The linear stability of three-dimensional (3D) vortices in rotating, stratified flows has been studied by analyzing the non-hydrostatic inviscid Boussinesq equations. We have focused on a widely-used model of geophysical and astrophysical vortices, which assumes an axisymmetric Gaussian structure for pressure anomalies in the horizontal and vertical directions. For a range of Rossby 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 structure of the most unstable eigenmodes have been numerically calculated and presented as a function of $Ro-Bu$. We have found neutrally-stable vortices only over a small 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 in general have slower growth rates compared to cyclones. In particular, the growth rate of the most unstable eigenmode for anticyclones in a large region of the parameter space (e.g., $Ro<0$ and $0.5 lesssim Bu lesssim 1.3$) is slower than $50$ turn-around times of the vortex (which often corresponds to several years for ocean eddies). For cyclones, the region with such slow growth rates is confined to $0<Ro<0.1$ and $0.5 lesssim Bu lesssim 1.3$. While most calculations have been done for $f/bar{N}=0.1$ (where $f$ and $bar{N}$ are the Coriolis and background Brunt-Vaisala frequencies), we have numerically verified and explained analytically, using non-dimensionalized equations, the insensitivity of the results to reducing $f/bar{N}$ to the more ocean-relevant value of $0.01$. The results of this paper provide a steppingstone to study the more complicated problems of the stability of geophysical (e.g., those in the atmospheres of giant planets) and astrophysical vortices (in accretion disks).
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