Model Reduction and Statistical Closure of Turbulent Dynamics

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

This project aims to develop and apply a new methodology of model reduction and statistical closure for deterministic, nonlinear dynamical systems. The overall objective of this work is to derive coarse-grained equations that approximate the effect of interactions between resolved and unresolved scales of motion from given, fine-grained, physical equations of motion. For a typical complex system, say a Hamiltonian system with many degrees of freedom, the method of reduction proceeds as follows: a vector of resolved variables is selected to describe the coherent, slow behavior of the system, and a parametric statistical model is canonically associated with this resolved vector; the predicted, or estimated, evolution of the mean resolved vector is defined by that path in the statistical parameter space which optimally fits the Liouville equation for the given dynamics. A cost functional is introduced to quantify the lack-of-fit of parameter paths; it is an information-theoretic metric on the Liouville residual containing weights that determine the adjustable constants in the ensuing closure. The desired equation for the mean resolved vector is derived by applying Hamilton-Jacobi theory to the optimization principle defining the best-fit closure. Recently the best-fit theory has been validated on a prototypical turbulent dynamics (truncated Burgers-Hopf equation), for which it produces a coarse-grained dynamics having novel dissipative effects and modified nonlinear interactions. Effective coarse-grained equations will be derived for barotropic and baroclinic quasi-geostrophic flows, and these closures will be tested against benchmark numerical simulations of fully-resolved ensembles. The best-fit approach will also be applied to represent approximate nonequilibrium spectra for these models, after first investigating these questions on some simpler turbulent wave dynamics. Mathematical models of many physical phenomena such as geophysical fluid dynamics, which are used to predict the motions of the Earth's atmosphere and oceans, typically display very complex, turbulent, behavior. It is therefore important to design appropriate reduced models of such complex systems that capture, at least approximately, the coherent behavior of the full system in many fewer variables. The P.I.'s research is directed towards developing general mathematical and statistical tools for constructing reduced models of this kind. Specifically, the research endeavors to simplify nonlinear dynamical models of the atmosphere and oceans by devising effective computational schemes in which unresolved turbulence is parameterized by statistical models that best fit the full equations of motion. A reduction method of this kind would furnish a mathematically justified technique for improving the performance of numerical models of the large-scale, low-frequency dynamics of weather and climate. Specifically, the work seeks to build up an understanding of model reduction and closure through a hierarchy of simplified, prototype problems examined using a systematic and robust methodology that draws on physical concepts from nonequilibrium statistical mechanics and thermodynamics, mathematical techniques from optimization theory and dynamical systems, as well as tools from statistics and information theory. A postdoctoral research associate will be hired to collaborate with the P.I., and the training of the postdoc in these interrelated topics will be an important outcome of the project.