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For rapidly rotating turbulent Rayleigh--B\'enard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like $Ra^{1/4}Ek^{2/3}$ where the Ekman number $Ek$ decreases with increasing rotation rate.
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arXiv:1911.09584v1 [physics.flu-dyn] 21 Nov 2019
Boundary Zonal Flow in Rotating Turbulent Rayleigh–B´enard Convection
Xuan Zhang1,Dennis P. M. van Gils1,2,Susanne Horn1,3,4, Marcel Wedi1, Lukas Zwirner1,
Guenter Ahlers1,5, Robert E. Ecke1,6, Stephan Weiss1,7, Eberhard Bodenschatz1,8,9, and Olga Shishkina1
1Max Planck Institute for Dynamics and Self-Organization, 37077 G¨ottingen, Germany
2Physics Fluids Group, J.M. Burgers Center for Fluid Dynamics,
University of Twente, 7500 AE Enschede, The Netherlands
3Department of Earth, Planetary, and Space Sciences, UCLA, CA 90095, USA
4Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, UK
5Department of Physics, University of California, Santa Barbara, CA 93106, USA
6Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
7Max Planck – University of Twente Center for Complex Fluid Dynamics
8Institute for Nonlinear Dynamics, Georg-August-University ottingen, 37073 G¨ottingen, Germany and
9Laboratory of Atomic and Solid-State Physics and Sibley School of Mechanical
and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
(Dated: November 22, 2019)
For rapidly rotating turbulent Rayleigh–B´enard convection in a slender cylindrical cell, experi-
ments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical
large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat
transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the
temperature is an anticyclonic traveling wave of mode one whose signature is a bimodal temperature
distribution near the radial boundary. The BZF width is found to scale like Ra1/4Ek 2/3where the
Ekman number Ek decreases with increasing rotation rate.
Turbulent fluid motion driven by buoyancy and influ-
enced by rotation is a common phenomenon in nature
and is important in many industrial applications. In the
widely studied laboratory realization of turbulent convec-
tion, Rayleigh–B´enard convection (RBC) [1, 2], a fluid
is confined in a convection cell with a heated bottom,
cooled top, and adiabatic vertical walls. For these condi-
tions, a large scale circulation (LSC) arises from cooper-
ative plume motion and is an important feature of turbu-
lent RBC [1]. The addition of rotation about a vertical
axis produces a different type of convection as thermal
plumes are transformed into thermal vortices, over some
regions of parameter space heat transport is enhanced
by Ekman pumping [3–10], and statistical measures of
vorticity and temperature fluctuations in the bulk are
strongly influenced [11–17]. A crucial aspect of rotation
is to suppress, for sufficiently rapid rotation rates, the
LSC of non-rotating convection [12, 13, 18, 19], although
the diameter-to-height aspect ratio Γ = D/H appears to
play some role in the nature of the suppression [20].
In RBC geometries with 1/2Γ2, the LSC usu-
ally spans the cell in a roll-like circulation of size H.
For rotating convection, the intrinsic linear scale of sep-
aration of vortices is reduced with increasing rotation
rate [21, 22], suggesting that one might reduce the geo-
metric aspect ratio, i.e., Γ <1 while maintaining a large
ratio of lateral cell size to linear scale [5]; such convec-
tion cells are being implemented in numerous new exper-
iments [23]. Thus, an important question about rotating
convection in slender cylindrical cells is whether there is a
global circulation that substantially influences the inter-
nal state of the system and carries appreciable global heat
transport. Direct numerical simulations (DNS) of rotat-
ing convection [24] in cylindrical geometry with Γ = 1,
inverse Rossby number 1/Ro = 2.78, Rayleigh number
Ra = 109and Prandtl number Pr = 6.4 (Ro,Ra and
Pr defined below) revealed a cyclonic azimuthal velocity
boundary-layer flow surrounding a core region of anti-
cyclonic circulation and localized near the cylinder side-
wall. The results were interpreted in the context of side-
wall Stewartson layers driven by active Ekman layers at
the top and bottom of the cell [25, 26].
