Vortex Stability in the Thermal Quasi-Geostrophic Dynamics

The stability of a circular vortex is studied in the thermal quasi-geostrophic (TQG) model. Several radial distributions of vorticity and buoyancy (temperature) are considered for the mean flow. First, the linear stability of these vortices is addressed. The linear problem is solved exactly for a simple flow, and two stability criteria are then derived for general mean flows. Then, the growth rate and most unstable wavenumbers of normal-mode perturbations are computed numerically for Gaussian and cubic exponential vortices, both for elliptical and higher mode perturbations. In TQG, contrary to usual QG, short waves can be linearly unstable on shallow vorticity profiles. Linearly, both stratification and bottom topography (under specific conditions) have a stabilizing role. In a second step, we use a numerical model of the nonlinear TQG equations. With a Gaussian vortex, we show the growth of small-scale perturbations during the vortex instability, as predicted by the linear analysis. In particular, for an unstable vortex with an elliptical perturbation, the final tripolar vortices can have a turbulent peripheral structure, when the ratio of mean buoyancy to mean velocity is large enough. The frontogenetic tendency indicates how small-scale features detach from the vortex core towards its periphery, and thus feed the turbulent peripheral vorticity. We confirm that stratification and topography have a stabilizing influence as shown by the linear theory. Then, by varying the vortex and perturbation characteristics, we classify the various possible nonlinear regimes. The numerical simulations show that the influence of the growing small-scale perturbations is to weaken the peripheral vortices formed by the instability, and by this, to stabilize the whole vortex. A finite radius of deformation and/or bottom topography also stabilize the vortex as predicted by linear theory. An extension of this work to stratified flows is finally recommended.