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 α. 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 /Ro/ is large and is stabilized for small and moderate /Ro/ 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 = α²Ro²=(4Fh/ is larger than unity. When Bu ≤ 0.7Ro + 0.1, he Gent-McWilliams instability changes into a mixed baroclinic-Gent-McWilliams instability. Shear instability for m=2 exists when F_h=α 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=α and Ro. Baroclinic instability develops when F_h/α/1 + 1/Ro/≥1.46 in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.