Axisymmetric vortices have served as models for hurricanes, tropical cyclones and tornadoes. However, most of what we know about the stability of such vortices pertains to homogeneous, nonrotating environments that are not a good model for planetary atmospheres. In this case, vortices exhibit ultraviolet catastrophe: a broadband instability such that the magnitude of the wavenumber to which most of the vortex’s energy is transferred is limited only by viscous effects. This implies a rapid and catastrophic breakdown into turbulence. In contrast, vortices in a rotating, stratified environment such as the Earth’s atmosphere have been found to break down in a more organized manner, forming lenses whose aspect ratio is governed by f/N, the ratio of rotation to buoyancy frequencies. Here, we describe investigations of turbulence and wave radiation in barotropic vortices via linear stability analysis and direct numerical simulation. A more detailed account is given in Smyth & McWilliams (1998).
A linear, inviscid perturbation theory has been developed for the case of barotropic vortices on a stratified, rotating environment. Results are summarized as:
_{}
Boundary conditions are derived to assure wellbehaved solutions in an infinite domain. This differential eigenvalue problem implicitly contains the dispersion relation _{} for linear normal modes.
Numerical solutions of this problem have revealed a wealth of information about instability and wave radiation in geophysical vortices. A typical eigenfunction structure is shown in figure 1. Energy is drawn from the vortex near the critical radius (where the azimuthal velocity matches the phase velocity of the mode) and fluxed radially in both directions. Far from the vortex core, the outward flux is readily identified as radially propagating waves. When stratification (rotation) is the dominant environmental influence, the flux has the form of gravity (inertial) waves. In other cases, the outward flux is weak, and perturbation energy is therefore trapped in the vortex core. This tends to result in stronger growth.
Figure 1: A typical radial eigenmode of the standard vortex profile. Solid and dashed curves represent the real and imaginary parts of the pressure eigenfunction. Shaded curves show the radial energy flux. The vertical dashed line indicates the critical radius 

Modes such as that shown in figure 1 have been studied over a broad region of parameter space. Parameter values included special cases in which only stratification (or only rotation) was present, cases in which both influences were present in equal degrees, and cases in which stratification was much stronger than rotation (as is the case in geophysical flows). Some results of these explorations are summarized in figure 2. There, the growth rate of the fastestgrowing instability is plotted as a function of vertical wavenumber and Rossby number. The Rossby number quantifies the rotation rate of the vortex as a multiple of the planetary rotation rate, with negative values corresponding to cyclonic rotation. In the case shown here, planetary rotation and stratification are balanced, so that f=N. Results for the geophysical case f/N=0.01 are qualitatively similar.
Normal modes of axisymmetric structure are summarized in figure 2a. The horizontal dashed lines indicate the theoretical limits for Rayleigh’s centrifugal instability. Outside these limits, the growth rate increases monotonically with increasing vertical wavenumber. This is the phenomenon we call ultraviolet catastrophe. In these flows, our inviscid perturbation theory predicts a rapid transition to turbulence as energy is transferred directly from the vortex to the smallest scales of motion.
The vortex also supports unstable modes with azimuthal wavenumbers 1 and 2.
Results for wavenumber 1 are shown in figure 2b. At large wavenumber, the
growth rates vary in a manner reminiscent of the axisymmetric case (figure 2a).
Rayleigh’s stability bounds prove useful for this case, although they were
derived for the axisymmetric case only. Outside these bounds, the ultraviolet
catastrophe is revealed as in the axisymmetric case. Near wavenumber unity, the
results are very different. Instability persists to relatively low wavenumbers
and to the limit of zero Rossby number. The latter corresponds to the
quasigeostrophic limit, and the island of instability near _{} is the instability
discovered previously by Gent & McWilliams (1986) using a quasigeostrophic
model.
Figure 2: Growth rate versus vertical wavenumber and Rossby number for a barotropic vortex in a stratified, rotating environment. The vertical wavenumber is scaled by the radius of the vortex. (a) axisymmetric modes; (b) modes with azimuthal wavenumber unity. Red circles indicate the cases shown in figure 3. 
In summary, we have identified two distinct regimes of instability in which flow evolution is dramatically different. The quasigeostrophically balanced regime applies to Rossby numbers of order unity and smaller. Here, instability is restricted to vertical wavenumbers such that the aspect ratio of the mode is near N/f (unity in this case). In the unbalanced regime (_{}), ultraviolet catastrophe is observed, and a rapid transition to turbulence is expected.
The predictions of the linear, inviscid perturbation theory described above are verified using direct numerical simulations of vortex evolution (figure 3). The two cases shown are indicated on figure 2b by small red circles. The first case shown lies in the quasigeostrophic regime. Scaleselective instability leads to the formation of lens vortices whose aspect ratio is near f/N. In the second case, which is just slightly outside the quasigeostrophic regime (figure 2b), the finiteamplitude effects of the ultraviolet catastrophe are clear. Smallscale disturbances grow rapidly, and the vortex is quickly annihilated.
Figure 3: Direct numerical simulations of unstable, barotropic cyclones in a rotating, stratified environment. The evolution of an enstrophy isosurface is shown. The two cases are identical except that background rotation and stratification are weaker by a factor of four in the second case. 
This research was sponsored by the National Science Foundation under grant ATM9617967.
References:
Gent, P.R. and J.C. McWilliams, 1986, The instability of barotropic circular vortices, Geophys. Astrophys. Fluid Dynamics 35, 209233.
Smyth, W.D. and J.C. McWilliams, 1998: Instability of an axisymmetric vortex in a stably stratified, rotating environment, Theo. Comp. Fluid Dyn. 11, 305322.