Linear and nonlinear statistical response theories with prototype applications to sensitivity analysis and statistical control of complex turbulent dynamical systems.

Statistical response theory provides an effective tool for the analysis and statistical prediction of high-dimensional complex turbulent systems involving a large number of unresolved unstable modes, for example, in climate change science. Recently, the linear and nonlinear response theories have shown promising developments in overcoming the curse-of-dimensionality in uncertain quantification and statistical control of turbulent systems by identifying the most sensitive response directions. We offer an extensive illustration of using the statistical response theory for a wide variety of challenging problems under a hierarchy of prototype models ranging from simple solvable equations to anisotropic geophysical turbulence. Directly applying the linear response operator for statistical responses is shown to only have limited skill for small perturbation ranges. For stronger nonlinearity and perturbations, a nonlinear reduced-order statistical model reduction strategy guaranteeing model fidelity and sensitivity provides a systematic framework to recover the multiscale variability in leading order statistics. The linear response operator is applied in the training phase for the optimal nonlinear model responses requiring only the unperturbed equilibrium statistics. The statistical response theory is further applied to the statistical control of inherently high-dimensional systems. The statistical response in the mean offers an efficient way to recover the control forcing from the statistical energy equation without the need to run the expensive model. Among all the testing examples, the statistical response strategy displays uniform robust skill in various dynamical regimes with distinct statistical features. Further applications of the statistical response theory include the prediction of extreme events and intermittency in turbulent passive transport and a rigorous saturation bound governing the total statistical growth from initial and external uncertainties.

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