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

大气和海洋的动力学表现出几个反复出现的大尺度模式，其中包括著名的厄尔尼诺-南方涛动（ENSO）。这种不规则气候模式的变化一直对人类有很大的影响；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提供可靠的预测，对我们的经济和社会都至关重要。然而，由于气候系统的复杂性，这些任务是地球科学的长期挑战。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性理解很重要的因素：普遍存在的随机波动对一些基本气候模型动力学的影响。特别是从噪声诱发现象的角度探讨了2015-2016年厄尔尼诺等极端变暖事件的发生机制。该方法主要依赖于研究人员及其同事最近开发的一种新颖的降维方法。在这个项目中获得的知识有望为设计更好的大尺度气候模式演变预测方法带来新的见解。研究生也参与了该项目的工作。本项目采用并进一步发展的降维方法是基于一种新的随机参数化技术，用于潜在的非线性随机偏微分方程的未解析小尺度动力学。该方法有几个显著特点：(i)参数化本质上是路径化的。它非常适合于人们不仅对统计量感兴趣，而且对轨迹明智的动力学行为感兴趣的情况，这就是气候动力学应用的情况。（二）小尺度动力学的参数化在简化的系统中特别导致外生记忆效应。即使在已知对其他传统方法具有挑战性的情况下，这种非马尔可夫特征也可以帮助实现良好的建模性能。该方法还提供了一种构造不同参数化的实用方法，并设计了一个简单的无量纲量，以便在对相应的简化系统进行数值模拟之前客观地比较这些参数化的技巧。所开发的框架也可以应用于确定性偏微分方程；该方法已成功地应用于相变研究、最优控制和噪声现象分析等领域。对于气候动力学的应用，目标是：(i)基于一些复杂的海洋-大气耦合模式的随机版本开发有用且易于使用的ENSO低维简化模型；（ii）使用这些简化模型研究不同类型噪声对ENSO动力学不规则性的影响。气候模式研究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。本项目开发的理论和计算工具具有通用性和灵活性，在非线性科学和工程中具有广泛的应用前景。研究生也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.