Abstract
The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two 'baroclinic' parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability.
Highlights
Internal waves are a major topic of scientific interest [1], playing a key role in climate and weather studies [2], [3], and mathematical models and their analysis play an important part in their understanding [4], [5]
Layered flow models in the geophysical context are considered in the long wave limit, where horizontal length scales are much larger than layer depths
In this paper we focus on the strongly nonlinear, non-dispersive case, and we study the dynamics of two layers of immiscible fluids, bounded below by a horizontal bottom and above by an unbounded layer of fluid, which is dynamically passive but has density, contributing hydrostatically to the pressure
Summary
Internal waves are a major topic of scientific interest [1], playing a key role in climate and weather studies [2], [3], and mathematical models and their analysis play an important part in their understanding [4], [5]. For two-layer flows with a free surface, the situation is similar: a wave-like initial condition might leave the hyperbolic region before it breaks [13], meaning that the Cauchy problem is ill-posed in the long-time (this is not to be confused with local well-posedness, which has been studied in [23]). It is shown that in the free surface case, the stability of the system depends only on two physical to ellipticity without the need for a four-dimensional space (as the PDE system is 4 × 4) and in this context, a numerical example of transition is shown This result is used to provide a precise and explicit criterion for nonlinear stability, which refines the ones provided by [13] and [20].
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