Mode−mode coupling theory of the heat convection threshold

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Abstract
The fluctuations of a horizontal fluid layer heated from below are analyzed, based on the hydrodynamic equations in the Boussinesq approximation including thermodynamic fluctuating forces. By means of a formalism, recently developed by Martin, Rose, and Siggia, a system of coupled nonlinear integral equations is derived for a set of correlation functions, response functions, and appropriate ’’self−energies.’’ The equations are solved approximately in several steps of simplification. In addition to results of an earlier linear theory, our results cover a narrow region near the convection threshold, where mode−mode coupling provides the only mechanism of energy dissipation. The fluctuation intensities, and the correlation times and correlation lengths, are calculated in that region. These quantities diverge at the convection threshold in the linearized theory, but remain finite, if mode−mode coupling is taken into account.