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Experimental observation of the geostrophic turbulence regime of rapidly rotating convection

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The competition between turbulent convection and global rotation in planetary and stellar interiors governs the transport of heat and tracers, as well as magnetic-field generation. These objects operate in dynamical regimes ranging from weakly rotating convection to the `geostrophic turbulence' regime of rapidly rotating convection. However, the latter regime has remained elusive in the laboratory, despite a worldwide effort to design ever-taller rotating convection cells over the last decade. Building on a recent experimental approach where convection is driven radiatively, we report heat transport measurements in quantitative agreement with this scaling regime, the experimental scaling-law being validated against direct numerical simulations (DNS) of the idealized setup. The scaling exponent from both experiments and DNS agrees well with the geostrophic turbulence prediction. The prefactor of the scaling-law is greater than the one diagnosed in previous idealized numerical studies, pointing to an unexpected sensitivity of the heat transport efficiency to the precise distribution of heat sources and sinks, which greatly varies from planets to stars.
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arXiv:2203.13860v1 [physics.flu-dyn] 25 Mar 2022
Experimental observation of the geostrophic
turbulence regime of rapidly rotating convection
Vincent Bouillauta, Benjamin Miquela, Keith Julienb, Sébastien Aumaîtrea, and Basile Galleta,1
aUniversité Paris-Saclay, CNRS, CEA, Service de Physique de l’Etat Condensé, 91191 Gif-sur-Yvette, France.; bDepartment of Applied Mathematics, University of Colorado,
Boulder, Colorado 80309, USA.
This manuscript was compiled on March 29, 2022
The competition between turbulent convection and global rotation
in planetary and stellar interiors governs the transport of heat and
tracers, as well as magnetic-field generation. These objects oper-
ate in dynamical regimes ranging from weakly rotating convection
to the ‘geostrophic turbulence’ regime of r apidly rotating convection.
However, the latter regime h as remained elu sive in the laboratory, de-
spite a worldwide effort to design ever-taller rotating convection cells
over the last decade. Building on a recent experimental approach
where convection is driven radiatively, we report heat transport mea-
surements in quantitative agreement with this scaling regime, the ex-
perimental scaling-law being validated against direct numerical sim-
ulations (DNS) of the idealized setup. The scaling exponent from
both experiments and DNS agrees well with the geostrophic turbu-
lence prediction. The prefactor of the scaling-law is greater than the
one diagnosed in previous idealized numerical studies, pointing to
an unexpected sensitivity of the heat transport efficiency to the pre-
cise distri bution of heat sou rces and sinks, which greatly varies from
planets to stars.
Turbulent convection |Geophysical and Astrophysical fluid dynamics |Rotating flows
The strong buoyancy gradients inside planets and stars drive tur-
bulent convective flows that are responsible for the efficient
transport of heat and tracers, as well as for the generation of the mag-
netic fields of these objects through the dynamo effect. This thermal
and/or compositional driving competes with the global rotation of the
astrophysical object: while moderate global rotation only affects the
largest flow structures (13), rapid global rotation greatly impedes
radial motion through the action of the Coriolis force, thereby re-
stricting the convective heat transfer (4,5). Because astrophysical
and geophysical flows operate at extreme parameter values, beyond
what will ever be achieved in laboratory experiments and numerical
simulations, the characterization of these highly complex flows pro-
ceeds through the experimental or numerical determination of the
constitutive equation, or scaling-law, that relates the turbulent heat
flux to the internal temperature gradients. Extrapolating this scaling-
law to the extreme parameter values of astrophysical objects sets the
effective transport coefficients, the turbulent energy dissipation rate,
the mixing efficiency and the power available to induce magnetic
field (4,611).
Within the Boussinesq approximation (12) and adopting a local
Cartesian geometry, the scaling-laws are cast in terms of the dimen-
sionless parameters that govern the system: the flux-based Rayleigh
number RaP=αgP H 4/ρCκ2νquantifies the strength of the heat
flux P, where Hdenotes the height of the fluid domain, αthe coef-
ficient of thermal expansion, gthe acceleration of gravity, κthe ther-
mal diffusivity, νthe kinematic viscosity, ρthe mean density and C
the specific heat capacity. The Nusselt number Nu = P H /ρCκT
measures the heat transport efficiency of the turbulent flow, as com-
pared to that of a steady motionless fluid, in terms of the typical
vertical temperature drop T. Finally, the magnitude of the Coriolis
force can be quantified through the Ekman number E = ν/2ΩH2, a
low value of Ecorresponding to a rapid global rotation rate .
