Accurate numerical models are essential to predict the complex evolution of rapidly changing sea ice conditions and study impacts on climate and navigation. However, sea ice models contain uncertainties associated with initial conditions and forcing (wind, ocean), as well as with parameter values, functional forms of the constitutive relations, and state variables themselves, all of which limit predictive capabilities. Due to the multiple types and scales of sea ice and the complex nonlinear mechanics and high dimensionality of differential equations, efficient ocean and sea ice probabilistic modeling, Bayesian inversion, and machine learning are challenging. In this work, we implement a deterministic 2D viscoplastic sea ice solver and derive and implement new sea ice probabilistic models based on the dynamically orthogonal (DO) equations.

We focus on the stochastic two-dimensional sea ice momentum equations with nonlinear viscoplastic constitutive law. We first implement and verify a deterministic 2D viscoplastic sea ice solver. Next, we derive the new stochastic Sea Ice Dynamically Orthogonal equations and develop numerical schemes for their solution. These equations and schemes preserve nonlinearities in the underlying spatiotemporal dynamics and evolve the non-Gaussianity of the statistics. We evaluate and illustrate the new stochastic sea ice modeling and schemes using idealized stochastic test cases. We employ two stochastic test cases with different types of sea ice: ice sheets and frozen ice cover with uncertain initial velocities. We showcase the ability to evolve non-Gaussian statistics and capture complex nonlinear dynamics efficiently. We study the convergence to the physical discretization, and stochastic convergence to the stochastic subspace size and coefficient samples. Finally, we assess and show significant computational and memory efficiency compared to the direct Monte Carlo method.

Sound waves are critical for a variety of underwater applications including communication, navigation, echo-sounding, environmental monitoring, and marine biology research. However, the incomplete knowledge of the ocean environment and acoustic parameters makes reliable acoustic predictions challenging. This is due to the sparse and heterogeneous data, as well as to the complex ocean physics and acoustics dynamics, multiscale interactions, and large dimensions. There are thus several sources of uncertainty in acoustic predictions. They include the ocean current, temperature, salinity, pressure, density, and sound speed fields, the bottom topography and seabed fields, the sources and receivers properties, and finally the model equations themselves. The goals of this thesis are to address these challenges. Specifically, we: (1) obtain, solve, and verify differential equations for efficient probabilistic underwater acoustic modeling in uncertain environments; (2) develop theory and implement algorithms for the Bayesian nonlinear inference and learning of the ocean, bathymetry, seabed, and acoustic fields and parameters using sparse data; and (3) demonstrate the new methodologies in a range of underwater acoustic applications and real sea experiments, showcasing new capabilities and leading to improved understanding.

In the first part, we derive, discretize, implement, and verify stochastic differential equations that (i) capture dominant input uncertainties in the environment (e.g., ocean, bathymetry, and seabed) and in the acoustic parameters (e.g. source location, frequency, and bandwidth), and (ii) predict the acoustic pressure fields and their probability distributions, respecting the nonlinear governing equations and non-Gaussian statistics. Starting from the acoustic Parabolic Equation (PE), we develop Dynamically Orthogonal (DO) differential equations for range-optimal acoustic uncertainty quantification. Using DO expansions for the input uncertainties, we develop the reduced-order DO-PEs theory for the Narrow-Angle PE (NAPE) and Pad\’e Wide-Angle PE (WAPE) stochastic partial differential equations (PDEs). We verify the discretized DO-PEs in new stochastic range-independent and range-dependent test cases, and demonstrate their advantages over state-of-the-art methods for uncertainty quantification and wave propagation in random media. Results show that a single DO-PE simulation can accurately predict stochastic range-dependent acoustic fields and their full non-Gaussian probability distributions, with computational savings of several orders of magnitude when compared to direct Monte Carlo methods.

In the second part, we extend recent nonlinear Bayesian data assimilation (DA) to the inference and learning of ocean-bathymetry-seabed-acoustic fields and parameters using sparse acoustic and oceanographic data. We combine the acoustic DO-PEs with Gaussian mixture models (GMMs) to predict probability densities in the DO subspace, allowing for efficient non-Gaussian estimation of state variables, parameters, and model functions themselves. The joint multidisciplinary estimation is enabled by state augmentation where the ocean-acoustic-bathymetry-seabed states and parameters are fit together to GMMs within the DO subspace. The new GMM-DO ocean acoustic inference system is validated by assimilating sparse data to infer the source depth, source frequency, and acoustic and environment fields and parameters in five new high-dimensional inference test cases based on state-of-the-art oceanographic and geoacoustic benchmarks. We evaluate the convergence to inference parameters and quantify the learning skill. Results show that our PDE-based Bayesian learning successfully captures non-Gaussian statistics and acoustic ambiguities. Using Bayes’ law, it provides accurate probability distributions for the multivariate quantities and enables principled learning from noisy, sparse, and indirect data.

In the final part, we integrate our acoustic DO-PEs and GMM-DO frameworks with the MSEAS primitive equation ocean modeling system to enable unprecedented probabilistic forecasting and learning of ocean physics and acoustic pressure and transmission loss (TL) fields, accounting for uncertainties in the ocean, acoustics, bathymetry, and seabed fields. We demonstrate the use of this system for low to mid-frequency propagation with real ocean data assimilation in three regions. The first sea experiment takes place in the western Mediterranean Sea where we showcase the system’s performance in predicting ocean and acoustic probability densities, and assimilating sparse TL and sound speed data for joint ocean physics-acoustics-source depth inversion in deep ocean conditions with steep ridges. In the second application, we simulate stochastic acoustic propagation in Massachusetts Bay around Stellwagen Bank and use our GMM-DO Bayesian inference system to assimilate TL data for acoustic and source depth inversion in shallow dynamics with strong internal waves. Finally, in the third experiment in the New York Bight, we employ our system as a novel probabilistic approach for broadband acoustic modeling and inversion. Overall, our results mark significant progress toward end-to-end ocean-acoustic systems for new ocean exploration and management, risk analysis, and advanced operations.

With the expanding availability of computational power, numerical modeling plays an increasingly pivotal role in the field of oceanography, enabling scientists to explore and understand ocean processes which are otherwise inaccessible or challenging to observe directly. It provides a crucial tool for investigating a range of phenomena from large-scale circulation patterns to small-scale turbulence, shaping our understanding of marine ecosystems, global climate, and weather patterns. However, this same wide range of spatiotemporal scales presents a distinct computational challenge in capturing physical interactions extending from the diffusive scale (millimeters, seconds) to planetary length scales spanning thousands of kilometers and time scales spanning millennia. Therefore, numerical and parameterization improvements have and will continue to define the state of the art in ocean modeling, in tandem with the integration of observational data and adaptive methods. As scientists strive to better understand multiscale ocean processes, the thirst for comprehensive simulations has proceeded apace with concomitant increases in computing power, and submesoscale resolutions where nonhydrostatic effects are important are progressively becoming approachable in ocean modeling. However, few realistic ocean circulation models presently have nonhydrostatic capability, and those that do overwhelmingly use low-order finite-difference and finite-volume methods, which are plagued by dispersive errors, and are arduous to utilize in general, especially on unstructured domains and in conjunction with adaptive numerical capabilities. High-order discontinuous Galerkin (DG) finite element methods (FEMs) allow for arbitrarily high-order solutions on unstructured meshes and often out-compete low-order models with respect to accuracy per computational cost, providing significant reduction of dispersion and dissipation errors over long-time integration horizons. These properties make DG-FEMs ideal for the next generation of ocean models, and, in this thesis, we develop a novel DG-FEM ocean model with the above longer-term vision and adaptive multiscale capabilities in mind.

Using a novel hybridizable discontinuous Galerkin (HDG) spatial discretization for both the hydrostatic and nonhydrostatic ocean equations with a free surface, we develop an accurate and efficient high-order finite element ocean model. We emphasize the stability and robustness properties of our schemes within a projection method discretization. We provide detailed benchmarking and performance comparisons for the parallelized implementation, tailored to the specifics of HDG finite element methods. We demonstrate that the model achieves optimal convergence, and is capable of accurately simulating nonhydrostatic behavior. We evaluate our simulations in diverse dynamical regimes including linear gravity waves, internal solitary waves, and the formation of Rayleigh-Taylor instabilities in the mixed layer. Motivated by investigating local nonhydrostatic submesoscale dynamics using realistic ocean simulation data, we develop schemes to initialize and nest the new DG-FEM model within a comprehensive hydrostatic ocean modeling system. Nested within such data-assimilative hydrostatic simulations in the Alboran Sea, we provide a demonstration of our new model’s ability to capture both hydrostatic and nonhydrostatic dynamics that arise in the presence of wind-forced instabilities in the upper ocean layers. We show that such a model can both validate and work in tandem with larger hydrostatic modeling systems, enabling multi-dynamics simulations and enhancing the predictive fidelity of ocean forecasts.

Next, as DG-FEM methods are well-suited to adaptive refinement, we develop a method to learn new adaptive mesh refinement strategies directly from numerical simulation by formulating the adaptive mesh refinement (AMR) process as a reinforcement learning problem. Finite element discretizations of problems in computational physics can usefully rely on adaptive mesh refinement to preferentially resolve regions containing important features during simulation. However, most spatial refinement strategies are heuristic and rely on domain-specific knowledge or trial-and-error. We treat the process of adaptive mesh refinement as a local, sequential decision-making problem under incomplete information, formulating AMR as a partially observable Markov decision process. Using a deep reinforcement learning (DRL) approach, we train policy networks for AMR strategy directly from numerical simulation. The training process does not require an exact solution or a high-fidelity ground truth to the partial differential equation (PDE) at hand, nor does it require a pre-computed training dataset. The local nature of our deep reinforcement learning approach allows the policy network to be trained inexpensively on much smaller problems than those on which they are deployed, and the DRL-AMR learning process we devise is not specific to any particular PDE, problem dimension, or numerical discretization. The RL policy networks, trained on simple examples, can generalize to more complex problems and can flexibly incorporate diverse problem physics. To that end, we apply the method to a range of PDEs relevant to fluid and ocean processes, using a variety of high-order discontinuous Galerkin and hybridizable discontinuous Galerkin finite element discretizations. We show that the resultant learned policies are competitive with common AMR heuristics and strike a favorable balance between accuracy and cost such that they often lead to a higher accuracy per problem degree of freedom, and are effective across a wide class of PDEs and problems.

Numerical modelling of ocean physics is essential for multiple applications such as scientific inquiry and climate change but also renewable energy, transport, autonomy, fisheries, water, harvesting, tourism, communication, conservation, planning, and security. However, the wide range of scales and interactions involved in ocean dynamics make numerical modelling challenging and expensive. Many regional ocean models resort to a hydrostatic (HS) approximation that significantly reduces the computational burden. However, a challenge is to capture and study local ocean phenomena involving complex dynamics over a broader range of scales, from regional to small scales, and resolving nonlinear internal waves, subduction, and overturning. Such dynamics require multi-resolution non-hydrostatic (NHS) ocean models. It is known that the main computational cost for NHS models arises from solving a globally coupled elliptic PDE for the NHS pressure. Optimally reducing these costs such that the NHS dynamics are resolved where needed is the motivation for this work.

We propose a new multi-dynamics model to decompose a domain into NHS and HS dynamic regions and solve the corresponding models in their subdomains, reducing the cost associated with the NHS pressure solution step. We extend a high-order NHS solver developed using the hybridizable discontinuous Galerkin (HDG) finite element methodology by taking advantage of the local and global HDG solvers for combining HS with NHS solvers. The multi-dynamics is derived, and the first version is implemented in the HDG framework to quantify computational costs and evaluate accuracy using several analyses. We first showcase results on Rayleigh Taylor instability-driven striations to evaluate computational savings and accuracy compared to the standard NHS HDG and finite-volume solvers. We highlight and discuss sensitivities and performance. Finally, we explore parameters that can be used to identify domain regions exhibiting NHS behaviour, allowing the algorithm to dynamically evolve the NHS and HS subdomains.

Advances in computational power have brought the possibility of realistically modeling our world with numerical simulations closer than ever. Nevertheless, our appetite for higher fidelity simulations and faster run times grows quickly; we will always grasp at what is just beyond our computational reach. No matter if we seek to understand biological, chemical, or physical systems, the bottleneck of scientific computing is almost always the same: high dimensionality. The goal of reduced-order modeling is to reduce the number of unknowns in a system while minimizing the loss of accuracy in the approximate solution. Ideal model-order reduction techniques are optimal compromises between computational tractability and solution fidelity. While there are plenty of such techniques to choose from, their widespread adoption remains to be seen due to several persistent challenges. Many methods are intrusive and difficult to implement, lack traditional numerical guarantees such as convergence and stability, and cannot adapt to unforeseen dynamics. We seek to promote the adoption of reduced-order models (ROMs) by creating non-intrusive, efficient, and dynamically adaptive algorithms that maintain the essential features and numerical guarantees of their full-order counterparts.

