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Turbulent dynamical systems characterized by both a high-dimensional phasespace and a large number of instabilities are ubiquitous among many complexsystems in science and engineering. The existence of a strange attractor in theturbulent systems containing a large number of positive Lyapunov exponentsresults in a rapid growth of small uncertainties, requiring naturally aprobabilistic characterization for the evolution of the turbulent system.Uncertainty quantification in turbulent dynamical systems is a grand challengewhere the goal is to obtain statistical estimates such as the change in meanand variance for key physical quantities in their nonlinear responses tochanges in external forcing parameters or uncertain initial data. One centralissue in contemporary research is the development of a systematic methodologythat can recover the crucial features of the natural system in statisticalequilibrium (model fidelity) and improve the imperfect model prediction skillin response to various external perturbations (model sensitivity). A generalmathematical framework to construct statistically accurate reduced-order modelsthat have skill in capturing the statistical variability in the principaldirections with largest energy of a general class of damped and forced complexturbulent dynamical systems is discussed here. The methods are developed undera universal class of turbulent dynamical systems with quadratic nonlinearitythat is representative in many applications in applied mathematics andengineering. The validity of general framework of reduced-order models isdemonstrated on instructive stochastic triad models. Recent applications totwo-layer baroclinic turbulence in the atmosphere and ocean with combinationsof turbulent jets and vortices are also surveyed.

Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems