PUBLICATIONS. Journal of Geophysical Research: Oceans

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1 PUBLICATIONS RESEARCH ARTICLE Key Points: Large-eddy simulation of solitary wave breaking is validated with laboratory data Simulation reveals that obliquely descending eddies are hairpin vortices Unless in shallow depth, near-bed and near-surface turbulence are not correlated Correspondence to: Z. Zhou, Citation: Zhou, Z., J. Sangermano, T.-J. Hsu, and F. C. K. Ting (214), A numerical investigation of wave-breakinginduced turbulent coherent structure under a solitary wave, J. Geophys. Res. Oceans, 119, , doi:1.12/ 214JC9854. Received 25 JAN 214 Accepted 1 SEP 214 Accepted article online 18 SEP 214 Published online 2 OCT 214 A numerical investigation of wave-breaking-induced turbulent coherent structure under a solitary wave Zheyu Zhou 1, Jacob Sangermano 1,2, Tian-Jian Hsu 1, and Francis C. K. Ting 3 1 Center for Applied Coastal Research, Department of Civil and Environmental Engineering, University of Delaware, Newark, Delaware, USA, 2 Key Environmental, Inc., 2 3rd Ave., Carnegie, Pennsylvania, USA, 3 Department of Civil and Environmental Engineering, South Dakota State University, Brookings, South Dakota, USA Abstract To better understand the effect of wave-breaking-induced turbulence on the bed, we report a 3-D large-eddy simulation (LES) study of a breaking solitary wave in spilling condition. Using a turbulenceresolving approach, we study the generation and the fate of wave-breaking-induced turbulent coherent structures, commonly known as obliquely descending eddies (ODEs). Specifically, we focus on how these eddies may impinge onto bed. The numerical model is implemented using an open-source CFD library of solvers, called OpenFOAM, where the incompressible 3-D filtered Navier-Stokes equations for the water and the air phases are solved with a finite volume scheme. The evolution of the water-air interfaces is approximated with a volume of fluid method. Using the dynamic Smagorinsky closure, the numerical model has been validated with wave flume experiments of solitary wave breaking over a 1/5 sloping beach. Simulation results show that during the initial overturning of the breaking wave, 2-D horizontal rollers are generated, accelerated, and further evolve into a couple of 3-D hairpin vortices. Some of these vortices are sufficiently intense to impinge onto the bed. These hairpin vortices possess counter-rotating and downburst features, which are key characteristics of ODEs observed by earlier laboratory studies using Particle Image Velocimetry. Model results also suggest that those ODEs that impinge onto bed can induce strong near-bed turbulence and bottom stress. The intensity and locations of these near-bed turbulent events could not be parameterized by near-surface (or depth integrated) turbulence unless in very shallow depth. 1. Introduction To improve the predictive skill of nearshore morphological evolution models, it is necessary to incorporate a robust parameterization of sediment transport fluxes associated with a range of known mechanisms, such as nonlinear wave statistics, undertow, wave-current interaction, bottom boundary layer streaming, and wave-breaking turbulence. Particularly, many field and laboratory studies of surf zone sediment transport provide strong evidence that wave-breaking-induced turbulence can approach seabed and cause large sediment suspension [e.g., Sato et al., 199; Jaffe and Sallenger, 1992; Ogston and Sternberg, 1995; Beach and Sternberg, 1996; Voulgaris and Collins, 2; Aagaard and Hughes, 21; Grasso et al., 212]. For example, field observation during Duck85 reported by Beach and Sternberg [1996] demonstrated that sediment suspension is enhanced by wave-breaking turbulence, but it is highly dependent on the locations in the surf zone and breaker type. During Duck94 field observation, Ogston and Sternberg [1995] further showed that breaking waves can induce an order of magnitude greater turbulence in the water column than that of bottom boundary layer alone. Motivated by these studies, many coastal evolution models parameterized enhanced turbulence level in the water column and the resulting suspended load via turbulent dissipation rate associated with surface breaking/roller [e.g., Kobayashi and Johnson, 21; Ruessink et al., 27]. However, several field and laboratory studies clearly demonstrated the highly intermittent nature of large near-bed turbulence events and sediment suspension events associated with wave-breaking turbulence [Jaffe and Rubin, 1996; Cox and Kobayashi, 2]. During a large-scale wave flume experiments at O. H. Hinsdale Wave Research Laboratory at Oregon State University, concurrently measured free-surface elevation, flow velocities, and sediment concentration at different locations in the water column near the breakpoint were analyzed. Using wavelet analysis [Scott et al., 29] and conditional probability [Yoon and Cox, 212] techniques, both studies found that near-bed turbulence and sediment suspension events are intermittent. From the perspective of local point measurement, there is only about 3% chance that steep wave breaking ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6952

