1999: Baroclinic instability in a two-layer model
with a free boundary and b effect, M. J. Olascoaga
and P. Ripa, J. Geophys. Res. vol. 104, No.
C10, pages 23,357-23,366.
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 Earth's curvature (the so-called b-effect) on the stability/instability problem. Conservation laws of energy, momentum and vorticity-related Casimirs, are used to establish non-linear stability conditions. A spectral analysis reveals that, unlike the case of Phillips problem, Earth's curvature can either strengthen or weaken the stability of the basic current, depending on the perturbation scale, the slope of the bottom relative to that of the interface, and the planetary b-effect relative to the topographic b-effect due to the slope of the interface. In particular, the maximal instability occurs in the limit of weak stratification, when the contributions of the planetary and the topographic b-effects to the potential vorticity compensate each other. The maximal unstable wave has an intermediate scale between the internal and the external deformation radii. Non-linear saturation bounds on unstable basics states are also determined using Shepherd's method. It is found that the most unstable wave can only be bounded by the total enstrophy of the system.