The Hydrodynamic Fluctuations near the Threshold of Bénard Convection

Abstract

The behaviors of the hydrodynamic fluctuations near the convection threshold are investigated on the basis of the nonlinear hydrodynamic equations with random forces. A nonlinear Langevin equation for the fundamental mode, which becomes unstable near the critical point, is obtained by taking into account the selection rule for the mode-mode coupling caused by the nonlinear interaction of hydrodynamic variables. It is found from the nonlinear Langevin equation that, when the Rayleigh number R exceeds its critical value R c , the convection associated with the fundamental mode sets in and its velocity varies as ( R - R c ) 1/2 and furthermore the second and the third modes depend on the Rayleigh number as ( R - R c ) and ( R - R c ) 3/2 , respectively. By solving the Fokker-Planck equation derived from the nonlinear Langevin equation, we obtain that the fluctuations of the fundamental mode become large as | R - R c | -1 near the threshold but remain finite at the critical point. It is shown that these results can be applied only to the fluid with the Plandtl number above unity and the behavior of the hydrodynamic variables near the convection threshold depends on whether the Prandtl number is below or above unity.