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\DOI{10.5802/crphys.219}
\datereceived{2024-05-03}
\daterevised{2024-10-17}
\dateaccepted{2024-10-17}
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\dateposted{2025-01-06}
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\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 aulaboratoire}
%\makeatother

\CDRsetmeta{articletype}{review}

\title{Geophysical flows over topography, a playground for laboratory experiments}

\alttitle{\'Ecoulements g\'eophysiques en pr\'esence de topographie, un
terrain de jeu pour la mod\'elisation exp\'erimentale}

\shortrunauthors

\author{\firstname{J\'er\'emie} \lastname{Vidal}\CDRorcid{0000-0002-3654-6633}}
\address{Universit\'e Grenoble Alpes, CNRS, ISTerre, 38000 Grenoble, France}
\email[J. Vidal]{jeremie.vidal@univ-grenoble-alpes.fr}

\author{\firstname{J\'er\^ome} \lastname{Noir}\CDRorcid{0000-0001-9977-0360}\IsCorresp}
\address{Institut f\"ur Geophysik, ETH Z\"urich, Sonneggstrasse 5, Z\"urich 8092, Switzerland}
\email[J. Noir]{jerome.noir@eaps.ethz.ch}

\author{\firstname{David} \lastname{C\'ebron}\CDRorcid{0000-0002-3579-8281}}
\addressSameAs{1}{Universit\'e Grenoble Alpes, CNRS, ISTerre, 38000 Grenoble, France}
\email[D. C\'ebron]{david.cebron@univ-grenoble-alpes.fr}

\author{\firstname{Fabian} \lastname{Burmann}\CDRorcid{0000-0001-8095-1081}}
\addressSameAs{2}{Institut f\"ur Geophysik, ETH Z\"urich, Sonneggstrasse 5, Z\"urich 8092, Switzerland}
\email[F. Burmann]{fabian.burmann@eaps.ethz.ch}

\author{\firstname{R\'emy} \lastname{Monville}\CDRorcid{0000-0002-9460-2293}}
\addressSameAs{1}{Universit\'e Grenoble Alpes, CNRS, ISTerre, 38000 Grenoble, France}
\email[R. Monville]{remy.monville@univ-grenoble-alpes.fr}

\author{\firstname{Vadim} \lastname{Giraud}\CDRorcid{0009-0003-6820-0148}}
\addressSameAs{2}{Institut f\"ur Geophysik, ETH Z\"urich, Sonneggstrasse 5, Z\"urich 8092, Switzerland}
\email[V. Giraud]{vadim.giraud@eaps.ethz.ch}

\author{\firstname{Yoann} \lastname{Charles}\CDRorcid{0000-0002-4861-8016}}
\addressSameAs{2}{Institut f\"ur Geophysik, ETH Z\"urich, Sonneggstrasse 5, Z\"urich 8092, Switzerland}
\address{Development of Advanced Engineering Solutions (DAES), Avenue des Grandes-Communes 8, Petit-Lancy 1213, Switzerland}

\thanks{European Research Council (ERC) (grant agreement No 847433,
THEIA project), ETH grant \#ETH-04 22-1, European Research Council (ERC)
(grant agreement No 833848, UEMHP project), French Academy of Sciences,
Electricit\'e de France, Labex OSUG@2020 (ANR10 LABX56)}

\keywords{\kwd{Topography}\kwd{Rotation}\kwd{Stratification}
\kwd{Geophysical flows}
\kwd{Planetary cores}\kwd{Subsurface oceans}}

\altkeywords{\kwd{Topographie}\kwd{Rotation}\kwd{Stratification}
\kwd{\'Ecoulements g\'eophysiques}\kwd{Noyaux
plan\'etaires}\kwd{Oc\'eans de subsurface}}

\begin{abstract}
Physicists face major challenges in modelling multi-scale phenomena
that are observed in geophysical flows (e.g.\ in the Earth's oceans and
atmosphere, or liquid planetary cores). In particular, complexities
arise because geophysical fluids are rotating and subject to density
variations, but also because the fluid boundaries have complex
geometries (e.g.\ the ocean floor) with wavelengths ranging from metres
to thousands of kilometres. Dynamical models of planetary fluid layers
are thus often constrained by observations, whose interpretation
necessitates a comprehensive understanding of the underlying physics.
To this end, geophysical studies often combine cutting-edge experiments
across a wide range of parameters, together with theory and numerical
simulations, to derive predictive scaling laws applicable for planetary
settings.  In this review, we discuss experimental efforts that have
contributed to our understanding of geophysical flows with topography.
More specifically, we focus on (i) the flow response to mechanical
(orbital) forcings in the presence of a large-scale (ellipsoidal)
topography, (ii) some effects of small-scale topography onto bulk flows
and boundary-layer dynamics, and (iii) the interaction between
convection and roughness. The geophysical context is briefly introduced
for each case, and some experimental perspectives are drawn.
\end{abstract}

