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Coherent vortices are one of the most common structures in geophysical and astrophysical fluid systems. On Earth, geostrophic eddies play important roles in the ocean kinetic energy balance (Ferrari & Wunsch, 2009). On Jupiter, the Great Red Spot (GRS) and other vortices have been observed for hundreds of years (Hooke, 1665; Dawes, 1857). On Saturn, its polar vortex shapes the hexagonal patterns at the north pole (Rostami et al., 2017). On Neptune, observations show dark spots emerging and fading away over the past 25 years (Wong et al., 2022).This dissertation focuses on the fluid dynamics of the Jovian vortices, given their abundance (Li et al., 2004; Sankar et al., 2022) and long observation history(Hooke, 1665; Cassini, 1666; Dawes, 1857). To study Jovian vortices, we develop a three-dimensional pseudo-spectral code, the Pseudo-spectraL ANElastic Three-dimensional Simulator (PLANETS), to simulate the anelastic equations of motion. Modeling planetary vortices is computationally challenging because the timestep dt is constrained by fast-dynamics (gravity waves, planetary rotation, zonal shear). We improve the semi-analytic method proposed by Barranco & Marcus (2006) to handle the complex environment of the Jovian atmosphere. We demonstrate that the code can efficiently model the fast-dynamics using a relatively large timestep dt without losing accuracy. More details of our equations of motion and numerical methods can be found in Chapter 2.Using the PLANETS, we systematically study the three-dimensional structure of Jovian vortices. We show that the model used in previous studies, characterized by a similar area and changing vorticity as a function of depth, is not a stable equilibrium. Instead, our findings indicate that a stable 3D Jovian vortex should feature a varying area and similar vorticity throughout its depth. We discuss the scaling analysis of the flow field based on our numerical simulations and present a semi-analytic relation of the shape of the vortices and how the convection zone of the lower layer affects the stability of the vortex. The study on the three-dimensional structure of Jovian vortices has been published in the Journal of Fluid Mechanics.We also use simulations to constrain the vertical structure of the Great Red Spot (GRS). Despite the long observational history, we have very little knowledge about the vertical structure of the GRS. We quantitatively reproduce vortices in quasi-equilibrium, consistent with the wind velocities observed by the Hubble Space Telescope (HST). We show thatthe new observational constraints from the temperature and velocity observations with the James Webb Space Telescope (JWST) severely reduce the possible range of allowable vortices that are stable solutions to the equations of motion and that also quantitatively fit the new and old observations. The allowable solutions provide precise values, with smalluncertainties, for the heights of the top of the Jovian convection zone, the HST and JWST cloud decks from which velocities are extracted, and the top and bottom of the GRS, along with its strength and vertical structure. We also show that our calculations are consistent with the gravity anomalies probed by the Juno mission.

Turbulent wakes are pervasive in man-made and natural environments. In the ocean and the atmosphere, these wakes interact with the background ambient stratification to give rise to a myriad of interesting phenomena, e.g., multistage decay of mean and turbulence, long-lived coherent structures, and the appearance of internal gravity waves, to name a few. With the rise in supercomputing power, high-fidelity numerical simulations have become an increasingly feasible way to investigate the phenomenology of these wakes. As these simulations become commonplace in research, there is an increased focus on the use of data-driven techniques to uncover the rich dynamics from the obtained datasets. This dissertation is an examination of turbulent wakes using data-driven techniques and numerical simulations. In the first part, spectral proper orthogonal decomposition is used to investigate a turbulent disk wake database at $\Rey = 5 \times 10^{4}$ and $\Fro = \infty$, $10$, $2$. We first study the evolution of the vortex shedding mode and double helix mode in the unstratified wake ($\Fro = \infty$), building on and refining the previous experimental studies. Thereafter, the SPOD analysis of the stratified wakes is performed that uncovers two new results: (a) coherence originating at the body gets stronger and lives longer with progressively increasing stratification levels and (b) for $\Fro \gtrsim 2$, vortex shedding is the dominant mechanism of internal gravity wave generation. In the second part of the work, large eddy simulations (LES) are used to investigate the flow past a prolate 6:1 spheroid. Firstly, high-resolution hybrid simulation is used to simulate the far wake of a 6:1 spheroid at 0-degree angle of attack and $\Rey = 10^{5}$, $\Fro = 2$ and $10$. The far wake is compared to the above-mentioned disk database. The spheroid wakes show differences in locations at which mean wake transitions take place. These differences are explained in light of energy budgets. Secondly, large eddy simulations of flow past a 6:1 spheroid at $\Rey = 5000$, $\Fro = \infty, 6, 1.9$, $1$, and a moderate angle of attack $\alpha = 10^{\circ}$ are carried out. Body forces, mean wake and vorticity dynamics, and flow spectra are analyzed in detail and presented in the dissertation.