Here we show through DNS and experimental mea-
surements for a cylindrical convection cell with Γ = 1/2
at large Ra and for a range of rotation rates from slow to
rapid that a wider (several times the Stewartson layer
width) annular flow, denoted a boundary zonal flow
(BZF), has profound effects on the overall flow structure
and on the spatial distribution of heat flux. In partic-
ular, this cyclonic zonal flow surrounds an anti-cyclonic
core. The BZF has alternating temperature sheets that
produce bimodal temperature distributions for radial po-
sitions r/R > 0.7 and that contribute greatly to the
overall heat transport; 60% of heat transport are carried
in the BZF. Although the location of the azimuthally-
averaged maximum cyclonic azimuthal velocity, the root-
mean-square (rms) vertical velocity fluctuations, and the
normalized vertical heat transport at the mid-plane are
consistent with a linear description of a Stewartson-layer
scaling [24], the dynamics of temperature, vertical ve-
locity and heat transport in the BZF are more complex
and interesting. The robustness of the BZF state as evi-
denced by its existence over 7 orders of magnitude in Ra
in DNS and experiment and over a range 1/2Γ2
and 0.1Pr 4.4 (results to be presented elsewhere)
suggests that this is an universal state of rotating con-
FIG. 1. Sidewall temperature PDFs, r/R = 1, for z/H = 1/4 (diamonds), z/H = 1/2 (circles), and z/H = 3/4 (squares), with
1/Ro = 0 (a, c) and 10 (b, d). Experimental measurements with Ra = 8 ×1012 (a, b) and DNS with Ra = 109(c, d), both
with Pr = 0.8. Bimodal Gaussian distributions (solid lines), the sum of two normal distributions (dashed lines), are observed
for rapid rotation (b, d). h·izsdenotes average in time and over all sensor positions at distance zfrom the hot plate.
FIG. 2. Horizontal cross-sections of time-averaged flow fields
(DNS), visualized with streamlines (arrows) and azimuthal
velocity huφit(colors) (a, b) at height of thermal BL, z=δθ
H/(2Nu) and (c, d) at the mid-plane, z=H/2, with Ra =
109and 1/Ro = 0.5 (a, c) and 10 (b, d). Blue (pink) color
indicates anticyclonic (cyclonic) motion. In (d), locations r=
r0of huφit= 0 (solid line) and r=rumax
φof the maximum of
huφit(dashed line) are shown.
vection that needs a physical understanding.
The dimensionless control parameters in rotating RBC
are the Rayleigh number Ra =αgH3/(κν), Prandtl
number Pr =ν/κ, cell aspect ratio Γ and Rossby number
Ro =αgH / (2ΩH) or, alternatively, Ekman number
Ek =ν/(2ΩH2). Here αis isobaric thermal expansion
coefficient, νkinematic viscosity, κfluid thermal diffu-
sivity, gacceleration of gravity, Ω angular rotation rate,
and ∆ temperature difference between the hotter bottom
and colder top plates. The main integral response param-
eters we consider is the Nusselt number Nu ≡ hFzit,V ,
where h·it,V denotes the time- and volume-averaging and
Fz(uz(TT0)κ∂zT)/(κ/H) is the normalized
vertical heat flux with uzbeing the vertical component
of the velocity and T0the average of the top and bottom
We present numerical and experimental results [27]
for rotating RBC in a Γ = 1/2 cylindrical cell and
1/Ro = 0, 0.5 and 10. The DNS used the goldfish code
[28, 29] with Pr = 0.8 and Ra = 109. The experiments
used pressurized sulfur hexafluoride (SF6) and were per-
formed over a large parameter space in the High Pressure
Convection Facility (HPCF, 2.24 m high) at the Max
Planck Institute for Dynamics and Self-Organization in
ottingen [30]. In the studied parameter range, the
Oberbeck–Boussinesq approximation is valid [31–33], and
the centrifugal force is negligible [8, 34, 35].
