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J. Fluid Mech. (2021), vol.913, A13, doi:10.1017/jfm.2020.1186

Generation of zonal ﬂows in convective systems

by travelling thermal waves

Philipp Reiter1,†, Xuan Zhang1, Rodion Stepanov2,3and Olga Shishkina1,†

1Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany

2Institute of Continuous Media Mechanics, Russian Academy of Science, Perm 614013, Russia

3Perm National Research Polytechnic University, Perm 614990, Russia

(Received 9 September 2020; revised 30 November 2020; accepted 29 December 2020)

This work addresses the effect of travelling thermal waves applied at the ﬂuid layer surface,

on the formation of global ﬂow structures in two-dimensional (2-D) and 3-D convective

systems. For a broad range of Rayleigh numbers (103≤Ra ≤107) and thermal wave

frequencies (10−4≤Ω≤100), we investigate ﬂows with and without imposed mean

temperature gradients. Our results conﬁrm that the travelling thermal waves can cause

zonal ﬂows, i.e. strong mean horizontal ﬂows. We show that the zonal ﬂows in diffusion

dominated regimes are driven purely by the Reynolds stresses and end up always travelling

retrograde. In convection dominated regimes, however, mean ﬂow advection, caused by

tilted convection cells, becomes dominant. This generally leads to prograde directed mean

zonal ﬂows. By means of direct numerical simulations we validate theoretical predictions

made for the diffusion dominated regime. Furthermore, we make use of the linear stability

analysis and explain the existence of the tilted convection cell mode. Our extensive 3-D

simulations support the results for 2-D ﬂows and thus provide further evidence for the

relevance of the ﬁndings for geophysical and astrophysical systems.

Key words: Bénard convection, turbulent convection, atmospheric ﬂows

1. Introduction

The problem of the generation of a mean (zonal) ﬂow in a ﬂuid layer due to a moving

heat source is an old one. Halley (1687) was probably the ﬁrst to perceive that the periodic

heating of the Earth’s surface, due to the Earth’s rotation, could be the reason for the

occurrence of zonal winds in the atmosphere. Nearly three centuries later, experiments

†Email addresses for correspondence: philipp.reiter@ds.mpg.de,olga.shishkina@ds.mpg.de

© The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article,

distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/

licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium,

provided the original work is properly cited. 913 A13-1

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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina

by Fultz et al. (1959), in which a Bunsen ﬂame was rotated around a cylinder ﬁlled with

water, veriﬁed Halley’s hypothesis. The moving ﬂame caused zonal ﬂows and the ﬂuid

started to move opposite to the direction of the ﬂame. Since then, several experimental

and theoretical studies have appeared, which illuminated this phenomenon.

Thus, Stern (1959) repeated Fultz’s experiments using a cylindrical annulus. His

observations conﬁrmed the previous result that the ﬂuid acquires a net vertical angular

momentum through the rotation of a ﬂame, this time despite the suppression of radial

currents in such a domain. Stern then provided a simple two-dimensional (2-D) model,

showing that the mean motion is maintained through the presence of the Reynolds

stresses. Davey (1967 ) extended Stern’s model and provided a theoretical explanation

that, in an enclosed domain, diffusion dominated ﬂows always acquire a net vertical

angular momentum in a direction opposite to the rotation of the heat source. His model

provided asymptotic scalings for the dependency of the time-and space-averaged mean

horizontal velocity, UxV, with the characteristic frequency of the moving heat source

Ω:UxV∼Ω1for Ω→0andUxV∼Ω−4for Ω→∞. The topic gained further

attention when Schubert & Whitehead (1969) suggested that the 4day retrograde rotation

of the Venus atmosphere might be driven by such a periodic thermal forcing. By using

a low Prandtl number (Pr) ﬂuid, they observed that the induced mean ﬂow rotated

rapidly and exceeded the rotation speed of the heat source, which was rotated below a

cylindrical annulus ﬁlled with mercury (Pr 1), by up to 4 times. This validated the

linear analysis by Davey, who predicted the speed of the ﬂuid to increase as Pr becomes

small. However, at this time, it became clear that the induced rapid mean ﬂows may

exceed the range of validity of Davey’s linear theory. Consequently, Whitehead (1972),

Young, Schubert & Torrance (1972 ) and Hinch & Schubert (1971) studied the inﬂuence

of weakly nonlinear contributions. They concluded that the small higher-order corrections

rather tend to suppress the induced retrograde zonal ﬂows and that the occurring secondary

rolls transport momentum in the direction of the moving heat source. It therefore seemed

unlikely that the mean ﬂows become much faster than the heat source phase speed, even

for small Pr, as soon as convective processes come into play.

The preceding analysis certainly lacked the complexity of convective ﬂows, and

therefore Malkus (1970), Davey (1967) and other authors anticipated that convective and

shear instabilities could alter the entire character of the solution. In particular, Thompson

(197 0) showed that the interaction of a mean shear with convection can lead to a tilt of the

convection rolls and thus to the transport of the momentum along the shear gradient,and

thereby ampliﬁes the mean shear ﬂow. In this scenario, the convective ﬂow is unstable to

the mean zonal ﬂow even in the absence of a modulated travelling temperature variation,

which suggests that the mean zonal ﬂows might be the rule and not the exception to

periodic ﬂows that are thermally or mechanically driven. However, the direction of this

mean zonal ﬂow would be solely determined by a spontaneous break of symmetry; it

could either move counter (retrograde) to the imposed travelling wave (TW) or in the

same directions as the TW (prograde).

The existence of mean ﬂow instabilities in internally heated convection and in rotating

Rayleigh–Bénard convection (RBC) (Ahlers, Grossmann & Lohse 2009) was studied

theoretically by Busse (1972,1983) and Howard & Krishnamurti (1986), but has not

been observed in laboratory experiments. In classical RBC, a zonal ﬂow, if imposed as

an initial ﬂow, can survive (Goluskin et al. 2014), but only if the ratio of the horizontal to

vertical extension of the domain is smaller than a certain value, see Wang et al. (2020b)

and Wang et al. (2020a). Also, several studies examined the effects of time-dependent

sinusoidal perturbations in RBC. Venezian (1969) showed that the onset of convection

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Generation of zonal ﬂows in convective systems

can be advanced or delayed by modulation, while Yang et al. (2020) and Niemela &

Sreenivasan (2008) demonstrated a strong increase of the global transport properties in

some cases.

Its general nature makes the travelling thermal wave problem appealing to study,

however, to our knowledge, there are only a few studies recently published that are related

to the original ‘moving ﬂame’ problem. Therefore, in the present study, we revisit the

existing theoretical models, speciﬁcally Davey’s model, and validate it by means of state

of the art direct numerical simulations (DNS). Furthermore, we study a set-up with a

non-vanishing vertical mean temperature gradient (as in RBC), to study the inﬂuence

of the travelling thermal wave on convection dominated ﬂows and discuss the absolute

strength and the direction of the induced zonal ﬂows. Despite the substantial advances over

the years, it remains unanswered whether the thermal TW problem is merely of academic

interest or, indeed, of practical relevance in the generation of geo- and astrophysical zonal

ﬂows (Yano, Talagrand & Drossart 2003; Maximenko, Bang & Sasaki 2005;Nadiga

2006). For this purpose, in § 4, we complement our analysis with thorough 3-D DNS.

For the sake of generality, we choose a classical RBC set-up. Ultimately, we analyse the

absolute angular momentum in 3-D ﬂows (respectively, horizontal velocity in 2-Dﬂows)

and provide insight into the mean ﬂow structures.