At the theoretical level, several arguments have been put forward
to predict the scaling-law for the heat transport efficiency of rota-
tionally constrained turbulent convection, as measured by the Nus-
selt number Nu. Central to these theories is the assumption that the
scaling relation between the turbulent heat flux and the internal tem-
perature gradient should not involve the tiny molecular diffusivities
κand ν. In the physics community, this assumption is sometimes
referred to as the existence of an ‘ultimate regime’ (13), while in the
astrophysical community it is often referred to as the ‘mixing-length’
regime, because the latter theory neglects molecular diffusivities at
the outset (6,14).
The second assumption is that the heat transport efficiency of the
flow depends only on the supercriticality of the system, i.e., on the
ratio of the Rayleigh number to the threshold Rayleigh number for
the emergence of thermal convection. This idea is put on firm analyt-
ical footing through careful asymptotic expansions of the equations
of thermal convection in the rapidly rotating limit (5,1517). When
combined, these two assumptions lead to the following scaling-law
for turbulent heat transport by rapidly rotating thermal convection
(see Ref. (4) for the initial derivation):
Nu = C × RaP
3/5E4/5Pr1/5,[1]
where Pr = ν/κ is the Prandtl number and Cis a dimension-
less prefactor. Equation [1] is referred to as the ‘geostrophic tur-
bulence’ scaling-law of rapidly rotating convection*. In terms of the
temperature-based Rayleigh number Ra = RaP/Nu, this scaling-
*Geostrophy refers to the large-scale balance between the Coriolis and pressure forces.
..
Significance Statement
Turbulent convection is the main process through which nature
moves fluids around, be it in deep planetary and stellar interi-
ors or in the external fluid layers of planets and their satellites.
Laboratory studies aim at reproducing the resulting fully turbu-
lent flows, with the goal of determining the effective transport
coefficients to be input into coarse geophysical or astrophysi-
cal models. Crucial to these applications is planetary or stel-
lar rotation, which competes with convective processes to set
the emergent transport properties. Building on a recent exper-
imental approach that bypasses the limitations of boundary-
forced convective flows, we report laboratory measurements
in quantitative agreement with the fully turbulent regime of ro-
tating convection.
1To whom correspondence should be addressed. E-mail: basile.gallet@cea.fr
www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX PNAS | March 29, 2022 | vol. XXX | no. XX | 1–7
DC motor
WIFI
microcontroller
water + dye
sapphire plate
water-cooled
thermal and IR
screens
metal-halide
spotlight
polyoxymethylene
diaphragm
T1
T2
P
Fig. 1. Radiatively driven rotating convection. A powerful spotlight shines from below at a mixture of water and dye. The resulting internal heat source decreases
exponentially with height over the absorption length , delivering a total heat flux P. The cylindrical tank is attached from above to a DC motor that imposes global rotation at
a rate (slight curvature of the top free surface not represented). Two thermocouples T1and T2measure the vertical temperature drop in the rotating frame, the data being
communicated through WIFI to a remote Arduino microcontroller. On the right-hand side is a DNS snapshot of the temperature field in horizontally periodic geometry devoid
of centrifugal and sidewall effects, highlighting the vertically elongated structures of rotating convection (RaP= 1012 , E = 2 ×106, Pr = 7,ℓ/H = 0.048, arbitrary
color scale ranging from blue for cool fluid to red for warm fluid).
