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 Experiment &amp; $Pr$ &amp; $Ra$ &amp; $Ek$ &amp; $Fr$ &amp; $Ro$ &amp; $\Gamma$\\
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\DOI{10.5802/crphys.233}
\datereceived{2024-09-07}
\daterevised{2025-01-13}
\dateaccepted{2025-01-13}
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\begin{document}

\begin{noXML}

\makeatletter
\def\TITREspecial{\relax}
\def\cdr@specialtitle@english{Geophysical and astrophysical fluid dynamics in the laboratory}
\def\cdr@specialtitle@french{Dynamique des fluides g\'eophysiques et astrophysiques au laboratoire}
\makeatother

\CDRsetmeta{articletype}{review}

\title{Seven decades of exploring planetary interiors with rotating
convection experiments}

\alttitle{Sept d\'ecennies d'exploration des int\'erieurs de plan\`etes
par les exp\'eriences de convection en rotation}

\author{\firstname{Alban} \lastname{Poth\'erat}\CDRorcid{0000-0001-8691-5241}\IsCorresp}
\address{Centre for Fluid and Complex Systems, Coventry University,
Mile Lane, CV1 2NL Coventry, UK}
\email[A. Poth\'erat]{alban.potherat@coventry.ac.uk}

\author{\firstname{Susanne} \lastname{Horn}\CDRorcid{0000-0002-7945-3250}}
\addressSameAs{1}{Centre for Fluid and Complex Systems, Coventry
University, Mile Lane, CV1 2NL Coventry, UK}
\email[S. Horn]{susanne.horn@coventry.ac.uk}

\keywords{\kwd{Rotating convection}\kwd{Experimental fluid mechanics}\kwd{Measurement techniques}\kwd{Planetary interiors}\kwd{Turbulent convection}}

\altkeywords{\kwd{Convection en rotation}
\kwd{M\'echanique des fluides expr\'erimentale}
\kwd{Techniques de mesure}
\kwd{Int\'erieurs de plan\`etes}
\kwd{Convection turbulente}}

\thanks{EPSRC (grant nos. EP/X010937/1 and EP/V047388/1), Leverhulme
Trust (grant no. RPG-2017-366), European Research Council (ERC) (grant no.
EP/X034402/1)}

\begin{abstract}
The interiors of all the planets in the solar system consist of layers,
most of which are made out of fluids. When these layers are subject to
superadiabatic temperature or compositional gradients, turbulent
convection takes place that transports heat and momentum. In addition,
planets are fast rotators. Thus, the key process that underpins
planetary evolution, the existence of dynamo action or lack thereof,
the observable flow patterns, and much more, is rotating convection.
Because planetary interiors are remote and inaccessible to direct
observation, experiments offer crucial, physically consistent models
capable of guiding our understanding and complementing numerical
simulations. If we can fully understand the fluid dynamics of the
laboratory model, we may eventually fully understand the original.
Experimentally reproducing rotating thermal convection relevant to
planetary interiors comes with very specific challenges, in particular,
modelling the central gravity field of a planet that is parallel to the
temperature gradient. Three distinct classes of experiments have been
developed to tackle this challenge. One approach consists of using an
alternative central force field such as the electric one. This comes
with the caveat that these forces are typically weaker than gravity and
require going to space. Another method entails rotating the device fast
enough so that the centrifugal force exceeds and effectively supersedes
Earth's gravity. This mimics the equatorial and lower latitude regions
of a planet. Lastly, insight into the polar and higher latitude regions
is gained by using the actual lab gravity aligned with the rotation
axis. These experiments have been continuously refined during the past
seven decades. Here, we review their evolution, from the early days of
visualising the onset patterns of convection, over central force field
experiments in spacecraft, ultrasound velocity measurements in liquid
metals, to the latest optical velocity mapping of rotating
magnetoconvection in sulphuric acid inside high-field magnets. We show
how innovative experimental design coupled with emerging experimental
techniques has advanced our understanding of planetary interiors and
helped us paint a more realistic, detailed picture of them, including
Earth's liquid metal outer core.
\end{abstract}

\begin{altabstract}
Les int\'erieurs de toutes les plan\`etes du syst\`eme solaire sont
constitu\'es de couches internes, dont la plupart sont fluides. Quand
ces couches sont sujettes \`a un gradient de temp\'erature ou de
composition chimique super-adiabatiques, un \'ecoulement convectif 
s'installe qui transporte chaleur et moment angulaire. De plus, les
plan\`etes tournent extr\^ement vite. Ainsi, le processus-cl\'e qui
sous-tend l'\'evolution des plan\`etes est la convection en rotation.
Alors que les int\'erieurs de plan\`etes sont eloign\'es et impropres
\`a l'observation directe, les exp\'eriences de laboratoire
pr\'esentent des mod\`eles coh\'erents \`a m\^eme de guider notre
compr\'ehension et de compl\'ementer les simulations num\'eriques. Si
nous parvenons \`a comprendre la m\'ecanique des fluides des mod\`eles
de laboratoire, nous finirons peut-\^etre par comprendre l'original
compl\`etement. Cependant, reproduire exp\'erimentalement une
convection en rotation pertinente pour les int\'erieurs de plan\`etes
pr\'esente une gageure en soi, ne serait-ce qu'au niveau de la
mod\'elisation de la gravit\'e centrale d'une plan\`ete et de son
gradient de temp\'erature parall\`ele. Trois classes d'exp\'eriences
distinctes sont apparues pour relever ce d\'efi. Une premi\`ere
approche consiste \`a utiliser une autre force centrale, telle que la
force de Coulomb. L'inconv\'enient est que ces forces sont typiquement
plus faibles que la gravit\'e ambiante et n\'ecessitent donc d'aller
dans l'espace. Une autre m\'ethode implique de faire tourner le
dispositif suffisamment vite pour que la force centrifuge d\'epasse et
m\^eme remplace la gravit\'e terrestre. Cette technique imite les
conditions des regions \'equatoriales et \`a faible latitude d'une
plan\`ete. Finalement, utiliser la v\'eritable gravit\'e terrestre
align\'ee avec l'axe de rotation nous renseigne sur les r\'egions
polaires et \`a haute latitude. Ces exp\'eriences ont \'et\'e
perfectionn\'ees au cours de plus de sept d\'ecenies. Nous parcourons
ici leur \'evolution, depuis les premi\`eres visualisations des motifs
au d\'epart de la convection, en passant par les exp\'eriences \`a
gravit\'e centrale conduites \`a bord d'engins spatiaux, l'av\`enement
de la v\'elocim\'etrie \`a ultrasons dans le m\'etaux liquides,
jusqu'aux derni\`eres cartographies de champs de vitesse par m\'ethodes
optiques dans des \'ecoulements de magnoconvection en rotation dans de
l'acide sulfurique \`a l'int\'erieur d'aimants \`a champs fort. Nous
montrons comment des concepts exp\'erimentaux innovants coupl\'es aux
techniques exp\'erimentales \'emergentes ont pouss\'e notre
compr\'ehension des int\'erieurs de plan\`etes et nous ont aid\'e \`a 
en d\'epeindre une image toujours plus d\'etaill\'ee et plus
r\'ealiste, en particulier du c{\oe}ur liquide de la Terre.
\end{altabstract}

\maketitle
\vspace*{-1pc}

\end{noXML}

\section{Introduction}

In 1906, Oldham~\cite{Oldham1906}, wrote

\begin{quote}
{Of all regions of the
earth none invites speculation more than that which lies beneath our
feet, and in none is speculation more dangerous: yet, apart from
speculation, it is little that we can say regarding the constitution of
the interior of the earth. [\ldots] The central substance of the earth
has been supposed to be fiery, fluid, solid, and gaseous in turn, till
geologists have turned in despair from the subject, and become inclined
to confine their attention to the outermost crust of the earth, leaving
its centre as a playground for mathematicians}.
\end{quote}
Then the seismograph
changed geophysics forever. Oldham analysed earthquake data and was
indeed able to see inside the Earth. He concluded that the Earth had a
mantle and core based on different travelling behaviours of seismic
waves. In 1936, Inge Lehmann further proved seismologically that there
was an inner solid core~\cite{Lehmann1936}.

Seismology provides us with an accurate measure of the core-mantle and
inner core boundaries (CMB, ICB) and unequivocal evidence that the
outer core is liquid. Recent advanced seismological techniques have
been revealing the detailed structure of Earth's mantle and inner core.
They also complement other approaches, e.g.\ numerical simulations,
high-pressure experiments or mineral physics
modelling~\cite{Billen2008, Olugboji2013, Mukhopadhyay2019, Long2013,
Stixrude2012, Brodholt2000}. The outer core is a very different story.
Its liquid nature makes it less amenable to seismology-based
constraints. The main reason lies in the outer core's comparatively low
viscosity and subsequent lack of shear strength. The outer core has a
viscosity of about $\nu_{\mathrm{oc}} \approx
10^{-6}~\mathrm{m}^2/\mathrm{s}$, which is about $10^{24}$ and
$10^{16}$ times lower than those of the mantle~\cite{Normand1977} and
the inner core~\cite{Jeanloz1988}, respectively.

The low viscosity of Earth's outer core has another crucial
consequence: Rotation matters. The relative importance of rotation is
expressed through the Ekman number, $Ek = \nu/(2 \Omega H^2)$, where
$\nu$ is the viscosity, $H$ is the layer thickness, and $\Omega$ is the
rotation rate. Earth's rotation rate is about $\Omega_{E} = 7.27 \times
10^{-5}~\mathrm{rad}/\mathrm{s}$, and the mantle, outer core and inner
core have respective thicknesses of $H_{\mathrm{m}} = 2885$~km, $H_{\mathrm{oc}}
= 2270$~km, and $H_{\mathrm{ic}} = 1216$~km. This gives $Ek_{\mathrm{m}} \approx
10^9$, $Ek_{\mathrm{oc}} \approx 10^{-15}$ and $Ek_{\mathrm{ic}}
\approx 47$, respectively. Throughout the entire interior of the Earth,
convection drives motion and transports heat and momentum. But the high
$Ek$ in the mantle and inner core means convection in those two regions
is practically unaffected by rotation, whereas the low $Ek$ in the
outer core implies that convection there is strongly constrained by
rotation~\cite{kunnen2021_jot, Ecke2023, aurnou2015_pepi, Aurnou2020}.
Another distinct feature of the outer core is that it is both made of
electrically conducting metal and liquid. Because of it, convection is
vigorous enough to power dynamo action and so generates and sustains
Earth's magnetic field.

In short, the outer core is dynamically very different compared to the
mantle or the inner core. It is the layer of the Earth that is least
accessible through conventional means, in particular, seismology.
Hence, the outer core is still largely a ``\textit{playground}'' for
mathematicians, numericists, and, importantly, experimentalists. This
by no means implies that it is all mere speculation or that we should
not study the problem. Instead, to account for the large uncertainties
on the actual conditions in Earth's interior, we need to create models
that are rigorously built upon the known principles of mathematics,
physics, and most crucially fluid dynamics. But as Willis puts
it~\cite{willis1894}, ``{[\ldots] it is less difficult to
imitate one of nature's processes than to understand either the
imitation or, through it, the original}''. In the study of outer core
dynamics, it is experiments that most reliably ensure that our
understanding is built on firm ground, and shall be the focus of this
review.\looseness=-1

We first discuss briefly the theoretical framework that is used to
understand both the laboratory experiments and the planetary core
processes these experiments aim to ``\textit{imitate}''. Then, we give
an overview of the major design principles and their historical
development. In the main part, we review individual experiments, the
challenges involved in experimental endeavours and the engineering
strategy to tackle them. We shall also see how boldly extrapolating
these models to the Earth as well as other planets encourages further
exploration along this path while flagging up its limitations at the
same time.{\vspace*{-.6pc}}

\subsection{Governing equations and control parameters} \label{sec1.1}

To understand the challenges in bringing planetary physics to life in a
small contraption at a laboratory scale, we shall start by summarising
the main theoretical ingredients that both are usually assumed to
share, i.e.\ rotating thermal convection. The most common and simplest
framework describing rotating convection experiments exploring
planetary interiors is that of the Navier--Stokes equations
complemented by the transport equation for temperature~$T$. The basic
configuration is the Rayleigh--B\'enard convection (RBC)
one~\cite{benard1900_rgspa, rayleigh1916}, made of a fluid layer,
heated from below and cooled from above (in the sense of gravity) with
a superadiabatic temperature gradient. The fluid in planetary cores
modelled by lab experiments is usually assumed to be Newtonian and to
be described within the Oberbeck--Boussinesq
approximation~\cite{Oberbeck1879, Boussinesq1903, Gray1976}. Thus, its
material properties, kinematic viscosity $\nu$ and thermal diffusivity
$\kappa$ are homogeneous, temperature- and pressure-independent. The
density is approximated as a Taylor expansion around $T = T_{\mathrm{m}}$ up to
the first order, i.e.\ it is assumed to be linearly dependent on the
temperature,
{\begin{equation} \label{eq:rho_oberbou}
\rho = \rho_{\mathrm{m}} (1 - \alpha (T-T_{\mathrm{m}})),
\end{equation}}\unskip
with $\alpha$ being the constant isobaric expansion coefficient,
$\alpha \equiv - ({1}/{\rho})\partial_T \rho|_p$. The subscript $m$
denotes the reference value, typically, the arithmetic mean temperature
between the top and bottom boundary temperatures, $T_{\mathrm{m}} = 
(T_{\mathrm{t}}+T_{\mathrm{b}})/2$,
and $\rho_{\mathrm{m}} = \rho(T_{\mathrm{m}})$. The second term 
in the \mbox{density} is small
except in the gravitational and centrifugal buoyancy
terms~\cite{Marques2007, Horn2018, Horn2021, Horn2019, busse1970_jfm}.
The temperature is the actual temperature in experiments and the
superadiabatic part of the temperature when considering planetary
interiors~\cite{Busse2002b}. Planetary rotation is assumed constant,
$\Omega\hat{\mathbi{e}}_z$, and the origin is placed at the centre of
the planet. The governing equations in dimensional form under these
assumptions are:
{\begin{eqnarray}
\nabla \cdot \mathbi{u} &amp;=&amp; 0, \label{eq:cont_dim}\Seqnsplit
D_{{t}} {\mathbi{u}} &amp;=&amp; \nu \nabla^2 {\mathbi{u}} - \nabla p + 2 \Omega
\mathbi{u} \times \hat{\mathbi{e}}_z - \Omega^2 r \alpha (T-T_{\mathrm{m}})
\hat{\mathbi{e}}_r + g \alpha (T -T_{\mathrm{m}}) \hat{\mathbi{e}}_g,
\label{eq:ns_dim}\Seqnsplit
D_{{t}} T &amp;=&amp; \kappa \nabla^2 {T}. \label{eq:energy_dim}
\end{eqnarray}}\unskip
The unit vector $\hat{\mathbi{e}}_g$ points in the opposite direction of
gravity, e.g.\ opposite to the spherical radial direction in spherical geometries
or opposite to the vertical direction in plane geometries, such as cylinders or
annuli; $\hat{\mathbi{e}}_r$ is the cylindrical radial unit vector
perpendicular to the direction of rotation~$\hat{\mathbi{e}}_z$.

These equations are more meaningful in nondimensional form. Choosing as
reference quantities, the fluid layer thickness~$H$, the temperature
difference between the hot and cold boundary, in the case of Earth, the
inner core boundary (ICB) and core-mantle boundary (CMB), $\Delta$, the
free-fall velocity $U_{\mathrm{ff}}= \sqrt{g\alpha H\Delta}$, and the
corresponding time and pressure scales, $H/U_{\mathrm{ff}}$ and $\rho_{\mathrm{m}}
U_{\mathrm{ff}}^2$, we have:
{\begin{eqnarray}
\nabla \cdot \mathbi{u} &amp;=&amp; 0, \label{eq:cont}\Seqnsplit
D_{t}\mathbi{u} &amp;=&amp; \sqrt{\frac{Pr}{Ra}}\, \nabla^2 {\mathbi{u}} -
\nabla p + \sqrt{\frac{Pr}{Ra\,Ek^2}}\, \mathbi{u} \times
\hat{\mathbi{e}}_z - \frac{Fr}{\gamma} Tr\,\hat{\mathbi{e}}_r +
T \hat{\mathbi{e}}_z, \label{eq:ns}\Seqnsplit
D_{{t}} T &amp;=&amp; \sqrt{\frac{1}{Ra\,Pr}}\,\nabla^2 T. \label{eq:energy}
\end{eqnarray}}\unskip
In this form, the problem is governed by the following nondimensional
control parameters: the Ekman number, which expresses the ratio of
viscous to Coriolis forces, the Rayleigh number, which is the ratio of
buoyancy to viscous forces, the Prandtl number, which is the ratio of
viscous to thermal diffusivities, the Froude number, which is the ratio
of centrifugal to gravitational forces, and the radius-to-height aspect
ratio (or alternatively, the diameter-to-height aspect ratio)
{\begin{equation}
 Ek= \frac{\nu}{2 \Omega H^2},\quad Ra=\frac{\alpha g \Delta H^3}
 {\nu\kappa},\quad Fr= \frac{\Omega^2 R}{g},\quad \gamma = \frac{R}{H},
 \quad \Gamma = \frac{2R}{H}.
\end{equation}}\unskip
The form of Equations~(\ref{eq:cont})--(\ref{eq:energy}) brings out
three further classical parameters, the free-fall-based Reynolds,
Rossby, and P\'eclet numbers,
{\begin{equation}
Re=\sqrt{\frac{Ra}{Pr}},\quad Ro=\sqrt{\frac{Ra\,Ek^2}{Pr}},\quad
Pe=\sqrt{Ra\,Pr}.
\end{equation}}\unskip
This Rossby number is often referred to as the convective Rossby number
to distinguish it from the output parameter. Similarly, also the
Reynolds and P\'eclet numbers can be output parameters when defined
with measured velocities~\cite{Aurnou2020}.{\vspace*{-.4pc}}

\subsection{Main physical processes} \label{sec:phys}

The Equations~(\ref{eq:ns_dim}) and~(\ref{eq:ns}) include both the
centrifugal and the standard gravitational buoyancy terms. Both act in
a very similar way, that is, denser (colder) fluid moves in the
direction of $\hat{\mathbi{e}}_r$ and $-\hat{\mathbi{e}}_g$, and
lighter (warmer) fluid moves in the direction of $-\hat{\mathbi{e}}_r$
and $\hat{\mathbi{e}}_g$, respectively. Specifically, in a plane
geometry, cold fluid moves radially outwards and downwards and warm
fluid moves radially inwards and upwards. The Froude number expresses
which of the buoyancy terms is dominant~\cite{Horn2018, Horn2019,
Horn2021, busse1970_jfm, Chandrasekhar1961}. In experiments, $Fr$ is
always non-zero and in some experiments, it is the design principle
that $Fr\gg1$ such that the centrifugal acceleration supersedes the
gravitational one and so becomes the effective gravity (see
Section~\ref{sec:cyl}).

In most theoretical and numerical studies, $Fr$ is set to zero. This is
justified in most planetary settings where centrifugal buoyancy is
negligible at the largest scale, e.g.\ for Earth's outer core
$Fr\simeq1.72\times10^{-3}$. In this case, the main effect of the
Coriolis force on convective and other flows can be seen by considering
the limit of fast rotation $Ek\rightarrow0$, $Ro\rightarrow0$ in the
governing equations: away from boundary layers, the main balance is
\textit{geostrophic}, i.e.\ between the Coriolis force and the pressure
gradient, so the curl of Equation~(\ref{eq:ns}) imposes that
{\begin{eqnarray}
\partial_z \mathbi{u} &amp;=&amp; \mathcal{O}(Ek,Ro)\rightarrow0,\Seqnsplit
\nabla_{r\phi}\cdot\mathbi{u} &amp;=&amp; \mathcal{O}(Ek,Ro)\rightarrow0,
\end{eqnarray}}\unskip
where $\phi$ is the azimuthal coordinate and $\nabla_{r\phi}$ is the
two-dimensional divergence in the $r$--$\phi$ plane. Hence, at the
leading order, the Coriolis force makes the flow quasi-two-dimensional
and quasi-horizontally solenoidal, a result known as the
{Taylor--Proudman Constraint (TPC)}{\break} \cite{thomson1887_pmjs,
proudman1916_prsa, greenspan1969}. It implies that a cylindrical fluid
parcel must conserve its height through motion, or equivalently, that
the flow must follow surfaces of constant height along the rotation
direction, called \textit{geostrophic surfaces}. Since the TPC
restricts possible fluid motions, the onset of stationary convection
occurs at higher critical Rayleigh numbers $Ra_c$ than without
rotation. In a plane Rayleigh--B\'enard configuration, $Ra_c \sim 
Ek^{-4/3}$, and the non-rotating convection cells at onset give way to
much thinner spiral cells, of length scale $\ell_c \sim
Ek^{1/3}$~\cite{Chandrasekhar1961}. Importantly, in low Prandtl number
fluids ($Pr &lt; 0.68$) such as liquid metals, the onset modes are
oscillatory~\cite{Chandrasekhar1961, clune1993_pre, goldstein1993_jfm,
Goldstein1994, Horn2017, Aurnou2018, Vogt2021}. The critical Rayleigh
number and length scales are then $Pr$-dependent, namely, $Ra_c \sim
(Ek/Pr)^{-4/3}$ and $\ell_c \sim (Ek/Pr)^{1/3}$. These plane layer
onset scalings hold asymptotically also in spherical and annular
geometries~\cite{roberts1968_prsa, busse1970_jfm, Barik2023,
Dormy2004}.