2. Methods

2.1. Direct numerical simulations

The governing equations in the Oberbeck–Boussinesq approximation for the dimensionless

velocity u,temperatureθand pressure pread as follows:

∂u/∂t+u·∇u+∇p=Pr/Ra∇2u+θez,(2.1)

∂θ/∂t+u·∇θ=1/√PrRa∇2θ, ∇·u=0.(2.2)

Here,tdenotes time and ezthe unit vector in the vertical direction. The equations have

been non-dimensionalised using the free-fall velocity uff ≡(αgˆ

H)1/2, the free-fall time

tff ≡ˆ

H/uff ,Δthe amplitude of the thermal TW and ˆ

Hthe cell height. The dimensionless

parameters Ra,Pr and the aspect ratio Γare deﬁned by

Ra ≡αgˆ

H3/(κν), Pr ≡ν/κ, Γ ≡ˆ

L/ˆ

H,(2.3a–c)

where ˆ

Lis the length of the domain, νis the kinematic viscosity, αthe isobaric thermal

expansion coefﬁcient, κthe thermal diffusivity and gthe acceleration due to gravity. This

set of equations is solved using the ﬁnite-volume code goldﬁsh (Shishkina et al. 2015;

Kooij et al. 2018), which employs a fourth-order discretisation scheme in space and a

third-order Runge–Kutta, or, alternatively, an Euler-leapfrog scheme in time. The code

runs on rectangular and cylindrical domains and has been advanced for a 2-D pencil

decomposition for highly parallel usage. The spatial grid resolution of the simulations

was chosen according to the minimum resolution requirements of Shishkina et al. (2010).

A stationary state is ensured by monitoring the volume-averaged, the wall-averaged and

the kinetic dissipation based Nusselt numbers.

In this study, the following notations are used: temporal averages are indicated by an

overline or by a capital letter, thus the Reynolds decomposition of the velocity reads

u=U+u, decomposing uinto its mean part Uand ﬂuctuating part u. Unless speciﬁcally

stated, time averages are carried out over a long period of time, however, in § 3.1.1,

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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina

the averaging period was deliberately restricted to only a few wave periods to achieve

a time scale separation. Further, the spatial averages are denoted by angular brackets

·, followed by the respective direction of the average, e.g. ·xdenotes an average in

x;·Vdenotes a volume average. And ultimately, the velocity vector deﬁnitions u≡

(ux,uy,uz)≡(u,v,w)are used interchangeably.

2.2. Theoretical model

Already the earliest models proposed by Stern (1959) and Davey (19 67 )gavea

considerable good understanding of the moving heat source problem. Although there are

more complex models (Stern 1971) based on adding higher-order nonlinear contributions

(Hinch & Schubert 1971; Busse 1972; Whitehead 19 72 ; Young et al. 1972), this section

focuses on revisiting the main arguments of Davey’s original work, which is expected to

give reasonably good results in the limit of small Ra. Besides, a more complete derivation

and concrete analytical solutions are provided in appendix A.

Given the linearised Navier–Stokes equations in two dimensions and averaging the

horizontal momentum equation in the periodic x-direction and over time t, one can derive

the following balance:

Pr/Ra∂2

zUx=∂zuwx+W∂zUx.(2.4)

Evidently, a mean zonal ﬂow Uxis maintained by the momentum transport due to the

Reynolds stress component uwand by mean advection through W∂zU. The theory further

advances by assuming that no vertical mean ﬂow exists (W=0), which reduces (2.4)to

the balance between viscous mean diffusion and Reynolds stress diffusion. Furthermore,

by neglecting convection and variations in x,thelinearised equations can be written as a

set of ordinary differential equations, that can be solved sequentially to ﬁnd uand wand

ultimately the Reynolds stress term uw. This procedure is shown in appendix A.Given

the Reynolds stress ﬁeld, (2.4) has to be integrated twice to obtain the mean zonal ﬂow

U(z). Integrating that proﬁle again ﬁnally gives the total mean zonal ﬂow UV, which

is an important measure of the amount of horizontal momentum or, respectively, angular

momentum in cylindrical systems, that is generated due to the moving heat source. The

last step can be solved numerically, however, following Davey (1967), the limiting relations

can be calculated explicitly

UxV=−π

2

k3Ra2Pr−2(Pr +1)

12! Ω+O(Ω 3)for Ω→0,(2.5)

UxV=−k3Ra−1/2Pr−3/2

256π4(Pr +1)Ω−4+O(Ω−9/2)for Ω→∞,(2.6)

where the horizontal TW, θ(x,t)=0.5cos(kx −2πΩt), is applied to the bottom and top

plate. We would like to add that this theoretical model is, as determined by its assumptions,

expected to be limited to diffusion dominated, small-Ra ﬂows. However, when momentum

and thermal advection take over, its validity remains questionable. We will show later that,

after the onset of convection, where eventually mean advection takes over, the neglect of

the W∂zU-contribution is no longer justiﬁed.

3. Two-dimensional convective system

As described by Stern (1959), the generation of a laminar zonal ﬂow by a TW can

be successfully explained in a 2-D system, which makes it a good starting point.

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Generation of zonal ﬂows in convective systems

θ

+1/2

–1/2

θ

+1

–1

Set-up A

0

z

H

0

z

H

0xL

0xL

Ω

θ =0θ =1

Set-up B

Ω

(a)

(b)

Figure 1. Sketch of the 2-D numerical set-up. The colour represents the dimensionless temperature

distribution for the purely conductive cases (Ω=0.1). The thermal wave is imposed at the top and bottom

plates, propagating to the right, in the positive x-direction. (a)Set-up A: no mean temperature gradient is

imposed between the top and the bottom. (b)Set-up B: with (unstably stratiﬁed) mean temperature gradient,

as in RBC. The ﬁgure shows temperature snapshots, while the time-averaged conduction temperature ﬁeld

depends linearly on z.

The temperature boundary conditions (BCs) are time and space dependent,

θ(x,z=0,t)=0.5[cos(x−2πΩt)+θ ],(3.1)

θ(x,z=H,t)=0.5[cos(x−2πΩt)−θ ].(3.2)

Here, Ωindicates the temporal frequency of the TW in free-fall time units. For example,

Ω=10−1describes a wave with a period of 10 free-fall time units τff ,andθ is

introduced as a control parameter for the strength of the mean temperature gradient.

In the following, two different set-ups are considered. In set-up A (ﬁgure 1a) – the one

originally examined by Davey (1967 ) – no mean temperature gradient exists (θ =0) and

the top and bottom plate temperatures are equal, whereas in set-up B (ﬁgure 1b) a mean,

unstable temperature gradient is applied (θ =1). For simplicity, the mean temperature

gradient is set equal to the amplitude of the thermal wave. In this set-up, effects of

convection are expected to become dominant. Averaged over time, this set-up resembles

RBC, therefore, it can be regarded as a spatially and temporally modulated variant of

RBC. Further, no-slip conditions are applied at the top and bottom plates, the x-direction

is periodic and the domain has length L=2πand height H=1. In upcoming studies, one

might introduce a second Rayleigh number based on the mean temperature gradient (as

in RBC), namely Raθ ≡αgθ ˆ

H3/(κν). However, in this work the connection to Ra is

simply Raθ =0forset-up A and Raθ =Ra for set-up B.

The overall focus in this study lies on variations of the zonal ﬂow with Ra and Ω. Thus,

the parameter space spans 103≤Ra ≤107and 10−4≤Ω≤100, while the aspect ratio

and Prandtl number are kept constant (Γ=2π,Pr =1). Exemplary temperature ﬁelds at

aﬁxedΩ=0.1areshowninﬁgure 2.

3.1. Results

The theoretical model, as presented in appendix A, aims to explain the generation of

the total mean momentum UxVforagivenRa and wave frequency Ω.Moreover,it

predicts that the generated mean momentum will be directed opposite, i.e. retrograde, to

the travelling thermal wave. In this section we study the validity of the model and reveal

its limitations.

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(a)

Ra = 103

Ra = 104

Ra = 105

Ra = 106

Ra = 107

0

z

H

0

z

H

0

z

H

0

z

H

0

z

H

0

(b)

0

xLxL

Ra = 103

Ra = 104

Ra = 105

Ra = 106

Ra = 107

Figure 2. Snapshots of the temperature ﬁeld θat a ﬁxed TW speed Ω=0.1 (propagating to the right). (a)

Set-up A and (b)set-up B. The plumes in set-up B travel either retrograde or prograde (see supplementary

movies available at https://doi.org/10.1017/jfm.2020.1186).