law becomes:
Nu = CRa ×Ra3/2E2Pr1/2,[2]
where the dimensionless prefactor is CRa =C5/2. Over the last
decade, several state-of-the-art laboratory experiments have been
developed to observe this extreme scaling regime and validate the
geostrophic turbulence scaling-law [1]: the TROCONVEX experi-
ment in Eindhoven (18), the rotating U-boot experiment in Göttin-
gen (19,20), the Trieste experiment at ICTP (21,22), and the Romag
and Nomag experiments at UCLA (23,24). The goal is to produce a
strongly turbulent convective flow in which rotational effects remain
predominant (hence the ever taller convective cells), while avoiding
parasitic centrifugal effects (25). These experiments are all based on
the Rayleigh-Bénard (RB) geometry, where a layer of fluid is con-
tained between a hot bottom plate and a cold top one. A particu-
larly challenging task then is to overcome the throttling effect of the
boundary layers near these two plates: fluid hardly moves there and
heat need be diffused away from those regions (26). Even though
asymptotic analysis indicates that heat transport should be controlled
by the bulk turbulent flow in rapidly rotating RB convection, labora-
tory realizations indicate that the boundary processes keep limiting
the heat transfer throughout the entire cell (27), bringing the molecu-
lar diffusivities back into play and preventing the observation of the
scaling-law [1] associated with the bulk rotating turbulent flow. Forty
years after its initial derivation (4) and despite a worldwide effort to
design ever taller convection cells, the geostrophic regime of rapidly
rotating convection still awaits experimental validation (28).
Recently, we introduced an innovative laboratory setup to over-
come the above-mentioned limitations of RB convection as a model
for bulk natural flows (29). Specifically, we used a combination of
radiative internal heating and effective internal cooling to bypass the
throttling boundary layers of traditional RB convection and achieve
the fully turbulent – or ‘ultimate’ – regime of non-rotating convec-
tion (2931). These recent experimental developments suggest an al-
ternative route to observe the geostrophic regime of rapidly rotating
turbulent convection in the laboratory: instead of trying to overcome
the throttling effect of the RB boundary layers through intense ther-
mal forcing, one can take advantage of the radiatively driven setup,
where these boundary layers are readily bypassed, and subject the
radiatively driven turbulent convective flow to rapid global rotation.
10-6 10-5 10-4
102
103
!/H = 0.048
1.3×1011
15cm
20cm
25cm
10cm
2.5×1010
3.5×1011
9×1011
H=
RaP
Fig. 2. Suppression of heat transport by global rotation. Heat transport efficiency
Nu as a function of the Ekman number E, for various fluid heights: blue, H= 10cm,
RaP2.5×1010; green, H= 15cm, RaP1.3×1011 ; red, H= 20cm,
RaP3.5×1011; black, H= 25cm, RaP9×1011 . The dimensionless
absorption length is ℓ/H = 0.024 (filled circles) or ℓ/H = 0.048 (open squares).
For fixed Hand , the mixing efficiency dramatically decreases with increasing ro-
tation rate (decreasing E). Errorbars are estimated from the values obtained for the
first and second halves of the measurement interval, see Methods and SI appendix.
The resulting experimental setup, sketched in Figure 1, is an evo-
lution over the non-rotating setup described in a previous publica-
2| www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Bouillaut et al.
tion (29). The apparatus consists of a cylindrical tank of radius
10cm with a transparent sapphire bottom boundary, filled with a light-
absorbing mixture of water and carbon-black dye. A powerful spot-
light located under a water-cooled IR-screening stage shines at the
tank from below. Absorption of light by the dye results in an inter-
nal heat source that decreases exponentially with height zmeasured
upwards from the bottom of the tank, transferring to the fluid a total
heat flux Pover an e-folding absorption length . This source term
causes the temperature at every location inside the tank to increase
linearly with time. Superposed to this linear drift are internal tem-
perature gradients that develop inside the tank and rapidly reach a
statistically steady state. As recalled in the Methods section, the in-
ternal temperature difference between any two points inside the tank
is then governed by a combination of the exponential radiative heat
source together with an effective uniform heat sink.
The experimental tank is attached from above to a DC motor that
drives global rotation at a constant rate [0; 85] rpm around the
vertical axis of the cylinder. Rotation results in a slight curvature of
the free surface: the relative variations in fluid height between center
and periphery reach ±20% and ±13% for the two most rapidly rotat-
ing and shallowest data points, but are below ±10% (and often much
below) for the remaining approximately 60 data points. On-board
temperature measurements are performed using two thermocouples,
one in contact with the bottom sapphire plate and one at z= 3H/4,
where Hdenotes the height of the free surface on the axis of the cylin-
drical cell, where the probes are located. The temperature signals are
transmitted through WIFI to a remote Arduino microcontroller to en-
sure live monitoring of the experimental runs.
