Spontaneous bending of a columnar vortex in stratified-rotating fluids

In a stably stratified and rotating fluids, an isolated asymmetric vortex can be unstable to a long-wavelength instability with an azimuthal wavenumber $m=1$ which is different from classical instabilities. This instability bends the vortex and leads to the formation of pancake vortices. It can be the most dangerous instability when the background rotation and the stratification are strong. In order to better characterize this bending instability, numerical and asymptotic analyses have been performed for wide range of Froude and Rossby numbers and various velocity profiles. The maximum growth rate increases with the Rossby numbers but is independent of the Froude number when it is below unity. When the Froude number is above unity, the growth rate decays abruptly because of critical layers. By means of an asymptotic stability analysis for long-wavelength, we show that necessary instability conditions for any Froude and Rossby numbers are that there exists a critical radius $r_c$ where the angular velocity is equal to the frequency $\Omega(r_c)=\omega$ and the vorticity gradient at the critical radius is positive $\zeta'(r_c) > 0$. These conditions are identical to those of the shear instability. Numerical results supporting these conditions will be presented.