Simple model for the Benard instability with horizontal flow near threshold

We present a simple model for Rayleigh-Benard convection with an imposed horizontal flow of Reynolds number R in a container of finite width and near threshold. The model consists of two coupled envelope equations representing longitudinal and transverse traveling convection rolls. Each equation is for a wave traveling in the direction of the imposed flow. For small R, transverse rolls are stable at the convective onset. For R>R*, longitudinal rolls bifurcate from the conduction state. For a range of cross-coupling strengths and for R>R*, we obtain a transition from longitudinal to transverse rolls as the Rayleigh number is increased. This transition occurs via states for which part of the system is occupied by longitudinal, and another by transverse rolls. The behavior is strongly influenced by the presence of noise since the system first becomes convectively unstable, and therefore noise-sustained structures can play an important role. We also show that for a range of parameters in the model, a mixed state (for which both envelopes assume a nonzero value at the same location) is possible over part of the cell in a finite geometry