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.

Understanding the complex dynamics of coastal upwelling is essential for coastal ocean dynamics, phytoplankton blooms, and pollution transport. Atmospheric-driven coastal upwelling often occurs when strong alongshore winds and the Coriolis force combine to displace warmer surface waters offshore, leading to upward motions of deeper cooler, nutrient-dense waters to replace these surface waters. Using the models of the MIT Multidisciplinary Simulation, Estimation, and Assimilation System (MSEAS) group, we conduct a large set of simulation sensitivity studies to determine which variables are dominant controls for upwelling events in the Monterey Bay region. Our motivations include determining the dominant atmospheric fluxes and the causes of high-frequency fluctuations found in ocean thermal balances. We focus on the first upwelling event from August 1-5, 2006 in Monterey Bay that occurred during the Monterey Bay 06 (MB06) at-sea experiment, for which MSEAS data-assimilative baseline simulations already existed.

Using the thermal energy (temperature), salinity and momentum (velocity) conservation equations, full ocean fields in the region as well as both control volume (flux) balances and local differential term-by-term balances for the upwelling event events were computed. The studies of ocean fields concentrate on specific depths: surface-0m, thermocline-30m and undercurrent-150m. Effects of differing atmospheric forcing contributions (wind stress, surface heating/cooling, and evaporation-precipitation) on these full fields and on the volume and term-by-term balances are analyzed. Tidal effects are quantified utilizing pairs of simulations in which tides are either included or not. Effects of data assimilation are also examined.

We find that the wind stress forcing is the most important dynamical parameter in explaining the extent and shape of the upwelling event. This is verified using our large set of sensitivity studies and examining the heat flux balances. The assimilation of data has also an impact because this first upwelling event occurs during the initialization. Tidal forcing and, to a lesser extent, the daily atmospheric and data assimilation cycles explain the higher frequency fluctuations found in the volume averaged time rate of change of thermal energy.

A new generation of efficient parallel, multi-scale, and interdisciplinary ocean models is required for better understanding and accurate predictions. The purpose of this thesis is to quantitatively identify promising numerical methods that are suitable to such predictions. In order to fulfill this purpose, current efforts towards creating new ocean models are reviewed, an understanding of the most promising methods used by other researchers is developed, the most promising existing methods are studied and applied to idealized cases, new methods are incubated and evaluated by solving test problems, and important numerical issues related to efficiency are examined.

The results of other research groups towards developing the second generation of ocean models are first reviewed. Next, the Discontinuous Galerkin (DG) method for solving advection-diffusion problems is described, including a discussion on schemes for solving higher order derivatives. The discrete formulation for advection-diffusion problems is detailed and implementation issues are discussed. The Hybrid Discontinuous Galerkin (HDG) Finite Element Method (FEM) is identified as a promising new numerical scheme for ocean simulations. For the first time, a DG FEM scheme is used to solve ocean biogeochemical advection-diffusion-reaction equations on a two- dimensional idealized domain, and p-adaptivity across constituents is examined. Each aspect of the numerical solution is examined separately, and p-adaptive strategies are explored. Finally, numerous solver-preconditioner combinations are benchmarked to identify an efficient solution method for inverting matrices, which is necessary for implicit time integration schemes. From our quantitative incubation of numerical schemes, a number of recommendations on the tools necessary to solve dynamical equations for multiscale ocean predictions are provided.