Hydrodynamic fluctuations near the convection instability

Hydrodynamic fluctuations of a horizontal liquid layer heated from below are considered in the vicinity of the point where convection sets in because of buoyancy. It is assumed that convection occurs in the form of nearly two-dimensional rolls. Close to the instability, the hydrodynamics (described in the Boussinesq approximation) is simplified considerably by the appearance of a slow mode which dominates the motion of all hydrodynamic variables. It is described by a slowly varying complex amplitude whose absolute value and phase describe the strength and the position of the convection rolls, respectively. Generalizing previous work by several authors, an approximate equation of motion is derived, which is satisfied by the slow variable. New in this analysis is the inclusion of fluctuating terms, which leads to a Langevin equation. The fluctuations are shown to satisfy a detailed-balance principle. Consequently, a generalized thermodynamic potential can be defined, which was discussed briefly in an earlier paper. It depends as a functional on the slow variable, which thereby assumes the role of an order parameter of the transition. I give a further evaluation of the hydrodynamic fluctuations for a horizontally unbounded liquid layer on both sides of and at the B\'enard point by using my potential and applying various approximations. For strictly two-dimensional flows (i.e., independent of one horizontal coordinate) I calculate the time-independent steady-state properties (coherence lengths) without any further approximation by relying on published numerical data obtained for one-dimensional Ginzburg-Landau fields. Dynamic steady-state properties (coherence times) for that case and fluctuations in the three-dimensional case are calculated in a quasilinear approximation which reproduces the time-independent results for two-dimensional flows reasonably well. In the purely heat-conducting region my results contain some earlier results of Zaitsev and Shliomis in lowest order. Large and long-lived fluctuations of velocity and temperature are shown to appear at the critical wave number as the liquid is brought near the convection instability. They are due to the random appearance and disappearance of convection cells. Their size and lifetime at the B\'enard point are only limited by nonlinear coupling of the critical modes to other passive modes. In the heat-convection region, the coupling to passive modes stabilizes the amplitude of the convection cells; only slow fluctuations of the positions of the rolls remain (for unbounded layers) and destroy the long-range order of the one-dimensional roll lattice, in agreement with well-known general theorems. If the B\'enard point is approached from this side, the stabilizing influence of the passive modes decreases and is efficient only for the large fluctuations at the B\'enard point. Approached from either side, the B\'enard point resembles the critical point of a Landau phase transition. The width of the region around the B\'enard point where the Landau description breaks down is calculated and found to be unobservably small in realistic liquids. An experimental check of these results, though very tedious, seems possible and very worthwhile.