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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 (10^3≤Ra≤10^7) and thermal wave frequencies (10^−4≤Ω≤10^0), 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.
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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 (103Ra 107) and thermal wave
frequencies (104Ω100), 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:UxVΩ1for Ω0andUxVΩ4for 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/Ra2u+θez,(2.1)
∂θ/t+u·∇θ=1/PrRa2θ, ∇·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 gˆ
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,(2.3ac)
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/Ra2
zUx=zuwx+WzUx.(2.4)
Evidently, a mean zonal flow Uxis maintained by the momentum transport due to the
Reynolds stress component uwand by mean advection through WzU. 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
k3Ra2Pr2(Pr +1)
12! Ω+O 3)for Ω0,(2.5)
UxV=−k3Ra1/2Pr3/2
256π4(Pr +1)Ω4+O9/2)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 WzU-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 =0θ =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(x2πΩt)+θ ],(3.1)
θ(x,z=H,t)=0.5[cos(x2πΩt)θ ].(3.2)
Here, Ωindicates the temporal frequency of the TW in free-fall time units. For example,
Ω=101describes a wave with a period of 10 free-fall time units τff ,andθ 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 Raθ αgθ ˆ
H3/(κν). However, in this work the connection to Ra is
simply Raθ =0forset-up A and Raθ =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 103Ra 107and 104Ω100, 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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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina
(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
WzU
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/Ra2
zUx=
zuwx+WzUx).
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 WzUstarts 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 Ω0.1. 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>101, 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 WzU. 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
(Rac1708). 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 σ0+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 (ac) 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
(τκΩ=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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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina
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 104to 103,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
Rcos2πΩt)+1,(4.1)
θ(ϕ,r,z=H,t)=0.5r
Rcos2πΩt)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 ΓD/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 Uϕrat 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 Ω101, 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
UxVΩ1for Ω0appears 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. Ω102, the mean values are highly significant and can induce zonal
flows of the same order of magnitude as the TW frequency, UϕV102. 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
(ae) and the corresponding snapshots of the temperature θ(fj). 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
rruϕ
r+1
r2
2uϕ
∂ϕ2+2uϕ
z2uϕ
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
z2Uϕr
r2
=u
ϕu
zr
z+UϕUzr
z+u
ϕu
r
rr+UrUϕ
rr
.(4.4)
Analysing the radial dependence, on the other hand, averaging over ϕand zgives
Pr
Ra 1
r2
2Uϕϕ,z
r2Uϕϕ,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=102, 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 11ce) 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 11ce), 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/R1(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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P. Reiter, X. Zhang, R. Stepanov and O. Shishkina
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=102and 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=101. The curves show the effects of
different imposed BCs and aspect ratios: baseline simulation (black, free-slip BCs, θr/Rand Γ=1),
sinusoidal radial temperature BCs (blue, free-slip BCs, θsin(πr/R)and Γ=1), no-slip (red, no-slip BCs,
θr/Rand Γ=1), Γ=0.2 (yellow, free-slip BCs, θr/Rand Γ=0.2) and Γ=2 (green, free-slip BCs,
θr/Rand Γ=2).
of the zonal flows. We do this for a single baseline simulation at Ra =105and Ω=101.
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 (θsin(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=3C. Given a mean temperature of 10 C,
the material properties of air are approximately κ=2.0×105m2s1,ν=1.4×
105m2s1and α=3.6×103K1.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
Hθ 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 Ω103. 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 Uϕ102(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 s1,
or equivalently 0.3kmh
1. However, locally, it could exceed this value (see figure 10)
multiple times, therefore speeds of 1 km h1are 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 UϕVΩ1for Ω0and
UϕVΩ4for 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.
