One of the most important problems in astrophysics is how angular momentum is transported in protoplanetary disks (PPDs - disks containing gas and dust orbit around newly-forming protostars). Collisional viscosity is believed to be insufficient for angular momentum transport. Therefore, turbulence enhanced transport are proposed. In addition, long-lived coherent vortices are also speculated to exist in PPDs, which could play an important role in completing star formation and building planets. Without instabilities, turbulence and vortices cannot form. In weak magnetized PPDs, magneto-rotational instability (MRI) operates to generate turbulence. However, regions known as ``dead zone", are cool and unionized to have MRI. This has led to intense theoretical and computational search for pure hydrodynamic instabilities.

A new hydrodynamic, finite amplitude instability has been discovered in linearly stable, rotating, stably-stratified, shear flows. The instability starts from a new family of critical layers - baroclinic critical layers. These critical layers, which are linear, neutrally stable eigenmodes in stratified shear flows, have singularity in their vertical velocities. Under the effect of rotation, these critical layers produce vortex layers. Vortex layers intensify by drawing energy from the background shear flows, and subsequently roll up to create new vortices, which in turn excite new critical layers. The whole process self-replicates until the whole domain is filled with large-volume, large amplitude vortices. Because this instability can occur in the dead zones of protoplanetary disks we refer it as zombie instability and these new class of vortices that self-replicate as zombie vortices. High resolution numerical simulations show this instability can be triggered by a variety of weak perturbations including small volume compact single vortex, a pair of vortices and noise. The threshold of the instability is determined by the Rossby number or vorticity of the initial perturbations. Energy analysis based on the zonal non-zonal decomposition of the energy shows energy that supplies the instability is extracted from the zonal flows. Vortex is responsible for the energy extraction process. Instability saturates when the all the space are taken by zombie vortices. The separation distance between zombie vortices is approximately the distance from critical layers with lowest stream-wise wave number to the perturbations. The flows at late time are determined only by the background parameters not their initial perturbations. Zombie instability is also discussed in a broader picture to show the dead zones of PPDs are not dead. Our numerical simulation suggest although zombie instability is a finite-amplitude instability, due to the large Reynolds number of the disk flows, it is effectively a linear instability. How zombie instability might lead to sufficient angular momentum transport is also discussed. Finally, we speculate there might not be a laminar Keplerian disks at all. The disk flows are essentially turbulence from the collapsing of gas cloud with possibility turbulent flows filled with zombie vortices. A newly developed semi-analytic method for flows with strong background shear is also presented to be an alternative to widely used shearing sheet method in the astrophysical community. The semi-analytic method is used for simulating internal inertial-gravity waves in rotating, stratified flows with and without shear. The method can also be generalized to systems with linear forcing terms.

The physical world is spatially three dimensional, hence understanding the physical and semantic properties of the three dimensional world is crucial to designing engineering systems that operate in and interact with its three dimensional surroundings. Recent advances in Deep Learning has shown great promise in offering a general methodology for creating algorithms that learns to map sensory inputs to desired outcomes by training on a repository of data. Such advances have been particularly salient in fields such as computer vision, where algorithms have achieved near-human or super-human performance in a range of traditionally difficult problems such as image classification, object detection and segmentation. However very often three dimensional problems have been reduced to two dimensional ones for simplicity. In traditional computer vision, inherently three dimensional objects are instead represented by images of their projections onto the two dimensional plane. In computational physics, three dimensional simulations are often simplified as two dimensional counterparts as the computational cost of 3D simulations are often times intractable for many applications. Such simplifications leave out essential information about the actual spatial relationships between objects and can result in inaccurate or erroneous predictions.

The focus of this dissertation is to address the challenges in designing deep learning algorithms and architectures that interact with three dimensional data. Conventional data representation and neural architectures in computer vision do not directly extend to three dimensional counterparts for various reasons. First, data representation for 3D objects is varied and diverse. For instance, in computer graphics and computation physics, geometries are usually represented as simplicial complexes (point clouds, wire frames, triangular meshes, volumetric tetrahedral meshes etc.) for efficiency. In robotics, however, the data representation is mainly determined by the form of the raw sensory inputs to the system, such as point clouds resulting from Lidar scans or Kinect sensors, RGB-D images from depth-enabled cameras, multiview images from binocular or multi-camera setup of the system. Panoramic or fisheye images are also becoming increasingly prevalent in UAVs (Unmanned Aerial Vehicles) and self-driving cars. Second, three dimensional data tend to be orders of magnitudes greater that its two dimensional counter part. Using Cartesian grid representation as an example, the storage complexity for 2D images is O(n^2) whereas for 3D objects that would be O(n^3). Therefore more efficient encoding and representation methods need to be brought forward to address such challenges. Last but not least, three dimensional data is much more costly to acquire (due to much higher costs of collection, and inaccessibility of such sensors), label (orienting and navigating 3D data on 2D screens requires engineering effort), store and process. In the sections that follow, I will discuss novel methodologies that address these aforementioned challenges, and present various useful applications in computer vision and computational physics that benefit from such methodologies.