We first consider the azimuthal variation of the tem-
perature measured by thermal probes at or near the
sidewall, a commonly used technique for parameteriz-
ing the LSC in RBC [18, 20, 36–38]. We measured
experimentally and in corresponding DNS the tempera-
ture at 8 equidistantly spaced azimuthal locations of the
sensors for each of 3 distances from the bottom plate:
z/H = 1/4, 1/2 and 3/4. The PDFs of the experimen-
tal data without rotation (1/Ro = 0, Ra = 8 ×1012) in
Fig. 1a show a distribution with a single peak and slight
asymmetry to hotter (colder) fluctuations for heights
smaller (larger) than z/H = 1/2, whereas the PDFs for
rapid rotation (1/Ro = 10, Fig. 1b), show a bimodal
distribution that is well fit by the sum of two Gaussian
distributions. The corresponding PDFs of the DNS data
(at Ra = 109) show the same qualitative transition from
a single peak without rotation to a bimodal distribution
in the rapidly rotating case with similar hot/cold asym-
metry for different z(Fig. 1c, d). To understand the
nature of the emergence of a bimodal distribution near
the radial boundary, we consider the DNS data in detail.
The LSC for non-rotating convection in cells with
1/2Γ2 and at large Ra extends throughout the
entire cell with a large roll-like circulation [39]. With
slow rotation, Coriolis forces induce anticyclonic motion
close to the plates owing to the diverging flow between the
FIG. 3. For Ra = 109, 1/Ro = 10: (a)huφit,φ versus zat r= 0.95R. The inset shows the same data for 0 zH/2 in a
log-plot. (b)huφit,φ versus rat z=H/2; radial zero crossing r=r0(solid line) and radial maximum r=rumax
φ(dashed line).
(c) Instantaneous thermal field at r=rumax
φversus zand φ.
LSC and the corner rolls. At the mid-plane, the LSC is
tilted with a small inward radial velocity component that
rotation spins up into cyclonic motion. These tendencies
are illustrated for 1/Ro = 0.5 in Figs. 2a, c, respectively,
where streamlines of time-averaged velocity are overlaid
on the field of azimuthal velocity. Fig. 2a shows fields
evaluated at the thermal BL height z=δθH/(2N u),
demonstrating the dominant anticyclonic flow near the
boundary. The situation is reversed at the mid-plane
(Fig. 2c) where cyclonic motion extends over almost the
entire cross sectional area.
For rapid rotation, viscous Ekman BLs near the plates
induce anticyclonic circulation with radial outflow in hor-
izontal planes as in Fig. 2b. The outflow is balanced by
the vertical velocity in an increasingly thin (with increas-
ing 1/Ro) annular region near the sidewall where cyclonic
vorticity is concentrated at the mid-plane, see Fig. 2d.
The core region, on the other hand, is strongly anticy-
clonic owing to the Taylor-Proudman effect [40, 41] that
tends to homogenize vertical motion. The circulation for
a rotating flow in a finite cylindrical container consists of
thin anticyclonic Ekman layers on top and bottom plates
and compensating Stewartson layers along the sidewalls
with up-flow from the bottom and down-flow from the
top [24, 42]. This classical BL analysis was successfully
applied to rotating convection [24] for a Γ = 1 cylindrical
cell with Pr = 6.4 and 108Ra 109in both exper-
iment and DNS. No evidence for a coherent large-scale
circulation for rapid rotation was found in those studies.
For our conditions, Pr = 0.8, Ra = 109, and 1/Ro =
10, we compute the time- and azimuthal-average az-
imuthal velocity huφit,φ (normalized by the free-fall ve-
locity u=pαg/R) as a function of height zfor fixed
r= 0.95Rand of radius rat fixed z=H/2. The height
dependence of huφit,φ, Fig. 3a, shows an anticyclonic
(negative) circulation close to the top and bottom plates
and an increasingly cyclonic (positive) circulation with
increasing (decreasing) zfrom the bottom (top) plate.
The radial dependence, Fig. 3b, demonstrates the sharp
localization of cyclonic motion near the sidewall as pa-
rameterized by the zero-crossing r0(solid line) and the
maximum rumax
φ(dashed line). Corresponding distances
from the sidewall are δ0=Rr0and δumax
where δumax
z(based on maximum of rms of uz).
zwas used to define the sidewall Stewartson layer
thickness in rotating convection [24], and our results for
huφitare consistent with that description. What was ab-
solutely not expected is the strong azimuthal variation
of the instantaneous temperature Tshown in Fig. 3c, a
feature that defines the global flow circulation, namely
the spatial distribution of the heat transport which is
the origin of the bimodal temperature distributions seen
in the experiments and DNS.
