An efficient two-layer landslide-tsunami numerical model: effects of momentum transfer validated with physical experiments of waves generated by granular landslides

A landslide-generated tsunami is a complex phenomenon that involves landslide dynamics, wave dynamics and their interaction. Numerous lives and infrastructures around the world are threatened by these events. Predictive numerical models are a suitable tool to assess this natural hazard. However, the complexity of this phenomenon 10 causes such models to be either computationally inefficient or unable to handle the overall process. Our model, which is based on shallow water equations, is developed to address these two problems. In our model, the two materials are treated as two different layers, and their interaction is resolved by momentum transfer inspired by elastic collision principles. The goal of this study is to demonstrate the validity of our model through benchmark tests based on physical experiments performed by Miller et al. (2017). A dry case is reproduced to validate the behaviour of the landslide propagation model 15 using different rheological laws and to determine which law performs the best. In addition, a wet case is reproduced to investigate the influence of different still water levels on both the landslide deposit and the generated waves. The numerical results are in good agreement with the physical experiments, thereby confirming the validity of our model, particularly concerning the novel momentum transfer approach.


Introduction 20
The generation of a tsunami by a landslide is a complex phenomenon that involves landslide dynamics, interactions between the landslide mass and a water body, propagation of a wave and its spread on the shore. A landslide could be either submarine, partially submerged or subaerial. Regions that combine steep slopes and water bodies are the most susceptible to landslidegenerated tsunamis. For instance, fjords (Åknes: Ganerød et al., 2008;Harbitz et al., 2014;Lacasse et al., 2008, Lituya Bay: Fritz et al., 2009, Slingerland and Voight, 1979, Weiss et al., 2009, volcanos in marine environments (Stromboli: Tinti et al., 25 2008, Réunion Island: Kelfoun et al., 2010, and regions such as lakes and reservoirs in mountainous areas are prone to this phenomenon (Chehalis Lake: Roberts et al., 2013, Vajont: Bosa andPetti, 2011;Slingerland and Voight, 1979;Ward and Day, 2011;Kafle et al., 2019; Lake Geneva: Kremer et al., 2012, Lake Lucerne: Schnellmann et al., 2006. On lowland water bodies, landslide-generated tsunamis are also encountered when particular conditions exist, such as quick clays or glacio-fluvial deposit slopes (Rissa: L'Heureux et al., 2012, Nicolet Landslide: Jaboyedoff et al., 2009Franz et al., 2015, Verbois 30 reservoir: Franz et al., 2016. Landslide-generated tsunamis severely threaten lives and infrastructures, as evidenced in Papua New Guinea in 1998, where a submarine landslide killed 2200 people (Tappin et al., 2008). To assess this hazard, predictive models must be used. These models can be separated into 3 different types: 1) empirical equations from physical models (Enet and Grilli, 2007;Heller et al., 2009., Fritz et al., 2004, Miller et al., 2017, Kamphuis and Bowering, 1970Slingerland and Voight, 1979), 2) physical 35 models reproducing site-specific setups (Åknes: Harbitz et al., 2014, Vajont: Ghetti, 1962in: Ghirotti et al. 2013, and 3) numerical models. Numerical models can be governed by different sets of equations, such as shallow water equations (Simpson and Castelltort, 2006;Harbitz et al., 2014;Franz et al. 2013Franz et al. , 2015Kelfoun 2011;Kelfoun et al., 2010;Mandli, 2013;Tinti and Tonini, 2013;Tinti et al. 2008), Boussinesq equations (Harbitz et al., 2014;Løvholt et al., 2013;, and RANS equations (Abadie et al., 2010;Clous & Abadie, 2019). 40 The assessment of natural hazards requires the use of predictive models, and in the case of numerical models, they need to be able to reproduce sufficiently well the studied phenomenon while being efficient in terms of computational resources and easy to use and to implement. These requirements are indeed necessary for a tool designed for applied studies. From this point of view, the models based on RANS equations fulfil the first requirements but are particularly slow to run (Abadie et al., 2010;Clous & Abadie, 2019). On the other hand, models based on approximations such as shallow water equations and Boussinesq 45 equations, and their free-surface nature, inherently imply a loss in the quality of the reproduction but permits to be used on standard computer with fast computational time. The difficulty to simulate the propagation of two different materials, the interaction between the landslide and the water, and the propagation on the shores using free surface models is often solved by coupled approaches (Tan et al., 2018;Ma et al., 2015;Harbitz et al., 2014;Løvholt et al., 2015) or with an advanced multiphase approach (Pudasaini, 2014;Kafle et al., 2019). These hybrid approaches are a good trade-off between reproduction 50 quality and efficiency but not regarding the ease of use. Kelfoun et al. (2010) presented the Volcflow model, which has the ability to handle such behaviour and to address the qualityefficiency-usability requirement. In this model, the momentum transfer is performed by a set of equations that take into account the form drag and the skin friction drag, modified from the methodology reported by Tinti et al. (2006). Their approach is an elegant way to solve this type of problem; however, this method also relies on complex assumptions linked with the 55 incompatibility between the free-surface nature of the model and the hydrodynamic shape of the sliding mass and its associated drag coefficient. Xiao et al. (2015) simulated momentum transfer through a so-called "drag-along" mechanism, whichis relevant, but requires the implementation of coefficients that are subject to interpretation. The two-phase mass flow model proposed by Pudasaini (2012) simulates landslides and tsunamis within a single framework (Pudasaini, 2014;Kafle et al., 2019). 60 The numerical model we propose in this study is a two-layer model that combines a landslide propagation model and a tsunami model. The proposed numerical model is based on the non-linear shallow water equations in an Eulerian framework of the flow field. The transfer of momentum between the two layers is performed by assuming a "perfect" collision, in the sense that that there is no dissipation, neither in momentum nor in kinetic energy. Thus, it appears to be appropriate to compute the momentum transfer between two materials of which each cell has a certain velocity and a constant mass (or constant shape 65 and constant density). This approach has the advantage of containing a limited number of coefficients to be implemented by the operator. The "lift-up" mechanism (i.e., the change in height of the water level due to bed rise) also contributes to wave generation.
The aim of this study is to test the whole model (i.e., landslide and tsunami) and more specifically to examine the transfer of momentum between the two materials. Miller et al. (2017) provided a relevant benchmark test that highlights the momentum 70 transfer through its effect on the granular flow deposit and on the amplitude of the generated wave. Moreover, the granular flow is gravitationally accelerated, which is a relevant aspect to test the behaviour of the numerical model. Miller et al. (2017) investigated landslide-generated tsunamis. In their study, the landslide, which consisted of a gravitationally accelerated granular flow, was simulated alongside the wave. The interaction between the two elements was of particular 75 interest, and their reciprocal effects were highlighted. The momentum transfer obviously affected the wave behaviour and also influenced the landslide deposit. Miller et al. (2017) and  described the flume at Queen's University in Kingston (ON), Canada, where their physical experiments were conducted. This apparatus consisted of a 6.7 m aluminium plate inclined at an angle of 30°, followed by a 33-m-long horizontal section, and ending in a 2.4-m-long smooth impermeable slope of 27°; the width of the 80 flume was 2.09 m. Nine different scenarios were tested, in which the water depth was varied from h = 0.05 to 0.5 m; one of these scenarios was tested without water. For each scenario, 0.34 m 3 (510 kg) of granular material was released from a box at the top of the slope. The granular material consisted of 3-mm-diameter ceramic beads, which had a material density of 2400 kg/m 3 , a bulk density of 1500 kg/m 3 , a static critical state friction angle of 33.7°, and a bed friction angle of 22°. The flat floor of the flume was composed of concrete. The bed friction angle was estimated as approximately 35°. The wave amplitudes 85 were determined by 5 probes, and the slide characteristics were captured with a high-speed camera (Cam 1).

Numerical model
The numerical model presented in this paper attempts to reproduce the experiments presented in Miller et al. (2017). The 90 models for both the tsunami and the landslide simulations are based on the non-linear shallow water equations. The two layers are computed simultaneously (i.e., in the same iteration). The landslide layer is computed first and is considered as a bed change from the water layer (at each time step). The transfer of momentum also occurs at every time step.