In this thesis, we derive and apply algorithms for dynamical reduced-order models. Many model-order reduction approaches project dynamical systems onto a fixed subspace obtained from either a simplification of the original equations, a set of known functions such as orthogonal polynomials, or a reduced basis of full-order simulations computed offline. However, if the true system exits the span of the prescribed subspace, such approaches quickly accumulate large errors. In contrast, dynamical ROMs adapt their subspaces as the system evolves. Geometrically, this amounts to integrating a dynamical system along a nonlinear manifold embedded in a full-order Euclidean space. We develop schemes that not only change subspaces at each discrete time step, but that change the subspace in between time steps for improved accuracy. Even further, our numerical schemes automatically detect when the dynamics depart the nonlinear manifold and may jump to a new nonlinear manifold that better captures the system state. For concreteness, we focus on a reduced-order modeling technique called the dynamical low-rank approximation (DLRA), a discrete analogue to the dynamically orthogonal (DO) differential equations. The DLRA evolves a low-rank system in time (or range) as an approximation to a full-rank system, and in contrast to many methods, the DLRA does not require an offline stage where full-order simulations are computed. It is also agnostic to the source of high dimensionality, whether it be the high resolution required, the large domain, or the stochasticity of the problem. These features make it a versatile tool suitable for a wide variety of problems. We evaluate, verify, and apply our new dynamical reduced-order models and schemes to a varied set of dynamical systems, including stochastic fluid flows and waves, videos and their dynamic compression, realistic ocean acoustics and underwater sound propagation with dynamic coordinate transforms, and stochastic reachability and time-optimal path planning.

The majority of this work is devoted to new adaptive integration schemes for the DLRA. We start by introducing perturbative retractions, which map arbitrary-rank matrices back to a manifold of fixed-rank matrices. They asymptotically approximate the truncated singular value decomposition at a greatly reduced cost while guaranteeing convergence to the best low-rank approximation in a fixed number of iterations. From these retractions, we develop the dynamically orthogonal Runge-Kutta (DORK) schemes, which change the subspace onto which the system’s dynamics are projected in between time steps. The DORK schemes are improved by using stable, optimal (so) perturbative retractions, resulting in the so-DORK schemes. They are more efficient, accurate, and stable than their predecessors. We also introduce gradient-descent (gd) retractions and the gd-DORK schemes, which tend to converge rapidly to the best low-rank approximation by recursively applying retractions. The DORK schemes may be made rank-adaptive and robust to rank overapproximation with either a pseudoinverse or by changing the gauge of the integration scheme. While the pseudoinverse technique accumulates slightly more error, it preserves mode continuity, a feature that changing the gauge lacks. Next, we derive an alternating-implicit (ai) linear low rank solver, which is used to create ai-DORK schemes. The ai-DORK schemes are a general-purpose family of implicit integration schemes that have the same algorithmic complexity as explicit schemes (provided some conditions on the dynamics), which vastly broadens the scope of problems that can be solved with the DLRA. This relieves stringent time-step restrictions and enables the DLRA to handle stiff systems. Furthermore, we develop a piecewise polynomial approximation using adaptive clustering in order to handle non-polynomial nonlinearities in reduced-order models. We thoroughly
test these numerical schemes on well-conditioned and ill-conditioned matrix differential equations; data-driven dynamical systems including videos; Schrödinger’s equation; a stochastic, viscous Burgers’ equation; a deterministic, two-dimensional, viscous Burgers’ equation; an advection-diffusion partial differential equation (PDE); a nonlinear, stochastic Fisher-KPP PDE; nonlinear, stochastic ray tracing; and a nonlinear, stochastic Hamilton-Jacobi-Bellman PDE for time-optimal path planning. We find that the reduced-order solutions may be made arbitrarily accurate using rank adaptive dynamical schemes that automatically track the true rank of the full-order simulation, and nonlinearities may be well-approximated by dynamically increasing the number of stochastic clusters, all at a greatly reduced computational cost.

In addition to DORK schemes, we create a tailor-made low-rank integration scheme for the narrow-angle parabolic wave equation called the low-rank split-step Fourier method. Acoustic simulations are often bottlenecked by the Nyquist criterion, which insists that we sample spatially at least twice per wavelength. To address this, our low-rank split-step Fourier method has an algorithmic complexity that scales sublinearly in the number of classical degrees of freedom, enabling vastly larger computational domains and higher frequencies. We demonstrate its efficacy on realistic ocean acoustics problems in Massachusetts Bay with sound speed fields obtained from our high-resolution ocean primitive equations modeling system. In comparing the low rank and full-rank simulations, we demonstrate that the dynamical low-rank method captures the full-rank features including three-dimensional acoustic energy propagation in complex ocean fields with internal waves and rapidly varying bathymetry.

Lastly, with tools from machine learning, we introduce learnable and automatically differentiable coordinate transforms. The compressibility of a system heavily depends on the choice of coordinates, and frequently a coordinate system is chosen for its simplicity rather than its efficiency. Our novel coordinate transforms are determined in a hands-off manner by minimizing a cost function that includes the environmental data expressed in terms of the non-constant coefficients and initial conditions of a PDE. Not only do we automatically obtain Jacobians and Hessians of the transforms, we also find coordinate systems that reduce the rank of solutions to PDEs. This improves the accuracy of the DLRA for the same cost as a typical low-rank simulation, and it accelerates the convergence in rank to the full-order solution. The coordinate transforms also enable low-rank domain decomposition, which is particularly useful in ocean acoustics where the water-seabed interface is discontinuous. We demonstrate this methodology on a first-order PDE with advection and a second-order PDE, the parabolic wave equation, using two examples. We first show acoustic propagation along a three-dimensional wedge and compare the accuracy of solutions computed in the original and transformed coordinate systems. We then show acoustic propagation in a realistic ocean environment over Stellwagen Bank in Massachusetts Bay with a
dynamic coordinate transform.

An important challenge in the problem of producing accurate forecasts of multiscale dynamics, including but not limited to weather prediction and ocean modeling, is that these dynamical systems are chaotic in nature. A hallmark of chaotic dynamical systems is that they are highly sensitive to small perturbations in the initial conditions and parameter values. As a result, even the best physics-based computational models, often derived from first principles but limited by varied sources of errors, have limited predictive capabilities for both shorter-term state forecasts and for important longer-term global characteristics of the true system. Observational data, however, provide an avenue to increase predictive capabilities by learning the physics missing from lower-fidelity computational models and reducing their various errors. Recent advances in machine learning, and specifically data-driven knowledge-based prediction, have made this a possibility but even state-of-the-art techniques in this area have not been able to produce short-term forecasts beyond a small multiple of the Lyapunov time of the system, even for simple chaotic systems such as the Lorenz 63 model. In this work, we develop a training framework to apply neural ordinary differential equation-based (nODE) closure models to correct errors in the equations of such dynamical systems. We first identify the key training parameters that have an outsize effect on the learning ability of the neural closure models. We then develop a novel learning algorithm, broadly consisting of adaptive tuning of these parameters, designing dynamic multi-loss objective functions, and an error-targeting batching process. We evaluate and showcase our methodology to the chaotic Balance Equations in an array of increasingly difficult learning settings: first, only the coefficient of one missing term in one perturbed equation; second, one entire missing term in on perturbed equation; third, two missing terms in two perturbed equations; and finally the previous but with a perturbation being two orders of magnitude larger than the state, thereby resulting in a completely different attractor. In each of these cases, our new multi-faceted training approach drastically increases both state-of-the-art state predictability (up to 15 Lyapunov times) and attractor-reproducibility. Finally, we validate our results by comparing them with the predictability limit of the chaotic BE system under different magnitudes of perturbations.

Onboard forecasting and data assimilation are challenging but essential for unmanned autonomous ocean platforms. Due to the numerous operational constraints for these platforms, efficient adaptive reduced-order models (ROMs) are needed. In this thesis, we first review existing approaches and then develop a new adaptive Dynamic Mode Decomposition (DMD)-based, data-driven, reduced-order model framework that provides onboard forecasting and data assimilation capabilities for bandwidth-disadvantaged autonomous ocean platforms. We refer to the new adaptive ROM as the incremental, stochastic Low-Rank Dynamic Mode Decomposition (iLRDMD) algorithm. Given a set of high-fidelity and high-dimensional stochastic forecasts computed in remote centers, this framework enables i) efficient and accurate send and receive of the high-fidelity forecasts, ii) incremental update of the onboard reduced-order model, iii) data-driven onboard forecasting, and iv) onboard ROM data assimilation and learning. We analyze the computational costs for the compression, communications, incremental updates, and onboard forecasts. We evaluate the adaptive ROM using a simple 2D flow behind an island, both as a test case to develop the method, and to investigate the parameter sensitivity and algorithmic design choices. We develop the extension of deterministic iLRDMD to stochastic applications with uncertain ocean forecasts. We then demonstrate the adaptive ROM on more complex ocean fields ranging from univariate 2D, univariate 3D, and multivariate 3D fields from multi-resolution, data-assimilative Multidisciplinary Simulation, Estimation, and Assimilation Systems (MSEAS) reanalyses, specifically from the real-time exercises in the Middle Atlantic Bight region. We also highlight our results using the Navy’s Hybrid Coordinate Ocean Model (HYCOM) forecasts in the North Atlantic region. We then apply the adaptive ROM onboard forecasting algorithm to interdisciplinary applications, showcasing adaptive reduced-order forecasts for onboard underwater acoustics computations and forecasts, as well as for exact time-optimal path-planning with autonomous surface vehicles.

For stochastic forecasting and data assimilation onboard the unmanned autonomous ocean platforms, we combine the stochastic ensemble DMD method with the Gaussian Mixture Model – Dynamically Orthogonal equations (GMM-DO) filter. The autonomous platforms can then perform principled Bayesian data assimilation onboard and learn from the limited and gappy ocean observation data and improve onboard estimates. We extend the DMD with the GMM-DO filter further by incorporating incremental DMD algorithms so that the stochastic ensemble DMD model itself is updated with new measurements. To address some of the inefficiencies in the first combination of the stochastic ensemble DMD with the GMM-DO filter, we further introduce the GMM-DMD algorithm. This algorithm not only uses the stochastic ensemble DMD as a computationally efficient forward model, but also employs the existing decomposition to fit the GMM to and perform Bayesian updates on. We demonstrate this incremental stochastic ensemble DMD with GMM-DO and GMMDMD using a real at-sea application in the Middle Atlantic Bight region. We employ a 300 member set of stochastic ensemble forecasts for the “Positioning System for Deep Ocean Navigation – Precision Ocean Interrogation, Navigation, and Timing” (POSYDON-POINT) sea experiment, and highlight the capabilities of reduced data assimilation using simulated twin experiments.

Complex dynamical models are used for prediction in many domains, and are useful to mitigate many of the grand challenges being faced by humanity, such as climate change, food security, and sustainability. However, because of computational costs, complexity of real-world phenomena, and limited understanding of the underlying processes involved, models are invariably approximate. The missing dynamics can manifest in the form of unresolved scales, inexact processes, or omitted variables; as the neglected and unresolved terms become important, the utility of model predictions diminishes. To address these challenges, we develop and apply novel scientific machine learning methods to learn unknown and discover missing dynamics in models of dynamical systems.

In our Bayesian approach, we develop an innovative stochastic partial differential equation (PDE) – based model learning framework for high-dimensional coupled biogeochemical-physical models. The framework only uses sparse observations to learn rigorously within and outside of the model space as well as in that of the states and parameters. It employs Dynamically Orthogonal (DO) differential equations for adaptive reduced-order stochastic evolution, and the Gaussian Mixture Model-DO (GMM-DO) filter for simultaneous nonlinear inference in the augmented space of state variables, parameters, and model equations. A first novelty is the Bayesian learning among compatible and embedded candidate models enabled by parameter estimation with special stochastic parameters. A second is the principled Bayesian discovery of new model functions empowered by stochastic piecewise polynomial approximation theory. Our new methodology not only seamlessly and rigorously discriminates between existing models, but also extrapolates out of the space of models to discover newer ones. In all cases, the results are generalizable and interpretable, and associated with probability distributions for all learned quantities. To showcase and quantify the learning performance, we complete both identical-twin and real-world data experiments in a multidisciplinary setting, for both filtering forward and smoothing backward in time. Motivated by active coastal ecosystems and fisheries, our identical twin experiments consist of lower-trophic-level marine ecosystem and fish models in a two-dimensional idealized domain with flow past a seamount representing upwelling due to a sill or strait. Experiments have varying levels of complexities due to different learning objectives and flow and ecosystem dynamics. We find that even when the advection is chaotic or stochastic from uncertain nonhydrostatic variable-density Boussinesq flows, our framework successfully discriminates among existing ecosystem candidate models and discovers new ones in the absence of prior knowledge, along with simultaneous state and parameter estimation. Our framework demonstrates interdisciplinary learning and crucially provides probability distributions for each learned quantity including the learned model functions. In the real-world data experiments, we configure a one-dimensional coupled physical-biological-carbonate model to simulate the state conditions encountered by a research cruise in the Gulf of Maine region in August, 2012. Using the observed ocean acidification data, we learn and discover a salinity based forcing term for the total alkalinity (TA) equation to account for changes in TA due to advection of water masses of different salinity caused by precipitation, riverine input, and other oceanographic processes. Simultaneously, we also estimate the multidisciplinary states and an uncertain parameter. Additionally, we develop new theory and techniques to improve uncertainty quantification using the DO methodology in multidisciplinary settings, so as to accurately handle stochastic boundary conditions, complex geometries, and the advection terms, and to augment the DO subspace as and when needed to capture the effects of the truncated modes accurately. Further, we discuss mutual-information-based observation planning to determine what, when, and where to measure to best achieve our learning objectives in resource-constrained environments.

Next, motivated by the presence of inherent delays in real-world systems and the Mori-Zwanzig formulation, we develop a novel delay-differential-equations-based deep learning framework to learn time-delayed closure parameterizations for missing dynamics. We find that our neural closure models increase the long-term predictive capabilities of existing models, and require smaller networks when using non-Markovian over Markovian closures. They efficiently represent truncated modes in reduced-order models, capture effects of subgrid-scale processes, and augment the simplification of complex physical-biogeochemical models. To empower our neural closure models framework with generalizability and interpretability, we further develop neural partial delay differential equations theory that augments low-fidelity models in their original PDE forms with both Markovian and non-Markovian closure terms parameterized with neural networks (NNs). For the first time, the melding of low-fidelity model and NNs with time-delays in the continuous spatiotemporal space followed by numerical discretization automatically provides interpretability and allows for generalizability to computational grid resolution, boundary conditions, initial conditions, and problem specific parameters. We derive the adjoint equations in the continuous form, thus, allowing implementation of our new methods across differentiable and non-differentiable computational physics codes, different machine learning frameworks, and also non-uniformly-spaced spatiotemporal training data. We also show that there exists an optimal amount of past information to incorporate, and provide methodology to learn it from data during the training process. Computational advantages associated with our frameworks are analyzed and discussed. Applications of our new neural closure modeling framework are not limited to the shown fluid and ocean experiments, but can be widely extended to other fields such as control theory, robotics, pharmacokinetic-pharmacodynamics, chemistry, economics, and biological regulatory systems.

For intelligent ocean exploration and sustainable ocean utilization, the need for smart autonomous underwater vehicles (AUVs), surface craft, and small aircraft is rapidly increasing. The challenge of creating time-optimal navigation routes for these vehicles has many applications, including ocean data collection, transportation and distribution of goods, naval operations, search and rescue, detecting marine pollution, ocean cleanup, conservation, and solar-wind-wave energy harvesting, among others. In this thesis, we employ the Massachusetts Institute of Technology – Multidisciplinary Simulation, Estimation, and Assimilation Systems (MIT-MSEAS) time-optimal path planning theory and schemes based on exact Hamilton–Jacobi partial differential equation (PDE) and Level Set methods to predict and study the sensitivity of reachable sets and time-optimal trajectories in the Portugal–Azores–Madeira region of the Northern Atlantic, for several types of missions and autonomous ocean vehicles. Specifically, using the MIT-MSEAS multi-resolution ocean modeling and data assimilation system to provide four-dimensional ocean currents in the region, we compute time-reachable sets and time-optimal paths for several missions, and examine the sensitivity to variations in vehicle type, speed, start time, voyage direction, and operating depths. Our real-data-driven multi-resolution simulation study illustrates how navigational paths vary with these parameters, and how ocean dynamics and variability in the Portuguese ocean regions affect the time optimization, as compared to direct voyages in the absence of any ocean currents. We also highlight effects of the Azores and Madeira archipelagos, differences between surface and bottom path planning, interception routes between vehicles of different speeds, and the utilization of arrival time fields in planning. Results showcase how principled path planning, integrating data-driven multi-resolution ocean modeling with exact reachability theory and numerical schemes, can assess the capabilities of ocean vehicles in the Portugal–Azores–Madeira ocean region, by predicting the fastest travel time, expected range, and optimal headings, for varied types of ocean missions.

There are many significant challenges for unmanned autonomous platforms at sea including predicting the likely scenarios for the ocean environment, quantifying regional uncertainties, and updating forecasts of the evolving dynamics using their observations. Due to the operational constraints such as onboard power, memory, bandwidth, and space limitations, efficient adaptive reduced order models (ROMs) are needed for onboard predictions. In the first part, several reduced order modeling schemes for regional ocean forecasting onboard autonomous platforms at sea are described, investigated, and evaluated. We find that Dynamic Mode Decomposition (DMD), a data-driven dimensionality reduction algorithm, can be used for accurate predictions for short periods in ocean environments. We evaluate DMD methods for ocean PE simulations by comparing and testing several schemes including domain splitting, adjusting training size, and utilizing 3D inputs. Three new approaches that combine uncertainty with DMD are also investigated and found to produce practical and accurate results, especially if we employ either an ensemble of DMD forecasts or the DMD of an ensemble of forecasts. We also demonstrate some results from projecting/compressing high-fidelity forecasts using schemes such as POD projection and K-SVD for sparse representation due to showing promise for distributing forecasts efficiently to remote vehicles. In the second part, we combine DMD methods with the GMM-DO filter to produce DMD forecasts with Bayesian data assimilation that can quickly and efficiently be computed onboard an autonomous platform. We compare the accuracy of our results to traditional DMD forecasts and DMD with Ensemble Kalman Filter (EnKF) forecast results and show that in Root Mean Square Error (RMSE) sense as well as error field sense, that the DMD with GMM-DO errors are smaller and the errors grow slower in time than the other mentioned schemes. We also showcase the DMD of the ensemble method with GMM-DO. We conclude that due to its accurate and computationally efficient results, it could be readily applied onboard autonomous platforms. Overall, our contributions developed and integrated stochastic DMD forecasts and efficient Bayesian GMM-DO updates of the DMD state and parameters, learning from the limited gappy observation data sets.

The past decade has seen an increasing use of autonomous vehicles (propelled AUVs, ocean gliders, solar-vehicles, etc.) in marine applications. For the operation of these vehicles, efficient methods for path planning are critical. Path planning, in the most general sense, corresponds to a set of rules to be provided to an autonomous robot for navigating from one configuration to another in some optimal fashion. Increasingly, having autonomous vehicles that optimally collect/harvest external fields from highly dynamic environments has grown in relevance. Autonomously maximizing the harvest in minimum time is our present path planning objective. Such optimization has numerous impactful applications. For instance, in the case of energy optimal path planning where long endurance and low power are crucial, it is important to be able to optimally harvest energy (solar, wind, wave, thermal, etc.) along the way and/or leverage the environment (winds, currents, etc.) to reduce energy expenditure. Similarly, autonomous marine cleanup or collection vehicles, tasked with harvesting plastic waste, oil spills, or seaweed fields, need to be able to plan paths that maximize the amount of material harvested in order to optimize the cleanup or collection process. In this work, we develop an exact partial differential equation-based methodology that predicts harvest-time optimal paths for autonomous vehicles navigating in dynamic environments. The governing differential equations solve the multi-objective optimization problem of navigating a vehicle autonomously in a highly dynamic flow field to any destination with the goal of minimizing travel time while also maximizing the amount harvested by the vehicle. Using Hamilton-Jacobi theory for reachability, our methodology computes the exact set of Pareto optimal solutions to the multi-objective path planning problem. This is completed by numerically solving a reachability problem for the autonomous vehicle in an augmented state space consisting of the vehicle’s position in physical space as well as its harvest state. Our approach is applicable to path planning in various environments, however we primarily present examples of navigating in dynamic ocean flows. The following cases, in particular, are studied. First, we validate our methodology using a benchmark case of planning paths through a region with a harvesting field present in a halfspace, as this case admits a semi-analytical solution that we compare to the results of our method. We next consider a more complex unsteady environment as we solve for harvest-time optimal missions in a quasi-geostrophic double-gyre ocean flow field. Following this, we provide harvest-time optimal paths to the highly relevant issue of collecting harmful algae blooms. Our final case considers an application to next generation offshore aquaculture technologies. In particular, we consider in this case path planning of an offshore moving fish farm that accounts for optimizing fish growth. Overall, we find that our exact planning equations and efficient schemes are promising to address several pressing challenges for our planet.

The present work focuses on applying the Dynamically Orthogonal Primitive Equations (DO-PE) for realistic high-resolution stochastic ocean forecasting in regions with complex ocean dynamics. In the first part, we identify and test a streamlined process to create multi-region initial conditions for the DO-PE framework, starting from temporally and spatially sparse historical data. The process presented allows us to start from a relatively small but relevant set of measured temperature and salinity historical vertical profiles (on the order of hundreds) and to generate a massive set of initial conditions (on the order of millions) in a stochastic subspace, while still ensuring that the initial statistics respect the physical processes, modeled complex dynamics, and uncertain initial conditions of the examined domain. To illustrate the methodology, two practical examples—one in the Gulf of Mexico and another in the Alboran Sea—are provided, along with a review of the ocean dynamics for each region. In the second part, we present a case study of three massive stochastic DO-PE forecasts, corresponding to ensembles of one million members, in the Gulf of Mexico region. We examine the effect of adding more dynamic DO modes (i.e., stochastic dimensions) and show that it tends to statistical convergence along with an enhancement of the uncertainty captured by the DO forecast realizations, both by increasing the variance of already existing features as well as by adding new uncertain features. We also use this case study to validate the DO-PE methodology for realistic high-resolution probabilistic ocean forecasting. We show good accuracy against equivalent deterministic simulations, starting from the same initial conditions and simulated with the same assumptions, setup, and original ocean model equations. Importantly, by comparing the reduced-order realizations against their deterministic counterparts, we show that the errors due to the DO subspace truncation are much smaller and growing slower than the fields themselves are evolving in time, both in the Root Mean Square Error (RMSE) sense as well as in the 3D multivariate ocean field sense. Based on these observations, we conclude that the DO-PE realizations closely match their full-order equivalents, thus enabling massive forecast ensembles with practically low numerical errors at a tractable computational cost.

Though computing power continues to grow quickly, our appetite to solve larger and larger problems grows just as fast. As a consequence, reduced-order modeling has become an essential technique in the computational scientist’s toolbox. By reducing the dimensionality of a system, we are able to obtain approximate solutions to otherwise intractable problems. And because the methodology we develop is sufficiently general, we may agnostically apply it to a plethora of problems, whether the high dimensionality arises due to the sheer size of the computational domain, the fine resolution we require, or stochasticity of the dynamics. In this thesis, we develop time integration schemes, called retractions, to efficiently evolve the dynamics of a system’s low-rank approximation. Through the study of differential geometry, we are able to analyze the error incurred at each time step. A novel, explicit, computationally inexpensive set of algorithms, which we call perturbative retractions, are proposed that converge to an ideal retraction that projects exactly to the manifold of fixed-rank matrices. Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low-rank approximation of the full-rank solution. We show that these high-order retractions significantly reduce the numerical error incurred over time when compared to a naive Euler forward retraction. Through test cases, we demonstrate their efficacy in the cases of matrix addition, real-time data compression, and deterministic and stochastic differential equations.

Transport of any material quantity due to background fields, i.e. advective transport, in fluid dynamical systems has been a widely studied problem. It is of crucial importance in classical fluid mechanics, geophysical flows, micro and nanofluidics, and biological flows. Even though mathematical models that thoroughly describe such transport exist, the inherent nonlinearities and the high dimensionality of complex fluid systems make it very challenging to develop the capabilities to accurately compute and characterize advective material transport. We systematically study the problems of predicting, uncovering, and learning the principal features of advective material transport in this work. The specific objectives of this thesis are to: (i) develop and apply new numerical methodologies to compute the solutions of advective transport equations with minimal errors and theoretical guarantees, (ii) propose and theoretically investigate novel criteria to detect sets of fluid parcels that remain the most coherent / incoherent throughout an extended time interval to quantify fluid mixing, and (iii) extend and develop new machine learning methods to infer and predict the transport features, given snapshot data about passive and active material transport.

The first part of this work deals with the development of the PDE-based ‘method of flow map composition’, which is a novel methodology to compute the solutions of the partial differential equation describing classical advective and advective-diffusive-reactive transport. The method of composition yields solutions almost devoid of numerical errors, and is readily parallelizable. It can compute more accurate solutions in less time than traditional numerical methods. We also complete a comprehensive theoretical analysis and analytically obtain the value of the numerical timestep that minimizes the net error. The method of flow map composition is extensively benchmarked and its applications are demonstrated in several analytical flow fields and realistic data-assimilative ocean plume simulations.

