Effective coastal boundary conditions for tsunami wave run-up over sloping bathymetry

Abstract

Abstract. An effective boundary condition (EBC) is introduced as a novel technique for predicting tsunami wave runup along the coast, and offshore wave reflections. Numerical modeling of tsunami propagation in the coastal zone has been a daunting task, since high accuracy is needed to capture aspects of wave propagation in the shallower areas. For example, there are complicated interactions between incoming and reflected waves due to the bathymetry and intrinsically nonlinear phenomena of wave propagation. If a fixed wall boundary condition is used at a certain shallow depth contour, the reflection properties can be unrealistic. To alleviate this, we explore a so-called effective boundary condition, 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 over real bathymetry using either the linear dispersive variational Boussinesq or the shallow water equations. We measure the incoming wave at this seaward boundary, and model the wave dynamics towards the shoreline analytically, based on nonlinear shallow water theory over bathymetry with a constant slope. We calculate the run-up heights at the shore and the reflection caused by the slope. The reflected wave is then influxed back into the simulation area using the EBC. 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. We verify 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.

Variational space–time (dis)continuous Galerkin method for nonlinear free surface water waves

“…The numerical validation with the MARIN experiments was somewhat limited because we used reflecting rather than absorbing boundary conditions at the laboratory beach. Interesting variational approaches to solve this boundary issue are: to use a new water wave model we developed [6,12], which includes the nonlinear water wave limit as well as wave breaking in the form of shallow water bores at the beach, or to employ effective, semi-analytical nonlinear boundary conditions such as developed and used in Kristina et al [22]. (A.7)…”

“…defining the jump term (22) in the limit of this narrow element shrinking to zero into node t n . The time integral of the other terms in the variational principle across such a narrow element vanishes in this limit.…”

An Efficient Two-Layer Non-Hydrostatic Model for Investigating Wave Run-Up Phenomena

“…Aceh is one of the worst affected areas by the 2004 Indian Ocean tsunami. We use the same bathymetry data as mentioned in [29] which located on the west coast of Aceh. Its location is about 100 km from the epicenter of the earthquake.…”