We show in Figure 2the Nusselt number based on the time-
averaged temperature difference Tbetween the two probes, for
experimental runs spanning 1.5 decades in RaPand 2.5 decades in
E, and two values of the dimensionless absorption length ℓ/H . The
dataset is provided in the Supplemental Information (SI) appendix,
together with estimates of the error bars. In a similar fashion to the
more standard RB system, for an approximately constant RaPan in-
crease in the global rotation rate leads to a dramatic drop in the heat
transport efficiency as measured by the Nusselt number Nu.
With the goal of establishing the turbulent nature of the flow and
assessing the independence of its transport properties with respect
to the molecular diffusivities, we form the ν- and κ-independent re-
duced Nusselt number N= Nu E/Pr, together with the composite
control parameter R= RaPE3/Pr2. The latter combination is the
only dimensionless control parameter if the diffusivities are to play
no roles (9,10,32). Ris also the cube of the so-called flux-based
convective Rossby number, identified as the main control parameter
of open ocean convection (17,33). We plot Nas a function of Rin
Figure 3(data points and estimates of the error provided in the SI).
In this representation, the dataset for a given value of ℓ/H collapses
onto a single master curve, which validates the fact that the molecular
diffusivities are irrelevant: we conclude that the present experimen-
tal setup achieves a ‘fully turbulent’ scaling regime, according to the
definition given at the outset. The collapse is particularly good for
rapid global rotation and slow global rotation – low and large R, re-
spectively – with a bit more scatter for intermediate values. For slow
global rotation (large R) the master curve gradually approaches the
scaling-law of radiatively driven non-rotating convection, reported in
previous publications (2931). This regime is associated with a large-
Rasymptote of the form N ∼ R1/3, represented in Figure 3: after
crossing out Efrom both sides of the scaling relation N ∼ R1/3one
As shown in the SI table, the temperature range varies between different data points, the conse-
quence being that RaPand Pr vary between different points of a constant-Hcurve in Figure 2.
The entire range of Pr spanned by the experimental data is 4.4Pr 6.7.
10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2
10-5
10-4
10-3
10-2
10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2
0
0.5
1
1.5
2
2.5
3
Fig. 3. Observation of the geostrophic turbulence regime. In terms of the diffusivity-
independent parameters Nand R, the data gathered for a given value of ℓ/H
collapse onto a master curve, which validates the ‘fully turbulent’ assumption. In the
rapidly rotating regime R.3×107the master curve displays a power-law be-
havior over one and a half decades in R, in excellent agreement with the prediction
N ∼ R3/5associated with the geostrophic turbulence scaling regime of rapidly ro-
tating convection (shown as an eye-guide, see Table 1for best-fit exponents). Same
symbols as in Figure 2for the experimental data. The triangles are DNS data for
RaP= 1012,Pr = 7 and ℓ/H = 0.048. Experimental and numerical error bars
are visible when larger than the symbol size. Bottom: same data compensated by
the geostrophic turbulence scaling prediction. An approximate plateau is observed
for R.3×107.
Bouillaut et al. PNAS | March 29, 2022 | vol. XXX | no. XX | 3
recovers the ultimate scaling-law of non-rotating convection, where
Nu is proportional to the square-root of Ra (2931). The approach to
that asymptotic behavior is clearly visible for ℓ/H = 0.024 at large
R, with a bit more scatter for ℓ/H = 0.048. More interestingly,
the focus of the present study is on the rapidly rotating regime that
arises for R.3×107. In this parameter range the master curve
follows a power-law behavior N ∼ Rβover one and a half decades.
The best-fit exponents βare given in Table 1. Over the last decade in
R, we measure β= 0.57 ±0.03 and β= 0.62 ±0.01, respectively,
for ℓ/H = 0.024 and ℓ/H = 0.048. These values are within 5%
of the theoretical exponent β= 3/5associated with the geostrophic
turbulence scaling-law [1].
While the flux-based parameter Ris the natural control parame-
ter of the present experiment, the reader accustomed to the standard
RB setup may be interested in characterizing the data in terms of
the Rayleigh number Ra based on the emergent temperature gradient.
