Research article 26 Feb 2021
Research article  26 Feb 2021
MultilayerHySEA model validation for landslidegenerated tsunamis – Part 1: Rigid slides
 ^{1}Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada, Facultad de Ciencias, Universidad de Málaga, 29080 Málaga, Spain
 ^{2}currently at: Departamento de Matemáticas, Campus de Rabanales, Universidad de Córdoba, 14071 Córdoba, Spain
 ^{1}Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada, Facultad de Ciencias, Universidad de Málaga, 29080 Málaga, Spain
 ^{2}currently at: Departamento de Matemáticas, Campus de Rabanales, Universidad de Córdoba, 14071 Córdoba, Spain
Correspondence: Jorge Macías (jmacias@uma.es)
Hide author detailsCorrespondence: Jorge Macías (jmacias@uma.es)
This paper is devoted to benchmarking the MultilayerHySEA model using laboratory experimental data for landslidegenerated tsunamis. This article deals with rigid slides, and the second part, in a companion paper, addresses granular slides. The US National Tsunami Hazard and Mitigation Program (NTHMP) has proposed the experimental data used and established for the NTHMP Landslide Benchmark Workshop, held in January 2017 at Galveston (Texas). The first three benchmark problems proposed in this workshop deal with rigid slides. Rigid slides must be simulated as a moving bottom topography, and, therefore, they must be modeled as a prescribed boundary condition. These three benchmarks are used here to validate the MultilayerHySEA model. This new HySEA model consists of an efficient hybrid finitevolume–finitedifference implementation on GPU architectures of a nonhydrostatic multilayer model. A brief description of model equations, dispersive properties, and the numerical scheme is included. The benchmarks are described and the numerical results compared against the labmeasured data for each of them. The specific aim is to validate this new code for tsunamis generated by rigid slides. Nevertheless, the overall objective of the current benchmarking effort is to produce a readytouse numerical tool for realworld landslidegenerated tsunami hazard assessment. This tool has already been used to reproduce the Port Valdez, Alaska, 1964 and Stromboli, Italy, 2002 events.
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Model development and benchmarking for earthquakeinduced tsunamis is a task addressed in the past and to which much effort and time has been dedicated. In particular, just to mention a couple of NTHMP efforts, the 2011 Galveston benchmarking workshop (Horrillo et al., 2015) and the 2015 Portland workshop for tsunami currents (Lynett et al., 2017) both focused on these topics. However, both model development and benchmarking efforts have advanced at a slower pace for landslidegenerated tsunamis. As examples we might mention the 2003 NSFsponsored landslide tsunami workshop organized in Hawaii and a similar followup workshop on Catalina Island in 2006 (Liu et al., 2008). Since then, no similar large comprehensive benchmarking workshop has been organized (Kirby et al., 2018).
In its 2019 strategic plan, the NTHMP required that all numerical tsunami inundation models to be used in hazard assessment studies in the United States be verified as accurate and consistent through a model benchmarking process. This mandate was fulfilled in 2011 but only for seismic tsunami sources and to a limited extent for idealized solid underwater landslides. However, recent work by various NTHMP states has shown that landslide tsunami hazard may in fact be greater than seismically induced hazard and may be also the dominant risk along significant parts of the US coastline (ten Brink et al., 2014).
As a result of this demonstrated gap, a set of candidate benchmarks was proposed to perform the required validation process. The selected benchmarks are based on a subset of available laboratory data sets for solid slide experiments and deformable slide experiments and include both submarine and subaerial slides. In order to complete this list of laboratory data, a benchmark based on a historic field event (Valdez, AK, 1964) was also chosen. The EDANYA group (https://www.uma.es/edanya, last access: 18 February 2021) from the University of Málaga participated in the workshop organized at Texas A&M University, Galveston (9–11 January 2017), presenting results for the benchmarking tests with two numerical codes: LandslideHySEA and MultilayerHySEA models. At Galveston, we presented numerical results for six out of the seven benchmark problems proposed, including the field case. The current work presents the numerical results obtained for the MultilayerHySEA model in the framework of the validation effort described above for the case of rigidslidegenerated tsunamis, whereas the benchmark problems dealing with granular slides are presented in a companion paper, Macías et al. (2021). A summary of the results for the field case at Port Valdez can be found in Macías et al. (2017).
