domain: spi sde

text:
Alexandre Paris

Philippe Heinrich

Stéphane Abadie

Landslide tsunamis: Comparison between depth-averaged and Navier-Stokes models

Two approaches are proposed to simulate tsunamis generated by granular landslides: a depth-averaged model, AVALANCHE, and laminar Navier-Stokes simulations using the OpenFOAM model. Both models are validated against two 2D benchmarks, a subaerial and a submerged one involving a triangle initial slide shape. In both models, the landslide is defined as a viscous fluid flowing downslope. Except in the first instants, both models can reproduce either the landslide behavior or the generated first water waves but can not reproduce simultaneously both the landslide and the water surface. There is an overlap in the optimal viscosity range between both models. Sensitivity studies are carried out by varying the slope angle and the landslide submergence for the subaerial benchmark. The largest waves are obtained for initial landslide position close to the free surface. The height of the generated waves increases linearly with the slope angle and the landslide Reynolds number in the depth-averaged model. The relationships are more complex in the Navier-Stokes model. For low slide Reynolds numbers and with an initial slide close to the free surface, both models produce similar waves. Nine additional cases are performed with a k -ϵ turbulence closure model and varying the submergence and the slope angle.

Introduction

Landslide tsunamis have been reported in coastal areas such as fjords [START_REF] Naranjo | Mass movement-induced tsunamis: main effects during the Patagonian Fjordland seismic crisis in Aisén (45°25'S), Chile[END_REF][START_REF] L'heureux | Identification of weak layers and their role for the stability of slopes at Finneidfjord, Northern Norway[END_REF], rivers [START_REF] Lefebvre | The September 5, 1987, landslide on the La Grande River, James Bay, Quebec, Canada[END_REF]) and volcanic islands [START_REF] Tinti | Numerical simulation of the landslide-induced tsunami of 1988 on Vulcano Island, Italy[END_REF][START_REF] Heinrich | Numerical simulation of the December 1997 Debris Avalanche in Montserrat, Lesser Antilles[END_REF][START_REF] Tinti | The landslides and tsunamis of the 30th of December 2002 in Stromboli analysed through numerical simulations[END_REF], in deep water [START_REF] Bondevik | The Storegga slide tsunami -comparing field observations with numerical simulations[END_REF] or in closed water basins such as lakes [START_REF] Wang | Numerical simulation of the December 4, 2007 landslide-generated tsunami in Chehalis Lake, Canada[END_REF][START_REF] Gylfadóttir | The 2014 Lake Askja rockslide-induced tsunami: Optimization of numerical tsunami model using observed data[END_REF]) and reservoirs [START_REF] Ward | The 1963 Landslide and Flood at Vaiont Reservoir Italy. A tsunami ball simulation[END_REF], Xiao et al. 2015). They constitute a major natural hazard for coastal populations and infrastructure and therefore a better understanding of the processes involved is required.

Since the early 2000s, the interest for landslide tsunamis has grown considerably. An extensive literature is available on historical cases and their diverse numerical simulations [START_REF] L'heureux | The 1888 shoreline landslide and tsunami in Trondheimsfjorden, central Norway[END_REF][START_REF] Ward | The 1963 Landslide and Flood at Vaiont Reservoir Italy. A tsunami ball simulation[END_REF][START_REF] Redfield | Analysis of the 2007 earthquake-induced Punta Cola rockslide and tsunami, Aysén Fjord[END_REF][START_REF] Franz | Testing a landslide-generated tsunami model. The case of the Nicolet landslide (Québec, Canada)[END_REF], Wang et al. 2015, Xiao et al. 2015[START_REF] Glimsdal | The 29th January 2014 submarine landslide at Statland, Norway-landslide dynamics, tsunami generation, and run-up[END_REF][START_REF] Huang | Analysis of the Tangjiaxi landslide-generated waves in the Zhexi Reservoir, china, by a granular flow coupling model[END_REF][START_REF] Gylfadóttir | The 2014 Lake Askja rockslide-induced tsunami: Optimization of numerical tsunami model using observed data[END_REF][START_REF] Haeussler | Submarine deposition of a subaerial landslide in Taan Fiord, Alaska[END_REF][START_REF] Grilli | Modelling of the tsunami from the December 22, 2018 lateral collapse of Anak Krakatau volcano in the Sunda Straits, Indonesia[END_REF]. Landslide tsunamis are rarely observed therefore their study is generally based on experimental or numerical models. Recent reviews of landslide tsunamis models can be found in Yavari-Ramshe & Ataie-Ashtiani (2016) and [START_REF] Kim | Three Dimensional Landslide Generated Tsunamis: Numerical and Physical Model Comparisons[END_REF].

The studies associated to the last significant reported event, the December 23, 2018 Anak Krakatau collapse, which generated a 80 to 100 m high tsunami in the near-field and 1 to 2 m water waves on the coasts of the Sunda Strait [START_REF] Grilli | Modelling of the tsunami from the December 22, 2018 lateral collapse of Anak Krakatau volcano in the Sunda Straits, Indonesia[END_REF], Paris et al. 2020[START_REF] Heidarzadeh | Numerical modeling of the subaerial landslide source of the 22 December 2018 Anak Krakatoa volcanic tsunami, Indonesia[END_REF][START_REF] Borrero | Field Survey and Numerical Modelling of the December 22, 2018 Anak Krakatau Tsunami[END_REF], illustrate the plurality of the numerical models usually employed. [START_REF] Grilli | Modelling of the tsunami from the December 22, 2018 lateral collapse of Anak Krakatau volcano in the Sunda Straits, Indonesia[END_REF] used a 3D model (NHWAVE) for the landslide tsunami generation and a 2D depth-integrated (2DH) model (FUNWAVE-TVD) for the tsunami propagation; [START_REF] Paris | The December 22, 2018 Anak Krakatau, Indonesia, landslide and tsunami: preliminary modeling results[END_REF], a unique 2DH model (AVALANCHE) for the landslide simulation, the tsunami generation and propagation; [START_REF] Heidarzadeh | Numerical modeling of the subaerial landslide source of the 22 December 2018 Anak Krakatoa volcanic tsunami, Indonesia[END_REF], a 2DH model (COMCOT) for the tsunami propagation with a simple initial elevation in order to introduce a generated water wave and Borrero et al. (2020) a 2DH model (pCOULWAVE) with an initial tsunami waveform derived from an analytical solution.

This highlights the existence of two methods to simulate landslide tsunamis. In the first one, the landslide is simulated separately and then introduced in a tsunami propagation model as an initial perturbation. In the second, landslide simulation, tsunami generation and propagation take place in the same model. The first approach allows to simulate the landslide with a more realistic 3D model before using a faster 2DH model for the tsunami propagation [START_REF] Grilli | Modelling of the tsunami from the December 22, 2018 lateral collapse of Anak Krakatau volcano in the Sunda Straits, Indonesia[END_REF], Abadie et al. 2020). The second approach has the advantage of being more flexible since only one code is needed. Moreover, the computational cost may be lower when both landslide and tsunami simulations are based on depth-averaged equations.

