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J. Fluid Mech. (2021), vol.915, A62, doi:10.1017/jfm.2021.74
Boundary zonal flows in rapidly rotating
turbulent thermal convection
Xuan Zhang1,,RobertE.Ecke
2,3and Olga Shishkina1,
1Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
2Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3Department of Physics, University of Washington, Seattle, WA 98195, USA
(Received 8 September 2020; revised 2 December 2020; accepted 19 January 2021)
Recently, in Zhang et al. (Phys. Rev. Lett., vol. 124, 2020, 084505), it was found that,in
rapidly rotating turbulent Rayleigh–Bénard convection in slender cylindrical containers
(with diameter-to-height aspect ratio Γ=1/2) filled with a small-Prandtl-number fluid
(Pr 0.8), the large-scale circulation is suppressed and a boundary zonal flow (BZF)
develops near the sidewall, characterized by a bimodal probability density function of the
temperature, cyclonic fluid motion and anticyclonic drift of the flow pattern (with respect
to the rotating frame). This BZF carries a disproportionate amount (>60 %) of the total
heat transport for Pr <1, but decreases rather abruptly for larger Pr to approximately
35 %. In this work, we show that the BZF is robust and appears in rapidly rotating turbulent
Rayleigh–Bénard convection in containers of different Γand over a broad range of Pr and
Ra. Direct numerical simulations for Prandtl number 0.1Pr 12.3, Rayleigh number
107Ra 5×109,inverse Ekman number 1051/Ek 107and Γ=1/3, 1/2, 3/4, 1
and 2 show that the BZF width δ0scales with the Rayleigh number Ra and Ekman number
Ek as δ0/HΓ0Pr{−1/4,0}Ra1/4Ek2/3({Pr <1,Pr >1})andwith the drift frequency
scales as ω/Ω Γ0Pr4/3Ra Ek5/3, where His the cell height and Ωthe angular rotation
rate. The mode number of the BZF is 1 for Γ1and2Γfor Γ={1,2}independent of
Ra and Pr. The BZF is quite reminiscent of wall mode states in rotating convection.
Key words: Bénard convection, rotating flows, rotating turbulence
1. Introduction
Turbulent convection driven by buoyancy and subject to background rotation is a
phenomenon of great relevance in many physical disciplines, especially in geo- and
Email addresses for correspondence: xuan.zhang@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. 915 A62-1
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X. Zhang, R. E. Ecke and O. Shishkina
astrophysics and also in engineering applications. In a model system of Rayleigh–Bénard
convection (RBC) (Bodenschatz, Pesch & Ahlers 2000; Ahlers, Grossmann & Lohse 2009;
Lohse & Xia 2010), a fluid is confined in a container where the bottom is heated, the
top is cooled and the vertical walls are adiabatic. The temperature inhomogeneity leads
to a fluid density variation, which, in the presence of gravity, produces convective fluid
motion. When the system rotates with respect to the vertical axis, significant modification
of the flow occurs owing to the rotational influence, including the suppression of the
onset of convection (Nakagawa & Frenzen 1955; Chandrasekhar 1961), the enhancement
or suppression of turbulent heat transport over different ranges of Rayleigh number Ra and
Prandtl number Pr (Rossby 1969; Pfotenhauer, Niemela & Donnelly 1987; Zhong, Ecke
& Steinberg 1993; Julien et al. 1996;Liu&Ecke1997), the transformation of thermal
plumes into thermal vortices with a rich variety of local structure dynamics (Boubnov &
Golitsyn 1986,1990; Hart, Kittelman & Ohlsen 2002; Vorobieff & Ecke 2002)and the
emergence of robust wall modes before the onset of the bulk mode (Buell & Catton 1983;
Pfotenhauer et al. 1987; Zhong, Ecke & Steinberg 1991; Ecke, Zhong & Knobloch 1992;
Goldstein et al. 1993; Herrmann & Busse 1993;Kuo&Cross1993).
The dimensionless control parameters in rotating RBC are the Rayleigh number Ra
αgΔH3/(κν), the Prandtl number Pr ν/κ, the Ekman number Ek ν/(2ΩH2)and the
diameter-to-height aspect ratio of the container, ΓD/H.Hereαdenotes the isobaric
thermal expansion coefficient, νthe kinematic viscosity, κthe thermal diffusivity of
the fluid, gthe acceleration due to gravity, Ωthe angular rotation rate, ΔT+T
the difference between the temperatures at the bottom (T+)andtop(T) plates, Hthe
distance between the isothermal plates (the cylinder height) and D2Rthe cylinder
diameter. The Rossby number Ro αgΔH/(2ΩH)=Ra/Pr Ek is another important
non-dimensional parameter that provides a measure of the balance between buoyancy and
rotation and is independent of dissipation coefficients.
The important global response parameter in thermal convection is the averaged total
heat transport between the bottom and top plates, described by the Nusselt number, Nu
(uzTzκ∂zTz)/(κΔ /H). Here, Tdenotes the temperature, uis the velocity field with
component uzin the vertical direction, and ·zdenotes the average in time and over a
horizontal cross-section at height zfrom the bottom.
Rotation has various effects on the structure of the convective flow and on the global
heat transport in the system. Rotation inhibits convection and causes an increase of the
critical RacEk4/3at which the quiescent fluid layer becomes unstable throughout the
layer (Nakagawa & Frenzen 1955; Chandrasekhar 1961; Rossby 1969; Lucas, Pfotenhauer
& Donnelly 1983; Zhong et al. 1993). In finite containers and at sufficiently large
rotation rates, however, a different instability occurs at lower RawEk1in the form of
anticyclonically drifting wall modes (Buell & Catton 1983; Pfotenhauer et al. 1987; Zhong
et al. 1991;Eckeet al. 1992; Goldstein et al. 1993; Herrmann & Busse 1993;Kuo&Cross
1993;Ning&Ecke1993; Zhong et al. 1993; Goldstein et al. 1994;Liu&Ecke1997,
1999; Zhang & Liao 2009; Favier & Knobloch 2020). The relative contribution of the wall
modes to the total heat transport depends on Γ(Rossby 1969; Pfotenhauer et al. 1987;
Ning & Ecke 1993; Zhong et al. 1993;Liu&Ecke1999) with decreasing contribution –
roughly as the perimeter-to-area ratio – with increasing Γ.
There are several regions of bulk rotating convection where rotation plays an important
role, namely a rotation-affected regime and a rotation-dominated regime. In the former,
where Ro 1, heat transport varies as a power law in Ra,i.e.Nu =A(Ek)Ra0.3, and can
be enhanced or weakly suppressed by rotation relative to the heat transport without rotation
(Rossby 1969; Zhong et al. 1991; Julien et al. 1996;Liu&Ecke1997;Kinget al. 2009;
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Boundary zonal flows in rotating turbulent convection
Liu & Ecke 2009; Zhong et al. 2009) depending on the range of Ra and Pr. In the latter
case, in which Ro 1, heat transport changes much more rapidly with Ra in what is
known as the geostrophic regime of rotating convection (Sakai 1997; Grooms et al. 2010;
Julien et al. 2012; Ecke & Niemela 2014; Stellmach et al. 2014; Cheng et al. 2020).