Depth-averaged models
The model is based on two-dimensional shallow water equations: 95 where H is the depth; u and v are the components of the depth-averaged velocity vector along the x and y directions, 100 respectively; and g is the gravitational acceleration. The source term S differs for the landslide (Eq. 8) and tsunami models (Eq. 19). Thus, the two formulations of the source term are specifically described in their respective sections (Sect. 3.1.1 and 3.1.2). The conservative discrete form is expressed as follows: eq. 4 where + 1 2 , is the intercell numerical flux corresponding to the intercell boundary at = + 1 2 between cells and + 1 105 and , + 1 2 is the intercell numerical flux corresponding to the intercell boundary at = + 1 2 between cells and j+1. The Lax-Friedrichs (LF) scheme defines these terms as follows (Franz et al., 2013;Toro, 2001): This numerical scheme is chosen because of its non-oscillatory behaviour, its capacity to withstand rough beds, and its 110 simplicity. See Franz et al. (2013) for more information concerning the choice of this numerical scheme.

Landslide model
The simulation of granular flow utilizes widely used rheological laws, among which the most commonly encountered are the Coulomb, Voellmy, and Bingham rheological laws (Iverson et al., 1997;Hungr and Evans, 1996;Longchamp et al. 2015;Pudasaini and Hutter, 2007;Pudasaini, 2012;Kelfoun, 2011, McDougall, 2006Pouliquen and Forterre, 2001). The continuum 115 equations used are the previously described equations. The source term S specifically governs the forces driving the landslide propagation: eq. 7 where ρs is the landslide bulk density, T is the total retarding stress, and MTs is the momentum transfer from the water to the sliding mass. The driving components of gravity GR and pressure term P are defined as follows (Pudasaini and Hutter, 2007): 120 eq. 9 where Hs is the landslide thickness, α is the bed slope angle, and K is the earth pressure coefficient. The density ρ is the apparent density of the landslide. This means that ρ is equal to the density of the slide ρs when the slide is subaerial and ρ = ρs-ρw (density of the water) when the slide is underwater (Kelfoun et al., 2010;Skvortsov, 2005). Since each term is divided by ρs 125 in Eq. (7), the relative nature of the density becomes effective. The total retarding stress T (Tx, Ty) is composed of the resisting force (frictional resistance) between the landslide and the ground. T differs depending on the chosen rheological law.
The Coulomb rheology can be used considering flow with either isotropic or anisotropic internal stresses. This difference is handled with the use of the earth pressure coefficient K (Hungr and McDougall, 2009;Kelfoun, 2011, Iverson andDenlinger, 2001;Pudasaini and Hutter, 2007). For the isotropic case, K is set to 1 (Kelfoun, 2011). In the anisotropic case, K designates 135 whether the flow is in compression (positive sign) or in dilatation (negative sign), for which the coefficients are denoted as Kpassive or Kactive (Hungr and McDougall, 2009;Pudasaini and Hutter, 2007), respectively. These terms are defined as follows: where the variable φint is the internal friction angle. 140 The Voellmy rheology Voel combines Coulomb frictional rheology with a turbulent behaviour: The first term is the Mohr-Coulomb frictional law, whereas the second term, which was originally presented by Voellmy (1955) for snow avalanches, acts as drag and increases the resistance with the square of the velocity. The turbulence coefficient ξ corresponds to the square of the Chézy coefficient, which is related to the Manning coefficient n by = 1 6 ⁄ ⁄ 145 (MacDougall, 2006). The turbulence coefficient presented in Kelfoun (2011) is equivalent to the inverse of the turbulence coefficient presented herein times g (1/(ξg)). However, the physical basis of the Voellmy rheology is questionable (Fisher et al., 2012).
The Bingham rheology combines plastic and viscous rheological laws and is defined as follows (Skvortsov, 2005): where T0 is the yield stress, which is the stress at which the slide to starts or stops to move; μ is the dynamic viscosity; and d1 is the relative thickness of the shear layer. The latter variable is used to mimic the behaviour of Bingham flow that contains two distinct layers: a solid layer (the plug zone) and a shear layer (the shear zone, d1). The determination of d1 can be performed 155 automatically (e.g., Skvortsov, 2005), but in this study, the use of a constant value provided better results.

