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GeoClaw-STRICHE: A coupled model for Sediment TRansport In Coastal Hazard Events

GeoClaw-STRICHE is designed for simulating the physical impacts of tsunami as it relates to erosion, transport and deposition. GeoClaw-STRICHE comprises of three components: (1) nonlinear shallow water equations; (2) advection-diffusion equation; (3) an equation for morphology updating. Multiple grain sizes and sediment layers are added into GeoClaw-STRICHE to simulate grain-size distribution and add the capability to develop grain-size trends from bottom to the top of a simulated deposit as well as along the inundation. Unlike previous models based on empirical equations or sediment concentration gradient, the standard Van Leer method is applied to calculate sediment flux. We tested and verified GeoClaw-STRICHE with flume experiment by \citet{johnson2016experimental} and data from the 2004 Indian Ocean tsunami in Kuala Meurisi as published in \citet{JGRF:JGRF786}. The comparison with experimental data shows GeoClaw-STRICHE's capability to simulate sediment thickness and grain-size distribution in experimental conditions. The agreement between experimental and model results is generally quite good, especially for sediment thickness, which builds confidence that sediment transport is correctly predicted by this model. The comparison with the data from the 2004 Indian Ocean tsunami reveals that the pattern of sediment thickness is well predicted and is of similar quality, if not better than the established computational models such as Delft3D.


Introduction
Recent tsunamis, such as the 2004 Indian Ocean tsunami and the 2011 Tohoku-oki tsunami, have caused approximately 420,000 deaths or missing reports and large destruction of properties in coastal communities (Apotsos et al., 2011b;Szczuciński et al., 2012). Unfortunately, tsunamis usually have long return period, which makes developing accurate tsunami hazard assessments very difficult. Therefore, to assess future events, the geologic record needs to be interrogated (Bourgeois and Minoura, 1997;Minoura et al., 2001;Jaffe and Gelfenbaum, 2002;Morton et al., 2007;Monecke et al., 2008;Szczuciński et al., 2012;Sugawara et al., 2013).
Tsunami deposits are the only recorders of past events (Huntington et al., 2007) and therefore play an important role in tsunami hazard assessments.
Due to the importance of tsunami deposit, laboratory, geological and numerical modeling studies of tsunami sediment transport have been an important (Tonkin et al., 2003;Young et al., 2009, and reference therein) but often overlooked research area. Most geological methods have focused on recognizing and reconstructing the tsunami recurrence interval and characteristics of the tsunami from geological records (Jaffe and Gelfenbaum, 2002;Burroughs and Tebbens, 2005). Post-tsunami field survey can be used to interpret tsunami deposit, infer the characteristics of tsunami flow as well as verify and validate numerical models (Gelfenbaum and Jaffe, 2003;Borrero, 2005;Fritz et al., 2006;Jaffe et al., 2006;Apotsos et al., 2011b;Tappin et al., 2012;Goto et al., 2014).
As mentioned earlier,numerical modeling of tsunami sediment transport is an important research area to improve our physical understanding of tsunami hydrodynamics and sedimentology. Numerical modeling of sediment transport during tsunamis can be approached in different ways. Inversion models (e.g. Moores advection model, Smith's model, Soulsbys model, TsuSedMod, TSUFLIND, TSUFLIND-EnKF) were developed to estimate the tsunami characteristics based on tsunami deposits (Moore et al., 2007;Smith et al., 2007;Soulsby et al., 2007;Jaffe and Gelfenbuam, 2007;Tang and Weiss, 2015;Wang et al., 2015;Tang et al., 2016). On the other hand, some forward numerical models (e.g. Delft3D, Xbeach, STM, C-HYDRO3D) have been developed to study time-varying tsunami sediment transport processes (Apotsos et al., 2011a;Li et al., 2012;Ontowirjo et al., 2013;Kihara and Matsuyama, 2011;Gusman et al., 2012). Three-dimensional models with high resolution like Delft3D and C-HYDRO3D can be applied to incorporate vertical flow velocities and vertical sediment concentration profiles (Van Rijn et al., 2004;Kihara and Matsuyama, 2011). However, three-dimension models require significant computational resources when they are employed to consider geophysically meaningful domain sizes (e.g., Sugawara et al., 2014). Therefore, two-dimensional models are better suited. In most of tsunami sediment transport models (C-HYDRO3D and STM), only sediment transport with uniform particle size can be simulated (Kihara and Matsuyama, 2011;Gusman et al., 2012). In order to study sedimentary structures, mixed particle sizes and multiple sediment layers need to be incorporated into the sediment transport models.
With regard to the calculation of sediment flux, three major methods can be used: empirical formulations (STM, Gusman et al., 2012), analytical approaches based on sediment concentration gradient (Xbeach and C-HYDRO3D, Roelvink et al., 2009;Kihara and Matsuyama, 2011), and numerical models qualified unbalance between depth-averaged concentration and equilibrium concentration (Xbeach and Deft3D, Roelvink et al., 2009;Van Rijn et al., 2004). All these approaches are restricted to relatively low sediment concentration gradient and small sediment flux condition, but they are inappropriate to apply in situations at which high sediment concentration gradient and large sediment flux occur. Hence, a two-dimensional, more comprehensive and robust sediment transport model with new sediment flux calculation method is needed. In this contribution, we present such a twodimensional fully coupled sediment transport model: GeoClaw-STRICHE. The critical shear velocity is employed to decide whether sediment particles in a given flow condition are entrained into the flow or not. The critical shear velocity, u b * ,cr , is estimated with the help of an iterative procedure that discretices the Shields diagram (see Weiss, 2008). The Rouse number, P , is used to determine the sediment transport condition for each grain-size class:

