We illustrate the use of our partial differential equations and super-accurate composition schemes for flow maps to quantify Lagrangian transports and non-advective dynamics in geophysical fluid flows. Flow maps are spatiotemporal fields governed by PDEs that correspond to an infinite number of classic trajectories governed by ODEs (the characteristics of the PDEs). We utilize our flow map predictions to extract dynamical regions and coherent structures, classify ocean processes, and inform classical geophysical fluid dynamics analyses. Our emphasis is on the use of spatiotemporal flow maps to help differentiate the advective transports from non-advective transformations of water masses and ocean features in four dimensions. Results are presented for real-time sea experiments with autonomous sensing platforms and advanced modeling systems in diverse ocean regions and dynamical regimes. They include the Nova Scotia Shelf-Slope and New England Seamount Chain regions, the Gulf of Mexico, and the Balearic and Alboran Seas in the western Mediterranean. Our differentiations directly highlight regions of higher shear and mixing, including the submesoscale features, the edges of meanders, eddies, filaments, and internal waves, and the regions undergoing strong vertical and helical-spiral motions.

The last two decades have seen the development of a few high-order nonhydrostatic (NHS) ocean models, but there is still a need to better understand the associated numerical properties. A key improvement in NHS models in comparison to hydrostatic models is their dispersive nature, which allows for accurate resolution of gravity waves. However, it has been shown (Vitousek 2011) that low-order finite difference and finite volume methods can suffer from large numerical dispersion, which hinders the resolution of such waves. Therefore, high-order methods pose an attractive alternative, and in this study, we explore the use of high-order hybridizable discontinuous Galerkin-based (HDG) schemes (Ueckermann 2016) to resolve NHS gravity waves. We quantify and compare the numerical dispersion and computational cost of these schemes to their low-order counterparts using idealized internal wave and bottom gravity current test cases. Additionally, the stability of NHS models in the presence of fast-moving free-surface gravity waves is crucial to their efficacy. To this end, we explore the stability and convergence of high-order HDG methods in the context of the NHS primitive ocean equations solved on skewed computational domains. Finally, using our high-order HDG NHS solver, we illustrate results from process studies of gravity-driven wave dynamics including (i) 3D internal wave propagation over complex bathymetry, (ii) tidally-forced oscillatory flow over seamounts and (iii) bottom gravity currents.

To simulate and study ocean phenomena involving complex dynamics over a wider range of scales, from regional to small scales (e.g., thousands of kilometers to meters), resolving submesoscale features, nonlinear internal waves, subduction, and overturning where they occur, non-hydrostatic (NHS) ocean models are needed, at least locally. The main computational burden for NHS models arises from solving a globally coupled 3D elliptic PDE for the NHS pressure. To address this challenge, we start with a high-order hybridizable discontinuous Galerkin (HDG) (Nguyen et al. 2009) finite element NHS ocean solver (Ueckermann and Lermusiaux 2016) that is well suited for multidynamics systems. We present a new adaptive algorithm to decompose a domain into NHS and HS dynamics subdomains and solve their corresponding equations, thereby reducing the cost associated with the NHS pressure solution step. The NHS/HS subdomains are adapted based on different numerical NHS estimators, such that NHS dynamics is used only where needed. We compare the performance of these NHS estimators and explore choices of boundary conditions imposed on the internal boundaries between subdomains of different dynamics. We evaluate and analyze the computational costs and accuracy of the adaptive NHS-HS solver using idealized NHS dynamics test cases: (i) idealized internal waves (Vitousek and Fringer 2011), (ii) tidally-forced oscillatory flow over seamounts, (iii) bottom gravity currents, and (iv) 3D internal wave propagation over complex bathymetry. We then complete more realistic NHS-HS simulations of Rayleigh-Taylor instability-driven subduction events by nesting with our MSEAS realistic and operational data-assimilative HS ocean modeling system. Finally, we discuss DG-FEM-based numerical techniques to stabilize and accelerate the high-order ocean solvers by leveraging the high aspect ratio characteristic of ocean domains.

Truncated fluid and ocean models omit subgrid physics and introduce numerical biases that degrade forecasts. We present a Bayesian, data-driven closure for 2-D finite-volume solvers that learns the dynamical discrepancy between low-resolution (LR) and high-resolution (HR) simulations. Using sparse variational Gaussian processes (GPs) and deep GPs, we map resolved features (local velocities and gradients) to a closure source term that corrects LR tendencies toward HR dynamics while quantifying predictive uncertainty. GPs can be well-suited to closure modeling in fluids because they encode smoothness/invariance via kernels, learn nonparametric mappings from data, and return uncertainty estimates alongside the mean correction. The trained GP is embedded intrusively into a numerical finite volume framework and evaluated online each coarse time step, keeping the closure consistent with the numerics.

We assess the approach on three test beds: (i) flow past a cylinder across multiple Reynolds numbers; (ii) tidally modulated flow past a cylinder with time-varying Reynolds number; and (iii) bottom gravity currents. Models are trained on HR downsamplings–LR pairs and then tested across different regimes. We evaluate performance by using field-wise errors and wake metrics: mean velocity profiles in the near and far wake, lift C_L and drag C_D coefficients, and Strouhal number St. Relative to LR baselines without closure, GP closures reduce L2 / L∞ errors of the resolved fields and bring mean velocity, C_D/C_L, and St closer to HR references across trained Reynolds numbers. The online GP closure adds negligible wall-clock cost relative to the fluid step, preserves the conservative finite-volume structure, and provides uncertainty estimates. Overall, these results demonstrate a practical, uncertainty-aware GP closure that improves coarse-grid fidelity for 2-D fluid and ocean flows, which could potentially be extended to 3-D ocean frameworks.

The Loop Current (LC), along with its associated meanders, eddies (LCEs), and cyclonic frontal eddies (LCFEs), plays a major role in the Gulf of Mexico and has been extensively studied over the past decade, with several field campaigns. During the recent 6-month collaborative GRand Adaptive Sampling Experiment (GRASE; April to September 2025), we employed our MIT Multidisciplinary Simulation, Estimation, and Assimilation Systems (MSEAS), including Error Subspace Statistical Estimation (ESSE) large-ensemble forecasting to provide real-time probabilistic forecasts. We describe the evolution of ocean features and evaluate the predictive skill of our forecasts compared to independent data. We present our probabilistic glider reachability and optimal path planning forecasts. This includes the use of reachability and heading forecasts for optimal deployment, feature sampling and tracking, and recovery of multiple gliders. We show that the actual glider tracks remain within our forecast reachability fronts and that headings could be followed in real-time. We demonstrate the use of our information-theoretic methodology for optimal adaptive sampling with gliders and floats, where we maximize information about specific future properties of the LC, LCEs, and LCFEs. We issued reachability forecasts for floats and modified 3D Lagrangian flow maps to account for float motions. We forecast float transports during several periods, highlighting how float deployment regions remain coherent or are being distorted, especially how the transport of floats on the edges of LCFEs can be affected by shear and turbulence. Lastly, we illustrate our real-time clustering of the large-ensemble probabilistic LCE forecasts and how we showed that an LCE detachment in June/early July 2025 was very unlikely. This work is in collaboration with the whole GRASE team.