General stability conditions for a multi-layer model

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Abstract
Sufficient conditions for the stability of steady solutions of a multi-layer model are found. The basic flow may be either parallel, axisymmetric or non-parallel. The lower boundary of the model may be either rigid, including the possibility of topography, or soft. The latter, ‘reduced gravity’, case represents an ideal situation in which the active layers are on top of an infinitely deep, motionless one.Two conditions are sufficient to assure the stability of the basic flow. It is conjectured that unstable flows for which only the first or second condition is violated decay through Rossby-like or Poincaré-like growing perturbations, respectively.In order to understand the meaning of both conditions, assume that a quite general O(α) ‘wave’ is superimposed on the basic flow: an O(α2) energy integral, δ2E can be calculated. This wave energy is neither conserved, because the wave might exchange energy with the O(a2) varying part of the the ‘mean flow’, nor positive definite, because the perturbation might lower the total energy by increasing the speed where it decreases the thickness, and vice versa. Now, the first condition determines that δ2E has an upper bound, and the second one implies that δ2E is positive definite; hence the stability of the basic solution. In the particular case of two-dimensional divergenceless flow, as well as for quasi-geostropic models, δ2E is a priori positive definite, and therefore the first condition suffices to guarantee the stability of the more basic solution. The conditions found here are indeed valid for more general perturbations, e.g. they prevent inertial (or 'symmetric’) instability, a phenomenon for which there is no distinction between wave and the varying part of the mean flow.