Stability and fluctuations of a spatially periodic convective flow

2014 Using a model equation we analyze the onset of convection for a roll instability. The wavelength increase follows from a variational approach. For certain wavelengths, phase diffusion turns out to be unstable against compression (or dilatation) and torsion of the rolls. Thermal phase fluctuations induce a tiny brownian motion of the structure which should be observable in carefully designed experiments such as Rayleigh-Bénard in liquid helium or Carr-Helfrich in nematic liquid crystals. Tome 40 No 23 1 er DECEMBRE 1979 LE JOURNAL DE PHYSIQUE LETTRES Classification Physics Abstracts 47.20 There are numerous examples of periodic spatial structures which occur in a system through a supercritical bifurcation and the Rayleigh-Benard (R. B.) thermoconvective instability [1] is perhaps the most famous one. In this paper we discuss some properties of the non-linear regime close to the instability threshold. For convenience we use the terminology of thermoconvective instabilities but our discussion has a more general character. We use a description of the bifurcation from the conductive state to the convective state in terms of a simple but plausible model equation which allows for a twodimensional spatial dependence of the structure which we think to be more universal than the dependence on the third dimension closely connected to the details of the hydrodynamic problem. It will be shown that : i) our model describes at least qualitatively the wavelength increase of the periodic structure in the vicinity of the instability threshold [2]. ii) For a system of parallel rolls, the dynamics of the phase variable which describes ~ a local translation of the rolls is governed by two diffusion coefficients Djj II and D 1. which depend on the wavevector of the underlying structure and on the distance to the threshold. Equations D II = 0 and D 1. = 0 give the marginal stability condition for that structure against the two seemingly possible types of perturbation close to the threshold. iii) Due to thermal noise [3] the phase fluctuates, which leads to a translational brownian motion of the structure. The corresponding diffusion coefficient will be evaluated and observational possibilities will be discussed. 1. The model equation and its stationary solution. The dynamics of a supercritical stationary bifurcation obey the Landau type equation [4] : where s depends on the constraint applied to the system (the vertical temperature gradient in the R.B. case) and measures the distance to the threshold (negative below and positive above). A is the amplitude of the convective state, the unstable Fourier component of the hydrodynamic velocity in the R.B. case. Owing to the symmetry (A ) -+ (A ) no term in A 2 is present in eq. (1). Article published online by EDP Sciences and available at http://dx.doi. org/10.1051/jphyslet:019790040023060900 L-610 JOURNAL DE PHYSIQUE LETTRES To describe the spatial structure of the unstable mode, one must turn to an equation more complicated than (1) but still simpler than the full set of hydrodynamic equations. As indicated above, the single quantity A now depends on two spatial coordinates. The required equation has to display translational and rotational invariance and must lead to a convective solution with a well defined wave-vector qo at threshold. In dimensionless notations the simplest candidate reads [5] where L1 = a~ + ay [6]. * Eq. (2) defines a functional gradient flow. Indeed it may be written as 3~ = bV[A ]/bA where the potential V reads : When e is positive, the null solution is linearly unstable against a certain class of perturbations. Close to the threshold, these unstable perturbations have a wave-vector q = qo + b with 16 1 611(s) and The unidimensional stationary periodic solution is expanded as