We then utilize the method of flow map composition to analyze Lagrangian material coherence in dynamic open domains. We develop new theory and schemes to efficiently predict the sets of fluid parcels that either remain the most or the least coherent over an extended amount of time. We also prove that these material sets are the ones to maximally resist advective stretching and diffusive transport. Thus, they are of significant importance in understanding the dynamics of fluid mixing and form the skeleton of material transport in unsteady fluid systems. The developed theory and numerical methods are utilized to analyze Lagrangian coherence in analytical and realistic scenarios. We emphasize realistic marine flows with multiple time-dependent inlets and outlets, and demonstrate applications in diverse dynamical regimes and several open ocean regions.

The final part of this work investigates the machine inference and prediction of the principal transport features from snapshot data about the transport of some material quantity. Our goals include machine learning the underlying advective transport features, coherent / incoherent sets, and attracting and repelling manifolds, given the snapshots of advective and advective-diffusive material fields. We also infer and predict high resolution transport features by optimally combining coarse resolution snapshot data with localized high resolution trajectory data. To achieve these goals, we use and extend recurrent neural networks, including a combination of long short-term memory networks with hypernetworks. We develop methods that leverage our knowledge of the physical system in the design and architecture of the neural network and enforce the known constraints that the results must satisfy (e.g. mass conservation) in the training loss function. This allows us to train the networks only with partial supervision, without samples of the expected output fields, and still infer and predict physically consistent quantities. The developed theory, methods, and computational software are analyzed, validated, and applied to a variety of analytical and realistic fluid flows, including high-resolution ocean transports in the Western Mediterranean Sea.

We develop an exact partial differential equation-based methodology that predicts time-energy optimal paths for autonomous vehicles navigating in dynamic environments. The differential equations solve the multi-objective optimization problem of navigating a vehicle autonomously in a dynamic flow field to any destination with the goal of minimizing travel time and energy use. Based on Hamilton-Jacobi theory for reachability and the level set method, the methodology computes the exact Pareto optimal solutions to the multi-objective path planning problem, numerically solving the equations governing time-energy reachability fronts and optimal paths. Our approach is applicable to path planning in various scenarios, however we primarily present examples of navigating in dynamic marine environments. First, we validate the methodology through a benchmark case of crossing a steady front (a highway flow) for which we compare our results to semi-analytical optimal path solutions. We then consider more complex unsteady environments and solve for time-energy optimal missions in a quasi-geostrophic double-gyre ocean flow field.

The rapidly-growing computational power and the increasing capability of uncertainty quantification, statistical inference, and machine learning have opened up new opportunities for utilizing data to assist, identify and refine physical models. In this thesis, we focus on Bayesian learning for a particular class of models: high-dimensional nonlinear dynamical systems, which have been commonly used to predict a wide range of transient phenomena including fluid flows, heat transfer, biogeochemical dynamics, and other advection-diffusion-reaction-based transport processes. Even though such models often express the differential form of fundamental laws, they commonly contain uncertainty in their initial and boundary values, parameters, forcing and even formulation. Learning such components from sparse observation data by principled Bayesian inference is very challenging due to the systems’ high-dimensionality and nonlinearity.

We systematically study the theoretical and algorithmic properties of a Bayesian learning methodology built upon previous efforts in our group to address this challenge. Our systematic study breaks down into the three hierarchical components of the Bayesian learning and we develop new numerical schemes for each. The first component is on uncertainty quantification for stochastic dynamical systems and fluid flows. We study dynamic low-rank approximations using the dynamically orthogonal (DO) equations including accuracy and computational costs, and develop new numerical schemes for re-orthonormalization, adaptive subspace augmentation, residual-driven closure, and stochastic Navier-Stokes integration. The second part is on Bayesian data assimilation, where we study the properties of and connections among the different families of nonlinear and non-Gaussian filters. We derive an ensemble square-root filter based on minimal-correction second-moment matching that works especially well under the adversity of small ensemble size, sparse observations and chaotic dynamics. We also obtain a localization technique for filtering with high-dimensional systems that can be applied to nonlinear non-Gaussian inference with both brute force Monte Carlo (MC) and reduced subspace modeling in a unified way. Furthermore, we develop a mutual-information-based adaptive sampling strategy for filtering to identify the most informative observations with respect to the state variables and/or parameters, utilizing the sub-modularity of mutual information due to the conditional independence of observation noise. The third part is on active Bayesian model learning, where we have a discrete set of candidate dynamical models and we infer the model formulation that best explains the data using principled Bayesian learning. To predict the observations that are most useful to learn the model formulation, we further extend the above adaptive sampling strategy to identify the data that are expected to be most informative with respect to both state variables and the uncertain model identity.

To investigate and showcase the effectiveness and efficiency of our theoretical and numerical advances for uncertainty quantification, Bayesian data assimilation, and active Bayesian learning with stochastic nonlinear high-dimensional dynamical systems, we apply our dynamic data-driven reduced subspace approach to several dynamical systems and compare our results against those of brute force MC and other existing methods. Specifically, we analyze our advances using several drastically different dynamical regimes modeled by the nonlinear Lorenz-96 ordinary differential equations as well as turbulent bottom gravity current dynamics modeled by the 2-D unsteady incompressible Reynolds-averaged Navier-Stokes (RANS) partial differential equations. We compare the accuracy, efficiency, and robustness of different methodologies and algorithms. With the Lorenz-96 system, we show how the performance differs under periodic, weakly chaotic, and very chaotic dynamics and under different observation layouts. With the bottom gravity current dynamics, we show how model parameters, domain geometries, initial fields, and boundary forcing formulations can be identified and how the Bayesian methodology performs when the candidate model space does not contain the true model. The results indicate that our active Bayesian learning framework can better infer the state variables and dynamical model identity with fewer observations than many alternative approaches in the literature.

Developing accurate and computationally efficient models for ocean acoustics is inherently challenging due to several factors including the complex physical processes and the need to provide results on a large range of scales. Furthermore, the ocean itself is an inherently dynamic environment within the multiple scales. Even if we could measure the exact properties at a specific instant, the ocean will continue to change in the smallest temporal scales, ever increasing the uncertainty in the ocean prediction. In this work, we explore ocean acoustic prediction from the basics of the wave equation and its derivation. We then explain the deterministic implementations of the Parabolic Equation, Ray Theory, and Level Sets methods for ocean acoustic computation. We investigate methods for evolving stochastic fields using direct Monte Carlo, Empirical Orthogonal Functions, and adaptive Dynamically Orthogonal (DO) differential equations. As we evaluate the potential of Reduced-Order Models for stochastic ocean acoustics prediction, for the first time, we derive and implement the stochastic DO differential equations for Ray Tracing (DO-Ray), starting from the differential equations of Ray theory. With a stochastic DO-Ray implementation, we can start from non-Gaussian environmental uncertainties and compute the stochastic acoustic ray fields in a reduced order fashion, all while preserving the complex statistics of the ocean environment and the nonlinear relations with stochastic ray tracing. We outline a deterministic Ray-Tracing model, validate our implementation, and perform Monte Carlo stochastic computation as a basis for comparison. We then present the stochastic DO-Ray methodology with detailed derivations. We develop varied algorithms and discuss implementation challenges and solutions, using again direct Monte Carlo for comparison. We apply the stochastic DO-Ray methodology to three idealized cases of stochastic sound-speed profiles (SSPs): constant-gradients, uncertain deep-sound channel, and a varied sonic layer depth. Through this implementation with non-Gaussian examples, we observe the ability to represent the stochastic ray trace field in a reduced order fashion.

We address the problem of finding the closest matrix Ũ to a given U under the constraint that a prescribed second-moment matrix P̃ must be matched, i.e. Ũ^TŨ=P̃. We obtain a closed-form formula for the unique global optimizer Ũ for the full-rank case, which is related to U by an SPD (symmetric positive definite) linear transform. This result is generalized to rank-deficient cases as well as to infinite dimensions. We highlight the geometric intuition behind the theory and study the problem’s rich connections to minimum congruence transform, generalized polar decomposition, optimal transport, and rank-deficient data assimilation. In the special case of P̃=I, minimum-correction second-moment matching reduces to the well-studied optimal orthonormalization problem. We investigate the general strategies for numerically computing the optimizer, analyze existing polar decomposition and matrix square root algorithms. More importantly, we modify and stabilize two Newton iterations previously deemed unstable for computing the matrix square root, which can now be used to efficiently compute both the orthogonal polar factor and the SPD square root. We then verify the higher performance of the various new algorithms using benchmark cases with randomly generated matrices. Lastly, we complete two applications for the stochastic Lorenz-96 dynamical system in a chaotic regime. In reduced subspace tracking using dynamically orthogonal equations, we maintain the numerical orthonormality and continuity of time-varying base vectors. In ensemble square root filtering for data assimilation, the prior samples are transformed into posterior ones by matching the covariance given by the Kalman update while also minimizing the corrections to the prior samples.

Grand challenges in ocean acoustic propagation are to accurately capture the dynamic environmental uncertainties and to predict the evolving probability density function of stochastic acoustic waves. This is due to the complex ocean physics and acoustics dynamics, nonlinearities, multiple scales, and large dimensions. There are several sources of uncertainty including: the initial and boundary conditions of the ocean physics and acoustic dynamics, the bathymetry and seabed fields; the models parameters; and, the models themselves. In the present work, we start addressing these challenges by deriving, implementing and verifying new optimally-reduced Dynamically Orthogonal (DO) differential equations that govern the propagation of stochastic acoustic waves for underwater sound propagation in an uncertain ocean environment. The developed methodology allows modeling environmental uncertainties in a rigorous probabilistic framework and predicting the uncertainties of acoustic fields, fully respecting the nonlinear governing equations and non-Gaussian statistics of the sound speed and acoustic state variables. The methodology is applied to the standard narrow-angle parabolic equation and is utilized to predict acoustic field uncertainties for three new stochastic idealized test cases: (1) an uncertain Pekeris waveguide with penetrable bottom, (2) an uncertain horizontal interface problem, and (3) an uncertain range-dependent sloping interface problem. For the first case, the solutions of the DO acoustic equations are validated against those obtained using standard Monte Carlo sampling. The second test case showcases results for predicting acoustic field probabilities due to uncertainties in the location of a sound speed channel. For the third test case, the advantages of the DO acoustic equations in predicting uncertainties in complex range-dependent environments are highlighted.

This work focuses on developing efficient and robust implementation methods for hybridizable discontinuous Galerkin (HDG) schemes for fluid and ocean dynamics. In the first part, we compare choices in weak formulations and their numerical consequences. We address details in making the leap from the mathematical formulation to the implementation, including the different spaces and mappings, discretization of the integral operators, boundary conditions, and assembly of the linear systems. We provide a flexible mapping procedure amenable to both quadrature-free and quadrature-based discretizations, and compare the accuracy of the two on different problem geometries. We verify the quadrature-free approach, demonstrating that optimal orders of convergence can be obtained, even on non-affine and curvilinear geometries. The second part of the work investigates the scalability of HDG schemes, identifying memory and time-to-solution bottlenecks. The form of the quadrature-free integral operators is exploited to develop a novel and efficient matrix-free approach to solving the global linear system that arises from HDG discretizations. Additional manipulations to improve numerical robustness are discussed. To mitigate the complexity of the implementation, we provide an automated and computationally efficient verification procedure for the HDG methodologies discussed, using a hierarchical approach to provide diagnostic information and isolate problems. Finally, challenges related to the effective visualization of high-order, discontinuous HDG-FEM data for fluid and ocean applications are illustrated and strategies are provided to address them.

Comprehensive spatiotemporal modeling and forecasting systems for ocean dynamics necessitate robust and efficient data delivery and visualization techniques. The multidisciplinary simulation, estimation, and assimilation systems group at MIT (MSEAS) focuses on capturing and predicting diverse ocean dynamics, including physics, acoustics, and biology on varied scales, thereby developing new methods for multi-resolution ocean prediction and analysis, including data generation and assimilation. The group has primarily used non-interactive ocean plots to visualize its simulated and measured data. Although these maps and sections allow for analysis of ocean physics and the underlying numerical schemes, more interactive maps provide more user control over depicted data, allowing easier study and pattern identification on multiple scales. Integrating static and geospatial data in dynamic visualization creates a heightened viewpoint for analysis, enhances ocean monitoring and prediction, and contributes to building scientific knowledge. This thesis focuses on explaining the motivation behind and the methodologies applied in designing these interactive maps.