\begin{altabstract}
Les \'ecoulements g\'eophysiques, par exemple dans les oc\'eans ou les
noyaux plan\'etaires liquides, pr\'esentent souvent une turbulence
caract\'eris\'ee par une multitude d'\'echelles de temps et d'espace.
Celles-ci r\'esultent notamment des effets de rotation globale et de
stratification en densit\'e, mais aussi des effets de paroi qui sont
souvent irr\'eguliers (e.g. la bathym\'etrie du plancher oc\'eanique).
Ainsi, peu d'\'etudes ont consid\'er\'e la mod\'elisation des
\'ecoulements g\'eophysiques avec rotation, stratification en densit\'e
et effets topographiques. En pratique, la meilleure approche consiste
souvent \`a combiner des exp\'eriences de laboratoire sur une large
gamme de param\`etres, ainsi que des travaux th\'eoriques et/ou
num\'eriques, pour ensuite extrapoler les r\'esultats aux conditions
g\'eophysiques. Dans cet article de revue, nous discutons les travaux
principalement exp\'erimentaux qui ont permis de mieux comprendre la
dynamique des \'ecoulements g\'eophysiques avec des effets
topographiques de paroi. Premi\`erement, nous d\'etaillons les
\'ecoulements engendr\'es par les for\c{c}ages orbitaux (e.g. la mar\'ee
ou la pr\'ecession) pour un fluide contenu dans un ellipso\"ide.
Ensuite, nous illustrons les effets d'une topographie de petite
\'echelle sur certains \'ecoulements en volume et de couche limite.
Enfin, nous discutons l'influence d'une paroi non lisse sur les
\'ecoulements engendr\'es par la convection. 
\end{altabstract}

\maketitle

\end{noXML}

\section{Introduction}

Fluid dynamics enjoys great popularity among geophysicists and
planetary scientists, which is not surprising considering the ubiquity
of fluid phenomena in the universe.  Many rocky planets possess shallow
thin fluid layers (e.g.\ oceans or atmospheres), as well as thicker ones
in depth (e.g.\ liquid cores or subsurface oceans in icy moons).  From a
physicist's perspective, geophysical fluid dynamics serves as both an
applied and pure science.  It is applied because we observe and predict
the effects of geophysical flows in our daily life (such as weather
forecasting, climate evolution, or Earth's magnetic field variations). 
Yet, it is also pure because most geophysical flows involve fundamental
fluid dynamics processes (like waves, hydrodynamic instabilities, and
turbulence) that are still not fully understood.

Indeed, several scientific questions currently challenge our
geophysical flow models, such as the origin of the ancient magnetic
fields on Earth~\cite{landeau2022sustaining} and the
Moon~\cite{wieczorek2023lunar}.  Similarly, controversies persist
regarding estimates of heat transport by turbulent convection, which is
key for understanding the long-term evolution of natural systems over
billions of years.  \rev{Actually, these questions arise because the
internal dynamics of planetary bodies remains barely known.} In the
middle of the nineteenth century, Hopkins~\cite{hopkins1839nutations}
proposed that one could probe planets' interiors from the variations of
their rotation. Indeed, if a planet or a moon has a hidden fluid layer,
it could be inferred by looking at the planet's response to orbital
perturbations, such as precession or nutations.  This pioneering idea
found many applications in the twentieth century, such as the
confirmation of the liquidity of the Earth's outer
core~\cite{jeffreys1957theory}, or the existence of a molten core in
Mercury~\cite{margot2007large} and in the Earth's
moon~\cite{yoder1981free}.   Nowadays, the increasing accuracy of the
Earth's and moon's rotations provides a unique opportunity to go beyond
the detection of their liquid cores to address the question of their
internal dynamics.  Meanwhile, space missions like the \textsc{esa}
mission \textsc{juice} or the \textsc{nasa} mission  \textsc{clipper},
which are dedicated to the icy moons of Jovian planets, will provide
unprecedented data to probe their subsurface oceans.  However, global
models based on Earth-like oceanic circulation are unlikely to be
directly applicable to these envelopes~\cite{soderlund2024physical}.
Specific models are thus still missing to properly model the fluid
dynamics within such moons.