Wakes of three-dimensional topographies in the abyssal ocean are sites of enhanced flow variability and mixing. To investigate the attendant vortex dynamics, internal waves and turbulence, a three-phased approach is adopted. In the first phase, a steady current impinges on a conical abyssal hill in a stratified environment. The sensitivity of the flow to boundary conditions (slip and partial-slip instead of no-slip) on the flat wall and on the obstacle surface is examined. Significant changes occur in the structure of lee vortices and wake turbulence when the boundary condition (BC) is changed. For instance, the boundary layer on the flat bottom in the no-slip case suppresses the unsteady behavior of the separated boundary layer behind the hill. In contrast, unsteady shedding of vortices from the body in a high Reynoldsnumber flow is captured by slip and partial-slip BCs. In the second phase, tidal modulations are superimposed on the steady flow to study the changes in vortex dynamics in the abyssal hill wake. The strength of the tidal modulationsrelative to the mean flow (R) and the tidal excursion number (Ex) are varied to reveal tidal synchronization of wake vortices. The ratio of the natural shedding frequency to the tidal frequency (f*) varies from 0.1 to 1 when Ex is varied, at R = 1. Wake vortices are observed at the subharmonics of the tidal frequency when 0.25 ≤ f* ≤ 1. Even weak tidal modulations (R ∼ O (0.1)) can alter the frequency of these wake vortices. Qualitative changes are observed in the spatial organization of the vortices, which influences form drag and dissipation. In the third phase, a statically unstable disturbance (originating from internal wave propagation) over a slope with inclination β and background buoyancy frequency N is considered. The energy exchange occurs between four energy reservoirs, namely the mean and turbulent components of kinetic energy (KE) and available potential energy (APE). When β is non-zero, a mean flow is initiated at a frequency of N sin β accompanied by an oscillatory energy exchange between the mean KE and APE reservoirs. The energy transfer between the mean and turbulent reservoirs of KE and APE is explored.

Many fundamental processes in oceanic transport and limnology occur in geophysical flows that are both local in space and transient in time, and that require equally space and time-resolved methods of analysis. The importance of providing physics-based, quantitative modeling of such flows has driven the development of numerical methods for geophysical fluid dynamics for over three decades. Here, we use direct numerical simulations to investigate a range of stratified, particle-laden flows that are accurately described by the three-dimensional Navier-Stokes equations for an incompressible flow in the Boussinesq limit. We firstly investigate the propagation, transport and mixing dynamics of density-driven gravity currents moving in stratified environments. We propose new models for the intrusion of a turbidity current into a linearly stratified ambient based on three-dimensional simulations. We then describe the interaction between a gravity-current and an internal wave and characterize a phenomenological change in the long-term effect of the interaction at a critical wave height. We then quantify the role of double-diffusive processes in the Dead Sea in Summer and their role in the seasonality of salt crystallization and deposition. We also describe large-scale double-diffusive instabilities that arise in high-Prandtl sedimentary double-diffusive systems such as linearly stratified particle-laden salt water. Finally, we quantify mixing induced by a swarm of small-scale self-propelled organisms migrating in a stratified ambient fluid. We compare the relative contribution to mixing by individual swimmers within the swarm to that of the large-scale motion produced by the collective motion of the swarm.

