Low-frequency noise from random dislocation motion in large convective systems.

Abstract

We investigate theoretically the low-frequency noise arising in layers of fluid of large horizontal extent subjected to the Rayleigh-B\'enard instability. Two models built on the phase-diffusion equation are investigated. In the first the phase of the rolls obeys a diffusion equation with a white-noise forcing. This corresponds to local agitation that does not result in nucleation or annihilation of rolls. It produces an ${f}^{\mathrm{\ensuremath{-}}1}$ noise for two-dimensional patterns and an ${f}^{\mathrm{\ensuremath{-}}3/2}$ noise in the one-dimensional case. The ${f}^{\mathrm{\ensuremath{-}}1}$ noise can be identified with noise observed in convective patterns very close to threshold. In the second model, roll patterns where a single dislocation performs a Brownian motion are investigated. It is shown that the corresponding stochastic phase equation can be solved for an infinite domain. The result is an ${f}^{\mathrm{\ensuremath{-}}2}$ noise, in agreement with recent experimental observations by Croquette, Le Gal, and Pocheau [Phys. Scr. T13, 135 (1986)]. The theoretical result is also valid for an arbitrary number of dislocations performing independent random walks.