Eady Baroclinic Instability of a Circular Vortex

The stability of two superposed buoyancy vortices is studied linearly in a two-level Surface Quasi-Geostrophic (SQG) model. The basic flow is chosen as two circular vortices with uniform buoyancy, coaxial, and the same radius. A perturbation with a single angular mode is added to this mean flow. The SQG equations linearized in perturbation around this basic flow form a two-dimensional ODE for which the normal and singular mode solutions are numerically computed. The instability of these two vortices depends on several parameters. The parameters varied here are: the vertical distance between the two levels and the two values of the vortex buoyancies (called vortex intensity hereafter); the other parameters remain fixed. For normal modes, the system is stable if the levels are sufficiently far from each other vertically, to prevent vertical interactions of the buoyancy patches. Stability is also reached if the layers are close to each other, but if the vortices have very different intensities, again preventing the resonance of Rossby waves around their contours. The system is unstable if the vortex intensities are similar and if the two levels are close to each other. The growth rates of the normal modes increase with the angular wave-number, also corresponding to shorter vertical distances. The growth rates of the singular modes depend more on the distance between the levels than on the ratio of the vortex intensities, at a short time; as expected, they converge towards the growth rates of the normal modes. This study remaining linear does not predict the final evolution of such unstable vortices. This nonlinear evolution will be studied in a sequel of this work