The amplitude equation for rotating Rayleigh–Bénard convection

The amplitude equation for rotating Rayleigh–Bénard convection is derived from the Boussinesq equations with the Coriolis force included. The vertical boundary conditions are no-slip, and the lateral boundary conditions are either periodic or rigid. In order to keep track of the mean flow, the full system of equations is considered, instead of a potential formulation. A multiple scales perturbation expansion in the control parameter $ϵ$ is performed, and appropriate solvability conditions are imposed. This leads to the usual amplitude equation at order ${ϵ}^{3∕2}$ , but a new rotation term enters at order ${ϵ}^{7∕4}$ . This rotation term will cause a change of phase with respect to time, whenever there is a gradient in the amplitude in the direction parallel to the rolls. As a result, rolls terminating perpendicularly to a wall will precess in the direction of rotation. The new rotation term will also cause stationary dislocations to glide perpendicular to the rolls. Amplitude equation results for a specific set of parameters are compared to numerical results from simulations of the full equations. Good agreement for mean flow, wall-induced precession, and dislocation glide is found.