Introduction¶

XBeach is an open-source numerical model which is originally developed to simulate hydrodynamic and morphodynamic processes and impacts on sandy coasts with a domain size of kilometers and on the time scale of storms. Since then, the model has been applied to other types of coasts and purposes.

The model includes the hydrodynamic processes of short wave transformation (refraction, shoaling and breaking), long wave (infragravity wave) transformation (generation, propagation and dissipation), wave-induced setup and unsteady currents, as well as overwash and inundation. The morphodynamic processes include bed load and suspended sediment transport, dune face avalanching, bed update and breaching. Effects of vegetation and of hard structures have been included. The model has been validated with a series of analytical, laboratory and field test cases using a standard set of parameter settings.

XBeach has two modes: a hydrostatic and a non-hydrostatic mode. In the hydrostatic mode, the short wave amplitude variation is solved separately from the long waves, currents and morphological change. This saves considerable computational time, with the expense that the phase of the short waves is not simulated. A more complete model is the non-hydrostatic model which solves all processes including short wave motions, but with more computational demand.

The original application, funded by the U.S. Corps of Engineers in the framework of the Morphos project and the U.S. Geological Survey, was to be able to assess hurricane impacts on sandy beaches. Since then with funding from the Dutch Public Works Department, the model has been extended, applied and validated for storm impacts on dune and urbanized coasts for the purpose of dune safety assessments. With support from the European Commission XBeach has been validated on a number of dissipative and reflective beaches bordering all regionall seas in the EU.

Beyond sandy coasts, the model has been applied to coral fringing and atoll reefs, in cooperation with and with funding by the University of Western Australia, the USGS and the Asian Development Bank. The model now also includes vegetative damping effects, with support of the U.S. Office of Naval Research.

The non-hydrostatic model has been developed initially by the TU Delft (as a prototype version of the SWASH model). For the purpose of simulating the morphodynamic processes on gravel beaches, the model was extended and validated with support from the University of Plymouth. In this mode, ship-induced waves can be simulated as well, demonstrating the flight that the model has taken since its first inception.

This development of XBeach could not have been possible without all of the above mentioned funding agencies and partners. It would also not have been possible without the enthusiastic, critical and constructive approach of all consultants, researchers, M.Sc. and Ph.D. students who have taken up XBeach, and made it into the tool that it is today.

This manual serves as an introduction to the model and a reference guide to its many functionalities, options and parameters. We sincerely hope that this document will help existing and new researchers apply the model for their purposes and advance our knowledge of coastal hydro- and morphodynamics.

Processes and model formulation¶

Domain and definitions¶

Coordinate system¶

XBeach uses a coordinate system where the computational x-axis is always oriented towards the coast, approximately perpendicular to the coastline, and the y-axis is alongshore, see Fig. 1 and Fig. 2. This coordinate system is defined in world coordinates. The grid size in x- and y-direction may be variable but the grid must be curvilinear. Alternatively, in case of a rectangular grid (a special case of a curvilinear grid) the user can provide coordinates in a local coordinate system that is oriented with respect to world coordinates (xw, yw) through an origin (xori, yori) and an orientation (alfa) as depicted in Fig. 1. The orientation is defined counter-clockwise w.r.t. the xw-axis (East).

Fig. 1 Rectangular coordinate system of XBeach

Grid set-up¶

The grid applied is a staggered grid, where the bed levels, water levels, water depths and concentrations are defined in cell centers, and velocities and sediment transports are defined in u- and v-points, viz. at the cell interfaces. In the wave energy balance, the energy, roller energy and radiation stress are defined at the cell centers, whereas the radiation stress gradients are defined at u- and v-points.

Velocities at the u- and v-points are denoted by the output variables uu and vv respectively; velocities u and v at the cell centers are obtained by interpolation and are for output purpose only. The water level, zs, and the bed level, zb, are both defined positive upward. uv and vu are the u-velocity at the v-grid point and the v-velocity at the u-grid point respectively. These are obtained by interpolation of the values of the velocities at the four surrounding grid points.

The model solves coupled 2D horizontal equations for wave propagation, flow, sediment transport and bottom changes, for varying (spectral) wave and flow boundary conditions.

Fig. 2 Curvilinear coordinate system of XBeach

Hydrodynamics options¶

XBeach was originally developed as a short-wave averaged but wave-group resolving model, allowing resolving the short wave variations on the wave group scale and the long waves associated with them. Since the original paper by [RRvD+09] a number of additional model options have been implemented, thereby allowing users to choose which time-scales to resolve:

Stationary wave model (keyword: wavemodel = stationary), efficiently solving wave-averaged equations but neglecting infragravity waves;
Surfbeat mode (instationary) (keyword: wavemodel = surfbeat), where the short wave variations on the wave group scale (short wave envelope) and the long waves associated with them are resolved;
Non-hydrostatic mode (wave-resolving) (keyword: wavemodel = nonh), where a combination of the non-linear shallow water equations with a pressure correction term is applied, allowing to model the propagation and decay of individual waves.

In the following these options are discussed in more detail. Important to note that all times in XBeach are prescribed on input in morphological time. If you apply a morphological acceleration factor (keyword: morfac) all input time series and other time parameters are divided internally by morfac. This way, you can specify the time series as real times, and vary the morfac without changing the rest of the input files (keyword: morfacopt = 1).

Fig. 3 Principle sketch of the relevant wave processes

Stationary mode¶

See also

The stationary mode is implemented in mod:wave_stationary_module.

In stationary mode the wave-group variations and thereby all infragravity motions are neglected. This is useful for conditions where the incident waves are relatively small and/or short, and these motions would be small anyway. The model equations are similar to HISWA ([HBH89]) but do not include wave growth or wave period variations. Processes that are resolved are wave propagation, directional spreading, shoaling, refraction, bottom dissipation and wave breaking, and a roller model is included; these processes are usually dominant in nearshore areas of limited extent. For the breaking dissipation we use the [BHBvW98] model, which is valid for wave-averaged modeling. The radiation stress gradients from the wave and roller model force the shallow water equations, drive currents and lead to wave setdown and setup. Additionally, wind and tidal forcing can be applied.

The mean return flow due to mass flux and roller is included in the model and affects the sediment transport, leading to an offshore contribution. To balance this, effects of wave asymmetry and skewness are included as well. Bed slope effects can further modify the cross-shore behavior. A limited number of model coefficients allow the user to calibrate the profile shape resulting from these interactions.

A typical application would be to model morphological changes during moderate wave conditions, often in combination with tides. The wave boundary conditions can be specified as constant (keyword: wbctype = stat) or as a time-series of wave conditions (keyword: wbctype = stat_table). Typical examples of such model applications are given below for tombolo formation behind an offshore breakwater (left panel) and development of an ebb delta at a tidal inlet (right panel). A big advantage of the stationary XBeach wave model over other models is that the lateral boundaries are entirely without disturbance if the coast is longshore uniform near these boundaries.

Fig. 4 Root-mean square wave height (left panels) and final bathymetry (right panels) for an offshore breakwater case (upper panels) and a tidal inlet with waves from 330 degrees (lower panels).

Surf beat mode (instationary)¶

See also

The surfbeat mode is implemented in mod:wave_instationary_module.

The short-wave motion is solved using the wave action equation which is a time-dependent forcing of the HISWA equations ([HBH89]). This equation solves the variation of short-waves envelope (wave height) on the scale of wave groups. It employs a dissipation model for use with wave groups ([Roe93][DRvD+12]) and a roller model ([Sve84][NRS90][SDeVriend94]) to represent momentum stored at the surface after breaking. These variations, through radiation stress gradients ([LHS62][LHS64]) exert a force on the water column and drive longer period waves (infragravity waves) and unsteady currents, which are solved by the nonlinear shallow water equations (e.g. [Phi77]). Thus, wave-driven currents (longshore current, rip currents and undertow), and wind-driven currents (stationary and uniform) for local wind set-up, long (infragravity) waves, and runup and rundown of long waves (swash) are included.

Using the surfbeat mode is necessary when the focus is on swash zone processes rather than time-averaged currents and setup. It is fully valid on dissipative beaches, where the short waves are mostly dissipated by the time they are near the shoreline. On intermediate beaches and during extreme events the swash motions are still predominantly in the infragravity band and so is the runup.

Under this surfbeat mode, several options are available, depending on the circumstances:

1D cross-shore; in this case the longshore gradients are ignored and the domain reduces to a single gridline (keyword: ny = 0). Within this mode the following options are available:

#. Retaining directional spreading (keyword: dtheta < thetamax – thetamin); this has a limited effect on the wave heights because of refraction, but can also allow obliquely incident waves and the resulting longshore currents;

#. Using a single directional bin (keyword: dtheta = thetamax – thetamin); this leads to perpendicular waves always and ignores refraction. If the keyword snells = 1 is applied, the mean wave direction is determined based on Snell’s law. In this case also longshore currents are generated.
2DH area; the model is solved on a curvilinear staggered grid (rectilinear is a special case). The incoming short wave energy will vary along the seaward boundary and in time, depending on the wave boundary conditions. This variation is propagated into the model domain. Within this mode the following options are available:
1. Resolving the wave refraction ’on the fly’ using the propagation in wave directional space. For large directional spreading or long distances this can lead to some smoothing of groupiness since the waves from different directions do not interfere but their energy is summed up. This option is possible for arbitrary bathymetry and any wave direction. The user must specify the width of the directional bins for the surfbeat mode (keyword: dtheta)
2. Solving the wave direction at regular intervals using the stationary solver, and then propagating the wave energy along the mean wave direction. This preserves the groupiness of the waves therefore leads to more forcing of the infragravity waves (keyword: single_dir = 1). The user must now specify a single directional bin for the instationary mode (dtheta = thetamax - thetamin) and a smaller bin size for the stationary solver (keyword: dtheta_s).
3. For schematic, longshore uniform cases the mean wave direction can also be computed using Snell’s law (keyword: snells = 1). This will then give comparable results to the single_dir option.

In the figures below some typical applications of 1D and 2D models are shown; a reproduction of a large-scale flume test, showing the ability of XBeach to model both short-wave (HF) and long-wave (LF) wave heights and velocities; and a recent 2DH simulation ([NLB+15]) of the impact of hurricane Sandy on Camp Osborne, Brick, NJ.

Fig. 5 Computed and observed hydrodynamic parameters for test 2E of the LIP11D experiment. Top left: bed level and mean water level. Top right: measured (dots) and computed

Fig. 6 Pre (left) and post-Sandy (right) in a three dimensional plot with both bed and water levels as simulated by XBeach ([NLB+15])

Non-hydrostatic mode (wave resolving)¶

See also

The non-hydrostatic mode is implemented in mod:nonh_module.

For non-hydrostatic XBeach calculations (keyword: wavemodel = nonh) depth-averaged flow due to waves and currents are computed using the non-linear shallow water equations, including a non-hydrostatic pressure. The depth-averaged normalized dynamic pressure ( $q$ ) is derived in a method similar to a one-layer version of the SWASH model ([ZSS11]). The depth averaged dynamic pressure is computed from the mean of the dynamic pressure at the surface and at the bed by assuming the dynamic pressure at the surface to be zero and a linear change over depth.

Under these formulations dispersive behavior is added to the long wave equations and the model can be used as a short-wave resolving model. Wave breaking is implemented by disabling the non-hydrostatic pressure term when waves exceed a certain steepness, after which the bore-like breaking implicit in the momentum-conserving shallow water equations takes over.

In case the non-hydrostatic mode is used, the short wave action balance is no longer required. This saves computation time. However, in the wave-resolving mode we need much higher spatial resolution and associated smaller time steps, making this mode much more computationally expensive than the surfbeat mode.

The main advantages of the non-hydrostatic mode are that the incident-band (short wave) runup and overwashing are included, which is especially important on steep slopes such as gravel beaches. Another advantage is that the wave asymmetry and skewness are resolved by the model and no approximate local model or empirical formulation is required for these terms. Finally, in cases where diffraction is a dominant process, wave-resolving modeling is needed as it is neglected in the short wave averaged mode. The XBeach-G formulations for gravel beaches ([MMP+14]) are based on the non-hydrostatic mode. Although sandy morphology can be simulated using the wave-resolving mode, it has not been extensively validated and it is likely that changes in the sediment transport formulations will be implemented in the near future.

An interesting recent application that has been validated for a number of cases concerns the modeling of primary waves generated by large ships, see Ship-induced wave motions.

Fig. 7 Measured (black) and modeled (red) time series of overtopping during BARDEX experiment, from [MMP+14].

Short wave action¶

Short wave action balance¶

See also

The short wave action balance is implemented in func:wave_instationary_module/wave_instationary.

The wave forcing in the shallow water momentum equation is obtained from a time dependent version of the wave action balance equation. Similar to Delft University’s (stationary) HISWA model ([HBH89]) the directional distribution of the action density is taken into account, whereas the frequency spectrum is represented by a frequency, best represented by the spectral parameter ${f}_{m-1,0}$ . The wave action balance (keyword: swave) is then given by:

(1) $\frac{\partial A}{\partial t} +\frac{\partial c_{x} A}{\partial x} +\frac{\partial c_{y} A}{\partial y} +\frac{\partial c_{\theta } A}{\partial \theta } =-\frac{D_{w} +D_{f} +D_{v} }{\sigma }$

In which the wave action $A$ is calculated as:

(2) $A(x,y,t,\theta )=\frac{S_{w} (x,y,t,\theta )}{\sigma (x,y,t)}$

where $\theta$ represents the angle of incidence with respect to the x-axis, ${S}_{w}$ represents the wave energy density in each directional bin and $\sigma$ the intrinsic wave frequency. The intrinsic frequency $\sigma$ and group velocity ${c}_{g}$ is obtained from the linear dispersion relation. The intrinsic frequency is for example obtained with:

(3) $\sigma =\sqrt{gk\tanh kh}$

The wave action propagation speeds in $x$ , $y$ and directional space are given by:

(4) $\begin{array}{l} {c_{x} (x,y,t,\theta )=c_{g} \cos (\theta )} \\ {c_{y} (x,y,t,\theta )=c_{g} \sin (\theta )} \\ {c_{\theta } (x,y,t,\theta )=\frac{\sigma }{\sinh 2kh} \left(\frac{\partial h}{\partial x} \sin \theta -\frac{\partial h}{\partial y} \cos \theta \right)} \end{array}$

where $h$ represents the local water depth and $k$ the wave number. The intrinsic wave frequency $\sigma$ is determined without wave current interaction (keyword: wci = 1, see Section 2.3.1.1), which means it is equal to the absolute radial frequency $\omega$ .

Wave-current interaction (wci)¶

See also

The wave-current interaction is implemented in func:wave_instationary_module/wave_instationary.

Wave-current interaction is the interaction between waves and the mean flow. The interaction implies an exchange of energy, so after the start of the interaction both the waves and the mean flow are affected by each other. This feature is especially of importance in gullies and rip-currents ([RMTS07]).

In XBeach this is taken into account by correcting the wave number $k$ with the use of Eikonal equations, which will have impact on the group and wave propagation speed (x, y, and directional space). The cross-shore and alongshore wave numbers, ${k}_{x}$ and ${k}_{y}$ , are defined according to equation (5). In these formulations the subscripts refer to the direction of the wave vector components.

(5) $\begin{array}{l} {k_{x} =k_{x}^{n-1} +k_{x}^{:} } \\ {k_{y} =k_{y}^{n-1} +k_{y}^{:} } \\ \end{array}$

Where subscript $n-1$ refers the wave number of the previous time step, $k_{x}^{:}$ and $k_{y}^{:}$ are the wave number corrections and ${k}_{x}$ and ${k}_{y}$ are the corrected wave numbers that take into account the presence of a current. The correction terms are determined with a second set of equations, the Eikonal equations:

(6) $\begin{array}{l} {\frac{\partial k_{x} }{\partial t} +\frac{\partial \omega }{\partial x} =0} \\ {\frac{\partial k_{y} }{\partial t} +\frac{\partial \omega }{\partial y} =0} \end{array}$

The wave number is then given by:

(7) $k=\sqrt{k_{x}^{2} +k_{y}^{2} }$

The absolute radial frequency $\omega$ is calculated with:

(8) $\omega =\sigma +k_{x} u^{L} +k_{y} v^{L}$

where ${u}^{L}$ and ${v}^{L}$ are the cross-shore and alongshore depth-averaged Lagrangian velocities respectively. The wave action propagation speed (in x and y direction) is given by:

(9) $\begin{array}{l} {c_{x} (x,y,t,\theta )=c_{g} \cos (\theta )+u^{L} } \\ {c_{y} (x,y,t,\theta )=c_{g} \sin (\theta )+v^{L} } \end{array}$

The propagation speed in directional ( $\theta$ ) space, where bottom refraction (first term) and current refraction (last two terms) are taken into account, is obtained from:

(10) $\begin{array}{r} {c_{\theta } (x,y,t,\theta )=\frac{\sigma }{\sinh 2kh} \left(\frac{\partial h}{\partial x} \sin \theta -\frac{\partial h}{\partial y} \cos \theta \right)+\cos \theta \left(\sin \theta \frac{\partial u}{\partial x} -\cos \theta \frac{\partial u}{\partial y} \right)} \\ {+\, \sin \theta \left(\sin \theta \frac{\partial v}{\partial x} -\cos \theta \frac{\partial v}{\partial y} \right)} \end{array}$

Dissipation¶

In XBeach there are three short wave dissipation processes that can be accounted for: wave breaking ( ${D}_{w}$ ), bottom friction ( ${D}_{f}$ ) and vegetation ( ${D}_{v}$ ). The three processes are explained in more detail in the following subsections.

Wave breaking¶

See also

Short wave dissipation by breaking is implemented in mod:roelvink_module.

Five different wave breaking formulations are implemented in XBeach. The formulations can be selected using the keyword break (Table 1).

Table 1 Different wave breaking formulations implemented¶
Wave breaking formula	Type of waves	keyword
Roelvink (1993a)	Instationary	roelvink1
Roelvink (1993a) extended	Instationary	roelvink2
Daly et al. (2010)	Instationary	roelvink_daly
Baldock et al. (1998)	Stationary	baldock
Janssen & Battjes (2007)	Stationary	janssen

For the surf beat approach the total wave energy dissipation, i.e. directionally integrated, due to wave breaking can be modeled according to [Roe93] (keyword: break = roelvink1). In the formulation of the dissipation due to wave breaking the idea is to calculate the dissipation with a fraction of breaking waves ( ${Q}_{b}$ ) multiplied by the dissipation per breaking event. In this formulation $\alpha$ is applied as wave dissipation coefficient of O (keyword: alpha), ${T}_{rep}$ is the representative wave period and ${E}_{w}$ is the energy of the wave. The fraction of wave breaking is determined with the root-mean-square wave height ( ${H}_{rms}$ ) and the maximum wave height ( ${H}_{max}$ ). The maximum wave height is calculated as ratio of the water depth ( $h$ ) plus a fraction of the wave height ( $\delta H_{rms}$ , keyword: delta) using a breaker index $\gamma$ (keyword: gamma). In the formulation for ${H}_{rms}$ the $\rho$ represents the water density and $g$ the gravitational constant. The total wave energy ${E}_{w}$ is calculated by integrating over the wave directional bins.

(11) $\begin{array}{c} {\bar{D}_{w} =2\frac{\alpha }{T_{rep} } Q_{b} E_{w} {\; }} \\ {Q_{b} {=1-exp}\left(-\left(\frac{H_{rms} }{H_{\max } } \right)^{n} \right),\quad H_{rms} =\sqrt{\frac{8E_{w} }{\rho g} } ,\quad H_{\max } =\gamma \cdot (h+\delta H_{rms} )} \\ {E_{w} (x,y,t)=\int _{0}^{2\pi }S_{w} (x,y,t,\theta )d\theta } \end{array}$

In variation of (11), one could also use another wave breaking formulation, presented in (12). This formulation is somewhat different than the formulation of [Roe93] and selected using keyword break = roelvink2. The main difference with the original formulation is that wave dissipation with break = roelvink2 is proportional to ${H}^{3} / h$ instead of ${H}^{2}$ .

(12) $\bar{D}_{w} =2\frac{\alpha }{T_{rep} } Q_{b} E_{w} \frac{H_{rms} }{h}$

Alternatively the formulation of [DRvD+10] states that waves are fully breaking if the wave height exceeds a threshold ( $\gamma$ ) and stop breaking if the wave height fall below another threshold ( $\gamma_{2}$ ). This formulation is selected by break = roelvink_daly and the second threshold, $\gamma_{2}$ , can be set using keyword gamma2.

(13) $\left\{\begin{array}{l} {Q_{b} =1\quad if\quad H_{rms} >\gamma h} \\ {Q_{b} =0\quad if\quad H_{rms} <\gamma _{2} h} \end{array}\right.$

In case of stationary waves [BHBvW98] is applied (keyword: break = baldock), which is presented in (14). In this breaking formulation the fraction breaking waves ${Q}_{b}$ and breaking wave height ${H}_{b}$ are calculated differently compared to the breaking formulations used for the non-stationary situation. In $\alpha$ is applied as wave dissipation coefficient, ${f}_{rep}$ represents a representative intrinsic frequency and $y$ is a calibration factor.

(14) $\begin{array}{l} {\bar{D}_{w} =\frac{1}{4} \alpha Q_{b} \rho gf_{rep} \left(H_{b}^{2} +H_{rms}^{2} \right)} \\ {Q_{b} =\exp \left[-\left(\frac{H_{b}^{2} }{H_{rms}^{2} } \right)\right]{\; ,\; \; }H_{b} =\frac{0.88}{k} \tanh \left[\frac{\gamma kh}{0.88} \right]} \end{array}$

Finally, it is possible to use the [JB07] formulation for wave breaking of stationary waves (keyword: break = janssen). This formulation is a revision of Baldock’s formulation.

(15) $\begin{array}{l} {\bar{D}_{w} =\frac{3\sqrt{\pi } \alpha f_{rep} \rho gH_{rms}^{3} }{16 h} Q_{b} } \\ {Q_{b} =1+\frac{4}{3\sqrt{\pi } } \left(R^{3} +\frac{3}{2} R\right)\exp \left(-R^{2} \right)-erf\left(R\right)} \\ {R=\frac{H_{b} }{H_{rms} } } \end{array}$

In both the instationary or stationary case the total wave dissipation is distributed proportionally over the wave directions with the formulation in (16).

(16) $D_{w} (x,y,t,\theta )=\frac{S_{w} (x,y,t,\theta )}{E_{w} (x,y,t)} \bar{D}_{w} (x,y,t)$

Bottom friction¶

See also

Short wave dissipation by bottom friction is implemented in func:wave_instationary_module/wave_instationary.

The short wave dissipation by bottom friction is modeled as

(17) $D_{f} =\frac{2}{3\pi } \rho f_{w} \left(\frac{\pi H_{rms} }{T_{m01} \sinh kh} \right)^{3}$

In (17) the ${f}_{w}$ is the short-wave friction coefficient. This value only affects the wave action equation and is unrelated to bed friction in the flow equation. Studies conducted on reefs (e.g. [LFK+07]) indicate that ${f}_{w}$ should be an order of magnitude (or more) larger than the friction coefficient for flow ( ${c}_{f}$ ) due to the dependency of wave frictional dissipation rates on the frequency of the motion.

The derivation of the short wave dissipation term is based time-averaged instantaneous bottom dissipation using the Johnson friction factor ${f}_{w}$ of the bed shear stress:

(18) $\tilde{D}_{f} =\left|\tau u\right|=\frac{1}{2} \rho f_{w} \left|\tilde{u}\right|^{3}$

The evaluation of the term $\left\langle\left|\tilde{u}\right|^{3}\right\rangle$ , the so-called third even velocity moment, depends on the situation. First we need expressions for the orbital velocity amplitude, which is expressed as:

(19) $u_{orb} =\frac{\pi H_{rms} }{T_{p} \sinh (kh)}$

In this formulation ${T}_{p}$ is the peak wave period, ${H}_{rms}$ is the root-mean-square wave height, $k$ is the wave number and $h$ is the local water depth.

If we consider the slowly-varying dissipation in wave groups, we need only to average over a single wave period and we can use a monochromatic (regular wave) expression. If we want to have the time-average dissipation over a full spectrum we get the best approximation from considering a linear Gaussian distribution. [GT85] give pragmatic expressions for both cases.

For the monochromatic case:

(20) $\left\langle \left|\tilde{u}\right|^{3} \right\rangle =1.20\left\langle \left|\tilde{u}\right|^{2} \right\rangle ^{3/2} =1.20\left(\frac{1}{2} u_{orb}^{2} \right)^{3/2} =0.42u_{orb}^{3}$

For the linear Gaussian approximation:

(21) $\left\langle \left|\tilde{u}\right|^{3} \right\rangle =1.60\left\langle \left|\tilde{u}\right|^{2} \right\rangle ^{3/2} =1.60\left(\frac{1}{2} u_{orb}^{2} \right)^{3/2} =0.57u_{orb}^{3}$

Combining (18) and orbital-velocity-monochromatic we get:

(22) $\left\langle \tilde{D}_{f} \right\rangle =0.21\rho f_{w} u_{orb}^{3}$

In XBeach the orbital velocity amplitude is computed as in orbital-velocity-monochromatic and the dissipation according to (22), which is correct for the case of instationary simulations on wave-group scale.

For the stationary case formulations (18) and (21) are similarly combined into:

(23) $\left\langle \tilde{D}_{f} \right\rangle =0.28\rho f_{w} u_{orb}^{3}$

Vegetation¶

See also

Short wave dissipation by vegetation is implemented in mod:vegetation_module.

The presence of aquatic vegetation within the area of wave propagation or wave breaking results in an additional dissipation mechanism for short waves. This is modeled using the approach of [ML04], which was adjusted by [SZB+12] to take into account vertically heterogeneous vegetation, see [vRVanTdVriesM+15]. The short wave dissipation due to vegetation is calculated as function of the local wave height and several vegetation parameters. The vegetation can be schematized in a number of vertical elements with each specific property. In this way the wave damping effect of vegetation such as mangrove trees, with a relatively dense root system but sparse stem area, can be modeled. The dissipation term is then computed as the sum of the dissipation per vegetation layer ([SZB+12]):

(24) $D_{v} =\sum _{i=1}^{n_{v} }D_{v,i}$

where ${D}_{v,i}$ is the dissipation by vegetation in vegetation layer $i$ and ${n}_{v}$ is the number of vegetation layers. The dissipation per layer is given by:

(25) $\begin{array}{l} {D_{v,i} =A_{v} \cdot \frac{\rho \widetilde{C}_{D,i} b_{v,i} N_{v,i} }{2\sqrt{\pi } } \left(\frac{kg}{2\sigma } \right)^{3} H_{rms} ^{3} ,{\; with}} \\ {A_{v} =\frac{\left(\sinh ^{3} k\alpha _{i} h-\sinh ^{3} k\alpha _{i-1} h\right)+3\left(\sinh k\alpha _{i} h-\sinh k\alpha _{i-1} h\right)}{3k\cosh ^{3} kh} } \end{array}$

where $\widetilde{C}_{D,i}$ is a (bulk) drag coefficient, ${b}_{v,i}$ is the vegetation stem diameter, ${N}_{v,i}$ is the vegetation density, and $\alpha_{i}$ is the relative vegetation height (= ${h}_{v} / h$ ) for layer $i$ . In case only one vegetation layer is specified, the plants are assumed to be vertically uniform, which would for example typically apply in case of modeling sea grass.

Radiation stresses¶

Given the spatial distribution of the wave action (and therefore wave energy) the radiation stresses can be evaluated by using linear wave theory as described by:

(26) $\begin{array}{l} {S_{xx,r} (x,y,t)=\int \cos ^{2} \theta S_{r} d\theta } \\ {S_{xy,r} (x,y,t)=S_{yx,r} (x,y,t)=\int \sin \theta \cos \theta S_{r} d\theta } \\ {S_{yy,r} (x,y,t)=\int \sin ^{2} \theta S_{r} d\theta } \end{array}$

Wave shape¶

See also

Wave shapes are implemented in func:morphevolution/RvR and func:morphevolution/vT.

The morphodynamic model considered is (short) wave averaged and resolves hydrodynamics associated with the wave group time scale. As a result the short wave shape is not solved for. However, as waves propagate from deep water onto beaches, their surface form and orbital water motion become increasingly non-linear because of the amplification of the higher harmonics.

There are two wave forms implemented to take this non-linearity into account:

A formulation of [RRvR12] based on a parameterization with the Ursell number. (keyword: waveform = ruessink_vanrijn)
A formulation of [vanTdVries09] based on the parameterized wave shape model of [RF81] (keyword: waveform = vanthiel)

The formulation of [RRvR12] relies on parameterizations for the non-linearity parameter $r$ and phase $\Phi$ . The parameterizations are based on a data set of 30.000+ field observations of the orbital skewness ${S}_{k}$ and asymmetry ${A}_{s}$ , collected under non-breaking and breaking wave conditions. The only variable parameter is the Ursell number, since according to [RRvR12] the Ursell that includes ${H}_{s}$ , $T$ and $h$ , describes the variability in ${S}_{k}$ and ${A}_{s}$ well. The Ursell number is calculated with the equation below.

(27) $U_{r} =\frac{3}{4} \frac{0.5H_{s} k}{(kh)^{3} }$

The value for the skewness and asymmetry is calculated with the use of a Boltzmann sigmoid. The skewness and asymmetry are a function of $\Psi$ . In the formulation of [RRvR12] the ${p}_{1:6}$ are used as parameterized factors on the data set of field observations.

(28) $\begin{array}{c} {B=p_{1} +\frac{p_{2} -p_{1} }{1+\exp \frac{p_{3} -\log Ur}{p_{4} } } } \\ {\psi =-90+90\tanh (p_{5} /Ur^{p_{6} } )} \\ {} \\ {S_{k} =B\cos \psi {\; \; \; and\; \; \; }A_{s} =B\sin \psi } \end{array}$

Alternatively, [vanTdVries09] utilized and extended the wave shape model of [RF81]. In this model the short wave shape is described by a Rienecker and Fenton lookup table with amongst others amplitudes non-linear components obtained with stream function theory. Therefore, wave skewness ( ${S}_{k}$ ) and asymmetry ( ${A}_{s})$ is computed in each grid point based on the water depth, dimensionless wave height and dimensionless wave period.

For $w$ equals one a skewed (Stokes) wave is obtained with high peaks and flat troughs whereas $w$ equals zero results in an asymmetric (saw tooth) wave with steep wave fronts. It is hypothesized that the weighting $w$ can be expressed as a function of wave skewness and asymmetry. The relation between the phase and the weighting is studied in more detail by [vanTdVries09] by varying $w$ between zero and one in small steps and computing the amplitudes with Rienecker and Fenton for a range of wave heights, wave periods and water depths. It is found that a unique relation between $w$ and $\Phi$ exists for any combination of wave height, wave period and water depth that is described by:

(29) $w=0.2719\ln \left(\left|\frac{\phi -1.8642}{0.2933-\phi } \right|\right)+0.5$

As explained in the next section, short-wave turbulence can be computed averaged over the bore interval ( ${T}_{bore}$ ). The bore interval is directly related to the wave shape and hence requires the weighting function $w$ is determined. For the formulation of [RRvR12] no exact wave shape is determined and therefore no bore interval can be calculated. Therefore this approach cannot be combined with bore averaged short-wave turbulence.

Turbulence¶

See also

Wave breaking induced turbulence is implemented in func:morphevolution/waveturb.

Wave breaking induced turbulence at the water surface has to be transported towards the bed in order to affect the up-stirring of sediment. [RS89] used an exponential decay model with the mixing length proportional to ${H}_{rms}$ to estimate the time averaged turbulence energy at the bed from turbulence at the water surface:

(30) $k_{b} =\frac{k}{\exp (h/H_{rms} )-1}$

where ${k}_{b}$ is turbulence variance at the bed and $k$ is the time averaged turbulence variance at the water surface.

There are three possibilities for the turbulence variance at the bed ( ${k}_{b}$ ) implemented into XBeach:

Wave averaged near-bed turbulence energy (keyword: turb = wave_averaged):

(31) $k_{b} =\frac{\overline{k_{s} }}{\exp (h/L_{mix} )-1}$
Bore-averaged near-bed turbulence energy [2] (keyword: turb = bore_averaged)

(32) $k_{b} =\frac{\overline{k_{s} }\cdot T_{rep} /T_{bore} }{\exp (h/L_{mix} )-1}$
Not taking into account the turbulence variance at the bed (keyword: turb = none)

Both formulations make use of the wave-averaged turbulence energy ( ${k}_{s}$ ) and a mixing length ( ${L}_{mix}$ ). The wave averaged turbulence energy at the surface is computed from the roller energy dissipation and following [Bat75] in which ${D}_{r}$ is roller dissipation:

(33) $\overline{k_{s} }=\left(D_{r} /\rho _{w} \right)^{2/3}$

The mixing length ( ${L}_{mix}$ ) is expressed as thickness of the surface roller near the water surface and depends on the roller volume ${A}_{r}$ ([Sve84]):

(34) $L_{mix} =\sqrt{A_{r} } =\sqrt{\frac{2E_{r} T_{rep} }{\rho _{w} c_{w} } }$

Roller energy balance¶

See also

The roller energy balance is implemented in func:wave_instationary_module/wave_instationary.

While the short wave action balance adequately describes the propagation and decay of organized wave energy, it has often been found that there is a delay between the point where the waves start to break (which is where you would expect the strongest radiation stress gradients to occur) and the point where the wave set-up and longshore current start to build. This transition zone effect is generally attributed to the temporary storage of shoreward momentum in the surface rollers. Several authors have analyzed the typical dimensions of such rollers and their effect on the radiation stress (e.g. [LHT74], [Sve84], [RS89], [NRS90], [Dei93], [SDeVriend94]).

The rollers can be represented as a blob of water with cross-sectional area A that slides down the front slope of a breaking wave. The roller exerts a shear stress on the water beneath it equal to:

(35) $\tau _{roller} =\frac{\rho gR}{L} \beta _{s}$

where $\beta_{s}$ is the slope of the breaking wave front, $R$ is the roller area and $L$ is the wave length. The roller has a kinetic energy equal to:

(36) $E_{r} =\frac{1}{2} \frac{\rho R(\overline{u_{roller}^{2} +w_{roller}^{2} })}{L}$

and a contribution to the radiation stress equal to:

(37) $S_{xx} =\frac{\rho R\overline{\left(u_{roller}^{2} -w_{roller}^{2} \right)}}{L}$

We can now formulate an energy balance for the roller as follows:

(38) $\frac{dE_{r} }{dt} =\frac{\partial E_{r} }{\partial t} +\frac{\partial E_{r} c\cos \theta }{\partial x} +\frac{\partial E_{r} c\sin \theta }{\partial y} =S-D$

where $S$ is the loss of organized wave motion due to breaking and $D$ is the dissipation. The latter is equal to the work done by the shear stress between the roller and the wave:

(39) $D_{r} =\tau _{roller} c_{g}$

Given the complex motion in the breaking waves, we can only give approximate estimates of the order of magnitude of the parameters in equations (35) till (39). Various authors have suggested that the velocity in the roller can be approximated as purely horizontal and equal to the wave celerity ${c}_{g}$ . In that case we get (for waves travelling in x-direction):

(40) $S_{xx,roller} =2E_{r}$

However, this must be seen as an (unrealistic) upper limit on the radiation stress contribution as this can only be valid for ${w}_{roller}=0$ . [NRS90] showed that the conceptual model of [RS89] would lead to a factor 0.22 instead of 2. However, a ratio in the order of 1 seems more realistic. [SDeVriend94] found a discrepancy between the roller shear stress derived from an energy balance and that derived from the momentum balance, in the order of a factor two. They explained this by a complicated analysis of the effect of water entering and leaving the roller, which led to a modification of the propagation term in the roller energy balance by a factor two. As this leads to the unphysical result that rollers would propagate at twice the wave celerity, we believe that the discrepancy must be sought in the ratio between roller energy and radiation stress contribution. Therefore we stick to the roller energy balance suggested by [NRS90] in equation (41) and the roller contribution to the radiation stress:

(41) $\begin{array}{l} {S_{xx,roller} \approx E_{r} \cos ^{2} \theta } \\ {S_{xy,roller} \approx E_{r} \cos \theta \sin \theta } \\ {S_{yy,roller} \approx E_{r} \sin ^{2} \theta } \end{array}$

This leads to an elegant and consistent distribution of the wave-induced forcing through the surfzone. To close the roller energy balance we need to express the dissipation of the roller as a function of ${E}_{r}$ . This can be done by introducing:

(42) $\overline{\left(u_{roller}^{2} +w_{roller}^{2} \right)}=\beta _{2} c_{g} ^{2}$

Combining this with equations (36) and (42) we then find:

(43) $D_{r} =2\beta _{s} \beta _{u} \frac{g}{c_{g} } E_{r}$

The coefficients $\beta_{s}$ and $\beta_{u}$ are usually lumped together into a single coefficient. This coefficient $\beta$ is in the O (keyword: beta), which may vary through the surf zone. The forcing of the longshore current by the radiation stress gradient can be derived from the wave and roller energy balances and :

(44) $\begin{array}{l} {F_{y} =-\frac{\partial S_{xy} }{\partial x} =-\frac{\partial }{\partial x} \left[\frac{c_{g} }{c} \left(E\cos \left(\theta \right)\sin \left(\theta \right)\right)+E_{r} \cos \left(\theta \right)\sin \left(\theta \right)\right]} \\ {\, \, \, \, \, \, \, \, \, \, \, =-\frac{\partial }{\partial x} \left[\frac{\sin \left(\theta \right)}{c} \left(Ec_{g} \cos \left(\theta \right)+E_{r} c\cos \left(\theta \right)\right)\right]} \\ {\, \, \, \, \, \, \, \, \, \, \, =-\left(Ec_{g} \cos \left(\theta \right)+E_{r} c\cos \left(\theta \right)\right)\frac{\partial }{\partial x} \left[\frac{\sin \left(\theta \right)}{c} \right]\, \, } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, -\frac{\sin \left(\theta \right)}{c} \frac{\partial }{\partial x} \left(Ec_{g} \cos \left(\theta \right)+E_{r} c\cos \left(\theta \right)\right)} \end{array}$

In a longshore uniform situation, according to Snell’s law, the first term on the right-hand side equals zero; the second term exactly equals the sum of the wave energy dissipation and the roller energy input and dissipation terms, so the forcing term reduces to:

(45) $F_{y} =\frac{D_{w} +(-D_{w} +D_{r} )}{c} \sin (\alpha )=\frac{D_{r} }{c} \sin (\alpha )$

Shallow water equations¶

See also

The shallow water equations are implemented in func:flow_timestep_module/flow.