This thesis demonstrates the use of exact equations to predict time-optimal mission plans for a marine vehicle that visits a number of locations in a given dynamic ocean current field. The missions demonstrated begin and end in the same location and visit a finite number of locations or waypoints in the minimal time; this problem bears close resemblance to that of the classic “traveling salesman,” albeit with the added complexity of a continuously changing flow field. The paths, or “legs,” between all goal waypoints are generated by numerically solving exact time-optimal path planning level-set differential equations. The equations grow a reachability front from the starting location in all directions. Whenever the front reaches a waypoint, a new reachability front is immediately started from that location. This process continues until one set of reachability fronts has reached all goal waypoints and has returned to the original location. The time-optimal path for the entire mission is then obtained by trajectory backtracking, going through the optimal set of reachability fields in reverse order. Due to the spatial and temporal dynamics, a varying start time results in different paths and durations for each leg and requires all permutations of travel to be calculated. Even though the method is very efficient and the optimal path can be computed serially in real-time for common naval operations, for additional computational speed, a high-performance computing cluster was used to solve the level set calculations in parallel. This method is first applied to several hypothetical missions. The method and distributed computational solver are then validated for naval applications using an operational multi-resolution ocean modeling system of real-world current fields for the complex Philippines Archipelago region. Because the method calculates the global optimum, it serves two purposes. It can be used in its present form to plan multi-waypoint missions offline in conjunction with a predictive ocean current modeling system, or it can be used as a litmus test for approximate future solutions to the traveling salesman problem in dynamic flow fields.

The coastal ocean is a prime example of multiscale nonlinear fluid dynamics. Ocean fields in such regions are complex, with multiple spatial and temporal scales and nonstationary heterogeneous statistics. Due to the limited measurements, there are multiple sources of uncertainties, including the initial conditions, boundary conditions, forcing, parameters, and even the model parameterizations and equations themselves. To reduce uncertainties and allow long-duration measurements, the energy consumption of ocean observing platforms need to be optimized. Predicting the distributions of reachable regions, time-optimal paths, and risk-optimal paths in uncertain, strong and dynamic flows is also essential for their optimal and safe operations. Motivated by the above needs, the objectives of this thesis are to develop and apply the theory, schemes, and computational systems for: (i) Dynamically Orthogonal ocean primitive-equations with a nonlinear free-surface, in order to quantify uncertainties and predict probabilities for four-dimensional (time and 3-d in space) coastal ocean states, respecting their nonlinear governing equations and non-Gaussian statistics; (ii) Stochastic Dynamically Orthogonal level-set optimization to rigorously incorporate realistic ocean flow forecasts and plan energy-optimal paths of autonomous agents in coastal regions; (iii) Probabilistic predictions of reachability, time-optimal paths and risk-optimal paths in uncertain, strong and dynamic flows.

For the first objective, we further develop and implement our Dynamically Orthogonal (DO) numerical schemes for idealized and realistic ocean primitive equations with a nonlinear free-surface. The theoretical extensions necessary for the free-surface are completed. DO schemes are researched and DO terms, functions, and operations are implemented, focusing on: state variable choices; DO norms; DO condition for flows with a dynamic free-surface; diagnostic DO equations for pressure, barotropic velocities and density terms; non-polynomial nonlinearities; semi-implicit time-stepping schemes; and re-orthonormalization consistent with leap-frog time marching. We apply the new DO schemes, as well as their theoretical extensions and efficient serial implementation to forecast idealized-to-realistic stochastic
coastal ocean dynamics. For the realistic simulations, probabilistic predictions for the Middle Atlantic Bight region, Northwest Atlantic, and northern Indian ocean are showcased.

For the second objective, we integrate data-driven ocean modeling with our stochastic DO level-set optimization to compute and study energy-optimal paths, speeds, and headings for ocean vehicles in the Middle Atlantic Bight region. We compute the energy-optimal paths from among exact time-optimal paths. For ocean currents, we utilize a data-assimilative multiscale re-analysis, combining observations with implicit two-way nested multi-resolution primitive-equation simulations of the tidal-to-mesoscale dynamics in the region. We solve the reduced-order stochastic DO level-set partial differential equations (PDEs) to compute the joint probability of minimum arrival-time, vehicle-speed time-series, and total energy utilized. For each arrival time, we then select the vehicle-speed time-series that minimize the total energy utilization from the marginal probability of vehicle-speed and total energy. The corresponding energy-optimal path and headings be obtained through a particle backtracking equation. For the missions considered, we analyze the effects of the regional tidal currents, strong wind events, coastal jets, shelfbreak front, and other local circulations on the energyoptimal paths.

For the third objective, we develop and apply stochastic level-set PDEs that govern the stochastic time-optimal reachability fronts and paths for vehicles in uncertain, strong, and dynamic flow fields. To solve these equations efficiently, we again employ their dynamically orthogonal reduced-order projections. We develop the theory and schemes for risk-optimal planning by combining decision theory with our stochastic time-optimal planning equations. The risk-optimal planning proceeds in three steps: (i) obtain predictions of the probability distribution of environmental flows, (ii) obtain predictions of the distribution of exact timeoptimal paths for the forecast flow distribution, and (iii) compute and minimize the risk of following these uncertain time-optimal paths. We utilize the new equations to complete stochastic reachability, time-optimal and risk-optimal path planning in varied stochastic quasi-geostrophic flows. The effects of the flow uncertainty on the reachability fronts and time-optimal paths is explained. The risks of following each exact time-optimal path is evaluated and risk-optimal paths are computed for different risk tolerance measures. Key properties of the risk-optimal planning are finally discussed.

Theoretically, the present methodologies are PDE-based and compute stochastic ocean fields, and optimal path predictions without heuristics. Computationally, they are several orders of magnitude faster than direct Monte Carlo.

Such technologies have several commercial and societal applications. Specifically, the probabilistic ocean predictions can be input to a technical decision aide for a sustainable fisheries co-management program in India, which has the potential to provide environment friendly livelihoods to millions of marginal fishermen. The risk-optimal path planning equations can be employed in real-time for efficient ship routing to reduce greenhouse gas emissions and save operational costs.

Ocean currents transport a variety of natural (e.g. water masses, phytoplankton, zooplankton, sediments, etc.) and man-made materials (e.g. pollutants, floating debris, particulate matter, etc.). Understanding such uncertain Lagrangian transport is imperative for reducing environmental damage due to natural hazards and for allowing rigorous risk analysis and effective search and rescue. While secondary variables and trajectories have classically been used for the analyses of such transports, Lagrangian Coherent Structures (LCSs) provide a robust and objective description of the important material lines. To ensure accurate and useful Lagrangian hazard scenario predictions and prevention, the first goal of this thesis is to obtain accurate probabilistic prediction of the underlying stochastic velocity fields using the Dynamically Orthogonal (DO) approach. The second goal is to merge data from both Eulerian and Lagrangian observations with predictions such that the whole information content of observations is utilized.

In the first part of this thesis, we develop high-order numerical schemes for the DO equations that ensure efficiency, accuracy, stability, and consistency between the Monte Carlo (MC) and DO solutions. We discuss the numerical challenges in applying the DO equations to the unsteady stochastic Navier-Stokes equations. In order to maintain consistent evaluation of advection terms, we utilize linear centered advection schemes with fully explicit and linear Shapiro filters. We then discuss how to combine the semi-implicit projection method with new high order implicit-explicit (IMEX) linear multi-step and multistage IMEX-RK time marching schemes for the coupled DO equations to ensure further stability and accuracy. We also review efficient numerical re-orthonormalization strategies during time marching. We showcase our results with stochastic test cases of stochastic passive tracer advection in a deterministic swirl flow, stochastic flow past a cylinder, and stochastic lid-driven cavity flow. We show that our schemes improve the consistency between reconstructed DO realizations and the corresponding MC realizations, and that we achieve the expected order of accuracy.

In the second part of the work, we first undertake a study of different Lagrangian instruments and outline how the DO methodology can be applied to obtain Lagrangian variables of stochastic flow maps and LCS in uncertain flows. We then review existing methods for Bayesian Lagrangian data assimilation (DA). Disadvantages of earlier methods include the use of approximate measurement models to directly link Lagrangian variables with Eulerian variables, the challenges in respecting the Lagrangian nature of variables, and the assumptions of linearity or of Gaussian statistics during prediction or assimilation. To overcome these, we discuss how the Gaussian Mixture Model (GMM) DO Filter can be extended to fully coupled Eulerian-Lagrangian data assimilation. We define an augmented state vector of the Eulerian and Lagrangian state variables that directly exploits the full mutual information and complete the Bayesian DA in the joint Eulerian-Lagrangian stochastic subspace. Results of such coupled Eulerian-Lagrangian DA are discussed using test cases based on a double gyre flow with random frequency.

In this work novel high-order hybridizable discontinuous Galerkin (HDG) projection methods are further developed for ocean dynamics and geophysical fluid predictions. We investigate the effects of the HDG stabilization parameter for both the momentum equation as well as tracer diffusion. We also make a correction to our singularity treatment algorithm for nailing down a numerically consistent and unique solution to the pressure Poisson equation with homogeneous Neumann boundary conditions everywhere along the boundary. Extensive numerical results using physically realistic ocean flows are presented to verify the HDG projection methods, including the formation of internal wave beams over a shallow but abrupt seamount, the generation of internal solitary waves from stratified oscillatory flow over steep topography, and the circulation of bottom gravity currents down a slope. Additionally, we investigate the implementation of open boundary conditions for finite element methods and present results in the context of our ocean simulations. Through this work we present the hybridizable discontinuous Galerkin projection methods as a viable and competitive alternative for large-scale, realistic ocean modeling.

Autonomous underwater vehicles (AUVs) are a valuable resource in several oceanic applications such as security, surveillance and data collection for ocean prediction. These vehicles typically travel at speeds comparable to ocean currents, and their movement is significantly affected by these dynamic currents. Further, the speed of currents may vary greatly with depth. Hence, path planning to generate safe and fast vehicle trajectories in such a three-dimensional environment becomes crucial for the successful operation of these vehicles. In addition, many marine vehicles can only move in specific directions and with a speed that is dependent on the direction of travel. Such constraints must be respected in order to plan safe and optimal paths.

Thus, our motivation in this thesis is to study path planning for vehicles with and without motion constraints in three-dimensional dynamic flow-fields. We utilize the time-optimal path planning methodology given by Lolla et al. (2012) for this purpose.

In this thesis, we first review some existing path planning methods (both in two and three-dimensional settings). Then, we discuss the theoretical basis of the rigorous partial differential equation based methodology that is utilized in order to plan safe and optimal paths. This is followed by an elaborate discussion about the application of this methodology to the various types of marine vehicles. We then look at the robust and accurate numerical methods developed in order to solve the governing equations for the path planning methodology with high accuracy in real ocean domains. We illustrate the working and capabilities of our path planning algorithm by means of a number of applications. First we study some benchmark examples with known analytical solutions. Second, we look at more complex flow-fields that analytically model different oceanic flows. Finally, we look at the path planning for different types of marine vehicles in a realistic ocean domain to illustrate the capabilities of the path planning methodology and the developed numerical framework.

This work focuses on developing theory and methodologies for the analysis of material transport in stochastic fluid flows. In a first part, two dominant classes of techniques for extracting Lagrangian Coherent Structures are reviewed and compared and some improvements are suggested for their pragmatic applications on realistic high-dimensional deterministic ocean velocity fields. In the stochastic case, estimating the uncertain Lagrangian motion can require to evaluate an ensemble of realizations of the flow map associated with a random velocity flow field, or equivalently realizations of the solution of a related transport partial differential equation. The Dynamically Orthogonal (DO) approximation is applied as an efficient model order reduction technique to solve this stochastic advection equation. With the goal of developing new rigorous reduced-order advection schemes, the second part of this work investigates the mathematical foundations of the method. Riemannian geometry providing an appropriate setting, a framework free of tensor notations is used to analyze the embedded geometry of three popular matrix manifolds, namely the fixed rank manifold, the Stiefel manifold and the isospectral manifold. Their extrinsic curvatures are characterized and computed through the study of the Weingarten map. As a spectacular by-product, explicit formulas are found for the differential of the truncated Singular Value Decomposition, of the Polar Decomposition, and of the eigenspaces of a time dependent symmetric matrix. Convergent gradient flows that achieve related algebraic operations are provided. A generalization of this framework to the non-Euclidean case is provided, allowing to derive analogous formulas and dynamical systems for tracking the eigenspaces of non-symmetric matrices. In the geometric setting, the DO approximation is a particular case of projected dynamical systems, that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. It is obtained that the error committed by the DO approximation is controlled under the minimal geometric condition that the
original solution stays close to the low-rank manifold. The last part of the work focuses on the practical implementation of the DO methodology for the stochastic advection equation. Fully linear, explicit central schemes are selected to ensure stability, accuracy and efficiency of the method. Riemannian matrix optimization is applied for the dynamic evaluation of the dominant SVD of a given matrix and is integrated to the DO time-stepping. Finally the technique is illustrated numerically on the uncertainty quantification of the Lagrangian motion of two bi-dimensional benchmark flows.

