This paper presents a rigorous, yet practical, method of multigrid data assimilation into regional structured-grid tidal models. The new inverse tidal nesting scheme, with nesting across multiple grids, is designed to provide a fit of the tidal dynamics to data in areas with highly complex bathymetry and coastline geometry. In these areas, computational constraints make it impractical to fully resolve local topographic and coastal features around all of the observation sites in a stand-alone computation. The proposed strategy consists of increasing the model resolution in multiple limited area domains around the observation locations where a representativeness error is detected in order to improve the representation of the measurements with respect to the dynamics. Multiple high-resolution nested domains are set up and data assimilation is carried out using these embedded nested computations. Every nested domain is coupled to the outer domain through the open boundary conditions (OBCs). Data inversion is carried out in a control space of the outer domain model. A level of generality is retained throughout the presentation with respect to the choice of the control space; however, a specific example of using the outer domain OBCs as the control space is provided, with other sensible choices discussed. In the forward scheme, the computations in the nested domains do not affect the solution in the outer domain. The subsequent inverse computations utilize the observation-minus-model residuals of the forward computations across these multiple nested domains in order to obtain the optimal values of parameters in the control space of the outer domain model. The inversion is carried out by propagating the uncertainty from the control space to model tidal fields at observation locations in the outer and in the nested domains using efficient low-rank error covariance representations. Subsequently, an analysis increment in the control space of the outer domain model is computed and the multigrid system is steered optimally towards observations while preserving a perfect dynamical balance. The method is illustrated using a real-world application in the context of the Philippines Strait Dynamics experiment.

We present the spatial and temporal variability of the acoustic field in Dabob Bay during the PLUSNet07 Exercise. The study uses a 4-D data-assimilative numerical ocean model to provide input to an acoustic propagation model. The ocean physics models (primitive-equations and tidal models), with CTD data assimilation, provided ocean predictions in the region. The output ocean forecasts had a 300m and 1-5m resolution in the horizontal and vertical directions, at 3-hour time intervals within a 15-day period. This environmental data, as the input to acoustic modeling, allowed for the prediction and study of the temporal variations of the acoustic field, as well as the varying spatial structures of the field. Using a one-way coupled-normal-mode code, along- and across-sections in the Dabob Bay acoustic field structures at 100, 400, and 900 Hz were forecasted and described twice-daily, for various source depths. Interesting propagation effects, such as acoustic fluctuations with respect to the source depth and frequency as a result of the regional ocean variability, wind forcing, and tidal effects are discussed. The novelty of this work lies in the possibility of accurate acoustic TL prediction in the littoral region by physically coupling the real-time ocean prediction system to real-time acoustic modeling.

The goal of adaptive sampling in the ocean is to predict the types and locations of additional ocean measurements that would be most useful to collect. Quantitatively, what is most useful is defined by an objective function and the goal is then to optimize this objective under the constraints of the available observing network. Examples of objectives are better oceanic understanding, to improve forecast quality, or to sample regions of high interest. This work provides a new path-planning scheme for the adaptive sampling problem. We define the path-planning problem in terms of an optimization framework and propose a method based on mixed integer linear programming (MILP). The mathematical goal is to find the vehicle path that maximizes the line integral of the uncertainty of field estimates along this path. Sampling this path can improve the accuracy of the field estimates the most. While achieving this objective, several constraints must be satisfied and are implemented. They relate to vehicle motion, intervehicle coordination, communication, collision avoidance, etc. The MILP formulation is quite powerful to handle different problem constraints and flexible enough to allow easy extensions of the problem. The formulation covers single- and multiple-vehicle cases as well as singleand multiple-day formulations. The need for a multiple-day formulation arises when the ocean sampling mission is optimized for several days ahead. We first introduce the details of the formulation, then elaborate on the objective function and constraints, and finally, present a varied set of examples to illustrate the applicability of the proposed method.

In this thesis, we explore the different methods for parameter estimation in straightforward diffusion problems and develop ideas and distributed computational schemes for the automated evaluation of physical and numerical parameters of ocean models. This is one step of “adaptive modeling”. Adaptive modeling consists of the automated adjustment of self-evaluating models in order to best represent an observed system. In the case of dynamic parameterizations, self-modifying schemes are used to learn the correct model for a particular regime as the physics change and evolve in time.

The parameter estimation methods are tested and evaluated on one-dimensional tracer diffusion problems. Existing state estimation methods and new filters, such as the unscented transform Kalman filter, are utilized in carrying out parameter estimation. These include the popular Extended Kalman Filter (EKF), the Ensemble Kalman Filter (EnKF) and other ensemble methods such as Error Subspace Statistical Estimation (ESSE) and Ensemble Adjustment Kalman Filter (EAKF), and the Unscented Kalman Filter (UKF). Among the aforementioned recursive state estimation methods, the so-called “adjoint method” is also applied to this simple study.