2 observed near the surface is associated with large sand suspension near the bed [Scott et al., 29]. Hence, it is not straightforward to correlate free-surface breaking information (e.g., roller dissipation) with bottom sand suspension. It would be useful to obtain concurrent data for surface wave-breaking and bottom suspension events (or near-bed high turbulence events) over a wide spatial coverage onshore of the breakpoint so that a more quantitative analysis on the spatial variability can be carried out. This study reports our first step to use a 3-D numerical simulation of the entire wave flume for this endeavor. The mechanism through which surface generated turbulence can be injected onto the bed is revealed via laboratory flume experiments. Pioneering work of Nadaoka et al. [1989] demonstrated the evolution of surface roller into 3-D obliquely descending eddies (ODEs), which effectively penetrate into the water column and are capable of approaching the bed. The dynamics of wave-breaking induced turbulent coherent structures were later quantified by several laboratory studies where clear evidences of ODEs interacting with the bed were reported [e.g., Kimmoun and Branger, 27; Ting, 26, 28; Ting and Nelson, 211; Huang et al., 21]. Using particle imaging velocimetry (PIV), a series of laboratory observations of Ting [26, 28, 213] and Ting and Nelsen [211] revealed that an ODE is consisted of two counter-rotating eddies with intense downward velocity fluctuations, called downburst. The size of these ODEs is on the order of the local water depth and hence the instantaneous flow field has strong spanwise variability. Motivated by the aforementioned laboratory observations on the existence of ODEs [e.g., Nadaoka et al., 1989], several numerical model studies have demonstrated that the generation and evolution of turbulent coherent structures under breaking waves can be captured by a 3-D large-eddy simulation (LES) approach [Christensen and Deigaard, 21; Christensen, 26; Watanabe et al., 25; Lubin et al., 26]. These numerical studies utilized standard finite difference or finite volume scheme with the volume of fluid (VOF) method to track free-surface evolution. More recently, the Smooth Particle Hydrodynamic (SPH) method was also utilized to better resolve the high free-surface curvature during breaking [Dalrymple and Rogers, 26]. Facilitated by GPUSPH [Herault et al., 21], Farahani and Dalrymple [213] was able to use several millions of SPH particles to resolve the generation and evolution of 3-D turbulent coherent structures under breaking waves. However, these studies do not focus on the fate of these ODEs near the bed. The purpose of this study is to validate a 3-D LES numerical model to simulate wave breaking over a slope and to further study the fate of wave-breaking-induced turbulent coherent structures approaching the bed. As a first step, this paper focuses on simulating solitary wave breaking over a sloping beach. Simulating solitary wave is less computationally intensive comparing to periodic wave conditions and hence it allows us to use higher numerical resolution. Moreover, Ting [26, 28] reported an extensive laboratory data set on solitary wave breaking over a 1/5 slope, which allows us to carry out comprehensive model validation, including the characteristics of ODEs. 2. Model Formulation In this study, a 3-D turbulence-resolving approach is adopted via large-eddy simulation (LES). Through filtering the Navier-Stokes equations, the flow field is separated into the resolved and the unresolved fields. With appropriate numerical resolution, the generation and the evolution of turbulent coherent structures are resolved. The effect of the unresolved small-scale turbulence is parameterized by an appropriate subgrid scale closure scheme. Using a finite volume scheme, a boundary-fit domain is used to establish the numerical wave flume in order to better resolve the interaction of wave-breaking turbulence with the bed. Volume of fluid method is used to track the evolution of air-water interface. The numerical implementation of the present 3-D model is achieved by OpenFOAM, an open-source C11 library of solvers for Computational Fluid Dynamics (CFD). It contains a Navier-Stokes equations solver based on a finite volume scheme for two immiscible fluids with an interface tracking capability [Hirt and Nichols, 1981], called interfoam [Rusche, 22; Klostermann et al., 212]. This open-source solver has recently been utilized by other coastal researchers to study nearshore hydrodynamics based on the Reynolds-averaged Navier-Stokes (RANS) approach [Jacobson et al., 211; Higuera et al., 213). Hence, the present study also validates the capability of this solver for turbulence-resolving simulation Governing Equations To simulate wave-breaking processes, the LES approach is employed in this study to balance the need of resolving a large portion of the turbulent flow energy in the model domain while parameterizing the ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6953

3 unresolved field with a subgrid closure in order to maintain a reasonable computational cost. In LES, the Navier-Stokes equations are filtered such that only motions with length scales greater than the filter scale are directly solved. Here the flow field is filtered numerically and the filter length D is defined as the characteristic length scale of the grid size: D 5ðDx Dy DzÞ 1=3 (1) where Dx; Dy; Dz are the grid size in streamwise, spanwise, and vertical directions, respectively. The filtered Navier-Stokes equations are written i 5 i j 52 u i u i u j 2u i u j 1gi j j x j where i, j 5 1, 2, 3, for the present 3-D flow field, u i is the filtered flow velocity, q is the fluid density, p is the fluid pressure, and m is the kinematic viscosity of the fluid. Here the overbar represents a filtered quantity, and the third term on the right-hand side of equation (3) is the gradient of subgrid stress tensor s ij, which is written as: s ij 5u i u j 2u i u j (4) A closure model is required for s ij (see section 2.2). In the present study, the fluid field consists of water and air and both phases are solved in the present study using the volume of fluid (VOF) method [Hirt and Nichols, 1981] and hence the general representation of fluid density q is written as: q5a 1 q 1 1ð12a 1 Þq 2 (5) where q 1 51 kg=m 3 is the density of water, q 2 51kg=m 3 is the density of air, a 1 is the volume fraction of water contained in a grid cell. The governing equation of the volume fraction a 1 in an immiscible twofluid system is written as [Hirt and Nichols, @x j ða 1 u 1i Þ5 (6) where u 1i represents the velocity of the water phase. When a grid cell is completely occupied by water (or air), a (or ). Grid cells representing the water-air interface are of a 1 ranging between zero and one. Consequently, the exact value of u 1i for a given interfacial grid cell cannot be obtained and requires further approximation (e.g., interpolation). This issue is discussed in section 2.3, where more detailed numerical implementation is discussed Subgrid Closure Model The dynamic Smagorinsky closure model based on the work of Germano et al. [1991] and modified by Lilly [1992] is utilized in the present study. The subgrid stress tensor is calculated as: s ij d ijs kk 52m ts 2S ij (7) where S ij 5 i i is the resolved strain-rate tensor and m ts is the subgrid scale viscosity. According to the standard Smagorinsky closure, the subgrid scale viscosity is assumed to be calculated by Pope [2]: m ts 5ðC s DÞ 2 js ij j (8) 1=2 where C s is the Smagorinsky coefficient and js ij j5 2S ij S ij is the magnitude of the strain-rate tensor. In the standard Smagorinsky closure, C s is specified as a constant of.167. Instead of maintaining a constant value of C s, the dynamic Smagorinsky model applies a second test filter D^, which is chosen to be 2D in the numerical model, to the resolved flow field [Lilly, 1992] and computes a test scale stress tensor. Subtraction of the subgrid scale stress tensor from the test grid scale stress tensor reveals the range of resolved motion between the two scales. A proper value of the dynamic Smagorinsky coefficient C s is then chosen to minimize the discrepancy. In this paper, simulation results are primarily presented by using the dynamic ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6954

