Stability of an isolated pancake vortex in continuously stratified-rotating fluids

This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number Ro, Froude number F-h, Reynolds number Re and aspect ratio alpha. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number vertical bar Ro vertical bar is large and is stabilized for small and moderate vertical bar Ro vertical bar when the generalized Rayleigh discriminant is positive everywhere. The Gent-McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber m = 1 when the Burger number Bu = alpha(2)Ro(2)/(4F(h)(2)) is larger than unity. When Bu less than or similar to 0.7Ro + 0.1, the Gent-McWilliams instability changes into a mixed baroclinic-Gent-McWilliams instability. Shear instability for m = 2 exists when F-h/alpha is below a threshold depending on Ro. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic-shear instability for small Bu. The main energy source for both baroclinic-shear and baroclinic-Gent-McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent-McWilliams instabilities. The growth rates of these four instabilities depend mostly on F-h/alpha and Ro. Baroclinic instability develops when F-h/alpha vertical bar 1 + 1/Ro vertical bar greater than or similar to 1.46 in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.