Finite-Size Effects Lead to Supercritical Bifurcations in Turbulent Rotating Rayleigh-Bénard Convection

Abstract

In turbulent thermal convection in cylindrical samples of aspect ratio Γ ≡ D/L (D is the diameter and L the height) the Nusselt number Nu is enhanced when the sample is rotated about its vertical axis, because of the formation of Ekman vortices that extract additional fluid out of thermal boundary layers at the top and bottom. We show from experiments and direct numerical simulations that the enhancement occurs only above a bifurcation point at a critical inverse Rossby number 1/Roc, with 1/Roc ∝ 1/Γ. We present a Ginzburg-Landau like model that explains the existence of a bifurcation at finite 1/Roc as a finite-size effect. The model yields the proportionality between 1/Roc and 1/Γ and is consistent with several other measured or computed system properties.PACS numbers: 47.27.te,47.32.Ef,47.20.Bp,47.27.ek Turbulence, by virtue of its vigorous fluctuations, is expected to sample all of phase space over wide parameter ranges. This viewpoint implies that there should not be any bifurcations between different turbulent states. Contrary to this, several cases of discontinuous transitions have been observed recently in turbulent systems [1][2][3]. When they occur, they are likely to be provoked either by changes in boundary conditions or boundarylayer structures, or by discontinuous changes in the largescale structures, as a parameter is varied.Recently some of us [4,5] reported on the effect of rotation about a vertical axis at a rate Ω on turbulent convection in a fluid heated from below and cooled from above (known as Rayleigh-Bénard convection or RBC; for recent reviews, see [6-8]). For a cylindrical sample of aspect ratio Γ ≡ D/L = 1.00 (D is the diameter and L the height) a supercritical bifurcation was found, both from experiments and from direct numerical simulation (DNS) of the Boussinesq equations of motion. At a finite Ω, as expressed by the inverse Rossby number 1/Ro ∝ Ω (to be defined explicitly below), there was a sharp transition from a state of nearly rotation-independent heat transport (as expressed by the Nusselt number Nu to be defined explicitly below) to one in which Nu was enhanced by an amount δNu(1/Ro). This is illustrated by the data shown in Fig. 1. The increase of Nu was attributed to Ekman pumping [4, 5, 9-15], i.e. to the formation of (cyclonic) vertical vortex tubes ("Ekman vortices"), which extract and vertically transport additional fluid from the boundary layers (BLs) and thereby enhance the heat transport. The bifurcation was located at a critical value 1/Ro c 0.40 [5]. The reason for the existence of the bifurcation at 1/Ro c > 0 hitherto had not been understood. While such bifurcations are common near the onset of RBC in the domain of pattern formation [16], their existence in the turbulent regime implies a paradigm shift.In this Letter we report on further experiments for samples with Γ = 2.00, 1.00, and 0.50 which (i) all show bifurcations between different turbulent states and (ii) reveal that 1/Ro c varies approximately in proportion to 1/Γ. We offer an explanation of these ...

Susceptibility divergence, phase transition and multistability of a highly turbulent closed flow

“…In other closed flows, the literature reports some examples of transitions occurring at Reynolds number of the order of 10 5 , a decade in Re beyond where the turbulence already looks like fully developed. First, in various type of turbulent fluid experiments, several authors report transitions and symmetry breaking at high Reynolds number (cf., e.g., references [4,5,6,7]). Moreover, in a similar von Kármán flow, de la Torre and Burguete [12,13] observed a bistability between two broken O(2)-symmetry states with z s /R = ±0.19.…”

“…Beside this well-established scenario, some experiments in closed flows have raised intriguing features proceeding during transition to turbulence [4,5,6,7,8,9,10,11]. For instance, Tabeling et al have evidenced a local peak in the flatness of velocity derivative in a von Kármán flow of helium around Re = 2 × 10 5 [8] that was claimed to exhibit some characteristics of a second-order phase transition [9] and suggested to be associated with the breakdown of small-scale vortical structures.…”

Turbulent convection in liquid metal with and without rotation

“…Of recent interest in these studies have been transitions between rotationally dominated and nonrotating turbulent states (2-10) and the enhancement of heat transport by rotation (6,8,(11)(12)(13)(14)(15). Numerical simulations of planetary dynamo action by rotating convection also concentrate on fluids with unit order Pr (16).…”

Experimental Study of Rayleigh-Bénard Convection in the Presence of Rotation

“…This region has been well studied Experimental Study of Rayleigh-Bé nard Convection in the Presence of Rotation Sridhar Balasubramanian and Robert E. Ecke [9], [14]. The geostrophic regime at moderate and lower Ra and the crossover to turbulent convection, however, are much less thoroughly investigated and are very relevant for geophysical applications.…”

“…Whereas thermal plumes are formed in long sheets and are swept across the cell by mean flow, rotation spins up these plumes into intense vortical structures. Furthermore, rotation is known to shorten the linear length scale dramatically as rotation is increased [14]. Additional ingredients introduced by rotation are the Ekman pumping or suction imposed by the differential rotation of the boundary and the interior flow and the dynamical constraints imposed by the Taylor-Proudman theorem for strongly rotating flows.…”