Ra

UxV

Ra

10–4 10–3 10–2 10–1 100

10–8

10–6

10–4

10–2

100

10–8

10–6

10–4

10–2

100

∼Ω

1

∼Ω

1

∼Ω

−4

∼Ω

−4

Ω

10–4 10–3 10–2 10–1 100

Ω

(a)(b)

Figure 3. Mean velocity of the zonal ﬂow vs. the wave frequency Ωfor Ra =103(blue), 104(orange), 105

(green), 106(red) and 107(black). Circles (stars) denote a retrograde (prograde) mean zonal ﬂow, the solid

lines of the corresponding colour show the results of the theoretical model by Davey (19 67 ). (a)Set-up A and

(b)set-up B.

Figure 3 shows the numerical data from the DNS together with the respective results

of the theoretical model, for different Ra. Worth noting ﬁrst is, that the maximum of the

theoretical model is located at a ﬁxed frequency, if the frequency is expressed in terms

of the diffusive time scale rather than the free-fall time scale Ωκ,max =Ω/√RaPr ≈

0.66. This indicates that the model predictions could be collapsed onto a single curve.

Nonetheless, this was avoided here for the sake of clarity.

We begin our discussion with the results of set-up A, shown in ﬁgure 3(a). The

theoretical model by Davey (1967), indicated by the solid lines, gives accurate results

for Ra =103and a good agreement for Ra =104, although, evidently, the model

systematically overestimates the mean momentum generation for higher Ra. In fact,

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Generation of zonal ﬂows in convective systems

0 0.2 0.4 0.6 0.8 1.0

–4

–2

0

2

4

(×10–4) (×10–3)

z

0 0.2 0.4 0.6 0.8 1.0

z

∂zuw

W∂zU

Theory

–2

–1

0

1

2

(b)(a)

Figure 4. Mean proﬁles of Reynolds stress vs. mean ﬂow advection contribution for Ra =104and Ω=0.1

for (a)θ =0and(b)θ =1, where mean advection dominates. The ﬂow in (a) moves retrograde due

to the Reynolds stress contribution, while the ﬂow in (b) shows a prograde mean ﬂow (√Pr/Ra∂2

zUx=

∂zuwx+W∂zUx).

this is consistent with Whitehead (1972), Young et al. (1972 )and Hinch & Schubert (1971)

who observed that corrections of higher-order nonlinear contributions tend to suppress

the induced retrograde zonal ﬂows. Also, it suggests that an induced mean ﬂow does not

strengthen itself, i.e. there is no positive feedback mechanism between the mean ﬂow

and Reynolds stresses. While all low Ra ﬂows and high Ra ﬂows in the limit of large

Ωare well predicted by the model, the large Ra ﬂows are mostly over predicted (except

Ra =107and Ω=0.1, the only ﬂow of that set-up that becomes truly turbulent, despite

similar initial conditions). Presumably, even more important is that some of the ﬂows

for Ra ≥105exhibit a positive/prograde mean ﬂow, indicated by a star symbol, which is

especially prevalent at small Ω.

Tu r n in g t h e fo c u s t o set-up B, shown in ﬁgure 3(b), the differences become even more

obvious, since adding a mean temperature gradient enhances the effects of convection

further. For Ra =103the picture is clear, as it is below the onset of convection Rac≈

1708 for classical RBC, even for the unbounded domains. The Reynolds stresses remain

dominant, which preserves the development of a mean ﬂow opposite to the TW direction.

However, for Ra ≥103, all but a few of the simulations end up with a prograde mean ﬂow

ﬁnal state. In order to understand the role of the mean ﬂow, we analyse the two terms on

the right-hand side of (2.4), which are presented in ﬁgure 4. The model neglects mean

advection, it only captures contributions of uw. As seen in ﬁgure 4(a), this is justiﬁed

for a ﬂow without strong convection effects and the model predictions agree well with

the Reynolds stresses obtained in the simulations. This is different from the situation in

ﬁgure 4(b), where obviously mean ﬂow advection W∂zUstarts to take over. The shape of

the mean ﬂow advection curve is antiphase to the Reynolds stress curve and contributes

the most. This explains the reversal of the mean ﬂow, from retrograde in ﬁgure 4(a)to

prograde in ﬁgure 4(b).

The underlying reason for that will be examined in more detail in the next section. But

brieﬂy, the main argument is that there exist two competing mechanisms, one induced by

the TW and the other induced by convection rolls, which act on different time scales. At

small Ra, as convection rolls move considerably slower, an average over a few TW time

periods can reliably separate both structures, so that the Reynolds stresses reﬂect mainly

the TW contributions, while the mean ﬁeld represents the convection rolls. Therefore,

the dominant mean ﬂow advection in ﬁgure 4(b) reﬂects the dominance of advection by

convection rolls as Ra increases.

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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina

0

0.2 Ra = 104

Etot

Ehor

2E

Ra = 105

Ehor

10−3 10010−3 10010−3 10010−3 100

0

0.2 Ra = 106

Ehor

Ω

2EEhor

Ω

0

0.4

Ehor Ehor

0

0.4

Etot

Ehor

Ω

Etot

Ehor

Ω

Ra = 104Ra = 105

Ra = 107Ra = 106Ra = 107

Etot

Etot

Etot

Etot

(b)(a)

Etot

Figure 5. Total kinetic energy Etot (black) and horizontal kinetic energy Ehor (blue) for (a)set-up A and

(b)set-up B.

A few more interesting observations can be deduced from ﬁgure 3(b). First, compared

to the theory, the simulations show signiﬁcantly larger values at high Ra. Apparently,

the mean zonal ﬂow can be substantially stronger than expected and its velocity can

exceed the TW phase velocity. Second, while the theory predicts the location of the

maximum zonal ﬂow at a constant diffusive time scale, the DNS indicates a coupling

with the free-fall time rather than with the diffusive time and the maximum is found in

the region 0.01 ≤Ω≤0.1. This is important, since natural ﬂows often fall within this

parameter range. We show this in the context of the Earth’s atmosphere in § 4.Finally

the instantaneous ﬁelds most often show three plumes (ﬁgure 2b), while a classical RBC

simulation with the same initial conditions would develop four plumes. Presumably, either

the sinusoidal temperature distribution at the plates, or a pre-existing shear ﬂow (before

Rayleigh–Taylor instabilities develop) reduces the number of plumes. On this basis, we

tested the linear stability of the Rayleigh–Taylor instabilities with an imposed shear ﬂow,

and found indeed that the wavelength of the most unstable mode decreases.

In ﬁgure 5 we show the total kinetic Etot =(u2

x+u2

z1/2

V)/2 and horizontal (zonal ﬂow)

kinetic energy Ehor =(u2

x1/2

V)/2 in order to elucidate the energetic impact of the present

zonal ﬂows and to evaluate the strength of the vertical and horizontal motions. Set-up A

(a) is clearly dominated by the horizontal kinetic energy throughout the whole parameter

range. For Ω>10−1, the kinetic energy drops close to zero and the temperature is

transported by conduction only above this limit. However, before the kinetic energy drops,

the curves show an energy enhancement. The location of the energy maximum coincides

with the maximum of the zonal ﬂow (ﬁgure 3a), which indicates that the zonal ﬂows can

have a signiﬁcant imprint in the energy of the system. Likewise, set-up B (ﬁgure 5b) is also

dominated by horizontal kinetic energy. However, obviously for larger Ra and larger Ω,

the magnitude of the vertical kinetic energy becomes increasingly important. This further

supports that the neglect of the vertical velocity component Wis eventually no longer

justiﬁed for these parameter regimes.

