A theoretical synthesis of forward reachability for minimum–time control of anisotropic vehicles operating in strong and dynamic flows is provided. The synthesis relies on the computation of the forward reachable set of states. Using ideas rooted in the theory of non–smooth calculus, we prove that this set is governed by the viscosity solution of an unsteady Hamilton–Jacobi (HJ) equation. We show that the minimum arrival time satisfies a static HJ equation, when a special local controllability condition holds. Results are exemplified by applications to a sailboat moving in a uniform wind–field and autonomous underwater gliders operating in the Sulu Archipelago.

This paper focuses on the extractions of Lagrangian Coherent Sets from realistic velocity fields obtained from ocean data and simulations, each of which can be highly resolved and non volume-preserving. We introduce two novel methods for computing two formulations of such sets. First, we propose a new “diffeomorphism-based” criterion to extract “rigid sets”, defined as sets over which the flow map acts approximately as a rigid transformation. Second, we develop a matrix-free methodology that provides a simple and efficient framework to compute “coherent sets” with operator methods. Both new methods and their resulting rigid sets and coherent sets are illustrated and compared using three numerically simulated flow examples, including a high-resolution realistic, submesoscale to large-scale dynamic ocean current field in the Palau Island region of the western Pacific Ocean.

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.