Problems in Spatiotemporal Chaos

Abstract

In this thesis we consider two problem areas involving spatiotemporally chaotic systems. In Part I we investigate data assimilation techniques applicable to large systems. Data assimilation refers to the process of estimating a system's state from a time series of measurements (which may be noisy or incomplete) in conjunction with a model for the system's time evolution. However, for practical reasons, the high dimensionality of large spatiotemporally chaotic systems prevents the use of classical data assimilation techniques such as the Kalman filter. Here, a recently developed data assimilation method, the local ensemble transform Kalman Filter (LETKF), designed to circumvent this difficulty is applied to \RaBen convection, a prototypical spatiotemporally chaotic laboratory system. Using this technique we are able to extract the full temperature and velocity fields from a time series of shadowgraphs from a Rayleigh-Benard convection experiment. The process of estimating fluid parameters is also investigated. The presented results suggest the potential usefulness of the LETKF technique to a broad class of laboratory experiments in which there is spatiotemporally chaotic behavior.

In Part II we study magnetic dynamo action in rotating electrically conducting fluids. In particular, we study how rotation effects the process of magnetic field growth (the dynamo effect) for a externally forced turbulent fluid. We solve the kinematic magnetohydrodynamic (MHD) equations with the addition of a Coriolis force in a periodic domain. Our results suggest that rotation is desirable for producing dynamo flows.