When humans or robots operate in complex dynamic environments, the planning of paths and the collection of observations are basic, indispensable problems. In the oceanic and atmospheric environments, the concurrent use of multiple mobile sensing platforms in unmanned missions is growing very rapidly. Opportunities for a paradigm shift in the science of autonomy involve the development of fundamental theories to optimally collect information, learn, collaborate and make decisions under uncertainty while persistently adapting to and utilizing the dynamic environment. To address such pressing needs, this thesis derives governing equations and develops rigorous methodologies for optimal path planning and optimal sampling using collaborative swarms of autonomous mobile platforms. The application focus is the coastal ocean where currents can be much larger than platform speeds, but the fundamental results also apply to other dynamic environments. We first undertake a theoretical synthesis of minimum-time control of vehicles operating in general dynamic flows. Using various ideas rooted in non-smooth calculus, we prove that an unsteady Hamilton-Jacobi equation governs the forward reachable sets in any type of Lipschitz-continuous flow. Next, we show that with a suitable modification to the Hamiltonian, the results can be rigorously generalized to perform time-optimal path planning with anisotropic motion constraints and with moving obstacles and unsafe ‘forbidden’ regions. We then derive a level-set methodology for distance-based coordination of swarms of vehicles operating in minimum time within strong and dynamic ocean currents. The results are illustrated for varied fluid and ocean flow simulations. Finally, the new path planning system is applied to swarms of vehicles operating in the complex geometry of the Philippine Archipelago, utilizing realistic multi-scale current predictions from a data-assimilative ocean modeling system. In the second part of the thesis, we derive a theory for adaptive sampling that exploits the governing nonlinear dynamics of the system and captures the non-Gaussian structure of the random state fields. Optimal observation locations are determined by maximizing the mutual information between the candidate observations and the variables of interest. We develop a novel Bayesian smoother for high-dimensional continuous stochastic fields governed by general nonlinear dynamics. This smoother combines the adaptive reduced-order Dynamically-Orthogonal equations with Gaussian Mixture Models, extending linearized Gaussian backward pass updates to a nonlinear, non-Gaussian setting. The Bayesian information transfer, both forward and backward in time, is efficiently carried out in the evolving dominant stochastic subspace. Building on the foundations of the smoother, we then derive an efficient technique to quantify the spatially and temporally varying mutual information field in general nonlinear dynamical systems. The globally optimal sequence of future sampling locations is rigorously determined by a novel dynamic programming approach that combines this computation of mutual information fields with the predictions of the forward reachable set. All the results are exemplified and their performance is quantitatively assessed using a variety of simulated fluid and ocean flows. The above novel theories and schemes are integrated so as to provide real-time computational intelligence for collaborative swarms of autonomous sensing vehicles. The integrated system guides groups of vehicles along predicted optimal trajectories and continuously improves field estimates as the observations predicted to be most informative are collected and assimilated. The optimal sampling locations and optimal trajectories are continuously forecast, all in an autonomous and coordinated fashion.

The primary contributions of this thesis include the first stages of development of a 2D, finitevolume, non-hydrostatic, sigma-coordinate code and beginning to apply the Dynamically Orthogonal field equations to study the sensitivity of internal tides to perturbations in the density field. First, we ensure that the 2D Finite Volume (2DFV) code that we use can accurately capture non-hydrostatic internal tides since these dynamics have not yet been carefully evaluated for accuracy in this framework. We find that, for low-aspect ratio topographies, the -coordinate mesh in the 2DFV code produces numerical artifacts near the bathymetry. To ameliorate these staircasing effects, and to develop the framework towards a moving mesh with free-surface dynamics, we have begun to implement a non-hydrostatic sigma-coordinate framework which significantly improves the representation of the internal tides for low-aspect ratio topographies. Finally we investigate the applicability of stochastic density perturbations in an internal tide field. We utilize the Dynamically Orthogonal field equations for this investigation because they achieve substantial model order reduction over ensemble Monte-Carlo methods.

The focus of this research is to study the uncertainties forecast by multi-resolution ocean models and quantify how those uncertainties affect the pressure fields estimated by coupled ocean models. The quantified uncertainty can then be used to provide enhanced sonar performance predictions for tactical decision aides. High fidelity robust modeling of the oceans can resolve various scale processes from tidal shifts to mesoscale phenomena. These ocean models can be coupled with acoustic models that account for variations in the ocean environment and complex bathymetry to yield accurate acoustic field representations that are both range and time independent. Utilizing the MIT Multidisciplinary Environmental Assimilation System (MSEAS) implicit two-way nested primitive-equation ocean model and Error Subspace Statistical Estimation scheme (ESSE), coupled with three-dimensional-inspace (3D) parabolic equation acoustic models, we conduct a study to understand and determine the effects of ocean state uncertainty on the acoustic transmission loss. The region of study is focused on the ocean waters surrounding Taiwan in the East China Sea. This region contains complex ocean dynamics and topography along the critical shelf-break region where the ocean acoustic interaction is driven by several uncertainties. The resulting ocean acoustic uncertainty is modeled and analyzed to quantify sonar performance and uncertainty characteristics with respect to submarine counter detection. Utilizing cluster based data analysis techniques, the relationship between the resulting acoustic field and the uncertainty in the ocean model can be characterized. Furthermore, the dynamic transitioning between the clustered acoustic states can be modeled as Markov processes. This analysis can be used to enhance not only submarine counter detection aides, but it may also be used for several applications to enhance understanding of the capabilities and behavior of uncertainties of acoustic systems operating in the complex ocean environment.

The pressure-correction projection method for the incompressible Navier-Stokes equation is approached as a preconditioned Richardson iterative method for the pressure- Schur complement equation. Typical pressure correction methods perform only one iteration and suffer from a splitting error that results in a spurious numerical boundary layer, and a limited order of convergence in time. We investigate the benefit of performing more than one iteration. We show that that not only performing more iterations attenuates the effects of the splitting error, but also that it can be more computationally efficient than reducing the time step, for the same level of accuracy. We also devise a stopping criterion that helps achieve a desired order of temporal convergence, and implement our method with multi-stage and multi-step time integration schemes. In order to further reduce the computational cost of our iterative method, we combine it with an Aitken acceleration scheme. Our theoretical results are validated and illustrated by numerical test cases for the Stokes and Navier-Stokes equations, using Implicit-Explicit Backwards Difference Formula and Runge-Kutta time integration solvers. The test cases comprises a now classical manufactured solution in the projection method literature and a modified version of a more recently proposed manufactured solution.

Path-planning has many applications, ranging from self-driving cars to flying drones, and to our daily commute to work. Path-planning for autonomous underwater vehicles presents an interesting problem: the ocean flow is dynamic and unsteady. Additionally, we may not have perfect knowledge of the ocean flow. Our goal is to develop a rigorous and computationally efficient methodology to perform path-planning in uncertain flow fields. We obtain new stochastic Dynamically Orthogonal (DO) Level Set equations to account for uncertainty in the flow field. We first review existing path-planning work: time-optimal path planning using the level set method, and energy-optimal path planning using stochastic DO level set equations. We build on these methods by treating the velocity field as a stochastic variable and deriving new stochastic DO level set equations. We use the new DO equations to simulate a simple canonical flow, the stochastic highway. We verify that our results are correct by comparing to corresponding Monte Carlo results. We explore novel methods of visualizing the results of the equations. Finally we apply our methodology to an idealized ocean simulation using Double-Gyre flows.

The growing use of autonomous underwater vehicles and underwater gliders for a variety of applications gives rise to new requirements in the operation of these vehicles. One such important requirement is optimization of energy required for undertaking missions that will enable longer endurance and lower operational costs. Our goal in this thesis is to develop a computationally efficient, and rigorous methodology that can predict energy-optimal paths from among all time-optimal paths to complete an underwater mission. For this, we develop rigorous a new stochastic Dynamically Orthogonal Level Set optimization methodology. In our thesis, after a review of existing path planning methodologies with a focus on energy optimality, we present the background of time-optimal path planning using the level set method. We then lay out the questions that inspired the present thesis, provide the goal of the current work and explain an extension of the time-optimal path planning methodology to the time-optimal path planning in the case of variable nominal engine thrust. We then proceed to state the problem statement formally. Thereafter, we develop the new methodology for solving the optimization problem through stochastic optimization and derive new Dynamically Orthogonal Level Set Field equations. We then carefully present different approaches to handle the non-polynomial non-linearity in the stochastic Level Set Hamilton-Jacobi equations and also discuss the computational efficiency of the algorithm. We then illustrate the inner-workings and nuances of our new stochastic DO level set energy optimal path planning algorithm through two simple, yet important, canonical steady flows that simulate a steady front and a steady eddy. We formulate a double energy-time minimization to obtain a semi-analytical energy optimal path for the steady front crossing test case and compare the results to these of our stochastic DO level set scheme. We then apply our methodology to an idealized ocean simulation using Double Gyre flows, and finally show an application with real ocean data for completing a mission in the Middle Atlantic Bight and New Jersey Shelf/Hudson Canyon region.

A path-planning methodology that takes into account sea state fields, specifically wind forcing, is discussed and exemplified in this thesis. This general methodology has been explored by the Multidisciplinary Simulation, Estimation, and Assimilation Systems group (MSEAS) at MIT, however this is the first instance of wind effects being taken into account. Previous research explored vessels and isotropy, where the nominal speed of the vessel is uniform in all directions. This thesis explores the non-isotropic case, where the maximum speed of the vessel varies with direction, such as a sailboat. Our goal in this work is to predict the time-optimal path between a set of coordinates, taking into account flow currents and wind speeds. This thesis reviews the literature on a modified level set method that governs the path in any continuous flow to minimize travel time. This new level set method, pioneered by MSEAS, evolves a front from the starting coordinate until any point on that front reaches the destination. The vehicles optimal path is then gained by solving a particle back tracking equation. This methodology is general and applicable to any vehicle, ranging from underwater vessels to aircraft, as it rigorously takes into account the advection effects due to any type of environmental flow fields such as time-dependent currents and dynamic wind fields.

Accurate modeling of physical and biogeochemical dynamics in coastal ocean regions is required for multiple scientific and societal applications, covering a wide range of time and space scales. However, in light of the strong nonlinearities observed in coastal regions and in biological processes, such modeling is challenging. An important subject that has been largely overlooked is the numerical requirements for regional ocean simulation studies. Major objectives of this thesis are to address such computational questions for non-hydrostatic multiscale flows and for biogeochemical interactions, and to derive and develop numerical schemes that meet these requirements, utilizing the latest advances in computational fluid dynamics. We are interested in studying nonlinear, transient, and multiscale ocean dynamics over complex geometries with steep bathymetry and intricate coastlines, from sub-mesoscales to basin-scales. These dynamical interests, when combined with our requirements for accurate, efficient and flexible ocean modeling, led us to develop new variable resolution, higher-order and non-hydrostatic ocean modeling schemes. Specifically, we derived, developed and applied new numerical schemes based on the novel hybrid discontinuous Galerkin (HDG) method in combination with projection methods. The new numerical schemes are first derived for the Navier-Stokes equations. To ensure mass conservation, we define numerical fluxes that are consistent with the discrete divergence equation. To improve stability and accuracy, we derive a consistent HDG stability parameter for the pressure-correction equation. We also apply a new boundary condition for the pressure-corrector, and show the form and origin of the projection method’s time-splitting error for a case with implicit diffusion and explicit advection. Our scheme is implemented for arbitrary, mixed-element unstructured grids using a novel quadrature-free integration method for a nodal basis, which is consistent with the HDG method. To prevent numerical oscillations, we design a selective high-order nodal limiter. We demonstrate the correctness of our new schemes using a tracer advection benchmark, a manufactured solution for the steady diffusion and stokes equations, and the 2D lock-exchange problem. These numerical schemes are then extended for non-hydrostatic, free-surface, variable-density regional ocean dynamics. The time-splitting procedure using projection methods is derived for non-hydrostatic or hydrostatic, and nonlinear free-surface or rigid-lid, versions of the model. We also derive consistent HDG stability parameters for the free-surface and non-hydrostatic pressure-corrector equations to ensure stability and accuracy. New boundary conditions for the free-surface-corrector and pressure-corrector are also introduced. We prove that these conditions lead to consistent boundary conditions for the free-surface and pressure proper. To ensure discrete mass conservation with a moving free-surface, we use an arbitrary LagrangianEulerian (ALE) moving mesh algorithm. These schemes are again verified, this time using a tidal flow problem with analytical solutions and a 3D lock-exchange benchmark. We apply our new numerical schemes to evaluate the numerical requirements of the coupled biological-physical dynamics. We find that higher-order schemes are more accurate at the same efficiency compared to lower-order (e.g. second-order) accurate schemes when modeling a biological patch. Due to decreased numerical dissipation, the higher-order schemes are capable of modeling biological patchiness over a sustained duration, while the lower-order schemes can lose significant biomass after a few non-dimensional times and can thus solve erroneous nonlinear dynamics. Finally, inspired by Stellwagen Bank in Massachusetts Bay, we study the effect of non-hydrostatic physics on biological productivity and phytoplankton fields for tidally-driven flows over an idealized bank. We find that the non-hydrostatic pressure and flows are important for biological dynamics, especially when flows are supercritical. That is, when the slope of the topography is larger than the slope of internal wave rays at the tidal frequency. The non-hydrostatic effects increase with increasing nonlinearity, both when the internal Froude number and criticality parameter increase. Even in cases where the instantaneous biological productivity is not largely modified, we find that the total biomass, spatial variability and patchiness of phytoplankton can be significantly altered by non-hydrostatic processes. Our ultimate dynamics motivation is to allow quantitative simulation studies of fundamental nonlinear biological-physical dynamics in coastal regions with complex bathymetric features such as straits, sills, ridges and shelfbreaks. This thesis develops the necessary numerical schemes that meet the stringent accuracy requirements for these types of flows and dynamics.