The landslide rheology is also a critical parameter in a landslide tsunami model. Numerical models considering rigid landslide [START_REF] Fuhrman | Tsunami generation, propagation and run-up with a high-order Boussinesq model[END_REF][START_REF] Ataie-Ashtiani | Numerical simulation of wave generated by landslide incidents in dam reservoirs[END_REF][START_REF] Bosa | Shallow water numerical model of the wave generated by the Vajont landslide[END_REF][START_REF] Dutykh | Boussinesq modeling of surface waves due to underwater landslides[END_REF] are opposed to deformable landslides models [START_REF] Serrano-Pacheco | A finite volume method for the simulation of the waves generated by landslides[END_REF][START_REF] Kelfoun | Landslide-generated tsunamis at Réunion Island[END_REF][START_REF] Glimsdal | The 29th January 2014 submarine landslide at Statland, Norway-landslide dynamics, tsunami generation, and run-up[END_REF][START_REF] Huang | Analysis of the Tangjiaxi landslide-generated waves in the Zhexi Reservoir, china, by a granular flow coupling model[END_REF]). Among the last category, the landslide can be defined, for example, as a granular flow following a Coulomb law friction [START_REF] Mangeney | Modeling of Debris Avalanche and Generated Water Waves: Application to Real and Potential Events in Montserrat[END_REF][START_REF] Kelfoun | Landslide-generated tsunamis at Réunion Island[END_REF][START_REF] Macías | The Al-Borani submarine landslide and associated tsunami. A modelling approach[END_REF], or as a viscous fluid, considering a Newtonian fluid [START_REF] Abadie | Numerical modeling of tsunami waves generated by the flank collapse of the Cumbre Vieja Volcano (La Palma, Canary Islands): Tsunami source and near field effects[END_REF][START_REF] Horrillo | A simplified 3-D Navier-Stokes numerical model for landslide-tsunami: Application to the Gulf of Mexico[END_REF] or non-Newtonian Bingham fluid [START_REF] Skvortsov | Numerical simulation of the landslide-generated tsunami in Kitimat Arm, British Columbia, Canada, 27[END_REF][START_REF] Serrano-Pacheco | A finite volume method for the simulation of the waves generated by landslides[END_REF]. Note that the last recent works introduced a two-phase formulation for a granular landslide combined with a volume of fluid model to capture the air-water interface [START_REF] Si | A general numerical model for surface waves generated by granular material intruding into a water body[END_REF][START_REF] Lee | Multi-phase flow simulation of impulsive waves generated by a sub-aerial granular landslide on an erodible slope[END_REF].

Comparison between 2DH and 3D models have been carried out for seismic tsunamis propagation [START_REF] Sogut | Performance Comparison of NAMI DANCE and FLOW-3D® Models in Tsunami Propagation, Inundation and Currents using NTHMP Benchmark Problems[END_REF], tsunami inundation (Qin et al. 2018a[START_REF] Horrillo | Performance Benchmarking Tsunami Models for NTHMP's Inundation Mapping Activities[END_REF][START_REF] Sogut | Performance Comparison of NAMI DANCE and FLOW-3D® Models in Tsunami Propagation, Inundation and Currents using NTHMP Benchmark Problems[END_REF] and generated coastal currents [START_REF] Lynett | Inter-model analysis of tsunami-induced coastal currents[END_REF][START_REF] Sogut | Performance Comparison of NAMI DANCE and FLOW-3D® Models in Tsunami Propagation, Inundation and Currents using NTHMP Benchmark Problems[END_REF]. In [START_REF] Kim | Three Dimensional Landslide Generated Tsunamis: Numerical and Physical Model Comparisons[END_REF], the authors question the applicability of 2DH models to landslide tsunamis, suggesting that 3D models taking into account the vertical acceleration would produce more accurate results. Over the past two decades, several 3D models based on Navier-Stokes equations have been developed, among them, THETIS [START_REF] Abadie | VOF/Navier-Stokes numerical modeling of surface waves generated by subaerial landslides[END_REF], FLUENT [START_REF] Biscarini | Computational fluid dynamics modelling of landslide generated water waves[END_REF], Fluidity [START_REF] Davies | Fluidity: A fully unstructured anisotropic adaptive mesh computational modeling framework for geodynamics[END_REF], NHWAVE [START_REF] Ma | Numerical simulation of tsunami waves generated by deformable submarine landslides[END_REF]), TSUNAMI3D [START_REF] Horrillo | A simplified 3-D Navier-Stokes numerical model for landslide-tsunami: Application to the Gulf of Mexico[END_REF], Splash3D [START_REF] Wu | Modeling the Slump-Type Landslide Tsunamis Part I: Developing a Three-Dimensional Bingham-Type Landslide Model[END_REF] and OpenFOAM [START_REF] Lee | Multi-dimensional rheology-based two-phase model for sediment transport and applications to sheet flow and pipeline scour[END_REF][START_REF] Si | A general numerical model for surface waves generated by granular material intruding into a water body[END_REF][START_REF] Romano | Tsunamis Generated by Submerged Landslides: Numerical Analysis of the Near-Field Wave Characteristics[END_REF][START_REF] Lee | Multi-phase flow simulation of impulsive waves generated by a sub-aerial granular landslide on an erodible slope[END_REF]. However, an extensive comparison between 2D and 3D models for landslide tsunamis generation is, to the knowledge of the authors of this present study, not available. Models validation often relies on analytical solutions, laboratory experiments or real cases. The literature reports experiments using rigid landslides, made of aluminium [START_REF] Liu | Runup and rundown generated by three-dimensional sliding masses[END_REF][START_REF] Enet | Experimental Study of Tsunami Generation by Three-Dimensional Rigid Underwater Landslides[END_REF][START_REF] Sue | Tsunami generation by submarine landslides: comparison of physical and numerical models[END_REF], rock [START_REF] Bolin | Experimental modeling of tsunamis generated by subaerial landslides: two case studies of the Three Gorges Reservoir, China[END_REF] or PVC [START_REF] Heller | On the effect of the water body geometry on landslide-tsunamis: Physical insight from laboratory tests and 2D to 3D wave parameter transformation[END_REF], deformable landslides made of rocks/gravel [START_REF] Mazzanti | The dynamics of coastal landslides: insights from laboratory experiments and theoretical analyses[END_REF][START_REF] Mohammed | Physical modeling of tsunamis generated by three-dimensional deformable granular landslides[END_REF][START_REF] Bregoli | Experimental studies on 3D impulse waves generated by rapid landslides and debris flows[END_REF][START_REF] Mcfall | Physical modelling of tsunamis generated by three-dimensional deformable granular landslides on planar and conical island slopes[END_REF], BaSO 4 /polypropylene grains [START_REF] Evers | Spatial Impulse Wave Generation and Propagation[END_REF], ceramics beads [START_REF] Mulligan | On the transfer of momentum from a granular landslide to a water wave[END_REF], glass beads [START_REF] Viroulet | Tsunami waves generated by cliff collapse: comparison between experiments and triphasic simulations[END_REF], Grilli et al. 2017) or simply water [START_REF] Bullard | An Enhanced Framework to Quantify the Shape of Impulse Waves Using Asymmetry[END_REF]. Experiments are either submerged or subaerial, on slopes generally ranging from 15° [START_REF] Enet | Experimental Study of Tsunami Generation by Three-Dimensional Rigid Underwater Landslides[END_REF][START_REF] Sue | Tsunami generation by submarine landslides: comparison of physical and numerical models[END_REF]) to 45° [START_REF] Heller | On the effect of the water body geometry on landslide-tsunamis: Physical insight from laboratory tests and 2D to 3D wave parameter transformation[END_REF][START_REF] Viroulet | Tsunami waves generated by cliff collapse: comparison between experiments and triphasic simulations[END_REF] or more (up to 70°for [START_REF] Bolin | Experimental modeling of tsunamis generated by subaerial landslides: two case studies of the Three Gorges Reservoir, China[END_REF]90°for Evers et al. (2019)). The main objective of the present study is to identify the application fields of depth-averaged models (represented here by AVALANCHE) by comparing the results of simulations over a large range of parameters with a Navier-Stokes model (here OpenFOAM, in laminar mode). To achieve this objective, both models are tested against two laboratory experiments: one subaerial [START_REF] Viroulet | Tsunami waves generated by cliff collapse: comparison between experiments and triphasic simulations[END_REF]) and one submerged [START_REF] Grilli | Modeling coastal tsunami hazard from submarine mass failures: effect of slide rheology, experimental validation, and case studies off the US East Coast[END_REF]. For these two benchmarks, the comparison between experiments and simulations is made for both the landslide behavior and the generated water waves. Then, the initial configuration of one of the experiment considered for benchmark is modified and simulations are realized by varying the slope angle and the depth of submergence of the slide. Finally, nine of the previous simulations are run in turbulent mode considering a k -ϵ model.

