The Universal Aspect Ratio of Vortices in Rotating Stratified Flows: Theory and Simulation
Abstract
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-Vaisala frequencies within the vortex
N
c
and in the background fluid outside the vortex
N
¯
, the Coriolis parameter
f
, and the Rossby number
Ro
of the vortex:
α
2
=Ro(1+Ro)
f
2
/(
N
2
c
−
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
α
has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have
N
2
c
>
N
¯
2
; weak anticyclones (with
|Ro|<1
) must have
N
2
c
<
N
¯
2
;andstronganticyclonesmusthave
N_c^2 > \bar{N}^2
.Weverifyourrelationfor
\alpha
withnumericalsimulationsofthethree−dimensionalBoussinesqequationsforawidevarietyofvortices,including:vorticesthatareinitiallyin(dissipationless)equilibriumandthenevolveduetoanimposedweakviscousdissipationordensityradiation;anticyclonescreatedbythegeostrophicadjustmentofapatchoflocallymixeddensity;cyclonescreatedbyfluidsuctionfromasmalllocalisedregion;vorticescreatedfromtheremnantsoftheviolentbreakupsofcolumnarvortices;andweaklynon−axisymmetricvortices.Thevaluesoftheaspectratiosofournumerically−computedvorticesvalidateourrelationshipfor
\alpha
,andgenerallytheydiffersignificantlyfromthevaluesobtainedfromthemuch−citedconjecturethat
\alpha = f/\bar{N}$ in quasi-geostrophic vortices.