Parameterization and Reduction for Nonlinear Stochastic Systems with Applications to Fluid Dynamics

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The dynamics of the oceans exhibits several large-scale persistent currents, including the Gulf stream and the Kuroshio in the middle latitudes as prominent examples. Together with other currents in low and high latitudes, they transfer substantial amounts of heat and momentum from the tropics to the polar regions, influencing local and global climate. The balmy jet of seawater also carries a great potential for producing clean offshore carbon-free energy. To understand the spatial and time variabilities of such currents is thus of vital importance for our society. In this project, the investigator will analyze the interplay between intrinsic nonlinearity and extrinsic stochastic forcing in shaping the observed variabilities. To disentangle such interactions and to analyze the impact of noise on dynamical and statistical behaviors of the governing systems are still grand challenges for many practical applications. To address these questions, the investigator will establish a new paradigm for the parameterization and the effective reduction of stochastically forced nonlinear dissipative equations, such as those governing large-scale oceanic flows. The proposed approach relies crucially on a dimension reduction methodology developed recently by the investigator and his colleagues. The knowledge gained in this project is expected to bring new understanding of the fundamental mechanisms of large-scale climate patterns. The award will also provide opportunities for the involvement of graduate students in this research.The dimension reduction methodology adopted and further developed in this project is based on a new stochastic parameterization technique for the unresolved small-scale dynamics of the underlying nonlinear stochastic partial differential equations. The investigator will derive explicit formulas that approximate the small-scale dynamics in terms of both the large-scale dynamics and the history of the noise path, leading thus to low-dimensional stochastic equations involving only large-scale variables. Such reduced equations are able to capture key dynamical features of the original stochastic systems and are much more accessible both theoretically and numerically. The impact of noise on both pattern formation in the classical Rayleigh-Benard convection and time-variability of the double-gyre wind-driven ocean circulation will be studied within the proposed theoretic framework. The parameterization formulas of unresolved small-scale dynamics are rigorously justified in the context of stochastic invariant manifolds. These formulas will be extended in this project to handle parameter regimes that are away from the onset of the first instability using a variational framework. The parameterization is pathwise in nature, which is very well suited for cases when one is not only interested in statistical quantities but also trajectory-wise dynamical behaviors. The formulas involve the history of the noise, which introduces memory into the corresponding reduced equations. This memory effect plays a fundamental role for the reduced equations to capture both qualitatively and quantitatively the dynamical and statistical features of the original system, and it has already been illustrated to be responsible for achieving good modeling performance even in situations that are known to be challenging for other traditional methods to operate. These reduced systems will help us understand better the impact of noise on the studied systems, which are otherwise computationally too expensive to obtain. By studying these reduced models subject to various types of noise, the proposed approach will also bring insights into possible ways of further improving the underlying stochastic models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.