  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].