For the low-frequency waves and mean flows we use the shallow water equations. To account for the wave induced mass-flux and the subsequent (return) flow these are cast into a depth-averaged Generalized Lagrangian Mean (GLM) formulation ([AM78], [WRG00]). In such a framework, the momentum and continuity equations are formulated in terms of the Lagrangian velocity ${u}^{L}$ which is defined as the distance a water particle travels in one wave period, divided by that period. This velocity is related to the Eulerian velocity (the short-wave-averaged velocity observed at a fixed point) by:

(46) $u^{L} =u^{E} +u^{S} \quad and\quad v^{L} =v^{E} +v^{S}$

where ${u}^{S}$ and ${v}^{S}$ represent the Stokes drift in x- and y-direction respectively ([Phi77]). The Strokes drift is calculated with in which the wave-group varying short wave energy ${E}_{w}$ and direction are obtained from the wave-action balance.

(47) $u^{S} =\frac{E_{w} \cos \theta }{\rho hc} \quad and\quad v^{S} =\frac{E_{w} \sin \theta }{\rho hc}$

The resulting GLM-momentum equations are given by:

(48) $\begin{array}{c} {\frac{\partial u^{L} }{\partial t} +u^{L} \frac{\partial u^{L} }{\partial x} +v^{L} \frac{\partial u^{L} }{\partial y} -f\, v^{L} \, -\, \nu _{h} \left(\frac{\partial ^{2} u^{L} }{\partial x^{2} } +\frac{\partial ^{2} u^{L} }{\partial y^{2} } \right)=\frac{\tau _{sx} }{\rho h} -\frac{\tau _{bx}^{E} }{\rho h} -g\frac{\partial \eta }{\partial x} +\frac{F_{x} }{\rho h} +\frac{F_{v,x} }{\rho h} } \\ {\frac{\partial v^{L} }{\partial t} +u^{L} \frac{\partial v^{L} }{\partial x} +v^{L} \frac{\partial v^{L} }{\partial y} +f\, u^{L} \, -\, \nu _{h} \left(\frac{\partial ^{2} v^{L} }{\partial x^{2} } +\frac{\partial ^{2} v^{L} }{\partial y^{2} } \right)=\frac{\tau _{sy} }{\rho h} -\frac{\tau _{by}^{E} }{\rho h} -g\frac{\partial \eta }{\partial y} +\frac{F_{y} }{\rho h} +\frac{F_{v,y} }{\rho h} } \\ {\frac{\partial \eta }{\partial t} +\frac{\partial hu^{L} }{\partial x} +\frac{\partial hv^{L} }{\partial y} =0} \end{array}$

where $\tau_{sx}$ and $\tau_{sy}$ are the wind shear stresses, $\tau_{bx}$ and $\tau_{by}$ are the bed shear stresses, $\eta$ is the water level, ${F}_{x}$ and ${F}_{y}$ are the wave-induced stresses, ${F}_{v,x}$ , and ${F}_{v,y}$ are the stresses induced by vegetation, $\nu_{h}$ is the horizontal viscosity and $f$ is the Coriolis coefficient. Note that the shear stress terms are calculated with the Eulerian velocities as experienced by the bed and not with the GLM velocities, as can be seen in .

Horizontal viscosity¶

See also

The Smagorinsky model is implemented in func:flow_timestep_module/visc_smagorinsky.

The horizontal viscosity ( ${v}_{h}$ ) is by default computed using the [Sma63] model to account for the exchange of horizontal momentum at spatial scales smaller than the computational grid size, which is given as:

(49) $v_{h} =c_{S} ^{2} 2^{\frac{1}{2} } \sqrt{\left(\frac{\delta u}{\delta x} \right)^{2} +\left(\frac{\delta v}{\delta y} \right)^{2} +\frac{1}{2} \left(\frac{\delta u}{\delta x} +\frac{\delta v}{\delta y} \right)^{2} } \Delta x\Delta y$

In ${c}_{S}$ is the Smagorinsky constant (keyword: nuh), set at 0.1 in all model simulations. It is also possible to use a user-defined value for the horizontal viscosity by turning off the Smagorinsky model (keyword: smag = 0) and specifying the value directly (also keyword: nuh).

Bed shear stress¶

See also

Bed shear stresses are implemented in mod:bedroughness_module.

The bed friction associated with mean currents and long waves is included via the formulation of the bed shear stress ( $\tau_{b}$ ). Using the approach of [RMF+01] the bed shear stress is calculated with:

(50) $\begin{array}{l} {\tau _{bx}^{E} =c_{f} \rho u_{E} \sqrt{\left(1.16u_{rms} \right)^{2} +\left(u_{E} +v_{E} \right)^{2} } } \\ {\tau _{by}^{E} =c_{f} \rho v_{E} \sqrt{\left(1.16u_{rms} \right)^{2} +\left(u_{E} +v_{E} \right)^{2} } } \end{array}$

There are 5 different formulations in order to determine the dimensionless bed friction coefficient ${c}_{f}$ (keyword: bedfriction) implemented in XBeach (Table 2).

Table 2 Different bed friction formulations implemented¶
Bed friction formulation	Relevant coefficient	keyword
Dimensionless friction coefficient	${c}_{f}$	cf
Chézy	C	chezy
Manning	n	manning
White-Colebrook	${k}_{s}$	white-colebrook
White-Colebrook grain size	D90	white-colebrook-grainsize

The dimensionless friction coefficient can be calculated from the Chézy value with equation (51). A typical Chézy value is in the order of $55 {m}^{1/2}/s$ .

(51) $c_{f} =\sqrt{\frac{g}{C^{2} } }$

In the Manning formulation the Manning coefficient ( $n$ ) must be specified. The dimensionless friction coefficient is calculated from equation (52). Manning can be seen as a depth-dependent Chézy value and a typical Manning value would be in the order of $0.02 s/{m}^{1/3}$ .

(52) $c_{f} =\sqrt{\frac{gn^{2} }{h^{1/12} } }$

In the White-Colebrook formulation the geometrical roughness of Nikuradse ( ${k}_{s}$ ) must be specified. The dimensionless friction coefficient is calculated from equation (53) The White-Colebrook formulation has al log relation with the water depth and a typical ${k}_{s}$ value would be in the order of 0.01 - 0.15 m.

(53) $c_{f} =\sqrt{\frac{g}{\left(18\log \left(\frac{12h}{k_{s} } \right)\right)^{2} } }$

The option of White-Colebrook based on the grain size is somewhat different than the other four formulations. This formulation is based on the relation between the ${D}_{90}$ of the top bed layer and the geometrical roughness of Nikuradse according to equation (54). The user does not have to specify a value for the bed friction coefficient.

(54) $c_{f} =\sqrt{\frac{g}{\left(18\log \left(\frac{12h}{3D_{90} } \right)\right)^{2} } }$

Values of the drag coefficient for different seabed sediment grain sizes (flat beds) and similarly for bed form scenarios have been empirically derived from field and laboratory data in previous studies for different bed friction coefficients. The value of the friction coefficient ( $C$ , ${c}_{f}$ , $n$ or ${k}_{s}$ ) can be defined with one single value (keyword: bedfriccoef) or for a separate value per grid cell (keyword: bedfricfile)

Damping by vegetation¶

See also

Infra-gravity wave damping by vegetation is implemented in mod:vegetation_module.

The presence of aquatic vegetation within the area of wave propagation or wave breaking may not only result in short wave dissipation (Dissipation), but also in damping of infragravity waves and/or mean flow. Since both long waves and mean flow are fully resolved with the nonlinear shallow water equations, the effect of vegetation can be modeled using a drag force (e.g. [DKH84]), which can be directly added to the momentum equations ([vRVanTdVriesM+15], equation (48)):

(55) $F_{v} =F_{D} =\frac{1}{2} \rho C_{D} b_{v} Nu\left|u\right|$

Where ${C}_{D}$ is a drag coefficient, ${b}_{v}$ is the vegetation stem diameter, $N$ is the vegetation density and $u$ is the wave or current related velocity. To take into account the velocity due to mean flow and infragravity waves, we use the Lagrangian velocity ( ${u}^{L}$ ) here. The vegetation-induced time varying drag force is then calculated as the sum of the vegetation-induced drag force per vegetation layer:

(56) $\begin{array}{l} {F_{v} (t)=\sum _{i=1}^{n_{v} }F_{v,i} (t)} \\ {F_{v,i} (t)=\frac{1}{2} \rho \widetilde{C}_{D,i} b_{v,i} N_{v,i} h_{v,i} u^{L} (t)\left|u^{L} (t)\right|} \end{array}$

where $\widetilde{C}_{D,i}$ is a (bulk) drag coefficient, ${b}_{v,i}$ is the vegetation stem diameter, ${n}_{v,i}$ is the vegetation density, and ${h}_{v,i}$ is the vegetation height for layer $i$ .

Wind¶

The first term on the right hand side of the momentum equations (equation (48)) represents the forcing due to the wind stress. These forcing terms due to the wind are formulated as:

(57) $\begin{array}{rcl} \tau _{sx} &=& \rho _{a} C_{d} W\left|W_{x} \right| \\ \tau _{sy} &=& \rho _{a} C_{d} W\left|W_{y} \right| \\ \end{array}$

where $\tau_{w}$ is wind stress, $\rho_{a}$ is density of air, ${C}_{d}$ is the wind drag the coefficient, $W$ is the wind velocity. The wind stress is turned off by default, and can be turned on specifying a constant wind velocity (keyword: windv) or by specifying a time varying wind file.

Non-hydrostatic pressure correction¶

See also

The non-hydrostatic pressure correction is implemented in mod:nonh_module.

For non-hydrostatic XBeach calculations (keyword: waveform = nonh) depth-averaged flow due to waves and currents are computed using the non-linear shallow water equations, including a non-hydrostatic pressure. The non-hydrostatic model accounts for all wave motions (including short waves) within the shallow water equations, so the wave action balance should be turned off (keyword: swave = 0). The depth-averaged normalized dynamic pressure ( $q$ ) is derived in a method similar to a one-layer version of the SWASH model ([ZSS11]). The depth averaged dynamic pressure is computed from the mean of the dynamic pressure at the surface and at the bed by assuming the dynamic pressure at the surface to be zero and a linear change over depth. In order to compute the normalized dynamic pressure at the bed, the contributions of advective and diffusive terms to the vertical momentum balance are assumed to be negligible.

(58) $\frac{\delta w}{\delta t} +\frac{\delta q}{\delta z} =0$

In $w$ is the vertical velocity and $z$ is the vertical coordinate. The vertical velocity at the bed is set by the kinematic boundary condition:

(59) $w_{b} =u\frac{\delta (\eta -h)}{\delta x}$

Combining the Keller-box method ([LS76]), as applied by [SZ03] for the description of the pressure gradient in the vertical, the dynamic pressure at the bed can be described by:

(60) $q_{b} =-\frac{h}{2} \left(\left. \frac{\delta q}{\delta z} \right|_{s} +\left. \frac{\delta q}{\delta z} \right|_{b} \right)$

Substituting in allows the vertical momentum balance at the surface to be described by:

(61) $\frac{\delta w_{s} }{\delta t} =2\frac{q_{b} }{h} -\frac{\delta w_{b} }{\delta t}$

In the subscript $s$ refers to the location at the surface. The dynamic pressure at the bed is subsequently solved by combining and the local continuity equation:

(62) $\frac{\delta u}{\delta x} +\frac{w_{s} -w_{b} }{h} =0$

In order to improve the computed location and magnitude of wave breaking, the hydrostatic front approximation (HFA) of [SJHS14] is applied, in which the pressure distribution under breaking bores is assumed to be hydrostatic. Following the recommendations of [SJHS14], we consider hydrostatic bores if $\frac{\delta \eta }{\delta t} >0.6$ and reform if $\frac{\delta \eta }{\delta t} <0.3$ . The values can respectively be changed with the keywords maxbrsteep and secbrsteep.

Although this method greatly oversimplifies the complex hydrodynamics of plunging waves, [MMP+14] shows that the application of this method provides sufficient skill to describe dominant characteristics of the flow, without requiring computationally expensive high-resolution discretization of the vertical and surface tracking of overturning waves.

Groundwater flow¶

See also

Groundwater flow is implemented in mod:groundwaterflow.

The groundwater module (keyword: gwflow = 1) in XBeach utilizes the principle of Darcy flow for laminar flow conditions and a parameterization of the Forchheimer equations for turbulent groundwater flow. The module includes a vertical interaction flow between the surface water and groundwater. This flow is assumed to be a magnitude smaller than the horizontal flow and is not incorporated in the momentum balance.

Continuity¶

In order to solve mass continuity in the groundwater model, the groundwater is assumed to be incompressible. Continuity is achieved by imposing a non-divergent flow field:

(63) $\nabla U=0$

where $U$ is the total specific discharge velocity vector, with components in the horizontal ( ${u}_{gw}$ , ${v}_{gw}$ ) and vertical ( ${w}_{gw}$ ) direction:

(64) $U=\left[\begin{array}{c} {u} \\ {v} \\ {w} \end{array}\right]$

Equation of motions¶

Laminar flow of an incompressible fluid through a homogeneous medium can be described using the well-known Law of [Dar56], valid for laminar flow conditions (keyword: gwscheme = laminar)

(65) $\begin{array}{c} {u_{gw} =-K\frac{\partial H}{\partial x} } \\ {v_{gw} =-K\frac{\partial H}{\partial y} } \\ {w_{gw} =-K\frac{\partial H}{\partial z} } \end{array}$

in which $K$ is the hydraulic conductivity of the medium (keyword: $kx$ $ky$ , $kz$ , for each horizontal and vertical direction) and $H$ is the hydraulic head.

In situations in which flow is not laminar, turbulent and inertial terms may become important. In these cases the user can specify XBeach to use a method (keyword: gwscheme = turbulent) that is comparable with the USGS MODFLOW-2005 groundwater model ([Har05]), in which the turbulent hydraulic conductivity is estimated based on the laminar hydraulic conductivity ( ${K}_{lam}$ ) and the Reynolds number at the start of turbulence ( ${Re}_{crit}$ ) ([Hal00]):

(66) $\begin{array}{c} {u_{gw} =-K\left(Re\right)\frac{\partial H}{\partial x} {\; \; \; in\; which\; }Re=\frac{\left|U\right|D_{50} }{n_{p} v} } \\ {K\left(Re\right)=\left\{\begin{array}{l} {K_{lam} \sqrt{\frac{Re_{crit} }{Re} } {\; \; \; if\; \; }Re{\; >\; }Re_{crit} {\; }} \\ {K_{lam} {\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; if\; \; }Re\le {\; }Re_{crit} {\; }} \end{array}\right. } \end{array}$

In the Reynolds number ( $Re$ ) is calculated using the median grain size ( ${D}_{50}$ ), the kinematic viscosity of water ( $\nu$ ) and the groundwater velocity in the pores ( $U/n_{p}$ ), where ${n}_{p}$ is the porosity. Similar expressions exist for the other two components of the groundwater flow.

The critical Reynolds number for the start of turbulence ( ${Re}_{crit}$ ) is specified by the user, based on in-situ or laboratory measurements, or expert judgment (keyword: gwReturb). Since the hydraulic conductivity in the turbulent regime is dependent on the local velocity, an iterative approach is taken to find the correct hydraulic conductivity and velocity.

Determination of the groundwater head¶

The XBeach groundwater model allows two methods to determine the groundwater head: a hydrostatic approach (keyword: gwnonh = 0) and a non-hydrostatic approach (keyword: gwnonh = 1).

Hydrostatic approach

In the hydrostatic approach, the groundwater head is computed as follows:

In cells where there is no surface water the groundwater head is set equal to the groundwater surface level $\eta_{gw}$ .
In cells where there is surface water, but the groundwater surface level $\eta_{gw}$ is more than ${d}_{wetlayer}$ (keyword: dwetlayer) below the surface of the bed, the groundwater head is set equal to the groundwater surface level.
In cells where there is surface water and the groundwater surface level $\eta_{gw}$ is equal to the surface of the bed, the groundwater head is set equal to the surface water level.
In cells where there is surface water and the groundwater surface level $\eta_{gw}$ is equal to or less than ${d}_{wetlayer}$ below the surface of the bed, the groundwater head is linearly weighted between that of the surface water level and the groundwater level, according to the distance from the groundwater surface to the surface of the bed.

It should be noted that the numerical parameter ${d}_{wetlayer}$ is required to ensure numerical stability of the hydrostatic groundwater model. Larger values of ${d}_{wetlayer}$ will increase numerical stability, at the expense of numerical accuracy.

Non-hydrostatic approach

Groundwater flow in the swash and surf zone has been shown to be non-hydrostatic (e.g., [LB00]; [LMHK07]). In order to capture this, it may be necessary in certain cases to reject the Dupuit–Forchheimer assumption of hydrostatic groundwater pressure.

In the non-hydrostatic approach, the groundwater head is not assumed to be constant in the vertical. Since XBeach is depth-averaged, the model cannot compute true vertical profiles of the groundwater head and velocity. In order to estimate of the groundwater head variation over the vertical, a quasi-3D modeling approach is applied, which is set by two boundary conditions and one non-hydrostatic shape assumption:

There is no exchange of groundwater between the aquifer and the impermeable layer below the aquifer.
The groundwater head at the upper surface of the groundwater is continuous with the head applied at the groundwater surface.
The shape of the non-hydrostatic head profile is parabolic (keyword: gwheadmodel = parabolic), implying that the vertical velocity increases or decreases linearly from the bottom of the aquifer to the upper surface of the groundwater, or the non-hydrostatic head profile is hyperbolic (keyword: gwheadmodel = exponential), cf., [RGE99].

The vertical groundwater head approximation can be solved for the three imposed conditions by a vertical head function, shown here for the parabolic head assumption. The depth-average value of the groundwater head is used to calculate the horizontal groundwater flux and is found by integrating the groundwater head approximation over the vertical:

(67) $\overline{H}=\frac{1}{h_{gw} } \int _{0}^{h_{gw} }H(\sigma )d\sigma =H_{bc} -\frac{2}{3} \beta h_{gw}^{1}$

In the mean vertical ground water head ( $H$ ) is calculated using the groundwater head imposed at the groundwater surface ( ${H}_{bc}$ ), the groundwater head parabolic curvature coefficient ( $\beta$ ) and the height of the groundwater level above the bottom of the aquifer ( ${h}_{gw}$ ).

The unknown curvature coefficient ( $\beta$ ) in the vertical groundwater head approximation is solved using the coupled equations for continuity and motion (equations (63) and (65)), thereby producing the depth-average horizontal groundwater head gradients and vertical head gradients at the groundwater surface.

Although the requirement for non-hydrostatic pressure has the benefit of being a more accurate representation of reality, and does not require the numerical smoothing parameter ${d}_{wetlayer}$ , resolving the non-hydrostatic pressure field can be computationally expensive, particularly in 2DH applications.

Exchange with surface water¶

In the groundwater model there are three mechanisms for the vertical exchange of groundwater and surface water: 1) submarine exchange, 2) infiltration and 3) exfiltration. The rate of exchange between the groundwater and surface water ( $S$ ) is given in terms of surface water volume, and is defined positive when water is exchanged from the surface water to the groundwater.

Infiltration and exfiltration can only occur in locations where the groundwater and surface water are not connected. Infiltration takes place when surface water covers an area in which the groundwater level is lower than the bed level. The flux of surface water into the bed is related to the pressure gradient across the wetting front.

(68) $\begin{array}{c} {S_{inf} =K\left(\frac{1}{\rho g} \frac{\left. p\right|^{z=\xi } }{\delta _{infill} } 1\right)} \\ {{in\; which\; }\delta _{infill} (t)=\int \frac{S}{n_{p} } dt } \end{array}$

In equation (68) the surface water-groundwater exchange flow of infiltration ( ${S}_{inf}$ ) is calculated using the effective hydraulic conductivity ( $K$ ), the surface water pressure at the bed ( $\left. p\right|^{z=\xi }$ ) and the thickness of the wetting front ( $\delta_{infill}$ ).

Since the groundwater model is depth-averaged and cannot track multiple layers of groundwater infiltrating into the bed, the wetting front thickness is reset to zero when there is no available surface water, the groundwater exceeds the surface of the bed, or the groundwater and the surface water become connected. In addition, all infiltrating surface water is instantaneously added to the groundwater volume, independent of the distance from the bed to the groundwater table. Since the groundwater model neglects the time lag between infiltration at the beach surface and connection with the groundwater table a phase error may occur in the groundwater response to swash dynamics

Exfiltration ( ${S}_{exf}$ ) occurs where the groundwater and surface water are not connected and the groundwater level exceeds the bed level. The rate of exfiltration is related to the rate of the groundwater level exceeding the bed level.

(69) $S_{exf} =n_{p} \frac{\delta (z_{b} -\eta _{gw} )}{\delta t}$

Submarine exchange ( ${S}_{sub}$ ) represents the high and low frequency infiltration and exfiltration through the bed due pressure gradients across the saturated bed. This process only takes place where the groundwater and surface water are connected. In the case of the non-hydrostatic groundwater model, the rate of submarine exchange is determined by the vertical specific discharge velocity at the interface between the groundwater and surface water. The value of this velocity can be found using the vertical derivative of the approximated groundwater head at the groundwater-surface water interface (shown for the parabolic head approximation).

(70) $S_{sub} =2\beta h_{gw} K$

In the case of the hydrostatic groundwater model, the difference between the surface water head and the groundwater head is used to drive submarine discharge when the groundwater level is less than ${d}_{wetlayer}$ from the bed surface.

While most beach systems can acceptably described through vertical exchange of surface water and groundwater, in cases of very steep permeable slopes (e.g., porous breakwaters), it is necessary to include the horizontal exchange of groundwater and surface water between neighboring cells (keyword: gwhorinfil = 1). In this case the horizontal head gradient between the surface water and groundwater across vertical interface between the cells is used to determine the horizontal exchange flux:

(71) $S_{hor} =-K\frac{\partial H_{s} }{\partial s} A$

where $\delta H_{s}$ is the head gradient between the surface water and groundwater in neighboring cell, $\delta s$ is the gradient distance, defined as the numerical grid size, and $A$ is the surface area through which the exchange takes place, defined as the difference in bed level between the neighboring cells.

Calculation of groundwater and surface water levels¶

Groundwater levels are updated through the continuity relation:

(72) $n_{p} \frac{\delta \eta _{gw} }{\delta t} =-\frac{\partial h_{gw} u_{gw} }{\partial x} -\frac{\partial h_{gw} v_{gw} }{\partial y} +S_{inf} +S_{exf} +S_{sub} +S_{hor}$

In these same areas the surface water level is modified to account for exchange fluxes:

(73) $\frac{\delta \eta }{\delta t} =-S_{inf} -S_{exf} -S_{sub} -S_{hor}$

Boundary conditions¶

Since the groundwater dynamics are described by a parabolic equation, the system of equations requires boundary conditions at all horizontal and vertical boundaries, as well as an initial condition:

A zero flux condition is imposed at the horizontal boundaries and bottom of the aquifer.
The initial condition for the solution is specified by the model user in terms of the initial groundwater head (keyword: gw0, or gw0file).

Sediment transport¶

See also

Sediment transport is implemented in func:morphevolution/transus.

Advection-diffusion equation¶

Sediment concentrations in the water column are modeled using a depth-averaged advection-diffusion scheme with a source-sink term based on equilibrium sediment concentrations ([GV83]):

(74) $\frac{\partial hC}{\partial t} +\frac{\partial hCu^{E} }{\partial x} +\frac{\partial hCv^{E} }{\partial y} +\frac{\partial }{\partial x} \left[D_{h} h\frac{\partial C}{\partial x} \right]+\frac{\partial }{\partial y} \left[D_{h} h\frac{\partial C}{\partial y} \right]=\frac{hC_{eq} -hC}{T_{s} }$

In $C$ represents the depth-averaged sediment concentration which varies on the wave-group time scale and ${D}_{h}$ is the sediment diffusion coefficient. The entrainment of the sediment is represented by an adaptation time ${T}_{s}$ , given by a simple approximation based on the local water depth $h$ and sediment fall velocity ${w}_{s}$ . A small value of ${T}_{s}$ corresponds to nearly instantaneous sediment response (keyword: Tsmin). The factor ${f}_{Ts}$ is a correction and calibration factor to take into account the fact that ${w}_{s}$ is determined on depth-averaged data (keyword: tsfac).

(75) $T_{s} =\max \left(f_{Ts} \frac{h}{w_{s} } ,T_{s,\min } \right)\;$

The entrainment or deposition of sediment is determined by the mismatch between the actual sediment concentration $C$ and the equilibrium concentration ${C}_{eq}$ thus representing the source term in the sediment transport equation.

General parameters¶

In the sediment transport formulations, the equilibrium sediment concentration ${C}_{eq}$ (for both the bed load and the suspended load) is related to the velocity magnitude ( ${v}_{mg}$ ), the orbital velocity ( ${u}_{rms}$ ) and the fall velocity ( ${w}_{s}$ ). This section elaborates how these are calculated. Important to note: XBeach calculates the equilibrium concentration for the bed and suspended load separately.

First of all the Eulerian , if long wave stirring is turned on (keyword: lws = 1), the velocity magnitude ${v}_{mg}$ is equal to the magnitude of the Eulerian velocity, as can be seen in .

(76) $v_{mg} =\sqrt{\left(u^{E} \right)^{2} +\left(v^{E} \right)^{2} }$

If wave stirring is turned off (keyword: lws = 0), the velocity magnitude will be determined by two terms: first of all a factor of the velocity magnitude of the previous time step ( ${v}_{mg}^{n-1}$ ) and secondly a current-averaged part. Averaging will be carried out based on a certain factor ${f}_{cats}$ (keyword: cats) of the representative wave period ${T}_{rep}$ .

(77) $v_{mg} =\left(1-\frac{dt}{f_{cats} T_{rep} } \right)v_{mg}^{n-1} +\frac{dt}{f_{cats} T_{rep} } \sqrt{\left(u^{E} \right)^{2} +\left(v^{E} \right)^{2} }$

Secondly, the , the ${u}_{rms}$ is obtained from the wave group varying wave energy using linear wave theory. In this formulation ${T}_{rep}$ is the representative wave period and the ${H}_{rms}$ is the root-mean-square wave height. In this equation the water depth is enhanced with a certain factor of the wave height (keyword: delta).

(78) $u_{rms} =\frac{\pi H_{rms} }{T_{rep} \sqrt{2} \sinh (k(h+\delta H_{rms} )}$

To account for wave breaking induced turbulence due to short waves, the orbital velocity is adjusted ([vanTdVries09]). In this formulation ${k}_{b}$ is the wave breaking induced turbulence due to short waves. The turbulence is approximated with an empirical formulation in XBeach.

(79) $u_{rms,2}^{2} =u_{rms}^{2} +1.45k_{b}$

Thirdly, the , the ${w}_{s}$ is calculated using the formulations of [Ahr00] which are derived based on a relationship suggested by [Hal81]:

(80) $w_{s} =\alpha _{1} \sqrt{\Delta gD_{50} } +\alpha _{2} \frac{\Delta gD_{50}^{2} }{\nu }$

(81) $\alpha _{1} =1.06\tanh \left(0.016A^{0.50} \exp \left(-120/A\right)\right)$

(82) $\alpha _{2} =0.055\tanh \left(12A^{-0.59} \exp \left(-0.0004A\right)\right)$

For high sediment concentrations, the fall velocity is reduced (keyword: fallvelred = 1) using the expression of [RZ54]:

(83) $w_{s,reduced} =\left(1-C\right)^{\alpha } w_{s}$

The exponent a is estimated using the equation of [Row87], which depends purely on the Reynolds particle number R:

(84) $\alpha =2.35\frac{2+0.175{R}^{3/4} }{1+0.175{R}^{3/4} }$

(85) ${R}=\frac{w_{s} D_{50} }{\nu }$

Transport formulations¶

See also

The sediment transport formulations are implemented in func:sedtransform

In the present version of XBeach, two sediment transport formulations are available. The formulae of the two formulations are presented in the following sections. For both methods the total equilibrium sediment concentration is calculated with equation (86). In this equation the minimum value of the equilibrium sediment concentration (for both bed load en suspended load) compared to the maximum allowed sediment concentration (keyword: cmax).

(86) $C_{eq} ={max\; }({min\; }(C_{eq,b} ,\frac{1}{2} C_{\max } )+{min\; }(C_{eq,s} ,\frac{1}{2} C_{\max } ),{\; }0)$

The transport formulations implemented into XBeach distinguishes bed load and suspended load transport. It is possible to in- and exclude these transports components (keywords: bed & sus, with bed = 1 will include bed load transport). There is also a possibility to compute the total bulk transport rather than bed and suspended load separately (keyword: bulk = 1). The bed load will be calculated if it is suspended transport. On top of that this switch will have impact on how the bed slope effect (see Section 2.7.6) will be calculated

Soulsby-Van Rijn¶

The first possible sediment transport formulation are the Soulsby-Van Rijn equations (keyword: form = soulsby_vanrijn) ([Sou97]; [vR85]). The equilibrium sediment concentrations are calculated according to:

(87) $\begin{array}{l} {C_{eq,b} =\frac{A_{sb} }{h} \left(\sqrt{v_{mg} ^{2} +0.018\frac{u_{rms,2} ^{2} }{C_{d} } } -U_{cr} \right)^{2.4} } \\ {C_{eq,s} =\frac{A_{ss} }{h} \left(\sqrt{v_{mg} ^{2} +0.018\frac{u_{rms,2} ^{2} }{C_{d} } } -U_{cr} \right)^{2.4} } \end{array}$

For which the bed load and suspended load coefficients are calculated with:

(88) $A_{sb} =0.005h\left(\frac{D_{50} }{h\Delta gD_{50} } \right)^{1.2} ,{\; }A_{ss} =0.012D_{50} \frac{D_{*}^{-0.6} }{(\Delta gD_{50} )^{1.2} }$

In which the dimensionless sediment diameter (D*) can be calculated with the following formulation. The $v$ is the kinematic viscosity based on the expression of Van Rijn and is a function of the water temperature. XBeach assumes a constant temperature of 20 degrees Celsius, this result in a constant kinematic viscosity of ${19}^{-6} {\rm m}^{2}/{\rm s}$ .

(89) $D_{*} =\left(\frac{\Delta g}{\nu ^{2} } \right)^{1/3} D_{50}$

The critical velocity ( ${U}_{cr}$ ) defines at which depth averaged velocity sediment motion is initiated:

(90) $U_{cr} =\left\{\begin{array}{l} {0.19D_{50}^{0.1} \log 10\left(\frac{4h}{D_{90} } \right){\; for\; }D_{50} \le 0.0005} \\ {8.5D_{50}^{0.6} \log 10\left(\frac{4h}{D_{90} } \right){\; for\; }D_{50} >0.05} \end{array}\right.$

Finally the drag coefficient ( ${C}_{d}$ ) is calculated with equation (91). A drag coefficient is used to determine the equilibrium sediment concentrations. On top of that [Sou97] gives a relation between the bed shear stress of the depth-averaged current speed.

(91) $C_{d} =\left(\frac{0.40}{\ln \left(\frac{\max (h,10z_{0} )}{z_{0} } \right)-1} \right)^{2}$

In this equation $z_0$ is used for the bed roughness length and is used as zero flow velocity level in the formulation of the sediment concentration. In XBeach this is a fixed value (keyword: z0), but [Sou97] argues there is a relation between the Nikuradse and kinematic viscosity.

Van Thiel-Van Rijn¶

The second possible sediment transport formulation are the Van Thiel-Van Rijn transport equations (keyword: form = vanthiel_vanrijn) ([vR07]; [vanTdVries09]). The major difference between the Soulsby – Van Rijn equations is twofold. First of all, there is no drag coefficient calculated anymore and secondly the critical velocity is determined by calculating separately the critical velocity for currents ( ${U}_{crc}$ ) according to [Shi36] and for waves ( ${U}_{crw}$ ) according to [KM75].

The equilibrium sediment concentrations are calculated according to

(92) $\begin{array}{l} {C_{eq,b} =\frac{A_{sb} }{h} \left(\sqrt{v_{mg} ^{2} +0.64u_{rms,2}^{2} } -U_{cr} \right)^{1.5} } \\ {C_{eq,s} =\frac{A_{ss} }{h} \left(\sqrt{v_{mg} ^{2} +0.64u_{rms,2}^{2} } -U_{cr} \right)^{2.4} } \end{array}$

For which the bed-load and suspended load coefficient are calculated with:

(93) $A_{sb} =0.015h\frac{\left(D_{50} /h\right)^{1.2} }{\left(\Delta gD_{50} \right)^{0.75} } ,{\; }A_{ss} =0.012D_{50} \frac{D_{*}^{-0.6} }{(\Delta gD_{50} )^{1.2} }$

The critical velocity is computed as weighted summation of the separate contributions by currents and waves ([vR07]):

(94) $U_{cr} =\beta U_{crc} +(1-\beta )U_{crw} {\; \; in\; which\; \; \; }\beta =\frac{v_{mg} }{v_{mg} +u_{rms} }$

The critical velocity for currents is based on [Shi36]:

(95) $U_{crc} =\left\{\begin{array}{l} {0.19D_{50}^{0.1} \log 10\left(\frac{4h}{D_{90} } \right){\; \; \; \; \; for\; }D_{50} \le 0.0005} \\ {8.5D_{50}^{0.6} \log 10\left(\frac{4h}{D_{90} } \right){\; \; \; \; \; \; \; for\; }D_{50} \le 0.002} \\ {1.3\sqrt{\Delta gD_{50} } \left(\frac{h}{D_{50} } \right)^{1/6} {\; \; \; \; \; \; \; for\; }D_{50} >0.0005} \end{array}\right.$

The critical velocity for waves is based on [KM75]:

(96) $U_{crw} =\left\{\begin{array}{l} {0.24(\Delta g)^{2/3} {\; }\left(D_{50} T_{rep} \right)^{1/3} {\; \; \; \; \; \; \; \; for\; \; }D_{50} <=0.0005} \\ {0.95(\Delta g)^{0.57} {\; }\left(D_{50} \right)^{0.43} T_{rep} ^{0.14} {\; \; for\; \; }D_{50} >0.0005} \end{array}\right.$

Van Rijn (1993)¶

The third possible sediment transport formulation are the Van Rijn (1993 equations (keyword: form = vanrijn1993) Van Rijn (1993) distinguishes between sediment transport below the reference height at which sediment is treated as bed-load transport and above the reference height which is treated as suspended-load.

