## 1. Introduction

[2] Recent assessments of tsunami hazards along the Gulf of Mexico (GOM) carried out by the U.S. Geological Survey (USGS) and the National Tsunami Hazard Mitigation Program (NTHMP) have identified underwater landslides as the primary potential source of tsunami generation [*ten Brink et al*., 2009; *Horrillo et al*., 2010]. Tsunami generation by underwater landslides depends on the geological characteristic of the slope materials and the triggering mechanism affecting the continental shelf. Common mechanisms to initiate an underwater landslide and the ensuing tsunami are: (a) earthquakes, (b) overpressure due to rapid deposition of soil sediments, (c) presence of weak soil layers, (d) wave loading on sea-bottom by storms or hurricanes, (e) build up of the excess pore water pressure, (f) gas hydrate dissociation by change of temperature or pressure, (g) groundwater seepage, and (h) slope oversteepening [*Hampton and Locat*, 1996; *Locat and Lee*, 2002; *Mason et al*., 2006]. Although a massive underwater landslide in the GOM is considered a potential hazard, the probability of such an event is quite low [*Dunbar and Weaver*, 2008]. The probability of occurrence is related to large ancient landslides which were probably active prior to 7000 years ago when large quantities of sediments were emptied into the GOM [*ten Brink et al*., 2009]. However, nowadays sediments continue to empty into the GOM mainly from the Mississippi river. The sediment supply contributes to slope steepening and also to the increasing of the excess pore water pressure in the underlying soils, which may lead to further landslide activities. Recent evidence from seismic records of small-scale energetic seismic-waves in the GOM have confirmed that there is a probability of recurrence [*Dellinger and Blum*, 2009].

[3] In the past century, the seriousness of this threat became evident after the 1929 Grand Banks underwater landslide event, which produced tsunami waves of 3–8 m high, killing 28 people along the Newfoundland coastline [*Cranford*, 2000; *Clague et al*., 2003]. Attempts to uncover the underlying physics came initially by a hand full of laboratory experiments [*Wiegel*, 1955; *Law and Brebner*, 1968; *Heinrich*, 1992; *Watts*, 1997]. These experiments used simple solid boxes sliding down incline planes. Further insight into the phenomenon was achieved with 2-D and 3-D experiments involving granular slide material on very steep slopes (fjord-like slopes) [*Huber*, 1980, 1982]. However, it was not until after the 1998 Papua New Guinea (PNG) tsunami that a thorough investigation of the underwater slide mechanisms and the generated tsunami was carried out in detail. This event claimed at least 2200 lives when waves up to 15 m high flooded the country's northern coast; this has been widely documented in e.g., *Tanioka and Ruff* [1998], *Kikuchi et al*. [1998], *Tanioka* [1999], *Sweet et al*. [1999], *Tappin et al*. [1999], *Kawata et al*. [1999], *Geist* [2000], *Heinrich et al*. [2000], *Tappin et al*. [2001], *Imamura and Hashi* [2002], *Synolakis et al*. [2002], *Satake and Tanioka* [2003], and it has served as the prelude for advanced landslide-tsunami investigations. However, the field data obtained from landslide-tsunami events are still very limited, so modelers depend heavily on laboratory experiments and analytical solutions for their research studies and numerical model validations. Other events of interest to the tsunami research community are the massive subaerial rockfall into Gilbert Inlet at the head of Lituya Bay, triggered by the earthquake on July, 1958, and the most recent landslide-tsunami occurred in the aftermath of Haiti earthquake on January, 2010 [*Fritz et al*., 2009, 2013].

