pressure gradient that produced a depth-uniform addition to the shear stress distribution ( Figure 3 B). The wave setup component was also adjusted so that the total shear stress was still linearly distributed with depth ( Figure 3 C). In this formulation, the total wave shear stress term had a maximum at the MWL about twice that of Dally and Dean (1984) and zero at the bed (Figure 3D). This resulted in a stronger net cross-shore shear force than Dally and Dean (1984) predicted. y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1A y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1B y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1C y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 D Figure 3. Svendsens description of the components of shear stress distribution. 21* = -/16dHgdx y/D is the normalized water depth. A) Wave motion. B) Surface roller. C) Setup. D) Resulting shear stress distribution. Dally and Dean (1984) and Svendsen (1984a) both used a control volume approach to describe the distribution of shear stress and omitted the transfer of momentum ( UW ) through the bottom of the control volume. In the presence of irregular waves, such as in the surf-zone, Deigaard and Fredsoe (1989) showed that this term was significant to the overall magnitude and distribution of the shear stress. They found that the UW term doubled the surface shear stress. The magnitude of the surface roller effect was also slightly higher in their calculations compared to Svendsen (1984a) As a result of the larger shear stresses due to wave motion and the surface roller, the balancing shear stress due to set-up was also larger. The net distribution (Figure 4) still increased linearly with

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14 height: with zero shear stress at the bed, and a maximum at the MWL almost three times that of Dally and Dean (1984) and one and a half times that of Svendsen (1984a) The reader is referred to Deigaard and Fredsoe (1989) for a more detailed comparison of the three theories presented. y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1A y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1B y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1C y/D -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 D Figure 4. Deigaard and Fredsoes description of the components of the shear stress distribution. y/D is the normalized water depth. 21* = -/16dHgdx A) Wave motion. B) Surface roller. C) Setup. D) Resulting shear stress distribution. Sobey and Thieke (1989) discussed the importance of depth-dependent forcing in the nearshore region. They clarified that although depth integration of the forcing was valid because it produced a net zero mass flux condition in the cross-shore direction, it did not include the cross-shore circulation produced by the wave induced mass-flux and undertow. Their wave action term ( 2(uw 2) ) in s xx (z), where ,uw were the time-averaged, u-, w-velocity fluctuations, was vertically uniform in agreement with Dally and Dean (1984) Their pressure gradient term, however, linearly increased from the trough to the MWL, where it reached a maximum. A complementary linear decrease in the apparent stress from MWL to the wave crest was also predicted (Figure 5). Rivero and Arcilla (1995) extended the work of Deigaard and Fredsoe (1989) and demonstrated that a sloping bottom bed had an important effect on the wave shear stress.

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15 Under dissipative wave conditions, the resulting distribution had a nonzero value at the bed equal to tanEh that linearly increased to a maximum of 1tan2 E EEhhx x at the MWL (Figure 6). troughcrest Figure 5. Sobey and Thiekes description of apparent radiation stress (s xx (z)) distribution. (E)(E)tanh2[xh(E)tan(E)xh]~~ Figure 6. Rivero and Arcillas description of the shear stress distribution for sloping topography. More recently, the work of Mellor (2003) addressed the vertical distribution of both the alongshore and cross-shore wave radiation stresses for surface waves ( Figure 7 ). His equations for radiation stress were:

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16 2cscccsccsscskkSkDEFFFFFFk (4) where: sinh(1),sinhsskDFkD cosh(1),sinhcskDFkD sinh(1),coshsckDFkD cosh(1).coshcckDFkD He described 22gaE D was the depth, was the Kronecker delta function, and was the transformed vertical coordinate such that = -1 when z = -h and = 0 at the surface. His resulting vertical distribution for s xx (z) differed from previous theory. As seen in Figure 7 A, s xx (z) linearly increased with depth. This was due to the F ss term that linearly decreased with depth and was subtracted from the other radiation stress components. When vertically integrated over the water depth, his equations were off by a factor of rho (1000 kg/m 3 ) in comparison to the depth averaged equations given by Mei (1989) Eq. 3-10. Vertical integration of his equations and the determination of the vertical distribution of each of his sub-terms (F ss F sc F cs and F cc ) were calculated separately. This was done to ensure that the resulting vertical distribution presented above was not an error in inputting the equations into the model. The vertical distributions of the sub-terms used in the model were similar to those presented in his paper. His s xy (z) term remained roughly depth-linear because the 2 nd term (with the Kronecker delta) in his equation only existed for normal (s xx (z), s yy (z)) forces. This representation of radiation stresses remains questionable for its purpose and validity in the nearshore. Putrevu and Svendsen (1999) and Zhao et al. (2003) also discussed effects

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17 of the local contribution of the radiation stress on lateral mixing within the surfzone but are not discussed in detail. x(m) z(m) 0 50 100 150 200 0 1 2 3 4A x(m) z(m) 0 50 100 150 200 0 1 2 3 4B Figure 7. Mellors description of the distribution of radiation stresses. A) s xx (z), B) s xy (z). The s xx (z) term affected the cross-shore flow and potentially the location of the alongshore current. In the formulations based on linear wave theory, the s xy (z) term did not include a depth-dependent pressure term as found in s xx (z). It was therefore uniformly distributed over depth, and drove the alongshore currents. The local variation of the radiation stresses over depth and the resulting cross-shore and alongshore currents could cause a lateral mixing effect that might dominate over turbulent mixing ( Putrevu and Svendsen, 1999 ). Chapter Contents A main purpose of our study was to investigate the effects of the vertical distribution of radiation stresses on the phase-averaged alongshore current position, the formation of shear instabilities, the net horizontal mixing, and to compare the influence

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18 of vertical forcing with those of previous theories. A variety of vertical distributions of the radiation stresses were tested in a 3D, phase-averaged, Large Eddy Simulation (LES) nearshore circulation model over an alongshore uniform barred beach bathymetry. The use of a 3D model allowed for the development, interaction, and analysis of cross-shore mixing by the undertow with the alongshore current. Mixing effects caused by large-scale shear instabilities, and those due to depth-dependent cross-shore circulation, were examined separately by utilizing alongshore domains of varying dimensions. The smaller [O (20 m)] domain inhibited the development of the fastest growing linearly unstable modes that produced large-scale [O (100 m)] shear waves. Longer domains [O (200 m)] were also modeled that permitted shear waves to be developed and allowed a determination of the relative contribution to cross-shore mixing of the two processes. The thesis is divided into five chapters. Chapter 2 reviews some previous model studies of alongshore current distributions and shear waves. Chapter 3 covers the 3D model problem setup, governing equations, and details of sub-models used in the numerical experiments. Chapter 4 provides a description of the simulations and the main results. The discussion focuses on the effects of the vertical distribution of the radiation stress and the response of the flows. Emphasis is given to the stability and structure of the alongshore current, the cross-shore current structure, and the effects of model approximations. The effect of domain length and the relative magnitude of lateral mixing caused by shear waves and radiation stress distributions are also addressed. Chapter 5 summarizes conclusions of the work, considers strengths and limitations of this approach, and outlines future directions.

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CHAPTER 2 MODEL COMPARISONS This chapter summarizes work of three previous model studies that focused on shear instabilities, and their effect on alongshore currents on barred beaches. Two of the models were 2D, and one was quasi-3D, a sophisticated hybrid approach. Two-Dimensional Alongshore Current Models for Barred Beaches Traditional 1and 2D phase-averaged models were unsuccessful at predicting the location of the peak alongshore current on barred beaches. Using linear wave theory, breaking wave parameterizations, and depth-averaged mean flow equations, they predicted peak currents at the two locations of maximum wave breaking intensity: the bar and the shoreline. Some field data, however, showed peak currents located in the trough region, an area where reduced breaking occurred ( Church et al., 1992 ). A probable contributing factor in this discrepancy was the effects of shear waves. In this section, two similar modeling approaches are discussed and summarized in Table 1. The first, by Slinn et al. (1998) was a qualitative examination of the shear instability problem, and the second, by zkan-Haller and Kirby (1999) was a more quantitative analysis with comparisons of their model results to field data from the SUPERDUCK experiment. The two models used slightly different governing equations, but both showed that shear waves could alter the alongshore current profile, and caused increased flow in the trough region. Both of the models were phase-averaged and worked with depth-averaged currents. For a full description of the models and their 19

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20 results, the reader is referred to Allen et al. (1996) Slinn et al. (1998) and zkan-Haller and Kirby (1999) In both models, as the frictional terms (C f ) were decreased, the alongshore currents became stronger and contained more perturbations. Results of varying the frictional terms are presented in Figure 8 These were obtained using a model similar to the one used by Slinn et al. (1998) A consequence of the more unsteady flows was the shedding of more vortices, and resulted in the cross-shore mixing of the alongshore current ( Figure 8 D). For high values of C f ( Figure 8 A), the currents were weak and stable. For moderate values ( Figure 8 B), the current remained relatively stable over time, with a small amount of horizontal spreading: peak values in the trough increased as currents over the bar decreased, but the location of the maxima did not change. As C f was decreased, the peak currents increased before instabilities appeared ( Figure 8 C, Figure 8 D). With lower friction coefficients, the current became strongly unstable. At the lowest value ( Figure 8 D), the current had the broadest cross-shore distribution due to more energetic instabilities that increased horizontal mixing. Slinn et al. (1998) found that lowering the bottom friction coefficient shifted energy in the frequency alongshore wave number (f k y ) spectra to smaller length and shorter timescales. zkan-Haller and Kirby (1999) also varied their horizontal mixing coefficient (M) in their diffusion terms to approximate diffusion caused by breaking waves or depth-varying currents. They found that by increasing the mixing coefficient, they generated less energetic instabilities at longer length scales. The model of zkan-Haller and Kirby (1999) was able to approximate alongshore currents and propagation speeds of the instabilities for the SUPERDUCK experiment. It should be noted,

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21 however, that peak alongshore currents for the days they simulated were still located near the nearshore bar and not in the trough. Their model was also not able to accurately match the observed frequency spectra, predicting more energy at lower frequencies than observed. 0 50 100 150 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2t=10t=20t=40t=60t=80t=100 V(m/s)x(m)Cf=0.01A 0 50 100 150 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2t=10t=20t=40t=60t=80t=100t=120t=140t=160t=180 V(m/s)x(m)Cf=0.006B 0 50 100 150 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2t=10t=20t=40t=60t=80t=100t=120t=140t=160t=180 V(m/s)x(m)Cf=0.003C 0 50 100 150 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2t=10t=20t=40t=60t=80t=100t=120t=140t=160t=180 V(m/s)x(m)Cf=0.0015D x(m) z(m) 0 50 100 150 200 0 1 2 3 4 Figure 8. Comparison of alongshore current profiles from 2D circulation for varying values of Cover time f Results were from a 2D model similar to the one used by Slinn et al. (1998) H o = 0.7 m, T p = 8 s, o = 30 o Bottom panel is the beach topography, h(x) and given in Eq.(5).

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22 In both models, the flows that were generated were sensitive to the free parameters. As the correct representative values of these parameters were unknown for field conditions, they were tuned to produce currents similar to field data. The free parameters could be seen as a weakness of the modeling approach because they influenced the outcome at the users discretion. Results of the alongshore current distribution ( Figure 8 ), and snapshots of vorticity (Figure 9) attained from a model similar to Slinn et al. (1998) are presented for comparison with the 3D model results. Wave parameters (H o = 0.7 m, T p = 8 s, o = 30 o ) and bathymetry were the same for the 2D and 3D models. A coefficient of friction equal to 0.0015 was used to generate currents in the 2D model with magnitudes comparable to the 3D results. This value generated a current with strong instabilities that caused considerable mixing. Larger values of C f produced weaker currents that became unstable at later times. The peak velocities before the current became unstable in this example were approximately 0.9 m/s over the bar, and about 1 m/s at the shoreline. x(m) y(m) 0 50 100 150 200 0 50 100 150 200t=15min x(m) y(m) 0 50 100 150 200 0 50 100 150 200t=25min x(m) y(m) 0 50 100 150 200 0 50 100 1500.0300.0260.0220.0180.0140.0100.0060.002-0.002-0.006-0.010-0.014-0.018-0.022-0.026-0.030 t=35min(1/s ) Figure 9. Vorticity contours from 2D model over time. Using a coefficient of friction (C f ) = 0.0015, the 2D results compared well with the 3D model. The current remained stable until about 20 min, then it began to meander (25 min), and evolved into large vortex structures (35 min) and remained in this condition.

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23 Quasi-3D Models: SHORECIRC Recent work by Zhao et al. (2003) analyzed 3D effects from shear waves using the quasi-3D nearshore circulation model, SHORECIRC. Their work focused on the importance of including the depth variation of the currents. Using the depth-integrated 2D equations, they included a quasi-3D dispersive mixing term that approximated the effect of depth-varying currents ( Putrevu and Svendsen, 1999 ). Their parameters are summarized in Table 2. The reader is referred to Zhao et al. (2003) for a more complete description of the model and their findings. The effect of breaking waves was parameterized with a roller model included in the radiation stress term and short wave-induced volume flux. The quasi-3D model approximated effects of vortex tilting, and depth-varying currents that produced horizontal mixing on the same order of magnitude as the shear wave component. The quasi-3D dispersive mixing term also included effects of depth-varying radiation stresses. Zhao et al. (2003) compared the quasi-3D simulation results to the 2D model results of zkan-Haller and Kirby (1999) They found that the quasi-3D model generated a more steady flow, localized shoreward of the bar. The flow contained a less energetic turbulent kinetic energy field that was attributed to the mixing terms being of the same order as the diffusion produced by the shear waves. zkan-Haller and Kirby (1999) found similar results when they increased their horizontal mixing coefficient. Summary The 2D and quasi-3D models had advantages because they produced reasonable approximations of the alongshore current structure efficiently or in a predictive, forecast manner. The use of relatively coarse grids and depth-integrated equations of motion, however, inhibited the development of 3D turbulence and depth-varying interactions

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24 between the alongshore and cross-shore currents. The utilization of a fully 3D model with a much finer grid resolution allowed for these properties to influence the alongshore current with fewer ad-hoc approximations. This allowed greater insight into properties that might be important to the dynamics of the surfzone.

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25 Table 1. Comparison of 2D models Property Slinn et al. (1998) zkan-Haller and Kirby (1999) Governing equations 440,,xyxtxyoytxyohuhvpuuuuvuuhpvVvuvvvvh where: h = local water depth, h(x) (u,v) = velocity components p = pressure o = constant fluid density = bottom friction coefficient = small biharmonic diffusion coefficient V = velocity due to wave forcing ''0,,txytxyxxxbxtxyyyybydudvuuuvugvuvvvg where: = wave-averaged free surface elevation d = h + (u,v) = velocity components in x and y direction respectively ,xy = wave forcing (see below) ,xy = momentum mixing terms ,bxby = bottom friction Wave forcing and wave breaking model ( b ) (,)()oVxyFhx where: V(x,y) is a function of xyS x b : Thornton and Guza (1986) 11,xyxxxySSdxdx b : Whitford (1988) Momentum mixing 41.25/ms for 2 m grid spacing, 42.5/ms for 2.5 m grid spacing. '21,xuvdddxxdyx '1yvddyx where : 1/3bMd 0 < M < 0.5, M = mixing coefficient Bottom friction = C f |u rms | where: |u rms |0.3 m/s is the wave orbital velocity 0.0020.012fC ,bxbyuvdd where: 2,0.0035,0.00314foformscucguHh Domain lengths Lx = 1000 m Ly = 1200 or 1280 m Lx = 550 m Ly = 16 x L max, where: L max = max2/k and ranges between 160 200 m. Boundary conditions Top: Rigid-Lid Lateral: periodic Shore: 0xxxxxxuuvv Offshore: 0xxxxxxuuvv Top: Free-surface Lateral: periodic Shore: free-slip Offshore: absorbing boundary

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26 Table 2. Quasi-3D model summary Property Zhao et al. (2003) Governing equations ()0,111()()0BVhtxSTLVhVVhghtxxxxx where: short-wave averaged free surface elevation h = still water depth are the x,y directions respectively V depth-uniform current S short-wave induced radiation stress L quasi-3D dispersive term T depth-integrated turbulent shear stress Wave forcing REF/DIF 1 is used as the wave driver to calculate wave radiation stresses and volume flux Momentum mixing tVVTh x x 1111()otwwh L VVdzuVuVdz where: 1/30,0.08||0.12wttfDuhh 0 1,V depth-varying current wu wave component velocity t free surface at the wave trough level Bottom friction ||2,BofucVh 0.0035fc Boundary conditions Top: free surface, phase-averaged Lateral: Periodic Shore: wall-boundary with free-slip condition Offshore: no-flux Domain lengths Lx = 788 m, Ly = 2490 m

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CHAPTER 3 PROBLEM AND MODEL SETUP The nonlinear dynamics of finite amplitude shear instabilities of alongshore currents over barred beach topography were studied using finite-difference, numerical experiments. The model was non-hydrostatic, wave phase-averaged, and fully 3D. It solved the transformed Navier-Stokes equations with a Large Eddy Simulation (LES) sub-grid scale closure on a curvilinear (-coordinate) grid. A principal advantage of our model was that it included dynamics of the undertow and depth varying currents that led to enhanced cross-shore mixing. Model Domain and Solution Method The model was formulated for alongshore-uniform bathymetry and included a shore-parallel sand bar. The depth profile was an approximate fit to topography measured at Duck, North Carolina, October 11, 1990, as part of the DELILAH field experiment ( Lippmann et al., 1999 ) given by: 25111111111(,)tanhtanhvccxxxvvvxbxbxbahxyaaeaaa 2 (5) where: a 1 = 2.97, defines the amplitude of the nearshore trough a 2 = 1.5, defines the amplitude of the offshore trough x v = (i-1)dx, 1 < i < nx-1 where i is the cross-shore counter b 1 = tan( 1 ), where 1 = 0.075, defines the nearshore beach slope b 2 = tan( 2 ), where 2 = 0.0064, defines the offshore beach slope 1 = b 1 /b 2 x c = 80, defines the cross-shore location of the nearshore bar 27

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28 Two alongshore domains were used. The short alongshore domain was 20 m in length and inhibited the production of alongshore generated shear waves so we could focus on other dynamics that might have influenced the alongshore current. The large alongshore domain extended to 200 m and allowed for large-scale shear wave generation and interaction with the alongshore current. Grid spacing was in part based upon results of Scotti et al. (1997) They found that a maximum aspect ratio between any two grid length scales of approximately 4:1 could be used for LES schemes on anisotropic grids without influencing the results. The effect of grid spacing is discussed in more detail in Chapter 4. Constraints in the grid generation scheme did not allow the depth of the cross-shore domain to go to zero, therefore the model domain was cut off at a depth of 0.5 m. The grid was clustered along the bottom and near the shoreline for higher resolution in areas of stronger variability ( Figure 10 ). The short domain was typically modeled with (nx, ny, nz) = (257, 33, 17) grid points, with an average aspect ratio of 3:1 between horizontal and vertical length scales. The long domain was modeled with (nx, ny, nz) = (257, 257, 17) grid points and also had an average aspect ratio between the horizontal and vertical length scales of approximately 3:1. Boundary Conditions Uncertainties associated with the bottom friction parameterization were reduced in this approach by modeling the bottom stress with a no-slip condition using a high-resolution vertically clustered grid with O (1-10 cm) scales near the seabed. The top boundary utilized the rigid-lid approximation and was modeled with a free-slip condition. The offshore and shoreline boundaries were modeled with a no-slip condition to aid in obtaining numerical stability. The lateral boundaries were periodic.

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29 Initial Conditions The model was initially at rest and was forced by the momentum input from the wave radiation stresses. Run-up time to steady-state flow was approximately 1 h of real time. x050100150 200 y05101520 z01234 Y X Z Figure 10. 3D outlay of physical grid used in the short domain simulations (x,y,z) are the cross-shore, alongshore, and vertical axes respectively and are measured in m. The long domain extended to 200 m in the alongshore (y) direction and had the same discretization in the x-z plane. Governing Equations The LES equations of motion for the Cartesian coordinate system were: 1iijiijxjiiuupuFtxxx (6) 0iiux (7) where were the cross-shore, alongshore, and vertical velocity components, p was pressure, t was time, was fluid density equal to 1028 kg/m (,,)iuuvw 3 and F x and F y were the horizontal body forces in the cross-shore and alongshore directions respectively and

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30 were equal to the vertical distribution of the radiation stress gradients, 1,xxSx and 1xyS x jijiijuuuu represented the LES sub-grid stresses written in terms of the resolved strain tensor, ijtijijS 23 where ijjiijxuxuS21 and the turbulent eddy viscosity, t were calculated with the Smagorinsky approach, (i.e., 1/222tijCSS ij where 3/4123kCC and 1.5kC was the Kolmogorov constant, and 1/3(xyz ) was the filter width based on the local grid spacing in each direction). The model used a boundary fitted curvilinear coordinate system that allowed variation of the geometry in two directions (x, z). To reduce the computational complexity, the model required the x-z curvilinear coordinate system to be orthogonal (), where the subscripts (, ) denoted partial differentiation between physical and computational space. Orthogonal grid intersection at the boundaries increased the accuracy of the boundary conditions with the interior solutions. To compute approximate solutions to the governing equations, the physical domain (x, y, z) that solved for the physical space velocities (u, v, w) was transformed to a cubic grid of regularly spaced grid points in computational space (, ) and solved for the contravariant velocities (U, v, W). This transformation allowed the use of easily coded numerical methods, but at an increased cost of more complicated transformed equations of motion. The mapping between physical and computational space was done through an orthogonal transformation of three successive elliptic boundary value problems following 0xzxz

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31 Winters et al. (2000) The bathymetry was uniform in the alongshore direction and grid spacing was constant in the y-direction. Transformation of velocities was done by: xxuUwzzW and, (8) 1xzzxUuWzxJ w.xz (9) where: ,xzJxz (10) Transformation of derivatives was done by: 1xzxzzzff x xffJ (11) After transformation, the governing equations to be solved were written as: 22221,11,21,321(),xxzxzxzzdUUUWWpFdissUdtJJ (12) 1(),ydvypFdissdt v (13) 22223,13,23,321(),xxzxzxzzdWUUWWpFdissWdtJJ (14) 1,,(())0,xzxzxzxzDUvWJUJWvJzy 1 (15) where: 1()dUvWdtty (16) 1,1()xzzxxzJ 1,22()xzzxxzJ 1,3()xzzxxzJ 3,1()xz x zxzJ 3,22()xz x zzxJ 3,3()xz x zzxJ ( Winters et al., 2000 )

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32 Diss (U, v, W) were dissipation terms implemented through the LES sub-grid-scale closure model described above, and were not given here in the transformed coordinate system for simplicity. D xz was the transformed divergence equation. A fourth-order compact low pass spatial filter ( Lele, 1992 ) was also applied to the contravariant velocity field as part of the sub-grid dissipation model to help maintain a resolved and stable simulation. Spatial derivatives were calculated with fourth-order compact differencing schemes. The pressure solver utilized a fourth-order finite difference scheme based on the multi-grid method. Time stepping was explicit and done to third-order accuracy with the Adams-Bashforth (AB3) method with a variable time step. Initial time stepping was done with the Euler and AB2 methods. Forcing Two-Dimensional Sub-model The alongshore current forcing conditions were coupled to the beach topography and specified surface wave conditions using the wave refraction model of Slinn et al. (2000) that implemented the wave energy dissipation model of Thornton and Guza ( 1983 1986 ). Deep water wave height, angle, and peak period were specified, and the waves were shoaled into the shoreline using linear theory. The majority of the effects of wave set-up were included through an induced cross-shore pressure gradient. Except for the simulation with Mellors equations (Case 13), the depth-integrated 2D radiation stress gradients were calculated for the shoaling wave as it propagated over the beach topography. These forces were then used as input to the 3D model and distributed throughout the water column following a variety of approximations introduced in Chapter 1, and discussed in more detail immediately below. The cross-shore distribution of the nondimensionalized forcing coupled to topography used in the simulations is

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33 presented in Figure 11 To implement the vertical distribution of radiation stresses following Mellor (2003) (Case 13), the individual wave parameters: wave energy, wave number, wave angle, and water depth were calculated at each grid point and used as input for his equations in a separate program. Three-Dimensional Forcing Model A variety of vertical distributions of the wave-induced radiation stress gradients were implemented to test their influence on the resulting alongshore and cross-shore current structures. Preliminary sensitivity tests of the model were performed with depth-uniform and linearly distributed forcing in both the alongshore and cross-shore directions. x(m) Fx Fy 0 50 100 150 200 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 -0.003 -0.0025 -0.002 -0.0015 -0.001 -0.0005 0 0.0005 FxcoupledFycoupled x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5 Figure 11. Cross-shore distribution of nondimensionalized wave radiation stress forcing coupled to topography. F= dS/dx, F= dS/dx, x = 0 is the shore boundary, x = 200 is the offshore boundary. The beach topography is given in the lower panel. x xx y xy Forcing Distributions The rigid-lid approximation used in the model represented the wave phase-averaged equations, and eliminated the vertical region between the MWL and the

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34 wave crest. All possible vertical distributions of the radiation stresses tested were only approximations because a portion of s xx (z) existed in that region, not contained in any phase-averaged model. The vertical distributions were also based on linear wave theory, but applied in the surfzone that was influenced by nonlinear effects. The vertical distribution of the cross-shore radiation stress in the surfzone was divided into two parts following Longuet-Higgins and Stewart (1964) Two-thirds of the total x-momentum forcing was distributed uniformly over the entire water column and represented the momentum due to wave orbital motion. The remaining one-third represented the excess pressure force due to the waves between the trough and crest of the wave. This portion was generally depth-dependent and various forms were implemented for the different tests. It was referred to as the pressure term. The alongshore momentum forcing, F y did not include the pressure term as in F x and was generally implemented as a depth-uniform force. The magnitude of the forcing was normalized by h(x) so that ()(,)xxxdSxFxzdzdx and ()(,)xyydSxFxzdzdx In Figure 12 to Figure 17 the true direction of F y pointed in the alongshore direction, but for graphing purposes, was presented as an onshore directed vector force above the 2D cross-shore profile. Case 8 used a depth-uniform distribution to represent the pressure term. It was added in over the top five grid points of the domain to represent the approximate depth of the wave trough ( Figure 12 A). This was a first attempt to simulate the vertical distribution of wave radiation stresses due to breaking waves. For Case 9, the pressure term was applied as a depth-linear force (increasing from zero at the bottom most point) over the top five grid points as illustrated in Figure 13A.

