Dynamics of defects in Rayleigh-Bénard convection

Abstract

The behavior of an extra roll extending into an otherwise regular convection pattern is studied as a function of Rayleigh number, Prandtl number, $P$, and wavelength, by means of a fully resolved numerical simulation of the Boussinesq equations with free-slip boundary conditions. For reduced Rayleigh numbers of order one or less and $P\gtrsim 40$, numerical simulations of the lowest-order amplitude equations reproduce the Boussinesq results semiquantitatively. In particular, we find that when this class of defects is stable, they move with constant velocity $v$, parallel to the roll axis and give rise to a slow modulation of the roll pattern of the form $f(x,y-vt)\mathrm{}$. Both $f$ and $v$ have been calculated analytically within a linearized theory. The envelope function $f$ depends in an essential way on $v$ such that the limit $v\to 0$ cannot be sensibly taken.