Quasigeostrophic Ellipsoidal Vortices in a Two-Dimensional Strain Field

Abstract

Vortex motions in a stably stratified rotating fluid are considered theoretically, based on the quasigeostrophic approximation. A class of exact staionary solution is obtained, which represents an ellipsoidal volume with uniform potential vorticity Q 0 embedded in a two-dimensional uniform strain field \({\hat e}\) with uniform background vorticity \(2{\hat \gamma}\). In a pure strain field (\({\hat \gamma}=0\)), stationary solutions are allowed for \(|{\hat e}/Q_0| \) about -0.1 in a simple shear flow (\({\hat \gamma}={\hat e}\)). We study the stability of these exact solutions against infinitesimal disturbances. In a pure strain field, highly elongated ellipsoids are shown to be unstable to Lame-modes whose order m is higher than 2. In a simple shear flow, a highly elongated ellipsoid whose major axis is perpendicular to the flow direction, is unstable, whereas any ellipsoidal vortex seems to be stable, if the major axis is pa...