On the stability of outcropping eddies in a constant-PV ocean

Stability analyses of baroclinic ocean eddies have previously demonstrated that eddies with no surface fronts are stable when the potential vorticity (PV) in the surrounding ocean is constant. This assumption of constant-PV ocean which might explain the long life span of observed ocean eddies is applied here in the stability analyses of two warm eddies that are characterized by surface fronts along the outcropping line of the interface that separates the eddy from the ocean. The first is an eddy in a solid body rotation and the second is a constant-PV eddy, both of them in a two-layer shallow-water model. The prescription of the basic state of constant PV in the lower layer couples the mean fields in the eddy and the surrounding ocean so that the mean flow in the eddy changes with the ocean depth. The calculation of the growth rates of small-amplitude wavelike perturbations are done numerically using a shooting to fitting point method, which enables a consistent incorporation of the regularity conditions at the centre, the surface front and infinity into the solution. In both eddies no unstable modes are found for wave numbers 2, 3 or 4. An analysis of the real phase speed shows that the coalescence of inertia-gravity waves in the two layers that generates instability in other settings is inhibited in the present model while Rossby waves are eliminated from the lower layer by the constant-PV assumption. The elimination of Rossby waves in the lower layer by the constant-PV assumption stabilizes the two eddies, both of which were previously shown to be highly unstable when the underlying ocean is assumed motionless.