The ocean is a complex, constantly changing, highly dynamical system. Prediction capabilities are constantly being improved in order to better understand and forecast ocean properties for applications in science, industry, and maritime interests. Our overarching goal is to better predict the ocean environment in regions of complex topography with a continental shelf, shelfbreak, canyons and steep slopes using the MIT Multidisciplinary Simulation, Estimation and Assimilation Systems (MSEAS) primitive-equation ocean model. We did this by focusing on the complex region surrounding Taiwan, and the period of time immediately following the passage of Typhoon Morakot. This area and period were studied extensively as part of the intense observation period during August – September 2009 of the joint U.S. – Taiwan program Quantifying, Predicting, and Exploiting Uncertainty Department Research Initiative (QPE DRI). Typhoon Morakot brought an unprecedented amount of rainfall within a very short time period and in this research, we model and study the effects of this rainfall on Taiwan’s coastal oceans as a result of river discharge. We do this through the use of a river discharge model and a bulk river-ocean mixing model. We complete a sensitivity study of the primitive-equation ocean model simulations to the different parameters of these models. By varying the shape, size, and depth of the bulk mixing model footprint, and examining the resulting impacts on ocean salinity forecasts, we are able to determine an optimal combination of salinity relaxation factors for highest accuracy.

This thesis develops and applies a procedure to generate high quality 2D meshes for any given ocean region with complex coastlines. The different criteria used in determining mesh element sizes for a given domain are discussed, especially sizing criteria that depend on local properties of the bathymetry and relevant dynamical scales. Two different smoothing techniques, Laplacian conditioning and targeted averaging, were applied to the fields involved in calculating the sizing matrix. The L^2 norm was used to quantify which technique had the greatest preservation of the original field. In both the reduced gradient and gradient cases, targeted averaging had a lower L^2 norm. The sizing matrices were used as inputs for two mesh generators, Distmesh and GMSH, and their meshing results were presented over a set of ocean domains in the Gulf of Maine and Massachusetts Bay region. Further research into the capabilities of each mesh generator are needed to provide a detailed evaluation. Mesh quality issues near coastlines revealed the need for small scale feature size recognition algorithms that could be implemented and studied in the future.

We know more about the dark side of the moon than the depths of the ocean. This is startling, considering how much more tangible the ocean is than space, and more importantly, how much more critical it is to the health and survival of humanity. Tens of billions of dollars are spent on manned and unmanned missions probing deeper into space, while 95% of Earth’s oceans remain unexplored. The result is a perilous dearth in knowledge about our planet at a time when rapid changes in our marine ecosystems profoundly affect its habitability.
The more intensive focus on space exploration is a historically recent phenomenon. For millennia until the mid-20th century, space and ocean exploration proceeded roughly at the same pace, driven by curiosity, military, and commerce. Both date back to early civilization when star-gazers scanned the skies, and sailors and free-divers scoured the seas. Since the 1960s when Don Walsh and Jacques Piccard descended to the deepest point on the ocean floor, and Neil Armstrong ascended to the moon, however, the trajectories of exploration diverged dramatically. Cold War-inspired geopolitical-military imperatives propelled space research to en extraordinary level, while ocean exploration stagnated in comparison. Moreover, although the Cold War ended more than 20 years ago, the disparity in effort remains vast despite evidence that accelerating changes in our marine ecosystems directly threatens our well being. Misconception about the relative importance of space and ocean exploration caused, and continues to sustain, this knowledge disparity to our peril.
In this thesis, we first review in section 2 the history of space and ocean exploration before the Cold War, when the pace of exploration in each sector was more or less comparable for thousands of years. We show in section 3, however, how the relative paces and trajectories of exploration diverged dramatically during the Cold War and continue to the present. In section 4 we seek to dispel the persistent misconceptions that have led to the disparity in resources allocated between space and ocean exploration, and argue for prioritizing ocean research. Finally, in section 5 we highlight the urgent imperative for expanding our understanding of the ocean.

p> The science of uncertainty quantification has gained a lot of attention over recent years. This is because models of real processes always contain some elements of uncertainty, and also because real systems can be better described using stochastic components. Stochastic models can therefore be utilized to provide a most informative prediction of possible future states of the system. In light of the multiple scales, nonlinearities and uncertainties in ocean dynamics, stochastic models can be most useful to describe ocean systems.

Uncertainty quantification schemes developed in recent years include order reduction methods (e.g. proper orthogonal decomposition (POD)), error subspace statistical estimation (ESSE), polynomial chaos (PC) schemes and dynamically orthogonal (DO) field equations. In this thesis, we focus our attention on DO and various PC schemes for quantifying and predicting uncertainty in systems with external stochastic forcing. We develop and implement these schemes in a generic stochastic solver for a class of non-autonomous linear and nonlinear dynamical systems. This class of systems encapsulates most systems encountered in classic nonlinear dynamics and ocean modeling, including flows modeled by Navier-Stokes equations. We first study systems with uncertainty in input parameters (e.g. stochastic decay models and Kraichnan-Orszag system) and then with external stochastic forcing (autonomous and non-autonomous self-engineered nonlinear systems). For time-integration of system dynamics, stochastic numerical schemes of varied order are employed and compared. Using our generic stochastic solver, the Monte Carlo, DO and polynomial chaos schemes are intercompared in terms of accuracy of solution and computational cost.

To allow accurate time-integration of uncertainty due to external stochastic forcing, we also derive two novel PC schemes, namely, the reduced space KLgPC scheme and the modified TDgPC (MTDgPC) scheme. We utilize a set of numerical examples to show that the two new PC schemes and the DO scheme can integrate both additive and multiplicative stochastic forcing over significant time intervals. For the final example, we consider shallow water ocean surface waves and the modeling of these waves by deterministic dynamics and stochastic forcing components. Specifically, we time-integrate the Korteweg-de Vries (KdV) equation with external stochastic forcing, comparing the performance of the DO and Monte Carlo schemes. We find that the DO scheme is computationally efficient to integrate uncertainty in such systems with external stochastic forcing.

A new methodology for Bayesian inference of stochastic dynamical models is developed. The methodology leverages the dynamically orthogonal (DO) evolution equations for reduced-dimension uncertainty evolution and the Gaussian mixture model DO filtering algorithm for nonlinear reduced-dimension state variable inference to perform parallelized computation of marginal likelihoods for multiple candidate models, enabling efficient Bayesian update of model distributions. The methodology also employs reduced-dimension state augmentation to accommodate models featuring uncertain parameters. The methodology is applied successfully to two high-dimensional, nonlinear simulated fluid and ocean systems. Successful joint inference of an uncertain spatial geometry, one uncertain model parameter, and 0(105) uncertain state variables is achieved for the first. Successful joint inference of an uncertain stochastic dynamical equation and 0(105) uncertain state variables is achieved for the second. Extensions to adaptive modeling and adaptive sampling are discussed.

In this thesis, we explore the use of stochastic Navier-Stokes equations through the Dynamically Orthogonal (DO) methodology developed at MIT in the Multidisciplinary Simulation, Estimation, and Assimilation Systems Group. Specifically, we examine the effects of the Reynolds number on stochastic fluid flows behind a square cylinder and evaluate computational schemes to do so. We review existing literature, examine our simulation results and validate the numerical solution. The thesis uses a novel open boundary condition formulation for DO stochastic Navier-Stokes equations, which allows the modeling of a wide range of random inlet boundary conditions with a single DO simulation of low stochastic dimensions, reducing computational costs by orders of magnitude. We first test the numerical convergence and validating the numerics. We then study the sensitivity of the results to several parameters, focusing for the dynamics on the sensitivity to the Reynolds number. For the method, we focus on the sensitivity to the: resolution of in the stochastic subspace, resolution in the physical space and number of open boundary conditions DO modes. Finally, we evaluate and study how key dynamical characteristics of the flow such as the recirculation length and the vortex shedding period vary with the Reynolds number.

James Cameron’s dive to the Challenger Deep in the Deepsea Challenger in March of 2012 marked the first time man had returned to the Mariana Trench since the Bathyscaphe Trieste’s 1960 dive. Currently little is known about the geological processes and ecosystems of the deep ocean. The Deepsea Challenger is equipped with a plethora of instrumentation to collect scientific data and samples. The development of the Deepsea Challenger has sparked a renewed interest in manned exploration of the deep ocean.
Due to the immense pressure at full ocean depth, a variety of advanced systems and materials are used on Cameron’s dive craft. This paper provides an overview of the many novel features of the Deepsea Challenger as well as related features of past vehicles that have reached the Challenger Deep. Four key areas of innovation are identified: buoyancy materials, pilot sphere construction/instrument housings, lighting, and battery power. An in depth review of technological development in these areas is provided, as well as a glimpse into future manned submersibles and their technologies of choice.

Data assimilation, as presented in this thesis, is the statistical merging of sparse observational data with computational models so as to optimally improve the probabilistic description of the field of interest, thereby reducing uncertainties. The centerpiece of this thesis is the introduction of a novel such scheme that overcomes prior shortcomings observed within the community. Adopting techniques prevalent in Machine Learning and Pattern Recognition, and building on the foundations of classical assimilation schemes, we introduce the GMM-DO filter: Data Assimilation with Gaussian mixture models using the Dynamically Orthogonal field equations.

We combine the use of Gaussian mixture models, the EM algorithm and the Bayesian Information Criterion to accurately approximate distributions based on Monte Carlo data in a framework that allows for efficient Bayesian inference. We give detailed descriptions of each of these techniques, supporting their application by recent literature. One novelty of the GMM-DO filter lies in coupling these concepts with an efficient representation of the evolving probabilistic description of the uncertain dynamical field: the Dynamically Orthogonal field equations. By limiting our attention to a dominant evolving stochastic subspace of the total state space, we bridge an important gap previously identified in the literature caused by the dimensionality of the state space.

We successfully apply the GMM-DO filter to two test cases: (1) the Double Well Diffusion Experiment and (2) the Sudden Expansion fluid flow. With the former, we prove the validity of utilizing Gaussian mixture models, the EM algorithm and the Bayesian Information Criterion in a dynamical systems setting. With the application of the GMM-DO filter to the two-dimensional Sudden Expansion fluid flow, we further show its applicability to realistic test cases of non-trivial dimensionality. The GMM-DO filter is shown to consistently capture and retain the far-from-Gaussian statistics that arise, both prior and posterior to the assimilation of data, resulting in its superior performance over contemporary filters. We present the GMM-DO filter as an efficient, data-driven assimilation scheme, focused on a dominant evolving stochastic subspace of the total state space, that respects nonlinear dynamics and captures non-Gaussian statistics, obviating the use of heuristic arguments.

In the past decades an increasing number of problems in continuum theory have been treated using stochastic dynamical theories. This is because dynamical systems governing real processes always contain some elements characterized by uncertainty or stochasticity. Uncertainties may arise in the system parameters, the boundary and initial conditions, and also in the external forcing processes. Also, many problems are treated through the stochastic framework due to the incomplete or partial understanding of the governing physical laws. In all of the above cases the existence of random perturbations, combined with the com- plex dynamical mechanisms of the system often leads to their rapid growth which causes distribution of energy to a broadband spectrum of scales both in space and time, making the system state particularly complex. Such problems are mainly described by Stochastic Partial Differential Equations and they arise in a number of areas including fluid mechanics, elasticity, and wave theory, describing phenomena such as turbulence, random vibrations, flow through porous media, and wave propagation through random media. This is but a partial listing of applications and it is clear that almost any phenomenon described by a field equation has an important subclass of problems that may profitably be treated from a stochastic point of view.

In this work, we develop a new methodology for the representation and evolution of the complete probabilistic response of infinite-dimensional, random, dynamical systems. More specifically, we derive an exact, closed set of evolution equations for general nonlinear continuous stochastic fields described by a Stochastic Partial Differential Equation. The derivation is based on a novel condition, the Dynamical Orthogonality (DO), on the representation of the solution. This condition is the “key” to overcome the redundancy issues of the full representation used while it does not restrict its generic features. Based on the DO condition we derive a system of field equations consisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal basis that describe the stochastic subspace where uncertainty “lives” as well as a system of Stochastic Differential Equations that defines how the uncertainty evolves in the time varying stochastic subspace. If additional restrictions are assumed on the form of the representation, we recover both the Proper-Orthogonal-Decomposition (POD) equations and the generalized Polynomial- Chaos (PC) equations; thus the new methodology generalizes these two approaches. For the efficient treatment of the strongly transient character on the systems described above we derive adaptive criteria for the variation of the stochastic dimensionality that characterizes the system response. Those criteria follow directly from the dynamical equations describing the system.