4 Smagorinsky closure. However, the performance of both the dynamic Smagorinsky model and the standard Smagorinsky model is evaluated by comparing with measured second-order turbulent statistics under a breaking solitary wave Numerical Implementation The filtered Navier-Stokes equations (2) and (3) are solved by the segregated pressure correction method following the procedure of Rusche [22] [see also Klostermann et al., 212]. A second-order implicit Crank- Nicholson scheme is used in time integration. In the present finite volume scheme, the convection term is converted into integrals over surfaces bounding each cell using Gauss theorem. Then the cell-face flux / f 5u!! f S, where subscript ðþf denotes face values and! S is the cell face, is obtained by a combination of upwind scheme and central difference scheme / f 5k/ ð P 2/ N Þ1/ N, in which subscripts ðþ P and ðþ N represent the present and the next adjacent cells, respectively, and k is a weighting factor calculated using a flux limiter from the limited scheme [Jasak et al., 1999] and weighting factor of a linear interpolation scheme [Berbervoić et al., 29]. In the present study, the flux limiter is a function of the ratio of the consecutive gradients of the variable of interest between two adjacent computational points, which is typical in the Total Variation Diminishing (TVD) approach [Berbervoić, 21; Klostermann et al., 212]. The diffusion term is discretized by approximating face values with central difference scheme, and evaluating the resulting flux through the face with nonorthogonal correction [Jasak, 1996]. The numerical model is fully parallelized with Message Passing Interface (MPI) Interface Tracking Method The volume of fluid (VOF) method is employed to track the free-surface (water-air interfaces). The volume fraction, a 1, is computed with equation (6). As mentioned previously, in order to solve equation (6), the velocity u 1i has to be determined. Instead of adopting the method proposed by Hirt and Nichols [1981], where the mixture velocity u 1i is estimated with a free-surface reconstruction scheme, a surface compression method is adopted here due to its computational efficiency. A detailed discussion on this method and a performance study is reported in Klostermann et al. [212]. However, a briefly overview of the interface compression scheme is discussed here. The mixture velocity u i is calculated by the velocity of the water phase and air phase: A relative velocity u ri between these two phases is defined as: Thus, equation (6) can be rewritten as: u i 5a 1 u 1i 1ð12a 1 Þu 2i (9) u ri 5u 1i 2u 2i 1r ð u ia 1 Þ1r½u r;i a 1 ð12a 1 ÞŠ5 (11) The third term on the left-hand side is called the interface compression term. Theoretically, this term can be considered as a necessary term to ensure accurate calculation of the interface location. Numerically, this interface compression term, once estimated appropriately, can maintain a minimal diffusion between cell faces. It should be pointed out that the advantage of this method is its computational efficiency because it avoids complex interface reconstruction. However, its accuracy must be also addressed. Klostermann et al. [212] carried out a comprehensive benchmark tests and validated the present numerical scheme with results computed from more accurate arbitrary Lagrangian-Eulerian scheme and Level-Set scheme for a rising bubble problem. It is found that the present scheme can achieve similar accuracy with these two more accurate schemes as long as surface tension effect is of minor importance. In this study, we simulate depth-limited surface wave breaking without considering surface tension effect and hence adopting the interface compression method is appropriate. Although the numerical scheme discussed here computes both the water phase and the air phase, the numerical resolution that we implemented in this study is not sufficient to resolve the complete air entrainment and bubble dynamics during the wave-breaking process. As we will discuss in section 3.1, the grid size used here is about 5 mm near the surface and hence it is fair to say that when bubble size is reduced to around 1 cm, bubble dynamics cannot be captured well and the effect of bubbles is neglected. The discretization scheme for the convection terms in equation (11) (second and third terms) is similar to that of equation (3) except that different flux limiters are used. For the second term, the limiter of Van Leer [1974] ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6955

5 is used, while for the interface compression term, we follow that adopted by Berbervoić [21] [see also Weller, 28; Deshpande et al., 212] Boundary Conditions In typical wave flume simulation, the numerical resolution is not sufficient to resolve the viscous sublayer and buffer layer. Hence, near-wall modeling is required. In the present study, we ensure the velocity follows the following semiempirical profile [Spalding, 1961]: z 1 5u ju E eju1 2 ju1 6 ju1 (12) where E59:8, j5:41, z 1 5zu s =m; u 1 5U=u s with U represents the velocity magnitude and u s is the friction velocity. For large z 1, equation (12) becomes the familiar logarithmic law of the wall while for small z 1,it reduces to linear profile of u in the viscous sublayer. Equation (12) is solved iteratively to obtain the friction velocity u s. The calculated friction velocity is used as bottom boundary condition for the momentum equations parallel to the bottom. Moreover, bottom boundary condition for subgrid scale viscosity is then calculated as: u2 s m ts 5 2m Although the near-wall modeling approach discussed here is originally developed for statistically steady flow, its applicability in oscillatory flow in the context of LES is demonstrated by Radhakrishnan and Piomelli [28] to be reasonably accurate. However, we must also point out that in the present application, wavebreaking turbulence can further interact with the bed and under this circumstance, the applicability of the present near-bed modeling formulation may become questionable. Solitary wave is sent in from the inlet of the numerical flume by specifying time series of free-surface elevation and streamwise and vertical velocity profiles based on the first-order solution of Lee et al. [1982]. This is implemented numerically via the groovybc toolbox [Gschaider, 29]. Following most of the turbulenceresolving simulation studies [e.g., Vittori and Verzicco, 1998; Lubin et al., 26; Ozdemir et al., 213], periodic boundary conditions are used for the two lateral boundaries in the numerical wave flume instead of the sidewalls. With periodic boundary conditions for two lateral boundaries, the width of the numerical flume is chosen to ensure that it is much larger than the characteristic size of the largest turbulent coherent structure. In general, the necessary domain width becomes smaller than the actual flume width and hence higher spatial resolution in the y direction can be used to better resolve flow turbulence. In addition, by implementing periodic boundary condition for two lateral boundaries, statistical flow quantities can be obtained by spatial averaging over the spanwise (y) direction to approximate the ensemble-averaged flow quantities. However, this procedure needs to be evaluated cautiously in the present application where the flow is statistically unsteady and inhomogeneous in the streamwise and vertical directions. We will discuss this issue more thoroughly in section Solitary Wave Breaking Over a 1/5 Slope Beach [Ting, 26] The capability of interfoam solver [Klostermann et al., 212] to simulate nearshore wave propagation in the Reynolds-averaged modeling framework has been carried out extensively by Jacobson et al. [211] and Higuera et al. [213]. We have also carried out model validation to assess the performance of the model in terms of wave generation and runup for nonbreaking and breaking solitary waves reported in Synolakis [1987]. Very good agreements are observed. Further details are referred to Sangermano [213]. In section 3.1, we validate the capability of the numerical model for 3-D LES through comparisons with measured free-surface elevation, mean, and turbulent flow statistics for solitary wave breaking over a slope reported by Ting [26, 28]. The performance of the dynamic Smagorinsky closure and the standard Smagorinsky closure is also discussed. In section 3.2, turbulence coherent structures simulated by the present model are further compared with PIV data reported in Ting [28] Model Validation The laboratory flume of Ting [26, 28] is of 25 m long by.9 m wide by.75 m deep with a constant slope of 1/5 extending the length of the flume (see Figure 1). A highly nonlinear solitary wave of incident ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6956