Tsunami model
For the tsunami model, the source term S includes a consistency term and a momentum transfer term and is defined as follows: eq. 18 where Hw is the water thickness, B is the bed elevation (including the thickness of the sliding mass), and MTw is the momentum 160 transfer from the slide to the water. The first term confers consistency to the model, which has been validated in Franz et al. (2013). The second term is the momentum transfer between the landslide and the water.
The wet-dry transition is realized by an ultrathin layer of water hmin that covers the whole topography (or the dry state). This permits the avoidance of zeros in the water depth array. Nevertheless, such a situation would lead to water flowing down the slopes after some iteration. Thus, to prevent this numerical artefact, the thin layer is governed by viscous equations (Turcotte 165 and Schubert, 2002):

eq. 20
A threshold Retr is set for a Reynolds number that determines which set of equations (between viscous equations and shallow 170 water equations) is used at each location in the tsunami model: = eq. 21 eq. 22

Momentum transfer
The interaction between the landslide and the water proposed by Kelfoun et al. (2010) is based on a formulation of drag both 175 normal and parallel to the displacement. This formulation involves two coefficients that need to be set manually, which is a manipulation this study aims to avoid. Moreover, they claim that their equation is a rewritten form of the equation presented in Tinti et al. (2006). However, in the latter, the depth of the landslide Hs is taken into account, whereas in Kelfoun et al. (2010), they account for the gradient of the landside depth.
In Xiao et al. (2015), the so-called "drag-along" approach also entails undesired user-defined coefficients from our point of 180 view and, when tried in our code, never fit the experiment data.
Regarding those two unsatisfying approaches, we decided to try an unconventional method. Based on a semi-empirical approach that fits the experimental data, the implementation of momentum transfer in our code is inspired by the rigid collision principle. This principle is relevant because 1) the kinetic energy of the system is conserved, 2) a true interaction between particles is not possible in a model based on shallow water equations (free surfaceno third dimension), and 3) the exchange 185 of momentum between the landslide mass and the water is perfectly symmetric.
As a reference case, we consider velocity changes during the elastic collision of two rigid bodies. The conservation of momentum in elastic collision is given by the following expression: As the kinetic energy is also conserved, the following constraint applies: where uw and us are the velocities for the 'water' and the 'slide' masses, respectively (subscript b = before collision and subscript a = after).
We assumed that the mass before collision remained constant after collision. Under this simplifying assumption, the two conservation equations can be used to solve for the two velocities after collision: 195 ) eq. 25 ) eq. 26 The discrete "finite control volume" masses m, having lateral lengths dx and dy, are defined as follows: = * * * eq. 27 = * * * eq. 28 where dt is the time discretization. Note that dx and dy are cancelled out. The right-hand sides of Eqs. (29) and (30) represent  205 the momentum exchange source terms during collision.
The momentum transfer during the collision reference case was modified by a 'shape factor' as a fitting parameter to reproduce the laboratory experiments from Miller et al. (2017), resulting in the following expressions: eq. 32 The shape factor SF is defined from experiments to match the wave amplitude and the landslide deposit. This shape factor consists of a non-dimensional value that depends on the ratio between the maximum landslide thickness at impact (recorded during the simulation by the numerical code) and the still water level h:

Results
The landslide and the tsunami models are computed in 2D (x and y), whereas the results, such as the landslide thickness or the 225 water elevation, can be represented visually in the third dimension (z), or in other words, in 2.5D, as illustrated in Fig. 3.

230
In this study, everything is computed in two-dimension, but the interpretations and the presentation of the results are done through longitudinal cross-sections across the centre of the numerical flume (Fig. 4). The Fig. 4 illustrate the generation of the wave: at 0 s, the landslide impacts the water; at 0.3 s, the landslide velocity is greater than the water velocity and the momentum is transferred; at 0.6 s, the front of the slide starts to stop while the back continues to transfer the momentum to the water; at 0.9, the momentum is nearly completed and the wave propagate along the flume. 235

Landslide
This section presents the results concerning the granular landslide. Furthermore, this section discusses the behaviour of the landslide propagation using different rheological laws and the effect of the water depth on the landslide deposit. 240