In the SI appendix, we thus plot Nas a function of the diffusivity-
free Rayleigh number Ra=Ra E2/Pr (also known as the square
of the temperature-based convective Rossby number (17)). This rep-
resentation is equivalent to the one in Figure 3, with an equally sat-
isfactory collapse of the dataset. The power-law fits reported in Ta-
ble 1translate into power-laws N Raγ
, where the exponent γis
within 12% of the theoretical prediction 3/2(γ= 1.33 ±0.14 and
γ= 1.63 ±0.07, respectively, for ℓ/H = 0.024 and ℓ/H = 0.048).
These values contrast with the scaling exponent γin the constant-
E scaling-law Nu Raγreported in laboratory studies of rotating
RB convection (see Ref. (34) for a recent review). According to
the literature, the RB exponent measured experimentally achieves a
value close to 1/3in the slowly rotating regime, in line with the ‘clas-
sical theory’ of non-rotating RB convection (26). For fast rotation
and moderate supercriticality, laboratory experiments typically enter
a transitional regime where the exponent γincreases sharply. An ex-
tension of the experimental data using Direct Numerical Simulations
(DNS) indicates that γeventually reaches a value ranging between
3and 4(23), the lower value 3being again associated with a ‘clas-
sical’ regime controlled by marginally stable boundary layers (35),
while the larger value 4has been attributed to Ekman pumping (36)
(see also Ref. (37) for a theoretical demonstration of increasing heat
transport exponents as a result of boundary layer pumping). By con-
trast, in the present experiment radiative heating bypasses the bound-
ary layers of standard rotating RB convection, thus circumventing the
limitations of this traditional setup and providing experimental obser-
vations in excellent agreement with the geostrophic scaling regime of
rapidly rotating turbulent convection.
As a side note, we stress the fact that the system operates far
above the instability threshold. In the rapidly rotating limit, con-
vection arises above a threshold value of the order of 15 for the re-
duced flux-based Rayleigh number RaPE4/3. We report the values
of RaPE4/3in the SI appendix: in the rapidly rotating regime this
parameter ranges between 1.5×103and 2.5×104, orders of mag-
nitude above its threshold value. This large distance from threshold
is confirmed by the large values of the Nusselt number in Figure 2,
which range between 102and 103. The collapse in Figure 3is thus
not a mere consequence of near-onset behavior, a phenomenon re-
ported in Cheng & Aurnou (38) for synthetic near-onset data. We
illustrate this point further in the bottom panel of Figure 3, where we
plot Ncompensated by the geostrophic turbulence scaling-law R3/5,
in semi-logarithmic coordinates. This rather stringent representation
confirms (i) the good collapse of the data and (ii) the existence of
a plateau at low R, in agreement with the geostrophic turbulence
One would probably need to reach larger Rto avoid any signature of the intermediate-Rscatter.
βR ≤ 3×107R ≤ 107
Experiments ℓ/H = 0.024 0.59 ±0.01 0.57 ±0.03
Experiments ℓ/H = 0.048 0.57 ±0.01 0.62 ±0.01
DNS ℓ/H = 0.048 0.601 ±0.002 0.601 ±0.002
Table 1. Best-fit exponent βfor laboratory and DNS data, to be
compared to the theoretical prediction 3/5associated with the
geostrophic turbulence scaling regime. The error ±σβis estimated
by propagating the error on log Ninto a standard deviation σβfor
the exponent.
scaling-law. By contrast, the near-onset data discussed by Cheng &
Aurnou would not display such a collapse onto a plateau in compen-
sated form, as illustrated in Figure S2 of the SI appendix. We fur-
ther emphasize this point in the SI appendix by plotting the Nusselt
number mutliplied by Pr – to collapse the various Pr data points,
according to the geostrophic turbulence scaling – as a function of the
reduced temperature-based Rayleigh number RaE4/3. The resulting
Figure S3 makes it clear that the present data do not correspond to
the near-onset behavior discussed by Cheng & Aurnou in the RB con-
text: they are associated with greater supercriticality – the latter be-
ing better estimated by the even greater reduced flux-based Rayleigh
number RaPE4/3in the present context – and a scaling exponent
compatible with the γ= 3/2geostrophic turbulence value (see SI
appendix for details).