3.1.1. Origin of prograde ﬂows in convection dominated ﬂows

In order to understand how prograde ﬂows can emerge, we looked at the route from small

to large Ra for a speciﬁc conﬁguration. Set-up B and Ω=0.1 is well suited for this

purpose, since the transition from a retrograde ﬂow to a prograde ﬂow appears early,

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Generation of zonal ﬂows in convective systems

100 200 300 400 500

–2.0

–1.5

–1.0

–0.5

0

(×10–4)

Ra = 1000

Ra = 2000

Ra = 3000

Ra = 4000

Ra = 5000

Convection rolls

Tilted rolls

t/tff

uxV

0

H

z

Retrograde

Prograde

Ra = 1000 Ra = 3000 Ra = 5000

Mean flow

Stability analysis

Most unstable mode

Θ

θ

ux

H

z

H

z

0

H

z

0

0xL

Uxx

Retrograde

ux

x

Prograde

(b)(a)

(c)

(d)

Figure 6. Path from a retrograde ﬂow to a prograde ﬂow. (a) Time evolution of the mean zonal ﬂow; Ra was

increased stepwise. For Ra ≥3000 convection rolls form; for Ra ≥4000 the rolls tilt signiﬁcantly, and the

mean zonal ﬂow becomes positive. (b)The uxproﬁles for Ra =1000,3000,5000. (c) Mean ﬂow extracted at

Ra =3000 (averaged over one TW period). (d) Result of the global stability analysis for the mean ﬂow of (c),

that becomes unstable for Ra ≥4000 to tilted convection rolls.

already below Ra =104(ﬁgure 3b). Thus, a simulation was initiated at Ra =1000 and

then Ra was progressively increased by 1000 each time after a steady state had settled.

The time evolution of the total mean zonal ﬂow is given in ﬁgure 6(a). At the lowest

Ra, the mean ﬂow is retrograde. Increasing Ra to 2000 enhances its strength further, as

anticipated. But already at Ra =3000 the zonal ﬂow breaks down and its vertical proﬁle,

as seen in ﬁgure 6(b), ﬂattens. Ultimately, at Ra ≥4000 this proﬁle ﬂips over and the total

zonal ﬂow turns into a prograde state.

As we have shown in the preceding analysis (ﬁgure 4b), in the presence of convection

cells, the mean zonal ﬂow can be fed by the base ﬂow itself, in particular it is fed by

the vertical advection of horizontal momentum W∂zU. Now, let us consider perfectly

symmetric convection cells; although locally, at a position in x, momentum may be

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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina

transported up- or downward, the symmetry, however, would balance this transport at

another location and the net transport would become zero. Therefore, there must be a

symmetry breaking in the convection cells, which correlates Wwith ∂zU. A possible

mechanism, even discussed in the context of the moving heat source problem, was

described by Thompson (1970) and theoretically analysed by Busse (1972), who showed

that, in a periodic domain, convection rolls can become unstable to a mean shear ﬂow.

This mean shear tilts the convection cells such that their asymmetric circulation maintains

a shear ﬂow. In the following this mean ﬂow instability will be called tilted cell instability.

Busse (1972) showed the existence of this instability on a analytic base ﬂow ﬁeld.

Differently, in the following we conduct a stability analysis on a base ﬂow extracted from

the DNS.

The ﬁrst rise of the curve in ﬁgure 6(a)atRa =3000 coincides with the observed

onset of convection, which is slightly delayed compared to classical, unmodulated RBC

(Rac≈1708). The convection cells at that point appear to be standing still, almost

unaffected from the TW and clearly orders of magnitudes slower than the TW. Therefore a

short time average, over one wave period, was applied to separate both time scales, which

results in the base ﬂow, as shown in ﬁgure 6(c). Based on this base ﬂow, a linear, temporal

stability analysis of the full 2-D linearised Navier–Stokes equations was conducted. Details

therefore are given in appendix C. While no unstable mode was detected for Ra =3000,

for Ra =4000 the mean ﬂow becomes unstable, to the mode presented in ﬁgure 6(d).

The growth rate of it is σ≈0+0.2i, suggesting no oscillatory behaviour (real part is

zero) but exponential temporal growth (imaginary part larger than zero). This mode shares

characteristics with the tilted cell instability described by Thompson (1970), in the sense

that the mode induces a mean shear ﬂow (see proﬁle in ﬁgure 6d). However, rather than

the ‘pure’ shear ﬂows as presented by Thompson (1970) and Busse (19 72 ,1983)witha

vanishing total net momentum when integrated vertically, the ﬂuctuation proﬁle found in

our study (ﬁgure 6don the right) shows a more directed ﬂow, negative in the vicinity of the

plates and stronger positive in the centre. And especially interesting, its momentum proﬁle

has a similar shape as the ﬁnal state solution of typical prograde ﬂows, e.g. the proﬁle on

the right in ﬁgure 6(b). A few more notes are necessary. The difference between the shape

of the mode found in this work, compared to the ones from Thompson and Busse might

be explained by different BCs, as both authors applied free-slip conditions at the plates, in

contrast to our no-slip conditions. In addition, in their seminal works and in the work of

Krishnamurti & Howard (1981), it was already remarked that the mean ﬂow transition is

caused by a spontaneous symmetry breaking and therefore the direction of the shear ﬂow

is somewhat arbitrary as it depends on the initial conditions. Indeed, a change in the grid

size of the stability analysis led to a most unstable mode with a reversed shear ﬂow proﬁle

compared to the mode shown in ﬁgure 6(d). And ﬁnally, even though in ﬁgure 6(a) tilted

rolls are shown to start later as convection rolls, it actually is likely that the convection

cells tilt as soon as convection sets in, it is just not clearly visible from the ﬂow ﬁelds at

that point.

In a nutshell, the mean ﬂow is unstable – even in the absence of a boundary temperature

modulation – to a mode with tilted convection cells and non-zero total mean horizontal

velocity. Both modes, prograde and retrograde, are found in the global stability analysis,

thus it remains unanswered why the DNS at high Ra almost exclusively end up moving

in the same direction as the TW. The disturbance velocity proﬁles resemble those of

the ﬁnal mean ﬂow velocity proﬁles, therefore, the presented mean ﬂow instability is

a plausible mechanism for the generation of moderate strong zonal ﬂows after onset

of convection, then dominating over the Reynolds stress mechanism, that is inherent to

diffusion dominated ﬂows.

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Generation of zonal ﬂows in convective systems

TW

TW TW

TW

TW

t/t

ff

0

700

0xxx

2π2π002π0ϕ2π0ϕ2π

(e)(b)(a)(c)(d)

Figure 7. Evolution of the temperature at mid-height z=H/2, for (a–c) 2-Dﬂowsand(d,e) 3-D RBC (r=

0.99R)ﬂows

;(a)Ω=0.01 , set-up A, (b)Ω=0.1, set-up A, (c)Ω=0.1, set-up B, (d)Ω=0.01, 3-D RBC

and (e)Ω=0.13-D RBC. All panels show Ra =107. The black solid line in each plot indicates the TW speed.

3.1.2. Space–time structures

The ﬂows found in this study revealed surprisingly rich formations. Therefore this part will

be completed with examples of some space–time structures that have been observed in the

2-D system and, already ahead of the next part, in the 3-D cylindrical system. In addition,

movies are provided as supplementary material.

In general, in two dimensions, as can be seen from ﬁgure 2, the temperature ﬁeld is

either symmetric around the horizontal mid-plane (set-up A), or not; in this case there exist

plumes (set-up B). In the latter case, there are usually three up- and three down-welling

plumes identiﬁable. In the 3-D case, the ﬂow consists of rising and falling plumes, which

together form a large scale circulation (LSC). If the TW propagates slowly (small Ω), the

plumes (two dimensions) or respectively the LSC plane (three dimensions) drift with the

same speed as the TW and both structures appear to be connected. However, as Ωincreases

and, hypothetically, the TW time scale τΩbecomes small compared to thermal diffusion τκ

(τκ/τΩ=√PrRaΩ), the plumes (two dimensions) or LSC (three dimensions) ‘break-off’

from the TW, forming two separate structures, acting on different time scales.