6 Figure 1. A schematic plot of the (a) numerical wave flume and (b) sensor locations similar to the laboratory flume experiment of Ting [26, 28]. wave height H 5.22 m is sent into still water with a flow depth of h 5.3 m at the wave maker. Twelve wave gauges (resistant type) placed along the length of the flume record the free-surface evolution. At a fixed streamwise location (7.37 m from the wave maker with local water depth of cm), five ADVs located at different elevations record time series of three component velocities at 25 Hz. The wave flume was replicated in the numerical model with minor modifications. In the interest of computational time constraints, the numerical wave flume dimensions are specified as 18.2 m long by.6 m wide by.6 m deep, while the 1/5 constant slope is maintained. Namely, the dry beach region at the right end of the flume is shortened. As we will present later, we primarily focus on the first 9 s of the experiment and the simulation results analyzed here are not affected by the shortened wave flume. The width of the numerical wave flume is.6 m, which is.3 m narrower than the physical wave flume. As we will demonstrate later, the largest size of the measured and predicted turbulent coherent structure is around 1 15 cm, which is much smaller than the modeled flume width. In this study, Reynolds decomposition is often used to separate the flow field into turbulence-averaged component and fluctuating component. For instance, the flow velocity is decomposed into: u i 5hu i i1u i (14) where hi represents the turbulence-average operator, hu i i is the turbulence-averaged velocity, and u i is the velocity fluctuations. To obtain the ensemble-averaged flow statistics, the same physical experiment is carried out 29 times for ADV measurements, at least 5 times for wave height measurements [Ting, 26], and 38 times for PIV measurements [Ting, 28]. Due to the highly transient and inhomogeneous nature of the problem, there are more approximations in obtaining the ensemble-averaged flow statistics in the numerical simulation. By implementing periodic boundary condition for two lateral boundaries in the simulation, the computed flow field can be averaged over the spanwise direction to obtain approximated flow statistics [e.g., Vittori and Verzicco 1998; Ozdemir et al., 213]. However, considering the largest size of the turbulent coherent structures and the limited width of the model domain in the present simulation, the calculated flow statistics may involve more uncertainties. Here Reynolds decomposition is adopted for some of the model-data comparisons to be discussed later. However, we should keep in mind that the measured statistics is based on averaging over about 3 ensembles while the simulated statistics are obtained via spanwise averaging that only accounts for a limited number of eddies. The uncertainties associated with spanwise averaging can be approximately estimated by examining spanwise-averaged velocity in the spanwise (transverse) direction (i.e., <v>). Theoretically, <v> should be zero in a statistically 2-D flume. We find that near the surface, <v> calculated by spanwise averaging has a RMS value of no more than.2 m/s (local peak value of no more than.8 m/s). Near the bed, <v> calculated by spanwise averaging has a RMS value of no more than.1 m/s (local peak value is below.4 m/s). As we will present next, the ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6957

7 <η> (m) <η> (m) <η> (m) <η> (m) time (s) Figure 2. Time series of free-surface elevation between the measured data (dashed curves) and simulation results (solid curves). (a) Time series at x 5.55, 2.2, and 3.2 m; (b) at x , 5.25, and 6.25 m; (c) at x , 8.25, and 9.25 m; (d) at x 5 1.3, 11.3, and 12.3 m. (a) (b) (c) (d) maximum streamwise mean velocity <u> is around.6 m/s and hence our spanwise averaging can be considered as reasonably accurate when estimating mean flow velocities. However, the uncertainties associated with RMS turbulent velocity fluctuation (second-order statistics) are expected to be larger. The model domain is composed of 2427, 8 and 8 grid points in the streamwise (x), spanwise (y), and vertical (z) directions, respectively. Hence, a total number of N 5 15,532,8 computational cells are used. Due to the sloping geometry of the wave flume (see Figure 1a), the vertical grid size Dz reduces in the positive x direction and hence Dx also reduces accordingly to keep the grid size ratios approximately constant. As a result, the largest (smallest) grid size is Dx max mm (Dx min mm) in the x direction and Dz max mm (Dz min 5 3 mm) in the z direction, respectively. A uniform grid size of Dy mm is specified in the spanwise direction. The central idea of LES is to resolve the generation and evolution of 3-D turbulent eddies, which tends to be more or less isotropic once these eddies cascade into smaller sizes. Therefore, it is necessary to avoid very large (or very small) aspect ratio of grid mesh sizes in different directions. Grid convergence tests have been carried out and the modeled turbulence-averaged free-surface elevation and streamwise/vertical velocities are insensitive to slight increase of grid size. In this paper, the origin of the streamwise (x), spanwise (y), and vertical (z) axes is defined at the wave maker, the right boundary of the flume (when facing onshore) and the initial still water level, respectively (see Figure 1a). Free-surface elevation is recorded at 12 locations (x 5.55, 2.2, 3.2, 4.25, 5.25, 6.25, 7.25, 8.25, 9.25, 1.3, 11.3, and 12.3 m) along the wave flume (see Figure 1b). The temporal evolutions of freesurface elevation between model results and measured data are shown in Figure 2. Moreover, snapshots of modeled and measured free-surface elevation throughout the entire wave flume are presented in Figure 3. Overall, the numerical model captures the temporal evolution of waveshape before and after the wave breaks. It is noted that initial discrepancies are present near the wave maker. Although the modeled wave height reaches the target wave height specified at the inlet boundary, the wave observed in the laboratory is slightly smaller. More importantly, the modeled wave reaches the largest wave height at the second wave gauge (x m) and starts to break as the wave propagates toward the third wave gauges (x m). On the other hand, in the physical experiment the largest wave height is recorded at the third wave gauge (x m). Consequently, the numerical model predicts the breakpoint at around x m, which can be up to 1 m seaward of that measured in the physical experiment. It is believed that the slightly early wave breaking predicted by the numerical model may be due to the wave generation method. In the physical experiment, a highly nonlinear solitary wave with initial wave height to water depth ratio of H/ h 5.73 is generated by a piston-type wave maker. On the other hand, in the numerical model, time series ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6958

8 z (m) (a) z (m) (b) z (m) (c) z (m) (d) x (m) Figure 3. Snapshots of free-surface elevation between the measured data (symbols) and simulation results (solid curves) at (a) t s, (b) t s, (c) t s, and (d) t s. of free-surface elevation and velocity profiles based on the analytical solution of Lee et al. [1982] are specified through the inflow boundary condition. It appears that this approach cannot generate a highly nonlinear and stable solitary wave as the laboratory counterpart. Further numerical experiments suggest higher order solitary wave solution [Jacobson et al., 211] does not improve this problem. Using a first-order time integration scheme triggers even earlier wave breaking. Because of the earlier breaking predicted by the numerical model, the overall breaking process may be more gradual than that in the physical experiment and the predicted wave height is slightly larger than the measured between gauges 5 and 1. However, the predicted wave decay rate and the waveshape are well captured by the numerical model. Measurements of flow velocity are taken at x m for five vertical locations where the local water depth is h cm. This is the location where intense turbulent coherent structures penetrate into the water column due to initial overturning of the breaking solitary wave. Figure 4 presents the measured and modeled averaged streamwise velocity and root-mean-square (RMS) velocity fluctuations in the streamwise, ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6959