Therefore, understanding the dynamics of planets' and moons' fluid
envelopes remains a central question in contemporary geophysics and
planetary sciences.  Before discussing recent progress on this topic,
we shall first establish the fundamental physical components necessary
for modelling these fluid envelopes.


\subsection{The key ingredients of geophysical fluid models}

Planetary fluid envelopes are unique by their large size, but also
because of the physical mechanisms governing their dynamics.   Notably,
they often experience \rev{the effects of} global rotation. The Earth
is currently rotating at an angular velocity $\Omega_s \simeq 7.3
\times 10^{-5}$~rad${\cdot}$s$^{-1}$, and was even rotating faster in the
past~\cite{farhat2022resonant}.  \rev{Global rotation is dynamically
important if, first, the dynamics is in a low-viscosity regime (i.e.\ 
with a rotation time scale much smaller than the viscous time scale). 
This leads to the definition of the Ekman number $E$ given by
{\begin{equation}
 E = \frac{\nu}{\Omega_s L^2},
\end{equation}}\unskip
where $\nu$ is the kinematic viscosity of the fluid, and $L$ is the
typical length scale of the system. The low-viscosity regime is thus
defined by $E\, \ll\, 1$.  Moreover, geophysical turbulence also depends on
the relative strength of the nonlinearities in the system.  We
introduce the Rossby number \textit{Ro} (comparing the rotation and nonlinear
advection time scales), which is related to the classical Reynolds
number \textit{Re} (which compares the viscous and nonlinear advection time
scales) as $Ro = \mathit{Re} E \, (L/\ell)^2$ with
\changeEqNo{2a,b}
{\begin{subequations}
\begin{equation}
 \mathit{Ro} = \frac{u_\ell}{\Omega_s \ell}, \quad \mathit{Re} = \frac{\ell u_\ell}{\nu},
\end{equation}
\end{subequations}}\unskip
\resetEqNo{2}\unskip
\noindent
where $u_\ell$ is the typical amplitude of the velocity at the flow
length scale $\ell$. The dynamics ignores rotation when $\mathit{Ro} \gg 1$,
whereas it is strongly shaped by rotational effects when $\mathit{Ro} \ll 1$. 
Note that the length scale $\ell$ introduced above is still undefined. 
Actually, since geophysical systems exhibit many length scales in the
range $0 &lt; \ell \,\leq\, L$, different dynamical regimes can be expected as
a function of the length scale (e.g.~\cite{nataf2024dynamic} for liquid
planetary cores). In practice, the Rossby number can have values from
$\mathit{Ro} \ll 1$ at large scales to $\mathit{Ro} \gtrsim 1$ at smaller scales. The
transition between these regimes is expected to be controlled by the
Zeman length scale  defined as~\cite{zeman1994note}
{\begin{equation}
 \ell_z \sim \left ( \frac{\epsilon}{\Omega_s^3} \right )^{1/2},
\end{equation}}\unskip
where $\epsilon$ is the mean energy dissipation rate per unit of mass. 
Turbulent structures with length scales $\ell \gg \ell_z$ are modified
by global rotation and anisotropic, whereas the turbulence is
unaffected by rotation and nearly isotropic when $\ell \ll
\ell_z$~\cite{mininni2012isotropization,delache2014scale}. If we
consider the largest scales (assuming $\ell \sim L$)}, typical values
are $\mathit{Ro} \sim 10^{-7}-10^{-6}$ for the Earth's liquid core with $E \sim
10^{-15}$ (assuming $u_\ell \sim 1$--10~km${\cdot}$y${}^{-1}$ for core flows
driven by convection~\cite{jones2015tog}, and the core radius ${\sim}$3480~km). 
This indicates \rev{that the Earth's liquid core} is in a
regime where viscous and advection effects are much smaller than
rotational ones.  Note also that a typical value for the Reynolds
number is then $\mathit{Re} \sim 10^{8}$ for the largest scales. \rev{The later
example clearly illustrates a common difficulty in geophysical
modelling, namely obtaining turbulent flows with $\mathit{Re} \gg 1$ in the
regime $\mathit{Ro} \ll 1$, which is only accessible by considering low enough
values of $E$.}

