Noise and pattern formation in periodically driven Rayleigh-Bénard convection

Abstract

We present a new model for periodically driven Rayleigh-Bénard convection with thermal noise, derived as a truncated vertical mode expansion of a mean field approximation to the Oberbeck-Boussinesq equations. The resulting model includes the continuous dependence on the horizontal wave number, and preserves the full symmetries of the hydrodynamic equations as well as their inertial character. The model is shown to reduce to a Swift-Hohenberg-like equation in the same limiting cases in which the Lorenz model reduces to an amplitude equation. The order-disorder transition experimentally observed in the recurrent pattern formation near the convective onset is studied by using both the present model and its above-mentioned limiting form, as well as a generalization of the amplitude equation for modulated driving introduced by Schmitt and Lücke ͓Phys. Rev. A 44, 4986 ͑1991͔͒ and the generalized Lorenz model previously introduced by the authors ͓Phys. Rev. E 55, R3824 ͑1997͔͒. We show that all these models agree with the experimental data much closer than previous models like the Swift-Hohenberg equation or the amplitude equation, though thermal noise alone still seems insufficient to lead to a precise fit. The relationship between these models is discussed, and it is shown that the inclusion of the continuous wave-number dependence, a consistent treatment of the driving time dependence, and the inclusion of inertial effects are all relevant to the formulation of a model describing equally well both the time-periodic and static driving cases. ͓S1063-651X͑97͒09812-7͔

Statistical dynamics of parametrically perturbed sine-square map

“…Noise is found to change the stability of nonlinear systems [23], induces vibrational and stochastic resonances [24] and fractal mean first passage limits [25]. Noise also reflects internal fluctuations [26,27] and phase transitions [28]. Hence, we consider a noise term η(n) introduced as perturbation in eq.…”

Additive noise may change the stability of nonlinear systems

“…Equation ( 11) is a natural extension of the stochastic Brazovskii model for the condensation of liquids [24] by high nonlinear orders, and the additive noise term reflects internal systems fluctuations, e.g. as in [25,26]. Moreover this extended model allows for the explanation of the interfaces between convection and conduction regions in binary fluids in the absence of noise [27].…”