Abstract
Internally generated vertical vorticity enters the lowest-order amplitude equations for free-slip boundaries in an essential way when the Prandtl number P is finite and the flow three dimensional. For parallel rolls the band of stable wavenumbers is substantially modified from what was previously believed to be correct. In particular there are no stable states for P<0.301. Numerical simulations for free boundaries and larger P suggest a mechanism through which the box size determines the critical Rayleigh number for noisy time dependent convection. A stability analysis of model amplitude equations for rigid boundaries agrees qualitatively with the numerical results of Cleaver or Busse for P≲O(1). There is now considerable continuity between the stability diagrams for rigid and free boundaries.
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