Upstream flow blocking is a distinguishing feature of stratified flows incident on dynamically tall mountain ridges. Blocking occurs as a consequence of the upstream propagation of long internal wave modes that are excited at the obstacle and which permanently modify the oncoming flow. When the ridge is infinite, the fluid upstream and below a `blocking level' is brought to stagnation. The resulting across-crest asymmetry combined with volume transport constraints causes the overflowing layer to accelerate and develop into a hydraulically controlled flow. The processes leading to the establishment of upstream blocking and hydraulic control occur on a characteristic short time scale. In the interior of a long, but finite ridge, a hydraulically controlled overflow similarly develops on a short time scale, while over a longer time scale, low-level horizontal flow splitting leads to the establishment of an upstream layer-wise potential flow beneath the blocking level. We demonstrate through numerical experiments that for sufficiently long ridges, crest control and streamwise asymmetry are seen on both the short and long time scales. We then proceed to quantify the overflow using the framework of stratified hydraulics.In a separate study, we investigate the dynamic stability of stratified flow configurations characteristic of blocked, topographically controlled downslope flows. The essential character of the base flow profiles considered is determined by the analytical solutions of Winters and Armi (2014). Their condition of optimal control necessitates a streamline bifurcation above the blocking location, which then naturally produces a stagnant isolating layer overlying an accelerating downslope flow. We show that the inclusion of the isolating layer is an essential component of the stability analysis. The spatial stability problem is also examined in order to estimate the downstream location where finite amplitude features might manifest in streamwise slowly-varying flows over topography. Finally, to expose the dynamical connection between topographic control and wave excitation aloft, we consider flow over dynamically tall ridges under stratification conditions that feature a strong density jump above crest level. We show that the height of the bifurcating streamline depends sensitively on the location of the step. Further, the question of whether or not the density interface remains flat or plunges across the crest as part of the hydraulically controlled flow is found to be directly related to the requirement of maintaining a subcritical overflow upstream. We also demonstrate that the top of the density interface acts as a `virtual topography' for the flow aloft and fundamentally controls the amplitude of the wave response there.

Within the present investigation, the broad span of applications of the vorticity-based modeling concept for stratified flows, based on the simultaneous use of horizontal and vertical momenta equations in the form of vorticity balance principle, is studied in detail. Towards this objective, this approach, originally introduced by Borden and Meiburg [Z. Borden and E. Meiburg, Phys. Fluids 25 (10), 101301 (2013); Z. Borden and E. Meiburg, J. Fluid Mech. 726, R1 (2013)], for gravity currents propagating into unstratified ambients and internal bores traveling at the interface of two-layer fluids, respectively, is extended to various well known stratified flow problems, in the following. These flows normally involve several fronts which can be analyzed according to the quasisteady conservation laws of mass and momentum by appropriate shift in the reference frame, or possibly unsteady sections for which the flow cannot be rendered quasisteady by any finite number of changes in the reference frames. The analyses of various flow components are then superimposed and matched to obtain the whole flow field. It is also demonstrated that under certain conditions the propagation of gravity currents (or intrusions) can lead to the formation of interfacial perturbations in the form of rarefaction waves or internal bores, which are a source of unsteadiness, and can substantially impact the flow dynamics as well as its energy budget.Enforcing the conservation laws for horizontal and vertical momenta concurrently, enables us to avoid employing energy-based closure assumptions invoked by previous peer models. Consequently, the assessment of flow energetics becomes plausible, which can be utilized to investigate the validity of the energy-related arguments made by other authors. Furthermore, the predictions of the current study obtained by detailed parametric studies are compared to the results of our two-dimensional direct numerical simulations as well as the theoretical and experimental findings of earlier investigations, where very good agreement is observed with regard to all flow properties.