Twenty years ago, at the beginning of the century, the challenge of solid block landslide modeling was taken by a number of researchers (Grilli and Watts, 1999, 2005; Grilli et al., 2002; Lynett and Liu, 2002; Watts et al., 2003, 2005; Wu, 2004; Liu et al., 2005), and laboratory experiments were developed for those cases and for tsunami model benchmarking (Enet and Grilli, 2007) (see also AtaieAshtiani and NajafiJilani, 2008). The benchmark problems (BPs) performed in the current work are based on the laboratory experiments of Grilli and Watts (2005) for BP1, Enet and Grilli (2007) for BP2, and Wu (2004) and Liu et al. (2005) for BP3. The basic reference for these three benchmarks, as well as for the three benchmarks related to granular slides and the Alaska field case (all of them proposed by the NTHMP), is Kirby et al. (2018). We highly recommend checking this reference for further details on benchmark descriptions, data provided for performing them, required benchmark items, and intermodel comparison. Finally, we would like to stress that the ultimate goal of our current benchmarking effort is to provide the tsunami community with an NTHMPapproved model for landslidegenerated tsunami hazard assessment, similarly to what we have done with the TsunamiHySEA model for the case of earthquakegenerated tsunamis (Macías et al., 2017, 2020a, c).
The HySEA (Hyperbolic Systems and Efficient Algorithms) software consists of a family of geophysical codes based on either singlelayer, twolayer stratified systems, or multilayer shallowwater models. HySEA codes^{1} have been under development by the EDANYA group from UMA (the University of Málaga) for more than a decade. These codes are in continuous evolution and undergo continuous upgrading, and they serve a wider scientific community every day. The first model we developed dealing with landslidegenerated tsunamis consisted of a stratified twolayer Savage–Hutter shallowwater model – the LandslideHySEA model. It was implemented based on the model described in Fernández et al. (2008) and was incorporated into the HySEA family. An initial validation of this code, comparing numerical results with the laboratory experiments of Heller and Hager (2011) and Fritz et al. (2001), can be found in SánchezLinares (2011). The 2018 numerical simulation of the 1958 Lituya Bay megatsunami with real topobathymetric data and encouraging results (GonzálezVida et al., 2019) represented a milestone in the verification process of this code. This validation effort was accomplished under a research contract with the NOAA Pacific Marine Environmental Laboratory (PMEL). The result of this project led the NCTR (NOAA Center for Tsunami Research) to adopt LandslideHySEA as the numerical code of choice to generate the initial conditions for the MOST model to be initialized in the case of a landslidegenerated tsunami scenario to be simulated. Further applications of LandslideHySEA can be found in de la Asunción et al. (2013), Macías et al. (2015), and Iglesias (2015).
The waves generated in the laboratory tests proposed in the NTHMP selected benchmarks are high frequency and dispersive, and the generated flows have a complex vertical structure. Therefore, the numerical model used must be able to reproduce such effects. This makes the twolayer LandslideHySEA model unsuitable for reproducing these experimental results as nonhydrostatic effects play an important role and a richer vertical structure is required. To address these requirements, the MultilayerHySEA model was very recently implemented, considering a stratified structure in the simulated fluid and including nonhydrostatic terms. A multilayer model is able to better approximate the vertical structure of a complex flow than a standard onelayer depthaveraged model. In particular, by increasing the number of layers, the linear dispersion relation of the model converges towards the exact dispersion relation from the Stokes linear theory (see FernándezNieto et al., 2018).
The MultilayerHySEA model implements one of the multilayer nonhydrostatic models of the family introduced and described in FernándezNieto et al. (2018) (model LDNH_{0}). The governing equations, which are obtained by a process of depth averaging, correspond to a semidiscretization for the vertical variable of the Euler equations following a standard Galerkin approach. The total pressure is decomposed into a sum of hydrostatic and nonhydrostatic pressures and is assumed to have a linear vertical profile. The horizontal and vertical velocities are assumed to have a constant vertical profile. At the discrete level on z, the total pressure matches at the interfaces and velocities satisfy a discrete jump condition (see Fernández et al., 2008, or Escalante et al., 2018a).
An alternative deduction for this system is performed in Escalante et al. (2018a) assuming linear vertical profiles for pressure and vertical velocity and a constant vertical profile for the horizontal velocity, as well as some extra hypotheses for the case of two layers. The proposed model admits an exact energy balance, and, when the number of layers increases, the linear dispersion relation of the linear model converges to that of Airy's theory (FernándezNieto et al., 2018). The model proposed in FernándezNieto et al. (2018) can be written in a compact form as
where, for $\mathit{\alpha}\in \mathit{\{}\mathrm{1},\phantom{\rule{0.33em}{0ex}}\mathrm{2},\mathrm{\dots},L\mathit{\}}$, the following notation is used:
where f denotes one of the generic variables of the system, i.e., u, w, and p, and, finally,
Total depth, h, is split along the vertical axis into L≥1 layers and $\mathrm{\Delta}s=\mathrm{1}/L$ (see Fig. 1). The variables u_{α} and w_{α} are the depthaveraged velocities in the x and z directions, respectively, t is time, and g is gravitational acceleration. The nonhydrostatic pressure at the interface ${z}_{\mathit{\alpha}+\mathrm{1}/\mathrm{2}}$ is denoted by ${p}_{\mathit{\alpha}+\mathrm{1}/\mathrm{2}}$. The water surface elevation measured from the stillwater level is $\mathit{\eta}=hH$, where H is the water depth when the water is at rest. Finally, τ is a friction law term, and the terms ${\mathrm{\Gamma}}_{\mathit{\alpha}+\mathrm{1}/\mathrm{2}}$ account for the mass transfer across interfaces and are defined by
In order to close the system of equations, the following boundary conditions are considered:
Note that the motion of the bottom surface can be taken into account as a boundary condition, imposing w_{0}≠0. Therefore, this model can simulate the interaction with a slide in the case that the motion of the bottom is prescribed by a function, given by a set of data, or simulated by a numerical model. In the present study, we are going to consider tests where the motion of the seafloor is given by a known function (the solid moving block).
