Loading content ...

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.

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.

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 Taiwanâ€™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.

Phadnis, A., 2013. *Uncertainty Quantification and Prediction for Non-autonomous Linear and Nonlinear Systems*. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September 2013.

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.

Lu, P., 2013. *Bayesian inference of stochastic dynamical models*. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, February 2013.

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.

Sondergaard, T., 2011. *Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations*. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September 2011.

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.

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.

Agarwal, A., 2009. *Statistical Field Estimation and Scale Estimation for Complex Coastal Regions and Archipelagos*. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, May 2009.

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.

Heubel, E., 2008. *Parameter Estimation and Adaptive Modeling Studies in Ocean Mixing*. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September 2008.

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.