In Section 2, we present the AVALANCHE and the OpenFOAM models. Section 3 is devoted to the description of the benchmarks, the numerical simulations and the sensitivity studies. Results are presented in Section 4 and a discussion is given in Section 5.

Models presentation

Depth-averaged model: AVALANCHE

AVALANCHE is a 2DH code designed to simulate the landslide and the generated tsunami. It is written in Fortran 90/95 and parallelized via the MPI library. The landslide can be modeled either as a granular flow, falling under gravity and following a Coulomb friction law, or as a viscous Newtonian fluid. The code can use a system of nested grids with a coarse grid over deep water regions and fine grids over coastal regions to model local effects of bathymetry. It has been successfully employed to simulate both subaerial [START_REF] Heinrich | Near-field modeling of the July 17, 1998 tsunami in Papua New Guinea[END_REF][START_REF] Hébert | Numerical modeling of the September 13, 1999 landslide and tsunami on Fatu Hiva Island (French Polynesia)[END_REF][START_REF] Friant | Numerical simulation of the last flank-collapse event of Montagne Pelée, Martinique, Lesser Antilles[END_REF][START_REF] Paris | Numerical modeling of the June 17, 2017 landslide and tsunami events in Karrat Fjord, west Greenland[END_REF], 2020) and submarine landslides [START_REF] Rodriguez | Owen Ridge deep-water submarine landslides: Implications for tsunami hazard along the Oman coast[END_REF][START_REF] Poupardin | The 1979 Submarine Landslide-Generated Tsunami in Mururoa, French Polynesia[END_REF].

The landslide part is governed by the following equations set adapted from the one-phase grain-flow model of [START_REF] Savage | The motion of a finite mass of granular material down a rough incline[END_REF] using here a viscous fluid (to be consistent with the rheological law of the Navier-Stokes model) and written in a coordinates system (x ′ ,y ′ ) parallel to the slope:

∂h s ∂t + ∂ ∂x ′ (h s u s ) + ∂ ∂y ′ (h s v s ) = 0 (1) ∂ ∂t (h s u s ) + ∂ ∂x ′ (h s u 2 s ) + ∂ ∂y ′ (h s u s v s ) = - 1 2 κ ∂ ∂x ′ (gh 2 s cos θ) + κgh s sin θ x + τ x ′ z ′ (z ′ =0) (2) ∂ ∂t (h s v s ) + ∂ ∂x ′ (h s v s u s ) + ∂ ∂y ′ (h s v 2 s ) = - 1 2 κ ∂ ∂y ′ (gh 2 s cos θ) + κgh s sin θ y + τ y ′ z ′ (z ′ =0) ′ (3)
where h s is the slide's thickness in a direction perpendicular to the slope, u = (u s , v s ) the depth-averaged velocity vector parallel to the bed, κ = 1 -ρ w /ρ s where ρ w and ρ s are the water and rock densities (for the subaerial part of the slide, κ is identically equal to 1), θ(x, y) the local steepest slope angle, θ x and θ y the slope angles along the x and y axes respectively, and τ z = -νρ w u/(h s ρ s ) the shear stress at the bed surface for a uniform profile of the horizontal velocity, where ν is the kinematic viscosity.

The tsunami propagation can be simulated by solving either the shallow water equations ( 4), ( 5), ( 6) or the Boussinesq equations of [START_REF] Løvholt | Oceanic propagation of a potential tsunami from the La Palma Island[END_REF] (7), ( 8), (9). A switch between the two sets of equations allows to use the shallow water equations for a few instants during the water wave generation in the nearfield and the Boussinesq model for the far-field propagation. Indeed, [START_REF] Audusse | Comparison of models for the simulation of landslide generated tsunamis[END_REF] showed that this combination of models improves the simulation of the generation process compared to using only the Boussinesq model.

Shallow water equations

∂η ∂t + ∂(hu) ∂x + ∂(hv) ∂y = - ∂d w ∂t (4) ∂u ∂t + u ∂u ∂x + v ∂u ∂y = -g ∂η ∂x + F x (5) ∂v ∂t + u ∂v ∂x + v ∂v ∂y = -g ∂η ∂y + F y ( 6
)
where η is the surface elevation, h=η+d the water column height with d the initial depth, d w the sea-bottom deformation due to the landslide, u and v the depth-averaged velocities along the x and y axes respectively and F x and F y the friction and Coriolis forces, which are assumed to be negligible in our modeling. where u t and v t the time-derivatives of u and v.

Boussinesq equations

∂η ∂t + ∂(hu) ∂x + ∂(hv) ∂y = - ∂d w ∂t (7) ∂u ∂t + u ∂u ∂x + v ∂u ∂y = -g ∂η ∂x + d 3 3 ∂ 2 u t ∂x 2 + ∂ 2 v t ∂x∂y + d 2 u t ∂ 2 d ∂x 2 +
The sea-bottom deformation due to the landslide is transmitted to the water surface by applying a filter in order to spread out the water deformation at the surface following [START_REF] Kelfoun | Landslide-generated tsunamis at Réunion Island[END_REF]:

∆d w = a × V b 2 × e -ln(π)× b h ( 10
)
where V is the volume displaced vertically at the bottom, a is a parameter that allows mass conservation in order that ∞ x=-∞ ∞ y=-∞ ∆d w dxdy = V , and b = x 2 + y 2 + h 2 is the distance between a given point (x, y, h) of the water surface and the point at the bottom (x = 0, y = 0, z = 0) where volume change occurs. This formula of the sudden elevation of the water is exact for a horizontal plane at the bottom and is assumed to be a good approximation in steep slopes environments. It is written in C++ and uses a system of modules that the user can assemble to build his own case and in which all the files can be modified. OpenFOAM is able to deal with incompressible/compressible turbulence, thermophysical, multiphase or wave modeling. It handles the mesh generation, the case simulation and the reading and postprocessing of the outputs. Among others, it was used by [START_REF] Qin | Three-dimensional modeling of tsunami forces on coastal communities[END_REF] to study the inundation and impact of a tsunami on a coastal city or by [START_REF] Honarmand | 3D numerical simulation of tsunami generation and propagation, case study: Makran tsunami generation and penetrating in Chabahar Bay[END_REF] to propagate the initial elevation generated by the Makran earthquake in 1945. [START_REF] Romano | Tsunamis Generated by Submerged Landslides: Numerical Analysis of the Near-Field Wave Characteristics[END_REF] also developed an approach to simulate tsunamis generated by a solid landslide using the Overset mesh technique. Considering a deformable landslide, [START_REF] Rauter | Numerical simulation of impulse wave generation by idealized landslides with OpenFOAM[END_REF] reproduced the experiments of [START_REF] Bullard | An Enhanced Framework to Quantify the Shape of Impulse Waves Using Asymmetry[END_REF] using the solver multiphaseInterFoam with two viscous fluids. Closer to the present work since it concerns the same benchmarks, [START_REF] Si | A general numerical model for surface waves generated by granular material intruding into a water body[END_REF], [START_REF] Yu | Multi-phase-flow modeling of underwater landslides on an inclined plane and consequently generated waves[END_REF] and [START_REF] Lee | Multi-phase flow simulation of impulsive waves generated by a sub-aerial granular landslide on an erodible slope[END_REF] used models based on OpenFOAM to simulate the subaerial experiments of [START_REF] Viroulet | Tsunami waves generated by cliff collapse: comparison between experiments and triphasic simulations[END_REF] and submerged experiments of [START_REF] Grilli | Modeling coastal tsunami hazard from submarine mass failures: effect of slide rheology, experimental validation, and case studies off the US East Coast[END_REF], respectively. The first authors introduced a granular temperature in the granular phase, while the second ones used a Navier-slip boundary condition adapted for granular flows.