[4] Recent landslide laboratory experiments [e.g., *Fritz*, 2002; *Grilli and Watts*, 2005; *Liu et al*., 2005; *Enet and Grilli*, 2005, 2007], produced a variety of empirical formulations [e.g., *Watts*, 1998, 2000; *Enet et al*., 2003; *Synolakis and Raichlen*, 2003; *Raichlen and Synolakis*, 2003; *Fritz et al*., 2004; *Lynett and Liu*, 2005; *Heller*, 2007; *Heller and Hager*, 2010], that together with several 1-D analytical solutions [e.g., *Noda*, 1970; *Hunt*, 1988; *Tinti and Bortolucci*, 2000; *Tinti et al*., 2001; *Okal and Synolakis*, 2003; *Liu et al*., 2003; *Pelinovsky*, 2003; *Haugen et al*., 2005; *Didenkulova et al*., 2010] and 2-D and 3-D analytical solutions [e.g., *Novikova and Ostrovsky*, 1978; *Pelinovsky and Poplavsky*, 1997; *Ward*, 2001] have proved to be essential in continuing developing, verifying, and validating landslide-tsunami numerical models. For instance, *Jiang and Leblond* [1992, 1993] developed a numerical model to simulate a deformable submarine landslide (mudslide) and the generated surface waves using nonlinear shallow water (SW) equations for both water waves and mudslide material. The numerical model fully coupled the mudslide and the water wave dynamics. *Imamura and Imteaz* [1995] and *Imteaz and Imamura* [2001] developed a numerical model for two-layer flows along a variable bottom by using the leap-frog finite difference scheme with a second-order truncation error for the solution of the SW equations. The landslide material was immiscible with uniform density and viscosity and the landslide motion was not prescribed but obtained using internally balanced forces. *Thomson et al*. [2001] modified a SW numerical model developed by *Fine et al*. [1998] to include arbitrary bottom topography and mudslide viscosity with full two-way interaction. The model was used to simulate the tsunami of 3 November 1994 in Skagway, Alaska. Concurrently, *Heinrich et al*. [2001] developed a SW numerical model to study the efficiency of deep water slumps in producing tsunami waves. The model was tested and validated by comparison with a numerical model that solves the Navier-Stokes (NS) equations. The SW mudslide phase included both, a non-Newtonian friction law and a basal friction coefficient. Through means of a sensitivity test and by applying it to a real tsunami event (PNG), it was concluded that the generated wave depends strongly on the constitutive law of the landslide rheology.

[5] *Assier-Rzadkiewicz et al*. [1977] simulated an underwater landslide using a 2-D fluid mechanics mixture model based on the NS equations. The mudslide material was considered as a viscoplastic fluid with rheological parameters, e.g., the diffusion and viscosity coefficients, the Bingham yield stress and the basal friction. The model was validated with analytical solutions and laboratory experiments documented in *Heinrich* [1992] for a viscous-Bingham flow and compared against a sliding-rigid box and a gravel slide laboratory experiment. They stressed the importance of the sediment rheology and the diffusion parameter in the wave dynamics. Later, *Grilli and Watts* [1999, 2005] and *Grilli et al*. [2010], applied fully nonlinear 2-D and 3-D potential flow (Boundary Element Method) simulations of underwater landslide-tsunamis to water wave generation. They assumed geometrically idealized landslide shapes, i.e., for the 2-D, a semiellipse or “bump” configuration and for the 3-D, a bi-Gaussian-shaped or “saucer” configuration. The landslide center of mass motion along the slope was prescribed based on a dynamic force balance using Newton's laws and some empirical coefficients based on theories or validated experimentally. The results obtained in this study were used to create landslide-tsunami sources for practical application of tsunami studies [*Tappin et al*., 2008].

[6] Another well-known numerical model is the SAGE hydrocode. SAGE has been used in many occasions by modelers to simulate landslide-induced tsunami, *Mader and Gittings* [2002, 2003] and *Gisler* [2006]. The code, originally developed by *Gittings* [1992] for Science Applications International, Los Alamos National Laboratory, is mainly suited in compressible multi-material simulations, e.g., meteorite impact, *Gisler et al*. [2004]. It solves the full set of compressible NS equations, including the equation of state and different constitutive models for material strength. An automatic adaptive Eulerian grid refinement is employed with a high-resolution Godunov scheme. The adaptive mesh can be refined locally where large gradients of certain physical properties of the fluid-flow exist, e.g., pressure, density, etc.

[7] *Liu et al*. [2005] implemented a numerical model to simulate a landslide-generated tsunami. The model solves the 3-D NS equations and is based on the Large Eddy Simulation diffusion mechanism. The Smagorinsky subgrid scale is employed for the turbulence closure. The volume of fluid (VOF) method is used to track the water free surface and the shoreline evolution. To test the model a laboratory experiment was carried out in a large scale wave tank by using a solid wedge sliding on a plane slope at one end of the tank [*Liu et al*., 2005; *Synolakis et al*., 2007].

[8] *Kowalik et al*. [2005a, 2005b] developed a 2-D NS model for waves generation by rigid and deformable moving objects. The standard VOF method was used to track the water free surface and the shoreline evolution. The first order VOF donor-acceptor technique for the fluid advection of *Hirt and Nichols* [1981] was used by reducing the centered difference approximation (second order) by means of the so-called parameter alpha, i.e., weighting the upstream derivative of the quantity being fluxed more than the downstream derivative. The model's capabilities to simulate a rigid underwater or subaerial landslide for the tsunami generation was achieved by including a dynamic fractional area-volume technique for the transient moving boundaries of the object within the Cartesian grid system. The model results were compared with SW analytical solutions (provided in *Synolakis et al*. [2007]) as well with the solutions obtained by using a SW numerical model. Large differences were observed between the two approaches (2-D NS versus SW) when nonhydrostatic effects were strong, mainly due to the fact that SW model and the SW analytical derivation inherently do not consider the vertical component of velocity/acceleration in their solutions. Later, *Horrillo* [2006] implemented and tested the model against a subaerial landslide laboratory experiment described in *Heinrich* [1992]. In this experiment, the 2-D NS model confirmed its capability to deal with complex wave kinematics at early stages of wave generation.

