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Abstract

Using direct numerical simulations, we study rotating Rayleigh-Bénard convection in a cylindrical cell with aspect ratio Γ=1/2, for Prandtl number 0.8, Ekman number 10−6, and Rayleigh numbers from the onset of wall modes to the geostrophic regime, an extremely important one in geophysical and astrophysical contexts. We connect linear wall-mode states that occur prior to the onset of bulk convection with the boundary zonal flow that coexists with turbulent bulk convection in the geostrophic regime through the continuity of length and timescales and of convective heat transport. We quantitatively collapse drift frequency, boundary length, and heat transport data from numerous sources over many orders of magnitude in Rayleigh and Ekman numbers. Elucidating the heat transport contributions of wall modes and of the boundary zonal flow are critical for characterizing the properties of the geostrophic regime of rotating convection in finite, physical containers and is crucial for connecting the geostrophic regime of laboratory convection with geophysical and astrophysical systems.
PHYSICAL REVIEW FLUIDS 7, L011501 (2022)
Letter
Connecting wall modes and boundary zonal flows in rotating
Rayleigh-Bénard convection
Robert E. Ecke ,1,2,3,*Xuan Zhang ,1,and Olga Shishkina 1,
1Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
2Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
3Department of Physics, University of Washington, Seattle, Washington 98195, USA
(Received 17 June 2021; accepted 7 December 2021; published 10 January 2022)
Using direct numerical simulations, we study rotating Rayleigh-Bénard convection in
a cylindrical cell with aspect ratio =1/2, for Prandtl number 0.8, Ekman number
106, and Rayleigh numbers from the onset of wall modes to the geostrophic regime,
an extremely important one in geophysical and astrophysical contexts. We connect linear
wall-mode states that occur prior to the onset of bulk convection with the boundary zonal
flow that coexists with turbulent bulk convection in the geostrophic regime through the
continuity of length and timescales and of convective heat transport. We quantitatively
collapse drift frequency, boundary length, and heat transport data from numerous sources
over many orders of magnitude in Rayleigh and Ekman numbers. Elucidating the heat
transport contributions of wall modes and of the boundary zonal flow are critical for
characterizing the properties of the geostrophic regime of rotating convection in finite,
physical containers and is crucial for connecting the geostrophic regime of laboratory
convection with geophysical and astrophysical systems.
DOI: 10.1103/PhysRevFluids.7.L011501
Rayleigh–Bénard convection with rotation (RRBC) about a vertical axis is a prototypical lab-
oratory realization of geophysical and astrophysical systems that combines buoyancy forcing and
rotation [110]. Much recent experimental [8,1114] and theoretical/numerical interest [1517]on
rotating convection has focused on the geostrophic regime where rotation dominates. In particular,
one is interested in the scaling of local and global characteristics of the convecting state. An
accessible and important global parameter is the normalized heat transport Nu for which there
are theoretical predictions of asymptotic models [1517] that provide insight into broader geo- and
astrophysical situations. There are significant experimental challenges [18] for making a compelling
comparison including reaching small Ek number with correspondingly large Ra. Consequently, the
geometry of experimental convection cells have tended toward small aspect ratio =D/H<1,
where Dand Hare the cell diameter and height, respectively. Recently several investigations
[1922] have revealed a boundary zonal flow (BZF) where an azimuthally-periodic, wall-localized
flow coexists, on average, with a turbulent bulk mode (here we define the BZF as the wall-localized
*ecke@lanl.gov
xuan.zhang@ds.mpg.de
Olga.Shishkina@ds.mpg.de
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0
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published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.
2469-990X/2022/7(1)/L011501(8) L011501-1 Published by the American Physical Society
ECKE, ZHANG, AND SHISHKINA
state in the presence of any bulk mode when rotation dominates or influences the convective state)
that contributes strongly to total heat transport. The BZF has features reminiscent of wall mode
states in RRBC that arise from a linear instability at smaller critical Rawthan Racof the bulk
mode and are characterized by an integer mode number (in periodic geometry), an anticyclonic
precession frequency, and a homogeneous time-independent (in the precessing frame) state [4,7,23
26]. A recent numerical study [22] provided evidence that the BZF was the nonlinear remnant
of wall modes. Here we establish unambiguously through direct numerical simulation (DNS) and
comparison among disparate data sets for the drift frequency that characterizes the average angular
precession frequency ωdof the modal pattern in the rotating frame, the radial length scale δ0defined
by the closest to the sidewall first zero crossing of the azimuthal velocity at the midplane, and the
heat transport Nu, that there is a continuous evolution of the wall mode states into the BZF which
coexists with the bulk convection modes. We also find that the wall mode contribution to the heat
transport plays an important role in determining the scaling of Nu in the geostrophic regime, a
crucial element in a proper comparison among experiment, DNS, and theory.
