The Universal Aspect Ratio of Vortices in Rotating Stratified Flows: Theory and Simulation

Abstract

We derive a relationship for the vortex aspect ratio $\alpha$ (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt-Vaisala frequencies within the vortex $N_c$ and in the background fluid outside the vortex $\bar{N}$, the Coriolis parameter $f$, and the Rossby number $Ro$ of the vortex: ${\alpha }^2=Ro(1+Ro)f^2/(N_c^2-{\bar{N}}^2)$. 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 $\alpha$ has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have $N_c^2>{\bar{N}}^2$; weak anticyclones (with $|Ro|<1$) must have $N_c^2<{\bar{N}}^2;andstronganticyclonesmusthave$N_c^2 > \bar{N}^2$.Weverifyourrelationfor$α$withnumericalsimulationsofthethree-dimensionalBoussinesqequationsforawidevarietyofvortices,including:vorticesthatareinitiallyin(dissipationless)equilibriumandthenevolveduetoanimposedweakviscousdissipationordensityradiation;anticyclonescreatedbythegeostrophicadjustmentofapatchoflocallymixeddensity;cyclonescreatedbyfluidsuctionfromasmalllocalisedregion;vorticescreatedfromtheremnantsoftheviolentbreakupsofcolumnarvortices;andweaklynon-axisymmetricvortices.Thevaluesoftheaspectratiosofournumerically-computedvorticesvalidateourrelationshipfor$α$,andgenerallytheydiffersignificantlyfromthevaluesobtainedfromthemuch-citedconjecturethat$α= f/\bar{N}$ in quasi-geostrophic vortices.

Submitted to the Journal of Fluid Mechanics. Also see the companion paper by Aubert et al. "The Universal Aspect Ratio of Vortices in Rotating Stratified Flows: Experiments and Observations" 2012

References in corpus (1)

• The Universal Aspect Ratio of Vortices in Rotating Stratifi?ed Flows: Experiments and Observations

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