At higher criticality, the TPC loosens, and the cellular structures
turn into convective Taylor columns, and then plumes and geostrophic
turbulence (see e.g.\ the recent reviews by
\mbox{Kunnen}~\cite{kunnen2021_jot} and Ecke and
Shishkina~\cite{ecke2023_arfm}). More generally, whenever inertia
antagonises the effect of rotation, i.e.\ when the Rossby number based
on the measured velocity $Ro_U$ is finite, the TPC is partly violated
and flow structures become anisotropic with a ratio of their axial to
transverse length scale governed by the balance between Coriolis and
inertia forces $l_z/l_{\perp}\sim Ro_U^{-1/3}${\break}
\cite{brons2020_jfm1, brons2020_jfm2, billant2021_jfm}, see the reviews
by Godeferd and Moisy~\cite{godeferd2015_amr} and
Biferale~\cite{biferale2021_jot} on how anisotropy determines rotating
turbulence. 

The Coriolis force may, however, have another important effect. It can
lead, at least locally, to a misalignment of the time-averaged
temperature gradient and gravity and therefore the main pressure
gradient. The misalignment drives a baroclinic flow (see the review by
Harlander within this special issue~\cite{harlander2024_crphys}). For
example, in Earth's core outside the polar region, baroclinic flows
naturally arise as the TPC aligns the flow with rotation rather than
gravity~\cite{aujogue2018_jfm, agrawal2024_gji}. Similarly, in an
experimental context or on a local planetary scale when $Fr &gt; 0$, there
is a flow even at arbitrarily low imposed temperature gradients. The
Coriolis force constrains the baroclinic motion into an azimuthal wind
$u_{\phi}$ which is controlled by the \textit{thermal wind balance}. In
cylindrical coordinates, and using the previous non-dimensionalisation,
it reads
{\begin{equation} \label{eq:tw_balance}
\partial_z u_{\phi} = Ro \left(\partial_r T + \frac{Fr}{\gamma} r
\partial_z T\right)
\end{equation}}\unskip
and derives from the curl of Equation~(\ref{eq:ns_dim}). The two
examples mentioned above are illustrations of the first and second
terms on the RHS of Equation~(\ref{eq:tw_balance}), respectively, being
significant.

\subsection{Planetary and experimental peculiarities} \label{sec1.3}

In the planetary context, the spherical shell geometry constrains the
expression of these processes. Since planets are extremely fast
rotators, the flow tends to follow geostrophic contours because of the
TPC. Spherical shells being axisymmetric, these geostrophic contours
are cylinders aligned with the rotation axis, so radial motions are
impeded whereas azimuthal flows are favoured by rotation. One of these
contours plays a particularly important role: the so-called
{tangent cylinder (TC)} extruded from the equatorial boundary of
the inner solid core along the rotation separates polar regions located
below the poles from equatorial regions (see
Figure~\ref{fig:geometries}a). If a fluid parcel was to cross the TC,
it would see either a doubling or a halving of the domain's height
along $\hat{\mathbi{e}}_z$, which the TPC strongly opposes. Hence the
TC acts as a mechanical boundary between the polar and equatorial
regions. For the same reason, radial motion near the equatorial plane
is very strongly suppressed by the jump in domain height there. Thus,
because of the Earth's solid inner core, the liquid outer core is split
into two regions with different types of rotating
convection~\cite{Gastine2023, agrawal2024_gji}.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:geometries}Sketches of the fundamental geometries
used for the study of rotating convection in planetary core convection.
The gravitational acceleration (or any proxy) $\mathbi{g}$ and the
temperature gradient $\nabla T$ and their respective directions are
indicated by dark green and lavender arrows. The angular rotation is
assumed to be vertical, $\bOmega = \Omega \hat{e}_z$, and the axis of
rotation is marked by the vertical black dash-dotted line. The
boundaries are demarcated by blue for the cold boundary $T_c$ and pink
for the hot boundary~$T_h$. (a)~Spherical shell; $R_{\mathrm{i}}$ is the inner
radius, $R_{\mathrm{o}}$ is the outer radius, and $H = R_{\mathrm{o}} - R_{\mathrm{i}}$ is the spherical
shell gap width. The grey cylinder of radius $R_{\mathrm{i}}$ is the tangent
cylinder (TC). The grey annulus is detailed in~(b), and the small grey
cylinder in the polar region is detailed in~(c). (b)~\textit{Busse
annulus}; $R_{\mathrm{i}}$ is the inner radius, $R_{\mathrm{o}}$ is the outer radius, $H =
R_{\mathrm{o}} - R_{\mathrm{i}}$ is the annulus gap width, $\bar{R} = R_{\mathrm{i}} + H/2 = R_{\mathrm{o}} - H/2$
is the mean radius, $L$ is the mean height, and $\chi$ is the sloping
angle of the endwalls. The endwalls are typically thermally insulating.
(c)~Cylinder; $R_{\mathrm{o}}$ is the radius, $H$ is the height of the cylinder.
The sidewalls are typically thermally{\break} insulating.}
\end{figure}

In the equatorial region outside the TC, the gravity is rather
perpendicular to the rotation. Near the equatorial plane, the gravity
and temperature gradients drive motion in the plane, and therefore
perpendicular to $\Omega\hat{\mathbi{e}}_z$. This motion is therefore
not as strongly opposed by the TPC. Instead the rotation tends to
elongate these cells along $\hat{\mathbi{e}}_z$ and so forms
quasi-geostrophic columns extending across the entire northern and
southern hemispheres, named \textit{Busse columns}, after Busse's
pioneering experimental work with the \textit{Busse annulus}
represented in Figure~\ref{fig:geometries}(b)~\cite{busse1976_sci}. The
variation in domain height in the equatorial region and in the Busse
annulus incurs a linear damping on the geostrophic structures called
$\beta$-effect~\cite{greenspan1969, pedlosky1987}. The reduced domain
height at low latitudes prevents Busse columns from entering that
region so they tend to remain close to the inner core. Nevertheless,
the $\beta$-effect incurred by the sloping outer boundary forces Busse
columns to oscillate radially and form \textit{thermal Rossby waves},
as sketched in Figure~\ref{fig:Rossbywaves}. Hindman and
Jain~\cite{Hindman2022} offer an elegant explanation of their basic 
mechanism: Considering a column of height~$L$, 

\begin{quote}
{If a spinning
column near the equator is pushed toward the rotation axis, the column
grows in height as the chord length of the column's axis increases. In
an incompressible fluid, this vortex stretching is accompanied by a
commensurate narrowing of the column to conserve mass. Subsequently, as
the column compresses laterally, the column must spin faster to
conserve angular momentum about its own axis~\cite{hide1966_prsa,
busse1970_jfm}. [\ldots] This conservation principle is enforced by the
constancy of the potential vorticity. As $L$ increases}, [the vorticity
component aligned with the rotation axis] {must also increase.
The resulting induced vorticity causes the neighbouring column to the
west to be pushed outward, away from the rotation axis, and the column
to the east to be pushed inward, toward the rotation axis. These newly
pushed columns conserve their own potential vorticity (i.e., angular
momentum) and induce spinning columns further down the belt to also
move inward and outward. The result is a prograde-propagating Rossby
wave where the spinning columns dance back and forth, toward and away
from the rotation axis}.
\end{quote}
One of the main design challenges in
imitating this phenomenology arises out of the vertical direction of
gravity in the lab, which makes it difficult to reproduce equatorial
convection in experiments using Earth's gravity. Here the centrifugal
forces comes to the experimentalist's rescue. Since the region outside
the TC is annular, it can be simulated by rotating an annulus with
sloping endwalls sufficiently fast for the centrifugal force to exert
an apparent radial gravity. This was the essence of Busse's
revolutionary idea~\cite{busse1974_jfm}.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:Rossbywaves}Sketch of the general propagating
behaviour of (incompressible) Rossby waves, adapted from
Ref.~\cite{Busse2009}. Visualised is part of the equatorial plane of a
rotating spherical shell. Initially, a series of fluid columns is at
rest and aligned along the mid-radius. If these columns are displaced
in a sinusoidal manner, then the vortices that move outwards 
(indicated by the green arrows) must reduce in length to conserve mass,
thus, they get squashed and acquire retrograde vorticity to conserve
potential vorticity. Similarly, the vortices that move inwards
(indicated by the purple arrows) must extend in length, they get
stretched and acquire prograde vorticity. The effect on the
neighbouring columns is that the columns to the west are pushed away
from the rotation axis and the columns to the east are pushed towards
the rotation axis (red arrows). This results in the phase of the
initial sinusoidal pattern propagating westwards (prograde) which is
known as the Rossby wave.}
\end{figure}

In the polar regions inside the TC, the temperature gradient, the
gravity and the rotation are mostly aligned, and topographic effects
are minor: the convection there is expected to develop somewhat
similarly to rotating RBC (RRBC). Experimentally, one would expect to
capture its main features by rotating a vertical cylindrical vessel
around its axis at a sufficiently large speed for the Coriolis force to
dominate, as in Figure~\ref{fig:geometries}(c). Unfortunately, at the
scale of the laboratory, the~centrifugal force becomes important at
rapid rotation, and even more so for cylinders of low aspect ratios
$\Gamma=2R_{\mathrm{o}}/H$~\cite{Horn2018}. Such a cylinder may represent the
entire TC or a smaller, concentric cylinder inside it, see
Figure~\ref{fig:geometries}(a,c). A further issue arises from the
nature of the TC: while it may be legitimate to model it with a rigid
boundary, the thermal boundary condition there is both unknown and
crucial: if adiabatic, isothermal surfaces remain horizontal all the
way across the cylinder, whereas if isothermal, very strong
baroclinicity drive a flow near the cylinder side
wall~\cite{aujogue2016_rsi, aujogue2018_jfm, agrawal2024_gji}. Since
there is no reason to think that either of these ideal cases applies to
planetary cores, actual TCs must be reproduced in experiments.

Besides topographic effects, solid boundaries in planets and rotating
experiments give rise to viscous boundary layers. Along boundaries
perpendicular to the rotation axis, Ekman layers, of thickness
$\mathcal{O}(Ek^{1/2})$ result from the balance between the viscous and
the Coriolis force~\cite{ekman1905_amaf, greenspan1969}. They become
extremely thin in regimes of rotation relevant to planets, and fully
resolving them curtails the range of Ekman numbers accessible to
numerical simulations. One of their most distinctive property is that
wall friction within the layer occurs at an angle with the bulk
velocity. This results from the progressive deflection of the flow by
the Coriolis force through the layer forming the so-called
\textit{Ekman spiral}. The second remarkable property of Ekman layers
is that local axial vorticity in the bulk $\omega_z$ pulls a local mass
flux out of the layer $u_z\simeq(1/2)\omega_zEk^{1/2}$. This secondary
flow, called \textit{Ekman pumping} is driven by the imbalance between
the radial pressure gradient induced in the Ekman layer by rotation in
the bulk, and the reduced centrifugal force due to low velocities
within the layer. This phenomenology even persists in the presence of
planetary or laboratory magnetic fields but follows different
scalings~\cite{acheson1973_arfm, potherat2000_jfm, davidson2002_ejmb}.

Boundary layers along vertical boundaries are \textit{Stewartson
layers}~\cite{stewartson1957_jfm, greenspan1969}. They exhibit a
sandwiched structure made of two layers. A layer of thickness
$\mathcal{O}(E^{1/4})$ arises from the balance between the Coriolis
force and viscous friction in planes perpendicular to the rotation,
while a thinner $\mathcal{O}(E^{1/3})$ is driven by secondary flows in
planes meridional to the rotation axis. Kunnen
et~al.~\cite{kunnen2013_jfm} offer a detailed analysis of their
structure in the context of rotating Rayleigh--B\'enard convection.

Vertical boundaries are also associated with specific convective
structures called \textit{wall modes}, with an onset at lower Rayleigh
numbers than the bulk modes~\cite{Zhong1991, Zhong1993,
herrmann1993_jfm, kuo1993_pre, goldstein1993_jfm, Goldstein1994,
zhang2009_jfm, Ecke2023b}. It is argued that their existence may be
connected to that of boundary zonal flows~\cite{Zhang2020, Zhang2021,
Vogt2021, ecke2022_prf}. These modes are robust to changes in the shape
of the vertical boundary~\cite{deWit2023, Terrien2023} and since they
compete with bulk modes relevant to planets, and efficiently carry
heat, their presence complicates the interpretation of experimental
results in geophysical context.

Finally, a key question that arises when experimentally modelling
either region is the choice of the working fluid, a choice that is
closely linked to the envisaged measurement techniques and the specific
aims pursued in building such devices. The Earth's outer core is mostly
made of liquid iron, whose physical properties can be estimated from
Iron's melting point~\cite{aubert2001_pepi, stacey1992}, and
high-pressure experiments~\cite{xu2021_epsl, morard2013_epsl,
gonzalez2023_jgr, pozzo2012_nat} $\rho\simeq
10^4~\mathrm{kg}/\mathrm{m}^3$, $\nu\simeq7\times
10^{-6}~\mathrm{m}^2/\mathrm{s}$, $\kappa\simeq4\times
10^{-6}~\mathrm{m}^2/\mathrm{s}$, $\alpha\simeq
10^{-5}~\mathrm{K}^{-1}$, so $0.1\leq
Pr\leq1$~\cite{schubert2011_pepi}. Liquid metals would therefore seem
the obvious choice but have only been used in a handful of experiments.
The main issue with liquid metals besides their direct or indirect
costs, toxicity, and chemical reactivity is that they are opaque and so
preclude any direct optical visualisation or any of the convenient
LASER-based techniques used in transparent fluids to map velocity and
temperature fields. For these reasons, high $Pr$ fluids such as water,
but also silicon oils and a few more exotic fluids, appear as cheap and
attractive alternatives to understand the interplay between convection
and rotation in planetary geometries. For the purpose of studying the
effect of magnetic fields, however, liquid metals are usually the fluid
of choice. However, neither liquid metals nor the high Prandtl fluids
mentioned above afford the combination of electromagnetic effects with
the advantage of optical visualisation. Sulphuric acid recently emerged
as a solution to do just that~\cite{andreev2013_jfm, moudjed2020_ef,
aujogue2016_rsi, potherat2023_prl}.

\subsection{Historical evolution through four classes of experiments}
\label{sec1.4}

The constraints imposed by terrestrial gravity led to four fundamental
classes of rotating convection experiments that explore planetary
interiors. However, with an outer core thickness of $H\simeq2.26\times
10^6~\mathrm{m}$, a rotation of one revolution per (Earth) day, a
mid-thickness gravity of $g\simeq
7~\mathrm{m}/\mathrm{s}^2$~\cite{schubert2011_pepi}, the regimes of the
outer core of Earth are too extreme for any laboratory experiment:
$Ek\simeq 10^{-15}$, $Ra\simeq 10^{22}$ to $10^{30}$,
$Ro\in[10^{-6},10^{-3}]$, $Re\in[10^{11},3\times10^{15}]$ and
$Pe\in[10^{21},10^{31}]$. Other planets offer no solace either as they
operate in just as extreme regimes~\cite{schubert2011_pepi,
aurnou2015_pepi}. Hence, experiments first focused on understanding the
fundamentals of convection (onset, regimes, heat flux) to later
extrapolate them to these extreme regimes, and incorporate some of the
more complex specifics of the Earth (inhomogeneities, magnetic field).
A timeline summarising the history of about seven decades of
experiments (1953--2024) is given in Figure~\ref{fig:timeline}, and a
large A3 version of it is provided as Supplementary Material.
Representations of the regimes of parameters in $(Ra,Ek)$ and in
$(Ra/Ra_c,Ek)$ spaces spanned by all four types of experiments are
shown in Figure~\ref{fig:parameters_space} along with projected Earth's
regimes. The details of dimensional and non-dimensional parameters for
experiments conducted by each team identified in the chronological
diagram of Figure~\ref{fig:timeline} is reported in
Tables~\ref{tab:dim_parameters} and~\ref{tab:nondim_parameters},
respectively.

In the following, we give a brief overview of the four fundamental
classes of rotating convection experiments.

\paragraph*{Central force field}

It is extremely hard to produce self-gravitating spheres in miniature
form within a terrestrial laboratory environment, thus,
proxy-gravitational forces have so far been the only way to generate
central force fields similar to Earth's gravity and suitable for
studying rotating convection. Examples are the dielectrophoretic force
which was utilised in two space experiments (GFFC,
1985--1999~\cite{hart1986_sci} and GeoFlow,
2008--2020~\cite{egbers1999_mst}) and very recently the pycnoclinic
acoustic radiation force which also works on Earth (Los Angeles,
2023~\cite{Koulakis2023}). These experiments are detailed in
Section~\ref{sec:central}.

\paragraph*{Cylindrical radial gravity}

This class of experiments simulates the regions outside the TC. Since
gravity is mostly cylindrical-radial there, these experiments use
annular vessels of various shapes: spherical, hemispherical,
cylindrical and variations thereupon, detailed in
Section~\ref{sec:cyl}. Guided by these constraints, the first
experiments were pioneered by Busse in the 1970s in a rotating annulus,
with slanted endwalls reproducing the $\beta$-effect and focussed
mostly on the onset of convection~\cite{busse1974_jfm}. Simpler
cylindrical annuli with flat endwalls were later used to study
inhomogeneous heat flux~\cite{sahoo2020_jfm} and ultimate
convection~\cite{jiang2020_sciadv}. The first and only experiment in a
full sphere in 1986 is due to Chamberlain~\cite{chamberlain1986_gafd}. 
Aside of this, spherical annuli became the geometry of choice for
groups in Los Angeles~\cite{busse1976_sci},
Cambridge~\cite{chamberlain1986_gafd}, Bayreuth~\cite{cordero1992_grl},
Baltimore~\cite{cardin1992_grl, sumita1999_sci},
Grenoble~\cite{aubert2001_pepi} and Maryland~\cite{shew2005_pepi}. The
mid-2000s mark the appearance of experiments on rotating
magneto-convection in spherical annuli in
Grenoble~\cite{gillet2007_jfm2} and Maryland~\cite{shew2005_pepi},
using gallium and sodium, respectively.

\paragraph*{Axial gravity}

Experiments relevant to the polar region started within the convection
community, rather than with geophysicists. The combination of axial
gravity and rotation, with the geometry of the geostrophic contour,
makes cylindrical vessels the obvious choice for this type of
experiment. Indeed the very first experiment on rotating convection was
conducted on the 20th of November 1953 in Chicago by Fultz, Nakagawa,
and Frenzen~\cite{Fultz1954}. Since then, cylinders have remained the
experimentalists' favourite to study rotating convection with axial
gravity. A very brief summary of the numerous experiments of this type
is given in Section~\ref{sec:axial}.

\paragraph*{Tangent cylinder}

The radial and axial strands met in 2003 when
Aurnou et~al.~\cite{aurnou2003_epsl} built the first experiment dedicated to
studying the TC, with a raised heater inside a hemispherical dome
filled with water. It wasn't until 2014 that this idea was pushed when
Aujogue et~al.\ used electrically conducting sulphuric acid and the
availability of high magnetic fields up to 10~T to produce the study of
magneto-rotating convection with PIV visualisations in a configuration
directly relevant to planets~\cite{aujogue2016_rsi, aujogue2018_jfm,
aujogue2016_phd, potherat2023_prl}. This was followed by a
much-improved rebuild using a cylinder instead of a
hemisphere~\cite{agrawal2024_gji} (see Section~\ref{sec:axial}).

