A Coupled-Mode Shallow-Water Model for Tidal Analysis: Internal Tide Reflection and Refraction by the Gulf Stream

A hydrostatic, coupled-mode, shallow-water model (CSW) is described and used to diagnose and simulate tidal dynamics in the greater Mid-Atlantic Bight region. The reduced-physics model incorporates realistic stratification and topography, internal tide forcing from a priori estimates of the surface tide, and advection terms that describe first-order interactions of internal tides with slowly varying mean flow and mean buoyancy fields and their respective shear. The model is validated via comparisons with semianalytic models and nonlinear primitive equation models in several idealized and realistic simulations that include internal tide interactions with topography and mean flows. Then, 24 simulations of internal tide generation and propagation in the greater Mid-Atlantic Bight region are used to diagnose significant internal tide interactions with the Gulf Stream. The simulations indicate that locally generated mode-one internal tides refract and/or reflect at the Gulf Stream. The redirected internal tides often reappear at the shelf break, where their onshore energy fluxes are intermittent (i.e., noncoherent with surface tide) because meanders in the Gulf Stream alter their precise location, phase, and amplitude. These results provide an explanation for anomalous onshore energy fluxes that were previously observed at the New Jersey shelf break and linked to the irregular generation of nonlinear internal waves.National Science Foundation (U.S.) (Grant OCE-1061160 (ShelfIT))National Science Foundation (U.S.) (Grant OCE-1060430)United States. Office of Naval Research (Grants N000 14-11-1-0701 (MURI- IODA))United States. Office of Naval Research (N00014-12-1-0944 (ONR6.2)

Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations. Part I. Theory and Scheme

This work introduces and derives an efficient, data-driven assimilation scheme, focused on a time-dependent stochastic subspace, that respects nonlinear dynamics and captures non-Gaussian statistics as it occurs. The motivation is to obtain a filter that is applicable to realistic geophysical applications but that also rigorously utilizes the governing dynamical equations with information theory and learning theory for efficient Bayesian data assimilation. Building on the foundations of classical filters, the underlying theory and algorithmic implementation of the new filter are developed and derived. The stochastic Dynamically Orthogonal (DO) field equations and their adaptive stochastic subspace are employed to predict prior probabilities for the full dynamical state, effectively approximating the Fokker-Planck equation. At assimilation times, the DO realizations are fit to semiparametric Gaussian mixture models (GMMs) using the Expectation-Maximization algorithm and the Bayesian Information Criterion. Bayes’ Law is then efficiently carried out analytically within the evolving stochastic subspace. The resulting GMM-DO filter is illustrated in a very simple example. Variations of the GMM-DO filter are also provided along with comparisons with related schemes.United States. Office of Naval Research (Grant N00014-08-1-1097)United States. Office of Naval Research (Grant 00014-09-1-0676)United States. Office of Naval Research (Grant N00014-08-1-0586

Bayesian Learning of Coupled Biogeochemical-Physical Models

Predictive dynamical models for marine ecosystems are used for a variety of needs. Due to sparse measurements and limited understanding of the myriad of ocean processes, there is however significant uncertainty. There is model uncertainty in the parameter values, functional forms with diverse parameterizations, level of complexity needed, and thus in the state fields. We develop a Bayesian model learning methodology that allows interpolation in the space of candidate models and discovery of new models from noisy, sparse, and indirect observations, all while estimating state fields and parameter values, as well as the joint PDFs of all learned quantities. We address the challenges of high-dimensional and multidisciplinary dynamics governed by PDEs by using state augmentation and the computationally efficient GMM-DO filter. Our innovations include stochastic formulation and complexity parameters to unify candidate models into a single general model as well as stochastic expansion parameters within piecewise function approximations to generate dense candidate model spaces. These innovations allow handling many compatible and embedded candidate models, possibly none of which are accurate, and learning elusive unknown functional forms. Our new methodology is generalizable, interpretable, and extrapolates out of the space of models to discover new ones. We perform a series of twin experiments based on flows past a ridge coupled with three-to-five component ecosystem models, including flows with chaotic advection. The probabilities of known, uncertain, and unknown model formulations, and of state fields and parameters, are updated jointly using Bayes' law. Non-Gaussian statistics, ambiguity, and biases are captured. The parameter values and model formulations that best explain the data are identified. When observations are sufficiently informative, model complexity and functions are discovered.Comment: 45 pages; 18 figures; 2 table

