MODELING WAVE DAMPING AND SEDIMENT TRANSPORT WITHIN A PATCH OF VEGETATION

Vegetation canopies control mean and turbulent flow structures as well as suspended sediment processes in the coastal wetlands. In this study, a three-dimensional hydrodynamic and sediment transport model is developed for studying flow/wave-vegetation-sediment interactions. The model is based on the non-hydrostatic model NHWAVE. The vegetation effects on turbulent flow are accounted for by introducing additional formulations associated with vegetationinduced drag and turbulence production in the governing equations. The sediment concentration is obtained by solving the advection-diffusion equation with sediment exchange at the bed. The turbulent flow and suspended sediment are simulated in a coupled manner. The model is validated against the laboratory measurements of partially vegetated open channel flows. It is shown that the model can well predict the vegetation effects on the flow field. The model is then employed to study nearshore sediment suspension influenced by a patch of vegetation, which is located in the surf zone. The turbulence generated by wave breaking is greatly damped by the vegetation patch, resulting in considerably less sediment pickup from the bottom in the surf zone. Within the vegetation patch, most suspended sediments are restricted in a thin layer near the bottom. The net sediment transport is in the shoreward direction, in contrast to the seaward net transport of sediments in the unvegetated surf zone.


INTRODUCTION
Aquatic vegetation play a significant role in controlling mean flow, turbulence as well as sediment processes.Due to the enhanced resistance associated with the plants, the mean flow is greatly reduced within the vegetation patch, compressing turbulence level (Nepf, 1999) and promoting sedimentation and the retention of suspended sediments (Lopez and Garcia, 1998;Sharp and James, 2006;Zong and Nepf, 2010).The resulting spatial depositional patterns are the key to the understanding of the long-term morphologic and ecological dynamics of tidal marshes and floodplains (Temmerman et al., 2005).In the coastal wetlands, sediment deposition determines the ability of tidal marshes to keep pace with the accelerated sea level rise.
Up to date, most of our knowledge on sediment transport processes and sedimentation within vegetation patches comes from the laboratory flume studies.For example, Abt et al. (1995) and Thornton et al. (1997) performed a series of laboratory experiments to assess the deposition and entrapment in riparian vegetation.Their results showed that the presence of the vegetation not only increases the deposition of sediment, but also the entrapment of sediment on the vegetative bases.It was determined that vegetation could retain 30 to 70 percent of the deposited sediments.They also found that the length and cross-sectional area of the vegetation are two dominant factors in controlling the ability of vegetation to entrap and retain sediments.Garcia et al. (1999) studied the effect of seagrass on the flow and sediment entrapment by conducting a series of experiments with the presence and absence of the seagrass.They showed that there is a direct relationship between the height of the plants and reduction of flow velocity.The amount of the particles trapped by seagrass is closely related to the surface area of the plants.Elliot (2000) investigated the removal of sediment in a channel by means of settling on the vegetation.It was theoretically and experimentally shown that the settling on vegetation can considerably enhance the sediment capture process.Sharpe and James (2005) performed a series of laboratory experiments to study how emergent vegetation promotes sediment deposition.They found that the suspended transport and the extent of longitudinal deposits from suspension within emergent stems is enhanced by increased flow depth and reduced by increased sediment grain size and stem density.Zong andNepf (2010, 2011) conducted laboratory experiments to investigate flow and sediment deposition in and around a finite patch of vegetation in an open channel.Their studies revealed the existence of a diverging flow at the leading edge of the vegetation patch and a shear layer with large-scale coherent vortices at the lateral interface between the patch and adjacent open channel, which have significant effects on sediment deposition within the patch.
Although the flume studies have enlarged our insights on vegetated flow and associated sediment processes, the flow-vegetation-sediment interactions on the landscape scale (i.e.length scale of 10 2 -10 3 m) are still poorly understood because field monitoring of three-dimensional flow field and sediment deposition within vegetation patches is extremely difficult.In this respect, numerical modeling may provide a powerful tool for investigating these complicated interactions in the field-scale.The traditional approach for modeling flow and sediment transport in vegetation canopy is to solve the depth-averaged shallow-water equations and to use an enhanced bottom roughness coefficient (i.e.Manning coefficient) for vegetated surfaces, for instance, Wu et al. (2005), Chen et al. (2007).Such an approach, however, does not account for the influence of vegetation over the entire water depth (Temmerman et al., 2005), thus is not able to accurately capture three-dimensional flow-vegetation interactions and associated suspended sediment processes.A more sophisticated approach is to solve the hydrodynamic and sediment transport equations by including additional formulations for drag and turbulence production induced by vegetation canopies.For example, Lopez and Garcia (1998) developed a 1DV model to study sediment suspension in the vegetated open channel.Their model accounted for vegetation effects by introducing drag terms in the momentum equations and additional turbulence production in k − equations.Temmerman et al. (2005) developed a three-dimensional vegetated flow and sediment transport model based on Delft3D.The strategy in their model was similar to that of Lopez and Garcia (1998).The model was applied to investigate the impact of vegetation on flow routing and sedimentation patterns in a field-scale tidal marsh.Sheng et al. (2012) developed a similar vegetated flow model based on CH3D and applied it to study the reduction of storm surge by vegetation canopies.These models are hydrostatic, and not capable of simulating wave-vegetation interactions.
In this paper, we aim to develop a 3D hydrodynamic and sediment transport model that is capable of simulating flow/wave-vegetation-sediment interactions.The model is based on the Non-Hydrostatic WAVE model NHWAVE, which was initially developed by Ma et al. (2012).The vegetation-induced turbulence production is simulated by a nonlinear k − turbulence model.The sediment concentration is obtained by solving the advection-diffusion equation with sediment exchange at the bed.The model is validated against the laboratory measurements of Pasche and Rouve (1985).Then the model is applied to study nearshore sediment suspension affected by a finite patch of vegetation.
The remainder of the paper is organized as follows.The mathematical models including the nonhydrostatic model NHWAVE, the nonlinear k − turbulence closure and sediment transport model are introduced in section 2. The nearshore sediment suspension and cross-shore sediment fluxes are also discussed in section 3. The conclusions are finally given in section 4. Ma et al. (2013a) have developed a 3D vegetated flow model that is capable of simulating wave damping induced by vegetation canopy.The model was based on the Non-Hydrostatic WAVE model NHWAVE, which was initially developed by Ma et al. (2012) and extensively used to study landslide-induced tsunami waves (Ma et al., 2013b), infragravity wave process in coral reef systems (Ma et al., 2014a) and wave interactions with porous structures (Ma et al., 2014b).NHWAVE solves the incompressible Reynolds Averaged Navier-Stokes equations in well-balanced conservative form, formulated in time-dependent surface and terrain-following σ coordinate.The model employs an arbitrary stretched vertical grid, which is able to accurately resolve bottom boundary layer and sediment suspension processes.The terrain-following σ-coordinate is defined as

