Reduced Stochastic Dynamics for Spatially Extended Systems

研究者和他的同事们研究了几个模拟大气/海洋动力学中更复杂、更现实系统行为的例子。本文的重点是理解各种低维混沌结构在多自由度全动力学中的重要性，并推导出能够正确捕捉快速尺度低维混沌相互作用的低维随机模型。在正压准地转方程的背景下，分析了非高斯小尺度过程的意义。对于科学和工程中的许多问题来说，一个非常重要的领域涉及为少数合适变量推导有效方程，这些变量捕获了具有许多未知数的大系统的本质。重要的例子包括大气/海洋耦合系统的演化、分子动力学中大蛋白质的折叠、长时间内空气污染的分布等。首先需要有效的方程，因为这些系统极大地压倒了直接的数值计算。此外，通常只有问题中的几个变量提供了所需的大部分信息。在上面的例子中，这些基本变量可能是美国的季节性平均温度，描述蛋白质折叠变化的几个角度，或者空气污染物从源头传播的主要方向。这项工作的主要目的是进一步推进最初为推导大气/海洋动力学中的有效方程而开发的随机模式缩减策略。为了对更现实的系统开发一种系统的方法，并验证该方法在各种情况下的适用性，研究了几个理想化的问题。

The investigator and his colleagues study several examples mimicking the behavior of more complex, realistic systems in atmosphere/ocean dynamics. The main focus here is on understanding the importance of various low-dimensional chaotic structures in full dynamics with many degrees of freedom and deriving low-dimensional stochastic models which correctly capture the interaction of the low-dimensional chaos with fast scales. In addition, significance of non-Gaussian small-scale processes is analyzed in the context of the Barotropic Quasi-Geostrophic Equations.An area of great importance for many problems in science and engineering involves derivation of effective equations for a small number of suitable variables capturing the essence of large systems with many unknowns. Important examples include evolution of the coupled atmosphere/ocean systems, folding of large proteins in molecular dynamics, distribution of air-pollution over a long period of time, etc. Effective equations are required first because these systems vastly overwhelm direct numerical computations. In addition, often only a few variables in the problem provide most of the needed information. In the above examples, these essential variables might be the seasonal average temperature in the US, a few angles describing the folding changes in the protein, or the primary direction for the spread of an air-pollutant from its source. The main aim of this work is to further advance the stochastic mode-reduction strategy originally developed for derivation of the effective equations in the atmosphere/ocean dynamics. Several idealized problems are examined in order to develop a systematic approach for more realistic systems and validate the applicability of the method in various settings.