Second, geophysical flows are often subject to buoyancy effects and
density stratification. The latter is usually characterised by the
Brunt--V\"ais\"al\"a (BV) frequency, denoted below by $N$. For a
compressible fluid, the square of the BV frequency is given
by~\cite{chaljub2004spectral}
%\changeEqNo{4}
{\begin{equation}
 N^2 = \frac{\boldsymbol{g}}{\rho} \boldsymbol{\cdot} \left ( \nabla \rho - \frac{\rho}{c_s^2} 
 \boldsymbol{g} \right ),
 \label{eq:BVN}
\end{equation}}\unskip
where $\boldsymbol{g}$ is the gravity field, $\rho$ is the fluid
density, and $c_s$ is the speed of sound. The BV frequency measures the
departure of the actual density gradient to the adiabatic (isentropic)
density gradient (given by $\rho \boldsymbol{g}/c_s^2$ in hydrostatic
equilibrium). As such, stably stratified fluids (e.g.\ a light fluid
over a dense one in laboratory conditions) are characterised by
$N^2&gt;0$, whereas convection can occur for unstable stratification with
$N^2&lt;0$. For example, the Earth's oceans are known to be stably
stratified in density. There is some variability as a function of the
location and the seasons~\cite{emery1984geographic}, but typical values
are $N \sim 10^{-3}$--$10^{-2}$~rad${\cdot}$s${}^{-1}$ in the upper ocean and $N
{\sim}10^{-5}$--$10^{-4}$~rad${\cdot}$s${}^{-1}$ in the deep ocean 
(Figure~\ref{fig:exmp}a). 
On the contrary, the Earth's liquid core is
convecting in bulk~\cite{jones2015tog} (which implies $N^2 \sim 0$),
but it might be either weakly stratified with $N/\Omega_s \leq
\mathcal{O}(1)$~\cite{buffett2014geomagnetic,gastine2020dynamo} or
strongly stratified with $N/\Omega_s \gg 1$~\cite{buffett2010chemical}
in a thin layer near the core-mantle boundary\footnote{This is the
interface between the liquid core made of molten iron, and the mantle
made of silicate rocks.} (\textsc{cmb}). Therefore, the two dynamical
regimes should be investigated.

\begin{figure}[tbp]
\includegraphics{fig01}
\caption{\label{fig:exmp}{(a)} Illustrative profiles of the Brunt--V\"ais\"al\"a
frequency $N$, as a function of depth, in the Earth's oceans (adapted
from~\cite{gent1985much}). Inset shows a typical profile in the deep
ocean in the western North Pacific (adapted from figure 3.6
in~\cite{talley2011descriptive}). {(b)} Spatial spectra of
topography, where $h_t$ is the height of the topography at wavelength
$\lambda_t$. Spectra deduced from the RMS of the observed power
spectrum at the Earth's and Mars'
surface\cite{rexer2015ultrahighdegree}, and a seismological model of
the Earth's \textsc{cmb} (obtained from body waves) is also
shown~\cite{koelemeijer2021consistent}. Blue region shows the (laminar)
Ekman boundary layer, of typical thickness
$\mathcal{O}(\sqrt{\nu/\Omega_s})$, at the Earth's \textsc{cmb}  or at
the bottom of the surface oceans (which have the same viscosity
$\nu\approx 10^{-6}$~m$^2{\cdot}$s$^{-1}$). Empirical law $h_t \propto
2 \times 10^{-4} \, \lambda_t$ (dash-dot line) for the RMS height at
the surface~\cite{ermakov2018power}, and $h_t \propto 8 \times 10^{-8}
\, \lambda_t^2$ (dotted line) for the \textsc{cmb} (from gravity RMS
spectrum\cite{puica2023analytical}).}
 \end{figure}

Both global rotation and stratification give rise to peculiar linear
and turbulent flows, which make geophysical systems so unique and
interesting.  Yet, we can also draw some analogies between these two
kinds of dynamics~\cite{veronis1970analogy}. For instance, they can
both sustain wave motions in the form of inertial waves with angular
frequencies bounded by $2 \Omega_s$ for rotating
fluids~\cite{greenspan1968theory}, or as internal (gravity) waves with
angular frequencies bounded by $N$ for stably stratified
fluids~\cite{mowbray1967theoretical}. In the Earth's atmosphere, 
internal gravity waves emitted over mountains carry energy from the
surface to the upper atmosphere, and participate in mixing
processes~\cite{legg2021mixing} and angular momentum exchange between
the atmosphere and the solid Earth ~\cite{wurtele1996atmospheric}.
Similarly, the radiation of internal (gravity) waves over the sea floor
bathymetry contributes to ocean mixing and tidal energy
dissipation~\cite{richet2018internal}. For the dynamics, rotating and
stratified flows are also often separated into bulk and thin
boundary-layer regions.  Finally, the associated turbulence is
\rev{strongly anisotropic} (e.g.~with
rotation~\cite{godeferd2015structure} and
stratification~\cite{cortet2024turbulence,lefauve2024cras}), which is
very different from the \rev{classical picture of isotropic Kolmogorov}
turbulence.