3.1 Linear dispersion relation
Some dispersive properties of the system (Eq. 1) are presented in this section, in particular, the phase and group velocities, as well as the linear shoaling. The first two properties are related to the propagation of dispersive wave trains and the last one to shoaling processes.
To obtain such properties, system Eq. (1) is linearized around the wateratrest steadystate solution. After that, a Stokestype Fourier analysis is carried out looking for firstorder planar wave solutions. This method constitutes a standard procedure to study systems that model dispersive water waves (see Escalante et al., 2018a; Lynett and Liu, 2004; Madsen and Sorensen, 1992; Schäffer and Madsen, 1995, and references therein). The phase and group velocities as well as the linear shoaling gradient are, respectively, defined as
where ω denotes the angular frequency, k the local wave number, and H the typical depth.
The measured quantities C, G, and γ are solely functions of the local wave number and the typical depth H. Thus, one can obtain the socalled linear dispersion relation of the three measured quantities. From the Airy wave theory, one can also obtain the corresponding linear dispersion relations that state the linear theory for the considered quantities (see Schäffer and Madsen, 1995, for the Airy reference formulae). For example, the expression for the phase velocity from Airy's theory is
The expressions of the phase velocity for the system (Eq. 1) are given in Table 1 for the nonlinear hydrostatic shallowwater system (SWE) and the MultilayerHySEA (nonhydrostatic) system with j≥1 layers (NH–jL). The last two columns contain Er_{C}(s) for s=5 and s=15, where Er_{C}(s) represents the maximum relative error of the phase velocity with respect to the Airy in a range $kH\in \left[\mathrm{0},s\right]$ in percent, i.e.,
The main goal when deriving dispersive shallowwater systems is to get the most accurate dispersive relations as possible, compared with the Airy wave theory, without highly increasing the complexity of the system. See Schäffer and Madsen (1995) for a review on stateoftheart dispersive models or a twolayer model with improved dispersive relations in Lynett and Liu (2004), as well as an enhanced twolayer nonhydrostatic pressure system in Escalante et al. (2018a). It has been shown (FernándezNieto et al., 2018) that increasing the number of layers leads to the convergence of the linear dispersion relation of the linear model to that of Airy's theory. Figure 2 shows this behavior and highlights the huge discrepancies between Airy's theory and the systems (SWE) and (NH1L). It is well known that waves generated by landslides might present high characteristic values for kH. For the (SWE) system, it is well known that it has an accurate phase velocity in a small range of kH and that this system is appropriate for long waves as tsunami waves but not for dispersive waves with higher values of kH. In the same vein, the onelayer nonhydrostatic pressure system (NH1L) can improve these results, but, again, poor linear dispersive results are achieved in a range of kH between 5 and 15. However, when the number of layers, L, is set to 3 (still a small value) the (Eq. 1) is in an excellent agreement with the Airy theory for kH up to 15. For the phase celerity, the percentage error is less than 0.62 %, and for the group velocity it is less than 1 % for kH smaller than 10 (see Fig. 2). Linear shoaling is also well reproduced in this same range.
The MultilayerHySEA model presents enhanced dispersive properties. In order to have similar dispersive results as the ones obtained here using a threelayer system, at least five layers are required for other similar multilayer models as the one presented in Bai and Cheung (2018). Furthermore, the results presented for the phase velocity with two layers in Table 1 show that the system proposed here produces smaller relative error for kH up to 15 compared with the twolayer system in Cui et al. (2014). That means that the MultilayerHySEA model can achieve better dispersive properties than models having similar or even more computational complexity.
3.2 Modeling of breaking waves and wetting and drying treatment
3.2.1 Modeling of breaking waves
In shallow areas the breaking of waves can be observed near the coast. As pointed out in Escalante et al. (2018a, b, 2019) and Roeber et al. (2010) among others, nonhydrostatic partialdifferentialequation (PDE) systems such as the one considered in this paper cannot describe this process without the inclusion of an additional term that accounts for the dissipation of the amount of energy required when breaking phenomena occur. In this work, we have implemented a simplified generalization of the breaking mechanism that was introduced in Escalante et al. (2018a) for the case of two layers. To do so, the vertical component of the stress tensor is depthaveraged on the vertical variable. Thus, adding the proposed integrated viscosity term to system Eq. (1), only the vertical momentum equation changes and reads for each $\mathit{\alpha}\in \mathit{\{}\mathrm{1},\mathrm{2},\mathrm{\dots},L\mathit{\}}$ as
where $\mathit{\varsigma}={\int}_{{z}_{\mathit{\alpha}\mathrm{1}/\mathrm{2}}}^{{z}_{\mathit{\alpha}\mathrm{1}/\mathrm{2}}}{\partial}_{z}\mathit{\nu}$ is the eddy viscosity. In this work, as in Escalante et al. (2018a) and Roeber et al. (2010), we choose ς to be
where B is an empirical parameter related to a breaking criteria to switch on and off this extra dissipation term. The definition of this empirical parameter is based on a quasiheuristic strategy to determine when the breaking occurs (see Escalante et al., 2018b; Roeber et al., 2010, and references therein).
Finally, a natural and simple extension of the criterion proposed by Roeber et al. (2010) is adopted
where
denote the flow speed at the onset and termination of the wavebreaking process and B_{1} and B_{2} are calibration coefficients. In this work, we use B_{1}=0.6 and B_{2}=0.15 for all the test cases studied. Finally, depending on the benchmark problem, we use $K\in \mathit{\{}\mathrm{2},\mathrm{10}\mathit{\}}.$
3.2.2 Wetting and drying treatment
For the computation of variables in areas of small water depth, a wet–dry treatment adapting the ideas described in Castro et al. (2005) is applied. The key elements for the numerical treatment of wet–dry fronts with emerging bottom topographies are based on the following.

The hydrostatic pressure terms ${\partial}_{x}\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}g{h}^{\mathrm{2}}\right)gh{\partial}_{x}H=gh{\partial}_{x}\mathit{\eta}$ at the horizontal velocity equations are modified for emerging bottoms to avoid spurious pressure forces (see Castro et al., 2005).

To overcome the difficulties due to large roundoff errors in computing the velocities u_{α},w_{α} from discharges for small values of h, we define the velocities analogously as in Kurganov and Petrova (2007), applying the desingularization formula
$$\begin{array}{}\text{(13)}& \begin{array}{rl}& {u}_{\mathit{\alpha}}={\displaystyle \frac{\sqrt{\mathrm{2}}hh{u}_{\mathit{\alpha}}}{\sqrt{{h}^{\mathrm{4}}+max({h}^{\mathrm{4}},{\mathit{\u03f5}}^{\mathrm{4}})}}},\\ & {w}_{\mathit{\alpha}}={\displaystyle \frac{\sqrt{\mathrm{2}}hh{w}_{\mathit{\alpha}}}{\sqrt{{h}^{\mathrm{4}}+max({h}^{\mathrm{4}},{\mathit{\u03f5}}^{\mathrm{4}})}}},\phantom{\rule{0.25em}{0ex}}\mathit{\alpha}\in \mathit{\{}\mathrm{1},\mathrm{2},\mathrm{\dots},L\mathit{\}},\end{array}\end{array}$$which gives the exact value of u_{α} and w_{α} for h≥ϵ and gives a smooth transition of u_{α} and w_{α} to zero when h tends to zero without truncation. In this work we set $\mathit{\u03f5}={\mathrm{10}}^{\mathrm{3}}$ for the numerical tests. A more detailed discussion about the desingularization formula can be seen in Kurganov and Petrova (2007).
We describe now the discretization of system Eq. (1) that follows the natural extension of the procedure described in Escalante et al. (2018a, b) for the one and twolayer nonhydrostatic system. The numerical scheme employed is based on a twostep projectioncorrection method, similar to the standard Chorin projection method for Navier–Stokes equations (Chorin, 1968). This is a standard procedure when dealing with dispersive systems (see Escalante et al., 2018a, b; Ma et al., 2012; Kazolea and Delis, 2013; Ricchiuto and Filippini, 2014, and references therein).
First, we shall solve the nonconservative hyperbolic underlying shallowwater (SW) system (Eq. 1) given by the compact equation
where the following compact notation has been used:
and B_{SW} is a matrix such that B_{SW}∂_{x}U contains the nonconservative products related to the mass transfer across interfaces appearing at the momentum equations.
Then, in a second step, nonhydrostatic terms given by the pressure vector correction term
as well as the divergence constraints at each layer
will be taken into account.
System Eq. (14) is discretized using a secondorder finitevolume polynomialviscositymatrix (PVM) positivepreserving wellbalanced pathconservative method (Castro and FernándezNieto, 2012). As usual, we consider a set of N finitevolume cells ${I}_{i}=[{x}_{i\mathrm{1}/\mathrm{2}},{x}_{i+\mathrm{1}/\mathrm{2}}]$ with constant lengths Δx and define
the cell average of the function U(x,t) on cell I_{i} at time t. Concerning nonhydrostatic terms, we consider midpoints x_{i} of each cell I_{i} and denote the point values of the function P at time t by
Nonhydrostatic terms will be approximated by secondorder compact finite differences.
Let us detail the time stepping procedure implemented. Assume given time steps Δt^{n} and denote ${t}^{n}={\sum}_{k\le n}\mathrm{\Delta}{t}^{k}$. To obtain secondorder accuracy in time, the twostage secondorder totalvariationdiminishing (TVD) Runge–Kutta scheme is adopted. At the kth stage, $k\in \mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}},$ the twostep projectioncorrection method is given by
where U^{(0)} is U at the time level t^{n}, ${\mathbf{U}}^{\left(\stackrel{\mathrm{\u0303}}{k}\right)}$ is an intermediate value in the twostep projectioncorrection method that contains the numerical solution of the hyperbolic system (Eq. 14) at the corresponding kth stage of the Runge–Kutta, and U^{(k)} is the kth stage estimate. After that, a final value of the solution at the t^{n+1} time level is obtained:
Observe that Eq. (20) requires, at each stage of the calculation respectively, the solving of a Poissonlike equation for each one of the variables contained in P^{(k)}. The resulting linear system is solved using an iterative Jacobi method combined with a scheduled relaxation (see Adsuara et al., 2016; Escalante et al., 2018a, b). Note that the usual Courant–Friedrichs–Lewy (CFL) restriction must be imposed for the computation of the time step Δt.
With respect to the breaking mechanism introduced in Sect. 3.2, these terms are semiimplicitly discretized at the end of the second step of the proposed numerical scheme, at each Runge–Kutta stage. The resulting numerical scheme is well balanced for the wateratrest solution and is linearly L^{∞} stable under the usual CFL condition related to the hydrostatic system. It is also worth mentioning that the numerical scheme is positive preserving and can deal with emerging topographies. Finally, its extension to 2D is straightforward. In this case, the computational domain is decomposed into subsets with a simple geometry, called cells or finite volumes. The numerical algorithm is well suited for its implementation in GPU architectures, as is shown in Castro et al. (2011). Furthermore, the compactness of the numerical stencil and the natural and the massive parallelization of the Jacobi method make it possible that the second step can also be implemented in GPUs (see Escalante et al., 2018a, b). That results in a high efficiency of the numerical code and much shorter computational times.
In this section, the numerical results obtained with the MultilayerHySEA model and the comparison with the measured lab data for waves generated by the movement of a rigid bottom surface or of a solid block are presented. In particular, BP1 deals with a 2D submarine solid slide, BP2 with a 3D submarine slide, and, finally, BP3 consists of two 3D slides, one partially submerged and a second one representing a completely submarine slide. In all these cases, a moving bottom condition has been used to model the solid block movement. Regarding the wavebreaking model, the breaking mechanism described in Sect. 3.2 was implemented, adopting B_{1}=0.6 and B_{2}=0.15 for all the benchmark problems, K=10 for the third benchmark, and K=2 for the rest. The description of all these benchmarks can be found in LTMBW (2017) and Kirby et al. (2018).
5.1 Benchmark problem 1: twodimensional submarine solid block
This benchmark problem is based on the 2D laboratory experiments of Grilli and Watts (2005), which were performed at the University of Rhode Island. Refer to the abovementioned work to get a detailed description of the present benchmark. Figure 3 depicts the sketch of the laboratory experiment design. The 2D slide model is semielliptical, leadloaded, and rolling down a smooth slope with a slope angle $\mathit{\theta}={\mathrm{15}}^{\circ}$ (2 mm above the slope), in between two vertical side walls, 20 cm apart. The water depth is h_{0} = 1.05 m over the flat bottom part. The slide dimensions were length B = 1 m, maximum thickness T=T_{ref} = 0.052 m, and width w = 0.2 m. The model initial submergence d was varied in experiments and the freesurface elevation recorded at four capacitance wave gauges installed at locations: x^{′} = 1.175, 1.475, 1.775, and 2.075 m, with the first location being nearly identical to ${x}_{g}^{\prime}$ = 1.168 m (where the tilde variables, such as x^{′}, mean than nondimensional units are used – see Table3).
