Linear instability of warm core, constant potential vorticity, eddies in a two-layer ocean

Linear instability of warm-core eddies of constant potential vorticity (PV) is studied in a two layer, finite depth, shallow-water ocean. The basic state flow in the constant PV eddy that obeys the gradient balance cannot be described by explicit expressions and can be solved only numerically. The various cases of gradient balance are classified by constructing a canonical formulation that relates any PV value to a value of the angular velocity that has to prevail near the centre of the constant PV eddy. The growth rates of perturbations imposed on the basic state are calculated for a variety of values of the (constant) PV and the depth of the surrounding ocean. The growth-rates, i.e. the eigenvalues, are calculated numerically by using a shooting to fitting point method which guarantees that the corresponding eigenfunctions are regular at all singular points. The maximal growth rates are contoured as functions of the PV and ocean depth for azimuthal wavenumbers 2 and 3 and the maximum of these growth rates is of the order of 1 day, which is similar to that of a solidly rotating eddy. The range of angular velocity and ocean depth where the constant PV eddy is unstable, however, is greatly reduced compared with that of a solidly rotating eddy. The instabilities found here are classified in terms of wave-wave interactions by comparing our results in each PV value with the known instabilities of the solidly rotating eddy with the same angular velocity. In the constant PV eddy the Baroclinic instability is filtered out and the range of angular velocity where the Hybrid instability exists is significantly reduced. All instabilities decay monotonically with the increase in ocean depth.