In our tests, the slope-limiter removed spurious numerical oscillations in tracer advection with sharp, discontinuous fronts. However, we noted additional diffusion in regions where the tracer fields were smooth. Our filtering approach, with an appropriately chosen strength, successfully removed spurious numerical oscillations for the same tests, but did not introduce additional diffusion where tracer fields were smooth (Ueckermann et. al., 2010). We obtained high-order accurate solutions of biological dynamics using our discontinuous Galerkin schemes (Fig. 1). We find that both quadrature-based and quadrature-free discretizations give accurate results in well-resolved regions, but the quadrature-based scheme has smaller errors in under-resolved regions. We show that low-order temporal discretizations allow rapidly growing numerical errors in biological fields. We find that if a spatial discretization (mesh resolution and polynomial degree) does not resolve the solution, oscillations due to discontinuities in tracer fields can be locally significant for both low and high-order discretizations. When the solution is sufficiently resolved, higher-order schemes on coarser grids perform better (higher accuracy, less dissipative) for the same cost than lower-order scheme on finer grids. This result applies to both passive and reactive tracers and is confirmed by quantitative analyses of truncation errors and smoothness of solution fields. To reduce oscillations in un-resolved regions, we develop a numerical filter that is active only when and where the solution is not smooth locally. Finally, our idealized simulations of biological patchiness (Fig. 2) reveal that higher-order numerical schemes can maintain patches for long-term integrations while lower order schemes are much too dissipative and cannot, even at very high resolutions. Implications for the use of simulations to better understand biological blooms, patchiness, and other nonlinear reactive dynamics in coastal regions with complex bathymetric features are considerable.

We found that the adaptive-size algorithm gave accurate results, and the new DO implementation was considerably more efficient (100-1000 time faster) than our first implementation. We apply the derived framework for the study of the statistical responses of the lid-driven cavity flow and "double gyre" circulation, which has elements of ocean, atmospheric and climate instability behaviors. In Figure 3 we present the mean flow (top - left) for the lid-driven cavity flow and for Reynolds number Re = 10000. For the considered Reynolds number, the flow is unstable causing a very small stochastic perturbation applied at the beginning to grow, leading to non-Gaussian statistics with strong spatial dependence. These properties are captured with our adaptive DO schemes. In Figure 4 we present the stochastic response of a double gyre flow driven by wind-stress forcing, constant in time and symmetric in space, in the presence of Coriolis acceleration. As in the previous case, instabilities lead to growth of an initially small stochastic perturbation. Our adaptive criteria allow for the efficient evolution of the stochastic subspace dimensionality. Moreover, we find that the mean flow preserve its symmetry properties while all the anti-symmetric part (which is due to the Coriolis terms) is captured by the stochastic modes. The bimodal symmetric structure of the probability density functions reveal the bi-stable states of the system, clearly illustrating that the system will "randomly" follow one of the two states in the bimodal distribution.

Results using idealized canonical flow fields are illustrated in Figs. 5a-5d. Such canonical flows could be very useful as rules-of-thumb for naval operations. The results for an example of complex flows are illustrated in Figure 6. The targets are i) sampling as far north as possible, and, ii) sampling the south-west corner of the domain, in both cases from a specified starting position. This figure shows that the optimally planned paths are the ones that reach the desired targets the fastest. Trajectories chosen assuming no flow-field do not reach their targets, and do not exist on the wavefront of best possible paths (e.g. see the magenta path). The green path illustrates the plan of a "smart oceanographer". After studying the flow field, it was assigned a due west heading for a certain amount of time, followed by a due south heading. The transition time between headings was determined by looking at the trajectory of the optimally planned path and flow fields. While this insightfully guessed green path does approach the target point, the red path still gets first to the target. This reveals that, though a very good guess could reach the desired targets in a quasi-optimal time, the calculated optimal path is still superior and is computed autonomously. We also show the evolution of the zero-level set superimposed with the magnitude of the local uncertainty. Future work includes finding paths that optimally utilize or reduce flow uncertainties. Another next step would be to predict the uncertainty of the level-sets themselves.

Traditional assimilation schemes partially retain the assumptions of linearity and Gaussianity in the analysis step, as in the Kalman Filter, and thus typically do not use information beyond the first two moments of the ensemble distribution. Consequently, much of the non-Gaussian structure may be lost during the analysis. One new scheme we are investigating is based on approximating the prior distribution by a Gaussian mixture. This is determined using the Expectation-Maximization (EM) algorithm and tested for optimality using the Bayesian Information Criterion. We consequently perform the analysis step by implementing update for each Gaussian mixture component, weighted by its likelihood as determined by the EM-algorithm. In the limit of large mixture complexity and ensemble members, this method may be shown to reproduce the Bayesian update.

