A mathematical study of meteo and landslide tsunamis: the Proudman resonance. (13th October 2015)

• Record Type: Journal Article
• Authors: Benjamin, Melinand
• Is Part Of: Nonlinearity. Volume 28:Number 11(2015:Nov.)
• Page Start: 4037
• Page End: 4080
• Publication Date: 2015-10-13
• Subjects: water waves equations -- quasilinear hyperbolic system -- asymptotic models -- dispersion estimates; 35Q35 -- 76B15 -- 76B03 -- 35L80 -- 35J65; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/28/11/4037
• Languages: English
• ISSNs: 0951-7715

Abstract

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 extends the result of Zakharov (1968 J. Appl. Mech. Tech. Phys .9 190–4)). 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.