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arXiv:1911.09584v1 [physics.flu-dyn] 21 Nov 2019

Boundary Zonal Flow in Rotating Turbulent Rayleigh–B´enard Convection

Xuan Zhang1,∗Dennis P. M. van Gils1,2,∗Susanne Horn1,3,4, Marcel Wedi1, Lukas Zwirner1,

Guenter Ahlers1,5, Robert E. Ecke1,6, Stephan Weiss1,7, Eberhard Bodenschatz1,8,9, and Olga Shishkina1†

1Max Planck Institute for Dynamics and Self-Organization, 37077 G¨ottingen, Germany

2Physics Fluids Group, J.M. Burgers Center for Fluid Dynamics,

University of Twente, 7500 AE Enschede, The Netherlands

3Department of Earth, Planetary, and Space Sciences, UCLA, CA 90095, USA

4Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, UK

5Department of Physics, University of California, Santa Barbara, CA 93106, USA

6Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

7Max Planck – University of Twente Center for Complex Fluid Dynamics

8Institute for Nonlinear Dynamics, Georg-August-University G¨ottingen, 37073 G¨ottingen, Germany and

9Laboratory of Atomic and Solid-State Physics and Sibley School of Mechanical

and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA

(Dated: November 22, 2019)

For rapidly rotating turbulent Rayleigh–B´enard convection in a slender cylindrical cell, experi-

ments and direct numerical simulations reveal a boundary zonal ﬂow (BZF) that replaces the classical

large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat

transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the

temperature is an anticyclonic traveling wave of mode one whose signature is a bimodal temperature

distribution near the radial boundary. The BZF width is found to scale like Ra1/4Ek 2/3where the

Ekman number Ek decreases with increasing rotation rate.

Turbulent ﬂuid motion driven by buoyancy and inﬂu-

enced by rotation is a common phenomenon in nature

and is important in many industrial applications. In the

widely studied laboratory realization of turbulent convec-

tion, Rayleigh–B´enard convection (RBC) [1, 2], a ﬂuid

is conﬁned in a convection cell with a heated bottom,

cooled top, and adiabatic vertical walls. For these condi-

tions, a large scale circulation (LSC) arises from cooper-

ative plume motion and is an important feature of turbu-

lent RBC [1]. The addition of rotation about a vertical

axis produces a diﬀerent type of convection as thermal

plumes are transformed into thermal vortices, over some

regions of parameter space heat transport is enhanced

by Ekman pumping [3–10], and statistical measures of

vorticity and temperature ﬂuctuations in the bulk are

strongly inﬂuenced [11–17]. A crucial aspect of rotation

is to suppress, for suﬃciently rapid rotation rates, the

LSC of non-rotating convection [12, 13, 18, 19], although

the diameter-to-height aspect ratio Γ = D/H appears to

play some role in the nature of the suppression [20].

In RBC geometries with 1/2≤Γ≤2, the LSC usu-

ally spans the cell in a roll-like circulation of size H.

For rotating convection, the intrinsic linear scale of sep-

aration of vortices is reduced with increasing rotation

rate [21, 22], suggesting that one might reduce the geo-

metric aspect ratio, i.e., Γ <1 while maintaining a large

ratio of lateral cell size to linear scale [5]; such convec-

tion cells are being implemented in numerous new exper-

iments [23]. Thus, an important question about rotating

convection in slender cylindrical cells is whether there is a

global circulation that substantially inﬂuences the inter-

nal state of the system and carries appreciable global heat

transport. Direct numerical simulations (DNS) of rotat-

ing convection [24] in cylindrical geometry with Γ = 1,

inverse Rossby number 1/Ro = 2.78, Rayleigh number

Ra = 109and Prandtl number Pr = 6.4 (Ro,Ra and

Pr deﬁned below) revealed a cyclonic azimuthal velocity

boundary-layer ﬂow surrounding a core region of anti-

cyclonic circulation and localized near the cylinder side-

wall. The results were interpreted in the context of side-

wall Stewartson layers driven by active Ekman layers at

the top and bottom of the cell [25, 26].

Here we show through DNS and experimental mea-

surements for a cylindrical convection cell with Γ = 1/2

at large Ra and for a range of rotation rates from slow to

rapid that a wider (several times the Stewartson layer

width) annular ﬂow, denoted a boundary zonal ﬂow

(BZF), has profound eﬀects on the overall ﬂow structure

and on the spatial distribution of heat ﬂux. In partic-

ular, this cyclonic zonal ﬂow surrounds an anti-cyclonic

core. The BZF has alternating temperature sheets that

produce bimodal temperature distributions for radial po-

sitions r/R > 0.7 and that contribute greatly to the

overall heat transport; 60% of heat transport are carried

in the BZF. Although the location of the azimuthally-

averaged maximum cyclonic azimuthal velocity, the root-

mean-square (rms) vertical velocity ﬂuctuations, and the

normalized vertical heat transport at the mid-plane are

consistent with a linear description of a Stewartson-layer

scaling [24], the dynamics of temperature, vertical ve-

locity and heat transport in the BZF are more complex

and interesting. The robustness of the BZF state as evi-

denced by its existence over 7 orders of magnitude in Ra

in DNS and experiment and over a range 1/2≤Γ≤2

and 0.1≤Pr ≤4.4 (results to be presented elsewhere)

suggests that this is an universal state of rotating con-

2

FIG. 1. Sidewall temperature PDFs, r/R = 1, for z/H = 1/4 (diamonds), z/H = 1/2 (circles), and z/H = 3/4 (squares), with

1/Ro = 0 (a, c) and 10 (b, d). Experimental measurements with Ra = 8 ×1012 (a, b) and DNS with Ra = 109(c, d), both

with Pr = 0.8. Bimodal Gaussian distributions (solid lines), the sum of two normal distributions (dashed lines), are observed

for rapid rotation (b, d). h·izsdenotes average in time and over all sensor positions at distance zfrom the hot plate.

FIG. 2. Horizontal cross-sections of time-averaged ﬂow ﬁelds

(DNS), visualized with streamlines (arrows) and azimuthal

velocity huφit(colors) (a, b) at height of thermal BL, z=δθ≡

H/(2Nu) and (c, d) at the mid-plane, z=H/2, with Ra =

109and 1/Ro = 0.5 (a, c) and 10 (b, d). Blue (pink) color

indicates anticyclonic (cyclonic) motion. In (d), locations r=

r0of huφit= 0 (solid line) and r=rumax

φof the maximum of

huφit(dashed line) are shown.

vection that needs a physical understanding.

The dimensionless control parameters in rotating RBC

are the Rayleigh number Ra =αg∆H3/(κν), Prandtl

number Pr =ν/κ, cell aspect ratio Γ and Rossby number

Ro =√αg∆H / (2ΩH) or, alternatively, Ekman number

Ek =ν/(2ΩH2). Here αis isobaric thermal expansion

coeﬃcient, νkinematic viscosity, κﬂuid thermal diﬀu-

sivity, gacceleration of gravity, Ω angular rotation rate,

and ∆ temperature diﬀerence between the hotter bottom

and colder top plates. The main integral response param-

eters we consider is the Nusselt number Nu ≡ hFzit,V ,

where h·it,V denotes the time- and volume-averaging and

Fz≡(uz(T−T0)−κ∂zT)/(κ∆/H) is the normalized

vertical heat ﬂux with uzbeing the vertical component

of the velocity and T0the average of the top and bottom

temperatures.

We present numerical and experimental results [27]

for rotating RBC in a Γ = 1/2 cylindrical cell and

1/Ro = 0, 0.5 and 10. The DNS used the goldfish code

[28, 29] with Pr = 0.8 and Ra = 109. The experiments

used pressurized sulfur hexaﬂuoride (SF6) and were per-

formed over a large parameter space in the High Pressure

Convection Facility (HPCF, 2.24 m high) at the Max

Planck Institute for Dynamics and Self-Organization in

G¨ottingen [30]. In the studied parameter range, the

Oberbeck–Boussinesq approximation is valid [31–33], and

the centrifugal force is negligible [8, 34, 35].

We ﬁrst consider the azimuthal variation of the tem-

perature measured by thermal probes at or near the

sidewall, a commonly used technique for parameteriz-

ing the LSC in RBC [18, 20, 36–38]. We measured

experimentally and in corresponding DNS the tempera-

ture at 8 equidistantly spaced azimuthal locations of the

sensors for each of 3 distances from the bottom plate:

z/H = 1/4, 1/2 and 3/4. The PDFs of the experimen-

tal data without rotation (1/Ro = 0, Ra = 8 ×1012) in

Fig. 1a show a distribution with a single peak and slight

asymmetry to hotter (colder) ﬂuctuations for heights

smaller (larger) than z/H = 1/2, whereas the PDFs for

rapid rotation (1/Ro = 10, Fig. 1b), show a bimodal

distribution that is well ﬁt by the sum of two Gaussian

distributions. The corresponding PDFs of the DNS data

(at Ra = 109) show the same qualitative transition from

a single peak without rotation to a bimodal distribution

in the rapidly rotating case with similar hot/cold asym-

metry for diﬀerent z(Fig. 1c, d). To understand the

nature of the emergence of a bimodal distribution near

the radial boundary, we consider the DNS data in detail.

The LSC for non-rotating convection in cells with

1/2≤Γ≤2 and at large Ra extends throughout the

entire cell with a large roll-like circulation [39]. With

slow rotation, Coriolis forces induce anticyclonic motion

close to the plates owing to the diverging ﬂow between the

3

FIG. 3. For Ra = 109, 1/Ro = 10: (a)huφit,φ versus zat r= 0.95R. The inset shows the same data for 0 ≤z≤H/2 in a

log-plot. (b)huφit,φ versus rat z=H/2; radial zero crossing r=r0(solid line) and radial maximum r=rumax

φ(dashed line).

(c) Instantaneous thermal ﬁeld at r=rumax

φversus zand φ.

LSC and the corner rolls. At the mid-plane, the LSC is

tilted with a small inward radial velocity component that

rotation spins up into cyclonic motion. These tendencies

are illustrated for 1/Ro = 0.5 in Figs. 2a, c, respectively,

where streamlines of time-averaged velocity are overlaid

on the ﬁeld of azimuthal velocity. Fig. 2a shows ﬁelds

evaluated at the thermal BL height z=δθ≡H/(2N u),

demonstrating the dominant anticyclonic ﬂow near the

boundary. The situation is reversed at the mid-plane

(Fig. 2c) where cyclonic motion extends over almost the

entire cross sectional area.

For rapid rotation, viscous Ekman BLs near the plates

induce anticyclonic circulation with radial outﬂow in hor-

izontal planes as in Fig. 2b. The outﬂow is balanced by

the vertical velocity in an increasingly thin (with increas-

ing 1/Ro) annular region near the sidewall where cyclonic

vorticity is concentrated at the mid-plane, see Fig. 2d.

The core region, on the other hand, is strongly anticy-

clonic owing to the Taylor-Proudman eﬀect [40, 41] that

tends to homogenize vertical motion. The circulation for

a rotating ﬂow in a ﬁnite cylindrical container consists of

thin anticyclonic Ekman layers on top and bottom plates

and compensating Stewartson layers along the sidewalls

with up-ﬂow from the bottom and down-ﬂow from the

top [24, 42]. This classical BL analysis was successfully

applied to rotating convection [24] for a Γ = 1 cylindrical

cell with Pr = 6.4 and 108≤Ra ≤109in both exper-

iment and DNS. No evidence for a coherent large-scale

circulation for rapid rotation was found in those studies.

For our conditions, Pr = 0.8, Ra = 109, and 1/Ro =

10, we compute the time- and azimuthal-average az-

imuthal velocity huφit,φ (normalized by the free-fall ve-

locity uﬀ=pαg∆/R) as a function of height zfor ﬁxed

r= 0.95Rand of radius rat ﬁxed z=H/2. The height

dependence of huφit,φ, Fig. 3a, shows an anticyclonic

(negative) circulation close to the top and bottom plates

and an increasingly cyclonic (positive) circulation with

increasing (decreasing) zfrom the bottom (top) plate.

The radial dependence, Fig. 3b, demonstrates the sharp

localization of cyclonic motion near the sidewall as pa-

rameterized by the zero-crossing r0(solid line) and the

maximum rumax

φ(dashed line). Corresponding distances

from the sidewall are δ0=R−r0and δumax

φ=R−rumax

φ

where δumax

φ≈δurms

z(based on maximum of rms of uz).

δurms

zwas used to deﬁne the sidewall Stewartson layer

thickness in rotating convection [24], and our results for

huφitare consistent with that description. What was ab-

solutely not expected is the strong azimuthal variation

of the instantaneous temperature Tshown in Fig. 3c, a

feature that deﬁnes the global ﬂow circulation, namely

the spatial distribution of the heat transport which is

the origin of the bimodal temperature distributions seen

in the experiments and DNS.