Navier-Stokes model: OpenFOAM

As [START_REF] Rauter | Numerical simulation of impulse wave generation by idealized landslides with OpenFOAM[END_REF], we use the Navier-Stokes equations to simulate three fluids (air, water and sediment). The continuity equation reads:

∇ • u = 0 ( 11
)
where u is the velocity, and the momentum equation reads:

∂ρu ∂t + ∇ • (ρuu) = -∇p + ∇ • (2(µ + µ t )D) + ρg ( 12
)
where D is the strain rate tensor, g the gravitational acceleration, p(x, t) the pressure field, and ρ, µ and µ t are the local fluid density, molecular dynamic viscosity and eddy viscosity, respectively and are defined as follows:

ρ = i α i (x, t)ρ i (13) µ = i α i (x, t)µ i (14)
µ t = ρC µ k 2 ϵ ( 15
)
where ϵ is the turbulent kinetic energy dissipation rate and k is the turbulent kinetic energy. They are implemented as follows:

∂ ∂t (αρϵ) + ∇ • (αρuϵ) -∇ 2 (αρD ϵ ϵ) = C 1 αρG ϵ k - 2 3 C 1 -C 3,RDT αρ∇ • uϵ -C 2 αρ ϵ k ϵ + S ϵ + S f vOptions (16) ∂ ∂t (αρk) + ∇ • (αρuk) -∇ 2 (αρD k k) = αρG - 2 3 αρ∇ • uk -αρ ϵ k k + S k + S f vOptions (17)
where G is the turbulent kinetic energy production rate due to the anisotropic part of the Reynolds-stress tensor (m 2 s -3 ), D ϵ is the effective diffusivity for ϵ, C 1 (s) and C 2 are two model coefficients, C 3,RDT is the rapid-distortion theory compression term coefficient, S ϵ is the internal source term for ϵ, S f vOptions are source terms introduced by fvOptions dictionary for ϵ, S k is the internal source term for k and S f vOptions are source terms introduced by fvOptions dictionary for k.

In Equations 13, 14, 16 and 17, phase indicator α i is defined as follows:

α i (x, t) =        1 if phase i is present at x, t 0 else (18)
and is calculated in a volumetric phase fraction equation:

∂α i ∂t + ∇ • (α i u) + j ∇ • (α i α j u r,ij ) = 0 ( 19
)
where u r,ij is the relative velocity between phases.

The following default values were used:

C µ = 0.09; C 1 = 1.44; C 2 = 1.92; C 3,RDT = 0
Here we use the multiphaseInterFoam solver with three viscous fluids, the air, the water and the landslide and the PIMPLE algorithm which combines PISO (Pressure Implicit with Splitting of Operator) and SIMPLE (Semi-Implicit Method for Pressure-Linked Equations). Cartesian grids are generated by blockMesh and refined by snappyHexMesh using triangular elements in the area along the slope. The landslide is modeled as a Newtonian non-miscible fluid. Python scripts are used to postprocess the results (local free surface time series, cross-sectional views of the landslide).

In Sections 3.2 and 3.3, and consistently with the depth-averaged model, the solver is used in laminar mode (µ t = 0 in Equation 12). This allows to focus the comparison on the main difference between the two models, namely flow depth-averaged (AVALANCHE) or fully resolved (OpenFOAM). Further, in Section 3.4, the influence of the turbulence is studied, considering a k -ϵ model in the Navier-Stokes model.

Simulations

We focus on two laboratory experiments to validate both models: a subaerial landslide [START_REF] Viroulet | Tsunami waves generated by cliff collapse: comparison between experiments and triphasic simulations[END_REF] and a submerged one [START_REF] Grilli | Modeling coastal tsunami hazard from submarine mass failures: effect of slide rheology, experimental validation, and case studies off the US East Coast[END_REF], both deformable (Figure 1).

Benchmarks description

Subaerial case

The experiment described in [START_REF] Viroulet | Tsunami waves generated by cliff collapse: comparison between experiments and triphasic simulations[END_REF] involves a subaerial landslide composed of 0.0015 m diameter glass beads, contained in a 0.11 m high isosceles right-angled triangle by a vertical polyvinyl plate.

The density of the beads is 2000 kg m -3 but the equivalent density of the slide (air and beads) is 1500 kg m -3 . Once the plate is raised, the beads collapse on a 45°slope into 0.15 m of water, in a 2.20 m long, 0.4 m high and 0.2 m wide tank. The evolution of the collapse is recorded by a high speed camera (100 frames Figure 1: Geometry andgauges locations for the subaerial (gauges at x=0.702, 1.002, 1.302 and1.602 m, top panel) andsubmerged (gauges at x=0.879 and1.879 m, bottom panel, andx=2.879 and3.879 m, not visible here) benchmarks [START_REF] Viroulet | Tsunami waves generated by cliff collapse: comparison between experiments and triphasic simulations[END_REF], Grilli et al. 2017). In both panels, the slide is represented in black, the slope in brown, the water in blue and the gauges with the red triangles.

0 0.5 1 1.5 2 -0.4 -0.3 -0.2 -0.1 0 0 0.5 1 1.5 2 X (m) -0.4 -0.3 -0.2 -0.1 0 Z (m) Subaerial Submarine
per second). The evolution of the generated water waves is recorded by four resistive gauges located at 0.45, 0.75, 1.05 and 1.35 m from the plate. [START_REF] Ionescu | Viscoplastic modeling of granular column collapse with pressure-dependent rheology[END_REF] proposed an approximation to evaluate the viscosity of the equivalent fluid following the µ(I)-rheology during the initial phase of sliding. Following this idea, [START_REF] Clous | Simulation of energy transfers in waves generated by granular slides[END_REF] found for this benchmark a value of 12.8 Pa.s, which will be used here in the simulations.

Submerged case

The experiment described in [START_REF] Grilli | Modeling coastal tsunami hazard from submarine mass failures: effect of slide rheology, experimental validation, and case studies off the US East Coast[END_REF] involves a submerged landslide composed of 0.004 m diameter glass beads, contained in a 0.12 m long, 0.085 m high right-angled triangle by a sluice gate. The density of the beads is 2500 kg m -3 leading to an equivalent density of 1951 kg m -3 . The collapse occurs on a 35°slope into 0.33 m of water, in a 6.27 m long and 0.25 m wide tank. In a similar way to the subaerial experiment, a high speed camera (1000 fps) records the landslide evolution, while gauges located at 0.6, 1.6, 2.6 and 3.6 m of the gate record the water waves time series.

Geometry and gauges locations of both cases are represented in Figure 1 (top panel for the subaerial benchmark, bottom panel for the submerged one).

Benchmarks simulations

Depth-averaged model

For both subaerial and submerged benchmarks, a sensitivity analysis is carried out to determine the effect of the viscosity µ in the range [0.01; 40] Pa.s. All the simulations are performed in a 1D cartesian grid of 0.001 m resolution in the x-direction. A convergence study was performed and the results did not show any significant differences between this resolution and a finer one. As in the experiment, a reflective right boundary condition is used for the submerged case. First of all, full shallow water simulations are computed to evaluate how fast the first generated wave separates from the slide. When the wave is considered to be not affected anymore by the landslide, we switch from shallow water equations to Boussinesq equations.