The strong variations in instantaneous temperature
shown in Fig. 3c organize into anticyclonic traveling
waves illustrated in the angle-time plot of T, Fig. 4a.
The BZF height is order H, Fig. 3c, but is increasingly
localized in the radial direction as the rotation rate in-
creases (Ro and Ek decrease) so that δ0/R 1. The
azimuthal mode of Tis highly correlated with a corre-
sponding mode of the vertical velocity, Fig. 4b, with a
resulting coherent mode-1 (m= 1) anticyclonic circu-
lation in φwith a warm up-flow on one side of the cell
balanced by a cool down-flow on the other side of the cell
(for Γ = 1,2, the dimensionless wave number m/Γ = 2
is independent of Γ, to be presented elsewhere). The
anticyclonic circulation is the speed of the anticyclonic
horizontal BL, suggesting that the thermal wave is an-
chored at the horizontal BLs so that it travels against
the cyclonic circulation near the sidewall. The coherence
between Tand uzleads to localization of vertical heat
flux near the sidewall shown in Fig. 4c where the heat
flux within the annular area defined by δ0is 60% of the
total heat flux.
We arrive at a compact description of the BZF. The
radial distances from the sidewall δurms
z, and δumax
of maxima of uz-rms, heat flux Fz, and uφ, respectively,
scale as Ek1/3, Fig. 5a, consistent with the expectations
of Ekman/Stewartson BL theory [24, 42]. On the other
hand, the cyclonic zone width δ0decreases more rapidly
with Ek, i.e., as E k2/3with a Ra1/4dependence (pre-
sented elsewhere). Thus, the inner layer is consistent
FIG. 4. For Ra = 109and 1/Ro = 10: (a, b) angle-time plots
at r=rumax
φ,z=H/2 of (a)Tand (b)uz; (c) normalized
time-averaged vertical heat flux hFzitat z=H/2. In (c),
location of rwhere h Fz|z=H/2it=Nu (dash-dotted line) and
locations r=r0of huφit= 0 (solid line) and r=rumax
φof the
maximum of huφit(dashed line) are shown. Color scale from
blue (min values) to pink (max values) ranges (a) between
the top and bottom temperatures, (b) in [u/2, u/2], (c)
from 0 to mid-plane magnitude of hFzit, which is 3.4Nu.
with Stewartson theory whereas the outer structure re-
flects the more complex character of interacting thermal
and velocity fields. The bimodal temperature distribu-
tion is now explained by the alternating thermal field.
We plot the radial dependence of the mean values of the
bimodal distributions (the bimodal PDFs are well fit by
the sum of two Gaussians) from the DNS for Ra = 109,
1/Ro = 10 in Fig. 5b. The unimodal distribution for
small r/R bifurcates sharply to a bimodal distribution for
r/R 0.72. The corresponding experimental measure-
ments do not provide data at intermediate r/R, but are
consistent (dashed curve) with a scaled BZF width based
on the scaling Ra1/4Ek 2/3. Finally, the transition value
of 1/Ro 2 from unimodal to bimodal distributions is
roughly independent of Ra as indicated in Fig. 5c.
Our observations provide insight into experimental re-
sults for Γ = 1/2 in water with Pr = 4.38 [20], where the
mode-1 LSC for non-rotating convection was reported
to transition into a then unknown state. Our BZF is
that unknown global mode. We conclude that the BZF
exists over a broad range of parameters 1/2Γ2,
0.1Pr 4.4, and 108Ra <1015 (details to be pub-
lished elsewhere). Here we presented details for Pr = 0.8
and Γ = 1/2 and for Ra spanning seven orders of magni-
tude [27]. A fully quantitative understanding remains a
challenge for the future.
The authors acknowledge support from the Deutsche
Forschungsgemeinschaft (DFG) through the Collabora-
tive Research Centre SFB 963 ”Astrophysical Flow In-
stabilities and Turbulence” and research grants Sh405/4-
1, Sh405/4-2, Sh405/8-1, Ho5890/1-1 and We5011/3-1,
from the LDRD program at Los Alamos National Labo-
ratory and by the Leibniz Supercomputing Centre (LRZ).