It proves insightful to compare the present experimental results
to existing numerical studies. DNS have been used as an extremely
valuable tool both to address rotating convection inside full or partial
spheres (39), but also to develop thought experiments in which one
can alter the exact equations and/or boundary conditions to identify
the mechanisms at play. This leads to idealized situations in which
the geostrophic scaling regime emerges. Some studies have consid-
ered stress-free boundary conditions instead of the no-slip bound-
aries of experimental tanks (40,41), some have used tailored internal
heat sources and sinks that conveniently vanish at the boundaries of
the domain (42,43), and some have focused on reduced sets of equa-
tions obtained through an asymptotic expansion of the rapidly rotat-
ing Boussinesq equations (15,16,44). DNS also offer an opportu-
nity to eliminate the potential biases of laboratory experiments. One
source of experimental bias is the centrifugal acceleration, which in-
creases with global rotation rate and distance from the rotation axis.
Horn and Aurnou (25) proposed the criterion 2H/g .1for cen-
trifugal effects to be negligible in standard rotating RB convection
(see also Refs. (45,46)). The precise threshold value on the right-
hand side of this inequality can probably be debated and requires fur-
ther investigation, specifically Cheng et al. (18) report experimental
measurements unaffected by centrifugal effects even when 2H/g is
as large as two. In the SI appendix, we provide the value of 2H/g
for all the experimental data points: this ratio never exceeds two, with
only three data points for which this ratio exceeds one (per value of
ℓ/H). A second distinction between the idealized horizontally un-
bounded convective layer and the finite-size experimental tanks is the
possible emergence of localized convective modes near the vertical
walls of the latter (4750). Wall modes have a lower onset than bulk
modes and can dominate the dynamics near the instability threshold
of the latter. However, they have also been shown to have a negligi-
ble impact on bulk heat transport in the turbulent regime (51). The
collapse of the various data points in Figure 3– which differ both in
terms of centrifugal ratio 2H/g and aspect ratio – is a first indica-
tion that the present measurements are not impacted by the centrifu-
gal acceleration nor the sidewalls.
4| www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Bouillaut et al.
With the goal of further validating the experimental results and
the subdominance of centrifugal, sidewall and non-Boussinesq ef-
fects, we have performed DNS of the present radiatively driven setup
in the idealized horizontally periodic plane layer geometry. The com-
bination of radiative heating and uniform internal cooling is imple-
mented in a pseudo-spectral code that solves the rotating Boussinesq
equations of thermal convection with a no-slip insulating bottom
boundary and a stress-free insulating top one (see Methods for de-
tails). We provide a snapshot of the temperature field in statistically
steady state in Figure 1, for RaP= 1012 , E = 2 ×106, Pr = 7
and ℓ/H = 0.048. This temperature field displays the typical ver-
tically elongated structures that characterize rapidly rotating convec-
tion (28), with a predominance of thin warm plumes emanating from
the heating region. The full numerical dataset consists in a sweep of
the Ekman number E for RaP= 1012, Pr = 7 and ℓ/H = 0.048,
the resulting data points being plotted in Figure 3. The error bars
on these data points, provided in the SI appendix, are much smaller
than the size of the symbols. The low-Rnumerical data again dis-
play a power-law behavior, with a best-fit exponent βwithin 0.5%
of the theoretical exponent 3/5associated with the geostrophic tur-
bulence scaling regime, see Table 1. The numerical data points lie
very close to the experimental ones for the same value of ℓ/H, the
reduced Nusselt number being slightly larger for the DNS data (by
approximately 20%), possibly as a consequence of the somewhat dif-
ferent geometries of the numerical and experimental setups. Overall,
the quantitative agreement between experiments and DNS, together
with the good collapse in Figure 3of data points obtained for various
aspect ratios and centrifugal ratios, indicates that the aforementioned
potential biases are subdominant in the present experiment. As far as
the present heat transport measurements are concerned, the central
region of the tank seems hardly affected by the centrifugal effects,
by the sidewalls, by the slight curvature of the free surface, or by
non-Boussinesq effects.
