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

Liu，DMS-1616450 Chekroun，DMS-1616981 大气和海洋的动力学表现出几种周期性的大尺度模式，其中包括著名的厄尔尼诺-南方涛动（ENSO），这是一个突出的例子。 这种不规则气候模式的变化一直对人类产生巨大影响;一些可能的灾难性后果包括不同地区的严重洪水或长期干旱，渔业崩溃，瘟疫和作物歉收。 因此，了解时间变化并对这种气候模式进行可靠的预测对我们的经济和社会都至关重要。 然而，由于我们气候系统的复杂性，这些任务是地球科学长期面临的挑战。 在这个项目中，研究人员和他们的同事研究了一个对这种预测理解很重要的因素：无处不在的随机波动对一些基本气候模型动态的影响。 特别是从噪声引起的现象的角度探讨了2015-2016年极端厄尔尼诺变暖事件的机制。 该方法依赖于一种新的降维方法，最近开发的研究人员和他们的同事。 预计该项目获得的知识将为设计更好的预测方法以预测大规模气候模式的演变带来新的见解。 研究生参与了该项目的工作。 在这个项目中采用和进一步发展的降维方法是基于一个新的随机参数化技术的未解决的小尺度动力学的基本非线性随机偏微分方程。 该方法有几个显著的特点：（i）参数化是路径的性质。 它非常适合于不仅对统计量感兴趣，而且对随机动力学行为感兴趣的情况，这是气候动力学应用的情况。 (ii)小尺度动力学的参数化导致特别是在减少系统的外源记忆效应。 这种非马尔可夫特征可以帮助实现良好的建模性能，即使在已知对其他传统方法具有挑战性的情况下也是如此。 (iii)该方法还提供了一种实用的方法来构建不同的参数化，并设计了一个简单的无量纲量来客观地比较这些参数化的技巧，然后对相应的简化系统进行数值模拟。 所开发的框架也可以应用于确定性偏微分方程;该方法已经成功地应用于包括相变研究、最优控制和噪声诱导现象分析在内的多个应用中。 在气候动力学应用方面，目标是：（i）根据一些复杂的海洋-大气耦合模式的随机版本，开发有用和易于使用的ENSO低维简化模式，（ii）使用这些简化模式调查不同类型的噪声对ENSO动力学不规则性的影响。 这种气候模式研究所固有的挑战有助于为方法的发展以及一般的参数化方案提供新的方向。 在这个项目中开发的理论和计算工具是通用的，灵活的，在非线性科学和工程中有广泛的应用。 研究生参与了该项目的工作。

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.