\subsection{The shape of planetary fluid envelopes}

To complete our picture of geophysical fluid envelopes,
it is often needed to account for boundary effects.  Planets' and
moons' fluid envelopes are near-spherical bodies, on which a broad
spectrum of topography wavelengths is superimposed (see
Figure~\ref{fig:exmp}b at the Earth's surface and \textsc{cmb}).  At
the leading order, the departure from the global spherical geometry is
elliptical due to the combined action of centrifugal distortion and
tides (as qualitatively expected from the theory of equilibrium figures
for rotating fluid masses~\cite{chandrasekhar1987ellipsoidal}).  For
the Earth's core, a typical value for polar ellipticity is ${\sim}$1/400
(as estimated from \textsc{lod} variations~\cite{davies2014strength}
and the nutations~\cite{dehant2022structure}). Similarly, the
equatorial ellipticity due to the lunar tides has the typical 
amplitude ${\sim}10^{-7}$--$10^{-6}$ over geologic time 
scales, as estimated from hydrostatic tidal
theory~\cite{farhat2022resonant}.  Contemplating the Earth's surface
and oceans' floor (Figure~\ref{fig:topomap}a), we observe that
topographic features exist at all scales ranging from a few kilometres
to a few hundred metres (i.e.\ meso-scales), but sub-metres structures
are also present (i.e.\ roughness, not shown here).  Meanwhile, mantle
dynamics and global seismology suggest the presence of bumps and ridges
at the Earth's \textsc{cmb}, with heights of a few kilometres and with
horizontal extents of 1--10$^3$~km (Figure~\ref{fig:topomap}b), as well
as small-scale roughness (e.g.\ due to thermo-chemical interactions
between the liquid core and the solid
mantle~\cite{narteau2001smallscale,le2006dissipation}). Fewer data are
available for the ocean's floor or bottom ice layer of Jovian
satellites. Nonetheless, while they are insufficient to derive a
precise picture, they all suggest that the interface is not
smooth~\cite{zebker2009size,iess2010gravity,nimmo2010shell}.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:topomap}Landscape of topographies affecting geophysical flows. Colour
bar shows the topography height (in km) on a map using a Mollweide
projection. {(a)} \textsc{etopo1} global relief model of the
Earth's surface, integrating continents' topography and oceans'
bathymetry. Data provided in~\cite{amante2009etopo1}. {(b)}
Meso-scale topography at the Earth's \textsc{cmb}. Mean model obtained
from seismological studies using body-wave and normal-mode inversions.
Only the largest-scale components with spherical harmonics degree $\leq
6$ are shown. Data provided in~\cite{koelemeijer2021consistent}.}
\end{figure}

In the context of this review paper, we define thin-layer systems (e.g.\ 
the Earth's oceans and atmosphere) as fluid envelopes with a depth much
smaller than the mean planetary radius, and thick-layer ones 
(e.g.\ planetary liquid cores and subsurface oceans) as envelopes 
whose depth is a large fraction of the radius.  Let us 
finally introduce the concept of roughness and meso-scale topography
(Figure~\ref{fig:roughnessMesoscales}). We refer below to roughness as
small-wavelength topographies with height of the order of (or smaller
than) the boundary-layer thickness. The latter is typically 
\mbox{${\sim}$0.1--10~m} for a laminar viscous boundary layer in the Earth's oceans or
in the Earth's core. On~the contrary, meso-scales are defined as
topographies with small to intermediate wavelengths, whose height is
large with respect to the Ekman-layer thickness but smaller than the large 
length scale $L$ (e.g.\ $100$~m to $100$~km). 

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:roughnessMesoscales}Illustration of the concept of {(a)} roughness and
{(b)} meso-scale topography in a planetary context.}
\end{figure}

\subsection{Geophysical modelling, a challenge for numerics and experiments}

Flows in geophysical fluid envelopes are mostly driven by two kinds
of forcings, namely thermal-solutal convection~\cite{jones2015tog} or
mechanical forcings through coupling with a rigid boundary (e.g.\ at a
\textsc{cmb}, or an icy crust in Jovian moons). In geophysical systems,
mechanical forcings typically result from orbital perturbations such as
precession, librations, or tides~\cite{le2015flows}.  Convective flows
in geophysical contexts are often characterised by \rev{slow} time
scales, while orbitally driven dynamics usually occur on time scales
comparable to the planetary rotation time scale.  This contrast gives
rise to distinct regimes of \rev{geophysical turbulence}.  Rapidly
rotating convection is expected to be quasi two-dimensional ($2$-D)
\rev{in the planetary regime $\mathit{Ro} \ll 1$}~\cite{nataf2024dynamic}.
\rev{In contrast, mechanical forcings can sustain anisotropic
turbulence combining three-dimensional ($3$-D) flows (e.g.\ waves) and
quasi $2$-D ones (e.g.\ mean zonal flows).} Since these phenomena are
often simultaneously present in natural systems, geophysical flows
often exhibit a wide range of time and length scales. Moreover,
geophysical systems are often characterised by rather extreme
parameters (Figure~\ref{fig:parameters}).  One of the difficulties in
geophysical studies is thus to account for such diverse phenomena, and
to approach the appropriate range of parameters.  \rev{In particular,
modelling high-\textit{Re} turbulent flows in the low-\textit{Ro} regime is very
challenging, as it requires lowering $E$ as much as possible.}  To this
end, numerical and experimental approaches can be employed, and their
pros and cons are summarised below.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig:parameters}Schematic regime diagrams for the dynamics of thick-layer
fluid systems. {(a)}~Parameter space as a function of the Rossby
number \textit{Ro} and the Ekman number $E$. It shows the huge gap between
parameters characterising geophysical systems, and those that are
accessible to experimental works and \textsc{dns} in global geometries.
\rev{Reduced QG models refer to models of rapidly rotating convection
in spherical
geometries~\cite{guervilly2019turbulent,barrois2022comparison}, in
which the flow is assumed quasi-invariant along the rotation axis.}
{(b)} Parameter space depending on $h_t/\lambda_t$ (where $h_t$
is the topography amplitude at the wavelength $\lambda_t$, see
Figure~\ref{fig:exmp}b) and \textit{Ro}.}
\end{figure}

\paragraph*{Numerics}
Direct numerical simulations (\textsc{dns}) usually offer a great
flexibility to model flows in global geometries, or to account for
phenomena that are difficult to control in laboratory conditions (e.g.\ 
dynamo action or thermal effects). However, the main disadvantage is
that probing the low-viscosity and rapidly rotating regime of
geophysical flows is beyond the capabilities of any \textsc{dns}, even
for the most advanced ones available today.  For instance, the largest
simulation of core convection struggled to consistently reach the value
$E \sim 10^{-7}$~\cite{schaeffer2017turbulent}. Moreover, it often
takes months to only model a few turnover time scales, such that the
long time scale of the flow remains currently out of reach in
\textsc{dns}. \rev{Fortunately, the expected almost $2$-D nature of
rapidly rotating convection~\cite{nataf2024dynamic} allows developing
reduced models that can efficiently probe the low-\textit{Ro} turbulent
regime~\cite{guervilly2019turbulent,barrois2022comparison}.} \rev{Yet,
we cannot easily adopt such an alternative route with topography.} 
Indeed, the numerical challenge is more pronounced when incorporating
topographic effects into the models.  Despite efforts to develop
efficient numerical methods capable of modelling topography (e.g.\ 
spectral-element methods), there remains a significant disparity of
many orders of magnitude from geophysical values (e.g.\ with $E \sim
10^{-5}$ for the most turbulent flows in global ellipsoidal
geometries~\cite{favier2015generation,grannan2017tidally}). 


