TY - JOUR
T1 - A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems
AU - Sapsis, Themistoklis P.
AU - Majda, Andrew J.
N1 - Funding Information:
The research of A. Majda is partially supported by NFS grant DMS-0456713 , NSF CMG grant DMS-1025468 , and ONR grants ONR-DRI N00014-10-1-0554 and N00014-11-1-0306 . T. Sapsis is supported as postdoctoral fellow on the first and last grant.
PY - 2013/6/1
Y1 - 2013/6/1
N2 - We develop a novel second-order closure methodology for uncertainty quantification 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 finite 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 modified 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 flux 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 specific 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. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation.
AB - We develop a novel second-order closure methodology for uncertainty quantification 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 finite 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 modified 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 flux 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 specific 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. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation.
KW - Modeling of nonlinear energy fluxes
KW - Quasilinear Gaussian closure
KW - Turbulent systems
KW - Uncertainty quantification
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U2 - 10.1016/j.physd.2013.02.009
DO - 10.1016/j.physd.2013.02.009
M3 - Article
AN - SCOPUS:84875830966
SN - 0167-2789
VL - 252
SP - 34
EP - 45
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
ER -