In this benchmark, two items remained not completely determined in the original description provided: the first one is related with the initialization of the numerical experiment, and the second one is related with how and where the solid moving block must stop. Other small issues related to the description of the benchmark were put forward in Macías et al. (2017) in our NTHMP report.
The motion of the rigid slide was prescribed as a function of time as
where ${S}_{\mathrm{0}}={u}_{t}^{\mathrm{2}}/{a}_{\mathrm{0}}=\mathrm{2.110}$ m, ${t}_{\mathrm{0}}={u}_{t}/{a}_{\mathrm{0}}=\mathrm{1.677}$ s, a_{0}=0.75 m/s^{2}, and u_{t}=1.258 m/s is the terminal velocity. Figure 4 shows the prescribed acceleration, velocity, and rigid slide displacement. In the laboratory experiment, the block is stopped at time $t={t}_{\mathrm{0}}=\mathrm{1.667}$ s, and we replicate this behavior in the numerical model. We also performed numerical experiments (not presented here) where the block continued moving at constant speed.
The benchmark here consists of using the above information on slide shape, submergence, and kinematics, together with reproducing the experimental setup to simulate surface elevations measured at the four wave gauges (average of two replicates of experiments provided).
Then, in order to reproduce the lab experiment, the interval $[\mathrm{1},\mathrm{10}]$ discretized with Δx=0.02 m is the computational domain considered. In the vertical, taking three layers seems to produce optimal results. Increasing the number of layers gives similar results, increasing the computational cost. The stability CFL number was set to 0.9 and g=9.81. The numerical simulation performed was 4 s long in real time. As boundary conditions, outflow conditions were imposed at $x=\mathrm{1},\phantom{\rule{0.33em}{0ex}}x=\mathrm{10}.$
In Fig. 5 the comparison of the numerical results with the filtered labmeasured data is presented. A good overall agreement between them can be observed. Some discrepancies can be seen after drawdown in all the gauges. This behavior could also be observed, except for the last gauge, in Grilli and Watts (2005) results. These authors explained that this behavior could be due to unwanted surface tension effects. Given this comparison, and considering the experimental variations and errors inherent to laboratory work and data processing, it can be concluded that the MultilayerHySEA model optimally performs the present benchmark test.
5.2 Benchmark problem 2: threedimensional submarine solid block
This second benchmark consists of a 3D extension of BP1. The longitudinal sketch of the experiment is the same as in Fig. 3. In the horizontal plane, cross sections are elliptic, and the plan view of the rigid slide, for the case d=61 mm, is presented in Fig. 6. It is based on the 3D laboratory experiments of Enet and Grilli (2007). The experiments were also performed at the University of Rhode Island in a water wave tank of width 3.6 m and length 30 m, with a stillwater depth of 1.5 m over the flat bottom portion. As in the previous benchmark, the angle of the plane slope where the slide slid down is θ=15^{∘}. The submarine slide model was built as a streamline Gaussianshaped aluminum body with an elliptical footprint (see Fig. 6), with downslope length b=0.395 m, crossslope width w=0.680 m, and maximum thickness T=0.082 m. Complete details about the analytic definition of the slide shape and the experimental setting can be found in Kirby et al. (2018) and in LTMBW (2017).
For the numerical simulations, the twodimensional computational domain $[\mathrm{1},\mathrm{10}]\times [\mathrm{1.8},\mathrm{1.8}]$ is considered and discretized with $\mathrm{\Delta}x=\mathrm{\Delta}y=\mathrm{0.02}\phantom{\rule{0.125em}{0ex}}m$. The number of layers was set to 3. Numerical tests were performed using more layers, and similar results were obtained. The CFL number was set to 0.9 and g=9.81. The simulated time was 6 s. As boundary conditions, rigid wall conditions were imposed at $y=\mathrm{1.8},\phantom{\rule{0.33em}{0ex}}y=\mathrm{1.8}$ and outflow conditions at $x=\mathrm{1},\phantom{\rule{0.33em}{0ex}}x=\mathrm{10}$.
The benchmark test proposed consists in reproducing the slide shape and complete experimental setup in and using the information about submergence and kinematics to replicate numerically Enet and Grilli's experiments for d=61 and d=120 mm. It is required to simulate surface elevations measured at the four wave gauges (average of two replicates of experiments) and present comparisons of the model with the experimental results.
Enet and Grilli (2007) performed experiments for seven initial submergence depths d. They are listed in Table 4, together with values of related slide parameters and some measured tsunami wave characteristics. Here, the numerical results corresponding to the two NTHMP required experiments (for d=61 and d=120 mm) will be presented first, then, as data for the seven experiments are provided, the comparison for the remaining five cases will also be presented.
