Analysis of Dispersive Landslide Tsunami Waves in the Lagrangian Framework
Abstract:Tsunamis’ inundation heights must be accurately and efficiently predicted for a timely evacuation of coastal populations exposed to such hazardous incidents. To achieve this, approximate models have been developed for an efficient estimation of the wave propagation and runup on the world shorelines. The accuracy of these approximate models is yet a matter of dispute in regard to how much dispersion and nonlinearity should be included, and to how well the runup phenomenon is resolved.
The linear shallow-water model-equation is widely used for runup predictions because it is computationally efficient. However, its lack of dispersive properties is known to adversely affect the correct prediction of wave height and arrival time. The Boussinesq set of equations considers weak dispersive and nonlinear effects and, despite being computationally more expensive, has a much better accuracy. The balance between dispersive and nonlinear effects is of significant importance for the problem of landslide generated tsunamis because such nonlinear waves have a relatively small horizontal length scale (wave length) compared to the domain of propagation. This renders dispersive effects a lot more pronounced than in the case of earthquake tsunamis.
Here we compare the runup predictions of a linear shallow-water, Boussinesq (weak dispersion and weak nonlinearity) and fully nonlinear Boussinesq model (weak dispersion, no assumption on nonlinearity) for various landslide tsunami scenarios. The equations are derived in the Lagrangian framework to allow for an accurate calculation of the runup. Contrary to Eulerian models, long-wave models in Lagrangian framework can be arranged to yield a system of partial-differential equations for the vertical and horizontal displacements of the free-surface. These evolutionary equations are then solved using a finite-difference scheme for time integration and spatial differentiation. The effect of a ridge on a long-wave train climbing up a beach is also discussed and specific features are highlighted.