Figure 7 shows the space–time structures of the temperature ﬁeld, evaluated at

mid-height, and in the 3-D case at mid-height and near the sidewall. The structures at

mid-height either (i) travel with the same speed (but a phase difference) as the thermal

wave (a,d), or travel with phase speeds different to the thermal wave and in this case

either (ii) retrograde (b,e) or (iii) prograde (c). Regime (i) is expected for small Ra and/or

small Ωparameters, (ii) is found for large Ra and large Ω, if no mean temperature is

present and (iii) exists in strongly convection dominated ﬂows for large Ra and large

Ω, especially if a mean temperature gradient is present. Furthermore, it is striking that

temperatures between the left and right regions in the vicinity of the plumes centre (hottest

or coldest regions in ﬁgure 7) do not necessarily ﬁll with the same temperature (c).

This gives further evidence of a mean ﬂow instability, as it features similarities of the

temperature ﬁeld of the unstable mode given in ﬁgure 6(d), due to which a plume loses

its horizontal symmetry. Considering the speed of the drifting plumes (b,c), we observe

initially exponential growth, as anticipated from an instability, followed by a, possibly,

nonlinear saturation.

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4. Three-dimensional convective systems

The preceding part, as most of the existing literature, is conﬁned to 2-D ﬂows. Now we will

discuss the moving heat source problem in the context of more complicated 3-D convective

ﬂows. In general, TW solutions are common amongst 3-D convective systems. Bensimon

et al. (1990), Kolodner, Bensimon & Surko (1988) and Kolodner & Surko (1988) observed

convection rolls propagating azimuthally in a large aspect ratio annulus near the onset of

convection. Their drift velocity was of the order of magnitude 10−4to 10−3,however,drift

velocity is not necessarily equal to the mean azimuthal ﬂow. Another kind of TW solution

in RBC systems are the spiral patterns found in large aspect ratio cells (Bodenschatz et al.

1991 ; Bodenschatz, Pesch & Ahlers 2000). These spirals are rotating in either direction,

although corotating spirals are more numerous (Cross & Tu 1995), and are known to

be coupled with an azimuthal mean ﬂow (Decker, Pesch & Weber 1994 ). Furthermore,

in rotating systems,TW structures are quite common (Knobloch & Silber 1990). These

structures are strongly geometry dependent (Wang et al. 2012) and known to induce mean

zonal ﬂows that propagate pro- and retrograde (Zhang et al. 2020).

Despite the vast literature on these phenomena, quantitative data on mean ﬂows that are

induced by external travelling thermal waves in 3-D ﬂows seem to be rare. Therefore our

main goal in this part is to gain insight into the strength and structure of such mean ﬂows,

and discuss whether their order of magnitude is relevant in natural ﬂows. For this purpose,

we took the paradigm convective system cylindrical RBC and studied it by means of DNS.

4.1. Numerical set-up: cylindrical RBC (Pr =1)

The set-up is essentially motivated by the original experiments of Fultz et al. (1959), where

a heat source rotated around a cylinder with the radius R(diameter D), except, in our case,

thermal waves travel at the bottom and top and a mean temperature gradient was applied,

as in set-up B of the previous part. In particular, the temperature distribution is linear in

the radial r-direction and consists of one wave period in ϕthat travels counterclockwise

θ(ϕ,r,z=0,t)=0.5r

Rcos(ϕ −2πΩt)+1,(4.1)

θ(ϕ,r,z=H,t)=0.5r

Rcos(ϕ −2πΩt)−1.(4.2)

Again, the mean temperature gradient, averaged over time, is the same as in classical RBC.

The cell is shown in ﬁgure 8(a). Furthermore, top and bottom plates are free slip (∂u/∂n=

0) and no-slip conditions are applied at the sidewall (u=0). All simulations are carried

out for the parameters Pr =1 and the aspect ratio Γ≡D/H=1. The rather large aspect

ratio is a sacriﬁce, in return, more simulations could be conducted and the parameter space

in the region of interest is well resolved, as shown in ﬁgure 8(b).

4.2. Results

Previously, we have shown that travelling thermal waves generate a mean horizontal, or,

synonymously, zonal ﬂow. The same can be observed in the cylindrical system, where a

zonal ﬂow now refers to non-vanishing azimuthal mean ﬂow. In the following, we evaluate

its strength and direction and discuss the results in the context of the 2-D results. As

no speciﬁc adjustments to the theoretical model have been made, from this point on, the

model results are intended to serve mainly as references to the previous results. A brief

remark beforehand: evaluating the time and volume average of uϕproves problematic,

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Generation of zonal ﬂows in convective systems

103104105106107

10–4

10–3

10–2

10–1

100

Ra

Ω

Ω

Ω(b)(a)

Figure 8. (a) Sketch of the cylindrical domain and imposed TW. (b) Studied parameter space. The mesh

sizes nr×nϕ×nzof the DNS are 48 ×130 ×98 for Ra =103,96×260 ×196 fo r Ra =104,10 5,106and

128 ×342 ×256 for Ra =107.

as often ﬂows are not purely pro- or retrograde. Therefore, rather than give precise

scaling laws, the primary purpose of the subsequent analysis is to explore the parameter

space, demonstrate the overall strength of the zonal ﬂows and ﬁnd the most critical wave

frequencies and determine the critical Ra above which the results deviate substantively

from the predictions.

Figure 9(a) shows the total mean azimuthal momentum UϕVand ﬁgure 9(b)shows

the value of Uϕr,ϕ at the mid-height. As before, circles denote a retrograde, stars a

prograde mean ﬂow and the solid lines are the 2-D model solutions from Davey (1967),

without modiﬁcations for no-slip walls. The obtained ﬂows for small Ra ≤105share

distinct features with the 2-D ﬂows. The mean momentum converges to the asymptotic

scalings, and, in fact, the data of ﬁgure 9(b) collapse under a transformation with Ra

remarkably well. For larger Ω, in particular Ω≥10−1, the most ﬂows are found to be

directed prograde, even for Ra =103, which is different from the 2-D case. And as in two

dimensions, the ﬂow structures reveal a transition in this Ω-region. As was discussed in

§3.1.2, the plane of the LSC drifts with the same speed as the TW (=Ω), if the TW

speed is small compared to thermal diffusion speed, and the LSC breaks off from the TW

at larger Ω, forming separate structures, acting on different time scales. It is in the regime

of this break-off above which a prograde ﬂow is present. This process hints towards a

similar mean ﬂow instability, as discussed in § 3.1.1, where the mean ﬂow is now a slow

LSC.

As Ra exceeds 105, turbulent ﬂuctuations increase and the data in ﬁgure 9 become

increasingly scattered. The asymptotic scalings are hardly determinable, even though

UxV∼Ω1for Ω→0appears still valid. The ﬂuctuations can exceed their mean values,

especially for small and large Ω. Despite the strong ﬂuctuations, in regions of maximal

zonal ﬂow, i.e. Ω≈10−2, the mean values are highly signiﬁcant and can induce zonal

ﬂows of the same order of magnitude as the TW frequency, UϕV≈10−2. Furthermore,

similarly to the 2-D case, in three dimensions, the zonal ﬂows at high Ra are most of the

time directed prograde, contrary to small Ra. From the vertical planes of the azimuthally

and time-averaged azimuthal velocity, shown in ﬁgure 10, the dominance of prograde

motion at large Ra becomes more obvious. Moreover, these ﬁgures reveal a complex,

inhomogeneous ﬂow, with strong differential rotation and poloidal mean velocities.

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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina

10–4 10–3 10–2 10–1 100

10–8

10–6

10–4

10–2

100

10–8

10–6

10–4

10–2

100

∼Ω

1

∼Ω

−4

Ra

Ω

10–4 10–3 10–2 10–1 100

Ω

rUϕV

(a)(b)

∼Ω

−4

Ra

rUϕr,ϕ (z = H/2)

∼Ω

1

Figure 9. (a) Time- and volume-averaged zonal ﬂow as a function of the heat source frequency Ω,(b) zonal

ﬂow at mid-height for 3-D RBC data; Ra =103(blue), 104(orange), 105(green), 106(red) and 107(black).