In Fig. 7 the comparison of the MultilayerHySEA model numerical results with measured data for the first case, d=61 mm, in the four gauges is presented. An excellent agreement can be observed between these time series. The comparisons for the second required case (d=120 mm) in the three gauges with data provided (gauge g_{3} was not available) are shown in Fig. 8. Good agreement can also be observed in this case. Finally, Fig. 9 shows the comparison for the five remaining cases provided by Enet and Grilli. In all cases (for all submergences), a good agreement between simulated results and measured lab data can be observed.
In Table 6, the execution times for simulations performed on an NVIDIA Tesla P100 GPU are presented. It can be observed that including nonhydrostatic terms in the nonlinear shallow water equations results in an increase in the computational time of 2.65 times. If a richer vertical structure is considered, then larger computational times are required. As in the examples for the two and threelayer systems, there is a 3.3 and 4.45 times increase in the computational effort.
Figure 10 shows the comparison, for the four models considered, of the numerical results obtained with the measured data at gauge g_{4} for the case d=189 mm. It can be observed that a model vertical structure considering only one layer is not enough to reproduce the observed data and that considering two and three layers in the model produces much better numerical results.
Moreover, Table 7 shows the time periods, T, of the time series data in Figs. 7, 8, and 9 for all the wave gauges. To obtain the values for the periods, we have computed the elapsed time between the first two wave troughs in each time series. We have omitted the measurement for wave gauge g_{1} because it was not clear how to measure the period in this case. Once the period from each time series was measured, we computed the wave number from the dispersion relation given from the Airy theory:
Table 8 shows the computed values kH by solving the dispersion relation (23). On the view of the computed kH values it can be stated that $kH\in [\mathrm{2.815},\mathrm{4.528}].$
Since multilayer models have good dispersion relation errors within this range of kH (see Table 1 and Fig. 2), this explains the aforementioned excellent agreement between the computed time series and the measured lab data.
Finally, although the phase velocity for the onelayer system shows an error bounded by only 3.02 % for $kH\in [\mathrm{0},\mathrm{5}]$ (see Table 1), it can be seen in Fig. 10 that the onelayer nonhydrostatic pressure system cannot represent the waves correctly. In contrast, the onelayer system tends to amplify waves. This behavior can be explained by observing the shoaling gradient for this model (see Fig. 2). The shoaling gradient verifies the ODE:
where A denotes the amplitude. Thus, it can be stated by inspecting the solutions of the ODE (24) that if the shoaling gradient of the model γ(kH) is underestimated with respect to the Airy theory (γ<γ_{Airy}), then in this case the solutions of the system tend to amplify waves for offshore wave propagation. The poor behavior shown by the onelayer system in some cases justifies the need to incorporate the improved multilayer model considered here.
5.3 Benchmark problem 3: threedimensional submarine/subaerial triangular solid block
This benchmark problem is based on the 3D laboratory experiment of Wu (2004) and Liu et al. (2005) for a series of triangular blocks of several aspect ratios moving down a plane slope into the water from a dry (subaerial) or wet (submarine) location. Figure 11 shows the schematic description of the setup for this benchmark in the case of a partially submerged block. Further details can be found in Kirby et al. (2018) and in LTMBW (2017). The laboratory experiments were conducted in a wave tank at Oregon State University of length 104 m, width 3.7 m, and depth 4.6 m.
A plane slope 1:2 (as the one shown in Fig. 11, upper panel) with $\mathit{\theta}=\mathrm{26.6}{}^{\circ}$ was located near one end of the tank and a dissipating beach in the other. In all the experiments, the water depth was h_{0}=2.44 m. The experiments retained for the present benchmark were all performed with a triangular block of length b=0.91 m, width w=0.61 m, and vertical front face $a=b/\mathrm{2}=\mathrm{0.455}$ m.
The block movement was provided by means of a polynomial fitting to measured data, giving the horizontal distance as
with $\mathit{\beta}=\mathrm{arctan}(\mathrm{1}/\mathrm{2})$ and ${x}_{(\mathrm{0},t=\mathrm{0})}=\mathrm{2}\mathrm{\Delta}$. The polynomial coefficients for the two cases proposed are given in Table 9.
For each case, measured freesurface elevations are given for two wave gauges placed at $(x,y)=(\mathrm{1.83},\mathrm{0})$ (in m) and $(x,y)=(\mathrm{1.2446},\mathrm{0.635})$, where x is the distance to the initial coastline and y is the distance to the central cross section (see location in Fig. 11, lower panel). Also measured runup for each case is given at runup gauges 2 and 3 in Fig. 11, lower panel, lying on the slope at a distance 0.305 and 0.611 m from the central cross section, respectively.
