Exact analytical solutions for elliptical vortices of the shallow-water equations

Abstract

It has long been known that the shallow-water equations in a rotating framework admit exact solutions for which the velocity components are linear and the height field is quadratic in the coordinate variables. The resulting 12 coefficients are, in general, functions of time and are governed by a set of nonlinear, coupled, ordinary differential equations. While previous analytical solutions have been plagued by a reduced number of degrees of freedom, the most general solution presented here contains 10 arbitrary constants of integration. These degrees of freedom can be sorted as follows: 4 are accounted for by inertial oscillations of the vortex' center of mass, 2 by an average circular structure (depth and radius), 2 by a circular pulsation mode (amplitude and phase), and 2 by 1 of 2 possible elliptical rotation modes (amplitude and phase). The complete solution with 12 degrees of freedom would be achieved if these 2 elliptical modes could be incorporated simultaneously. Unfortunately, the solution that includes the nonlinear interaction between the elliptical rotation modes has yet to be found. The present solution sheds some ***light on the behavior of time-dependent elliptical warm-core rings as observed in the ocean. It can be used as the basis for further theoretical investigations and testing of numerical models.