Despite considerable previous work, the spatial distribution of flow and heat transport
in confined geometries has not been well studied for high Ra and low Ro when one
is significantly above the onset of bulk convection but still highly affected by rotation.
Recently, Zhang et al. (2020) demonstrated in direct numerical simulations (DNS) and
experiments that a boundary zonal flow (BZF) develops near the vertical wall of a
slender cylindrical container (Γ=1/2) in rapidly rotating turbulent RBC for Pr =0.8
(pressurized gas SF6)and over broad ranges of Ra (Ra =109in DNS and for 1011 Ra
1014 in experiments) and Ek (106Ek 105in the DNS and for 3 ×108Ek
3×106in experiments). The BZF becomes the dominant mean flow structure in the cell
for Ro 1, at which the large-scale mean circulation (termed the large-scale circulation;
LSC) vanishes (Vorobieff & Ecke 2002; Kunnen et al. 2008; Weiss & Ahlers 2011a,b).
Further, it contributes a disproportionately large fraction of the total heat transport.
Another group (de Wit et al. 2020) also showed the existence of the BZF and its strong
influence on heat transport using DNS for Pr =5 (water) and Γ=1/5forEk =107
in the range 5 ×1010 Ra 5×1011 . Thus, the BZF has been observed in different
fluids, in cells of different aspect ratios and over a wide range of parameter values. Given
the strongly enhanced heat transport in the BZF region (de Wit et al. 2020; Zhang et al.
2020), it is important to explore the BZF in detail. Here we investigate the robustness of
the BZF with respect to Pr and to Γin the geostrophic regime; we do not address here the
transition from the low rotation state to the BZF.
Recently, Favier & Knobloch (2020) demonstrated for Ek =106through DNS that the
linear wall modes of rotating convection (Buell & Catton 1983; Zhong et al. 1991;Ecke
et al. 1992; Goldstein et al. 1993; Herrmann & Busse 1993;Kuo&Cross1993;Ning&
Ecke 1993; Zhong et al. 1993;Liu&Ecke1997,1999; Sánchez-Álvarez et al. 2005; Horn
&Schmid2017; Aurnou et al. 2018) evolve with increasing Ra and appear to be robust
with respect to the emergence of bulk convection even with well-developed turbulence.
They suggested that the BZF may be the nonlinear evolution of wall modes, an idea that
we address briefly but that requires significantly more analysis and comparison than can
be included here.
In the present work, a series of DNS is carried out to study the robustness and the scaling
properties of the BZF with respect to Rayleigh number Ra, Ekman number Ek, Prandtl
number Pr and cell aspect ratio Γ. We explore the extended scalings of the characteristics
of the BZF, including the width of the BZF, the drift frequency of the BZF and the heat
transport within the BZF in terms of these non-dimensional parameters. We first present
our numerical methods, then discuss the results of our calculations and conclude with our
main findings.
2. Numerical method
We present results of DNS of rotating RBC in a cylindrical cell obtained using the
GOLDFISH code (Shishkina et al. 2015; Kooij et al. 2018)forRa up to 5 ×109and Ek
down to 107. In the DNS, the Oberbeck–Boussinesq approximation is assumed as in
Horn & Shishkina (2014). Centrifugal force effects are neglected since the Froude number
in experiments is typically small (see Zhong et al. 2009; Horn & Shishkina 2015).
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X. Zhang, R. E. Ecke and O. Shishkina
The governing equations based on the Oberbeck–Boussinesq approximation are
∇·u=0,(2.1)
tu+(u·∇)u=−1
ρ
p+ν2u2Ω×u+α(TT0)gez,(2.2)
tT+(u·∇)T=κ2T.(2.3)
Here, u=(ur,uφ,uz)is the velocity with radial, azimuthal and vertical coordinates,
respectively, ρis the density, pis the reduced pressure, Ω=Ωezis the angular
rotation-rate vector, Tis the temperature with T0=(T++T)/2, and ezis the unit vector
in the vertical direction. The applied boundary conditions are no slip for the velocity on all
surfaces, constant temperature for the top and bottom plates and adiabatic for the sidewall.
To non-dimensionalize the governing equations, we use Δ=T+Tas the temperature
scale, the cylinder height Has the length scale and the free-fall velocity αgΔHas the
velocity scale (the corresponding time scale is τff =H/(αgΔ)).
To evaluate the grid requirements for the simulations, we consider the thermal and
velocity boundary layers (BLs) near solid boundaries. The thickness of the BLs near the
heated and cooled plates are calculated as
δth =H/(2Nu). (2.4)
This is the standard way to define the thermal BL thickness under the assumption of
pure conductive heat transport within this layer (cf. Ahlers et al. 2009). The viscous
BL thicknesses near the plates (δu) and near the sidewall (δsw) are defined as the
distances from the corresponding walls to the location where the maxima of, respectively,
ur2t,φ,r+uφ2t,r(z)and uφ2t,φ,z+uz2t,φ,z(r)are obtained. The velocity
components are all averaged in time and over the surface parallel to the corresponding
wall. The same criterion was used previously in studies of the sidewall layers in rotating
convection (see Kunnen et al. 2011).
The computational grids are set to be sufficiently fine to resolve the mean Kolmogorov
microscales (Shishkina et al. 2010) in the bulk and within the BLs (see table 2 in the
Appendix). Grid nodes are clustered near the walls to resolve thermal and velocity BLs,
resulting in grids that are non-equidistant in both the radial and vertical directions. As
rotation increases, the viscous BL gets thinner (Kunnen et al. 2008; Stevens, Verzicco &
Lohse 2010; Horn & Shishkina 2015) so more points are required near boundaries: we
take at least seven points within each BL. The details of all simulated parameters and the
corresponding grid resolution are listed in table 2 along with a benchmark comparison
between Nu data from these simulations and from experimental data in compressed gases
with similar Pr from Wedi et al. (2021); the agreement is excellent. To explore the
robustness of the BZF with respect to Ra,Pr and Γ, we conducted simulations in three
groups, i.e. in every group we vary only one parameter while keeping the others fixed.
The specific parameter ranges are shown in table 1 (also included in several figures with
Ra =109and Pr =0.8 are data in the range 0.51/Ro 5 from Zhang et al. (2020);
the calculation details for those values are included in the Appendix).
3. Results
3.1. Boundary zonal flow structure
Our goal here is to explore the robustness of the BZF with respect to variations of
control parameters. We follow closely the approach and characterization presented in
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Boundary zonal flows in rotating turbulent convection
ΓPr Ra (107)1/Ro Ek (106)
1/2 0.8 5–500 10 1.3–13
1/2 0.1–12.3 10 10 3.2–35
1/3–2 0.8 10 10 8.9
1/2 0.8 100 5.6–33.3 0.85–5.1
Tab le 1. Ran ge s of Γ,Ra,Pr,Ro1and Ek. For details, see the Appendix.
Tt
T+
1/Ro = 0.5 1/Ro = 10
T
PP PP
(e)(b)(a)(c)(d)(f)
Figure 1. Isosurfaces of instantaneous temperature T(a)and time-averaged flow fields (b,c),visualized by
streamlines (arrows) and temperature (colours), for Ra =109and 1/Ro =10, in vertical orthogonal planes P
(b,e)and P(c,f). In the case of weak rotation (ac),Pis the plane of the LSC (b). Averaging in (b,c)is
conducted over 1000 free-fall time units. For strong rotation (e,f), mean radial and axial velocity magnitudes
are approximately 10-fold smaller than those for weak rotation (b,c).
Zhang et al. (2020) but focus on the geostrophic regime where the BZF is well developed.
After presenting our main results, we consider the BZF with respect to wall mode
structures. We begin with the influence of rotation on the overall temperature and velocity
fields in the cell. In figure 1, for particular cases of 1/Ro =0.5 (weak rotation) and 1/Ro =
10 (fast rotation), three-dimensional instantaneous temperature distributions (figure 1a,d)
and two-dimensional vertical cross-sections (figure 1b,c,e,f) of the time-averaged flow
fields are shown. The two-dimensional views are taken in a plane P(figure 1b,e), which
in the case of a weak rotation is the LSC plane, and additionally in a plane Pthat is
perpendicular to P(figure 1c,f). For slow rotation, an LSC spanning the entire cell with
two secondary corner rolls is observed in Pwhereas a four-roll structure is seen in P,
typical of classical RBC at large Ra and for Γ1(
see e.g. Shishkina, Wagner & Horn
2014; Zwirner et al. 2020). Near the plates, the LSC and the secondary corner flows move
the fluid towards the sidewall (figure 1b) so the Coriolis acceleration (2Ωez×u) induces
anticyclonic fluid motion close to the plates.
In the central part of the cell, at z=H/2, the radial component of the mean velocity,
urt, always points towards the cell centre (figure 1a,b). Therefore, Coriolis acceleration
results in cyclonic fluid motion in the central part of the cell, as is also observed in the
time-averaged azimuthal velocity field uφin figure 2(a). Cases at higher rotation rates
are shown in figures 1(df)and2(see also Kunnen et al. 2011). For both small and
large rotation rates, the presence of viscous BLs near the plates implies anticyclonic
motion of the fluid there. For strong rotation, the subject of this paper, with high and
constant angular velocity Ω, the fluid velocity becomes more uniform along ezowing
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X. Zhang, R. E. Ecke and O. Shishkina
Cyclonic
Anticyclonic
Ω
(b)(a)(c)(d)
Figure 2. Time-averaged fields uφtfor Pr =0.8, Γ=1/2, Ra =109and (a)1/Ro =0.5, (b)1/Ro =2,
(c)1/Ro =10 and (d)1/Ro =20.
to the Taylor–Proudman constraint with larger components of lateral velocity compared
to the vertical component as in figure 1(e,f). Thus, anticyclonic fluid motion not only is
present in the vicinity of the plates, but also involves more and more fluid volume with
increasing Ro1. With increasing rotation rate, anticyclonic motion grows from the plates
towards the cell centre whereas cyclonic motion at z=H/2 remains near the sidewall and
becomes increasingly more localized there (figure 2c,d).
As introduced in Zhang et al. (2020), the BZF in rapidly rotating turbulent convection is
characterized by an anticyclonic bulk flow, cyclonic vortices clustering near the sidewall
and anticyclonic drift of thermal plumes (see figures 3a,band 4). These structures are
associated with the bimodal temperature probability density functions (p.d.f.s) obtained
in the measurements and DNS near the sidewall (Zhang et al. 2020;Wediet al.
2021). The radial location r0where the mean fluid motion at z/H=1/2 changes from
anticyclonic to cyclonic as indicated by the solid line in figure 3 (see also inset of
figure 6abelow) is used to describe the width of the BZF δ0=Rr0. As one might
expect, vertical coherence of the BZF is enhanced by strong rotation. In figure 4,
time–angle plots of the temperature at three different heights show that the drift frequency
ω=2πR(dφ(rumax
φ)/dt)/(2πR/m)=mdφ(rumax
φ)/dtis quite constant along zwithout
significant phase differences, i.e. the BZF maintains good vertical coherence. Here, the
mode number mequals 1 and dφ(rumax
φ)/dtdenotes the angular velocity of the temperature
drift at r=rumax
φ, where the maximum of the time-averaged azimuthal velocity is obtained.
In the lower half of the cell, for z=H/4, warm plumes dominate, so the warm regions
(pink stripes) are wider, whereas in the upper half of the cell, for z=3H/4, cold plumes
dominate, resulting in wider cooler regions (blue stripes). Similarly, figures 2(c,d)and
5(a,d) show that the zonal flow develops away from the top and bottom plates and extends
vertically throughout the bulk. Figure 5 illustrates that, owing to the drift, time-averaged
fields in the vertical plane average to zero and do not capture important features of the
flow motion, in particular, the uzfield. The averaged u2
z, however, does retain important
information about the locations of the Stewartson ‘1/3’ and ‘1/4’ layers (dashed lines) and
the BZF (solid line).
3.2. Contribution to heat transport
An important and unexpected property of the BZF in rotating RBC is its disproportionately
large contribution to the heat transport in the system. Figures 3(a)and6show that the
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Boundary zonal flows in rotating turbulent convection
Fztωz
ωz
max 0ωz
max
01 Fz
max
(b)(a)
Figure 3. For Ra =109,1/Ro =10, Pr =0.8, Γ=1/2and z=H/2: (a,b)horizontal cross-sections of (a)
time-averaged vertical heat flux Fztand (b)instantaneous vertical component of vorticity ωz(negative values
correspond to anticyclonic fluid motion), together with two-dimensional streamlines. The solid line indicates
the radial position r0that defines the BZF by the condition uφ(r0,z=H/2)t=0. In (a), the dash-dotted
line (inner circle) and the dashed line (outer circle) are, respectively, the radial locations of Fzt=1 (global
averaged heat flux) and umax
φ, the maximum of the time-averaged azimuthal velocity.
(a)(b)(c)
T
0
T+
200
150
100
50
0002π2π2π0
φφφ
t/H/(αgΔ)
Figure 4. For Pr =0.8, Γ=1/2, Ra =109,1/Ro =20 and r=R: time evolution of temperature
distribution (space–time plot of temperature) at height (a)z=H/4, (b)z=H/2and (c)z=3H/4.
averaged heat flux inside the BZF is much stronger than in the region outside the BZF. To
be clear about the averaging,wedene
Fi(r,z)(uiTκ∂iT)/(κΔ/H), i=r,z,(3.1)
Nu(r,t)φ(2π)12π
0
Fz(r,z=H/2)dφ, (3.2)
Nu(t)V(πR2H)12π
0R
0H
0
Fz(r,z)rdrdφdz,(3.3)
Nu(t)BZF (π(R2r2
0))12π
0R
r0
Fz(r,z=H/2)rdrdφ, (3.4)
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X. Zhang, R. E. Ecke and O. Shishkina
0
ur
2
uz
2
uz
uφ
101010
z/H
1
1
0
0.25
0.75 1.00 1.00
0.75 0.75 1.00
r/R
z/H
0.75
(e)(g)(h)
(b)(a)(c)(d)
(f)
0 maxmin 0 max
Figure 5. Time-averaged flow fields in a vertical plane, for Ra =109,1/Ro =10, Pr =0.8and Γ=1/2.