The twodimensional computational domain $[\mathrm{2},\mathrm{6}]\times [\mathrm{2},\mathrm{2}]$ is discretized with $\mathrm{\Delta}\phantom{\rule{0.125em}{0ex}}x=\mathrm{\Delta}\phantom{\rule{0.125em}{0ex}}y=\mathrm{0.025}\phantom{\rule{0.33em}{0ex}}m$ and the number of layers was set up to 3. Numerical experiments using a higher number of layers were performed, obtaining similar results. The stability CFL number was set to 0.9 and g=9.81. The simulated time was 4 s. The same boundary conditions, as in the previous case, were imposed.
The numerical results obtained for the subaerial test case are presented in Figs. 12 and 13. Figure 12 depicts the comparison for the time series at the wave gauges and Fig. 13 at the runup gauges. The same comparison has been performed for the submerged test case, and it is presented in Figs. 14 and 15. The agreement for the wave gauges is quite good for WG1 in both cases. For WG2, just in front of the block, an overshoot after the first depression wave is observed in both cases. This feature is related to the turbulent nature of the experiment. Note that although a turbulent model is not considered here, we have noted that the breaking criteria helps to dissipate energy associated with this turbulent process. For the runup, the qualitative agreement is quite good, with the larger discrepancies in RG3 for the submarine test case.
Validation of numerical models is a first unavoidable step before their use as predictive tools. This requirement is even more necessary when the developed models are going to be used for risk assessment in natural events where human lives are involved. The present work is the first step in this task for the MultilayerHySEA model, a novel dispersive multilayer model of the HySEA suite developed at the University of Malaga. This model considers a stratified vertical structure and includes nonhydrostatic terms; this is done in order to include the dispersive effects in the propagation of the waves in a homogeneous, inviscid, and incompressible fluid. The numerical scheme implemented combines a highly robust and efficient finitevolume pathconservative scheme for the underlying hyperbolic system and finite differences for the discretization of the nonhydrostatic terms. In order to increase numerical efficiency, the numerical model is implemented to run in GPU architectures – in particular in NVIDIA graphics cards and using CUDA language. In the case of the traditional SW nondispersive model, this kind of implementation produces an extremely efficient and fast code (Macías et al., 2020c). Increasing the number of layers in SW models provides an enhanced vertical resolution and, at the same time, increases the computational cost. Despite this, from a computational point of view, the twolayer nonhydrostatic code presents a good computational efficiency, and computing times with respect to the onelayer SWE GPU code are absolutely reasonable, being only from 2 to 2.5 larger than for the onelayer case. In the numerical simulations performed in the present work, the nonhydrostatic wallclock times are always below 4.45 times those for the traditional SWE HySEA model, for a number of vertical layers up to three. The numerical scheme presented here and the corresponding multilayer SW water model proposed is highly efficient and is able to model dispersive effects with a low computational cost.
Regarding model results, they show a good agreement with the experimental data for the three benchmark problems studied in the present work. In particular, for BP2, but this also occurs for the other two benchmark problems, we have shown that a onelayer, hydrostatic or nonhydrostatic, model is not able to reproduce the complexity in the observed lab data considered in the proposed benchmarks. The waves to be modeled in the test cases proposed here are high in frequency and dispersive. Hence, it is at least necessary that a twolayer structure and nonhydrostatic terms in the model be used in order to capture the dynamics of the generated waves. As noted in Kirby et al. (2018) and in view of the results presented here, the nonhydrostatic multilayer model discussed in this work can adequately represent the physics and behavior of the waves generated at a reasonable low computational cost.
The numerical code used to perform the numerical simulations in this paper is available at the HySEA codes web page at https://edanya.uma.es/hysea/index.php/download (Macías, 2021).
All the data used in the present work and necessary to reproduce the setup of the numerical experiments as well as the laboratorymeasured data to compare with can be downloaded from LTMBW (2017) at the web site http://www1.udel.edu/kirby/landslide/. Finally, the NetCDF files containing the numerical results obtained with the MultilayerHySEA code can be found and downloaded from https://doi.org/10.17632/xtfzrbvcb2.3 (Macías et al., 2020b).
JM led the HySEA codes benchmarking effort undertaken by the EDANYA group, wrote most of the paper, reviewed and edited it, and assisted in the numerical experiments and in their setup. CE implemented the numerical code and performed all the numerical experiments; he also contributed to writing the manuscript. JM and CE did all the figures. MC significantly contributed to the design and implementation of the numerical code.
The authors declare that they have no conflict of interest.
The authors are indebted to Diego Arcas (PMEL/NOAA) and Victor Huérfano (PRSN) for supporting our participation in the 2017 Galveston workshop and to the MMS of the NTHMP for kindly inviting us to that event.
This research has been supported by the Spanish Government–FEDER (MEGAFLOW project, grant no. RTI2018096064BC21) and the Junta de Andalucía–FEDER (grant no. UMA18Federja161).
This paper was edited by Maria Ana Baptista and reviewed by two anonymous referees.
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