Modeling landslide-generated water waves
with long-wave equations
Long-wave equations (e.g., shallow water equations, Boussinesq-type equations) are approximate water wave equations based on the assumption that the wavelength is much larger than the typical water depth. They are often employed to study tsunamis, due to the low computational cost and the fact that the most damaging tsunami waves are often very long waves, whose wavelengths are much larger than the typical water depths. This project seeks to establish a set of criteria that guides the selection of the simplest long-wave equations to be used to model tsunamis generated by a landslide.
Various sets of long-wave equations were rederived to ensure consistency in the formulation and to allow for a deforming bottom boundary (i.e., a non-porous landslide). A shock-capturing numerical model was developed to solve the various long-wave equations in a consistent manner. As a result of the consistency, for a given problem, the numerical results predicted by different wave equations can be directly compared, and the performance of each set of long-wave equations can be directly ranked against each other.
In addition, analytical solutions based on linear wave theory (linear and fully dispersive, as well as the linear shallow water equations) were derived, for water waves generated by a landslide, a moving atmospheric pressure, an earthquake, or due to an initial water surface displacement. Simple analytical expressions that reveal the scaling relations between the forcing function and the generated waves were obtained, which help us better understand the physical processes.
Utilizing the insights provided by the analytical solutions, we proposed a set of criteria, which can be estimated a priori, to predict how well the assumptions behind each set of long-wave equations would hold in a given landslide wave problem. The performance of these criteria were then tested against a large number of numerical simulations based on different long-wave equations. These criteria are of practical value because they provide means to quickly predict whether an approximate water wave model applies in a landslide wave problem.
We are currently working on generalizing the criteria (which are based on idealized scenarios) so that they apply to a wider range of landslide wave scenarios.
