The phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers

Abstract

We derive the phase diffusion and mean drift equations for the Oberbeck–Boussinesq equations in large-aspect-ratio containers. We are able to recover all the long-wave instability boundaries (Eckhaus, zigzag, skew-varicose) of straight parallel rolls found previously by Busse and his colleagues. Moreover, the development of the skew-varicose instability can be followed and it becomes clear how the mean drift field conspires to enhance the necking of phase contours necessary for the production of dislocation pairs. We can calculate the wavenumber selected by curved patterns and find very close agreement with the dominant wavenumbers observed by Heutmaker & Gollub at Prandtl number 2.5, and by Steinberg, Ahlers & Cannell at Prandtl number 6.1. We find a new instability, the focus instability, which causes circular target patterns to destabilize and which, at sufficiently large Rayleigh numbers, may play a major role in the onset of time dependence. Further, we predict the values of the Rayleigh number at which the time-dependent but spatially ordered patterns will become spatially disordered. The key difficulty in obtaining these equations is the fact that the phase diffusion equation appears as a solvability condition at order ε (the inverse aspect ratio) whereas the mean drift equation is the solvability condition at order ε2. Therefore, we had to use extremely robust inversion methods to solve the singular equations at order ε and the techniques we use should prove to be invaluable in a wide range of similar situations. Finally, we discuss the introduction of the amplitude as an active order parameter near pattern defects, such as dislocations and foci.