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35 This forcing was a closer approximation of the vertical distribution of wave radiation stresses obtained from linear wave theory, with the approximation that the portion above the MWL was a mirror image in the vertical distribution to the portion between the trough and the MWL. It was therefore folded back on its mirror image below the MWL and doubled the force distribution in the trough to MWL region. x(m) z(m) 0 50 100 150 200 0 1 2 3 4B x(m) z(m) 0 50 100 150 200 0 1 2 3 4A Figure 12. Case 8: Sample vertical distribution of nondimensionalized forcing. A) F x B) F y Forcing distribution was applied over the cross-shore domain and varied in magnitude according to Figure 11 The distribution in Case 10 was another attempt to more accurately model the varying wave trough depth in the cross-shore. The -coordinate grid stretched to fill the water column, with grid spacing more refined in regions associated with smaller wave amplitudes. For example, over the bar crest, the wave trough depth was approximately 0.3 m and the vertical grid spacing was approximately 0.1 m. For the majority of the grid points in the cross-shore direction, the depth of the wave trough was approximately three vertical grid spaces. Case 10 had uniform forcing occurring over the top three grid points instead of five (Figure 14A).

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36 x(m) z(m) 0 50 100 150 200 0 1 2 3 4B x(m) z(m) 0 50 100 150 200 0 1 2 3 4A Figure 13. Case 9: Sample vertical distribution of nondimensionalized forcing. A) F x B) F y Forcing distribution was applied over the cross-shore domain and varied in magnitude according to Figure 11 x(m) z(m) 0 50 100 150 200 0 1 2 3 4B x(m) z(m) 0 50 100 150 200 0 1 2 3 4A Figure 14. Case 10: Sample vertical distribution of nondimensionalized forcing. A) F x B) F y Forcing distribution was applied over the cross-shore domain and varied in magnitude according to Figure 11

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37 The forcing in Case 11 ( Figure 15 A, B) combined the linear distribution of x-momentum from Case 9, but with the linearly increasing force applied over the top three grid points (Case 10). x(m) z(m) 0 50 100 150 200 0 1 2 3 4B x(m) z(m) 0 50 100 150 200 0 1 2 3 4A Figure 15. Case 11: Sample vertical distribution of nondimensionalized forcing. A) F x B) F y Forcing distribution was applied over the cross-shore domain and varied in magnitude according to Figure 11 With the rigid-lid approximation used in our model, the moving free surface was removed or wave-phase averaged out. Instead of doubling the pressure term in the trough-MWL region as was done in Cases 8-11, the forcing for Case 12 was broken down into three parts: two-thirds of the total forcing was depth-uniform over the entire water column, one-sixth of the total forcing was a depth-linear force over the top three grid points, and a surface stress at the top grid point accounted for the remaining one-sixth of the total wave radiation stress force above the MWL ( Figure 16 A). The forcing in Case 13 was based on the equations of Mellor (2003) As is shown in Figure 17 the vertical distribution of forcing in the x-momentum equation was quite different from the other approaches. Since Mellors work focused on large scale,

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38 deep-ocean modeling, his equations for the radiation stress terms might not have been a suitable description of surfzone radiation stress distributions. The full equations of momentum presented in his work, however, did account for the transfer of momentum through dissipation in other terms (Personal Communication, 2004). These other terms were not used in the current model because we focused on the cross-shore derivatives of the vertical distribution of radiation stresses only. As seen in Figure 17 A, the direction of the resulting force due to radiation stress gradients in the x-direction produced variable forcing in both the vertical profile, and in direction (note at x = 125 m, the resulting force was offshore directed). x(m) z(m) 0 50 100 150 200 0 1 2 3 4B x(m) z(m) 0 50 100 150 200 0 1 2 3 4A Figure 16. Case 12: Sample vertical distribution of nondimensionalized forcing. A) F x B) F y. Forcing distribution was applied over the cross-shore domain and varied in magnitude according to Figure 11 These vertical distributions of wave radiation stresses followed the basic guidance arising from linear theory. Breaking waves in the surfzone have more complex dynamics than contained in linear theory. Hence, other vertically variable distributions of F x and F y may be more accurate approximations of natural conditions. The vertical distribution of

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39 radiation stresses due to nonlinear breaking waves is a topic of ongoing research. The results presented below should be considered a sensitivity study to variation of s xx (z) rather than as the most accurate method to model the surfzone with a 3D model. x(m) z(m) 0 50 100 150 200 0 1 2 3 4A x(m) z(m) 0 50 100 150 200 0 1 2 3 4B Figure 17. Case 13: Sample vertical distribution of nondimensionalized forcing. A) F x B) F y Forcing distribution was applied over the cross-shore domain. Scope of the Model The current model simulations pursued process-based studies concentrating on surfzone dynamics. While the model was a rational advance to investigate new physical interactions in the surfzone, it still contained several necessary approximations that limited the scope of utility of the model. Results were qualitatively and semi-quantitatively similar to field and lab data, however, several limitations inherent in the model hindered direct comparison with field data. The surfzone is a highly dynamic area where many different physical processes interact and influence each other. The wave breaking model that was implemented is widely accepted, but was still an approximate representation of forcing. Our model did

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40 not include the effects of mass-flux/Stokes drift, wave-current interaction, or a wave roller, each of which would affect the quantitative predictions of the current profile. Phase averaging of the waves removed the time-variance of the forcing, and the mass-flux, and modified the total bottom drag applied to the mean currents. The boundary conditions imposed also influenced several processes. The shoreline boundary cut-off at a depth of approximately 0.5 m was a crude approximation and influenced the inner jet region. The no-slip condition imposed for stability purposes on the cross-shore boundaries also manipulated fluid velocities in those areas. The rigid-lid approximation on the top boundary removed the MWL to crest area where a portion of the wave stresses could have been applied. Approximating this force as a surface stress was limited by the free-slip condition. Bottom friction in LES was another inelegant compromise, both because the wave component was omitted, and no wall model was used. The grid aspect ratios require further testing to increase confidence that convergence was reached and was no longer affecting the fluid dynamics. The alongshore periodic boundary conditions forced only alongshore wavelengths of integral numbers of periods to be present and did not permit features longer than the domain length to arise. The model is in its development stages with many of the above mentioned limitations being considered. Increasing domain lengths in the alongshore and cross-shore directions will decrease boundary effects and include more of the inner jet. Increased grid resolution will improve the modeling of bottom friction and other LES effects. Higher grid resolution may also increase stability and allow for an improved representation of the free-slip boundary condition to be applied on the cross-shore

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41 boundaries. The inclusion of mass flux and alternative methods of approximating the forces above the mean water level are currently being developed. The inclusion of a wave roller is presently available in the model if desired. It was desired to pursue simplified process-oriented studies to isolate interactions of certain key physical processes. A consequence of this approach was that other important processes were suppressed making quantitative comparison with field data difficult. The main utility of the 3D model was the simultaneous resolution of both alongshore and cross-shore low frequency currents. Even with its inherent weaknesses and engineering approximations it was able to contribute new understanding to the dynamics of nearshore currents.

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CHAPTER 4 SIMULATIONS Sixteen cases are discussed to address the effects of grid resolution, vertically dependent forcing, and alongshore domain length. Table 3 lists the case numbers, the domain size, what was tested, and the grid used. All runs used a deepwater wave angle ( o ) of 30 o offshore boundary wave height (H o ) of 0.7 m, and peak period (T p ) of 8 s. Table 3. Summary of simulations Case number Domain size (x,y) (m) Grid resolution (nx, ny, nz) Average aspect ratio (H:V) Effect tested 1 (200, 198) (129, 129, 33) 11 : 1 Vertical forcing, domain length 2 (200, 198) (129, 129, 33) 11 : 1 Vertical forcing, domain length 3 (200, 20) (257, 33, 33) 6 : 1 Grid resolution 4 (200, 20) (257, 33, 17) 3 : 1 Grid resolution 5 (200, 20) (129, 17, 33) 11 : 1 Grid resolution 6 (200, 20) (129, 17, 17) 6 : 1 Grid resolution 7 (200, 20) (513, 33, 33) 3 : 1 Grid resolution 8 (200, 20) (257, 33, 17) 3 : 1 Vertical forcing 9 (200, 20) (257, 33, 17) 3 : 1 Vertical forcing 10 (200, 20) (257, 33, 17) 3 : 1 Vertical forcing 11 (200, 20) (257, 33, 17) 3 : 1 Vertical forcing 12 (200, 20) (257, 33, 17) 3 : 1 Vertical forcing 13 (200, 20) (257, 33, 17) 3 : 1 Vertical forcing 14 (200, 198) (257, 257, 17) 3 : 1 Domain length 15 (200, 198) (257, 257, 17) 3 : 1 Domain length 16 (200, 198) (257, 257, 17) 3 : 1 Domain length Simulations were compiled using the Intel Fortran Compiler (IFC) and ran on Intel Pentium 4 (2.6 GHz) to Intel Xeon (3.06 GHz) single processors. The model was computationally expensive based on grid resolution, domain size, and forcing distribution. Depth-uniform forcing simulations ran at faster speeds because they were more stable. Cases 1 and 2 produced 60 min of data in approximately 6 weeks of CPU time. Cases 3-7 required between 4 and 12 weeks of CPU time to compute 30 min of data. Cases 8-12 averaged 9 weeks of CPU time for 60 min of data. The model could 42

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43 produce results at approximate rates of 3.5 min/CPU week (Case 15 and 16) and 6 min/CPU week for Case 14 on the long domains. The model also was run in parallel on an SGI-Origin 3400. Running a simulation in parallel on four processors decreased the computational time by approximately a factor of two compared to a single processor Intel computer. A disadvantage of the parallel platform was that batch limits were set to allow a simulation to run for a maximum of five days at a time. Thus, a simulation had to be restarted manually several times to be completed. There were additional inevitable turn-around time losses between the completion of one portion of a simulation and the next opportunity to restart it. An advantage of the dedicated single processor Intel computers was that they could be run continuously for extended periods. It could also be noted that the University of Florida, Civil and Coastal Engineering Department's 20 processor SGI Origin 3400 cost approximately $400,000 to purchase and maintain during 2001-2003, compared to single or dual processor Intel computer costs of under $2,000. The cost to benefit ratio appeared to favor the single processor platforms. All the simulations produced reasonable velocities for the given wave parameters. Representative time series for u, v, and wvelocity are given for two locations within the surfzone. The first, at x = 80 m, was located at the bar. The second time series was taken at x = 40 m, and was situated in the middle of the bar trough. The data presented was for Case 11. Both time series were taken at mid-depth and mid-way of the alongshore domain (y = 10 m). U and w fluctuated about a mean, while the time series of v showed the spin-up and migration of the alongshore current into the trough region by the time lag between the two signals.

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44 t(min) u(m/s) 10 20 30 40 50 -0.2 -0.1 0 0.1 0.2 0.3 0.4uatx=80uatx=40 Figure 18. Time series for u-velocity. Data was from Case 11, sampled at (x o, y o z o ) = (80, 10, 1.1) (red) and (x o, y o z o ) = (40, 10, 1.5) (green). All co-ordinates were in m, where z o was measured positive downwards from the top boundary. t(min) w(m/s) 10 20 30 40 50 -0.15 -0.1 -0.05 0 0.05 0.1 0.15watx=80watx=40 Figure 19. Time series for w-velocity. Data was from Case 11, sampled at (x o, y o z o ) = (80, 10, 1.1) (red) and (x o, y o z o ) = (40, 10, 1.5) (green). All co-ordinates were in m, where z o was measured positive downwards from the top boundary. All the simulations produced reasonable magnitudes of the mean alongshore currents for the given parameters. Since the bottom friction was modeled using a no-slip condition, ad-hoc tuning of the bottom friction coefficient (C f ) could not be done to adjust the strength of the alongshore current. In addition, with the phase-averaged model,

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45 the bottom friction did not feel the time-varying wave bottom boundary layer where fDragCuuu w This was significant because u w could be relatively large. The omission of u w produced a no-slip drag that was biased lower than field conditions. This offered one explanation for why the mean currents generated by the model might have been slightly higher than observed for the same wave conditions in the field. t(min) v(m/s) 10 20 30 40 50 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0vatx=80vatx=40 Figure 20. Time series for v-velocity. Data was from Case 11, sampled at (x o, y o z o ) = (80, 10, 1.1) (red) and (x o, y o z o ) = (40, 10, 1.5) (green). All co-ordinates were in m, where z o was measured positive downwards from the top boundary. The first simulations discussed were a comparison of forcing the model with two vertical distributions over the water column: a depth-uniform (Case 1) or a linearly depth-varying force (Case 2) (Figure 21) in both the alongshore and cross-shore directions based on Fredsoe and Deigaard (1992) These runs produced a number of new and interesting results. First, instabilities of the alongshore current developed much sooner for depth-dependent forcing. Second, the depth-averaged current was much stronger before breaking down into instability and turbulence for depth-uniform forcing. Third, the alongshore wavelength of the initial

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46 instabilities was longer (of order 100s of meters) for the depth-uniform current, but on the order of 10s of meters for the depth-dependent current. A comparison for the alongshore and depth-averaged alongshore currents is given in Figure 22 The strong peak currents at the shore and over the bar were present in the depth-uniform case (Figure 22 A) until the current became unstable (20 min). After this time, the current laterally spread across the surfzone, but a strong current was still present at the shoreline. In Figure 22 B, the depth-dependent forcing did not produce distinct alongshore current peaks over the bar, but instead the current was much weaker, and was spread across the surfzone with local maxima shoreward of the bar. Figure 21. Case 1 and 2: Vertical distribution of forcing used in the simulations. Case 1 (A) and 2 (B). These results showed that the response of the alongshore current was sensitive to the vertical distribution of forcing. The cross-shore distributions of the mean alongshore current profiles revealed effects of 3D mixing and preferential cross-shore advection. The physical interpretation had two major components. The first was that much stronger cross-shore circulation developed for the depth-dependent forcing. When F x was depth-uniform, very little cross-shore circulation was produced; rather, a barotropic

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47 cross-shore pressure gradient balanced the forcing. When F x was depth-dependent, however, a strong undertow developed rapidly and was compensated by a shoreward mass flux in the top half of the water column ( Figure 2 3 ). x(m) z(m) 0 50 100 150 200 0 1 2 3 4 Figure 22. Comparison of depth and alongshore averaged currents for the initial simulations. A) Depth-uniform forcing, B) Depth-dependent forcing. Note: vertical axes used different scales for A and B. Bottom panel is the beach profile. One reason this was dynamically important to the depth-averaged alongshore current was that F y and V(z) were also depth-dependent in this case. This was partly due to the no-slip boundary condition on V at the sea bed and the free-slip condition at the surface. Hence, there was a much stronger alongshore current in the top half of the water column than in the bottom, as seen in nature. Thus, the faster surface current was advected shoreward (into the trough) while the weaker, bottom current drifted offshore. The net effect was a shoreward shift of the alongshore current maximum that became balanced in the trough. This was a similar response to having included a roller model in

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48 the wave sub-model to shift the location of wave momentum input shoreward. A roller model was not incorporated into the formulations of the Thornton-Guza (T-G, 1983) wave breaking model that was used to generate input radiation stresses for these cases. Figure 23. A typical vertical profile of cross-shore velocity vectors produced by depth linear forcing. Vectors were plotted to show the extent of the undertow along the bottom boundary. The reference vector above represented 1 grid space/unit magnitude. The vertical distribution of forcing also affected the stability of the alongshore current. In the depth-uniform case (Figure 24), linear growth of the alongshore current produced finite large scale instabilities similar to 2D models. Depth-linear forcing (Figure 25), however, produced small-scale disturbances in the alongshore current at much earlier times. Figure 24. Surface vorticity contours for depth-uniform forcing as a function of time.

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49 Figure 25. Surface vorticity contours for depth-dependent forcing as a function of time. In summary, these initial simulations showed that depth-uniform forcing produced results comparable to previous 2D works. Depth-linear forcing, however, produced very different results for the vorticity structures, and provided the motivation to investigate the effects of vertical forcing and grid resolution further. The Effect of Grid Resolution The initial simulations were done on a grid with a high horizontal to vertical aspect ratio that exceeded the recommendations of Scotti et al. (1997) for LES closure. A series of tests were run to determine adequate grid resolution, and to verify the generality of the results. A total of five different grids were tested using depth-linear forcing as in Case 2 on a 200 m by 20 m by depth-varying grid. The 20 m domain length was used to reduce computational time, and was valid because the turbulent structures seen in Case 2 were on the order of 6 m and could be adequately captured on the smaller domain. Case 5 had the same aspect ratio as the original tests conducted, while Case 4 followed the recommendations of Scotti et al. (1997) and Case 7 was within close proximity of his upper bound recommendation of a 4:1 grid aspect ratio. Alongshore and cross-shore currents, as well as depth-averaged vorticity fields, were compared to study grid resolution effects on flow characteristics. Each test was run to approximately 30 min.

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50 Alongshore-Averaged Alongshore Currents Comparing the vertical distribution of alongshore currents showed effects of the grid resolution and aspect ratio. Velocity contours comparing runs (Figure 26) showed that they were qualitatively similar to first-order observation. x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=15min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=25minv(m/s) A x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=15min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50. 00 -0. 11 -0. 22 -0. 33 -0. 44 -0. 56 -0. 67 -0. 78 -0. 89 -1. 00 t=25minv(m/s) B x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=15min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50. 00 -0. 11 -0. 22 -0. 33 -0. 44 -0. 56 -0. 67 -0. 78 -0. 89 -1. 00 t=25minv(m/s) C x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=15min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50. 00 -0. 11 -0. 22 -0. 33 -0. 44 -0. 56 -0. 67 -0. 78 -0. 89 -1. 00 t=25minv(m/s) D x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=15min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min E x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 50.00-0.11-0.22-0.33-0.44-0.56-0.67-0.78-0.89-1.00 t=25minv(m/s) Figure 26. Cases 3-7: Contours of the alongshore-averaged alongshore velocity (v) for variable grid sizes in the x-z plane. Plots are given in 5 min intervals to show progression of current migration. A) Case 3. B) Case 4. C) Case 5. D) Case 6. E) Case 7.

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51 The high aspect ratio grid of Case 5 showed preferential cross-shore mixing at earlier time periods. The alongshore current spread across more of the surfzone, and showed a stronger jet at the shoreline between 0 and 10 min than was present in other runs. Currents generated over the offshore bar were also weaker due to the increased lateral mixing. Case 6 produced somewhat stronger alongshore currents over the bar, and within the surfzone. Cases 3 and 4 were comparable in magnitude but varied slightly in cross-shore position. Cases 4 and 7 were comparable in cross-shore distribution but Case 7 had slightly weaker currents over the bar. Alongshore and Depth-Averaged Alongshore Currents The analysis of alongshore, and depth-averaged alongshore currents showed the cross-shore distribution of the current. At earlier time (Figure 27), there was little difference between the cross-shore distributions. At later times (Figure 28), however, Cases 3, 5, and 6 produced somewhat stronger alongshore currents at the bar trough and shoreline, while Cases 4 and 7 were comparable to each other. x(m) v(m/s) 0 50 100 150 20 0 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Case3Case4Case5Case6Case7 x(m) z(m) 0 50 100 150 20 0 0 1 2 3 4 5 Figure 27. Depth and alongshore-averaged alongshore currents at t = 5 min.

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52 x(m) v(m/s) 0 50 100 150 200 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Case3Case4Case5Case6Case7 x(m) z(m) 0 50 100 150 20 0 0 1 2 3 4 5 Figure 28. Depth and alongshore-averaged alongshore currents at t = 25 min. Alongshore-Averaged Cross-shore Currents Time, and alongshore-averaged cross-shore velocity (, ) vector plots (Figure 29 to Figure 33 ) gave a qualitative feel for the cross-shore currents. The vertical axis of the grid was greatly exaggerated by approximately a factor of 40:1 compared to the horizontal scale. Therefore the direction of the velocity vectors gave a different impression of the flow depending on the manner chosen for plotting. The results were presented in two manners, but conveyed the same information. Vectors were plotted relative to grid units/magnitude scale (A), and specified the vector length as the number of grid units per unit vector magnitude, or relative to cm/magnitude scale (B) and, specified the vector length as the number of screen centimeters per unit of vector magnitude. Each option provided a different sense of the cross-shore currents. Using the vectors plotted as in A, the correct direction of the velocity vectors was given. In B, the correct magnitude was given. In both options, a reference vector equal to the relative scale was included. The grid resolution in the x-z plane varied, however, only 64 x 8 vectors were plotted to allow more space for larger vectors.

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53 Cases 3, 5, and 6 produced stronger vertical mixing over the bar (A) as well as stronger cross-shore currents over the bar and at the shoreline (B). Case 6 had very strong downwards velocities at the shoreline. Cases 4 and 7 produced similar results. The strongest cross-shore flow occurred over the bar and along the bottom at the shoreline edge. x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1A x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1B Figure 29. Case 3: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to grid units/magnitude scale. B) relative to cm/magnitude scale. x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1A x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1B Figure 30. Case 4: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to grid units/magnitude scale. B) relative to cm/magnitude scale.

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54 x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1A x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1B Figure 31. Case 5: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to grid units/magnitude scale. B) relative to cm/magnitude scale. x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1A x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1B Figure 32. Case 6: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to grid units/magnitude scale. B) relative to cm/magnitude scale. An alternate assessment of the magnitude of cross-shore velocities was done by comparing the depth and alongshore-averaged term (Figure 34). It was clear here that the magnitude of was much higher (approximately a factor of 2) over the bar, and at the shoreline for Cases 5, and 6 compared to Cases 4, and 7.