\paragraph*{Experiments}
As stated by K. S. Thorne and R. D. Blandford in their textbook
\emph{Modern Classical Physics}~\cite{thorne2017modern}, fluid dynamics
is historically an experimental science. Given the aforementioned
limitations of \textsc{dns} for rapidly rotating flows with topography,
experiments still appear as a valuable alternative nowadays.  In short,
experiments do not suffer from the difficulty of resolving a broad
range of time and spatial scales, and they are not limited to weakly
nonlinear regimes.  Typically, a metre-size experiment using water and
rotating up to $60$~\textsc{rpm} corresponds to the values $E\sim10^{-7}$
and $\mathit{Ro}\sim 10^{-3}$--10.  Another benefit of experiments is that very
different shapes can be chosen for the fluid container.  For that
reason, they have proven invaluable to investigate ellipsoidal
geometries or cavities with topographical features. Briefly speaking,
experiments must be designed with care in order to (i) ensure that the
desired physical processes to investigate do control the dynamics, and
(ii) provide sufficient quantitative information to test the models or
derive scaling laws.  Both aspects are paramount to successful
experimental investigations, and they must be well-designed from the
very beginning of the experimental development.  A detailed discussion
of experimental techniques is beyond the scope of the present review,
but we can outline for the non-expert readers what is achievable by
using non-intrusive measurements.  There are different techniques to
measure the velocity, pressure, temperature, or density. They all
provide partial views of the system, in contrast with \textsc{dns}
giving access to all quantities in the fluid domain. For instance,
routinely used optical techniques in recent
experiments~\cite{sous2013friction,le2019experimental}, such as
Particle Image Velocimetry (\textsc{piv}), typically allow recovering
$2$-D maps of the velocity fields with certain spatial and temporal
resolutions.  With a typical mid-range setup, we can expect to measure
velocities $u_\ell \sim 10^{-3}$--1~m${\cdot}$s$^{-1}$, with a resolution of a
few millimetres over a view field of tens of centimetres at a sampling
frequency ${\sim}$10$^2$~Hz.  Dedicated highly customised \textsc{piv}
systems, which are much more costly, can only marginally extend this
range by a factor ${\sim}$10.  In some peculiar experimental setups,
$3$-D reconstructions are also possible but difficult to perform
(especially in rapidly rotating experiments).  In opaque fluids (e.g.\ 
liquid metal), acoustic techniques must be employed, such as Ultrasonic
Doppler Velocimetry (\textsc{udv})~\cite{noir2001experimental} or time
of flight~\cite{burmann2022low}, which only offer $1$-D profiles of a
single velocity component (with roughly the same resolution range and
slightly lower sampling frequency).  Temperature, density, or pressure
are often limited to point measurements. More elaborated techniques,
such as Laser Induced Fluorescence
(\textsc{lif})~\cite{lherm2022rayleigh} or synthetic
Schlieren~\cite{sutherland1999visualization,dalziel2000whole}, can
provide $2$-D maps but at the expense of a complex setup that is not
always possible to mount on rotating devices. 

\rev{Contrary to non-rotating regimes (i.e.\ $\mathit{Ro} \to \infty$), for which
laboratory experiments are miles ahead of \textsc{dns} for turbulent
modelling, it appears that \textsc{dns} and experiments are rather
close in terms of accessible parameters for rotating flows.  Since the
high-\textit{Re} and low-\textit{Ro} regimes are still largely unexplored, both
numerical and experimental approaches are worth considering.} Moreover,
since the extreme parameters relevant to planetary fluid envelopes
are (and will remain) out of reach of both \textsc{dns} and
experiments, we must resort to dimensional arguments and scaling laws
for geophysical applications. This requires a systematic exploration of
the parameter space and, then, the results must be extrapolated using
asymptotic laws constrained by the available data. To this end,
experiments are well suited to explore more complex small-wavelength
geometries and, sometimes, extreme dynamical regimes than \textsc{dns},
but at the expense of limited quantitative information (in particular
at low forcing).  On the other hand, \textsc{dns} are suitable to study
global geometries in laminar and less turbulent regimes, but with
well-controlled physics and with access to all resolved quantities in
the entire volume.  Consequently, they often allow exploring different
regimes in the parameter space (Figure~\ref{fig:parameters}), which
makes the two approaches complementary to infer robust scaling laws for
geophysical applications.

\subsection{Outline of the paper}

Motivated by some applications discussed above, we will describe
experimental works that have significantly contributed to our
understanding of geophysical flows with topography. We will only
describe \textsc{dns} and theory when required for interpreting 
laboratory experiments. Nonetheless, we point out that numerical and
theoretical works remain essential to make progress in our
understanding of these geophysical questions. As such, they would
deserve an independent review.  It is only through the combination of
all three approaches that we shall gain insights into the geophysical
dynamics of thick fluid envelopes.