Navier-Stokes model

For both landslides, the (x, z) 0.005 m resolution mesh generated by blockMesh is refined on one layer along the slopes and the bottom with triangular elements by snappyHexMesh. The resolution in x is coarser than that of the depth-averaged model due to the higher computational cost, but it is nevertheless sufficient to ensure convergence. For both benchmarks, pseudo 3D grids are composed of one cell in the transversal y-direction. Sensitivity studies are conducted on the viscosity µ (with values in the range [0.01; 50] Pa.s) and the lowerWall (slope and bottom) boundary condition (i.e. noSlip and partialSlip). The latter is defined through a coefficient α between 0 and 1. If u 0 is the tangential velocity on the wall and u 1 the tangential velocity of the adjacent cell, α is defined as follows: u 0 = (1 -α)u 1 , α being defined as α = d/(d + 2λ) where λ is the slip length and d the cell size. With these definitions, a noSlip boundary condition corresponds to a α value of 1. In the following, a 0.1 partialSlip condition will refer to a partialSlip condition with α=0.1.

Analysis

The comparison of the free surface signals obtained from the simulations (f 1 ) and the experiments (f 2 ) is carried out based on the Q parameter, built on Sobolev norms [START_REF] Perlin | A robust quantitative comparison criterion of two signals based on the Sobolev norm of their difference[END_REF]:

Q(f 1 , f 2 ) = |F 1 (ω) -F 2 (ω)| 2 dω 1/2 |F 1 (ω)| 2 dω 1/2 + |F 2 (ω)| 2 dω 1/2 (20)
where f 1 , f 2 are the two temporal signals and F 1 (ω), F 2 (ω) the associated Fourier transforms.

This parameter Q gives a measure of the phase shift of the two signals. For identical time series, Q=0, whereas Q=1 for two ideal (sinusoidal) signals separated by a half-period (e.g. sin(x) and sin(x + π)).

As regards the landslide evolution, comparisons between simulations and experiments are made in a qualitative way simply by overlapping simulated and observed snapshots of slice views of the landslide at different instants.

Sensitivity studies on the landslide characteristics

The simulation of the two benchmarks and the comparison with the data allow to select a set of optimal parameters for each code (viscosity, type of equations for the depth-averaged model, slip condition for the Navier-Stokes model). These parameters will then be used in a sensitivity study consisting in varying the slope and the submergence considered in the first benchmark (Figure 2), keeping the same shape, volume, gauges locations and water depth (i.e., a triangle, 0.00605*1 m 3 , x=0.702, 1.002, 1.302, 1.602 and 0.15 m, respectively).

In the sensitivity study, simulations are performed with slope angles ranging from 20 to 60°with 5°step.

Considering the reference point for the submergence as the foot of the landslide, we also tested five values of relative submergences S from -1 to +1, a submergence S +1 being a raise of one landslide length along the slope (Figure 2). The initial geometry of the subaerial benchmark does not allow to simulate a submergence deeper than -1 without changing the water depth.

We chose to keep the same volume for all the simulations. Therefore, the portion of slide under water is not identical for submerged cases and different slopes. Consequently, comparing a same submergence for different slope angles will be delicate and we can only study the influence of the submergence for a given slope. Re s ∼ 20 (in laminar cases).

Sensitivity study on the turbulence

Dimensionless variables

The analysis of the results of the sensitivity studies is performed using the following dimensionless variables:

• The dimensionless wave amplitude H/h where H is the amplitude of the wave and h the water depth (over the flat bottom part).

• The dimensionless velocity V s /V 0 where V s is the landslide velocity and V 0 = √ 2gd 0 a characteristic velocity with d 0 the landslide thickness at t=0 s. V 0 approximates the initial fluid velocity appearing in the bottom part of the slide (here considering no energy loss). The landslide viscosity µ is linked to the Reynolds number through the following relationship:

Re s = ρ s V 0 d 0 µ (21)
where ρ s is the landslide density.

Results

As indicated in Section 3.2, we recall that the following results were obtained by simulations involving a combination of shallow water and Boussinesq models.

Subaerial case

Depth-averaged model

Water waves

Figure 3 shows that the best average Q-value for the four gauges is obtained with a viscosity µ of 12.8

Pa.s. This value corresponds to the µ(I)-rheology viscosity calculated by [START_REF] Clous | Simulation of energy transfers in waves generated by granular slides[END_REF] following [START_REF] Ionescu | Viscoplastic modeling of granular column collapse with pressure-dependent rheology[END_REF]. In a general way, in the "far-field" (Gauge 4), Q-values are lower than those in the "near-field" (Gauge 1), showing that the depth-averaged model is more accurate in the far-field.

For viscosity values between 6 and 15 Pa.s, results are similar with low Q-values. This is illustrated by the analysis of the free surface elevation time series at the four gauges (Figure 4) which shows similar results, close to the experimental water waves whatever the viscosity taken in this range. Nevertheless, we note that if the characteristics of the leading wave (period and height) are reproduced accurately, the fit is lower for the following water waves. Landslide Considering the landslide itself, the simulation with a viscosity value µ of 12.8 Pa.s gives satisfactory results, as shown in Figure 5. Due to the landslide model hypothesis, the experimental front shape is not properly reproduced but the quantity of material entering into the water at different times is very similar in the simulation and in the experiment. Moreover, as shown by [START_REF] Clous | Simulation of energy transfers in waves generated by granular slides[END_REF], most of the energy transfer between the landslide and the wave occurs within the first instants (before t=0.3s) in this case, so it is not essential to reproduce perfectly the landslide behavior afterward to properly simulate the water waves generated.

For lower and higher values of viscosity, Q-values increase, probably due to incorrect landslide velocities, as supported by Figure 6, which illustrates the variety of the average slide velocity observed in the simulations. Before the plateau, the noSlip condition gives better results (lower Q-values) whereas best values are obtained after the plateau for the 0.1 partialSlip condition. This stresses the importance of the slide average velocity at the beginning of the process which is controlled both by viscosity and bottom boundary condition.

Navier-Stokes model

Water waves

Figure 8 shows indeed that different combinations of viscosity values and boundary conditions give similar results regarding the leading wave.

Figure 7B shows that, as for the depth-averaged model, Q-values in the far-field (Gauge 4) are lower than those computed in the near-field (Gauge 1).

Landslide

With a viscosity of 12.8 Pa.s, the slide kinematics is only correct up to t=0.2s (Figure 9). After this time, the landslide is too slow independently of the bottom boundary condition. This behavior differs from the one obtained with the depth-averaged model (Figure 5). To reproduce the experimental landslide behavior with the Navier-Stokes model, at least the overall trajectory of the gravity center, a viscosity value µ of 1 Pa.s with a 0.1 partialSlip boundary condition (Figure 10) has to be used, but in this case the landslide front is too large, producing incorrect waves (more than two times larger that the experimental data). In the submerged landslide case, the same shape for the Q-values curve as that of the subaerial benchmark is observed, a trough and a plateau of optimal viscosity. The best performances are obtained with viscosity values between 0.1 and 1 Pa.s (Figure 11). Overall, considering the four gauges, a viscosity of 1 Pa.s gives the best results.

Submerged case

Surface elevations are globally well reproduced using a viscosity value µ between 0.1 and 1 Pa.s (Figure 12). This is especially true for the first wave, the next waves being slightly too high compared to the experiments, especially at Gauge 1 (top panel in Figure 12).

Landslide

With µ=1 Pa.s (Figure 13), the simulated landslide matches the experimental data up to t=0.17s (middle left panel, Figure 13) and deviates afterward due to larger velocities. With µ=3 Pa.s, the landslide behavior is better reproduced (Figure 14), the simulation being close to the experimental data up to t=0.47s (bottom left panel).

Figure 15 shows times landslides touch the bottom and start to slow down in cases µ=0.01, 0.1, 0.5 and 1 Pa.s (peaks of corresponding lines in Figure 15) whereas for µ=5 or 10 Pa.s, the landslide starts to slow down before reaching the bottom which likely affects waves generation.