Global analysis of Navier–Stokes and Boussinesq stochastic flows using dynamical orthogonality

We provide a new framework for the study of fluid flows presenting complex uncertain behaviour. Our approach is based on the stochastic reduction and analysis of the governing equations using the dynamically orthogonal field equations. By numerically solving these equations, we evolve in a fully coupled way the mean flow and the statistical and spatial characteristics of the stochastic fluctuations. This set of equations is formulated for the general case of stochastic boundary conditions and allows for the application of projection methods that considerably reduce the computational cost. We analyse the transformation of energy from stochastic modes to mean dynamics, and vice versa, by deriving exact expressions that quantify the interaction among different components of the flow. The developed framework is illustrated through specific flows in unstable regimes. In particular, we consider the flow behind a disk and the Rayleigh–Bénard convection, for which we construct bifurcation diagrams that describe the variation of the response as well as the energy transfers for different parameters associated with the considered flows. We reveal the low dimensionality of the underlying stochastic attractor.United States. Office of Naval Research (Grant N00014-08-1-1097 (ONR6.1))United States. Office of Naval Research (Grant N00014-08-1-0586 (QPE))United States. Office of Naval Research (Grant N00014-09-1-0676 (Science of Autonomy - A-MISSION))United States. Office of Naval Research (Grant N00014-12-1-0944 (ONR6.2))Natural Sciences and Engineering Research Council of Canad

Adaptive Sampling Using Fleets of Gliders in the Presence of Fixed Buoys: a Prototype Built Upon the MyOcean Service

In the last decade the use of fleets of gliders has proven to be an effective way for sampling the ocean for long-duration missions (order of months). In a previous study [1] a method for adaptive sampling the ocean using fleets of gliders based on the use of a clustering algorithm has been introduced. The key ideas were: i) build a 2D mesh grid over the synoptic uncertainty of the ocean field to sample with “knots” having density proportional to the level of the uncertainty; ii) group this set of knots using a clustering algorithm, i.e. the Fuzzy C-Means (a fuzzy variant of the well-known K-Means algorithm). The centroids are the next way-points for the gliders. However, that method assumed all-maneuverable assets. In this study we extend it by exploiting the existence of non-maneuverable assets, i.e. fixed buoys (a situation that frequently occurs in real scenarios) and by considering time-dependent uncertainty, i.e. aiming to reach the way-points at time t such that the uncertainty at future times is minimized. The first essential idea is to consider the positions of fixed buoys as part of the centroids to obtain from the clustering algorithm: the remaining centroids to be computed will be considered as the next positions where to send each glider. By using the clustering algorithm described in [2], called “Partially Provided Centroids Fuzzy C-Means” (PPC-FCM), we have been able to exploit the presence of fixed buoys by sending the gliders in regions not already covered by the buoys/floats. This allows a better distribution (lower overlapping) of the sensing assets, with respect to the direct use of the standard Fuzzy C-Means, uninformed of the presence of the buoys. The second idea is to replace the synoptic uncertainty field by the field of mutual information between the way points at time t and a selected future time. We have built a prototype of this novel adaptive sampling scheme for mixed assets (maneuverable and non-maneuverable) that automatically retrieves ocean forecasts (currents, temperature, salinity, etc.) from MyOcean services. In addition, the prototype comes with a graphical user interfaces that facilitates the selection of the region of interest for data download. Once the data have been downloaded with low efforts, the (PPC-FCM) algorithm is run to get the next gliders way-points. The procedure is then repeated any time new forecasts are available. Our tool will be even more effective if MyOcean forecast products in future releases contain, other than the expected (mean) value of the field of interest obtained from forecasting models, a measure of the associated uncertainty, such as standard deviations. By including this uncertainty estimate, glider mission planners would have valuable information on where to send the assets in order to reduce the uncertainty as much as possible. REFERENCES [1] Cococcioni, et. al., «SONGs: Self Organizing Network of Gliders for Adaptive Sampling of the Ocean», Maritime Rapid Environmental Assessment Conference, October 18-22, Lerici, Italy, 2010 [2] Cococcioni, «Clustering in the presence of partially provided centroids: a fuzzy approach», Technical Report, Department of Information Engineering, Pisa, 2014