Finally, real data is examined for the applicability of such schemes in real-time fore- casting using the MIT Multidisciplinary Simulation, Estimation, and Assimilation System (MSEAS). The MSEAS model currently contains the free surface hydrostatic primitive equation model from the Harvard Ocean Prediction System (HOPS), a barotropic tidal prediction scheme, and an objective analysis scheme, among other models and developing routines. The experiment chosen for this study is one which involved the Monterey Bay region off the coast of California in 2006 (MB06). Accurate vertical mixing parameterizations are essential in this well known upwelling region of the Pacific. In this realistic case, parallel computing will be utilized by scripting code runs in C-shell. The performance of the simulations with different parameters is evaluated quantitatively using Pattern Correlation Coefficient, Root Mean Squared error, and bias error. Comparisons quantitatively determined the most adequate model setup.

In 1969, the Journal of Computational Physics published a seminal article by K. Bryan presenting the first ocean general circulation model. Since then, many numerical studies of the World Ocean, as well as regional or coastal flows, used models directly or indirectly inspired by the work of Bryan and his colleagues. A number of these models have evolved into highly modular and versatile computational systems, including multiple physical modules and options as well as varied biogeochemical, ecosystem and acoustics modeling capabilities. Several modeling systems are now well-documented tools, which are widely used in research institutions and various organizations around the world. The list of such modeling systems is large and too long to be summarized in this editorial. Over the last three decades, significant progress has been made in the parameterization of subgrid-scale processes, in data assimilation methodologies and in boundary condition schemes, as well as in the efficient implementation of algorithms on fast vector and subsequently parallel computers, allowing higher and higher resolution in space and time. However, many of today’s popular modeling systems can still be regarded as members of the first generation of ocean models: at their core, rather similar geophysical fluid dynamics equations are solved numerically using a conservative finite-difference method on a structured grid. Today, several aspects of structured-grid models could benefit from significant upgrades, learning from major advances in computational fluid dynamics. In particular, the use of a structured grid limits the flexibility in the spatial resolution and does not allow one to take full advantage of numerical algorithms such as finite volumes and finite elements, which can achieve their best performance when implemented on unstructured meshes. Even though many of today’s complex marine modeling and data assimilation systems have evolved significantly since Bryan’s prototype, it would be challenging to modify them step-by-step from a structured-grid approach to an unstructured-grid one. Therefore, novel marine model design research is underway, paving the way for the second generation of ocean modeling systems. It is difficult to predict today if this new generation of ocean models will achieve its chief objective: widening the range of resolved scales of motion with increased efficiencies and accuracies, possibly allowing multi-resolution, multi-scale, and multidynamics numerical simulations of marine flows, all occurring seamlessly within distributed computing environments. In fact, hybrid approaches merging the advantages of structured and unstructured-grid modeling may be the way forward. Whether or not unstructured mesh approaches will prevail is all the more difficult to predict now that structured mesh modelers have developed powerful solutions for increasing the resolution when and where needed. For instance, grid embedding is still a popular and useful method for enhancing model resolution. It can involve multiply nested domains and allows the relatively straightforward use of different dynamics or models in each domain. Research is also underway for developing multigrid, wavelet, and other multi-scale decompositions for the numerical solution of dynamical equations but also for the study of results, model evaluation or data assimilation. This special issue presents a number of examples of the abovementioned developments. Ringler et al. examine the potential of spherical centroidal Voronoi tessellations for performing multi-resolution simulations; they apply this method to the Greenland ice sheet and the North Atlantic Ocean. Lambrechts et al. present a triangular mesh generation system and its applications to the World Ocean and various shelf seas, including the Great Barrier Reef, Australia. Finite element models on unstructured grids are described and utilized in several manuscripts. Bellafiore et al. study the Adriatic Sea and the Lagoon of Venice, while Jones and Davies simulate tides and storm surges along the western coast of Britain. Danilov et al. assess two finite element discretizations, i.e., a continuous element and a non-conforming one, and compare the results of these discretizations with those of a finite-difference model. In Harig et al., the tsunami generated by the great Sumatra-Andaman earthquake of 26 December 2004 is simulated by means of a finite element model. Comparisons are carried out with various types of data as well as with the results of a structured mesh model using a nested structured-grid system. A nested-grid ocean circulation model is also employed by Yang and Sheng to carry out a process study on the Inner Scotian Shelf, Canada, focusing on the circulation induced by a tropical storm. Debreu and Blayo present a detailed review of two-way embedding algorithms for structured-grid models. Finally, Logutov develops a multi-scale assimilation scheme for tidal data within the framework of a multiply nested structured-grid barotropic tidal modeling approach. As illustrated by these manuscripts, the next generation of ocean modelers is motivated by a wide range of research opportunities over a rich spectrum of needs. Future progress will involve fundamental and applied numerical and computational research as well as new multi-scale geophysical fluid modeling. Domains of ongoing interest range from estuaries to the global ocean, including coastal regions and shelf seas. New multi-scale modeling of physical as well as biological, chemical or interdisciplinary processes will flourish in the coming decades. We are grateful to the authors for their contributions and to the chief-editor for his support in this endeavor. We are thankful to the reviewers for their time and help in assessing the manuscripts submitted to this special issue. Eric Deleersnijder is a Research associate with the Belgian National Fund for Scientific Research (FNRS); he is indebted to the Communaut Francaise de Belgique for its support through contract ARC 04/09-316. Pierre Lermusiaux is grateful to the Office of Naval Research for support under grant N00014-08-1-1097 to the Massachusetts Institute of Technology.