A Statistically Accurate Modi…ed Quasilinear Gaussian Closure for Uncertainty Quanti…cation in Turbulent Dynamical Systems

Wednesday, March 13, 2013 - 3:00pm - 3:30pm
Keller 3-180
Themistoklis Sapsis (Courant Institute of Mathematical Sciences)
We develop a novel second-order closure methodology for uncertainty quanti…cation in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the …nite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We …first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modi…ed quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy ‡ux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable …fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a speci…c toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes.

(Joint work with A. Majda)
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