A note on the stability theory of buoyancy-driven ocean currents over a sloping bottom

In a previous paper, a new model was derived describing the baroclinic dynamics of buoyancy-driven ocean currents over a sloping bottom. In particular, a normal mode stability analysis and a general stability analysis based on an appropriately constrained energy invariant were presented. However, these two sets of stability results were not identical to each other. Here, we show that the normal-mode stability results previously described may be derived from a general stability analysis based on an appropriately constrained linear momentum invariant. In addition, we establish conditions for the nonlinear stability in the sense of Liapunov of these flows. The analysis presented here eliminates the need to introduce a Poincare inequality between the perturbation energy and the enstrophy which the previous analysis was forced to assume. Relaxing this assumption means the present analysis is applicable to a much larger range of flow geometries and is therefore a substantially stronger result.