Stability criterion for the centrifugal instability of surface intensified anticyclones

We investigate the linear stability of intense baroclinic anticyclones, with a particular focus on the centrifugal (inertial) instability. Various vertical and radial velocity profiles are studied. The vertical profiles are such that the velocity is maximum at the surface. These profiles correspond to oceanic eddies such as submesoscale mixed-layer eddies or intense mesoscale eddies in the upper thermocline. The results show that the main characteristics of the centrifugal instability (growth rate, vertical wavelength) depend weakly on the baroclinic structure of the anticyclone. The dominant azimuthal wavenumber is m=2 for small Burger number (Bu) and m=1 for higher Bu, where Bu is the square root of the ratio of the deformation radius R_d over the characteristic eddy radius R_max where the velocity is maximum. The marginal stability limits of the centrifugal instability for the different velocity profiles collapse approximately on a single curve in the parameter space (Ro, Bu), where Ro=V_max/(fR_max) is the Rossby number, with V_max being the maximum velocity. By means of an asymptotic analysis for short vertical wavelength, an explicit prediction for the marginal stability limit is derived for a wide range of velocity profiles. We then suggest to use, for most of oceanic anticyclones, the instability criterion valid for a Gaussian eddy: √Bu=R_d/R_max ≤(0:23/√Ek)(Ro + 0:3)²/ √|Ro|, where Ek = n/fH² is the Ekman number, H is the eddy depth, and v is the turbulent viscosity at the ocean surface. Some baroclinic anticyclones can remain stable even if they have a core region of negative absolute vorticity provided that they are small enough. This formula explains the few observations of intense anticyclonic eddies having a negative core vorticity around -1.5f.