On the short-wavelength stabilities of some geophysical flows

Abstract

This paper is a survey of the short-wavelength stability method for rotating flows. Additional complications such as stratification in the flow or the presence of non-conservative body forces are considered too. This method is applied to the specific study of some exact geophysical flows. For Gerstner-like geophysical flows one can identify perturbations in certain directions as a source of instabilities with an exponentially growing amplitude, the growth rate of the instabilities depending on the steepness of the travelling wave profile. On the other hand, for certain physically realistic velocity profiles, steady flows moving only in the azimuthal direction, with no variation in this direction, are locally stable to the short-wavelength perturbations.This article is part of the theme issue 'Nonlinear water waves'.

Steady periodic equatorial water waves with vorticity

“…Starting with this pioneering paper, recently some essential results have been achieved. We refer the reader to [10,24,28,29,30,32,33] for the study of exact solutions and instability, and [5,31,34,41] for the related properties of the periodic geophysical water flows with vorticity. In this paper, we will prove the existence of steady two-dimensional periodic equatorial water waves in the f -plane approximation and the underlying current has a very general vorticity.…”

Exact solution and instability for geophysical waves at arbitrary latitude

“…A rigorous mathematical approach to the problem of stability for general three-dimensional inviscid incompressible flows is the short-wavelength method, which was developed independently by Bayly [1], Friedlander & Vishik [18] and Lifschitz & Hameiri [42]. For Gerstner-like geophysical surface waves in various settings, the short wavelength instability analysis is suitable and elegant; these solutions have been shown to be unstable when the travelling wave profiles are steep enough, the critical steepness being very close to 1 3 -instability results were established in [9,16,19,28,29,33,34] (see also the survey [35]).…”

Geophysical internal equatorial waves of extreme form

“…Equatorially trapped surface waves have been investigated in [7,13,14] while surface waves in the presence of underlying currents have been treated in [22,25]. Internal geophysical flows are investigated in [10,4,33,32,36,27], while instabilities in such flows are addressed in [12,26,18,24].…”