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:timeline}Timeline of the laboratory experiments
relevant for planetary core convection from 1953 till 2024. As starting
point the year of the date of submission of the first publication, or,
where available, the date of the first experimental run is used, the
endpoint is the year of the publication of the last publication of the
experimental data. The experiments are either referred to by the name
of the device or by the place they were conducted if the
experimentalists were less creative with their naming convention than
with the experimental design. The main experimentalist(s) are also
given together with a selection of the relevant publications. The
background colours indicate the class of experiment: pink---axial
gravity, i.e.\ either cylinder or cuboid; blue---cylindrical radial
gravity, i.e.\ cylindrical or spherical annulus, hemispherical or
spherical shell; green---central force fields, i.e.\ sphere and
hemisphere (the dotted areas indicate when the space missions took
place); purple---tangent cylinder geometry. The working fluids in the
experiments are indicated by the different droplets. If rotating
magnetoconvection experiments were performed, a magnet symbol is added.
An A3-scale version of the figure is also provided as a Supplementary
File.}
\end{figure}

\begin{figure}[t!]
\includegraphics{fig04}
\caption{\label{fig:parameters_space}Parameter space spanned by the
four types of experiments modelling rotating convection in the Earth's
outer liquid core, and parameter range where the actual Earth's outer
core is currently believed to operate. While in $(Ra,Ek)$ space, all
experiments sit very far from realistic geophysical regimes,
especially, in terms of Rayleigh numbers, their levels of criticality
$Ra/Ra_c$ may well reach the Earth's. The data represented here is given in
Tables~\ref{tab:dim_parameters} and~\ref{tab:nondim_parameters}.}
\end{figure}

\begin{sidewaystable}[p!]%tab1
\caption{\label{tab:dim_parameters}Rotation rate $\Omega$, temperature
difference $\Delta$, inner radius $R_{\mathrm{i}}$ (heater radius in TC
geometries), outer radius $R_{\mathrm{o}}$ (corresponds to radius, i.e.\ half the
diameter $D$ in cylinders, outer vessel radius in TC geometries),
height $H$ (corresponds to gap width in spherical and other annulus
experiments). No parameters are given in Busse and Carrigan~\cite{busse1976_sci}. The
parameters for the experiments by Putterman et~al.~\cite{Koulakis2023}
are based on acoustic values.}
\tabcolsep=11.1pt
\begin{tabular}{lccccc}
\thead
Experiment &amp; $\Omega$ (rad/s) &amp; $\Delta$ (K) &amp; $R_{\mathrm{i}}$ (cm) &amp; $R_{\mathrm{o}}$ (cm) &amp; $H$ (cm)\\
\endthead
&amp; \multicolumn{5}{c}{Central force field}\\
Hart et~al.\ (GFFC)~\cite{hart1986_sci, hart1986_jfm, Hart1999, hart1996_3mg}\raisebox{.8pc}{} &amp; [0.03, 3.14] &amp; [0.2, 20] &amp; 2.402 &amp; 3.300 &amp; 0.908\\
Egbers et~al.\ (GeoFlow)~\cite{egbers1999_mst, egbers2003_asr, Futterer2010, Futterer2012, Futterer2013, Zaussinger2017, Zaussinger2020} &amp; $\{0.05, 5, 10\}$ &amp; [0.1, 10] &amp; 1.35 &amp; 2.70 &amp; 1.35\\
Putterman et~al.~\cite{Koulakis2023} &amp; 106.8 &amp; - &amp; - &amp; 3 &amp; 3{\vspace*{6pt}}\\
&amp; \multicolumn{5}{c}{Axial gravity}{\vspace*{6pt}}\\
Nakagawa et~al.~\cite{Fultz1955, Nakagawa1955, Nakagawa1957, Nakagawa1959, Goroff1960} &amp; 0.52, 1.05 &amp; n/a &amp; - &amp; 7.25, 10.5, 12.5 &amp; 3, 5\\
Nakagawa et~al.~\cite{Fultz1954} &amp; [0.26, $\uppi$] &amp; n/a &amp; - &amp; 7 &amp; [4, 8]\\
Koschmieder~\cite{Koschmieder1967} &amp; [$\uppi/2$, $2\uppi$] &amp; [1.25, 2.5] &amp; - &amp; 10.5 &amp; 3\\
Rossby~\cite{Rossby1969} &amp; [0.1, 4.83] &amp; n/a &amp; - &amp; 11.1125 &amp; 0.5, 1, 2, 3.5\\
Donnelly et~al.~\cite{Lucas1983, Pfotenhauer1984, Pfotenhauer1987} &amp; [0.11, 1.981] &amp; [2.1, 3.6] &amp; 0.81, 1.25 &amp; 0.16, 0.25\\
Boubnov and Golitsyn~\cite{Boubnov1986, Boubnov1990} &amp; [0.1, $2\uppi$] &amp; - &amp; - &amp; 8.5 &amp; [2, 24]\\
Fernando~\cite{Fernando1991} &amp; [0.07, 1] &amp; - &amp; - &amp; 30 (cube) &amp; 50\\
Ecke et~al.~\cite{Liu1997, Zhong1991, Zhong1993, Ning1993, Liu2009, Vorobieff1998} &amp; [$\uppi/50$, 4.9] &amp; n/a &amp; - &amp; 5, 6.35, 3.65 (square) &amp; 5, 12.7, 9.4 (square)\\
Sakai~\cite{Sakai1997} &amp; [0.52, 2.1] &amp; n/a &amp; - &amp; 10 &amp; 3, 6, 9\\
Aurnou and Olson~\cite{Aurnou2001} &amp; [$7.3\times10^{-3}$, $\uppi$] &amp; n/a &amp; - &amp; 7.6 (square) &amp; 1.85, 2.5\\
Aurnou et~al.\ (RoMag) \cite{King2009, Cheng2015, Aurnou2018, King2013, Vogt2021, Grannan2022} &amp; [0.7, 3.1] &amp; [0.4, 60] &amp; - &amp; 20 &amp; $\{5, 10, 20, 50\}$\\
Kunnen et~al.~\cite{Kunnen2008, Kunnen2010, Rajaei2016, Joshi2016, Rajaei2017} &amp; [0.01, 4.1] &amp; n/a &amp; - &amp; 10, 11.5, 12.5 &amp; 10, 11.5, 12.5\\
Ahlers et~al.~\cite{Zhong2009, Weiss2010, Weiss2011} &amp; n/a &amp; n/a &amp; - &amp; 24.8 &amp; 24.8, 49.5\\
Niemela and Ecke~\cite{Niemela2010, Ecke2014} &amp; [0.03, 0.2] &amp; [0.03, 0.2] &amp; - &amp; 50 &amp; 100\\
Niemela and Ecke~\cite{Niemela2010, Ecke2014} &amp; [0.05, 1.8] &amp; [0.4, 25] &amp; - &amp; $\{20, 60\}$ &amp; $\{10, 20, 40, 80, 180\}$\\
Bodenschatz et~al.\ (U-Boot) \cite{Zhang2020, Wedi2021} &amp; [0.002, 1.3] &amp; [1, 10.6] &amp; - &amp; 110 &amp; $\{110, 220\}$\\
\splittabular
Kunnen et~al.\ (TROCONVEX) \cite{cheng2018_gafd, Cheng2020, Kunnen2021}\raisebox{.8pc}{} &amp; [0.15, 2.2] &amp; [1, 25] &amp; - &amp; 40 &amp; $\{80, 200, 300, 400\}$\\
Abbate and Aurnou (Calimero) \cite{Abbate2023} &amp; [$8.4 \times 10^{-2}$, 3.7] &amp; [2, 55] &amp; - &amp; 9.64 &amp; $\{19.05, 38.10, 80.01\}$\\
Zhong et~al.~\cite{Ding2021, Lu2021, Shi2020} &amp; n/a &amp; n/a &amp; - &amp; 12 &amp; [6.3, 60]\\
Xia et~al.~\cite{Hu2022} &amp; [0.63, 6.1] &amp; [1, 34] &amp; - &amp; 9.8 &amp; 9.8{\vspace*{6pt}}\\
&amp; \multicolumn{5}{c}{Cylindrical radial gravity}{\vspace*{6pt}}\\
Busse et~al.~\cite{busse1974_jfm, Busse1982, Azouni1986} &amp; n/a &amp;n/a &amp;n/a &amp;n/a &amp; n/a \\
Busse and Carrigan~\cite{busse1976_sci, carrigan1983_jfm} &amp; n/a &amp;n/a &amp;n/a &amp;n/a &amp; n/a \\
Chamberlain and Carrigan~\cite{chamberlain1986_gafd} &amp; [16, 80] &amp; 7 &amp; 0 &amp; 7.51 &amp; 7.51\\
Cordero and Busse~\cite{cordero1992_grl, Cordero1993} &amp; 17.80, 19.89 &amp; [0.7, 3.7] &amp; 5 &amp; 6.35, 8.94 &amp; 1.35, 2.94\\
Cardin and Olson~\cite{cardin1992_grl, cardin1994_pepi} &amp; [26.2, 41.9] &amp; [0, 10] &amp; 5 &amp; 15 &amp; 10\\
Busse et~al.~\cite{Jaletzky2000, Westerburg2003} &amp; n/a &amp;n/a &amp;n/a &amp;n/a &amp; n/a \\
Sumita and Olson~\cite{sumita1999_sci, sumita2000_pepi, sumita2002_jgr, sumita2003_jfm} &amp; 21.57 &amp; [0, 10] &amp; 5 &amp; 15 &amp; 10\\
Aubert et~al.\ (Water)~\cite{aubert2001_pepi} &amp; [20.9, 83.8] &amp; [0, 25] &amp; 4 &amp; 11 &amp; 7\\
Aubert et~al.\ (Gallium)~\cite{aubert2001_pepi} &amp; [41.8, 83.8] &amp; [0, 30] &amp; 4 &amp; 11 &amp; 7\\
Gillet et~al.\ (Water)~\cite{gillet2007_jfm1} &amp; [20.9, 83.8] &amp; [8, 33] &amp; 4 &amp; 11 &amp; 7\\
Gillet et~al.\ (Gallium)~\cite{gillet2007_jfm1, gillet2007_jfm2} &amp; [20.9, 83.8] &amp; [8, 33] &amp; 4 &amp; 11 &amp; 7\\
Sahoo and Sreenivasan \cite{sahoo2020_epsl, sahoo2020_jfm} &amp; 31.31 &amp; - &amp; 0.051 &amp; 0.142 &amp; 0.092\\
Sun et~al.~\cite{jiang2020_sciadv, jiang2022_prl} &amp; [12, 74] &amp; n/a &amp; 12 &amp; 12 &amp; 12{\vspace*{6pt}}\\
&amp; \multicolumn{5}{c}{Tangent cylinder}{\vspace*{6pt}}\\
Aurnou and Olson~\cite{aurnou2003_epsl} &amp; [$4\times10^{-2}$, 19.5] &amp; - &amp; 5 &amp; 15.2 &amp; [11.6, 15.2]\\
Poth\'erat et~al.\ (LEE1)~\cite{aujogue2016_rsi, aujogue2018_jfm, potherat2023_prl} &amp; [1.52, 6.28] &amp; [0.3, 25] &amp; 5 &amp; 13.8, 14.25 &amp; 12\\
Poth\'erat et~al.\ (LEE2)~\cite{agrawal2024_gji} &amp; 0, [0.75, 6.28] &amp; [0.3, 23] &amp; 7.75 &amp; 14.3 &amp; 11
\botline
\end{tabular}
\vspace*{-33.5pc}
\end{sidewaystable}

\begin{sidewaystable}[p!]%tab2
\fontsize{8.56}{11}\selectfont\tabcolsep3.8pt
\caption{\label{tab:nondim_parameters}Prandtl number $Pr = \nu/\kappa$,
Rayleigh number, Ekman number $Ek = \nu/(2\Omega H^2)$, Froude number
$Fr = \Omega^2 R_{\mathrm{o}}/g$ (where $R_{\mathrm{o}} = D/2$ in cylindrical geometries),
and the (convective) Rossby number $Ro = \sqrt{Ek^2Ra/Pr}$. The aspect
ratio $\Gamma = R_{\mathrm{i}}/(R_{\mathrm{o}} - R_{\mathrm{i}})$ for spheres and $\Gamma = 2 R_{\mathrm{o}} /H$
for cylinders. Note that $Pr$, $Ra$, $Ek$, $Fr$, $Ro$ are only
approximate due to the temperature-dependence of the material properties.
For terrestrial gravity we use $g = 9.8~\mathrm{m}/\mathrm{s}^2$. Most
values for cylindrical geometries were taken from the review by Cheng
et~al.~\cite{cheng2018_gafd}. No parameters are given in Busse and
Carrigan (1976). In the experiments by Boubnov and
Golitsyn~\cite{Boubnov1986, Boubnov1990}, Fernando~\cite{Fernando1991}
and Sahoo and Sreenivasan~\cite{sahoo2020_epsl, sahoo2020_jfm}
convection is driven by imposing a heat flux instead of a temperature
difference. Not all
values in this table are taken from the original papers, some are
estimated and may not represent actual measurement points}
\begin{tabular}{lcccccc}
\thead
Experiment &amp; $Pr$ &amp; $Ra$ &amp; $Ek$ &amp; $Fr$ &amp; $Ro$ &amp; $\Gamma$\\
\endthead
Hart et~al.\ (GFFC)~\cite{hart1986_sci, hart1986_jfm, Hart1999, hart1996_3mg}\raisebox{.8pc}{} &amp; 8.4 &amp; [$4.1\times10^3$, $4.3 \times 10^5$] &amp; [$1.3\times 10^{-3}$, $1.5\times10^{-1}$] &amp; [$1.7\times10^{-5}$, 0.26] &amp; [0.03, 33] &amp; 2.65\\
\parbox[t]{3.5cm}{\raggedright Egbers et~al.\ (GeoFlow) \cite{egbers1999_mst, egbers2003_asr, Futterer2010, Futterer2012, Futterer2013, Zaussinger2017, Zaussinger2020}}{\vspace*{3pt}} &amp; $\{64, 125, 178\}$ &amp; [$5.6\times10^2$, $2.2 \times 10^5$] &amp; [$2.6 \times 10^{-3}$, $7.6 \times 10^{-3}$] &amp; [$5.1\times 10^{-5}$, 2.55] &amp; [0.07, 3.1] &amp; [17, 75.0]\\
Putterman et~al.~\cite{Koulakis2023} &amp; 0.15 &amp; $3 \times 10^6$ &amp; $6 \times 10^{-4}$ &amp; 0.026 &amp; 2.7 &amp; -{\vspace*{8pt}}\\
\parbox[t]{3.5cm}{\raggedright Nakagawa et~al.\ (mercury) \cite{Fultz1954, Fultz1955, Nakagawa1957, Nakagawa1959, Goroff1960}}{\vspace*{3pt}} &amp; 0.024 &amp; [$10^4$, $1.27\times10^{11}$] &amp; [$10^{-6}$, $3.46\times10^{-4}$] &amp; $[0.36, 8.1] \times10^{-3}$ &amp; [0.13, 1.2] &amp; 2.8, 2.9\\
Nakagawa et~al.\ (water)~\cite{Nakagawa1955} &amp; 6.4 &amp; [$10^5$, $9\times10^8$] &amp; [$7\times10^{-6}$, $10^{-3}$] &amp; [$7\times10^{-3}$, 1] &amp; [$6.24\times10^{-4}$, $8.32\times10^{-2}$] &amp; [1.66, 15]\\
Koschmieder~\cite{Koschmieder1967} &amp; 881 &amp; $[9.9, 20]\times10^2$ &amp; $[1.34, 5.38]\times10^{-2}$ &amp; [0.025, 0.4] &amp; $[1.4, 8.1]\times10^{-2}$ &amp; 20.6\\
Rossby~\cite{Rossby1969} &amp; [0.025, 200] &amp; [$2\times10^3$, $6\times10^6$] &amp; [$1.6\times10^{-5}$, $1.8\times10^{-2}$] &amp; [$1.1\times10^{-4}$, 0.26] &amp; [$5\times10^{-5}$, 283] &amp; 6.4, 11.1, 22.2, 44.5\\
Donnelly et~al.~\cite{Lucas1983, Pfotenhauer1984, Pfotenhauer1987} &amp; [0.49, 0.76] &amp; [$10^3$, $4\times10^6$] &amp; [0.03, $1.7\times10^{-4}$] &amp; [$9.2\times10^{-5}$, 3.36] &amp; [0.016, 4.28] &amp; 3.2, 4.9, 7.8\\
Ecke et~al.~\cite{Liu1997, Zhong1991, Zhong1993, Ning1993, Liu2009, Vorobieff1998} &amp; [3, 7] &amp; [$2\times10^3$, $5\times10^8$] &amp; 0, $[0.1,1.4\times10^{-5}]$ &amp; [$1.1\times10^{-5}$, 0.66] &amp; [$1.17\times10^{-4}$, 8.9] &amp; 1, 2, 5, 0.78 (square)\\
Sakai~\cite{Sakai1997} (square vessel) &amp; 6.4 &amp; [$10^6$, $10^9$] &amp; [$1.4\times10^{-5}$, $1.1\times10^{-3}$] &amp; [$2.8\times10^{-1}$, 0.1] &amp; [$1.1\times10^{-2}$, 0.44] &amp; 2.2, 3.3, 6.7\\
Aurnou and Olson~\cite{Aurnou2001} &amp; 0.023 &amp; [$1.5\times10^3$, $2.9\times10^4$] &amp; [$3.2\times10^{-3}$, $3.2\times10^{-2}$] &amp; [$4.15\times10^{-7}$, $7.6\times10^{-2}$] &amp; [0.11, 8.1] &amp; 4 (square)\\
\parbox[t]{3.5cm}{\raggedright Aurnou et~al.\ (RoMag) \mbox{\cite{King2009, Cheng2015, Aurnou2018, King2013, Vogt2021, Grannan2022}}}{\vspace*{3pt}} &amp; 0.025 &amp; [$1.4\times10^4$, $2.1\times 10^9$] &amp; [$2.3 \times 10^{-7}$, $1 \times 10^{-4}$] &amp; [$10^{-2}$, 0.2] &amp; [$1.7\times 10^{-4}$, 3.0] &amp; $\{0.4,1,2,4\}$\\
Kunnen et~al.~\cite{Kunnen2008, Kunnen2010, Rajaei2016, Joshi2016, Rajaei2017} &amp; [5.68, 7.85] &amp; [$1.28\times10^8$, $3.5\times10^9$] &amp; [$1.17\times10^{-6}$, $8.7\times10^{-4}$] &amp; [$1.17\times10^{-6}$, 0.56] &amp; [0.023, 11.5] &amp; 1\\
Ahlers et~al.~\cite{Zhong2009, Weiss2010, Weiss2011} &amp; [4.38, 6.26] &amp; [$2.37\times10^8$, $7.2\times10^{10}$] &amp; [$10^{-6}$, $3.36\times10^{-3}$] &amp; [$3.3\times10^{-7}$, 0.05] &amp; [0.06, 22.2] &amp; 0.5, 1, 2\\
Niemela and Ecke~\cite{Niemela2010, Ecke2014} &amp; 0.7 &amp; [$4 \times10^{9}$, $4\times10^{11}$] &amp; [$2\times 10^{-7}$, $3 \times 10^{-5}$] &amp; [$4.6\times10^{-5}$, $2.0\times10^{-3}$] &amp; [0.07, 5.0] &amp; 0.5\\
\sesplittabular
Aurnou et~al.\ (NoMag)~\cite{Hawkins2023}\raisebox{.8pc}{} &amp; [4, 7] &amp; [$10^5$, $10^{13}$] &amp; [$10^{-8}$, $10^{-2}$] &amp; [$5.1\times10^{-5}$, 0.20] &amp; [$10^{-6}$, $10^3$] &amp; \parbox[t]{2.55cm}{\centering \{0.11, 0.25, 0.33, 0.5, 0.75, 1.5, 2, 3, 6\}}{\vspace*{3pt}}\\
\parbox[t]{3.5cm}{\raggedright Bodenschatz et~al.\ (U-Boot) \cite{Zhang2020, Wedi2021}}{\vspace*{3pt}} &amp; 0.8\raisebox{.8pc}{} &amp; [$8\times 10^9$, $8\times 10^{14}$] &amp; [$1 \times 10^{-8}$, $3\times 10^{-5}$] &amp; [$4.48\times10^{-7}$, 0.19] &amp; [0.07, 42] &amp; $\{0.5, 1\}$\\
\parbox[t]{3.8cm}{\raggedright Kunnen et~al.\ (TROCONVEX) \cite{cheng2018_gafd, Cheng2020, Kunnen2021}}{\vspace*{3pt}} &amp; [2, 7] &amp; [$1 \times10^{10}$, $7\times 10^{13}$] &amp; [$5 \times 10^{-9}$, $3\times10^{-6}$] &amp; [$9.2\times10^{-4}$, 0.20] &amp; [$2 \times 10^{-4}$, 30] &amp; $\{0.1, 0.13, 0.2, 0.5\}$\\
\parbox[t]{3.8cm}{\raggedright Abbate and Aurnou~(Calimero) \cite{Abbate2023}}{\vspace*{3pt}} &amp; $\{6,41, 206, 993\}$ &amp; [$10^8$, $10^{12}$] &amp; [$10^{-7}$, $10^{-4}$] &amp; $\{6.9 \times 10^{-5},0.13\}$ &amp; [0.02, 2.5] &amp; $\{0.24, 0.51, 1.0\}$\\
Zhong et~al.~\cite{Ding2021, Lu2021, Shi2020} &amp; 4.38 &amp; [$1.4\times10^7$, $2.9\times10^{11}$] &amp; [$2\times10^{-7}$, $1.3\times10^{-4}$] &amp; [0.005, 0.31] &amp; [$1.2\times10^{-6}$, 6.94] &amp; 0.4, 0.5, 1, 2, 3.8\\
Xia et~al.~\cite{Hu2022} &amp; 4.34 &amp; [$2.8\times10^8$, $9.5\times10^9$] &amp; [$1.4\times10^{-6}$, $1.4\times 10^{-5}$] &amp; [0.004, 0.363] &amp; [0.025, 1] &amp; 1{\vspace*{8pt}}\\
Chamberlain and Carrigan~\cite{chamberlain1986_gafd} &amp; 6.5 &amp; [$3.3\times10^5$, $2.6\times10^6$] &amp; [$10^{-6}$, $5.0\times10^{-6}$] &amp; [2.0, 49] &amp; [$2.3\times10^{-4}$, $3.2\times10^{-3}$] &amp; 1\\
Cordero and Busse~\cite{cordero1992_grl, Cordero1993} &amp; 6.2 &amp; [$5.3\times10^5$, $5.1\times10^6$] &amp; $3.3\times10^{-6}$, $5.6\times10^{-6}$ &amp; 2.1, 3.6 &amp; $9.5\times10^{-4}$, $5.1\times10^{-3}$ &amp; 1.7, 3.7\\
Cardin and Olson~\cite{cardin1992_grl, cardin1994_pepi} &amp; 7 &amp; $4.5\times10^9$ &amp; [$1.2\times10^{-6}$, $1.9\times10^{-6}$] &amp; [11, 24] &amp; ${\leq}4.8\times10^{-2}$ &amp; 0.5\\
Sumita and Olson~\cite{sumita1999_sci, sumita2000_pepi, sumita2002_jgr, sumita2003_jfm} &amp; 7 &amp; [$4.5\times10^8$, $1.2\times10^9$] &amp; $2.4\times10^{-6}$ &amp; 7.0 &amp; [$1.9\times10^{-2}$, $3.8\times10^{-2}$] &amp; 0.5\\
Aubert et~al.\ (water)~\cite{aubert2001_pepi} &amp; 7 &amp; [$2.9\times10^7$, $8.6\times10^9$] &amp; [$1.2\times10^{-6}$, $4.9\times10^{-6}$] &amp; [4.9, 79] &amp; [0, $1.7\times10^{-1}$] &amp; 0.64\\
Aubert et~al.\ (gallium)~\cite{aubert2001_pepi} &amp; 0.025 &amp; [$1.2\times10^7$, $2.6\times10^8$] &amp; [$3.6\times10^{-7}$, $7.2\times10^{-7}$] &amp; [20, 79] &amp; [0, $7.4\times10^{-2}$] &amp; 0.64\\
Gillet et~al.\ (water)~\cite{gillet2007_jfm1} &amp; 7 &amp; [$3.3\times10^8$, $9.1\times10^9$] &amp; [$1.2\times10^{-6}$, $4.9\times10^{-6}$] &amp; [4.9, 79] &amp; [$8.4\times10^{-3}$, $1.8\times10^{-1}$] &amp; 0.64\\
Gillet et~al.\ (gallium)~\cite{gillet2007_jfm1, gillet2007_jfm2} &amp; 0.025 &amp; [$2.4\times10^7$, $2.3\times10^8$] &amp; [$4.9\times10^{-7}$, $1.5\times10^{-6}$] &amp; [4.9, 79] &amp; [$1.5\times10^{-2}$, $1.4\times10^{-1}$] &amp; 0.64\\
Sun et~al.~\cite{jiang2020_sciadv, jiang2022_prl} &amp; 4.34 &amp; [$10^6$, $3.7\times10^{12}$] &amp; [$6.7\times10^{-7}$, $2.4\times10^{-2}$] &amp; [$5\times10^{-8}$, 80] &amp; [$2.5\times10^{-3}$, 100] &amp; 1{\vspace*{8pt}}\\
Aurnou and Olson~\cite{aurnou2003_epsl} &amp; 7 &amp; [$3\times10^6$, $3\times10^{10}$] &amp; [$10^{-5}$, $9\times10^{-4}$] &amp; [$1.5\times10^{-5}$, $4.8\times10^{-2}$] &amp; [$6.6\times10^{-3}$, 59] &amp; [0.66, 0.86] (0.33)\\
Poth\'erat et~al.\ (LEE1)~\cite{aujogue2016_rsi, aujogue2018_jfm, potherat2023_prl} &amp; $\{7, 12\}$ &amp; [$1.4\times10^7$, $2.9\times10^9$] &amp; [$1.3\times10^{-6}$, $4.5\times10^{-5}$] &amp; [$2.8\times10^{-2}$, 0.44] &amp; [$3.2\times10^{-2}$, 0.58] &amp; 0.8 (0.35)\\
Poth\'erat et~al.\ (LEE2)~\cite{agrawal2024_gji} &amp; 6.4 &amp; [$2.1\times10^7$, $1.1\times10^9$] &amp; [$3.5\times10^{-6}$, $4.2\times10^{-5}$] &amp; 0, [$3.1\times10^{-3}$, 0.44] &amp; [$6.3\times10^{-3}$, 0.29] &amp; 1.1
\botline
\end{tabular}
\vspace*{-33pc}
\end{sidewaystable}