The bed-load transport is computed with

(97) ${Sb} = 0.006\rho {}_s{w_s}{D_{50}}{M^{0.5}}{M_e}^{0.7}$

In which the sediment mobility number due to waves and currents ( ( ${M}$ ) ) can be calculated with the following formulation with ( ${v}_{e}$ ) being the effective velocity. The excess sediment mobility number ( ( ${M}_{e}$ ) ) is computed with the difference between the effective and critical velocity ( ( ${v}_{cr}$ ) ).

(98) $M = \frac{{{v_e}^2}}{{(s - 1)g{D_{50}}}}$

For the suspended-load, first the reference concentration is calculated in accordance with ([VanRijn84])

(99) ${c_a} = 0.015{\rho _s}\frac{{{D_{50}}{T_a}^{1.5}}}{{\alpha {D_*}^{0.3}}}$

Secondly, the concentration profile is resolved by calculating the bed-shear stresses due to waves and currents and estimating the concentration profile where the combined bed shear stress exceeds the critical bed shear stress. Depth-averaged mixing due to waves and currents.

Effects of wave non-linearity¶

Effects of wave skewness and asymmetry are accounted for in the advection-diffusion equation, repeated here:

(100) $\begin{array}{c} {\frac{\partial hC}{\partial t} +\frac{\partial hC(u^{E} -u_{a} \sin \theta _{m} )}{\partial x} +\frac{\partial hC(v^{E} -u_{a} \cos \theta _{m} )}{\partial y} } \\ {+\frac{\partial }{\partial x} \left[D_{h} h\frac{\partial C}{\partial x} \right]+\frac{\partial }{\partial y} \left[D_{h} h\frac{\partial C}{\partial y} \right]=\frac{hC_{eq} -hC}{T_{s} } } \end{array}$

XBeach considers the wave energy of short waves as averaged over their length, and hence does not simulate the wave shape. A discretization of the wave skewness and asymmetry was introduced by [vanTdVries09], to affect the sediment advection velocity. In this equation ${u}_{a}$ is calculated as function of wave skewness ( ${S}_{k}$ ), wave asymmetry parameter ( ${A}_{s}$ ), root-mean square velocity ${u}_{rms}$ and two calibration factor ${f}_{Sk}$ and ${f}_{As}$ (keyword: facSk & facAs), see equation (101). To set both values one can use the keyword: facua. The method to determine the skewness and asymmetry is described in section 2.3.4. A higher value for ${u}_{a}$ will simulate a stronger onshore sediment transport component.

(101) $u_{a} =(f_{Sk} S_{k} -f_{As} A_{s} )u_{rms}$

Hindered erosion by dilatancy¶

Under overwash and breaching conditions (high flow velocities and large bed level variations in time), dilatancy might hinder the erosion rates ([dV14]). To account for this effect, the theory of [vR10] could be applied (keyword: dilatancy = 1), reducing the critical Shields parameter at high flow velocities:

(102) $\theta _{cr}^{adjusted} =\theta _{cr} \left(1+\frac{v_{e} }{k_{l} } \frac{n_{l} -n_{0} }{1-n_{l} } \frac{A}{\Delta } \right)$

In this equation, ${v}_{e}$ refers to the erosion velocity, ${k}_{l}$ is the permeability, $n0$ is the porosity prior, ${n}_{l}$ is the porosity in the sheared zone (keyword: pormax) and the parameter $A$ (keyword: rheeA) is equal to 3/4 for single particles and approximately 1.7 for a continuum.

The larger the permeability of the bed, the smaller the dilatancy effect. [vR10] suggests using the equation proposed by [dA87]:

(103) $k_{l} =\frac{g}{160\nu } D_{15}^{2} \frac{n_{0}^{3} }{\left(1-n_{0}^{2} \right)}$

Finally, the erosion velocity ${v}_{e}$ , is the velocity at which the bottom level decreases:

(104) $v_{e} =\left\{\begin{array}{l} {-\frac{dz_{b} }{dt} {\; \; \; \; \; \; \; \; \; \; if\; \; \; \; \; \; \; \; \; \; \; }\frac{dz_{b} }{dt} <0} \\ {{\; \; \; \; }0{\; \; \; \; \; \; \; \; \; \; \; else}\; \; \; \; \; \; \; \; \; \; \; } \end{array}\right.$

Bed slope effect¶

The bed slope affects the sediment transport in various ways ([WvRVanOrmondt+07]):

The bed slope influences the local near-bed flow velocity;
The bed slope may change the transport rate once the sediment is in motion;
The bed slope may change the transport direction once the sediment is in motion;
The bed slope will change the threshold conditions for initiation of motion.

The influence of the bed slope on the local hydrodynamics is not considered in XBeach.

Two possible expressions are implemented to change the magnitude of the sediment transport. The first method is the default one in XBeach:

(105) $\begin{array}{l} {q_{x,slope} =q_{x} -\alpha hC\sqrt{\left(u^{L} \right)^{2} +\left(v^{L} \right)^{2} } \frac{\partial z_{b} }{\partial x} } \\ {q_{y,slope} =q_{y} -\alpha hC\sqrt{\left(u^{L} \right)^{2} +\left(v^{L} \right)^{2} } \frac{\partial z_{b} }{\partial y} \; \; \; \; \; \; \; } \end{array}$

This method could be applied on either the total sediment transport (keyword: bdslpeffmag = roelvink_total) or only on the bed load transport (keyword: bdslpeffmag = roelvink_bed). The second method is based on the engineering formula of [Sou97]:

(106) $q_{slope} =q\left(1-\alpha \frac{\partial z_{b} }{\partial s} \right)\;$

Also this method could be applied on the total transport (keyword: bdslpeffmag = soulsby_total) or on the bed load transport only (keyword: bdslpeffmag = soulsby_total). To change the direction of the bed load transport, the expressions of [vB47] and [TvMS95] could be used (keyword: bdslpeffdir = talmon):

(107) $\tan \left(\alpha _{\psi ,new} \right)=\frac{\sin \left(\alpha _{\psi } \right)-f\left(\theta \right)\frac{dz_{b} }{dy} }{\cos \left(\alpha _{\psi } \right)-f\left(\theta \right)\frac{dz_{b} }{dx} }$

(108) $f\left(\theta \right)=\frac{1}{9\left(D_{50} /h\right)^{0.3} \theta ^{0.5} }$

(109) $\begin{array}{l} {q_{b,x} =\left|q_{b} \right|\cos \left(\alpha _{\psi ,new} \right)} \\ {q_{b,y} =\left|q_{b} \right|\sin \left(\alpha _{\psi ,new} \right)} \end{array}$

Finally, it is possible to adjust the initiation of motion criteria for the total transport (keyword: bdslpeffini = total) or the bed load transport only (keyword: bdslpeffini = bed) through ([Sou97]):

(110) $\theta _{cr}^{adjusted} =\theta _{cr} \frac{\cos \left(\psi \right)\sin \left(\beta \right)+\sqrt{\cos ^{2} \left(\beta \right)\tan ^{2} \left(\phi _{i} \right)-\sin ^{2} \left(\psi \right)\sin ^{2} \left(\beta \right)} }{\tan \left(\phi _{i} \right)}$

In this equation is $\psi$ the difference in angle between the flow direction and the on-slope directed vector, $\beta$ the bed slope and ${\phi}_{i}$ the angle of repose.

[dV14] provides a detailed overview on how the bed slope and flow direction are calculated and how the bed slope effect is combined with the dilatancy concept if the adjustment to the initiation of motion is considered.

It is also possible to prescribe a given bed slope. The result is that the swash zone profile is teased towards a given bermslope. This functionality Works in surfbeat mode (where H/h>1) and in stationary mode (where h<1m) and have been tested for profiles Praia de Faro (keyword: bermslope = desired slope).

Bottom updating¶

See also

Bed updating is implemented in func:morphevolution/bed_update.

Due to sediment fluxes¶

Based on the gradients in the sediment transport the bed level changes according to:

(111) $\frac{\partial z_{b} }{\partial t} +\frac{f_{mor} }{\left(1-p\right)} \left(\frac{\partial q_{x} }{\partial x} +\frac{\partial q_{y} }{\partial y} \right)=0$

In $\rho$ is the porosity, ${f}_{mor}$ (keyword: morfac) is a morphological acceleration factor of O(1-10) ([RRT04]) and ${q}_{x}$ and ${q}_{y}$ represent the sediment transport rates in x- and y-direction respectively. Sediment transport can be activated with the keyword: sedtrans.

The morphological acceleration factor speeds up the morphological time scale relative to the hydrodynamic timescale. It means that if you have a simulation of 10 minutes with a morfac of 6 you effectively simulate the morphological evolution over one hour. There are now two ways in which you can input the time-varying parameters in combination with morfac:

All times are prescribed on input in morphological time. If you apply a morfac all input time series and other time parameters are divided internally by morfac. This is determined with keyword morfacopt = 1. If you now specify a morfac of 6, the model just runs for 10 (hydrodynamic) minutes each hour, during which the bottom changes per step are multiplied by a factor 6. This of course saves a factor of 6 in computation time.

This method is appropriate for short-term simulations with extreme events. This approach is only valid as long as the water level changes that are now accelerated by morfac do not modify the hydrodynamics too much. This is the case if the tide is perpendicular to the coast and the vertical variations do not lead to significant currents. If you have an alongshore tidal current, as is the case in shallow seas, you cannot apply this method because you would affect the inertia terms and thus modify the tidal currents.

Alternatively you run the model over, say, over a tidal cycle, and apply the morfac without modifying the time parameters. This means you leave all the hydrodynamic parameters unchanged and just exaggerate what happens within a tidal cycle. As long as the evolution over a single tidal cycle is limited, the mean evolution over a tidal cycle using a morfac is very similar to running morfac tidal cycles without morfac. See [Roe06] for a more detailed description of this approach. This option is enabled with keyword: morfacopt = 0.

This method is more appropriate for longer-term simulations with not too extreme events.

Avalanching¶

To account for the slumping of sandy material from the dune face to the foreshore during storm-induced dune erosion avalanching (keyword: avalanching) is introduced to update the bed evolution. Avalanching is introduced via the use of a critical bed slope for both the dry and wet area (keyword: wetslp and dryslp). It is considered that inundated areas are much more prone to slumping and therefore two separate critical slopes for dry and wet points are used. The default values are 1 and 0.3 respectively. When this critical slope is exceeded, material is exchanged between the adjacent cells to the amount needed to bring the slope back to the critical slope.

(112) $\left|\frac{\partial z_{b} }{\partial x} \right|>m_{cr}$

To prevent the generation of large shockwaves due to sudden changes of the bottom level, bottom updating due to avalanching has been limited to a maximum speed of ${v}_{av,max}$ (keyword: dzmax). Equation (113) shows the resulting bed level change within one time step.

(113) $\begin{array}{l} {\Delta z_{b} =\min \left(\, \, \, \, \left(\left|\frac{\partial z_{b} }{\partial x} \right|-m_{cr} \right)\Delta x,\, \, \, \, v_{av,\max } \Delta t\right)\quad ,\frac{\partial z_{b} }{\partial x} >0} \\ {\Delta z_{b} =\max \left(-\left(\left|\frac{\partial z_{b} }{\partial x} \right|-m_{cr} \right)\Delta x,-v_{av,\max } \Delta t\right)\quad ,\frac{\partial z_{b} }{\partial x} <0} \end{array}$

Bed composition¶

If the effect of different sediment fractions, sorting and armoring are of importance, a bed composition constituting multiple sediment fractions can be defined. Each sediment fraction is characterized by a median grain size ( ${D}_{50}$ ) and possible a ${D}_{15}$ and ${D}_{90}$ as well. When using multiple sediment fractions, multiple bed layers are needed as well to describe the vertical distribution of the sediment fractions in the bed.

By specifying multiple bed layers, XBeach can keep track of the different sediment fractions both in the horizontal and in the vertical. Coarse sediments may be deposited on top of fine sediment after which erosion of the coarse sediment is needed to expose the fine sediment again, effectively armoring the bed. Three types of bed layers are distinguished: 1) the top layers 2) the variable or “breathing” layer and 3) the bottom layers. The top layer is the only layer that interacts with the water column and can be eroded, but preserves its thickness. The bottom layers are layers of constant thickness that move with the top layer. A single variable or “breathing” layer is defined that adapts its thickness to the erosion and sedimentation of the bed. For example: if a grid cell is eroded, particular fractions of sediment are removed from the top layer, but the top layer preserves its thickness and thus it takes the same volume of sediment, likely of different composition than the eroded sediment, from the layer below. If this layer is a top layer as well, the thickness is preserved and again the same volume of sediment is taken form a lower bed layer. This continues until the variable or “breathing” layer is reached. This layer adapts its thickness to the amount of erosion. If the thickness of the layer becomes too small, the variable layer is merged with an adjacent bottom layer and a new bottom layer is defined underneath the existing ones to ensure a constant number of bed layers. Reversely, if a grid cell is accreting, the thickness of the variable layer will be increased and with sufficient increase the variable layer will be split in two effectively creating a new bottom layer. The lowest existing bottom layer is then discarded to ensure a constant number of bed layers. The “breathing” layer can be the upper or bottom layer in which case the top layer or bottom layer class does not exist. The thickness of the different layer classes can be set separately (keyword: dzg1, dzg2 and dzg3) or at once (keyword: dzg).

Each grid cell in XBeach holds its own sediment distribution and the sediment transport formulations are used differentiate between fractions. Therefore the distribution of sediment may change over time and processes like armoring and sorting can be simulated. Due to the shifting of sediment between bed layers numerical mixing of sediment occurs. Choosing bed layer thicknesses that are in balance with the expected erosion and deposition during the simulation should keep the numerical mixing to a minimum. A bed layer thickness that is too large will result in relatively uniform behavior, while a bed layer thickness that is too small will result in a lot of shifting and thus numerical mixing (Fig. 8).

Fig. 8 Visualization of the diffusion that occurs when XBeach calculates sediment compositions. After sedimentation of fine sediment on top of coarser material it is uniformly mixed over the whole layer. Subsequent erosion erodes both the fines as the coarser material. To avoid this phenomenon, layers should not be too thick ([vdZ14]).

Ship-induced wave motions¶

See also

Ship-induced wave motions are implemented in mod:ship_module.

A relatively new application field for XBeach is the generation and propagation of waves induced by sailing vessels. This functionality has been implemented recently ([ZRVL13]), and has currently been used in several studies (e.g. [ZRZvW14], [dJRB13]), showing very good results.

For computing ship-induced waves the non-hydrostatic version of XBeach is used. A moving ship is represented as a pressure head that moves along a pre-defined track through the model domain. The ship is defined on a separate grid, where the ship draft is specified per grid point. Each computational time step the ship draft is interpolated from the ship grid to the global grid, where the ship volume is kept constant. Then the water pressure head in each global grid cell is updated based on the interpolated ship draft. By moving the pressure fields, the waves are generated and will propagate further through the global domain.

In Fig. 9 an example XBeach setup for ship waves is shown. The ship track is user defined and can, for instance, be obtained from the Automatic Identification System (AIS) for marine traffic. In this example, the model results were compared with measurements taken at Bath, The Netherlands. A filtered time series of the measured and computed water level is shown in Fig. 10. The time series was filtered to focus on the computation of the primary ship wave.

Fig. 9 Example XBeach setup (left) and result (right) for a ship wave simulation in the Scheldt Estuary (The Netherlands). The ship track (red dashed line) is user-defined, and the measurement location is indicated (magenta dot).

Fig. 10 Example XBeach result for ship-induced waves. Measurements are taken at Bath, in the Scheldt Estuary, The Netherlands ([SHvdWT11]).

In addition to the propagation of ship-induced waves, XBeach computes the forces and moments acting on the ship body. With this functionality, passing ship effects can be analyzed (e.g. [ZRVL13]).

Gravel (XBeach-G)¶

XBeach-G was a branch of the main XBeach development that has been developed to simulate storm impacts on gravel beaches. The development of XBeach-G have been taking place as a joint collaboration between Plymouth University and Deltares, as part of the EPSRC-funded NUPSIG-project. The relevant XBeach-G formulations have been merged back into to the XBeach trunk for the XBeachX release (fall 2017). XBeach-G formulations can be activated with the keyword useXBeachGSettings = 1).

XBeach-G uses the one-layer, depth-averaged, non-hydrostatic extension to the XBeach model (nonh=1), which is similar to the SWASH model ([ZSS11]) to solve wave-by-wave flow and surface elevation variations due to short waves in intermediate and shallow water depths. This is particularly important for application on gravel beaches, where due to steep slopes swash motion is mainly at incident wave frequencies, and infragravity wave motion, which dominates the inner surf and swash zone on sandy beaches during storms, is of secondary importance. To correctly account for upper swash infiltration losses and exfiltration effects on lower swash hydrodynamics on gravel beaches, XBeach-G computes groundwater dynamics and the exchange between groundwater and surface water using the XBeach groundwater model (gwflow = 1). Again, interaction between swash flows and the beach groundwater table are considered particularly important on gravel beaches due to the relatively large hydraulic conductivity of the sediment, while on sandy beaches this process is of significantly less importance. Finally, gravel sediment transport processes (McCall & Van Rijn) have been included in XBeach-G to simulate the morphodynamics of gravel beaches during storms. These transport processes are currently under further development and validation.

For more information on XBeach-G, download the PhD thesis of McCall (2015) on URL: http://hdl.handle.net/10026.1/3929.

Boundary conditions¶

Waves¶

See also

Wave boundary conditions are implemented in mod:waveparams. The latest functionalities, like spatially varying spectral conditions are implemented in mod:spectral_wave_bc_module.

XBeach allows users to include two different options for wave boundary conditions in the model. These wave boundary conditions can be applied only at the seaward boundary (keyword: wbctype). First of all, in Spectral conditions the method to specify wave spectra is discussed. Secondly, in Non-spectral conditions the method to apply non-spectra, such as stationary wave conditions or time-series is elaborated. In Lateral boundary conditions the lateral boundary conditions for waves are discussed. There is currently not a possibility to force waves on the landward boundary of a model.

Spectral conditions¶

The most-used wave boundary condition in XBeach is a spectral type. The input description of spectral wave boundary conditions can be found in Spectral wave boundary conditions. XBeach allows the user to define these with three possibilities:

: With this option you define the boundary condition as parametric spectral input. The parameters (i.e. the spectral shape, the wave period and the directional spreading) can be specified. The option is especially handy when there is no nested model or measured spectrum. Here are two options:
1. Specify a single parametric spectrum (keyword wbctype = jons).
2. Specify a series of parametric spectra (keyword wbctype = jons_table).
: In this case the two-dimensional (frequency-direction) output by the spectral wave model SWAN (.sp2 files) can be specified. (keyword wbctype = swan). This option is especially convenient when nesting XBeach into a SWAN model.
: In this case a more general type spectrum can be specified. (keyword wbctype = vardens). This option is often used when a measured spectrum is available.

Non-spectral conditions¶

XBeach also allows the user to define non-spectral wave boundary conditions. This is a variation of both wave conditions without wave groups and time series. The input description of non-spectral wave boundary conditions can be found in Non-spectral wave boundary conditions. XBeach allows the user to define these with two possibilities:

This means that a uniform and constant wave energy is specified, based on the given values of , , direction and power of the directional distribution function. The station boundary condition will contain wave groups. Here there are two options:
1. Specify a single sea state (keyword wbctype = stat)
2. Specify a series of sea states (keyword wbctype = stat_table)
. The user can also specify the variation in time of the wave energy. There are three options:
1. First-order time series of waves (keyword wbctype = ts_1). XBeach will calculate the bound long wave based on the theory of [LHS64]).
2. Second-order time series of waves (keyword wbctype = ts_2). The bound long wave is specified by the user via a long wave elevation.
3. It is also possible to specify a variation in time of the horizontal velocity, vertical velocity and the free surface elevation (keyword: wbctype = ts_nonh). Last two terms are optional in this boundary conditions type.

Special conditions¶

Besides clear spectral or non-spectral wave boundary conditions, there are also three special boundary condition types implemented in XBeach.

(keyword wbctype = bichrom). In this case, XBeach will be forced with regular wave groups as the two short-wave components force one difference (infragravity) wave period. The user needs to specify not only variables of the stationary situation but also a wave period for the long wave. This wave period will be used to calculate the long wave based on the theory of [LHS64]. The bichromatic boundary condition is the most simplified form of a wave spectrum.
(keyword wbctype = off). This is a simple no wave action boundary condition. It still allows for a tidal record to be specified, however this trough the zs0file parameter.
(keyword: wbctype = reuse). If the user does not wish to recalculate boundary condition files or specifically wants to reuse the boundary condition files of another XBeach simulation should be used. No further wave boundary condition data need be given. Obviously, the calculation grid should remain the same between runs, as the angles and number of grid points are embedded in the boundary condition files.

Lateral boundary conditions¶

There are two options to set the lateral boundaries for the wave model:

Neumann boundaries (keyword: lateralwave = neumann): here the longshore gradient is set to zero.
Wave crest boundaries (keyword: lateralwave = wavecrest). here the gradient in the wave energy along the wave crest is set to zero.

For the stationary wave mode (keyword: wavemodel = stationary) this is the only option. It allows a correct representation of the wave propagation near the lateral boundaries, without the usual shadow zones in e.g. SWAN. By neglecting the longshore gradients, the model automatically computes a consistent 1D solution.

For the surfbeat mode (keyword: wavemodel = surfbeat), Neumann leads to shadow zones, not so much in the wave height, but in the groupiness; the ’blobs’ propagating in the mean wave direction turn into elongated, longshore uniform patches. To reduce this effect, the gradient along the wave crests of the wave energy can be set to zero, instead of the longshore gradient (keyword: lateralwave = wavecrest). This way the crests of the wave groups have approximately the right orientation, though the along-crest groupiness also disappears. In the wavecrest case, the wave refraction may be overestimated leading to somewhat too large longshore currents. The effects of both boundary conditions are shown in Fig. 11.

Fig. 11 Effect of the lateral wave boundary conditions on root-mean square wave height patterns (top) and longshore velocity (bottom) for the Delilah test case. In this figure the left panels are used for simulations with Neumann boundaries and the right panel with the wavecrest boundary.

Shallow water equations¶

Offshore boundary¶

Typically, an offshore or lateral boundary is an artificial boundary which has no physical meaning. On the offshore boundary wave and flow conditions are imposed. In the domain waves and currents will be generated which need to pass through the offshore boundary to the deep sea with minimal reflection. One way to do this is to impose a weakly reflective-type boundary condition (absorbing-generating), but there are also other possibilities implemented in XBeach (keyword: front). This method can be applied in 1D or 2D, is recommended and therefore the default value for XBeach.

In XBeach, there are two options with regard to the offshore absorbing-generating boundary condition. With the parameter setting front = abs1d a simple one-dimensional absorbing-generating boundary condition is activated. This option allows for a time-varying water level (surge and/or infragravity waves) to be specified at the boundary while allowing any waves propagating perpendicularly towards the boundary to be absorbed (i.e., passed through the boundary with a minimum of reflection. It is therefore only useful for 1D (flume like) simulations.

With option front = abs2d (default value) the formulation by [vDS97] is activated which in turn is based on [VSO81] and is based on the ‘Method of Characteristics’. This boundary condition allows for obliquely-incident and obliquely-reflected waves to pass through the boundary. It is possible to account for situations with boundary-perpendicular and boundary-parallel currents. In order to differentiate between the particle velocities, the keyword epsi must be set. This parameter control a simple Kalman-update filter which controls which part of the particle velocity is assumed to be part of the current and which part is wave-related. By default XBeach computes the value for epsi automatically using offshore boundary conditions (keyword: epsi = -1).

There are three other possibilities implemented besides the absorbing-generating boundary conditions:

No flux wall (keyword: front = wall). This boundary condition type is a simple no flux boundary condition.
Water level specification (keyword: front = wlevel). This boundary sets the water level at a prescribed value. This can be constant or time-varying. With this option the outgoing long waves are not absorbed.
Boundary condition for the non-hydrostatic option (keyword: : front = nonh_1d). This boundary condition is required for non-hydrostatic simulations.
Radiation boundary condition (keyword: front = waveflume). This boundary uses a continuity relation at the front boundary. This means that no net water can come into the model domain. The wave flume boundary condition is especially useful in lab experiments with a large set-up (e.g. coral reefs).

Lateral boundaries¶

Lateral boundaries are the boundaries perpendicular to the coastline. Usually these are artificial, because the model domain is limited but the physical coast will continue. At these boundaries (keywords: left & right) we need to prescribe information about the area beyond the numerical model domain in such a way that the boundary condition does not influence the results in an adverse way. One way to do this is to prescribe a so-called “no-gradient” or Neumann boundaries (XBeach default), which state that there is locally no change in surface elevation and velocity, but there are also other possibilities implemented into XBeach. This method is recommended and is therefore the default value for XBeach. Each lateral boundary is a separate condition, so it is possible to mix different type of lateral boundary per side.

Neumann boundary conditions are activated where the longshore water level gradient is prescribed. The alongshore gradient is prescribed by the difference in specified water levels at the offshore corner points, divided by the alongshore length of the domain. This type of Neumann boundary condition has been shown to work quite well with (quasi-) stationary situations, where the coast can be assumed to be uniform alongshore outside the model domain. So far we have found that also in case of obliquely incident wave groups this kind of boundary conditions appears to give reasonable results when a shadow zone is taken into account. This means that regions where the boundary conditions are not fully enforced the results are not taken into account. Neumann boundaries can be individually defined (keyword: left = neumann).

There are three other possibilities implemented besides the absorbing-generating boundary conditions:

Simple no-flux boundary conditions can also be applied (keyword: left = wall). Wall boundary conditions will result in a zero velocity at the lateral boundary.
Velocity at the boundary will be calculated from NLSWE, but only include the advective terms (keywords: left = no_advec). The effect is that only terms that decrease the velocity will be taken into account. The result is an intermediate form between a full Neumann boundary and a wall boundary.
Velocity at the boundary will simply be copied from the adjacent cell in the model domain (keyword: left = neumann_v).

Tide and surge¶

XBeach can take in up to four time-vary tidal signals to be applied to the four boundaries (offshore-right, offshore-left, backshore-left, backshore-right). A time-varying water level signal is read into XBeach by reading the specified file in zs0file. The input signal will be interpolated to the local time step of the simulation; therefore the signals only need to be long enough and temporally-fine enough to resolve the water level phenomenon of interest (i.e. tide variations, surge event).

There are now four options for handling the tidal and/or surge contribution to the boundaries:

Uniform water level (keyword: tideloc = 0)
One time-varying water level signal (keyword: tideloc = 1)
Two time-varying water level signals, which requires point of application indication. (keyword: tideloc = 2)
Four time-varying water level signals (keyword: tideloc = 4)

For the option with a uniform water level the value specified in the params.txt is applied in the complete model domain (keyword: zs0). For the option with one time-varying water level signal the specified water level is applied (keyword: zs0file = <name_of_your_time_series_file>) to the offshore boundary and a fixed value is applied at the backshore boundary (keyword: zs0). For the option with two time-varying water level signals two water level signals are read from the zs0file. Note: one tidal record is applied to both sea corners and one tidal record to both land corners. This means there is no alongshore variation. An alongshore variation can be applied when applying four time-varying water level signals.

River and point discharge¶

The effect of a river outflow or other discharges can be simulated with XBeach. Multiple discharge locations can be designated. At a discharge location the discharge orifice is defined as well as the discharge time series in ${\rm m}^{3}/{\rm s}$ . The discharge orifice always constitutes an uninterrupted series of full grid abreast cell borders. It is not possible to define a discharge over half a grid cell nor is it possible to define a single discharge through grid cell borders that are either not adjacent or are not abreast.

At each time step the model sets the discharge and velocities at the grid cell borders that constitute the discharge orifice, which can be computed given the size of the discharge orifice and discharge time series. The discharge is positive in positive x or y direction. An exception is made when discharges are defined at the domain border. In that case the discharge is positive towards the domain (influx).

When a discharge is defined with a zero size orifice the discharge is assumed to be in vertical direction where a positive discharge is into the domain (influx). In these cases the discharge is linked to the closest grid cell center and at each time step mass according to the discharge time series is added. No momentum is added in case of a vertical discharge.

Sediment transport¶

The boundary conditions for sediment transport are Neumann boundaries everywhere, implying that the cross-boundary gradients in the advection-diffusion equation are set to zero, as well as the gradients of the bed load transports in that direction. Cross-shore profile changes due to cross-shore transport gradients are possible, allowing the boundary to smoothly follow the rest of the model. Still, it is good modeling practice to have the boundaries away from the area of interest.

Input description¶

General¶

Upon running the XBeach executable xbeach.exe, the file params.txt in the current working directory will be read. The params.txt file contains grid and bathymetry info, wave input, flow input, morphological input, etc. in the form of keyword/value pairs. Each keyword/value pair may contain an actual model parameter or refer to another file with additional information on the model setup. If a params.txt file cannot be found then XBeach will not run.

In the params.txt file there can be a single keyword/value pair per line. The keywords can be specified in any order. A keyword/value pair is separated by an equal sign (=). Each line containing an equal sign is interpreted as a keyword/value pair. Reversely, any lines without an equal sign are ignored and may be used for comments. Only a few keywords are required for the model to run, others have default values that are used in case the keyword is not mentioned in the params.txt file. The essential parameters for a simulation with a JONSWAP spectrum are listed below:

A . This can both in XBeach format (separate x and y files; keyword: xfile and yfile) or Delft3D (one single xy file; keyword: xyfile). On top of that the user needs to specify the width of each domain (keyword: nx and ny)
A file (keyword: depfile) that matches with the grid you specified at 1)
A (keyword: tstop) in seconds
A for short waves and rollers. The grid is determined by a minimum and maximum angle and width per bin (keywords: thetamin, thetamax and dtheta).
A (keyword: wbctype = jons). With a separate file containing the variables of the parametric (keyword: bcfile).

It is strongly recommended to specify as few parameters explicitly as possible and rely on the defaults for the other parameters. When running XBeach, a file called xbeach.log is created, which lists all the parameters set through the params.txt file but also all parameters not set, for which the defaults are used. When the user starts the model, it generates a file named XBlog.txt. In this file all the different keyword available are determined. When no keyword is defined the default value will be applied.

This chapter describes the possibilities of the params.txt file and any auxiliary information files that are called from the params.txt file. The tables in this chapter contain a description of the keywords, the default values, its units and recommended value ranges, while the formats for additional input files are described in the relevant sections. Keyword marked with an astrix (*) are essential for XBeach to run. Keywords marked with a plus (+) are considered advanced expert options and should not be used for regular applications of XBeach.

In this chapter, any references to keywords refer to keywords that can be used in the params.txt file. Also any references to time indications are in seconds unless stated otherwise. A typical params.txt file for a 1D XBeach model is:

params.txt

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% XBeach parameter settings input file                                %%%
%%%                                                                     %%%
%%% date: 01-Jan-2015 12:00                                             %%%
%%% function: xb_write_params                                           %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Grid parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

depfile = bed.dep
posdwn = 0
nx = 265
ny = 0
alfa = 0
vardx = 1
xfile = x.grd
yfile = y.grd
thetamin = -90
thetamax = 90
dtheta = 15
thetanaut = 0

%%% Model time %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

tstop = 3600

%%% Physical constants %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

rho = 1025

%%% Tide boundary conditions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

tideloc = 2
zs0file = tide.txt

%%% Wave boundary condition parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

wbctype = jons
bcfile = filelist.txt

%%% Output variables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

outputformat = netcdf
tint = 3600
tstart = 0
nglobalvar = 3
zb
zs
H

Physical processes¶

XBeach supports a variety of physical processes from generic, like waves and flow, to very specific, like ship motions and point discharge. Each process can be switched on or off. The commonly used processes are turned on by default. The table below lists the keywords used to switch on or off physical processes in XBeach.

Table 3 Overview of available keywords related to physical processes¶
parameter	description	default	range
advection	Include advection in flow solver	1	0 - 1
avalanching	Turn on avalanching	1	0 - 1
bchwiz	Turn on beachwizard	0	0 - 1
cyclic	Turn on cyclic boundary conditions	0	0 - 1
flow	Turn on flow calculation	1	0 - 1
gwflow+	Turn on groundwater flow	0	0 - 1
lwave	Turn on short wave forcing on nlsw equations and boundary conditions	1	0 - 1
morphology	Turn on morphology	1	0 - 1
nonh+	Turn on non-hydrostatic pressure: 0 = nswe, 1 = nsw + non-hydrostatic pressure compensation stelling & zijlema, 2003	0	0 - 1
q3d	Turn on quasi-3d sediment transport	0	0 - 1
sedtrans	Turn on sediment transport	1	0 - 1
setbathy	Turn on timeseries of prescribed bathy input	0	0 - 1
ships+	Turn on ship waves	0	0 - 1
single_dir+	Turn on stationary model for refraction, surfbeat based on mean direction	0	0 - 1
snells+	Turn on snell’s law for wave refraction	0	0 - 1
swave	Turn on short waves	1	0 - 1
swrunup	Turn on short wave runup	0	0 - 1
vegetation+	Turn on interaction of waves and flow with vegetation	0	0 - 1
viscosity	Include viscosity in flow solver	1	0 - 1
wavemodel	Stationary (0), surfbeat (1) or non-hydrostatic (2)	surfbeat	stationary, surfbeat, nonh
wind	Include wind in flow solver	1	0 - 1

Grid and bathymetry¶

XBeach’ spatial grid size is defined by the keywords nx and ny. Here nx are the number of grid points in the cross-shore direction and ny the number in the alongshore direction. The size of the computational grid will be nx +1 by ny +1 cells large. The initial bathymetry is provided using a separate file that is referred to by the depfile keyword, which has to have a size of [nx+1, ny+1]. This file contains an initial bed level for each grid cell where each line corresponds to a transect in x-direction (cross-shore). The values are positive down by default (so e.g. a value of ‘10’ is 10 meters depth), but this can be changed using the posdwn keyword.

Three main types of XBeach grids are supported: fast 1D, 1D and 2DH. Fast 1D grids have a single alongshore grid cell and thus a value ny = 0 and thus a single row (ny+1=1) in the depfile. The 1D grids have 3 alongshore grid cells and thus a value ny = 2 and three rows in the depfile. The 2DH grids have more than 3 alongshore grid cells, a value ny >2 and as many rows in the depfile. In general, the bathymetry file has the following space-separated format:

bed.dep

<z 1,1> <z 2,1> <z 3,1> ... <z nx,1> <z nx+1,1>
<z 1,2> <z 2,2> <z 3,2> ... <z nx,2> <z nx+1,2>
<z 1,3> <z 2,3> <z 3,3> ... <z nx,3> <z nx+1,3>

...

<z 1,ny> <z 2,ny> <z 3,ny> ... <z nx,ny> <z nx+1,ny>
<z 1,ny+1> <z 2,ny+1> <z 3,ny+1> ... <z nx,ny+1> <z nx+1,ny+1>

XBeach spatial grids can be equidistant or non-equidistant. In the former case the grid size is defined by the keywords dx and dy. In the latter case the keyword vardx should be set to 1 and x- and y-coordinates of the grid cells should be provided through the files referenced by the xfile and yfile keywords. These files take exactly the same format as the depfile file where all coordinates along the x-direction are in one row and each row represents a cell in y-direction. XBeach grids are defined in a coordinate system of choice and can be either rectangular or curvilinear grids as discussed in Domain and definitions.

Delft3D grids created with tools like RFGRID are also supported. To use Delft3D grids, choose gridform = delft3d and provide a grid file via the keyword xyfile. The format of Delft3D grids is not described here, but can be found in the Delft3D manual ([Del11]). Also forced updating of bathymetries is supported as described in Bed update.

