Blending Modified Gaussian Closure and Non-Gaussian Reduced Subspace Methods for Turbulent Dynamical Systems
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Turbulent dynamical systems are characterized by persistent instabilities which are balanced by nonlinear dynamics that continuously transfer energy to the stable modes. To model this complex statistical equilibrium in the context of uncertainty quantification all dynamical components (unstable modes, nonlinear energy transfers, and stable modes) are equally crucial. Thus, order-reduction methods present important limitations. On the other hand uncertainty quantification methods based on the tuning of the non-linear energy fluxes using steady-state information (such as the modified quasilinear Gaussian (MQG) closure) may present discrepancies in extreme excitation scenarios. In this paper we derive a blended framework that links inexpensive second-order uncertainty quantification schemes that model the full space (such as MQG) with high order statistical models in specific reduced-order subspaces. The coupling occurs in the energy transfer level by (i) correcting the nonlinear energy fluxes in the full space using reduced subspace statistics, and (ii) by modifying the reduced-order equations in the subspace using information from the full space model. The results are illustrated in two strongly unstable systems under extreme excitations. The blended method allows for the correct prediction of the second-order statistics in the full space and also the correct modeling of the higher-order statistics in reduced-order subspaces.
KeywordsBlended stochastic methods Modified quasilinear Gaussian closure Dynamical orthogonality Uncertainty quantification in reduced order subspaces
Mathematics Subject Classification34F05 60H10
We would like to thank the referees for useful comments and suggestions. The research of A. Majda is partially supported by NSF grant DMS-0456713, NSF CMG grant DMS-1025468, and ONR grants ONR-DRI N00014-10-1-0554, N00014-11-1-0306, and ONR-MURI N00014-12-1-0912. T. Sapsis is supported as postdoctoral fellow by the first and third grants.
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