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55 x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1A x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5 1B Figure 33. Case 7: Time-averaged vertical profiles in the cross-shore of the alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to grid units/magnitude scale. B) relative to cm/magnitude scale. x(m) u2 0 50 100 150 200 0 0.02 0.04 0.06 0.08 0.1 0.12Case3Case4Case5Case6Case7 b x(m) z(m) 0 50 100 150 20 0 0 1 2 3 4 5 Figure 34. Comparison of the depth-averaged term as a function of cross-shore location. Vorticity Comparison of vorticity structures of all the runs showed no major differences. This led to the conclusion that the small-scale structures seen in Case 2 with depth-linear forcing were not dependent on grid resolution or aspect ratio. The results also supported

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56 that the 3D effects produced by vertical forcing caused the highly chaotic nature of the flow, and not resolution issues related to the grid aspect ratio or LES on an anisotropic grid. Conclusions of Grid Testing The high aspect ratios between the horizontal and vertical length scales led to increased flow in the cross-shore direction. Case 4 was considered the best numerical experiment because it had the smallest grid aspect ratio. The error difference between Case(i) and Case 4 was defined as 222()(4)()(4)uiuErroriu where 2u represented the depth, cross-shore, and time averaged u 2 term. The values for the error between the cases were E(3) = 0.32, E(5) = 0.66, E(6) = 0.49, E(7) = 0.13. The overall alongshore currents were more similar than the cross-shore exchanges (Figure 27, Figure 28), and the results obtained for Cases 1 and 2 were representative of results that would be obtained on higher resolution grids. After comparing the criteria for the five simulations, the grid from Case 4 with 257 by 33 by 17 grid points was used for additional experiments on the 20 m alongshore domain. This grid fell within the recommendations given by Scotti et al. (1997) and produced characteristic flow response. Results for this grid were comparable to Case 7 that had a much higher resolution grid. It was advantageous to use the coarser grid to decrease computational time from months to weeks. The Effect of Vertical Distribution Based on the literature review of radiation stresses in the surfzone, a variety of vertical distributions were tested to determine the effects of depth-varying forcing on the

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57 resulting flow structures and alongshore current (Table 3, and Chapter 3). Six distributions were tested on the small domain (x, y, z) = (200 m, 20 m, variable). Three distributions were tested on the large domain (x, y, z) = (200 m, 198 m, variable) to include effects of larger-scale alongshore current instabilities. Cross-shore Flow The vertical distribution of momentum forcing affected the cross-shore velocities. Stronger cross-shore flow was produced when forcing was concentrated over fewer grid points at the top of the domain (Case 8 vs. Case 10). Figure 35 C and Figure 36 B showed that concentrating the force over the top three grid points also pushed the peak cross-shore flow to the shoreward side of the offshore bar. Depth-linear forcing decreased cross-shore velocities when compared to depth-uniform forcing over the same number of grid points. This was an unexpected result and attributed to the free-slip condition enforced on the top boundary that set the U-velocity at the boundary by extrapolating from the four points below it at the end of each time step. The unfortunate misapplication of a shear stress at the top boundary produced alongshore currents over the bar, another effect that was attributed to the use of the incompatible free-slip boundary condition ( Figure 36 A). There were three cases in Table 3 that suffered from this implementation; Cases 11, 12 and 16. Cases 11 and 16 ended up putting approximately 1/5 th of the total F x as a depth uniform force at the surface region. They were retained in the study to illustrate the influence of the long domain and shear waves on the current profile in Case 16. Mellors equations ( Figure 36 C) produced an opposite cross-shore circulation pattern, with the shoreward flow along the bottom contour, and the resulting off-shore flow along the surface. Since this cross-shore circulation pattern was contrary to field and laboratory observations, his equations werent considered useful

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58 under these circumstances, and were not pursued further in our study. In Figure 35 C, the strongest currents were slightly shoreward. In Figure 36 C, the currents were weaker and opposite in direction, with the onshore flow (blue) along the bottom and the off-shore directed flow (red) along the top boundary. x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5A x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5B x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 50.200.170.150.120.090.070.040.01-0.01-0.04-0.07-0.09-0.12-0.15-0.17-0.20 u(m/ s) C Figure 35. Cases 8-10: Comparison of alongshore and time-averaged u-velocity as a function of depth and location in the cross-shore domain. A) Case 8. B) Case 9. C) Case 10. The dark blue contours indicate strong on-shore flow and the red contours represent the off-shore directed undertow. x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5A x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 5B x(m) z( m ) 0 50 100 150 200 0 1 2 3 4 50.200.170.150.120.090.070.040.01-0.01-0.04-0.07-0.09-0.12-0.15-0.17-0.20 u(m/s ) C Figure 36. Cases 11-13: Comparison of alongshore and time-averaged u-velocity as a function of depth and location in the cross-shore domain. A) Case 11. B) Case 12. C) Case 13. The dark blue contours in A and B indicate strong on-shore flow and the red contours represent the off-shore directed undertow. Another aspect of the influence of cross-shore mixing on currents was the effect of vertical mixing. Various 2D and 3D models have assumed hydrostatic flow (u >> w),

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59 which was a valid approximation resulting in predominantly 2D (x-y) currents. The vertical velocities (w), however, were actually greater than u in various positions of the cross-shore, especially off-shore of the bar. These vertical currents resembled the strong vertical mixing in vortex trains seen by Li and Dalrymple (1998) (Figure 37). The non-hydrostatic approach in the 3D model produced similar vertical eddies offshore of the bar, with the ratio of u:w in the range of 0.5 at some locations (Figure 38 and Figure 39 ). Vertical mixing was significant, and a non-hydrostatic model was necessary to produce that type of cross-shore current and mixing. x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10.86min Figure 37. Vector plot of alongshore-averaged u-w velocities for Case 7 u:w = 0.5 at (x o z o ) = (138 m, 3.2 m). x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.020.0150.010.0050-0.005-0.01-0.015-0.02-0.025-0.03-0.035-0.04-0.045-0.05-0.055-0.06 t=10.86minw(m/s) Figure 38. Alongshore-averaged w-velocity contours for Case 7, t = 10.86 min. Red contours indicate upwards flow. Blue contours represent a downwards directed velocity.

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60 x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.020.010.01-0.00-0.01-0.02-0.02-0.03-0.04-0.05-0.05-0.06 t=10.86minu(m/s) Figure 39. Alongshore-averaged u-velocity contours for Case 7, t = 10.86 min. Red contours represent offshore velocity and blue represent onshore directed velocity. Alongshore Currents Each forcing distribution influenced the alongshore current to some degree. Each distribution consisted of depth-uniform y-momentum forcing, and had two-thirds of the x-momentum inputted as a depth-uniform force. The remaining one-third of the total x-momentum was varied in the five runs. It was primarily this portion that affected the cross-shore current, and the eventual location of the alongshore current maximum. When a force was located over a smaller vertical area, the alongshore current was spread further into the trough. A depth-uniform distribution, compared to a depth-linear distribution, ( Figure 42 vs. Figure 43 ) also pushed the alongshore current further into the trough. The depth-linear forcing produced stronger currents at the shoreline (approximately 40 to 50%) and over the bar (5 to 10%) ( Figure 43 ) compared to the depth-uniform distribution ( Figure 42 ). Applying a shear stress at the top ( Figure 44 ) produced strong currents at the shoreline, and reduced spreading of the current across the surfzone. Figure 40 to Figure 44 show the cross-shore distribution of the alongshore, and depth-averaged alongshore current over time for the five cases. By 60 min, the current

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61 over the bar had begun to stabilize while the current in the trough was still increasing. The shoreward shift of the peak alongshore current was compared for the five runs in Figure 45. The apparent noise of the alongshore current was due to the small alongshore domain length that the currents were averaged over. Over longer domain lengths, the currents had a smoother profile because the small-scale instabilities were averaged over a larger scale. x(m) v(m/s) 0 50 100 150 20 0 -1.5 -1 -0.5 0t=10t=20t=30t=40t=50t=60 x(m) z(m) 0 50 100 150 20 0 0 1 2 3 4 5 Figure 40. Case 8: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period Time progression shows the migration of the current shoreward of the bar (x = 80 m). By 60 min, the peak alongshore current at the shoreline seen in Cases 8 and 9 was not visible in Case 10. The current was more uniform in the cross-shore. Concentrating the forcing over fewer grid points at the surface had smoothed out the current. The peak velocity remained around 1 m/s. Time progression showed the migration of the current shoreward of the bar (x = 80 m) for Case 11. By 60 min, the current over the bar had begun to stabilize while the current in the trough was still increasing. A distinct peak still existed at the shoreline.

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62 Time progression showed the current did not migrate substantially shoreward of the bar, with distinct peaks at the shoreline, and over the bar for Case 12. By 60 min, the current over the bar was starting to stabilize while the current in the trough was still increasing. Peak velocities exceeded 1 m/s. x(m) z(m) 0 50 100 150 20 0 0 1 2 3 4 5 x(m) v(m/s) 0 50 100 150 20 0 -1.5 -1 -0.5 0t=10t=20t=30t=40t=50t=60 Figure 41. Case 9: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period. Time progression showed the migration of the current shoreward of the bar (x = 80 m). x(m) z(m) 0 50 100 150 20 0 0 1 2 3 4 5 x(m) v(m/s) 0 50 100 150 20 0 -1.5 -1 -0.5 0t=10t=20t=30t=40t=50t=60 Figure 42. Case 10: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period Time progression showed the migration of the current shoreward of the bar (x = 80 m).

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63 x(m) z(m) 0 50 100 150 20 0 0 1 2 3 4 5 x(m) v(m/s) 0 50 100 150 20 0 -1.5 -1 -0.5 0t=10t=20t=30t=40t=50t=60 Figure 43. Case 11: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period. x(m) z(m) 0 50 100 150 20 0 0 1 2 3 4 5 x(m) v(m/s) 0 50 100 150 20 0 -1.5 -1 -0.5 0t=10t=20t=30t=40t=50t=60 Figure 44. Case 12: Cross-shore distribution of the alongshore and depth-averaged alongshore current over a 60 min time period The results of the alongshore and depth-averaged alongshore currents showed the influence of 3D forcing. Since the forcing was not entirely depth uniform in the x-direction, the alongshore momentum located over the bar was advected shoreward into the trough, without the influence of large-scale shear instabilities or a surface roller. The most effective form of cross-shore spreading of the alongshore current was produced by

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64 Case 10 (Figure 45, Figure 46). Comparison of the depth-averaged current showed the migration of the peak from over the bar into the trough (Figure 45). x(m) t(min) 0 20 40 60 80 100 120 140 160 180 200 10 20 30 40 50 60Case8Case9Case10Case11Case12 x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5 Figure 45. Cross-shore position as a function of time of the peak alongshore current for the 5 vertical distsributions tested. t(min) v(m/s) 0 20 40 60 -1.2 -1 -0.8 -0.6 -0.4 -0.2Case8Case9Case10Case11Case12 Figure 46. Peak velocity of the alongshore current as a function of time. To better understand the differences caused by the vertical forcing, the vertical distribution of the alongshore current was studied. Figure 47 to Figure 50 compared the effects of vertical forcing on the alongshore current over a 60 min interval for each run. In each plot, a boundary layer existed along the bottom (light blue) where the alongshore

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65 current was insignificant. This coincided with the areas of stronger cross-shore currents. Peak currents occurred slightly below the surface (darker yellow and red areas). x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=15minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=25min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=30minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=35min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=40min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=45minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=50min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=55min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=60minv(m/s) Figure 47. Case 8: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v) Time progression at 5 min intervals showed the migration of the current over the bar into the trough area.

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66 In Figure 48, peak alongshore currents (red) were stronger over the same time intervals compared to Figure 47. x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=15minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=25min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=30minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=35min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=40min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=45minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=50min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=55min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=60minv(m/s) Figure 48. Case 9: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v) Time progression at 5 min intervals showed the migration of the current over the bar into the trough area.

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67 In Figure 49, a boundary layer along the bottom (light blue) existed. The alongshore current was weaker, and more spread out in the cross-shore over the same time periods for the other examples. x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=15minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=25min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=30minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=35min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=40min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=45minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=50min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=55min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=60minv(m/s) Figure 49. Case 10: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v). Time progression at 5 min intervals showed the migration of the current over the bar into the trough area.

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68 Linearly distributing the pressure term over the top 3 grid points produced cross-shore flow at the top boundary at early time (5 min) for Case 11. Alongshore currents slowly migrated into the trough at similar rates to the other forcing distributions. x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=15minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=25min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=30minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=35min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=40min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=45minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=50min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=55min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=60minv(m/s) Figure 50. Case 11: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v). Time progression at 5 min intervals showed the migration of the current over the bar into the trough area.

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69 x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=5min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=10min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=15minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=20min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=25min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=30minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=35min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=40min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=45minv(m/s) x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=50min x( m ) z(m) 0 50 100 150 200 0 1 2 3 4 5t=55min x(m) z(m) 0 50 100 150 200 0 1 2 3 4 50.0 0 -0.1 1 -0.2 2 -0.3 3 -0.4 4 -0.5 6 -0.6 7 -0.7 8 -0.8 9 -1.0 0 t=60minv(m/s) Figure 51. Case 12: Vertical cross-section contour plots of the alongshore-averaged alongshore current (v). Time progression at 5 min intervals showed the migration of the current over the bar into the trough area. Case 12 examined the affect of applying a shear stress at the surface. At t = 5 min, the effect of the shear stress was apparent with a stronger shoreward migration of the

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70 surface current. Currents were stronger, and less spread out in the cross-shore over time compared to other examples. A boundary layer still existed, but was less apparent. Vorticity Plots of the vertical vorticity in depth integrated flow models had been used in previous 2D studies as a means to observe migrating large scale features in the mean flow. The 3D vertical vorticity contour plots, however, included significant small scale features that dominated the principal impression conveyed ( Figure 52 A). Due to the inherent three-dimensionality of vortex dynamics, and the fine resolution of the grid that allowed turbulence to be calculated at much smaller length scales than previous 2D studies, direct vorticity comparisons were not particularly useful for large scale flow features. Contour plots and velocity vectors of alongshore velocity ( Figure 52 B) provided a more readily interpreted and informative tool for alongshore current flow analysis, and confirmed the mean alongshore current was not simply chaotic, as the vorticity contours had indicated. x(m) y(m) 0 50 100 150 200 0 5 10 15 200.010 0 0.008 0 0.006 0 0.004 0 0.002 0 0.000 0 -0.002 0 -0.004 0 -0.006 0 -0.008 0 -0.010 0 (1/s ) A x y 0 50 100 150 200 0 5 10 15 200.00-0.08-0.16-0.24-0.32-0.40-0.48-0.56-0.64-0.72-0.80-0.88-0.96-1.04-1.12-1.20 1v(m/s ) B Figure 52. Comparison of surface vorticity component () and alongshore surface current (v) contours in the cross-shore (x) and alongshore (y) domain. z A) Vorticity contours showed small scale features that were highly erratic and suggested a very unstable current. B) Velocity contours and vectors of the same data showed a strong current that had peaks at the shoreline and around the bar. Snapshot was of Case 11 at t = 50 min.

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71 Small scale perturbations seen as vorticity patches with horizontal length scales on the order of 2 m were very apparent in the cases examined in Figure 52 and Figure 53 Similar small scale vortex structures were not produced in 2D model studies and further understanding of what properties influenced these structures in 3D were needed. Two potential influences other than the vertical distribution of forcing were examined: spatial grid size and velocity filtering. Two-dimensional models did not produce the same vorticity structures as seen in the 3D model for a number of reasons. They used coarser grids, with horizontal grid spacing in the range of 2 to 5 m which removed the ability to calculate small-scale turbulence that might have been present. As well, a 2D, depth-averaged flow could have never produced 3D turbulence. Instead of a cascade of turbulence from larger to smaller scales, a 2D flow would have produced eddies that coupled with each other to form larger eddies, known as an inverse cascade of energy. Horizontal vorticity ( x y ) was also not produced in 2D flows. In a 3D flow, these turbulent eddies could have been stretched and tilted, which could have influenced vertical vortex structures ( z ). To study the effect of horizontal grid resolution on the vorticity contours, the model output velocities were spatially averaged to a 5 m grid resolution and then vertical vorticity was calculated from the averaged velocity. The 3D model had a spatially varying, high resolution grid that was able to calculate turbulence at scales on the order of 0.2 m in the nearshore to 1 m at the offshore. By taking an average velocity over a 5 m by 5 m spatial grid, the small scale perturbations were removed. The effective grid resolution was reduced from (nx, ny) = (257, 33) to (nx, ny) = (40, 4). Spatial averaging the velocities with a coarser resolution smoothed out the vorticity contours ( Figure 53 A).

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72 This result demonstrated one of the influences of grid resolution on the resulting vorticity contours. Comparing Case 1 and Case 14 (discussed later) that both used depth-uniform forcing on the 200 m x 198 m physical domain at different horizontal grid resolution also showed that the finer grid produced smaller scale turbulent features. The omission of 3D turbulence and the use of coarse grids found in 2D models did not permit the presence of small-scale turbulent features as was seen in our results. Time averaging over a variety of time intervals was done to find the mean vorticity field. When the vorticity data were averaged over long time periods, as in Figure 53 C, the associated vorticity contour plots were smoothed out. Time averaging over small time periods, however, did not smooth out the contour plots. x(m) y(m) 0 50 100 150 200 0 5 10 15 20A x(m) y(m) 0 50 100 150 200 0 5 10 15 20B x(m) y(m) 0 50 100 150 200 0 5 10 15 200.010 0 0.008 0 0.006 0 0.004 0 0.002 0 0.000 0 -0.002 0 -0.004 0 -0.006 0 -0.008 0 -0.010 0 (1/ s) C Figure 53. Sample comparison of spatially averaged, original vorticity, and time-averaged contours at vertical grid point nz-1 Plots were generated from Case 11 data, t = 50 min. A) vorticity contours for spatially averaged data at 5 m grid spacing. B) original vorticity plot on a variable grid. C) time averaged (50 min) vorticity contours of the same simulation. Time averaging of the data over long time periods showed the mean vorticity and smoothed out the contours. Energy spectra of the uand v-velocities were used to determine the frequency band at which the majority of the energy was concentrated. When significant energy was concentrated at lower frequencies, it indicated that the large scale features were more important to the transportation of energy. (E.g., alongshore currents with shear waves

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73 had a peak energy concentration of about 1 m 2 /s 2 located around 0.01 Hz.) Spectra of the currents during run-up time of the model were generated at various locations of x and z within the domain. Results showed that the energy was concentrated at low frequencies. In the spectral analysis of the depth-averaged currents at x = 40 m in the cross-shore (center of the bar trough), the u-velocity spectrum had a peak at 0.047 Hz (Figure 54), and the v-velocity spectrum had a peak at 0.031 Hz (Figure 55). The time and depth-averaged uand v-velocities were 0.004 m/s, and -0.52 m/s respectively. The length scales of the peak energy were determined by using the following equation: 22(/)()()ppEmsLengthmfHz where subscript (p) indicated the peak value. The length scale of the peak energy structures in the u-energy spectrum was approximately 2 m, with fluctuation velocities around 0.10 m/s. The length scale in the v-energy spectrum was approximately 6 m, with velocities around -0.2 m/s. The energy spectra showed that a significant amount of energy in the velocity field was contained within the small-scale features shown. This was also verified by taking the RMS of the peak v-velocity, pavgavg(V-V)V which was equal to almost 50% of the mean flow at this cross-shore location. Spectral analysis on equilibrated flow (t > 60 min) produced slightly different results because the effects of run-up and current migration were not present. The purpose of the filtering, however, was to determine the energy distribution in the currents at times when small scale instabilities were first appearing. Low-pass filtering the velocities at these frequencies to remove high frequency fluctuations was done to determine the effect (Figure 56 and Figure 57). Filtering above the peak frequencies was an attempt to remove high-frequency noise that might have

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74 obscured trends in the data. In this particular example, filtering did not remove the small scale perturbations from the velocity signal because they were where the bulk of the energy of the current lied. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0. 5 0 0.002 0.004 0.006 0.008 0.01 0.012 f (Hz)energy (m2/s2) Figure 54. Depth-averaged U(xo,yo,t)-velocity energy density spectrum for Case 8 where x o = 40 m, y o = 10 m, t =12-66 min. Data was sampled at 1 Hz, totaling 3200 data points. Peak frequency = 0.0469 Hz. Maximum energy = 0.0105 m 2 /s 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 energy (m 2 /s 2 ) f (Hz) Figure 55. Depth-averaged V(xo,yo,t)-velocity energy density spectrum for Case 8 where x o = 40 m, y o = 10 m, t = 12-66 min. Data was sampled at 1 Hz, totaling 3200 data points. Peak frequency = 0.0313 Hz. Maximum energy = 0.0386 m 2 /s 2

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75 2400 2450 2500 2550 2600 2650 2700 275 0 .08 .06 .04 .02 0 0.02 0.04 0.06 v (m/s)t (sec) Figure 56. Time sample of velocity fluctuations (black) and the low-pass filtered (red) depth-averaged u-velocity at 0.03 Hz taken from energy spectrum data U(x o ,y o ,t) for Case 8, where x o = 40 m, y o = 10 m, t = 40-46 min. 0 500 1000 1500 2000 2500 3000 3500 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 t (sec) v (m/s) v filtered v Figure 57. Time sample of raw data (black) and the low-pass filtered (red) depth-averaged v-velocity at 0.046 Hz V(x o ,y o ,z o ,t) for Case 8, where x o = 75 m, y o = 10 m, t = 12-66 min.

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76 Effect of Domain Length Two domain lengths were used for the simulations. The 20 m domain prohibited the development of large scale alongshore shear instabilities [O (100 m)] that could have influenced the cross-shore diffusion of the alongshore current. The 20 m domain restricted the focus of the study on the effects of the vertical distribution of the forcing. To verify that this assumption was correct, simulations with depth-uniform forcing in both directions were run on 20 m and 200 m domains. The alongshore current remained stable on the 20 m domain for the length of the run. On the 200 m domain, however, the current became unstable over time, and broke down into features with alongshore wavelengths of approximately 100 m, similar to those presented in previous 2D model results ( Slinn et al. (1998) and zkan-Haller and Kirby (1999) ). These tests verified that domain length influenced the stability of the alongshore current. The extent of the domain lengths influence on the alongshore currents generated was dependent on the vertical distribution of forcing. Three vertical distributions were tested. Case 14 was forced with a depth-uniform distribution. Case 15 used the vertical distribution from Case 10, and Case 16 used the distribution from Case 11. When the domain length was increased and large scale instabilities were allowed to influence the alongshore current, they spread the mean alongshore current laterally. The instability of ()Vx was most visible when the model was forced by a depth-uniform force (Figure 58). Without the large-scale instabilities, the current continued to grow over the bar. The large-scale shear instabilities also decreased the maximum mean alongshore current velocities through increased cross-shore mixing. This was qualitatively visible in Figure 58 to Figure 60 that compare the short and long domain results of the alongshore

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77 depth-averaged alongshore currents for the various forcing distributions. When F x was not entirely uniform as in Case 14, the alongshore current was spread laterally at earlier times. It appeared that the instability imposed by the vertical shear of the cross-shore currents influenced the stability of the large-scale features of the alongshore current as well, and caused the shear waves to develop at earlier times. The short domain produced stronger currents over the bar. The alongshore current was more widely spread across the cross-shore with the long domain that included the effects of the large scale instabilities. A strong jet remained at the shoreline for both domain lengths that indicated the large scale motions had minimal effect on the currents at that location. Large-scale shear instabilities on the order of 100 m were still present in the 200 m domain when forcing was not depth uniform. Vertical vorticity contours for Cases 14 -16 are compared in Figure 61 to Figure 63 x(m) v(m/s) 0 50 100 150 200 -2 -1.5 -1 -0.5 0515253035455152530 t(min) x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5 Figure 58. Influence of domain length on alongshore currents for depth-uniform forcing: Alongshore and depth-averaged alongshore current profiles as a function of cross-shore position and time. Solid lines represent the 200 m alongshore domain. Dashed lines represent the 20 m alongshore domain.

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78 x(m) v(m/s) 0 50 100 150 20 0 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 051015205101520 t(min) x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5 Figure 59. Influence of domain length on alongshore currents for Case 10 forcing: Alongshore and depth-averaged alongshore current profiles as a function of cross-shore position and time. Solid lines represent the 200 m alongshore domain. Dashed lines represent the 20 m alongshore domain. x(m) z(m) 0 50 100 150 200 0 1 2 3 4 5 x(m) v(m/s) 0 50 100 150 200 -1 -0.8 -0.6 -0.4 -0.2 05101520253051015202530 t(min) Figure 60. Influence of domain length on alongshore currents for Case 11 forcing: Alongshore and depth-averaged alongshore current profiles as a function of cross-shore position and time. Solid lines are 200 m alongshore domain. Dashed lines are for 20 m alongshore domain.

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79 Figure 61. Case 12: Contours of the depth-averaged vorticity as a function of time The current became unstable around 20 min and produced large scale instabilities (t = 25 min) that broke down into smaller scale turbulence (t = 45 min). Figure 62. Case 15: Contours of the depth-averaged vorticity as a function of time. The current became unstable at the shore almost immediately (t = 5 min) and large scale instabilities were less visible.

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80 Figure 63. Case 16: Contours of the depth-averaged vorticity as a function of time. The current at the shoreline became unstable at early time, while the current over the bar became unstable around 15 min, and evolved into smaller scale turbulence (t = 25 min). The large-scale [O (70 m)] instabilities apparent in Case 16 (t = 15 min) resulted from the instability of the alongshore current similar to the depth-uniform case. With the depth-linear forcing in the top three grid points, the effect of the top grid point was reduced because of the free-slip condition. This removed a portion of the depth-varied forcing, and put a larger influence on the depth-uniform component. This resulted in a more stable alongshore current over the bar that was strongly influenced by the alongshore shear instabilities on the current. Although the small-scale vorticity features obscured the presence of the larger scale shear waves, they were still present, and influenced the alongshore current position over time. In effect, the depth-dependent forcing pushed the peak alongshore current velocities into the trough, and the shear waves spread the current horizontally, with preferential spreading in the shoreward direction for depth-dependent forcing.

