Baroclinic instability in a two‐layer model with a free boundary and β effect

Abstract

The classical Phillips problem of baroclinic instability is generalized, allowing for free deformations of the bottom boundary. The simplicity of the model is exploited to analyze the effects of the variation of the Coriolis parameter with latitude (the so‐called β effect) on the stability/instability problem. Conservation laws of energy, momentum, and vorticity‐related Casimirs are used to establish nonlinear stability conditions. A spectral analysis reveals that unlike the case of Phillips problem, the β effect can either strengthen or weaken the stability of the basic current, depending on the perturbation scale and the slope of the bottom relative to that of the interface. In particular, the maximal instability occurs in the limit of weak stratification when the planetary and the topographic β effects compensate each other. The maximal unstable wave has an intermediate scale between the internal and the external deformation radii. Nonlinear saturation bounds on unstable basics states are also determined using Shepherd's method. It is found that the enstrophy of the most unstable wave can only be bounded by the total enstrophy of the system.