LadHyX, Ecole Polytechnique, Palaiseau
The stability of vortices in a stratified and rotating fluid is studied numerically and theoretically in order to better understand the dynamics of vortices in the oceans and atmosphere. In the first part of the work, the stability of a columnar vertical axisymmetric vortex in stratified-rotating fluids is analysed. For strong stratification and rapid background rotation and some vortex profiles such as a Gaussian angular velocity, the dominant instability is a special instability which bends and slices the vortex into pancake-shaped vortices. Both numerical and asymptotic stability analyses for long-wavelength are conducted to better understand this instability called “Gent-McWilliams instability”. It is shown that it originates from a destabilization of the long-wavelength bending mode by a critical layer at the radius where the frequency of the mode is equal to the angular velocity of the vortex. A necessary condition for instability is that the base vorticity gradient is positive at the critical radius. In the second part, the stability of an axisymmetric pancake-shape vortex with a Gaussian angular velocity in both radial and vertical directions is analyzed. Different types of instability have been identified depending on the intensity of the stratification, the rotation of the fluid and the aspect ratio r of the vortex. The case of a strongly stratified non-rotating fluid is first considered. Centrifugal and shear instabilities are shown to have similar characteristics as for columnar vortices. The centrifugal instability occurs when the buoyancy Reynolds number ReFh (where Re is the Reynolds number and Fh the Froude number) is above a threshold. The shear instability for m = 2 develops only when the vertical Froude number Fh/r is low: when the thickness of the vortex is larger than the cut-off wavelength for an equivalent columnar vortex. Baroclinic and gravitational instabilities due to the deformations of the density field in the vortex core are also observed. The gravitational instability occurs when the total density gradient is positive Fh/r > 1.5. Just below this threshold, the baroclinic instability occurs in a small range of vertical Froude numbers. A simple model shows that it cannot develop for low Fh/r because of confinement effects.
Strongly stratified rotating fluids are
next considered. The centrifugal instability becomes stabilized for
small Rossby number Ro, in agreement with the generalized Rayleigh
criterion. An instability with an azimuthal wavenumber
m = 1 similar to the Gent-McWilliams instability of a columnar vortex
occurs for small Fh and Ro. The occurrence of the shear instability
continues to be governed by confinement effects.
However, mixed instabilities: shear-baroclinic and baroclinic Gent-McWilliams instabilities are also observed when Ro is of order unity. In the presence of rotation, the baroclinic instability develops when Fh/r(1 + 1/|Ro|) >1.43, i.e. only when the Burger number based on the absolute rotation of the vortex is below a threshold.