Collaborative Research: Non-Markovian Reduction of Nonlinear Stochastic Partial Differential Equations, and Applications to Climate Dynamics

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Liu, DMS-1616450Chekroun, DMS-1616981 The dynamics of the atmosphere and oceans exhibits several recurrent large-scale patterns, which include the well-known El Nino-Southern Oscillation (ENSO) as a prominent example. The variability of such irregular climate patterns has always had a large impact on humans; some possible disastrous consequences include heavy flooding or extended drought in different regions, collapse of fisheries, plagues, and crop failure. To understand the time variability and to provide robust prediction of such climate patterns are thus of vital importance -- both for our economy and for society. These tasks are, however, long-standing challenges in geosciences due to the complexity of our climate system. In this project, the investigators and their colleagues study a factor important for such predictive understandings: the effect of ubiquitous random fluctuations on the dynamics of some fundamental climate models. In particular, the mechanism of extreme El Nino warming events such as the 2015-2016 one is explored from the perspective of noise-induced phenomena. The approach relies crucially on a novel dimension reduction methodology developed recently by the investigators and their colleagues. The knowledge gained in this project is expected to bring new insights into the design of better prediction methods for the evolution of large-scale climate patterns. Graduate students are involved in the work of the project. 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 approach has several distinctive features: (i) The parameterization is pathwise in nature. It is very well suited for cases when one is not only interested in statistical quantities but also trajectory-wise dynamical behaviors, which is the case for the applications to climate dynamics. (ii) The parameterization of the small-scale dynamics leads in particular to exogenous memory effects in the reduced systems. This non-Markovian feature can help achieve good modeling performance even in situations that are known to be challenging for other traditional methods to operate. (iii) A practical way to construct different parameterizations is also offered within the approach, and a simple non-dimensional quantity is designed to compare objectively the skills of these parameterizations prior to numerical simulations of the corresponding reduced systems. The developed framework can be applied to deterministic partial differential equations as well; and the method has already been successfully used in several applications including the study of phase transitions, optimal control, and the analysis of noise-induced phenomena. For the applications to climate dynamics, the goals are: (i) to develop useful and easy-to-use low-dimensional reduced models for ENSO based on stochastic versions of some sophisticated coupled ocean-atmosphere models, and (ii) to use these reduced models to investigate the impact of different types of noise on the irregularity of ENSO dynamics. The challenges inherent to this study of climate models help provide new directions for the development of the methodology as well as of parameterization schemes in general. The theoretical and computational tools developed in this project are general, flexible, and have a broad range of applications in nonlinear sciences and engineering. Graduate students are involved in the work of the project.