Linear instability of constant PV cold-core eddies in a two-layer ocean

The linear instability of a constant potential vorticity cold-core eddy is studied in a two-layer shallow water model. The basic state consists of the nonlinear gradient balance with uniform potential vorticity, and in a cold-core eddy these two relations have no analytic solution. Numerical solutions of the basic state structure provide a functional relation between the maximum tangential velocity and the ratio of the eddy's maximal depth and its radius. The maximal growth rates of small amplitude wavelike disturbances are found numerically by employing the shooting to fitting point method, applied in a way that guarantees the regularity of the solutions at the singular points: infinity, eddy centre and the radius where the interface outcrops. Our results show that wave-numbers 2, 3 and 4 are unstable with growth rates of the order of 1 day at small ocean depth and that the growth rates decay with the ocean depth. The instabilities result from the interaction between a lower-layer Rossby wave and an upper-layer Frontal wave. An energy equation for the perturbations shows that the decay of the growth rates with the increase in ocean depth results from the decay of both the Reynolds Stress in the upper layer and the baroclinic energy conversion between the two layers. The decay of the Reynolds Stress with ocean depth was not found in studies of ocean currents which explains the fact that eddies become stable in a 11/2-layer model while currents can be unstable at large ocean depth. Comparison with long-lived eddies in the ocean suggests that the current model should be made more realistic before it can be applied to observed eddies.