The dimensionless control parameters in RRBC are the Rayleigh number Ra =αgH3/(κν),
Prandtl number Pr =ν/κ, Ekman number Ek =ν/(2H2), and cell aspect ratio where αis iso-
baric thermal expansion coefficient, νkinematic viscosity, κfluid thermal diffusivity, gacceleration
of gravity, angular rotation rate, and the temperature difference between horizontal confining
plates. The global response of the system is the normalized heat transport Nu, and the time and
length scales of the wall modes and of the BZF are the normalized precession frequency ωd=ω/
and radial localization length scale δ0/H. We present data for Ek =106,Pr=0.8, =1/2, and
2×107Ra 5×109that spans the wall mode onset at Raw=2.8×107through the onset of
bulk convection at Rac9×108. We use our results on this system over wider ranges of Ek and Ra
[19,20] with data from other experiments and DNS [4,5,7,13,21,22] to test our proposed power-law
scalings.
The regimes of rotating convection in finite containers are wall-mode states at the lowest Ra
Raw, a transition at Racto a bulk state of rotating convection [1] in which rotation dominates over
buoyancy in geostrophic balance [27], a state with Ra >Ragwhere rotation and buoyancy are of
equal importance, and the buoyancy dominated state where rotation becomes unimportant at the
highest Ra >Rat. The first instability from the no-convection base state is to wall modes with
critical Rayleigh number Raw31.8Ek1+46.5Ek2/3[28].
To emphasize the role of these wall modes, we plot the boundaries of rotating convection regimes
in Fig. 1in a parameter space of Ra/Rawand Ek. The transition to bulk rotating convection would
occur in an infinite system via linear instability at Rac(8.7–9.6Ek1/6)Ek4/3[1,29,30]. In the
presence of sidewalls, however, the transition to a bulk convection state depends on and on the
nonlinear state of the wall modes because of the non-zero base state [4] with Nu >1.
Whereas at modest Ek 105the onset of bulk convection Rac/Raw10, for smaller Ek there
is an expanding and more nonlinear range of wall modes. For large enough Ra at fixed Ek, buoyancy
dominates over rotation, and the transition to this regime for Ek 106and Pr <1 is identified
empirically as RatEk2[10,13,20]. In the region Rac<Ra <Rat, a BZF has been identified in
both experimental and numerical studies [13,1921]. Near Rac, Nu rises rapidly [2,4,8,14,17] over
a range Rac<Rag=cRac,3c6 before a transition region where Nu increases less rapidly
with Ra, finally approaching the buoyancy dominated approximate scaling Nu Ra1/3beginning
at Rat. Given the paucity of data and the unknown influences of wall-mode contributions to the
heat transport in most cases, the dependence of Ragon Ek is fairly uncertain and the line we
draw in Fig. 1is a suggestive one based on Refs. [4,14,17]. To understand experiments and DNS
in realistic, confined convection cells, it is crucial to characterize the role of wall modes on the
nonlinear evolution from no convection into the geostrophic regime and to connect the wall modes
with the BZF that exists in the turbulent geostrophic regime [19,21,22]. That is our task here.
A qualitative understanding of the evolution of the state of rotating convection can be gained by
considering instantaneous midplane horizontal cross sections of temperature fields and associated
streamlines and corresponding vertical temperature fields at the sidewall boundary (r=0.98R).
L011501-2
CONNECTING WALL MODES AND BOUNDARY ZONAL
108106104
100
102
104
106
Buoyancy
dominated
Ra =Ra
t
Wall
modes
Ra =R
a
c
Ra = Raw
Ra =Ra
g
Geostrophic
Ek
Ra/Raw
Pr=0.8, Γ = 1/2 [19–20] Pr=0.8, Γ = 1/2
Pr=5.2, Γ = 1/5[21] Pr=1.0, Γ = 3/2 [22]
Pr=6.4, Γ = 2,5,10 [4, 5, 7] Pr=0.8, Γ = 1/2 [13]
FIG. 1. Phase diagram of states of rotating Rayleigh–Bénard convection: Ra/Rawvs Ek. Boundaries are
for Raw,Ra
c,Ra
g,andRa
tdefined in the text.
