Date Awarded


Document Type


Degree Name

Doctor of Philosophy (Ph.D.)




George M Vahala


The constrained decimation scheme (CDS) is applied to a turbulence model. The CDS is a statistical turbulence theory formulated in 1985 by Robert Kraichnan; it seeks to correctly describe the statistical behavior of a system using only a small sample of the actual dynamics. The full set of dynamical quantities is partitioned into groups, within each of which the statistical properties must be uniform. Each statistical symmetry group is then decimated down to a small sample set of explicit dynamics. The statistical effects of the implicit dynamics outside the sample set are modelled by stochastic forces.;These forces are not totally random; they must satisfy statistical constraints in the following way: Full-system statistical moments are calculated by interpolation among sample-set moments; the stochastic forces are adjusted by an iterative process until decimated-system moments match these calculated full-system moments. Formally, the entire infinite heirarchy of moments describing the system statistics should be constrained. In practice, a small number of low-order moment constraints are enforced; these moments are chosen on the basis of physical insights and known properties of the system.;The system studied in this work is the Betchov model--a large set of coupled, quadratically nonlinear ordinary differential equations with random coupling coefficients. This turbulence model was originally devised to study another statistical theory, the direct interaction approximation (DIA). By design of the Betchov system, the DIA solution for statistical autocorrelation is easy to obtain numerically. This permits comparison of CDS results with DIA results for Betchov systems too large to be solved in full.;The Betchov system is decimated and solved under two sets of statistical constraints. Under the first set, basic statistical properties of the full Betchov system are reproduced for modest decimation strengths (ratios of full-system size to decimated-system size); however, problems arise at stronger decimation. These problems are solved by the second set of constraints. The second constraint set is intimately related to the DIA; that relationship is shown, and results from the CDS under those constraints are shown to approach the DIA results as the decimation strength increases.



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