Numerical simulation of tsunami waves generated by deformable submarine landslides
Introduction
Landslides, or submarine mass failures (SMFs), are presently thought to be one of the major mechanisms for tsunami generation in coastal areas (Masson et al., 2006). Owing to the large volume involved, landslides can generate very large and energetic surface waves (Abadie et al., 2012), producing high wave run-up along the coast. For example, submarine mass failure is considered as one of the major sources for the 1998 Papua New Guinea tsunami that caused great loss in human life (Synolakis et al., 2002, Tappin et al., 2001, Tappin et al., 2002). Compared to seismogenic tsunami, landslide induced tsunami waves feature relatively shorter wavelengths, and hence frequency dispersion effects can be significant or even dominant in the wave evolution process. Interactions between the landslide and the associated waves are strong and will affect the characteristics of both (Jiang and Leblond, 1992, Assier-Rzadkiewicz et al., 1997). Numerical simulation of landslide tsunamis has to take these factors into consideration.
Tremendous effort has been devoted to simulating landslide tsunami generation in the last several decades. Various computational models have been employed using different levels of simplification; for example, shallow water theory (Harbitz, 1992), fully nonlinear potential flow (Grilli and Watts, 1999, Grilli and Watts, 2005, Grilli et al., 2002), Boussinesq equations (Lynett and Liu, 2003, Watts et al., 2003, Fuhrman and Madsen, 2009, Zhou and Teng, 2010), non-hydrostatic wave equations (Ma et al., 2012) and Navier–Stokes equations (Heinrich, 1992, Liu et al., 2005; Yuk et al., 2006, Ataie-Ashtiani and Shobeyri, 2008, Abadie et al., 2010, Montagna et al., 2011; Horrilo Horrillo et al. (2013)). Most of these models treat the landslide as a rigid solid with prescribed slide shape and behavior. The slide motion, which is generally accounted for through a moving boundary condition, is specified based on a dynamic force balance on the sliding mass involving weight, buoyancy, friction, hydrodynamic drag and inertia forces (Enet Grilli and Watts, 2005). An exception is the model of Abadie et al. (2010), in which the coupling between the rigid slide and water is implicitly computed, rather than specifying known slide kinematics. All of these models are able to accurately reproduce water waves generated by rigid sliding objects as observed in the laboratory environment. As summarized by Abadie et al., 2010, Abadie et al., 2012, however, the methodology employed in these models has severe limitations in application to real cases, where landslides are always deformable. Due to its time-varying 3D geometry, rheology as well as slide–water interactions, the deformable landslide is much more complex than a rigid slide. It is impossible to prescribe slide kinematics a priori. In this sense, more advanced models are needed, which allow the slide to deform and are capable of describing the two-way coupling between the slide and surrounding water (Jiang and Leblond, 1992, Jiang and Leblond, 1993, Abadie et al., 2012).
Attempts have been made to develop more sophisticated numerical models for deformable landslides. Most of the existing deformable models were based on long-wave approximation. The landslides were modeled as either rheological materials or granular flow. For example, Jiang and Leblond, 1992, Jiang and Leblond, 1993 developed a model to study the coupling between a deformable submarine landslide and associated tsunami waves based on the assumption that the slide material is not diluted while flowing downslope. The long-wave approximation was adopted for both water waves and the landslide. The slide flow was assumed to be laminar with a parabolic distribution of the horizontal velocity. They applied the model to study the wave characteristics and the parameters dominating the slide-wave interactions. Their model is able to capture the slide motion and tsunami wave generation. Imran et al. (2001) proposed a 1D two-layer numerical model (BING1D) describing the downslope development of submarine debris flows. Their model incorporates three rheological models as user defined alternatives. Similar to Jiang and Leblond (1993), the long wave approximation is adopted and the flow is assumed to remain laminar throughout the computation. Watts and Grilli (2003) employed the BING model to study the underwater landslide shape, motion and deformation at early times. Recently, Kelfoun and Druitt, 2005, Kelfoun et al., 2010, Giachetti et al., 2011 developed a depth-averaged granular flow model, which was coupled with a shallow water flow model to simulate tsunamis generated by large debris avalanches. The long-wave approximation has intrinsic limitations which prevent these models from being applied to most submarine landslides where vertical accelerations and frequency dispersion are not negligible. To avoid the long-wave assumption, Navier–Stokes solvers with advanced free surface capturing schemes such as the Volume-of-Fluid (VOF) method and smoothed particle hydrodynamics (SPH) approach have been proposed to simulate tsunami wave generation by deformable landslides. Assier-Rzadkiewicz et al. (1997) proposed a 2D sediment–water mixture model for submarine landslides based on Navier–Stokes equations. In their model, the free surface motion was captured by a volume of fluid (VOF) approach. The dense part was considered as a Bingham fluid, and the dispersed part was modeled as an ideal fluid. The model was applied to simulate a laboratory landslide. The results showed that the model could reproduce the water waves generated by the landslide in reasonable accuracy. Ataie-Ashtiani and Shobeyri (2008) developed a similar model using meshless smoothed particle hydrodynamics (SPH) method. Their model could also predict landslide-induced wave generation by adjusting the rheology of the mud. These models have not been applied to simulate marine landslides at large scales, which are different from the landslides at laboratory scales. Abadie et al. (2012) employed a 3D multi-fluid Navier–Stokes model THETIS to simulate tsunami waves generated by the potential collapse of the west flank of the Cumbre Vieja Volcano (CVV), Canary Island, Spain. The free surface and slide–water interface were captured by the VOF algorithm. The deformable landslide was considered as an inviscid fluid with a constant density. The model was successfully applied to study CVV tsunami generation. Horrillo et al. (2013) developed a simplified 3D Navier–Stokes model for full scale landslide scenario in the Gulf of Mexico, the East-Breaks underwater landslide. Their model used a simplified and relatively diffusive free surface capturing scheme to speed up the simulations.
