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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 (1–3), 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 (4,6–11).

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 /ρCκ∆T

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 Ω.

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 (5,15–17). 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

3/5E4/5Pr−1/5,[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

Ω

P

ℓ

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 ℓ, delivering a total heat ﬂux P. The cylindrical tank is attached from above to a DC motor that imposes global rotation at

a rate Ω(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 ×10−6, Pr = 7,ℓ/H = 0.048, arbitrary

color scale ranging from blue for cool ﬂuid to red for warm ﬂuid).

law becomes:

Nu = CRa ×Ra3/2E2Pr−1/2,[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 (29–31). 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

!ℓ/H = 0.048

•ℓ/H = 0.024

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,

RaP≃2.5×1010; green, H= 15cm, RaP≃1.3×1011 ; red, H= 20cm,

RaP≃3.5×1011; black, H= 25cm, RaP≃9×1011 . The dimensionless

absorption length is ℓ/H = 0.024 (ﬁlled circles) or ℓ/H = 0.048 (open squares).

For ﬁxed Hand ℓ, 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 ℓ. 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 Ω∈[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 RaP†an 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 (29–31). 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 4.4≤Pr ≤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.3×10−7the 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.3×10−7.

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 (29–31). 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.3×10−7. 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 ≤ 3×10−7R ≤ 10−7

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 Ω2H/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 Ω2H/g is

as large as two. In the SI appendix, we provide the value of Ω2H/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 (47–50). 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 Ω2H/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 ×10−6, 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 ) = ρCκ∇2T+P

ℓe−z/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 1−e−H/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=κ∇2θ+S(z),[5]

where the source/sink term S(z)is:

S(z) = P

ρC 1

ℓe−z/ℓ −1−e−H/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+ν∇2u,[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 ℓ. The tank is set into uniform rotation at a rate

Ω. 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 L⊥of 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 =τmeas/τcorr 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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