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

We derive a relationship for the vortex aspect ratio α (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 Nc and in the background fluid outside the vortex N̄ , the Coriolis parameter f and the Rossby number Ro of the vortex: α ² = Ro(1+ Ro)f² / ( Nc² - N̄ 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 α has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have Nc² > N̄ ² ; weak anticyclones (with Ro < 1 ) must have Nc² < N̄ ² ; and strong anticyclones must have Nc² > N̄ ² . We verify our relation for α 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 α , and generally they differ significantly from the values obtained from the much-cited conjecture that α = f/ N̄ in quasi-geostrophic vortices.