Abstract
Abstract. An effective boundary condition (EBC) is introduced as a novel technique for predicting tsunami wave run-up 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.
Highlights
Shallow water equations are widely used in the modeling of tsunamis, since their wavelengths are far greater than the depth of the ocean
In the deep ocean for x ∈ [B, L] with horizontal coordinate x and seaward boundary point x = B, denoted as the simulation area, we model the wave propagation numerically using a linear model
In the coastal zone for x ∈ [xs(t), B] with shoreline position xs(t) < B, denoted as the model area, we model the wave propagation analytically using a nonlinear model by approximating the bathymetry as a planar beach
Summary
Shallow water equations are widely used in the modeling of tsunamis, since their wavelengths (typically 200 km) are far greater than the depth of the ocean (typically 2–3 km). In the deep ocean for x ∈ [B, L] with horizontal coordinate x and seaward boundary point x = B, denoted as the simulation area, we model the wave propagation numerically using a linear model. The shoreline position and wave reflection in the model area (sloping region) are determined using an analytical solution of the nonlinear shallow water equations (NSWE) following the approach of Antuono and Brocchini (2010) for unbroken waves. The novelty of our approach is the utilization of an observation operator at the boundary x = B to calculate the incoming wave elevation towards the shore from the numerical solution of the LSWE in the simulation area. Giving the water wave oscillations at this hard wall at x = B, the maximum run-up height of tsunami waves at the coast is subsequently calculated separately by employing a linear approach.
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