9 Figure 4. Comparison of (a) averaged streamwise velocity <u>, RMS velocity fluctuations (cm/s) in Figure 4b streamwise, (c) spanwise, and (d) vertical directions between the measured data (dots) and simulation results with dynamic Smagorinsky closure (solid curves) at 11 cm above the bed (z/h 5.72). spanwise, and vertical directions at 11 cm above the bed (z cm). This location is quite close to the free-surface (z/h 5.72) and we can observe a rapid increase of RMS velocity fluctuations in all three components right after the breaking wavefront passes. As we will illustrate more clearly in section 4, this rapid increase of turbulent fluctuations is associated with the penetration of 3-D turbulent coherent structures into the water column. The predicted RMS velocity fluctuations are more or less similar to those measured in terms of the temporal evolution while the peak magnitude is slightly under-predicted. The discrepancies may be due to slightly earlier breaking predicted in the numerical simulation (see Figure 2). The streamwise turbulent velocity fluctuation is only slightly larger than the spanwise and vertical components, confirming a highly 3-D turbulent flow field under the breaking solitary wave. At 7 cm above the bed (z cm, or z/h 5.46), the numerical model predicts a double-peaked feature in RMS turbulent velocity fluctuations, which is not seen in the measured data (see Figure 5). Since the peaks of turbulent velocity fluctuations are directly associated with the turbulent coherent structures, the limited number of ensembles used here in calculating the statistics of the simulation results may play a role (i.e., spanwise averaging over limited width only covers a very small number of eddies). Beyond this, however, the trend in RMS turbulent velocity fluctuations from reaching the peak value at t s to subsequent decay is well captured by the numerical model. Further approaching the bed at z cm (only 3 cm above the bed or z/h 5.2, see Figure 6), the numerical model predicts sufficiently intense turbulent ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 696

10 Figure 5. Comparison of (a) averaged streamwise velocity <u>, RMS velocity fluctuations (cm/s) in Figure 5b streamwise, (c) spanwise, and (d) vertical directions between the measured data (dots) and simulation results with dynamic Smagorinsky closure (solid curves) at 7 cm above the bed (z/h 5.46). fluctuations at t s which is comparable to measured data. Near the free-surface, the RMS velocity fluctuation is dominated by the streamwise component and followed by the vertical component. However, the streamwise component of RMS turbulent velocity fluctuations appears to decay rapidly closer to the bed. On the other hand, the corresponding decay rate for the vertical component of RMS turbulent velocity fluctuation is milder. Particularly, we can still observe sufficiently strong vertical component of velocity fluctuation at only 3 cm above the bed (see Figure 6d). This feature is associated with major turbulent coherent structures generated by the breaking solitary wave and will be further elucidated in section 4. Simulation results presented so far are computed based on the dynamic Smagorinsky closure. Numerical simulation is also carried out using the less sophisticated standard Smagorinsky closure. In terms of the predicted velocity fluctuations, differences are observed between these two subgrid closures. For conciseness, simulated turbulent fluctuation statistics are presented in terms of turbulence kinetic energy hki (per unit mass): hki5 1 2 ðhu 2 i1hv 2 i1hw 2 iþ (15) Figure 7 presents the time series of turbulence kinetic energy k at all three elevations for simulation results using two different subgrid closures and measured data. Using the standard Smagorinsky closure, the peak value of hki in the upper water column during the breaking wave passage is slightly overpredicted (see t s in Figure 7b). Nevertheless, measured data and both model results show peaks of hki around this early stage of breaking, indicating the passage of turbulent coherent structures in the upper water column at this cross-shore location (x m). More notably, results from the standard Smagorinsky closure predict very weak turbulence close to the bed at this early stage (see t s in Figure 7d), which does not resemble that predicted by dynamic Smagorinsky closure and measured data. In fact, standard Smagorinsky closure predicts high turbulence at 3 cm above the bed to occur at a later time (see t s in Figure 7d). Visual observation of the simulation results suggests that this is due to high near-bed turbulence occurs at about 1 m seaward of the sensor location, which is then advected landward and hence captured later by the sensor located at x m. This feature also does not resemble measured time series. Overall, simulation results using dynamics Smagorinsky agree better with the measured data. Hence, our subsequent analysis will be based on results obtained by dynamics Smagorinsky closure Turbulent Coherent Structure Formation and Characterization To investigate the generation, evolution and fate of turbulent coherent structures under the breaking solitary wave, we adopt the k 2 -method suggested by Jeong and Hussein [1995]. The k 2 is defined as the eigenvalues of the symmetric tensor S 2 1 X 2 with S and X representing the symmetric part and antisymmetric part of the velocity gradient tensor. Figure 8 presents the instantaneous images of iso-surfaces of air-water ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6961

11 Figure 6. Comparison of (a) averaged streamwise velocity <u>, RMS velocity fluctuations (cm/s) in Figure 6b streamwise, (c) spanwise, and (d) vertical directions between the measured data (dots) and simulation results with dynamic Smagorinsky closure (solid curves) at 3 cm above the bed (z/h 5.2). interface (represented by blue surfaces with a 5.5) and iso-surfaces of k and 22 (represented by gray surfaces; viewed upward from the bottom) during the initial stage of wave breaking (t s). The flow field visualized here is under the location where the initial wave overturnning occurs. The streamwise length of the field of view (FOV) is around 2 m and the spanwise length of the FOV covers the entire width of the numerical wave flume (.6 m). The first defining pattern exhibited here is the formation of twodimensional spanwise vortices (x y ) around surface rollers due to wave overturning (see Figure 8a). These vortices are not yet associated with turbulent motions, but more or less intense 2-D rotational flow fields. As flow further accelerates at t s (Figure 8b), the resulting high shear rate triggers the formation of rip vortices around the roller and the overall vortex structures evolve into more significant 3-D features. This straining continues to break the monolithic two-dimensional vortex structures into complete threedimensional turbulent coherent structures at t s (see Figure 8c). The generation of 2-D roller and its evolution into 3-D turbulent coherent structures simulated here are similar to earlier numerical studies of Watanabe et al. [25] and Christensen [26] for periodic waves. At t s, the flow field is populated with a large amount of smaller coherent structures with local irregularities in terms of shape and intensity. Meanwhile, a couple of the most intense coherent structures start to descend and penetrate further into the water column, e.g., see the eddy identified by the red box in Figures 8d and 9. Since the width of the flume is 6 cm, the size of this 3-D coherent structure can be visually estimated to be around 1 cm. Figure 9 further shows the subsequent evolution of the eddy identified in Figure 8d. By observing this 3-D eddy from a different angle, it is evident that this intense turbulent coherent structure is what commonly known as a hairpin vortex. Similar vortex structure under a broken solitary wave, which was named as the reversed horseshoe vortex, is also reported in the recent numerical simulation study of Farahani and ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6962