The Minimum Divergence Filter is an assimilation scheme which uses general background probability density functions. The framework proves particularly useful in cases where the background distribution exhibits multimodal behavior, e.g. as for the Kuroshio. At the analysis step, the prior distribution is determined as that which minimizes the Kullback-Leibler (K-L) divergence among a set of distributions consistent with specific moments of the background distribution. We have implemented the scheme in a tracking scenario using the "Double Well Diffusion Experiment", assuming a Gaussian Mixture. Work is underway to extend the schemes to more complex cases.

Using our 2D Finite Volume codes, we tested of our new DA schemes on simple double-well problems as well as on several 2D flows. One 2D flow example is a sudden expansion in a pipe which forms re-circulation eddies and instabilities similar to those of ocean jets. Figs. 7a and 7b plot snapshots of the results at two arbitrary, but adjacent, time instants. The transition in time, during which data is assimilated, clearly shows the ability of the new filter to capture the true solution via appropriate assimilation of the limited data. Fig. 8 further shows how the new DA scheme captures the non-Gaussian structure of the probability density function. Preliminary results show improvements over DA schemes using Gaussian-based analyses, including the Ensemble Kalman Filter.

A manuscript on the oceanographic and atmospheric conditions on the continental shelf north of the Monterey Bay during August 2006 was submitted (Ramp et al, 2011), showing that multiscale 2-way nested modeling systems were most accurate for the region allowing for example to capture some of the cross-shelf transports. A review of the multiple scales and features occurring in the California Current System was submitted (Gangopadhyay et al, 2011), including the development of a Feature-Oriented Regional Modeling System for the region. A special edition of Ocean dynamics on multi-scale modeling of coastal, shelf and global ocean dynamics was completed (Deleersnijder et al, 2010). A publication on "Many Task Computing" for multidisciplinary real-time uncertainty prediction and data assimilation was written and accepted (Evangelinos et al., 2010) and earlier results published (Evangelinos et al., 2009). The dynamic coupling of models and measurements for a Dynamic Data-Driven Application System (DDDAS) is described based on examples in Massachusetts Bay, and Monterey Bay and the California Current System (Patrikalakis et al., 2009).

Our study of 2-way nesting schemes showed that for nesting with free surfaces, the most accurate schemes are those with the strongest implicit couplings among grids, especially for velocity (Haley and Lermusiaux, 2010). For the simulations studied, the scheme with the least amount of fine grid feedback (Fig. 9) had differences between the barotropic velocity estimates of O(10) cm/s, with the structures of the difference field organized on the (sub)mesoscale. Conversely, the scheme with the most feedback (Fig. 10) had the smallest discrepancies, much smaller than 1 cm/s except near steep topography or strong dynamical gradients where differences reach 1 cm/s: the finer grid is needed there to represent this variability.

With our theoretical truncation error analysis, we revealed the benefits of additional feedback from the fine to coarse domains. We proved that coarse domain estimates which are made up from averages of fine domain estimates retain the truncation error of the fine grid. Even for a second order scheme with only a 3:1 refinement, this equates to an order of magnitude reduction in the truncation error at these coarse domain points. A corollary of this analysis is that the improvement from the fine grid feedback can be guaranteed only if the feedback algorithm (in our case volume-averaging) has at least the same order of accuracy as that of the overall discretization.

In our three realistic simulations, we resolved large domains with multiscale dynamics, including steep bathymetries and strong tidal flows in shallow seas. In each case, we found that without our new discrete PE model, or without nested grids, predictions do not match the ocean data. Specifically, in the middle Atlantic Bight (Fig. 11), we compared nested estimates to mooring data not assimilated in simulations (Fig. 12). Using twin experiments between a 3km resolution "stand-alone" domain and the same domain with a two-way nested 1km resolution domain, we showed that the addition of the nested sub-domain removes large biases and RMSEs, O(10-15) cm/s, in the model velocities when compared to the mooring data. The 2-way nesting scheme is found especially needed during tropical storm Ernesto. It is also required for future studies of internal tide propagation.

In our study of the multiscale dynamics in the Philipines (Lermusiaux et al, 2011), our oceanographic findings include: the biogeochemical features forecast in real-time, the multiscale circulation features and their characterization, the evolution of flow fields within three major straits, the estimation of transports to and from the Sulu Sea and the corresponding balances, and finally, an investigation of multiscale mechanisms involved in the formation of the deep Sulu Sea water.