Asymptotic scale-dependent stability of surface quasi-geostrophic vortices: semi-analytic results

The scale-dependent stability of surface quasi-geostrophic (SQG) vortices is studied both analytically and numerically. In particular, we study the sensitivity of the stability of SQG vortices on a nondimensional number σ, namely the square root of the Burger number, which sets the transition scale between different dynamical regimes corresponding to local and nonlocal dynamics. We analyse the stability of two different examples. The first example is given by a Rankine vortex, characterised by constant buoyancy. For this case, asymptotic analysis suggests that the frequencies of the perturbations at scales smaller than the transition scale show a σ-1 dependence. At scales larger than the transition scale, the frequencies scale instead like σ-2. The second example consists of a Rankine vortex shielded by a filament characterised by a different value of constant buoyancy. For this example we study the dispersion relation for the perturbations for the cases in which the inner vortex and the outer filament have different asymptotic properties behaviour.