Apart for the spatial grid, XBeach also uses a directional grid for short waves and rollers. The grid is determined by a minimum and maximum angle and a directional bin size using the keywords thetamin, thetamax and dtheta respectively. The thetamin and thetamax angles are either defined according to the Cartesian convention (angle w.r.t. the computational x-axis) or according to the nautical convention (angle w.r.t. deg. N, so from W is 270 deg. N). The convention is chosen using the keyword thetanaut (thetanaut = 0 for Cartesian and thetanaut = 1 for Nautical)

Examples of typical input for a non-equidistant, fast 1D XBeach model, together with the params.txt example at the start of this chapter, are:

depfile = bed.dep

-20.00 -20.00 -19.90 -19.80 -19.70 ... 14 14 15 15 15

xfile = x.grd

0.00 10.00 20.00 30.00 40.00 ... 1992.00 1994.00 1996.00 1998.00 2000.00

yfile = y.grd

0.00 0.00 0.00 0.00 0.00 0.00 0.00 ... 0.00 0.00 0.00 0.00 0.00 0.00 0.00

All keywords related to grid and bathymetry input are listed in the following table:

Table 4 Overview of available keywords related to grid parameters¶
parameter	description	default	range	units
alfa	Angle of x-axis from east	0.0	0.0 - 360.0	deg
depfile*	Name of the input bathymetry file			<file>
dtheta*	Directional resolution	10.0	0.1 - 180.0	deg
dtheta_s*	Directional resolution in case of stationary refraction	10.0	0.1 - 20.0	deg
dx*	Regular grid spacing in x-direction	-1.0	0.0 - 1000000000.0	m
dy*	Regular grid spacing in y-direction	-1.0	0.0 - 1000000000.0	m
gridform	Grid definition format	xbeach	xbeach, delft3d
nx*	Number of computational cell corners in x-direction	50	2 - 10000
ny*	Number of computational cell corners in y-direction	2	0 - 10000
nz	Number of computational cells in z-direction	1	1 - 100
posdwn	Bathymetry is specified positive down (1) or positive up (-1)	1.0	-1.0 - 1.0
thetamax*	Higher directional limit (angle w.r.t computational x-axis)	90.0	-360.0 - 360.0	deg
thetamin*	Lower directional limit (angle w.r.t computational x-axis)	-90.0	-360.0 - 360.0	deg
thetanaut	Switch to specify thetamin and thetamax in nautical convention rather than cartesian	0	0 - 1
vardx	Switch for variable grid spacing	0	0 - 1
xfile	Name of the file containing x-coordinates of the calculation grid			<file>
xori	X-coordinate of origin of axis	0.0	-1000000000.0 - 1000000000.0	m
xyfile*	Name of the file containing delft3d xy-coordinates of the calculation grid			<file>
yfile	Name of the file containing y-coordinates of the calculation grid			<file>
yori	Y-coordinate of origin of axis	0.0	-1000000000.0 - 1000000000.0	m

Waves input¶

An XBeach model is generally forced by waves on its offshore boundary. These waves are described by the wave boundary conditions discussed in this section. The details of the wave motions within the model are described by the wave numerics in terms of the wave action balance (see Short wave action balance), wave dissipation model (see 0) and wave roller model (see Roller energy balance)

XBeach supports a variety of wave boundary condition types that are divided in two main groups: stationary and spectral boundary conditions. The wbctype keyword can be used to select one particular type of wave boundary conditions. Waves gives an overview of all types of wave boundary conditions available for XBeach. Fig. 12 can be used to help determine what type of wave boundary conditions is appropriate for your case. Each wave boundary condition type is explained in the following subsections. Note that most spectral wave boundary conditions can vary both in space and time using a FILELIST and/or LOCLIST construction as described in Temporally and/or spatially varying wave boundary conditions.

Table 5 Overview of available keywords related to wave boundary condition parameters¶
parameter	description	default	range	units
Hrms	Hrms wave height for instat = stat, bichrom, ts_1 or ts_2	1.0	0.0 - 10.0	m
Tlong	Wave group period for case instat = bichrom	80.0	20.0 - 300.0	s
Trep	Representative wave period for instat = stat, bichrom, ts_1 or ts_2	10.0	1.0 - 20.0	s
bclwonly	Switch to run boundary conditions with long waves only	0	0 - 1
dir0	Mean wave direction for instat = stat, bichrom, ts_1 or ts_2 (nautical convention)	270.0	-360.0 - 360.0	deg
instat	Old wave boundary condition type	bichrom	stat, bichrom, ts_1, ts_2, jons, swan, vardens, reuse, ts_nonh, off, stat_table, jons_table
lateralwave	Switch for lateral boundary at left	neumann	neumann, wavecrest, cyclic
m	Power in cos^m directional distribution for instat = stat, bichrom, ts_1 or ts_2	10	2 - 128
nmax+	Maximum ratio of cg/c for computing long wave boundary conditions	0.8	0.5 - 1.0
taper	Spin-up time of wave boundary conditions, in morphological time	100.0	0.0 - 1000.0	s
wbctype	New wave boundary condition type	params	params, parametric, swan, vardens, off, jonstable, reuse, ts_1, ts_2, ts_nonh

Fig. 12 Decision tree for selecting the appropriate type of wave boundary conditions

Spectral wave boundary conditions¶

Spectral wave boundary conditions are enabled using wbctype values jons, swan, vardens or jons_table. The conditions are defined in separate files referenced from the params.txt file using the bcfile keyword. A spectral wave boundary condition describes a spectrum shape that XBeach uses to generate a (random) wave time series. The length and resolution of the generated time series is determined by the keywords rt and dtbc respectively. XBeach will reuse the generated time series until the simulation is completed. The resolution of the time series should be enough to accurately represent the bound long wave, but need not be as small as the time step used in XBeach.

An overview of all keywords relevant for spectral wave boundary conditions is given in the table below. The necessary file formats for each type of spectral wave boundary condition is explained in the following subsections.

Table 6 Overview of available keywords related to wave-spectrum boundary condition parameters¶
parameter	description	default	range	units
Tm01switch+	Switch to enable tm01 rather than tm-10	0	0 - 1
bcfile	Name of spectrum file			<file>
correctHm0	Switch to enable hm0 correction	1	0 - 1
dtbc+	Timestep used to describe time series of wave energy and long wave flux at offshore boundary (not affected by morfac)	1.0	0.1 - 2.0	s
dthetaS_XB+	The (counter-clockwise) angle in the degrees needed to rotate from the x-axis in swan to the x-axis pointing east	0.0	-360.0 - 360.0	deg
fcutoff+	Low-freq cutoff frequency for instat = jons, swan or vardens boundary conditions	0.0	0.0 - 40.0	Hz
nonhspectrum+	Spectrum format for wave action balance of nonhydrostatic waves	0	0 - 1
nspectrumloc+	Number of input spectrum locations	1	1 - par%ny+1
nspr	Switch to enable long wave direction forced into centres of short wave bins	0	0 - 1
random+	Switch to enable random seed for instat = jons, swan or vardens boundary conditions	1	0 - 1
rt	Duration of wave spectrum at offshore boundary, in morphological time	min(3600.d0,par%tstop)	1200.0 - 7200.0	s
sprdthr+	Threshold ratio to maximum value of s above which spectrum densities are read in	0.08	0.0 - 1.0
trepfac+	Compute mean wave period over energy band: par%trepfac*maxval(sf) for instat jons, swan or vardens; converges to tm01 for trepfac = 0.0 and	0.01	0.0 - 1.0
wbcversion	Version of wave boundary conditions	3	1 - 3

JONSWAP wave spectra¶

JONSWAP spectrum input is enabled using wbctype = jons. A JONSWAP wave spectrum is parametrically defined in a file that is referenced using the bcfile keyword. This file contains a single parameter per line in arbitrary order. The parameters that can be defined are listed in Table 7. All variables are optional. If no value is given, the default value as specified in the table is used. It is advised not to specify the keyword dfj and allow XBeach to calculate the default value.

A typical JONSWAP definition file looks as follows:

jonswap.txt

Hm0 = 0.8
Tp = 8
mainang = 285.
gammajsp = 3.3
s = 10.
fnyq = 0.3

For the definitions see the table below.

It is possible to use an alternative file format for time-varying JONSWAP spectra. To enable this option use the wbctype value jons_table. In this case, each line in the spectrum definition file contains a parametric definition of a spectrum, like in a regular JONSWAP definition file, plus the duration for which that spectrum is used during the simulation. XBeach does not reuse time-varying spectrum files. Therefore the total duration of all spectra should at least match the duration of the simulation. The name of the file can be chosen freely, but the file format is fixed as follows and all parameters should be present in all lines:

jonswap.txt

<Hm0> <Tp> <mainang> <gammajsp> <s> <duration> <dtbc>

Note that we refer to the keywords used in a regular JONSWAP definition file in this example, with three differences: 1) the peak period rather than the peak frequency is defined 2) the duration is added (similar to rt in params.txt) 3) the time resolution is added (similar to dtbc in params.txt). The duration and boundary condition time step in this file overrules rt and dtbc in params.txt. This format is also used for time-varying stationary wave boundary conditions as described in Temporally and/or spatially varying wave boundary conditions. As an example, the JONSWAP spectrum definition file presented above would look as follows if the significant wave height should be increased with 0.2 m every hour:

jonswap.txt

8 8. 285. 3.3 10. 0.3 3600. 1
0 8. 285. 3.3 10. 0.3 3600. 1
2 8. 285. 3.3 10. 0.3 3600. 1

A more generic way of providing time-varying spectral wave boundary conditions is using a FILELIST construction as described in Temporally and/or spatially varying wave boundary conditions. This approach is compatible with all spectral wave boundary condition types as well as spatially varying boundary conditions as described in the same section.

The parameter s in the JONSWAP spectrum definition is related to the directional spreading (in deg.) through the following relation $\sigma =\sqrt{\frac{2}{s+1} } \, s=\frac{2}{\sigma ^{2} } -1$ . Here $\sigma$ is the directional spreading in radians and s the JONSWAP spreading parameter.

Fig. 13 Effect a variation in s for the direction spreading of wave energy

Table 7 Overview of available keywords in JONSWAP definition file¶
parameter	description	default	range
Hm0	Hm0 of the wave spectrum, significant wave height [m]	0.0	0.0 - 5.0
fp	Peak frequency of the wave spectrum [s-1]	0.08	0.06258 - 0.4
gammajsp	Peak enhancement factor in the JONSWAP expression [-]	3.3	1.0 - 5.0
s	Directional spreading coefficient, law [-]	10.0	1.0 - 1000.0
mainang	Main wave angle (nautical convention) []	270.0	180.0 - 360.0
fnyq	Highest frequency used to create JONSWAP spectrum [s-1]	0.3	0.2 - 1.0
dfj	Step size frequency used to create JONSWAP spectrum [s-1]	fnyq/200	fnyq/1000 - fnyq/20

SWAN wave spectra¶

XBeach can read standard SWAN 2D variance density or energy density output files (+.sp2 files) as specified in the SWAN v40.51 manual. This option is enabled using wbctype = swan in params.txt and a reference to the spectrum file via the keyword bcfile. XBeach assumes the directional information in the SWAN file is according to the nautical convention. If the file uses the Cartesian convention for directions, the user must specify the angle in degrees to rotate the x-axis in SWAN to the x-axis in XBeach (by the Cartesian convention). This value is specified in params.txt using the keyword dthetaS_XB.

Note that time-varying and spatially varying SWAN spectra can be provided using the FILELIST and LOCLIST constructions as described in Temporally and/or spatially varying wave boundary conditions.

An example of a 2D SWAN spectrum is given below:

swan.txt

SWAN 1 Swan standard spectral file
$ Data produced by SWAN version 40.51
$ Project:’projname’ ; run number:’runnum’
LOCATIONS locations in x-y-space
number of locations
22 0.00
RFREQ relative frequencies in Hz
number of frequencies
0545
0622
0710
0810
0924
1055
1204
1375
1569
1791
2045
2334
2664
3040
3470
3961
4522
5161
5891
6724
7675
8761
0000
CDIR spectral Cartesian directions in degr 12 number of directions
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
QUANT 1 number of quantities in table VaDens variance densities in m2/Hz/degr m2/Hz/degr unit
-0.9900E+02 exception value
FACTOR
675611E-06
242 574 956 1288 1482 1481 1286 957 579 244 51
610 1443 2402 3238 3725 3724 3234 2406 1454 613 128
1287 3054 5084 6846 7872 7869 6837 5091 3076 1295 271
3152 7463 12402 16712 19229 19221 16690 12419 7518 3172 662
6159 14608 24275 32688 37618 37603 32644 24309 14716 6198 1296
10989 26020 43341 58358 67109 67080 58281 43401 26213 11058 2317
15922 37712 62733 84492 97150 97110 84380 62820 37991 16021 3349
16230 38440 63939 86109 99010 98969 85995 64027 38724 16331 3410
9612 22730 37790 50909 58529 58505 50841 37843 22898 9672 2018
3178 7538 12535 16892 19440 19432 16870 12552 7594 3198 669
479 1135 1890 2542 2924 2923 2539 1892 1144 482 101
11 26 43 57 66 66 57 43 26 11 2
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0

Variance density spectra¶

2D spectral information that is not in SWAN format can be provided using a formatted variance density spectrum file and wbctype = vardens. The spectrum file itself is again referenced using the keyword bcfile. The contents of the file must adhere to a specific format:

vardens.txt

<number of frequencies (n)>
<frequency 1>
<frequency 2>
<frequency 3>

...

<frequency n-1>
<frequency n>
<number of directions (m)>
<directions 1>
<directions 2>
<directions 3>

...

<directions m-1>
<directions m>
<variance density 1,1> <variance density 2,1> ... <variance density m,1>
<variance density 1,2> <variance density 2,2> ... <variance density m,2>

...

<variance density 1,n> <variance density 2,n> ... <variance density m,n>

Note that the directions must be defined according to the Cartesian convention and in the coordinate system used by XBeach. In this coordinate system ${0}^\circ$ corresponds to waves travelling in the direction of the x-axis, while ${90}^\circ$ corresponds to the direction of the y-axis. Also, the directions must be defined in increasing order. Time-varying and spatially varying variance density spectra can be provided using the FILELIST and LOCLIST constructions as described in Temporally and/or spatially varying wave boundary conditions.

An example of a formatted variance density file is given below:

vardens.txt

15
0418
0477
0545
0622
0710
0810
0924
1055
1204
1375
1569
1791
2045
2334
2664
13
-180.0000
-150.0000
-120.0000
-90.0000
-60.0000
-30.0000
0000
0000
0000
0000
0000
0000
0000
0 0 0 0 0 0 0 0 0 0 0
242 574 956 1288 1482 1481 1286 957 579 244 51
610 1443 2402 3238 3725 3724 3234 2406 1454 613 128
1287 3054 5084 6846 7872 7869 6837 5091 3076 1295 271
3152 7463 12402 16712 19229 19221 16690 12419 7518 3172 662
6159 14608 24275 32688 37618 37603 32644 24309 14716 6198 1296
10989 26020 43341 58358 67109 67080 58281 43401 26213 11058 2317
15922 37712 62733 84492 97150 97110 84380 62820 37991 16021 3349
16230 38440 63939 86109 99010 98969 85995 64027 38724 16331 3410
9612 22730 37790 50909 58529 58505 50841 37843 22898 9672 2018
3178 7538 12535 16892 19440 19432 16870 12552 7594 3198 669
479 1135 1890 2542 2924 2923 2539 1892 1144 482 101
11 26 43 57 66 66 57 43 26 11 2
0 0 1 1 1 1 1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0

Non-spectral wave boundary conditions¶

Stationary wave boundary conditions are enabled using wbctype values stat, ts_1, ts_2, ts_nonh or stat_table. The conditions are generally defined within the params.txt file directly using the keywords described in the table below. In addition, in case of wbctype values ts_1 or ts_2 the file bc/gen.ezs should be present that describes the infragravity wave forcing generated outside of XBeach. For the wbctype ts_nonh also a separate file is needed. More information about these files in Non-spectral wave boundary conditions.

Stationary wave boundary conditions¶

Only in case of wbctype = stat_table the time-varying stationary wave boundary conditions are fully described in an external file referenced by the bcfile keyword. The bcfile keyword is part of the spectral wave boundary condition input and also the referenced file is designed for time-varying spectral input in the form of JONSWAP spectra (see Spectral wave boundary conditions). In stationary mode only the relevant data from this file is used and irrelevant data like gamma and dfj are discarded.

Table 8 Overview of available keywords related to wave boundary condition parameters¶
parameter	description	default	range	units
Hrms	Hrms wave height for instat = stat, bichrom, ts_1 or ts_2	1.0	0.0 - 10.0	m
Tlong	Wave group period for case instat = bichrom	80.0	20.0 - 300.0	s
Trep	Representative wave period for instat = stat, bichrom, ts_1 or ts_2	10.0	1.0 - 20.0	s
bclwonly	Switch to run boundary conditions with long waves only	0	0 - 1
dir0	Mean wave direction for instat = stat, bichrom, ts_1 or ts_2 (nautical convention)	270.0	-360.0 - 360.0	deg
instat	Old wave boundary condition type	bichrom	stat, bichrom, ts_1, ts_2, jons, swan, vardens, reuse, ts_nonh, off, stat_table, jons_table
lateralwave	Switch for lateral boundary at left	neumann	neumann, wavecrest, cyclic
m	Power in cos^m directional distribution for instat = stat, bichrom, ts_1 or ts_2	10	2 - 128
nmax+	Maximum ratio of cg/c for computing long wave boundary conditions	0.8	0.5 - 1.0
taper	Spin-up time of wave boundary conditions, in morphological time	100.0	0.0 - 1000.0	s
wbctype	New wave boundary condition type	params	params, parametric, swan, vardens, off, jonstable, reuse, ts_1, ts_2, ts_nonh

Time series¶

The wave boundary condition types of ts_1 and ts_2 need a separate file containing short wave energy and free surface elevation (including long wave motions). The format of this file is as follows:

bc/gen.ezs

<time 1> <zs 1> <E 1>
<time 1> <zs 2> <E 2>
<time 2> <zs 3> <E 3>

...

The wave boundary condition type of ts_nonh also needs a separate file in order to run the simulation. This file, however, needs to contain free surface elevations and velocities (both in u and v).

Boun_u.bcf

<scalar/vector>
<number of variables>
<variables: t,U,Zs,W>

(114) $\left. \begin{array}{cccccccccc} {t^{0} } & {U_{1}^{0} } & {\cdots } & {U_{J}^{0} } & {\eta _{1}^{0} } & {\cdots } & {\eta _{J}^{0} } & {W_{1}^{0} } & {\cdots } & {W_{J}^{0} } \\ {\vdots } & {\vdots } & {} & {\vdots } & {\vdots } & {} & {\vdots } & {\vdots } & {} & {\vdots } \\ {t^{n} } & {U_{1}^{N} } & {\cdots } & {U_{J}^{N} } & {\eta _{1}^{N} } & {\cdots } & {\eta _{J}^{N} } & {W_{1}^{N} } & {\cdots } & {W_{J}^{N} } \end{array}\right\}{[Data]}$

Special types of wave boundary conditions¶

Two special types of wave boundary conditions are available that makes XBeach skip the generation of new wave time series. The first is wbctype = off which simply does not provide any wave forcing on the model and hence no wave action in the model.

The second is wbctype = reuse which makes XBeach reuse wave time series that were generated during a previous simulation. This can be a simulation using the same or a different model as long as the computational grids are identical. In order to reuse boundary conditions, all relevant files should be copied to the current working directory of the model (where the params.txt file is located). Relevant files are the ebcflist.bcf and qbcflist.bcf files and all files referenced therein. Generally, the referenced files have E_ and q_ prefixes. No further wave boundary condition data need be given in params.txt.

On top of that bichromatic waves are also supported. Currently the same input parameters as non-spectral waves (see Non-spectral wave boundary conditions) are required; however there are planes to elaborate the input parameters of the bichromatic wave in order to specify individual frequency.

Temporally and/or spatially varying wave boundary conditions¶

Time-varying spectral wave boundary conditions can be defined by feeding in multiple spectrum definition files rather than a single definition file. In addition, the duration for which these spectra should occur needs to be defined.

To make use of this option, the user must specify a regular wbctype value for spectral wave boundary conditions (jons, swan or vardens), but instead of referencing a single spectrum definition file using the bcfile keyword, an extra file listing all spectrum definition files is now referenced.

The first word in this extra file must be the keyword FILELIST. In the following lines, each line contains the duration of this wave spectrum condition in seconds (similar to rt in params.txt), the required time step in this boundary condition file in seconds (similar to dtbc in params.txt) and the name of the spectral definition file used to generate these boundary conditions. The duration and boundary condition time step in this file overrules rt and dtbc in params.txt. XBeach does not reuse time-varying spectrum files. Therefore the total duration of all spectra should at least match the duration of the simulation.

A typical input file contains the following:

filelist.txt

FILELIST
0.2 jonswap1.inp
0.2 jonswap1.inp
0.2 jonswap2.inp
0.2 jonswap3.inp
0.2 jonswap2.inp
0.2 jonswap4.inp

Similar to time-varying spectral wave boundary conditions, also spatially varying wave boundary conditions can be defined using a similar construction. In order to apply spatially varying spectra on the offshore boundary, the user must specify set the keywords wbcversion = 3 and nspectrumloc = <ns> in params.txt where <ns> is the number of locations in which a spectrum is defined. By default the number of defined spectra is one.

Similar to time-varying spectral wave boundary conditions, its spatially varying sibling uses an extra file listing all relevant spectrum definition files. The first word in this extra file must be the keyword LOCLIST. This line should be followed by one line per spectrum definition location containing the world x-coordinate and world y-coordinate of the location that the input spectrum should apply, and the name of the file containing spectral wave information.

A typical input file for a run with three JONSWAP spectra contains the following:

loclist.txt

LOCLIST
0. jonswap1.inp
100. jonswap2.inp
200. jonswap3.inp

Note that it is not possible to use a mix of JONSWAP, SWAN and variance density files in either a FILELIST or a LOCLIST construction. It is also not possible to vary dthetaS_XB between files in case of non-nautical SWAN spectra. However, it is possible to combine FILELIST and LOCLIST files by referencing FILELIST files from the LOCLIST file. In this case all FILELIST files should adhere to the same time discretization, so the duration and time step values should be constant over al FILELIST files as well as the number of wave spectra definitions.

The user is reminded that along the offshore boundary of the model, the wave energy, rather than the wave height, is interpolated linearly between input spectra without consideration of the physical aspects of the intermediate bathymetry. In cases with large gradients in wave energy, direction or period, the user should specify sufficient wave spectra for the model to accurately represent changes in offshore wave conditions.

Flow, tide and surge input¶

An XBeach model needs flow boundary conditions on all boundaries of the model domain. Moreover, on each boundary tidal elevations and/ or surges may be imposed. The flow boundary conditions and time-varying tide or surge input are discussed in this section. The details on how the flow is computed within the model are described in the sections on bed friction and viscosity parameters (see Shallow water equations).

Flow boundary conditions¶

Flow boundary conditions need to be specified on all sides of the domain. We will differentiate between the offshore, lateral and landward boundaries that are set using the keywords front, back and left / right, respectively. Table 9, Table 10 and Table 11 give an overview of the available flow boundary condition types for each of these boundaries.

The keyword freewave can be used to switch from bound to free long waves, which can be useful when time series of the free long wave incident on the offshore boundary need to be specified. The file bc/gen.ezs can be used to describe the free long waves at the offshore boundary as discussed in Waves.

Table 9 Overview of available offshore flow boundary condition types¶
front	description
abs1d	absorbing-generating (weakly-reflective) boundary in 1D
abs2d	absorbing-generating (weakly-reflective) boundary in 2D
wall	no flux wall
wlevel	water level specification (from file)
nonh_1d	boundary condition for non-hydrostatic option
waveflume	boundary condition for flume experiments based on a continuity relation

Table 10 Overview of available landward flow boundary condition types¶
back	description
wall	no flux wall
abs1d	absorbing-generating (weakly-reflective) boundary in 1D
abs2d	absorbing-generating (weakly-reflective) boundary in 2D
wlevel	water level specification (from file)

Table 11 Overview of available lateral flow boundary condition types¶
left right	description
wall	no flux wall
neumann	Neumann boundary condition (constant water level gradient)
neumann_v	velocity is determined by the adjacent cell
no_advec	Neumann boundary condition, but only the advective terms are taken into account. Intermediate form between wall and neumann

Table 12 Overview of available keywords related to flow boundary condition parameters¶
parameter	description	default	range
ARC+	Switch for active reflection compensation at seaward boundary	1	0 - 1
back	Switch for boundary at bay side	abs_2d	wall, abs_1d, abs_2d
epsi+	Ratio of mean current to time varying current through offshore boundary	-1.0	-1.0 - 0.2
freewave+	Switch for free wave propagation 0 = use cg (default); 1 = use sqrt(gh) in instat = ts_2	0	0 - 1
front	Switch for seaward flow boundary	abs_2d	abs_1d, abs_2d, wall, nonh_1d, waveflume
left	Switch for lateral boundary at ny+1	neumann	neumann, wall, no_advec, neumann_v, abs_1d
nc	Smoothing distance for estimating umean (defined as nr of cells)	par%ny+1	1 - par%ny+1
order+	Switch for order of wave steering,first order wave steering (short wave energy only), second oder wave steering (bound long wave corresponding to short wave forcing is added)	2.0	1.0 - 2.0
right	Switch for lateral boundary at 0	neumann	neumann, wall, no_advec, neumann_v, abs_1d
tidetype+	Switch for offfshore boundary, velocity boundary or instant water level boundary	velocity	instant, velocity

Time-varying tide/surge¶

Time-varying tidal (or surge) signals can be applied all four boundaries in a number of ways.

The number of tidal signals is determined by the keyword tideloc that can take the values 0, 1, 2 or 4. Specifying three tidal signals is not an option. Setting tideloc = 0 disables the time-varying tide/surge option. In this case a constant and uniform water level is used specified by the keyword zs0. With tideloc = 1 the specified tidal record is specified on all four corners of the domain and interpolated along the boundaries.

Using tideloc = 2, two tidal signals are specified and there are two options available: 1) the first signal is imposed on the offshore boundary and the second on the landward boundary or 2) the first signal is imposed on the left lateral boundary and the second on the right lateral boundary. The choice between the two options is made using the keyword paulrevere where a value 0 indicates the first option and a value 1 indicates the second option. Also in the case of two tidal signals the signals are spatially interpolated along the boundaries.

Using tideloc = 4, four tide/surge signals are to be specified on each corner of the model domain and spatially interpolated along the boundaries. The first signal is imposed to the left offshore boundary seen from sea (x=1,y=1) and the others according to a clockwise rotation. Therefore the columns in the zs0file must follow the order of: (x=1,y=1), (x=1,y=N), (x=N,y=N), (x=N,y=1).

The length of the tidal signals is determined by the keyword tidelen. This is the number of water levels specified in the file referenced with the zs0file keyword. The tidal signal will be interpolated to the local time step of the XBeach simulation; therefore the resolution of the signals only needs to be enough to resolve the water level phenomenon of interest (i.e. tide variations, surge event). The tidal signals are not re-used, therefore the signal should be at least as long as the simulation time.

The zs0file file must adhere to the following format where the last three columns are optional depending on the value of tideloc and tlen represents the value of tidelen:

tide.txt

<time 1> <zs 1,1> [<zs 2,1> [<zs 3,1> <zs 4,1>]]
<time 2> <zs 1,2> [<zs 2,2> [<zs 3,2> <zs 4,2>]]
<time 3> <zs 1,3> [<zs 2,3> [<zs 3,3> <zs 4,3>]]

...

<time tlen> <zs 1,tlen> [<zs 2,tlen> [<zs 3,tlen> <zs 4,tlen>]]

In case of a single tidal signal, the signal is imposed on both offshore corners of the domain, while a constant water level defined by the keyword zs0 is imposed on the landward corners.

Table 13 Overview of available keywords related to tide boundary conditions¶
parameter	description	default	range	units
paulrevere	Specifies tide on sea and land or two sea points if tideloc = 2	land	land, sea
tideloc	Number of corner points on which a tide time series is specified	0	0 - 4
zs0file	Name of tide boundary condition series			<file>

Water level (dam break)¶

Water levels can be imposed on the model boundaries as explained in Time-varying tide/surge after which the shallow water equations force the water body in the model domain. Specific applications may require the initialization of the entire water body in the model domain at the start of the simulation. For example, an initial significant gradient in the water level that “collapses” at the start of the simulation may simulate a dam break. The initialization of the water level in the model domain is governed by the keywords listed in the table below.

The keyword zsinitfile references an external file describing the initial water levels in the entire model domain. The file should have the same format as the bathymetry input files described in Grid and bathymetry.

Table 14 Overview of available keywords related to initial conditions¶
parameter	description	default	range	units
hotstartflow+	Switch for hotstart flow conditions with pressure gradient balanced by wind and bed stress	0	0 - 1
zs0	Inital water level	0.0	-5.0 - 5.0	m
zsinitfile	Name of inital water level file			<file>

Wind input¶

Spatially-uniform winds can parametrically defined using the keywords windv and windth that represent the wind velocity and direction (nautical convention) respectively. Time-varying winds can be defined in an external file referenced by the windfile keyword. The file should adhere to the format indicated below. The total length of the time series is automatically determined and should be at least as long as the simulation time.

wind.txt

<time 1> <windv 1> <windth 1>
<time 2> <windv 2> <windth 2>
<time 3> <windv 3> <windth 3>

...

The table below gives an overview of all keywords related to the wind:

Table 15 Overview of available keywords related to wind parameters¶
parameter	description	default	range	units
Cd+	Wind drag coefficient	0.002	0.0001 - 0.01
rhoa+	Air density	1.25	1.0 - 2.0	kgm^-3
windfile	Name of file with non-stationary wind data			<file>
windth	Nautical wind direction, in case of stationary wind	270.0	-360.0 - 360.0	deg
windv	Wind velocity, in case of stationary wind	0.0	0.0 - 200.0	ms^-1

Sediment input¶

The sediment input determines the (initial) composition of the bed and the detail in which processes related to sediment sorting are resolved. This is different from how the sediment transport processes are handled in the model itself and that are described in Sediment transport and Bottom updating

The simplest situation is an XBeach simulation with uniform sediment. In this case it is sufficient to specify the uniform grain size using the keyword D50 indicating the median grain size. The effects of a specific sediment distribution can be parametrically defined by additionally specifying values for D15 and D90 and optionally the bed composition can be fine-tuned by specifying the porosity and sediment density using the keywords por and rhos respectively. In this case no sorting of sediment will be simulated.

If the effect of different sediment fractions, sorting and armoring are of importance, multiple sediment fractions can be defined. The number of sediment fraction is determined by the keyword ngd. For each sediment fraction a value for D50, and optionally D15 and D90, should be defined separated by a space. Moreover, when using multiple sediment fractions, multiple bed layers are needed as well. The number of bed layers can be defined using the keyword nd.

Three types of bed layers are distinguished: 1) the top layer 2) the variable or “breathing” layer and 3) the bottom layers. At least one of each type of bed layer is needed, which makes that at least three bed layers are required (see Bed composition). Each bed layer has a thickness. Choosing bed layer thicknesses that are in balance with the expected erosion and deposition during the simulation should keep the numerical mixing to a minimum. A bed layer thickness that is too large will result in relatively uniform behavior, while a bed layer thickness that is too small will result in a lot of shifting and thus numerical mixing. The bed layer thicknesses are determined by the three keywords dzg1, dzg2 and dzg3 for the top, variable and bottom layers respectively.

Apart from the discretization of the grain size distribution and the vertical structure of the bed, the initial bed composition needs to be defined. The bed composition is defined using external files that are not explicitly referenced from params.txt, but are assumed to be located in the working directory of the model (next to params.txt). There is one file for each sediment fraction specified by ngd. The file corresponding to the first sediment fraction is named gdist1.inp, the second gdist2.inp, et cetera.

The bed composition files hold information on how much sediment of a specific fraction is in each grid cell and bed layer at the start of the simulation. The values are a volumetric fraction that implies that they should add up to unity over all fractions. For example, if a specific grid cell is filled with the first sediment fraction only, the value corresponding to this grid cell will be one in the gdist1.inp file and zero in all others. Alternatively, if we defined five sediment fractions and a specific grid cell is filled equally with all fractions, the value corresponding to this grid cell will be 1/5 = 0.2 in all files. The gidst< N >.inp files are formatted comparable to the bathymetry files (see Grid and bathymetry), but now holds values over the three dimensions x (nx+1), y (ny+1) and the bed layers (nd). The file format is as follows:

gdist1.inp

<p 1,1,1> <p 1,2,1> <p 1,3,1> ... <p 1,nx,1> <p 1,nx+1,1>
<p 1,1,2> <p 1,2,2> <p 1,3,2> ... <p 1,nx,2> <p 1,nx+1,2>
<p 1,1,3> <p 1,2,3> <p 1,3,3> ... <p 1,nx,3> <p 1,nx+1,3>

...

<p 1,1,ny> <p 1,2,ny> <p 1,3,ny> ... <p 1,nx,ny> <p 1,nx+1,ny>
<p 1,1,ny+1> <p 1,2,ny+1> <p 1,3,ny+1> ... <p 1,nx,ny+1> <p 1,nx+1,ny+1>

...

<p 2,1,1> <p 2,2,1> <p 2,3,1> ... <p 2,nx,1> <p 2,nx+1,1>
<p 2,1,2> <p 2,2,2> <p 2,3,2> ... <p 2,nx,2> <p 2,nx+1,2>
<p 2,1,3> <p 2,2,3> <p 2,3,3> ... <p 2,nx,3> <p 2,nx+1,3>

...

<p 2,1,ny> <p 2,2,ny> <p 2,3,ny> ... <p 2,nx,ny> <p 2,nx+1,ny>
<p 2,1,ny+1> <p 2,2,ny+1> <p 2,3,ny+1> ... <p 2,nx,ny+1> <p 2,nx+1,ny+1>

...

<p nd,1,1> <p nd,2,1> <p nd,3,1> ... <p nd,nx,1> <p nd,nx+1,1>
<p nd,1,2> <p nd,2,2> <p nd,3,2> ... <p nd,nx,2> <p nd,nx+1,2>
<p nd,1,3> <p nd,2,3> <p nd,3,3> ... <p nd,nx,3> <p nd,nx+1,3>

...

<p nd,1,ny> <p nd,2,ny> <p nd,3,ny> ... <p nd,nx,ny> <p nd,nx+1,ny>
<p nd,1,ny+1> <p nd,2,ny+1> ... <p nd,nx,ny+1> <p nd,nx+1,ny+1>

The table below gives an overview of all keywords related to working with multiple sediment fractions and bed layers:

Table 16 Overview of available keywords related to bed composition parameters¶
parameter	description	default	range	units
dzg1+	Thickness of top sediment class layers	par%dzg1	0.01 - 1.0	m
dzg2+	Nominal thickness of variable sediment class layer	par%dzg1	0.01 - 1.0	m
dzg3+	Thickness of bottom sediment class layers	par%dzg1	0.01 - 1.0	m
nd+	Number of computational layers in the bed	3	3 - 1000
ngd	Number of sediment classes	1	1 - 20
por	Porosity	0.4	0.3 - 0.5
rhos	Solid sediment density (no pores)	2650.0	2400.0 - 2800.0	kgm^-3
ws	Average fall velocity (is computed in morphevolution)	0.02d0		m/s

Vegetation input¶

Short wave dissipation, long wave dissipation and flow interaction due to vegetation is supported. The user can define multiple vegetation species. The number of vegetation species is set by the keyword nveg. Furthermore, two files should be created and specified in the params.txt-file: a vegetation characteristics file (keyword: veggiefile) and a vegetation location file (keyword: veggiemapfile).

The veggiefile is a text file listing the names of the vegetation characteristics files that should be created for every individual vegetation species that should be accounted for. These property files contain the vegetation parameters nsec, ah, Cd, bv and N that represent the number of vertical sections, height of vegetation section relative to the bed , the drag coefficient, stem diameter and vegetation density per vegetation section, respectively. An example of a set of files describing two different vegetation species is given below.

veggiefile.txt

seagrass.txt
mangrove.txt

seagrass.txt

ah = 0.2
Cd = 1.0
bv = 0.02
N = 1200

mangrove.txt

nsec = 3
ah = 0.5 0.8 1.3
Cd = 2.0 1.0 2.0
bv = 0.05 0.15 0.1
N = 1000 50 500

The nsec keyword in the species property file allows the user to define multiple height segments of the species with different properties. The height per vegetation section is defined relative to the bed level. For all properties, the values are given from bottom to top. A definition sketch is given in Fig. 14.

Fig. 14 Definition sketch of vegetation specification in XBeach (example for mangrove type vegetation schematized in three vertical sections).

Finally, the veggiemapfile indicates in what grid cell which vegetation species can be found. The format of this file is similar to the bathymetry files described in Grid and bathymetry, but the values are integers referring to a species where 1 refers to the first listed species, 2 to the second, et cetera. A zero indicates no vegetation at that particular location. The use of non-integer values in this file will result in an error.

