Optimal two-layer approximation for continuous density stratification

Abstract

Two-layer fluids of finite depth under gravity are the simplest configuration capable of supporting internal wave motion. The extent to which such systems can be used to provide quantitative information on smoothly stratified fluids, in configurations relevant for geophysical applications, is analysed and a model of practical interest derived. The model is based on long-wave asymptotic expansions and on first-principle criteria for an optimal choice of effective two-layer parameters for the incompressible, smoothly stratified Euler equations. The accuracy of the model is extensively tested, via fully resolved numerical computations, on the class of travelling wave solutions supported by smooth stratification systems. It is found that, despite the severe restrictions posed by the discrete two-layer density assumption, solitary wave solutions corresponding to experimentally realizable parametric values can be accurately predicted, in both wave and fluid parcel markers, such as phase speed and density fields, respectively. Thanks to this analysis, explicit closed-form solutions are available for all relevant physical quantities. The agreement between the simple, optimized two-layer model and the parent smooth-stratification Euler system persists up to extreme cases, such as that of internal fronts, and even up to thicknesses of the pycnocline comparable to that of the effective layers.