Reduced Stochastic Dynamics for Spatially Extended Systems

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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.