Over the last four decades the use of numerical flow models in oceanography has vastly increased. Models are run operationally for regional locations, ocean basins, and the entire earth. In addition, specialized research models targeting specific processes and areas are routinely produced. These models are often coupled with biological and chemical models for research into biological-physical and biogeochemical-physical interactions. The role of some models is to create conditions close to reality, in a deterministic sense, whereas others have the role of imitating mean behavior or fluctuation behavior. The role of yet another family of models is to alter conditions from reality to study the ramifications, examples being interdisciplinary climate models [1-3]. All of these models provide full access to time- evolving three-dimensional fields (4-D fields) for process studies, or for predictive purposes. There is strong motivation for using these models for ocean acoustic studies. Suitably formulated models can include the important flow and water-mass features of the ocean, with the important features covering a wide dynamic range. Each feature has its own acoustic propagation or scattering signature, with some signatures having an interfering effect on underwater acoustic activities. The signature can be in the temporal domain, the spatial domain, or both. An important part of ocean acoustics research at this time is identifying which processes are dominant at specific times and places, and models are well suited to this. Significant acoustic effects of water-column and seafloor features occur in concert. However, they have traditionally been studied individually, sometimes in idealized or very simple form. Despite the isolation of the processes, many of these studies have been very successful. Examples are the analysis of the Pekeris waveguide [4], adiabatic mode propagation in a smoothly varying waveguide [5], and propagation through idealized internal waves [6-8]. The state of our knowledge now demands that the full complexity be analyzed, as can be done using the ocean models. Initial efforts that have coupled four-dimensional ocean fields with 2D acoustics modeling include data assimilation and uncertainty studies [9, 10], end-to-end computations [11], real-time at-sea predictions [12] and coupled adaptive sampling [13]. In the present work, a specific focus is on 3D acoustic effects coupled to 4D ocean predictions. We have thus motivated the use of oceanographic flow models as a straightforward approach for objective and comprehensive study of sound propagation in realistic environments, which we refer to as coupled ocean/acoustics modeling. The alternative of investigating the overall effects of simultaneously occurring feature types by constructing idealized process models with multiple features (straight line internal waves in two-layer fluid over a uniformly sloped bottom and one eddy, for example) is likely to lack objectivity or completeness. In fact, such feature models are mainly utilized to initialize ocean models or describe/assimilate specific features [14]. Coupled ocean/acoustics modeling can have high value, under the condition that the synthesized environments are sufficiently inclusive, representative, and accurate. This is a nontrivial condition; many challenges remain for flow models in terms of boundary conditions and data assimilation, resolution of near-boundary effects and mixing effects, and three-dimensional nonlinear gravity waves with hydrostatic pressure. Note that making acoustic propagation predictions, without analysis of the behavior or the mechanisms at work, is a byproduct of coupled ocean-acoustic modeling. Coupled ocean/acoustics modeling is becoming more common. Nevertheless, the approach is relatively recent and the best research path to take at this time deserves discussion. In this paper we discuss the potential of this method, and inform the discussion with some example computations from recent work in the Mid Atlantic Bight.

In the past decades an increasing number of problems in continuum theory have been treated using stochastic dynamical theories. This is because dynamical systems governing real processes always contain some elements characterized by uncertainty or stochasticity. Uncertainties may arise in the system parameters, the boundary and initial conditions, and also in the external forcing processes. Also, many problems are treated through the stochastic framework due to the incomplete or partial understanding of the governing physical laws. In all of the above cases the existence of random perturbations, combined with the com- plex dynamical mechanisms of the system often leads to their rapid growth which causes distribution of energy to a broadband spectrum of scales both in space and time, making the system state particularly complex. Such problems are mainly described by Stochastic Partial Differential Equations and they arise in a number of areas including fluid mechanics, elasticity, and wave theory, describing phenomena such as turbulence, random vibrations, flow through porous media, and wave propagation through random media. This is but a partial listing of applications and it is clear that almost any phenomenon described by a field equation has an important subclass of problems that may profitably be treated from a stochastic point of view.

In this work, we develop a new methodology for the representation and evolution of the complete probabilistic response of infinite-dimensional, random, dynamical systems. More specifically, we derive an exact, closed set of evolution equations for general nonlinear continuous stochastic fields described by a Stochastic Partial Differential Equation. The derivation is based on a novel condition, the Dynamical Orthogonality (DO), on the representation of the solution. This condition is the “key” to overcome the redundancy issues of the full representation used while it does not restrict its generic features. Based on the DO condition 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 uncertainty “lives” as well as a system of Stochastic Differential Equations that defines how the uncertainty evolves in the time varying stochastic subspace. If additional restrictions are assumed on the form of the representation, we recover both the Proper-Orthogonal-Decomposition (POD) equations and the generalized Polynomial- Chaos (PC) equations; thus the new methodology generalizes these two approaches. For the efficient treatment of the strongly transient character on the systems described above we derive adaptive criteria for the variation of the stochastic dimensionality that characterizes the system response. Those criteria follow directly from the dynamical equations describing the system.

We illustrate and validate this novel technique by solving the 2D stochastic Navier-Stokes equations in various geometries and compare with direct Monte Carlo simulations. We also apply the derived framework for the study of the statistical responses of an idealized “double gyre” model, which has elements of ocean, atmospheric and climate instability behaviors.

Finally, we use our new stochastic description for flow fields to study the motion of inertial particles in flows with uncertainties. Inertial or finite-size particles in fluid flows are commonly encountered in nature (e.g., contaminant dispersion in the ocean and atmosphere) as well as in technological applications (e.g., chemical systems involving particulate reactant mixing). As it has been observed both numerically and experimentally, their dynamics can differ markedly from infinitesimal particle dynamics. Here we use recent results from stochastic singular perturbation theory in combination with the DO representation of the random flow, in order to derive a reduced order inertial equation that will describe efficiently the stochastic dynamics of inertial particles in arbitrary random flows.