MATHEMATICAL MODELS Governing Equations
where U = (Du, Dv, Dw) T .(u, v, w) are velocity components in (x, y, z) directions.ω is the vertical velocity in σ coordinate.The fluxes in the momentum equations are The first three source terms on the right hand side of equation ( 2) are contributions from hydrostatic pressure, dynamic pressure and turbulent diffusion, respectively.These terms can be formulated as The fourth term is the baroclinic pressure gradient forcing, which is generated by the spatial variation of water density.In the following section, we will show that the suspended sediment may change the density of water-sediment mixture.The last term on the right-hand side of equation ( 2) accounts for the vegetationinduced drag force and inertia force.For rigid circular cylinders (used in the current study), the vegetation patch can be described by the following parameters: the cylinder diameter d, the number of cylinder per unit area n, the frontal area per unit volume λ = nd, and the average solid volume fraction of the patch φ = n(πd 2 /4).The vegetation patch imposes friction and form drag force on the flow, which are usually modeled together as where u is the velocity vector.C D is the drag coefficient.For rigid cylinders, the typical value of C D is 1.0-1.5.For oscillatory flow, inertia force is not negligible and is given by where C M is the virtual mass coefficient.Thus, the vegetation-induced source term in equation ( 2) can be written as To solve the water depth D, we integrate the continuity equation ( 1) from σ = 0 to 1.By using the boundary conditions at the bottom and surface for ω, we may obtain the equation for free surface movement.

