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Numerical Schemes for Dynamically Orthogonal Equations of Stochastic Fluid and Ocean Flows

Ueckermann, M.P., P.F.J. Lermusiaux and T.P. Sapsis, 2013. Numerical Schemes for Dynamically Orthogonal Equations of Stochastic Fluid and Ocean Flows. J. Comp. Phys., 233, 272-294, doi: 10.1016/j.jcp.2012.08.041.

The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multiscale, intermittent and non- homogeneous fluid and ocean flows. The Dynamically Orthogonal (DO) field equations provide an efficient time-dependent adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier-Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit projection methods are developed for the mean and for the orthonormal modes that define a basis for the evolving DO subspace, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with Total Variation Diminishing schemes for the advection terms. Other results specific to the DO equations include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size in- stead of quadratic; (ii) symmetric Total Variation Diminishing-based advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. To verify the correctness of our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.

Quantifying, predicting, and exploiting uncertainties in marine environments

Rixen, M., P.F.J. Lermusiaux and J. Osler, 2012. Quantifying, Predicting and Exploiting Uncertainties in Marine Environments, Ocean Dynamics, 62(3):495–499, doi: 10.1007/s10236-012-0526-8.

Following the scientific, technical and field trial initiatives ongoing since the Maritime Rapid Environmental Assessment (MREA) conferences in 2003, 2004 and 2007, the MREA10 conference provided a timely opportunity to review the progress on various aspects of MREA, with a particular emphasis on marine environmental uncertainty management. A key objective of the conference was to review the present state-of-the art in Quantifying, Predicting and Exploiting (QPE) marine environmental uncertainties. The integration of emerging environmental monitoring and modeling techniques into data assimilation streams and their subsequent exploitation at an operational level involves a complex chain of non-linear uncertainty transfers, including human factors. Accordingly the themes for the MREA10 conference were selected to develop a better understanding of uncertainty, from its inception in the properties being measured and instrumentation employed, to its eventual impact in the applications that rely upon environmental information.

Contributions from the scientific community were encouraged on all aspects of environmental uncertainties: their quantification, prediction, understanding and exploitation. Contributions from operational communities, the consumers of environmental information who have to cope with uncertainty, were also encouraged. All temporal and spatial scales were relevant: tactical, operational, and strategic, including uncertainty studies for topics with long-term implications. Manuscripts reporting new technical and theoretical developments in MREA, but acknowledging effects of uncertainties to be accounted for in future research, were also included.

The response was excellent with 87 oral presentations (11 of which were invited keynote speakers) and 24 poster presentations during the conference. A subset of these presentations was submitted to this topical issue and 22 manuscripts have been published by Ocean Dynamics.

Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty

Sapsis, T.P. and P.F.J. Lermusiaux, 2012. Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty. Physica D, 241(1), 60-76, doi:10.1016/j.physd.2011.10.001.

We estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations, within the context of dynamically orthogonal (DO) field equations. Transient nonlinear dynamics, irregular data and non-stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation. To efficiently quantify uncertainties in such systems, it is essential to vary the dimensionality of the stochastic subspace with time. An objective here is to provide criteria to do so, working directly with the original equations of the dynamical system under study and its DO representation. We first analyze the scaling of the computational cost of these DO equations with the stochastic dimensionality and show that unlike many other stochastic methods the DO equations do not suffer from the curse of dimensionality. Subsequently, we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i) stability arguments and ii) Bayesian data updates. We then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2D stochastic fluid flows, specifically a double-gyre wind-driven circulation and a lid-driven cavity flow in a basin. In these two applications, we focus on the growth of uncertainty due to internal instabilities in deterministic flows. We consider a range of flow conditions described by varied Reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions.

Dynamically orthogonal field equations for continuous stochastic dynamical systems

Sapsis, T.P. and P.F.J. Lermusiaux, 2009. Dynamically orthogonal field equations for continuous stochastic dynamical systems. Physica D, 238, 2347-2360, doi:10.1016/j.physd.2009.09.017.

In this work we derive an exact, closed set of evolution equations for general continuous stochastic fields described by a Stochastic Partial Differential Equation (SPDE). By hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component, we derive a system of field equations consisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal basis that describe the stochastic subspace where the stochasticity `lives’ as well as a system of Stochastic Differential Equations that defines how the stochasticity evolves in the time varying stochastic subspace. These new evolution equations are derived directly from the original SPDE, using nothing more than a dynamically orthogonal condition on the representation of the solution. If additional restrictions are assumed on the form of the representation, we recover both the Proper Orthogonal Decomposition equations and the generalized Polynomial Chaos equations. We apply this novel methodology to two cases of two-dimensional viscous fluid flows described by the NavierStokes equations and we compare our results with Monte Carlo simulations.

Uncertainty Estimation and Prediction for Interdisciplinary Ocean Dynamics

Lermusiaux, P.F.J., 2006. Uncertainty Estimation and Prediction for Interdisciplinary Ocean Dynamics. Refereed manuscript, Special issue on "Uncertainty Quantification". J. Glimm and G. Karniadakis, Eds. Journal of Computational Physics, 217, 176-199. doi: 10.1016/j.jcp.2006.02.010.

Scientific computations for the quantification, estimation and prediction of uncertainties for ocean dynamics are developed and exemplified. Primary characteristics of ocean data, models and uncertainties are reviewed and quantitative data assimilation concepts defined. Challenges involved in realistic data-driven simulations of uncertainties for four-dimensional interdisciplinary ocean processes are emphasized. Equations governing uncertainties in the Bayesian probabilistic sense are summarized. Stochastic forcing formulations are introduced and a new stochastic-deterministic ocean model is presented. The computational methodology and numerical system, Error Subspace Statistical Estimation, that is used for the efficient estimation and prediction of oceanic uncertainties based on these equations is then outlined. Capabilities of the ESSE system are illustrated in three data-assimilative applications: estimation of uncertainties for physical-biogeochemical fields, transfers of ocean physics uncertainties to acoustics, and real-time stochastic ensemble predictions with assimilation of a wide range of data types. Relationships with other modern uncertainty quantification schemes and promising research directions are discussed.