In summary, the following files should be created when the effect of vegetation is modeled:

1 x Veggiemapfile: file similar to bathymetry file containing 0, 1 etc. (up to nveg)
1 x Veggiefile: list of file names of vegetation property files per species
nveg x vegetation property file(s): describing vegetation properties per species

Below the relevant keywords in the params.txt are given. In addition, the keyword vegetation should be set to 1.

Table 17 Overview of available keywords related to vegetation parameters¶
parameter	description	default	range
nveg	Number of vegetation species	-123
vegcanflo	Include incanopy flow [1] or not [0]	0	0 - 1
veggiefile	Name of veggie species list file
veggiemapfile	Name of veggie species map file
vegnonlin	Include non-linear wave effect [1] or not [0]	0	0 - 1
veguntow	Include undertow in phase-averaged vegetati	1	0 - 1

Discharge input¶

Discharge of water at the model boundaries or directly in the model domain is defined along specific grid sections. The keywords ndischarge and ntdischarge define the number of discharge sections and the length of the discharge time series respectively. The disch_loc_file keyword references a file that defines the discharge sections. Each line in this file corresponds to a grid section and each line contains four numbers being the start and end coordinates of the section. The file is formatted as follows, where ndisch refers to the keyword ndischarge:

disch_loc.txt

<x_start 1> <y_start 1> <x_end 1> <y_end 1>
<x_start 2> <y_start 2> <x_end 2> <y_end 2>
<x_start 3> <y_start 3> <x_end 3> <y_end 3>

...

<x_start ndisch> <y_start ndisch> <x_end ndisch> <y_end ndisch >

The world coordinates specified in this file must be chosen such that they are close to the desired grid cell borders, since the grid cell borders are eventually used as discharge section. Discharge sections can be located along grid cell borders that are either oriented cross-shore or alongshore, but not a combination of the two. In a regular grid this implies that either the start or end x-coordinates are equal, or the start and end y-coordinates are equal. Alternatively, both are equal. In this case a vertical discharge from above is assumed, rather than a horizontal discharge. Vertical discharges only add mass and no momentum to the water body.

The keyword disch_timeseries_file references a file defining the time series imposed on the discharge locations. The file lists the timings in the first column and a discharge value in ${\rm m}^{3}/{\rm s}$ for each discharge section as follows, where ntdisch refers to the keyword ntdischarge:

disch_timeseries.txt

<t 1> <Q 1,1> <Q 2,1> ... <Q ndisch,1>
<t 2> <Q 1,2> <Q 2,2> ... <Q ndisch,2>
<t 3> <Q 1,3> <Q 2,3> ... <Q ndisch,3>

...

<t ntdisch> <Q 1,ntdisch> <Q 2,ntdisch> ... <Q ndisch,ntdisch >

Discharges defined at the domain borders are positive in direction towards the domain (influx). Discharges defined in the domain itself are positive in direction of the positive x or y direction. Vertical discharges are positive into the domain (influx).

Fig. 15 Possible discharge orifices. A discharge orifice is defined as line in between two points (red). The resulting discharge orifice constitutes out of full abreast grid cells (green). The discharge direction is in positive grid direction (s or n), except at the domain border where the discharge is an influx in the domain (green arrows).

The table below gives an overview of all keywords related to discharges:

Table 18 Overview of available keywords related to flow discharge boundary conditions¶
parameter	description	default	range	units
disch_loc_file+	Name of discharge locations file			<file>
disch_timeseries_file+	Name of discharge timeseries file			<file>
ndischarge+	Number of discharge locations	par%ndischarge	0 - 100
ntdischarge+	Length of discharge time series	par%ntdischarge	0 - 100

Drifters input¶

Drifters can be deployed during the model simulation by specifying the number of drifters using the keyword ndrifter and the location, start and end time of the drifter deployment in a separate file referenced by the drifterfile keyword. The file format is as follows:

drifter.txt

<x 1> <y 1> <t_start 1> <t_end 1>
<x 2> <y 2> <t_start 2> <t_end 2>
<x 3> <y 3> <t_start 3> <t_end 3>

...

<x ndrifter> <y ndrifter> <t_start ndrifter> <t_end ndrifter>

The table below gives an overview of all keywords related to drifters:

Table 19 Overview of available keywords related to drifters parameters¶
parameter	description	default	range	units
drifterfile	Name of drifter data file			<file>
ndrifter	Number of drifers	par%ndrifter	0 - 50

Ship-induced wave motions¶

Ship waves can be simulated by defining the ships’ geometries and trajectories in a collection of files. The user can define multiple ships. The number of ships is set by the keyword nship. In the file referenced by the keyword shipfile each ship is given a name. The properties of each ship are summarized in another textfile with the name of the ship (shipname.txt). This properties file defines the parameters for the discretization of the ships geometry. The ship grid is determined by the keywords dx, dy, nx, and ny, and the ship geometry itself is given in a separate file, referenced bythe keyword shipgeom. This file contains the ship draft per ship grid point, and should have a size of nx + 1 by ny + 1. The center of gravity of the ship is also defined in the ship properties file using the keywords xCG, yCG and zCG. The ships trajectory is defined in a file referenced from the ship properties file by the keyword shiptrack. Each row in this file contains a time, x- and y-coordinate indicating the ships position as function of time.

To avoid numerical problems, the full ship track should be within the model domain (i.e. a vessel cannot sail through the model boundary). Furthermore, it is advised to start the ship track with a very low velocity and gradually increase the sailing speed. When not taking into account such spin up period, the impact of the ship on the water level can be too abrupt, resulting in unrealistic wave patterns. In addition, it is advised to maintain a relatively deep edge of a few grid cells at both model boundaries (i.e. front and back side).

Another way to avoid numerical issues at the initialization of a ship simulation is to use the ’flying’ option, which can be specified in the ship file (flying = 1). In case the option flying is enabled, also a z-coordinate is defined in the shiptrack-file indicating the vertical position of the ship. In this way, the vessel can ‘land’ on the water with its correct sailing speed, thereby avoiding unwanted disturbance to the water level. Also, the ship can ‘fly out’ of the model domain before reaching the back boundary. By using this method the spin-up time can be reduced considerably and both inflow and outflow boundaries are unaffected by the ship

The two keywords compute_force and compute_motion enable the computation of forces on the ship and the ships motions due to wave forcing respectively; the latter has not been implemented yet. Forces on a ship in motion may be unreliable due to near-field effects; forces on a ship at rest are much more reliable. An example of ship definition files is:

shipfile.txt

containership.txt
oiltanker.txt

containership.txt

dx = 10
dy = 10
nx = 30
ny = 10
shipgeom = container_geom.dep
xCG = 120
yCG = 50
zCG = 30
shiptrack = container_track.txt
flying = 1
compute_force = 1
compute_motion = 1

pannamax_geom.txt

<z 0,0> <z 1,0> <z 2,0> <z 3,0> ... <z nx,0> <z nx+1,0>
<z 0,1> <z 1,1> <z 2,1> <z 3,1> ... <z nx,1> <z nx+1,1>

...

<z 0,ny> <z 1,ny> <z 2,ny> <z 3,ny> ... <z nx,ny> <z nx+1,ny>
<z 0,ny+1> <z 1,ny+1> <z 2,ny+1> <z 3,ny+1> ... <z nx,ny+1> <z nx+1,ny+1>

pannamax_track.txt

<t 1> <x 1> <y 1> <z 1>
<t 2> <x 2> <y 2> <z 2>
<t 3> <x 3> <y 3> <z 3>

...

oiltanker.txt

dx = 2
dy = 2
nx = 20
ny = 4
shipgeom = tanker_geom.dep
xCG = 20
yCG = 40
zCG = 1.5
shiptrack = tanker_track.txt
flying = 0

tanker_track.txt

<t 1> <x 1> <y 1>
<t 2> <x 2> <y 2>
<t 3> <x 3> <y 3>

...

Table 20 Overview of available keywords related to ship parameters¶
parameter	description	default	range	units
nship+	Number of ships	-123
shipfile	Name of ship data file			<file>

Output selection¶

Output selection determines what data computed by XBeach is written to a file in terms of location and time and in what format. The output types, output times and output formats supported by XBeach are explained in more detail in the following subsections. The table below gives an overview of all keywords related to model output:

Table 21 Overview of available keywords related to output variables¶
parameter	description	default	range	units
ncfilename+	Xbeach netcdf output file name			<file>
nglobalvar	Number of global output variables (as specified by user)	-1	-1 - 20
nmeanvar	Number of mean, min, max, var output variables	0	0 - 15
npoints	Number of output point locations	0	0 - 50
npointvar	Number of point output variables	0	0 - 50
nrugauge	Number of output runup gauge locations	0	0 - 50
nrugdepth+	Number of depths to compute runup in runup gauge	1	1 - 10
outputformat+	Output file format	fortran	fortran, netcdf, debug
outputprecision	Switch between single and double precision output in netcdf	double	single, double
timings+	Switch enable progress output to screen	1	0 - 1
tintc+	Interval time of cross section output	-123		s
tintg	Interval time of global output	1.0	0.01 - par%tstop-par%tstart	s
tintm	Interval time of mean, var, max, min output	par%tstop-par%tstart	1.0 - par%tstop-par%tstart	s
tintp	Interval time of point and runup gauge output	1.0	0.01 - par%tstop-par%tstart	s
tsglobal+	Name of file containing timings of global output
tsmean+	Name of file containing timings of mean, max, min and var output
tspoints+	Name of file containing timings of point output
tstart	Start time of output, in morphological time	0.0	0.0 - par%tstop	s

Output types¶

XBeach supports four different types of output: 1) instantaneous spatial output 2) time-averaged spatial output 3) fixed point output or 4) run-up gauge output. In principle any variable in XBeach can be outputted as long as it is part of the spaceparams structure defined in variables.f90 in the XBeach source code. An overview of all currently supported parameters in this file is presented in Table 22.

The amount of output variables used for each type is determined by the keywords nglobalvar, nmeanvar, npoints and nrugauge. Each of these keywords takes a number indicating the number of parameters or locations that should be written to file. If any of the keywords is set to zero, the output type is effectively disabled. If nglovalvar is set to -1 then a standard set of output variables is used, being H, zs, zs0, zb, hh, u, v, ue, ve, urms, Fc, Fy, ccg, ceqsg, ceqbg, Susg, Svsg, E, R, D and DR. If nglobalvar is not set it defaults to -1. The lines in the params.txt file immediately following these keywords determine what parameters or locations are used, as will be explained in more detail in the following subsections.

Table 22 Overview of available keywords related to *¶
parameter	description	units
As	Asymmetry of short waves
BR	Maximum wave surface slope used in roller dissipation formulation
Cdrag	Vegetation drag coefficient
D	Dissipation	W/m2
D15	D15 grain diameters for all sediment classses	m
D50	D50 grain diameters for all sediment classses	m
D50top	Friction coefficient flow
D90	D90 grain diameters for all sediment classses	m
D90top	Friction coefficient flow
DR	Roller energy dissipation	W/m2
Dc	Diffusion coefficient	m2/s
Df	Dissipation rate due to bed friction	W/m2
Dp	Dissipation rate in the swash due to transformation of kinetic wave energy to potential wave energy	W/m2
Dveg	Dissipation due to short wave attenuation by vegetation	W/m2
E	Wave energy	Nm/m2
Fvegu	X-forcing due to long wave attenuation by vegetation	N/m2
Fvegv	Y-forcing due to long wave attenuation by vegetation	N/m2
Fx	Wave force, x-component	N/m2
Fy	Wave force, y-component	N/m2
H	Hrms wave height based on instantaneous wave energy	m
Hrunup	Short wave height used in runup formulation	m
L1	Wave length (used in dispersion relation)	m
Qb	Fraction breaking waves
R	Roller energy	Nm/m2
Sk	Skewness of short waves
Subg	Bed sediment transport for each sediment class (excluding pores), x-component	m2/s
Susg	Suspended sediment transport for each sediment class (excluding pores), x-component	m2/s
Sutot	Sediment transport integrated over bed load and suspended and for all sediment grains, x-component	m2/s
Svbg	Bed sediment transport for each sediment class (excluding pores), y-component	m2/s
Svsg	Suspended sediment transport for each sediment class (excluding pores), y-component	m2/s
Svtot	Sediment transport integrated over bed load and suspended and for all sediment grains, y-component	m2/s
Sxx	Radiation stress, x-component	N/m
Sxy	Radiation stress, y-component	N/m
Syy	Radiation stress, y-component	N/m
Tbore	Wave period interval associated with breaking induced turbulence	s
Tsg	Sediment response time for each sediment class	s
alfau	Grid orientation at u-point	rad
alfav	Grid orientation at v-point	rad
alfaz	Grid orientation at z-point	rad
bedfriccoef	Dimensional/dimensionless input bed friction coefficient; depends on value of parameter bedfriction
bi	Incoming bound long wave	m
breaking	Indicator whether cell has breaking nonh waves
bwalpha	Beachwizard weighting factor
c	Wave celerity	m/s
ca	Reference concentration	m3/m3
ccg	Depth-averaged suspended concentration for each sediment fraction	m3/m3
cctot	Sediment concentration integrated over bed load and suspended and for all sediment grains	m3/m3
ccz	Concentration profile	m3/m3
ceqbg	Depth-averaged bed equilibrium concentration for each sediment class	m3/m3
ceqsg	Depth-averaged suspended equilibrium concentration for each sediment class	m3/m3
cf	Friction coefficient flow
cfu	Friction coefficient flow in u-points
cfv	Friction coefficient flow in v-points
cg	Group velocity	m/s
cgx	Group velocity, x-component	m/s
cgx_s	Group velocity, x-component	m/s
cgy	Group velocity, y-component	m/s
cgy_s	Group velocity, y-component	m/s
cobs	Beachwizard observed wave celerity	m/s
costh	Cos of wave angles relative to grid direction
costh_s	Cos of wave angles relative to grid direction
ctheta	Wave celerity theta-direction (refraction)	rad/s
ctheta_s	Wave celerity theta-direction (refraction)	rad/s
cx	Wave celerity, x-component	m/s
cy	Wave celerity, y-component	m/s
dU	U-velocity difference between two vertical layers (reduced 2-layer non-hydrostatic model)	m2/s2
dUi	Velocity difference at boundary due to (short) waves	m/s
dV	V-velocity difference between two vertical layers (reduced 2-layer non-hydrostatic model)	m2/s2
dassim	Beachwizard depth change	m
dcbdx	Bed concentration gradient x-dir.	kg/m3/m
dcbdy	Bed concentration gradient y-dir.	kg/m3/m
dcmdo	Beachwizard computed minus observed dissipation	W/m2
dcsdx	Suspended concentration gradient x-dir.	kg/m3/m
dcsdy	Suspended concentration gradient y-dir.	kg/m3/m
depo_ex	Explicit bed deposition rate per fraction	m/s
depo_im	Implicit bed deposition rate per fraction	m/s
dinfil	Infiltration layer depth used in quasi-vertical flow model for groundwater	m
dnc	Grid distance in n-direction, centered around c-point	m
dnu	Grid distance in n-direction, centered around u-point	m
dnv	Grid distance in n-direction, centered around v-point	m
dnz	Grid distance in n-direction, centered around z-point (=eta-point)	m
dobs	Beachwizard observed dissipation	W/m2
dsc	Grid distance in s-direction, centered around c-point	m
dsdnui	Inverse of grid cell surface, centered around u-point	1/m2
dsdnvi	Inverse of grid cell surface, centered around v-point	1/m2
dsdnzi	Inverse of grid cell surface, centered around z-point	1/m2
dsu	Grid distance in s-direction, centered around u-point	m
dsv	Grid distance in s-direction, centered around v-point	m
dsz	Grid distance in s-direction, centered around z-point (=eta-point)	m
dzav	Total bed level change due to avalanching	m
dzbdt	Rate of change bed level	m/s
dzbdx	Bed level gradient in x-direction
dzbdy	Bed level gradient in y-direction
dzbed	No description
dzbnow	Bed level change in current time step	m
dzs0dn	Alongshore water level gradient due to tide alone
dzsdt	Rate of change water level	m/s
dzsdx	Water surface gradient in x-direction	m/s
dzsdy	Water surface gradient in y-direction	m/s
ee	Directionally distributed wave energy	J/m2/rad
ee_s	Directionally distributed wave energy	J/m2/rad
ero	Bed erosion rate per fraction	m/s
fw	Wave friction coefficient
gw0back	Boundary condition back boundary for groundwater head	m
gwbottom	Level of the bottom of the aquifer	m
gwcurv	Curvature coefficient of groundwater head function
gwhead	Groundwater head (differs from gwlevel)	m
gwheadb	Groundwater head at bottom (differs from gwlevel)	m
gwheight	Vertical size of aquifer through which groundwater can flow	m
gwlevel	Groundwater table (min(zb,gwhead))	m
gwqx	Groundwater discharge in x-direction	m/s
gwqy	Groundwater discharge in y-direction	m/s
gwu	Groundwater flow in x-direction	m/s
gwv	Groundwater flow in y-direction	m/s
gww	Groundwater flow in z-direction (interaction between surface and ground water)	m/s
hh	Water depth	m
hhw	Water depth used in all wave computations, includes h*par%delta	m
hhwcins	Water depth used in wave instationary computation in case of wci	m
hhws	Water depth used in wave stationary computation (and single_dir wave directions)	m
hold	Water depth previous time step	m
hu	Water depth in u-points	m
hum	Water depth in u-points	m
hv	Water depth in v-points	m
hvm	Water depth in v-points	m
idrift	Drifter x-coordinate in grid space
infil	Rate of exchange of water between surface and groundwater (positive from sea to groundwater)	m/s
istruct	Location of revetments toe
iwl	Location of water line (including long wave runup)
jdrift	Drifter y-coordinate in grid space
k	Wave number	rad/m
kb	Near bed turbulence intensity due to depth induces breaking	m2/s2
kturb	Depth averaged turbulence intensity due to long wave breaking	m2/s2
maxzs	Maximum elevation in simulation	m
minzs	Minimum elevation in simulation	m
n	Ratio group velocity/wave celerity
nd	Number of bed layers (can be different for each computational cell)
ndist	Cum. distance from right boundary along n-direction	m
nuh	Horizontal viscosity coefficient	m2/s
nutz	Turbulence viscosity
pbbed	No description
pdisch	Discharge locations
ph	Pressure head due to ship	m
pntdisch	Point discharge locations (no momentum)
pres	Normalized dynamic pressure	m2/s2
qdisch	Discharges	m2/s
qx	Discharge in u-points, x-component	m2/s
qy	Discharge in u-points, y-component	m2/s
refA	Reference level	m
rolthick	Long wave roller thickness	m
rr	Directionally distributed roller energy	J/m2/rad
runup	Short wave runup height	m
sdist	Cum. distance from offshore boundary along s-direction	m
sedcal	Equilibrium sediment concentartion factor for each sediment class
sedero	Cum. sedimentation/erosion	m
setbathy	Prescribed bed levels	m
shipFx	Force on ship in x-direction	N
shipFy	Force on ship in y-direction	N
shipFz	Force on ship in z-direction	N
shipMx	Moment on ship around x-axis	Nm
shipMy	Moment on ship around y-axis	Nm
shipMz	Moment on ship around z-axis	Nm
shipchi	Turning angle arround y-axis	deg
shipphi	Turning angle arround x-axis	deg
shippsi	Turning angle arround z-axis	deg
shipxCG	X-coordinate of ship center of gravity	m
shipyCG	Y-coordinate of ship center of gravity	m
shipzCG	Z-coordinate of ship center of gravity	m
shobs	Beachwizard observed shoreline	m
sig2prior	Beachwizard prior std squared	m2
sigm	Mean frequency	rad/s
sigt	Relative frequency	rad/s
sigz	Vertical distribution of sigma layers q3d
sinth	Sin of wave angles relative to grid direction
sinth_s	Sin of wave angles relative to grid direction
strucslope	Slope of structure
structdepth	Depth of structure in relation to instantaneous bed level	m
taubx	Bed shear stress, x-component	N/m2
taubx_add	Additional bed shear stress due to boundary layer effects, x-component	N/m2
tauby	Bed shear stress, y-component	N/m2
tauby_add	Additional bed shear stress due to boundary layer effects, y-component	N/m2
tdisch	Discharge time series
tdriftb	Drifter release time	s
tdrifte	Drifter retrieval time	s
thet	Wave angles	rad
thet_s	Wave angles	rad
theta	Wave angles directional distribution w.r.t. comp. x-axis	rad
theta_s	Wave angles directional distribution w.r.t. comp. x-axis	rad
thetamean	Mean wave angle	rad
tideinpt	Input time of input tidal signal	s
tideinpz	Input tidal signal	m
tsetbathy	Points in time of prescibed bed levels	s
u	Glm velocity in cell centre, x-component	m/s
ua	Time averaged flow velocity due to wave assymetry	m/s
ucrcal	Calibration factor for u critical for each sediment class
ududx	Advection	m2/s2
udvdx	Advection	m2/s2
ue	Eulerian velocity in cell centre, x-component	m/s
ue_sed	Advection velocity sediment in cell centre, x-component	m/s
ueu	Eulerian velocity in u-points, x-component	m/s
ui	Incident bound wave velocity in, x-component	m/s
umean	Longterm mean velocity at bnds in u-points, x-component	m/s
umwci	Velocity (time-averaged) for wci, x-component	m/s
ur	Reflected velocity at bnds in u-points	m/s
urepb	Representative flow velocity for sediment advection and diffusion, x-component	m/s
ureps	Representative flow velocity for sediment advection and diffusion, x-component	m/s
urms	Orbital velocity	m/s
usd	Return flow due to roller after breaker delay	m/s
ust	Stokes drift	m/s
ustr	Return flow due to roller	m/s
ustz	Stokes velocity (q3d)
uu	Glm velocity in u-points, x-component	m/s
uv	Glm velocity in v-points, x-component	m/s
uwcins	U-velocity used in wave stationary computation in case of wci	m/s
uwf	Stokes drift, x-component	m/s
uws	U-velocity used in wave stationary computation (and single_dir wave directions)	m/s
uz	Velocity (q3d) ksi-comp
v	Glm velocity in cell centre, y-component	m/s
vdudy	Advection	m2/s2
vdvdy	Advection	m2/s2
ve	Eulerian velocity in cell centre, y-component	m/s
ve_sed	Advection velocity sediment in cell centre, y-component	m/s
vegtype	Vegetation type index
vev	Eulerian velocity in u-points, y-component	m/s
vi	Incident bound wave velocity in, y-component	m/s
viscu	Viscosity	m2/s2
viscv	Viscosity	m2/s2
vmag	Velocity magnitude in cell centre	m/s
vmageu	Eulerian velocity magnitude u-points	m/s
vmagev	Eulerian velocity magnitude v-points	m/s
vmagu	Glm velocity magnitude u-points	m/s
vmagv	Glm velocity magnitude v-points	m/s
vmean	Longterm mean velocity at bnds in u-points, y-component	m/s
vmwci	Velocity (time-averaged) for wci, y-component	m/s
vrepb	Representative flow velocity for sediment advection and diffusion, y-component	m/s
vreps	Representative flow velocity for sediment advection and diffusion, y-component	m/s
vu	Glm velocity in u-points, y-component	m/s
vv	Glm velocity in v-points, y-component	m/s
vwcins	V-velocity used in wave stationary computation in case of wci	m/s
vwf	Stokes drift, y-component	m/s
vws	V-velocity used in wave stationary computation (and single_dir wave directions)	m/s
vz	Velocity (q3d) eta-comp
wb	Vertical velocity at the bottom	m/s
wete	Mask wet/dry wave-points
wetu	Mask wet/dry u-points
wetv	Mask wet/dry v-points
wetz	Mask wet/dry eta-points
wi	Vertical velocity at boundary due to (short) waves	m/s
winddirts	Input wind direction	deg_nautical
windinpt	Input time of input wind signal	s
windnv	Wind velocity in n direction in v point at current time step	m/s
windsu	Wind velocity in s direction in u point at current time step	m/s
windvelts	Input wind velocity	m/s
windxts	Time series of input wind velocity (not s direction), x-component	m/s
windyts	Time series of input wind velocity (not n direction), y-component	m/s
wm	Mean abs frequency	rad/s
ws	Vertical velocity at the free surface	m/s
wscrit	Critial vertical velocity at the free surface for breaking	m/s
x	X-coord. original cmp. grid	m
xHrunup	Location at which short wave height for runup is taken	m
xu	X-coord. comp. grid u-points	m
xv	X-coord. comp. grid v-points	m
xyzs01	Global xy coordinates of corner (x=1,y=1)
xyzs02	Global xy coordinates of corner (x=1,y=n)
xyzs03	Global xy coordinates of corner (x=n,y=n)
xyzs04	Global xy coordinates of corner (x=n,y=1)
xz	X-coord. comp. grid (positive shoreward, perp. to coastline)	m
y	Y-coord. original comp. grid	m
yu	Y-coord. comp. grid u-points	m
yv	Y-coord. comp. grid v-points	m
yz	Y-coord. comp. grid	m
z0bed	No description
zb	Bed level	m
zb0	Initial bed level	m
zbobs	Beachwizard observed depth	m
zi	Surface elevation at boundary due to (short) waves	m
zs	Water level	m
zs0	Water level due to tide alone	m
zs0fac	Relative weight of offshore boundary and bay boundary for each grid point is stored in zs0fac
zs1	Water level minus tide	m
zswci	Waterlevel (time-averaged) for wci	m

Instantaneous spatial output¶

Instantaneous spatial output describes the instantaneous state of variables across the entire model domain at various points in time. To make use of this option the user must specify the number of output variables required using the nglobalvar keyword in params.txt, immediately followed by the names of the requested variables on a separate line each. The output of three instantaneous grids can look as follows:

params.txt

nglobalvar = 3
zs
zb
H

Time-averaged spatial output¶

Time-averaged spatial output describes the time-averaged state of variables across the entire model domain at various points in time. The user can define the averaging period in params.txt. To make use of this option the user must specify the number of output variables required using the nmeanvar keyword in params.txt, immediately followed by the names of the requested variables on a separate line each. The output of two time-averaged grids may look as follows:

params.txt

nmeanvar = 2
u
v

Fixed point output¶

Fixed point output allows the user to select one or more locations for which a time series of data is stored. This output describes a time-series of one or more variables at one point in the model domain. To make use of this option, the user must specify the number of output locations using the npoints keyword in params.txt, immediately followed by one line per output location describing the location coordinates given as the x-coordinate and y-coordinate and in world coordinates. XBeach will link the output location to the nearest computational point.

The user can specify the number and selection of output variables for all points (and run-up gauges, discussed in the following section) using the npointvar keyword in params.txt to specify the number of output variables, immediately followed by the names of the requested variables on a separate line each.

Fixed point output significantly reduces the amount of data written to file in each time step and is therefore particularly suitable for high temporal resolution output.

An example with two output locations is given below. The first point is located on the offshore boundary (x = 0.0) and somewhere in the middle of the model domain in y-direction (y = 800.0). The second point is located on the lateral boundary (y = 1600.0) and somewhere in the middle of the domain in x-direction (x = 2000.0). Both locations have four output variables: H, zs, zb and D.

params.txt

npoints = 2
0. 800.
2000. 1600.
npointvar = 4
H
zs
zb
D

Run-up gauge output¶

Run-up gauge output describes a time-series of a number of variables at the (moving) waterline. In this case XBeach scans in an x-directional transect defined by the user for the location of the waterline. Output information is recorded for this moving point. This is particularly useful to keep track of run-up levels in cross-shore transects.

The definition of run-up gauges is similar to the definition of fixed point output. The user needs to specify the number of run-up gauges using the nrugauge keyword in params.txt, immediately followed by one line per run-up gauge location describing the coordinates of the initial location of the run-up gauge. XBeach will subsequently link the initial run-up gauge location to the nearest computational cross-shore transect rather than just the nearest computational point.

Run-up gauges share their selection of output variables with regular point output. However, in the case of run-up gauges, XBeach will automatically also include the variables xw, yw and zs to the point output variables, if these variables were not specified using the npointvar keyword in params.txt. Note that the user should refer to the pointvars.idx output file to check order of output variables for points and run-up gauges.

An example of a run-up gauge input is given below. The run-up gauge is initially located on the offshore boundary (x = 0.0) and somewhere in the middle of the model domain in y-direction (y = 800.0). The run-up gauge will display standard output variables (xw, yw and zs, as well as any output variables specified by the npointvar keyword).

params.txt

nrugauge = 1
0. 800.

Output times¶

The user may determine the output times for regular spatial output variables, time averaged spatial variables and point location variables individually. Run-up gauge output and fixed point output are given at the same moments in time. For all three types of output the user may choose to either state a fixed interval time at which output is given or supply an external file containing times at which output should be given or a combination of both.

Output at fixed intervals¶

The user should define a point in time after the start of the simulation at which the first output is generated for fixed interval output. The user can do this by using the tstart keyword in params.txt. All output that is being generated at fixed intervals uses tstart as their base. The interval for instantaneous spatial output is given by the tintg keyword. The keywords for the interval of time-averaged spatial output and point output are tintm and tintp respectively, where tintp is used both for fixed point and run-up gauge output. Note that tintg, tintm and tintp supersede the older tint parameter that is valid for all types of output. The default value of tintg is one second. If tintp or tintm is not stated, but output is declared (npoints, nrugauge or nmeanvar is stated larger than zero), XBeach assumes the same output interval as tintg. An example of the definition of fixed intervals is given below.

params.txt

tstart = 100.
tintg = 100.
tintp = 2.
tintm = 3600.

In the case of instantaneous spatial output and point output, the first output is given at tstart. In the case of time-averaged spatial variables, the first output is given at tstart+tintm. This output represents the average condition over the interval between tstart and tstart+tintm.

Output times defined by external file¶

The user is given the option to have output at a set of points in time that are not separated by regular intervals. In this case the user must supply an additional file for each output type. The user specifies the name of the output time series file for instantaneous spatial output using the tsglobal keyword. The keywords for time series files for time-averaged spatial output and point output are tsmean and tspoint respectively, where tspoint is again used for both fixed point and run-up gauge output. All time series files must contain on the first line the number of output times followed by every output time on a new line. An example of such irregular output time definition is given below.

params.txt

tsglobal= time_series1.txt
tspoints = time_series2.txt
tsmean= time_series3.txt

time_series1.txt

In the case of instantaneous spatial output and point output, the first output is given at the first stated point in time. In the case of time-averaged spatial variables, the first output is given at the second stated point in time. This output represents the average condition over the interval between first and second stated point in time. Subsequent averaging is done over every interval.

Combinations of fixed internal and external files¶

The user is allowed to define certain types of output using fixed intervals and others using external files. The use of an external file supersedes the use of fixed intervals. Note that tstart will only apply to output of fixed interval type. An example of mixing fixed and varying output time intervals is given below.

param.txt

tstart = 100.
tintg = 100.
tspoints = time_series2.txt
tintm = 3600.

Output format¶

XBeach supports two types of output: 1) Fortran binary and 2) netCDF. The output format used is determined by the keyword outputformat. The use of netCDF output is more convenient since all output (and input) is stored in a single, easy accessible file. Also the netCDF file format is compatible with many programming languages (e.g. Matlab, Python) as well as many visualization tools (e.g. QuickPlot, Morphan). It should be noted that the support for output types in netCDF could be limited for recent functionalities of the XBeach model.

Fortran binary¶

Output files in Fortran binary format are bare matrix dumps of XBeach’ computational matrices. At each output time, one such matrix block is added to the output file. These files can generally be read by binary read functions, like fread in Matlab and the struct package in Python.

Output files written in Fortran binary format are given the name < variable >.dat, for example zs.dat, for instantaneous spatial output. The only exception is that files containing information about the wave height of the short waves are called hrms.dat instead of H.dat to maintain backward compatibility. Time-averaged spatial output is stored similarly, but the file names have a suffix indicating the type of averaging < variable >_mean.dat. For time-averaged spatial output also the variance, minimum and maximum values are stored using the suffixes _var, _min and _max respectively.

All data corresponding to fixed point locations will be stored in files called point< NNN >.dat. < NNN > represents a number between 001 and 999 corresponding to the order in which the points are declared in params.txt. The data files are plain text and contain one row for each output time step. The first position on each row is the time at which the output is given. The subsequent positions in the row are the instantaneous values of the variables at the given point. The order of the variables is equal to the order in which they are defined for that point in params.txt. Data corresponding to run-up gauge locations are stored in the same format as fixed point output, but the files are named rugau< NNN >.dat.

An extra file called dims.dat is always written at the start of the simulation in Fortran binary output mode. This file contains the dimensions of the XBeach model. It simply states the following dimensions in order: nt (number of output time steps), nx (number of grid cells in x-direction), ny (number of grid cells in y-direction), ngd (number of sediment fractions), nd (number of bed layers), ntp (number of point output time steps), ntm (number of time-averaged output time steps). Subsequently, the irregular time series are stored, if applicable: tsglobal (irregular output times), tspoints (irregular point output times) and tsmean (irregular time-averaged output times). Similarly, a file xy.dat is written containing the x- and y- coordinates of the full computational grid.

netCDF¶

All data in netCDF output is stored in a single output file. By default this file is named xboutput.nc, but this name can be chosen freely using the keyword ncfilename. The netCDF file holds all output data, dimensions and input data in a single file. It should be noted that netCDF files can hold a multiple time axes. The temporal unit can be specified in the params.txt file using the keyword tunits. This unit does not affect calculations and is only used for output. An example of the layout of the netcdf file is given below:

xboutput.nc (structure only, no real contents)

netcdf xboutput {
   dimensions:
      x = 565 ;
      y = 101 ;
      wave_angle = 9 ;
      bed_layers = 3 ;
      sediment_classes = 1 ;
      inout = 2 ;
      globaltime = 2 ;
      tidetime = 435 ;
      tidecorners = 2 ;
      windtime = 2 ;
   variables:
      double x(x) ;
         x:units = “m” ;
         x:long_name = “local x coordinate” ;
      double y(y) ;
         y:units = “m” ;
         y:long_name = “local y coordinate” ;
      double globaltime(globaltime) ;
         globaltime:units = “s” ;
      double H(globaltime, y, x) ;
         H:units = “m” ;
         H:long_name = “wave height” ;
      double zs(globaltime, y, x) ;
         zs:units = “m” ;
         zs:long_name = “water level” ;
      double zb(globaltime, y, x) ;
         zb:units = “m” ;
         zb:long_name = “bed level” ;
      double ue(globaltime, y, x) ;
         ue:units = “m/s” ;
}

Time parameters¶

In all XBeach simulations the hydrodynamic simulation starts at time 0. Model output can be postponed until the time specified by the keyword tstart. The simulation stops at the time specified by tstop. The time step used in the hydrodynamic simulation is determined based on a given maximum Courant number using the keyword CFL. The table below gives an overview of all keywords related to time management:

Table 23 Overview of available keywords related to model time¶
parameter	description	default	range	units
CFL	Maximum courant-friedrichs-lewy number	0.7	0.1 - 0.9
defuse	Turn on timestep explosion prevention mechanism	1	0 - 1
dtset	Fixed timestep, overrides use of cfl	0.0	0.001 - 100.0
maxdtfac	Maximum increase/decrease in time stp in explosion prevention mechanism	500.0	100.0 - 1000.0
tstop*	Stop time of simulation, in morphological time	2000.0	1.0 - 1000000.0	s
tunits+	Time units in udunits format (seconds since 1970-01-01 00:00:00.00 +1:00)	s

Appendices¶

Hands on exercises (based on basic XBeach of the Delft Software Days 2014)¶

The hands-on exercises can be downloaded via subversion. Subversion is a well-known version management system that allows you to always have the most recent source code at hand. It also allows developers to commit changes to the source code, without interfering with other developers. In order to use Subversion, you will need a Subversion client. A well-known client for Windows is Tortoise. If you have registered, you can download the source code via the following URL: https://svn.oss.deltares.nl/repos/xbeach/Courses/DSD_2014/Examples – Basic. For the tools like Quickplot and Quickin the Delft3D environment is needed.

Dune erosion at Delfland, Netherlands (1D)¶

The first case we will run is a relative simple 1D case. It concerns a profile along the Dutch coast and the hydraulic boundary conditions are based on the 1953 storm surge that caused substantial flooding in the Netherlands.