Dry case
The dry case investigates the propagation of the granular material using various rheological laws. The rheological laws implemented herein are the Voellmy, Coulomb (flow with isotropic and anisotropic internal stresses) and Bingham rheological models. The velocities, the thicknesses and the deposit shapes obtained through the numerical simulation are compared to those data obtained from the physical experiment to identify and select the best solution. 245 The Bingham rheology is set by best fit with a shear zone relative thickness d1 of 0.6, a yield stress T0 of 12 Pa, and a dynamic viscosity μ of 1.6 Pa•s.
Concerning the Voellmy rheology, the determination of the turbulence coefficient ζ is performed by trial and error to obtain 250 the best fit (back analysis). Thus, the turbulence coefficient ζ, as described in Evans (1996) andin McDougall (2006), is set to 250 m/s 2 . The bed friction angle φ bed of 22°, given in Miller et al. (2017), was reduced to 11°. Indeed, the Voellmy rheology uses significantly lower values (Hungr and Evans, 1996). This study uses the same ratio (~0.5) between "classical" φ bed and "Voellmy" φ bed as the one presented in Hungr and Evans (1996) for cases with similar friction angles and turbulence coefficients ("Voellmy" φ bed = 11°). 255 Regarding the two Coulomb models, we use the bed friction angle φ bed of 22°, as measured in the physical experiment (Miller et al, 2017). In addition, the anisotropic Coulomb rheological model considers the internal friction angle φ int , which is 33.7° (Miller et al, 2017).
In Miller et al. (2017), the velocity and the thickness of the landslide at impact are estimated through high-speed camera footage analysis with a still water depth h of 0.25 m. To measure the same variables of the simulated granular flow, the values 260 are recorded using a corresponding window (Cam 1, Fig. 1). Figure 5 shows the temporal evolution of the flow thickness and velocity captured at the numerical equivalent location of Cam 1 (Fig. 1). The numerical models do not fit the physical simulation very well. This poor fit can be explained by the diffuse nature of a granular flow boundary that is not replicable by the shallow water assumption (continuum mechanics) and by the absence of expansion in the numerical moving mass. Nevertheless, the results from the numerical and physical models are 265 close enough to validate globally the different numerical models without allowing their discrimination. Consequently, analysing the deposit (Fig. 6) is the way to identify best fitting rheological model. The Bingham rheological model does not correctly reproduce the shape of a granular deposit. The Voellmy model performs better than the Bingham model in this respect, but in comparison with the two Coulomb frictional models, the Voellmy model is not satisfactory.
Indeed, the two Coulomb rheological models (anisotropic and isotropic) fit the best with the observed deposit, which was 275 expected because frictional rheological laws are typically developed to describe granular flows. The rear parts of the deposits are correctly located, whereas the fronts are slightly too distant. However, this imperfection is negligible and could be attributed to numerical diffusion. The deposit simulated with the isotropic Coulomb model is slightly closer to the real deposit than that simulated with the anisotropic model; this method has the advantage of being simple (only the bed friction angle φ bed is implemented). 280 Since the velocities and the thickness are of a realistic magnitude, the deposit shapes of the two Coulomb models correctly reproduce the real case. Furthermore, due to the ease of implementation, the isotropic Coulomb model is the rheological model of choice. In Sect. 4.1.2, this model is used to study the wet cases. 285

Wet cases
This section investigates the interaction between the landslide and the water. More precisely, this section investigates the effect of the momentum transfer on the deposit shape for different water levels. Figure 7a shows the results for still water depths h of 0.05, 0.08, and 0.1 m. The deposits resulting from the numerical simulation (solid lines) are compared with the physical model observations (dashed lines), which shows a rather good similarity when focusing on the height of the piles. The 290 numerical deposition shape for a still water depth of 0.05 m fits well the physical shape, also regarding the spread. Concerning still water depths h of 0.08 and the 0.1 m, the numerical granular flows stop further than the real flows.
At still water depths of 0.17, 0.2, and 0.25 m (Fig. 7b), the numerical and physical results are in good agreement; however, the "tails" and the apexes of the deposits are located slightly farther away in the numerical simulation. In contrast, for still water depths h of 0.38 and 0.5 m (Fig. 8), the deposit shapes obtained by the numerical simulations stop slightly ahead of the real deposits. Nevertheless, the deposits are of equivalent heights. The momentum transfer acts correctly on the granular flow as the global correspondence between the numerical and physical deposition pattern is good. In fact, it is the combination of the momentum transfer (Eqs. 31 and 32) with the relative density ρ (Eqs. 7 to 11) that performs well. This is highlighted by the simulated granular landslide without momentum transfer, which 305 travels excessively far (Figure 8, red line). The travel distance in this case is even greater than that in the dry case (the result of the isotropic Coulomb model presented in Fig. 6) due to the effect of the relative density ρ. Indeed, the "drop in density" when the granular flow enters the water body reduces the total retarding stress T in particular (Eqs. 10 and 11; alongside P (Eq. 9) and GR (Eq. 8), which is negligible on flat surfaces). It is worth noting that without momentum transfer or relative density, the model would lead to the same deposit as the dry case. 310

