A mathematical study of meteo and landslide tsunamis : The Proudman resonance

TL;DR

This paper investigates Proudman resonance phenomena in shallow waters and landslide tsunamis using mathematical models, analyzing complex factors like variable bottoms and deeper waters, supported by theoretical proofs and numerical simulations.

Contribution

It introduces a mathematical framework for analyzing Proudman resonance in complex scenarios, including variable bottoms and deeper waters, extending previous physics-based studies.

Findings

Proved local well-posedness of water waves equations with moving bottom

Justified linear asymptotic models for tsunami amplitude analysis

Performed numerical simulations supporting theoretical results

Abstract

In this paper, we want to understand the Proudman resonance. It is a resonant respond in shallow waters of a water body on a traveling atmospheric disturbance when the speed of the disturbance is close to the typical water wave velocity. We show here that the same kind of resonance exists for landslide tsunamis and we propose a mathematical approach to investigate these phenomena based on the derivation, justification and analysis of relevant asymptotic models. This approach allows us to investigate more complex phenomena that are not dealt with in the physics literature such as the influence of a variable bottom or the generalization of the Proudman resonance in deeper waters. First, we prove a local well-posedness of the water waves equations with a moving bottom and a non constant pressure at the surface taking into account the dependence of small physical parameters and we show that these equations are a Hamiltonian system (which extend the result of Zakharov [33]). Then, we justify some linear asymptotic models in order to study the Proudman resonance and submarine landslide tsunamis; we study the linear water waves equations and dispersion estimates allow us to investigate the amplitude of the sea level. To complete these asymptotic models, we add some numerical simulations.