When the density of a gravitationally stable fluid depends on a fast diffusing scalar and a slowly diffusing scalar of opposite contribution to the stability, ‘double diffusive’ instabilities may develop and drive convection. When the slow diffuser settles under gravity, as is for instance the case for small sediment particles in water, settling-driven double-diffusive instabilities can additionally occur. Such instabilities are relevant in a variety of naturally occurring settings, such as particle-laden river discharges, or underground inflows in lakes. Inspired by the dynamics of the more traditional thermohaline double-diffusive instabilities, we ask whether large-scale ‘mean-field’ instabilities can develop as a result of sedimentary double-diffusive convection. We first apply the mean-field instability theory of Traxler et al. (J. Fluid Mech., vol. 677, 2011, pp. 530–553) to high-Prandtl-number fluids, and find that these are unstable to Radko's layering instability, yet collectively stable. We then extend the theory of Traxler et al. (2011) to include settling and study its impact on the development of the collective instability. We find that two distinct regimes exist. At low settling velocities, the double-diffusive turbulence in the fingering regime is relatively unaffected by settling and remains stable to the classical collective instability. It is, however, unstable to a new instability in which large-scale gravity waves are excited by the phase shift between the salinity and particle concentration fields. At higher settling velocities, the double-diffusive turbulence is substantially affected by settling, and becomes unstable to the classic collective instability. Our findings, validated by direct numerical simulations, reveal new opportunities to observe settling-driven layering in laboratory and field experiments.

Funder: John Simon Guggenheim Memorial Foundation; doi: http://dx.doi.org/10.13039/100005851, Funder: Einstein Stiftung Berlin; doi: http://dx.doi.org/10.13039/501100006188, We establish the local and global well-posedness of strong solutions to the two- and three-dimensional anelastic equations of stratified viscous flows. In this model, the interaction of the density profile with the velocity field is taken into account, and the density background profile is permitted to have physical vacuum singularity. The existing time of the solutions is infinite in two dimensions, with general initial data, and in three dimensions with small initial data.

We consider the slow motion generated when a body is set into motion relative to an incompressible, inviscid, non-diffusive rotating stratified fluid, showing that there is generated in general a topographic Rossby wave which leads to non-decaying fluctuations in the lift on the obstacle and a fluctuating non-zero drag. The problem is relevant to the flow patterns and forces excited when slow oceanic flows cross bottom topography, and suggests a mechanism for slow fluctuations observed in laboratory experiments.

We extend the vorticity-based modelling approach of Borden & Meiburg (Phys. Fluids, vol. 25 (10), 2013, 101301) to non-Boussinesq gravity currents and derive an analytical expression for the Froude number without the need for an energy closure or any assumptions about the pressure. The Froude-number expression we obtain reduces to the correct form in the Boussinesq limit and agrees closely with simulation data. Via detailed comparisons with simulation results, we furthermore assess the validity of three key assumptions underlying both our as well as earlier models: (i) steady-state flow in the moving reference frame; (ii) inviscid flow; and (iii) horizontal flow sufficiently far in front of and behind the current. The current approach does not require an assumption of zero velocity in the current.

The measurements taken during the Vertical Transport and Mixing Experiment (VTMX, October, 2000) on a northeastern slope of Salt Lake Valley, Utah, were used to calculate the statistics of velocity fluctuations in a katabatic gravity current in the absence of synoptic forcing. The data from ultrasonic anemometer-thermometers placed at elevations 4.5 and 13.9 m were used. The contributions of small-scale turbulence and waves were isolated by applying a high-pass digital (Elliptical) filter, whereupon the filtered quantities were identified as small-scale turbulence and the rest as internal gravity waves. Internal waves were found to play a role not only at canonical large gradient Richardson numbers $$(\overline{\hbox {Ri}_\mathrm{g} } >1)$$ , but sometimes at smaller values $$(0.1 < \overline{\hbox {Ri}_\mathrm{g}}1$$ , the use of both scaling types is questionable, although filtered data showed a tendency to follow local scaling. A relationship between z/L and $$\overline{\hbox {Ri}_\mathrm{g} }$$ was identified. Eddy diffusivities of momentum $$\hbox {K}_\mathrm{M}$$ and heat $$\hbox {K}_\mathrm{H}$$ were dependent on wave activities, notably when $$\overline{\hbox {Ri}_\mathrm{g} } > 1$$ . The ratio $$\hbox {K}_{\mathrm{H}}/\hbox {K}_{\mathrm{M}}$$ dropped well below unity at high $$\overline{\hbox {Ri}_\mathrm{g} }$$ , in consonance with previous laboratory stratified shear layer measurements as well as other field observations.