These two authors contributed equally.
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Full-text available
We present measurements of temperature fluctuations in turbulent rotating Rayleigh-Bénard convection. The temperature variance exhibits power-law dependence on the fluid height outside the thermal boundary layers irrespective of the rotating rates. Rotations increase the magnitudes of temperature variance, but reduce the skewness and kurtosis, leading to Gaussian-like temperature distributions. We derive a general theoretical expression for all statistical moments of temperature in terms of the dynamical properties of the thermal plumes, based on the findings that both the amplitude and time width of thermal plumes are log-normally distributed. Our model replicates the statistical properties of the temperature fluctuations and reveals the physical origin of their rotation dependence. Rotations increase the temperature amplitude of thermal plumes by virtue of the Ekman pumping process, but reduce the variations of the plume amplitude in time, presumably through the suppression of turbulent mixing between the plumes and the ambient fluid.
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Many geophysical and astrophysical phenomena are driven by turbulent fluid dynamics, containing behaviors separated by tens of orders of magnitude in scale. While direct simulations have made large strides toward understanding geophysical systems, such models still inhabit modest ranges of the governing parameters that are difficult to extrapolate to planetary settings. The canonical problem of rotating Rayleigh-Bénard convection provides an alternate approach - isolating the fundamental physics in a reduced setting. Theoretical studies and asymptotically-reduced simulations in rotating convection have unveiled a variety of flow behaviors likely relevant to natural systems, but still inaccessible to direct simulation. In lieu of this, several new large-scale rotating convection devices have been designed to characterize such behaviors. It is essential to predict how this potential influx of new data will mesh with existing results. Surprisingly, a coherent framework of predictions for extreme rotating convection has not yet been elucidated. In this study, we combine asymptotic predictions, laboratory and numerical results, and experimental constraints to build a heuristic framework for cross-comparison between a broad range of rotating convection studies. We categorize the diverse field of existing predictions in the context of asymptotic flow regimes. We then consider the physical constraints that determine the points of intersection between flow behavior predictions and experimental accessibility. Applying this framework to several upcoming devices demonstrates that laboratory studies may soon be able to characterize geophysically-relevant flow regimes. These new data may transform our understanding of geophysical and astrophysical turbulence, and the conceptual framework developed herein should provide the theoretical infrastructure needed for meaningful discussion of these results.
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Computational codes for direct numerical simulations of Rayleigh–Bénard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at Ra=10⁸ and Pr=1 in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code (AFID/RBFLOW) and a specialized fourth-order finite-volume code (GOLDFISH) are compared with a general purpose finite-volume approach (OPENFOAM) and a general purpose spectral-element code (NEK5000). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID/RBFLOW and GOLDFISH are found to excel in efficiency, outperforming the general purpose flow solvers NEK5000 and OPENFOAM by an order of magnitude with an error on the Nusselt number Nu below 5%. However, we find that Nu alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using NEK5000 clearly indicate inadequate flow resolution even when Nu is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes.
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Rapidly rotating Rayleigh-B\'enard convection is studied by combining results from direct numerical simulations (DNS), laboratory experiments and asymptotic modeling. The asymptotic theory is shown to provide a good description of the bulk dynamics at low, but finite Rossby number. However, large deviations from the asymptotically predicted heat transfer scaling are found, with laboratory experiments and DNS consistently yielding much larger Nusselt numbers than expected. These deviations are traced down to dynamically active Ekman boundary layers, which are shown to play an integral part in controlling heat transfer even for Ekman numbers as small as $10^{-7}$. By adding an analytical parameterization of the Ekman transport to simulations using stress-free boundary conditions, we demonstrate that the heat transfer jumps from values broadly compatible with the asymptotic theory to states of strongly increased heat transfer, in good quantitative agreement with no-slip DNS and compatible with the experimental data. Finally, similarly to non-rotating convection, we find no single scaling behavior, but instead that multiple well-defined dynamical regimes exist in rapidly-rotating convection systems.
