Baroclinic instability in a two-layer model with a free boundary and beta effect

The classical Phillips problem of baroclinic instability is generalized, al lowing 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 beta effect) on the stability/instab ility 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 beta effect can either strengthen or weaken the stability of the basic current, dependi ng on the perturbation scale and the slope of the bottom relative to that o f the interface. In particular, the maximal instability occurs in the limit of weak stratification when the planetary and the topographic beta effects compensate each other. The maximal unstable wave has an intermediate scale between the internal and the external deformation radii. Nonlinear saturat ion bounds on unstable basics states are also determined using Shepherd's m ethod. It is found that the enstrophy of the most unstable wave can only, b e bounded by the total enstrophy of the system.