Asymptotic Shallow Water Models for Internal Waves in a Two-Fluid System with a Free Surface

2009/06/30 by Vincent Duchêne, Vincent Duchene · 43 citations
Earth and Planetary Sciences · Mathematics · #Boundary value problem #Coastal and Marine Dynamics #Convergence (economics) #Dirichlet distribution #Euler equations #Euler system #Euler's formula #Free surface #Geometry #Internal wave #Mathematical analysis #Mathematics #Mechanics #Navier-Stokes equation solutions #Nonlinear system #Ocean Waves and Remote Sensing #Physics #Surface (topology) #Thermodynamics #Wavelength #Waves and shallow water #math.AP

paper · pdf · doi:10.1137/090761100

published in SIAM Journal on Mathematical Analysis 42(5), 2229-2260 (Society for Industrial and Applied Mathematics) · 32 pages, 4 figures

openalex publication_date 2010/01/01 · arxiv created 2010/10/13 · arxiv updated 2021/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper, we derive asymptotic models for the propagation of two- and three-dimensional gravity waves at the free surface and the interface between two layers of immiscible fluids of different densities over an uneven bottom. We assume the thickness of the upper and lower fluids to be of comparable size and small compared to the characteristic wavelength of the system (shallow water regimes). Following a method introduced by Bona, Lannes, and Saut [J. Math. Pures Appl. (9), 89 (2008), pp. 538–566] based on the expansion of the involved Dirichlet-to-Neumann operators, we are able to give a rigorous justification of classical models for weakly and strongly nonlinear waves, as well as interesting new ones. In particular, we derive linearly well-posed systems in the so-called Boussinesq/Boussinesq regime. Furthermore, we establish the consistency of the full Euler system with these models and deduce the convergence of the solutions.

Citations

• Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory
• The Cauchy problem for quasi-linear symmetric hyperbolic systems
• Stability of periodic waves of finite amplitude on the surface of a deep fluid
• Well-posedness of the Cauchy problem for models of large amplitude internal waves
• On 2D Rayleigh–Taylor instabilities

Cited by

• DECOUPLED AND UNIDIRECTIONAL ASYMPTOTIC MODELS FOR THE PROPAGATION OF INTERNAL WAVES
• Asymptotic models for the generation of internal waves by a moving ship, and the dead-water phenomenon
• Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation
• Shallow water asymptotic models for the propagation of internal waves