In some sense, our experiment follows a strategy similar to Barker
et al. (42) while proposing a situation that can be realized in the lab-
oratory. It thus comes as a surprise that the heat transport efficiency
measured in the present experiment is significantly greater than the
one reported in the idealized numerical setups of Barker et al. (42),
Stellmach et al. (40) and Julien et al. (16) in Cartesian geometry, and
in Gastine et al. (39) in spherical geometry: the experimentally mea-
sured value of the prefactor Cis approximately twice as large as the
value extracted from Barker et al. (42), it is six times greater than
the value reported in Julien et al. (16) and three times larger than the
value reported in Gastine et al. (39) (the latter in spherical geome-
try). In terms of the prefactor appearing in the scaling-law [2], this
translates respectively into an experimentally measured CRa that is
approximately six times greater than the value extracted from Barker
et al. (42), sixty times greater than the value reported in Julien et
al. (16), and twenty times greater than the value reported in Gas-
tine et al. (39). This points to an unexpected sensitivity of the heat
transport efficiency of rapidly rotating turbulent convection to the pre-
cise spatial distribution of heat sources and sinks. We confirmed this
conclusion experimentally by doubling the value of the absorption
length ℓ/H, from ℓ/H = 0.024 to ℓ/H = 0.048: this change in the
geometry of the heat source leads to an increase in the prefactor C
by approximately 30%, and an approximate doubling of the prefac-
tor CRa. Beyond the observation of the geostrophic turbulence heat
transport scaling-law [1], our laboratory setup thus offers a unique
experimental opportunity to determine the dependence of the prefac-
tor on the distribution of heat sources and sinks, which greatly varies
from planets to stars.
Materials and Methods
Radiative heating and effective uniform cooling. Within the framework
of the Boussinesq approximation, and denoting the temperature and velocity
fields as Tand u, respectively, the temperature equation for the radiatively
heated fluid reads:
t(ρCT ) + u·(ρC T ) = ρCκ2T+P
ez/ℓ ,[3]
where the radiative heating term – the last term on the right-hand side – results
from Beer-Lambert’s law. In this expression, zdenotes the vertical coordinate
measured upwards from the bottom of the tank and Pis the total heat flux.
The boundaries are thermally insulating: T·n= 0 at all boundaries,
with nthe unit vector normal to the boundary. Denoting as T(t)the spatial
average of the temperature field inside the fluid domain, the spatial average
of equation [3] yields:
dT(t)
dt=P
ρCH 1eH/ℓ .[4]
The spatially averaged temperature increases linearly with time. Once the
system reaches a quasi-stationary drifting state, the temperature everywhere
inside the tank drifts at a mean rate given by the right-hand side of [4]. We
can thus extract the power Pfrom the drift of the timeseries.
Consider now the deviation from the spatial mean, θ(x, t) = T(x, t)
T(t). We form the equation for θby subtracting equation [4] from equation
[3]/ρC:
tθ+u·θ=κ2θ+S(z),[5]
where the source/sink term S(z)is:
S(z) = P
ρC 1
ez/ℓ 1eH/ℓ
H.[6]
The second term inside the parenthesis is an effective cooling term associated
with the secular heating of the body of fluid. It balances the heating term on
average over the domain but has a different spatial structure. Equation [5] is
coupled to the rotating Navier-Stokes equation:
tu+ (u·)u+ 2Ωez×u=p+αgθ ez+ν2u,[7]
where the generalized pressure term absorbs the contribution from the mean
temperature T(t)and the centrifugal acceleration has been neglected.
The set of equations [5-7] corresponds to the standard equations of ro-
tating Boussinesq convection, with internal heating decreasing exponentially
with height and uniform cooling at an equal and opposite rate. The solutions
to this set of equations reach a statistically steady state, and the temperature
difference θrealized by [5-7] is equal to the temperature difference Tof
the initial setup.
