Reduced Stochastic Dynamics for Spatially Extended Systems

University Of Houston, Houston TX

Investigators

Abstract

The investigator and his colleagues study several examples mimicking the behavior of more complex, realistic systems in atmosphere/ocean dynamics. The main focus here is on understanding the importance of various low-dimensional chaotic structures in full dynamics with many degrees of freedom and deriving low-dimensional stochastic models which correctly capture the interaction of the low-dimensional chaos with fast scales. In addition, significance of non-Gaussian small-scale processes is analyzed in the context of the Barotropic Quasi-Geostrophic Equations. An area of great importance for many problems in science and engineering involves derivation of effective equations for a small number of suitable variables capturing the essence of large systems with many unknowns. Important examples include evolution of the coupled atmosphere/ocean systems, folding of large proteins in molecular dynamics, distribution of air-pollution over a long period of time, etc. Effective equations are required first because these systems vastly overwhelm direct numerical computations. In addition, often only a few variables in the problem provide most of the needed information. In the above examples, these essential variables might be the seasonal average temperature in the US, a few angles describing the folding changes in the protein, or the primary direction for the spread of an air-pollutant from its source. The main aim of this work is to further advance the stochastic mode-reduction strategy originally developed for derivation of the effective equations in the atmosphere/ocean dynamics. Several idealized problems are examined in order to develop a systematic approach for more realistic systems and validate the applicability of the method in various settings.