UC Berkeley

Large coherent vortices are abundant in geophysical and astrophysical flows. They play significant roles in the Earth's oceans and atmosphere, the atmosphere of gas giants, such as Jupiter, and the protoplanetary disks around forming stars. These vortices are essentially three-dimensional (3D) and baroclinic, and their dynamics are strongly influenced by the rotation and density stratification of their environments. This work focuses on improving our understanding of the physics of 3D baroclinic vortices in rotating and continuously stratified flows using 3D spectral simulations of the Boussinesq equations, as well as simplified mathematical models. The first chapter discusses the big picture and summarizes the results of this work.

In Chapter 2, we derive a relationship for the aspect ratio (i.e., vertical half-thickness over horizontal length scale) of steady and slowly-evolving baroclinic vortices in rotating stratified fluids. We show that the aspect ratio is a function of the Brunt-Vaisala frequencies within the vortex and outside the vortex, the Coriolis parameter, and the Rossby number of the vortex. This equation is basically the gradient-wind equation integrated over the vortex, and is significantly different from the previously proposed scaling laws that find the aspect ratio to be only a function of the properties of the background flow, and independent of the dynamics of the vortex. Our 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 non-uniform background density gradient. The relation for the aspect ratio has many consequences for quasi-equilibrium vortices in rotating stratified flows. For example, cyclones must have interiors more stratified than the background flow (i.e., super-stratified), and weak anticyclones must have interiors less stratified than the background (i.e., sub-stratified). In addition, this equation is useful to infer the height and internal stratification of some astrophysical and geophysical vortices because direct measurements of their vertical structures are difficult. We verify our relation for the aspect ratio with numerical simulations for a wide variety of families of vortices, including: vortices that are initially in (dissipationless) equilibrium and then evolve due to an imposed weak viscous dissipation or density radiation; anticyclones created by the geostrophic adjustment of a patch of locally-mixed density; cyclones created by fluid suction from a small localized region; vortices created from the remnants of the violent breakups of columnar vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios of our numerically-computed vortices validate our theoretically-derived relationship for aspect ratio, and generally they differ significantly from the values obtained from the much-cited conjecture that the aspect ratio of quasi-geostrophic vortices is equal to the ratio of the Coriolis parameter to the Brunt-Vaisala frequency of the background flow.

In Chapter 3, we show numerically and experimentally that localized suction in rotating continuously stratified flows produces three-dimensional baroclinic cyclones. As expected from Chapter 2, the interiors of these cyclones are super-stratified. Suction, modeled as a small spherical sink in the simulations, creates an anisotropic flow toward the sink with directional dependence changing with the ratio of the Coriolis parameter to the Brunt-Vaisala frequency. Around the sink, this flow generates cyclonic vorticity and deflects isopycnals so that the interior of the cyclone becomes super-stratified. The super-stratified region is visualized in the companion experiments that we helped to design and analyze using the synthetic schlieren technique. Once the suction stops, the cyclones decay due to viscous dissipation in the simulations and experiments. The numerical results show that the vertical velocity of viscously decaying cyclones flows away from the cyclone's midplane, while the radial velocity flows toward the cyclone's center. This observation is explained based on the cyclo-geostrophic balance. This vertical velocity mixes the flow inside and outside of cyclone and reduces the super-stratification. We speculate that the predominance of anticyclones in geophysical and astrophysical flows is due to the fact that anticyclones require sub-stratification, which occurs naturally by mixing, while cyclones require super-stratification.

In Chapter 4, we show that a previously unknown instability creates space-filling lattices of 3D turbulent baroclinic vortices in linearly-stable, rotating, stratified shear flows. The instability starts from a newly discovered family of easily-excited critical layers. This new family, named the baroclinic critical layer, has singular vertical velocities; the traditional family of (barotropic) critical layer has singular stream-wise velocities and is hard to excite. In our simulations, the baroclinic critical layers in rotating stably-stratified linear shear are excited by small-volume, small-amplitude vortices or waves. The excited baroclinic critical layers then intensify by drawing energy from the background shear and roll-up into large coherent 3D vortices that excite new critical layers and vortices. The vortices self-similarly replicate to create lattices of turbulent vortices. These vortices persist for all time and are called zombie vortices because they can occur in the dead zones of protoplanetary disks. The self-replication of zombie vortices can de-stabilize the otherwise linearly and finite-amplitude stable Keplerian shear and lead to the formation of stars and planets.