PAGE 94

CHAPTER 5 SUMMARY AND DISCUSSION Findings and Results A 3D LES model was adapted to simulate alongshore currents forced by breaking waves. It produced strong alongshore currents in the trough area without the use of a roller model, alongshore pressure gradients, alongshore shear waves, or ad-hoc tuning of the bottom friction coefficient. The 3D model allowed the effect of vertical forcing on alongshore current structure to be studied on a small domain [O (20 m)]. The development and migration of alongshore currents were analyzed as the alongshore current spun up to steady state. The model did not specify the bottom friction and achieved reasonable, balanced values of ()Vx for the specified wave field conditions. The effect of grid resolution, vertically dependent forcing, and alongshore domain length were all examined. The LES closure scheme was sensitive to grid aspect ratios, and overall grid resolution. Aspect ratios less than 4:1 generated reproducible currents at varying grid resolutions. Cross-shore flow was most affected by grid aspect ratio. Grid cells that were long and flat produced preferential undertow. The overall magnitude and location of the mean alongshore current, however, was not as sensitive to grid aspect ratio. The five cases tested produced similar mean alongshore current profiles in the cross-shore. The ideal vertical distribution of the radiation stress gradients was dependent on the situation considered. In a phase-averaged model, assumptions were made for the forcing terms that existed outside the modeling domain. Depth-uniform forcing produced 81

PAGE 95

82 alongshore current and shear instabilities similar to 2D models. Non-uniform vertical forcing, however, produced small scale structures that were not seen in 2D models. Vertical forcing effectively produced cross-shore currents that mixed the alongshore current horizontally: currents over the bar were pushed shoreward by wave breaking stresses, and currents at the shore were pulled seaward into the trough by the resulting undertow. Case 10 that had the pressure term uniformly distributed over the top three grid points, was the most effective means of mixing the alongshore current, and produced reasonable peak velocities [O (1 m/s)] in the trough. Case 12 produced the least satisfactory results. It implemented 1/6 th of F x as a surface shear stress. The effect on the location of the alongshore current was minimized, however, by the co-implementation of the free-slip boundary condition at the surface. The alongshore current maxima that resulted remained over the bar, and at the shoreline, with maximum velocities exceeding 1.5 m/s. It was learned that the methods of implementing the surface stress forcing were incompatible with the free-slip boundary condition. Alternate boundary conditions that will allow a surface stress can be implemented in future work. The use of a 3D, non-hydrostatic model produced strong vertical mixing in the cross-shore. Vortex structures similar to those from lab results were produced in the model. Turbulence is described as a 3D phenomenon and its effects on fluid dynamics are best modeled in a 3D system. Some previous two-dimensional models attempted to parameterize effects of 3D turbulence by assuming isotropic horizontal diffusion. Non-isotropic net cross-shore mixing will always be present as long as there is cross-shore circulation and the alongshore currents are not entirely depth-uniform. This always occurs in nature because of the bottom boundary layer and cross-shore

PAGE 96

83 non-uniformity of currents that are advected horizontally by vertically dependent forcing. The 3D model also captured another important coupling between horizontal and vertical structures of the flow. Small-scale turbulence generated in the vertical ( x and y ) could be stretched and tilted. These evolved into small scale vertical turbulent structures ( z ) that produced enhanced cross-shore diffusion. Complementary lines of investigation, such as the formulations in the quasi-3D implementation of SHORECIRC ( Zhao et al., 2003 ), had considered net effects of the undertow on depth-averaged currents. They did not observe current behavior qualitatively similar to results presented here. This was explained by the fact that the quasi-3D information was used primarily to estimate an isotropic horizontal diffusion coefficient. This was a good approximation if the alongshore current had been depth-uniform because the shoreward mass-flux in the top half of the water column would have carried equal amounts of alongshore momentum as the undertow had carried offshore. If the alongshore current was depth-dependent, however, this approximation was no longer adequate. Depth-dependence of the currents occurred because of several factors. These could have included depth-dependent momentum input, a thick boundary layer developed by bottom friction, or from ongoing depth-dependent cross-shore diffusion. The net effect of 3D cross-shore circulation produced non-isotropic, preferentially shoreward diffusion of the alongshore currents. Increasing the domain length influenced the alongshore velocity profiles. Peak currents were decreased, and the profile was laterally spread further across the cross-shore domain. Domain lengths on the order of 200 m produced large scale shear waves that horizontally diffused the mean alongshore current across the surfzone. This

PAGE 97

84 feature was most noticeable with depth-uniform forcing. The magnitude of the influence of the large scale motions was dependent on the vertical distribution of forcing. When a greater portion of the forcing was depth-uniform over the water depth, as in Case 14, the large scale instabilities were the main source of horizontal mixing of the alongshore current. In Case 15, however, the effect of large scale motions was not as apparent. Horizontal diffusion of the current was present due to large scale motions not captured in the small domain, but large structures were obscured by the small-scale instabilities produced by the vertical distribution of forcing. Including the effects of vertically dependent forcing and large scale shear instabilities produced mean alongshore currents that were weaker, and centered in the bar trough. The omission of either of these effects produced alternative alongshore current distributions. The omission of vertically dependent forcing led to weaker, more stable currents centered over the bar, whereas the omission of shear waves led to stronger currents centered in the bar trough. Future Work A great deal remains to be done on this line of work. Linear stability analysis of the three velocity profiles should show that the growth rates of instabilities of the vertical shear are more rapid than growth rates from the horizontal shear. Model improvement continues with the addition of mass flux to future simulations. Alternative vertical distributions of the alongshore and cross-shore forcing based on complimentary model and laboratory studies to include effects of breaking and nonlinear waves should also be examined. Simulations with different wave conditions (H o T p o ) should also be pursued and produce different alongshore current dynamics and increase understanding on the relative importance of including large scale shear instabilities and vertically distributed forcing.

PAGE 98

85 Larger domains and finer grid testing should also improve the statistics of the model results. Larger domains will more effectively include alongshore instabilities and reduce the wall effects on the offshore boundary. Finer grids, especially in the vertical, will better resolve turbulence closer to the boundaries, but may affect the stability of the pressure solver. Comparing model results to available field data from the DELILAH experiment could be used to help validate the model.

PAGE 99

LIST OF REFERENCES Allen, J.S., P.A. Newberger, and R.A. Holman, Nonlinear shear instabilities of alongshore currents on plane beaches, Journal of Fluid Mechanics, 310, 181-213, 1996. Bowen A.J., and R.A. Holman, Shear instabilities of the mean longshore-current, 1, theory, Journal Of Geophysical Research, 94,18023-18030, 1989. Church, J.C., E.B. Thornton, and J. Oltman-Shay, Mixing by shear instabilities of the longshore current, Coastal Engineering 1992, Chapter 230, 1992. Dally, W.R., and R.G. Dean, Suspended sediment transport and beach evolution, Journal of Waterway, Port, Coastal and Ocean Engineering, 110,15-33, American Society of Civil Engineers,1984. Deigaard, R., A note on the three-dimensional shear stress distribution in a surf zone, Coastal Engineering, 20, 157-171, 1993. Deigaard, R., and J. Fredsoe, Shear stress distribution in dissipative water waves, Coastal Engineering, 13, 357-378, 1989. Dodd, N., On the destabilization of a longshore current on a plane beach: Bottom shear stress, critical conditions, and onset of instability, Journal of Geophysical Research, 99, 811-824, 1994. Dodd, N., V. Iranzo, and A.J.H.M. Reniers, Shear instabilities of wave-drive alongshore currents, Reviews of Geophysics, 38, 437-463, 2000. Dodd, N., and E.B. Thornton, Growth and energetics of shear waves in the nearshore, Journal of Geophysical Research, 95,16,975-16,083, 1990. Falques, A., and V. Iranzo, Numerical simulation of vorticity waves in the nearshore, Journal of Geophysical Research, 99, 825-841, 1994. Faria, A.F.G., E.B. Thornton, T.C. Lippmann, and T.P. Stanton, Undertow over a barred beach. Journal of Geophysical Research, 105, 16,999-17,010, 2000. Fredsoe, J. and R. Deigaard, Mechanics of Coastal Sediment Transport, 369 pp., World Scientific Publishing, River Edge, NJ, 1992. 86

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87 Greenwood, B., and D.J. Sherman, Longshore current profiles and lateral mixing across the surf zone of a barred nearshore, Coastal Engineering, 10, 149-168, 1986. Haines, J.W., and A.H. Sallenger, Vertical structure of mean cross-shore currents across a barred surf zone, Journal of Geophysical Research, 99, 14,223-14,242, 1994. Hansen, J.B., and I.A. Svendsen, A theoretical and experimental study of undertow, Proceedings of the 19 th International Conference of Coastal Engineering, Houston, TX, ch.151, pp. 2246-2262, 1984. Holman, R.A., Nearshore processes, Reviews of Geophysics, Supplement, 1237-1247, 1995. Lele, S. K., Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics, 103, 16-42, 1992. Li, L., and R.A. Dalrymple, Instabilities of the undertow, Journal of Fluid Mechanics, 369, pp175-190, 1998. Lippmann, T.C., T.H.C. Herbers, and E.B. Thornton, Gravity and shear wave contributions to nearshore infragravity motions, Journal of Physical Oceanography, 29, 231-239, 1999. Lippmann, T.C., E.B. Thornton, and A.J.H.M. Reniers, Wave stress and longshore current on barred profiles, Coastal Dynamics '95: International Conference on Coastal Research in Terms of Large Scale Experiments, edited by W.R. Dally, 401-412, American Society of Civil Engineers, Reston, VA, 1995. Longuet-Higgins, M.S., Longshore currents generated by obliquely incident sea waves, 1. Journal of Geophysical Research, 75, 6778-6789, 1970. Longuet-Higgins, M.S., and R.W. Stewart, Radiation stresses in water waves; a physical discussion, with applications, Deep-Sea Research, 11, pp.529-562, 1964. Matsunaga, N., K. Takehara, and Y. Awaya, Coherent eddies induced by breakers on a sloping bed, Proceedings of the 21 st International Coastal Engineering Conference, edited by B.L. Edge, pp.234-245, American Society of Civil Engineers, Reston, VA, 1989. Matsunaga, N., K. Takehara, and Y. Awaya, The offshore vortex train, Journal of Fluid Mechanics, 276, 113-124, 1994. Mei, C.C., The Applied Dynamics of Ocean Surface Waves, 465 pp., World Scientific Publishing, River Edge, NJ, 1989. Mellor, G., The three-dimensional current and surface wave equations, Journal of Physical Oceanography, 33, 1978-1989, 2003.

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88 Noyes, T.J., R.T. Guza, S. Elgar, and T.H.C. Herbers, Field observations of shear waves in the surf zone, Journal of Geophysical Research, 109, CO1031, 2004. Oltman-Shay, J., P.A. Howd, and W.A. Birkemeier, Shear instabilities of the mean longshore current, 2. Field data, Journal of Geophysical Research, 94, 18031-18042, 1989. zkan-Haller, H.T., Nonlinear Evolution of Shear Instabilities of the Longshore Current, Ph.D. dissertation, University of Delaware, Newark, DE, pp. 268, 1997. zkan-Haller, H.T., and J.T. Kirby, Nonlinear evolution of shear instabilities of the longshore current: A comparison of observations and computations, Journal of Geophysical Research, 104, 25953-25984, 1999. Putrevu, U., J. Oltman-Shay, and I.A. Svendsen, Effect of alongshore nonuniformities on longshore current predictions, Journal of Geophysical Research, 100, 16119-16130, 1995. Putrevu, U., and I.A. Svendsen, A mixing mechanism in the nearshore region, Proceedings of the 23 rd International Conference on Coastal Engineering, edited by B.L. Edge, 2758-2771, American Society of Civil Engineering, Reston, VA, 1992. Putrevu, U., and I. A. Svendsen, Three-dimensional dispersion of momentum in wave-induced nearshore currents, European Journal of Mechanics, 409-427, 1999. Rayleigh, J.W.S., On the stability, or instability, of certain fluid motions, Proceedings of the London Mathematical Society, 11, 57-70, 1880. Reniers, A.J.H.M., and J. A. Battjes, A laboratory study of longshore currents over barred and nonbarred beaches, Coastal Engineering, 30, 1-22, 1997. Rivero, F.J., and A.S. Arcilla, On the vertical distribution of uw Coastal Engineering, 25, 137-152, 1995. Scotti, A., C. Meneveau, and M. Fatica, Dynamic Smagorinsky model on anisotropic grid, Physics of Fluids, 9, 1856-1858, 1997. Slinn, D. N., J.S. Allen, and R.A. Holman, Alongshore currents over variable beach topography, Journal of Geophysical Research, 105, 16,971-16,998, 2000. Slinn, D.N., J.S. Allen, P.A. Newberger, and R.A. Holman, Nonlinear shear instabilities of alongshore currents over barred beaches, Journal of Geophysical Research, 103, 18,357-18,379, 1998. Sobey, R.J., and R.J. Thieke, Mean flow circulation equations for shoaling and breaking waves, Journal of Engineering Mechanics, 115, 285-303, 1989.

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89 Stive, M., and H. Wind, Cross-shore mean flow in the surf zone, Coastal Engineering, 10, 325-340, 1986. Svendsen, I.A., Mass flux and undertow in a surf zone, Coastal Engineering, 8, 347-366, 1984a. Svendsen, I.A., Wave heights and set-up in a surf-zone, Coastal Engineering, 8, 303-329, 1984b. Svendsen, I.A., and J.B. Hansen, Cross-shore currents in surf-zone modelling. Coastal Engineering, 12, 23-42, 1988. Svendsen, I.A., and R.S. Lorenz, Velocities in combined undertow and longshore currents, Coastal Engineering, 13, 55-79, 1989. Svendsen, I.A., H.A. Schaffer, and J.B. Hansen, The interaction between undertow and the boundary layer flow on a beach, Journal of Geophysical Research, 92, pp. 11845-11856, 1987. Thornton, E.B. and R.T. Guza, Transformation of wave height distribution, Journal of Geophysical Research, 88, 5929-5938, 1983. Thornton, E.B. and R.T. Guza, Surf zone longshore currents and random waves: Field data and models, Journal of Physical Oceanography, 16, 1165-1178, 1986. Van Dongeren, A.R., and I.A. Svendsen, Nonlinear and quasi 3-D effects in leaky infragravity waves, Coastal Engineering, 41, pp.467-496, 2000. Winters, K.B., H.E. Seim, and T.D. Finnigan, Simulation of non-hydrostatic, density-stratified flow in irregular domains. International Journal for Numerical Methods in Fluids, 32, 263-284, 2000. Zhao, Q., I.A. Svendsen, and K. Haas, Three-dimensional effects in shear waves, Journal of Geophysical Research, 108, 3270, 2003.

PAGE 103

BIOGRAPHICAL SKETCH Kristen Splinter was born in Kingston, Ontario, Canada, on February 12, 1979. She grew up under the influence of hard-working parents. Her environmentally conscious mother taught her to appreciate the delicate balance of nature. Having spent most of her time in close proximity to water, she grew a desire to protect the animals that rely on it for life. After graduating from high school in 1998, she pursued a degree in Civil-Environmental Engineering at Queens University at Kingston (located in Kingston, Ontario, Canada). Throughout this time, she continued to participate in environmental causes, and focused her work on water quality and coastal resources. The Natural Science and Engineering Research Council of Canada (NSERC) funded her undergraduate research, studying water resource management using wetland technology. In 2002, she attained the degree of Bachelor of Science in Engineering with First Class Honors. Upon completion of her undergraduate degree, she wanted to continue her research career, and gain a greater understanding of coastal processes. She chose to attend the University of Florida, for its program and sunny location. Here she pursued the degree of Master of Science in Coastal and Oceanographic Engineering, under the guidance of her advisor, Dr. Donald N. Slinn. Having not tired of learning about coastal issues, and wanting a deeper scientific background in her related fields of study, she plans to attain her Ph.D. in Oceanography, under the guidance of Drs. Robert Holman and Tuba zkan-Haller at Oregon State University starting in September of 2004. 90


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EFFECTS OF THREE-DIMENSIONAL FORCING ON
ALONGSHORE CURRENTS: A COMPARATIVE STUDY













By

KRISTEN D. SPLINTER


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


2004

































Copyright 2004

by

Kristen D. Splinter















This thesis is dedicated to my parents, for their support and guidance throughout my
education.















ACKNOWLEDGMENTS

I would like to thank my parents for their continual support throughout my

education; pushing me to excel, and letting me find my way. I thank my sister, Karen,

and my parents, for always being there when I needed someone to talk to.

I would like to extend my appreciation to my advisor, Dr. Donald N. Slinn, for his

patience and guidance throughout the process. I thank Drs. Robert Thieke (University of

Florida), and Todd Holland and Joe Calantoni (Navel Research Laboratory, Stennis

Space Center, MS) for giving me the opportunity to help with field research. It brought a

new level of understanding to my work. I also thank Dr. Kraig Winters (Scripps Institute

of Oceanography, San Diego, CA) for the use of his computer programs and for his

continued support throughout. I would also like to thank Drs. Robert Dean and Robert

Thieke for serving on my supervisory committee.

I thank all of the students and faculty who have helped me along the way; Brian

Barr, Jodi Eshleman, Jamie MacMahan, Jon Miller, Robert Weaver, and Bret Webb.

Their guidance and friendship throughout this process, along with their patience with my

computer problems and various dilemmas have been greatly appreciated. Finally, I thank

all of my housemates and friends in Gainesville who made me feel welcome and part of a

family so far away from home.















TABLE OF CONTENTS
Page

A C K N O W L E D G M EN T S ......... ............................................................................... iv

L IST O F T A B L E S .... ................................ .. ........................... .............. vii

LIST OF FIGURES ............. ............ ...... ............ ......... viii

ABSTRACT ................................... ...... .............. xii

CHAPTER

1 IN TR O D U C TIO N ........................ .... ........................ ........ ..... ................

M ethods of Cross-Shore M ixing......................................... ......................... 2
Shear Instabilities........................................... ......... ........ .. .. ........ .. 2
Cross-Shore Circulation: Mass Flux and Undertow........................................5
Surface R oilers ................................................... .... ...............
Alongshore Pressure Gradients....................................................................... 9
Vertical Distribution of Wave Stresses....................... ..... ..............10
Chapter Contents ............................................... .. ..... ................. 17

2 M ODEL COM PARISON S........................................................ ............... 19

Two-Dimensional Alongshore Current Models for Barred Beaches...................19
Quasi-3D M odels: SH ORECIRC ........................................ ....... ............... 23
Summary ................... .......................... ... ...... 23

3 PROBLEM AND MODEL SETUP ........................................... ............... 27

M odel Dom ain and Solution M ethod ....................................... ............... 27
B oundary C conditions ............................................... ............................ 28
Initial C conditions ......................... ...................... .. .. ...... ........... 29
G governing E quations ......................... .. .................... .. ...... ........... 29
Forcing ..................................... ............... 32
Two-Dimensional Sub-model ....... ................. ...............32
Three-D im ensional Forcing M odel....................................... ............... 33
Scope of the M odel ......... ... ..................... ........ .................. 39







v










4 S IM U L A T IO N S .......................................................................... ................ .. 4 2

The Effect of Grid Resolution............................................... 49
Alongshore-Averaged Alongshore Currents..................................................50
Alongshore and Depth-Averaged Alongshore Currents..............................51
Alongshore-Averaged Cross-shore Currents................................................52
Vorticity ............... ......... ....... ........ ... .... ............ 55
Conclusions of Grid Testing .. .. ..................... .................56
The Effect of Vertical Distribution.............................. ...............56
Cross-shore Flow ............. .. .................... .. ...................... 57
A longshore C urrents ............................................... ............................. 60
Vorticity ............... ......... ......... ......... 70
Effect of Domain Length ........... .............. .......................... 76

5 SUM M ARY AND DISCUSSION ..................................... ........................ 81

F findings and R results ......... ... ...... .... .. ...... ...................................... 8 1
Future Work ........... .. ............................ 84

6 LIST OF REFERENCES ......... .................................... .... ...................... 86

7 BIOGRAPHICAL SKETCH ........................................ ........................... 90
















LIST OF TABLES

Table page

1 Com prison of 2D m odels................ ......... .................................... .. ............. 25

2 Q uasi-3D m odel sum m ary ............................................. ............................. 26

3 Sum m ary of sim ulations............ ................... ........................ ............... 42















LIST OF FIGURES


Figure page

1 Longuet-Higgins and Stewart's description of the vertical distribution of the
total radiation stresses ......... .... .... ...... .... .... .......... .... .. ...... ... 12

2 Dean and Daily's description of the components of shear stress distribution..........12

3 Svendsen's description of the components of shear stress distribution ................. 13

4 Deigaard and Fredsoe's description of the components of the shear stress
distribution .........................................................................14

5 Sobey and Thieke's description of apparent radiation stress (sxx(z))
distribution .........................................................................15

6 Rivero and Arcilla's description of the shear stress distribution for sloping
top ography ..................................................... ................. 15

7 Mellor's description of the distribution of radiation stresses............................... 17

8 Comparison of alongshore current profiles from 2D circulation for varying
values of Cf over tim e........ ....... ............... ..... .... .. ............ 21

9 Vorticity contours from 2D model over time. .................................. .................22

10 3D outlay of physical grid used in the short domain simulations...........................29

11 Cross-shore distribution of nondimensionalized wave radiation stress
forcing coupled to topography. .......................................................................... 33

12 Case 8: Sample vertical distribution of nondimensionalized forcing ...................35

13 Case 9: Sample vertical distribution of nondimensionalized forcing ...................36

14 Case 10: Sample vertical distribution of nondimensionalized forcing ..................36

15 Case 11: Sample vertical distribution of nondimensionalized forcing ..................37

16 Case 12: Sample vertical distribution of nondimensionalized forcing ...................38

17 Case 13: Sample vertical distribution of nondimensionalized forcing ...................39









18 Tim e series for u-velocity. ...... ........................... ........................................ 44

19 Tim e series for w -velocity. .............................................. ............................. 44

20 Tim e series for v-velocity. ...... ........................... ........................................ 45

21 Case 1 and 2: Vertical distribution of forcing used in the simulations. Case
1 (A ) and 2 (B)..................................................................... .... .. ....... 46

22 Comparison of depth and alongshore averaged currents for the initial
sim ulations. .........................................................................47

23 A typical vertical profile of cross-shore velocity vectors produced by depth
linear forcing ........................................................................48

24 Surface vorticity contours for depth-uniform forcing as a function of time.............48

25 Surface vorticity contours for depth-dependent forcing as a function of time. ........49

26 Cases 3-7: Contours of the alongshore-averaged alongshore velocity (v) for
variable grid sizes in the x-z plane ..................................... ........ ............... 50

27 Depth and alongshore-averaged alongshore currents at t = 5 min............................51

28 Depth and alongshore-averaged alongshore currents at t = 25 min..........................52

29 Case 3: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors..................................53

30 Case 4: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors..................................53

31 Case 5: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors..................................54

32 Case 6: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors..................................54

33 Case 7: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors..................................55

34 Comparison of the depth-averaged term as a function of cross-shore
lo c atio n ............................. ........................................................ ............... 5 5

35 Cases 8-10: Comparison of alongshore and time-averaged u-velocity as a
function of depth and location in the cross-shore domain................................58

36 Cases 11-13: Comparison of alongshore and time-averaged u-velocity as a
function of depth and location in the cross-shore domain................................58









37 Vector plot of alongshore-averaged u-w velocities for Case 7.............................59

39 Alongshore-averaged u-velocity contours for Case 7, t = 10.86 min.......................60

40 Case 8: Cross-shore distribution of the alongshore and depth-averaged
alongshore current over a 60 min time period ......................................... 61

41 Case 9: Cross-shore distribution of the alongshore and depth-averaged
alongshore current over a 60 min time period. ............. .................................... 62

42 Case 10: Cross-shore distribution of the alongshore and depth-averaged
alongshore current over a 60 min time period ......................................... 62

43 Case 11: Cross-shore distribution of the alongshore and depth-averaged
alongshore current over a 60 min time period. ............. .................................... 63

44 Case 12: Cross-shore distribution of the alongshore and depth-averaged
alongshore current over a 60 min time period ......................................... 63

45 Cross-shore position as a function of time of the peak alongshore current for
the 5 vertical distsributions tested.......... .. .............. .................... ............... 64

46 Peak velocity of the alongshore current as a function of time ..............................64

47 Case 8: Vertical cross-section contour plots of the alongshore-averaged
alongshore current (v) ....... ....... .... ... ... .... ...... ..... .. ............... .. 65

48. Case 9: Vertical cross-section contour plots of the alongshore-averaged
alongshore current (v) ...... ...................................................... .. ...... ...... 66

49 Case 10: Vertical cross-section contour plots of the alongshore-averaged
alongshore current (v). ................................... ............. ....... .....67

50 Case 11: Vertical cross-section contour plots of the alongshore-averaged
alongshore current (v). ................................... ............. ....... .....68

51 Case 12: Vertical cross-section contour plots of the alongshore-averaged
alongshore current (v). ................................... ............. ....... .....69

52 Comparison of surface vorticity component (coz) and alongshore surface
current (v) contours in the cross-shore (x) and alongshore (y) domain .............70

53 Sample comparison of spatially averaged, original vorticity, and
time-averaged contours at vertical grid point nz-1 .......................................... 72

54 Depth-averaged U(xo,yo,t)-velocity energy density spectrum for Case 8................74

55 Depth-averaged V(xo,yo,t)-velocity energy density spectrum for Case 8................74









56 Time sample of velocity fluctuations (black) and the low-pass filtered (red)
depth-averaged u-velocity at 0.03 Hz taken from energy spectrum data ..........75

57 Time sample of raw data (black) and the low-pass filtered (red)
depth-averaged v-velocity at 0.046 Hz ....................... ... ... .............75

58 Influence of domain length on alongshore currents for depth-uniform
forcing: Alongshore and depth-averaged alongshore current profiles as a
function of cross-shore position and tim e ................. ................. ............... 77

59 Influence of domain length on alongshore currents for Case 10 forcing:
Alongshore and depth-averaged alongshore current profiles as a function
of cross-shore position and tim e. ............................................. ............... 78

60 Influence of domain length on alongshore currents for Case 11 forcing:
Alongshore and depth-averaged alongshore current profiles as a function
of cross-shore position and tim e. ............................................. ............... 78

61 Case 12: Contours of the depth-averaged vorticity as a function of time.................79

62 Case 15: Contours of the depth-averaged vorticity as a function of time................79

63 Case 16: Contours of the depth-averaged vorticity as a function of time................80













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

EFFECTS OF THREE-DIMENSIONAL FORCING ON
ALONGSHORE CURRENTS: A COMPARATIVE STUDY

By

Kristen D. Splinter

August 2004

Chair: Donald N. Slinn
Major Department: Civil and Coastal Engineering

Our study introduced a new 3D, time-dependent, non-hydrostatic, Large Eddy

Simulation (LES) nearshore circulation model. It was wave-phase-averaged, in a

curvilinear, bottom conforming, a-coordinate system; with a rigid lid that was capable of

examining depth-dependent, low-frequency, and nearshore current response to breaking

waves in the surf zone. A principal advantage of our model was that it included

dynamics of the undertow and vertical mixing that led to enhanced cross-shore mixing.

Uncertainties associated with modeling bottom friction were reduced because bottom

stress was modeled with the no-slip condition by using a high-resolution vertically

clustered grid with O (1-10 cm) scales near the seabed. The LES turbulence closure

scheme produced reasonable time-averaged alongshore and cross-shore vertical velocity

profiles.