Navier-Stokes model

Water waves

Figure 16A shows again that optimal results are similarly obtained for the noSlip and 0.4 partialSlip conditions at viscosity values between 1 and 5 Pa.s. While with the 0.1 partialSlip condition, best Qvalues are obtained for higher viscosity values. Considering the 0.4 partialSlip condition (Figure 16B), the best results are obtained with a viscosity value of 3 Pa.s. Moreover, as previously seen, waves are better Figure 17 presents the comparison between the Navier-Stokes model simulations and the experimental data for the three best combination of parameters (i.e., viscosity value and bottom boundary condition).

Surprisingly, the behavior of the model is very similar at the four gauges whatever the combination. At Gauge 1 (top panel), the peak of the first wave is well reproduced whereas the following waves are attenuated.

The latter are better reproduced on the three other gauges. Hence, to generate the same wave, a full slip condition (0.1 partialSlip) requires a higher viscosity value (µ=8 Pa.s) than a medium slip condition (0.4 partialSlip, µ=3 Pa.s) or a noSlip condition (µ=5 Pa.s).

Landslide

Although simulations with the following combination (0.4 partialSlip and µ=3 Pa.s) give good results regarding the waves, they do not reproduce the landslide correctly (Figure 18). A lower viscosity value (µ=0.1 Pa.s) is actually needed to better reproduce the landslide motion at the first instants (i.e., until t=0.17 s in Figure 19). Beyond t=0.17 s, even a quasi full slip condition (i.e., 0.1 partialSlip, in orange) is not sufficient to reach the position of the experiment slide front. The difference along the slope between the observed and the simulated locations of the landslide front increases progressively and reaches 7.2 cm at t=0.32 s and about 17 cm at t=0.62 s (Figure 19). The difference observed could be explained by the absence of water incorporation within the landslide.

Additionally, whatever the viscosity value and regardless of the boundary condition at the bed, a 5-10 cm high flow front forms slowing down the whole landslide.

Sensitivity studies on the landslide characteristics

This section presents a sensitivity analysis aiming at better understanding the influence of the slope angle and the landslide submergence, on the first generated wave amplitude. The main objective is to define the sets of parameters for which both models give similar results and the ones leading to differences. Results are presented in dimensionless variables (see Section 3.5 which introduces the landslide Reynolds number Re s ).

The viscosity in each model has been chosen so that the first water wave is similar in both models and close to the two previous experimental data. This is achieved with: a Re s ∼ 20 (µ=12.8 Pa.s) (and a 0.4 partialSlip condition for the Navier-Stokes model). We also decided to include a second Reynolds number closer to the one used in our submerged benchmark simulations: Re s ∼ 250 (µ=1 Pa.s).

With the depth-averaged model, the following results are obtained (Figure 20):

• For both Reynolds values, the higher the slope angle, the larger the first generated wave, following quasi linear relationships.

• For any slope angles and for both Reynolds, when the landslide starts near or just above the water surface (submergences S -0.5 and S 0 ) the generated wave is larger than with the other submergences tested. Submergences S -0.5 and S 0 give very similar results, except from 50°for Re s ∼ 250. B) and for submergences S -0.5 (red), S 0 (green), S +0.5 (blue) and S +1 (orange). Horizontal pink lines mark an interval of ± 15% within which results with both models are considered similar. Pink circles correspond to the cases the closest to the benchmarks simulated in Sections 4.2 (A) and 4.1 (B). • With the large slide Reynolds number Re s (250), the water waves generated are about two times higher than those obtained with a lower Re s (Figure 20).

The following main results are obtained with the Navier-Stokes model (Figure 21):

• As for the depth-averaged model, the wave amplitude increases quasi linearly with the bed slope, except for the submergence S -1 for which the relationship is reversed, and the submergence S -0.5 , for which the amplitude of the first wave reaches an asymptotic value for slopes larger than 45°(Figure 21).

• We see again that with an initial landslide position crossing the water surface or just above, higher generated water waves are generated compared to more distant initial positions.

• Overall, as for the depth-averaged model, the wave amplitude is two times larger with Re s ∼ 250 (Figure 20) than with a lower Re s .

Differences of water heights between the two models are presented in Figure 22. In this figure, the two following statements can be made:

• With the higher Re s , the models give different results, whatever the slope angles or the submergence.

Overall, the wave amplitudes obtained with the depth-averaged model are smaller than those obtained with the Navier-Stokes model, except for negative initial submergences and large slope angles. As expected, results are similar for the combination corresponding to the submerged benchmark (Section 4.2) represented by the pink circle in Figure 22(A).

• With the lower Re s , the agreement is much better except for the negative submergence as previously and large slopes (above 45°) for which the depth-averaged model gives higher waves.

Figure 23 shows the sensitivity of both models to the Reynolds number for a given submergence and slope.

Below Re s ∼ 20, both models tend to give similar results, whereas above this value, the depth-averaged model produces smaller water waves. Nevertheless, the difference between the two models does not increase with the Reynolds but rather reaches a constant value.

Sensitivity study on the turbulence

This section presents a sensitivity analysis on the turbulence using a k -ϵ model, three different slope angles (35, 45 and 60°) and three different submergences (S -1 , S 0 and S +1 ).

Results of simulated waves in laminar and turbulent modes are shown on Figure 24. Turbulence is shown to play a complex role in some cases enhancing the wave field in others reducing it. A careful investigation of the fields evolution during the simulations allows interesting observations. First, the slide dynamics with and without turbulence modeling is very similar. Most of the time, the additional viscosity within the slide due to turbulence induces a very slight delay and momentum decrease when slide is transferring energy to the free surface. In some cases, this difference does not change the wave field. This happens when there is only a delay of the process but not a change in the direction slide is pushing water (Figure 24A for S +1 ).

In others, the difference in the wave amplitude is very large as in Figure 24C for S +1 . In this case, the turbulent slide shows an increase of about 10% of its front thickness due to the slightly different interaction with water at the penetration instant compared to the laminar case. This difference induces a larger horizontal momentum transfer to water and subsequently larger waves.

The contrary can occur as in the submarine case shown Figure 24B for S -1 . In this case, the wave is generated immediately and the initial shape has a critical effect. In this particular situation, the laminar case induces more horizontal momentum transfer that the turbulent case.

We also note that when the slide dynamics and especially the ratio between horizontal and vertical momentum transfer is comparable, even though turbulence is present in water with an additional viscosity of about 100 times the molecular value, this does not change significantly the wave field.

Discussion

Benchmarks

General results

Both the depth-averaged and the Navier-Stokes models can reproduce the benchmarks waves using their own sets of parameters with approximately the same accuracy. 3, 7, 11 and 16. We note that the waves in the far-field are even less sensitive to the viscosity.

Poor landslide reproduction especially for the Navier-Stokes model

The optimal parameters to get the correct generated first wave generally do not allow to reproduce the exact measured landslide behavior for both benchmarks and with both models. Hence, as already mentioned by [START_REF] Grilli | Modeling coastal tsunami hazard from submarine mass failures: effect of slide rheology, experimental validation, and case studies off the US East Coast[END_REF], the accurate modeling of water waves does not depend on details of the landslide deformation. Moreover, although the equations governing the landslide motion in the depth-averaged model do not make possible to reproduce the bulbous shape of the landslide front, the model allows to obtain correct landslide velocity and quantity of materials entering water for the subaerial benchmark (Figure 5).

As regards the submerged benchmark, the landslide simulated by the depth-averaged model is a bit faster than the experimental one (Figure 13). By contrast, the Navier-Stokes model produces a bulbous front that progressively develops while traveling down the slope. This bulbous front is due to a vortex generated at the water/slide interface. In this case, the simulated slide is much slower than in the experiment.

Calibration of the viscosity value

In every practical case, the major issue is to determine the viscosity value. For the subaerial case, the approximation of [START_REF] Ionescu | Viscoplastic modeling of granular column collapse with pressure-dependent rheology[END_REF] seems to be valid and both models reproduce the experimental results.

