Effective Coastal Boundary Conditions for Dispersive Tsunami Propagation

We aim to improve the techniques to predict tsunami wave heights along the coast. The modeling of tsunamis with the shallow water equations has been very successful, but often shortcomings arise, for example because wave dispersion is neglected. To bypass the latter shortcoming, we use the (linearized) variational Boussinesq model derived by Klopman et al. [12]. Another shortcoming is that complicated interactions between incoming and reflected waves near the shore are usually simplified by a fixed wall or absorbing boundary condition at a certain shallow depth contour. To alleviate this, we explore and present a so-called effective boundary condition (EBC), developed here in one spatial dimension. From the deep ocean to a seaward boundary, i.e., in the simulation area, we model wave propagation numerically. We measure the incoming wave at this seaward boundary, and model the wave dynamics towards the shoreline analytically, based on shallow water theory and the Wentzel-Kramer-Brillouin (WKB) approximation, as well as extensions to the dispersive, Boussinesq model. The coupling between the dynamics in the two areas, respectively treated numerically and analytically, is handled using variational principles, which leads to (approximate) conservation of the overall energy in both areas. The reflected wave is then influxed back into the simulation area using the EBC in a discrete, variational, finite element formulation. We verify and validate our approach in a series of numerical test cases of increasing complexity, including a case akin to tsunami propagation to the coastline at Aceh, Sumatra, Indonesia.