Detailed experimental protocol. An experimental run consists of the fol-
lowing steps: the tank is filled with 7oC water mixed with carbon-black dye
to obtain a target value of . The tank is set into uniform rotation at a rate
. After an initial waiting period, for the fluid to achieve solid body rotation,
the 2500W metal-halide spotlight is turned on. Two thermocouples horizon-
tally centered inside the tank give access to the temperature at heights z= 0
and z= 3H/4. The corresponding temperature signals are measured by an
Arduino microcontroller and transmitted through WIFI to a second Arduino
microcontroller, which allows for live monitoring of the signals. An example
of timeseries is provided in Figure 4. After an initial transient phase the sys-
tem settles in a quasi-stationary state characterized by a linear drift of the two
timeseries at an equal rate (visible for t&1500s in Figure 4), together with
a statistically steady temperature difference between the two probes. The fact
that the two timeseries drift at a constant and equal rate is a first indication
that thermal losses are negligible. We determine the input heat flux Pfrom
the drift rate of the two signals using relation [4]. The time-average of the
temperature difference between z= 0 and z= 3H/4yields the temperature
drop T. This average is performed over the boxed time interval in Figure 4.
The dimensionless parameters are computed using the fluid properties eval-
uated for the mean bottom temperature over that interval. To quantify the
error associated with the slow temporal drift of the various fluid properties
(diffusivities, thermal expansion, etc) we also compute the various quantities
and dimensionless parameters using the first and second halves of the boxed
region, denoted respectively as sub-region I and sub-region II in Figure 4. For
each of the two sub-regions, we average the temperature difference over the
Bouillaut et al. PNAS | March 29, 2022 | vol. XXX | no. XX | 5
0 500 1000 1500 2000 2500 3000
-5
0
5
10
15
20
25
30
35
Fig. 4. Raw signals from thermocouples T1(red, z= 0) and T2(blue, z= 3H/4)
as a function of time tfor H= 25cm, Ω = 30rpm and ℓ/H = 0.048. Also
shown are the instantaneous temperature drop between the two probes (black), and
room temperature (green). The solid box indicates the total measurement inter val,
separated by a dashed-line into two sub-inter vals I and II.
sub-interval and we compute the dimensionless parameters using the mean
bottom temperature inside the sub-interval. The corresponding values are re-
ported in the SI with a subscript I or II depending on the sub-interval. Also
reported are the initial and final bottom temperatures of the boxed measure-
ment interval, denoted as Tstart and Tend, respectively. The error bars in
Figures 2and 3correspond to the values obtained by restricting attention to
a single sub-interval. When estimating the best-fit exponent β, we first com-
pute the root-mean-square error on log Nover the range of Rof interest (of
the order of 5%), before propagating this error into a standard deviation σβ
for the best-fit exponent β.
Direct numerical simulations. We solve equations [5-7] inside a horizon-
tally periodic domain with the pseudo-spectral solver Coral (52), previously
used for non-rotating convective flows (31) and validated against both ana-
lytical results (53) and solutions computed with the Dedalus software (54).
The bottom boundary is insulating and no-slip, while the top boundary is in-
sulating and stress-free. Depending on the Ekman number, the horizontal
extent Lof the domain is set to 0.4Hor 0.5H, to account for the variation
in the characteristic horizontal scale of the rotating flow. The equations are
discretized on a grid containing (Nx, Ny, Nz) = (441,441,576) points,
which corresponds to 296 alias-free Fourier modes in the horizontal direc-
tions and 384 Chebyshev polynomials along the vertical. The initial condi-
tion is chosen as either small amplitude noise or a checkpoint from a previous
simulation with smaller supercriticality. We restrict attention to the statisti-
cally steady state that arises after the initial transient. We denote as τmeas
the duration of integration in this statistically steady state and we focus on the
difference between the horizontally averaged temperatures at z= 0 (bottom
boundary) and z= 3H/4: the time-average of the resulting signal yields the
temperature drop from which Nis inferred. The standard deviation σof the
signal and its correlation time τcorr (time-lag of the first zero of the autoco-
variance function) allow us to estimate the statistical error on N. Following
e.g. Ref. (18), we compute the number of ‘effectively independent realiza-
tions’ Neff =τmeascorr before estimating the statistical error σNon the
mean temperature drop as σ/Neff. The resulting error bars are provided
in the SI appendix, together with the values of Nobtained by averaging over
only the first or second half of the signal, denoted as NIand NII, respectively.
ACKNOWLEDGMENTS. This research is supported by the European
Research Council under grant agreement FLAVE 757239. The numerical
study was performed using HPC resources from GENCI-CINES and TGCC
(grant 2020-A0082A10803 and grant 2021-A0102A10803). KJ acknowl-
edges support from the National Science Foundation, grant DMS-2009319.
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