Vortices in rotating and stratified flows: aspect ratio and sustainability

In linear stratifications, vortices have a typical flat shape that appears to be not only a compromise between the rotation and the stratification of the background flow through their Coriolis parameter f and buoyancy frequency \hbox{$\bar{N}$}, but also involves the buoyancy frequency Nc within the vortices and their Rossby number Ro. We derive an analytical solution for the self-similar ellipsoidal shape of the vortices and the law for their aspect ratio. From this law, we show that long-lived vortices must necessary be either weakly stratified anticyclones or superstratified cyclones (which is less likely to occur). These predictions are experimentally and numerically verified and agree with published measurements for Jovian vortices and ocean meddies. This approach can be applied to a gaussian stratification to give good insights of the shape of vortices in protoplanetary disks and their sustainability