New instabilities in two-dimensional rotating convection and magnetoconvection

We describe a new instability of rolls in three related convection problems. The instability results in a large-scale modulation of the amplitude of the convection (in contrast to a phase instability such as that of Eckhaus) and may be supercritical, with strongly amplitude-modulated rolls predicted at onset. Essential to the instability is the presence of a conserved quantity (whose existence is contingent upon an appropriate choice of boundary conditions), which gives rise to a corresponding slowly evolving large-scale mode. In rotating convection between stress-free boundaries, the conserved quantity is the velocity component along the axes of the rolls; in magnetoconvection it is the magnetic flux through the layer. In rotating magnetoconvection both quantities may be conserved. In each case the appropriate amplitude equations to describe convection near onset consist of a modulation equation of Ginzburg–Landau type for the amplitude of the rolls, coupled to a large-scale modulation equation for each conserved quantity. The large-scale mode(s), although damped according to linear theory, can destabilise the rolls at onset. It is found that rolls are unstable for sufficiently small Prandtl number in rotating convection, and for sufficiently small magnetic diffusivity in magnetoconvection. Throughout we consider only the onset of convection through a stationary bifurcation, but we find, remarkably, that in rotating magnetoconvection stable travelling waves may be found at onset due to this new instability.