\section{Central force field: outer space and elsewhere}
\label{sec:central}
\subsection{Outer space: the geophysical fluid flow cell (GFFC) and
the GeoFlow experiments} \label{sec2.1}

Spherical geometries with a central force field may at first glance be
essential for creating a laboratory model of Earth's outer core, and
other geophysical and astrophysical systems. \mbox{Unfortunately}, on
the ground, the terrestrial gravitational potential prevents such a
setup. Thus, realistic experiments where the angle between the Coriolis
and buoyancy force varies, are not straightforwardly realisable. We
first focus on the two experiments that went to space to escape the
terrestrial limitations: the {Geophysical Fluid Flow Cell
(GFFC)}~\cite{hart1986_sci, Hill1982, hart1986_jfm, hart1996_3mg} and
the {GeoFlow} experiment~\cite{egbers1999_mst, egbers2003_asr,
Futterer2010, Futterer2012, Futterer2013, Zaussinger2017,
Zaussinger2020}, shown in Figure~\ref{fig:space}. Neither experiment
specifically aimed at understanding Earth's core, but both constitute
unique laboratory experiments of \mbox{(hemi-)spherical} rotating convection.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig:space}Left panels~(a--f): the Geophysical Fluid
Flow Cell (GFFC) experiment; (a)~Space Shuttle Challenger with
Spacelab~3 that flew GFFC to space; (b)~sketch of the cross-section of
the GFFC test cell~\cite{hart1986_jfm}; (c)~rotating turntable with
power supply, circulation pumps, test cell, and associated
circuitry~\cite{hart1986_jfm}; (d)~the full GFFC experiment including
optical imaging system~\cite{Hill1982}; (e,f)~unwrapped convection
planforms showing banana cells and pronounced polar disturbances, the
tangent cylinder is at 43{\textdegree}, (e)~shows north--south fringes
(sensitive to latitudinal temperature gradients) and (f)~shows
east--west fringes (sensitive to longitudinal temperature
gradients)~\cite{Hart1999}. Right panels (g--j):~the GeoFlow
experiment; (g)~photo of the experimental setup with the International
Space Station (ISS), image by ESA; (h,i)~Wollaston Shearing
Interferometry which visualises the thermal gradients based on the
changes of the temperature-dependent refractive index, the columnar
cells are coloured using automatic pattern
recognition~\cite{Zaussinger2020}, (h)~spiralling columnar cells and
(i)~\mbox{almost} straight cells; (j)~the full GeoFlow experiment with
adaption optics and rotary tray~\cite{egbers2003_asr}.}
\end{figure}

The main underlying principle of both the space experiments is that if
an electric potential is applied to a dielectric fluid, the neutral
non-charged molecules are polarised, and a force acts in the direction
of the strongest electric field region~\cite{egbers1999_mst,
hart1986_jfm}. The idea was originally suggested by
Smylie~\cite{Smylie1966, Chandra1972} but Hart
et~al.~\cite{hart1986_sci} later independently realised it for the first
time in a spherical experiment. These polarisation forces are also
known as dielectrophoretic forces and are density- and
permittivity-dependent. They act in a very similar manner to
traditional buoyancy forces resulting from either the gravitational or
centrifugal potential, but here the potential is $\mathbi{E} \cdot
\mathbi{E}$, with $\mathbi{E}$ being the electric field. The key
advantage is that in spherical geometries these forces act effectively
like a central gravity. Hence, in a rotating experiment, the angle
between the electro-hydrodynamic ``gravity'' vector and the rotation
vector vary latitudinally in a similar manner as in planetary and
stellar interiors.

More specifically, one can approximate the functional dependence of the
permittivity to the first order, as for the density in the
Oberbeck--Boussinesq approximation~(\ref{eq:rho_oberbou}):
{\begin{equation}
\epsilon = \epsilon_{\mathrm{m}}(1 - \gamma (T-T_{\mathrm{m}})),
\end{equation}}\unskip
where $\epsilon_{\mathrm{m}}$ is the ambient permittivity and $\gamma$ is the
dielectric variability. The values for the different fluids used in the space
experiment are given in Table~\ref{tab:spaceparameters}. Since in all
fluids $\gamma \approx \alpha$, the resulting system can be treated
with an equivalent version of the Oberbeck--Boussinesq
approximation~\cite{Oberbeck1879, Boussinesq1903}. Further, if the
electrical field $\mathbi{E}$ behaves like a spherical
capacitor~\cite{Futterer2013}, we have
{\begin{equation}
\mathbi{E}(\mathbi{r}) = \frac{R_{\mathrm{i}} R_{\mathrm{o}}}{R_{\mathrm{o}} - R_{\mathrm{i}}} \frac{1}{r^2}
V_{\mathrm{rms}} \hat{\mathbi{e}}_r,
\end{equation}}\unskip
where $V_{\mathrm{rms}}$ is the applied AC voltage to the shells. The
resulting electro-hydrodynamic ``gravity'' is then
{\begin{equation}
\mathbi{F}_{e} = g_{e} \gamma T \hat{\mathbi{e}}_r\quad
\mbox{with}\ g_{e}(r) = \frac{2 \epsilon_{\mathrm{m}} V_{\mathrm{rms}}^2}{\rho_{\mathrm{m}}}
\left(\frac{R_{\mathrm{i}} R_{\mathrm{o}}}{R_{\mathrm{o}} - R_{\mathrm{i}}}\right)^2\frac{1}{r^5}.
\end{equation}}\unskip
Thus, gravity varies radially but falls off much steeper with $1/r^5$
compared to planetary gravities, closer to $r$ or $1/r^2$. This,
however, is argued to be only of secondary
importance~\cite{hart1986_jfm} even though the inner boundary gravity
is much stronger than at the outer boundary, i.e.\ $g_{\mathrm{i}} \gg g_{\mathrm{o}}$. These
values are given in Table~\ref{tab:spaceparameters}. $Ra$ is the usual
Rayleigh number but is defined using the outer value $g_{\mathrm{o}}$. The typical
values of $g_{\mathrm{o}}$ are much smaller than Earth's gravity. This demands
microgravity and severely constrains rotation rates and vessel sizes to
small values.

\begin{table}[t!]%tab3
\fontsize{9.7}{11.7}\selectfont
\caption{\label{tab:spaceparameters}Material properties used in the
space experiments, $\rho_{\mathrm{m}}$ is the mean density, $\epsilon_{\mathrm{m}}$ the mean
permittivity, $\alpha$ the expansion coefficient, $\gamma$ the
dielectric variability, $V_{\mathrm{rms}}$ the (maximum) applied a.c.\ 
voltage, and $g_{\mathrm{o}}$ and $g_{\mathrm{i}}$ the values of the electro-hydrodynamic
``gravity'' at the outer and inner radius of the (hemi)spherical shell,
respectively. The silicone oil Dow Corning 0.65~cSt 200 Fluid was used
in the {GFFC} experiment, silicone oil M5 in the
{GeoFlow~I} experiments, and 1-nonanol at different mean
temperatures during the {GeoFlow~II} experiments. Note that
$\epsilon_0 = 8.854 \times
10^{-12}~\mathrm{CV}^{-1}{\cdot}\mathrm{m}^{-1}$ is the vacuum
permittivity. Taken from Refs.~\cite{hart1986_jfm}
and~\cite{Futterer2013, Zaussinger2017}.}
\begin{tabular}{cccccccc}
\thead
Fluid &amp;
\parbox[t]{1cm}{\centering $\rho_{\mathrm{m}}$ (kg/m$^3$)} &amp; $\epsilon_{\mathrm{m}}$ &amp;
\parbox[t]{.7cm}{\centering $\alpha$ (K$^{-1}$)} &amp;
\parbox[t]{.7cm}{\centering $\gamma$ (K$^{-1}$)} &amp;
\parbox[t]{1cm}{\centering $V_{\mathrm{rms}}$ (kV)} &amp;
\parbox[t]{1cm}{\centering $g_{\mathrm{o}}$ (m/s$^2$)} &amp;
\parbox[t]{1cm}{\centering $g_{\mathrm{i}}$ (m/s$^2$)}\vspace*{2pt}\\
\endthead
Silicone oil DC\raisebox{.8pc}{} 0.65~cs &amp; 760 &amp; 2.5$\epsilon_0$ &amp; $1.34 \times 10^{-3}$ &amp; $1.29 \times 10^{-3}$ &amp; 10 &amp; 1.13 &amp; \05.55\\
Silicone oil M5 &amp; 920 &amp; 2.70$\epsilon_0$ &amp; $1.08 \times 10^{-3}$ &amp; $1.07 \times 10^{-3}$ &amp; 10 &amp; 0.26 &amp; \08.54\\
1-nonanol (20.0~{\textdegree}C) &amp; 820 &amp; 8.60$\epsilon_0$ &amp; $9.45 \times 10^{-4}$ &amp; $2.95 \times 10^{-3}$ &amp; 6.5 &amp; 0.40 &amp; 12.76\\
1-nonanol (30.5~{\textdegree}C) &amp; 822 &amp; 8.33$\epsilon_0$ &amp; $9.45 \times 10^{-4}$ &amp; $2.46 \times 10^{-3}$ &amp; 6.5 &amp; 0.39 &amp; 12.32
\botline
\end{tabular}
\end{table}

The {Geophysical Fluid Flow Cell (GFFC)} flew to space for the
first time on the 29th of April 1985 together with two squirrel
monkeys, 24 rats and seven astronauts~\cite{Hill1982, hart1986_sci,
hart1986_jfm}. The principal investigator was John Hart of the
University of Colorado at Boulder, it was managed by NASA's Marshall
Space Flight Center. The experiment was conducted in the Spacelab, a
microgravity laboratory developed by ESA and flown by the NASA space
shuttle Challenger. The specific mission was Spacelab~3 (STS 51-B; 29
April 6--May 6, 1985), the first in which a strict low-gravity
environment was maintained in orbit, here in a low Earth orbit of
400~km altitude. {GFFC}'s second space flight in 1995 was with
the US Microgravity Laboratory-2 (USML-2) on board the space shuttle
Columbia (STS-73; 20 October--5 November 1995)~\cite{Hart1999,
hart1996_3mg}. The first mission provided about 110~h of experimental
data mainly in the form of about 50,000 film images. During the second
mission, 29 separate 6-h runs (174~h in total) were carried out.

The set-up itself is a spherical shell rotating at up to 3~rad/s, but
only its northern hemisphere was used in the experiment, as depicted in
Figure~\ref{fig:space}. The inner shell was made of polished nickel of
radius $R_{\mathrm{i}} = 2.402$~cm and the outer northern hemispherical shell of
radius $R_{\mathrm{o}} = 3.300$~cm was a 1~cm thick transparent sapphire dome,
with electrically conducting coating on the inner side acting as an
electrode. An AC voltage $V_{\mathrm{rms}} = 10~\mathrm{kV}\pm 0.5\%$
at 300~Hz was applied across the gap. This ensured the absence of
conductive electric currents that may lead to other and undesired fluid
instabilities, as the associated period was short compared to the
charges' relaxation time of the working fluid. The southern hemisphere
was filled with Teflon to avoid the effect of non-radial electric
fields occurring in the vicinity of the inner sphere's mechanical
support. The working fluid was a dielectric silicone oil Dow Corning 
with $Pr = 8.4$. The inner sphere was heated and the outer sphere was
cooled with a computer-regulated temperature gradient of up to
$20~\mathrm{K}\pm 0.1~\mathrm{K}$. Individual heater elements also
allowed to study differential, latitudinally varying heating, and
several experiments were run with a hotter north pole (a cooler pole
would have obscured visualisations). During the second flight, problems
with the temperature regulation led to a temperature drift of about
10\% to 20\% and overheating occurred. This limited the data
interpretability, especially for the planned different latitudinal
heating scenarios. As optical techniques shadowgraph and back-focus
Schlieren visualisations were used, which give information about the
temperature structures. Successive images from the fixed camera only
allowed flow reconstruction in weakly time-dependent flows but not for
more chaotic and turbulent flows. During the second flight, an
additional video camera was installed, but it failed at the 19th run.
Originally, it was also planned to use spiropyrane, a photochromic dye
that is activated by UV light, as a flow tracer for velocity
measurements during the Spacelab~3 mission. However, the dye degraded
over the time of 1.5 years when it was not accessible during
integration into the space module, thus, not permitting the extraction
of accurate velocity data.

The {GeoFlow} experiment was very similar in conception, and
was integrated into the Fluid Science Laboratory of the European
Columbus module on the International Space Station (ISS). This crucial
difference allowed for much longer, extensive campaigns with more than
2500~h of scientific run time~\cite{egbers1999_mst, egbers2003_asr,
Futterer2010, Futterer2012, Futterer2013, Zaussinger2017,
Zaussinger2020}. It allowed daily data transfer from the orbit and
thereby almost real-time data analysis. The {GeoFlow} experiment
went to space for the first time on 7 August 2008 and stayed there
until January 2009 ({GeoFlow~I}), with three follow-up missions
(March 2011--May 2012; December 2012--May 2013; November 2016--February
2017; ({GeoFlow~II, IIb, IIc}). The head of the {GeoFlow}
Team was Christoph Egbers from the Brandenburg University of Technology
Cottbus-Senftenberg. Unlike {GFFC}, {GeoFlow} was a full
sphere with inherent limitations due to the heating supply shaft at the
south pole~\cite{egbers1999_mst, egbers2003_asr}. The device had an
inner radius of $R_{\mathrm{i}} = 1.35$~cm and an outer radius of $R_{\mathrm{o}} = 2.70$~cm
with the possible rotation rates of $\Omega = 63$~rad/s. The working
fluid of {GeoFlow~I} was the silicone oil M5 with $Pr = 64$. For
{GeoFlow~II}, the team improved the olfactory aspect and used
the alcohol 1-nonanol, known in the cosmetics industry for its lemon
fragrance, with $Pr = 125$ and $Pr = 178$. Most of the results were for
very slow rotation rates that were mainly there to ensure the
visualisation of the full latitudinal direction and more relevant to
mantle convection~\cite{Futterer2010, Futterer2012, Futterer2013,
Zaussinger2017}. Only the {GeoFlow~IIc} experiment focused on
rotating convection with parameters comparable to the {GFFC}
experiment, with a minimum $Ek = 2.64 \times 10^{-3}$, and maximum $Ra
= 1.59 \times 10^5$. The fluid 1-nonanol also allowed them to study
both convection induced by the temperature difference between the
spherical shells and due to the internal dielectric
heating~\cite{Zaussinger2020}. The flow visualisation technique used
was Wollaston prism shearing interferometry which showed only
projections of the thermal structure since safety and weight
limitations did not permit the use of tracer particles or larger
optical systems~\cite{Zaussinger2020}. Exemplar pictures are shown in
Figure~\ref{fig:space}.

Both experiments, {GFFC} and {GeoFlow} studied the
stability properties, the formation of patterns and the transition
between states, including chaos and turbulence. Many of the results had
rather high values of $Ek$ and $Ro$, as well as low values of $Ra$, 
thus, they are practically in a rotation-unaffected
regime~\cite{Aurnou2020, kunnen2021_jot, Ecke2023}. These flows show a
tessellated pattern, also found in more recent numerical simulations of
non-rotating convection~\cite{Gastine2015, Featherstone2016}. The
{GeoFlow} experiments also show mantle-like plumes influenced by
non-Oberbeck--Boussinesq effects~\mbox{\cite{Horn2013, Futterer2010,
Futterer2012, Futterer2013}}.

In the rotating regime, the main result was the confirmation of the
existence of columnar convection in radial gravity (``banana
cells''~\cite{busse1970_jfm}) in the equatorial region. Experiments
also captured their drifting in the prograde direction, illustrated in
Figure~\ref{fig:space}~\cite{hart1986_jfm, hart1986_sci,
Zaussinger2020}. In both experiments, columnar convection was very
robust, likely because of the relatively high Prandtl
number~\cite{Julien2012, Stevens2010b, Horn2014, Horn2015,
kunnen2021_jot, Abbate2023}. The experiments, especially {GFFC},
also showed that at higher supercriticalities, retrograde mid-latitude
convection modes develop, the TC becomes convectively active, and later
on these mid-latitude and polar modes become unstable. Likely, there
were two effects at play here, a loosening of the rotational
constraint, i.e.\ moving closer to the non-rotating regime expected at
higher supercriticality~\cite{agrawal2024_gji}. The unique specificity
of {GFFC} was that they could also observe the interactions of
the columnar equatorial cells with the retrograde-propagating
mid-latitude convection modes, which can only be captured with
spherical radial gravity. These interactions initially caused a
wavering, and finally the erosion of the columnar equatorial cells.
They also found that the virtual boundary of the tangent cylinder broke
down, with columnar convection only within 20{\textdegree} of the
equator whereas the tangent cylinder extended up to 43{\textdegree}.
The no-slip boundary conditions (in addition to the not-very-extreme
control parameters) were likely the reason for not observing any zonal
flows. However, these boundary conditions are arguably also more
relevant for Earth's core than for gas giants. Many of the results are
indeed reminiscent of more modern numerical spherical simulations of
rotating thermal convection~\cite{Gastine2016, Featherstone2016,
Featherstone2023}. They further also showed that spherical shell
thermal convection may be hysteretic, finding sometimes the same and
sometimes different flow patterns depending on whether they increased
or decreased $Ra$~\cite{hart1996_3mg, Hart1999}. The {GFFC}
experiment also explored latitudinal inhomogeneous heating. They
observed spiralling convection when the north pole was heated, which
was interpreted as evidence for baroclinic waves. They showed that with
increased differential heating, the interaction of the equatorial
columnar cells with mid-latitude waves leads to triangular waves and
ultimately, with turbulent structures moving downward from the pole to
full turbulence.\looseness=-1

\subsection{\ldots\,and elsewhere: Kelvin forces and pycnoclinic
acoustic gravity} \label{sec2.2}

The main limitations of the space experiments' applicability to
planetary core convection are the relatively low $Ra$ and high $Ek$
required to minimise centrifugal effects. Their other shortcoming is
that the poor electric conductivity of dielectric materials renders
the study of MHD effects practically infeasible.

Having said that, it may be worthwhile to embrace Coriolis-centrifugal
convection, and so thanks to the combination of $\Omega^2\mathbi{r}$
and~$\mathbi{g}$, spot tornado-like vortices as they occur in planetary
and stellar atmospheres~\cite{Luna2015, Horn2018, horn2019_prf,
Horn2021, Ding2021, Kuniyoshi2023, Tziotziou2023}.