Theoretical Background
(1) P > 2.5: all grains travel as bed load; (2) 1.2 < P < 2.5: parts of sediment can travel as suspended load and rest will travel as bed load (the percentages for bed and suspended load linearly depends on critical velocity); (3) P < 1.2: all grains travel in suspended condition. The critical velocity for bed load is given by: where z 0 is the bottom roughness from MacWilliams (2004). The eddy viscosity profile from Gelfenbaum and Smith (1986) is given by: in which z is the elevation above bed. The critical velocity for suspended load is: and The combined bed and suspended load sediment concentration is updated by solving the advection-diffusion equation (Galappatti and Vreug-denhil, 1985): in which C is the depth-averaged sediment concentration, D h denotes to the sediment diffusion coefficient, T s refers to the adaptation time, and C eq is the equilibrium sediment concentration.

Transport formulations
Two methods to calculate the equilibrium sediment concentration methods are made available in the current version of STRICHE. For both methods, the total equilibrium sediment concentration should be between maximum allowed sediment concentration, C max and 0. The formulae to determine the equilibrium sediment concentration are given in the following sections. It should be noted that both of these two methods are modified to distinguish between bed and suspend load. The first one is modified based on Soulsby-Van Rijn equations (Van Rijn, 1984;Soulsby, 1997). In this method, the equilibrium sediment concentration is calculated by: The suspended load and bed load coefficient are calculated by: where D * is dimensionless grain size class. The critical velocity for bed load and suspended load are calculated based on the method presented in section 2.1.1. When STRICHE deals with multiple grain-size classes, the median value of grain-size distribution, D 50 , in these equations is replaced by diameter of each grain size class. Parameter v mg is the magnitude of the Eulerian velocity, and U rms denotes the root-mean-squared velocity obtained from linear wave theory. For the drag coefficient C d is given by: The second method used to calculate the equilibrium concentration is (11) The suspended load and bed load coefficient are calculated with:

Sediment settling velocity and density effect
The calculation of the settling velocity w s is based on Hallermeier (1981) and Ahrens (2000). For high sediment concentration, the fall velocity is reduced: in which C is the total volume sediment concentration. Exponent a 1 is estimated with the help of Rowe (1987), which depends on the Reynolds particle number, R: Then bulk density of the fluid becomes a function of the water density and the sediment that is suspended in it (Warner et al., 2008): In this equation, ρ water is water density, and C m denotes sediment concentration for grain-size class m. ρ s,m is sediment density for grain-size class m. N is the total number of grain-size classes. This enables STRICHE to simulate sediment transport in high sediment concentration condition and help to extend the density stratification in the future.

