Baroclinic instability in the Eady model for two coupled flows

The linear instability of baroclinic flows in two superimposed and coupled, immiscible fluids is studied theoretically. These flows are westerlies, and they are thermally or mechanically coupled. The fluids are stably stratified, internally and mutually (the upper fluid is lighter than the lower fluid). In each fluid, two-level surface quasi-geostrophy governs the evolution of the perturbed westerly flow. The perturbations are horizontal normal modes. Firstly, the models are not coupled and the flow instability in each fluid is validated separately against the results of the classical Eady model of baroclinic instability. Secondly, the two fluids are thermally and/or mechanically coupled. With thermal coupling, and for meridionally uniform perturbations, a new mode of instability appears for long waves. This pair of unstable modes converges towards the modes of the uncoupled fluids at medium wavelengths. For perturbations with a non trivial meridional structure, the thermal coupling essentially damps the instability. For an upper flow with a larger deformation radius than in the lower flow, the growth rates of the perturbation are therefore more strongly altered in the former than in the latter. With mechanical coupling, the instability is essentially damped at large to medium scales, while the short-wave cut-off is extended towards smaller waves. When the fluids are both thermally and mechanically coupled, these effects add up. This very idealised study is a first step towards studying more realistic cases.