Linear instability of barotropic submesoscale coherent vortices observed in the ocean

The linear instability of circular quasigeostrophic vortices with horizontal cross sections of angular velocity identical to those that fit the observations of submesoscale eddies in the ocean is investigated analytically and numerically. The theoretical necessary conditions for instability are formulated as a single condition on the mean potential vorticity or mean angular velocity rather than the streamfunction, which is more readily applicable to oceanic lenses, eddies, and meddies. It is shown that the suggested cross sections that best fit the observed angular velocity of several long-lived vortices are all unstable to small, wavelike perturbations, and that the e-folding time for perturbation growth at all wavenumbers and cross sections is on the order of 1 day. For all cross sections considered, azimuthal wavenumber 1 is stable while all higher azimuthal wavenumbers, as well as all vertical wavenumbers, are unstable. The main contribution to the instability comes from the jump in potential vorticity at the radius of maximum angular velocity. When the potential vorticity is continuous at this radius, the growth rates become smaller and the details of potential vorticity distribution become important. The fast growth rates obtained by the numerical calculations clearly emphasize the insufficient spatial resolution of existing observations for deciphering the exact velocity cross sections of submesoscale oceanic vortices; especially near the radius of maximum angular velocity.