The investigator undertakes a theoretical and numerical study of strong stochastic effects in mathematical fluid dynamics, specifically in the area of wave-mean interaction theory. Here "strong" refers to effects that substantially change the overall dynamics of the fluid system when compared to the standard deterministic setting. In general, wave-mean interaction theory describes the nonlinear interactions between small-scale waves and the large-scale mean flows on which the waves are propagating. There exists a substantial body of classical wave-mean interaction that is based on deterministic waves, but very little has been done on the stochastic version of the problem. The present work combines a number of projects aimed at extending the classical theory in this direction. Examples include the use of stochastic theory for wave dynamics and nonlinear wave breaking, for two-dimensional wave-vortex turbulence, and for the derivation of effective mean-flow equations for stratified flows subject to high-frequency oscillatory forcing.

Interactions between waves and mean currents include the driving of long-shore currents and vortical motions such as rip currents by breaking waves on a beach (important for civil engineering and naval operations), the creation of clear-air turbulence by breaking waves in the stratosphere (important for aviation and the environment), and the driving of the global air circulation by breaking waves in the mesosphere, above 60km altitude or so (important for climate evolution). These processes are far too small in spatial scale to be resolvable in numerical models for atmosphere-ocean dynamics, which means that in these models their impact must be put in by hand based on theory and observations. This is called the "parametrization" problem for unresolvable processes, and even on the biggest supercomputers it will remain a bottleneck problem for decades to come. Now, in the classical "deterministic" theory in this area, the prevailing conditions are assumed to be simple and perfectly known. However, in reality, the prevailing conditions are often complex and poorly known. Stochastic theory addresses this by allowing for uncertain, or random, components of the situation. This project brings modern stochastic theory to bear on the kind of wave-mean interaction problems that are relevant to atmosphere-ocean science. There are two principal project aims: first, finding new effects that are missed by the deterministic theory; and second, laying the foundation for a more realistic representation of wave-mean interactions in numerical models for use in climate prediction, aviation, and near-shore civil engineering.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Michael H. Steuerwalt
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New York University
New York
United States
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