Comparing approaches for numerical modelling of tsunami generation by deformable submarine slides

Tsunami generated by submarine slides are arguably an under-considered risk in comparison to earthquake-generated tsunami. Numerical simulations of submarine slide-generated waves can be used to identify the important factors in determining wave characteristics. Here we use Fluidity, an open source finite element code, to simulate waves generated by deformable submarine slides. Fluidity uses flexible unstructured meshes combined with adaptivity which alters the mesh topology and resolution based on the simulation state, focussing or reducing resolution, when and where it is required. Fluidity also allows a number of different numerical approaches to be taken to simulate submarine slide deformation, free-surface representation, and wave generation within the same numerical framework. In this work we use a multi-material approach, considering either two materials (slide and water with a free surface) or three materials (slide, water and air), as well as a sediment model (sediment, water and free surface) approach. In all cases the slide is treated as a viscous fluid. Our results are shown to be consistent with laboratory experiments using a deformable submarine slide, and demonstrate good agreement when compared with other numerical models. The three different approaches for simulating submarine slide dynamics and tsunami wave generation produce similar waveforms and slide deformation geometries. However, each has its own merits depending on the application. Mesh adaptivity is shown to be able to reduce the computational cost without compromising the accuracy of results.

long run-out distance. When slide run-out distance is relatively short com-199 pared to the slide length, the velocity of the slide becomes more important. 200 They further concluded that rapid deformation during the initial accelera-    Here, Fluidity is used to solve the single phase incompressible Navier-Stokes equations: where u is the velocity vector, t represents time, p is pressure, µ is the 221 dynamic viscosity, ρ is the density, and for this work we assume that we are 222 in a coordinate system where g, the gravitational acceleration, acts in the z 223 direction: k = (0, 0, 1) T .
The settling velocity, u si is the hindered sinking velocity, which depends 242 on the sediment concentration. Here, due to the high density of the slide, 243 the sinking velocity is negligible and thus ignored. κ is the diffusivity of the 244 sediment and here is set to a small value, 10 −6 m 2 s −1 .

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In this work we assume a single sediment class and denote its concentra- where ρ s is the density of the individual sediment particles and ρ w is the Since, from (4), one of the volume fraction fields (here always water) can 275 be recovered from the others using n ϕ − 1 advection equations of the form where ρ i and µ i represent the constituent densities and viscosities of the    The MM2FS approach (section 3.1.2) has many similarities to the SEDFS 326 approach, but with a different underlying finite element pair, and the use of pressure is discretised using the same approximation as the volume fraction 337 fields, i.e. using the P1CV discretisation. The same pressure space is also 338 used as the test space for the continuity equation (1b). The consistency with 339 the volume fraction discretisation leads to a method that is both bounded 340 and conservative (Wilson, 2009). For the MM2FS approach, a P1CV based 341 method is not available for the combined pressure and free-surface field. In 342 this case, we therefore combine the P0 velocity discretisation with a piece-343 wise linear (P1) discretisation for pressure and free surface. As a result the 344 volume fraction discretisation is not conservative. However, for the cases 345 studied here the amount of conservation loss was negligible.

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In MM3, the interface between water and slide is dealt with as for MM2FS.

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The interface between air and water is also handled using a compressive lim- where ε ϕwater is a constant user-defined weight for ϕ water . Based on sensi-366 tivity studies, in this work ϕ water alone was used to construct M , to ensure 367 the interfaces between materials were well resolved. |H ϕwater | is the Hessian

Vertically aligned adaptivity
For relatively high aspect ratio problems it has been found that maintain-399 ing columns of elements in the vertical direction has advantages for stability.

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Fully unstructured meshes without any alignment in the vertical direction, 401 can give rise to artificial horizontal gradients of fields that only vary vertically.

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For instance, in the MM3 approach, the initial air-water interface should be Finally the nodes are connected into cells.

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Since the test cases considered here are only two-dimensional, both the 415 horizontal mesh, and the vertical meshes (columns of nodes) below each 416 surface node, are one-dimensional and mesh adaptivity is straight-forward.

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First we obtain the desired new edge lengths ∆x i by projecting the metric 418 in the appropriate direction given by a unit vectorê, and using the following 419 relation: This expresses the fact that the optimal edge when measured with the metric 421 should have length one.
Next, the old mesh co-ordinates are mapped x →x from physical space 423 to a so called metric space using: where ∆x i is the desired edge length between nodes x i and x i−1 . Regions of 425 the old mesh that require adaptation will give node spacings in metric space space, respectively, the interpolation is given by:    the air and water as a discontinuity in volume fraction, and is therefore able 562 to continue simulating the wave evolution after breaking and back-fill occurs.

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This is shown in Figure 4.

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For MM3 simulations, an adaptive mesh (e.g. Figure 6) was used to 577 dynamically increase spatial resolution in regions of interest and decrease 578 spatial resolution away from these regions. In the MM3 adaptive simulations 579 described in Table 1, the mesh adapted to the volume fraction of water. This  Table 1.        An adaptive mesh (e.g., a section of which is shown in Figure 11) was 706 used to increase spatial resolution at the interfaces between slide and water, 707 and water and air. Coarser spatial resolution can be seen with increasing