Circles (stars) denote a retrograde (prograde) mean zonal ﬂow, the solid lines of the corresponding colour show

the results of the theoretical model by Davey (19 67).

Ra = 103

H

00

z

rR

0rR

0rR

0rR

0rR

Ra = 104Ra = 105Ra = 106Ra = 107

–2 ×10–4 2×10–4

0–2×10–3 2×10–3

0–1×10–2 1×10–2

0–2×10–2 2×10–2

0–3×10–2 3×10–2

0

(e)(b)(a)(c)(d)

(j)(g)( f)(h)(i)

Figure 10. For a ﬁxed TW frequency Ω=0.01. The azimuthally averaged mean azimuthal velocity Uϕϕ

(a–e) and the corresponding snapshots of the temperature θ(f–j). As Ra increases, the core zonal ﬂow becomes

ﬁrst stronger retrograde (Ra =104,105), then switches its state to a prograde ﬂow originating from the sidewall

(Ra ≥106), while still increasing its strength (see colour bar).

4.2.1. Vertical and radial momentum transport

In the following, we assess the contributing terms of the mean ﬂow azimuthal momentum

equation. For clarity, let us write the equation for uϕexplicitly

∂tuϕ+1

r

∂ruϕur

∂r+1

r

∂uϕuϕ

∂ϕ +∂uϕuz

∂z

=−1

r

∂p

∂ϕ +Pr

Ra 1

r

∂

∂rr∂uϕ

∂r+1

r2

∂2uϕ

∂ϕ2+∂2uϕ

∂z2−uϕ

r2+2

r2

∂ur

∂ϕ .(4.3)

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Generation of zonal ﬂows in convective systems

First, we consider how Uϕchanges in the vertical direction and, second, how it changes

radially. Therefore, decomposing (4.3) into its mean and ﬂuctuating components, and

averaging over ϕand rgives the following balance:

Pr

Ra ∂2Uϕr,ϕ

∂z2−Uϕr,ϕ

r2

=∂u

ϕu

zr,ϕ

∂z+∂UϕUzr,ϕ

∂z+u

ϕu

r

rr,ϕ +UrUϕ

rr,ϕ

.(4.4)

Analysing the radial dependence, on the other hand, averaging over ϕand zgives

Pr

Ra 1

r2

∂2Uϕϕ,z

∂r2−Uϕϕ,z

r2

=1

r

∂ru

ϕu

rϕ,z

∂r+1

r

∂rUϕUrϕ,z

∂r+u

ϕu

r

rϕ,z+UrUϕ

rϕ,z

.(4.5)

The right-hand side terms of these equations are evaluated for Ω=10−2, which are

shown in ﬁgure 11. We ensured that, in the simulations,the data were averaged over an

integer number of TW periods, to prevent artefacts of the TW in the mean ﬁelds (the exact

time values can be found in the supplementary material). When we compare the individual

mean velocities for (a)Ra =103and (b)Ra =104, it becomes clear that the mean ﬁeld

transport in both, vertical and radial, directions is rather negligible. Hence, the nonlinear

Reynolds stress sustains the mean zonal ﬂow, just as in the 2-D case for small Ra (see

ﬁgure 4a), as expected (Stern 1959; Davey 1967). The small mean ﬁeld contributions even

reinforce the zonal ﬂow, since the shape of the mean advection curves matches the shape

of the Reynolds stress curve. Comparing further the vertical and radial transports, we ﬁnd

that the former dominates the latter one by an order of magnitude. This proves that,in

this case, the neglect of the radial currents, as suggested by Stern (1959), is justiﬁed, and

therefore the mean momentum scalings (ﬁgure 9) match remarkably well with their 2-D

analogue (ﬁgure 3), and the difference in the prefactors can presumably be explained by

the different velocity BCs.

The situation for larger Ra (ﬁgure 11c–e) is vastly different. First, the problem becomes

considerably three-dimensional and the radial transport now reaches the same order of

magnitude as the vertical transport (e.g. ﬁgure 11c–e), which suggests that the validity

of the 2-D analogy at large Ra is no longer justiﬁed. Furthermore, the mean ﬁeld

advection contributions, which can be partially seen from ﬁgure 10, increase signiﬁcantly.

As a matter of fact, locally, the mean ﬁeld advection can even exceed the Reynolds

stress contributions. Furthermore, whereas for small Ra, vertical and radial momentum

transports are present throughout the whole domain, at large Ra it becomes strongly

conﬁned to the boundaries. In particular, the vertical transport peaks close to the top

and bottom boundaries and is less pronounced in the centre. The radial transport, on

the other hand, shows an interesting feature in the region 0.95 ≤r/R≤1(ﬁgure 11d,e).

All terms are simultaneously positive, which causes an enhanced zonal transport close

to the sidewall. This may explain why a prograde ﬂow ﬁrst appears close to the sidewall

(ﬁgure 10,Ra =106)and,from there, spreads further inwards (ﬁgure 10,Ra =107).

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Vertical balance

–2

0

2

(×10−4)

(×10−3) (×10−3)

(×10−3) (×10−3)

(×10−2) (×10−2)

(×10−2) (×10−2)

(×10−4)

–5

0

5

–5

0

5

0

Radial balance

–2

0

2

–5

0

5

–5

0

5

–1

0

1

–1

0

1

–1

0

1

–1

0

1

Total

Total

0.1

zH rR

(e)

(b)

(a)

(c)

(d)

∂uϕ

uz

r,ϕ

∂z

∂UϕUzr,ϕ

∂z

UrUϕ

rϕ,z

UrUϕ

rr,ϕ

1

r

∂rUϕUrϕ,z

∂r

1

r

∂ruϕ

urϕ,z

∂r

uϕ

ur

rr,ϕ

uϕ

ur

rϕ,z

Figure 11. Components of the vertical momentum transport, (4.4), (left) and the radial momentum transport,

(4.5), (right). Parameters: Ω=10−2and Ra:(a)10

3,(b)10

4,(c)10

5,(d)10

6and (e)10

7.

4.2.2. Sensitivity to the BCs and aspect ratio

The systems studied in this paper allow many variations of the velocity and temperature

boundary conditions as well as geometrical characteristics of the system. Discussing all

of them goes beyond the scope of a single study. Nevertheless, in order to provide some

preliminary intuition, we examine selected variations and their effects on the generation

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Generation of zonal ﬂows in convective systems

0 0.01

0

z

H

Base

θ(r)

No-slip

×100

Γ = 2

Γ = 0.2

rUϕr,ϕ

Figure 12. Mean angular momentum proﬁle for Ra =105and Ω=10−1. The curves show the effects of

different imposed BCs and aspect ratios: baseline simulation (black, free-slip BCs, θ∼r/Rand Γ=1),

sinusoidal radial temperature BCs (blue, free-slip BCs, θ∼sin(πr/R)and Γ=1), no-slip (red, no-slip BCs,

θ∼r/Rand Γ=1), Γ=0.2 (yellow, free-slip BCs, θ∼r/Rand Γ=0.2) and Γ=2 (green, free-slip BCs,

θ∼r/Rand Γ=2).

of the zonal ﬂows. We do this for a single baseline simulation at Ra =105and Ω=10−1.

The mean angular momentum proﬁles are shown in ﬁgure 12.

First, we consider the effects of the aspect ratio. From classical RBC, it is known that

zonal ﬂow properties depend strongly on Γ(Wang et al. 2020a). In our case, a decrease of

the aspect ratio from Γ=1toΓ=0.2 (slender cell) weakens the zonal ﬂow considerably

by a factor of 100. Furthermore, the zonal ﬂow becomes conﬁned to the top and bottom

plates, while no zonal ﬂow is observed in the centre of the cell. On the other hand,

increasing the aspect ratio to Γ=2 has only minor impact on the zonal ﬂow. We must

note that for the case of Γ=0.2, convection has yet not started and subsequent studies

would be necessary to conclusively elucidate on the aspect ratio dependence.