[9] It is noteworthy that even though the SW approximation is relatively accurate in many practical tsunami applications, e.g., cosiesmic-sources in which the resulting waves are usually in the shallow water regime (long waves), it is still doubtful when this approximation is applied to landslide-tsunamis because the landslide motion usually presents large vertical velocity and acceleration which are important for the wave kinematics and free surface evolution. The physical aspect on the wave kinematics is even more critical at early stage of the landslide motion or tsunami generation [*Grilli et al*., 2002; *Fritz et al*., 2003a, 2003b; *Kowalik et al*., 2005a]. In addition, the departing or out-going waves, usually fit in the intermediated depth regime as they reach deeper water from the generation region. Simultaneously, the back-going waves evolve as highly dispersive in the shoaling process toward the coastline.

[10] *Abadie et al*. [2010] reported on the application and experimental validation of a multiple fluid NS model, THETIS, for waves generated by idealized slide geometries or deforming slides. The model treated all computational domain regions, i.e., water, air, and slide, as Newtonian fluids. Instead of specifying the slide kinematics, a penalty method was employed to force implicitly the two-way coupling between the rigid slide and the air or water phase. The model has been validated using analytical solutions and several laboratory experiments from previous studies, including the 3-D landslide experiment described in *Liu et al*. [2005] and *Synolakis et al*. [2007].

[11] Application of numerical models to develop practical tsunami hazard/mitigation products, for example tsunami inundation maps, requires model testing over a variety of benchmark problems to ensure model results match expected values within a minimal margin of errors. In addition, tsunami numerical models need to be continuously tested with new releases or updated versions, or new sets of suited laboratory/tsunami-field data that have become available. Therefore, one of the main objectives of this work is to validate the simplified 3-D NS version derived from the work described in *Kowalik et al*. [2005a, 2005b] and *Horrillo* [2006]. The simplified 3-D NS tsunami model is specifically customized for tsunami calculations and it is dubbed TSUNAMI3D for Tsunami Solution Using Navier-Stokes Algorithm with Multiple Interfaces. The model was initially developed at the University of Alaska Fairbanks and improved later in Texas A&M University at Galveston. The model is further tested in this study using the laboratory setup described in *Liu et al*. [2005] and *Synolakis et al*. [2007] as 3-D tsunami generation by underwater landslides, see also *National Tsunami Hazard Mitigation Program (NTHMP)* [2012].

[12] It is well known that full 3-D numerical models are highly computationally intensive and require a considerable amount of computer resources. Therefore, the simplified 3-D NS model has been conceived to overcome the computational burden that is common in 3-D tsunami simulations. The simplification is derived from the large aspect ratio (horizontal/vertical scale) of the tsunami wave and the selected computational cell size required to construct an efficient 3-D grid. The large aspect ratio of the tsunami wave requires also a large grid aspect ratio to reduce runtime and memory usage. However, the grid aspect ratio should be smaller than the aspect ratio of the tsunami wave to facilitate the fluid surface reconstruction. The standard VOF algorithm, the donor-acceptor technique of *Hirt and Nichols* [1981], has been simplified to account for the large aspect ratio of the grid. The pressure term is split in two components, hydrostatic and nonhydrostatic. In addition, this study discusses the effect of using a sharp interface between the mudslide material and the water for a full-scale landslide event in the GOM. In this particular experiment, it is thought that excessive diffusion of certain physical properties (e.g., the averaged-density at a given cell having the water-mudslide interface) originated by the low resolution necessary for efficient numerical computation, might not affect considerably the generated (initial) tsunami wave configuration. To confirm this assertion, a 2-D numerical experiment in *x*, *z* (horizontal and vertical) axes is carried out using the simplified 3-D NS model and compared with the commercial Computational Fluid Dynamic (CFD) model FLOW3D. The commercial model uses a diffusive interface between mudslide and water; on the other hand, the simplified 3-D NS model utilizes a sharp (not diffusive) interface condition. Last, a large scale 3-D numerical simulation is carried out for the ancient GOM's East-Breaks landslide by using the simplified model to calculate the early stages of the tsunami wave propagation.