The ranges of variables are, respectively: (a,b,e,f)from 0.17 to 0.17; and (c,d,g,h)from 0 to 0.0289.
Rf≡NuBZF,t/NuV,t,(3.5)
Rh(NuBZF,tπ(R2r2
0))/(NuV,tπR2)=R2r2
0
R2NuBZF,t/NuV,t,(3.6)
where r0=Rδ0. The quantity Rfis the ratio of the mean vertical heat flux within
the BZF to the vertical heat flux averaged over the whole cell. The quantity Rhreflects
the portion of the heat transported through the BZF compared to the total transported
heat. Especially, in figure 6(a), the time- and φ-averaged radial profile at the mid-height
for Ra =109,Pr =0.8and Γ=1/2 shows a significant peak of heat transport inside
the BZF, and the peak amplitude increases dramatically as rotation becomes stronger.
Thus, although the width of the BZF shrinks with increasing rotation, thereby reducing
the effective area of the BZF with respect to the whole domain, as shown in figure 6(b),
the increasing magnitude of the peak makes the heat transport carried by the BZF quite
significant. Note that the annular BZF region of width δ0is smaller than the positive
contribution to the heat transport, as shown in the inset of figure 6(a).
Figure 6(c) reveals that the enhancement of the local heat transfer within the BZF
increases more rapidly when rotation is very strong (1/Ro 10). As a result of these
properties, the heat transport carried by the BZF for these parameter values is always more
than 60 % of the total heat transport at fast rotation (see figure 6d). Note, however, that the
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Boundary zonal flows in rotating turbulent convection
0 0.2 0.4 0.6 0.8 1.0
0
2
4
r
Nuφ,t(r)/NuV,t
1/Ro = 3.3
1/Ro = 5
1/Ro = 5.56
1/Ro = 6.67
1/Ro = 8.33
1/Ro = 10
1/Ro = 12.5
1/Ro = 16.67
1/Ro = 20
00.2 0.4 0.6 0.8 1.0
–0.10
–0.05
0
0.05
0.10
r
uφφ,t(r)/uff
1/Ro = 5
1/Ro = 10
1/Ro = 20
100101102100101102
0
0.1
0.2
0.3
0.4
0.5
1/Ro 1/Ro
100101102
1/Ro
A0
0
2
4
6
8
Rf
0
0.2
0.4
0.6
0.8
1.0
Rh
10–1 100101
0
0.2
0.4
0.6
0.8
1.0
Pr
R
h
(b)
(a)
(c)(d)
Figure 6. (a)Radial profiles of normalized time- and φ-averaged heat flux Nuφ,t(r)/NuV,tat z=H/2, for
different rotation rates. The inset shows the radial profiles of time- and φ-averaged uφ, where solid lines pass
through uφφ,t=0 (radial location corresponds to r0). (b)Ratio of BZF area to the total area at z=H/2, i.e.
A0=(R2r2
0)/R2.(c)Ratio of mean vertical heat flux inside the BZF to mean global heat flux, i.e. Rf(3.5).
(d)Ratio of heat transported inside the BZF (solid circles) or in an extended zone R2δ0<r<R(open
circles) to total transported heat, i.e. Rh(R
h)(3.6). For all panels Ra =109,Pr =0.8and Γ=1/2.
effect of the BZF on the heat transport extends over a wider range r<r0;oversomerange,
Nu is actually negative (see figure 6a),implying an anticorrelation of vertical velocity and
buoyancy, i.e. warm fluid going down or cooler fluid moving up. If we modify the annular
averaging to take into account the decreased Nu region as well as the inner structure of
the BZF, i.e. we average over the extended region R2δ0rR,wegettheratioR
h,
which is also shown in figure 6(d) (open symbols) where one sees an even larger fractional
contribution.
We also consider the dependence of the heat transport ratio Rhas a function of Pr (see
inset of figure 6d). Interestingly, for Pr <1wefind0.6<Rh<0.7, whereas for Pr >1
we have 0.3<Rh<0.4, with a quite sharp transition for Pr 1. The origin of this rather
sharp change emphasizes the important role that Pr plays, perhaps through the competition
between thermal and viscous BLs. Finally, comparing our computation of the total Nu
with increasing rotation with that of Wedi et al. (2021)(seefigure 13 in the Appendix),
we conclude, given the close agreement, that the contribution of the BZF affects both
measures of Nu substantially and needs to be taken into account when considering the
scaling of geostrophic heat transport in experiments and also in DNS with no-slip sidewall
boundary conditions (see also de Wit et al. 2020).
3.3. Dependence on Ra, Pr and Γ
We first discuss the qualitative robustness of the BZF with respect to Ra,Pr and Γbefore
we consider its quantitative spatial and temporal properties. We demonstrate the character
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X. Zhang, R. E. Ecke and O. Shishkina
(a)(b)(c)
T
0
T+
200
150
100
50
200
150
100
50
200
150
100
50
00
2π2π2π
0
φφφ
t/H/(αgΔ)
Figure 7. Space–time plots of temperature Tat the sidewall, r=R,andat half-height, z=H/2, for
Ra =108,1/Ro =10, Γ=1/2, and (a)Pr =0.1, (b)Pr =0.8and (c)Pr =4.38.
of the BZF with respect to variations of Pr and Γby considering time–angle plots of
temperature Tat z=H/2andr=R.Figure 7(a) shows that the BZF exists in the flows at
different Pr =0.1,0.8and 4.38 (also for Pr =0.25,0.5,2,3,7and 12.3, not shown),
i.e. from small to large Pr. Although there are some quantitative differences among the
three cases, they all qualitatively demonstrate the existence of the BZF for more than two
decades of Pr.
The qualitative dependence of the BZF on the aspect ratio Γis shown in figure 8 for
three different aspect ratios: Γ=1/2,1and 2. The BZF is present in all three cases,
has the same scaling of BZF width when scaled by H,i.e.δ0/His independent of Γ
(see figure 9dinset), and has a drift period (in units of free-fall time τff =H/(αgΔ) =
τκPr1/2Ra1/2, where τκ=H2is the thermal diffusion time) of approximately 70.
The quantitative scaling of the drift frequency is analysed later, and the data are tabulated
in the Appendix (see table 2). The wavelength λof the travelling BZF mode is independent
of Γfor these three values in a straightforward way, as seen in figure 8, namely λ/H=
π/2, so that the number of wavelengths around the circumference is m=2Γand the
wavenumber is k=2π/λ=4/H. We note, however, that this relationship is for a limited
number of values of Γand control parameters Ra and Ro. Thus, we make no strong
claims to its generality. Indeed, there is already evidence from de Wit et al. (2020)that
for Γ=1/5 one gets m=1/=2Γ, and we made additional measurements with Γ=1/3
and 3/4 that also yield m=1. We conjecture that, owing to periodic azimuthal symmetry,
mwill take on only integer values, similar to the situation for wall mode states (Ecke
et al. 1992; Goldstein et al. 1993;Ning&Ecke1993; Zhong et al. 1993;Liu&Ecke
1999) in cylindrical convection cells. Because of this periodic constraint, one cannot have
m<1, so small aspect ratios with Γ1havem=1. We also note that the mode-number
dependence on Γof the BZF is similar to that of the Γdependence of linear wall state
mode number, i.e. m3Γ(Goldstein et al. 1993; Herrmann & Busse 1993;Kuo&Cross
1993;Ning&Ecke1993;Liu&Ecke1999; Zhang & Liao 2009). Given that our states
have values of Ra that are 10–100 times greater than the linear wall mode onset Raw,this
difference is not unreasonable and the correspondence is very suggestive. In particular,
a range of mode numbers are stable near onset (Ning & Ecke 1993; Zhong et al. 1993;
Liu & Ecke 1999) owing to the azimuthally periodic boundary conditions. Significantly
above onset there seems to be a selection towards lower mode numbers: for example,
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Boundary zonal flows in rotating turbulent convection
(a)(b)(c)
T
0
T+
200
150
100
50
200
150
100
50
200
150
100
50
00
2π2π2π
0
φφφ
t/H/(αgΔ)
Figure 8. Space–time plots of temperature Tat the sidewall, r=R,andat half-height, z=H/2, for
Ra =108,1/Ro =10, Pr =0.8, and (a=1/2, (b=1and (c=2.
Zhong et al. (1993, figures 3 and 8) with Γ=2 show stable wall modes with m=4,5,6
and 7 near onset but only the m=4and 5 modes persist for higher Ra, which yields
m=2Γand m=2.5Γ, respectively, consistent with our results for the BZF (see also
Favier & Knobloch 2020).
3.4. Spatial and temporal scales
We next consider the quantitative dependence of the different layer widths on Ra,Ek
and Pr, looking for a universal scaling of the form δ/HPrξRaβEkγ.Infigure 9(a),
the dependence of δ0/Hon Ek for Ra =109,Pr =0.8and 2 <1/Ro <20 is shown to
be consistent with an Ek2/3scaling,whereas the widths based on other measures scale
closely as Ek1/3,i.e.γtakes on values of 2/3 and 1/3 for BZF width and velocity layer
widths, respectively. (Because the statistical uncertainty in our reported exponents is of
the order of 5%–10%, we report fractional scalings consistent with the data to within
these uncertainties; they are not intended to denote exact results.) As mentioned in Zhang
et al. (2020), the BZF is characterized by bimodal temperature p.d.f.s near the sidewall.
This property was used in both DNS and experimental measurements to identify the
BZF over a wide range of Ra. Here, we conduct a more detailed analysis of the DNS
data to explore how the width of the BZF changes with Ra. We compute the width at
fixed Ro =Ra1/2Pr1/2Ek so Ek =Ro Ra1/2Pr1/2. To determine the scaling with Ra at
fixed Ro =1/10, we have that δ/HRaβγ/2. By multiplying by Raγ/2we obtain the
scaling exponent β.Infigure 9(b), we plot 0/H)Ra1/3and (δ/H)Ra1/6corresponding
to γvalues of 2/3 and 1/3, respectively. From this plot, we obtain values for βof 1/4 and
0, respectively. Similarly for the dependence on Pr,weplotinfigure 9(c) the corrected
quantities (δ/H)Prγ/2, which yields δ0/Hscalings for ξof 1/4forPr <1and0for
Pr >1. The other layer widths based on uφ,uzand Fzare independent of Pr for Pr <1
but do not collapse for Pr >1. The separation of the different widths for Pr >1 suggests
some interesting behaviour not captured by our scaling ansatz.
Finally, we can collapse all the data for BZF width onto a single scaling curve by plotting
in figure 9(d)δ
0/H=δ0/H(Pr{1/4,0}Ra1/4)versus Ek (to compact the different scalings
with Pr we denote them as Pr{1/4,0}for scaling with Pr <1andPr >1, respectively)
so that we can conclude that δ0/HPr{−1/4,0}Ra1/4Ek2/3. The results at one set of
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X. Zhang, R. E. Ecke and O. Shishkina
10−6 10−5
10−2
10−1.5
Ek
2/3
Ek
1/3
Ek
δ/H
107108 1091010 1011
10−1
100
101
102
103
Ra
1/4
Ra0
Ra
10–1 100101102103
10−2
10−1
Pr
−1/4
Pr0
Pr
(δ/H)Prγ/2
(δ/H)Raγ/2
10−6 10−5
10−4
10−3
10−2
Ek
2/3
Ek
δ0
/H
Ro varies
Ra varies
Pr varies
Γ varies
0.5 1.0 1.5 2.0
0
0.2
0.4
0.6
0.8
1.0
Γ
103δ0
/H
(c)
(e)
(d)
(a)(b)
10−6 10–5
0.6
0.8
1.0
1.2
1.4
Ek
(δ
0/H)/(0.85Ek2/3)
δ0
δurms
z
δFz
δumax
φ
max
δ0
δurms
z
δFz
δumax
φ
max
δ0
δurms
z
δFz
δumax
φ
max
Figure 9. (a)Scaling with Ek of characteristic widths δ/HEkγ(for δ0,the distance from the vertical wall to
the location where uφt=0), for Ra =109,Pr =0.8and Γ=1/2. For δ0/H,one obtains γ2/3, whereas
for other δ/H,one has γ=1/3. (b)Scaling with Ra of compensated width Raγ/2δ0/Hfor fixed 1/Ro =10
and Pr =0.8. (c)Scaling with Pr of compensated width Prγ/2δ0/Hfor Ra =108and 1/Ro =10. (d)Scaling
with Ek of normalized BZF width δ
0/H=Ra1/4Pr1/4δ0/H(for Pr <1) and δ
0/H=Ra1/4Pr0δ0/H(for
Pr >1); for compactness, we write the two scalings with Pr as Pr{−1/4,0}. The inset shows δ
0/Hversus Γ.
(e)Compensated plot of BZF width 0/H)/(0.85Pr{−1/4,0}Ra1/4Ek2/3)versus Ek (all data from table 2 are
shown; open symbols are cases with larger statistical uncertainty owing to shorter averaging time).
parameter values {Ra,Pr,Ro}are independent of Γ(see figure 9dinset), which implies
that δ0/HΓ0(other dependences on Γare not ruled out for other parameter values,
although it is reasonable to assume it to be general in the absence of other data). Thus, we
plot in figure 9(d) all the data with different Γ,Pr,Ra and Ek to obtain scalings
δ0/H0.85Γ0Pr1/4Ra1/4Ek2/3,for Pr <1,(3.7)
δ0/H0.85Γ0Pr0Ra1/4Ek2/3,for Pr >1.(3.8)
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Boundary zonal flows in rotating turbulent convection
1071081091010
106
107
108
109
Ra
Ra
ωdPr4/3Ek5/3
ωdRa–1Ek5/3
ωdPr4/3Ra1
10−1 100101
10−3
10−2
10−1
100
Pr4/3
Pr
10−6 10−5
10−12
10−11
10−10
10−9
Ek5/3
Ek
(b)(a)(c)
Figure 10. Scalings of ωd:(a)data scaled by Pr4/3Ek5/3showing Ra scaling;(b)data scaled by Ra1Ek5/3
showing Pr4/3scaling;and(c)data scaled by Pr4/3Ra1showing Ek5/3scaling (cases at different Ra (grey
squares), at different Pr (red triangles) and at different Ro (blue circles)).
We plot in figure 9(e) the scaled BZF width 0/H)/(0.85Γ0Pr{−1/4,0}Ra1/4Ek2/3). One
sees that the data scatter randomly within ±10 %, quite good agreement.
The BZF drifts anticyclonically, the same as the direction of travelling wall modes of
rotating convection (Zhong et al. 1991;Eckeet al. 1992; Herrmann & Busse 1993;Kuo
&Cross1993). We plot in figure 10(a) the drift frequency ωdω/Ω versus Ra showing
scaling as Ra and in figure 10(b) versus Pr showing scaling as Pr4/3(data in both are
corrected for constant-Ro conditions). In figure 10(c), we scale out the dependence on Ra
and Pr,i.e.ωdRa1Pr4/3, and observe reasonable collapse with Ek5/3scaling. From the
cases listed in table 2, we get the frequency scaling in terms of Ra,Pr,Γand Ek as
ωd0.03Γ0Pr4/3Ra Ek5/3.(3.9)
The linear dependence on Ra is consistent with earlier results (Horn & Schmid 2017;
Favier & Knobloch 2020;deWitet al. 2020) and suggests that there is a correspondence
between the states we observe and the nonlinear manifestation of linear wall mode states.
The scalings we have determined for ωdwith Ek and Pr will be useful in making a more
quantitative comparison with the wall mode hypothesis among datasets with different Ek
and Pr. Such an analysis is beyond the scope of the present work and will be presented
elsewhere. These scalings, of course, depend on the definition of the time unit. Using the
free-fall time or the vertical thermal diffusion time, respectively, we obtain
ω/αgΔ/H0.015Γ0Pr5/6Ra1/2Ek2/3,(3.10)
ω/(κ/H2)0.015Γ0Pr1/3Ra Ek2/3,(3.11)
which both show the same Ek scaling as δ0,i.e.Ek2/3(see figure 9a). For the three choices
of time scale, the drift frequency decreases with increasing Pr for all Pr as opposed to
the scaling of δ0/H, which has different scaling for small and large Pr (see figures 9cand
12b).
As reported in Zhang et al. (2020)andshownhereinfigure 3, the thermal structures drift
anticyclonically, opposite to the azimuthal velocity, which is cyclonic near the sidewall,
as shown in figure 2(bd). We show in figure 11(a) that the drift frequency decreases
as rotation increases with a scaling Ek2/3.Infigure 11(b), we show that the near-plate
azimuthal velocity upeak
φis also anticyclonic and shows the same scaling behaviour with
Ek (see figure 10b) as the BZF width and drift frequency. Based on this observation, we
believe that the drift characteristics of the BZF are determined not only by the presence of
the vertical wall but also by the near-plate region.
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X. Zhang, R. E. Ecke and O. Shishkina
10−6 10−5 10−6 10−5
0.04
0.08
0.12
0.16
0.20
Ek
2/3
Ek
2/3
Ek
0.01
0.02
0.03
0.04
0.05
Ek
uφ
peak/
αgΔH
ω/αgΔ/H
(b)(a)
Figure 11. For fixed Ra =109and rotation rates, 1/Ro =5.6, 6.7, 8.3, 10, 12.5, 16.7 and 20: (a)drift
frequency ωof BZF;and (b)maximum absolute value of uφnear plates (mean value of two maxima). In
both panels Pr =0.8and Γ=1/2.
10–8 10–7 10–6 10–5 10–4
107
108
109
1010
No
convection
Wall
modes
Geostrophic
bulk
Ekw
Ekc
Ekt
Ek
Ra
10–4
10–6
10–8
10–1
10–2
δ
0
min
Ek
1/3
δ0 Ra1/4 Ek2/3
δ
urms
Ek
1/3
δ
0
max
Ek
1/6
Ekt
Ekc
Ekw
No convection
Buoyancy - dominated
Wall
modes
Geostrophic
Ek
δ/H
(b)(a)
z
Figure 12. (a)The RaPr phase diagram. Different rotating convective states are labelled. The dashed
horizontal line corresponds to Ra =109. Critical Ek values Ekw,Ekcand Ektare indicated. The Ek data (solid
circles) correspond to values in (b).(b)Widths of BZF δ0and Stewartson Ek1/3layer δrms
uzat Ra =109
(DNS) versus Ek. Vertical dashed lines (black, blue and red,respectively) are the critical Ekman numbers for
onset of wall modes (Ekw), onset of bulk convection (Chandrasekhar 1961; Niiler & Bisshopp 1965)(Ekc), and
transition to rotation-dominated regimes (Ekt)forPr =0.8, Ra =10 9and Γ=1/2.
Finally, we consider the range of Ra and Ek in which the BZF is observed in this
study. There are three regions defined by the onset of wall mode convection Raw
32Ek1, the onset of bulk convection Rac=AEk4/3and the transition from geostrophic
convection (Grooms et al. 2010; Julien et al. 2012)tobuoyancy-dominated convection
Rat=Pr Ro2
tEk2, where Rot1(seefigure 13 in the Appendix) is the transition Rossby
number out of the geostrophic regime (Julien et al. 1996;Kinget al. 2009;Liu&Ecke
2009; Weiss & Ahlers 2011b) as indicated in the RaEk phase diagram in figure 12(a).
Our data fall solely within the geostrophic regime of bulk convection, but we include the
other zones for context. According to Chandrasekhar (1961) (see also Clune & Knobloch
1993), the critical Rayleigh number for the onset of convection is RacEk4/3with a
prefactor Athat is weakly dependent on Ek, in the range 6–8.7 (Chandrasekhar 1961; Niiler
& Bisshopp 1965); we use a value of 7.5 consistent with our range of Ek.
To illustrate one aspect of this range, we consider the BZF width δ0/Hversus
Ek for Ra =109in figure 12(b). A path of constant Ra =109(figure 12a)
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Boundary zonal flows in rotating turbulent convection
10–3 10–2 10–1 100101103
102
0.4
0.6
0.8
1.0
1.2
Pr Ro2
Nu/Nu0
DNS Ra = 1 × 108(SF6)
DNS Ra = 1 × 109(SF6)
Exp. Ra = 5 × 109(N2)
Exp. Ra = 7 × 109(N2)
Exp. Ra = 1 × 1010(N2)
Exp. Ra = 2 × 1010(N2)
Exp. Ra = 8 × 1012(SF6)
Exp. Ra = 2 × 1013(SF6)
Exp. Ra = 5 × 1013(SF6)
Exp. Ra = 8 × 1013(SF6)
Exp. Ra = 1 × 1014(SF6)
Exp. Ra = 8 × 1014(SF6)
Figure 13. Double-logarithmicscaleplotofNu/Nu0versus PrRo2.The horizontal line indicates Nu/Nu0=1;
the vertical dashed line indicates the value Rot, i.e. a transition between buoyancy-dominated convection at
larger Ro (Nu Nu0)andtherotation-dominated regime at smaller Ro (Nu <Nu0). Experimental data are
from Wedi et al. (2021).
