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The linear stability of three-dimensional (3D) vortices in rotating,stratified flows has been studied by analyzing the non-hydrostatic inviscidBoussinesq equations. We have focused on a widely-used model of geophysical andastrophysical vortices, which assumes an axisymmetric Gaussian structure forpressure anomalies in the horizontal and vertical directions. For a range ofRossby number ($-0.5 < Ro < 0.5$) and Burger number ($0.02 < Bu < 2.3$)relevant to observed long-lived vortices, the growth rate and spatial structureof the most unstable eigenmodes have been numerically calculated and presentedas a function of $Ro-Bu$. We have found neutrally-stable vortices only over asmall region of the $Ro-Bu$ parameter space: cyclones with $Ro \sim 0.02-0.05$and $Bu \sim 0.85-0.95$. However, we have also found that anticyclones ingeneral have slower growth rates compared to cyclones. In particular, thegrowth rate of the most unstable eigenmode for anticyclones in a large regionof the parameter space (e.g., $Ro<0$ and $0.5 \lesssim Bu \lesssim 1.3$) isslower than $50$ turn-around times of the vortex (which often corresponds toseveral years for ocean eddies). For cyclones, the region with such slow growthrates is confined to $0<Ro<0.1$ and $0.5 \lesssim Bu \lesssim 1.3$. While mostcalculations have been done for $f/\bar{N}=0.1$ (where $f$ and $\bar{N}$ arethe Coriolis and background Brunt-V\"ais\"al\"a frequencies), we havenumerically verified and explained analytically, using non-dimensionalizedequations, the insensitivity of the results to reducing $f/\bar{N}$ to the moreocean-relevant value of $0.01$. The results of this paper provide asteppingstone to study the more complicated problems of the stability ofgeophysical (e.g., those in the atmospheres of giant planets) and astrophysicalvortices (in accretion disks).

Stability of 3D Gaussian vortices in an unbounded, rotating, vertically-stratified, Boussinesq flow: Linear analysis