Analytical Solutions for Circular Stratified Eddies of the Reduced-Gravity Shallow-Water Equations

Abstract

Abstract A set of new analytical nonstationary solutions of the nonlinear, reduced-gravity shallow-water equations on an f plane in a vertically stably stratified active environment is presented. The solutions, which describe the dynamics of inertially pulsating surface as well as intermediate lens-like stratified vortices, represent an extension of previous analytical solutions to more realistic vortex shapes and structures of the vortex swirl velocity fields in the presence of an arbitrary stable vertical stratification within the active environment. To elucidate aspects of the novelty of the new set of solutions, examples are presented referring to a vertically stratified surface vortex and to a vertically stratified intermediate vortex, both characterized by azimuthal velocity fields that are nonlinear functions of the radius and by layer shapes that largely deviate from paraboloidal shapes. First, a solution describing a five-layer surface “warm core” eddy is analyzed and characteristics resembling characteristics observed in geophysical surface vortices are revealed: Within each layer, the obtained azimuthal velocity shows a realistic distribution, as its maximum is located far from the vortex rim, where it is negligible. Moreover, the obtained azimuthal velocity is largest in the surface layer and decreases monotonically toward the deeper layers. Integral properties of the new stratified solutions significantly differ from the corresponding properties of equivalent, known homogeneous solutions. Significant differences are found, for instance, between the shape of a five-layer vortex and that of its homogenous counterpart having the same mean azimuthal velocity structure. Second, a solution referring to a five-layer intermediate “meddy like” vortex is analyzed: while in each layer the obtained azimuthal velocity is maximum in the interior part of the vortex and decreases toward its center and its periphery, its magnitude is largest in the intermediate layer and it decreases toward the surface and toward the bottom layer, which are characteristics resembling characteristics of observed meddies. The quoted examples demonstrate that the new solutions add substantial realism to the analytical description of oceanic nonlinear geophysical vortices. In the context of the nonlinear reduced-gravity shallow-water equations on an f plane they seem to represent the most general analytical features achievable, which refer to vertically stratified circular vortices characterized by linear radial velocity fields.