The effects of the BC variations on the formation of zonal ﬂows can be formulated

as follows. No-slip conditions at the top and bottom plates lead to a slightly weaker,

but qualitatively similar zonal ﬂow. Likewise, replacing the linear radial temperature

distribution at the plates by a sinusoidal distribution (θ∼sin(πr/R)) shows still a

qualitatively similar angular momentum proﬁle, although the strength of the zonal ﬂow

in the centre of the cell increases by a factor of approximately 1.5. This indicates that the

system is rather sensitive to variations of the temperature BCs.

4.3. Example: atmospheric boundary layer

Finally, we would like to illustrate the strength of the induced zonal ﬂows on a concrete

example. Assume an atmospheric boundary layer with a height of ˆ

H=500 m and

a vertical temperature difference of T=3◦C. Given a mean temperature of 10 ◦C,

the material properties of air are approximately κ=2.0×10−5m2s−1,ν=1.4×

10−5m2s−1and α=3.6×10−3K−1.Fromthat,weﬁndPr ≈0.7andRa ≈10 16 and

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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina

the free-fall units uff ≡αgˆ

Hθ ≈7ms

−1,tff ≡ˆ

H/uff ≈70 s. This system is exposed

to a travelling thermal wave through the solar radiation with a period of 24 h, or, in

dimensionless units Ω≈10−3. For simplicity, we say, the day and night difference is also

approximately 3 ◦C, which is likely to be a rather conservative estimate. Our study does

not conclusively show how the zonal ﬂows scale up to Ra =1016 , but the results suggest

a saturation at higher Ra, therefore we proceed using the maximum order of magnitude,

which is Uϕ≈10−2(for the given Ωit might be smaller). With these values, the thermal

variation of the Earth’s surface would induce a prevailing zonal ﬂow of around 0.07 m s−1,

or equivalently 0.3kmh

−1. However, locally, it could exceed this value (see ﬁgure 10)

multiple times, therefore speeds of 1 km h−1are conceivable. Nevertheless, the variance

of this estimate is rather high. Subsequent studies have to examine the inﬂuence of Ra,Pr

and the geometry, in order to make more conﬁdent statements about natural systems.

5. Conclusions

We have explored the original moving heat source problem by means of DNS in 2-D and

3-D systems, for varying Rayleigh numbers Ra and travelling thermal wave frequency

Ω. In the seminal works of Fultz et al. (1959)andStern(1959), it was discovered that a

system subjected to such a TW generates Reynolds stresses, which induce a large scale

mean horizontal, or equivalently zonal, ﬂow directed counter to the propagating thermal

wave. Therefore, in the ﬁrst part, we revisited the theoretical model proposed by Davey

(1967) and found excellent agreement with the theory for low Ra ﬂows, where even the

absolute magnitude of the zonal ﬂows is reproduced remarkably well. As Ra increases,

the theoretical model overestimates the DNS data, which is consistent with the effects of

higher-order nonlinear contributions (Hinch & Schubert 1971; Whitehead 1972; Young

et al. 1972).

However, when an unstable mean temperature gradient is added to the system, the ﬂows

deviate substantially from the initial predictions and often reverse their direction to a

prograde moving zonal ﬂow. Such a behaviour was theorised before to be the result of

a mean ﬂow instability caused by the tilt of convection cells (Thompson 1970 ; Busse

1972,1983). Therefore, we have conducted a global linear stability analysis of a base ﬂow

near onset of convection and conﬁrmed this hypothesis. The most unstable mode can give

rise to a reverse of the horizontal velocity proﬁle. Despite the strong plausibility, that

this mean ﬂow instability is the dominating mechanism at large Ra, the question remains

open as to why prograde ﬂow are more numerous than retrograde ﬂows, while the mean

ﬂow instability suggests a spontaneous break of symmetry and therefore a more balanced

distribution. In this context, it would be interesting to study in the future the interaction

between the TW induced and convection rolls induced ﬁelds.

In the second part we have examined the moving heat source problem in the context

of a 3-D cylindrical RBC system. The asymptotic scalings UϕV∼Ω1for Ω→0and

UϕV∼Ω−4for Ω→∞of the 2-D theoretical model (Davey 19 67 ) still hold in this

system, especially at small Ra. An analysis of the vertical and radial momentum transport

contributions suggests that the radial transport is negligible at small Ra (which justiﬁes a

2-D approximation), but becomes relevant as Ra increases. Furthermore, again, large Ra is

found to predominantly induce a prograde mean zonal ﬂow. This gives more evidence that

the prograde prevalence is likely not fully explained by the mean ﬂow instability picture

and further studies are required to explain its origin.

913 A13-18

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Generation of zonal ﬂows in convective systems

The studied problem is sufﬁciently general and can be extended to more complicated

systems (Whitehead 1975; Shukla et al. 1981; Mamou et al. 1996). A more generalised

theoretical framework already exists, which includes the inﬂuence of a basic stability

and rotation (Stern 1971; Chawla & Purushothaman 1983), however, as this study

showed, the theoretical models most often cannot fully explain the phenomena in

convection dominated systems. Furthermore, the moving heat source problem might help

to understand the ubiquitous structures present in rotating systems. In rotating RBC

systems, the ﬂow structures near the sidewall (Favier & Knobloch 2020; Zhang et al.

2020) are similar to a certain extent to those structures due to the imposed TW.

Ultimately, this study also revealed that the estimates of the order of magnitudes are

still afﬂicted with too large variances to make reliable statements about natural systems.

A naive approach showed that atmospheric currents, caused by solar radiation and the

Earth’s rotation, can actually generate prevailing zonal ﬂows of approximately 1.0kmh

−1.

However, the variance of this estimate is rather high, it therefore is pivotal for subsequent

studies to examine the sensitivities with Ra,Pr and the geometry in greater detail.

Supplementary material and movies. Supplementary material and movies are available at https://doi.org/

10.1017/jfm.2020.1186.

Acknowledgements. The authors would like to thank the Max-Planck HPC Teams in Göttingen and Munich

for their generous technical support and additional computational resources.

Funding. We acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) under the grant

Sh405/10 and Sh405/7 (SPP 1881 ‘Turbulent Superstructures’) and the Leibniz Supercomputing Centre (LRZ).

Declaration of interests. The authors report no conﬂict of interest.

Author ORCIDs.

Philipp Reiter https://orcid.org/0000-0003-3656- 1099;

Olga Shishkina https://orcid.org/0000-0002-6773-6464.

Appendix A. Theory for diﬀusion dominated ﬂows

We follow the theory of Davey (1967), but solve the equations in a more general

way, to allow for ﬂexibility in the chosen BCs; for more details, the reader is referred

to Davey (1967) or Kelly & Vreeman (19 70). Neglecting the mean vertical velocity

component, assuming the mean horizontal velocity to be independent of xand neglecting

the contributions from the mean temperature ﬁeld ¯

θ,thelinearised, non-dimensionalised

Navier–Stokes equations in two dimensions read

∂tu+(U+u)∂xu+w∂z(U+u)=−∂xp+ν∗∂2U

∂z2+∂2u

∂x2+∂2u

∂z2,(A1)

∂tw+(U+u)∂xw+w∂z(w)=−∂zp+ν∗∂2w

∂x2+∂2w

∂z2+θ,(A2)

∂xu+∂zw=0.(A3)

Here, uand ware, respectively, the horizontal and vertical components of the velocity

ﬂuctuations with respect to their time averages, i.e. Uand W=0, and θis the

temperature ﬂuctuation. For non-dimensionalisation we have used the free-fall velocity

uff ≡(αgˆ

H)1/2, the height ˆ

Hand the amplitude of the thermal TW, Δ,sothatν∗=

√Pr/Ra. Let us consider a single wave mode in the horizontal x-direction and in time t,

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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina

e.g.

w(x,z,t)=1

2ˆw(z)exp(+i(kx −2πΩt)) +ˆw∗(z)exp(−i(kx −2πΩt)),(A4)

u(x,z,t)=−∂zwdx=i

2k∂zˆw(z)exp(+i(kx −2πΩt))

−∂zˆw∗(z)exp(−i(kx −2πΩt)),(A5)

θ(x,z,t)=1

2ˆ

θ(z)exp(+i(kx −2πΩt)) +ˆ

θ∗(z)exp(−i(kx −2πΩt)),(A6)

where the asterisk denotes the complex conjugate of a function. We will consider two BCs

(different scenarios), Scenario 1describes aset-up where two travelling thermal waves

are imposed at the top and the bottom (without any phase difference). This case was

considered in the present work. Scenario 2, on the other hand, describes a set-up where the

thermal wave travels only at the bottom, while the dimensionless top temperature equals

zero.