Wave
This section investigates the momentum transfer between the slide and the generated wave This effect is analysed for different still water depths h (0.05, 0.1, 0.2, and 0.5 m) through probes located at different distances from the bottom of the slope (2.3, 15, and 23 m; Fig. 9). Concerning the case with the smallest still water depth (h = 0.05 m), the numerical simulation reproduces the wave observed in the physical experiment very well in terms of amplitude and timing at each probe. Note that the simulated 315 wave is taller than the real wave in the very near field (2.3 m gauge). For a still water depth h of 0.1 m, the timing is good at the 2.3 m gauge, but, as previously described, the numerical wave is taller. Concerning the gauges at 15 and 23 m, the wave celerity is faster and its amplitude is smaller in the numerical simulation than in the physical experiment. Moreover, the wave train observed in the physical model is non-existent in the numerical model. Except for the equivalence in amplitude in the near field, the same observations apply for a still water depth h of 0.2 m. For a still water depth of 0.2 m, a reflected wave is 320 present at the 23 m gauge after approximately 23 s. The numerically simulated wave arrives slightly earlier than the observed wave. Concerning the case of a still water depth h of 0.5 m, the simulated wave is slightly smaller than the real wave, and the reflected wave (at 28, 23, and 18 s) is visible at the 3 gauge locations with a good correspondence in terms of time.  Heller and Hager (2010) proposed a relationship between the landslide characteristics and the near-field maximum amplitude of the generated wave through the concept of the impulse product parameter P. The impulse produce parameter includes the governing parameters related to the landslide and the still water depth. The maximum wave amplitude can be predicted as a 330 function of P through Eq. (37). This approach is relevant to our study because it inherently considers the momentum transfer occurring during wave generation. The following values are captured at the impact zone (Cam 1, Fig. 1) for the sliding mass and in the near-field area for the wave. The relative maximum near-field wave amplitude Am is defined by the following expression: = 0.25 1.4 0.8 eq. 34 335 where S is the relative landslide thickness and Fr is the Froude number, which are defined as follows:

Impulse product parameter
The relationships between the impulse product parameter and the Froude number and the relative landslide thickness were found empirically through a large set of tests based on different landslide setups (Fritz et al., 2004). The impulse product 340 parameter P defined by Heller and Hager (2010) is expressed as follows: where M is the relative landslide mass, which is defined by the following expression: where ms is the landslide mass and b is the flume width. The near-field relationship between the P and Am is defined as follows 345 (Heller and Hager, 2010):  (Fritz et al., 2004;Heller and Hager, 2010) falls between those limits for P<9. 350 The near-field relationships between P and Am obtained in the present study correspond very well with those obtained by Miller 355 et al. (2017) (Fig. 10). On the other hand, for P<9 (as originally presented in Heller and Hager, 2010), the results of this study are located within a range of ±30%.

Computation time
All the simulations are performed on a conventional computer. It is an Acer Aspire V17 Nitro. The computations are performed on an Intel® Core ™ i7-7700HQ CPU @ 2.80GHz with 16 Go of RAM. For a complete simulation, i.e. simulation that include 360 landslide and wave propagation, for a duration of 30 s (as presented in Fig. 9), the computation time is of 16'650 s. The cell size is 0.01 x 0.01 m and the grid size is 45.2 m x 2.1 m (949'000 cells).

