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For rapidly rotating turbulent Rayleigh--B\'enard convection in a slender cylindrical cell, experiments 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 $Ra^{1/4}Ek^{2/3}$ where the Ekman number $Ek$ decreases with increasing rotation rate.
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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 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 1/2Γ2, 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 1/2Γ2
and 0.1Pr 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 =αgH3/(κν), Prandtl
number Pr =ν/κ, cell aspect ratio Γ and Rossby number
Ro =αgH / (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(TT0)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
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
1/2Γ2 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 zH/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=δθH/(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 108Ra 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 u=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=Rr0and δumax
φ=Rrumax
φ
where δumax
φδurms
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 [u/2, u/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 1/2Γ2,
0.1Pr 4.4, and 108Ra <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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