Abstract
A mode-coupling theory of free convection in the neighborhood of the linear convective instability is presented. A projection operator formalism is used to eliminate the fast hydrodynamic modes from the equations of motion. A non-linear Langevin equation involving only the soft convective mode is obtained. The N-order expansion scheme developed by Machta and Oppenheim, and the diagrammatic expansion of the resolvent operator developed by Lim and Kavassalis, are used to btain expressions for the mode-coupling corrections to the random force correlation up to second order in the mode-coupling parameter. The effect of this correction on the steady state soft mode correlation is estimated. A self-consistent approximate evaluation of the steady state soft mode correlation valid to all orders in the mode-coupling parameter and very close to the instability threshold is also presented. These non-classical mode-coupling effects are nevertheless too small in magnitude to be observed in a macroscopic experiment.
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