The strong variations in instantaneous temperature

shown in Fig. 3c organize into anticyclonic traveling

waves illustrated in the angle-time plot of T, Fig. 4a.

The BZF height is order H, Fig. 3c, but is increasingly

localized in the radial direction as the rotation rate in-

creases (Ro and Ek decrease) so that δ0/R ≪1. The

azimuthal mode of Tis highly correlated with a corre-

sponding mode of the vertical velocity, Fig. 4b, with a

resulting coherent mode-1 (m= 1) anticyclonic circu-

lation in φwith a warm up-ﬂow on one side of the cell

balanced by a cool down-ﬂow on the other side of the cell

(for Γ = 1,2, the dimensionless wave number m/Γ = 2

is independent of Γ, to be presented elsewhere). The

anticyclonic circulation is the speed of the anticyclonic

horizontal BL, suggesting that the thermal wave is an-

chored at the horizontal BLs so that it travels against

the cyclonic circulation near the sidewall. The coherence

between Tand uzleads to localization of vertical heat

ﬂux near the sidewall shown in Fig. 4c where the heat

ﬂux within the annular area deﬁned by δ0is ≈60% of the

total heat ﬂux.

We arrive at a compact description of the BZF. The

radial distances from the sidewall δurms

z,δFmax

z, and δumax

φ

of maxima of uz-rms, heat ﬂux Fz, and uφ, respectively,

scale as Ek1/3, Fig. 5a, consistent with the expectations

of Ekman/Stewartson BL theory [24, 42]. On the other

hand, the cyclonic zone width δ0decreases more rapidly

with Ek, i.e., as E k2/3with a Ra1/4dependence (pre-

sented elsewhere). Thus, the inner layer is consistent

4

FIG. 4. For Ra = 109and 1/Ro = 10: (a, b) angle-time plots

at r=rumax

φ,z=H/2 of (a)Tand (b)uz; (c) normalized

time-averaged vertical heat ﬂux hFzitat z=H/2. In (c),

location of rwhere h Fz|z=H/2it=Nu (dash-dotted line) and

locations r=r0of huφit= 0 (solid line) and r=rumax

φof the

maximum of huφit(dashed line) are shown. Color scale from

blue (min values) to pink (max values) ranges (a) between

the top and bottom temperatures, (b) in [−uﬀ/2, uﬀ/2], (c)

from 0 to mid-plane magnitude of hFzit, which is ≈3.4Nu.

with Stewartson theory whereas the outer structure re-

ﬂects the more complex character of interacting thermal

and velocity ﬁelds. The bimodal temperature distribu-

tion is now explained by the alternating thermal ﬁeld.

We plot the radial dependence of the mean values of the

bimodal distributions (the bimodal PDFs are well ﬁt by

the sum of two Gaussians) from the DNS for Ra = 109,

1/Ro = 10 in Fig. 5b. The unimodal distribution for

small r/R bifurcates sharply to a bimodal distribution for

r/R ≈0.72. The corresponding experimental measure-

ments do not provide data at intermediate r/R, but are

consistent (dashed curve) with a scaled BZF width based

on the scaling Ra1/4Ek 2/3. Finally, the transition value

of 1/Ro ≈2 from unimodal to bimodal distributions is

roughly independent of Ra as indicated in Fig. 5c.

Our observations provide insight into experimental re-

sults for Γ = 1/2 in water with Pr = 4.38 [20], where the

mode-1 LSC for non-rotating convection was reported

to transition into a then unknown state. Our BZF is

that unknown global mode. We conclude that the BZF

exists over a broad range of parameters 1/2≤Γ≤2,

0.1≤Pr ≤4.4, and 108≤Ra <1015 (details to be pub-

lished elsewhere). Here we presented details for Pr = 0.8

and Γ = 1/2 and for Ra spanning seven orders of magni-

tude [27]. A fully quantitative understanding remains a

challenge for the future.

The authors acknowledge support from the Deutsche

Forschungsgemeinschaft (DFG) through the Collabora-

tive Research Centre SFB 963 ”Astrophysical Flow In-

stabilities and Turbulence” and research grants Sh405/4-

1, Sh405/4-2, Sh405/8-1, Ho5890/1-1 and We5011/3-1,

from the LDRD program at Los Alamos National Labo-

ratory and by the Leibniz Supercomputing Centre (LRZ).

∗These two authors contributed equally.

†Olga.Shishkina@ds.mpg.de

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