Sediment Flux
In STRICHE, the standard Van Leer method (Van Leer, 1997) and the Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL) scheme (Van Leer, 1979) are modified and applied to calculate sediment flux. The Van Leer method is a finite volume method that can provide highly accurate numerical solutions, especially for the solutions include shocks, discontinuities, or large gradient. First, the left and right states of the wave speed are determined on the cell surface (i + 1/2, j) in which cell surface (i + 1/2, j) denotes the surface between cells (i, j) and (i + 1, j) by: h l i+1/2,j and h r i+1/2,j refer to the left and right states of water depth on the cell surface. The right and left states of the variables are decided by the MUSCL scheme. For the first order, Also the left and right states of Froude number are given by: U l i+1/2,j and U r i+1/2,j represent the left and right states of the depth-averaged velocity at the cell surface. After that, the positive and negative Froude numbers are given by: And then parameter α and β are calculated: int is a function to evaluate the integer part from the variable, and sign is a function to return the value of the first variable with the sign of the second variable. The modified Froude number is determined: Finally, the sediment flux at this cell surface is: in which C l i+1/2,j and C r i+1/2,j are the left and right states of sediment concentration on cell surface. In order to reduce non-physical dissipation in this model, we apply sediment flux limiter in the second order simulation. Then sediment flux in y direction is calculated with the same method.

Sediment Flux Limiter
To reduce non-physical dissipation, limit the influence of topography change and get high order accuracy in this model, we apply sediment flux limiter and MUSCL scheme based on Van Leer (1979): It should be noted that the sediment flux limiter limits the sediment flux by modifying the sediment concentration. Parameter ε is the order of accuracy.
k is decided by the type of scheme. The flux limiter is given by: Where sediment concentration gradient r The flux limiter function, Ψ, is based on Van Albada et al. (1982).

Morphology Update
During deposition or erosion, the surface elevation, Z b , is updated by: in which q x and q y are sediment flux in horizontal direction and f mor is a morphological acceleration factor from Reniers et al. (2004). To account for the slumping of sediment during tsunami, avalanching scheme from Roelvink et al. (2009) is incorporated. When the slope between two adjacent grid cells exceeds the critical slope, sediment is exchanged between these cells to reduce the slope below the critical slope.

Sediment Classes
In order to be able to reproduce vertical grain-size trends, different grainsize classes need to be implemented as well as different layers of erodible bed sediments. The grain-size distributions are represented by discrete grain-size classes. It should be noted that natural sediment are most likely mixtures of cohesive and non-cohesive sediments. While sediment entrainment is affected by cohesive sediments, the influence of cohesive material on non-cohesive sediments is not trivial. A dynamical model to take cohesive sediments into account is still under development and will be included in future versions of STRICHE.

Sediment Layers
The thickness of each layer of the erodible bed is user-defined, and the top one is considered the active layer. Sediment in erodible sediment layers can be transported by suspended load and/or bed load throughout the entire simulation. However, only the sediment in the active layer is available for transport at any given time step. It is assumed that in one time step not more than the thickness of the active layer can be eroded (see

Model algorithm
For the coupling between GeoClaw and STRICHE, a two-way coupling but separately solving system is utilized. The hydrodynamic model, sediment transport model, and morphodynamic model are constructed as three separate modules in GeoClaw-STRICHE. STRICHE includes the sediment transport model and morphodynamic model (Fig. 2). In each time step, the hydrodynamic model (GeoClaw) outputs hydrodynamic condition to the sediment transport model. The sediment transport model passes sediment fluxes to the morphodynamic model. In turn, STRICHE returns topography information to GeoClaw and change the source term of shallow water equations (Fig. 2).

