Congratulations to Sydney Sroka on her recent graduation! Sydney received an SM from Mechanical Engineering for her research on “Internal Tides Near Steep Topographies” with our MSEAS group at MIT.

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.

A novel hydrostatic coupled-mode shallow water model (CSW) is developed and used to simulate tides in the greater Middle Atlantic Bight region. The model incorporates realistic stratification and topography, an internal tide generating function (ITGF) that provides internal tide forcing from existing surface tide parameters, and dynamical terms that describe linearized wave- mean-flow and mean-density interactions. Several idealized and realistic simulations are used to verify the model. These verification simulations include internal-tide interactions involving topographic coupling and mean-flow coupling, and comparisons with other simpler and more complex nonlinear primitive-equation models. Then, twenty-four simulations of internal tide generation and propagation in the greater Middle Atlantic Bight region are used to identify significant internal-tide interactions with the Gulf Stream. The simulations indicate that locally generated mode-1 internal tides can refract and/or reflect at the Gulf Stream. The redirected internal tides often re-appear at the shelfbreak, where they produce onshore energy fluxes that are intermittent (i.e., noncoherent) because meanders in the Gulf Stream alter their precise location, phase, and amplitude. These results provide an explanation for the anomalous onshore energy fluxes previously observed at the New Jersey Shelfbreak and linked with the generation of nonlinear internal waves.

Internal tides in the Middle Atlantic Bight region are noticeably influenced by the presence of the shelfbreak front and the Gulf Stream. To identify the dominant interactions of these waves with subtidal flows, vertical-mode momentum and energy partial differential equations are derived for small-amplitude waves in a horizontally and vertically sheared mean flow and in a horizontally and vertically variable density field. First, the energy balances are examined in idealized simulations with mode-1 internal tides propagating across and along the Gulf Stream. Next, the fully-nonlinear dynamics of regional tide-mean flow interactions are simulated with a primitive equation model, which incorporates realistic summer mesoscale features and atmospheric forcing. The summer shelfbreak front, which has horizontally variable stratification, decreases topographic internal-tide generation by about 10% and alters the wavelengths and arrival times of locally generated mode-1 internal tides on the shelf and in the abyss. The (sub)-mesoscale variability at the front and on the shelf, as well as the summer stratification itself, also alter the internal tide propagation. The Gulf Stream produces anomalous regions of O(20 mW m^-2) mode-1 internal-tide energy-flux divergence, which are explained by mean-flow terms in the mode-1 energy balance. Advection explains most tide-mean flow interaction, suggesting that geometric wave theory predicts mode-1 reflection and refraction at the Gulf Stream. Geometric theory predicts that offshore-propagating mode-1 internal tides that strike the Gulf Stream at oblique angles (more than thirty degrees from normal) are reflected back to the coastal ocean, preventing their radiation into the central North Atlantic.