TY - JOUR

T1 - Low-dimensional reduced-order models for statistical response and uncertainty quantification

T2 - Barotropic turbulence with topography

AU - Qi, Di

AU - Majda, Andrew J.

N1 - Funding Information:
This research of the Andrew Majda is partially supported by the Office of Naval Research through MURI N00014-16-1-2161 and DARPA through W911NF-15-1-0636 . Di Qi is supported as a graduate research assistant on these grants. The authors are grateful about the comments from the reviewers that have helped to improve this manuscript.

PY - 2017/3/15

Y1 - 2017/3/15

N2 - A low-dimensional reduced-order statistical closure model is developed for quantifying the uncertainty to changes in forcing in a barotropic turbulent system with topography involving interactions between small-scale motions and a large-scale mean flow. Imperfect model sensitivity is improved through a recent mathematical strategy for calibrating model errors in a training phase, where information theory and linear statistical response theory are combined in a systematic fashion to achieve the optimal model parameters. Statistical theories about a Gaussian invariant measure and the exact statistical energy equations are also developed for the truncated barotropic equations that can be used to improve the imperfect model prediction skill. A stringent paradigm model of 57 degrees of freedom is used to display the feasibility of the reduced-order methods. This simple model creates large-scale zonal mean flow shifting directions from westward to eastward jets with an abrupt change in amplitude when perturbations are applied, and prototype blocked and unblocked patterns can be generated in this simple model similar to the real natural system. Principal statistical responses in mean and variance can be captured by the reduced-order models with desirable accuracy and efficiency with only 3 resolved modes. An even more challenging regime with non-Gaussian equilibrium statistics using the fluctuation equations is also tested in the reduced-order models with accurate prediction using the first 5 resolved modes. These reduced-order models also show potential for uncertainty quantification and prediction in more complex realistic geophysical turbulent dynamical systems.

AB - A low-dimensional reduced-order statistical closure model is developed for quantifying the uncertainty to changes in forcing in a barotropic turbulent system with topography involving interactions between small-scale motions and a large-scale mean flow. Imperfect model sensitivity is improved through a recent mathematical strategy for calibrating model errors in a training phase, where information theory and linear statistical response theory are combined in a systematic fashion to achieve the optimal model parameters. Statistical theories about a Gaussian invariant measure and the exact statistical energy equations are also developed for the truncated barotropic equations that can be used to improve the imperfect model prediction skill. A stringent paradigm model of 57 degrees of freedom is used to display the feasibility of the reduced-order methods. This simple model creates large-scale zonal mean flow shifting directions from westward to eastward jets with an abrupt change in amplitude when perturbations are applied, and prototype blocked and unblocked patterns can be generated in this simple model similar to the real natural system. Principal statistical responses in mean and variance can be captured by the reduced-order models with desirable accuracy and efficiency with only 3 resolved modes. An even more challenging regime with non-Gaussian equilibrium statistics using the fluctuation equations is also tested in the reduced-order models with accurate prediction using the first 5 resolved modes. These reduced-order models also show potential for uncertainty quantification and prediction in more complex realistic geophysical turbulent dynamical systems.

KW - Reduced-order methods

KW - Statistical responses

KW - Topographic barotropic system

KW - Uncertainty quantification

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U2 - 10.1016/j.physd.2016.11.006

DO - 10.1016/j.physd.2016.11.006

M3 - Article

AN - SCOPUS:85008339419

VL - 343

SP - 7

EP - 27

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -