In this paper, we illustrate the use of our partial differential equations for flow maps to quantify Lagrangian transports and non-advective dynamics in geophysical fluid flows. 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. Preliminary 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, Gulf of Mexico, and Balearic and Alboran Seas in the western Mediterranean. Our differentiations directly highlight regions of higher shear and mixing, including the edges of meanders, eddies, filaments, and internal waves, and the regions undergoing strong vertical or spiral motions.

To cleanup marine plastics, accurate modeling is needed. We outline and illustrate a new partial-differential-equation methodology for characterizing and modeling plastic transports in time and space (4D), showcasing results for Massachusetts Bay. We couple our primitive equation model for ocean dynamics with our composition based advection for Lagrangian transport. We show that the ocean physics predictions have skill by comparison with synoptic data. We predict the fate of plastics originating from four sources: rivers, beach and nearshore, local Bay, and remote offshore. We analyze the transport patterns and the regions where plastics accumulate, comparing results with and without plastic settling. Simulations agree with existing debris and plastics data. They also show new results: (i) Currents set-up by wind events strongly affect floating plastics. Winds can for example prevent Merrimack outflows reaching the Bay; (ii) There is significant chaotic stirring between nearshore and offshore floating plastics as explained by ridges of Lagrangian Coherent Structures (LCSs); (iii) With 4D plastic motions and settling, plastics from the Merrimack and nearshore regions can settle to the seabed before offshore advection; (iv) Internal waves and tides can bring plastics downward and out of main currents, leading to settling to the deep bottom. (v) Attractive LCSs ridges are frequent in the northern Cape Cod Bay, west of the South Shore, and southern Stellwagen Bank. They lead to plastic accumulation and sinking along thin subduction zones.

To understand the dynamics and health of marine ecosystems such as lagoons and coral reefs as well as to understand the impact of human activities on these systems, it is imperative to predict the residence times of water masses and connectivity between ocean domains. In the present work, we consider the pristine lagoons and coral reefs of the Red Sea as an example of such sensitive ecosystems, with a large number of marine species, many of which are unique to the region. To study the residence times and connectivity patterns, we make use of recent advances in dynamic three-dimensional Lagrangian analyses using partial differential equations. Specifically, we extend and apply our novel efficient flow map composition scheme to predict the time needed for any particular water parcel to leave the domain of interest (i.e., a lagoon) as well as the time for any particular water parcel to enter that domain. These spatiotemporal residence time fields along with four-dimensional Lagrangian metrics such as finite time Lyapunov exponent (FTLE) fields provide a quantitative description of the Lagrangian pathways and connectivity patterns of lagoons in the Red Sea.

Recent advances in probabilistic forecasting of regional ocean dynamics, and stochastic optimal path planning with massive ensembles motivate principled analysis of their large datasets. Specifically, stochastic time-optimal path planning in strong, dynamic and uncertain ocean flows produces a massive dataset of the stochastic distribution of exact timeoptimal trajectories. To synthesize such big data and draw insights, we apply machine learning and data mining algorithms. Particularly, clustering of the time-optimal trajectories is important to describe their PDFs, identify representative paths, and compute and optimize risk of following these paths. In the present paper, we explore the use of hierarchical clustering algorithms along with a dissimilarity matrix computed from the pairwise discrete Frechet distance between all the optimal trajectories. We apply the algorithms to two datasets of massive ensembles of vehicle trajectories in a stochastic flow past a circular island and stochastic wind driven double gyre flow. These paths are computed by solving our dynamically orthogonal level set equations. Hierarchical clustering is applied to the two datasets, and results are qualitatively and quantitatively analyzed.

