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 is somewhat simplistic because wave dispersion is neglected. To bypass this shortcoming, we use the (linearized) variational Boussinesq model derived by Klopman et al. [J. Fluid Mech. 657, 36--63, 2010]. Another shortcoming is that the complicated interactions between incoming and reflected waves near the shore are usually simplified by a fixed wall boundary condition at a certain shallow depth contour. To alleviate this shortcoming, we explore and present in one spatial dimension a so-called effective boundary condition (EBC). From the deep ocean to the seaward boundary, i.e., the simulation area, we model wave propagation numerically. Given the measurements of the incoming wave at the seaward boundary, we 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 reflected wave is then influxed back into the simulation area using the EBC. The coupling between the two areas, one done numerically and one analytically, via the EBC is handled using variational principles, to preserve the overall energy in both areas. 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.