Model Validation
In this section, we provide two validation cases that highlight the capabilities of coupled model GeoClaw-STRICHE. The first case is a flume experiment of tsunami sediment transport processes in an open channel.
This case demonstrates the ability of GeoClaw-STRICHE to reproduce sediment thickness and grain-size distribution in experimental conditions. The second case is a real application with complex topography from the 2004 Indian Ocean tsunami. This case demonstrates the capability of GeoClaw-STRICHE to simulate sediment thickness in a realistic application with complex bathymetry.

Flume Experiment Case
GeoClaw-STRICHE is tested by simulating several open channel cases from Johnson et al. (2016). These laboratory experiments originally were conducted to evaluate the tsunami inversion model assumptions and qualify uncertainties. These experiment cases were set in a 32 m long, 0.5 m wide and 0.8 m high water tank at the University of Texas at Austin (Fig. 3). There was a smooth bed without slope and a lift gate to control input flow. The first three cases ran with the same grain-size distribution of the sediments ( Fig. 4a, source 1) and three different initial water depths in the water tank (0, 10 and 19 cm). Case IV, V and VI set with the same initial water depth (8 cm) and three different grain-size distributions in sediment (Fig. 4a). The sediment was located between 0.5m and 2.0m from the lift gate, and had a trapezoidal cross section that was 1.5m long and 0.15m high (Fig. 3). When the gate was lifted, the flow eroded the sand dune like a tsunami would erode a coastal dune during propagation.
[ Figure 3 Fig. 3) by using ultrasonic transducers. Flow velocity was measured by two side-looking Notrek Vectrino ADVs located 19.3 m from the lift gate (Fig. 3). We calculated mean value and smoothed the original data including water depth and velocity to make them easy to compare with simulation data. After sediment deposition and water drainage, sediment thickness was measured every 25 cm. Grain-size distribution was measured every meter at 1/8 Φ resolution by an imagedbased Retsch Camsizer. The sediment samples used to analyze grain-size distribution included the entire thickness of deposit in the sample location.
[ Figure 4 about here.] For this application, 2D depth-averaged simulations including bed load and suspended load by GeoClaw-STRICHE were carried out. The water depth at Headbox (Fig. 4b) was treated as a boundary condition in simulations. All simulations started with a sand dune located within 0.5m to 2.0m m from the gate (Fig. 3). Grain-size distributions of sediment were discretized with ten grain-size classes. The grain-size distributions for the experiment are depicted in Fig. 4a. The sediment layer in this case will be only separated to one erodible layer and hard structure in the model runs.
[ that the shallowest water depth achieved from the case I (Fig. 5-I), and deepest one achieved from case III (Fig. 5-III) [ Figure 6 about here.] Flow velocities were measured by ADVs at 19.3m from the gate. As there is little difference between velocities measured at 9 cm and 15 cm above the bed, the velocities measured in the experiment are considered as depthaveraged velocity (Johnson et al., 2016). It should be noted that the Froude number for case III is based on the flow depth at 18.7 m downstream. The flow velocity was highest right after the bore and then decreased gradually (Fig.6b). The model slightly overestimates the Froude number in the first five seconds of case III (Fig. 6a). Also the model results follow a similar pattern, but there is an obvious overprediction after 15s for both velocity and Froude number (Fig. 6).
[ Figure 7 about here.] Figure 7 compares measured and calculated sediment thickness. In case I, most of the sediment dune was eroded by the flow (Fig. 7-I (Fig. 7). The thickest sediment thickness generally occurred between 1m and 6m from the gate (Fig. 7). [