yields Ekw32Ra1=3.2×108,Ekc=(ARa
1)3/4=8×107and Ekt=RotPr1/2
Ra1/2=2.8×105. Here the subscripts ‘w’, ‘c’and‘t’ correspond, respectively, to the
onset of wall mode, bulk convection and transition from rotation to buoyancy-dominated
regime. These values are indicated by vertical dashed lines in figure 12(b). Knowing the
dependence of the critical Racand Ek, and using the relations (3.7)and (3.8), we can
evaluate the smallest possible δ0for any fixed Ek,i.e.δmin
0Ra1/4
cEk2/3Ek1/3(see
δmin
0in figure 12b). Connecting these onset points, we obtain the black line in the diagram,
which is parallel to the Stewartson ‘1/3’ layer scaling. The gap between these two black
solid lines depends slightly on A, but the ratio of the BZF width to the Stewartson layer
width is constant (based on δrms
uz) at the onset of convection (the fixed gap). Thus, although
the BZF width decreases faster than the Stewartson layer as rotation increases, there is
no crossing of the BZF boundary and the boundary of the Stewartson layer at extremely
fast rotation because bulk convection ceases before they can cross. Note that all the data
considered here fall within the geostrophic range of rotating convection; what happens in
the wall mode region is not addressed.
The other bound on the BZF scaling depends on when rotation becomes significant.
An estimate is made based on a plot of Nu/Nu0versus Pr Ro2for our DNS and for
experimental data from Wedi et al. (2021)(see figure 13 in the Appendix), where
thedataforRa from 108to 1014 merge onto a single curve. Here Nu0is the Nusselt
number in the non-rotating case. Using an empirical estimate Rot1 for the onset of the
rotation-dominated regimes, i.e. the geostrophic regime, we get an estimate for the largest
possible δ0,foranyEk (grey line in figure 12, that is, δmax
0Ra1/4
tEk2/3Ro1/2
tEk1/6
Ek1/6). (The value Pr Ro2
t1 is the onset in figure 13, but in the experiments Pr varies
from 0.7 to 0.9 and in the DNS Pr =0.8; thus here we take Pr =1, which gives Rot1,
for simplicity.)
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X. Zhang, R. E. Ecke and O. Shishkina
It is remarkable that the BZF regime is confined by these two critical lines (Ek1/6and
Ek1/3) and the range confined in between gets broader for higher Ra.Inotherwords,
at low Ra, the BZF is only observed over a small range of rotation rates. At large Ra,
the BZF exists over a much broader range of rotation rates (Zhang et al. 2020;Wedi
et al. 2021). For any fixed Ra, the BZF exists in a certain Ek range that is determined
by the grey and black lines in figure 12(b) and the BZF width changes as δ0Ek2/3
over that range. How the BZF contributes to the heat transport relative to the contribution
of the laterally unbounded system in the geostrophic regime remains an open question.
Further, the connection between the BZF and linear wall modes requires additional work
to understand the relationship between the two convective states.
4. Conclusion
The BZF is found to be an important flow structure in rapidly rotating turbulent
Rayleigh–Bénard convection in the geostrophic regime and is robust over considerable
ranges of Ra,Ek,Pr and Γ. The main structure, drift of plume pairs, is found to be a
m=2Γmode for the choices of Γ=1/2, 1and 2; additional values of Γ=1/3and 3/4
yield m=1, suggesting mode 1 for Γ1. In addition, the BZF carries a large portion of
the total heat; its contribution to the total heat transport is approximately 60 % of the heat
transport at fast rotation, Ro <0.1, and for Pr <1. For Pr >1, the BZF heat transport
contribution drops to approximately 35%. Understanding this important contribution to
the heat transport is essential in analysing experiments in rotating convection in the
geostrophic regime.
The scaling of the BZF width δ0depends on Pr,Ra and Ek as δ0/H
Γ0Pr{−1/4,0}Ra1/4Ek2/3(that is, Pr1/4for small-to-moderate Pr and independent of
Pr for large Pr). The universal scaling of the BZF and the sidewall BLsisvery
clean for Pr <1 but the BZF is less coherent for Pr >1 and the sidewall BL widths
behave differently for those conditions. Further, the sharp decrease in the BZF heat
transport contribution similarly marks a transition to a perhaps more complex BZF state
for Pr >1. The drift frequency of the BZF shows scaling ω/Ω Γ0Pr4/3Ra Ek5/3,
indicating that the drift frequency decreases significantly as Pr increases, is proportional
to Ra and decreases rapidly with increasing rotation (decreasing Ek). Interestingly,
ωseems to be more robust than δwith respect to changes in Pr. Finally, the BZF
shares qualitative and some quantitative characteristics with linear wall modes,and
establishing the connection between these two states will be an important area of future
research.
Acknowledgements. The authors would like to thank E. Bodenschatz, D. Lohse, M. Wedi and S. Weiss for
fruitful discussions, cooperation and support. The authors acknowledge the Leibniz Supercomputing Centre
(LRZ) for providing computing time.
Funding. This work was supported by the Deutsche Forschungsgemeinschaft (X.Z. and O.S., grant number
Sh405/8, Sh405/7 (SPP 1881 Turbulent ‘Superstructures’)); and the Los Alamos National Laboratory LDRD
program under the auspices of the US Department of Energy (R.E.E.).
Declaration of interests. The authors report no conflict of interest.
Author ORCIDs.
Xuan Zhang https://orcid.org/0000-0001-5684-3064;
Robert E. Ecke https://orcid.org/0000-0001-7772-5876;
Olga Shishkina https://orcid.org/0000-0002-6773-6464.
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Boundary zonal flows in rotating turbulent convection
ΓPr Ra 1/Ro tavgff NrNφNzNth NuNsw
uδth/Hδu/Hδsw
u/Hmax(hk)
1/2 0.85.0×10710 440 90 256 380 43 49 16 3.5×1024.5×1022.3×1020.63
1.0×10810 500 100 256 380 39 50 17 3.0×1024.9×1022.0×1020.79
5.0×10810 370 128 320 620 46 89 17 1.6×1025.4×1021.4×1020.67
1.0×10910 1130 192 512 820 37 101 18 1.2×1025.3×1021.3×1020.85
5.0×10910 200 256 620 860 39 118 27 7.3×1035.1×1029.4×1031.24
1/2 0.11.0×10810 750 256 256 512 70 60 33 7.3×1025.7×1021.2×1021.10
0.25 10 80 100 256 380 39 50 17 5.1×1025.2×1021.9×1020.77
0.5 10 250 100 256 380 39 50 17 4.2×1024.9×1021.7×1020.92
0.8 10 500 100 256 380 39 50 17 3.0×1024.9×1022.0×1020.79
1 10 450 100 256 380 37 50 17 2.8×1024.8×1022.0×1020.72
2 10 500 100 256 380 29 53 22 1.8×1025.3×1022.9×1020.80
3 10 600 100 256 380 26 53 35 1.5×1025.2×1023.4×1020.83
4.38 10 430 100 256 380 24 50 16 1.3×1024.8×1021.6×1010.86
7 10 540 100 256 380 24 46 49 1.3×1024.2×1028.9×1020.88
12.3 10 420 100 256 380 23 41 54 1.2×1023.3×1021.0×1010.88
1/3 0.81.0×10810 810 96 256 320 28 34 22 2.8×1023.8×1012.0×1020.59
1/2 10 500 100 256 380 39 50 17 3.0×1024.8×1022.0×1020.79
3/4 10 830 128 256 380 38 53 17 3.4×1025.8×1012.0×1020.91
1 10 1480 180 320 320 22 36 15 3.5×1026.5×1022.0×1020.90
2 10 1500 256 320 320 25 46 14 4.0×1029.0×1022.1×1021.24
Table 2. For caption see next page.
