Abstract
Wave reflection and wave trapping can lead to long wave run-up resonance. After reviewing the theory of run-up resonance in the framework of the linear shallow water equations, we perform numerical simulations of periodic waves incident on a linearly sloping beach in the framework of the nonlinear shallow water equations. Three different types of boundary conditions are tested: fully reflective boundary, relaxation zone, and influx transparent boundary. The effect of the boundary condition on wave run-up is investigated. For the fully reflective boundary condition, it is found that resonant regimes do exist for certain values of the frequency of the incoming wave, which is consistent with theoretical results. The influx transparent boundary condition does not lead to run-up resonance. Finally, by decomposing the left- and right-going waves into a multi-reflection system, we find that the relaxation zone can lead to run-up resonance depending on the length of the relaxation zone.
Highlights
Long wave run-up is difficult to observe in nature in real-time due to the catastrophic consequences it usually leads to
By decomposing the left- and right-going waves into a multi-reflection system, we find that the relaxation zone can lead to run-up resonance depending on the length of the relaxation zone
To emphasize the effect of the relaxation zone, we show the free-surface elevation time series with the use of the other boundary conditions described above
Summary
Long wave run-up is difficult to observe in nature in real-time due to the catastrophic consequences it usually leads to. One generally uses an open boundary condition seaward, which is a numerical artifact that lies in a limited computational domain to solve a mathematical problem that has no boundary at all. Such open boundary conditions are usually called “radiation”, “absorbing”, “non-reflecting”, or “far-field” boundary conditions They all allow waves to leave the truncated domain, avoiding spurious reflections that may pollute the solution in the interior of the computational domain of interest. Run-up on a sloping beach with the use of an open boundary condition at the seaward boundary matches the linear analytical solution that shows no resonance at all. For the fully reflective boundary condition, it is found that resonant regimes do exist for certain values of the frequency of the incoming wave, which is consistent with theoretical results. The applicability of the results to tsunami science is briefly discussed
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