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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

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.

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 [1–10]. Much recent experimental [8,11–14] and theoretical/numerical interest [15–17]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 [15–17] 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

[19–22] 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

International license. Further distribution of this work must maintain attribution to the author(s) and the

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 =10−6,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 Rac≈9×108. 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 Raw≈31.8Ek−1+46.5Ek−2/3[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)Ek−4/3[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 10−5the 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 10−6and Pr <1 is identiﬁed

empirically as Rat∼Ek−2[10,13,20]. In the region Rac<Ra <Rat, a BZF has been identiﬁed in

both experimental and numerical studies [13,19–21]. 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 …

10−810−610−4

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/Raw−1 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 a≈1.54 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=10−6. 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 Rac≈7.8×108for Ek =10−6[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 =10−6and =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/Raw−1 as the abscissa and plot the quantities Nu −1,

ωd−ωdc(inset shows expected linear dependence of ωd), and (δ0/H)Ek−2/3(the factor Ek−2/3

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)Pr−1[25,32]. The trends show the nonlinear evolution of the wall mode states with

increasing Ra.

FIG. 4. (a) Nusselt number vs ,(b)(ωd−ωdc)Ek−5/3(inset linear dependence on near onset), and

(c) (δ0/H)Ek−2/3vs [31].

L011501-5

ECKE, ZHANG, AND SHISHKINA

FIG. 5. Scaled (a) boundary mode drift frequency (ωd−ωdc)Ek−5/3Pr4/3≈0.022(Ra −Raw)and(b)side-

wall boundary length scale (δ0/H)(Ra −Raw)−1/6≈1.0Ek2/3.

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 (ωd−ωdc)Ek−5/3Pr4/3≈

0.022(Ra −Raw) (consistent with [20]) and (δ0/H)(Ra −Raw)−1/6≈4.7Ek2/3(the data for Ek =

10−6are 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 ωd−ωdc, respectively. The collapse over almost 10 decades in

Ra −Raw[Fig. 5(a)] showing ωd−ωdc∼Ra −Rawand over 4 decades in Ek showing δ0∼Ek2/3,

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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ECKE, ZHANG, AND SHISHKINA

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