AMC-SS: Stochastic Waves and Wave-Mean Interactions

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BuhlerDMS-0604519 The investigator undertakes a theoretical and numericalstudy of strong stochastic effects in mathematical fluiddynamics, specifically in the area of wave-mean interactiontheory. Here "strong" refers to effects that substantially changethe overall dynamics of the fluid system when compared to thestandard deterministic setting. In general, wave-meaninteraction theory describes the nonlinear interactions betweensmall-scale waves and the large-scale mean flows on which thewaves are propagating. There exists a substantial body ofclassical wave-mean interaction that is based on deterministicwaves, but very little has been done on the stochastic version ofthe problem. The present work combines a number of projectsaimed at extending the classical theory in this direction. Examples include the use of stochastic theory for wave dynamicsand nonlinear wave breaking, for two-dimensional wave-vortexturbulence, and for the derivation of effective mean-flowequations for stratified flows subject to high-frequencyoscillatory forcing. Interactions between waves and mean currents include thedriving of long-shore currents and vortical motions such as ripcurrents by breaking waves on a beach (important for civilengineering and naval operations), the creation of clear-airturbulence by breaking waves in the stratosphere (important foraviation and the environment), and the driving of the global aircirculation by breaking waves in the mesosphere, above 60kmaltitude or so (important for climate evolution). Theseprocesses are far too small in spatial scale to be resolvable innumerical models for atmosphere-ocean dynamics, which means thatin these models their impact must be put in by hand based ontheory and observations. This is called the "parametrization"problem for unresolvable processes, and even on the biggestsupercomputers it will remain a bottleneck problem for decades tocome. Now, in the classical "deterministic" theory in this area,the prevailing conditions are assumed to be simple and perfectlyknown. However, in reality, the prevailing conditions are oftencomplex and poorly known. Stochastic theory addresses this byallowing for uncertain, or random, components of the situation. This project brings modern stochastic theory to bear on the kindof wave-mean interaction problems that are relevant toatmosphere-ocean science. There are two principal project aims:first, finding new effects that are missed by the deterministictheory; and second, laying the foundation for a more realisticrepresentation of wave-mean interactions in numerical models foruse in climate prediction, aviation, and near-shore civilengineering.