Ferrofluids offer an alternative route to creating a central force field,
with the benefit that some of them are also electrically conducting.
Experiments of this kind have been conducted in non-rotating
thermo-magnetic convection where spherical gravity is mimicked using
permanent magnets as an inner core, and a ferrofluid fills the outer
core~\cite{Rosensweig1999, frueh2005_nlpg}. However, neither rotating
convection nor magnetoconvection have so far been realised. The idea is
here that a magnetised fluid is attracted towards higher magnetic
fields. This attracting body force is referred to as Kelvin force and
mimics buoyancy. The magnetisation depends on both the magnetic field
and the temperature, a colder fluid is attracted more strongly than a
warmer fluid. This is expressed through the pyromagnetic coefficient
which plays the role of the expansion coefficient.

Acoustic gravity is another clever way of creating thermal rotating
convection with a central force field in an Earth-bound laboratory
setting~\cite{Koulakis2023}: Sound, when averaged over many cycles,
exerts a force on density gradients in a gas. The so-called pycnoclinic
acoustic radiation force $-\langle v_{\mathrm{ac}}^2\rangle \nabla
\bar{\rho}/2$ yields an acoustic gravity of $\mathbi{g}_{\mathrm{ac}} =
\nabla \langle v_{\mathrm{ac}}^2\rangle/2$~\cite{Koulakis2018,
Koulakis2021}. Here, $\bar{\rho}$ is the time-averaged density and
$v_{\mathrm{ac}}$ is the acoustic velocity. 
Koulakis et~al.~\cite{Koulakis2023} use a rotating spherical plasma bulb with a radius of
1.5~cm filled with a weakly ionized sulfur gas. The gas is heated
volumetrically by microwaves. The rotation is essential to guarantee
quiescent conditions of the plasma. Then, the amplitude modulation of
the microwave power generates a high amplitude, spherically symmetric
acoustic standing wave. This creates an acoustic gravity that can
briefly reach values of up to 1000 times Earth's gravity, so
centrifugal effects become negligible. Gravity changes sign roughly at
half the bulb radius, so confining the convective zone to the outer
region, as in stars and Earth's core. While the parameters are only
slightly more extreme than the space experiments, density differences
of a factor of about two make it possible to study compressible
rotating spherical convection too.

\section{Cylindrical radial gravity} \label{sec:cyl}

Given the difficulties in modelling spherical rotating thermal
convection with central force fields, geophysicists, planetary
scientists and fluid dynamicists had to resort to alternative means to
model planetary core convection. A very successful branch of laboratory
research in this regard relied on centrifugal buoyancy to mimic
equatorial and low-latitude convection.

\subsection{The Busse annulus} \label{sec3.1}

The cylindrical annulus geometry was first conceptualised as a
laboratory device with direct application to Earth's core by Busse in
1970~\cite{busse1970_jfm}. The underlying idea here is to use the
cylindrically radially outward-directed centrifugal force and then cool
the inner boundary and heat the outer one. This ensures that buoyancy
acts in the right direction, as sketched in
Figure~\ref{fig:geometries}. Only the product of gravity and
temperature gradient matters physically. Busse let the centrifugal
acceleration exceed Earth's gravity by a factor of two or
three~\cite{Busse2009} and envisioned slanted boundaries, where the
height varies with cylindrical radial distance from the vertical
rotation axis. The resulting $\beta$-effect creates slowly drifting
columnar thermal Rossby waves. These are nowadays also known as
\textit{Busse} columns and have been argued to exist in planetary
cores~\cite{busse1970_jfm}. Busse first derived a simple analytical
model for the onset Rayleigh number, drift frequency and wave number in
the small gap approximation with asymptotically small endwall slopes.
He then materialised his theoretical ideas into an innovative
laboratory setup~\cite{busse1974_jfm}. The concept of the \textit{Busse
annulus} turns the arduous three-dimensional spherical problem into a
tractable two-dimensional one and offers simple access to the spherical
rotating convection problem.

Busse (and collaborators) subsequently fleshed out the original
idea~\cite{busse1974_jfm, Azouni1986, Busse1982, Schnaubelt1992,
Busse1986a, Busse1986b, Or1987, Westerburg2003, Jaletzky2000, Tao2006,
Auer1995}, including the application to
magnetoconvection~\cite{Busse1993, Kurt2007} and the
geodynamo~\cite{Busse1975} (see also the reviews~\cite{Busse1994,
Busse1997, Busse2002, Busse2009} and Section~\ref{sec:sphere} on the
spherical geometry). The \textit{Busse Annulus} has been subject to
many theoretical and numerical studies up to recent
days~\cite{Bassom1996, Oishi2022, Teed2012, Wicht2018, Calkins2013,
Barik2023}. In particular, Calkins et~al.~\cite{Calkins2013} lifted the
original restriction to small slopes and near-criticality. He
introduced an asymptotically reduced three-dimensional set of equations
for quasi-geostrophic convection, thus, establishing a closer
connection to spheres and spherical shells. 

Here, we focus on the experimental aspects of the \textit{Busse
Annulus}~\cite{busse1974_jfm, Busse1982, Azouni1986, Busse1997,
Jaletzky2000}. A sketch of the first laboratory
realisation~\cite{busse1974_jfm} is shown in
Figure~\ref{fig:busseannulus}(a). Two concentric cylinders were mounted
on two circular end plates. The outer cylinder was heated through a
circulating thermostatically controlled reservoir enclosed in an outer
nearly cubical Plexiglas box. The inner cylinder was cooled by
circulating water through two centred hollow shafts mounted on the end
plates, and ball-and-socket couplings connected the stainless steel
shaft to the cooling tubes. Baffles at the inside of the end plates
ensured an efficient distribution towards the inner annulus wall. The
shafts also served as rotation axis, and rotation rates of up to
400~rpm (41.9~rad/s) were achieved. The temperature difference was
measured using a thermocouple in each of the water baths. Since the
experiment concerned the onset of convection, typical temperature
differences were less than 1~K. The inner cylinder was made of
aluminium and three different sizes were used, $R_{\mathrm{i}} =
\{13.34~\mathrm{cm}, 3.81~\mathrm{cm}, 3.18~\mathrm{cm}\}$. The outer
cylinder was made of acrylic plastic (Plexiglas) to allow for optical
access, and the outer Plexiglas cube helped to avoid optical
distortions. The used radii were $R_{\mathrm{o}} = \{13.94~\mathrm{cm},
4.775~\mathrm{cm}, 4.74~\mathrm{cm}\}$. The endwalls were made of
Teflon and three different sloping angles were used: $\chi = 0$
(corresponding to horizontal plates) and $\chi =
22.5\text{\textdegree}$ and 45{\textdegree}. The mean heights were $L =
\{28.6~\mathrm{cm}, 6.97~\mathrm{cm}, 6.69~\mathrm{cm},
1.96~\mathrm{cm}, 0.95~\mathrm{cm}, 0.45~\mathrm{cm}\}$. The working
fluids were silicon-oil and water. Small amounts of Kalliroscope were
used for flow visualisation. These are nearly neutrally buoyant
particles that align with the shear. Stroboscopic light was used to
make the shear patterns visible and to measure the rotation rate. The
onset of convection was determined visually. 

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig:busseannulus}(a)~Schematic vertical cross-section
of Busse and Carrigan's original annulus apparatus, adapted from
Ref.~\cite{busse1974_jfm}. The northeast lines pattern indicates the
rotating parts and the crosshatch dotted pattern indicates the
non-rotating parts, in particular, the Plexiglas cube. The radius and
height of the inner and outer cylinders were adaptable and varied
during the experimental campaigns. (b)~Photograph of two annuli stacked
on top of each other, scale is in cm, adapted from
Ref.~\cite{busse1974_jfm}. The upper part shows an annulus with
constant height, showing strong convection columns. The lower part
shows an annulus with conical endwalls, that results in an axisymmetric
flow at the same parameters. The working fluid was water at
30~{\textdegree}C and Kalliroscope was used for visualisation.}
\end{figure}

Quantitative measurements involved measuring a buoyancy parameter $B$
versus an inverse Ekman number $Ek = \nu/(\Omega L^2)$ (bar the
factor~2 in both cases). In modern terminology, their buoyancy
parameter is in fact the square of the (convective) Rossby number. This
can be seen by recalling that $Ro=
\tau_{\mathrm{rot}}/\tau_{\mathrm{buoy}}$, i.e.\ the ratio of a
rotational time scale, $\tau_{\mathrm{rot}} = 1/\Omega$ and the
buoyancy time scale, $\tau_{\mathrm{buoy}} = H/\sqrt{\alpha \Delta
\Omega^2 \bar{R} H}$, defined in analogy to the usual free-fall time
scale, with the temperature difference $\Delta =T_h - T_c$, $\Omega^2
\bar{R}$ being used as centrifugal gravity, and the characteristic
length scale being the gap width~$H$. Thus, we have
{\begin{equation}
Ro^2 = B= \frac{\tau_{\mathrm{rot}}^2}{\tau_{\mathrm{buoy}}^2} =
\frac{\alpha \Delta \Omega^2 \bar{R} H}{H^2 \Omega^2} = \frac{\alpha
\Delta \bar{R}}{H}.
\end{equation}}\unskip
At the onset of convection, Busse and Carrigan saw the first
experimental evidence of the retrograde drifting columns that were
slightly tilted and helical. Their results showed decent agreement with
the small gap linear theory~\cite{busse1970_jfm}. Qualitatively, it was
also found that the wavelength of the instability decreased with
increasing rotation rate in the sloping endwall cases. The
instabilities in this case were typically the superposition of several
waves and the columns showed beating phenomena. Dedicated experiments
sought hysteresis and subcritical convection but found none of them.
The $\beta$-effect arising out of the interplay of the Coriolis force
with the top and bottom conical endwalls was found to inhibit the
convective instability compared to the flat case that develops
columns, see Figure~\ref{fig:busseannulus}(b). At high rotation rates
the experiment strikingly exhibits geostrophic columns, in agreement
with the TPC, which imposes that the flow be $z$-independent in these
regimes.

\paragraph*{Zonal flows}

For the observations of zonal flows, Busse and Hood modified the Busse
annulus ($L = 12.6$~cm, $H = 2.46$~cm, $\bar{R} = 3.51$~cm) to include
radial curvature of top and bottom boundaries, either convex or
concave~\cite{Busse1982}. The mean flow was measured by releasing
electrolytic dye into the working fluid (water) by frequent pulsed
currents, following a method devised by Baker~\cite{Baker1966}, and
taking photographs at intervals of~10~s.

A similar experiment by Azouni et~al.~\cite{Azouni1986} used mercury
and water as as working fluids. The outer cylinder was replaced by
anodized aluminium to avoid the thermal loss through the acrylic
sidewall ($L = 14.48$~cm, $\bar{R} = 3.917$~cm, $H = 3.917$~cm and
$\chi = 47.4\text{\textdegree}$). They measured drift rates and
amplitudes of the convection columns by combining and correlating five
thermistor probes in the equatorial plane and spaced in azimuth. Three
were embedded in the inner cylinder flush with the outer surface, two
others were glued to the outer cylinder. 

Earth's gravity creates a thermal wind and a meridional circulation
scaling as $\Omega^{-1}$ and $\Omega^{-1/2}$, respectively. The thermal
wind is antisymmetric relative to the equatorial plane, being
retrograde in the lower part and prograde in the upper part, thus, it
does not affect the dye injected in the equatorial plane too much. The
meridional circulation is noticeable, on the other hand. It leads to an
upflow near the hot outer wall and a downflow near the cold inner wall,
thus, there is an additional prograde zonal flow at the outer wall and
a retrograde flow at the inner one. 

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig:diffrot}Creation of mean zonal flows in spherical
shell and annuli geometries. (a)~Convective columns, initially at rest;
(b)~columns get tilted, here in the prograde direction, e.g.\ by small
fluctuations which create Reynolds stresses. Prograde momentum (purple
arrows) is carried outwards and retrograde momentum (green arrows) is
transported inwards. (c)~Resulting prograde flow at the outer boundary
and retrograde flows at the inner boundary increases the initial tilt
leading to a feedback process sustaining a mean flow.}
\end{figure}

Both experiments confirmed the existence of a mean zonal flow. The main
differential rotation was argued to stem from a mean flow instability
due to Reynolds stresses generated through strong mean zonal shear,
shown schematically in Figure~\ref{fig:diffrot}. The convective columns
in Figure~\ref{fig:diffrot}(a) get slightly tilted, e.g.\ by small
fluctuations. This creates Reynolds stresses $\langle u_r' u_{\phi}'
\rangle$, where $u_r'$ and $u_{\phi}'$ are the fluctuating part of the
radial and azimuthal velocity component, respectively. Depending on the
initial tilt, either direction of the evolving mean flow is possible.
If tilted in the prograde direction, Figure~\ref{fig:diffrot}(b), then
prograde momentum (purple arrows) is carried outwards and retrograde
momentum (green arrows) is transported inwards. Thus, the flow is
retrograde at the inner boundary and prograde at the outer boundary.
The resulting flow increases the initial tilt leading to a feedback
process sustaining a mean flow, Figure~\ref{fig:diffrot}(c). Thus,
there is a differential rotation in which the outer fluid rotates
faster than the inner one. If the columns are tilted in the opposite
direction, the mean flow direction is reversed. For similar reasons,
for convex endwalls, thermal Rossby waves propagate faster on the
outside than on the inside, thus, the columns spiral outwards and the
mean flow is retrograde at the inner boundary and prograde on the outer
boundary. The opposite happens for concave endwalls~\cite{Busse1982,
Busse2009}. Viscous stresses oppose the differential rotation and lead
to an equilibrium. The differential rotation was argued to be rather
insensitive to the cylindrical annulus sidewall, and, hence should also
be found in spherical shells~\cite{Busse2009}. Azouni
et~al.~\cite{Azouni1986} found drift rates in agreement with Busse's
simplified theory, whereas the amplitudes decreased with increasing
Rayleigh number.

\paragraph*{More recent developments}

Busse continued investigating flows in the \textit{Busse annulus}. He
and his collaborators found quasi-geostrophic chaotic states in
air~\cite{Busse1997}. They also devised an elegant method to visualise
convective patterns using thermochromatic liquid crystals in high-$Pr$
fluids~\cite{Jaletzky2000}. At such high $Pr$, the effects of rotation
were much lower. They found different types of patterns, such as knots
and hexaroll, which are predicted theoretically~\cite{Auer1995}, but
also patterns that still lack a theoretical description such as oblique
rolls.

The \textit{Busse annulus} without sloping endwalls was later
replicated by Jiang et~al.~\cite{jiang2020_sciadv, jiang2022_prl}; they
rediscovered the centrifugal force as a proxy for gravity and called it
supergravity. Their achievable parameters were comparable to Busse's
original design, with rotation rates between 211~rpm to 705~rpm,
focussing on the strongly supercritical regime. While they also
rediscovered the mean flow instability, the constant radial height does
not permit e.g.\ Rossby waves and thus, makes the experiment less
geophysically relevant.

\subsection{Spheres, spherical annuli and hemispherical shells}
\label{sec:sphere}

Spherical geometries are a natural improvement of the \textit{Busse
annuli}, as they capture a more realistic $\beta$-effect and, in some
setups, the Stewartson layers wrapped around the
TC~\cite{greenspan1969, kunnen2013_jfm}. Here too, centrifugal gravity
is used to mimic the gravitational field in the equatorial region,
still at the expense of the polar regions. As for the \textit{Busse
annulus}, since the centrifugal gravity points outwards, the
temperature difference has to be reversed with the outer spherical wall
held at constant hot temperature using either
water~\cite{carrigan1983_jfm} or air~\cite{cardin1994_pepi}.

\looseness=-1
In spherical shell geometries, the radial temperature gradient is
controlled by cold fluid (water, or even Kerosene~\cite{shew2005_pepi})
circulating through the inner sphere. A solid shaft must be fitted to
connect the inner sphere to the cooling circuit and to hold the solid
sphere at the centre of the outer sphere. This, however, finalises the
sacrifice of the polar regions. Because of this shaft, we refer to
setups of this category as the \textit{spherical annulus}
configuration. Another non-planetary feature of the spherical shells is
the misalignment of the temperature gradient and gravity that increases
with latitude. This source of baroclinicity combined with the Coriolis
force incurs an artificial azimuthal wind, and results must therefore
be interpreted keeping its influence in mind~\cite{carrigan1983_jfm}.

Several experiments were built on this principle with variants trying
to address some of its shortcomings. These experiments are classics of
geophysical rotating convection and have been extensively
reviewed~\cite{olson2011_pepi, olson2013_areps, cardin2015_tg}. We
shall only recall their main outcomes and highlight some specificities
of the experimental approach itself.

\paragraph*{Experiments at moderate \textit{Pr}: water and oil}

The first spherical annulus experiment was designed by Busse and
Carrigan~\cite{busse1976_sci, carrigan1983_jfm}, whose original sketch
is reproduced in Figure~\ref{fig:sphere_setups}(a). The working fluids
were water and ethylene glycol and the inner core radius was varied by
using different inner spheres. High rotations up to 1000~rpm ensured
the gravity was effectively cylindrical radial. The outer sphere (inner
radius 10~cm) was made of transparent plastic and heated by a
temperature-controlled water bath. The flow was diagnosed by seeding
the fluid with reflective platelets that align with the shear and
taking snapshots through the outer sphere. From these visualisations,
the authors were able to detect the onset of convection and
characterise the size of the onset structures. They mostly recovered
Busse's linear theory for the onset of convection in a rotating shell
with central gravity in the small-gap limit and identified the onset
structures as being Rossby waves originating near the inner sphere, at
$Ra_c\sim Ek^{-4/3}$, see original \mbox{visualisations} in
Figure~\ref{fig:sphere_patterns}(a). The critical wavenumber of the
Rossby waves follows the theoretical scaling law but with a lower
prefactor. A discrepancy between the experiment and the small gap
theory arises due to the presence of Stewartson layers developing along
the TC. The Stewartson layers tend to stabilise the Rossby waves
because they are precisely located where the waves originate. This
effect becomes negligible in the limit $Ek\rightarrow0$.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig:sphere_setups}Evolution of the spherical shell
experiments since 1976: (a)~1972--1983: Busse and Carrigan's original
spherical shell convection device with spherical inner cold
boundary~\cite{busse1976_sci, carrigan1983_jfm}, (b)~1984--1986:
Chamberlain and Carrigan's modification of Carrigan and Busse's
original setup, with no inner sphere and linearly increasing outer wall
temperature~\cite{chamberlain1986_gafd}, (c)~1990--1993: Cordero and
Busse's hemispherical experiment~\cite{cordero1992_grl},
(d)~1001--1994: Cardin and Olson (1994)'s air-cooled
setup~\cite{cardin1994_pepi}, (e)~1999--2003: Sumita and Olson's
modified version of Cardin and Olson's setup with a hemispherical
annulus~\cite{sumita1999_sci, sumita2000_pepi, sumita2002_jgr,
sumita2003_jfm}, (f)~1999--2007: Aubert and collaborators' water and
liquid metal experiment~\cite{aubert2001_pepi, gillet2007_jfm1}.}
\end{figure}

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig:sphere_patterns}Patterns of rotating convection in
a spherical gap by order of criticality: (a)~Rossby waves at the onset
of convection originally observed by Busse and Carrigan (side
view)~\cite{busse1976_sci}, (b)~spiralling columns at
$\widetilde{R}a=5.9$ (top view through the transparent top lid of the
hemispherical vessel)~\cite{sumita2000_pepi}, (c)~dual convection at
$\widetilde{R}a=19.3$ (top view as for~(b))~\cite{sumita2000_pepi},
(d)~chaotic convection at $\widetilde{R}a=50$ (side
view)~\cite{cardin1994_pepi}, (e)~same as~(d), but top view.}
{\vspace*{-5pt}}
\end{figure}

Chamberlain and Carrigan~\cite{chamberlain1986_gafd} later modified the
setup, by removing the inner sphere and thereby the Stewartson
layers~\cite{chamberlain1986_gafd}, see
Figure~\ref{fig:sphere_setups}(b). The outer bath temperature was
controlled to increase linearly in time to maintain a constant
temperature gradient in the absence of inner cooling. After a short
transient, the lag in the temperature response of the fluid inside the
sphere leads to a constant temperature gradient there. The radial
temperature profile is then the same as if the heat source was
distributed in volume, up to a background temperature linearly
increasing in time. This radial temperature profile was monitored by
radially aligned temperature probes. The critical Rayleigh numbers were
found in better agreement with Robert's linear theory for an internally
heated sphere~\cite{roberts1968_prsa} than Busse's small-gap linear
theory~\cite{busse1970_jfm}. As in the previous experiment, however,
the centrifugal wind incurred by the high-latitude baroclinicity
prevented the authors from observing the retrograde drift of the Rossby
waves. This led them to suggest experiments using a cylindrical annulus
to eliminate that effect as Busse and collaborators
did~\cite{Busse1982, Azouni1986}.

The first measurement of the drift of Rossby waves with a spherical
geometry came from Cordero and Busse~\cite{cordero1992_grl} using a
hemispherical shell. Baroclinic effects were minimised by rotating the
hemisphere at a speed where the sum of gravity and centrifugal forces
produced an apparent gravity with parabolic iso-values nearly parallel
to the curvature of the sphere. The outer sphere was made of glass and
heated by an external thermostatically-controlled bath of water. The
inner sphere was made of brass with an internal water-cooling circuit. 
Using the same visualisation method as in spherical
experiments~\cite{busse1976_sci} and thermistors, the authors captured
the variations of the decay of the drift velocity with the Rayleigh
number as well as the appearance of secondary rolls of larger
wavelength near the outer sphere.

The next series of spherical annulus experiments was initiated by
Cardin and Olson~\cite{cardin1992_grl, cardin1994_pepi}, see
Figure~\ref{fig:sphere_setups}(c). They also used water but with a
slightly larger outer sphere of radius 15~cm. Unlike previous spherical
annuli, the outer temperature was thermostatically controlled at room
temperature by the air current induced by the sphere's rotation. The
rotation up to 400~rpm was somewhat lower than in Carrigan and Busse's
experiment, so the proxy-gravity was not entirely cylindrical-radial
but also had a significant vertical component. A new visualisation
system relied on snapshots synchronised with a strobe light
illuminating vertical planes and a fluid seeded with rheoscopic flakes
to identify the structures. In the horizontal planes, the flow was
visualised by releasing dye at the inner sphere. The new visualisation
system made it possible to investigate the regimes beyond the onset of
convection. They found that for moderate and higher supercriticality,
the classical Rossby waves give way to a chaotic columnar regime.
The~flow is quasi-geostrophic, made of irregularly distributed columns
extending over the entire shell height. Its basic pattern is made of
retrograde vortices produced at the inner core boundary extending
outward into prograde spiralling streets of retrograde vortices (see
Figure~\ref{fig:sphere_patterns}(d)). This regime is consistent with
equatorial mirror-symmetry~\cite{hulot1990_jgg} and non-periodic
magnetic flux~\cite{gubbins1987_nat} inferred from geomagnetic data.
Scaling up experimental velocities in this regime to Earth parameters
yields estimates of 0.1~cm/s for the convective velocity, consistent
with estimates from geomagnetic variations~\cite{bloxham1989_ptrsa}.
However, the extrapolated column size of 20~km-radius columns is
unobservable and inconsistent with the large-scale geomagnetic data.
Cardin and Olson attributed this discrepancy to the action of the
Lorentz force in the core.

The transition to the chaotic regime and the fully developed stages of
convection were further investigated by Sumita and
Olson~\cite{sumita2000_pepi}. They modified Cardin and Olson's setup
into a lower hemisphere, with copper inner and outer boundaries, closed
at the equatorial plane with a transparent Plexiglas lid. The copper
parts enabled a better control of the temperature with a more
homogenous distribution at the boundary. The lid offered a better
visualisation window and its near-thermally insulating properties left
the mostly radial conducting thermal gradient unaffected. The
temperature was monitored by thermistors inserted at adjustable depths
through the lid. They rotated the experiment at a fixed speed of
206~rpm to generate a parabolic gravity iso-potential, with radial
gravity seven times greater at the equator than at the pole. Under
these conditions, Sumita and Olson observed the progressive filling of
the gap by Rossby waves drifting in the retrograde direction for
$\widetilde{Ra}=Ra/Ra_c\leq 8$, see
Figure~\ref{fig:sphere_patterns}(b), as well as a radial increase in
their azimuthal wave number: the effect is driven by the increasing
slope enforcing an ever stronger TPC suppressing radial motion. At
higher forcing $\widetilde{Ra} &gt; 8$, this effect splits the flow into a
turbulent inner region and strongly suppressed motion in the outer
region where Rossby waves are expelled, a regime they coined 
\textit{dual convection}, illustrated in
Figure~\ref{fig:sphere_patterns}(c). 

Further experiments in the same setup using silicon oil and water as
working fluid enabled Sumita and Olson to explore much higher levels of
supercriticality, up to $\widetilde{Ra}=612$~\cite{sumita2003_jfm}. In
these regimes, the turbulent convection homogenises the temperatures to
a nearly isothermal state with a thin boundary layer near the inner
boundary and the heat transfer follows a scaling law of the form $Nu\sim
Ra^{0.4}$. This law, and a comparison of experimental thermal
fluctuations to theory~\cite{cardin1994_pepi} led the authors to
identify this regime as geostrophically turbulent.

\paragraph*{Experiments at low \textit{Pr}: liquid metals}

The advent of Ultrasound Doppler Velocimetry (UDV)~\cite{brito2001_ef,
baker2017_ef, Vogt2021}, which unlike the name suggests does not rely
on the Doppler effect~\cite{signalprocessing}, made it possible to
measure velocities in the bulk flow of core fluid-like liquid metals.

Aubert and collaborators~\cite{aubert2001_pepi} took advantage of this
technology to perform experiments in a spherical annulus geometry with
a thick central shaft, and no inner sphere, see
Figure~\ref{fig:sphere_setups}(f). The shaft provided cooling through a
central cylindrical copper wall, so in planetary terms, Aubert et~al.'s
experiment had a cylindrical core and a solid, impermeable TC. They
performed experiments both in water and gallium. With water, direct
visualisations with Kalliroscope were obtained through the outer sphere
made of transparent plastic and maintained at thermostated room
temperature by air currents as in Cardin and Olson's
experiment~\cite{cardin1994_pepi}. With gallium, optical visualisation
is not possible and the outer sphere was replaced by a copper sphere
heated by a resisting wire wrapped around it. They measured velocities
in both fluids across the gap with two probes along two directions
within the equatorial plane. From these two measurements, they are able
to reconstruct radial (convective) and azimuthal (zonal) velocities.
For both fluids, the maximum convective velocity in the turbulent
regime scales as $u_r\sim (\alpha g Q/\rho
C_p\Omega^3(R_{\mathrm{o}}-R_{\mathrm{i}})^4)^{2/5}$, a law that can be recovered by assuming
a triple balance between Coriolis, inertia and the Archimedean
(buoyancy) force, called CIA balance. Here, $Q$ is the heat flux, and
$C_p$ is the heat capacity at constant pressure of the fluid.
Velocities rescaled by this law are higher in gallium than water for
the same criticality. The reason is the two-order-of-magnitude lower
$Pr$ of gallium. These high velocities drive a strong zonal flow, 
which forms as the condensate of an inverse energy cascade fed by the
convective scales~\cite{gillet2007_jfm1}. The condensate materialises
at Rhines' scale, which results from the balance between the
$\beta$-effect and Reynolds stresses~\cite{rhines1975_jfm}. Energy
dissipation at this scale and for the larger scales of turbulence
occurs by Ekman friction. However, bold extrapolation of this
phenomenology to Earth leads to overestimated velocities by an order of
magnitude and too small scales, compared to geomagnetic data. The
authors attributed the mismatch, again, to the absence of a magnetic
field in their experiments.

Gillet and collaborators conducted further experiments with the same
setup up to $\widetilde{R}a=80$ focusing on the zonal
flow~\cite{gillet2007_jfm1}. In the viscous regime of water
experiments, the zonal flow scales with the onset convective scales. In
the inertial regime reached with gallium, by contrast, they recovered
the phenomenology of Aubert et~al.~\cite{aubert2001_pepi} with a zonal
flow scaling as Rhines' scale. Based on this phenomenology, they showed
that at these larger criticalities, the average zonal flow
$\overline{U}_{\phi}$ scales with the \textit{rms} radial convective
flow $\widetilde{U}_r$ as $\overline{U}_{\phi}\sim
\widetilde{U}_r^{4/3}$. Unlike early experiments at low criticality in
water, where the zonal flow expected from the drift of Rossby waves is
weak, the zonal flow observed in this regime is at least an order of
magnitude faster than the parasitic thermal wind incurred by the
misalignment of temperature gradients and gravity. Hence, in these
turbulent regimes at low $Pr$, the spherical annulus produces a more
geophysically relevant zonal flow than most of the water experiments.

Zonal flows controlled by the Rhines scales were also recovered in the
\textit{Coraboloid} experiment recently conducted at
UCLA~\cite{lonner2022_jgrp}: the $\beta$-effect was obtained by
rotating water in a 37.25~cm diameter cylindrical tank with a free
surface. Rotations speeds of 35~rpm, 50~rpm and 60~rpm lead to a
parabolic deformation of the free surface. Convection in this
``paraboloid core'' was driven by a cold 10.2~cm cylinder placed at the
centre of the tank spanning the entire fluid height. Zonal flows form
in all cases but are more stable at 60~rpm. The Rhines scale varies
little between 50~rpm and 60~rpm but centrifugal buoyancy only
dominates vertical buoyancy at 60~rpm. The authors concluded that
centrifugal buoyancy stabilised the zonal flow, which is consistent
with Gillet et~al.'s~\cite{gillet2007_jfm1} experiments in liquid
metal. The properties of jets in a geophysical context are extensively
reviewed by Read et~al.~\cite{read2024_crphys}.

Lastly, Shew and Lathrop~\cite{shew2005_pepi} built a large spherical
annulus (external diameter), $D = 60$~cm with magnetic runs and dynamo
applications in mind. A much larger setup was later built more
specifically to create a dynamo, which is still under
development~\cite{lathrop2011_pd}. However, the Shew and Lathrop
experiment remains to date the only rotating convection experiment
running with liquid sodium. Measurements rely on thermocouples immersed
within the fluid, and velocities are obtained by correlating
neighbouring thermocouples and using Taylor's hypothesis. They focussed
on the turbulent regime and observed small-scale convective motions
with a strong, large-scale retrograde azimuthal flow. The local
intensity of the zonal flow intensity fits a scaling of
$u_{\theta}\simeq 3.5 \Omega\alpha D\Delta$ obtained from a simple
balance between Buoyancy and the Coriolis force. As one would expect,
they observe developed turbulence with weak convective heat transport
and a near-diffusive temperature profile. They measured Reynolds
numbers in the range $Re\sim 10^3$--$10^4$, and a total radial heat
flux following a $Nu\sim Ek^{-1/3} Ra$ scaling, indicating a rotation-dominated
heat transfer~\cite{kunnen2021_jot, ecke2023_arfm}. Extrapolating these
measurements to Earth leads to an estimated Rayleigh number of $Ra\sim
10^{23}$ and convective velocities comparable to the azimuthal flow of
$\simeq 2\times10^{-4}$~m/s, from which they estimated the magnetic
Reynolds number in the equatorial regions of the Earth to be
$Rm\simeq2\times10^2$. They identified a ``knee'' in the energy spectra
where energy was injected by convection into the flow, which,
extrapolated to Earth, corresponds to a turnover time of 30 days, and a
size of 1~km, again well below any observable scale. They also derived
an estimate for the Joule dissipation in the Earth's core in the range
of{\break} 10~GW to 10~TW.

\subsection{Rotating magnetoconvection in spherical annulus
experiments} \label{sec:sphere_mhd}

Extrapolating rotating convection experiments to Earth yields estimates
that are inconsistent with geomagnetic data, for which the Lorentz
force usually takes the blame. It was therefore natural for rotating
magnetoconvection experiments to emerge. There are essentially three
strategies to go about these experiments: letting the fluid motion
itself generate its own magnetic field, putting a magnet around the
experiment, or bringing the experiment to a magnet.

The first strategy, dynamo experiments, however, poses such a challenge
on its own that no working concept of a convective dynamo experiment
has been put forward yet. There are even claims that such experiments
are impossible at laboratory scale~\cite{Tilgner2000}. Time will
tell\,{\ldots}

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig:spheremhd_setup}Liquid metal experiments:
(a)~2003--2005: Shew and Lathrop's sodium experiment with a Helmholtz
coil generating an axial magnetic field~\cite{shew2005_pepi},
(b)~2001--2007: Gillet et~al.'s Gallium experiment with their ``hairy
magnet'' made of a long toroidal wire wrapped around the shell,
generating an azimuthal magnetic field of up to
0.03~T~\cite{gillet2007_jfm2}.}
{\vspace*{-.15pc}}
\end{figure}

The second strategy has been pursued to date by only two
magnetoconvection experiments with a spherical geometry by Shew and
Lathrop and Gillet et~al.~\cite{shew2005_pepi, gillet2007_jfm2}, 
both represented on Figure~\ref{fig:spheremhd_setup}. These experiments rely on the 
rotating convection devices previously built by
the Maryland and Grenoble groups~\cite{shew2005_pepi, aubert2001_pepi,
gillet2007_jfm2}. The main drawback of this strategy is that only
relatively weak magnets can be transported and fitted around existing
setups, and thus, deliver limited magnetic fields. In the geophysical
context, the Lorentz force is typically assessed relative to the
Coriolis force by the Elsasser number $\Lambda=\sigma B_0^2/(2\rho
\Omega)$, which for the Earth is deemed to lie between 0.1 and
100~\cite{cardin1994_pepi, schubert2011_pepi, roberts2013_rpp,
horn2022_prsa, aurnou2015_pepi}. Shew and Lathrop~\cite{shew2005_pepi}
placed a 3~mT Helmholtz coil around their spherical annulus providing
$\Lambda\leq1.9\times10^{-4}$ and saw a global decrease in heat flux
with increasing magnetic field. Gillet and
collaborators~\cite{gillet2007_jfm2}, created a ``hairy magnet'', see
Figure~\ref{fig:spheremhd_setup}, by wrapping 444 turns of a wire
around meridional planes and through the central shaft of their
spherical annulus to generate an azimuthal field of up to 0.03~T, and
$\Lambda\leq9.94\times10^{-2}$. They found a zonal flow following the
non-magnetic phenomenology~\cite{gillet2007_jfm2}, but with larger
azimuthal length scales and magnetically controlled zonal velocity
scaling as $\bar{U}\sim (\widetilde{U}_r
\widetilde{U}_\theta)^{2/3}\sim \widetilde{U}_r
(l_\theta/l_\beta)^{2/3}$. Hence, both experiments showed the
stabilising effect predicted at low $\Lambda$~\cite{fearn1979_prsa,
soward1979_pepi}. Their relevance to the Earth is, however,
questionable considering the Lorentz force may become destabilising at
greater values of $\Lambda$ around unity~\cite{Chandrasekhar1961,
eltayeb1972_prsa, eltayeb1975_jfm, fearn1979_prsa, aujogue2015_pf,
horn2022_prsa}.

Finally, the third strategy, and the power of large magnets it affords,
has only been exploited in set-ups targeting the polar region, namely
LEE1 and LEE2, which took advantage of fields up to 10~T. These are
discussed in Section~\ref{sec:TC}.{\vspace*{-.3pc}}

\subsection{Experiments with inhomogeneous heating} \label{sec3.4}

In most experiments, great efforts are deployed to keep homogeneous
temperature boundary conditions, so as to preserve the ideal character
and to make the results universal. Yet, there is evidence that 
large-scale magnetic flux anomalies are driven by large-scale
inhomogeneities at the CMB~\cite{gubbins1986_grl, bloxham1987_nat,
johnson1998_grl} and that they may drive magnetic field
reversals~\cite{glatzmaier1999_nat, kutzner2004_grl, wicht2010_ssr,
olson2014_pepi, wicht2016_pepi, terranova2024_sr, tarduno2015_natcomm}.
Heat flux heterogeneity at the CMB may thus play a role in the interior
dynamics. Hart~\cite{hart1996_3mg, Hart1999} made the first attempt at
studying them but suffered from limitations inherent to space
experiments, when the heat source failed, as discussed in
Section~\ref{sec:central}. Since then, two experiments studied their
effect on rotating convection systematically.

Sumita and Olson~\cite{sumita1999_sci, sumita2002_jgr} adapted their
experiment~\cite{sumita2000_pepi, sumita2003_jfm} and created a local
thermal inhomogeneity with a small rectangular heater attached to the
outer boundary. Its size and latitudinal position were varied. This
extra heating models a local cold anomaly of the Earth's CMB where the
outward heat flux locally is higher since the temperature gradient in
the experiment is reversed compared to the Earth's outer core. The
total heat flux was monitored by comparing the inlet and outlet
temperatures of the inner sphere's cooling shaft. The intensity of the
anomaly was quantified by the ratio of excess heat flux at the
rectangular heater $q_h$ to the flux at the inner boundary~$Q$,
$Q^*=q_h/Q$. The heterogeneity drives a local prograde (eastwards)
flow. Two different regimes exist: For $Q^*&lt; 0.7$, in the so-called
\textit{local locking} regime, this flow remains localised near the
outer boundary. For stronger inhomogeneities, in the \textit{global
locking} regime, a large-scale spiral with a sharp front develops
across the whole gap, see Figure~\ref{fig:spiral}. 
The front separates warm and cold regions and is
accompanied by a thin jet linking the outer and inner boundaries, which
is responsible for an increase in global heat flux and large
inhomogeneities at the inner boundary. The regimes are practically
independent of the size of the heater but depend on its latitudinal
position: at low latitudes, the prograde flow forms but does not
develop into a front, whereas at higher latitudes, the front enters a
regime of periodic formation and destruction. Sumita and Olson
estimated the heat flux beneath East Asia to be $Q^*\simeq2$, based on
the CMB temperature and its variations~\cite{boehler2000_rg,
castle2000_jgr, garnero2000_areps} and seismic
models~\cite{vanderhilst1999_sci}. For this value, global locking is
possible, so they proposed that ``{the existence of this front
in the core may explain the Pacific quiet zone in the secular variation
of the geomagnetic field and the longitudinally heterogeneous structure
of the solid inner core.}''

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig:spiral}Flow patterns produced by localised heating
of the outer boundary of a hemispherical shell~\cite{sumita2002_jgr}.
(a)~Inner core spiral developing in the global locking mechanism as a
result of localised heating at the outer hemisphere, visualised by
injection of dye~\cite{sumita2002_jgr}. (b)~The front forming the
spiral results from the collision of the eastward and westward jets
originating at the heating anomaly, a mechanism suspected to take place
in the Earth's core (right).}
\end{figure}

Sahoo and Sreenivasan~\cite{sahoo2020_epsl, sahoo2020_jfm} built a
cylindrical annulus filled with water (height 37~cm, inner/outer radii
5~cm/14.2~cm), with flat, thermally insulating top and bottom
boundaries, mostly radial centrifugal force and PIV in horizontal
planes~\cite{sahoo2020_epsl, sahoo2020_jfm}. At the inner boundary the
temperature is imposed, while at the outer boundary, the heat flux is
imposed by the heater fitted around the annulus. Unlike Sumita and
Olson's experiment, the inhomogeneity is not local but takes the form
of an azimuthal variation of boundary heat flux controlled by dividing
the external heater into four independently controlled $\uppi/2$
sectors. The system spanned inhomogeneities in the range $0\leq
Q^*\leq2$. Here $q_h$ is the amplitude of the azimuthal flux variation.

Experiments were conducted with alternating high and low fluxes every
$\uppi$ (one-fold symmetry) and every $\uppi/2$ (two-fold symmetry).
Both configurations lead to a slight reduction of the \mbox{critical}
flux-based Rayleigh number for the onset of convection. 
Examples of vorticity fields are represented on Figure~\ref{fig:sahoo2020_piv}.
The flow consists of counter-rotating vortex pairs separated by thin
downwellings similar to Sumita and Olson. These large scales are
accompanied by small-scale motion. With a two-fold variation, the flow
becomes homogeneous above about 30 times the critical Rayleigh number,
most likely as a result of the strong azimuthal flow that exists in
this regime. Homogenisation was not observed with a one-fold variation
as convection in the low-flux sector is not able to transport motion
across it. Sahoo and Sreenivasan interpreted the magnetic field
inhomogeneities as signatures of CMB inhomogeneities to estimate that
$Q^*&gt;2$ for the Earth, implying regions of subadiabatic heat flux.
Achieving these in the experiments, would, however, require adding
active cooling to the outer wall. Nevertheless, such inhomogeneities in
the Earth's magnetic field suggest that homogenisation does not take
place in the equatorial regions of the core. This offers a potential
method to constrain the Rayleigh number in the Earth's outer core.\looseness=-1

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig:sahoo2020_piv}Horizontal velocity vectors (arrows)
and shaded contours of axial vorticity (s$^{-1}$) in horizontal planes
of Sahoo and Sreenivasan's experiment~\cite{sahoo2020_jfm} in an
annulus with a heated outer boundary with two or four sectors of
independently adjustable heat flux (yellow and green colours indicate
regions or high and low heat flux respectively). Plots are averaged in
time, and shown for different levels of supercriticality
$\widetilde{Ra}$. (a--c)~One-fold variations with $Q^*=0.7$,
(d--f)~two-fold variation at $Q^*=1$ showing regionalisation of
convection, (g--i)~two-fold variation with $Q^*=2$, homogenisation of
convection takes place at sufficiently high forcing under the effect of
the azimuthal flow.}
\end{figure}

\section{The default setup for axial gravity: cylinders} \label{sec:axial}

Experiments in geometries where the natural terrestrial gravity, the
temperature gradient and the imposed rotation are aligned
(Figure~\ref{fig:geometries}(c)) capture the part of the planet that
the experiments with cylindrical radial gravity neglected---the polar
region. The default set-up is a straight cylinder, the shape that
naturally arises in rapidly rotating systems. Only three experiments
were conducted in box geometries~\cite{Sakai1997, Boubnov1986,
Fernando1991} which creates the undesired issue of developing corner
flows that have no planetary equivalent. These Cartesian geometries,
however, ease visual access in experiments.

Convection in rotating spherical convection onsets in the equatorial
region and thus was initially thought to be the most relevant for
planetary interiors. Nowadays, it is, however, clear that the polar
region is at least equally important in the more supercritical regimes
where planets reside~\cite{Gastine2023, Schaeffer2017}. Thus, the
importance of some of the classical rotating convection experiments
with axial gravity to geophysical processes is sometimes only
emphasised more strongly in hindsight~\cite{horn2022_prsa, Horn2024,
roberts2013_rpp, King2013, King2015}. Nonetheless, as visualised in
Figure~\ref{fig:timeline}, this configuration has been the most common
and enduring one since the beginning of rotating convection experiments
and shows no signs of losing its popularity. There are several recent
thorough reviews detailing the experiments, their results and
more~\cite{cheng2018_gafd, aurnou2015_pepi, kunnen2021_jot,
ecke2023_arfm} so that here we keep the discussion very brief.