During submarine landslides, strong free surface deformation, large vertical acceleration and non-hydrostatic pressure may occur. As discussed by Abadie et al. (2012), these phenomena may significantly affect energy transfer between slide and surrounding water, and can only be modeled by 3D Navier–Stokes models. The Navier–Stokes solvers discussed above use either VOF algorithm or SPH method to simulate free surface, which are computationally expensive. This paper describes a new submarine landslide model based on a Non-Hydrostatic WAVE model (NHWAVE). The slide is considered as water–sediment mixture, which can be diffused and diluted during its movement. The dense plume is driven by the baroclinic pressure forcing, which is introduced by the spatial density variation. The current model is anticipated to be able to better represent the deformable landslide than that of Abadie et al. (2012), in which the submarine landslide was modeled as inviscid dense fluid.
The present landslide model is still a simplified one as the particle–particle interactions are not considered. The inter-particulate stresses may slow down the slide motion. These processes will be implemented in the future. Comparing with the existing landslide models based on VOF or SPH approaches, another major advantage of the current model is the computational efficiency because (1) less vertical layers are required to capture the landslide motion and (2) free surface is directly solved. Therefore, the current model is practically feasible for modeling 3D large-scale submarine mass failure in the ocean.
The paper is organized as follows. Section 2 introduces the model formulations and numerical scheme. Section 3 presents the model validation using laboratory measurements on turbidity currents. Section 4 applies the model to study tsunami waves generated by landslides at both laboratory and large scales. Section 5 gives the conclusions of the paper.
Section snippets
Formulation
The model we employed in this study is the Non-Hydrostatic WAVE model NHWAVE, which was recently developed by Ma et al. (2012) to study the propagation of fully dispersive, fully nonlinear surface waves in complex 3D coastal environments as well as tsunami wave generation by a prescribed bottom motion. NHWAVE solves the incompressible Navier–Stokes equations in well-balanced conservative form, formulated in time-dependent, surface and terrain-following coordinates. The governing equations are
Lock-exchange problem
The model was first tested against an exchange flow using the parameters of direct numerical simulations (Härtel et al., 2000) and several nonhydrostatic studies (Fringer et al., 2006, Lai et al., 2010). The computational domain is two-dimensional with length m and depth m, which is discretized by grid cells with m. The horizontal and vertical molecular viscosities are m2/s. The eddy viscosities are zero. The sediment settling velocity is assumed to be
Waves generated by a small-scale landslide
In this section, we present a numerical simulation of water waves generated by a laboratory landslide. The experiments were described in Assier-Rzadkiewicz et al. (1997). A series of experiments were conducted by allowing a mass of sand to slide freely down an inclined plane with varying slope and sediment diameter. We have chosen to simulate the case of a volume of 63,000 cm3 of coarse gravel sliding down a 45° slope. The model setup is the same as that of Assier-Rzadkiewicz et al. (1997),
Conclusions
This paper presents a new submarine landslide model base on the Non-Hydrostatic WAVE model (NHWAVE) (Ma et al., 2012). The model solves free surface elevation directly, which makes it more efficient than VOF and SPH models. The landslide is simulated as water–sediment mixture, which can be diffused and diluted during its movement. The dense plume is driven by the baroclinic pressure forcing, which is introduced by the spatial density variation. The model is validated by the laboratory
Acknowledgments
G. Ma acknowledges the support from the Old Dominion University research foundation, Project 993092. J.T. Kirby and F. Shi acknowledge the support of the National Tsunami Hazard Mitigation Program (NOAA), Grant NA10NWS4670010. The authors are indebted to four anonymous reviewers for constructive comments and thorough reviews of the paper.
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