Time-optimal path planning in dynamic flows using level set equations: realistic applications

The level set methodology for time-optimal path planning is employed to predict collision-free and fastest-time trajectories for swarms of underwater vehicles deployed in the Philippine Archipelago region. To simulate the multiscale ocean flows in this complex region, a data-assimilative primitive-equation ocean modeling system is employed with telescoping domains that are interconnected by implicit two-way nesting. These data-driven multiresolution simulations provide a realistic flow environment, including variable large-scale currents, strong jets, eddies, wind-driven currents, and tides. The properties and capabilities of the rigorous level set methodology are illustrated and assessed quantitatively for several vehicle types and mission scenarios. Feasibility studies of all-to-all broadcast missions, leading to minimal time transmission between source and receiver locations, are performed using a large number of vehicles. The results with gliders and faster propelled vehicles are compared. Reachability studies, i.e., determining the boundaries of regions that can be reached by vehicles for exploratory missions, are then exemplified and analyzed. Finally, the methodology is used to determine the optimal strategies for fastest-time pick up of deployed gliders by means of underway surface vessels or stationary platforms. The results highlight the complex effects of multiscale flows on the optimal paths, the need to utilize the ocean environment for more efficient autonomous missions, and the benefits of including ocean forecasts in the planning of time-optimal paths.United States. Office of Naval Research (Grant N00014-09-1-0676 (Science of Autonomy - A-MISSION))United States. Office of Naval Research (Grant N00014-07-1-0473 (PhilEx))United States. Office of Naval Research (Grant N00014-12-1-0944 (ONR6.2))United States. Office of Naval Research (Grant N00014-13-1-0518 (Multi-DA)

Graph-Search and Differential Equations for Time-Optimal Vessel Route Planning in Dynamic Ocean Waves

Time-optimal paths are evaluated by VISIR (\u201cdis- coVerIng Safe and effIcient Routes\u201d), a graph-search ship routing model, with respect to the solution of the fundamental differential equations governing optimal paths in a dynamic wind-wave environment. The evaluation exercise makes use of identical setups: topological constraints, dynamic wave environmental conditions, and vessel-ocean parametrizations, while advection by external currents is not considered. The emphasis is on predicting the time-optimal ship headings and Speeds Through Water constrained by dynamic ocean wave fields. VISIR upgrades regarding angular resolution, time-interpolation, and static nav- igational safety constraints are introduced. The deviations of the graph-search results relative to the solution of the exact differential equations in both the path duration and length are assessed. They are found to be of the order of the discretization errors, with VISIR\u2019s solution converging to that of the differential equation for sufficient resolution

Time-optimal path planning in dynamic flows using level set equations: theory and schemes

We develop an accurate partial differential equation-based methodology that predicts the time-optimal paths of autonomous vehicles navigating in any continuous, strong, and dynamic ocean currents, obviating the need for heuristics. The goal is to predict a sequence of steering directions so that vehicles can best utilize or avoid currents to minimize their travel time. Inspired by the level set method, we derive and demonstrate that a modified level set equation governs the time-optimal path in any continuous flow. We show that our algorithm is computationally efficient and apply it to a number of experiments. First, we validate our approach through a simple benchmark application in a Rankine vortex flow for which an analytical solution is available. Next, we apply our methodology to more complex, simulated flow fields such as unsteady double-gyre flows driven by wind stress and flows behind a circular island. These examples show that time-optimal paths for multiple vehicles can be planned even in the presence of complex flows in domains with obstacles. Finally, we present and support through illustrations several remarks that describe specific features of our methodology.United States. Office of Naval Research (Grant N00014-09-1-0676 (Science of Autonomy - A-MISSION))United States. Office of Naval Research (Grant N00014-12-1-0944 (ONR6.2))Natural Sciences and Engineering Research Council of Canada (Postgraduate Fellowship