Sediment Model
The suspended sediment concentration can be computed from the convection-diffusion equation, which is formulated in σ coordinate as follows.
where C is the concentration of suspended sediment and w s is sediment settling velocity.Note that w s is not affected by the transformation of coordinates from the Cartesian coordinate system into the σ coordinate system (Wang and Pinardi, 2002).σ h and σ v are horizontal and vertical Schmidt numbers for sediment, respectively.To solve the sediment transport equation, no-flux boundary condition is employed at the free surface. (ν At the bottom, mass exchange of suspended sediment occurs, resulting in sediment erosion and deposition.This mass exchange can be expresses as the sum of a downward flux w s C and an upward flux due to various suspension mechanisms, which is usually described by a pickup function. In this paper, we adopt the sediment pickup formula suggested by van Rijn (1984).
in which the critical Shields number is θ c = 0.05.The Shields number θ is related to the bottom shear stress τ b , which is derived by matching the velocities with the "logarithmic law of the wall".
where u b is the current velocity in the first grid point nearest the bottom, C f is the drag coefficient.
The suspended sediment and flow are simulated in a coupled manner (Ma et al., 2014c).Three effects of suspended sediment on turbulent flow field are considered: (1) baroclinic forcing effect; (2) turbulence damping effect and (3) bottom boundary layer effect.The suspended sediment may change the density of water-sediment mixture, which can be formulated as where ρ 0 is the reference water density and ρ s is the sediment density.From the above equation, we see that the spatial variation of suspended sediment concentration can produce the baroclinic pressure gradient forcing S ρ .In addition, the suspended sediment creates a stably stratified water column, damping fluid turbulence and reducing turbulent diffusivity, which will be considered in the turbulence closure.The suspension of sediment also produces a stably stratified bottom boundary layer.Consequently, the drag coefficient and bottom shear stress are reduced.These effects are accounted for through the modification of bottom boundary condition.

Turbulence Closure
A nonlinear k − turbulence model has been implemented into NHWAVE to simulate turbulent flow.In this model, the turbulent eddy viscosity is calculated as Within the vegetation patch, mean flow is greatly reduced by the vegetation drag.The large-scale mean kinetic energy is converted to small-scale turbulent kinetic energy (Nepf, 1999).The turbulent diffusivity would also be reduced due to the downward shifting of turbulent length scale (Nepf, 1999).These processes are modeled by introducing vegetation-induced turbulence production terms in the k − equations.The turbulence production generated by vegetation is related to the drag force and can be formulated as Thus, the k − equations written in conservative forms read where σ k , σ , C 1 , C 2 , C µ are empirical coefficients (Rodi, 1987).
P s and P b are shear and buoyancy productions, which are described as and in which the Reynolds stress u i u j is calculated by a nonlinear model proposed by Lin and Liu (1998) for breaking waves, which is given by where C d , C 1 , C 2 and C 3 are empirical coefficients as given by Lin and Liu (1998) as well as Ma et al. (2011). where The drag-related coefficients C f k and C f in equations ( 15) and ( 16) deserve more discussions.Shimizu and Tsujimoto (1994) chose values of C f k = 0.07 and C f = 0.16 based on calibration for the vegetated open channel flow.Lopez and Garcia (1998) discussed them based on a theoretical argument, and suggested that the weighing coefficient C f k should be set to unity if the total turbulence kinetic energy is to be modeled.
In the following applications, we select the theoretical values suggested by Lopez and Garcia (1998).The coefficient C 3 is chosen to be zero for stably stratified water column (Snyder and Hsu, 2011).

Numerical Schemes and Boundary Conditions
A combined finite-volume and finite-difference scheme with a Godunov-type method is employed to discretize equations ( 1) and (2) (Ma et al., 2012;Shi et al., 2012).In order to apply Godunov-type scheme, all the variables are defined at the cell centers except that dynamic pressure p is defined at the verticallyfacing cell faces.The momentum equations are solved by a second-order Godunov-type finite volume method.The HLL approximate Riemann solver is used to estimate fluxes at the cell faces.
For sediment transport and k − equations, the convective fluxes are determined by the hybrid linear/parabolic approximation (HLPA) scheme (Zhu, 1991), which has approximately second-order accuracy in space.Two-stage second-order nonlinear Strong Stability-Preserving (SSP) Runge-Kutta scheme (Gottlieb et al., 2001) was adopted for adaptive time stepping.Uniform gridding is used in the horizontal direction while gridding in the vertical direction is generalized to be non-uniform in order to capture the bottom and surface boundary layers when desired.The Poisson equation for the pressure field, described in Ma et al. (2012), is discretized by the finite-difference method, resulting in an asymmetric coefficient matrix with a total of 15 diagonal lines.The linear system is solved using the high performance preconditioner HYPRE software library.The model is fully parallelized using Message Passing Interface (MPI) with non-blocking communication.Further details of the numerical method may be found in Ma et al. (2012).
To solve the equations, boundary conditions are required for all the physical boundaries.Specifically, at the free surface, we have Dynamic pressure is zero at the free surface.For the k − model, zero gradients of k and are imposed.
At the bottom, the normal velocity and the tangential stress are prescribed.The normal velocity w is imposed through the kinematic boundary condition.
For the horizontal velocities, bottom shear stresses are considered.
where C f is the bed drag coefficient, which can be specified or computed from the law of the wall for fully rough, turbulent flow.For stably stratified flow, it is formulated as (Wang, 2002) where A is the empirical constant (=5.5) (Wang, 2002), R f is the flux Richardson number, referred to as an index of the vertical density stratification.∆z 1 = 0.5D∆σ 1 is the thickness of the cell above the bed, and k s is the bottom roughness height and u b is the velocity at the cell above the bed.The boundary conditions for k and become  (1985).The cross section consists of a main channel and a vegetated floodplain joined by a transition slope.

