  Convection in a vertical channel subjected to a horizontal temperature
gradient is numerically investigated. Previous numerical simulations reveal a
variety of behaviors: steady, time-periodic and chaotic. We extend previous
numerical investigations by constructing stable and unstable branches of
equilibria and periodic orbits of the underlying Oberbeck-Boussinesq equations
by parametric continuation. In a narrow domain of vertical aspect ratio ten,
the observed flow structures are dominated by the competition between three and
four co-rotating rolls. We identify the invariant solution branches underlying
the observed flow dynamics, characterize the bifurcations creating those
branches and link the observed bifurcation scenarios to the symmetries of the
primary branches. Specifically, we show that D4 symmetry dictates that several
intermediate branches bifurcate simultaneously from the four-roll branch, while
D3 symmetry requires that the intersection of these intermediate branches with
the three-roll branch be transcritical. We observe other manifestations of the
competition between three and four rolls, in which the symmetry in time or in
the transverse direction is broken, leading to limit cycles or wavy rolls,
respectively. Our work highlights the interest of combining numerical
simulations, bifurcation theory, and group theory, in order to understand the
transitions between and origin of flow patterns.
