Finite amplitude cellular convection

Abstract

When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.

High Rayleigh number convection with double diffusive fingers

“…The important point is that the exponents obey 4γ 1 + 3γ 2 = −1 so that eq. (11) is compatible with the general picture of fingers The velocity near onset is small so that the Reynolds number is small, but experiments and weakly nonlinear analysis 19 show that velocity is proportional to (Ra − Ra crit ) 1/2 ∝ g 1/2 .…”

Dynamics of Langmuir circulation in oceanic surface layers

“…'=Vyl z)e i k This methodology is similar to the finite-amplitude convection results of Malkus and Veronis (1958). * Their paper assumed that the shape and relationship between the density and streamfunction perturbation of the finite amplitude modes was given by the neutrally stable solution at the critical Rayleigh number.…”

Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio

“…In this paper we establish that this complexity is not caused by singularity formation but our ultimate goal is a theoretical understanding and a quantitative capture of the (small) mean flows. It is known that wave-number distortion, roll curvature and the mean flow make straight convection rolls become unstable [38,44,1,25]. These effects have been successfully modeled by Decker and Pesch [18] and simulated.…”

“…These solutions, form the Busse B-attractors. There are seven possibilities, described by the amplitude equations of Stuart, Watson and Landau [51,53], and the various cases arising from cubic terms of these equations have been described in [38,44,1]. Namely, up to rotation through an angle that is fixed by the initial conditions, one can have straight rolls with three orientations separated by 120 o .…”