12 <u> (cm/s) <k> (cm 2 /s 2 ) <k> (cm 2 /s 2 ) <k> (cm 2 /s 2 ) (a) (b) (c) (d) time (s) Figure 7. Comparison of (a) averaged streamwise velocity u and (b d) turbulent kinetic energy k among measured data (red dots), numerical results with dynamic Smagorinsky closure (black solid curve), and numerical results with standard Smagorinsky closure (blue dashed curve) at (b) 11 cm above the bed (z/h 5.72); (c) 7 cm above the bed (z/h 5.46); (d) 3 cm above the bed (z/h 5.2). Dalrymple [213] using a SPH approach. Hairpin vortices are commonly observed in turbulent shear flows as a 2-D roller evolves into complete 3-D turbulent coherent structures. For example, a thorough discussion on the generation of hairpin vortices in bottom boundary layer can be found in Schlichting and Gersten [23]. Unlike those observed in the bottom boundary layer where the hairpin vortices have to move upward due to the existence of bottom solid boundary, the hairpin vortices generated under breaking waves must descend due to the existence of upper freesurface boundary. At t s (see Figure 9a), the distance between the bottom tip of the vortex and the bed is 8.25 cm. The angle between the principal axis of the vortex and the streamwise (x) axis is around 35. Only.4 s later (see Figure 9b), the bottom tip of the vortex is located at around.25 cm above the bed. Hence, this hairpin vortex approaches the bed rapidly. At this moment, the angle between the principal axis of the vortex and the streamwise (x) axis increases slightly to 38. In about.6 s from its initial appearance as a mature hairpin vortex, the vortex already collapases onto the bed at t s (see Figure 9c) and finally gets completely dissipated at t s (not shown). It is important to mention that there are other similar hairpin vortices generated in the water column during the entire breaking processes. In more landward location (shallower depth), the angle between the principal axis of the vortex and the streamwise (x) axis is observed to increase to about However, not all of the vortices are sufficiently intense to reach the bed. More quantitative discussion on the interaction of these hairpin vortices with the bed will be presented in section 4. As mentioned previously, obliquely descending eddies (ODEs) are of particular interest in the study of breaking waves because of their capability of elevating mixing in the water column and their potential of interacting with the bed, both of which encourage significant sediment suspension. Qualitative visualizaiton of vortices using k 2 method indicates a couple of 3-D turbulent coherent strucutures indeed approach the bottom. In the present simulation without the effect of sidewalls, the so-called ODEs appear to be hairpin vortices generated due to shear instability. More quantative analyses are needed in order to diagnose whether the predicted turbulent coherent structures are similar to those observed in prior laboratory observations [e.g., Nadaoka et al., 1989; Ting, 28; Huang et al., 21]. With this in mind, it is worthwhile to study the turbulent kinetic energy and vorticities associated with the turbulent p ffiffiffiffiffiffiffiffiffi coherent structures. Here we identify p turbulence ffiffiffiffiffi through spanwise-averaged turbulent intensity ( 2hki ) and instantaneous turbulent intensity ( 2k ) at a given xz plane where a large downward turbulent coherent structure is identified. The instantaneous turbulent kinetic energy (per unit mass) k is computed from the instantaneous velocity fluctutions in all three directions: k5 1 2 ðu 2 1v 2 1w 2 Þ (16) p ffiffiffiffiffiffiffiffiffi pffiffiffiffiffi In Figure 1, snapshots of 2hki, 2k and instantaneous vorticities in all three directions are shown at t s. Relative to the turbulent coherent structure that we discussed previously, t s corresponds to the later stage where three intense hairpin vortices already impinge onto the bed and the solitary wavefront develops into more mature bore with p a ffiffiffiffiffiffiffiffiffi sawtooth shape (see x m surface guage data in Figure 2c). The instantaneous turbulent intensity ( 2hki ) and components of vorticity (x) are shown in the xz plane located at y 5.3 m (i.e., center-plane). By examining the averaged turbulent intensity (Figure 1a), it ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6963

13 Figure 8. Turbulent coherenet structures under the breaking solitary wave at different time (a d) visualized by k 2 method. The blue isosurfaces are the free-surface represented by a 5.5. The gray iso-surfaces represent k for Figure 8a and k for Figures 8b 8d. Each plot shows the entire width of the computational domain, which is.6 m. is clear that the most intense region of the wave breaking is the surface roller. The trailing region that is once part of the wave crest also maintains some level of turbulence but the intensity is about half p of ffiffiffiffiffiffiffiffiffi that near the crest at this instant. Near x and 7.5 m, notable level of averaged turbulent intensity p 2hki ffiffiffiffiffi can be observed throughout the water column. The distribution of instantanous turbulent intensity 2k at the center-plane (Figure 1b) is qualitatively similar to that p of ffiffiffiffiffiffiffiffiffi averaged p turbulent intensity except some locally very high turbulence spots. The difference between 2hki and ffiffiffiffiffi 2k distributions simply refects the spanwise variation and three-dimensoinal nature of the broken wavefield. The local turbulence at a given xz plane can be several times larger than the spanwise-averaged values. Looking at the vorticity associated with these high turbulent intensity regions, all three components similarly display bottom interactions. Near x and 7.5 m where turbulence impinges onto the bottom, we observe a region of positive vorticity adjacent to region of negative vorticity. This is consistent with the counter-rotating vortical motion observed in Ting [28] associated with ODEs. In terms of the y component of instantaneous vorticity x y, its magnitude exceeds 2 3 s 21 under the broken wave, which is of similar magnitude to that reported in Chang and Liu [1998]. Attributing a vortex with a highly turbulent region likewise lends evidence that the bottom interaction is related to wave-breaking-induced coherent structures, and not simply background incoherent turbulence. This interesting feature is further investigated in more details next. Using the particle imaging velocimetry (PIV), Ting [26, 28] measures detailed flow field of ODEs near the bed. Several main features of ODEs are revealed in Ting [28] that can be used to diagnose the hairpin-shaped turbulent coherent structures predicted by the present numerical simulaiton. The PIV images captured by Ting [28] show ODEs at around the middle of the water column. The observed ODEs are composed of a counter-rotating, vertically oriented vortex pair. Through the cores of these two vortices is an associated uprush or downrush of fluid, noted by volumes of strong positive and negative turbulent vertical velocity fluctuation. Due to the proximity of the uprush and downrush of fluid, a high shear region is produced, which in turn shows high levels of turbulent kinetic energy. Here we present numerical results for instantaneous z direction vorticity x z, vertical velocity fluctuations w, and TKE k by tracking the hairpin vortex identified previously in Figure 9 for two different vertical locations in the water column (see Figures 11 and 12). In Figure 11, the vortex identified is located in the upper water ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6964

