The Phase Diffusion and Mean Drift Equations for Convection at Finite Rayleigh Numbers in Large Containers I

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, zig-zag, skew-varicose) of straight parallel rolls found previously by Busse and his colleagues. We can calculate the wavenumber selected by curved patterns and find very close agreement with the dominant wavenumbers observed by Heutmaker and Gollub at Prandtl number 2.5 and by Steinberg, Ahlers and 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, plays 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. In addition to providing more details of comparisons between theory and experiment, we plan to make these techniques and the program to implement them available in paper II.