Rayleigh-Benard Convection: Dynamics and Structure in the Physical Space
  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]. 
Tian Ma, Shouhong Wang