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.