A Linear Analysis of the Transition Curve for the Baroclinic Annulus

Abstract

A rotating cylindrical annulus of an incompressible fluid with horizontal density gradients is studied by the use of numerical models. Steady axisymmetric states are calculated using the full Navier-Stokes equations for a broad range of thermal Rossby number (Ror) and Taylor number (Ta). These states are tested for stability to nonaxisymmetric perturbations by the use of a model based upon the linearized hydrostatic primitive equations. The results include a prediction of the transition curve, the curve separating axisymmetric flow and nonaxisymmetric flow. This predicted curve is very close to that observed in the laboratory. The structure and energetics of the fastest growing eigenmodes are examined. It is found that the structure of the linear wave, for one point in the nonaxisymmetric regime, has only small differences from the nonlinear wave calculated by Williams. The structures of the waves at this and other points are similar to the classic Eady wave, except near the extreme lower part of the transition curve. There, the waves have little structure with height, and the present models fail to predict the cutoff of nonaxisymmetric flow, probably due to the assumption that the upper surface is flat. In all regions in parameter space, the eddy kinetic energy generation was found to be baroclinic in nature. Large static stability of the basic state is important in suppressing the generation of eddy potential energy near the upper part of the transition curve but not near the lower part, in agreement with previous theoretical results. Dissipation of the eddies is important near all boundaries.