A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

Abstract

We introduce a new class of Green-Naghdi type models for the propagation of internal waves between two (1 + 1)-dimensional layers of homogeneous, immiscible, ideal, incompressible, irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original bi-layer Green-Naghdi model, and in particular to manage high-frequency Kelvin-Helmholtz instabilities, while maintaining its precision in the sense of consistency. Our models preserve the Hamiltonian structure, symmetry groups and conserved quantities of the original model. We provide a rigorous justification of a class of our models thanks to consistency, well-posedness and stability results. These results apply in particular to the original Green-Naghdi model as well as to the Saint-Venant (hydrostatic shallow-water) system with surface tension.

Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

“…Under reasonable assumptions on F 1 , F 2 , and for sufficiently regular ζ, A F γ,δ [ζ] defines a well-defined, symmetric, positive definite operator [23]. We may thus introduce the variable 4) and write…”

“…Due to the weak density contrast, the observed solitary waves typically have much larger amplitude than their surface counterpart, hence the bilayer extension of the GreenNaghdi system introduced by [17,35,38], often called Miyata-Choi-Camassa model, is a very natural choice. It however suffers from strong Kelvin-Helmholtz instabilities-in fact stronger than the ones of the bilayer extension of the water waves system for large frequencies-and the work in [23] was motivated by taming these instabilities. The modified bilayer system reads ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂ t ζ + ∂ x w = 0, …”

“…However, the validity of all these models relies on the hypothesis that the depth of the layer is thin compared with the horizontal wavelength of the flow and, as expected, the models do not describe accurately the behavior of medium or short waves. In order to tackle this issue, one of the authors has recently proposed in [23] a new family of models: ⎧ ⎪ ⎨ ⎪ ⎩ ∂ t ζ + ∂ x w = 0,…”

“…The study of [23] is not restricted to surface propagation, but is rather dedicated to the propagation of internal waves at the interface between two immiscible fluids, confined above and below by rigid, impermeable and flat boundaries. Such a configuration appears naturally as a model for the ocean, as salinity and temperature may induce sharp density stratification, so that internal solitary waves are observed in many places [27,29,40].…”

“…Note that compared to equations (7)- (9) in [23] we have scaled the variables so that the shallowness parameter μ and amplitude parameter do not appear in the equations. This is for notational convenience since the parameters do not play a direct role in our results.…”