The model was formulated for alongshore-uniform bathymetry, and included a

shore-parallel sand bar. Forcing conditions for the alongshore currents in the model were

coupled to beach topography and specified surface wave conditions, using the









Thornton-Guza (1983,1986) wave breaking sub-model. The model included wave set-up

through an induced cross-shore pressure gradient. A variety of vertical distributions of

the wave-induced radiation stresses were tested. Results indicated that vertical profiles

for undertow, mean alongshore current profiles, and dynamics of alongshore-current

shear instabilities depended on the particular approximation chosen for the forcing.

In contrast to simulations using a depth-averaged 2D model, the initial instabilities

developed on the vertical shear in the water column and produced relatively short

wavelength disturbances that caused the flow to transition to turbulence before

organizing into larger scale flow features. The strength, stability, and location of the

alongshore current were dependent on the vertical distribution of forcing and domain

length. Depth-varying forcing suppressed the development of the large-scale shear

instabilities and influenced the time-dependent effects of the unsteady flows. Short

domains [O (20 m)] were dominated by the effects of the vertical distribution of forcing

that produced stronger mean alongshore currents with peaks located shoreward of the

nearshore bar. Long domains [O (200 m)], however, included the effects of shear waves,

and produced mean alongshore currents with broader cross-shore profiles, and less

distinct peaks located further in the bar trough.

Including the effects of anisotropic mixing, vertically dependent forcing, and

large-scale shear instabilities produced mean alongshore currents that were of reasonable

magnitude centered in the bar trough of a barred beach system.














CHAPTER 1
INTRODUCTION

The circulation patterns of the surfzone are important because they affect the

alongshore and cross-shore movement of sediment, and the strength of the overall

currents. Two distinct types of idealized nearshore circulation models are commonly

used to investigate aspects of surfzone dynamics. Two-dimensional (2D) cross-shore

(x-z) circulation models are often used to study sediment transport and undertow. These

generally assume alongshore uniformity, and are sensitive to the vertical distribution of

wave shear stresses. The second class of models simulate 2D alongshore (x-y) currents.

These generally consider variations of the current and topography in both the alongshore

and cross-shore directions, but are approximated by a depth-averaged current. Some

alongshore current models have proven useful for predicting the current structure on

plane beaches for certain wave conditions, but are less useful for a barred beach system.

According to wave radiation stress theory (Longuet-Higgins, 1970), alongshore

currents are produced near the location of wave breaking. In a barred beach system, this

commonly occurred near the bar crest, and at the shoreline. Some field observations,

however, showed that a mean current maximum developed in the trough region, a

location where reduced breaking occurred (Church et al., 1992). Our purpose was to gain

a better understanding of the physics that influenced alongshore currents.

Most contemporary nearshore circulation numerical models (Allen et al., 1996;

Slinn et al., 1998, 2000; Ozkan-Haller and Kirby, 1999; Van Dongeren and Svendsen,

2000) used various forms of the 2D depth-integrated flow approximation. Although









valid, because the water was shallow, O (1 m), compared to its horizontal dimensions,

O (100 m), this approximation may have obscured several important three-dimensional

(3D) features. Two such features were:

* A depth-dependent vertical distribution of the cross-shore radiation stress could
have produced shoreward momentum flux in the upper portion of the water
column, and a resulting return flow in the lower portion of the water column,
known as undertow.

* Under some conditions, breaking waves could have effectively mixed the
momentum across the water depth at relatively high frequency.

Together these processes might have led to enhanced cross-shore diffusion rates that

could have decreased the mean cross-shore shear of the alongshore current, and shifted

the peak current toward the bar trough.

Ongoing research to explain the shoreward shift of the alongshore current on a

barred beach has focused on four main topics: shear waves, cross-shore circulation, wave

rollers, and alongshore pressure gradients. These topics and aspects of the vertical

distribution of the wave radiation stresses are reviewed next.

Methods of Cross-Shore Mixing

Shear Instabilities

Strong alongshore currents on the order of 1 m/s have been generated in the

surfzone by the alongshore momentum flux caused by obliquely incident surface gravity

waves. The transfer of momentum from the wave field to the mean current was

characterized by the cross-shore gradient of the wave radiation stress (e.g., dSxy/dx)

(Longuet-Higgins, 1970). Greenwood and Sherman (1986) discussed lateral mixing of

alongshore current profiles over barred beaches, and suggested that for a plane beach

solution, the presence of bars would enhance lateral mixing by increasing the cross-shore

velocity gradients. Observations at Duck, North Carolina by Oltman-Shay et al. (1989)









showed the presence of alongshore propagating velocity perturbations associated with the

presence of shear in the alongshore currents. Ozkan-Haller and Kirby (1999) and Slinn et

al. (1998, 2000) showed that the nonlinear evolution of shear instabilities provided

significant horizontal mixing of alongshore momentum that affected the cross-shore

distribution of the alongshore current. The stability and structure of shear instabilities

depended on several factors. These included magnitude and cross-shore profile of the

alongshore current, beach topography, presence of nonzero gradients of the horizontal

Reynolds stresses (Dodd and Thornton, 1990), and bed shear stress (Dodd et al., 2000).

Shear instabilities were distinguished from gravity waves because their periods

[O (102-103 s)], were too long for their associated wave lengths [O (102 m)] (Oltman-Shay

et al., 1989) based on the dispersion relationship. (E.g., a 100 m long wave in 5 m of

water should have had a period of approximately 14.5 s). Shear waves were most

pronounced in areas where strong alongshore O (1 m/s) currents were present, and had

total RMS shear wave fluctuations varying between 10 and 40% of the maximum

alongshore current (Noyes et al., 2004). They were generated as the alongshore current

grew in strength and became unstable, which caused the alongshore current to meander in

a sinusoidal pattern, and first appeared as an alongshore progressive wave feature

(Bowen and Holman, 1989). They were often strongest, and most prevalent on barred

beach profiles where wave breaking over the bar generated a strong shear in the

alongshore current, and enhanced the growth rate of the shear waves (Dodd et al., 2000).

Field observations by Noyes et al. (2004) also suggested that these instabilities

were generated in the region where the greatest shear occurred, just seaward of the

location of the maximum alongshore velocity. The instabilities were characterized by an









exchange of energy from the mean current to a perturbation velocity field. Cross-shore

mixing of alongshore momentum through the Reynolds stresses usually occurred when

the cross-shore gradient of the depth-averaged alongshore velocity went through an

inflection point (Dodd and Thornton (1990) and Rayleigh (1880)).

Research to explain the field observations of shear instabilities has been based on

linear theory, weakly nonlinear theory, and nonlinear analysis of shear wave growth.

Bowen and Holman (1989) used linear stability analysis of depth-integrated currents to

show that these disturbances could have been produced by instabilities developed by the

cross-shore shear of the alongshore current. Allen et al. (1996) used a nonlinear model to

show that the finite amplitude behavior of unstable flows on plane beaches could be

described by the ratio of frictional to advective terms (AQ). As the friction factor was

decreased, the flows changed from weakly unstable, to chaotic. They also examined the

effect of different alongshore domain lengths, and concluded that shear wave instabilities

initially grew at the wave length corresponding to the fastest-growing linearly unstable

mode (2i/ ko ). As domain length increased, however, the initial disturbances of the flow

merged, and their length scales evolved into longer wave length propagating disturbances

when AQ was moderate to large (Allen et al., 1996).

Slinn et al. (1998) extended the work of Allen et al. (1996) by applying coupled

forcing to a barred beach profile. They found that short wave-length disturbances

continued to dominate the velocity spectrum even for large values of AQ. They presented

four different flow regimes that were attained by altering the bottom friction. As bottom

friction was decreased, instabilities of the current changed from stable shear waves, to

fluctuating vortex patches, to shedding vortex pairs, and finally to turbulent shear flows.









Finite amplitude structures generated were independent of domain length for a barred

beach system, and remained at length scales predicted by linear theory: on the order of

100 to 200 m. The strength of these fluctuations increased as bottom friction decreased.

This agreed well with Dodd (1994) and Falques and Iranzo (1994) who found that

increasing the bed shear stress and eddy viscosity reduced the current velocity and

dampened the shear instabilities.

Once the alongshore instabilities grew to finite amplitude, they continued to evolve

and shed vortices from the mean current. Bowen and Holman (1989) suggested that this

might be an important feature of cross-shore mixing in this environment, because shear

instabilities might provide rates of turbulent diffusion up to ten times greater than those

produced by breaking gravity waves.

Cross-Shore Circulation: Mass Flux and Undertow

Cross-shore circulation was usually excluded from 2D depth-averaged alongshore

current models because of the approximation that alongshore and cross-shore currents

were approximately vertically uniform. Although this approximation was often sufficient

for large-scale nearshore models, cross-shore circulation existed and could potentially

have influenced the alongshore current. Instead, mass flux and undertow were studied in

2D cross-shore circulation models. Cross-shore circulation was driven by the onshore

component of the wave radiation stress gradient, dSxx/dx. The undertow was largely a

result of the local variation between the depth-dependent radiation stress and the

depth-uniform horizontal pressure gradient due to set-up. These were in equilibrium with

each other over the water depth (Svendsen, 1984a), but unbalanced throughout the water

column. As the waves traveled through the surfzone and breaking occurred, their wave

heights decreased, and resulted in a decreased shoreward mass flux. This produced a









downwards directed flow (Svendsen, 1984a) from the free surface, and also encouraged

cross-shore circulation. The mass flux above the trough level from the shoaling waves

pushed the alongshore current near the surface shoreward, while the alongshore current

near the bottom was pulled seaward by the increased return flow in the undertow.

Svendsen and Lorenz (1989) noted that in the surfzone, the wave-averaged cross-shore

current always had a shoreward velocity component near the free surface, and an off-

shore directed component near the bottom.

Results from cross-shore circulation models were sensitive to boundary conditions

and integral constraints. Cross-shore models tended to invoke:

Depth-integrated mass balance (net zero mass flux).

A no-slip condition on the bottom boundary (Svendsen and Hansen, 1988).

The choice of the turbulent eddy-viscosity term in these models also contributed to

the predicted undertow profile. Haines and Sallenger (1994) found that the

eddy-viscosity varied significantly over the cross-shore in the surfzone, and also varied in

the vertical near the top and bottom boundaries. Faria et al. (2000) disagreed, and found

that a depth-dependent eddy viscosity did not reduce the differences between model

results and field data. Using a constant eddy-viscosity inside and outside the boundary

layer, with the no-slip bottom boundary condition, however, yielded onshore velocities in

the bottom boundary layer. This result, however, contradicted field data from Hansen

and Svendsen (1984) and Haines and Sallenger (1994) who observed a strong undertow

jet over the bar during their field experiments. Two modeling approaches have been

developed to address this problem. Stive and Wind (1996) suggested replacing the

no-slip condition at the bottom with a specified shear stress at the wave trough level.









Svendsen et al. (1987) proceeded with the no-slip condition, but applied two eddy

viscosity terms: with a smaller one in the boundary layer than in the interior of the fluid.

Several lab experiments have been conducted to study the undertow profile.

Matsunaga et al. (1988, 1994) found instabilities in the off-shore directed flow seaward

of the breaker line. These instabilities, which they termed a vortex train, consisted of a

solitary layer of large eddies rotating about a horizontal, alongshore axis. Larger scale

lab experiments conducted by Li and Dalrymple (1998) showed two layers of oppositely

rotating vortex trains. These vortex trains caused vertical mixing of the cross-shore

current and could potentially affect the alongshore current as well.

Faria et al. (2000) presented model results for the vertical structure of the undertow.

The undertow was not uniform with depth, over the cross-shore, or steady in time.

Undertow flows were approximately parabolic in the vertical, and varied in magnitude

with water depth. In the inner trough zone, they found the return flow was weak with

very little vertical structure. Maximum offshore flows occurred at the shoreward face,

and on top of the bar, which coincided with areas of intense wave breaking. Seaward of

the bar, they found that the undertow was nearly depth uniform. The variation of the

undertow across the surfzone could induce cross-shore mixing of the alongshore current.

The changing strength of the undertow across a barred beach profile was explained

by channel flow theory and mass conservation. As the cross-sectional area decreased in

shallow water, the flow velocity must have increased to conserve volumetric flow rate.

The same concept held on the offshore side of the bar where undertow currents

decreased. Putrevu and Svendsen (1992) suggested that the vertical profile of the

undertow could have caused considerable shearing of the alongshore current in the









cross-shore direction. Including effects of variable cross-shore circulation could

influence the position of the alongshore current.

Surface Rollers

The standard wave radiation stress theory assumed an instantaneous transfer of

momentum from breaking waves to the mean water column. The wave roller was

introduced to represent the intermediate process that gradually released energy and

momentum from the breaking wave bore into the water column over some distance. The

wave roller effectively redistributed the incident wave stress over the surfzone

(Lippmann et al., 1995) and was most noticeable on a barred beach (Svendsen, 1984b),

where multiple breakpoints were common.

The transfer of momentum was described as a two-step process. The momentum

from a breaking wave was supplied to a turbulent roller that in turn transferred

momentum through a surface shear stress (Deigaard, 1993) to the water column at a finite

rate. The roller was an approximation to the front face of a breaking wave, where

turbulent fluid was conveyed at the wave celerity (Lippmann et al., 1995). In their

model, wave asymmetry, and phase speed determined the spatial distribution and

intensity of the roller energy dissipation. Waves with smaller asymmetry, or increased

local water depth, resulted in a shoreward shift of the break point, and an increased

distance that the roller traveled. Together these effects shifted the current towards the

trough region. The spatial lag of momentum input into the mean currents by the roller

was an efficient means for offsetting the current maximum from over the bar into the

trough.

Although rollers were able to shift the momentum input shoreward of the bar, their

effect alone did not accurately predict the cross-shore distribution of alongshore currents









(Lippmann et al., 1995). Reniers and Battjes (1997) presented laboratory results of

alongshore currents over barred beaches that showed:

* The roller contribution helped to describe the cross-shore distribution of the
observed shear stress.

* Lateral mixing alone had no significant influence on the location of the maximum
velocity.

The concept that emerged was that the velocity profile was shifted shoreward by the

roller, and spread horizontally by the effect of lateral mixing.

Alongshore Pressure Gradients

In the same paper, Reniers and Battjes (1997) also suggested that alongshore

pressure gradients were important for predicting alongshore current location. Laboratory

experiments of wave driven currents over a barred beach in the absence of alongshore

pressure gradients produced alongshore currents near the regions of breaking; over the

bar, and at the shoreline. They hypothesized that alongshore currents located in the

trough were explained by alongshore pressure gradients in the field due to uneven

topography in the alongshore direction.

Putrevu et al. (1995) conducted model studies that examined the importance of

including the effects of alongshore non-uniformities on alongshore currents. Variable

beach topography in the alongshore direction affected the breaker height and location of

wave breaking in the cross-shore. Including the forcing from alongshore pressure

gradients had a significant impact on the alongshore current location. They found that

this component could be almost equal in magnitude to the forcing due to the alongshore

component of the radiation stress term in some cases. The effect of the alongshore









pressure gradient was dependent on the angle of wave incidence: it increased as waves

approached closer to shore normal.

Slinn et al. (2000) coupled topography and radiation stress gradients over

alongshore variable topography in numerical model experiments to test this hypothesis.

Their results showed that currents tended to follow contours of constant depth. Variable

points of wave breaking caused momentum to be inputted at different cross-shore

locations depending on topography. The alongshore pressure gradients and the radiation

stress terms counteracted each other to some degree. They concluded that alongshore

pressure gradients alone were insufficient to produce peak currents in the trough. They

concluded, however, that topographic variability in the alongshore had a significant

influence on nearshore circulation.

Vertical Distribution of Wave Stresses

In depth-integrated, 2D, circulation models on a uniform coast with straight and

parallel contours, alongshore currents were forced by the cross-shore gradients of the

depth integrated wave radiation stresses, dSxy/dx. With the advancement to quasi-3D and

3D modeling techniques, the distribution of the wave shear stresses within the water

column became important. Earlier work describing the vertical distribution of wave

stresses was simplified to examine shore normal waves, where dSx/dx = 0. The only

dynamic wave force present in these models was dSxx/dx, which drove set-up and

cross-shore circulation.

Longuet-Higgins and Stewart (1964) defined the relationship between the

depth-dependent radiations stress, sxx(z) and the depth-integrated value that was used in











2D circulation models as S,= f s, (z)dz. The wave radiation stress, Sxx, was divided
h

into three parts, so S = S1 + S_ + S The breakdown of each component was:



S f pu2dz= pu2dz (1)
-h -h
0 0 0
S = (p- po)dz (p -p )dz (-pw)d (2)
-h h h

S = pdz = pg2 (3)
0
0
In shallow water, S(1 + S' = pu2dz because u2 >> w2. This term was due to the
-h

momentum of the wave orbital motion, and was equal to the total energy density of the

waves, E. It was distributed evenly over the water column and accounted for two-thirds

of the total radiation stress. S.) was the pressure term due to the waves, and was equal

to 1/2 E. It accounted for one-third of the total force, and was centered about the mean

water level (MWL) (Figure 1). Sxy lacked the pressure terms due to the waves, and was


given by the relationship S = puvdz.
-h

Deigaard and Fredsoe (1989) compared their findings to the works of Dally and

Dean (1984), and Svendsen (1984a), for the vertical variation of the wave induced shear

stress that included components of radiation stress, wave roller effects, and set-up. Their

analyses were based on linear shallow water wave theory over a horizontal bottom. Dally

and Dean (1984) modeled the vertical distribution of radiation stress following the

approach of Longuet-Higgins and Stewart (1964). The resulting shear stress distribution

was split up into the three components: wave motion, surface roller, and set-up. Dally









and Dean (1984) did not include the effect of a surface roller. They assumed the set-up

balanced the radiation stress gradient, and the result was a linear distribution of the shear

stress across the water column, and a depth-uniform distribution of the radiation stress

below the trough. The shear stress had a maximum at the MWL and was zero at the bed

(Figure 2).


Figure 1. Longuet-Higgins and Stewart's description of the vertical distribution of the
total radiation stresses. A) sxx(z). B) sxy(z)

A B C D


Figure 2. Dean and Dally's description of the components of shear stress distribution.
S1 dH2
y/D is the normalized water depth. z* = -r/ pg A) Wave motion.
16 dx)
B) Surface roller. C) Setup. D) Resulting shear stress distribution.

Svendsen's (1984a) results were similar to Dally and Dean (1984). He used the

same linear shallow water wave theory, but included the effect of a surface roller. The









surface roller added an additional component to the pressure gradient that produced a

depth-uniform addition to the shear stress distribution (Figure 3B). The wave setup

component was also adjusted so that the total shear stress was still linearly distributed

with depth (Figure 3C). In this formulation, the total wave shear stress term had a

maximum at the MWL about twice that of Dally and Dean (1984) and zero at the bed

(Figure 3D). This resulted in a stronger net cross-shore shear force than Dally and Dean

(1984) predicted.

A B C D

O8 O8 O8 iO







Figure 3. Svendsen's description of the components of shear stress distribution.
r* = -/ pg-j y/D is the normalized water depth. A) Wave
1 6 dx )
motion. B) Surface roller. C) Setup. D) Resulting shear stress distribution.

Dally and Dean (1984) and Svendsen (1984a) both used a control volume approach

to describe the distribution of shear stress and omitted the transfer of momentum (UW )

through the bottom of the control volume. In the presence of irregular waves, such as in

the surf-zone, Deigaard and Fredsoe (1989) showed that this term was significant to the

overall magnitude and distribution of the shear stress. They found that the UW term

doubled the surface shear stress. The magnitude of the surface roller effect was also

slightly higher in their calculations compared to Svendsen (1984a). As a result of the

larger shear stresses due to wave motion and the surface roller, the balancing shear stress

due to set-up was also larger. The net distribution (Figure 4) still increased linearly with










height: with zero shear stress at the bed, and a maximum at the MWL almost three times

that of Dally and Dean (1984) and one and a half times that of Svendsen (1984a). The

reader is referred to Deigaard and Fredsoe (1989) for a more detailed comparison of the

three theories presented.

A B C D





02 02 02 02
i 1 2 3 4 4 2 1 o1 '2 o4 3' 2 1 o1 2 3 3 2 1 04


Figure 4. Deigaard and Fredsoe's description of the components of the shear stress

distribution. y/D is the normalized water depth. r* = -/ pg- dH
16 dx )
A) Wave motion. B) Surface roller. C) Setup. D) Resulting shear stress
distribution.

Sobey and Thieke (1989) discussed the importance of depth-dependent forcing in

the nearshore region. They clarified that although depth integration of the forcing was

valid because it produced a net zero mass flux condition in the cross-shore direction, it

did not include the cross-shore circulation produced by the wave induced mass-flux and

-2 -2
undertow. Their wave action term ((pu pw )) in sxx(z), where u, w were the

time-averaged, u-, w-velocity fluctuations, was vertically uniform in agreement with

Dally and Dean (1984). Their pressure gradient term, however, linearly increased from

the trough to the MWL, where it reached a maximum. A complementary linear decrease

in the apparent stress from MWL to the wave crest was also predicted (Figure 5).

Rivero and Arcilla (1995) extended the work of Deigaard and Fredsoe (1989) and

demonstrated that a sloping bottom bed had an important effect on the wave shear stress.









Under dissipative wave conditions, the resulting distribution had a nonzero value at the

(E
bed equal to tan / that linearly increased to a maximum of
ph

EtF 1 iE) E Fh
Stan8- [ c at the MWL (Figure 6).
ph 2 _x p p Ox

crest -r


hrougJh


Figure 5. Sobey and Thieke's description of apparent radiation stress (sxx(z)) distribution.


Figure 6. Rivero and Arcilla's description of the shear stress distribution for sloping
topography.

More recently, the work of Mellor (2003) addressed the vertical distribution of both

the alongshore and cross-shore wave radiation stresses for surface waves (Figure 7). His

equations for radiation stress were:









Fkk 8
S = kDE I FF +k (F Fc (4)

where:
S sinh kD(1 + )) cosh kD(1 +
sinh kD sinh kD
S sinh kD(1+) F cosh kD(1+ )
coshkD coshkD

He described E = D was the depth, c5, was the Kronecker delta function, and


; was the transformed vertical coordinate such that = -1 when z = -h and = 0 at the

surface.

His resulting vertical distribution for sxx(z) differed from previous theory. As

seen in Figure 7A, sxx(z) linearly increased with depth. This was due to the Fss term that

linearly decreased with depth and was subtracted from the other radiation stress

components. When vertically integrated over the water depth, his equations were off by a

factor of rho (1000 kg/m3) in comparison to the depth averaged equations given by Mei

(1989) Eq. 3-10. Vertical integration of his equations and the determination of the

vertical distribution of each of his sub-terms (Fss, Fso, Fes, and Fee) were calculated

separately. This was done to ensure that the resulting vertical distribution presented

above was not an error in inputting the equations into the model. The vertical

distributions of the sub-terms used in the model were similar to those presented in his

paper. His Sxy(z) term remained roughly depth-linear because the 2nd term (with the

Kronecker delta) in his equation only existed for normal (sxx(z), syy(z)) forces. This

representation of radiation stresses remains questionable for its purpose and validity in

the nearshore. Putrevu and Svendsen (1999) and Zhao et al. (2003) also discussed effects









of the local contribution of the radiation stress on lateral mixing within the surfzone but

are not discussed in detail.

A








B










Figure 7. Mellor's description of the distribution of radiation stresses. A) sxx(z), B)
xy(z).

The sxx(z) term affected the cross-shore flow and potentially the location of the

alongshore current. In the formulations based on linear wave theory, the s,(z) term did

not include a depth-dependent pressure term as found in sxx(z). It was therefore

uniformly distributed over depth, and drove the alongshore currents. The local variation

of the radiation stresses over depth and the resulting cross-shore and alongshore currents

could cause a lateral mixing effect that might dominate over turbulent mixing (Putrevu

and Svendsen, 1999).

Chapter Contents

A main purpose of our study was to investigate the effects of the vertical

distribution of radiation stresses on the phase-averaged alongshore current position, the

formation of shear instabilities, the net horizontal mixing, and to compare the influence









of vertical forcing with those of previous theories. A variety of vertical distributions of

the radiation stresses were tested in a 3D, phase-averaged, Large Eddy Simulation (LES)

nearshore circulation model over an alongshore uniform barred beach bathymetry. The

use of a 3D model allowed for the development, interaction, and analysis of cross-shore

mixing by the undertow with the alongshore current. Mixing effects caused by

large-scale shear instabilities, and those due to depth-dependent cross-shore circulation,

were examined separately by utilizing alongshore domains of varying dimensions. The

smaller [O (20 m)] domain inhibited the development of the fastest growing linearly

unstable modes that produced large-scale [O (100 m)] shear waves. Longer domains

[O (200 m)] were also modeled that permitted shear waves to be developed and allowed a

determination of the relative contribution to cross-shore mixing of the two processes.

The thesis is divided into five chapters. Chapter 2 reviews some previous model

studies of alongshore current distributions and shear waves. Chapter 3 covers the 3D

model problem setup, governing equations, and details of sub-models used in the

numerical experiments. Chapter 4 provides a description of the simulations and the main

results. The discussion focuses on the effects of the vertical distribution of the radiation

stress and the response of the flows. Emphasis is given to the stability and structure of

the alongshore current, the cross-shore current structure, and the effects of model

approximations. The effect of domain length and the relative magnitude of lateral mixing

caused by shear waves and radiation stress distributions are also addressed. Chapter 5

summarizes conclusions of the work, considers strengths and limitations of this approach,

and outlines future directions.