14 Figure 9. The fate of a hairpin-shaped turbulent coherent structure (see the red dashed box) under the breaking solitary wave at different time (a c) visualized by k 2 method. The blue iso-surfaces are the free-surface represented by a 5.5. The gray iso-surfaces represent k Each plot shows the entire width of the computational domain, which is.6 m. column (z 5 9 mm above the bed) at t s. According to x z shown in Figure 11a, the counter-rotating feature with negative (positive) z vorticity on the right side (left side) is evident. This counter-rotating feature corresponds to the two legs of the haripin vortex identified previously. A very strong downrush of flow velocity fluctuation exceeding.4 m/s is observed in the core (see Figure 11b). Region of high instantaneous TKE is also located near the downrush region (Figure 11c). Hence, the downrush of hairpin vortices is the main mechanism delivering surface generated turbulence into the water column. By further examining simulation results for this specific hairpin vortex in the lower water column (3 9 cm above the bed) at slightly later time, we observe that the intensity of downburst, the size and the intensity of the vortex are more or less unchanged (not shown). However, significant change of these vortex features are found near the bed. At 3 cm above the bed, we start to observe notable decay of downburst velocity fluctuation to around.3 m/s as well as the instantaneous TKE (not shown). Most notably, at 1 cm above the bed, there is clearly interaction between the vortex and the bottom (see Figure 12). First, the vorticity at the core of the vortex decays by about 3% while the characteristic lengh of the vortex increase to about 15 cm (see Figure 12a). A lower magnitude of downrush flow velocity fluctuation of.25 m/s is observed in the core, which is clearly surrounded by less intense but noteworthy positive (uprush) velocity fluctuations (see Figure 12b). As a result, the intensity of instantaneous TKE also drops to more than half of that in the middle of the water column. Main features of the ODEs, namely counter-rotating and downburst, as well as the characteristic size (1 cm) of the vortex predicted by the present numerical simulation are similar to the PIV observation reported by Ting [28]. Numerical results also indicate that the so-called obliquely descending eddies are the downrush of hairpin vortices generated under breaking waves. Because the downburst fluctuating velocity of the ODE remains to be around.25 m/s at 1 cm above the bed, a rapid decay and interaction with the bed is expeceted within the bottom 1 cm. Hence, we obtain evidence of turbulent coherent structures generated by breaking waves capable of impinging onto the bed. For all the ODEs that are sufficiently large (characteristic size larger than 7 cm) to be resolved by the simulation, we observe that these ODEs always have the shape of reversed hairpin with a pair of counter-rotating vortices. Inside these vortices, the downburst feature is also observed although the intensity of downward velocity fluctuation may vary. Finally, we like to also point out that the present simulation ignores sidewall effect and the generation of ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6965

15 Figure 1. A snapshot of the simulated flow field at t s. (a) Spanwise-averaged turbulent intensity, (b) instantaneous turbulent intensity at a given x-z plane (y 5.3 m), and (c e) vorticity in the streamwise, spanwise, and vertical directions, respectively. ODEs is solely due to shear instability. There are evidences from laboratory observation that the two sidewalls may play an important role in bending the rollers and encourages the generation of ODEs [Ting, 26, 28]. On the other hand, it appears that regardless of the generation mechanisms, once the initial instability advances into the nonlinear stage in sufficiently high Reynolds number, the resulting 3-D turbulent coherent structures are of similar characteristics. 4. Discussion Fate of Obliquely Descending Eddies (ODEs) Given the evidence of ODEs interacting with the bed, further investigation into the detail of this process is warranted. An advantage of the present 3-D numerical simulation is that a complete map of turbulent intensity at various levels in the water pffiffiffiffiffi column (xy plane) can be conveniently obtained. Here we present the instantaneous turbulent intensity 2k at the nearest-bed resolved grid point (Figures 13a1 13a5) during the passage of breaking solitary wave along with the corresponding near-surface instantaneous turbulent intensity (Figures 13b1 13b5). At t s, two spots near the bed between x 5 6mandx56.5 m and one spot at x m display high turbulent intensity related to ODE impingement (see Figure 13a1). At t s, turbulent intensity at these three spots start to decay and disintegrate into smaller and less coherent turbulent fluctuations (Figure 13a2). These three eddies are the turbulent events originated from the surface rollers during the initial energetic free-surface overturning. Generally, when eddies are sufficiently energetic to impinge onto the bed, they quickly diffuse outward from the focused area as energy is dissipated. For the first two intense eddies located between x 5 6mandx56.5 m, the overall duration of interaction with the bed is around 2. s. However, the third eddy located at x m is of shorter duration of around 1. s. After the impingement of the first three intense eddies, many more eddies further impinge onto the bed bewteen about x 5 8 m and x 5 12 m (see Figures 13a3 and 13a4). However, the size of these spots is slightly smaller. These eddies are due to subsequently less intense breaking of a spilling breaker, which is consistent with the diminished length scale of these turbulent coherent structures. This area likely represents effects related to the interval of breaking between the energetic free-surface turnover, and the less energetic depth-limited breaking region (landward of x 5 12 m). When the wavefront propagates into shallow depth at x m (see Figure 13a5), where the local water depth is no more than 5 cm, a distinct depth-limited breaking region can be identified. In the depth-limited region, turbulent coherent structures are clearly much smaller in size. It also lacks sufficient strength comparing to those discussed previously. However, the water depth in this region decreases to the point where the ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6966

