The three-dimensional development of the shearing instability of convection

Two-dimensional convection can become unstable to a mean shear flow. In three dimensions, with periodic boundary conditions in the two horizontal directions, this instability can cause the alignment of convection rolls to alternate between the x and y axes. Rolls with their axes in the y-direction become unstable to a shear flow in the x-direction that tilts and suppresses the rolls, but this flow does not affect rolls whose axes are aligned with it. New rolls, orthogonal to the original rolls, can grow, until they in turn become unstable to the shear flow instability. This behaviour is illustrated both through numerical simulations and through low-order models, and the sequence of local and global bifurcations is determined.