CHAPTER 2
MODEL COMPARISONS

This chapter summarizes work of three previous model studies that focused on

shear instabilities, and their effect on alongshore currents on barred beaches. Two of the

models were 2D, and one was quasi-3D, a sophisticated hybrid approach.

Two-Dimensional Alongshore Current Models for Barred Beaches

Traditional 1- and 2D phase-averaged models were unsuccessful at predicting the

location of the peak alongshore current on barred beaches. Using linear wave theory,

breaking wave parameterizations, and depth-averaged mean flow equations, they

predicted peak currents at the two locations of maximum wave breaking intensity: the bar

and the shoreline. Some field data, however, showed peak currents located in the trough

region, an area where reduced breaking occurred (Church et al., 1992). A probable

contributing factor in this discrepancy was the effects of shear waves.

In this section, two similar modeling approaches are discussed and summarized in

Table 1. The first, by Slinn et al. (1998), was a qualitative examination of the shear

instability problem, and the second, by Ozkan-Haller and Kirby (1999) was a more

quantitative analysis with comparisons of their model results to field data from the

SUPERDUCK experiment. The two models used slightly different governing equations,

but both showed that shear waves could alter the alongshore current profile, and caused

increased flow in the trough region. Both of the models were phase-averaged and

worked with depth-averaged currents. For a full description of the models and their









results, the reader is referred to Allen et al. (1996), Slinn et al. (1998), and Ozkan-Haller

and Kirby (1999).

In both models, as the frictional terms (Cf) were decreased, the alongshore currents

became stronger and contained more perturbations. Results of varying the frictional

terms are presented in Figure 8. These were obtained using a model similar to the one

used by Slinn et al. (1998). A consequence of the more unsteady flows was the shedding

of more vortices, and resulted in the cross-shore mixing of the alongshore current (Figure

8D). For high values of Cf (Figure 8A), the currents were weak and stable. For moderate

values (Figure 8B), the current remained relatively stable over time, with a small amount

of horizontal spreading: peak values in the trough increased as currents over the bar

decreased, but the location of the maxima did not change. As Cf was decreased, the peak

currents increased before instabilities appeared (Figure 8C, Figure 8D). With lower

friction coefficients, the current became strongly unstable. At the lowest value (Figure

8D), the current had the broadest cross-shore distribution due to more energetic

instabilities that increased horizontal mixing.

Slinn et al. (1998) found that lowering the bottom friction coefficient shifted energy

in the frequency alongshore wave number (f ky) spectra to smaller length and shorter

timescales. Ozkan-Haller and Kirby (1999) also varied their horizontal mixing

coefficient (M) in their diffusion terms to approximate diffusion caused by breaking

waves or depth-varying currents. They found that by increasing the mixing coefficient,

they generated less energetic instabilities at longer length scales. The model of Ozkan-

Haller and Kirby (1999) was able to approximate alongshore currents and propagation

speeds of the instabilities for the SUPERDUCK experiment. It should be noted,














however, that peak alongshore currents for the days they simulated were still located near


the nearshore bar and not in the trough. Their model was also not able to accurately


match the observed frequency spectra, predicting more energy at lower frequencies than


observed.


A
12
11
1
09
08


06
05
04
03
02
01
0



C
12
11
1
09
08

V(m)7
06
05
04
03
02
01


I 50 10
x m)


150 2(


Figure 8. Comparison of alongshore current profiles from 2D circulation for varying

values of Cf over time. Results were from a 2D model similar to the one

used by Slinn et al. (1998). Ho = 0.7 m, Tp = 8 s, 0o = 300. Bottom panel is

the beach topography, h(x) and given in Eq.(5).


C,= 0.01
-- t=10
--- t=20
t=40
--- t=60
-- t= 80
t= 100











In both models, the flows that were generated were sensitive to the free parameters.


As the correct representative values of these parameters were unknown for field


conditions, they were tuned to produce currents similar to field data. The free parameters


could be seen as a weakness of the modeling approach because they influenced the


outcome at the user's discretion.


Results of the alongshore current distribution (Figure 8), and snapshots of vorticity


(Figure 9) attained from a model similar to Slinn et al. (1998) are presented for


comparison with the 3D model results. Wave parameters (H = 0.7 m, Tp = 8 s, 0o = 300)


and bathymetry were the same for the 2D and 3D models. A coefficient of friction equal


to 0.0015 was used to generate currents in the 2D model with magnitudes comparable to


the 3D results. This value generated a current with strong instabilities that caused


considerable mixing. Larger values of Cf produced weaker currents that became unstable


at later times. The peak velocities before the current became unstable in this example


were approximately 0.9 m/s over the bar, and about 1 m/s at the shoreline.


t= 15min t= 25 min t= 35min (1/s)

0030
e 00300026
0022
0018
0014
0006
0002









(Cf) = 0.0015, the 2D results compared well with the 3D model. The current
-0 002
-0006
-0010
-0014
5 50 0018
-0022
-0026
-0030
x (m) x(m) x (m)


Figure 9. Vorticity contours from 2D model over time. Using a coefficient of friction
(Cf) = 0.0015, the 2D results compared well with the 3D model. The current
remained stable until about 20 min, then it began to meander (25 min), and
evolved into large vortex structures (35 min) and remained in this condition.









Quasi-3D Models: SHORECIRC

Recent work by Zhao et al. (2003) analyzed 3D effects from shear waves using the

quasi-3D nearshore circulation model, SHORECIRC. Their work focused on the

importance of including the depth variation of the currents. Using the depth-integrated

2D equations, they included a quasi-3D dispersive mixing term that approximated the

effect of depth-varying currents (Putrevu and Svendsen, 1999). Their parameters are

summarized in Table 2. The reader is referred to Zhao et al. (2003) for a more complete

description of the model and their findings.

The effect of breaking waves was parameterized with a roller model included in the

radiation stress term and short wave-induced volume flux. The quasi-3D model

approximated effects of vortex tilting, and depth-varying currents that produced

horizontal mixing on the same order of magnitude as the shear wave component. The

quasi-3D dispersive mixing term also included effects of depth-varying radiation stresses.

Zhao et al. (2003) compared the quasi-3D simulation results to the 2D model results of

Ozkan-Haller and Kirby (1999). They found that the quasi-3D model generated a more

steady flow, localized shoreward of the bar. The flow contained a less energetic turbulent

kinetic energy field that was attributed to the mixing terms being of the same order as the

diffusion produced by the shear waves. Ozkan-Haller and Kirby (1999) found similar

results when they increased their horizontal mixing coefficient.

Summary

The 2D and quasi-3D models had advantages because they produced reasonable

approximations of the alongshore current structure efficiently or in a predictive, forecast

manner. The use of relatively coarse grids and depth-integrated equations of motion,

however, inhibited the development of 3D turbulence and depth-varying interactions






24


between the alongshore and cross-shore currents. The utilization of a fully 3D model

with a much finer grid resolution allowed for these properties to influence the alongshore

current with fewer ad-hoc approximations. This allowed greater insight into properties

that might be important to the dynamics of the surfzone.










Table 1. Comparison of 2D models
Property Slinn et al. (1998)
Governing (hu) +(hv)y 0,
equations


Ozkan-Haller and Kirby (1999)
7, +(du) +(dv) = 0,


Ut + UUx + Vuy
ut + uu + VV

vt +ux +yv


vV4u,


v-V
---V
h


where:
h = local water depth, h(x)
(u,v) = velocity components
p = pressure
po = constant fluid density
S= bottom friction coefficient
v = small biharmonic diffusion coefficient
V = velocity due to wave forcing
- PopV(x, y)


Wave forcing
and wave
breaking


h(x)


, W1Cre:.


model (sb) aS
V(x,y) is a function of

bb: Thornton and Guza (1986)
Momentum v = 1.25m4 / s for 2 m grid spacing,
mixing
v = 2.5m4 / s for 2.5 m grid spacing.


Ut + UUx + Vu

t + uv+vv :


-gr + X + x -T,

gu+y + Ty + Ty- by


where:
V = wave-averaged free surface elevation
d=h+f
(u,v) = velocity components in x and y
direction respectively
x = wave forcing (see below)
z' ",' = momentum mixing terms
,Tbx,> by = bottom friction


1 8S ,
Spd ax )'


Fb: Whitford (1988)


2 0 (d u +
Sd x ax

wi a dv av
r = dv- ,
Sd y O x'
where :


pd Kxj


-- dv-
d Oy cx


p = Cf\u.1
where:
lurs = 0.3 m/s is the wave orbital velocity
0.002 < Cf <0.012


Lx = 1000 m
Ly = 1200 or 1280 m


Top: Rigid-Lid
Lateral: periodic
Shore: u = = v =v v =
Offshore: u = uxx =


v=Md b, < M < 0.5,

M = mixing coefficient

d u by = v, where:
bx bY d

2
/ = -fo, cf = 0.0035,0.003

uZ H

0 4 h r
Lx = 550 m
Ly = 16 x Lmx, where:
Lmax= 2" / kmax and ranges between 160
200 m.
Top: Free-surface
Lateral: periodic
Shore: free-slip
Offshore: absorbing boundary


Bottom
friction


Domain
lengths


Boundary
conditions











Table 2. Quasi-3D
Property
Governing
equations


Wave forcing

Momentum mixing


model summary
Zhao et al. (2003)
a4 a
+ ( Vh)= 0,
at +x =

8 8a
(Vh) + (Vffh)+
at x,


1 as#, 1 aTfl
P ax, p ax,


1 aLf
P C'X,


+ gh P+
axC P


where:
S= short-wave averaged free surface elevation
h = still water depth
a, ) = are the x,y directions respectively
V = depth-uniform current
S a = short-wave induced radiation stress
Lp, = quasi-3D dispersive term
Tp = depth-integrated turbulent shear stress
REF/DIF 1 is used as the wave driver to calculate wave radiation stresses and
volume flux

T, = phvO +


L#a =

where:


Bottom friction

Boundary
conditions


Domain lengths


, =0.08 | uo I h+0.1h D +v,

Vla,p = depth-varying current
uw = wave component velocity
t = free surface at the wave trough level
2 1 uo
r = cf, c = 0.0035
Top: free surface, phase-averaged
Lateral: Periodic
Shore: wall-boundary with free-slip condition
Offshore: no-flux
Lx = 788 m,
Ly = 2490 m


P VlcVldz + J (u + u














CHAPTER 3
PROBLEM AND MODEL SETUP

The nonlinear dynamics of finite amplitude shear instabilities of alongshore

currents over barred beach topography were studied using finite-difference, numerical

experiments. The model was non-hydrostatic, wave phase-averaged, and fully 3D. It

solved the transformed Navier-Stokes equations with a Large Eddy Simulation (LES)

sub-grid scale closure on a curvilinear (o-coordinate) grid. A principal advantage of our

model was that it included dynamics of the undertow and depth varying currents that led

to enhanced cross-shore mixing.

Model Domain and Solution Method

The model was formulated for alongshore-uniform bathymetry and included a

shore-parallel sand bar. The depth profile was an approximate fit to topography

measured at Duck, North Carolina, October 11, 1990, as part of the DELILAH field

experiment (Lippmann et al., 1999) given by:


h(x, y) = a1 tanh x + a- b tanh xb1' -ae a C (5)
a a) I a a)
where:

al = 2.97, defines the amplitude of the nearshore trough
a2 = 1.5, defines the amplitude of the offshore trough
x, = (i-1)dx, 1 < i < nx-1 where i is the cross-shore counter
bl = tan(fi), where fl = 0.075, defines the nearshore beach slope
b2= tan(p2), where 2 = 0.0064, defines the offshore beach slope
yi = b1/b2
xc= 80, defines the cross-shore location of the nearshore bar









Two alongshore domains were used. The short alongshore domain was 20 m in

length and inhibited the production of alongshore generated shear waves so we could

focus on other dynamics that might have influenced the alongshore current. The large

alongshore domain extended to 200 m and allowed for large-scale shear wave generation

and interaction with the alongshore current.

Grid spacing was in part based upon results of Scotti et al. (1997). They found that

a maximum aspect ratio between any two grid length scales of approximately 4:1 could

be used for LES schemes on anisotropic grids without influencing the results. The effect

of grid spacing is discussed in more detail in Chapter 4. Constraints in the grid

generation scheme did not allow the depth of the cross-shore domain to go to zero,

therefore the model domain was cut off at a depth of 0.5 m. The grid was clustered along

the bottom and near the shoreline for higher resolution in areas of stronger variability

(Figure 10). The short domain was typically modeled with (nx, ny, nz) = (257, 33, 17)

grid points, with an average aspect ratio of 3:1 between horizontal and vertical length

scales. The long domain was modeled with (nx, ny, nz) = (257, 257, 17) grid points and

also had an average aspect ratio between the horizontal and vertical length scales of

approximately 3:1.

Boundary Conditions

Uncertainties associated with the bottom friction parameterization were reduced in

this approach by modeling the bottom stress with a no-slip condition using a

high-resolution vertically clustered grid with O (1-10 cm) scales near the seabed. The top

boundary utilized the rigid-lid approximation and was modeled with a free-slip condition.

The offshore and shoreline boundaries were modeled with a no-slip condition to aid in

obtaining numerical stability. The lateral boundaries were periodic.









Initial Conditions

The model was initially at rest and was forced by the momentum input from the

wave radiation stresses. Run-up time to steady-state flow was approximately 1 h of real

time.






4

3

2




0
20
0 15
50 X 150 200

Figure 10. 3D outlay of physical grid used in the short domain simulations. (x,y,z) are
the cross-shore, alongshore, and vertical axes respectively and are
measured in m. The long domain extended to 200 m in the alongshore (y)
direction and had the same discretization in the x-z plane.


Governing Equations

The LES equations of motion for the Cartesian coordinate system were:

au au, 1 ap a t,
'+u -' + -+ F (6)
at ] x, p x, ax,

=, 0, (7)
dx,
where u, = (u, v, w) were the cross-shore, alongshore, and vertical velocity components, p

was pressure, t was time, p was fluid density equal to 1028 kg/m3, and Fx and Fy were

the horizontal body forces in the cross-shore and alongshore directions respectively and










1
were equal to the vertical distribution of the radiation stress gradients, S, and
p 9x

18
S r, uuj u, uj represented the LES sub-grid stresses written in terms of the


3 1 (u au,
resolved strain tensor, r, -2v, Sj where S + and the turbulent
3 2 9x axc

eddy viscosity, vt, were calculated with the Smagorinsky approach,

(i.e., v, = (CA) 2(2SS)2 where C = (C,)3 and Ck = 1.5 was the Kolmogorov

constant, and A = (AxAYA)3 was the filter width based on the local grid spacing in each

direction).

The model used a boundary fitted curvilinear coordinate system that allowed

variation of the geometry in two directions (x, z). To reduce the computational

complexity, the model required the x-z curvilinear coordinate system to be orthogonal

(xz, + x z = 0), where the subscripts (, g) denoted partial differentiation between

physical and computational space. Orthogonal grid intersection at the boundaries

increased the accuracy of the boundary conditions with the interior solutions. To

compute approximate solutions to the governing equations, the physical domain (x, y, z)

that solved for the physical space velocities (u, v, w) was transformed to a cubic grid of

regularly spaced grid points in computational space (, r, C) and solved for the

contravariant velocities (U, v, W). This transformation allowed the use of easily coded

numerical methods, but at an increased cost of more complicated transformed equations

of motion. The mapping between physical and computational space was done through an

orthogonal transformation of three successive elliptic boundary value problems following









Winters et al. (2000). The bathymetry was uniform in the alongshore direction and grid

spacing was constant in the y-direction.

Transformation of velocities was done by:

~u ~ x X, x ~ U ~
= ][] and, (

[U] 1 [z -x, [u
W = Jx -zr x w
where:
J= (j ,)= x z,-x zX (1

Transformation of derivatives was done by:

A ft1 -,


After transformation, the governing equations to be solved were written as:


0)


1)


d I (x_+z_) (z
-U + ,8,IU2 +,1UW,+ a,3W W I p-+
dt p J2 J=
d 1
Sv = -- y,p, + F +diss(v),


tW+Ai +A,UW+A p3 j- 2 P +
1 a a 1

D= [U,v, W] ( (J U)+(J W)) +-- = 0,
J" a aOz y', 07


where:
-= +U-a +( )v- +W a
dt at 9^ y, ar 9a

1(z'x -xrz 8) 2 2(zrx -
J. J

,1- (xz ),2 = 2(xz -z
Jx J
(Winters et al., 2000)


xz X)
Xzx )


S+ diss(U),


F + diss(W),


(z'x- xzX' )
Jx
(xz zx )
Jx


(12)

(13)

(14)

(15)


(16)









Diss (U, v, W) were dissipation terms implemented through the LES sub-grid-scale

closure model described above, and were not given here in the transformed coordinate

system for simplicity. Dx was the transformed divergence equation.

A fourth-order compact low pass spatial filter (Lele, 1992) was also applied to the

contravariant velocity field as part of the sub-grid dissipation model to help maintain a

resolved and stable simulation. Spatial derivatives were calculated with fourth-order

compact differencing schemes. The pressure solver utilized a fourth-order finite

difference scheme based on the multi-grid method. Time stepping was explicit and done

to third-order accuracy with the Adams-Bashforth (AB3) method with a variable time

step. Initial time stepping was done with the Euler and AB2 methods.

Forcing

Two-Dimensional Sub-model

The alongshore current forcing conditions were coupled to the beach topography

and specified surface wave conditions using the wave refraction model of Slinn et al.

(2000) that implemented the wave energy dissipation model of Thornton and Guza (1983,

1986). Deep water wave height, angle, and peak period were specified, and the waves

were shoaled into the shoreline using linear theory. The majority of the effects of wave

set-up were included through an induced cross-shore pressure gradient. Except for the

simulation with Mellor's equations (Case 13), the depth-integrated 2D radiation stress

gradients were calculated for the shoaling wave as it propagated over the beach

topography. These forces were then used as input to the 3D model and distributed

throughout the water column following a variety of approximations introduced in

Chapter 1, and discussed in more detail immediately below. The cross-shore distribution

of the nondimensionalized forcing coupled to topography used in the simulations is











presented in Figure 11. To implement the vertical distribution of radiation stresses


following Mellor (2003) (Case 13), the individual wave parameters: wave energy, wave


number, wave angle, and water depth were calculated at each grid point and used as input


for his equations in a separate program.


Three-Dimensional Forcing Model


A variety of vertical distributions of the wave-induced radiation stress gradients


were implemented to test their influence on the resulting alongshore and cross-shore


current structures. Preliminary sensitivity tests of the model were performed with


depth-uniform and linearly distributed forcing in both the alongshore and cross-shore


directions.



-0 035 -00
03 -00025
-0 025
-0002
-002 -
-0 0015
-0 01

\ -00005
-0 005 -\ 0 0
0 0
0005 o00005
0 50 100 150 200
x(m)










Figure 11. Cross-shore distribution of nondimensionalized wave radiation stress forcing
coupled to topography. Fx= dSxx/dx, Fy= dSxy/dx, x = 0 is the shore
boundary, x = 200 is the offshore boundary. The beach topography is
given in the lower panel.


Forcing Distributions


The rigid-lid approximation used in the model represented the wave


phase-averaged equations, and eliminated the vertical region between the MWL and the









wave crest. All possible vertical distributions of the radiation stresses tested were only

approximations because a portion of sxx(z) existed in that region, not contained in any

phase-averaged model. The vertical distributions were also based on linear wave theory,

but applied in the surfzone that was influenced by nonlinear effects.

The vertical distribution of the cross-shore radiation stress in the surfzone was

divided into two parts following Longuet-Higgins and Stewart (1964). Two-thirds of the

total x-momentum forcing was distributed uniformly over the entire water column and

represented the momentum due to wave orbital motion. The remaining one-third

represented the excess pressure force due to the waves between the trough and crest of

the wave. This portion was generally depth-dependent and various forms were

implemented for the different tests. It was referred to as the pressure term. The

alongshore momentum forcing, Fy, did not include the pressure term as in Fx, and was

generally implemented as a depth-uniform force. The magnitude of the forcing was

dS, (x) dS, (x)
normalized by h(x) so that \F(x, z)dz = and F (x, z)dz= In Figure
dx In Figure
dx dx

12 to Figure 17, the true direction of Fy pointed in the alongshore direction, but for

graphing purposes, was presented as an onshore directed vector force above the 2D cross-

shore profile.

Case 8 used a depth-uniform distribution to represent the pressure term. It was

added in over the top five grid points of the domain to represent the approximate depth of

the wave trough (Figure 12A). This was a first attempt to simulate the vertical

distribution of wave radiation stresses due to breaking waves.

For Case 9, the pressure term was applied as a depth-linear force (increasing from

zero at the bottom most point) over the top five grid points as illustrated in Figure 13A.










This forcing was a closer approximation of the vertical distribution of wave radiation

stresses obtained from linear wave theory, with the approximation that the portion above

the MWL was a mirror image in the vertical distribution to the portion between the

trough and the MWL. It was therefore folded back on its mirror image below the MWL

and doubled the force distribution in the trough to MWL region.

A
















o0o xl(oo) 1o0 2oo



B) Fy. Forcing distribution was applied over the cross-shore domain and
varied in magnitude according to Figure 11.

The distribution in Case 10 was another attempt to more accurately model the

varying wave trough depth in the cross-shore. The a-coordinate grid stretched to fill the

water column, with grid spacing more refined in regions associated with smaller wave

amplitudes. For example, over the bar crest, the wave trough depth was approximately

0.3 m and the vertical grid spacing was approximately 0.1 m. For the majority of the grid

points in the cross-shore direction, the depth of the wave trough was approximately three

vertical grid spaces. Case 10 had uniform forcing occurring over the top three grid

points instead of five (Figure 14A).








36



A






















o loo 5 2
x (m)


Figure 13. Case 9: Sample vertical distribution of nondimensionalized forcing. A) Fx.
B) Fy. Forcing distribution was applied over the cross-shore domain and
varied in magnitude according to Figure 11.


A









0 0 5 10o 150 200
S(m)















Figure 14. Case 10: Sample vertical distribution of nondimensionalized forcing. A) Fx.
B) Fy. Forcing distribution was applied over the cross-shore domain and
varied in magnitude according to Figure 11.
x(M)


Fiur 4. Cae1:Sml etcldsrbto fnniesoaie ocn.A ,

B)F.Frigdsrbto a ple vrtecossoedmi n










The forcing in Case 11 (Figure 15A, B) combined the linear distribution of

x-momentum from Case 9, but with the linearly increasing force applied over the top

three grid points (Case 10).

A












050 15 200




S(m)

Figure 15. Case 11: Sample vertical distribution of nondimensionalized forcing. A) Fx.
B) Fy. Forcing distribution was applied over the cross-shore domain and
varied in magnitude according to Figure 11.

With the rigid-lid approximation used in our model, the moving free surface was

removed or wave-phase averaged out. Instead of doubling the pressure term in the

trough-MWL region as was done in Cases 8-11, the forcing for Case 12 was broken down

into three parts: two-thirds of the total forcing was depth-uniform over the entire water

column, one-sixth of the total forcing was a depth-linear force over the top three grid

points, and a surface stress at the top grid point accounted for the remaining one-sixth of

the total wave radiation stress force above the MWL (Figure 16A).

The forcing in Case 13 was based on the equations of Mellor (2003). As is shown

in Figure 17, the vertical distribution of forcing in the x-momentum equation was quite

different from the other approaches. Since Mellor's work focused on large scale,










deep-ocean modeling, his equations for the radiation stress terms might not have been a

suitable description of surfzone radiation stress distributions. The full equations of

momentum presented in his work, however, did account for the transfer of momentum

through dissipation in other terms (Personal Communication, 2004). These other terms

were not used in the current model because we focused on the cross-shore derivatives of

the vertical distribution of radiation stresses only. As seen in Figure 17A, the direction of

the resulting force due to radiation stress gradients in the x-direction produced variable

forcing in both the vertical profile, and in direction (note at x = 125 m, the resulting force

was offshore directed).

A







0 o0 15 200









S(m)

Figure 16. Case 12: Sample vertical distribution of nondimensionalized forcing. A) Fx.
B) Fy. Forcing distribution was applied over the cross-shore domain and
varied in magnitude according to Figure 11.

These vertical distributions of wave radiation stresses followed the basic guidance

arising from linear theory. Breaking waves in the surfzone have more complex dynamics

than contained in linear theory. Hence, other vertically variable distributions of Fx and Fy

may be more accurate approximations of natural conditions. The vertical distribution of










radiation stresses due to nonlinear breaking waves is a topic of ongoing research. The

results presented below should be considered a sensitivity study to variation of sxx(z)

rather than as the most accurate method to model the surfzone with a 3D model.

A







x (m)

B









Figure 17. Case 13: Sample vertical distribution of nondimensionalized forcing. A) Fx,
B) Fy. Forcing distribution was applied over the cross-shore domain.

Scope of the Model

The current model simulations pursued process-based studies concentrating on

surfzone dynamics. While the model was a rational advance to investigate new physical

interactions in the surfzone, it still contained several necessary approximations that

limited the scope of utility of the model. Results were qualitatively and semi-

quantitatively similar to field and lab data, however, several limitations inherent in the

model hindered direct comparison with field data.

The surfzone is a highly dynamic area where many different physical processes

interact and influence each other. The wave breaking model that was implemented is

widely accepted, but was still an approximate representation of forcing. Our model did









not include the effects of mass-flux/Stoke's drift, wave-current interaction, or a wave

roller, each of which would affect the quantitative predictions of the current profile.

Phase averaging of the waves removed the time-variance of the forcing, and the

mass-flux, and modified the total bottom drag applied to the mean currents.