With advances in engineering and technology, mining the deep sea for untapped rare metal resources from the bottom of the ocean has recently become economically viable. However, extracting these metal ores from the seabed creates plumes of fine particles that are deposited at various depths within the ocean, and these may be extremely harmful to the marine ecosystems and its components. Thus, for sustainable management, it is of utmost importance to carefully monitor and predict the impact of such harmful activities including plume dispersion on the marine environment. To forecast the plume dispersion in real-time, data-driven ocean modeling has to be coupled with accurate, efficient, and rigorous sediment plume transport computations. The goal of the present paper is to demonstrate the real-time applications of our coupled 3D-and-time data-driven ocean modeling and plume transport forecasting system. Here, the region of focus is the southern California bight, where the PLUMEX 2018 deep sea mining real-time sea experiment was recently conducted (23 Feb – 5 Mar, 2018). Specifically, we demonstrate the improved capabilities of the multiscale MSEAS primitive equation ocean modeling system to capture the complex oceanic phenomenon in the region of interest, the application of the novel method of composition to efficiently and accurately compute the transport of sediment plumes in 3D+1 domains, and the portability of our software and prediction system to different operational regions and its potential in estimating the environmental impacts of deep sea mining activities, ultimately aiding sustainable management and science-based regulations.

A pressing environmental question facing the ocean is the potential impact of possible deep-sea mining activities. This work presents our initial results in developing an ocean and plume modeling system for the Bismark Sea where deep sea mining operations will probably take place. We employ the MSEAS modeling system to both simulate the ocean and to downscale initial conditions from a global system (HYCOM) and tidal forcing from the global TPXO-8 Atlas. We found that at least 1.5 km resolution was needed to adequately resolve the multiscale flow fields. In St. Georges channel, the interaction between the tides, background currents, and underlying density fields increased the subtidal flows. Comparing to historical transport estimates, we showed that tidal forcing is needed to maintain the correct subtidal transport through that Channel. Comparisons with past simulations and measured currents all showed good agreement between the MSEAS hindcasts. Quantitative comparisons made between our hindcasts and independent synoptic ARGO profiles showed that the hindcasts beat persistence by 33% to 50%. These comparisons demonstrated that the MSEAS current estimates were useful for assessing plume advection. Our Lagrangian transport and coherence analyses indicate that the specific location and time of the releases can have a big impact on their dispersal. Our results suggest that ocean mining plumes can be best mitigated by managing releases in accord with such ocean modeling and Lagrangian transport forecasts. Real-time integrated mining-modeling-sampling is likely to provide the most effective mitigation strategies.

The observation, computation and study of “Lagrangian Coherent Structures” (LCS) in turbulent geophysical flows have been active areas of research in fluid mechanics for the last 30 years. Growing evidence for the existence of LCSs in geophysical flows (e.g., eddies, oscillating jets, chaotic mixing) and other fluid flows (e.g., separation prole at the surface of an airfoil, entrainment and detrainment by a vortex) generates an increasing interest for the extraction and understanding of these structures as well as their properties. In parallel, realistic ocean modeling with dense data assimilation has developed in the past decades and is now able to provide accurate nowcasts and predictions of ocean flow fields to study coherent structures. Robust numerical methods and sufficiently fast hardware are now available to compute real-time forecasts of oceanographic states and render associated coherent structures. It is therefore natural to expect the direct predictions of LCSs based on these advanced models. The impact of uncertainties on the coherent structures is becoming an increasingly important question for practical applications. The transfer of these uncertainties from the ocean state to the LCSs is an unexplored but intriguing scientific problem. These two questions are the motivation and focus of this presentation. Using the classic formalism of continuous-discrete estimation [1], the spatially discretized dynamics of the ocean state vector x and observations are described by (1a) dx =M(x; t) + d yok (1b) = H(xk; tk) + k where M and H are the model and measurement model operator, respectively. The stochastic forcings d and k are Wiener/Brownian motion processes,   N(0;Q(t)), and white Gaussian sequences, k  N(0;Rk), respectively. In other words, Efd(t)d T (t)g := Q(t) dt. The initial conditions are also uncertain and x(t0) is random with a prior PDF, p(x(t0)), i.e. x(t0) = bx0 + n(0) with n(0) random. Of course, vectors and operators in Eqs. (1a-b) are multivariate which impacts the PDFs: e.g. their moments are also multivariate. The estimation problem at time t consists of combining all available information on x(t), the dynamics and data (Eqs. 1a-b), their prior distributions and the initial conditions p(x(t0)). Defining the set of all observations prior to time t by yt