RESULTS AND DISCUSSIONS Flow in A Straight Channel with Vegetated Floodplain
The model is firstly validated against laboratory measurements of Pasche and Rouve (1985), who performed a series of experiments in a fixed-bed flume to investigate horizontal velocity distributions in a compound channel.The channel cross section contained a vegetated floodplain and a main channel joined by a transitional slope (Figure 1).The diameter of the rigid cylinder (modeled vegetation) is d =12 mm.The width of the cross section is 1.0 m.The vegetated floodplain is 0.5 m wide.The bed slope is 0.05%.The flow regime is steady and subcritical with flow rate of 0.0345 m 3 /s.
We conducted simulations of two experiments with different vegetation densities n.The corresponding solid volume fractions φ = n(πd 2 /4) are 1.26% and 2.53%, respectively.In the simulations, free-slip boundary conditions are imposed at the lateral walls.The vegetation drag coefficient is chosen as the typical value of 1.20.Forty grid cells are employed in the lateral direction to resolve the cross-section distribution of flow velocity.Figure 2 shows the model-data comparisons of lateral distributions of depth-averaged streamwise velocities for two experiments.We see that the simulations agree well with the measurements, except the near-wall regions.The discrepancy is primarily caused by the free-slip boundary conditions we imposed at the lateral walls.At this moment, the model does not have the function to deal with vertical wall boundaries.Away from the walls, the lateral variations of the flow velocity are well captured by the model.Particularly, the model is able to simulate the mean flow reduction induced by vegetation in the floodplain.With increased vegetation density (or solid volume fraction), the mean flow in the floodplain as well as the main channel is reduced.

Nearshore Wave Damping and Sediment Suspension Influenced by A Patch of Vegetation
We will employ the model to examine how vegetation influences nearshore sediment suspension in this section.The computational domain is shown in figure 3(a), which is a wave flume with 12 m long and a beach of 1/20 slope.The bottom is filled with sand of specific gravity s = 2.65 and diameter of d 50 = 0.15 mm.The bottom roughness is k s = 5.5d 50 = 0.825 mm.The domain is discretized by 800 grid cells with grid size of ∆x = 0.015 m.50 layers with uniform grid size are used in the vertical direction.The incident wave height is H = 9.9 cm, wave period is T = 1.35 s.The water depth in the flat bottom is h = 39.5 cm.The model setup is similar to the laboratory experiment of Sato et al. (1990).We conducted two numerical experiments: one with a patch of vegetation covering the beach from x = 8 m to x = 10 m, and the other without vegetation.The vegetation density λ is 30 m −1 .The vegetative drag coefficient C D is taken as 0.5, which represents the upper range of values seen for the similarly scaled experimental data (Dubi and Torum, 1997).
Figure 3(a) shows the simulated instantaneous surface elevation with the vegetation patch, from which we can clearly see the attenuation of waves inside the vegetation patch.The direct comparison of wave heights with and without vegetation patch is given in figure 3(b).Similar to the findings of Ma et al. (2014c), the wave height is greatly reduced by the patch.Consequently, the cross-shore gradient of wave height in the offshore part of the patch is greater than that without vegetation, possibly resulting in an increase of wave setup.In the onshore part of the patch, the cross-shore gradient of wave height is smaller than that without vegetation, which would produce a smaller wave setup.From the wave height distribution, it is noticed that the wave breaking happens around x b = 8.0 m, indicating that the vegetation patch is located   Figure 4 compares the predicted wave-averaged turbulent kinetic energy with and without vegetation patch at three stations in the surf zone.It has been widely recognized that the vegetation canopy can greatly reduce mean flow through enhancing resistance to the flow field, consequently compressing turbulence level (Nepf, 1999).In the surf zone, wave breaking generated turbulence is generally intense.However, the vegetation patch can dramatically damp turbulence, resulting in the turbulent kinetic energy (TKE) at least one order of magnitude smaller than that in the unvegetated surf zone as seen in figure 4. It seems that the wave breaking is completely inhibited by the vegetation canopy.The damping of turbulence within the patch would directly affect sediment suspension from the bottom as displayed in figure 5, which compares the vertical profiles of wave-averaged suspended sediment concentration with and without vegetation patch at three stations in the surf zone.To show the model's capability, we also compare the numerical results with laboratory measurements of Sato et al. (1990) in the case of no vegetation patch.In this simulation, the vertical Schmidt number for sediment σ v is taken as 0.5.We can see that the simulated vertical profiles of wave-averaged suspended sediment concentration agree well with the measurements, indicating that the model is capable of capturing nearshore sediment suspension.With the vegetation patch, the wave-averaged suspended sediment concentration is considerably smaller, indicating that much less sediments are picked up from the bottom.Most of these sediments are restricted in the bottom boundary layer with the thickness of about 1 cm.
The cross-shore sediment fluxes with and without vegetation are examined in figure 6.Following Henderson et al. (2004), we present three sediment fluxes: the mean Eulerian sediment flux < uC >, the "current-generated" flux < u >< C > and the "wave-generated" flux < (u− < u >)(C− < C >) >.Clearly, the former one equals the sum of the latter two fluxes.Similar to the findings in Ma et al. (2013b), in the surf zone without vegetation, the mean Eulerian sediment flux is seaward, which is contributed by both the "current-generated" flux and "wave-generated" flux.This is due to the strong return flow (undertow) in the surf zone, which generates offshore sediment flux.With the vegetation patch, suspended sediments are mostly restricted in the bottom boundary layer (figure 5).The sediment fluxes are also limited in a thin layer near the bottom.Because wave breaking is inhibited by the vegetation canopy, the undertow is relatively weak.Therefore, the offshore-directed "current-generated" flux is small.The mean Eulerian sediment flux is shoreward, which is mostly contributed by the "wave-generated" flux due to the skewness and asymmetry of the waves.These sediment fluxes are very similar to that outside the surf zone (Ma et al., 2014c).The shoreward Eulerian sediment flux indicates onshore net sediment transport within the vegetation patch.