16 Figure 11. A x-y plane snapshot (view from the top) of an obliquely descending eddy at t s and 9 mm above the bed. (a) Instantaneous vorticity in the z direction (s 21 ), (b) vertical velocity fluctuation (m/s), and (c) instantaneous turbulent kinetic energy (m 2 /s 2 ). required energy to impinge onto the bed is small, and high level of TKE is present at the bed in much denser configurations. The turbulent spots at the bed are small but highly populated, representing a more chaotic flow field. Many nearshore hydrodynamic models are based on a depth-integrated approach with appropriate parameterization of surface wave breaking, for example, through a roller dissipaiton rate. These depthintegrated models are not designed to resolve the penetration of surface generated turbulence into the water column and interaction with the bed. Hence, any description of enhanced turbulence near the bed would need to be parameterized by surface-generated turbulence. The question becomes whether the level of surface-generated turbulence is a good indicator for near-bed turbulence associated with wave breaking. This issue is investigated here for spilling breaker of solitary wave by plotting the corresponding near-surface instantaneous turbulent intensity in Figure 13 (bottom). During the initial stage of the wave overturnning where three large ODEs are generated (see Figures 13b1 and 13b2), there is very weak resemblance between the near-bed instantaneous turbulent intensity and the near-surface turbulent intensity. The region of high surface turbulence is located near the wavefront, while the region of high near-bed turbulence due to ODE impingement is located somewhat behind the wavefront. This is because the impingement of ODEs is highly localized and there is about 1 2 s time lag between the generation and penetration of these ODEs. On the other hand, in the depthlimited region where the flow depth is rather shallow, we can start to see some degree of resemblance between the nearbed and the near-surface turbulent intensities (Figures 13a5 and 13b5). To better quantify the similarities and differences of those turbulent quantities at different level in the water column, we further present the cross-shore evolution of spanwise-averaged near-bed TKE hki b, spanwise-averaged surface TKE hki s, and depth-averaged TKE K (taking depth-average of hki throughout the water column) for the early stage and the late stage of the simulation (see Figure 14). It is ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6967

17 clear that at the early stage of the breaking right after the initial overturnning occurs, both surface TKE hki s and depthaveraged TKE K peak near the wavefront with very large magnitudes (see their large values at x m). However, the near-bed TKE hki b at this moment only has two smaller peaks refecting the three intense ODEs located at around x m and x m (see Figure 13a1). More careful examination shows that these two peaks shown in the nearbed TKE hki b can also be seen as two local peaks in hki s and K, but the overall distributions of hki s and K are dominated by high turbulence at the wavefront. On the other hand, at the later stage of the simulation as the bore enters the shallow region (see Figure 14b), we start to see that all three turbulent quantities hki b, hki s, and K start to become similar. They all have the largest value near the bore front and then decay seaward. Therefore, simulation results suggest that for a spilling solitary wave breaking, turbulence level associated with nearsurface value is not a good indicator for bottom turbulence unless when the flow depth is very shallow. Beyond demonstrating that there is indeed significant bottom interaction due to turbulent coherent structures under a breaking solitary wave, it is also useful to understand what implications this interaction may have on bottom sediment suspension. Although the present simulation does not include sediment transport, insights to bottom interaction due to ODEs may be examined via bottom shear stress. Our particular interest is that if ODEs can leave such significant near-bed TKE signature shown in Figures 13a1 and 13a5, their influence on bottom shear stress must be notable. However, it is also reminded that in the present simuation, the numerical resolution is not sufficiently fine to resolve the viscous sublayer and the buffer layer and hence near-wall modeling is adopted. Essentially, the nearest resolved instantaneous velocities are substituted into equation (12) to obtain the bottom friction velocity u. The instantaneous bottom stress Figure 12. A x-y plane snapshot (view from the top) of an obliquely descending eddy at t s and 1 mm above the bed. (a) Instantaneous vorticity in the z direction (s 21 ), (b) vertical velocity fluctuation (m/s), and (c) instantaneous turbulent kinetic energy (m 2 /s 2 ). is then obtained as s b 5qu 2. The velocity distribution represented by equation (12) is obtained semiempirically for statistically steady, fully developed bottom boundary layer flow. Uncertainties arise when applying ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6968

18 Figure 13. Snapshots of (a1 a5) instantaneous turbulent intensity (m/s) near the bed, and (b1 b5) near-surface instantaneous turbulent intensity from t with.8 s interval. such formulation under breaking waves. Nevertheless, bottom stress estimated this way is still presented to understand, at least qualitatively, the magnitude of enhanced bottom stress due to ODEs. Figure 15 present the snapshots of instantaneous bottom stress during the passage of the breaking solitary wave. It is clear that there are two sources of bottom stress. Large near-bed velocity due to the passage of solitary wave causes major bottom stress with a magnitude around Pa. For example, at t s, this region is located between x and 9. m (see Figure 15a) and its subsequent location can be identified ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 6969

19 <k> b, <k> s, K (m 2 /s 2 ) (a) (b) Distance from wave maker (m) Figure 14. Cross-shore evolution of spanwise-averaged near-bed TKE hki b (red-solid curve), spanwise-averaged surface TKE hki s (blue-dashed curve), and depthaveraged TKE K (black-dotted curve) at (a) t s and (b) t s. by following the progressing wavefront. This type of bottom stress is induced directly by the wave motion, not the ODEs. On the other hand, signatures of large bottom stress directly correlated with ODEs interacting with the bed can also be identified. For example, at around t s three spots of large instantaneous bottom stress are clearly seen behind the wavefront with a peak magnitude exceeding.8 Pa. The locations of these three spots are almost identical to those shown in Figure 13a1, for instantaneous nearbed TKE. Notice that at this early stage of wave breaking, the nearbed turbulence and bottom stress have significant spanwise variability in the flume. In many flume and field studies [e.g., Jaffe and Rubin, 1996; Cox and Koboyashi, 2; Scott et al., 29], they have noted this intermittent feature which reflects in the measured time series of turbulent velocity fluctuations. As the wavefront moves further landward passing x 5 1 m, we observe that the intensity of instantaneous bottom stress caused by the wave motion starts to decay (see Figures 15c 15e). This is expected as the local flow depth becomes small (smaller than 1 cm) and the wave is already broken. Landward of x 5 12 m, the most intense events in terms of bottom stress begins to occur, indicated by the crimson colored blotches on the plot (see Figure 15e). The flow depth at this location is only 6 cm, roughly equivalent to the local wave height. Although surface-generated turbulence (bore turbulence) is less intense here, the flow depth Figure 15. Snapshots of instantaneous bottom shear stress (Pa) throughout the numerical wave flume where wave-breaking turbulence approaches the bed. (a e) Snapshots from t with.8 s interval. ZHOU ET AL. VC 214. American Geophysical Union. All Rights Reserved. 697