We illustrate and validate this novel technique by solving the 2D stochastic Navier-Stokes equations in various geometries and compare with direct Monte Carlo simulations. We also apply the derived framework for the study of the statistical responses of an idealized “double gyre” model, which has elements of ocean, atmospheric and climate instability behaviors.

Finally, we use our new stochastic description for flow fields to study the motion of inertial particles in flows with uncertainties. Inertial or finite-size particles in fluid flows are commonly encountered in nature (e.g., contaminant dispersion in the ocean and atmosphere) as well as in technological applications (e.g., chemical systems involving particulate reactant mixing). As it has been observed both numerically and experimentally, their dynamics can differ markedly from infinitesimal particle dynamics. Here we use recent results from stochastic singular perturbation theory in combination with the DO representation of the random flow, in order to derive a reduced order inertial equation that will describe efficiently the stochastic dynamics of inertial particles in arbitrary random flows.

Understanding the complex dynamics of coastal upwelling is essential for coastal ocean dynamics, phytoplankton blooms, and pollution transport. Atmospheric- driven coastal upwelling often occurs when strong alongshore winds and the Coriolis force combine to displace warmer surface waters offshore, leading to upward motions of deeper cooler, nutrient-dense waters to replace these surface waters. Using the models of the MIT Multidisciplinary Simulation, Estimation, and Assimilation System (MSEAS) group, we conduct a large set of simulation sensitivity studies to determine which variables are dominant controls for upwelling events in the Monterey Bay region. Our motivations include determining the dominant atmospheric fluxes and the causes of high-frequency fluctuations found in ocean thermal balances. We focus on the first upwelling event from August 1- 5, 2006 in Monterey Bay that occurred during the Monterey Bay 06 (MB06) at-sea experiment, for which MSEAS data-assimilative baseline simulations already existed.

Using the thermal energy (temperature), salinity and momentum (velocity) conservation equations, full ocean fields in the region as well as both control volume (flux) balances and local differential term-by-term balances for the upwelling event events were computed. The studies of ocean fields concentrate on specific depths: surface-0m, thermocline-30m and undercurrent-150m. Effects of differing atmospheric forcing contributions (wind stress, surface heating/cooling, and evaporation-precipitation) on these full fields and on the volume and term-by-term balances are analyzed. Tidal effects are quantified utilizing pairs of simulations in which tides are either included or not. Effects of data assimilation are also examined.

We find that the wind stress forcing is the most important dynamical parameter in explaining the extent and shape of the upwelling event. This is verified using our large set of sensitivity studies and examining the heat flux balances. The assimilation of data has also an impact because this first upwelling event occurs during the initialization. Tidal forcing and, to a lesser extent, the daily atmospheric and data assimilation cycles explain the higher frequency fluctuations found in the volume averaged time rate of change of thermal energy.

A new generation of efficient parallel, multi-scale, and interdisciplinary ocean models is required for better understanding and accurate predictions. The purpose of this thesis is to quantitatively identify promising numerical methods that are suitable to such predictions. In order to fulfill this purpose, current efforts towards creating new ocean models are reviewed, an understanding of the most promising methods used by other researchers is developed, the most promising existing methods are studied and applied to idealized cases, new methods are incubated and evaluated by solving test problems, and important numerical issues related to efficiency are examined. The results of other research groups towards developing the second generation of ocean models are first reviewed. Next, the Discontinuous Galerkin (DG) method for solving advection-diffusion problems is described, including a discussion on schemes for solving higher order derivatives. The discrete formulation for advection-diffusion problems is detailed and implementation issues are discussed. The Hybrid Discon- tinuous Galerkin (HDG) Finite Element Method (FEM) is identified as a promising new numerical scheme for ocean simulations. For the first time, a DG FEM scheme is used to solve ocean biogeochemical advection-diffusion-reaction equations on a two- dimensional idealized domain, and p-adaptivity across constituents is examined. Each aspect of the numerical solution is examined separately, and p-adaptive strategies are explored. Finally, numerous solver-preconditioner combinations are benchmarked to identify an efficient solution method for inverting matrices, which is necessary for implicit time integration schemes. From our quantitative incubation of numerical schemes, a number of recommendations on the tools necessary to solve dynamical equations for multiscale ocean predictions are provided.

A fundamental requirement in realistic computational geophysical fluid dynamics is the optimal estimation of gridded fields and of spatial-temporal scales directly from the spatially irregular and multivariate data sets that are collected by varied instruments and sampling schemes. In this work, we derive and utilize new schemes for the mapping and dynamical inference of ocean fields in complex multiply-connected domains, study the computational properties of our new mapping schemes, and derive and investigate new schemes for adaptive estimation of spatial and temporal scales.

Objective Analysis (OA) is the statistical estimation of fields using the Bayesian- based Gauss-Markov theorem, i.e. the update step of the Kalman Filter. The existing multi-scale OA approach of the Multidisciplinary Simulation, Estimation and Assimilation System consists of the successive utilization of Kalman update steps, one for each scale and for each correlation across scales. In the present work, the approach is extended to field mapping in complex, multiply-connected, coastal regions and archipelagos. A reasonably accurate correlation function often requires an estimate of the distance between data and model points, without going across complex land- forms. New methods for OA based on estimating the length of optimal shortest sea paths using the Level Set Method (LSM) and Fast Marching Method (FMM) are derived, implemented and utilized in general idealized and realistic ocean cases. Our new methodologies could improve widely-used gridded databases such as the climatological gridded fields of the World Ocean Atlas (WOA) since these oceanic maps were computed without accounting for coastline constraints. A new FMM-based methodology for the estimation of absolute velocity under geostrophic balance in complicated domains is also outlined. Our new schemes are compared with other approaches, including the use of stochastically forced differential equations (SDE). We find that our FMM-based scheme for complex, multiply-connected, coastal regions is more efficient and accurate than the SDE approach. We also show that the field maps obtained using our FMM-based scheme do not require postprocessing (smoothing) of fields. The computational properties of the new mapping schemes are studied in detail. We find that higher-order schemes improve the accuracy of distance estimates. We also show that the covariance matrices we estimate are not necessarily positive definite because the Weiner Khinchin and Bochner relationships for positive definiteness are only valid for convex simply-connected domains. Several approaches to overcome this issue are discussed and qualitatively evaluated. The solutions we propose include introducing a small process noise or reducing the covariance matrix based on the dominant singular value decomposition. We have also developed and utilized novel methodologies for the adaptive estimation of spatial-temporal scales from irregularly spaced ocean data. The three novel methodologies are based on the use of structure functions, short term Fourier transform and second generation wavelets. To our knowledge, this is the first time that adaptive methodologies for the spatial-temporal scale estimation are proposed. The ultimate goal of all these methods would be to create maps of spatial and temporal scales that evolve as new ocean data are fed to the scheme. This would potentially be a significant advance to the ocean community for better understanding and sampling of ocean processes.

In this thesis, we conduct research toward understanding coupled physics-biology processes in ocean straits. Our focus is on new analytical studies and higher-order simulations of idealized dynamics that are relevant to generic biological processes. The details of coupled physics-biology models are reviewed and an in-depth global equilibrium and local stability analysis of a Nutrient-Phytoplankton-Zooplankton (NPZ) model is performed. This analysis includes parameter studies and methods to evaluate parameter sensitivity, especially in the case where some system parameters are unknown. As an initial step toward investigating the interaction between physics and biology in ocean straits, we develop and verify a new coupled physics-biology model for two-dimensional idealized physical processes including tides and apply it to the San Bernardino Strait in the Philippine Archipelago. This two-dimensional numerical model is created on a structured grid using operator splitting and masking. This model is able to accurately represent biology for various physical flows, including advection-dominated flows over discontinuities, by using the Weighted Essentially Non-Oscillatory (WENO) scheme. The numerical model is verified against a Discontinuous-Galerkin (DG) numerical scheme on an unstructured grid. Several simulations of tidal flow are completed using bathymetry and flow magnitudes com- parable to those found in the San Bernardino Strait with different sets of parameters, tidal periods, and levels of diffusion. Results are discussed and compared to those of a three-dimensional modeling system. New results include: new methods for analyzing stability, the robust two-dimensional model designed to best represent advection-dominant flows with minimal numerical diffusion and computational time, and a novel technique to initialize three-dimensional biology fields using satellite data. Additionally, application of the two-dimensional model with tidal forcing to the San Bernardino Strait reveals that flow frequencies have strong influence on biology, as very fast oscillations act to stabilize biology in the water column, while slower frequencies provide sufficient transport for increased biological activity.

In this thesis, we explore the different methods for parameter estimation in straightforward diffusion problems and develop ideas and distributed computational schemes for the automated evaluation of physical and numerical parameters of ocean models. This is one step of “adaptive modeling”. Adaptive modeling consists of the automated adjustment of self-evaluating models in order to best represent an observed system. In the case of dynamic parameterizations, self-modifying schemes are used to learn the correct model for a particular regime as the physics change and evolve in time.

The parameter estimation methods are tested and evaluated on one-dimensional tracer diffusion problems. Existing state estimation methods and new filters, such as the unscented transform Kalman filter, are utilized in carrying out parameter estimation. These include the popular Extended Kalman Filter (EKF), the Ensemble Kalman Filter (EnKF) and other ensemble methods such as Error Subspace Statistical Estimation (ESSE) and Ensemble Adjustment Kalman Filter (EAKF), and the Unscented Kalman Filter (UKF). Among the aforementioned recursive state estimation methods, the so-called “adjoint method” is also applied to this simple study.

Finally, real data is examined for the applicability of such schemes in real-time fore- casting using the MIT Multidisciplinary Simulation, Estimation, and Assimilation System (MSEAS). The MSEAS model currently contains the free surface hydrostatic primitive equation model from the Harvard Ocean Prediction System (HOPS), a barotropic tidal prediction scheme, and an objective analysis scheme, among other models and developing routines. The experiment chosen for this study is one which involved the Monterey Bay region off the coast of California in 2006 (MB06). Accurate vertical mixing parameterizations are essential in this well known upwelling region of the Pacific. In this realistic case, parallel computing will be utilized by scripting code runs in C-shell. The performance of the simulations with different parameters is evaluated quantitatively using Pattern Correlation Coefficient, Root Mean Squared error, and bias error. Comparisons quantitatively determined the most adequate model setup.

In shallow water, a large part of underwater acoustic prediction uncertainties are in- duced by sub-meso-to-small scale oceanographic variabilities. Conventional oceano- graphic measurements for capturing such ocean-acoustic environmental variabilities face the classical conflict between resolution and coverage. The Adaptive Rapid En- vironmental Assessment (AREA) project was proposed to resolve this conflict by optimizing the location of in-situ measurements in an adaptive manner. In this thesis, ideas, concepts and performance limits in AREA are clarified. Both an engineering and a mathematical model for AREA are developed. A modularized AREA simulator was developed and implemented in C++. Philosophies in AREA are discussed. Presumptions about the ocean are made to bridge the gap between the viewpoint in the oceanography community, where the ocean environment is consid- ered to be a deterministic but very complicated system, and that of the underwater acoustic community, where the ocean environment is treated as a random system. At present, how to optimally locate the in-situ measurements made by a single AUV carrying a CTD (conductivity, temperature and depth) sensor is considered in AREA. In this thesis, the AUV path planning is modeled as a Shortest Path problem. However, due to the sound velocity correlation effect, the size of this problem can be very large. A method is developed to simplify the graph for a fast solution. As a significant step, a linear approximation for acoustic Transmission Loss (TL) is investigated numerically and analytically. In addition to following a predetermined path, an AUV can also adaptively gener- ate its path on-board. This adaptive on-board AUV routing problem is modeled using Dynamic Programming (DP) in this thesis. A method based on an optimized prede- termined path is developed to reduce the size of the DP problem and approximately yet efficiently solve it using Pattern Recognition. As a special case, a thermocline- oriented AUV yoyo control and control parameter optimization methods for AREA are also developed. 2 Finally, some AUV control algorithms for capturing fronts are developed. A frame- work for real-time TL forecasts is developed. This is the first time that TL forecasts have been linked with ocean forecasts in real-time. All of the above ideas and methods developed were tested in two experiments, FAF05 in the northern Tyrrhenian Sea in 2005 and MB06 in Monterey Bay, CA in 2006. The latter MB06 sea exercise was a major field experiment sponsored by the Office of Naval Research and the thesis compiles significant findings from this effort.