Effective coastal boundary conditions for tsunami simulations

Abstract

Numerical modeling of tsunami propagation at the coastal zone has been a daunting task since high accuracy is needed to capture aspects of wave propagation in the more shallow areas. For example, there are complicated interactions between incoming and reflected waves due to the bathymetry, the run-up and run-down flooding phenomena at the beaches or (other) man-made structures that form the coastline, and intrinsically nonlinear phenomena of wave propagation. Numerical modeling of tsunamis with nested methods in shallower areas is computationally expensive and difficult to use in the operational practice. Meanwhile, if a fixed wall boundary condition is used at a certain shallow depth contour, the reflection properties can be unrealistic. To alleviate this, we develop a so-called effective boundary condition as a novel technique to predict tsunami wave run-up along the coast and offshore wave reflections. The general idea of the effective boundary condition is as follows. From the deep ocean to a seaward boundary, i.e., in the simulation area, the wave propagation is modeled numerically over real bathymetry using either nondispersive, linear, shallow water equations or the dispersive, linear, variational Boussinesq model. The incoming wave is measured at this seaward boundary, and the wave dynamics towards the shoreline and the reflection caused by the bathymetry are modeled analytically. The reflected wave is then influxed back into the simulation area using the effective boundary condition. The location of this seaward boundary point is determined by assessing when nonlinearity starts to play a role in the wave propagation. The modeling of wave dynamics towards the shoreline is achieved by employing the analytical solution of (i) linear shallow water equations and (ii) nonlinear shallow water equations. The linear approach is started with the simplest case, that is flat bathymetry with closed wall boundary condition. Further, a slowly varying bathymetry case is considered. The analytical solution is based on linear shallow water theory and the Wentzel-Kramer-Brillouin approximation, as well as extensions to the dispersive Boussinesq model. Subsequently, in the nonlinear approach, the coastal bathymetry is approximated by using a mean, planar beach. The run-up heights at the shore and the reflection caused by the slope are then modeled based on nonlinear shallow water theory over sloping bathymetry. The coupling between the numerical and analytic dynamics in the two areas is handled using variational principles, which leads to (approximate) conservation of the overall energy in both areas. The numerical solution in the simulation area is based on a variational finite element method. Verifications of the effective boundary condition technique and implementation are done in a series of numerical test cases of increasing complexity, including a case akin to tsunami propagation to the coastline at Aceh, Sumatra, Indonesia. The comparisons show that the effective boundary condition method gives a good prediction of the wave arriving at the shoreline as well as the wave reflection, based only on the information of the wave signal at this seaward boundary point. The computational times needed in simulations using the effective boundary condition implementation show a reduction compared to times required for corresponding full numerical simulations.