You can work on the following assignments:

Go to the folder “Examples\DelflandStorm” and double click the file “run_model.bat”. The simulation will start. The model will run for a few minutes, but in the meantime you can already work on question 2 to 5.
Open params.txt in which you specify the model input files and settings. Check the number of grid-points in x-direction (keyword: nx) and y-direction (keyword: ny). Check the filenames in which you specify the wave conditions (keyword: bcfile) and the storm surge level (SSL) (keyword: zs0file).
Do the wave conditions change during the simulation? What is/are the wave height(s) and wave period(s) applied in the simulation?
Does the storm surge level change during the simulation? What is the maximum surge height in the simulation. Surge height is defined with respect to the mean sea level (MSL)?
What is the simulation time (keyword: tstop)? Do we apply a morphological acceleration factor (keyword: morfac)? What variables are stored as output and with what time interval? How much hydrodynamic time is simulated?
Probably the simulation has finished. When you start the model, it generates a file named XBlog.txt. Open this file and check what is stored in the file. What was the total simulation time?
To check out the simulation results we make use of the Quickplot tool (A brief tutorial is attached to this document). You can start Quickplot via the Delft3D environment we installed (Start Programs Deltares Delft3D Delft3D). In the Delft 3D menu choose Utilities Quickplot. Choose Files of type “NetCDF files and GRIB files” and open “xboutput.nc” in the simulation folder.
Use the Quickplot tutorial and try to make an animation in which you plot short wave height (H), water level (including long wave variations, zs) and bed level (zb) as function of time.
Plot the offshore water level as function of time. Also open the file “tide.tek” (Tekal data files format), which contains the imposed surge level. Did the model correctly simulate the imposed surge level?
Copy all model files to a new folder named “superfast”. Edit params.txt and set ny=0 (instead of ny=2), and run the model. What is the simulation time compare to the original simulation?
Compare simulation results for the “superfast” and “default” simulation. Are these the same? What option will you use in the future?

Nourishment scenarios near Kijkduin, Holland (1D)¶

This case concerns the exploration of a nourishment strategy near Kijkduin along the Holland coast in the Netherlands. At this location a mega nourishment of 21 ${\rm Mm}^{3}$ named the Sand Engine was constructed. In this case we will explore to what extent nourishments can reduce the (dune and beach) erosion during a storm event.

You can work on the following assignments:

Go to the folder “Examples\Nourishment case” and double click on the file “runall.bat”. This batch file will run three simulations sequentially in which the profile configuration varies and corresponds with the undisturbed profile (folder reference), a shoreface nourishment (folder shoreface) and a beach nourishment (folder beach) respectively. Each model will run for a few minutes. While running you can already answer question 2 to 6.
For the reference case open the params.txt in which you specify model input files and settings. Check the number of grid-points in x-direction (keyword: nx) and y-direction (keyword: ny). How many directional wave bins are defined and what is their width (keywords: thetamin, thetamax, dtheta).
Do the wave conditions change during the simulation? What is/are the wave height(s) and wave period(s) applied in the simulation?
Does the storm surge level change during the simulation? What is the maximum surge height in the simulation. Surge height is defined with respect to the mean sea level (MSL)?
What is the simulation time (keyword: tstop)? Do we apply a morphological acceleration factor (keyword: morfac)? What variables are stored as output and with what time interval? How much hydrodynamic time is simulated?
Probably the simulation has finished. When you start the model, it generates a file named XBlog.txt. Open this file and check what is stored in the file. What was the total simulation time?
Inspect the initial bathymetries of each simulation with QUICKPLOT. Choose Files of type “NetCDF files and GRIB files” and open “xboutput.nc” in the simulation folder).
1. At what cross-shore position were the shoreface nourishment and beach nourishment placed?
2. What is the (average) thickness of the nourishments?
3. Is the volume of the nourishments comparable?
4. Plot the reference profile with markers; does the grid resolution vary in cross-shore direction?
Use the Quickplot tutorial and try to make an animation in which you plot short wave height (H), water level (including long wave variations, zs) and bed level (zb) as function of time.
Plot the offshore water level as function of time. Also open the file “tide.tek” (Tekal data files format), which contains the imposed surge level. Did the model correctly simulate the imposed surge level?
Inspect the final bathymetries of each simulation.
1. What is the dune face retreat in the three simulations you have carried out?
2. Where does the eroded sediment form the dunes deposit?
3. What nourishment type is most effective in reducing the impact of a storm and do you have an explanation for this?
In the folder “banquette” you find a final simulation in which a special beach nourishment type is evaluated named a banquette. This beach nourishment has a highly elevated flat area that connects to the dune foot on which beach restaurants can be build.
1. Run the model and compare in Quickplot the banquette design with the beach nourishment design we have evaluated before. Do you expect more or less erosion?
2. Check your hypothesis by comparing the final profile of the banquette simulation to the other simulations.
3. What would be your approach to further reduce beach and dune erosion?

Overwash at Santa Rosa Island , USA (2DH)¶

This case concerns overwash at Santa Rosa island in the Gulf of Mexico during hurricane Ivan in 2004.

You can work on the following assignments.

For the reference case open the params.txt in which you specify model input files and settings. Check the number of grid-points in x-direction (keyword: nx) and y-direction (keyword: ny). How many directional wave bins are defined and what is their width (keywords: thetamin, thetamax, dtheta).
In this simulation the grid is specified in Delft3D format. Open Quickin in the Delft 3D menu (Grid Quickin) and use the brief tutorial to read in the grid and bathymetry. Does the grid resolution vary in cross-shore direction? And in longshore direction? What are the minimum dx and dy? Why can the grid be coarse offshore?
How many wave conditions do we apply in this simulation? What is the offshore mean wave direction? Does the surge level change in the simulation?
What is the simulation time (hydrodynamic and morphologic)?
Inspect the model results and make an animation of the short wave height (H) and the water levels (including long wave, zs). Describe what is happening.
For the water levels set the color limits manual between -0.5 and 3.5.
Make an animation of cumulative sedimentation/erosion. Describe what is happening.
For the sedimentation/erosion set the color limits manual between -3 and 3
Look at the mean flow field. Plot the flow field in colored vectors. Where are the flow velocities highest and what is the direction of the flow (cross-shore or longshore)? Is there (also) a longshore current present and what is its intensity?

If you have time left feel free to:

Narrow or broaden the imposed spectrum by changing the parameter directional spreading (s) in ‘jonswap.inp’ (you could for example set s = 100 and s = 2 respectively). Make animations of the instantaneous short wave height to see what is happening to the size of the wave groups.
Design a nourishment in Quickin to reduce the impact of the storm on Santa Rosa Island. Change the depth file in params.txt to make a simulation with the updated bathymetry.

Yanchep perched beach and natural breakwater (2DH)¶

This case is an example of a beach 60km north of Perth most commonly known as Yanchep lagoon. Many beaches in WA like Yanchep are fronted by shallow reef and here we are investigating the effects of the reef on the morphodynamics.

You can work on the following assignments:

Go to the folder “Examples\YanchepBeach” and double click the file “run_model.bat”. The simulation will start (and will run about 15 minutes).
Meanwhile, inspect the bathymetry file and the structure file (using Quickin). What is the depth in the lagoon? Is the reef enclosing the lagoon below or above the model initial water level? What is the wave height at the boundary condition?
Use Quickplot and try to make an animation in which you plot short wave height (H), water level (including long wave variations) (zs) and Eulerian velocities (ue and ve) as function of time. What happens in the lagoon?
Use Quickplot and try to make an animation of cumulative sedimentation/erosion. What happens in the lagoon?
How is the lagoon affected by the mean water level? Increase or decrease the mean water level condition (‘tide.tx’), run the model again (maybe for a shorter time by reducing keyword: tstop). How are the circulation and sediment transport affected?
What would happen if the lagoon was open at the southern end? Open the structure file (keyword: ne_layer = reef.dep) with the Quickin tool and modify it to allow the southern end of the lagoon to be eroded. Modify the param.txt file to use this new structure file and run the model. Alternatively, remove the reef from the bathymetry and rerun the model without the structure file, by setting the keyword struct = 0.

If you still have time;

Reefs are very rough what happens in the model when the friction is increased? Reduce the Chezy roughness and increase the value of ${f}_{w}$ . Rerun the model what do you observe?
Is wave/current interaction (keyword: wci = 1) switched on? Rerun the model with the wave/current switch on/off. Compare the output with model you ran previously. How much effect do you see on the morphology?

Advanced model coefficients¶

In General the main input parameters and files required by XBeach to start a simulation are explained. It explained how the user can switch on and off specific processes and how the user can define the model initial and boundary conditions. XBeach offers, however, many more parameters to fine-tune the simulation of different processes. These parameters are listed in the following subsections grouped by process. Most parameters are not relevant for the average XBeach user. Parameters marked with a plus (+) are considered advanced options that are recommended to stay untouched unless you know what you are doing.

Wave numerics¶

The parameters listed in the table below involve the numerical aspects of the wave action balance that solves the wave propagation in the model. The keyword scheme can be used to set the numerical scheme. To overcome the undesired effects of steepening of wave groups we implemented a correction to the second-order upwind scheme according to ([BW76]), which implies a small additional diffusion term which is a function of time step and group velocity. By default Warming and Beam (1976) is used.

Table 24 Overview of available keywords related to wave numerics parameters¶
parameter	description	default	range	units
maxerror+	Maximum wave height error in wave stationary iteration	0.0005	1e-05 - 0.001	m
maxiter+	Maximum number of iterations in wave stationary	500	2 - 1000
scheme+	Numerical scheme for wave propagation	warmbeam	upwind_1, lax_wendroff, upwind_2, warmbeam
swkhmin	Minimum kh value to include in wave action balance, lower included in nlswe (default -1.d0)	-0.01	-0.01 - 0.35
wavint	Interval between wave module calls (only in stationary wave mode)	60.0	1.0 - 3600.0	s

Wave dissipation¶

The parameters listed in the table below involve the wave dissipation process. For instationary model runs use either break = roelvink1, roelvink2 or roelvink_daly. Note that the standard value gamma = 0.55 and n = 10 was calibrated for option break = roelvink1. For break = roelvink2 the wave dissipation is proportional to ${H}^{3}/h$ instead of ${H}^{2}$ ; this affects the calibration. For stationary runs the break = baldock option is suitable. The break = roelvink_daly option is a model in which waves start and stop breaking. Reducing gammax will reduce wave heights in very shallow water, probably 2 is a reasonable value.

Table 25 Overview of available keywords related to wave breaking parameters¶
parameter	description	default	range	units
alpha+	Wave dissipation coefficient in roelvink formulation	1.0	0.5 - 2.0
break	Type of breaker formulation	roelvink2	roelvink1, baldock, roelvink2, roelvink_daly, janssen
breakerdelay+	Switch to enable breaker delay model	1.0	0.0 - 3.0
delta+	Fraction of wave height to add to water depth	0.0	0.0 - 1.0
facrun	Calibration coefficient for short wave runup	1.0	0.0 - 2.0
facsd	Fraction of the local wave length to use for shoaling delay depth	1.0	0.0 - 2.0
fwcutoff	Depth greater than which the bed friction factor is not applied	1000.0	0.0 - 1000.0
gamma	Breaker parameter in baldock or roelvink formulation	0.55	0.4 - 0.9
gamma2	End of breaking parameter in roelvink daly formulation	0.3	0.0 - 0.5
gammax+	Maximum ratio wave height to water depth	2.0	0.4 - 5.0
n+	Power in roelvink dissipation model	10.0	5.0 - 20.0
shoaldelay	Switch to enable shoaling delay	0	0 - 1
wavfriccoef	Wave friction coefficient	-123
wavfricfile	Wave friction file			<file>

Rollers¶

The parameters listed in the table below involve the wave roller model. Using the roller model will give a shoreward shift in wave-induced setup, return flow and alongshore current. This shift becomes greater for lower beta values.

Table 26 Overview of available keywords related to roller parameters¶
parameter	description	default	range
beta+	Breaker slope coefficient in roller model	0.1	0.05 - 0.3
rfb+	Switch to feed back maximum wave surface slope in roller energy balance, otherwise rfb = par%beta	0	0 - 1
roller+	Switch to enable roller model	1	0 - 1

Wave-current interaction¶

The parameters listed in the table below involve the process of wave-current interaction. With the switch wci one can turn off or on the wave-current interaction, the wave current interaction will result in a feedback of currents on the wave propagation. On top of that, hwci limits the computation of wave-current interaction in very shallow water where the procedure may not converge.

Table 27 Overview of available keywords related to wave-current interaction parameters¶
parameter	description	default	range	units
cats+	Current averaging time scale for wci, in terms of mean wave periods	4.0	1.0 - 50.0	Trep
hwci+	Minimum depth until which wave-current interaction is used	0.1	0.001 - 1.0	m
hwcimax+	Maximum depth until which wave-current interaction is used	100.0	0.01 - 100.0	m
wci	Turns on wave-current interaction	0	0 - 1

Bed friction and viscosity¶

The parameters listed in the table below involve the settings for bed friction and viscosity influencing the flow in XBeach. The bed friction is influenced by the dimensionless friction coefficient cf or other formulation like the dimensional Chézy or Manning. The bed friction formulation applied needs to be determined with the keyword bedfriction. It is possible both to define one value (keyword: bedfriccoef) or to apply, spatially varying values for the bed friction. A spatial varying friction can be provided through an external file referenced via the keyword bedfricfile. The file has the same format as the bathymetry file explained in Grid and bathymetry.

The horizontal viscosity is composed of an overall background viscosity nuh and a viscosity depending on the roller dissipation tuned by nuhfac. In the alongshore direction the viscosity may be multiplied by a factor nuhv to account for additional advective mixing. It is also possible to use a user-defined value for the horizontal viscosity (keyword smag = 0)

Table 28 Overview of available keywords related to flow parameters¶
parameter	description	default	range	units
bedfriccoef	Bed friction coefficient	0.01	3.5e-05 - 0.9
bedfricfile	Bed friction file (only valid with values of c)			<file>
bedfriction	Bed friction formulation	chezy	chezy, cf, white-colebrook, manning, white-colebrook-grainsize
friction_acceleration	Turn on or off the effect of acceleration on bed roughness (morrison)	none	none, mccall, nielsen
friction_infiltration	Turn on or off the effect of infiltration on bed roughness (conley and inman)	0	0 - 1
friction_turbulence	Turn on or off the effect of turbulence on bed roughness (reniers van thiel)	0	0 - 1
gamma_turb	Calibration factor for turbulence contribution to bed roughness	1.0	0.0 - 2.0
nuh	Horizontal background viscosity	0.1	0.0 - 1.0	m^2s^-1
nuhfac+	Viscosity switch for roller induced turbulent horizontal viscosity	1.0	0.0 - 1.0
nuhv	Longshore viscosity enhancement factor, following svendsen (?)	1.0	1.0 - 20.0
smag+	Switch for smagorinsky subgrid model for viscocity	1	0 - 1

Flow numerics¶

The parameters listed in the table below involve the numerical aspects of the shallow water equations that solve the water motions in the model. Especially in very shallow water some processes need to be limited to avoid unrealistic behavior. For example hmin prevents very strong return flows or high concentrations and the eps determines whether points are dry or wet and can be taken quite small.

Table 29 Overview of available keywords related to flow numerics parameters¶
parameter	description	default	range	units
eps	Threshold water depth above which cells are considered wet	0.005	0.001 - 0.1	m
eps_sd	Threshold velocity difference to determine conservation of energy head versus momentum	0.5	0.0 - 1.0	m/s
hmin	Threshold water depth above which stokes drift is included	0.2	0.001 - 1.0	m
oldhu	Switch to enable old hu calculation	0	0 - 1
secorder+	Use second order corrections to advection/non-linear terms based on maccormack scheme	0	0 - 1
umin	Threshold velocity for upwind velocity detection and for vmag2 in equilibrium sediment concentration	0.0	0.0 - 0.2	m/s

Sediment transport¶

The parameters listed in the table below involve the process of sediment transport. The keywords facAs and facSk determine the effect of the wave form on the sediment transport, this is especially important in the nearshore. The facua is an alias setting in which both parameters can be varied at once. The wave form model itself is selected using the keyword waveform. Processes like short- and long-wave stirring and turbulence can be switched on or off using the keywords sws, lws and lwt. Several options for calibrating the sediment transport formulations are available as well as keywords to incorporate the bed slope effect.

Table 30 Overview of available keywords related to sediment transport parameters¶
parameter	description	default	range	units
BRfac+	Calibration factor surface slope	1.0	0.0 - 1.0
Tbfac+	Calibration factor for bore interval tbore: tbore = tbfac*tbore	1.0	0.0 - 1.0
Tsmin+	Minimum adaptation time scale in advection diffusion equation sediment	0.5	0.01 - 10.0	s
bdslpeffdir	Modify the direction of the sediment transport based on the bed slope	none	none, talmon
bdslpeffdirfac	Calibration factor in the modification of the direction	1.0	0.0 - 2.0
bdslpeffini	Modify the critical shields parameter based on the bed slope	none	none, total, bed
bdslpeffmag	Modify the magnitude of the sediment transport based on the bed slope, uses facsl	roelvink_total	none, roelvink_total, roelvink_bed, soulsby_total, soulsby_bed
bed+	Calibration factor for bed transports	1	0 - 1
bermslope	Swash zone slope for (semi-) reflective beaches	0.0	0.0 - 1.0
betad+	Dissipation parameter long wave breaking turbulence	1.0	0.0 - 10.0
bulk+	Switch to compute bulk transport rather than bed and suspended load separately	0	0 - 1
ci+	Mass coefficient in shields inertia term	1.0	0.5 - 1.5
cm	Mass coefficient in shields inertia term	1.5	0.0 - 3.0
dilatancy	Switch to reduce critical shields number due dilatancy	0	0 - 1
facAs+	Calibration factor time averaged flows due to wave asymmetry	0.1	0.0 - 1.0
facDc+	Option to control sediment diffusion coefficient	1.0	0.0 - 1.0
facSk+	Calibration factor time averaged flows due to wave skewness	0.1	0.0 - 1.0
facsl+	Factor bedslope effect	1.6	0.0 - 1.6
facua+	Calibration factor time averaged flows due to wave skewness and asymmetry	0.1	0.0 - 1.0
fallvelred	Switch to reduce fall velocity for high concentrations	0	0 - 1
form	Equilibrium sediment concentration formulation	vanthiel_vanrijn	soulsby_vanrijn, vanthiel_vanrijn, vanrijn1993
jetfac	Option to mimic turbulence production near revetments	0.0	0.0 - 1.0
lws+	Switch to enable long wave stirring	1	0 - 1
lwt+	Switch to enable long wave turbulence	0	0 - 1
phit+	Phase lag angle in nielsen transport equation	25.0	0.0 - 90.0
pormax	Max porosity used in the experession of van rhee	0.5	0.3 - 0.6
reposeangle	Angle of internal friction	30.0	0.0 - 45.0	deg
rheeA	A parameter in the van rhee expression	0.75	0.75 - 2.0
smax+	Maximum shields parameter for equilibrium sediment concentration acc. diane foster	-1.0	-1.0 - 3.0
sus+	Calibration factor for suspensions transports	1	0 - 1
sws+	Switch to enable short wave and roller stirring and undertow	1	0 - 1
tsfac+	Coefficient determining ts = tsfac * h/ws in sediment source term	0.1	0.01 - 1.0
turb+	Switch to include short wave turbulence	wave_averaged	none, wave_averaged, bore_averaged
turbadv+	Switch to activate turbulence advection model for short and or long wave turbulence	none	none, lagrangian, eulerian
waveform	Wave shape model	vanthiel	ruessink_vanrijn, vanthiel
z0+	Zero flow velocity level in soulsby and van rijn (1997) sediment concentration	0.006	0.0001 - 0.05	m

Sediment transport numerics¶

The parameters listed in the table below involve the numerical aspects of sediment transport that are all considered advanced options. For example the maximum allowed sediment concentration can be varied with the keyword cmax. It is however not recommended varying these settings.

Table 31 Overview of available keywords related to sediment transport numerics parameters¶
parameter	description	default	range
cmax+	Maximum allowed sediment concentration	0.1	0.0 - 1.0
sourcesink+	Switch to enable source-sink terms to calculate bed level change rather than suspended transport gradients	0	0 - 1
thetanum+	Coefficient determining whether upwind (1) or central scheme (0.5) is used.	1.0	0.5 - 1.0

Quasi-3D sediment transport¶

The parameters listed in the table below involve the tuning of quasi-3D sediment transport, if enabled. The most important setting is the kmax in which the user specifies the number of layers used in the quasi 3D sediment model.

Table 32 Overview of available keywords related to q3d sediment transport parameters¶
parameter	description	default	range
deltar	Estimated ripple height	0.025	0.001 - 1.0
kmax	Number of sigma layers in quasi-3d model; kmax = 1 is without vertical structure of flow and suspensions	100	1 - 1000
rwave	User-defined wave roughness adjustment factor	2.0	0.1 - 10.0
sigfac	Dsig scales with log(sigfac)	1.3	0.0 - 10.0
vicmol	Molecular viscosity	1e-06	0.0 - 0.001
vonkar	Von karman constant	0.4	0.01 - 1.0

Morphology¶

The parameters listed in the table below involve the morphological processes. The dryslp and wetslp keyword define the critical avalanching slope above and below water respectively. If the bed exceeds the relevant critical slope it collapses and slides downward (avalanching). To reduce the impact of these landslides the maximum bed level change due to avalanching is limited by the dzmax value. Which of the two slopes is applied to a grid cell is determined by the hswitch keyword.

The keyword morfac enables the user to decouple the hydrodynamical and the morphological time. This is suitable for situations where the morphological process is much slower than the hydrodynamic process. The factor defined by the morfac keyword is applied to all morphological change. A morfac = 10 therefore results in 10 times more erosion and deposition in a given time step than usual. The simulation time is however then shortened with the same factor to obtain an approximate result more quickly. The user can prevent the simulation time to be adapted to the morfac value by setting morfacopt to zero. The keywords morstart and morstop let the user enable the morphological processes in XBeach only for a particular period during the (hydrodynamic) simulation. These options can be useful if a spin-up time is needed for the hydrodynamics.

The struct and ne_layer keywords enable the user to specify non-erodible structures in the model. To switch on non-erodible structures use struct = 1. The location of the structures is specified in an external file referenced by the ne_layer keyword. The file has the same format as the bathymetry file explained in Grid and bathymetry. The values of the file define the thickness of the erodible layer on top of the non-erodible layer. A ne_layer file with only zeros therefore defines a fully non-erodible bathymetry and a file with only tens means a erodible layer of 10 meters. Only at the grid cells where the value in the ne_layer file is larger than zero erosion can occur. Non-erodible layers are infinitely deep and thus no erosion underneath these layers can occur.

Table 33 Overview of available keywords related to morphology parameters¶
parameter	description	default	range	units
dryslp	Critical avalanching slope above water (dz/dx and dz/dy)	1.0	0.1 - 2.0
dzmax+	Maximum bed level change due to avalanching	0.05	0.0 - 1.0	m/s/m
hswitch+	Water depth at which is switched from wetslp to dryslp	0.1	0.01 - 1.0	m
lsgrad	Factor to include longshore transport gradient in 1d simulationsdsy/dy=lsgrad*sy; dimension 1/length scale of longshore gradients	0.0	-0.1 - 0.1	1/m
morfac	Morphological acceleration factor	1.0	0.0 - 1000.0
morfacopt+	Switch to adjusting output times for morfac	1	0 - 1
morstart	Start time morphology, in morphological time	0.0	0.0 - par%tstop	s
morstop	Stop time morphology, in morphological time	2000.0	0.0 - 10000000.0	s
ne_layer	Name of file containing thickness of the erodible layer			<file>
struct	Switch for enabling hard structures	0	0 - 1
wetslp	Critical avalanching slope under water (dz/dx and dz/dy)	0.3	0.1 - 1.0

Bed update¶

The parameters listed in the table below involve the settings for the bed update process especially in the case multiple sediment fractions and bed layers are involved. The frac_dz, split and merge keywords determine the fraction of the variable bed layer thickness at which the layer is split or merged respectively with the surrounding bottom layers. The variable layer is chosen using the nd_var keyword.

Pre-defined bed evolution can be used with the keywords setbathy that is used to turn user-imposed bed level change on, nsetbathy that determines the number of imposed bed level conditions and setbathyfile that references a file that specifies the time series of imposed bed levels. If the setbathy option is used, XBeach will automatically interpolate the bed level at each computational time step from the imposed bed level time series.

The setbathyfile file must contain a time series (of length nsetbathy) of the bed level at every grid point in the XBeach model. The format of the setbathyfile file is such that each imposed bed level starts with the time at which the bed level should be applied (in seconds relative to the start of the simulation), followed on a new line by the bed level in each grid cell in an identical manner to the normal initial bathymetry file (e.g., one line per transect in the x-direction). The following imposed bed level condition starts on a new line with the time at which the condition is imposed and the bed levels at that time, etc. An example of a setbathyfile is given below.

setbathyfile.dep (generalized)

<t 0>
<z 1,1> <z 2,1> <z 3,1> ... <z nx,1> <z nx+1,1>
<z 1,2> <z 2,2> <z 3,2> ... <z nx,2> <z nx+1,2>
<z 1,3> <z 2,3> <z 3,3> ... <z nx,3> <z nx+1,3>

...

<z 1,ny> <z 2,ny> <z 3,ny> ... <z nx,ny> <z nx+1,ny>
<z 1,ny+1> <z 2,ny+1> <z 3,ny+1> ... <z nx,ny+1> <z nx+1,ny+1>
<t 1>
<z 1,1> <z 2,1> <z 3,1> ... <z nx,1> <z nx+1,1>
<z 1,2> <z 2,2> <z 3,2> ... <z nx,2> <z nx+1,2>
<z 1,3> <z 2,3> <z 3,3> ... <z nx,3> <z nx+1,3>

...

<z 1,ny> <z 2,ny> <z 3,ny> ... <z nx,ny> <z nx+1,ny>
<z 1,ny+1> <z 2,ny+1> <z 3,ny+1> ... <z nx,ny+1> <z nx+1,ny+1>

...

...

<t nsetbathy>
<z 1,1> <z 2,1> <z 3,1> ... <z nx,1> <z nx+1,1>
<z 1,2> <z 2,2> <z 3,2> ... <z nx,2> <z nx+1,2>
<z 1,3> <z 2,3> <z 3,3> ... <z nx,3> <z nx+1,3>

...

<z 1,ny> <z 2,ny> <z 3,ny> ... <z nx,ny> <z nx+1,ny>
<z 1,ny+1> <z 2,ny+1> <z 3,ny+1> ... <z nx,ny+1> <z nx+1,ny+1>

setbathyfile.dep (short 1D example, nx = 20, ny = 0, nsetbathy = 4)

  -20 -18 -16 -14 -12 -10 -8 -6 -5 -4 -3   -2   -1    -0.5  0    0.5 1   2   3   4   5
-20 -18 -16 -14 -12 -10 -8 -6 -5 -4 -3   -1.8 -0.75 -0.25 0    0.2 0.7 2   3   4   5
-20 -18 -16 -14 -12 -10 -8 -6 -5 -4 -2.5 -1.6 -0.5  -0.3  0    0.1 0.6 1.5 3   4   5
-20 -18 -16 -14 -12 -10 -8 -6 -5 -4 -2   -1.4 -0.3   0    0.05 0.1 0.6 1   2.2 3.5 5

Note that if the setbathy option is used, the initial bed level is derived from an interpolation of the setbathyfile file time series, not from the depfile file. It is strongly advised to turn of the computation of morphological updating (keyword: morphology = 0) if the setbathy option is used, as the computed morphological change will be overridden by the imposed morphological change.

Table 34 Overview of available keywords related to bed update numerics parameters¶
parameter	description	default	range	units
frac_dz+	Relative thickness to split time step for bed updating	0.7	0.5 - 0.98
merge+	Merge threshold for variable sediment layer (ratio to nominal thickness)	0.01	0.005 - 0.1
nd_var+	Index of layer with variable thickness	2	1 - par%nd
nsetbathy+	Number of prescribed bed updates	1	1 - 1000
setbathyfile*+	Name of prescribed bed update file			<file>
split+	Split threshold for variable sediment layer (ratio to nominal thickness)	1.01	1.005 - 1.1

Groundwater flow¶

The parameters listed in the table below involve the process of groundwater flow. The vertical permeability coefficient in the vertical can be set differently than that of the horizontal using the keywords kz and kz respectively. The initial bed level of the aquifer is read from an external file referenced by the aquiferbotfile keyword and the initial groundwater head can be set to either a uniform value using the gw0 keyword or to spatially varying values using an external file referenced by the gw0file keyword. Both files have the same format as the bathymetry file explained in Grid and bathymetry.

Table 35 Overview of available keywords related to groundwater parameters¶
parameter	description	default	range	units
aquiferbot+	Level of uniform aquifer bottom	-10.0	-100.0 - 100.0	m
aquiferbotfile+	Name of the aquifer bottom file			<file>
dwetlayer+	Thickness of the top soil layer interacting more freely with the surface water	0.1	0.01 - 1.0	m
gw0+	Level initial groundwater level	0.0	-5.0 - 5.0	m
gw0file+	Name of initial groundwater level file			<file>
gwReturb+	Reynolds number for start of turbulent flow in case of gwscheme = turbulent	100.0	1.0 - 600.0
gwfastsolve	Reduce full 2d non-hydrostatic solution to quasi-explicit in longshore direction	0	0 - 1
gwheadmodel+	Model to use for vertical groundwater head	parabolic	parabolic, exponential
gwhorinfil+	Switch to include horizontal infiltration from surface water to groundwater	0	0 - 1
gwnonh+	Switch to turn on or off non-hydrostatic pressure for groundwater	0	0 - 1
gwscheme+	Scheme for momentum equation	laminar	laminar, turbulent
kx+	Darcy-flow permeability coefficient in x-direction	0.0001	1e-05 - 0.1	ms^-1
ky+	Darcy-flow permeability coefficient in y-direction	0.0001	1e-05 - 0.1	ms^-1
kz+	Darcy-flow permeability coefficient in z-direction	0.0001	1e-05 - 0.1	ms^-1

Non-hydrostatic correction¶

The parameters listed in the table below involve the settings for the non-hydrostatic option (keyword: wavemodel = nonh). These are all considered advanced options and it is thus recommended not to change these

Table 36 Overview of available keywords related to non-hydrostatic correction parameters¶
parameter	description	default	range	units
Topt+	Absolute period to optimize coefficient	10.0	1.0 - 20.0	s
breakviscfac+	Factor to increase viscosity during breaking	1.5	1.0 - 3.0
breakvisclen+	Ratio between local depth and length scale in extra breaking viscosity	1.0	0.75 - 3.0
dispc+	Coefficient in front of the vertical pressure gradient	-1.0	0.1 - 2.0	?
kdmin+	Minimum value of kd (pi/dx > min(kd))	0.0	0.0 - 0.05
maxbrsteep+	Maximum wave steepness criterium	0.4	0.3 - 0.8
nhbreaker+	Non-hydrostatic breaker model	2	0 - 2
nhlay+	Layer distribution in the nonhydrostatic model, default = 0.33	0.33	0.0 - 1.0
nonhq3d	Turn on or off the the reduced two-layer nonhydrostatic model, default = 0	0	0 - 1
reformsteep+	Wave steepness criterium to reform after breaking	0.25d0*par%maxbrsteep	0.0 - 0.95d0*par%maxbrsteep
secbrsteep+	Secondary maximum wave steepness criterium	0.5d0*par%maxbrsteep	0.0 - 0.95d0*par%maxbrsteep
solver+	Solver used to solve the linear system	tridiag	sip, tridiag
solver_acc+	Accuracy with respect to the right-hand side usedin the following termination criterion: \|\|b-ax \|\| < acc*\|\|b\|\|	0.005	1e-05 - 0.1
solver_maxit+	Maximum number of iterations in the linear sip solver	30	1 - 1000
solver_urelax+	Underrelaxation parameter	0.92	0.5 - 0.99

Physical constants¶

The parameters listed in the table below involve physical constants used by XBeach. The gravitational acceleration and density of water are universally used coefficient. The depthscale is a factor in order to set different cut-off values like eps and hswitch. A value of the depthscale lower than one means the cut-off values will increase.

Table 37 Overview of available keywords related to physical constants¶
parameter	description	default	range	units
depthscale+	Depthscale of (lab)test simulated, affects eps, hmin, hswitch and dzmax	1.0	1.0 - 200.0
g	Gravitational acceleration	9.81	9.7 - 9.9	ms^-2
rho	Density of water	1025.0	1000.0 - 1040.0	kgm^-3

Coriolis force¶

The parameters listed in the table below involve the settings for incorporating the effect of Coriolis on the shallow water equations. The keywords are universally used coefficients.

Table 38 Overview of available keywords related to coriolis force parameters¶
parameter	description	default	range	units
lat+	Latitude at model location for computing coriolis	0.0	-90.0 - 90.0	deg
wearth+	Angular velocity of earth calculated as: 1/rotation_time (in hours)	1.d0/24.d0	0.0 - 1.0	hour^-1

MPI¶

The parameters listed in the table below involve the settings for parallelization of XBeach. When running XBeach in parallel mode, the model domain is subdivided in sub models and each sub model is then computed on a separate core. This will increase the computational speed of the model. The sub models only exchange information over their boundaries when necessary. The MPI parameters determine how the model domain is subdivided. The keyword mpiboundary can be set to auto, x, y or man. In auto mode the model domain is subdivided such that the internal boundary is smallest. In x or y mode the model domain is subdivided in sub models extending to either the full alongshore or the full cross-shore extent of the model domain. In man mode the model domain is manually subdivided using the values specified with the mmpi and nmpi keywords. The number of sub models is not determined by XBeach itself, but by the MPI wrapper (e.g. MPICH2 or OpenMPI). It is important to note that information about slopes isn’t exchanged over the boundaries. Therefore the avalanching algorithm will not function over MPI boundaries.

Table 39 Overview of available keywords related to mpi parameters¶
parameter	description	default	range
mmpi+	Number of domains in cross-shore direction when manually specifying mpi domains	2	1 - 100
mpiboundary+	Fix mpi boundaries along y-lines, x-lines, use manual defined domains or find shortest boundary automatically	auto	auto, x, y, man
nmpi+	Number of domains in alongshore direction when manually specifying mpi domains	4	1 - 100

Output projection¶

The parameters listed in the table below involve the projection of the model output. These settings do not influence the model results in anyway. The rotate keyword can be used to rotate the model output with an angle specified by the keyword alfa. The projection string can hold a string specifying the coordinate reference system used and is stored in the netCDF output file as metadata.

Table 40 Overview of available keywords related to output projection¶
parameter	description	default	range
projection+	Projection string
remdryoutput	Remove dry output points from output data of zs etc.	1	0 - 1
rotate	Rotate output as postprocessing with given angle	1	0 - 1

Numerical implementation¶

Grid set-up¶

The new implementation utilizes a curvilinear, staggered grid where depths, water levels, wave action and sediment concentrations are given in the cell centers (denoted by subscript z) and velocities and sediment fluxes at the cell interfaces (denoted by subscript u or v). In Fig. 16 the z, u, v and c (corner) points with the same numbering are shown. The grid directions are named s and n; grid distances are denoted by $\Delta s$ and $\Delta n$ , with subscripts referring to the point where they are defined. A finite-volume approach is utilized where mass, momentum and wave action are strictly conserved. In the middle panel of Fig. 16, the control volume for the mass balance is shown with the corresponding grid distances around the u- and v-points. The right panel explains the numbering of the fluxes Q and the volume V.

Fig. 16 Location of staggered grid points (left panel); definition of grid distances (middle) and terms in volume balance (right)

Wave action balance¶

Surfbeat solver¶

The time-varying wave action balance solved in XBeach is as follows:

(115) $\frac{\partial E}{\partial t} +\frac{\partial EC_{g,u} }{\partial s} +\frac{\partial EC_{g,v} }{\partial n} +\frac{\partial EC_{\vartheta } }{\partial \vartheta } =-Sink$

Where $E$ is the wave energy or wave action, $C_g$ is the group velocity, $C_{\vartheta}$ the refraction speed in theta-space and $Sink$ refers to effects of wave breaking and bottom friction.

Again, the advection terms are the only ones affected by the curvilinear scheme so we will discuss their treatment in detail. The control volume is the same as for the mass balance. In equation (116) the procedure to compute the wave energy fluxes across the cell boundaries is outlined. All variables should also have an index $itheta$ referring to the directional grid, but for brevity these are omitted here.

