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

Abstract

In this paper, we derive asymptotic models for the propagation of two and threedimensional 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 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.

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

“…Let us recall here briefly the governing equations of our system (see [16] for more details). The velocity fields of an irrotational flow can be expressed as gradients of potentials (that we call φ 1 for the upper fluid and φ 2 for the lower fluid).…”

“…there exists h min > 0 such that ∀x ∈ R, h 1 (x) ≡ 1 + 1 η 1 (x) ≥ h min > 0 and h 2 (x) ≡ 1 δ + 2 η 2 (x) ≥ h min > 0. (1.3) This assumption is necessary in order to obtain the consistency of the full Euler system (1.1) with the Boussinesq/Boussinesq model (2.1), as seen in [16], and recalled in Proposition 2.1 below. We do not always precise this assumption thereafter.…”

“…As said in the introduction, the starting point of our study is the Boussinesq/Boussinesq model, obtained from (1.1) thanks to an asymptotic expansion of the operators G 1 , G 2 and H (see [16]). In order to simplify the notations, we assume that the small parameters of the long wave regime are equal (the general case can easily be obtained by modifying some constants) and set…”

“…with a 1 ∈ R and a 2 , b 1 ≥ 0, and one recovers the three-parameter family of Boussinesq/Boussinesq systems introduced in [16]. Then, as it has been achieved in [5], one can also use the classical BBM-trick [2], and inherit new choices as parameters.…”

“…The governing equations of such a system, that we call "full Euler", have been obtained in [16]; we briefly recall the system in Section 1.3 below. This system is strongly nonlinear, and remains complex, for its direct study as well as for numerical computations.…”