The universal aspect ratio of vortices in rotating stratified flows: theory and simulation

We derive a relationship for the vortex aspect ratio $\ensuremath{\alpha} $ (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt–Väisälä frequencies within the vortex ${N}_{c} $ and in the background fluid outside the vortex $\bar {N} $, the Coriolis parameter $f$ and the Rossby number $\mathit{Ro}$ of the vortex: ${\ensuremath{\alpha} }^{2} = \mathit{Ro}(1+ \mathit{Ro}){f}^{2} / ({ N}_{c}^{2} \ensuremath{-} {\bar {N} }^{2} )$. This relation is valid for cyclones and anticyclones in either the cyclostrophic or geostrophic regimes; it works with vortices in Boussinesq fluids or ideal gases, and the background density gradient need not be uniform. Our relation for $\ensuremath{\alpha} $ has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have ${ N}_{c}^{2} \gt {\bar {N} }^{2} $; weak anticyclones (with $\vert \mathit{Ro}\vert \lt 1$) must have ${ N}_{c}^{2} \lt {\bar {N} }^{2} $; and strong anticyclones must have ${ N}_{c}^{2} \gt {\bar {N} }^{2} $. We verify our relation for $\ensuremath{\alpha} $ with numerical simulations of the three-dimensional Boussinesq equations for a wide variety of vortices, including: vortices that are initially in (dissipationless) equilibrium and then evolve due to an imposed weak viscous dissipation or density radiation; anticyclones created by the geostrophic adjustment of a patch of locally mixed density; cyclones created by fluid suction from a small localized region; vortices created from the remnants of the violent breakups of columnar vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios of our numerically computed vortices validate our relationship for $\ensuremath{\alpha} $, and generally they differ significantly from the values obtained from the much-cited conjecture that $\ensuremath{\alpha} = f/ \bar {N} $ in quasi-geostrophic vortices.