Multilayer-HySEA model validation for landslide generated tsunamis. Part II Granular slides

The overall objective of the present work is to benchmark the novel Multilayer-HySEA model using laboratory experiment data for landslide generated tsunamis. In particular, this second part of the work deals with granular slides, while the ﬁrst part, in a companion paper, considers rigid slides. The experimental data used have been proposed by the US National Tsunami Hazard and Mitigation Program (NTHMP) and established for the NTHMP Landslide Benchmark Workshop, held in January 2017 at Galveston. Three of the seven benchmark problems proposed in that workshop dealt with tsunamis generated by rigid slides and are collected in the companion paper (Mac´ıas et al., 2020). Another three benchmarks considered tsunamis generated by granular slides. They are the subject of the present study. In order to reproduce the laboratory experiments dealing with granular slides, two models need to be coupled, one for the granular slide and a second one for the water dynamics. The coupled model used consists of a new and eﬃcient hybrid ﬁnite volume/ﬁnite diﬀerence implementation on GPU architectures of a non-hydrostatic multilayer model coupled with a Savage-Hutter model. A brief description of model equations and the numerical scheme is included. The dispersive properties of the multilayer model can be found in the companion paper. Then, results for the three NTHMP benchmark problems dealing with tsunamis generated by granular slides are presented with

iments were developed for those cases and for tsunami model benchmarking. 68 In contrast, some early models (e.g., Heinrich (1992  The basic reference for these three benchmarks, but also the three ones related 96 to solid slides and the Alaska field case, all of them proposed by the NTHMP, 97 is Kirby et al. (2018). That is a key reference for readers interested in the 98 benchmarking initiative in which the present work is based on.
where g is the gravity acceleration (g = 9.81 m/s 2 ); H(x) is the non-erodible (do 104 not evolve in time) bathymetry measured from a given reference level; z s (x, t) 105 represents the thickness of the layer of granular material at each point x at time 106 t; h(x, t) is the total water depth; η(x, t) denotes the free surface (measured 107 form the same fixed reference level used for the bathymetry, for example, the 108 mean sea surface) and is given by η = h + z s − H; u(x, t) and u s (x, t) are fluid and the granular layer. Finally, here we will consider τ P (x, t) as the friction 113 term given in Pouliquen and Forterre (2002) to be described more precisely in 114 the next section.

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System (1) presents the difficulty of considering the complete coupling be-116 tween sediment and water, including the corresponding coupled pressure terms.

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That makes its numerical approximation more complex. Moreover, it makes 118 also difficult to consider its natural extension to non-hydrostatic flows.

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Now, if ∂ x η is neglected in the momentum equation of the granular material, 120 that is, the fluctuation of pressure due to the variations of the free-surface are 121 neglected in the momentum equation of the granular material, then the following 122 weakly-coupled system could be obtained: where the first system is the standard one-layer shallow-water system and the 125 second one is the one layer reduced-gravity Savage-Hutter model ( coupling between these two models is performed through the boundary con-139 ditions at their interface. The parameter r represents the ratio of densities 140 between the ambient fluid and the granular material. Usually where ρ s stands for the typical density of the granular material, ρ f is the density 142 of the fluid (ρ s > ρ f ), and ϕ represents the porosity (0 ≤ ϕ < 1). In the 143 present work, the porosity, ϕ, is supposed to be constant in space and time 144 and, therefore, the ratio r is also constant. This ratio ranges from 0 to 1 (i.e. 145 0 < r < 1) and, even on a uniform material is difficult to estimate as it depends for α ∈ {1, 2, . . . , L}, with L the number of layers and where the following 161 notation has been used: where f denotes one of the generic variables of the system, i.e., u, w and p; 163 ∆s = 1/L and, finally, H(x) is the unchanged non-erodible bathymetry measured from the same fixed 171 reference level. τ α = 0, for α > 1 and τ 1 is given by where τ b stands for an classical Manning-type parameterization for the bottom 173 shear stress and, in our case, is given by and n a (u s −u 1 ) accounts for the friction between the fluid and the granular layer.

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The latest two terms are only present at the lowest layer (α = 1). Finally, for for the lowest layer (α = 1), given by It should be noted that both models, the hydrodynamic model described here the Froude number F r = us √ gzs . The friction law is given by: Folterre friction law is due to the buoyancy effects, which must be taken into 218 account only in the case that the granular material layer is submerged in the 219 fluid. Otherwise, this term must be replaced by 1.

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System (3) can be written in the following compact form: Analogously, the multi-layer non-hydrostatic shallow-water system (5) can also 225 be expressed in a similar way: where https://doi.org/10.5194/nhess-2020-172 Preprint. Discussion started: 15 September 2020 c Author(s) 2020. CC BY 4.0 License. terms: The non-hydrostatic corrections in the momentum equations are given by and finally, the operator related with the incompressibility condition at each 231 layer is given by: The discretization of systems (6) and (7) becomes difficult. In the present work, technique. Initially, the systems (6) and (7) are expressed as the following non-236 conservative hyperbolic system: Then, the non-hydrostatic pressure corrections p 1/2 , · · · , p L−1/2 at the vertical 244 interfaces are computed from which requires the discretization of an elliptic operator that is done using stan-246 dard second-order central finite differences. This results in a linear system than 247 in our case it is solved using an iterative Scheduled Jacobi method (see Adsuara  underwater. Figure 2 shows a schematic picture of the experiment set-up. The    Finally d s was set to 1.5 · 10 −3 and 10 · 10 −3 depending on the test case. Figure   337 6 shows the comparison for Case 1. In this case, the numerical results show an 338 very good agreement when compared with lab measured data and, in particular, 339 the two leading waves are very well captured. Figure 7 shows the comparison 340 for Case 2. In this case, the agreement is good, but larger differences between 341 model and lab measurements can be observed. Figure 8 shows the location of the  The vertical structure of the fluid layer is modeled using three layers. Similar 369 results were obtained with 2 layers.

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In the beginning, the slide box is driven using four pneumatic pistons. Here wave-gauge in coordinates (r, θ • ) are given more precisely in Table 1. Before r 5.12 8.5 14 24.1 3.9 5.12 8.5 3.9 5.12 Table 1: Location of the 9 waves gauges referenced to the toe's slope.