Large coherent vortices are the abundant features of geophysical and astrophysical turbulent flows. Here we explore and investigate a widely-used model for these vortices, that uses an axisymmetric Gaussian structure for pressure distribution of the vortices. We discuss the linear stability of the vortices, their evolution in long-term, and their robustness in the ocean. These results provide us with information, for example of the efficient mixing and transport, that can occur because of the vortex dynamics in the ocean; which can redistribute momentum, heat, and salt, in an otherwise stably-stratified ocean. The first chapter discusses the big picture and summarizes the results of this work.
In the first part of this work (i.e., in chapter 2 of the thesis), the linear stability of three-dimensional (3D) vortices in rotating, stratified flows is 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 $0
In the second part of this work (i.e., in chapter 3 of the thesis), the evolution and finite-amplitude stability of 3D vortices has been studied with an initial value code in a rotating stratified flow. Focusing on axisymmetric Gaussian vortices, the chapter 2 analysis showed, by analyzing the vortices' linear stability, that anticyclones have slower growth rates than those of cyclones. Here we examine the nonlinear stability for vortices that have growth rates faster than 50 turnaround times of the vortex, and for different finite amplitude perturbations (i.e., with different types, and/or amplitudes), for a range of Rossby number ($-0.5 < Ro < 0.5$) and Burger number ($0.07 < Bu < 2$), that are again relevant to the observations. We demonstrate that despite these vortices' fast growth rates, the perturbations quickly plateau - at a constant value, and the vortices usually have small-sized attracting basins (with only one vortex splitting radially, i.e., into tripoles), and for almost all cases we examined the initial and final equilibria remain close. For the non-splitting cases, the vortex's core always remains close to a Gaussian state. All the calculations here have been done for $f/\bar{N}=0.1$, and effects of changing $f/\bar{N}$ is therefore forwarded to a future study. While we mostly use enstrophy measures to describe the vortex dynamics (i.e., as they evolve towards their final equilibria), we also show, for the vortices with close initial unperturbed and final equilibria, that their long-term evolution is represented entirely by a simple analytical solution, with the deviations from this representation shown to be small (that is after the flow reaches quasi-steady state). It should be noted as well that, while generally a fast instability growth rate can destroy a vortex via nonlinear finite-amplitude instabilities, it also is possible that the perturbations quickly saturate, such that the initial \textit{unperturbed} and final equilibria remain close to each other. Our goal here has been to demonstrate the latter. Our explanation for the equilibria's robustness does not require a direct forcing mechanism; it only involves damping of velocity and density far from the initial conditions' position and at our numerically implemented sponge layers.