Figure 2shows these fields for several Ra [32]; also labeled is the reduced Ra defined as =
Ra/Raw1 taking the experimental value Raw=2.8×107. Very close to onset (0.07), the
flow is organized as a mode-1 state with symmetric upwelling warmer (red) and downwelling cooler
regions (blue), an overall anticyclonic rotation at the midplane, and a sinusoidal mean-temperature
isotherm in the vertical field as shown in Fig. 2(a). The confined geometry of =1/2 means that
the wall-mode temperature amplitude is not localised near the sidewall as in larger [5,7] but has
an almost linear variation across the diameter as in Fig. 2(a). Thus, when the bulk mode appears at
higher Ra, it grows from a nonzero base state.
With increasing Ra, the wall-mode state becomes more nonlinear but time-independent (in a
frame corotating with the retrograde traveling wall mode) for Ra 4×108. The state presented
in Fig. 2(b) for Ra =5×108shows the more complex horizontal temperature field and flow
circulation and the strongly nonlinear square-wave-like vertical profile with forward/backward
(left/right) asymmetry; it is also weakly time dependent indicating a wall-mode transition to an
oscillatory state. For larger Ra, Fig. 2(c), the streamlines are irregular, indicating unsteady flow and
thermal inhomogeneity appears in the interior. One sees vertical striations arising from the influence
of aperiodic time-dependent bulk modes interacting with the wall mode.
The wall mode state is characterized by four main properties that we consider here: the heat
transport Nu, the precession frequency ω, the azimuthal mode number, and the radial distribution of
heat transport or azimuthal velocity uφ. The azimuthal mode number is 1 because of small =1/2.
Previously we demonstrated that for the BZF m=1for3/4 and m=2for =1or2[20].
Our data show continuity from wall mode to BZF for this .
We first consider the heat transport and its contributions from the wall mode, from the bulk
state, and from the BZF. In Fig. 3(a), we show Nu versus Ra that covers the wall mode regime
3×107<Ra <5×108, a transition region 5 ×108<Ra <9×109, and the onset of strong
bulk modes coexisting with remnant sidewall-localized modes, i.e., a BZF. The inset shows linear
growth of the wall mode heat transport near onset consistent with the expected scaling Nu 1=a
with a1.54 and Raw=2.8×107(compared to the theoretical value 3.2×107for an insulating
sidewall and a planar (as opposed to a curved wall in cylindrical geometry) wall [25,32]). As the wall
modes become more nonlinear, Nu increases less rapidly and approaches an inflection point around
L011501-3
ECKE, ZHANG, AND SHISHKINA
FIG. 2. Instantaneous temperature fields (left—horizontal at z=H/2 with streamlines; right—vertical at
r=0.98R)forEk=106. Corresponding Ra and :(a)3×107, 0.071, (b) 5 ×108, 17, (c) 1 ×109, 35.
Directions of rotation and wall mode precession are shown.
Ra 5×108where a weak signature of time-dependent convection can be detected (vertical bars
denote root-mean-square fluctuations). At slightly higher Ra 7×108, bulk modes begin to grow
as demonstrated in instantaneous horizontal and vertical slices. At Ra 109, Nu increases rapidly as
bulk convection and wall localized convection act together. In Figs. 3(b) and 3(c), total Nu increases
roughly linearly from an effective Nuoff 9 and Rac=8.9×108(compared to linear-stability
prediction Rac7.8×108for Ek =106[29]). Given the nonlinear base state created by the wall
modes, the correspondence for the onset of bulk convection is good.
To further explore the relative contributions of the wall localized states and the bulk state, we
divide up the heat transport according to a radial separation r0(normalized by cell radius R). We
denote the portion from 0 to r0as contributing to the bulk state whereas the remaining portion from
r0to 1 is attributed to wall states. In the language of Ref. [19], the wall portion is the nonlinear
L011501-4
CONNECTING WALL MODES AND BOUNDARY ZONAL
FIG. 3. Nu vs Ra for Pr =0.8, Ek =106and =1/2. (a) Time-dependent wall modes Ra <5×108
and onset of bulk convection for Ra 9×108. Vertical bars are standard deviations of Nu fluctuations.