For the submerged benchmark, the viscosity values considering in this study are around 10 times lower than those for the subaerial benchmark. A simple extrapolation of these results to other cases is however dangerous. Indeed, this approach requires intrinsic landslide parameters (µ(I) parameters, grain density and grain diameter) to which access is conditioned to a field survey. Moreover, the grain size in the submerged benchmark is about four times greater than the one in the subaerial benchmark, so there is more water between the grains in the submerged case. Because of this water, the landslide becomes more fluid and a lower viscosity is required.

Sensitivity study

Overall results

Energy transfers

For both models, the generated waves are higher if the initial landslide position is close to the surface and this may be explained by the combination of the slide front height and the landslide velocity. As shown in Figure 25, the deeper the landslide, the smaller the velocity, so the largest velocities are always calculated for subaerial cases. However, for both models, the most suberial cases result in thinner landslide fronts at the free surface since these landslides stretch rapidly along the slope (see Figure 26 for an illustration with a slope angle of 45°and three submergences, S +1 , S 0 and S -1 ) and the duration close to the free surface is also shorter. In consequences, the energy transfer is reduced for these landslides.

Slope effect

The landslide velocity in the depth-averaged model increases with the slope angle for both Reynolds, which leads to higher water waves for large slope angles (Figure 25A). The behavior is more complex in the case of the Navier-Stokes model (Figure 25B).

Slide Reynolds

Results in Figure 20 and 21 show that wave amplitudes strongly depend on the Reynolds number. For a given submergence and a given slope angle, wave amplitudes are about two times larger with Re s ∼ 250 compared to Re s ∼ 20. In the case of the depth-averaged model, landslide velocities are about two times larger for Re s ∼ 250 (Figure 25A) which results in larger wave amplitudes. In the case of the Navier-Stokes model, the variation of slide velocities with the slope is more complex (Figure 25B) and the slide shape plays a stronger role in energy transfers.

Models concordance

Importance of the first instants

For a given slope angle, both models produce similar waves with Re s ∼ 20 if the landslide initial position is close to the surface or just above (submergences S 0 and S +0.5 ). In these cases, there is no clear difference of landslide behavior in the first instants (see for example the results of the subaerial benchmark in Section 4.1), and the energy transfers should be similar. It is also the case for submergences S +1 and S -0.5 if the slope angle stays below 45°.

Differences between models

For both Re s , if the landslide velocities are similar (e.g. subaerial cases for Re s ∼ 250 in Figure 25 A), the waves are larger with the Navier-Stokes model because the depth-averaged model landslide is thinner inherently. Moreover, for small slope angles, the landslide in the depth-averaged model is more likely to spread on both sides of its initial position.

For cases where landslide velocities are larger with the depth-averaged model (e.g. submerged cases for slope angles above 40-45°in Figure 25), the waves are larger too. In these cases, the landslide in the Navier-Stokes model is slowed down by the bulbous shape of the landslide front.

Finally, for large slope angles, the depth-averaged model produces higher waves for the highest submergence, although the average landslide velocity is lower than the one calculated by the Navier-Stokes model (Figure 25). Further analysis of the Navier-Stokes simulations shows that this averaged velocity is not representative of the velocity of the landslide front. In these cases, the front is slowed down by the bottom friction which results in lower water wave amplitudes.

Study limitations

The reproduction of the benchmarks could be improved by using more sophisticated landslide rheology (e.g. a granular rheology instead of a viscous fluid) or boundary conditions but beyond the idea of a perfect reproduction, we were looking for fast solutions in order to use both models in tsunami early warning systems.

We note that numerous sets of parameters can lead to very similar water waves and landslide behaviors.

We did not address the influence of the shape of the initial landslide in this study. The triangular shape of the landslide is favorable to the generation of large water waves. However, the slide in the first instants violates the shallow water hypothesis. Further studies with different shapes (such as a more stretched rectangle or an ellipsoid) would be required to reinforce our conclusions.

As seen previously, the initial landslides for the submergence S -1 were not fully submerged for small slope angles. Due to the benchmarks geometry, we could not define deeper submergences but it would be definitely interesting to study.

All the Navier-Stokes simulations were conducted in laminar mode. The sensitivity study using a k -ϵ model showed that the role of turbulence is complex, depending on the submergence and the slope.

Conclusions

The following main conclusions can be drawn from this study:

• Viscous depth-averaged and Navier-Stokes models can reproduce waves generated by granular slides by calibrating the viscosity but generally the slide dynamics is not properly calculated, especially for the Navier-Stokes model.

• Far enough from the generation area, the wave field is less sensitive to the viscosity value.

• The optimal viscosity range is lower in the depth-averaged model, but there is an overlap between the two models ranges which allows to use the same value with comparable results.

• With this triangle initial slide shape, in general the largest waves are generated when the initial slide is close to the free surface. For higher or lower submergences, the waves are smaller.

• In the depth-averaged model, waves increase continuously with the slope and the slide Reynolds number.

The Navier-Stokes model behavior is more complex.

• The concordance of the two models is strongly dependent on the initial submergence and the slide Reynolds number. When the latter is around 20 (i.e. relatively high viscosity influence on the slide dynamics), both models behave similarly in terms of waves generated for initial slides crossing or above the free surface. When the slide Reynolds increases, the difference between the two models is much more pronounced.

• The influence of the turbulence on the generated waves is the result of a combination of initial submergence and slope angle, and everything is played out in the very first instants.

  OpenFOAM (Open-source Field Operation And Manipulation) is an open source CFD software developed by OpenCFD Ltd since 2004. It allows to simulate multiphase flow, solving the 3D Navier-Stokes equations.

Figure 2 :

 2 Figure2: Initial slides positions (in black) for slopes (in brown) of 60°(A), 45°(B) and 30°(C) and different relative submergences (S -1 , S -0.5 , S 0 , S +0.5 and S +1 , in red). The blue line is the initial water surface.

  Now, we propose to show the influence of the turbulence modeling basically by turning on and off a k -ϵ model (same as[START_REF] Rauter | Numerical simulation of impulse wave generation by idealized landslides with OpenFOAM[END_REF]) in the Navier-Stokes simulations. As shown in the previous part, the molecular viscosities required in the slide for the simulations to compare favorably with granular experiments are quite high and the slide Reynolds numbers correspondingly low. Therefore, the turbulence should be quite limited in the slide but plays a role in water and at the interface between the slide and water. Nine cases have been simulated with the Navier-Stokes model in both laminar and turbulent modes to investigate the influence of k -ϵ involving the slope angles of 35, 45 and 60°, three submergences (S -1 , S 0 and S +1 ) with

Figure 3 :

 3 Figure 3: Subaerial benchmark: Q-values calculated for surface elevation time series from the depth-averaged model as functions of the viscosity µ (Pa.s) at Gauges 1 (black) and 4 (blue). Red curve represents the average Q-value for the four gauges. Pink lines define a zone where results are considered satisfactory in average.

Figure 4 :

 4 Figure 4: Subaerial benchmark: comparison of surface elevation time series between the experiments (black triangles) and the depth-averaged model simulations using slide viscosity µ of 8 Pa.s (red), 10 Pa.s (green) and 12.8 Pa.s (blue).

Figure 5 :Figure 6 :

 56 Figure 5: Subaerial benchmark: comparison of slide contours between experiments (red squares) and the depth-averaged model simulation with µ=12.8 Pa.s (black circles). From left to right and up to down, contours are represented every 0.1 s from t=0 to t=0.5 s. The blue line represents the initial water surface. The brown line represents the slope.

Figure 7 :

 7 Figure 7: Subaerial benchmark: Q-values calculated for surface elevation time series from the Navier-Stokes model as functions of the viscosity µ (Pa.s). A. Average Q-values for the four gauges calculated using the three slip boundary conditions (noSlip (black), 0.4 partialSlip (red) and 0.1 partialSlip (green)). B. Q-values calculated for Gauges 1 (black) and 4 (blue) and average Q-values for the four gauges (red) using the 0.4 partialSlip condition.