The component of the group velocity normal to the cell boundary, at the cell boundary, is interpolated from the two adjacent cell center points. Depending on the direction of this component, the wave energy at the cell boundary is computed using linear extrapolation based on the two upwind points, taking into account their grid distances. This second order upwind discretization preserves the propagation of wave groups with little numerical diffusion.

(116) $\begin{array}{l} {Cg_{u,u}^{i,j} =\frac{1}{2} (Cg_{u}^{i,j} +Cg_{u}^{i+1,j} )} \\ {E_{u}^{i,j} =E^{i,j} +\frac{1}{2} \Delta s_{u}^{i,j} \frac{E^{i,j} -E^{i-1,j} }{\Delta s_{u}^{i-1,j} } =} \\ {=\left(\left(\Delta s_{u}^{i-1,j} +\frac{1}{2} \Delta s_{u}^{i,j} \right)E^{i,j} -\frac{1}{2} \Delta s_{u}^{i,j} E^{i-1,j} \right)/\Delta s_{u}^{i-1,j} \, \, \, \, \, ,Cg_{u,u}^{i,j} >0} \\ {E_{u}^{i,j} =E^{i+1,j} -\frac{1}{2} \Delta s_{u}^{i,j} \frac{E^{i+2,j} -E^{i+1,j} }{\Delta s_{u}^{i+1,j} } =} \\ {=\left(\left(\Delta s_{u}^{i+1,j} +\frac{1}{2} \Delta s_{u}^{i,j} \right)E^{i+1,j} -\frac{1}{2} \Delta s_{u}^{i,j} E^{i+2,j} \right)/\Delta s_{u}^{i+1,j} \, \, \, \, \, ,Cg_{u,u}^{i,j} <0} \\ {} \\ {Flux_{u}^{i,j} =Cg_{u,u}^{i,j} E_{u}^{i,j} \Delta n_{u}^{i,j} } \end{array}$

The other three fluxes are computed in a similar way; for brevity we will not present all formulations.

The time integration is explicit and the same as in the original implementation. The advection in u- and v-direction is computed simply by adding the four fluxes and dividing by the cell area. This procedure guarantees conservation of wave energy.

The procedure for the roller energy balance is identical to that for the wave energy balance and will not be repeated here.

Stationary solver¶

In the stationary solver the wave energy and roller energy balances are solved line by line, from the seaward boundary landward. For each line the automatic timestep is computed and the quasi-time-dependent balance is solved until convergence or the maximum number of iterations is reached, after which the solver moves to the next line.

The iteration is controlled by the parameters maxiter and maxerror.

Shallow water equations¶

Mass balance equation¶

The mass balance reads as follows:

(117) $\frac{\partial V}{\partial t} =Q_{u}^{i,j} -Q_{u}^{i-1,j} +Q_{v}^{i,j} -Q_{v}^{i,j-1}$

This is discretized according to:

(118) $\begin{array}{l} {A_{z}^{i,j} \frac{z_{s}^{i,j,n+1} -z_{s}^{i,j,n} }{\Delta t} =u_{u}^{i,j,n+1/2} h_{u}^{i,j,n} \Delta n_{u}^{i,j} -u_{u}^{i-1,j,n+1/2} h_{u}^{i-1,j,n} \Delta n_{u}^{i-1,j} } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, +v_{v}^{i,j,n+1/2} h_{v}^{i,j,n} \Delta s_{v}^{i,j} -v_{v}^{i,j-1,n+1/2} h_{v}^{i,j-1,n} \Delta s_{v}^{i,j-1} } \end{array}$

Here, ${A}_{z}$ is the area of the cell around the cell centre, ${z}_{s}$ is the surface elevation, ${u}_{u}$ is the u-velocity in the u-point, ${h}_{u}$ the water depth in the u-point and ${v}_{v}$ the v-velocity in the v-point. The indices $i,j$ refer to the grid number in u resp. v direction; the index $n$ refers to the time step.

Momentum balance equation¶

Second, we will outline the derivation of the u-momentum balance. The control volume is given in Fig. 17. It is centeredaround the u-point. We now consider the rate of change of the momentum in the local u-direction as follows:

(119) $\frac{d\left(Vu\right)}{dt} -\sum Q_{in} u_{in} +\sum Q_{out} u +Vg\frac{\partial z_{s} }{\partial s} +A\frac{\tau _{b,u} }{\rho } =A\frac{\tau _{s,u} }{\rho } +A\frac{F_{u} }{\rho }$

where V is the cell volume, u the velocity in local grid direction, Q the fluxes, $\rho$ the density, g acceleration of gravity, $\tau_{b,u}, \tau_{b,u}, F_{u}$ the bed shear stress, wind shear stress and wave force in u-direction. We consider that the outgoing fluxes carry the velocity inside the cell, $u$ , and that ${u}_{in}$ is determined at each inflow boundary by interpolation, reconstructing the component in the same direction as $u$ .

The volume balance for the same volume reads:

(120) $\frac{dV}{dt} -\sum Q_{in} + \sum Q_{out} =0$

By multiplying the volume balance by $u$ , subtracting it from the momentum balance and dividing the result by $V$ we arrive at the following equation:

(121) $\frac{du}{dt} +\frac{\sum Q_{in} \left(u-u_{in} \right) }{Ah_{um} } +g\frac{\partial z_{s} }{\partial s} +\frac{\tau _{b,s} }{\rho h_{um} } =\frac{\tau _{s,u} }{\rho h_{um} } +\frac{F_{u} }{\rho h_{um} }$

where $A$ is the cell area and ${h}_{um}$ is the average depth of the cell around the u-point. The procedure for the second term (the others are straightforward) now boils down to integrating (only) the incoming fluxes over the interfaces and multiplying them with the difference between $u$ in the cell and the component of velocity in the same direction at the upwind cell.

Fig. 17 Control volume u-momentum balance and definition of fluxes

In the following three quations the procedure for computing the u-momentum balance is outlined. The discharges in the u-points are computed by multiplying the velocity in the u- or v-point by the water depth at that point. These discharges are then interpolated to the borders of the control volume around the u-point. The difference $\Delta \alpha$ in grid orientation between the incoming cell and the u-point is computed and used to compute the component of the incoming velocity in the local u-direction, from the left and right side of the control volume.

(122) $\begin{array}{rclcrcl} q_u^{i,j} &=& u_u^{i,j,n-1/2} h_u^{i,j,n} &~& ~ &~& ~ \\ q_{in}^{left} &=& \frac{1}{2} \left( q_u^{i,j} + q_u^{i-1,j} \right ) &,& q_{in}^{right} &=& -\frac{1}{2} \left( q_u^{i+1,j} + q_u^{i,j} \right ) \\ \Delta \alpha^{left} &=& \alpha_u^{i,j} - \alpha_u^{i-1,j} &,& \Delta \alpha^{right} &=& \alpha_u^{i+1,j} - \alpha_u^{i,j} \\ u_{in}^{left} &=& u_u^{i-1,j} \cos \left( \Delta \alpha \right ) - v_u^{i-1,j} \sin \left( \Delta \alpha \right ) &,& u_{in}^{right} &=& u_u^{i+1,j} \cos \left( \Delta \alpha \right ) - v_u^{i+1,j} \sin \left( \Delta \alpha \right ) \\ \end{array}$

The same is done for the top and bottom of the control volume, based on the discharges in v-direction:

(123) $\begin{array}{rclcrcl} q_v^{i,j} &=& v_v^{i,j,n-1/2} h_v^{i,j,n} &~& ~ &~& \\ q_{in}^{bottom} &=& \frac{1}{2} \left( q_v^{i,j-1} + q_v^{i+1,j-1} \right ) &,& q_{in}^{top} &=& -\frac{1}{2} \left( q_v^{i,j} + q_v^{i+1,j} \right ) \\ \Delta \alpha ^{bottom} &=& \alpha_u^{i,j} - \alpha_u^{i,j-1} &,& \Delta \alpha ^{top} &=& \alpha_u^{i,j+1} - \alpha_u^{i,j} \\ u_{in}^{bottom} &=& u_u^{i,j-1} \cos \left( \Delta \alpha \right ) - v_u^{i,j-1} \sin \left( \Delta \alpha \right ) &,& u_{in}^{top} &=& u_u^{i,j+1} \cos \left( \Delta \alpha \right ) - v_u^{i,j+1} \sin \left( \Delta \alpha \right ) \\ \end{array}$

Finally, the advective term in the momentum balance is given as:

(124) $\begin{array}{l} {\left(\frac{\sum Q_{in} \left(u-u_{in} \right) }{Ah_{um} } \right)^{i,j} =\, {max(q}_{in}^{left} ,0){(}u_{u}^{i,j} {-}u_{in}^{left} {)}\frac{\Delta n_{z}^{i,j} }{h_{um}^{i,j} A_{u}^{i,j} } \, \, \, } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, +\, \, {max(q}_{in}^{right} ,0){(}u_{u}^{i,j} {-}u_{in}^{right} {)}\frac{\Delta n_{z}^{i+1,j} }{h_{um}^{i,j} A_{u}^{i,j} } \, } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, +\, {max(q}_{in}^{bottom} ,0){(}u_{u}^{i,j} {-}u_{in}^{bottom} {)}\frac{\Delta s_{c}^{i,j-1} }{h_{um}^{i,j} A_{u}^{i,j} } } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, +\, \, {max(q}_{in}^{top} ,0){(}u_{u}^{i,j} {-}u_{in}^{top} {)}\frac{\Delta s_{c}^{i,j} }{h_{um}^{i,j} A_{u}^{i,j} } \, } \end{array}$

Time integration scheme¶

The time integration of the mass and momentum balance equations is combined in an explicit leap-frog scheme, as depicted in Fig. 18. The velocities (in the ’-’ points) are updated using the momentum balance, the water levels are updated using the mass balance. The water level gradients influence the momentum balance and the velocities and derived discharges affect the mass balance. Because of the leap-frog scheme these influences are always computed at the half time step level, which makes the scheme second order accurate.

Fig. 18 Leap-frog time integration scheme

Using this straightforward finite volume approach, complicated transformations of the equations are avoided and the solution scheme remains transparent. It is also completely compatible with the original rectilinear implementation and is even slightly more efficient.

Groundwater flow¶

In order to solve the equations in xx, the spatial and temporal domain of the groundwater system is split into the same spatial grid and time steps as the XBeach surface water model it is coupled to. At each time step in the numerical model, the depth average groundwater head is calculated in the center of the groundwater cells, and the fluxes (specific discharge, submarine exchange, infiltration and exfiltration) are calculated on the cell interfaces

At the start of the time step, every cell is evaluated whether the groundwater and surface water are connected:

(125) $\kappa _{i,j} =\eta _{gw,i,j} \ge z_{b,i,j} -\varepsilon \wedge \eta _{i,j} \ge z_{b,i,j} +\varepsilon$

In equation (125) $\varepsilon$ is a numerical smoothing constant used to deal with numerical round off errors near the bed (equal to the parameter dwetlayer in the case of hydrostatic groundwater flow, and to the parameter eps in the case of non-hydrostatic groundwater flow), and $i$ and $j$ represent cross-shore and longshore coordinates in the numerical solution grid, respectively. Infiltration is calculated in cells where the groundwater and surface water are not connected and there exists surface water. As shown in equation (68) the infiltration rate is a function of the thickness of the wetting front, which is zero at the start of infiltration, and increases as a function of the infiltration rate. The equations for the infiltration rate and the thickness of the wetting front are approximated by first-order schemes, in which the wetting front is updated using a backward-Euler scheme, which ensures numerical stability:

(126) $\begin{array}{c} {S_{inf,i,j}^{n} =K_{i,j} \left(\frac{1}{\rho g} \frac{\left. p_{lpf,i,j} \right|^{z=z_{b} } }{\delta _{infill,i,j}^{n} } 1\right)} \\ {{\; }\delta _{infill,i,j}^{n} =\delta _{infill,i,j}^{n-1} +S_{inf,i,j}^{n} \frac{\Delta t}{n_{p} } } \end{array}$

In the pressure term ${p}_{lpf}$ is the surface water pressure at the bed, which in the case of non-hydrostatic surface water flow is high-pass filtered at $4/{T}_{rep}$ , the superscript $n$ corresponds to the time step number and $\Delta t$ is the size of the time step. The infiltration rate in the coupled relationship can be solved through substitution:

(127) $S_{inf,i,j}^{n} =\frac{-\delta _{infill,i,j}^{n-1} +\frac{\Delta t}{n_{p} K_{i,j} } +\sqrt{\left(\delta _{infill,i,j}^{n-1} \right)^{2} +\frac{2}{n_{p} } \frac{\Delta t}{\delta _{infill,i,j}^{n-1} K_{i,j} } +\frac{4\rho g}{n_{p} } \frac{\Delta t}{p_{i,j} \left|^{z=\xi } \right. K_{i,j} } +\frac{\Delta t^{2} }{K_{_{i,j} }^{2} } } }{\frac{2\Delta t}{n_{p} } }$

At the end of infiltration, i.e. when the groundwater and surface water become connected or there is no surface water left, the wetting front thickness is reset to zero. If the infiltration rate exceeds the Reynolds number for the start of turbulence, the local hydraulic conductivity is updated using the local Reynolds number:

(128) $K_{i,j} =K_{lam} \sqrt{\frac{Re_{crit} }{\max (Re_{i,j} ,Re_{crit} )} }$

XBeach iterates until a minimum threshold difference between iterations is found for and . Infiltration in one time step is limited to the amount of surface water available in the cell and to the amount of water required to raise the groundwater level to the level of the bed:

(129) $S_{inf,i,j}^{n} =\min \left(S_{inf,i,j}^{n} ,\frac{\eta _{i,j-} z_{b,i,j} }{\Delta t} ,\frac{z_{b,i,j} -\eta _{gw,i,j-} }{n_{p} \Delta t} \right)$

If during infiltration the groundwater level reaches the bed level, the fraction of the time step required to do so is estimated (x) and the remaining fraction is used in the submarine exchange.

(130) $\begin{array}{c} {\Lambda _{t,i,j} =\frac{n_{p} \left(z_{b,i,j} -\eta _{gw,i,j} \right)}{K_{i,j} \left(\frac{1}{\rho g} \frac{\left. p_{i,j} \right|^{z=\xi } }{z_{b,i,j} -\eta _{gw,i,j} } +1\right)} \frac{1}{\Delta t} } \\ {} \\ {0\le \Lambda _{t,i,j} \le 1} \end{array}$

Exfiltration is calculated in cells where the groundwater and surface water are not connected and the groundwater level exceeds the bed level:

(131) $S_{exf,i,j} =n_{p} \frac{z_{b,i,j} -\eta _{gw,i,j} }{\Delta t}$

Horizontal infiltration and exfiltration (keyword: gwhorinfil = 1) is computed across the numerical vertical interface between the groundwater domain and the surface water domain in adjoining cells. The direction of exchange is determined by the head gradient, and the bed level slope direction:

(132) $\begin{array}{l} {S_{i,j,hor}^{u} =\left\{\begin{array}{cc} {K_{_{i,j} }^{u} \left(\frac{\eta _{i,j} +\frac{q_{i,j} }{g} -H_{gw,i+1,j} }{\Delta x} \right)\left(z_{b,i+1,j} -z_{b,i,j} \right)} & {z_{b,i+1,j} >z_{b,i,j} \wedge \eta _{i,j} >eps} \\ {K_{_{i,j} }^{u} \left(\frac{\eta _{i+1,j} +\frac{q_{i+1,j} }{g} -H_{gw,i,j} }{\Delta x} \right)\left(z_{b,i,j} -z_{b,i+1,j} \right)} & {z_{b,i+1,j} <z_{b,i,j} \wedge \eta _{i+1,j} >eps} \end{array}\right. } \\ {S_{i,j,hor}^{v} =\left\{\begin{array}{cc} {K_{_{i,j} }^{v} \left(\frac{\eta _{i,j} +\frac{q_{i,j} }{g} -H_{gw,i,j+1} }{\Delta y} \right)\left(z_{b,i,j+1} -z_{b,i,j} \right)} & {z_{b,i,j+1} >z_{b,i,j} \wedge \eta _{i,j} >eps} \\ {K_{_{i,j} }^{v} \left(\frac{\eta _{i,j+1} +\frac{q_{i,j+1} }{g} -H_{gw,i,j} }{\Delta y} \right)\left(z_{b,i,j} -z_{b,i,j+1} \right)} & {z_{b,i,j+1} <z_{b,i,j} \wedge \eta _{i,j+1} >eps} \end{array}\right. } \end{array}$

After infiltration and exfiltration have been calculated, the groundwater level and surface water level are updated:

(133) $\begin{array}{l} {\eta _{gw,i,j}^{n+\frac{1}{2} } =\eta _{gw,i,j}^{n} +\frac{\Delta t}{n_{p} } \left(S_{\inf } +S_{exf} +S_{hor} \right)} \\ {\eta _{i,j}^{n+\frac{1}{2} } =\eta _{i,j}^{n} +\Delta t\left(-S_{\inf } -S_{exf} -S_{hor} \right)} \end{array}$

All updated cells are subsequently re-evaluated on whether the surface water and groundwater are connected or unconnected

The cell height at the center of the groundwater cells ( $\Delta z_{H,i,j}$ ) is calculated from the groundwater level and the bottom of the aquifer in the center of the cell, whereas the cell heights at the horizontal cell interfaces are calculated using an upwind procedure:

(134) $\begin{array}{c} {\Delta z_{H,i,j} =\eta _{gw,i,j} -z_{aquifier,i,j} =h_{gw,i,j} } \\ {} \\ {\Delta z_{u,i,j} =\left\{\begin{array}{l} {\Delta z_{H,i+1,j} {\; \; if\; }q_{qw,i,j}^{x} <0} \\ {\Delta z_{H,i,j} {\; \; \; \; if\; }q_{qw,i,j}^{x} \ge 0} \end{array}\right. } \\ {\Delta z_{v,i,j} =\left\{\begin{array}{l} {\Delta z_{H,i+1,j} {\; \; if\; }q_{qw,i,j}^{y} <0} \\ {\Delta z_{H,i,j} {\; \; \; \; if\; }q_{qw,i,j}^{y} \ge 0} \end{array}\right. } \end{array}$

In ${z}_{aquifer}$ is the level of the bottom of the aquifer. As described in Sediment transport, the head applied on the top boundary of the groundwater domain ( ${H}_{bc}$ ) depends on whether the groundwater and surface water are connected or unconnected:

(135) $\begin{array}{l} {H_{bc,i,j} =\left(1-\kappa _{r,i,j} \right)\eta _{gw,i,j} +\kappa _{r,i,j} \left(z_{b,i,j} {+}\frac{\left. p_{i,j} \right|^{z=z_{b} } }{\rho g} \right)} \\ {\begin{array}{ccc} {\kappa _{r,i,j} =1-\frac{z_{b,i,j} -\eta _{gw,i,j} }{\varepsilon } } & {} & {0\le \kappa _{r,i,j} \le 1} \end{array}} \end{array}$

In the parameter $\kappa_{r}$ is the relative numerical ‘connectedness’ of the groundwater and surface water head, determined by linear interpolation across the numerical smoothing constant $\varepsilon$ .

In the case of hydrostatic groundwater flow, the groundwater head in each cell is set equal to the head applied on the top boundary of the groundwater domain ( ${H}_{bc}$ ) and the horizontal groundwater flux is computed from the groundwater head gradient:

(136) $\begin{array}{l} {q_{gw,i,j}^{x} =-K_{u,i,j} \Delta z_{u,i} \frac{H_{i+1,j} -H_{i,j} }{\Delta x_{u,i,j} } =-K_{u,i,j} \Delta z_{u,i} \frac{H_{bc,i+1,j} -H_{bc,i,j} }{\Delta x_{u,i,j} } } \\ {q_{gw,i,j}^{y} =-K_{v,i,j} \Delta z_{v,i} \frac{H_{i+1,j} -H_{i,j} }{\Delta y_{v,i,j} } =-K_{v,i,j} \Delta z_{v,i} \frac{H_{bc,i+1,j} -H_{bc,i,j} }{\Delta y_{v,i,j} } } \end{array}$

In the superscripts $x$ and $y$ refer to the components of the variable in the cross-shore and longshore direction, respectively, and the subscripts $u$ and $v$ refer to variables approximated at the horizontal cell interfaces in the cross-shore and longshore direction, respectively.

In the case of non-hydrostatic groundwater flow, the horizontal specific discharge on each cell interface can be found through an approximation of the non-hydrostatic groundwater head gradient:

(137) $\begin{array}{l} {q_{gw,i,j}^{x} =-K_{u,i,j} \Delta z_{u,i} \frac{H_{i+1,j} -H_{i,j} }{\Delta x_{u,i,j} } =-K_{u,i,j} \Delta z_{u,i} \frac{H_{bc,i+1,j} -\frac{2}{3} \beta _{i+1,j} \Delta z_{H,i+1,j}^{2} -H_{bc,i,j} +\frac{2}{3} \beta _{i,j} \Delta z_{H,i,j}^{2} }{\Delta x_{u,i,j} } } \\ {q_{gw,i,j}^{y} =-K_{v,i,j} \Delta z_{v,i} \frac{H_{i+1,j} -H_{i,j} }{\Delta y_{v,i,j} } =-K_{v,i,j} \Delta z_{v,i} \frac{H_{bc,i+1,j} -\frac{2}{3} \beta _{i+1,j} \Delta z_{H,i+1,j}^{2} -H_{bc,i,j} +\frac{2}{3} \beta _{i,j} \Delta z_{H,i,j}^{2} }{\Delta y_{v,i,j} } } \end{array}$

In the subscript $H$ refers to variables approximated at the cell centers. The hydraulic conductivity may be different at each cell interface and is therefore computed at every interface where every K is calculated separately.

Continuity in the groundwater cell is found following:

(138) $q_{gw,i-1,j}^{x} -q_{gw,i,j}^{x} +q_{gw,i,j-1}^{y} -q_{gw,i,j}^{y} -q_{gw,i,j}^{z} =0$

In the variable ${q}^{z}$ refers to the vertical groundwater discharge (e.g., submarine exchange if connected to the surface water, or groundwater level fluctuations if the groundwater is not connected to the surface water).

In the case of hydrostatic groundwater flow, the variable ${q}^{z}$ can be solved through the known variables ${q}^{x}$ and ${q}^{y}$ . However, in the case of non-hydrostatic groundwater flow, all variables in contain an unknown value for the groundwater pressure head, described in terms of a known head at the surface of the groundwater ( ${H}_{bc}$ ) and the unknown curvature of the vertical groundwater head function ( $\beta$ ). Since water is incompressible, the groundwater pressure must be solved for all cells simultaneously using matrix algebra:

(139) $Ax+b=0$

In A is a matrix containing coefficients for the horizontal and vertical specific discharge, x is a vector containing the unknown groundwater head curvature, and b contains the known forcing terms. For a one dimensional cross-shore case, A is reduced to a tridiagonal matrix. The vector of known forcing consists of the numerical gradients in the contribution of the head applied on the top boundary of the groundwater domain to the horizontal specific discharge.

In the one dimensional case, the solution to the tridiagonal matrix A can be computed using the efficient Thomas algorithm ([Tho49]). In the two dimensional case, matrix A contains two additional diagonals that are not placed along the main diagonal, and vector b contains additional forcing terms from the alongshore contribution. The solution to the two dimensional case requires a more complex and less computationally efficient matrix solver. In this case the Strongly Implicit Procedure ([Sto68]) is used in a manner similar to [ZSS11]. The horizontal and vertical groundwater fluxes are calculated using the solution of x plus and .

Since in both hydrostatic and non-hydrostatic groundwater flow some local velocities may exceed the critical Reynolds number for the start of turbulence ( ${Re}_{crit}$ ), the turbulent hydraulic conductivity ( $K$ ) is updated using the local Reynolds number. The solution to and the update of the turbulent hydraulic conductivity are iterated until a minimum threshold difference between iterations is found. Note that this approach is only used is the turbulent groundwater model is selected (keyword: gwscheme = turbulent).

The (iterated) solution for the specific vertical discharge is used to update the groundwater level and surface water level:

(140) $\begin{array}{rcl} {\eta _{_{gw,i,j} }^{n+1} } & {=} & {\left\{\begin{array}{l} {\eta _{_{gw,i,j} }^{n+\frac{1}{2} } {\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }C_{i,j} } \\ {\eta _{_{gw,i,j} }^{n+\frac{1}{2} } +\frac{\Delta t}{n_{p} } \frac{q_{i,j}^{z} }{\Delta x_{H,i,j} } {\; \; \; }\neg C_{i,j} {\; \; \; }} \end{array}\right. } \\ {\eta _{_{i,j} }^{n+1} } & {=} & {\left\{\begin{array}{l} {\eta _{_{i,j} }^{n+\frac{1}{2} } +\Delta t\frac{q_{i,j}^{z} }{\Delta x_{H,i,j} } {\; \; \; \; \; }C_{i,j} } \\ {\eta _{_{i,j} }^{n+\frac{1}{2} } {\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }\neg C_{i,j} } \end{array}\right. } \end{array}$

If the groundwater and surface water are connected, and the submarine exchange from the surface water to the groundwater estimated in is greater than the amount of surface water available in the cell, continuity is enforced by lowering the groundwater level to compensate for the lack of permeating water:

(141) $\eta _{_{gw,i,j} }^{n+1} =\eta _{_{gw,i,j} }^{n+1} +\frac{1}{n_{p} } \left(\eta _{_{i,j} }^{n+\frac{1}{2} } -z_{b,i,j} -\Delta t\frac{q_{i,j}^{z} }{\Delta x_{H,i,j} } \right){\; if\; \; }\kappa _{i,j} \wedge \eta _{i,j}^{n+\frac{1}{2} } -z_{b,i,j} <\Delta t\frac{q_{i,j}^{z} }{\Delta x_{H,i,j} }$

Sediment transport¶

The advection-diffusion equation for suspended sediment is the basis for the sediment transport computations in XBeach. The partial differential equation to solve is:

(142) $\frac{\partial hc}{\partial t} +\nabla \vec{S}_{s} =\frac{hc_{eq} -hc}{T_{s} } =-\frac{hc}{T_{s} } +Ero$

Here $c$ is the depth-averaged concentration, ${c}_{eq}$ the equilibrium concentration, $Ts$ a typical timescale proportional to water depth divided by fall velocity. As is often done to increase robustness, we treat the erosion term explicitly but take an implicit scheme for the sedimentation term:

(143) $\frac{{c}_{{z}}^{{i,j,n+1}} h_{h}^{i,j,n+1} -h_{h}^{i,j,n} {c}_{z}^{{i,j,n}} }{\Delta {t}} {\; =\; }-\frac{h_{h}^{i,j,n+1} {c}_{z}^{{i,j,n+1}} }{{T}_{{s}}^{{i,j}} } -\left(\nabla \vec{S}_{s} \right)^{i,j,n} {+\; \; }Ero^{i,j,n}$

This can be rewritten as:

(144) ${c}_{{c}}^{{i,j,n+1}} {\; =\; }\frac{{T}_{{s}}^{{i,j}} \Delta {t}}{h_{h}^{i,j,n+1} \left({T}_{{s}}^{{i,j}} +\Delta {t}\right)} \left(\frac{h_{h}^{i,j,n} {c}_{{c}}^{{i,j,n}} }{\Delta t} -\left(\nabla \vec{S}_{s} \right)^{i,j,n} {+}Ero^{i,j,n} \right)$

The sediment transport gradient is discretized in a similar way as the mass balance:

(145) $\left(\nabla \vec{S}_{s} \right)^{i,j} =\frac{\left(S_{u,s}^{i,j} \Delta n_{u}^{i,j} -S_{u,s}^{i-1,j} \Delta n_{u}^{i-1,j} +S_{v,s}^{i,j} \Delta s_{v}^{i,j} -S_{v,s}^{i,j-1} \Delta s_{v}^{i,j-1} \right)}{A_{z}^{i,j} }$

The sediment transports in the u- points contain an advective term, a diffusive term and a bed slope term:

(146) ${S}_{{u,s}} {=c}_{u} {u}_{{rep,s}} {h}_{{u}} {-D}_{{c}} {h}_{{u}} \frac{\partial c}{\partial s} {-f}_{slope} {c}_{{u}} \left|v\right|_{u} {h}_{{u}} \frac{\partial z_{b} }{\partial s} {)}$

Here ${u}_{rep,s}$ is a representative velocity for suspended transport, which contains contributions due to return flow, wave skewness and wave asymmetry; ${D}_{c}$ is a horizontal diffusion coefficient and ${f}_{slope}$ a coefficient. In discretized form the expression for the suspended transport in the u-point is:

(147) $S_{u,s}^{i,j} = c_u^{i,j} u_{rep,s}^{i,j} h_u^{i,j} -D_c^{i,j} h_u^{i,j} \frac{c_c^{i+1,j} -c_c^{i,j} }{\Delta s_u^{i,j} } -f_{slope} c_u^{i,j} \left|v\right|_u^{i,j} h_u^{i,j} \frac{z_b^{i+1,j} -z_b^{i,j} }{\Delta s_u^{i,j} } )$

The concentrations in the u-points are computed with a $\theta$ -method, where $\theta=1$ means a fully upwind approximation, and $\theta=0.5$ a central scheme. In practice, we mostly use the upwind approximation for its robustness.

(148) $\begin{array}{l} {c_{u}^{i,j} =\theta c_{z}^{i,j} +\left(1-\theta \right)c_{z}^{i+1,j} \, \, \, \, \, \, \, ,u_{rep,s}^{i,j} >0} \\ {c_{u}^{i,j} =\left(1-\theta \right)c_{z}^{i,j} +\theta c_{z}^{i+1,j} \, \, \, \, \, \, \, ,u_{rep,s}^{i,j} <0} \end{array}$

The erosion and deposition terms, which may also be used in the bed updating, are finally computed from:

(149) $\begin{array}{l} {Ero^{i,j,n} {=\; h}_{{h}}^{i,j,n} {c}_{{eq}}^{{i,j,n}} {/T}_{{s}}^{{i,j}} } \\ {Depo^{i,j} ={h}_{{h}}^{i,j,n+1} {c}_{z}^{{i,j,n+1}} {/T}_{{s}}^{{i,j}} } \end{array}$

The evaluation of the bedload transport takes place in the same way as in the previous versions of XBeach, except for the fact that the directions are taken in local grid direction, and will not be repeated here.

Bottom updating schemes¶

Two alternative formulations are available for the bed updating: one where the bottom changes are computed based on the gradients of suspended and bed load transport, equation (150) , and one where the changes due to suspended transport are accounted for through the erosion and deposition terms, equation (151).

(150) $\left(1-n_{p} \right)\frac{\partial z_{b} }{\partial t} +MF\left(\nabla \overrightarrow{S}_{s} +\nabla \overrightarrow{S}_{b} \right)=0$

(151) $\left(1-n_{p} \right)\frac{\partial z_{b} }{\partial t} +MF\left(Ero-Depo+\nabla \overrightarrow{S}_{b} \right)=0$

In both cases $MF$ is the morphological factor used to accelerate morphological changes. In the first case, the sediment in the bottom is conserved in all cases, but changes in the amount of sediment in the water are not considered; one can also say that the sediment in suspension is added to the bottom sediment. In the second case, the storage of sediment in the water is accounted for, but will be distorted in cases of high $MF$ . Since under most circumstances the real effect of the storage in the water phase is small we prefer the first formulation which guarantees mass conservation in the bottom. Both formulations are calculated using an explicit scheme:

(152) $\Delta z_{b} ^{i,j,n+1} =\Delta t\frac{MF}{\left(1-n_{p} \right)} \left(\left(\nabla \overrightarrow{S}_{s} \right)^{i,j,n} +\left(\nabla \overrightarrow{S}_{b} \right)^{i,j,n} \right)$

(153) $\Delta z_{b} ^{i,j,n+1} =\Delta t\frac{MF}{(1-n_{p} )} \left(Ero^{i,j,n} -Depo^{i,j,n} +\left(\nabla \overrightarrow{S}_{b} \right)^{i,j,n} \right)$

Avalanching¶

XBeach implements avalanching as described in section Avalanching. It first calculates bed level change due to avalanching in the cross-shore dimension. It then calculates the slopes in alongshore direction and bottom change due to avalanching in this direction. To avoid disrupted sediment balance XBeach does not calculate bottom change due to avalanching at the boundary grid cells. Consequently XBeach cannot calculate avalanching at the boundary between two MPI domains.

Bed composition¶

The bed is discretized into layers with mass $M(i,j)$ in which $i$ refers to the layer number and $j$ to the sediment class. The mass fraction per sediment class $p$ , layer thickness $\Delta$ and bed level ${z}_{b}$ are defined by:

(154) $\begin{array}{l} {p(i,j)=\frac{M(i,j)}{\sum _{j=1}^{J}M(i,j) } } \\ {\Delta (i)=\frac{1}{\left(1-n_{p} \right)\rho _{s} } \sum _{j=1}^{J}M(i,j) } \\ {z_{b} =z_{0} +\sum _{i=1}^{I}\Delta (i) } \end{array}$

with porosity ${n}_{p}$ and sediment density $\rho$ . The level ${z}_{0}$ is the lowest point of the array of bed layers.

Due to bed load transport, sediment is exchanged between the top layer and the four horizontally neighboring top layers. Exchange with the water column and the top layer is due to erosion rate $E$ and deposition rate $D$ . A mixed Eulerian/Lagrangian framework is proposed. Within the set of layers, one layer is defined as the variable layer. This is the only layer that has a variable total mass. All other layers have a constant total mass, which implies for a constant porosity a constant thickness. Above the variable layer, the layers move with the bed level (Lagrangian): upwards in case of aggradation and downwards in case of degradation. This vertical movement gives an advective flux with advection velocity equal to the bed level change $A=dz/dt$ . The variable layer is the transition to the lower layers, which are passive. The number of layers below the variable layer has thus no influence on the computation time. Note that diffusive processes within the bed are not considered yet. These could lead to fluxes between the layers below the variable layer.

The mass balance for the top layer can now be defined by:

(155) $\begin{array}{l} {\frac{\partial M(1,j)}{\partial t} =dy\left\{p(1,j)S_{b} (j)\right\}_{W} -dy\left\{p(1,j)S_{b} (j)\right\}_{E} } \\ {\quad \quad \quad \quad dx\left\{p(1,j)S_{b} (j)\right\}_{S} -dx\left\{p(1,j)S_{b} (j)\right\}_{N} } \\ {\quad \quad \quad \quad -dxdyp(1,j)E(j)+dxdyD(j)+dxdy\delta Ap_{bot} (1,j)} \\ {} \\ {A=\sum _{j=1}^{J}\left\{\begin{array}{l} {dy\left\{p(1,j)S_{b} (j)\right\}_{W} -dy\left\{p(1,j)S_{b} (j)\right\}_{E} } \\ {\quad \quad dx\left\{p(1,j)S_{b} (j)\right\}_{S} -dx\left\{p(1,j)S_{b} (j)\right\}_{N} } \\ {\quad \quad -dxdyp(1,j)E(j)+dxdyD(j)} \end{array}\right\} } \end{array}$

in which ${S}_{b}$ is the bed-load transport (e.g. Meyer-Peter-Muller), based on the sediment properties of the specific class. The dimensions of the grid cell are defined by $dx$ and $dy$ . The subscripts W, E, S and N refer to West, East, South and North indicating the four vertical faces of the bed cell. The horizontal faces are indicated with bot for the bottom of the cell and ceil for the ceiling of the cell. As the fraction p is not defined at the faces but in the cell centers, the upstream fraction is required. For the bed load fluxes, the velocity direction used. For the vertical advection term, the upstream value is based on the bed level variation: in case of aggradation the value in the top layer is used and in case of degradation the value of the second layer is used. If the top layer is the variable layer, there is no advective flux: $\delta=0$ otherwise $\delta=1$ .

The mass balance for the layers in between the top layer and the variable layer is:

(156) $\frac{\partial M(i,j)}{\partial t} =dxdyA\left(p_{bot} (i,j)-p_{ceil} (i,j)\right)$

and for the variable layer, it reads:

(157) $\frac{\partial M(i,j)}{\partial t} =Ap_{ceil} (i,j)$

In order to avoid too thin a or too thick a variable layer, the variable layer is merged or split. If the thickness is smaller than the critical value $\Delta_{merge}$ , the variable layer is merged with the lower layer. To keep the same number of cells, a cell is added at the bottom of the array, implying that ${z}_{0} = z_{0} - \Delta$ . Similarly, the variable is split into two layers if the critical value $\Delta_{split}$ is exceeded. Then, the array is shifted upwards: ${z}_{0} = z_{0} + \Delta$ .