The boundary conditions imposed also influenced several processes. The shoreline

boundary cut-off at a depth of approximately 0.5 m was a crude approximation and

influenced the inner jet region. The no-slip condition imposed for stability purposes on

the cross-shore boundaries also manipulated fluid velocities in those areas. The rigid-lid

approximation on the top boundary removed the MWL to crest area where a portion of

the wave stresses could have been applied. Approximating this force as a surface stress

was limited by the free-slip condition. Bottom friction in LES was another inelegant

compromise, both because the wave component was omitted, and no wall model was

used. The grid aspect ratios require further testing to increase confidence that

convergence was reached and was no longer affecting the fluid dynamics. The

alongshore periodic boundary conditions forced only alongshore wavelengths of integral

numbers of periods to be present and did not permit features longer than the domain

length to arise.

The model is in its development stages with many of the above mentioned

limitations being considered. Increasing domain lengths in the alongshore and

cross-shore directions will decrease boundary effects and include more of the inner jet.

Increased grid resolution will improve the modeling of bottom friction and other LES

effects. Higher grid resolution may also increase stability and allow for an improved

representation of the free-slip boundary condition to be applied on the cross-shore









boundaries. The inclusion of mass flux and alternative methods of approximating the

forces above the mean water level are currently being developed. The inclusion of a

wave roller is presently available in the model if desired.

It was desired to pursue simplified process-oriented studies to isolate interactions of

certain key physical processes. A consequence of this approach was that other important

processes were suppressed making quantitative comparison with field data difficult. The

main utility of the 3D model was the simultaneous resolution of both alongshore and

cross-shore low frequency currents. Even with its inherent weaknesses and engineering

approximations it was able to contribute new understanding to the dynamics of nearshore

currents.















CHAPTER 4
SIMULATIONS

Sixteen cases are discussed to address the effects of grid resolution, vertically

dependent forcing, and alongshore domain length. Table 3 lists the case numbers, the

domain size, what was tested, and the grid used. All runs used a deepwater wave angle

(ao) of 300, offshore boundary wave height (Ho) of 0.7 m, and peak period (Tp) of 8 s.

Table 3. Summary of simulations
Case number Domain size Grid resolution Average aspect Effect tested
(x,y) (m) (nx, ny, nz) ratio (H:V)
1 (200, 198) (129, 129, 33) 11: 1 Vertical forcing, domain length
2 (200, 198) (129, 129, 33) 11: 1 Vertical forcing, domain length
3 (200, 20) (257, 33, 33) 6: 1 Grid resolution
4 (200, 20) (257, 33, 17) 3 : 1 Grid resolution
5 (200, 20) (129, 17,33) 11: 1 Grid resolution
6 (200, 20) (129, 17, 17) 6: 1 Grid resolution
7 (200, 20) (513, 33, 33) 3 :1 Grid resolution
8 (200, 20) (257, 33, 17) 3: 1 Vertical forcing
9 (200, 20) (257, 33, 17) 3: 1 Vertical forcing
10 (200, 20) (257, 33, 17) 3: 1 Vertical forcing
11 (200, 20) (257, 33, 17) 3: 1 Vertical forcing
12 (200, 20) (257, 33, 17) 3: 1 Vertical forcing
13 (200, 20) (257, 33, 17) 3: 1 Vertical forcing
14 (200, 198) (257, 257, 17) 3 :1 Domain length
15 (200, 198) (257, 257, 17) 3 :1 Domain length
16 (200, 198) (257, 257, 17) 3 :1 Domain length

Simulations were compiled using the Intel Fortran Compiler (IFC) and ran on Intel

Pentium 4 (2.6 GHz) to Intel Xeon (3.06 GHz) single processors. The model was

computationally expensive based on grid resolution, domain size, and forcing

distribution. Depth-uniform forcing simulations ran at faster speeds because they were

more stable. Cases 1 and 2 produced 60 min of data in approximately 6 weeks of CPU

time. Cases 3-7 required between 4 and 12 weeks of CPU time to compute 30 min of

data. Cases 8-12 averaged 9 weeks of CPU time for 60 min of data. The model could









produce results at approximate rates of 3.5 min/CPU week (Case 15 and 16) and

6 min/CPU week for Case 14 on the long domains.

The model also was run in parallel on an SGI-Origin 3400. Running a simulation

in parallel on four processors decreased the computational time by approximately a factor

of two compared to a single processor Intel computer. A disadvantage of the parallel

platform was that batch limits were set to allow a simulation to run for a maximum of

five days at a time. Thus, a simulation had to be restarted manually several times to be

completed. There were additional inevitable turn-around time losses between the

completion of one portion of a simulation and the next opportunity to restart it. An

advantage of the dedicated single processor Intel computers was that they could be run

continuously for extended periods. It could also be noted that the University of Florida,

Civil and Coastal Engineering Department's 20 processor SGI Origin 3400 cost

approximately $400,000 to purchase and maintain during 2001-2003, compared to single

or dual processor Intel computer costs of under $2,000. The cost to benefit ratio

appeared to favor the single processor platforms.

All the simulations produced reasonable velocities for the given wave parameters.

Representative time series for u, v, and w- velocity are given for two locations within the

surfzone. The first, at x = 80 m, was located at the bar. The second time series was taken

at x = 40 m, and was situated in the middle of the bar trough. The data presented was for

Case 11. Both time series were taken at mid-depth and mid-way of the alongshore

domain (y = 10 m). U and w fluctuated about a mean, while the time series ofv showed

the spin-up and migration of the alongshore current into the trough region by the time lag

between the two signals.















0.4
u at x = 80
u at x =40
0.3


0.2


E 0.1


0


-0.1


-0.2
10 20 30 40 50
t (min)



Figure 18. Time series for u-velocity. Data was from Case 11, sampled at (xo, yo, Zo) =
(80, 10, 1.1) (red) and (xoyo, ,o) = (40, 10, 1.5) (green). All co-ordinates
were in m, where Zo was measured positive downwards from the top
boundary.


0.15 -
w at x = 80
w at x = 40
0.1


0.05

O-
E 0


-0.05


-0.1


-0.15
15 10 20 30 40 50
t (min)



Figure 19. Time series for w-velocity. Data was from Case 11, sampled at (xo, yo, Zo) =
(80, 10, 1.1) (red) and (xo, yo, Zo) = (40, 10, 1.5) (green). All co-ordinates
were in m, where Zo was measured positive downwards from the top
boundary.


All the simulations produced reasonable magnitudes of the mean alongshore


currents for the given parameters. Since the bottom friction was modeled using a no-slip


condition, ad-hoc tuning of the bottom friction coefficient (Cf) could not be done to


adjust the strength of the alongshore current. In addition, with the phase-averaged model,











the bottom friction did not feel the time-varying wave bottom boundary layer where


Drag = Cfu 2+u+ This was significant because w, could be relatively large. The


omission of u, produced a no-slip drag that was biased lower than field conditions. This


offered one explanation for why the mean currents generated by the model might have


been slightly higher than observed for the same wave conditions in the field.




1.2
-1. -v at x = 80
-1.1 1 a v at x = 40
-1
-0.9
-0.8
-0.7

-0.6

-0.4
-0.3
-0.2
-0.1
O
10 20 30 40 50
t (min)


Figure 20. Time series for v-velocity. Data was from Case 11, sampled at (xo,yo, z,o)=
(80, 10, 1.1) (red) and (x,) yo, z,) = (40, 10, 1.5) (green). All co-ordinates
were in m, where Zo was measured positive downwards from the top
boundary.


The first simulations discussed were a comparison of forcing the model with two


vertical distributions over the water column: a depth-uniform (Case 1) or a linearly


depth-varying force (Case 2) (Figure 21) in both the alongshore and cross-shore


directions based on Fredsoe and Deigaard (1992).


These runs produced a number of new and interesting results. First, instabilities


of the alongshore current developed much sooner for depth-dependent forcing. Second,


the depth-averaged current was much stronger before breaking down into instability and


turbulence for depth-uniform forcing. Third, the alongshore wavelength of the initial









instabilities was longer (of order 100's of meters) for the depth-uniform current, but on

the order of 10's of meters for the depth-dependent current. A comparison for the

alongshore and depth-averaged alongshore currents is given in Figure 22. The strong

peak currents at the shore and over the bar were present in the depth-uniform case (Figure

22A) until the current became unstable (20 min). After this time, the current laterally

spread across the surfzone, but a strong current was still present at the shoreline. In

Figure 22B, the depth-dependent forcing did not produce distinct alongshore current

peaks over the bar, but instead the current was much weaker, and was spread across the

surfzone with local maxima shoreward of the bar.

A B















Figure 21. Case 1 and 2: Vertical distribution of forcing used in the simulations. Case 1
(A) and 2 (B).

These results showed that the response of the alongshore current was sensitive to the

vertical distribution of forcing. The cross-shore distributions of the mean alongshore

current profiles revealed effects of 3D mixing and preferential cross-shore advection.

The physical interpretation had two major components. The first was that much stronger

cross-shore circulation developed for the depth-dependent forcing. When Fx was

depth-uniform, very little cross-shore circulation was produced; rather, a barotropic










cross-shore pressure gradient balanced the forcing. When Fx was depth-dependent,

however, a strong undertow developed rapidly and was compensated by a shoreward

mass flux in the top half of the water column (Figure 23).


A B
tT t--- 0.50
-- t- 10.00
N-t et-l 0,49 "'- -- t20.01
0.01 t- 30,00
-- t- 40.00
-- 20.01 ----- t- 50.01
t- 30.00
S--- 40.00



















Figure 22. Comparison of depth and alongshore averaged currents for the initial
simulations. A) Depth-uniform forcing, B) Depth-dependent forcing.
Note: vertical axes used different scales for A and B. Bottom panel is the
beach profile.

One reason this was dynamically important to the depth-averaged alongshore

current was that Fy and V(z) were also depth-dependent in this case. This was partly due

to the no-slip boundary condition on Vat the sea bed and the free-slip condition at the

surface. Hence, there was a much stronger alongshore current in the top half of the water

column than in the bottom, as seen in nature. Thus, the faster surface current was

advected shoreward (into the trough) while the weaker, bottom current drifted offshore.

The net effect was a shoreward shift of the alongshore current maximum that became

balanced in the trough. This was a similar response to having included a roller model in







48


the wave sub-model to shift the location of wave momentum input shoreward. A roller

model was not incorporated into the formulations of the Thornton-Guza (T-G, 1983)

wave breaking model that was used to generate input radiation stresses for these cases.











1-1
( I





x (m)

Figure 23. A typical vertical profile of cross-shore velocity vectors produced by depth
linear forcing. Vectors were plotted to show the extent of the undertow
along the bottom boundary. The reference vector above represented 1 grid
space/unit magnitude.

The vertical distribution of forcing also affected the stability of the alongshore

current. In the depth-uniform case (Figure 24), linear growth of the alongshore current

produced finite large scale instabilities similar to 2D models. Depth-linear forcing

(Figure 25), however, produced small-scale disturbances in the alongshore current at

much earlier times.


U~ I U ~
.1 1
.~ oOLC
A U


0 I A
A, f


Figure 24. Surface vorticity contours for depth-uniform forcing as a function of time.



















Figure 25. Surface vorticity contours for depth-dependent forcing as a function of time.

In summary, these initial simulations showed that depth-uniform forcing produced

results comparable to previous 2D works. Depth-linear forcing, however, produced very

different results for the vorticity structures, and provided the motivation to investigate the

effects of vertical forcing and grid resolution further.

The Effect of Grid Resolution

The initial simulations were done on a grid with a high horizontal to vertical aspect

ratio that exceeded the recommendations of Scotti et al. (1997) for LES closure. A series

of tests were run to determine adequate grid resolution, and to verify the generality of the

results. A total of five different grids were tested using depth-linear forcing as in Case 2

on a 200 m by 20 m by depth-varying grid. The 20 m domain length was used to reduce

computational time, and was valid because the turbulent structures seen in Case 2 were

on the order of 6 m and could be adequately captured on the smaller domain.

Case 5 had the same aspect ratio as the original tests conducted, while Case 4

followed the recommendations of Scotti et al. (1997), and Case 7 was within close

proximity of his upper bound recommendation of a 4:1 grid aspect ratio. Alongshore and

cross-shore currents, as well as depth-averaged vorticity fields, were compared to study

grid resolution effects on flow characteristics. Each test was run to approximately

30 min.
30 mmn.
















Alongshore-Averaged Alongshore Currents



Comparing the vertical distribution of alongshore currents showed effects of the



grid resolution and aspect ratio. Velocity contours comparing runs (Figure 26) showed



that they were qualitatively similar to first-order observation.



t 5 min t=10 min t= 15 min t = 20 min t = 25 min



A o






oo oi 415o 150 200 11 M


t= 15 min


t= 15 min


t = 20 min
5

4







u 0o 100

t = 20 min


t = 15 min t= 20 min


t = 25 min v(ms)


o3o
044
o 5
2 -
o6,
o 11


ou o 1 ... ..-100 o


t= 25 min v(m/s)
oon
4 0 ll
0 33
0 44


10 7

oo 50 100 1...0 200 o


t = 25 min (,)


U 5 5 5









0 A .100 1 200 50 0. s.10 200 0.0 10o 10 200 0o so 1o 10 200 0. 0 1 10 200


E t 5 min t= 10 min t= 15 min t= 20 min t= 25 min (m/s)


I nO






0 0 s 0 10 0 200. 1001 .. 0 0 100 0 200 u s 0 ... ... 2 00




Figure 26. Cases 3-7: Contours of the alongshore-averaged alongshore velocity (v) for

variable grid sizes in the x-z plane. Plots are given in 5 min intervals to

show progression of current migration. A) Case 3. B) Case 4. C) Case 5.

D) Case 6. E) Case 7.


t = 10 min


t = 10 min


oot
o l
o2:






0




o




06
7


oo


t = 5 min t = 10 min









The high aspect ratio grid of Case 5 showed preferential cross-shore mixing at

earlier time periods. The alongshore current spread across more of the surfzone, and

showed a stronger jet at the shoreline between 0 and 10 min than was present in other

runs. Currents generated over the offshore bar were also weaker due to the increased

lateral mixing. Case 6 produced somewhat stronger alongshore currents over the bar, and

within the surfzone. Cases 3 and 4 were comparable in magnitude but varied slightly in

cross-shore position. Cases 4 and 7 were comparable in cross-shore distribution but

Case 7 had slightly weaker currents over the bar.

Alongshore and Depth-Averaged Alongshore Currents

The analysis of alongshore, and depth-averaged alongshore currents showed the

cross-shore distribution of the current. At earlier time (Figure 27), there was little

difference between the cross-shore distributions. At later times (Figure 28), however,

Cases 3, 5, and 6 produced somewhat stronger alongshore currents at the bar trough and

shoreline, while Cases 4 and 7 were comparable to each other.




08 Case7
i :L


Figure 27. Depth and alongshore-averaged alongshore currents at t = 5 min.











S2Case3





02
.. 50 1oo 150 200








Figure 28. Depth and alongshore-averaged alongshore currents at t = 25 min.

Alongshore-Averaged Cross-shore Currents

Time, and alongshore-averaged cross-shore velocity (, ) vector plots

(Figure 29 to Figure 33) gave a qualitative feel for the cross-shore currents. The vertical

axis of the grid was greatly exaggerated by approximately a factor of 40:1 compared to

the horizontal scale. Therefore the direction of the velocity vectors gave a different

impression of the flow depending on the manner chosen for plotting. The results were

presented in two manners, but conveyed the same information. Vectors were plotted

relative to grid units/magnitude scale (A), and specified the vector length as the number

of grid units per unit vector magnitude, or relative to cm/magnitude scale (B) and,

specified the vector length as the number of screen centimeters per unit of vector

magnitude. Each option provided a different sense of the cross-shore currents. Using the

vectors plotted as in A, the correct direction of the velocity vectors was given. In B, the

correct magnitude was given. In both options, a reference vector equal to the relative

scale was included. The grid resolution in the x-z plane varied, however, only 64 x 8

vectors were plotted to allow more space for larger vectors.







53


Cases 3, 5, and 6 produced stronger vertical mixing over the bar (A) as well as

stronger cross-shore currents over the bar and at the shoreline (B). Case 6 had very

strong downwards velocities at the shoreline. Cases 4 and 7 produced similar results.

The strongest cross-shore flow occurred over the bar and along the bottom at the

shoreline edge.

A B


50 100
x(m)


150 200


Figure 29. Case 3: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to
grid units/magnitude scale. B) relative to cm/magnitude scale.


Figure 30. Case 4: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to grid
units/magnitude scale. B) relative to cm/magnitude scale.


--- ------























x (m) x (m)


Figure 31. Case 5: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to grid
units/magnitude scale. B) relative to cm/magnitude scale.


Figure 32. Case 6: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to grid
units/magnitude scale. B) relative to cm/magnitude scale.

An alternate assessment of the magnitude of cross-shore velocities was done by

comparing the depth and alongshore-averaged term (Figure 34). It was clear here

that the magnitude of was much higher (approximately a factor of 2) over the bar,


and at the shoreline for Cases 5, and 6 compared to Cases 4, and 7.













A B
5 //- 5, I-



1 1






2 2 -



1 0


0 50 100 150 200 0 50 100 150 200
x (m) x (m)



Figure 33. Case 7: Time-averaged vertical profiles in the cross-shore of the
alongshore-averaged cross-shore velocity (u,w) vectors. A) relative to
grid units/magnitude scale. B) relative to cm/magnitude scale.






,, 1
01 Case3
Case4
Case5
Case6

008

006

004

002

0 50 100 150 200














Figure 34. Comparison of the depth-averaged term as a function of cross-shore
location.


Vorticity


Comparison of vorticity structures of all the runs showed no major differences.


This led to the conclusion that the small-scale structures seen in Case 2 with depth-linear


forcing were not dependent on grid resolution or aspect ratio. The results also supported









that the 3D effects produced by vertical forcing caused the highly chaotic nature of the

flow, and not resolution issues related to the grid aspect ratio or LES on an anisotropic

grid.

Conclusions of Grid Testing

The high aspect ratios between the horizontal and vertical length scales led to

increased flow in the cross-shore direction. Case 4 was considered the best numerical

experiment because it had the smallest grid aspect ratio. The error difference between


< u" 2 )-< U2 >(4)
Case(i) and Case 4 was defined as Error(i) = u2 >) <2 where (u2)
< u2 >(4)

represented the depth, cross-shore, and time averaged u2 term. The values for the error

between the cases were E(3) = 0.32, E(5) = 0.66, E(6) = 0.49, E(7) = 0.13. The overall

alongshore currents were more similar than the cross-shore exchanges (Figure 27, Figure

28), and the results obtained for Cases 1 and 2 were representative of results that would

be obtained on higher resolution grids.

After comparing the criteria for the five simulations, the grid from Case 4 with 257

by 33 by 17 grid points was used for additional experiments on the 20 m alongshore

domain. This grid fell within the recommendations given by Scotti et al. (1997) and

produced characteristic flow response. Results for this grid were comparable to Case 7

that had a much higher resolution grid. It was advantageous to use the coarser grid to

decrease computational time from months to weeks.

The Effect of Vertical Distribution

Based on the literature review of radiation stresses in the surfzone, a variety of

vertical distributions were tested to determine the effects of depth-varying forcing on the









resulting flow structures and alongshore current (Table 3, and Chapter 3). Six

distributions were tested on the small domain (x, y, z) = (200 m, 20 m, variable). Three

distributions were tested on the large domain (x, y, z) = (200 m, 198 m, variable) to

include effects of larger-scale alongshore current instabilities.

Cross-shore Flow

The vertical distribution of momentum forcing affected the cross-shore velocities.

Stronger cross-shore flow was produced when forcing was concentrated over fewer grid

points at the top of the domain (Case 8 vs. Case 10). Figure 35C and Figure 36B showed

that concentrating the force over the top three grid points also pushed the peak

cross-shore flow to the shoreward side of the offshore bar. Depth-linear forcing

decreased cross-shore velocities when compared to depth-uniform forcing over the same

number of grid points. This was an unexpected result and attributed to the free-slip

condition enforced on the top boundary that set the U-velocity at the boundary by

extrapolating from the four points below it at the end of each time step. The unfortunate

misapplication of a shear stress at the top boundary produced alongshore currents over

the bar, another effect that was attributed to the use of the incompatible free-slip

boundary condition (Figure 36A). There were three cases in Table 3 that suffered from

this implementation; Cases 11, 12 and 16. Cases 11 and 16 ended up putting

approximately 1/5th of the total Fx as a depth uniform force at the surface region. They

were retained in the study to illustrate the influence of the long domain and shear waves

on the current profile in Case 16. Mellor's equations (Figure 36C) produced an opposite

cross-shore circulation pattern, with the shoreward flow along the bottom contour, and

the resulting off-shore flow along the surface. Since this cross-shore circulation pattern

was contrary to field and laboratory observations, his equations weren't considered useful










under these circumstances, and were not pursued further in our study. In Figure 35C, the

strongest currents were slightly shoreward. In Figure 36C, the currents were weaker and

opposite in direction, with the onshore flow (blue) along the bottom and the off-shore

directed flow (red) along the top boundary.


Figure 35. Cases 8-10: Comparison of alongshore and time-averaged u-velocity as a
function of depth and location in the cross-shore domain. A) Case 8. B)
Case 9. C) Case 10. The dark blue contours indicate strong on-shore flow
and the red contours represent the off-shore directed undertow.

A B C

u(mis)




0 0.


0 100oo 150 200 1 50 100 150 200
xim) () (mm)


Figure 36. Cases 11-13: Comparison of alongshore and time-averaged u-velocity as a
function of depth and location in the cross-shore domain. A) Case 11. B)
Case 12. C) Case 13. The dark blue contours in A and B indicate strong
on-shore flow and the red contours represent the off-shore directed
undertow.

Another aspect of the influence of cross-shore mixing on currents was the effect of

vertical mixing. Various 2D and 3D models have assumed hydrostatic flow (u >> w),











which was a valid approximation resulting in predominantly 2D (x-y) currents. The


vertical velocities (w), however, were actually greater than u in various positions of the


cross-shore, especially off-shore of the bar. These vertical currents resembled the strong


vertical mixing in vortex trains seen by Li and Dalrymple (1998) (Figure 37). The


non-hydrostatic approach in the 3D model produced similar vertical eddies offshore of


the bar, with the ratio of u:w in the range of 0.5 at some locations (Figure 38 and Figure


39). Vertical mixing was significant, and a non-hydrostatic model was necessary to


produce that type of cross-shore current and mixing.


t= 10.86 min

5 i










0 50 100 150 200
x(m)


Figure 37. Vector plot of alongshore-averaged u-w velocities for Case 7. u:w = 0.5 at
(xo, z,) (138 m, 3.2 m).


t= 10.86 min
5 -






2o o
o !I .. 1 I I o 1 w(mls)



--

i
0 50 100 150 200
x (m)


Figure 38. Alongshore-averaged w-velocity contours for Case 7, t = 10.86 min. Red
contours indicate upwards flow. Blue contours represent a downwards
directed velocity.











t= 10.86 min
5-

4 u (m/s)



2 00 00o o



0 50 100 150 200


Figure 39. Alongshore-averaged u-velocity contours for Case 7, t = 10.86 min. Red
contours represent offshore velocity and blue represent onshore directed
velocity.

Alongshore Currents

Each forcing distribution influenced the alongshore current to some degree. Each

distribution consisted of depth-uniform y-momentum forcing, and had two-thirds of the

x-momentum inputted as a depth-uniform force. The remaining one-third of the total

x-momentum was varied in the five runs. It was primarily this portion that affected the

cross-shore current, and the eventual location of the alongshore current maximum. When

a force was located over a smaller vertical area, the alongshore current was spread further

into the trough. A depth-uniform distribution, compared to a depth-linear distribution,

(Figure 42 vs. Figure 43) also pushed the alongshore current further into the trough. The

depth-linear forcing produced stronger currents at the shoreline (approximately 40 to

50%) and over the bar (5 to 10%) (Figure 43) compared to the depth-uniform distribution

(Figure 42). Applying a shear stress at the top (Figure 44) produced strong currents at the

shoreline, and reduced spreading of the current across the surfzone.

Figure 40 to Figure 44 show the cross-shore distribution of the alongshore, and

depth-averaged alongshore current over time for the five cases. By 60 min, the current










over the bar had begun to stabilize while the current in the trough was still increasing.

The shoreward shift of the peak alongshore current was compared for the five runs in

Figure 45. The apparent noise of the alongshore current was due to the small alongshore

domain length that the currents were averaged over. Over longer domain lengths, the

currents had a smoother profile because the small-scale instabilities were averaged over a

larger scale.











0 50 100 150 200
x 20







Figure 40. Case 8: Cross-shore distribution of the alongshore and depth-averaged
alongshore current over a 60 min time period. Time progression shows
the migration of the current shoreward of the bar (x = 80 m).

By 60 min, the peak alongshore current at the shoreline seen in Cases 8 and 9 was

not visible in Case 10. The current was more uniform in the cross-shore. Concentrating

the forcing over fewer grid points at the surface had smoothed out the current. The peak

velocity remained around 1 m/s.

Time progression showed the migration of the current shoreward of the bar

(x = 80 m) for Case 11. By 60 min, the current over the bar had begun to stabilize while

the current in the trough was still increasing. A distinct peak still existed at the shoreline.









62




Time progression showed the current did not migrate substantially shoreward of the


bar, with distinct peaks at the shoreline, and over the bar for Case 12. By 60 min, the


current over the bar was starting to stabilize while the current in the trough was still


increasing. Peak velocities exceeded 1 m/s.