Figure

  Figure 7A highlights the existence of a plateau between viscosity values of 12.8 and 30 Pa.s (pink lines) where Q-values are satisfactory. Note that similar values are obtained for the three tested boundary conditions in this viscosity range.

Figure 8 :

 8 Figure 8: Subaerial benchmark: comparison of surface elevation time series (m) between the experiments (black triangles) and the Navier-Stokes model simulations using viscosity µ of 15 Pa.s with a noSlip boundary condition (red), 22 Pa.s with a 0.4 partialSlip condition (green) and 30 Pa.s with a 0.1 partialSlip condition (blue).

Figure 9 :Figure 10 :

 910 Figure 9: Subaerial benchmark: comparison of slide contours between experiments (red squares) and the Navier-Stokes model simulations with µ=12.8 Pa.s and noSlip (green), 0.4 partialSlip (black) or 0.1 partialSlip (orange) slip conditions. From left to right and up to down, contours are represented every 0.1 s from t=0 to t=0.5s. The blue line represents the initial water surface. The brown line represents the slope.

Figure 11 :Figure 12 :

 1112 Figure11: Submerged benchmark: Q-values calculated for surface elevation time series from the depthaveraged model as functions of the viscosity µ at Gauges 1 (black) and 4 (blue). Red curve represents the average Q-value for the four gauges. Pink lines define a zone where results are considered satisfactory in average. X-axis is logarithmic.

Figure 13 :Figure 14 :Figure 15 :

 131415 Figure13: Submerged benchmark: comparison of slide contours between experiments (red squares) and the depth-averaged model simulation with µ=1 Pa.s (black circles). From left to right and up to down, contours are represented at t=0, 0.02, 0.17, 0.32, 0.47 and 0.62 s. The blue line represents the initial water surface. The brown line represents the slope.

Figure 16 :

 16 Figure16: Submerged benchmark: Q-values calculated for surface elevation time series from the Navier-Stokes model as functions of the viscosity µ. A. Average Q-values for the four gauges calculated using the three slip boundary conditions (noSlip (black), 0.4 partialSlip (red) and 0.1 partialSlip (green)). B. Q-values calculated for Gauges 1 (black) and 4 (blue) and average Q-values for the four gauges (red) using the 0.4 partialSlip condition. X-axis is logarithmic.

Figure 17 :

 17 Figure 17: Submerged benchmark: comparison of surface elevation time series (m) between the experiments (black triangles) and the Navier-Stokes model simulations using viscosity µ of 5 Pa.s with a noSlip boundary condition (red), 3 Pa.s with a 0.4 partialSlip condition (green) and 8 Pa.s with a 0.1 partialSlip condition (blue).

Figure 18 :Figure 19 :

 1819 Figure18: Submerged benchmark: comparison of slide contours between experiments (red squares) and the Navier-Stokes model simulations with µ=3 Pa.s and noSlip (green), 0.1 partialSlip (orange) or 0.4 partialSlip (black) slip conditions. From left to right and up to down, contours are represented at t=0, 0.02, 0.17, 0.32, 0.47 and 0.62 s. The blue line represents the initial water surface. The brown line represents the slope.

Figure 20 :

 20 Figure20: Amplitude of the first wave (m), measured at Gauge 1, as a function of the slope angle (in degree) for the depth-averaged model, five submergences (S -1 in black, S -0.5 in red, S 0 in green, S +0.5 in blue and S +1 in orange) and two slide Reynolds numbers Re s : 250 (A) and 20 (B).

Figure 21 :Figure 22 :

 2122 Figure21: Amplitude of the first wave (m), measured at Gauge 1, as a function of the slope angle (in degree) for the Navier-Stokes model, five submergences (S -1 in black, S -0.5 in red, S 0 in green, S +0.5 in blue and S +1 in orange) and two slide Reynolds numbers Re s : 250 (A) and 20 (B).

Figure 23 :

 23 Figure 23: Dimensionless maximal water wave amplitude computed by the depth-averaged model (black) and the Navier-Stokes model (red) as a function of the slide Reynolds number. The submergence is S 0 and the slope 35°.

Figure 24 :

 24 Figure 24: Comparison of surface elevation time series (m) between laminar (black) and k-ϵ (red) simulations with the Navier-Stokes model using Re s ∼ 20 with a noSlip boundary condition for submergences S -1 (top panels), S 0 (middle panels) and S +1 (bottom panels) and for slope angles of 35°(A), 45°(B) and 60°(C).

Figure 25 :

 25 Figure 25: Landslide velocity (m.s -1 ) as a function of the slope angle (in degree) for the depth-averaged model (A) and the Navier-Stokes model (B), five submergences (S -1 in black, S -0.5 in red, S 0 in green, S +0.5 in blue and S +1 in orange) and two Reynolds numbers Re s : 20 (plus) and 250 (circles).

Figure 26 :

 26 Figure26: Snapshots of initial (in black) and entering or just entered into the water (in red) landslides for a slope angle of 45°and three submergences, S +1 (top panels), S 0 (middle panels) and S -1 (bottom panels), for the depth-averaged model (left panels) and the Navier-Stokes model (right panels) and Re s ∼20. The blue line represents the initial water surface. The brown line represents the slope.

Figures

  Figures 3, 7, 11 and 16. We note that the waves in the far-field are even less sensitive to the viscosity.

  u t

														∂ 2 d ∂x∂y	+ d	∂d ∂x	∂u t ∂x	+	d 2	∂d ∂x	∂v t ∂y	+	∂d ∂y	∂v t ∂x	+ F x (8)
	∂v ∂t	+ u	∂v ∂x	+ v	∂v ∂y	= -g	∂η ∂y	+	d 3 3	∂ 2 v t ∂y 2 +	∂ 2 u t ∂x∂y
										+	d 2	v t	∂ 2 d ∂y 2 + v t	∂ 2 d ∂x∂y	+ d	∂d ∂y	∂v t ∂y	+	d 2	∂d ∂x	∂u t ∂y	+	∂d ∂y	∂u t ∂x	+ F y (9)

Table 1 :

 1 Summary table of viscosity µ and Reynolds number Re s values (in [ ]) that give the best results relatively to the two benchmarks studied.

		Subaerial	Submerged
	Depth-averaged	6-15 [16-40]	0.1-1 [200-2000]
	Navier-Stokes	12.8-30 [8-20]	1-10 [20-200]

  The viscosity is generally lower in the depthaveraged model than in the Navier-Stokes model, as summarized in Table1. However, both models share viscosity values in their intervals of validity: between 12 and 15 Pa.s for the subaerial benchmark (higher than[START_REF] Viroulet | Tsunami waves generated by cliff collapse: comparison between experiments and triphasic simulations[END_REF], 5 Pa.s, but similar to[START_REF] Clous | Simulation of energy transfers in waves generated by granular slides[END_REF], 10 Pa.s) and around 1 Pa.s for the submerged benchmark (higher than[START_REF] Grilli | Modeling coastal tsunami hazard from submarine mass failures: effect of slide rheology, experimental validation, and case studies off the US East Coast[END_REF], 0.01 Pa.s, but similar to[START_REF] Clous | Simulation of energy transfers in waves generated by granular slides[END_REF], 1 Pa.s). This difference between both models is likely due both to the nature of the equations formulations in the depth-averaged model and to the boundary condition that leads to higher viscosity values in the Navier-Stokes model.

5.1.2 Relative low influence of the slide viscosity on the waves

Similar water surface elevations are calculated for a range of viscosity values (Sections 4.1 and 4.2 and Table

1

). The relative low sensitivity to viscosity value is illustrated by the plateau of lower Q-value depictable in

Acknowledgements

We thank the two anonymous reviewers for their constructive comments which improved the quality of this work.