Evolution of unstable shear layers in a rotating fluid

Abstract

The time development of non-axisymmetric disturbances on a shear layer in a uniformly rotating fluid is studied theoretically. It is assumed that the Rossby number, Ekman number, shear-layer length scale, and initial disturbance amplitude are small, and that disturbances grow if vorticity transfer from the shear flow exceeds Ekman-layer vorticity dissipation, a mechanism investigated by Busse (1968). Expressions for the ultimate instability amplitude are determined for specified relations among the small parameters, using the volume-integrated energy equation, the shape assumption, and approximations for the spatial dependence of disturbances. Disturbance growth is limited by modification of the initially-unstable shear-layer profile, and the eventual amplitude is shown to be fairly insensitive to the specific form of the profile. Using data from observations by Hide & Titman, the predicted maximum velocity amplitude of the non-axisymmetric motions for their experiments is approximately one-quarter of the velocity of the shear flow at the point of maximum gradient.