Equilibrium and Nonequilibrium Statistical Theories of Turbulent Geophysical Flows

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

The research project applies modern methods in analysis, computation and probability to the modeling turbulent flows arising in geophysical fluid dynamics. Specifically, the projects investigates statistical theories of coherent structures in turbulence, which in a model atmosphere or ocean usually take the form of long-lived, large-scale jets and vortices. The approach builds on recent developments in the equilibrium statistical theory of such structures, which has now matured to the point that it can be justified mathematically, implemented computationally and applied succesfully to geophysical models. A prime example is the recent model of the zonal jets and vortical spots in the atmosphere of Jupiter, whose predictions agree remarkably well with observational data. In light of these developments, the goals of the project are two-fold. First, the equilibrium statistical theory of coherent structures is elaborated for increasingly realistic geophysical models, such as multi-layer quasi-geostrophic models and shallow-water models. The physical implications of the theory are investigated by computing families of equilibrium structures and deriving nonlinear stability theorems for them. Second, a novel approach to statistical closure is developed from the corresponding nonequilibrium theory. Unlike traditional closure schemes for fluid turbulence, this new methodology derives macroscopic equations for some specified resolved variables by conditioning random paths of microstates on the ensemble-averaged dynamics for those resolved variables. A combined theoretical and computational investigation of this approximation to nonequilibrium behavior is undertaken for some prototype problems. In the context of model geophysical systems with damping and driving, this approach is envisioned as a general procedure for deriving effective subgrid-scale parametrizations of unresolved eddies. Turbulent fluid flow remains one of the unsolved puzzles of physical science. A better theoretical understanding of turbulence is needed as a basis for the computational simulation of almost all natural fluid motions. This is especially true of geophysical fluid flows -- the motions of the Earth's oceans and atmosphere -- which involve complex motions over a wide range of scales, from meters up to the planetary size. Every modern computer code used in weather forecasting or climate prediction requires special, but often unreliable, assumptions about how the small-scale turbulent motions affect the computed large-scale behavior. The research conducted in this project addresses the general issue of modeling a complex fluid flow -- a mathematical prototype of an atmosphere or ocean -- in such a way that its predominant large-scale features can be captured reliably without resolving the full complexity of its small-scale motions. In particular, the work seeks to develop the mathematical and computational tools necessary to predict the behavior of modeled geophysical fluid systems which exhibit organized features on large scales but disordered and random motions on a range of small scales. To do so, the project draws on sophisticated techniques from statistical physics to construct theoretical models of complex systems of this kind, and thereby to provide efficient and reliable methods for computing their expected or most probable behavior. A recent example of this approach is the remarkably successful explanation of the persistent jetstreams and vortices, such as the Great Red Spot, in the atmosphere of the giant planet Jupiter, which for the first time shows quantitative and qualitative agreement between mathematical theory and NASA spacecraft observations. In the context of Earth's atmosphere and oceans, such theoretical models and computational methods can used as building blocks in predictions about the long-term trends in the ocean-atmosphere system.