Development of a 3D numerical wave tank for modeling tsunami generation by underwater landslides

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Abstract

A three-dimensional (3D) numerical wave tank (NWT) solving fully nonlinear potential flow theory, with a higher-order boundary element method (BEM), is modified to simulate tsunami generation by underwater landslides. New features are added to the NWT to model underwater landslide geometry and motion and specify corresponding boundary conditions in the BEM model. In particular, a new snake absorbing piston boundary condition is implemented to remove reflection from the onshore and offshore boundaries of the NWT. Model results are favorably compared to recent laboratory experiments. Sensitivity analyses of numerical results to the width and length of the discretization are conducted, to determine optimal numerical parameters. The effect of landslide width on tsunami generated is estimated. Results show that the two-dimensional approximation is applicable when the ratio of landslide width over landslide length is greater than 2. Numerical accuracy is examined and found to be excellent in all cases.

Introduction

Tsunamis generated by underwater landslides appear to be one of the major coastal hazards for moderate earthquakes [12], [13]. Whereas tsunamis generated by direct coseismic displacement are usually relatively small in height and correlate with moment magnitude, landslide tsunamis are only limited in height by the landslide vertical displacement [10], [15], [16]. Since underwater landslides are usually triggered on the continental slope, their displacement may reach several thousand meters and thus produce huge tsunamis, offering little time for warning due to their proximity to shore [8], [17]. There is evidence, for instance, that the large tsunami originated near Unimak Island along the Aleutian Trench in 1946 was caused by a giant ≈200 km3 underwater landslide, triggered by a magnitude Ms=7.1 earthquake [2]. The landslide headscarp was on the shallow continental shelf in 150 m water depth, and the landslide mass moved over a 4° mean slope, down to the 6000 m deep Aleutian Terrace, where parts of it apparently stopped. Coastal runup for this tsunami reached 35 m above sea level at Scotch Cap lighthouse, right onshore of the landslide area.

Predicting landslide tsunamis requires complex numerical models which must accurately represent both landslide and bottom geometry, and the nonlinear interactions between landslide motion and surface wave field. Such a model has been demonstrated by Grilli and Watts [8], in their implementation of a two-dimensional (2D) numerical model for underwater landslides, based on a higher-order boundary element method (BEM), i.e. a numerical wave tank (NWT).2 Reviews of the literature to date regarding tsunamis generated by underwater landslides and their numerical modeling can be found in the latter paper and in Watts and Grilli [19].

Here, we describe the current implementation, validation, and simulation of tsunami generation by underwater landslides in the three-dimensional (3D) NWT developed by Grilli et al. [3], [4]. Fully nonlinear potential flow equations are solved in this NWT based on a higher-order BEM and an explicit time stepping scheme. Wave overturning can be modeled if it occurs in the computations (Fig. 1). Grilli et al. validated their 3D-NWT for solitary wave shoaling and breaking over slopes, by comparing results both to experiments and to an earlier numerical solution. The agreement was excellent and it was found that a high degree of accuracy could be obtained in the NWT through careful discretization of the simulation domain.

Various improvements were made to this 3D-NWT, as part of the present work, to efficiently and accurately simulate tsunamis caused by underwater landslides. Open boundary conditions were implemented and validated for solitary wave propagation over constant depth. These conditions extend to 3D, the piston-like boundary condition used by Clément [1] and Grilli and Horrillo [5]. The landslide shape and kinematics were modeled on a way similar to Grilli and Watts' [8] 2D model, by assuming a smooth initial shape for the landslide, moving down a planar slope (Fig. 2).

Once the relationship of result accuracy versus spatio-temporal discretization is assessed, numerical experiments can be performed in the 3D-NWT for specified initial and boundary conditions, herein for underwater landslides. This is an important point: the exact nature of wave generation is both known and controlled. Different motions of the same submerged body can produce very different waves, and such waves can be directly related to the input parameters of the motion. This is the basis of the wavemaker formalism introduced by Watts [16], [17].

NWTs enable many outputs to be obtained with minimal error, and in virtually no setup time (free surface profiles, numerical wave gages, runup, etc.). Here, however, as done in earlier 2D studies [8], we will usually represent results of the 3D-NWT by a characteristic wave amplitude calculated above the initial landslide position, at the location of maximum landslide thickness (defined at horizontal location (xg,0) in the following; Fig. 4). Our characteristic wave amplitude is thus an implicit function of the underwater landslide shape and motion input parameters.

Section snippets

Governing equations and boundary conditions

Equations for fully nonlinear potential flows with a free surface, which are solved in the 3D-NWT, are summarized below. The velocity potential, defined as φ(x,t), describes inviscid irrotational 3D flows in Cartesian coordinates x=(x,y,z), with z the vertical upward direction (and z=0 at the undisturbed free surface; Fig. 3). The velocity is defined by u=φ=(u,v,w).

Continuity in the fluid domain Ω(t), with boundary Γ(t), is a Laplace's equation for the potential2φ=0inΩ(t)Green's second

Validation of snake AP boundary

The new AP boundary was validated by propagating a fully nonlinear solitary wave over constant depth h0 in the 3D-NWT. The initial wave height is 0.3h0 and the initial wave shape, potential, and normal velocity are specified on the free surface based on Tanaka's [11] method. The NWT is 15h0 long and discretized with 20 elements over x, four elements over y and four elements over z. One AP boundary is initially located at the far extremity at x′=x/h0=15. Due to the 2D geometry and boundary

Effect of domain length

For simplicity, quasi-2D simulations, i.e. cases with no variation in the y-direction (w=∞), are modeled in order to investigate the influence of the total domain length on tsunami characteristic amplitude. The domain width is set to w0=B for all calculations.

The total domain length L0 is set to multiples of the theoretical wavelength λ0 (0.6–3). As indicated before, a constant grid density of 20 elements per theoretical wavelength is used in the x-direction, which provides the same BEM

Effect of domain width

When performing numerical calculations, it is desirable to minimize the influence of numerical parameters on results. In particular, here, it is important to verify at which distance, yw0/2, the impermeable lateral boundaries have to be located to avoid perturbing the tsunami generation process, i.e. changing the characteristic tsunami amplitude, through reflection, while maintaining a realistic size for the computational domain.

To do so, landslides of different widths W=B, 2B and 3B are

Effect of landslide width

Landslides of various widths W=B/2 to 3B are modeled and the resulting characteristic amplitudes compared. According to the above findings, calculations use w0=2W and L0=λ0. Fig. 12 shows results of these computations. Essentially, we see an increase in characteristic amplitude with increasing landslide width, i.e. landslide aspect ratio W/B. The large initial increase with W/B, however, becomes slower for W/B>2, where the amplitude eventually approaches its 2D value of ≃6m asymptotically.

Conclusions

Realistic simulations of tsunami generation by submarine landslides can be obtained by performing wave tank experiments. However, a well-validated numerical model can often provide as accurate or even more accurate results, in some cases, than experiments, in a much shorter time. This is important for assessing tsunami hazard in specific situations.

Here, landslide tsunami generation mechanisms were explored in a 3D NWT solving fully nonlinear potential flow equations, using a higher-order BEM.

Acknowledgements

This work was supported by Grant CMS0100223 of the US National Science Foundation.

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