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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+wz(U+u)=−xp+ν2U
z2+2u
x2+2u
z2,(A1)
tw+(U+u)∂xw+wz(w)=−zp+ν2w
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 gˆ
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)) w(z)exp(i(kx 2πΩt)),(A4)
u(x,z,t)=−zwdx=i
2kzˆw(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=κ2θ
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
dz2λ2ˆ
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/2λz)
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
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Generation of zonal flows in convective systems
larger than the height of the cell (kH 1) to obtain
4ˆw
z4α22ˆw
z2=k2ˆ
θ, α2=2πiΩ
ν.(A9)
For ˆw|z=1/2w|z=−1/2=zˆw|z=1/2=zˆw|z=−1/2=0, the solution to (A9)is
ˆw(z)=c1
α2coshz)+c2
α2sinhz)+c3z+c4+c5cosh(λz)+c6sinh(λz). (A10)
Scenario 1
A=k2
2νλ2(λ2α2),
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
4νλ2(λ2α2),
c1=−λαAtanh(λ/2)
sinh(α/2),
c2=αAλ
tanh(λ/2)2
(2/α) sinh(α/2)cosh/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):
νd2U
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 ) and Re ),normalised by their values at
Ω=103. 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) and Re)
is observed. However, a small Re enhancement is present at Ω102.
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)eiωtand 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
z1
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
xD2
z,(C2)
K2D=UDx+WDz+1/RaPr D2
xD2
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.
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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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... Temperature contours near the onset of convection are shown for some Γ, as obtained from the linearized DNS. See details in [5][6][7][8] and the Supplemental Material [9]. Most data are for Γ ¼ 1 and 1=2, which form the shape of this dependence. ...
... From (8), (7), and (15) we finally obtain Ra ≥ ΛðNu − 1Þ, which, when combined with (13), implies that close to the onset of convection, the Nusselt number behaves as ...
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To study turbulent thermal convection, one often chooses a Rayleigh-Bénard flow configuration, where a fluid is confined between a heated bottom plate, a cooled top plate of the same shape, and insulated vertical sidewalls. When designing a Rayleigh-Bénard setup, for specified fluid properties under Oberbeck-Boussinesq conditions, the maximal size of the plates (diameter or area), and maximal temperature difference between the plates, Δmax, one ponders: Which shape of the plates and aspect ratio Γ of the container (ratio between its horizontal and vertical extensions) would be optimal? In this article, we aim to answer this question, where under the optimal container shape, we understand such a shape, which maximizes the range between the maximal accessible Rayleigh number and the critical Rayleigh number for the onset of convection in the considered setup, Rac,Γ. First we prove that Rac,Γ∝(1+cuΓ−2)(1+cθΓ−2), for some cu>0 and cθ>0. This holds for all containers with no-slip boundaries, which have a shape of a right cylinder, whose bounding plates are convex domains, not necessarily circular. Furthermore, we derive accurate estimates of Rac,Γ, under the assumption that in the expansions (in terms of the Laplace eigenfunctions) of the velocity and reduced temperature at the onset of convection, the contributions of the constant-sign eigenfunctions vanish, both in the vertical and at least in one horizontal direction. With that we derive Rac,Γ≈(2π)4(1+cuΓ−2)(1+cθΓ−2), where cu and cθ are determined by the container shape and boundary conditions for the velocity and temperature, respectively. In particular, for circular cylindrical containers with no-slip and insulated sidewalls, we have cu=j112/π2≈1.49 and cθ=(j̃11)2/π2≈0.34, where j11 and j̃11 are the first positive roots of the Bessel function J1 of the first kind or its derivative, respectively. For parallelepiped containers with the ratios Γx and Γy, Γy≤Γx≡Γ, of the side lengths of the rectangular plates to the cell height, for no-slip and insulated sidewalls we obtain Rac,Γ≈(2π)4(1+Γx−2)(1+Γx−2/4+Γy−2/4). Our approach is essentially different to the linear stability analysis, however, both methods lead to similar results. For Γ≲4.4, the