In Chapter One, I will present an in-depth overview of these challenges that arise from 3D data, as well as a detailed discussion of existing disciplines from computational physics, climate science, to computer vision that can benefit from progress in 3D Geometric Deep Learning. In Chapter Two, I will present a novel and general methodology for differentiably rasterizing unstructured geometric representations in the form of simplicial complexes. This can serve as a geometry layer within deep neural networks that allows a natural extension of Convolutional Neural Network (CNN) based architectures to a vast collection of 3D representations. In Chapter Three, I will introduce a methodology for natively performing convolutions on the spherical manifold, which is the underlying geometric representation for signals from a range of disciplines, from climate science to panoramic vision. Our methodology is efficient to compute, and naturally prevents the distortion related problems that arise from directly using CNNs on equirectangular projections of spherical signals. Finally, in Chapter Four, I will present a novel continuous implicit 3D representation for large scenes and large physical systems that can allow us to leverage localized learned geometric priors for 3D reconstruction tasks. Moreover, a simple extension to this representation allows us to inject Partial Differential Equation (PDE) constraints within physical systems, in order to facilitate a physics-informed and physics-abiding deep learning architecture.

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.

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.

We present a new design method, which we call design-by-morphing, for the optimal design of the shape of an object. The surface of one or more objects (or the sub-objects from which it is composed) is represented as a truncated series of exponentially-convergent spectral basis functions multiplied by spectral coefficients. A morphed object or sub-object is obtained from a new set of spectral coefficients, which are a weighted average of the spectral coefficients of the original objects or sub-objects from which it is morphed. Optimized designs are created by choosing the weights such that a cost function of the new morphed shape is minimized. The boundaries of an object and the interfaces between sub-objects can be forced to satisfy geometric constraints on their shapes, slopes, curvature, etc. With these constraints, sub-objects can be seamlessly attached to each other to create a complex object. Because design-by-morphing has the flexibility to adjust independently the weights of sub-objects, users or an automated algorithm can choose some subset of sub-objects to be optimized while preserving or restricting the changes of other sub-objects. Our design-by-morphing method can be automated and is computationally efficient, so it requires much less human input than traditional design methods and is therefore not only inexpensive but also free from human bias in finding optimal designs that are radical and non-intuitive.

We have applied optimization via design-by- morphing to aircraft and a turbine-99 draft tube, and reduced drag-to-lift ratio and maximized mean pressure recovery factor by 23.1% and 10.9%, respectively. We believe that this optimization method is applicable to a wide variety of engineering applications in which the performance of an object depends on the aerodynamic or hydrodynamic

The early-time growth of an individual vortex in a wake vortex pair is explored, focusing on its linear stability and transient growth, which are meaningful for understanding the lifespan of vortices and mitigating hazards from wake turbulence. This study synthesises two parts: linear stability analysis and transient growth analysis. The linear stability of wake vortices is examined in both inviscid and viscous contexts, using the Batchelor or Lamb-Oseen vortex as a base vortex model. Linear stability analysis determines the growth rates of infinitesimal perturbations in the vortex flow by linearising the incompressible Navier-Stokes or Euler equations around the base vortex, leading to an eigenvalue problem that discloses the vortex's stability properties. Inviscid analysis, with zero viscosity, successfully reveals a continuous spectrum associated with critical-layer singularities, causing discontinuity in perturbation velocity in the inviscid limit and requiring careful numerical parameter tuning to avoid under-resolved, or so-called spurious, solutions. Non-zero viscosity alters both the discrete and continuous spectra, necessitating numerical resolution conforming to the Reynolds number to the one-third power scaling law to resolve the newly discovered viscous critical-layer eigenmode family. The transient growth phenomenon is then explored through non-modal analysis to understand how initial perturbations, as combinations of numerous eigenmodes, can experience significant transient growth even in linearly stable vortices. The transient growth formalism is applied to the linearised Navier-Stokes equations. Optimal perturbations that maximise energy amplification over finite time intervals are found, to which the viscous critical-layer eigenmode family serves as the main contributor. Numerical simulations quantify the growth of perturbations considering nonlinearity, showing that while nonlinearity may slightly alter transient dynamics, overall trends align with formalism predictions. Lastly, motivated by contrails around wake vortices, the research investigates particle-initiated transient growth, where perturbations evolve through particle-vortex interactions. Two-way coupled particle-vortex simulations demonstrate continual perturbation development, presenting primary growth patterns as captured in the transient growth formalism. This investigation implies the practicability of the transient growth phenomenon.

Perturbation methods are used to calculate nonlinear interaction

of waves, however most analyses skip the question as to whether

the zeroth order solutions exist. The dispersion

relation for internal gravity waves does not relate the

magnitude of the wave vector and its frequency, rather it relates

the frequency and direction of the wave vector. Thus,

spatially columnated beams of internal waves are made of a

continuum of plane waves with different wavelengths, but the same magnitude of

frequency. For two parent beams to create a daughter, the

plane waves within the parent and daughter beams must obey the

triad condition (the spatial wave vector of the daughter equals

the sum of the parents' vectors, and temporal frequency of the

daughter equals the sum of the parents'

frequencies) and the dispersion relation. Contrary to what is

assumed implicitly, these conditions cannot always be satisfied.

If they could, then the interaction of two

beams of gravity waves would produce 8 daughter beams, consisting

of two St. Andrew's crosses (each with 4 beams). The beams in one

cross have a frequency equal to the sum of the frequencies of the

parents and the beams in the other have a frequency equal to the

difference. At least two daughter beams cannot exist for each cross according to the selection rules derived in this work.

Similar selection rules are obtained for the interaction among inertia-gravity conical waves.

When two parent conical waves intersect, unlike the two-dimensional beams, the interaction area is not a single point.

The intersection produces spatial continuous curves and there are no more than two intersection points in any horizontal plane with a normal vector parallel to gravity direction.

Each intersection point is acted like the collision point of two-dimensional beam-beam interaction.

As a result, at most two harmonic beams with a frequency either equal to the sum of the frequencies of the

parents or to the difference, not a conical wave, are produced from the nonlinear interaction.