Rayleigh-Benard对流：物理空间中的动力学和结构
Rayleigh-Benard Convection: Dynamics and Structure in the Physical Space
摘要：
这篇文章的主要目的是一个研究计划的一部分，以连接流体流动的动力学与结构及其在物理空间中的过渡。作为问题的原型，我们研究了二维Rayleigh-Benard对流。该分析基于两个最近开发的非线性理论：不可压缩流的几何理论[10]和非线性动力系统（有限维和无限维）的分叉和稳定性理论[9]。在文献[8]中，我们证明了当Rayleigh数R超过第一临界Rayleigh数R_c时，Rayleigh-Benard问题从基态分叉到吸引子A_R，而与线性问题的特征值R_c的重数无关。本文除了对分叉吸引子A_R进行了分类外，还对物理空间中解的结构及其变迁进行了分类，从而得出了两种不同的流动结构：纯卷流和被横槽流分开的卷流的存在性和稳定性。尽管在其他物理背景下（如大气动力学中的Branstator-Kushnir波）观察到了这种结构，但似乎没有仔细检查横通道流分离的卷结构[1，7]。
英文摘要：
The main objective of this article is part of a research program to link the dynamics of fluid flows with the structure and its transitions in the physical spaces. As a prototype of problem and to demonstrate the main ideas, we study the two-dimensional Rayleigh-Benard convection. The analysis is based on two recently developed nonlinear theories: geometric theory for incompressible flows [10] and the bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) [9]. We have shown in [8] that the Rayleigh-Benard problem bifurcates from the basic state to an attractor A_R when the Rayleigh number R crosses the first critical Rayleigh number R_c for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue R_c for the linear problem. In this article, in addition to a classification of the bifurcated attractor A_R, the structure and its transitions of the solutions in the physical space is classified, leading to the existence and stability of two different flows structures: pure rolls and rolls separated by a cross the channel flow. It appears that the structure with rolls separated by a cross channel flow has not been carefully examined although it has been observed in other physical contexts such as the Branstator-Kushnir waves in the atmospheric dynamics [1,7].
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