We critically analyse the different ways to evaluate the dependence of the Nusselt number ( $\mathit{Nu}$ ) on the Rayleigh number ( $\mathit{Ra}$ ) in measurements of the heat transport in turbulent Rayleigh–Bénard convection under general non-Oberbeck–Boussinesq conditions and show the sensitivity of this dependence to the choice of the reference temperature at which the fluid properties are evaluated. For the case when the fluid properties depend significantly on the temperature and any pressure dependence is insignificant we propose a method to estimate the centre temperature. The theoretical predictions show very good agreement with the Göttingen measurements by He et al. ( New J. Phys. , vol. 14, 2012, 063030). We further show too the values of the normalized heat transport $\mathit{Nu}/\mathit{Ra}^{1/3}$ are independent of whether they are evaluated in the whole convection cell or in the lower or upper part of the cell if the correct reference temperatures are used.
Centrifugal buoyancy affects all rotating turbulent convection phenomena, but is conventionally ignored in rotating convection studies. Here, we include centrifugal buoyancy to investigate what we call Coriolis-centrifugal convection (C3), characterizing two so far unexplored regimes, one where the flow is in quasicyclostrophic balance (QC regime) and another where the flow is in a triple balance between pressure gradient, Coriolis and centrifugal buoyancy forces (CC regime). The transition to centrifugally dominated dynamics occurs when the Froude number Fr equals the radius-to-height aspect ratio γ. Hence, turbulent convection experiments with small γ may encounter centrifugal effects at lower Fr than traditionally expected. Further, we show analytically that the direct effect of centrifugal buoyancy yields a reduction of the Nusselt number Nu. However, indirectly, it can cause a simultaneous increase of the viscous dissipation and thereby Nu through a change of the flow morphology. These direct and indirect effects yield a net Nu suppression in the CC regime and a net Nu enhancement in the QC regime. In addition, we demonstrate that C3 may provide a simplified, yet self-consistent, model system for tornadoes, hurricanes, and typhoons.
We propose a recipe to calculate accurately the Nusselt number Nu in turbulent Rayleigh-Bénard convection, using the measured total heat flux q and known parameters of the fluid and convection cell. More precisely, we present a method to compute the conductive heat flux q̂, which is a normalization of q in the definition of Nu, for conditions where the fluid parameters may vary strongly across the fluid layer. We show that in the Oberbeck-Boussinesq approximation and also when the thermal conductivity depends exclusively on the temperature, the value of q̂ is determined by simple explicit formulas. For a general non-Oberbeck-Boussinesq (NOB) case we propose an iterative procedure to compute q̂. Using our procedure, we critically analyze some already conducted and some hypothetical experiments and show how q̂ is influenced by the NOB effects.
We report a new thermal boundary layer equation for turbulent Rayleigh–Bénard convection for Prandtl number Pr>1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr≳1 and (b) Pr≫1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr=4.38 (water) and Pr=2547.9 (glycerol), respectively.
We consider rotating Rayleigh–Bénard convection of a fluid with a Prandtl number of \$\mathit{Pr}=0.8\$ in a cylindrical cell with an aspect ratio \${\it\Gamma}=1/2\$. Direct numerical simulations (DNS) were performed for the Rayleigh number range \$10^{5}\leqslant \mathit{Ra}\leqslant 10^{9}\$ and the inverse Rossby number range \$0\leqslant 1/\mathit{Ro}\leqslant 20\$. We propose a method to capture regime transitions based on the decomposition of the velocity field into toroidal and poloidal parts. We identify four different regimes. First, a buoyancy-dominated regime occurring while the toroidal energy \$e_{tor}\$ is not affected by rotation and remains equal to that in the non-rotating case, \$e_{tor}^{0}\$. Second, a rotation-influenced regime, starting at rotation rates where \$e_{tor}>e_{tor}^{0}\$ and ending at a critical inverse Rossby number \$1/\mathit{Ro}_{cr}\$ that is determined by the balance of the toroidal and poloidal energy, \$e_{tor}=e_{pol}\$. Third, a rotation-dominated regime, where the toroidal energy \$e_{tor}\$ is larger than both \$e_{pol}\$ and \$e_{tor}^{0}\$. Fourth, a geostrophic regime for high rotation rates where the toroidal energy drops below the value for non-rotating convection.