Oscillations of the large-scale circulation in turbulent Rayleigh...

Under consideration for publication in J. Fluid Mech. 1

Oscillations of the large-scale circulation inturbulent Rayleigh-Benard convection

Denis Funfschilling† , Eric Brown, and Guenter Ahlers

Department of Physics and iQUEST,University of California, Santa Barbara, CA 93106

(Received 7 May 2007)

Measurements that determine the nature and amplitude of the oscillatory mode of thelarge-scale circulation (LSC) of turbulent Rayleigh-Benard convection are presented.They cover the Rayleigh-number range 108 <∼ R <∼ 1011 and Prandtl-number range4.4 <∼ σ <∼ 29. For cylindrical samples of aspect ratio Γ = 1 the mode consists of an az-imuthal twist of the near-vertical LSC circulation plane, with the top and bottom halvesof the plane oscillating out of phase by half a cycle. The data for Γ = 1 and σ = 4.4showed that the oscillation amplitude varied irregularly in time, yielding a Gaussianprobability distribution centered at zero for the displacement angle. This result can bedescribed well by the equation of motion of a stochastically driven damped harmonicoscillator. The power spectrum of the LSC orientation had a peak at finite frequencywith a quality factor Q ≃ 5, nearly independent of R. For samples with Γ ≥ 2 we did notfind this mode, but there remained a characteristic periodic signal that was detectablein the density of plumes ρp above the bottom-plate center. Measurements of ρp revealeda strong dependence on the Rayleigh number R and the aspect ratio Γ that could berepresented by ρp ∼ Γ2.7±0.3R2.0±0.3.

1. Introduction

Understanding turbulent Rayleigh-Benard convection (RBC) in a fluid heated frombelow [see, for instance, Siggia (1994), Kadanoff (2001), Ahlers, Grossmann & Lohse(2002)] remains one of the challenging problems in nonlinear physics. It is well establishedthat a major component of the dynamics of this system is a large-scale circulation (LSC)[Krishnamurty & Howard (1981), Sano et al. (1989), Castaing et al. (1989), Cilibertoet al. (1997), Qiu & Tong (2001a), Funfschilling & Ahlers (2004), Sun et al. (2005a), Tsujiet al. (2005)]. The LSC plays an important role in many natural phenomena, includingatmospheric and oceanic convection [see, for instance, Doorn et al. (2000)] where it hasan impact on climate, and convection in the outer core of the Earth [see, for instance,Glatzmaier et al. (1999)] where it is responsible for the generation of the magnetic field.

In this paper we consider cylindrical samples, primarily with aspect ratio Γ ≡ D/L ≃ 1(D is the diameter and L the height). For these the LSC consists of a single convectionroll, with both down-flow and up-flow near the side wall but at azimuthal locations θthat differ by π. An interesting dynamical-systems property of the LSC is a twistingazimuthal oscillation mode with frequency f0 = ω0/2π. The spatial nature of this modehas become apparent only recently [Funfschilling & Ahlers (2004), Resagk et al. (2006)].It is a time-periodic twist of the circulation plane of the LSC that consists, at a given

† Present address: LSGC CNRS - GROUPE ENSIC, BP 451, 54001 Nancy Cedex, France

2 Denis Funfschilling, Eric Brown, and Guenter Ahlers

moment, of a rotation in opposite azimuthal directions in the top and the bottom halfof the sample (Funfschilling & Ahlers (2004)). In the present paper the existence ofthis mode is confirmed by quite different measurements, its spatio-temporal structure isfurther illuminated, and measurements of its properties are extended to a larger Rayleigh-and Prandtl-number range.

Although the geometrical features of this mode were unclear until recently, a charac-teristic frequency was measured much earlier in a number of single-point determinationsof the temperature or the velocity [Heslot et al. (1987), Castaing et al. (1989), Cilib-erto et al. (1997), Takeshita et al. (1996), Cioni et al. (1997), Qiu et al. (2000), Qiu& Tong (2001a), Qiu & Tong (2001b), Niemela et al. (2001), Qiu & Tong (2002), Qiuet al. (2004)]. Depending on the nature of the probe, and the probe location within thesample, such a measurement could determine either the inverse turnover time 1/T of theLSC or its azimuthal oscillation frequency f0. In either case, it was found that, withinexperimental resolution, f0 is equal to 1/T over a wide parameter range (Qiu & Tong(2002), Funfschilling & Ahlers (2004)). The reason for this equality, and indeed for theexistence of the oscillations, is not known at this time, but this synchronization maysuggest a physical origin of the oscillation.

We find that this mode has oscillating but gradually decaying time-correlation func-tions and a Lorentzian peak in the height-dependent power spectrum (or structure factor)S(z, f) of the LSC orientation θ(z, t). For Γ = 1 and σ = 4.4 we present results for theheight S0, center frequency f0, half-width σf , and total power P = πσfS0 of this peakas a function of the Rayleigh number

R =αg∆TL

κν(1.1)

(α is the isobaric thermal expansion coefficient, g the acceleration of gravity, and ∆Tthe applied temperature difference). Although the half-width increases with R, the centerfrequency increases also with the result that the quality factor Q = f0/(2σf) ≃ 5 remainsnearly independent of R. This suggests that the oscillations will survive up to values ofR well above those available in the laboratory, unless a dramatic event intervenes andinvalidates a simple power-law extrapolation of the R-dependences of f0 and σf .

Further we find that this oscillatory mode exists only for aspect ratios near one. Forlarger Γ we do not find the azimuthal oscillation, but instead observe that the plumedensity above the center portion of the bottom plate oscillates periodically. Thus, asthe aspect ratio changes, the nature of the time-periodic mode of the LSC changes itscharacter.

2. Experimental apparatus and methods

2.1. The samples

Measurements using water at a mean temperature of 40◦C (Prandtl number σ ≡ ν/κ =4.38 and ν = 6.7 × 10−7 m2/sec) as the fluid were made for the two cylindrical sampleswith Γ ≃ 1 described by Brown et al. (2005) as the “medium” and the “large” sample. Thetop and bottom plates of both were made of copper. A plexiglas side wall had a thicknessof 0.32 (0.63) cm for the medium (large) sample. The samples had L(D) = 24.76 (24.81)and 50.61 (49.69) cm respectively. Rayleigh numbers were determined from temperaturemeasurements with five thermometers imbedded in each of the top and bottom plate.

Measurements using methanol with σ = 6.0 and ν = 5.79×10−7 m2/sec or 2-propanolwith σ = 28.9 and ν = 1.77 × 10−6 m2/sec as the fluid were made in three smallercylindrical samples. They each had a 6.35 mm thick aluminum bottom plate with a

Oscillations of the large-scale circulation in turbulent Rayleigh-Benard convection 3

Figure 1. The location of the side-wall thermometers. For the top view only the thermometerlocations for the set located a distance L/2 above the bottom plate (on the horizontal mid-plane,or z = 0) are shown. The side view indicates the location of all three sets.

mirror top surface and a 3.15 mm thick sapphire top plate. A high-density polyethyleneside wall was sealed to the top and bottom plates by ethylenepropylene O-rings. One ofthe smaller samples had Γ ≃ 1 with L (D) = 8.74 (8.66) and had been used before by Xuet al. (2000) and by Funfschilling & Ahlers (2004). We shall call it the “small” sample.The other two had L (D) = 8.97 (4.48) and 8.96 (3.00) cm respectively. They will bereferred to as the Γ ≃ 2 and Γ ≃ 3 sample respectively. The bottom-plate temperaturewas determined with several thermistors imbedded in the plate. The top temperaturewas inferred from measurements of the temperature of the bath that circulated abovethe top sapphire plate.

2.2. Sidewall-temperature measurements

For the medium and large sample, three sets of eight thermistors each, equally spacedaround the circumference at the three vertical positions −L/4, 0, and L/4 (we takethe origin of the vertical axis at the horizontal midplane of the sample) and labeledi = 0, ..., 7, etc. as shown in Fig. 1, were imbedded in small holes drilled horizontallyinto but not penetrating the side wall. The thermistors were able to sense the adjacentfluid temperature without interfering with delicate fluid-flow structures. We measuredthe temperature of each thermistor with a sampling period of about 2.5 seconds. Sincethe LSC carried warm (cold) fluid from the bottom (top) plate up (down) the side wall,these thermistors detected the location of the upflow (downflow) of the LSC by indicatinga relatively high (low) temperature.

To determine the orientation and strength of the LSC, we fit the function

Ti = T0 + δ cos

(

iπ

4− θ0

)

, i = 0, . . . , 7 , (2.1)

separately at each time step, to the eight temperature readings obtained from the ther-mistors at height 0. The fit parameter δ is a measure of the temperature amplitude ofthe LSC and θ0 is the azimuthal orientation of the plane of the LSC circulation. Asdefined here, the orientation θ0 is on the side of the sample where the LSC is warm andup-flowing and is measured relative to the location of thermometer zero. We calculatedorientations θt and θb and amplitudes δt and δb for the top and bottom levels at L/4 and−L/4 separately by the same method as for the middle row. Results from this method


Figure 2. (a): Background-image-divided shadowgraph image of the small sample (Γ = 1) with2-propanol at a mean temperature of 40◦C. The Rayleigh number was R = 7.8 × 108. (b): abinary rendering of the image on the left obtained by applying a threshold and inverting thegrey scale. The white circle illustrates the area used for the plume-density measurements.

of determining the orientation and strength of the LSC were reported in several previouspublications (Brown et al. (2005), Ahlers et al. (2005), Brown & Ahlers (2006a), Brown& Ahlers (2006b)).

2.3. Shadowgraph measurements

For the small, Γ ≃ 2, and Γ ≃ 3 samples the LSC movement and plume density closeto but above (below) the bottom (top) plate were examined by means of shadowgraphy(deBruyn et al. (1996)) as described in part already by Funfschilling & Ahlers (2004).Figure 2a shows a typical image obtained for the small sample. This image had alreadybeen divided by a background image taken with ∆T = 0, and the ratio was re-scaled tocover a suitable grey range. An elongated, narrow dark stripe like the one near the upperright of the image corresponds to a stripe of relatively warm fluid. For most of our workthe optics was adjusted so as to emphasize the relatively warm plumes near the bottomplate. Cold plumes near the top plate are also discernible in the figure, but provide arelatively weaker bright signal. The two types of plumes are readily separated by imageprocessing.

When the sample axis is parallel to the gravitational acceleration, the LSC orientationθ0 meanders nearly randomly in the azimuthal direction (Sun et al. (2005b), Brown &Ahlers (2006a), Brown & Ahlers (2006b), Xi et al. (2006)) on time scales much longerthan the oscillation period. Some measurements were made under these conditions, butfor others the sample was tilted by an angle β ≃ 1◦ relative to gravity. This led toa strongly preferred azimuthal orientation of the LSC circulation-plane [Ahlers et al.

(2005), Brown et al. (2005), Brown & Ahlers (2006a)]. In Fig. 3 the coordinate systemrelative to the tilt direction is illustrated. For the small tilt angle used the shadowgraphmeasurements did not resolve any influence of the tilt on the properties of the oscillatorymode.

2.3.1. Plume-velocity measurements

For the small sample (Γ = 1) the speed sp and direction θp of the plume motion wasdetermined from cross-correlation functions C(r, t) =< I(r, t)I(r, t+ δt) >. Here r is thetwo-dimensional position vector, t the time, δt the time difference between two images,and < ... > indicated a spatial average. The method is illustrated in Fig. 4. Two images


Figure 3. Schematic illustration of the coordinates used when the sample was tilted by anangle β ≃ 1◦ relative to gravity.

separated in time by δt = 0.9 s (a and b) are shown. A central square (c) was extractedand converted to a one-bit image showing the plumes as black and most of the rest ofthe sample as white (d and e). The cross-correlation function between two such centralsquares, separated in time by various values indicated in the figure, are shown in the rightpart of Fig. 4. For δt = 0.0 s one has the auto-correlation function which has a maximumat r = 0. Various time delays give various displacements of this peak, by an amountδr which, together with δt, gives the plume velocity vp. The magnitude of vp gives theplume speed sp, and its direction is taken as the direction θp of the plume motion. It isassumed that the LSC orientation is equal to the direction of the plume motion. Timeseries typically contained from 8200 to 20200 images.

2.3.2. Plume-density measurements

In order to determine the plume density ρp, a threshold equal to half the averageintensity was applied to the background-divided images and all pixels with values above(below) the threshold value were set to zero (one). This isolated the warm plumes nearthe bottom plate, and the fraction of the pixels in the plumes (i.e.. the fractional areaoccupied by them) could be determined. This density was calculated using a centralcircular area of radius about half that of the cell as shown in Fig. 2b, but the resultswere not very sensitive to the precise area chosen. Although ρp depended somewhaton the intensity cutoff, its time and Rayleigh-number dependence and its probabilitydistribution-function showed no significant change over a wide cutoff range.

3. Experimental results for Aspect ratio Γ = 1

3.1. Medium and large sample

In the small sample, the frequency f0 of temporal oscillations in the azimuthal direction ofthe motion of plumes across the bottom plate was previously observed by shadowgraphy(Funfschilling & Ahlers (2004)). We can observe the same oscillations in the medium andlarge samples by looking at time series of the LSC orientations θt, θ0, and θb at differentheights. A short section of a much longer time series at the three heights is shown in


Figure 4. (a) and (b): Two background-divided and re-scaled shadowgraph images separatedin time by δt = 0.9 s. (c): A central square section of (a). (d) and (e): one-bit rendering of thesquare sections from (a) and (b). Right part: Time-correlation functions between images like (d)and (e), with several different time delays δt ranging from zero to 1.5 s (the origin is indicated bya small black dot). The fluid was methanol at a mean temperature of 40◦C and R = 4.6 × 108.

Fig. 5 for R = 3× 1010 in the large sample. While the orientation given by each data setcontains erratic motion, there is a tendency for the orientation at the top and bottomplanes to oscillate out of phase with each other around the mid-plane orientation.

Power spectra of the tree time series for θi(t), i = t, 0, b are shown in Fig. 6. Forthe mid-plane (θ0) the spectrum gives no hint of oscillations, except, curiously, whenthe sample is tilted by several degrees (this is not shown). Instead the spectrum decaysin proportion to f−1.25 over more than two decades of f as shown by the solid line inthe figure. In contrast to this, the top- and bottom-plane signal reveal, in addition tothe power-law background, a pronounced peak centered near f0 ≃ 8 × 10−3 Hz for theexample of Fig. 6.

The oscillations of θm,t about θ0 are shown more clearly by the deviations θt − θ0 andθb − θ0 of the top- and bottom-plane orientations from the mid-plane orientation; theseare shown in Fig. 7 for the data of Fig. 5. To reveal the nature of the oscillations morequantitatively, we compute the cross-correlation functions

gi,j(τ) = 〈[θi(t) − θ0(t)] × [θj(t + τ) − θ0(t + τ)]〉 (3.1)

where i and j can each refer to either the top- or bottom-plane orientations θt or θb, andwhere 〈...〉 without a subscript represents a time average. Normalized by

√

gi,i(0)gj,j(0),these are shown in Fig. 8 for R = 3 × 1010 and the large sample. The oscillations in theauto-correlation functions confirm that the top- and bottom-plane orientations oscillate,and the lack of a large background in the auto-correlations confirms that the oscillation


0 500 1000

0.2

0.4

0.6

t (s, arbitrary origin)

(θb,

θ0, a

nd θ

t) / 2

π

Figure 5. The large-scale-circulation orientations θb, θ0, and θt calculated separately for the3 sets of thermistors at R = 3 × 1010 in the large sample. Dotted line: θb from the bottomplane at a distance L/4 above the sample bottom (z = −L/4). Solid triangles: θ0 from the midplane at a distance L/2 above the sample bottom (z = 0). Solid line: θt from the top plane at adistance 3L/4 above the sample bottom (z = L/4). One can see that the top- and bottom-planeorientations oscillate, out of phase with each other, around the mid-plane orientation.

10−3 10−2 10−1

101

102

103

104

f (s−1)

pow

er (

rad2 s

)

Figure 6. The smoothed power spectra of the three angles θi(t) for R = 3×1010 from the largesample. Open triangles: θ0. Solid line: θt. Dotted line: θb. Dashed line: power-law fit to the datafor θ0 over the range 10−3 ≤ f ≤ 0.05. That fit yielded an exponent of −1.25.

is really around the mid-plane orientation θ0 and not around some fixed orientation.The cross-correlation similarly shows oscillations, and the minimum of gt,b(τ) at τ = 0indicates that the top and bottom row oscillations are out-of-phase with each other byhalf a period. The oscillations shown here lead to the same picture of an azimuthaltwisting oscillation of the LSC as found by Funfschilling & Ahlers (2004).


0 500 1000

−0.2

0.0

0.2

t (s, arbitrary origin)

(θi−

θ0)

/2π

Figure 7. The large-scale-circulation orientations with the meandering of θ0 subtracted out atR = 3× 1010 in the large sample. Dotted line: θb − θ0 from the bottom plane. Solid line: θt − θ0

from the top plane. One can see that the top- and bottom-plane orientations oscillate, out ofphase with each other, around the mid-plane orientation.

In Fig. 9 we show a schematic diagram that illustrates the nature of the oscillatingmode, with the top and bottom out of phase. This mode can be described by

θ(z, t) = θ0(t) + f(z)θ′(t) (3.2)

where θ(z, t) is the height-dependent orientation of the circulation plane of the LSC. Ifwe choose the origin of the vertical axis at the horizontal mid-plane as shown in thefigure, then the oscillation amplitude becomes an odd function of z, i.e. f(z) = −f(−z).We set f(L/4) = 1 to so that θ′(t) = θi(t) − θ0(t) as shown in Fig. 7.

Elsewhere we showed that the background dynamics given by θ0(t) is primarily diffu-sive, albeit interrupted by various relatively sudden re-orientation events (Brown et al.

(2005), Brown & Ahlers (2006a)). The twist angle θ′(t) also has an irregular time depen-dence. In Fig. 10 we show the probability distribution pθ′ as a function of θ′ = θi − θ0

with i = t, b. As shown by the solid line in the figure, the peak of pθ′ can be fitted well bya Gaussian function. Further away from the peak there are deviations from the Gaussianform, with the experimental data yielding a somewhat larger pθ′ . If the twist could berepresented by a simple oscillation with θ′(t) = A cos(ωt) with a constant (time indepen-dent) A, then one would expect pθ′ to have maxima at θ′ = ±A. In contrast to this, themeasurements indicate a peak at θ′ = 0. This is consistent with a time dependent A(t)with a probability distribution pA of A that is symmetric about A = 0. A simple modelfor θ′(t) that reproduces a Gaussian pθ′ is given by the stochastic differential equationdescribing a damped driven harmonic oscillator

θ′ = −bθ′ − ω2

0θ′ + g(t) (3.3)

where b is a damping coefficient, ω is the known angular oscillation frequency, and g(t)is a Gaussian distributed noise term. This equation produces oscillatory motion in ap-propriate parameter ranges, and it is known that diffusion in a harmonic well produces aGaussian position distribution [Gitterman (2005)]. As shown by the solid line in Fig. 10,


−400 −200 0 200 400

0.0

0.5

1.0

τ (s)

g i,j(

τ) /

[gi,i

(0)g

j,j(0

)]1/

2

Figure 8. Normalized correlation functions gi,j(τ )/√

gi,i(0)gj,j(0) for the top-, mid-, and bot-

tom-row LSC orientations for R = 3 × 1010. Open triangles: cross-correlation between top andbottom rows (i = t, j = b). Dotted line: auto-correlation of bottom row (i = j = b). Solid line:auto-correlation of top row (i = j = t). The oscillations are very clear, with the minimum ofgt,b(τ ) at τ = 0 indicating that the top- and bottom-plane orientations oscillate out-of-phasewith each other.

Figure 9. Schematic diagram of the LSC oscillation at time t0 when the oscillation is at anextremum (left figure), and at time t0 + T /2 (i.e. half an oscillation period later) (right figure).

a Gaussian function provides an excellent fit to the data except in the tails of pθ′ . Whilewe have no physical reason for using the model Eq. 3.3, it is suggestive that it is similarin mathematical form to others used to describe the LSC dynamics (Brown & Ahlers(2006b), Brown & Ahlers (2007)).

The random time dependence of θ′ also leads to the decay of the correlation functions


–0.2 0.0 0.210–2

10–1

100

101

θ’/2π

p(θ’

/2π)

Figure 10. The probability-distribution pθ′ for θ′ = θb − θ0 (solid circles) and for θ′ = θt − θ0

(open circles) for R = 3× 1010 in the large sample. The solid line is a fit of a Gaussian functionnear the peak. The dashed line was obtained from a numerical solution of Eq. 3.3 as describedin the text.

0.003 0.0100

4

8

12

f (s−1)

Si,j

( r

ad2 s

)

Figure 11. The smoothed power spectra Si,j(f) of θ′ obtained from the Fourier transforms ofeach of the three correlation functions gi,j(τ ) for R = 3× 1010 from the large sample. Triangles:St,b. Circles: St,t. Squares: Sb,b. Lines: A Lorentzian function with a exponential background isfit to the peak for St,b (dashed line), St,t (solid line), and Sb,b (dotted line).

shown in Fig. 8. In order to provide a quantitative characterization of θ′(t), we show asmoothed power spectrum (the square of the modulus of the Fourier transform, or the“structure factor”) Si,j(f) of θt − θ0 and θb − θ0 in Fig. 11. In addition, the Fouriertransform of gt,b(t) is shown. Particularly for Si,j(f) one sees that there remain smallbackgrounds that are monotonically decreasing with f , although these backgrounds aremuch smaller than those seen in Fig. 6 for the spectra of θi. In addition to the backgroundthere is a strong peak at f = f0 for all three spectra. Its finite width is attributable to


108 109 1010 1011

0.01

0.05

10−5

10−4

101

102

103

104

S0ν

/L2

σ fL2 /ν

and

f 0L2 /ν

PL

= π

σ fS

0

R

(a)

(b)

(c)

Figure 12. WHY ARE THE f0 DATA NOT AT THE SAME R VALUES AS THESIGMA DATA? Fit paramters from Eq. 3.4 for the power spectrum. These results are forthe bottom plane at z = −L/4. Solid symbols: medium sample. Open symbols: large sample.(a): dimensionless peak height S0ν/L2. The dotted line corresponds to a power law with anexponent of -0.29. (b): dimensionless peak half-width σfL2/ν (circles) and center frequency f0

(squares). The solid (dashed) line corresponds to a power law with an exponent of 0.55 (0.50).(c) The total power given by the integral of the area of the peak in frequency space. The dottedline corresponds to a power law with an exponent of 0.44.

the distribution of A(t). We fitted Si,j(f) to an exponential background and a Lorentzianpeak, i.e. to

Si,j(f) =S0

1 + [(f − f0)/σf ]2+ a exp

−f

b(3.4)

over the range 0.5f0 < f < 1.5f0 with fitting parameters S0, f0, σf , a, and b. In Fig. 11one sees that the Lorentzian shape gives a good representation of the data.

In Fig. 12 we show the Rayleigh-number dependence of the scaled (i.e. dimensionless)fit parameters S0ν/L2 (the peak height), σfL2/ν (the peak half-width), and PL = πS0σf

(the total power under the Lorentzian peak excluding the background) for the bottom-plane data for θ′ = θb − θ0. The fit parameters for the top-plane data and for the cross-correlation data are similar, even though the background for the cross-correlation data ismuch weaker. One sees that the width of the peak increases and the height decreases withincreasing R. For the dimensionless height the data from the two samples are consistentwith a common power law only when data for relatively small R are considered for each


(a) (b)

(c) (d)

Figure 13. Shadowgraph images for the small Γ = 1 sample with methanol at a meantemperature of 40◦C. (a): R = 5.7 × 107. (b): R = 1.2 × 108. (c): R = 2.3 × 108. (d):

R = 4.6 × 108.

sample. At larger R there are strong negative deviations in both cases. It is not clear tous what physical phenomenon would produce this effect. Deviations from the Boussinesqapproximation (Ahlers et al. (2006), Ahlers et al. (2007)) come to mind; however, forother properties such as the center temperatures of the samples these effects come intoplay only at much larger values of the applied temperature differences. The deviationsfrom a common power law seen for the peak height of course are revealed also in the totalpower PL as seen in Fig. 12c. Using only the data at the smallest R for each sample, onemight argue for a power-law dependence of PL with an exponent near 0.44 as suggestedby the dotted line. On the other hand, an R-independent value for PL, different for thetwo samples, is suggested by the larger-R data.

A question of interest is whether the LSC and/or its oscillations disappear at large R.Besides looking at the total power of the oscillations, one might consider the mode tobe unresolvable when the width 2σf of the peak becomes significantly greater than thefrequency f0 of the mode, i.e. when the quality factor Q = f0/(2σf ) becomes significantlyless than one. To make this comparison, we also show f0 in Fig. 12b. A power-law fitto the half-width for R > 3 × 109 yields σf ∝ R0.55, whereas such a fit to f0 yieldsf0 ∝ R0.50 (Brown et al. (2007)). In our range of R the half-width is about 1/10 of thefrequency as seen in Fig. 11, i.e. Q ≃ 5. The closeness of the effective exponents for thetwo parameters implies that an extrapolation of the width to larger R remains smallcompared to the extrapolated frequency over a very large range of R, and thus that Qis nearly independent of R. This suggests that the twisting oscillation will not die out atany R obtainable in the laboratory due to the spectral width becoming larger than thepeak frequency unless the observed R-dependences change dramatically at higher R.


3.2. Small sample

3.2.1. On the nature of plumes

The nature of the relatively warm (cold) localized volumes of fluid, also known as“plumes”, that are emitted by the bottom (top) boundary layer has been investigated nu-merous times, in most cases by using the shadowgraph method. This method detects theintegral along the path of observation of the refractive-index variations orthogonal to thatpath. When viewed from above (Tanaka & Miyara (1980), Zocchi et al. (1990), Gluckmanet al. (1993), Vorobie & Ecke (2002), Funfschilling & Ahlers (2004), Haramina & Tilgner(2004), Puthenveettil & Arakeri (2005), Zhou et al. (2007)) plumes have generally giventhe appearance of structures that are localized in only one horizontal direction and ex-tended in an orthogonal horizontal direction. These structures often are referred to as“sheet” plumes, although this term may be misleading because their two-dimensionalextent does not seem to be established. We prefer to call them “line” plumes because itappears more likely to us that they are one-dimensional excitations in the marginally-stable boundary layers. Examples of plume visualizations taken from above were shownalready in Figs. 2 and 4. Four additional examples are shown in Fig. 13 for severalvalues of R, and mpeg movies showing the dynamics for these examples are availableelsewhere (?). A plume near the bottom plate appears as a narrow elongated relativelydark stripe. Since, to lowest order, the shadowgraph method under certain conditionsyields the vertical average of the horizontal Laplacian of the refractive-index (and thus ofthe temperature) field (Rasenat et al. (1989), deBruyn et al. (1996), Trainoff & Cannell(2002)), such a dark stripe is generally framed on each side by an even more narrowlight stripe. This entire structure then corresponds to a line plume. Analogous brightlines flanked by dark ones correspond to cold line plumes near the top, but in our workthese were de-emphasized by appropriate adjustment of the optics. Since the structureis viewed from above, we can have no direct information about its vertical extent. Thus,it could indeed be a sheet-like structure with significant vertical extent, or it could be aline structure with a predominantly horizontal orientation. However, these structures aregenerally not observed in shadowgraphs taken in the horizontal direction (Zocchi et al.

(1990), Moses et al. (1991), Moses et al. (1993), Ciliberto et al. (1997), Zhang et al.

(1997), Du & Tong (1998), Qiu & Tong (2001b), Shang et al. (2003), Xi et al. (2004)),where plumes usually have the appearance of lines, often emanating from and extendingaway from the boundary layer and terminating in a mushroom cap. We shall refer tothese structures as “mushroom plumes”. Since line plumes do not appear in lateral ob-servations, it appears that they are confined to the immediate vicinity of the bottom ortop boundary layer where visualization from the side becomes difficult. The mushroomplumes are found in transverse observations primarily in the lower or upper portion of thesample, but often well separated from the bottom or top boundary layer. They are sweptlaterally by the LSC toward the side wall. Near the side wall they rise or fall due to theirbuoyancy. From all of these observations the picture emerges that plumes are born as lineexcitations of the boundary layers, extended in one horizontal direction only and havinginitially very little if any vertical extent. These excitations are also swept laterally and invery close proximity to the boundary layer by the LSC. In the presence of the LSC theyorient themselves with their long axis in the direction of the LSC. This alignment phe-nomenon is well known in other contexts, where convection rolls align their axes with aprevailing wind or in the direction of a shear (for a review, see for instance Kelly (1994)).At some point along their path, and especially when impinging upon the side wall, lineplumes separate from the boundary layer and evolve into mushroom plumes with theircharacteristic mushroom heads. This process leaves the sample center relatively free of


0 200 400 600

−1

0

1

2 θ

p( t

)

0 100 200 300

0.0

0.5

1.0

Cq(

t )

0 200 400 6000.0

0.1

0.2

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ρp(

t )

0 100 200 300

0.0

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Cr(

t )

time [s]

(a)

(b)

(c)

(d)

0 100 200 300

0 200 400 600

0 100 200 300time [s]

0 200 400 600

(e)

(f)

(g)

(h)

Figure 14. The plume direction θp and density ρp, and their correlation functions Cθ(τ ) andCρ(τ ), as a function of time t or τ . (a) to (d): Γ = 1 with 2-propanol and R = 7.8 × 108. (e) to(h): Γ = 2 with 2-propanol and R = 1.5 × 108.

plumes. Consequently observations from the top and not too close to the side wall revealprimarily the line plumes very near the top and bottom boundaries.

3.2.2. Plume-direction oscillation and plume density

As reported before from experiments using methanol (Funfschilling & Ahlers (2004)),the azimuthal direction θp(t) of the lateral motion of plumes near the top or bottomplate oscillates in time. This is illustrated in Fig. 14a and b for a Rayleigh number of7.8× 108 using 2-propanol as the fluid. There (a) gives a short segment of a much longertime series of θp(t). The signal is quite noisy because occasionally there are too few or noplumes present and the calculated cross-correlation function between successive images(see Fig. 4) does not reflect plume motion. Although some algorithm could have beendeveloped to eliminate these spurious points, it turned out that the time correlation-function

Cθ(τ) = 〈θp(t) × θp(t + τ)〉t (3.5)

shown in the second panel (b) of the figure clearly reveals the oscillations and shows thatthey remain coherent over a long time period. We assume that the plume motion is slavedto the motion of the LSC and that θp closely reflects the LSC flow-direction θ(±L/2, t)(see Eq. 3.2) near the top and bottom plates. Indeed the same oscillations have been foundalso in velocity measurements using air (Resagk et al. (2006)) as the fluid. Simultaneousmeasurements for methanol using the top (bright) and bottom (dark) plumes revealed


−2 0 20.01

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(θp(t) − < θp(t)>)/σθp

PD

F o

f θp

−4 −2 0 2 40.01

0.10

1.00

( ρp − < ρp>) / σρp

PD

F o

f ρp

Figure 15. The probability-density functions (PDF) of θp and ρp for Γ = 1 and 2-propanol.The horizontal axes were scaled by the square roots of the variances σθp

and σρpof the data.

The Rayleigh nimber was R = 7.8 × 108. The solid lines are fits of a Gaussian function to thedata.

that the oscillations near the top and bottom plate are out of phase with each other by π(Funfschilling & Ahlers (2004)), consistent with the side-wall temperature-measurementsreported above in Sect. 3.1 and shown in Fig. 8.

The results for θp lead to a power spectrum (not shown here) similar to those of θt

and θb in Fig. 6, with a pronounced peak centered at a characteristic frequency f0 whichcorresponds to the inverse turn-over time of the LSC. We shall postpone the discussion ofthe Reynolds numbers corresponding to f0 to a later publication where we shall includeall of our measurements, using methanol, 2-propanol, and water.

The density of plumes ρp(t) for the example of Fig. 14 is illustrated in panel (c). Onesees that ρp varied irregularly with time, with no obvious oscillatory component. This isborne out by the time auto-correlation function of ρp, which is shown in panel (d) of thefigure and which meanders non-periodically about zero.

In Fig. 15 we show the probability-density functions (PDF) of θp(t) and of the densityρp(t) for the example of Fig. 14. These functions show an unremarkable Gaussian dis-tribution. Such a distribution has been noted (Zhou et al. (2007)) with regard to otherfeatures of plumes in turbulent RBC.

The oscillation amplitudes of θp can in principle be estimated from the power spectrumof θp in a manner similar to that employed to obtain the data in Fig. 12 for θt and θb.However, the signal-to-noise ratio was considerably smaller for this method then it was forthe sidewall-temperature measurements, and no definitive statement about the Rayleigh-number dependence of the amplitude could be made. Thus we only mention that typicalvalues were in the range from 0.3 to 0.7 radian for 5 × 108 < R < 3 × 109, both for2-propanol and for methanol.

4. Results for Aspect ratios Γ > 1

For the aspect ratios Γ = 2 and Γ = 3 the oscillations of θb were very weak or absententirely. This is illustrated in Fig. 14 (e) to (h) for Γ = 2 and 2-propanol. The toppanel (e) shows part of a time series of θp. No oscillations are apparent to the eye. Thesecond panel (f) gives the corresponding correlation function. Whereas it shows a hint ofoscillations at early times, these decay within a few cycles. This stands in stark contrastto the Γ = 1 case where Cθ shows little or no decay over the time interval shown. On the


107 108 1090.01

0.10

1.00

R

ρ p

107 108 109

G2.7 R

Figure 16. Left panel: The plume density above the bottom plate as a function of the Rayleighnumber R. Solid circles: Γ = 1. Solid squares: Γ = 2. Open circles: Γ = 3. Right panel: The samedata plotted as a function of Γ2.7R. Solid line: a power law with an exponent of 2.0.

other hand, for Γ = 2 the density of plumes (panel g) clearly oscillates. These ocillationsare more clearly revealed by the correlation function in panel (h). This behavior againstands in contrast to the Γ = 1 case where Cρ showed no periodic component at all. Theoscillations of ρp were found for all Rayleigh numbers above R ≃ 2× 107 for aspect ratio2 and R ≃ 1.5 × 107 for aspect ratio 3. The oscillation amplitudes of the densities werequite large (typically 50 percent of the mean value < ρp >). In spite of the difference inthe temporal structure between Γ = 1 on the one hand and Γ = 2 and 3 on the other,the PDFs of both θp and ρp were very similar for the two categories. We do not have anyinformation about the geometrical features of the oscillating mode for the larger Γ cases.However, the difference presumably is associated with a major change in the structureof the flow.

Finally, in the left part of Fig. 16 we show the density of the plumes above the bottomplate as a function of R for all three samples with 2-propanol. The data for the threeaspect ratios fall on well separated curves. Interestingly, they essentially collapse onto asingle curve when plotted against Γ2.7R, revealing a strong Γ dependence. A powerlawscaling of ρp with R can be expected only when ρp << 1 because the density has an upperlimit of unity by virtue of its definition. The R-range of the data over which ρp << 1is rather small, and thus no effective exponent can be established with great confidence.However, the solid line in Fig. 16 is drawn with a slope of 2, suggesting an exponent inthe vicinity of 2.

5. Summary and conclusions

In this paper we presented a detailed account of shadowgraph measurements, in partreported briefly before (Funfschilling & Ahlers (2004)), of the azimuthal orientationsθp of plume motions across the top and bottom plates of turbulent samples of fluid incylindrical containers and heated from below. These data are for aspect ratios Γ fromone to three. Data for methanol with a Prandtl number σ = 6.0 and 2-propanol withσ = 29 did not reveal any significant dependence on σ. It is argued that θp reflects theorientation of the LSC that is sweeping the plumes across the plates. For Γ = 1 thedata revealed an azimuthal LSC oscillation that had been shown before (Funfschilling& Ahlers (2004)) to be a twisting mode of the LSC circulation plane, with the top andbottom halves oscillating our of phase by π. For Γ = 2 and 3 this mode was not presentor extremely weak.

In addition to the results for θp, shadowgraph measurements for the plume density ρp


near the center of the bottom plate are reported. For Γ = 1, ρp varied irregularly in timebut did not reveal the oscillations found in θp. Interestingly, for the larger Γ = 2 and3 (where θp did not oscillate) ρp had a strong oscillating component. The geometricalnature of the mode leading to an oscillation of ρp and only irregular meandering of θp isnot yet revealed by the data.

For Γ = 1 the typical oscillation amplitude of θp was in the range from 0.2 to 0.7radian. For all three aspect ratios, the shadowgraph measurements revealed that bothρp and θp had Gaussian probability distributions regardless of the absence or presenceof oscillations. For small ρp (say ρp <∼ 0.2) the plume density for all three Γ had a strongRayleigh-number dependence, suggesting approximately ρp ∼ R2.0±0.2, but at large Rρp saturated near ρp ≃ 0.4 (we note that by virtue of its definition ρp < 1). There alsowas a strong Γ dependence of ρp(R). The data for all three Γ could be collapsed onto aunique curve when plotted against Γ2.7±0.3R.

We also presented new measurements of the LSC orientation θ(z, t) by a very differ-ent method. These measurements are for two cylindrical samples with Γ = 1 over theRayleigh-number range 108 <∼ R <∼ 1011 and for σ = 4.4. In this work we determinedθb = θ(−L/4, t), θ0 = θ(0, t), and θt = θ(+L/4, t) (we take the origin of the z axis atthe horizontal mid-plane) from measurements of the azimuthal dependence of the side-wall temperatures. The results confirm the shadowgraph observations in that they alsoshow a twisting mode of the LSC circulation plane, with θb(t) and θt(t) oscillating outof phase by π and θ0 revealing no oscillatory contribution at all. Although each angle byitself shows a diffusive time evolution, with an oscillatory component added for θt andθb, the differences θ′i = θi − θ0, i = p, t reveal almost no meandering background at all,indicating that the oscillations are about θ0 rather than about some other fixed angle.

For a purely sinusoidal oscillation of θ′ with constant amplitude A one expects aprobability distribution pθ′ of θ′ that is strongly peaked at θ′ = ±A. Instead, we foundthat pθ′ has a maximum at θ′ = 0 and that it has a Gaussian shape when |θ′| is not toolarge. This result could be reproduced by the stochastic second-order differential equationthat describes a damped harmonic oscillator driven by random noise. This equation forθ′(t) is reminiscent of the equations that we discussed elsewhere (Brown & Ahlers (2006b),Brown & Ahlers (2007)) to explain the rapid re-orientations of the LSC orientationand the influence of Earth’s Coriolis force on the LSC. In those equations the physicalorigins of all terms were clear, and the small-scale turbulent background was representedby a phenomenological additive noise term corresponding to experimentally determineddiffusivities. In the present case we assume that the needed stochastic term likewiserepresents the background fluctuations, but we do not have a physical explanation forthe origin of the inertial term that is needed in order to reproduce the oscillations.

The oscillations of θb and θt lead to a peak at finite frequency of the power spectrumof θi − θ0 (i = t, b), with only a small broad background contribution. The peak could bedescribed by a Lorentzian function. We presented data for the height S0, half-width σf ,center frequency f0, and total power PL = πS0σf of this peak. We found that the heightS0 of the peak decreased and that the width σf increased with increasing R. Althoughboth f0 and σf increase with R, the quality factor f0/(2σf ) is nearly independent of R,suggesting that the oscillatory mode will survive up to R-values well beyond that of thepresent data unless a dramatic change in the behavior of the system intervenes.

An interesting question is how the oscillatory mode comes into existence as R increasesfrom small values. We do not have data at sufficiently small R to provide an answer,but this issue was addressed by Qiu & Tong (2001b) using measurements at σ = 5.4.From velocity and temperature data at single points in the sample, and of correlationfunctions between them, they found a sharp transition at Rc ≃ 5 × 107 from a small-R


state without coherent oscillations to another at larger R that exhibited the oscillations.At Rc the frequency observed by these authors had a finite value, suggesting that theoscillatory mode evolves via a Hopf bifurcation. Qiu & Tong (2001b) did not focus on theamplitude of the mode near onset. Additional measurements will be required in order toclarify this important aspect of the LSC.

6. Acknowledgments

This work was supported by the US National Science Foundation through GrantDMR07-02111.

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