CONCLUSIONS
In this paper, we have developed a three-dimensional hydrodynamic and sediment transport for studying flow/wave-vegetation-sediment interactions.The vegetation effects on the mean flow and turbulence were accounted for by introducing additional formulations for drag and turbulence production induced by vegetation canopy.The suspended sediment concentration was obtained by solving the advection-diffusion equation with sediment exchange at the bed.The turbulent flow and suspended sediment are simulated in a coupled manner by considering sediment-induced stratification.The model was validated by the laboratory measurements of partially vegetated open channel flows of Pasche and Rouve (1985).Numerical results showed that the model is capable of simulating vegetated open channel flow.
The model was then applied to study nearshore sediment suspension influenced by a patch of vegetation.It was found that the vegetation canopy could dramatically damp turbulence in the surf zone, resulting in considerably less sediment suspension.Within the vegetation patch, most suspended sediment are restricted in the wave boundary layer.The net sediment transport is shoreward, in contrast to the seaward net sediment transport in the unvegetated surf zone.This shoreward net sediment flux is mostly contributed by the wave-generated flux due to the skewness and asymmetry of the waves.

Figure 1 :
Figure 1: Cross-section geometry and vegetation arrangement in the experiments of Pasche and Rouve(1985).The cross section consists of a main channel and a vegetated floodplain joined by a transition slope.

Figure 2 :
Figure2: Model-data comparisons of lateral distributions of depth-averaged streamwise flow velocities for two experiments ofPasche and Rouve (1985) with different solid volume φ.Lines: simulations; circles: measurements.

Figure 3 :
Figure 3: (a) Model setup and simulated instantaneous free surface with vegetation.The patch of the emergent vegetation covers the region from x = 8 m to x = 10 m.(b) Comparison of simulated cross-shore distributions of wave height with (dashed line) and without (solid line) vegetation patch.

Figure 4 :
Figure 4: Comparisons of vertical profiles of wave-averaged turbulent kinetic energy with (dashed lines) and without (solid lines) vegetation patch at three stations.(a) x − x b = 0.378 m; (b) x − x b = 0.540 m and (c) x − x b = 0.729 m. x b is the location of breaking point.

Figure 5 :
Figure 5: Comparisons of vertical profiles of wave-averaged suspended sediment concentration with (dashed lines) and without (solid lines) vegetation patch at three stations.(a) x − x b = 0.378 m; (b) x − x b = 0.540 m and (c) x − x b = 0.729 m. x b is the location of breaking point.Circles represent laboratory measurements.

Figure 6 :
Figure 6: Cross-shore volumetric sediment flux (a) without vegetation and (b) with vegetation at station x − x b = 0.729 m.Solid lines: Eulerian mean sediment flux; dashed lines: mean current-induced sediment flux; dash-dotted lines: wave-induced sediment flux y, t), h is water depth, η is surface elevation.