The main objective of this article is part of a research program to link the dynamics of fluid flows with the structure of these fluid flows in physical space and the transitions of this
structure. To demonstrate the main ideas, we study the two-dimensional Rayleigh-Bénard convection, which serves as a prototype problem. The analysis is based on two recently developed nonlinear theories: geometric theory for incompressible flows [T. Ma and S. Wang, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 119, 2005] and bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) [T. Ma and S. Wang, World Scientific, 2005]. We have shown in [T. Ma and S. Wang, Commun. Math. Sci., 2(2), 159–183, 2004] that the Rayleigh-Bénard problem bifurcates from the basic state to an attractor AR when the Rayleigh number R crosses the first critical Rayleigh number Rc for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue Rc for the linear problem. In this article, in addition to a classification of the bifurcated attractor AR, the structure of the solutions in physical space and the transitions of this structure are 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 atmospheric dynamics [G.W. Branstator, J. Atmos. Sci., 44, 2310–2323, 1987] and [K. Kushnir, J. Atmos. Sci., 44, 2727–2742, 1987].