As the bed level update is explicit, the timestep is limited. A conservative estimate can be made by assuming that no more mass can be eroded than available in the top layer:

(158) $\begin{array}{l} {dy\left\{p(1,j)S_{b} (j)\right\}_{W/E} +dx\left\{p(1,j)S_{b} (j)\right\}_{S/N} +dxdyp(1,j)E(j)<\frac{M(1,j)}{dt} } \\ {dt<\frac{dxdy\left(1-n_{p} \right)\rho _{s} \Delta (1)}{dyS_{b,W/E} (1)+dyS_{b,N/S} (1)+dxdyE(1)} } \end{array}$

The transport rate depends on the direction of the transport. The transport rates and erosion rates should be based on the formulation for the smallest fraction: $j=1$ . Note that the fraction p falls out. This time step restriction is less severe than the one for shallow water flow. Only in case of very top layers and/or the use of a morphological factor, this time step restriction might be relevant.

Boundary conditions¶

At the start of the XBeach simulation, XBeach checks whether non-stationary varying wave boundary conditions are to be used. If this is the case, it next checks whether the wave spectrum of the wave boundary conditions is to change over time, or remain constant. If the wave spectrum is to remain constant, XBeach will only read from one input file to generate wave boundary conditions. If the wave spectrum is to vary in time, XBeach reads from multiple files.

Whether or not the wave spectrum of the boundary conditions changes over time, the XBeach module requires a record length during which the current wave spectral parameters are applied. For the duration of the record length, boundary conditions are calculated at every boundary condition file time step. These time steps are not required to be the same as the time steps in the XBeach main program; XBeach will interpolate where necessary. The boundary condition time steps should therefore only be small enough to accurately describe the incoming bound long waves. The statistical data for the generation of the wave boundary conditions is read from user-specified files. The XBeach module tapers the beginning and end of the boundary condition file. This is done to ensure smooth transitions from one boundary condition file to the next.

The combination of a large record length and a small time step lead to large demands on the system memory. If the memory requirement is too large XBeach will shut down. The user must choose to either enlarge the boundary condition time step, or to reduce the record length. In case of the latter, several boundary condition files can be generated and read sequentially. It is unwise however to reduce the record length too much, as then the transitions between the boundary condition files may affect the model results.

Every time the XBeach wave boundary condition module is run, it outputs data to the local directory. Metadata about the wave boundary conditions are stored in list files: ebcflist.bcf and qbcflist.bcf. The main XBeach program uses the list files to know how and when to read and generate boundary condition files. The actual incoming short-wave energy and long-wave mass flux data is stored in other files. These files have E_ and q_ prefixes. The main XBeach program uses these files for the actual forcing along the offshore edge.

Non-hydrostatic¶

Global continuity equation¶

As was outlined in the previous chapter the global continuity equation, which describes the relation between the free surface and the depth averaged discharge, is given by

(159) $\frac{\partial \eta }{\partial t} +\frac{\partial }{\partial x} \left(UH\right)+\frac{\partial }{\partial x} \left(VH\right)=0$

A simple semi-discretisation of using central differences for the space derivative and using the Hansen scheme for the coupling between velocity and free surface results in

(160) $\frac{\eta_{i,j}^{\*} - \eta_{i,j}^{n}}{\Delta t} + \frac{{}^{x} q_{i+{\tfrac{1}{2}},j}^{\*} - {}^{x} q_{i-{\tfrac{1}{2}},j}^{\*}}{\Delta x} + \frac{{}^{y} q_{i,j+{\tfrac{1}{2}}}^{\*} - {}^{y} q_{i,j-{\tfrac{1}{2}}}^{\*}}{\Delta y} = 0$

With ${}^{x} q_{i+{\tfrac{1}{2}} ,j}^{\*} = H_{i+{\tfrac{1}{2}} ,j}^{n} U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} }$ , ${}^{y} q_{i,j+{\tfrac{1}{2}} }^{\*} =H_{i,j+{\tfrac{1}{2}} }^{n} V_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} }$ and the water depth is defined by a first order accurate upwind interpolation

(161) $H_{i+{\tfrac{1}{2}} ,j}^{n} =\left\{\begin{array}{c} {\zeta _{i,j}^{n} +d_{i,j} {\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }if{\; }U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } >0} \\ {\zeta _{i+1,j}^{n} +d_{i+1,j} {\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }if{\; }U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } <0} \\ {\max \left(\zeta _{i,j}^{n} ,\zeta _{i+1,j}^{n} \right)+\min \left(d_{i} ,d_{i+1,j} \right){\; }if{\; }U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } =0} \end{array}\right. {\; }$

The resulting scheme is only first order accurate by virtue of the upwind interpolations and mass conservative. When first order computations are considered accurate enough $\eta_{i,j}^{n+1}$ is set to $\eta_{i,j}^{n\*}$ . For higher order accuracy the first order prediction is corrected using a limited version of the McCormack scheme. The corrector step reads

(162) $\frac{\eta _{i,j}^{n+1} -\eta _{i,j}^{*} }{\Delta t} +\frac{{}^{x} \Delta q_{i+{\tfrac{1}{2}} ,j}^{*} -{}^{x} \Delta q_{i-{\tfrac{1}{2}} ,j}^{*} }{\Delta x} +\frac{{}^{y} \Delta q_{i,j+{\tfrac{1}{2}} }^{*} -{}^{y} \Delta q_{i,j-{\tfrac{1}{2}} }^{*} }{\Delta y} =0$

With ${}^{x} \Delta q_{i+{\tfrac{1}{2}} ,j}^{n\*} = U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } \Delta H_{i+{\tfrac{1}{2}} ,j}$ and $\Delta H_{i+{\tfrac{1}{2}} ,j}$ is given for positive flow as

(163) $\Delta H_{i+{\tfrac{1}{2}} ,j}^{n\*} ={\tfrac{1}{2}} \psi \left(^{\zeta } r_{i+{\tfrac{1}{2}} } \right)\left(\zeta _{i+1,j}^{n\*} -\zeta _{i,j}^{n} \right){\; \; \; \; \; \; \; \; \; \; }^{\zeta } {r}_{{i+}{\tfrac{{1}}{{2}}} } =\frac{\zeta _{i,j}^{n\*} -\zeta _{i-1,j}^{n} }{\zeta _{i+1,j}^{n\*} -\zeta _{i,j}^{n} } {\; \; \; \; \; \; \; \; \; \; }\psi \left(r\right)=\max \left(0,\min \left(r,1\right)\right)$

Here $\psi \left(r\right)$ denotes the minmod limiter. Similar expression can be constructed for negative flow. The expression for ${}^{y} \Delta q_{i,j+{\tfrac{1}{2}} }^{n\*}$ and $\Delta H_{i,j+{\tfrac{1}{2}} }$ are obtained in a similar manner. Note that the total flux ${}^{x} q_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} }$ at the cell boundaries thus reads

(164) ${}^{x} q_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } ={}^{x} q_{i+{\tfrac{1}{2}} ,j}^{\*} +{}^{x} \Delta q_{i+{\tfrac{1}{2}} ,j}^{\*} {,\; \; \; \; }{}^{y} q_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } ={}^{y} q_{i,j+{\tfrac{1}{2}} }^{\*} +{}^{y} \Delta q_{i,j+{\tfrac{1}{2}} }^{\*} {\; }$

The predictor-corrector set is second order accurate in regions where the solution is smooth, and reduces locally to first order accuracy near discontinuities. Furthermore, the method remains mass conservative. Note that other flux limiters can be used instead of the minmod limiter. However, as the minmod limiter performed adequately, this has not been investigated. (for an overview of flux limiters see [Hir07])

Local continuity equation¶

The depth averaged local continuity equation is given by

(165) $\frac{\partial HU}{\partial x} +\frac{\partial HV}{\partial y} +\left. w\right|_{z=\eta } -\left. u\frac{\partial z}{\partial x} \right|_{z=-\zeta } -\left. v\frac{\partial z}{\partial y} \right|_{z=-\zeta } =0$

This equation is discretized using central differences

(166) $\frac{H_{i+{\tfrac{1}{2}} ,j}^{n+1} U_{i+{\tfrac{1}{2}} ,j}^{n+1{\tfrac{1}{2}} } -H_{i-{\tfrac{1}{2}} ,j}^{n+1} U_{i-{\tfrac{1}{2}} ,j}^{n+1{\tfrac{1}{2}} } }{\Delta x} +\frac{H_{i+{\tfrac{1}{2}} ,j}^{n+1} V_{i,j+{\tfrac{1}{2}} }^{n+1{\tfrac{1}{2}} } -H_{i,j-{\tfrac{1}{2}} }^{n+1} V_{i,j-{\tfrac{1}{2}} }^{n+1{\tfrac{1}{2}} } }{\Delta y} +w_{i,j,s}^{n+1{\tfrac{1}{2}} } -\bar{U}_{i,j}^{n+1{\tfrac{1}{2}} } \frac{\eta _{i+{\tfrac{1}{2}} ,j}^{n+1} -\eta _{i-{\tfrac{1}{2}} ,j}^{n+1} }{\Delta x} -\bar{V}_{i,j}^{n+1{\tfrac{1}{2}} } \frac{\eta _{i,j+{\tfrac{1}{2}} }^{n+1} -\eta _{i,j-{\tfrac{1}{2}} }^{n+1} }{\Delta x} =0$

Missing grid variables $\eta _{i+{\tfrac{1}{2}} ,j}^{n+1} ,\eta _{i,j+{\tfrac{1}{2}} }^{n+1}$ are approximated with upwind interpolation. Because there is no separate time evolution equation for the pressure the local continuity equation will be used to setup a discrete set of poison type equations in which the pressures are the only unknown quantities.

Horizontal Momentum¶

To obtain a conservative discretisation of the momentum equation the approach from [SZ03] is followed. However, to improve the accuracy of the method the combined space-time discretisation of the advection is done using a variant of the [Mac69] is used. This scheme consists of a first order predictor step and a flux limited corrector step. The hydrostatic pressure is integrated using the midpoint rule and central differences, while the source terms and the turbulent stresses are integrated using an explicit Euler time integration. Formally the time integration is therefore first order accurate, but in regions where the turbulent stresses are negligible the scheme is of almost second order accuracy.

The depth averaged horizontal momentum equation for $HU$ is given by

(167) $\frac{\partial }{\partial t} \left(HU\right)+\frac{\partial }{\partial x} \left(HU^{2} +{\tfrac{1}{2}} gH^{2} +H\bar{p}-\tau _{xx} \right)+\frac{\partial }{\partial y} \left(HUV-\tau _{yx} \right)=gH\frac{\partial d}{\partial x} -p\frac{\partial d}{\partial x} +S_{x}$

A first order accurate predictor step in time and space is then given as

(168) $\begin{array}{l} {\frac{\left(HU\right)_{i+{\tfrac{1}{2}} ,j}^{*} -\left(HU\right)_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\Delta t} +\frac{{}^{x} \bar{q}_{i+1,j}^{n+{\tfrac{1}{2}} } U_{i+1,j}^{n+{\tfrac{1}{2}} } -{}^{x} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } U_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta x} +\frac{{}^{y} \bar{q}_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } U_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } -{}^{y} \bar{q}_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } U_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } }{\Delta y} } \\ {+g\frac{\left(H^{2} \right)_{i+1,j}^{n+1} -\left(H^{2} \right)_{i,j}^{n+1} }{2\Delta x} =g\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n} \frac{d_{i+{\tfrac{1}{2}} ,j} -d_{i-{\tfrac{1}{2}} ,j} }{\Delta x} +{}^{x} {Pr}_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } +{}^{x} {S}_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } +{}^{x} {T}_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } } \end{array}$

Here Pr represents a discretisation of the dynamic pressure; T the effect of (turbulent) viscosity and S includes all other source terms. The discretisation of the (turbulent) viscous terms is given by central differences:

(169) $\begin{array}{rcl} {{}^{x} T_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } } & {=} & {\frac{2}{\Delta x_{i+{\tfrac{1}{2}} } } \left[\nu _{i+1,j}^{n} H_{i+1,j}^{n+1} \frac{U_{i+1{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } -U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\Delta x_{i+1} } -\nu _{i,j}^{n} H_{i,j}^{n+1} \frac{U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } -U_{i-{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\Delta x_{i} } \right]} \\ {} & {} & {+\frac{1}{\Delta y_{i} } \left[\bar{\bar{\nu }}_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \bar{\bar{H}}_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \frac{U_{i+{\tfrac{1}{2}} ,j+1}^{n+1} -U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\Delta y_{i+{\tfrac{1}{2}} } } -\bar{\bar{\nu }}_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \bar{\bar{H}}_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{n+1} \frac{U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } -U_{i+{\tfrac{1}{2}} ,j-1}^{n+{\tfrac{1}{2}} } }{\Delta y_{i-{\tfrac{1}{2}} } } \right]} \\ {} & {} & {+\frac{1}{\Delta y_{i} } \left[\bar{\bar{\nu }}_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \bar{\bar{H}}_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \frac{V_{i+1,j+{\tfrac{1}{2}} }^{n+1} -V_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } }{\Delta x_{i+{\tfrac{1}{2}} } } -\bar{\bar{\nu }}_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \bar{\bar{H}}_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{n+1} \frac{V_{i+1,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } -V_{i,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } }{\Delta x_{i+{\tfrac{1}{2}} } } \right]} \end{array}$

Here $\bar{\bar{\nu }}_{i+{\tfrac{1}{2}} ,j+1}^{n+{\tfrac{1}{2}} }$ and $\bar{\bar{H}}_{i+{\tfrac{1}{2}} ,j+1{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} }$ are obtained from the surrounding points by simple linear interpolation.

Due to the incompressible flow assumption the dynamic pressure does not have a separate time evolution equation, but instead it satisfies an elliptical equation in space. As such its effect cannot be calculated explicitly using values at the previous time level. However to improve the accuracy of the predictor step the effect of the dynamic pressure is included explicitly. To do this first the unknown pressure is decomposed as:

(170) $p_{i,j}^{n+1{\tfrac{1}{2}} } =p_{i,j}^{n+{\tfrac{1}{2}} } +\Delta p_{i,j}^{n+1{\tfrac{1}{2}} }$

where the difference in pressure $\Delta p_{i,j}^{n+1{\tfrac{1}{2}} }$ is generally small. In the predictor step the effect of the pressure is included explicitly using $p_{i,j}^{n+{\tfrac{1}{2}} }$ . In the corrector step the full Poisson equation is then solved for $\Delta p_{i,j}^{n+1{\tfrac{1}{2}} }$ . The pressure term in the predictor step is thus given as

(171) ${}^{{x}} {Pr}_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } {=}\frac{H_{i+1,j}^{n+1} \bar{p}_{i+1,j}^{n+{\tfrac{1}{2}} } -H_{i,j}^{n+1} \bar{p}_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta x} -p_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } \frac{d_{i+{\tfrac{1}{2}} ,j} -d_{i-{\tfrac{1}{2}} ,j} }{\Delta x} =\frac{\left(\eta _{i+1,j}^{n+1} +d_{i,j}^{n+1} \right)p_{i+1,j}^{n+{\tfrac{1}{2}} } -\left(\eta _{i,j}^{n+1} -d_{i+1,j}^{n+1} \right)p_{i,j}^{n+{\tfrac{1}{2}} } }{2\Delta x}$

Here $\bar{p}_{i+1,j}^{n+{\tfrac{1}{2}} }$ represents the average pressure over the vertical which is approximated with $\bar{p}_{i+1,j}^{n+{\tfrac{1}{2}} } ={\tfrac{1}{2}} p_{i+1,j}^{n+{\tfrac{1}{2}} }$ , in which $p_{i+1,j}^{n+{\tfrac{1}{2}} }$ is the pressure at the bottom. Furthermore $p_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} }$ is given as $p_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } ={\tfrac{1}{2}} \left(p_{i+1,j}^{n+{\tfrac{1}{2}} } +p_{i,j}^{n+{\tfrac{1}{2}} } \right)$ .

Currently is formulated with the depth integrated momentum as the primitive variable, and not the depth averaged velocity. To reformulate in terms of $U$ we use the method by [SZ03]. First note that $\left(HU\right)_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} }$ and $\left(HU\right)_{i+{\tfrac{1}{2}} ,j}^{\*}$ are approximated as $\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n} U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} }$ and $\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} U_{i+{\tfrac{1}{2}} ,j}^{\*}$ . Now using $\left(HU\right)_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} }$ is equivalent to:

(172) $\left(HU\right)_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } =\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } -U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } \Delta t\frac{{}^{x} \bar{q}_{i+1,j}^{n+{\tfrac{1}{2}} } -{}^{x} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta x} -U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } \Delta t\frac{{}^{y} \bar{q}_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } -{}^{y} \bar{q}_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } }{\Delta y}$

(173) $\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} ={\tfrac{1}{2}} \left(H_{i+1,j}^{n+1} +H_{i,j}^{n+1} \right){,\; \; }{}^{x} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } ={\tfrac{1}{2}} \left({}^{x} q_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } +{}^{x} q_{i-{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } \right){,\; \; \; }{}^{y} \bar{q}_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } ={\tfrac{1}{2}} \left({}^{y} q_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } +{}^{y} q_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \right)$

Substituting into the full expressions (including those for $V_{i,j+{\tfrac{1}{2}} }^{\*}$ ) become:

(174) $\begin{array}{r} {\frac{U_{i+{\tfrac{1}{2}} ,j}^{*} -U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\Delta t} +\frac{{}^{x} \bar{q}_{i+1,j}^{n+{\tfrac{1}{2}} } U_{i+1,j}^{n+{\tfrac{1}{2}} } -{}^{x} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } U_{i,j}^{n+{\tfrac{1}{2}} } }{\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} \Delta x} +\frac{{}^{y} \bar{q}_{i,j+1}^{n+{\tfrac{1}{2}} } U_{i,j+1}^{n+{\tfrac{1}{2}} } -{}^{y} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } U_{i,j}^{n+{\tfrac{1}{2}} } }{\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} \Delta y} -\frac{U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} } \frac{{}^{x} \bar{q}_{i+1,j}^{n+{\tfrac{1}{2}} } -{}^{x} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta x} } \\ {-\frac{U_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} } \frac{{}^{y} \bar{q}_{i,j+1}^{n+{\tfrac{1}{2}} } -{}^{y} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta y} +g\frac{\eta _{i+1,j}^{n+1} -\eta _{i,j}^{n+1} }{\Delta x} =\frac{^{{x}} {Pr}_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } +{}^{x} {S}_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } +{}^{x} {T}_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} } } \\ {\frac{V_{i,j+{\tfrac{1}{2}} }^{*} -V_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } }{\Delta t} +\frac{{}^{y} \bar{q}_{i,j+1}^{n+{\tfrac{1}{2}} } V_{i,j+1}^{n+{\tfrac{1}{2}} } -{}^{y} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } V_{i,j}^{n+{\tfrac{1}{2}} } }{\bar{H}_{i,j+{\tfrac{1}{2}} }^{n+1} \Delta y} +\frac{{}^{x} \bar{q}_{i+1,j}^{n+{\tfrac{1}{2}} } V_{i+1,j}^{n+{\tfrac{1}{2}} } -{}^{x} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } V_{i,j}^{n+{\tfrac{1}{2}} } }{\bar{H}_{i,j+{\tfrac{1}{2}} }^{n+1} \Delta x} -\frac{V_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\bar{H}_{i,j+{\tfrac{1}{2}} }^{n+1} } \frac{{}^{y} \bar{q}_{i,j+1}^{n+{\tfrac{1}{2}} } -{}^{y} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta y} } \\ {\frac{V_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } }{\bar{H}_{i,j+{\tfrac{1}{2}} }^{n+1} } \frac{{}^{x} \bar{q}_{i+1,j}^{n+{\tfrac{1}{2}} } -{}^{x} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta x} +g\frac{\eta _{i,j+1}^{n+1} -\eta _{i,j}^{n+1} }{\Delta x} =\frac{{}^{y} {Pr}_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } +{}^{y} {S}_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } +{}^{y} {T}_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } }{\bar{H}_{i,j+{\tfrac{1}{2}} }^{n+1} } } \end{array}$

Where we again use a first order upwind interpolation for $U_{i+1,j}^{n+{\tfrac{1}{2}} }$ and $U_{i,j+1}^{n+{\tfrac{1}{2}} }$ . This is exactly the approximation used by [SZ03] and is fully momentum conservative.

The predictor step is first order accurate in both space and time due to the use of upwind approximations for and Euler explicit time integration for the advective terms, and first order time integration for the source/viscous terms. This level of accuracy is acceptable near shore, where strong non-linearity (wave breaking, flooding and drying) will force the use of small steps in space and time anyway. However, in the region where waves only slowly change (e.g. shoaling/refraction on mild slopes), the first order approximations suffer from significant numerical damping. To improve the accuracy of the numerical model in these regions a corrector step is implemented after the predictor step.

The corrector step is given by:

(175) $\begin{array}{l} {\frac{\left(HU\right)_{i+{\tfrac{1}{2}} ,j}^{n+1{\tfrac{1}{2}} } -\left(HU\right)_{i+{\tfrac{1}{2}} ,j}^{n*} }{\Delta t} +\frac{{}^{x} \bar{q}_{i+1,j}^{n+{\tfrac{1}{2}} } \Delta U_{i+1,j}^{} -{}^{x} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } \Delta U_{i,j}^{} }{\Delta x} +\frac{{}^{y} \bar{q}_{i,j+1}^{n+{\tfrac{1}{2}} } \Delta U_{i,j+1}^{} -{}^{y} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } \Delta U_{i,j}^{} }{\Delta y} +} \\ {\frac{\left(\eta _{i+1,j}^{n+1} +d_{i,j}^{n+1} \right)\Delta p_{i+1,j}^{n+1{\tfrac{1}{2}} } -\left(\eta _{i,j}^{n+1} -d_{i+1,j}^{n+1} \right)\Delta p_{i,j}^{n+1{\tfrac{1}{2}} } }{2\Delta x} =0} \end{array}$

Or, when formulated in terms of the depth averaged velocity

(176) $\begin{array}{r} {\frac{U_{i+{\tfrac{1}{2}} ,j}^{n+1{\tfrac{1}{2}} } -U_{i+{\tfrac{1}{2}} ,j}^{n*} }{\Delta t} +\frac{{}^{x} \bar{q}_{i+1,j}^{n+{\tfrac{1}{2}} } \Delta U_{i+1,j}^{} -{}^{x} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } \Delta U_{i,j}^{n*} }{\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} \Delta x} +\frac{{}^{y} \bar{q}_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \Delta U_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{} -{}^{y} \bar{q}_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \Delta U_{i+{\tfrac{1}{2}} ,j-{\tfrac{1}{2}} }^{} }{\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} \Delta y} +...} \\ {...+\frac{\left(\eta _{i+1,j}^{n+1} +d_{i,j}^{n+1} \right)\Delta p_{i+1,j}^{n+1{\tfrac{1}{2}} } -\left(\eta _{i,j}^{n+1} -d_{i+1,j}^{n+1} \right)\Delta p_{i,j}^{n+1{\tfrac{1}{2}} } }{2\bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} \Delta x} =0} \\ {\frac{V_{i,j+{\tfrac{1}{2}} }^{n+1{\tfrac{1}{2}} } -V_{i,j+{\tfrac{1}{2}} }^{n*} }{\Delta t} +\frac{{}^{y} \bar{q}_{i,j+1}^{n+{\tfrac{1}{2}} } \Delta V_{i,j+1}^{} -{}^{y} \bar{q}_{i,j}^{n+{\tfrac{1}{2}} } \Delta V_{i,j}^{} }{\bar{H}_{i,j+{\tfrac{1}{2}} }^{n+1} \Delta y} +\frac{{}^{x} \bar{q}_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \Delta V_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{} -{}^{y} \bar{q}_{i-{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \Delta V_{i-{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }^{} }{\bar{H}_{i,j+{\tfrac{1}{2}} }^{n+1} \Delta y} +...} \\ {...+\frac{\left(\eta _{i,j+1}^{n+1} +d_{i,j}^{n+1} \right)\Delta p_{i,j+1}^{n+1{\tfrac{1}{2}} } -\left(\eta _{i,j}^{n+1} -d_{i,j+1}^{n+1} \right)\Delta p_{i,j}^{n+1{\tfrac{1}{2}} } }{2\bar{H}_{i,j+{\tfrac{1}{2}} }^{n+1} \Delta x} =0} \end{array}$

The values of $\Delta U_{i+1,j}^{n\*}$ are obtained from slope limited expressions. For positive flow these read:

(177) $\Delta U_{i,j}^{n*} ={\tfrac{1}{2}} \psi \left(^{{u}} {r}_{{i+}{\tfrac{{1}}{{2}}} }^{} \right)\left(U_{i+{\tfrac{1}{2}} ,j}^{*} -U_{i-{\tfrac{1}{2}} ,j}^{n} \right){\; \; \; \; \; }^{{u}} {r}_{{i+}{\tfrac{{1}}{{2}}} }^{} =\frac{U_{i-{\tfrac{1}{2}} ,j}^{*} -U_{i-1{\tfrac{1}{2}} ,j}^{n} }{U_{i+{\tfrac{1}{2}} ,j}^{*} -U_{i-{\tfrac{1}{2}} ,j}^{n} } {\; \; \; \; \; \; \; \; if\; q}_{{i},j}^{{n+}{\tfrac{{1}}{{2}}} } >0$

Where $\psi$ again denotes the minmod limiter. Similar expressions can be constructed for $\Delta U_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }$ , $\Delta V_{i,j}$ and $\Delta V_{i+{\tfrac{1}{2}} ,j+{\tfrac{1}{2}} }$ .

The predictor-corrector set is second order accurate in regions where the solution is smooth, and reduces to first order accuracy near sharp gradients in the solutions to avoid unwanted oscillations. Furthermore, the method remains momentum conservative.

Vertical momentum equations¶

The vertical momentum equation is discretized in a similar manner to the horizontal momentum equations using the McCormack scheme. In terms of the depth averaged vertical velocity the predictor step is:

(178) $\begin{array}{r} {\frac{\bar{W}_{i,j}^{*} -\bar{W}_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta t} +\frac{{}^{x} q_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } \bar{W}_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } -{}^{x} q_{i-{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } \bar{W}_{i-{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{H_{i,j}^{n+1} \Delta x} -\frac{\bar{W}_{i,j}^{n+{\tfrac{1}{2}} } }{H_{i,j}^{n+1} } \frac{{}^{x} q_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } -{}^{x} q_{i-{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } }{\Delta x} +\frac{{}^{y} q_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \bar{W}_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } -{}^{y} q_{i,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \bar{W}_{i,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } }{H_{i,j}^{n+1} \Delta y} } \\ {-\frac{\bar{W}_{i,j}^{n+{\tfrac{1}{2}} } }{H_{i,j}^{n+1} } \frac{{}^{y} q_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } -{}^{y} q_{i,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } }{\Delta y} +\frac{p_{i,j,1}^{n+{\tfrac{1}{2}} } -p_{i,j,0}^{n+{\tfrac{1}{2}} } }{H_{i,j}^{n+1} } =\frac{{}^{w} {S}_{i,j}^{n+{\tfrac{1}{2}} } +{}^{w} {T}_{i,j}^{n+{\tfrac{1}{2}} } }{H_{i,j}^{n+1} } } \end{array}$

The pressures are defined on the cell faces and therefore do not have to be interpolated. Furthermore, we can exactly set the dynamic pressure at the free surface $p_{i,j,1}^{n+{\tfrac{1}{2}} }$ to zero. The vertical velocities are defined on the cell faces and therefore the depth averaged velocity $\bar{W}_{i,j}^{n+{\tfrac{1}{2}} }$ needs to be expressed in terms of the bottom and surface velocities. Using a simple central approximation gives

(179) $\bar{W}_{i,j}^{n+{\tfrac{1}{2}} } ={\tfrac{1}{2}} \left(w_{i,j,1}^{n+{\tfrac{1}{2}} } +w_{i,j,0}^{n+{\tfrac{1}{2}} } \right),{\; \; \; \; \; \; }\bar{W}_{i,j}^{*} ={\tfrac{1}{2}} \left(w_{i,j,1}^{*} +w_{i,j,0}^{*} \right)$

At the bottom the kinematic boundary condition is used for the vertical velocity:

(180) $w_{i,j,0}^{*} ={\tfrac{1}{2}} \left(U_{i+{\tfrac{1}{2}} ,j}^{*} +U_{i-{\tfrac{1}{2}} ,j}^{*} \right)\frac{d_{i+{\tfrac{1}{2}} ,j} -d_{i-{\tfrac{1}{2}} ,j} }{\Delta x_{i} } +{\tfrac{1}{2}} \left(V_{i,j+{\tfrac{1}{2}} }^{*} +V_{i,j-{\tfrac{1}{2}} }^{*} \right)\frac{d_{i,j+{\tfrac{1}{2}} } -d_{i,j-{\tfrac{1}{2}} } }{\Delta x_{j} }$

Horizontal interpolation of $\bar{W}_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} }$ and $\bar{W}_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} }$ is done using first order upwind similar to . The turbulent stresses are again approximated using a central scheme as

(181) $\begin{array}{rcl} {{}^{w} T_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } } & {=} & {\frac{1}{\Delta x_{i} } \left[{}^{x} \bar{\nu }_{i+{\tfrac{1}{2}} ,j}^{n} {}^{x} \bar{H}_{i+{\tfrac{1}{2}} ,j}^{n+1} \frac{W_{i+1,j}^{n+{\tfrac{1}{2}} } -W_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta x_{i+{\tfrac{1}{2}} } } -{}^{x} \bar{\nu }_{i-{\tfrac{1}{2}} ,j}^{n} {}^{x} \bar{H}_{i-{\tfrac{1}{2}} ,j}^{n+1} \frac{W_{i,j}^{n+{\tfrac{1}{2}} } -W_{i-1,j}^{n+{\tfrac{1}{2}} } }{\Delta x_{i+{\tfrac{1}{2}} } } \right]} \\ {} & {} & {+\frac{1}{\Delta y_{i} } \left[{}^{y} \bar{\nu }_{i,j+{\tfrac{1}{2}} }^{n} {}^{y} \bar{H}_{i,j+{\tfrac{1}{2}} }^{n+1} \frac{W_{i,j+1}^{n+{\tfrac{1}{2}} } -W_{i,j}^{n+{\tfrac{1}{2}} } }{\Delta x_{j+{\tfrac{1}{2}} } } -{}^{y} \bar{\nu }_{i,j-{\tfrac{1}{2}} }^{n} {}^{y} \bar{H}_{i,j-{\tfrac{1}{2}} }^{n+1} \frac{W_{i,j}^{n+{\tfrac{1}{2}} } -W_{i,j-1}^{n+{\tfrac{1}{2}} } }{\Delta y_{j+{\tfrac{1}{2}} } } \right]} \end{array}$

Thus combining, and explicit expressions for $w_{i,j,1}^{\*}$ and $w_{i,j,0}^{\*}$ are obtained.

The predicted values are again corrected using a variant of the McCormack scheme and including the pressure difference implicitly gives the corrector step:

(182) $\frac{\bar{W}_{i,j}^{n+1{\tfrac{1}{2}} } -\bar{W}_{i,j,1}^{*} }{\Delta t} +\frac{{}^{x} q_{i+{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } \Delta \bar{W}_{i+{\tfrac{1}{2}} ,j}^{} -{}^{x} q_{i-{\tfrac{1}{2}} ,j}^{n+{\tfrac{1}{2}} } \Delta \bar{W}_{i-{\tfrac{1}{2}} ,j}^{} }{H_{i,j}^{n+1} \Delta x} +\frac{{}^{y} q_{i,j+{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \Delta \bar{W}_{i,j+{\tfrac{1}{2}} }^{} -{}^{y} q_{i,j-{\tfrac{1}{2}} }^{n+{\tfrac{1}{2}} } \Delta \bar{W}_{i,j-{\tfrac{1}{2}} }^{} }{H_{i,j}^{n+1} \Delta y} -\frac{\Delta p_{i,j}^{n+1{\tfrac{1}{2}} } }{H_{i,j}^{n+1} } =0$

Where $\Delta \bar{W}_{i+{\tfrac{1}{2}} ,j}$ and $\Delta \bar{W}_{i,j+{\tfrac{1}{2}} }$ are obtained using relations similar to . Note that similar to $\bar{W}_{i,j}^{n+1{\tfrac{1}{2}} } ={\tfrac{1}{2}} \left(w_{i,j.1}^{n+1{\tfrac{1}{2}} } +w_{i,j,0}^{n+1{\tfrac{1}{2}} } \right)$ and again the kinematic boundary conditions is substituted for $w_{i,j,0}^{n+1{\tfrac{1}{2}} }$ .

The discrete vertical momentum balance of and looks very different from the relations found in [ZS05], [ZS08] and [Smi08]. This is mainly due to the application of the McCormack scheme for the advection. The discretisation of the pressure term is numerically fully equivalent to either the Keller box scheme as used in [ZS05], [ZS08] or the Hermetian relation used in [Smi08].

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(1, 2, 3) R. T. McCall, Gerhard Masselink, T. G. Poate, J. a. Roelvink, L. P. Almeida, M. Davidson, and P. E. Russell. Modelling storm hydrodynamics on gravel beaches with XBeach-G. Coastal Engineering, 91:231–250, 2014. URL: http://dx.doi.org/10.1016/j.coastaleng.2014.06.007, doi:10.1016/j.coastaleng.2014.06.007.

[ML04]

F. J. Mendez and I. J. Losada. An empirical model to estimate the propagation of random breaking and nonbreaking waves over vegetation fields. Coastal Engineering, 51:103–118, 2004. doi:10.1016/j.coastaleng.2003.11.003.

[NRS90]

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[NLB+15]

(1, 2) C. M. Nederhoff, Q. J. Lodder, Marien Boers, J. P. Den Bieman, and J. K. Miller. Modeling the effects of hard structures on dune erosion and overwash - a case study of the impact of Hurricane Sandy on the New Jersey coast. Proceedings Coastal Sediments, San Diego, CA, 2015.

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[RRT04]

A. J. H. M. Reniers, J. A. Roelvink, and E. B. Thornton. Morphodynamic modeling of an embayed beach under wave group forcing. Journal of Geophysical Research, 109:1–22, 2004. doi:10.1029/2002JC001586.

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(1, 2, 3) J. A. Roelvink. Dissipation in random wave group incident on a beach. Coastal Engineering, 19:127–150, 1993.

[Roe06]

J. A. Roelvink. Coastal morphodynamic evolution techniques. Coastal Engineering, 53:277–287, 2006. doi:10.1016/j.coastaleng.2005.10.015.

[RRvD+09]

J. A. Roelvink, A. J. H. M Reniers, A. R. van Dongeren, J. S. M. van Thiel de Vries, R. T. McCall, and Jamie Lescinski. Modelling storm impacts on beaches, dunes and barrier islands. Coastal Engineering, 56(11-12):1133–1152, November 2009. URL: http://linkinghub.elsevier.com/retrieve/pii/S0378383909001252, doi:10.1016/j.coastaleng.2009.08.006.

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[Row87]

P.N. Rowe. A convenient empirical equation for estimation of the richardson-zaki exponent. Chemical Engineering Science, 42(11):2795 – 2796, 1987.

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[RRvR12]

(1, 2, 3, 4, 5) B. G. Ruessink, G. Ramaekers, and L. C. van Rijn. On the parameterization of the free-stream non-linear wave orbital motion in nearshore morphodynamic models. Coastal Engineering, 65:56–63, July 2012. URL: http://linkinghub.elsevier.com/retrieve/pii/S0378383912000488, doi:10.1016/j.coastaleng.2012.03.006.

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[Sou97]

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[WvRVanOrmondt+07]

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[ZRVL13]

(1, 2) M. Zhou, J. A. Roelvink, H. J. Verheij, and H. Ligteringen. Study of Passing Ship Effects along a Bank by Delft3D-FLOW and XBeach. International Workshop on Nautical Traffic Models 2013, Delft, The Netherlands, July 5-7, 2013. Delft University of Technology, 2013.

[ZRZvW14]

M. Zhou, J. A. Roelvink, Z. Zou, and H. J. van Wijhe. Effects of Passing Ship With a Drift Angle on a Moored Ship. ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering, 2014.

[ZS05]

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Footnotes

[2]	Currently, this formulation is only possible when the wave shape formulation of [vanTdVries09] is applied, see Wave shape.