derived Rac,Γ is larger than Jeffreys' result Rac,∞J≈1708 for an unbounded layer, which was obtained with linear stability analysis of the normal modes restricted to the consideration of a single perturbation wave in the horizontal direction. In the limit Γ→∞, the difference between Rac,Γ→∞=(2π)4 for laterally confined containers and Jeffreys' Rac,∞J for an unbounded layer is about 8.8%. We further show that in Rayleigh-Bénard experiments, the optimal rectangular plates are squares, while among all convex plane domains, circles seem to match the optimal shape of the plates. The optimal Γ is independent of Δmax and of the fluid properties. For the adiabatic sidewalls, the optimal Γ is slightly smaller than 1/2 (for cylinder, about 0.46), which means that the intuitive choice of Γ=1/2 in most Rayleigh-Bénard experiments is right and justified. For the given plate diameter D and maximal temperature difference Δmax, the maximal attainable Rayleigh number range is about 3.5 orders of magnitudes smaller than the order of the Rayleigh number based on D and Δmax. Deviations from the optimal Γ lead to a reduction of the attainable range, namely, as log10(Γ) for Γ→0 and as log10(Γ−3) for Γ→∞. Our theory shows that the relevant length scale in Rayleigh-Bénard convection in containers with no-slip boundaries is ℓ∼D/Γ2+cu=H/1+cu/Γ2. This means that in the limit Γ→∞, ℓ equals the cell height H, while for Γ→0, it is rather the plate diameter D.
... In contrast, in regime II b , the strongest horizontal fluid motions appear in the bulk, see, e.g. the strong zonal flow motions in the central region for Ra = 10 12 , as shown in figure 3(c). We remark that the zonal flow has been found in many geo-and astrophysical flows (Yano, Talagrand & Drossart 2003;Heimpel, Aurnou & Wicht 2005;Nadiga 2006), and it has also been extensively studied in RBC (Goluskin et al. 2014;Wang et al. 2020a;Zhang et al. 2020;Reiter et al. 2021). It is remarkable and interesting to also observe zonal flows in the high-Ra VC system. ...
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Rayleigh-Bénard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimensional (2-D) RB convection and the other one three-dimensional (3-D) RB convection with a rotating axis parallel to the plate, which for strong rotation mimics 2-D RB convection. For the 2-D simulations, we explore the parameter range of Rayleigh numbers Ra from 10 7 to 10 9 and Prandtl numbers Pr from 1 to 100. The effect of the width-to-height aspect ratio Γ is investigated for 1 Γ 128. We show that zonal flow, which was observed, for example, by Goluskin et al. (J. Fluid. Mech., vol. 759, 2014, pp. 360-385) for Γ = 2, is only stable when Γ is smaller than a critical value, which depends on Ra and Pr. The regime in which only zonal flow can exist is called the first regime in this study. With increasing Γ , we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger Γ , in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. How many convection rolls form (or in other words, what the mean aspect ratio of an individual roll is), depends on the initial conditions and on Ra and Pr. For instance, for Ra = 10 8 and Pr = 10, the aspect ratio Γ r of an individual, statistically stable convection roll can vary in a large range between 16/11 and 64. A convection roll with a large aspect ratio of Γ r = 64, or more generally already with Γ r 10, can be seen as 'localized' zonal flow, and indeed carries over various properties of the global zonal flow. For the 3-D simulations, we fix Ra = 10 7 and Pr = 0.71, and compare the flow for Γ = 8 and Γ = 16. We first show that with increasing rotation rate both the flow structures and global quantities like the Nusselt number Nu and the Reynolds number Re increasingly behave like in the 2-D case. We then demonstrate that with increasing † Email address for correspondence: d.lohse@utwente.nl https://doi.org/10.1017/jfm.2020.793 Downloaded from https://www.cambridge.org/core. IP address: 130.89.108.130, on 28 Oct 2020 at 13:42:58, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. 905 A21-2 Q. Wang and others aspect ratio Γ , zonal flow, which was observed for small Γ = 2π by von Hardenberg et al. (Phys. Rev. Lett., vol. 15, 2015, 134501), completely disappears for Γ = 16. For such large Γ , only convection roll states are statistically stable. In-between, here for medium aspect ratio Γ = 8, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2-D case.