Quantifying Uncertainties in Ocean Predictions

Lermusiaux, P.F.J., C.-S. Chiu, G.G. Gawarkiewicz, P. Abbot, A.R. Robinson, R.N. Miller, P.J. Haley, W.G. Leslie, S.J. Majumdar, A. Pang and F. Lekien, 2006. Quantifying Uncertainties in Ocean Predictions. Refereed invited manuscript. Oceanography, Special issue on "Advances in Computational Oceanography", T. Paluszkiewicz and S. Harper (Office of Naval Research), Eds., 19, 1, 92-105, doi: 10.5670/oceanog.2006.93.

A multitude of physical and biological processes occur in the ocean over a wide range of temporal and spatial scales. Many of these processes are nonlinear and highly variable, and involve interactions across several scales and oceanic disciplines. For example, sound propagation is influenced by physical and biological properties of the water column and by the seabed. From observations and conservation laws, ocean scientists formulate models that aim to explain and predict dynamics of the sea. This formulation is intricate because it is challenging to observe the ocean on a sustained basis and to transform basic laws into generic but usable models. There are imperfections in both data and model estimates. It is important to quantify such uncertainties to understand limitations and identify the research needed to increase accuracies, which will lead to fundamental progress. There are several sources of uncertainties in ocean modeling. First, to simplify models (thereby reducing computational expenses), explicit calculations are only performed on a restricted range of spatial and temporal scales (referred to as the “scale window”) (Nihoul and Djenidi, 1998). Influences of scales outside this window are neglected, parameterized, or provided at boundaries. Such simplifications and scale reductions are a source of error. Second, uncertainties also arise from the limited knowledge of processes within the scale window, which leads to approximate representations or parameterizations. Third, ocean data are required for model initialization and parameter values; however, raw measurements are limited in coverage and accuracy, and they are often processed with the aim of extracting information within a predetermined scale window. Initial conditions and model parameters are thus inexact. Fourth, models of interactions between the ocean and Earth system are approximate and ocean boundary conditions are inexact. For example, effects of uncertain atmospheric fluxes can dominate oceanic uncertainty. Fifth, miscalculations occur due to numerical implementations. All of the above leads to differences between the actual values (unknown) and the measured or modeled values of physical, biological, and geo-acoustical fields and properties.

Transfer of uncertainties through physical-acoustical-sonar end-to-end systems: A conceptual basis

Robinson, A.R., P. Abbot, P.F.J. Lermusiaux and L. Dillman, 2002. Transfer of uncertainties through physical-acoustical-sonar end-to-end systems: A conceptual basis. In "Acoustic Variability, 2002:. N.G. Pace and F.B. Jensen (Eds.), SACLANTCEN. Kluwer Academic Press, 603-610.

An interdisciplinary team of scientists is collaborating to enhance the understanding of the uncertainty in the ocean environment, including the sea bottom, and characterize its impact on tactical system performance. To accomplish these goals quantitatively an end-to-end system approach is necessary. The conceptual basis of this approach and the framework of the end-to-end system, including its components, is the subject of this presentation. Specifically, we present a generic approach to characterize variabilities and uncertainties arising from regional scales and processes, construct uncertainty models for a generic sonar system, and transfer uncertainties from the acoustic environment to the sonar and its signal processing. Illustrative examples are presented to highlight recent progress toward the development of the methodology and components of the system.

Evolving the subspace of the three-dimensional multiscale ocean variability: Massachusetts Bay

Lermusiaux, P.F.J., 2001. Evolving the subspace of the three-dimensional multiscale ocean variability: Massachusetts Bay. Journal of Marine Systems, Special issue on "Three-dimensional ocean circulation: Lagrangian measurements and diagnostic analyses", 29/1-4, 385-422, doi: 10.1016/S0924-7963(01)00025-2.

A data and dynamics driven approach to estimate, decompose, organize and analyze the evolving three-dimensional variability of ocean fields is outlined. Variability refers here to the statistics of the differences between ocean states and a reference state. In general, these statistics evolve in time and space. For a first endeavor, the variability subspace defined by the dominant eigendecomposition of a normalized form of the variability covariance is evolved. A multiscale methodology for its initialization and forecast is outlined. It combines data and primitive equation dynamics within a Monte-Carlo approach. The methodology is applied to part of a multidisciplinary experiment that occurred in Massachusetts Bay in late summer and early fall of 1998. For a 4-day time period, the three-dimensional and multivariate properties of the variability standard deviations and dominant eigenvectors are studied. Two variability patterns are discussed in detail. One relates to a displacement of the Gulf of Maine coastal current offshore from Cape Ann, with the creation of adjacent mesoscale recirculation cells. The other relates to a Bay-wide coastal upwelling mode from Barnstable Harbor to Gloucester in response to strong southerly winds. Snapshots and tendencies of physical fields and trajectories of simulated Lagrangian drifters are employed to diagnose and illustrate the use of the dominant variability covariance. The variability subspace is shown to guide the dynamical analysis of the physical fields. For the stratified conditions, it is found that strong wind events can alter the structures of the buoyancy flow and that circulation features are more variable than previously described, on multiple scales. In several locations, the factors estimated to be important include some or all of the atmospheric and surface pressure forcings, and associated Ekman transports and downwelling/upwelling processes, the Coriolis force, the pressure force, inertia and mixing.