Arguably the first systematic and quantitative experiments of rotating
convection was a series of mercury experiments conducted by Fultz,
Nakagawa, Frenzen and Goroff in the years 1953 to 1960 at the
University of Chicago~\cite{Fultz1954, Fultz1955, Nakagawa1955,
Nakagawa1957, Nakagawa1959, Goroff1960}, the first one conducted on the
20 November 1953 as shown in Figure~\ref{fig:cylexp}(a). (A few earlier
rotating experiments were also conducted by Nakagawa and Fultz at the
University of Tokyo and Chicago, respectively, in water and air,
however, the upper surface was cooled by evaporation which is a
slightly different fluid dynamics problem from the
Rayleigh--B\'enard-like set-up we consider throughout this
review~\cite{Nakagawa1955}.) These seminal experiments were very
comprehensive. Since they were conducted in a liquid metal with $Pr
\approx 0.025$, oscillatory convection was naturally included.
Moreover, Nakagawa had access to an electromagnet of a discarded
36.5~in cyclotron which had been reconditioned at the Enrico Fermi
Institute for Nuclear Studies at the University of Chicago with a field
strength of up to 1.3~T, shown in
Figure~\ref{fig:cylexp}(b)~\cite{Chandrasekhar1957}. This allowed him
to also study rotating magnetoconvection. These experiments were mainly
concerned with the flow behaviour close to the \mbox{onset} of
convection and the validation of linear stability results derived by
Chandrasekhar and{\break} Elbert~\cite{Chandrasekhar1961,
horn2022_prsa,Horn2024}.

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig:cylexp}(a)~The first experiment of rotating
convection in Chicago on the 20 November 1953~\cite{Fultz1954}. (b)~The
hydromagnetic laboratory at the University of Chicago where Nakagawa
did his seminal rotating convection
experiments~\cite{Chandrasekhar1957}. (c)~Visualisation of columns in
rotating convection in water using thermotropic liquid-crystal capsules
by Sakai, top and side view (top/bottom)~\cite{Sakai1997}.
(d)~Visualisation of chaotic vortices in rotating convection in water
using aluminium powder on the top surface by Boubnov and
Golitsyn~\cite{Boubnov1986}. (e)~Visualisation of wall modes in water
using shadowgraph by Zhong et~al.~\cite{Zhong1991}. (f)~Visualisation
of the flow regimes in turbulent rotating convection in water using
rheoscopic particles and illumination with a vertical light sheet; left
to right: convective Taylor columns, plumes, geostrophic turbulence,
rotationally influenced turbulence by Cheng et~al.~\cite{Cheng2020}.}
\end{figure}

The focus on relatively low supercriticalities remained until the end
of the last century. The most common working fluid was
water~\cite{Rossby1969, Boubnov1986, Boubnov1990, Fernando1991,
Liu1997, Zhong1991, Zhong1993, Ning1993, Vorobieff1998, Sakai1997}
which allows direct optical access and visualisation, but prohibits
oscillatory convection. The same applies to silicone oil as tested by
Koschmieder~\cite{Koschmieder1967} and Rossby~\cite{Rossby1969}. These earlier
experiments established the formation of regular cell patterns and
convective Taylor columns as the typical structures of steady bulk
convection, see Figure~\ref{fig:cylexp}(c), but also found irregular
vortex patterns; an example is shown Figure~\ref{fig:cylexp}(d). Also,
the heat transport and its scaling with the control parameters were
investigated, especially in the few visually opaque experiments (either
due to the fluid or by design) by Rossby in mercury~\cite{Rossby1969},
Donnelly and collaborators in cryogenic helium-I~\cite{Lucas1983,
Pfotenhauer1984, Pfotenhauer1987}, and Aurnou and Olson in
gallium~\cite{Aurnou2001}. None of the experiments was likely extreme
enough to capture the geostrophic turbulent heat transport scaling one
may expect in Earth's outer core or planetary interior, though.

Likely the most crucial finding in that period, however, was the
discovery of wall modes by Ecke and collaborators, shown in
Figure~\ref{fig:cylexp}(e)~\cite{Zhong1991, Zhong1993, Ecke2023b}. Wall
modes explained why the onset of convection as determined e.g.\ by heat
transport measurements did not agree with Chandrasekhar's predictions
but was indeed at lower $Ra$. The destabilising effect of the wall and the
breaking of the symmetry results in structures that travel along the
periphery of the rotating convection vessel and exponentially decay
towards its centre. Initially thought to be mainly an onset phenomenon
and an experimental restriction, wall modes are now known to be much
more persistent and transform non-linearly into a boundary-zonal flow
affecting also strongly supercritical settings~\cite{Zhang2020,
Wedi2022, Zhang2021, Shishkina2020, Vogt2021, Favier2020, deWit2023,
Terrien2023}. Moreover, they may also exist along the virtual boundary
of the tangent cylinder in Earth's outer core, as we will discuss in
the next{\break} Section~\ref{sec:TC}.

In our current 21st century, the goal has changed and is now rather to
push turbulent rotating convection to the extreme, such as in the
TROCONVEX experiment in
Figure~\ref{fig:cylexp}(f)~\cite{kunnen2021_jot, Cheng2020}. The most
popular working fluid is still water~\cite{King2009, Cheng2015,
Liu2009,Cheng2020, Zhong2009, Weiss2010, Weiss2011, Kunnen2008,
Kunnen2010, Rajaei2016, Joshi2016, Rajaei2017, cheng2018_gafd,
Cheng2020, Kunnen2021, Ding2021, Lu2021, Hu2022, Shi2020}, but also
cryogenic helium~\cite{Niemela2010, Ecke2014}, silicone
oils~\cite{Abbate2023}, sucrose solution~\cite{King2009},
gallium~\cite{Aurnou2018, King2013, Vogt2021, Grannan2022}, and
pressurised SF$_{6}$~\cite{Zhang2020, Wedi2021}. The connection to
geophysics and planetary physics is sought much more explicitly, in
particular, in the Calimero (Californian Model of Earth's Rotation),
\mbox{NoMag}, \mbox{RoMag} (all three in Los Angeles, Aurnou and
collaborators~\cite{Abbate2023, Hawkins2023, King2009}) and TROCONVEX
(Eindhoven, Kunnen and collaborators~\cite{cheng2018_gafd})
experiments. Notably, RoMag, is the only operative liquid metal
rotating magnetoconvection experiment with thus, the closest
resemblance to planetary core fluid. Thus, this meant establishing the
possible regimes of geostrophic convection (see
Figure~\ref{fig:cylexp}(e), convective Taylor columns, plumes,
geostrophic turbulence, rotationally influenced turbulence), testing
various scalings for the heat and momentum transport, length scales, the
importance of centrifugal effects and much more. Several of these
experiments in China, the USA and The Netherlands, are still active and
running and producing results (see Figure~\ref{fig:timeline}), thus, we
may expect more to come that will enrich our understanding of the polar
physics in planetary interiors in the future.

\section{Tangent cylinder dynamics} \label{sec:TC}

Rotating convection in cylinders, reviewed in Section~\ref{sec:axial},
may represent the dynamics in the polar regions. However, those
experiments do likely not fully capture the entirety of the polar
region that extends up to the virtual tangent cylinder (TC) boundary,
see Figure~\ref{fig:geometries}(a). The sidewall boundary conditions in
cylindrical experiments are usually impermeable and adiabatic. The
boundary conditions of planetary tangent cylinders are, however, likely
neither. The most recent data from the SWARM mission clearly shows a
strong jet wrapped around the TC and, importantly, meandering in and
out of the TC~\cite{finlay2023_nat}. Similarly, dynamo simulations at
extreme enough parameters show flows across the
TC~\cite{Schaeffer2017}.

Thus, several major questions arise: First, under which conditions can
the {Taylor--Proudman constraint} (TPC, see
Section~\ref{sec:phys}) be violated such that the TC boundary becomes
permeable? Second, how much heat is transferred between the polar and
equatorial regions across the TC? Finally, given such drastic
differences in boundary conditions, how well do the experiments in
cylinders truly represent polar convection?

\subsection{Aurnou et~al.'s TC: what's new in Baltimore?}

Aurnou and collaborators~\cite{aurnou2003_epsl} undertook the first
attempt at addressing these questions in their Baltimore lab. They
built a setup with a hemispherical vessel (15.2~cm diameter) filled
with water, sitting in a cylindrical bath also filled with water
ensuring a cold, approximately homogeneous temperature at its outer
boundary (Figure~\ref{fig:tc_setups}(a)). A heater shaped like a hockey
puck (10~cm-diameter, 3.6~cm high) rested at the flat bottom of the
hemisphere. The heater achieved two functions: first, it delivered a
constant heat flux $q$ into the fluid. Second, the outer edge of the
puck formed a discontinuity in the height of the fluid domain in the
same way as a planetary solid core does. The choice of a puck shape for
the heater, instead of a spherical one was motivated by the difference
in gravitational fields between the Earth and the experiment: in the
experiment, the axial gravity would misalign with a spherical boundary
of near-constant temperature and drive unwanted baroclinic flows in its
vicinity. By contrast, the flat upper surface of the heater is
everywhere perpendicular to the gravity, and so eliminates that
problem. Hence, in the lab, pucks are a better representation of the
Earth's inner core than spheres.

\begin{figure}
\includegraphics{fig14}
\caption{\label{fig:tc_setups}Sketches of the three experiments
reproducing the TC geometry: (a)~2001--2003: rotating vessel of Aurnou
et~al.\ showing the puck-shaped heater inside the hemispherical dome
filled with water~\cite{aurnou2003_epsl}, (b)~2013--2019: rotating part
and static support of LEE1 set against a section of the Earth
illustrating how the inner core, ICB and CMB are modelled in the
experiment~\cite{aujogue2016_rsi, aujogue2018_jfm, potherat2023_prl},
(c)~2019--: rotating part of LEE2, with cameras, mirrors and optical
slip ring, where the fluid vessel is now cylindrical instead of
hemispherical. The two large black cameras near the top and the two
mirrors fitted at the bottom are part of the PIV system for vertical
planes. The vertically travelling carriage near the bottom holds the
lens generating the horizontal LASER plane for PIV in horizontal planes
of adjustable height~\cite{agrawal2024_gji}.}
\end{figure}

Aurnou and collaborators used dye visualisation and thermistors
embedded near the heater surface to measure heat fluxes and velocities.
They identified five successive regimes based on the flow structure
observed as the thermal forcing ramps up during a transient experiment.
These regimes map to those in cylindrical vessels but for specificities
inherent to the TC geometry. At low thermal forcing, no convection is
detected inside or outside the TC but non-vertical residual thermal
gradients drive a very weak thermal wind that suppresses convection and
thus increases the critical Rayleigh number for the onset of convection
inside the TC. Increasing the thermal forcing leads to a ring of
vortices attached to the outside of the TC. These vortices result from
baroclinic instability and remain confined outside the TC due to the
TPC. Inside the TC, convection is still absent. At higher thermal
forcing, convection sets in within the TC at a slightly higher critical
Rayleigh number than in cylindrical vessels with adiabatic sidewalls
because of the weak thermal wind. The onset structures are helical,
prograde near the heater and retrograde near the top. They ignite near
the TC boundary and their motion is confined within the TC by the TPC
as visualised during a transient experiment in
Figure~\ref{fig:tc_hemisphere}(b). Further increasing the thermal
forcing, but keeping strong rotation, the helical structures fill the
entire TC, in a regime that would nowadays be referred to as
``rotation-dominated columnar regime''~\cite{kunnen2021_jot,
Julien2012}. When rotation is less dominant over buoyancy, the flow
structures do not extend over the full height of the volume but remain
strongly affected by rotation~\cite{kunnen2021_jot, Julien2012}.

\begin{figure}
\includegraphics{fig15}
\caption{\label{fig:tc_hemisphere}Dye visualisations showing the
regimes of rotating convection inside a TC during the instationary
phase following the ignition of thermal forcing from a state of solid
body rotation by Aurnou et~al.~\cite{aurnou2003_epsl}. Top view of
(a)~the onset of convection and (b)~helical plumes at $Ra =
9.01\times10^9$; $Ek = 9.65\times10^{-5}$, (c)~side view of 
baroclinic instability developing across the TC boundary, (d)~side view
of fully developed convection inside the TC before the onset of
baroclinic instability at ($Ra = 1.10\times10^{10}$, $Ek =
4.11\times10^{-5}$). Arrows indicate the direction of rotation.
(e)~Plumes arising near the TC boundary at the onset of convection ($Ra
= 4.44\times10^9$, $Ek = 4.26\times10^{-5}$).}
\end{figure}

The presence of baroclinicity in background rotation drives a strong
azimuthal wind with maximum velocity $U_{\phi}^{\mathrm{max}}$ near the
inner TC boundary. The scaling of its intensity with the convective
buoyancy flux $q_{\mathrm{B}}=\alpha g q/(\rho C_p)$, as $U_{\phi}^{\mathrm{max}}
\sim 2.05(q_{\mathrm{B}}/\Omega)^{0.52\pm 0.03}$ confirms this mechanism. The
authors suggest that this mechanism and its confinement to polar
regions within the TC may explain the polar vortices inferred there
from the 1980 Magsat and 2000 Oersted satellite
missions~\cite{hulot2002_nat}. Extrapolation of their experimental
observation to Earth leads to estimates of a few km for the diameter of
columnar vortices within the TC and a flux Rayleigh number of the order
of~$10^{31}$. They do stress however that testing this thermal wind
scaling for the Earth would require incorporating such effects as the
Lorentz force, compressibility and other possible effects.

\subsection{Little Earth experiment I (LEE1)} \label{sec5.2}

Poth\'erat and collaborators used design ideas similar to Aurnou
et~al.~\cite{aurnou2003_epsl} in the {Little Earth Experiment}
(LEE1), a hemispherical dome of diameter 28.5~cm, and a puck-shaped
heater 2.25~cm in height and 10~cm in diameter. They introduced two
novelties with drastic consequences for the experimental realisation
(Figure~\ref{fig:tc_setups}(b)). First, they incorporated PIV
measurements to capture the detail of the time-dependent velocity field
in horizontal planes and in one meridional plane. Second, LEE1 targeted
magnetostrophic regimes with an Elsasser number of $\Lambda\sim1$~\cite{aujogue2016_rsi,
aujogue2018_jfm, potherat2023_prl}.

Combining optical methods and strong MHD effects requires a working
fluid that is both transparent and electrically conducting. The fluid
with the highest electrical conductivity satisfying these constraints
is sulphuric acid at 30\% mass concentration, of conductivity
$\sigma=86$~S/m, i.e.\ $10^4$ times less than a typical liquid metal.
Luckily, sulphuric acid is also the least dangerous amongst potential
candidates (hydrofluoric acid, ionic liquids etc.). Since sulphuric
acid is typically 5 to 10 times less dense than a liquid metal, it is
easy to see that to reach $\Lambda\sim1$ at $Ek\sim10^{-6}$, magnetic
fields of the order of 10~T are needed. To complicate matters, the
magnetic field must pervade a sufficiently large volume to host a fluid
dynamics experiment with its instrumentation i.e.\ a cylindrical volume
of about 30~cm diameter over about 1~m. At the time of LEE1's
operation, only one magnet in the world was capable of delivering such
a magnetic field: the 12~MW M10 magnet of the Grenoble High Magnetic
Field Laboratory (LNCMI-G) with up to 10~T in a
376~mm-diameter/2~m-long warm bore (Nowadays, LNCMI-G offers up to 18~T
in their new hybrid magnet~\cite{pugnat2022_ieee}). Using such a large
magnet meant that only one strategy was reasonably possible for the
design of LEE1 out of the three outlined in
Section~\ref{sec:sphere_mhd}: LEE1 had to be built to fit into the
existing magnet. This imposed further severe constraints on the design.
Materials rotating within the bore had to be non-conducting to avoid
Foucault currents, and those in contact with the working fluid had to
possess long-term resistance to highly concentrated acids. Electronic
equipment and electric motors cannot operate in high magnetic fields
and have to be deported either far above or far below the magnet. The
space between the vessel and the inner wall of the magnet bore is only
a few cm. Thus, visualisation from the side had to rely on small
concave mirrors providing sufficient angular access to capture the
entire height of the TC. Finally, LEE1's motor was located about 0.5~m
underneath the bottom of the magnet. The acquisition system (camera,
laptop controlling PIV LASERS and recording thermocouple signals) was
placed on a rotating platform about 2~m above the top of the magnet. In
total, LEE1 was approximately 4.5~m in height and linked by a structure
mostly made of different plastics.

The heater was made of a plastic heat exchanger circulated by
ethylene--glycol heated in the static frame. The top of the heater
delivering heat to the acid was made of SHAPAL, a heat-conducting,
electrically insulating and acid-resistant ceramic. This design
avoided electric parts but its thermal inertia prevented any feedback
control of the temperature difference.

\paragraph*{Non-magnetic rotating convection}

LEE1 was first operated in water to focus on non-magnetic rotating
convection in the TC. In the entire range of parameters investigated,
the flow was mostly confined within the TC, i.e.\ the TPC was
sufficiently well satisfied for the TC boundary to be practically
impermeable, with only some weak motion detected outside the TC, see
Figure~\ref{fig:lee1_piv}(a,d,g). The onset of convection follows
similar laws to the onset of planar layer convection, but with slightly
different prefactors: close for the critical Rayleigh number $Ra_c =
(32.3 \pm 4) Ek^{-1.29\pm0.05}$, but lower for the critical wavenumber
$k_c=(0.58\pm 0.08) Ek^{-0.32\pm0.05}$. Past the onset, both wall modes
and bulk modes are observed, which implies that the TC boundary acts
as a solid impermeable wall in conditions relatively close to
onset~\cite{goldstein1993_jfm}. Unlike in cylindrical tanks, however,
wall modes were not detected below the onset of bulk convection. A
possible explanation is that the TPC does not erect a boundary when the
flow is still. In other words, below the onset of convection, the TC
does not exist as a wall, so neither do wall modes. With increasing
supercriticality, the flow follows a complex sequence of patterns to
arrive at a state dominated by a single retrograde vortex for
$\widetilde{R}a\simeq 10$. At high latitude, this vortex is associated
with a retrograde thermal wind located \textit{inside} the TC, whose
intensity follows a scaling nearly identical to that found by Aurnou
et~al.~\cite{aurnou2003_epsl} $Ro = (5.33\pm 0.3) (Nu\, Ra\, Ek^3
Pr^{-2})^{0.51\pm0.04}$. The heat flux shows a clear transition between
regimes of rotation-dominated convection scaling as $Nu\simeq 0.38 Ek^2
Ra^{1.58}$ and buoyancy-dominated convection with $Nu\simeq 0.2
Ek^{0.33}$ which resembles convection in cylinders and plane
layers~\cite{kunnen2021_jot, ecke2023_arfm}.

\begin{figure}
\includegraphics{fig16}
\caption{\label{fig:lee1_piv}PIV visualisation of the non-magnetic and
MHD flow patterns inside the TC in LEE1~\cite{aujogue2016_phd,
aujogue2018_jfm} showing vorticity (blue denotes negative vorticity, 
red positive vorticity) and velocity (arrows)
averaged in time in a plane located at 51{\textdegree} latitude
(i.e.~just below the point where the TC meets the hemispherical vessel
wall). Values of $\widetilde{Ra}$ are calculated with respect to the
critical Rayleigh number for $\Lambda=0$, even for cases where
$\Lambda&gt;0$. The red circle indicates the position of the TC boundary
whereas the blue circle corresponds to the vessel wall. The
non-magnetic case (a,d,g)~exhibits large, merged columns in the weakly
supercritical regime. For higher criticality, wall modes are visible
and for the highest criticality, a large central retrograde vortex
exists at the centre. At $\Lambda=0.15$~(b,e,h), the effect of the
magnetic field is not pronounced and the flow patterns resemble those
at $\Lambda=0$. For $\Lambda=0.33$~(c,f,i), a large off-centre
retrograde structure forms a much lower criticality, with a
zero-vorticity ``eye'' in its centre.}
{\vspace*{-.3pc}}
\end{figure}

\paragraph*{Magneto-rotating convection}

Unlike liquid planetary cores, LEE1 operates in the quasi-static MHD
regime where the feedback of the flow onto the magnetic field is
negligible, so the magnetic field is imposed, and constant, and dynamo
processes are excluded~\cite{roberts1967}. The acid's low conductivity
also precludes the occurrence of Alfv\'en waves or magneto-Coriolis
waves, even though these can exist a low magnetic Reynolds numbers in
liquid metals~\cite{schmitt2008_jfm, lalloz2024_jfm} and bear relevance
to those in planetary cores~\cite{gillet2010_nat, gillet2022_pnas}.
Hence, LEE1 focuses on capturing the feedback of the magnetic field on
convection rather than the reverse. 

The magnetic regimes differ from the nonmagnetic ones in a number of
ways, see Figure~\ref{fig:lee1_piv}. At the most fundamental level, the
Coriolis force now competes with the Lorentz force, so the TPC no
longer applies. Instead, a {Magnetic Taylor--Proudman
constraint} (MTPC) imposes a kinematic relation between the flow along
the TC boundary ($u_{\phi}$) and the flow through it ($u_r$):
{\begin{equation} \label{eq:kinematic_mpt_rtheta}
\Delta_{rz}\langle u_r\rangle_{t,\phi}+ \Lambda\partial^2_{zz}\langle
u_{\phi}\rangle_{t,\phi}=0.
\end{equation}}\unskip
This theory is confirmed using PIV in two horizontal planes at low and
high latitudes showing that a meridional flow through the TC boundary
follows the MTPC~\cite{potherat2023_prl}. As a result of this
reorganisation of the flow, the azimuthal flow does not follow the
thermal wind scaling and increases significantly with~$\Lambda$.
Applying the MTPC to the Earth would require a more general form to
incorporate the complex geometry of the Earth's magnetic field.
Nonetheless, the MTPC provides at least a plausible mechanism to
account for the meandering of the zonal flow in and out of the TC
inferred from the SWARM data~\cite{finlay2023_nat}, and previous
simulations suggesting that the TPC was broken at the TC
boundary~\cite{Schaeffer2017, cao2018_pnas}.

The main effect of the Lorentz force inside the TC is a suppression of
heat transfer. At $\Lambda&gt;0.2$, Nusselt numbers are systematically
much lower than their counterparts at $\Lambda=0$, but scale with a
greater power of $Ra$ (up to~1). Unlike in the non-magnetic case, the
highest thermal forcing available, $Ra\sim 10^{9}$, is insufficient to
reach a transition to a buoyancy-dominated
regime~\cite{aujogue2016_phd}. This does not imply, however, that the
presence of a magnetic field always suppresses heat transfer in
general. In spherical shells for example, dynamo simulations at $Pr=1$
show enhancement of heat transfer at
$Ek\lesssim10^{-5}$~\cite{yadav2016_gafd} and the trend observed in
LEE1 does not exclude that the imposed magnetic field could lead to
heat transfer enhancement in the TC at higher Rayleigh numbers than
accessible in the experiment. This just goes to show how much is left
to understand the basics of rotating magneto-convection. Detecting the
onset of rotating magneto-convection inside the TC would require far
more magnet time than possible in M10 but several states of convection
are detected when increasing the thermal forcing. Near the onset of
convection within the TC, both bulk and wall modes display similar
structures to the non-magnetic case. As the thermal forcing is
increased these are progressively replaced by a large retrograde vortex
occupying the middle of the TC, consistent with the very strong
azimuthal flows observed there, see Figure~\ref{fig:lee1_piv}. This
structure however appears at comparably lower thermal forcing, and much
higher intensity than in the non-magnetic case and displays off-centre
maximum vorticity~\cite{aujogue2016_phd}. These results suggest that
the axial magnetic field favours the emergence of a large polar vortex,
again a plausible, if not directly applicable mechanism for the
emergence of polar vortices in the Earth's outer
core~\cite{hulot2002_nat}.

\subsection{Little Earth experiment II} \label{sec5.3}

LEE2 is a complete rebuild of LEE1 with much greater control of both
the rotation and heating, reduced vibrations and a 10-fold increase in
PIV resolution. Its new vessel focuses on the TC itself so the heater
diameter was increased and the outer dome was replaced by a cylinder of
diameter 22~cm with a puck-shaped heater of diameter 15~cm and
yielding a TC height of $H=14.3$~cm. Thus, the region outside the TC is
now a rather thin rectangular annulus. The outer wall is still cooled
by a bath of water at a constant temperature, which imposes a cold
temperature along the top and its side walls, i.e.\ only 3.5~cm away
from the TC boundary. This creates a much greater source of
baroclinicity there than in LEE1 to potentially break away from the
TPC, and so attain regimes of higher inertia that would otherwise
require higher thermal forcing. 

Inside the new heater, the fluid heat exchanger was replaced by two
resistive wires joined together in a spiral shape, and fed with
opposite currents to cancel out the total Lorentz force due to the
heating current inside the high magnetic field. This made it possible
to implement a feedback control loop on the heater to control the
temperature difference between any pair of thermistors placed inside
the device. Four were located at the top of the heater surface. Two
pairs were placed inside and outside of the side wall and the top wall.
Measurement inside the vessel provided a more accurate input for the
control of the temperature drop within the fluid. With this system,
the axial temperature difference between the top of the heater and the
underside of the top wall could be kept constant for longer periods of
time and controlled to ${\pm}$0.1~K. The two thermistor pairs also
measured the local heat flux across the outer cylinder top and side
walls. 

The new PIV system benefited from a movable vertical LASER plane whose
axial position could be remotely controlled during operation, without
having to restart the experiment: this made it possible to scan the
flow using several PIV planes during the same run. All signals were
passed to the static frame via an optical slip ring capable of
absorbing the bandwidth of the PIV cameras. These were located further
away from the bore than in LEE1 to be able to operate at fields up to
12~T. These changes made LEE2 considerably heavier, so the new, more
powerful motor too had to be placed further away from the magnet and
the total height of the device reached 6.5~m. LEE2 could also be
operated outside M10 in a shorter non-magnetic configuration (2.5~m
tall).

LEE2's high PIV resolution made it possible to map the regimes of
convection against the classification established for classical
rotating RBC during the 2010's~\cite{Julien2012, kunnen2021_jot}. By
order of criticality, the flow starts with a regime without convection
within the TC, but with a weak toroidal flow in the outer annulus
driven by baroclinicity. At slightly higher forcing, the outer flow
becomes unstable to baroclinic instabilities and exhibits a ring of
co-rotating vortices similar to the ``rim instabilities'' identified by
Aurnou et~al.~\cite{aurnou2003_epsl}, and visible in
Figure~\ref{fig:lee2_snapshots}(a). Since this region has a constant
height, these vortices are not Rossby waves, which they nevertheless
resemble. Convection in the TC sets in at approximately the critical
Rayleigh number for plane layer rotating RBC and displays the same
cellular regime at the onset with the same length scale
(Figure~\ref{fig:lee2_snapshots}(b)). For slightly higher thermal
forcing, the cellular network breaks into quasi-geostrophic columns
with a symmetric spiral structure of prograde vorticity near the bottom
(hot) wall and retrograde vorticity at the top (cold) wall. The
structure is surrounded by a sheath of opposite vorticity
(Figure~\ref{fig:lee2_snapshots}(c)). Up to this point, the TC boundary
acts essentially as an impermeable wall separating the flow inside and
outside the TC with the exception of thin jets induced by baroclinic
vortices locally traversing the TC boundary near the top, where
baroclinicity is strongest. Further increasing the thermal forcing,
convection within the TC enters a plume regime, where columns become
three-dimensional, and lose their outer sheath of opposite vorticity.
More jets cross the TC near the top and the TC boundary becomes locally
permeable over practically the whole height but a small region near the
bottom (Figure~\ref{fig:lee2_snapshots}(d)). Even higher forcing
disrupts the plumes and the flow displays turbulent features with a
wide range of scales from small, 3D scales to a small number (two to
four) of large vortices. The global balance of force suggests these may
still be quasi-geostrophic (QG). However, they appear to be driven by
strong baroclinic jets originating outside the TC. In this regime, the
TPC is broken over the entire cylinder height but a strong zonal flow
persists at its location (Figure~\ref{fig:lee2_snapshots}(e)).

\begin{figure}
\includegraphics{fig17}
\caption{\label{fig:lee2_snapshots}Snapshots of axial vorticity
obtained from non-simultaneous PIV measurements in LEE2 at $z=H/6$,
$H/3$, $H/2$, $2H/3$, $5H/6$ (from bottom to
top)~\cite{agrawal2024_gji}. The pink circle marks the location of the
TC boundary. Regimes encountered by order of criticality
$\widetilde{Ra}$ are similar to those encountered in rotating
convection in a cylinder with a solid sidewall. However, the
transitions between them take place at lower criticality because of
inertia incurred by the baroclinic flow outside the TC. In particular,
quasi-geostrophic turbulence appears at $\widetilde{Ra}=37$
{versus} $\widetilde{Ra}\simeq80$ in Aguirre-Guzman et~al.'s
work~\cite{aguirre-guzman2021_jfm}. The baroclinic flow incurs a local
breakup of the TPC visible through structures straddling the TC
boundary as low as $\widetilde{Ra}=2.9$ (in the plane at $z=5/6H$).}
\end{figure}

These regimes broadly follow those found in classical rotating
RBC~\cite{kunnen2021_jot, ecke2023_arfm}. A key feature of the TC
geometry is that these regimes become more and more distorted by
baroclinically-induced inertia, which progressively loosens the TPC.
The breakup of the TC boundary also enables the heat flux to escape
laterally, and thus to bypass the suppression of axial heat transfer by
rotation within the TC. As a result, the transition between them takes
place at significantly lower thermal forcing. A further effect of the
strong baroclinicity at the TC boundary is the suppression of wall
modes which, unlike in cylindrical vessels appear in none of these
regimes. The absence of wall modes and the fact that transitions take
place over a narrower range of critical Rayleigh numbers implies that
QG turbulence and large-scale vortices~\cite{Julien2012, favier2014_pf,
guervilly2014_jfm, stellmach2014_prl, aguirre-guzman2021_jfm} are more
easily reachable in TC geometries with strong baroclinicity.

These results suggest that inertia induced by baroclinicity, or
otherwise, near the TC boundary in the Earth or other planets may
favour the emergence of large structures at lower levels of
criticality. This could potentially explain the large scales observed
in the polar regions. Hence, inertia such as that induced by
baroclinicity offers a second possible route for the breakup of the TC
besides magnetic effects. This mechanism too could favour the
meandering of zonal flows in and out of the TC. LEE2 is poised to
examine the competition between these mechanisms in detail with future
campaigns in high magnetic fields at LNCMI-G.

\section{Discussion} \label{sec6}

In the end, what does the evolution of experiments modelling rotating
convection in planetary interiors over the last 70 years tell us? For
sure, experiments are hard, cumbersome, and inherently limited by the
abysmal gap that separates the relative comfort of the laboratory from
realistic planetary conditions: the gravity points in the wrong
direction, the fluids are either too viscous or too opaque, the
experiments are of the wrong shape, cannot rotate fast enough, reliable
measurements are extremely difficult to obtain and the list goes
on\,{\ldots} Yet, none of these hurdles have deterred the ingenuity of
experimentalists. From Nakagawa's early experiments in
cylinders~\cite{Nakagawa1955, Nakagawa1957, Nakagawa1959}, to modelling
equatorial regions with an annulus~\cite{busse1974_jfm} and the solid
core with a hockey puck~\cite{aurnou2003_epsl} and the more recent
mapping velocity fields in vessels with a tangent
cylinder~\cite{aujogue2016_rsi, aujogue2018_jfm, potherat2023_prl},
they have kept focussing on targeted aspects of planetary interiors:
they isolated equatorial and polar regions to obtain Earth-like gravity
at least locally, they worked with transparent fluids to reveal the
first flow patterns of the outer core~\cite{busse1976_sci}, be it at
expense of some of the low-$Pr$ convection patterns~\cite{Aurnou2018},
they simulated radial gravity in space at the cost of limited ranges of
parameters~\cite{hart1986_sci, hart1986_jfm, hart1996_3mg}, etc. None
of these experiments is a realistic representation of the Earth's outer
core, but each of them has revealed key bits of knowledge, which
accumulated, and confronted with geophysical data, simulations and
mathematical models, have, little by little, shaped our current
understanding of planetary cores. The key to their success is that
however incomplete, experiments are the best proxy for actual physical
processes taking place in planetary interiors, provided the relevant
processes can be identified and characterised.

Yet, at a time when numerical simulations have become so accessible and
so powerful, and when ``realistic'' or ``Earth-like'' simulations are
emerging~\cite{aubert2023_gji}, it is worth asking whether there is
still a place for cumbersome laboratory experiments. The answer lies in
the respective merits of both approaches: simulations in extreme
regimes will remain time-intensive, at least as costly as experiments
and not easily repeatable for the foreseeable future. Unlike
experiments, they do offer access to the entire time-dependent field of
variables. Unlimited accessibility to an ocean of data is only
useful when knowing where to look, and experiments on more targeted
processes covering wide parameter ranges are ideally placed to point in
the right direction. So are the analysis of real geophysical data and
simple mathematical theories. Furthermore, even when simulations are
perfectly resolved at all scales and all time scales, a very hard
condition to meet, the results are still only as good as the
mathematical model they are solving. Experiments, on the other hand,
are no facsimile: they return the real physics of all the processes
they incorporate and so provide a real-life test of the mathematical
and numerical models. So the answer is yes, experiments are crucial and
will continue to improve, just like numerical simulations, and just
like the oldest methods in the field: geophysical observation and
simple mathematical theories. None of these approaches work anywhere
near as well in isolation as when they can draw on the progress and
feedback of the other three. 

The main issue faced by experimentalists and theoreticians alike is
that of the extrapolation of their results to the Earth or other
planets. Given the limitations of numerics and experiments and the
large uncertainties on the actual conditions in planetary interiors,
attempting a precise model combining all potentially relevant processes
would certainly be in vain. Stevenson finds solace in an elegant way
out of this conundrum~\cite{Stevenson1985}, citing a similar point
made by Urey about the chemical composition of planets~\cite{urey1952}:

\begin{quote}
{What Urey is doing is giving a license to study ill-posed
problems. By ill-posed, I mean that the issue can be stated with
precision but the knowledge needed to settle the issue is not yet
entirely adequate. [{\ldots}] The important lesson that we learn is
this: It is not so important that you get the right answer: it is most
important that you ask the right question.}
\end{quote}
In this spirit, the
question we may be able to ask may not concern so much the exact nature
of the convection in planetary interiors but rather: ``What processes
arising from the interplay of convection and rotation may we expect to
play a role there?''. Seven decades of exploring planetary interiors
with experiments on rotating convection have certainly provided a
wealth of answers to that question but by no means all of them.

So what lies ahead for experiments? The colossal progress of
experiments certainly owes to the audacious ideas of their creators,
but not only. They also constantly drew on technological and scientific
advances from other fields: from the dawn of spacecraft in the 1960s it
became possible to escape lab gravity twenty years
later~\cite{hart1986_sci}. LASERs and digital signal processing led to
the advent of optical velocity mapping, ultrasound velocimetry made it
possible to measure velocity profiles in opaque
metals~\cite{brito2001_ef, gillet2007_jfm2, baker2017_ef}. At the same
time, experimental design has incorporated more and more complex
aspects of outer core dynamics: cylinders became slanted annuli, then
spheres, then spheres with a puck. Heating became inhomogeneous,
internal~\cite{bouillaut2021_pnas, Zaussinger2020,
chamberlain1986_gafd} and now magnetic fields are becoming an
increasingly important part of the mix.

What is next is certainly found at the crossing of what is currently
missing or insufficiently explored in terms of physical processes and
what is becoming technologically possible. It is the biased view of the
authors that the greatest scope for progress may lie in incorporating
MHD effects in the rotating convection experiments. Certainly, the
dynamo problem has been the focus of significant experimental
endeavours, whose main focus was to find under which conditions a flow
could spontaneously generate a magnetic field. The physical processes
reproduced in these experiments are largely inspired by potential
mechanisms thought to be driving them in planets and stars. Convection
is the best candidate for the Earth and all planets of the Solar system
but tidal forces exerted by the Earth and precession of the Moon's
rotation axis may have driven a past dynamo on the
Moon~\cite{shea2012_sci, stys2020_jgr}. For Ganymede, iron snow
produced by compositional precession is a possible dynamo
mechanism~\cite{christensen2015_icarus}. Yet, successful dynamos to
date are all mechanically driven: the Karlsruhe dynamo forces a flow
through pipes shaped to guarantee the dynamo~\cite{stieglitz2001_pf}.
In the Riga dynamo, a turbine produces a double helix flow
structure~\cite{gailitis2000_prl}, and in the \mbox{Von K\'arman Sodium (VKS)}
dynamo, magnetised counter-rotating impellers generate an unstable von
K\'arman flow and produce the dynamo~\cite{monchaux2007_prl,
berhanu2007_epl}. The upcoming DRESDYN dynamo project in Dresden will
spin a 8~m$^3$ cylinder filled with Sodium along two axes to produce a
precession-driven dynamo~\cite{stefani2019_gafd, stefani2024_crphys}.
This will potentially be the first dynamo driven by the same process as
a geophysical or astrophysical dynamo. Hence, the generic character of
this question and the magnitude of the challenge involved in building
experimental dynamos have pushed aside the question of how actual
planetary dynamos work~\cite{petrelis2024_crphys}. So understandably,
none of the dynamo experiments directly model actual planetary dynamos
and the playground of understanding planetary dynamos has been occupied
by numericists~\cite{glatzmaier1999_nat, Schaeffer2017}. For planets,
the question is to find which flow within the outer core produces a
magnetic field consistent with observations. Crucially, the associated
Lorentz force due to the magnetic field may drastically alter the
answers to that question. In fact, the effects of the Earth's field are
often invoked when extrapolating rotating convection experiments to
Earth~\cite{cardin1994_pepi, aubert2001_pepi, aurnou2003_epsl}. Yet,
the Lorentz force is still subject to debate: proponents of the
geostrophic scenario argue that it remains much smaller than the
Coriolis force~\cite{schwaiger2019_gji, aubert2020_gji}, while others
argue that a magnetostrophic scenario where both forces are comparable
is required to explain the flow patterns inferred from geomagnetic
data~\cite{dormy2016_jfm, horn2022_prsa}. Here the availability of
large electromagnets offers scope for experiments in magnetorotating
convection to advance these questions. These possibilities are only
starting to be explored: fields up to 18~T in sufficiently large
volumes to host laboratory experiments~\cite{pugnat2022_ieee} even make
it possible to reach extreme regimes where Alfv\'en waves may interact
with convection~\cite{roberts2000_jfm, lalloz2024_jfm}. Such
experiments may also benefit from decades of accumulated knowledge on
liquid metal MHD turbulence~\cite{alemany1979_jm, messadek2002_jfm,
alboussiere1999_etfs, eckert2001_ijhff, potherat2014_jfm,
potherat2017_prf, baker2018_prl} and MHD convection tracing back to
Nakagawa's first experiments~\cite{Nakagawa1955, davoust1999_jfm,
Aurnou2001, vogt2018_prf}. Such large magnets combined with measurement
techniques developed for transparent electrolytes~\cite{andreev2003_pf,
andreev2013_jfm, alferenok2013_mst, moudjed2020_ef} now even offer the
possibility of mapping velocity and pressure fields in magnetorotating
convection at similar $Ek$ to non-MHD experiments but with Elsasser
numbers within the range expected for the Earth (between 0.1 and
10)~\cite{aujogue2016_rsi, potherat2023_prl}. So perhaps now is the
time for Hide's 1958 wisdom to finally come true~\cite{hide1958_pta}:
{\vspace*{4pt}}
\begin{quote}
{Now it is clear that extensions in several directions,
especially into hydromagnetics, will have to be made if results of any
geomagnetic interest are to be obtained.}
\end{quote}
{\vspace*{4pt}}
Other, non-MHD processes that have been well investigated outside the
context of convection in the outer core can be integrated into rotating
convection experiments to capture essential elements of the outer core
dynamics: solidification and re-melting of the inner core releases or
absorbs lighter elements into/from the liquid outer core, and so incurs
significant compositional buoyancy~\cite{alboussiere2009_nat}. Only one
experiment focused on compositional convection in the context of the
inner core~\cite{cardin1992_grl}. It is sometimes dismissed on the
ground that the large Schmidt numbers associated with it justify
mimicking convection there with high Prandtl number thermal convection.
This analogy may not capture well the crucial conditions at the core
boundaries, and even less the interplay between thermal and
compositional convection.

Variations of physical properties too may play several roles. In gas
giants, they can lead to compressibility effects or to a radial
regionalisation of convection forming layers with strongly
differentiated convective patterns~\cite{leconte2012_aa,
gastine2021_icarus, moore2022_jgr, fuentes2024_apjl, horn2024_epsl}.
While less significant in the liquid core of Earth-like planets,
density variations and the variation of other physical properties may
still produce local stratification capable of locally suppressing
convection: the presence of such a stratified layer has indeed been
theorised near the core-mantle boundary~\cite{buffett2010_jgr}.

The role of the centrifugal force is somewhat less obvious: it remains
very small at the scale of planetary cores, but may act at the scale of
crucial convection patterns forming there\cite{Horn2021}: tornadoes
forming in the Sun offer a reminder not to discard the centrifugal
force too quickly in large systems\cite{Luna2015, Kuniyoshi2023}.

Another seven decades will surely tell which of these, or other avenues
convection experiments will travel along, but there is little doubt
that even then, they will still be advancing in the journey to the
outer core of the Earth.{\vspace*{-.15pc}}

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.
{\vspace*{-.15pc}}

\section*{Funding}

AP is supported by EPSRC (grant EP/X010937/1) and the Leverhulme Trust
(grant RPG-2017-366). SH is supported by the UKRI Horizon Europe
guarantee, selected by the European Research Council (ERC), grant no.
EP/X034402/1 (MAGNADO). SH further received funding from EPSRC, grant
no. EP/V047388/1.
{\vspace*{-.15pc}}

\CDRGrant[EPSRC]{EP/X010937/1}
\CDRGrant[LT]{RPG-2017-366}
\CDRGrant[ERC]{EP/X034402/1}
\CDRGrant[EPSRC]{EP/V047388/1}

\section*{Acknowledgements}

We would like to thank John Brothold for his inspiring talk at the 18th
SEDI conference at Great Barrington, MA. The authors acknowledge the
relaxed atmosphere around the time of the XXXIIIrd Zappanale in Bad
Doberan, Germany, which allowed them to fully focus on this review, and
perhaps sipped into some of the wording therein, thusly prompting the
question: ``{\it Does humour belong in} [Science]?''.
{\vspace*{-.15pc}}

\back{}

\section*{Supplementary data}

Supporting information for this article is available on the journal's
website under \printDOI\ or from the author.

\CDRsupplementaryTwotypes{supplementary-material}{\cdrattach{crphys-233-suppl.pdf}}

\printbibliography
\refinput{crphys20240723-reference.tex}

\end{document}
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