Short-dashed line indicates linear fit to the data near onset (inset Nu 1vsnear onset). Nuoff is the amount
contributed by wall modes at bulk convection onset. [(b), (c)] Larger range of Ra with total Nu (solid, black)
vs Ra/Racand contributions averaged over regions defined by r/Rr0(bulk modes) and r/R>r0(wall
modes/BZF). Black squares are the data from Ref. [20]forfixedRa=109. Solid lines are linear fits for
Ra/Rac5.
BZF whereas the bulk convection is rotation dominated and in the geostrophic regime (similar
decompositions [20,21] have demonstrated relative wall and bulk contributions to Nu in the BZF
region) . Although the quantitative split between different regions depends on the choice of r0,
the trends are unambiguous. Both the BZF and bulk portions grow roughly linearly for Ra/Rac
5. Nevertheless, the BZF remains larger throughout the range studied here. The bulk/geostrophic
contribution to Nu over the range Ra/Rac5 is comparable to the asymptotic equation prediction
for Pr =1[16] whereas the much larger total Nu is similar in magnitude to measurements in the
same Ek range for =0.4 and Pr =4.4[14]; the contribution of wall-mode and BZF modes to the
heat transport are important for comparisons with theoretical predictions.
We present in Figs. 4(a),4(b) and 4(c), respectively, the Ra dependence of Nu, ωd, and δ0in
the wall mode region. We use =Ra/Raw1 as the abscissa and plot the quantities Nu 1,
ωdωdc(inset shows expected linear dependence of ωd), and (δ0/H)Ek2/3(the factor Ek2/3
nicely collapses the data for Pr =0.8 and Pr =6.4[4,5,7]), respectively. Here we use the asymptotic
linear stability result for a planar wall [28] for the precession frequency at onset ωdc(132 Ek
1465 Ek4/3)Pr1[25,32]. The trends show the nonlinear evolution of the wall mode states with
increasing Ra.
FIG. 4. (a) Nusselt number vs ,(b)(ωdωdc)Ek5/3(inset linear dependence on near onset), and
(c) (δ0/H)Ek2/3vs [31].
L011501-5
ECKE, ZHANG, AND SHISHKINA
FIG. 5. Scaled (a) boundary mode drift frequency (ωdωdc)Ek5/3Pr4/30.022(Ra Raw)and(b)side-
wall boundary length scale (δ0/H)(Ra Raw)1/61.0Ek2/3.
We now show trends in ωdand δ0extending over many decades in Ra and Ek that spans the
wall mode region and the geostrophic region coexisting with the BZF using data from our previous
simulations on this system [19,20] as well as from others for which these quantities have been
measured [4,13,21,22]. In Figs. 5(a) and 5(b) we show scaled quantities (ωdωdc)Ek5/3Pr4/3
0.022(Ra Raw) (consistent with [20]) and (δ0/H)(Ra Raw)1/64.7Ek2/3(the data for Ek =
106are consistent with the scalings determined by us elsewhere [19,20] and with other data
[4,5,7,13,21,22]). The Ra and ωddependences are corrected for their finite values at the onset of
wall modes using Ra Rawand ωdωdc, respectively. The collapse over almost 10 decades in
Ra Raw[Fig. 5(a)] showing ωdωdcRa Rawand over 4 decades in Ek showing δ0Ek2/3,
see Fig. 5(b), establishes the connection between the wall modes and the BZF.
In conclusion, we have demonstrated the continuity of measures of time, length, and heat
transport from the wall mode state through the appearance of bulk modes coexisting with the
BZF, unambigulously connecting the two states. We show that by accounting for the wall mode
contribution Nuoff and by considering the radial distribution of Nu (see also Ref. [21]), the bulk
Nu in the geostrophic regime can be extracted with a scaling consistent with a linear dependence
L011501-6
CONNECTING WALL MODES AND BOUNDARY ZONAL
for Ra/Rac5. Our analysis of Nu data accounting for the wall-mode/BZF contributions seems
essential in considering the Nu scaling of rotating convection in the rapidly rotating regime for finite
geometries.
The authors acknowledge support from the Deutsche Forschungsgemeinschaft (DFG), SPP 1881
“Turbulent Superstructures” and Grants No. Sh405/7 and No. Sh405/8, from the LDRD program
at Los Alamos National Laboratory and by the Leibniz Supercomputing Centre (LRZ).
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L011501-8
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