Figure 41. Case 9: Cross-shore distribution of the alongshore and depth-averaged

alongshore current over a 60 min time period. Time progression showed

the migration of the current shoreward of the bar (x = 80 m).


Figure 42. Case 10: Cross-shore distribution of the alongshore and depth-averaged

alongshore current over a 60 min time period. Time progression showed

the migration of the current shoreward of the bar (x = 80 m).


---e 10
e 20
e 30
e to
e 50
e 60


e 10
e 20
e 30
e to
e 50
e 60













































Figure 43. Case 11: Cross-shore distribution of the alongshore and depth-averaged

alongshore current over a 60 min time period.


Figure 44. Case 12: Cross-shore distribution of the alongshore and depth-averaged

alongshore current over a 60 min time period.



The results of the alongshore and depth-averaged alongshore currents showed the



influence of 3D forcing. Since the forcing was not entirely depth uniform in the



x-direction, the alongshore momentum located over the bar was advected shoreward into



the trough, without the influence of large-scale shear instabilities or a surface roller. The



most effective form of cross-shore spreading of the alongshore current was produced by


---e 10
e 20
e 30
e to
e 50
e 60


~e 10
e 20
e 30
e to
e 50
e 60







64



Case 10 (Figure 45, Figure 46). Comparison of the depth-averaged current showed the


migration of the peak from over the bar into the trough (Figure 45).

























Figure 45. Cross-shore position as a function of time of the peak alongshore current for
the 5 vertical distsributions tested.
1 2-

















4 -00
30








20























S20 40 60
t(min)


Figure 46. Peak velocity of the alongshore current as a function of time.

To better understand the differences caused by the vertical forcing, the vertical


distribution of the alongshore current was studied. Figure 47 to Figure 50 compared the


effects of vertical forcing on the alongshore current over a 60 min interval for each run.


In each plot, a boundary layer existed along the bottom (light blue) where the alongshore
















current was insignificant. This coincided with the areas of stronger cross-shore currents.



Peak currents occurred slightly below the surface (darker yellow and red areas).


t= 5 min


50 100 150 200
x(m)

S 20 min
















50 100 150 200
x(m)

1= 35 min
















50 100 150 200

5x(m)
t= 50 min


50 100 150
x(m)


1= 10 min
5















0 50 100 150 200
x (m)

= 25 min
















rrm)

00 40 min
0 50 100 150 200

x (m)










255 min
5r-A










0 50 100 150 200
x (min)

t 55 min


t= 15 min











-




50 100 150
x(m)

t= 30 min


50 100 150


1= 45 min
















50 100 150
x(m)

1= 60 min












' p



50 100 150
x(m)


Figure 47. Case 8: Vertical cross-section contour plots of the alongshore-averaged

alongshore current (v). Time progression at 5 min intervals showed the

migration of the current over the bar into the trough area.


0.00
0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1.00



(mI/s)
0.00
0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1.00
0


(mis)
0.00
0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1.00





0.00
0.11
0.22
0.33
0.44
0.56
0.67
0.78
S0.89
1.00


'I









66




In Figure 48, peak alongshore currents (red) were stronger over the same time



intervals compared to Figure 47.


t= 5 min


50 100 150
x (m)

1= 20 min


t= 10 min


S 50 100 150 200
x (m)

t= 25 min


t= 15 min













50 100 150
x (m)

1= 30 min


2


50 100 150
S (m)

t= 35 min


t= 40 min t= 45 min


II


50 100 150 200
x (m)

1t 50 min


3 ,







0 50 100 150 200
x (m)

1t 55 min


50 100 150 200
x(m)


Figure 48. Case 9: Vertical cross-section contour plots of the alongshore-averaged

alongshore current (v). Time progression at 5 min intervals showed the

migration of the current over the bar into the trough area.


50 100 150
x (m)

t= 60 min









67




In Figure 49, a boundary layer along the bottom (light blue) existed. The



alongshore current was weaker, and more spread out in the cross-shore over the same


time periods for the other examples.


t= 5 min


50 100 150
x(m)

t= 20 min













50 100 150
x(m)

t= 35 min













50 100 150
x 5 m

t= 50 min


t= 10 min


S 50 100 150 200
x (m)

t- 25 min













I 50 100 150 200
x (m)

S 40 min


t= 15 min













50 100 150
S (m)

t= 30 min


t= 45 min


t- 55 min t= 60 min


Figure 49. Case 10: Vertical cross-section contour plots of the alongshore-averaged

alongshore current (v). Time progression at 5 min intervals showed the

migration of the current over the bar into the trough area.








68



Linearly distributing the pressure term over the top 3 grid points produced


cross-shore flow at the top boundary at early time (5 min) for Case 11. Alongshore


currents slowly migrated into the trough at similar rates to the other forcing distributions.


t= 5 min


50 100 150
x (m)

1= 20 min


I


. 1 v w v


50 100 150
x (m)


t= 35 min


50 100 150
x (m)
1= 50 min


t= 10 min


t= 25 min












S 50 100 150 200
x (m)

t 40 min


t= 15 min












50 100 150
x (m)
t= 30 min












50 100 150
x (m)
t= 45 min












50 100 150
x (m)


t= 55 min t= 60 min


100 150 200 0 50 100 150
x (m) x (m)


Figure 50. Case 11: Vertical cross-section contour plots of the alongshore-averaged

alongshore current (v). Time progression at 5 min intervals showed the
migration of the current over the bar into the trough area.


50 100 150
x (m)
















t= 10 min


50 100 150
x (m)

t- 20 min












50 100 150
x (im)

t= 35 min













50 100 150
x 5 m

t= 50 min


1'U1


50 100 150
x(m)


t= 25 min






'N;


50 100 150 200
t 40mi
40 min


I IUA









S 50 100 150
S (m)

t= 55 min


t= 5 min


Figure 51. Case 12: Vertical cross-section contour plots of the alongshore-averaged

alongshore current (v). Time progression at 5 min intervals showed the

migration of the current over the bar into the trough area.



Case 12 examined the affect of applying a shear stress at the surface. At t = 5 min,



the effect of the shear stress was apparent with a stronger shoreward migration of the


50 100 150
x (m)

t= 30 min












50 100 150
x (m)

t= 45 min








I'




50 100 150
x(m)

1= 60 min













50 100 150
x(m)


0.00
0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1.00




0.00
0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1.00




0.00
0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1.00
00,11



0.00

0.22
0.33
0.44
0.56
0.67
0.78
0.89
1.00


t= 15 min


\. i







70


surface current. Currents were stronger, and less spread out in the cross-shore over time


compared to other examples. A boundary layer still existed, but was less apparent.


Vorticity

Plots of the vertical vorticity in depth integrated flow models had been used in


previous 2D studies as a means to observe migrating large scale features in the mean


flow. The 3D vertical vorticity contour plots, however, included significant small scale


features that dominated the principal impression conveyed (Figure 52A). Due to the


inherent three-dimensionality of vortex dynamics, and the fine resolution of the grid that


allowed turbulence to be calculated at much smaller length scales than previous 2D


studies, direct vorticity comparisons were not particularly useful for large scale flow


features. Contour plots and velocity vectors of alongshore velocity (Figure 52B)


provided a more readily interpreted and informative tool for alongshore current flow


analysis, and confirmed the mean alongshore current was not simply chaotic, as the


vorticity contours had indicated.


A B
20 (l/s) :- 1 v(mis)
0.0100
0.0080
15 0.0060
0.0040
0.0020
10 0.0000
-0.0020
0.0040
0.0060
-0.0080 /
-0.0100

0 50 100 150 200
x(m) x


Figure 52. Comparison of surface vorticity component (coz) and alongshore surface
current (v) contours in the cross-shore (x) and alongshore (y) domain. A)
Vorticity contours showed small scale features that were highly erratic and
suggested a very unstable current. B) Velocity contours and vectors of the
same data showed a strong current that had peaks at the shoreline and
around the bar. Snapshot was of Case 11 at t = 50 min.










Small scale perturbations seen as vorticity patches with horizontal length scales on

the order of 2 m were very apparent in the cases examined in Figure 52 and Figure 53.

Similar small scale vortex structures were not produced in 2D model studies and further

understanding of what properties influenced these structures in 3D were needed. Two

potential influences other than the vertical distribution of forcing were examined: spatial

grid size and velocity filtering.

Two-dimensional models did not produce the same vorticity structures as seen in

the 3D model for a number of reasons. They used coarser grids, with horizontal grid

spacing in the range of 2 to 5 m which removed the ability to calculate small-scale

turbulence that might have been present. As well, a 2D, depth-averaged flow could have

never produced 3D turbulence. Instead of a cascade of turbulence from larger to smaller

scales, a 2D flow would have produced eddies that coupled with each other to form larger

eddies, known as an inverse cascade of energy. Horizontal vorticity (ox, coy) was also not

produced in 2D flows. In a 3D flow, these turbulent eddies could have been stretched

and tilted, which could have influenced vertical vortex structures (coz).

To study the effect of horizontal grid resolution on the vorticity contours, the model

output velocities were spatially averaged to a 5 m grid resolution and then vertical

vorticity was calculated from the averaged velocity. The 3D model had a spatially

varying, high resolution grid that was able to calculate turbulence at scales on the order of

0.2 m in the nearshore to 1 m at the offshore. By taking an average velocity over a 5 m

by 5 m spatial grid, the small scale perturbations were removed. The effective grid

resolution was reduced from (nx, ny) = (257, 33) to (nx, ny) = (40, 4). Spatial averaging

the velocities with a coarser resolution smoothed out the vorticity contours (Figure 53A).










This result demonstrated one of the influences of grid resolution on the resulting vorticity

contours. Comparing Case 1 and Case 14 (discussed later) that both used depth-uniform

forcing on the 200 m x 198 m physical domain at different horizontal grid resolution also

showed that the finer grid produced smaller scale turbulent features. The omission of 3D

turbulence and the use of coarse grids found in 2D models did not permit the presence of

small-scale turbulent features as was seen in our results. Time averaging over a variety

of time intervals was done to find the mean vorticity field. When the vorticity data were

averaged over long time periods, as in Figure 53C, the associated vorticity contour plots

were smoothed out. Time averaging over small time periods, however, did not smooth

out the contour plots.

A B C




10 S1 0 0
0 0080

0 0020








Figure 53. Sample comparison of spatially averaged, original vorticity, and time-
averaged contours at vertical grid point nz-1. Plots were generated from
Case 11 data, t = 50 min. A) vorticity contours for spatially averaged data
at 5 m grid spacing. B) original vorticity plot on a variable grid. C) time
averaged (50 min) vorticity contours of the same simulation. Time
averaging of the data over long time periods showed the mean vorticity
and smoothed out the contours.

Energy spectra of the u- and v-velocities were used to determine the frequency

band at which the majority of the energy was concentrated. When significant energy was

concentrated at lower frequencies, it indicated that the large scale features were more

important to the transportation of energy. (E.g., alongshore currents with shear waves









had a peak energy concentration of about 1 m2/s2 located around 0.01 Hz.) Spectra of the

currents during run-up time of the model were generated at various locations of x and z

within the domain. Results showed that the energy was concentrated at low frequencies.

In the spectral analysis of the depth-averaged currents at x = 40 m in the cross-shore

(center of the bar trough), the u-velocity spectrum had a peak at 0.047 Hz (Figure 54),

and the v-velocity spectrum had a peak at 0.031 Hz (Figure 55). The time and depth-

averaged u- and v-velocities were 0.004 m/s, and -0.52 m/s respectively. The length

scales of the peak energy were determined by using the following equation:

eE (m2 /s2)
Length(m)= where subscript (p) indicated the peak value. The length
f (Hz)

scale of the peak energy structures in the u-energy spectrum was approximately 2 m, with

fluctuation velocities around 0.10 m/s. The length scale in the v-energy spectrum was

approximately 6 m, with velocities around -0.2 m/s. The energy spectra showed that a

significant amount of energy in the velocity field was contained within the small-scale

features shown. This was also verified by taking the RMS of the peak v-velocity,

Vp -av, Vag which was equal to almost 50% of the mean flow at this cross-shore


location. Spectral analysis on equilibrated flow (t > 60 min) produced slightly different

results because the effects of run-up and current migration were not present. The purpose

of the filtering, however, was to determine the energy distribution in the currents at times

when small scale instabilities were first appearing.

Low-pass filtering the velocities at these frequencies to remove high frequency

fluctuations was done to determine the effect (Figure 56 and Figure 57). Filtering above

the peak frequencies was an attempt to remove high-frequency noise that might have







74


obscured trends in the data. In this particular example, filtering did not remove the small

scale perturbations from the velocity signal because they were where the bulk of the

energy of the current lied.


0 005 01 015 02 025
f(Hz)


03 035 04 045 05


Figure 54. Depth-averaged U(xo,yo,t)-velocity energy density spectrum for Case 8,
where Xo = 40 m, yo = 10 m, t =12-66 min. Data was sampled at 1 Hz,
totaling 3200 data points. Peak frequency = 0.0469 Hz. Maximum energy
= 0.0105 m2/s2.


0 035


0005


f(Hz)


Figure 55. Depth-averaged V(xo,yo,t)-velocity energy density spectrum for Case 8,
where Xo = 40 m, yo = 10 m, t = 12-66 min. Data was sampled at 1 Hz,
totaling 3200 data points. Peak frequency = 0.0313 Hz. Maximum energy
= 0.0386 m2/s2.



















004


002


-0 02


-0 06


2400 2450 2500 2550 2600 2650 2700 2750
t (sec)


Figure 56. Time sample of velocity fluctuations (black) and the low-pass filtered (red)
depth-averaged u-velocity at 0.03 Hz taken from energy spectrum data.

U(xo,yo,t) for Case 8, where o = 40 m, yo = 10 m, t = 40-46 min.





filtered
-01

-02


-03 -


-04

05 -


500 1000 1500 2000 2500 3000 3500


Figure 57. Time sample of raw data (black) and the low-pass filtered (red)
depth-averaged v-velocity at 0.046 Hz. V(xo,yo,zo,t) for Case 8, where
o = 75 m, y = 10 m, t = 12-66 min.









Effect of Domain Length

Two domain lengths were used for the simulations. The 20 m domain prohibited

the development of large scale alongshore shear instabilities [O (100 m)] that could have

influenced the cross-shore diffusion of the alongshore current. The 20 m domain

restricted the focus of the study on the effects of the vertical distribution of the forcing.

To verify that this assumption was correct, simulations with depth-uniform forcing in

both directions were run on 20 m and 200 m domains. The alongshore current remained

stable on the 20 m domain for the length of the run. On the 200 m domain, however, the

current became unstable over time, and broke down into features with alongshore

wavelengths of approximately 100 m, similar to those presented in previous 2D model

results (Slinn et al. (1998) and Ozkan-Haller and Kirby (1999)). These tests verified that

domain length influenced the stability of the alongshore current. The extent of the

domain length's influence on the alongshore currents generated was dependent on the

vertical distribution of forcing. Three vertical distributions were tested. Case 14 was

forced with a depth-uniform distribution. Case 15 used the vertical distribution from

Case 10, and Case 16 used the distribution from Case 11.

When the domain length was increased and large scale instabilities were allowed to

influence the alongshore current, they spread the mean alongshore current laterally. The

instability of V(x) was most visible when the model was forced by a depth-uniform force

(Figure 58). Without the large-scale instabilities, the current continued to grow over the

bar. The large-scale shear instabilities also decreased the maximum mean alongshore

current velocities through increased cross-shore mixing. This was qualitatively visible in

Figure 58 to Figure 60 that compare the short and long domain results of the alongshore







77


depth-averaged alongshore currents for the various forcing distributions. When Fx was

not entirely uniform as in Case 14, the alongshore current was spread laterally at earlier

times. It appeared that the instability imposed by the vertical shear of the cross-shore

currents influenced the stability of the large-scale features of the alongshore current as

well, and caused the shear waves to develop at earlier times.

The short domain produced stronger currents over the bar. The alongshore current

was more widely spread across the cross-shore with the long domain that included the

effects of the large scale instabilities. A strong jet remained at the shoreline for both

domain lengths that indicated the large scale motions had minimal effect on the currents

at that location.

Large-scale shear instabilities on the order of 100 m were still present in the 200 m

domain when forcing was not depth uniform. Vertical vorticity contours for Cases 14 -16

are compared in Figure 61 to Figure 63.


-2
t(min)
25






0 0 50 100 150 200
x (m)








Figure 58. Influence of domain length on alongshore currents for depth-uniform forcing:
Alongshore and depth-averaged alongshore current profiles as a function
of cross-shore position and time. Solid lines represent the 200 m
alongshore domain. Dashed lines represent the 20 m alongshore domain.



















-1

-09
-08

-07
-06

-05
-04

-03
-02
-01


t(min)
-5
-10
15
20
205
10
15
20


Figure 59. Influence of domain length on alongshore currents for Case 10 forcing:

Alongshore and depth-averaged alongshore current profiles as a function

of cross-shore position and time. Solid lines represent the 200 m

alongshore domain. Dashed lines represent the 20 m alongshore domain.


t(min)
-5
10
15
20
-25
30
---5
....10
--- 15
20
--- 25
--- 30


Figure 60. Influence of domain length on alongshore currents for Case 11 forcing:

Alongshore and depth-averaged alongshore current profiles as a function

of cross-shore position and time. Solid lines are 200 m alongshore

domain. Dashed lines are for 20 m alongshore domain.










1= 15 min


1= 35 mln 1= 45 mm


-.2 -0.01(
-0.02D -0.013


I I
-0.007 0.000


0.07


O.DI3 0.020


Figure 61. Case 12: Contours of the depth-averaged vorticity as a function of time. The
current became unstable around 20 min and produced large scale
instabilities (t = 25 min) that broke down into smaller scale turbulence
(t = 45 min).


1= 5 min


1= 15 rin


1= 20 mm


-0.020 -0.015


-0.007 0.000


O.D07


0.013 0.020


Figure 62. Case 15: Contours of the depth-averaged vorticity as a function of time. The
current became unstable at the shore almost immediately (t = 5 min) and
large scale instabilities were less visible.


1= 5 rin


1= 25 min











1=5s 1in 15 (min 1= 25 mmin











T lts) n
fro 77. ... I I .. .-a t I case. W-A .h




-0.02D -0.013 -0.007 0.000 O.D0 0.Ds13 0.020

Figure 63. Case 16: Contours of the depth-averaged vorticity as a function of time. The
current at the shoreline became unstable at early time, while the current
over the bar became unstable around 15 min, and evolved into smaller
scale turbulence (t = 25 min).

The large-scale [0 (70 m)] instabilities apparent in Case 16 (t = 15 min) resulted

from the instability of the alongshore current similar to the depth-uniform case. With the

depth-linear forcing in the top three grid points, the effect of the top grid point was

reduced because of the free-slip condition. This removed a portion of the depth-varied

forcing, and put a larger influence on the depth-uniform component. This resulted in a

more stable alongshore current over the bar that was strongly influenced by the

alongshore shear instabilities on the current. Although the small-scale vorticity features

obscured the presence of the larger scale shear waves, they were still present, and

influenced the alongshore current position over time. In effect, the depth-dependent

forcing pushed the peak alongshore current velocities into the trough, and the shear

waves spread the current horizontally, with preferential spreading in the shoreward

direction for depth-dependent forcing.














CHAPTER 5
SUMMARY AND DISCUSSION

Findings and Results

A 3D LES model was adapted to simulate alongshore currents forced by breaking

waves. It produced strong alongshore currents in the trough area without the use of a

roller model, alongshore pressure gradients, alongshore shear waves, or ad-hoc tuning of

the bottom friction coefficient. The 3D model allowed the effect of vertical forcing on

alongshore current structure to be studied on a small domain [O (20 m)]. The

development and migration of alongshore currents were analyzed as the alongshore

current spun up to steady state. The model did not specify the bottom friction and

achieved reasonable, balanced values of V(x) for the specified wave field conditions.

The effect of grid resolution, vertically dependent forcing, and alongshore domain length

were all examined.

The LES closure scheme was sensitive to grid aspect ratios, and overall grid

resolution. Aspect ratios less than 4:1 generated reproducible currents at varying grid

resolutions. Cross-shore flow was most affected by grid aspect ratio. Grid cells that

were long and flat produced preferential undertow. The overall magnitude and location

of the mean alongshore current, however, was not as sensitive to grid aspect ratio. The

five cases tested produced similar mean alongshore current profiles in the cross-shore.

The ideal vertical distribution of the radiation stress gradients was dependent on the

situation considered. In a phase-averaged model, assumptions were made for the forcing

terms that existed outside the modeling domain. Depth-uniform forcing produced









alongshore current and shear instabilities similar to 2D models. Non-uniform vertical

forcing, however, produced small scale structures that were not seen in 2D models.

Vertical forcing effectively produced cross-shore currents that mixed the alongshore

current horizontally: currents over the bar were pushed shoreward by wave breaking

stresses, and currents at the shore were pulled seaward into the trough by the resulting

undertow. Case 10 that had the pressure term uniformly distributed over the top three

grid points, was the most effective means of mixing the alongshore current, and produced

reasonable peak velocities [O (1 m/s)] in the trough. Case 12 produced the least

satisfactory results. It implemented 1/6th of Fx as a surface shear stress. The effect on the

location of the alongshore current was minimized, however, by the co-implementation of

the free-slip boundary condition at the surface. The alongshore current maxima that

resulted remained over the bar, and at the shoreline, with maximum velocities exceeding

1.5 m/s. It was learned that the methods of implementing the surface stress forcing were

incompatible with the free-slip boundary condition. Alternate boundary conditions that

will allow a surface stress can be implemented in future work.

The use of a 3D, non-hydrostatic model produced strong vertical mixing in the

cross-shore. Vortex structures similar to those from lab results were produced in the

model. Turbulence is described as a 3D phenomenon and its effects on fluid dynamics

are best modeled in a 3D system. Some previous two-dimensional models attempted to

parameterize effects of 3D turbulence by assuming isotropic horizontal diffusion.

Non-isotropic net cross-shore mixing will always be present as long as there is

cross-shore circulation and the alongshore currents are not entirely depth-uniform. This

always occurs in nature because of the bottom boundary layer and cross-shore









non-uniformity of currents that are advected horizontally by vertically dependent forcing.

The 3D model also captured another important coupling between horizontal and vertical

structures of the flow. Small-scale turbulence generated in the vertical (cox and coy) could

be stretched and tilted. These evolved into small scale vertical turbulent structures (COz)

that produced enhanced cross-shore diffusion.

Complementary lines of investigation, such as the formulations in the quasi-3D

implementation of SHORECIRC (Zhao et al., 2003), had considered net effects of the

undertow on depth-averaged currents. They did not observe current behavior

qualitatively similar to results presented here. This was explained by the fact that the

quasi-3D information was used primarily to estimate an isotropic horizontal diffusion

coefficient. This was a good approximation if the alongshore current had been

depth-uniform because the shoreward mass-flux in the top half of the water column

would have carried equal amounts of alongshore momentum as the undertow had carried

offshore. If the alongshore current was depth-dependent, however, this approximation

was no longer adequate. Depth-dependence of the currents occurred because of several

factors. These could have included depth-dependent momentum input, a thick boundary

layer developed by bottom friction, or from ongoing depth-dependent cross-shore

diffusion. The net effect of 3D cross-shore circulation produced non-isotropic,

preferentially shoreward diffusion of the alongshore currents.

Increasing the domain length influenced the alongshore velocity profiles. Peak

currents were decreased, and the profile was laterally spread further across the

cross-shore domain. Domain lengths on the order of 200 m produced large scale shear

waves that horizontally diffused the mean alongshore current across the surfzone. This









feature was most noticeable with depth-uniform forcing. The magnitude of the influence

of the large scale motions was dependent on the vertical distribution of forcing. When a

greater portion of the forcing was depth-uniform over the water depth, as in Case 14, the

large scale instabilities were the main source of horizontal mixing of the alongshore

current. In Case 15, however, the effect of large scale motions was not as apparent.

Horizontal diffusion of the current was present due to large scale motions not captured in

the small domain, but large structures were obscured by the small-scale instabilities

produced by the vertical distribution of forcing. Including the effects of vertically

dependent forcing and large scale shear instabilities produced mean alongshore currents

that were weaker, and centered in the bar trough. The omission of either of these effects

produced alternative alongshore current distributions. The omission of vertically

dependent forcing led to weaker, more stable currents centered over the bar, whereas the

omission of shear waves led to stronger currents centered in the bar trough.

Future Work

A great deal remains to be done on this line of work. Linear stability analysis of

the three velocity profiles should show that the growth rates of instabilities of the vertical

shear are more rapid than growth rates from the horizontal shear. Model improvement

continues with the addition of mass flux to future simulations. Alternative vertical

distributions of the alongshore and cross-shore forcing based on complimentary model

and laboratory studies to include effects of breaking and nonlinear waves should also be

examined. Simulations with different wave conditions (Ho, Tp, 0o) should also be

pursued and produce different alongshore current dynamics and increase understanding

on the relative importance of including large scale shear instabilities and vertically

distributed forcing.






85


Larger domains and finer grid testing should also improve the statistics of the

model results. Larger domains will more effectively include alongshore instabilities and

reduce the wall effects on the offshore boundary. Finer grids, especially in the vertical,

will better resolve turbulence closer to the boundaries, but may affect the stability of the

pressure solver. Comparing model results to available field data from the DELILAH

experiment could be used to help validate the model.
















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