Baroclinic instability of frontal geostrophic currents over a slope

The Phillips problem of baroclinic instability is generalized in a frontal geostrophic model. The configuration used here is a two-layer flow (with quasigeostrophic upper-layer current) over a sloping bottom. Baroclinic instability in the frontal model has a single unstable mode, corresponding to isobaths and isopycnals sloping in the same direction, contrary to the quasigeostrophic model, which has two unstable modes. In physical terms, this is explained by the absence of relative vorticity in the lower (frontal) layer. Indeed, the frontal geostrophic model can be related to the quasigeostrophic model in the limit of very small thickness of the lower layer, implying that potential vorticity reduces to vortex stretching in this layer. This stability study is then extended to unsteady flows. In the frontal geostrophic model, a mean flow oscillation can stabilize an unstable steady flow; it can destabilize a stable steady flow only for a discrete spectrum of low frequencies. In this case, the model equations reduce to the Mathieu equation, the properties of which are well known.