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X. Zhang, R. E. Ecke and O. Shishkina
ΓPr Ra 1/Ro tavgff NrNφNzNth NuNsw
uδth/Hδu/Hδsw
u/Hmax(hk)
1/2 0.81.0×1090.5 965 192 512 768 14 78 42 7.8×1035.6×1023.4×1020.94
2 1020 192 512 768 14 87 28 7.8×1036.5×1022.1×1020.94
2.5 1410 192 512 768 14 88 28 7.7×1036.6×1022.1×1020.94
3.3 1400 192 512 768 15 90 26 8.0×1036.8×1022.0×1020.94
5 1630 192 512 768 16 88 22 8.8×1036.5×1021.6×1020.92
5.6 480 180 280 680 34 105 25 9.0×10 36.5×1021.6×1021.13
6.7 480 180 280 680 35 102 23 9.8×10 36.3×1021.4×1021.10
8.3 500 180 280 680 38 99 23 1.1×1025.8×10 21.4×1021.07
10 1130 192 512 820 37 101 18 1.2×1025.3×1021.3×1020.85
12.5 400 128 320 620 45 83 15 1.5×1024.8×1021.2×1021.20
16.7 400 128 320 620 54 73 13 2.1×1023.8×1021.1×1021.10
20 400 128 320 620 63 65 12 2.8×1023.0×1029.5×1031.03
28.6 40 128 320 620 83 49 12 4.8×1021.8×1029.5×1030.90
33.3 37 128 320 620 86 48 11 5.0×1021.7×1028.5×1030.89
Table 2 (cntd). Details of the DNS: including the time of statistical averaging, tavg, normalized with the free-fall time τff ; the number of nodes Nr,Nφand Nzin the
directions r,φand z, respectively; the numbers of nodes within the thermal BL Nth (near the plates), within the viscous BL Nu(near the plates), and within the viscous BL
Nsw
u(near the sidewall); the relative thicknesses of the viscous BL δu/Hand the thermal BL near the plates δth/H, and the viscous BL near the sidewall δsw
u/H;andthe
maximal value of the ratio of the mesh size to the mean Kolmogorov microscale, max(hk).
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Boundary zonal flows in rotating turbulent convection
ΓPr Ra Ek 1/Ro ωκωff ωdω
d
1/2 0.85.0×1071.3×10510 6.3×1011.0×1021.3×1021.9×1010
1.0×1088.9×10610 1.3×1021.5×1021.8×1021.4×1010
5.0×1084.0×10610 3.3×1021.7×1022.1×1023.1×1011
1.0×1092.8×10610 5.7×1022.0×1022.5×1021.9×1011
5.0×1091.3×10610 1.7×1032.6×1023.3×1024.9×1012
1/2 0.11.0×1083.2×10610 1.1×1023.3×1024.2×1021.9×1011
0.25 5.0×10610 1.0×1022.0×1022.5×1024.0×1011
0.57.1×10610 1.2×1021.7×1022.1×1028.3×1011
0.88.9×10510 1.3×1021.5×1021.8×1021.4×1010
1.01.0×10510 9.1×1019.1×1031.1×1021.1×1010
2.01.4×10510 1.1×1028.0×1031.0×1022.5×1010
3.01.7×10510 1.2×1026.7×1038.4×1033.6×1010
4.38 2.1×10510 1.0×1025.0×1036.3×1034.5×1010
7.02.6×10510 1.2×1024.5×1035.7×1037.6×1010
12.33.5×10510 1.4×1024.0×1035.0×1031.4×109
1/2 0.81.0×1088.9×10610 1.3×1021.5×1021.8×1021.4×1010
1101.4×1021.6×1022.0×1021.5×1010
2101.4×1021.6×1022.0×1021.5×1010
1/2 0.81.0×1095.1×1065.67.1×1022.5×1025.7×1024.2×1011
4.2×1066.76.6×1022.3×1024.4×1023.3×1011
3.4×1068.36.3×1022.2×1023.4×1022.5×1011
2.8×10610 5.7×1022.0×1022.5×1021.9×1011
2.3×10612.53.8×1021.3×1021.3×1021.0×1011
1.7×10616.73.1×1021.1×1028.4×1036.2×1012
1.4×10620 2.6×1029.1×1035.7×1034.2×1012
9.9×10728.62.2×1027.7×1033.4×1032.5×1013
8.5×10733.31.7×1025.9×1032.2×1031.6×1013
Table 3. Values of Γ,Ra,Pr,Ek,Ro1,ωκ=ωH2,ωff =ω(H/(gαΔ))1/2,ωd=ω/Ω and
ω
d=ωdPr4/3Ra1.(ForΓ=1/3 and 3/4, there are insufficient data to determine ω.)
Appendix
We tabulate here a full characterization of the parameters in the DNS (see table 2)and
compare our results for Nu with experimental data from Wedi et al. (2021). The excellent
agreement is a strong indication that our DNS are fully resolved. We also include the
resulting numerical values of the BZF drift frequency for different choices of time scale
(see table 3), namely ωκ,ωff and ωdfor different parameter values Γ,Pr,Ra,Ek and
1/Ro.
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... We can remark that the onset of wall modes prior to bulk modes is similar to a certain extent to the occurrence of the wall modes in rotational Rayleigh-Bénard convection prior to the onset of bulk convection (see [78][79][80][81][82][83][84][85][86][87][88][89]). When a cylindrical Rayleigh-Bénard cell is rapidly rotated along its axis, convection is certainly suppressed, and larger, compared to the nonrotating case, Rayleigh numbers should be achieved in order to enforce a fluid motion. ...
... Based on the results of (almost) Oberbeck-Boussinesq experiments and direct numerical simulations of Rayleigh-Bénard convection in cylindrical containers [59,87,89,107,, let us now illustrate that the derived relevant length scale in Rayleigh-Bénard convection is , Eq. (125), and the corresponding Rayleigh number is Ra , Eq. (134). For cylindrical Rayleigh-Bénard convection cells, c u is defined by Eq. (103), which in combination with Eq. (135) gives the relevant scaling quantity ...
... In Fig. 14 we present the results [59,87,89,107, for fluids with Prandtl numbers ranging from 0.7 to about 6, which correspond to the most popular fluids of air and water at room temperature. The range of the considered aspect ratios of the cylindrical Rayleigh-Bénard cell used in these experiments is very broad, from 1/32 to 32. Figure 14(a) shows the dependence of the compensated Nusselt number, (Nu − 1) / Ra 1/3 , on the Rayleigh number Ra. ...