Discussion
The numerical model displays a taller wave in the near field, which can be explained by the fact that the model does not reproduce the breaking of the wave. This lack also implies that the model cannot consider the impact crater as long as the 365 steepness of the water surface is not steeper than sub-vertical. This discrepancy is inherent to the shallow water model and its two-dimensional nature.
The physical experiments produce wave trains for water levels of 0.1 and 0.2 m. This phenomenon is not reproduced by the numerical model because of the absence of breaking in the unstable numerical waves, (Miller et al. 2017;Ruffini et al. 2019) along with the lack of frequency dispersion (Ruffini et al. 2019). These mismatches are due to the incomplete physics inherent 370 to shallow water equations approximations.
On the other hand, the fronts of the waves are very different. The "excess" volume of water at the front of the numerical wave is also partially explained by the lack of energy dissipation that would occur during breaking. On average, the simulated water level located at the wave train "match" the trough and the crests. For those cases, the imperfect reproductions are, however, sufficiently close in terms of celerity and volume to be considered relevant. This consideration was in addition confirmed by 375 the good match of the reflected wave (e.g. Fig. 9, h=0.2, x=25, 23 s).
The general observation of the evolution of the wave (Fig. 9) shows that the decay occurring in the physical experiment is present in the numerical simulation. This fact also strengthens the general validity of the whole numerical model.
Inherently, as the impulse product parameter values obtained through a wide set of experiments (Heller and Hager, 2010;Miller et al., 2017) fall into an envelope of ±30%, our near-field results, which also fall into these limits, strongly confirm the 380 validity of our model and our momentum transfer approach.

Conclusions
This study aimed to validate our two-layer landslide-generated tsunami numerical model, based on the shallow water equations, particularly concerning the momentum transfer between the landslide and the water. This is performed through the reproduction of the physical experiments conducted by Miller et al. (2017). A dry case is simulated to document the behaviour 385 of the landslide propagation model using different rheological laws. In addition, a wet case is reproduced to investigate the influence of different still water levels on both the landslide deposit and the generated waves.
The dry case shows that the two Coulomb rheological models (flow with isotropic or anisotropic internal stresses) correctly reproduce the deposit observed in the physical model studied by Miller et al. (2017). The isotropic Coulomb model is the simplest and easiest to implement and is chosen to study the wet case. 390 The numerical simulation of the wet case investigates the abilities of the model to correctly handle momentum transfer. This case focuses on both the effect of the water on the landslide deposit and the effect of the landslide on the resulting wave. These effects are investigated through different water levels, and it appears that the landslide deposit obtained by the numerical simulations fits well with the physical model observations. On the other hand, the numerical waves behave similarly to the waves in the physical model. Despite imperfections, the combined results from investigating these two effects permits us to 395 consider that, overall, the model effectively handles the complex phenomenon occurring during the interaction between the landslide and the water. In addition, the choice to transfer the momentum through the simple "perfect" collision principle is verified to be relevant.
A comparison involving the impulse product parameter particularly highlights that our model satisfactorily reproduces the physical experiment of Miller et al. (2017). The values of Am versus P presented in Heller and Hager (2010) are based on their 400 211 experiments in addition with 137 experiments performed by Fritz et al. (2002) and86 by Zweifel (2004). Hence, the validity of our model is further strengthened by the fact that the results of our model also fit well with those experiments. Finally, our model is validated by a benchmark test performed herein, as this approach is very simple to implement and is very efficient in terms of computational resources. Therefore, we consider our model as a tool of choice for the assessment of landslide-generated tsunami hazards. 405 Author contributions. MF and MJ conceived the project. MF MJ RM and AT designed the benchmark tests. MF and YP developed the model code, RM ant AT provided the data and MF carried out the simulations. MF prepared the manuscript with review and editing from all co-authors.

410
Competing interests. The authors declare that they have no conflict of interest.
Acknowledgements. The authors are thankful to Dr. Shiva P. Pudasaini for sharing his valuable insights about landslides impacting water bodies, particularly on the associated complexity of momentum transfer. We acknowledge two anonymous reviewers for their constructive comments. 415

Notation
The following symbols are used in this paper: