Quasigeostrophic, Tilted Spheroidal Vortices

Abstract

Vortex motions in a stably stratified rotating fluid are considered theoretically, based on the quasigeostrophic approximation. A class of exact solution is obtained, which represents a tilted spheroidal volume with uniform potential vorticity embedded in an otherwise quiescent fluid of infinite extent. The spheroid rotates steadily about the vertical axis, where the angular velocity is independent of the inclination angle of the axis of symmetry from the vertical axis. We study the stability of these solutions against infinitesimal disturbances. Highly elongated prolate spheroids are shown to be unstable if the inclination angle is large. In contrast, oblate spheroids of certain aspect ratio are unstable even if the inclination angle is very small. This instability is caused by resonace between the tilted rotation and other disturbance modes.