Step 1:calculate ˆ

θ(z).

Neglecting dissipation in x, all convective terms and mean temperature contributions,

the linearised non-dimensional energy equation reads

∂tθ=κ∗∂2θ

∂z2,(A7)

where κ∗=1/√RaPr. This, together with (A6), leads to the following equation for the

wave amplitude equation ˆ

θ(z):

d2ˆ

θ

dz2−λ2ˆ

θ=0;λ2=2πiΩ

κ∗.(A8)

The solution to (A8), for the two scenarios is

Scenario 1

For ˆ

θ|z=−1/2=ˆ

θ|z=1/2=1

2:

ˆ

θ(z)=cosh(λz)

2cosh(λ/2).

Scenario 2

For ˆ

θ|z=−1/2=1

2,ˆ

θ|z=1/2=0:

ˆ

θ(z)=sinh(λ/2−λz)

2 sinh(λ).

Step 2:calculate ˆw(z).

Eliminate the pressure term by cross-differentiation of (A1)and(A2), substitute

(A4)–(A6), neglect convective terms and assume that the thermal wavelength is much

913 A13-20

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Generation of zonal ﬂows in convective systems

larger than the height of the cell (kH 1) to obtain

∂4ˆw

∂z4−α2∂2ˆw

∂z2=k2ˆ

θ, α2=2πiΩ

ν∗.(A9)

For ˆw|z=1/2=ˆw|z=−1/2=∂zˆw|z=1/2=∂zˆw|z=−1/2=0, the solution to (A9)is

ˆw(z)=c1

α2cosh(αz)+c2

α2sinh(αz)+c3z+c4+c5cosh(λz)+c6sinh(λz). (A10)

Scenario 1

A=k2

2ν∗λ2(λ2−α2),

c1=−λαAtanh(λ/2)

sinh(α/2),

c2=0,

c3=0,

c4=Aλ

α

tanh(λ/2)

tanh(α/2)−1,

c5=A

cosh(λ/2),

c6=0.

Scenario 2

A=k2

4ν∗λ2(λ2−α2),

c1=−λαAtanh(λ/2)

sinh(α/2),

c2=−αAλ

tanh(λ/2)−2

(2/α) sinh(α/2)−cosh(α/2),

c3=−c2

αcosh(α/2)+λA

tanh(λ/2),

c4=Aλ

α

tanh(λ/2)

tanh(α/2)−1,

c5=A

cosh(λ/2),

c6=−A

sinh(λ/2).

Step 3:calculate U(z).

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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina

Averaging equation (A1) over time and over one wavelength in x, we obtain the

following equation for the mean ﬂow U(z):

ν∗d2U

dz2=d

dz(uw), (A11)

which can be solved via numerical integration using the no-slip BCs at the plates.

In addition, in the supplementary material we provide a Python code snippet, which

gives the solution for the various quantities ˆ

θ, ˆu,ˆw,uw.Notethatzruns from −1/2to

1/2 and there is a singularity for Pr =1, which can be avoided by choosing a value very

close to one or could be resolved by L’Hôpital’s rule.

Appendix B. Heat and momentum transport

The Nusselt number Nu and Reynolds number Re, based on the wind velocity, are deﬁned

as

Nu ≡−∂¯

θ

∂zz=0A

,Re ≡Ra

Pr u2V,(B1a,b)

where Adenotes the horizontal plane for the cylinder or, respectively, the x-direction for

the 2-D simulations. Figure 13 shows Nu(Ω ) and Re(Ω ),normalised by their values at

Ω=10−3. Their exact values are given in the supplementary material. The 2-Dsystem

(ﬁgure 13a,b) shows a signiﬁcant heat and momentum transport enhancement for certain

TW speeds Ω, especially for large Ra.Forthe3-D cylindrical system (ﬁgure 13c), no

clear correlation between the zonal ﬂow maximum (see ﬁgure 9)andNu(Ω) and Re(Ω)

is observed. However, a small Re enhancement is present at Ω≈10−2.

Appendix C. Linear stability analysis

In § 3.1.1 a temporal linear stability analysis was conducted to identify the most unstable

eigenmode of the 2-D linearised Navier–Stokes equations, where a wave-like form was

considered only in time. Thus, any ﬂow quantity φ(x,z,t)is represented as φ(x,z,t)=

ˆ

φ(x,z)e−iωtand the system of equations for the horizontal velocity u, the vertical velocity

w, the pressure pand the temperature θreads

⎡

⎢

⎢

⎢

⎢

⎢

⎣

L2D+DxUD

zUD

x0

DxWL

2D+DzWD

z−1

DxDz00

Dx¯

θDz¯

θ0K2D

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎣

ˆu

ˆv

ˆp

ˆ

θ

⎤

⎥

⎥

⎦=ω⎡

⎢

⎢

⎣

i000

0i00

0000

000i

⎤

⎥

⎥

⎦

⎡

⎢

⎢

⎣

ˆu

ˆv

ˆp

ˆ

θ

⎤

⎥

⎥

⎦

,(C1)

where

L2D=UDx+WDz+Pr/Ra −D2

x−D2

z,(C2)

K2D=UDx+WDz+1/√RaPr −D2

x−D2

z.(C3)

The overline represents the mean ﬁeld quantity. In our study we applied the Chebyshev

method to approximate the vertical gradient (Dz)andthe Fourier method for the horizontal

gradient (Dx). Conveniently, the corresponding differentiation matrices are available open

source, e.g. we used the Python package dmsuite.

913 A13-22

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Generation of zonal ﬂows in convective systems

0

0.5

1.0

0

0.5

1.0

Re/ReΩ = 10–3

Nu/NuΩ = 10–3

Nu/NuΩ = 10–3

Nu/NuΩ = 10–3

Re/ReΩ = 10–3

Re/ReΩ = 10–3

0.9

1.0

1.1

1.2

0

0.5

1.0

0.7

0.8

0.9

1.0

Ω

0

0.5

1.0

Ω

10–3 10–2 10–1 10010–3 10–2 10–1 100

(b)

(a)

(c)

Figure 13. Normalised Nu and Re vs. Ωfor (a) 2-D set-up A, (b) 2-D set-up B and (c) 3-D cylinder;

Ra =103(•,blue),10

4(•,orange),10

5(•, green), 106(•,red)and10

7(•,black).

The linear set of (C1)issolvedasageneralised eigenvalue problem of the form Aˆ

φ=

ωBˆ

φ, where the eigenvectors φ(x,z,t)represent the wave amplitudes and the eigenvalues

ωtheir respective temporal behaviour. The matrices,of size 4 ×Nx×Nz,areverylarge

and therefore an iterative solver has to be used (e.g. Python’s scipy.eigs). The code has

been validated by solving the Blasius boundary layer, pipe ﬂow and Rayleigh–Taylor

instabilities in one and two dimensions, and in closed and periodic domains. For all cases

we have found excellent agreement with results in the literature.

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