Noncoincidence of separatrices in two-layer modons

Abstract

Modons, i.e., localized steady-state solutions to the nonlinear equations of potential vorticity conservation, represent a paradigm for coherent structures in geophysical flows. A characteristic property of the baroclinic modons suggested until the present time is that, in such solutions, the boundary of the trapped-fluid area, the separatrix, is essentially independent of depth. In this paper, the existence of translating two-layer modons with noncoincident separatrices in the layers is demonstrated. The solutions are constructed by a combination of analytical and numerical methods. The exterior fields (outside the separatrices), governed by linear equations, are given analytically by explicit formulas; implicitly, these formulas also determine the separatrices. The interior fields are then determined using an iterative numerical algorithm for solving nonlinear partial differential equations. Applying this semianalytical approach, a variety of modon solutions are produced. Among them, the smooth modons (i.e., those with continuous vorticity) are of special geophysical significance. The existence of solutions of even higher smoothness, the so-called supersmooth modons marked by continuity of both the vorticity and its derivatives at the separatrices, is shown. The supersmooth modon possesses a number of features typical of the baroclinic vortical pairs observed earlier in numerical simulations.