The 2004 Indian Ocean Tsunami Case
Validation data retrieved in a controlled environment does not exist for complicated topographies. Therefore, field data play a crucial role in model validation. In this section, a real case from Kuala Meurisi after the 2004 Indian Ocean tsunami is presented. Nearshore bathymetry, onshore topography and incident wave characteristics were taken from Apotsos et al. (2011a, ; red dotten line in Fig. 9a represents the pre-tsunami onland topography).
A grid spacing of 10 m is used in this simulation. The erodible sediment layer is 5 m thick with five grain-size classes (-1 to 4 in φ scale) with equal percentages.
[ Figure 9 about here.] The GeoClaw-STRICHE results are compared with the field data from Kuala Meurisi and respective model results computed with Delft3D in Fig.   9. Figure  There are obvious overpredictions after 50s for both velocity and the Froude number in Case III (Fig. 6). While this fact can arguably be ascribed to the imperfect model, we note that it is also possible that the data suffer a large uncertainty due to the fact that ADVs can return erroneous data if the flow is really shallow or a large number of air bubbles are present. Note that the Froude number in this case is calculated based on 18.7 m's flow depth and 19.1 m's velocity, which may also contribute to the overprediction. The thickness trends are not significantly different between case IV to VI in both experimental and model results, which indicates the grain-size distribution did not significantly control the thickness distribution for this experimental condition. However, sediment transport in the dry land condition seems to be much more difficult to simulate compared to other cases due to strong turbulence. As a result the calculated thicknesses are deviated from experiment results in Fig. 7-I. The D 10 contains apparent difference in case V and VI between observation and model results (Fig. 8). It is likely that underestimations in the fluid dynamics from model have more influence on large particles than small ones.

Model Limitations and Future works
The GeoClaw-STRICHE has proven its capability of simulating sediment transport by comparing experimental, field and results from Delft3D with our model. However, there are still some limitations for the present version of GeoClaw-STRICHE. A significant assumption of GeoClaw-STRICHE is that tsunami sediments are represented with several grain-size classes. As a result, the increasing number of sediment grain-size classes will significantly increase the computational load. Meanwhile, GeoClaw-STRICHE can only simulate sand sediment transport. However, tsunamis can have the power to transport almost all types of sediment from mud to boulder. Another important assumption of GeoClaw-STRICHE is that sediment will be picked up when shear velocity is larger than the critical shear velocity. In this model, we use Shield diagram to estimate the critical shear velocity for simplicity.
The Shield diagram can only deal with particle size between 9 to -4 Φ. Also the critical shear velocity from Shield diagram usually is considered as a rough estimate. At the same time, the depth-averaged velocity for sediment movement is calculated based on water depth, shear velocity and eddy viscosity profile. The choice of eddy viscosity profile will extremely influence the velocity estimation.
Future model improvement will be developed in an open source, community development approach. In order to solve computational load problems, the OpenMP and/or MPI (Message Passing Interface) should be implemented to increase calculation efficiency. Algorithms to represent cohesive sediment, gravel, and boulder sediment transport, as well as density stratification are under development. We also plan to investigate alternative approaches to calculate settling velocity, shear velocity, critical shear velocity, and sediment flux.

Conclusions
In this paper, we propose a tsunami sediment transport model, STRICHE.
STRICHE is fully coupled with GeoClaw for computing the hydrodynamics (combined model is referred to as GeoClaw-STRICHE). STRICHE's features include multiple sediment grain-size classes and sediment layers. The sediment concentration is computed by an advection-diffusion equation from Galappatti and Vreugdenhil (1985). The standard Van Leer method is applied for calculating sediment flux. To avoid the nonphysical model insta-bility, a flux limiter is used. The bed-updating module controls the topography change and the mass balance between two neighbor cells. GeoClaw-STRICHE tracks sediment thickness and properties for each morphological step and updates topography information for fluid dynamic module. The quality of the matches between experimental, field and numerical results, show that STRICHE is capable of reliably simulate sediment transport during coastal hazard events, such as tsunamis. The coupling with other hydrodynamic models, for example to consider storm waves and surge, will show its capabilities for general coastal hazard flooding. Furthermore, in its current version, but especially with the planned improvement, STRICHE is and will more so be able in the future to be applied on only to modern cases, but also to coastal hazard flooding event in the past where the only physical evidence is the deposits.