The manuscript is divided as follows. We first describe in
Section~\ref{sec:ellipsoid} the effects of large-scale ellipsoidal
topographies on the flow dynamics driven by mechanical (orbital)
forcings. Next, we discuss in Section~\ref{sec:smallscale} how a
small-wavelength topography can modify rotating flows, with a
particular focus on spin-up experiments with meso-scale topographies
and the transition to turbulent (Ekman) boundary layers. We then
consider turbulent convection with small-wavelength topography in
Section~\ref{sec:convection}. We briefly discuss in
Section~\ref{sec:stratification} why density stratification is
challenging to account for with global rotation in experimental
conditions, before ending the manuscript in Section~\ref{sec:ccl}.

\section{Flow dynamics with a large-scale (ellipsoidal) topography}
\label{sec:ellipsoid}

We first consider the effects of large-wavelength topographies on the
flow dynamics of thick-layer envelopes.  Given the broad range of the
subject, we have chosen to focus on flows driven by mechanical
(orbital) forcings in liquid-filled ellipsoids without wall roughness,
and we also discuss the case of spherical containers when appropriate
(as the flow properties can be similar, e.g.\ for precession). For
geophysical applications, the mechanical forcings that have been mainly
considered in the literature are 
\begin{itemize}
\item tidal forcing,
\item precession (i.e.\ a fluid-filled cavity whose axis of rotation is
itself rotating around a secondary axis, which is fixed in the inertial
frame),
\item librations (i.e.\ oscillations of the figure axes of an object
with respect to a fixed, mean rotation axis), which can be either in
longitude (i.e.\ an azimuthal harmonic modulation of the rotation rate
of the cavity) or in latitude.
\end{itemize}
\rev{As discussed below}, these forcings are capable of driving
\rev{flow instabilities} and (possibly) \rev{anisotropic} turbulence in
the bulk of the cavity, through either \rev{topographic} or viscous
couplings at the boundary. Hence, \rev{they are worth considering for
geophysical applications.}

Investigating turbulent flows in ellipsoidal or spherical geometries is
still very challenging, especially in the low-viscosity regime relevant
to most geophysical systems (Figure~\ref{fig:parameters}a).  Starting
with the seminal work of Malkus~\cite{malkus1968precession},  flows
driven in ellipsoids by orbital forcings have received increasing
attention, especially using experimental and theoretical approaches. Most
studies have considered water-filled ellipsoids
(Figure~\ref{fig:manipellip}), but a few have also employed a liquid
metal to explore hydromagnetic
effects~\cite{lacaze2006magnetic,herreman2009effects}. A detailed
discussion of all these studies is beyond the scope of the present
paper. We refer the readers to~\cite{le2015flows} for an exhaustive
review of the particularities of every forcing, and
to~\cite{le2022fluid} for a summary of the different experiments. 
Here, we first emphasise that experimental studies of mechanically
driven flows in full ellipsoids (i.e.\ without an inner core) share some
common points that can be understood in light of theoretical progress. 
Secondly, we discuss whether the experimental results can be
extrapolated to planetary conditions, and outline some experimental
perspectives to build more realistic\break models. 

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig:manipellip}Illustrative experiments of mechanically driven flows in
water-filled ellipsoids, which can go down to $E \sim 10^{-6}$ and $\mathit{Ro}
\sim 10^{-2}$. {(a)} Tidal forcing. The fluid cavity, rotating
at the angular velocity $\Omega_s$, is elliptically deformed by the
action of two rollers rotating at $\Omega_{\mathrm{orb}}$. Adapted
from~\cite{grannan2017tidally}. {(b)} Precession. The
ellipsoidal cavity is rotating around its $z$-axis at the angular
velocity $\Omega_s \boldsymbol{1}_z$, which it is itself rotating at
${\bOmega}_p$ around another axis that is fixed in the
inertial frame. The two axes are tilted by the precession angle
$\alpha$ (here at $\uppi/2$). Adapted
from~\cite{burmann2022experimental}. {(c)} Longitudinal
librations. The cavity is rotating at the angular velocity $\Omega_s
\left [1 + \epsilon \sin (\omega_l t) \right ]$, where $\Omega_s$ is
the angular velocity of the fluid, $\epsilon$ is the libration
amplitude, and $\omega_l$ is the libration (angular) frequency. Adapted
from~\cite{le2019experimental}.}
\end{figure}

\subsection{The full ellipsoid, a playground for joint experimental and theoretical works}
\label{subsec:fullellipsoid}

\subsubsection{Inertial modes, a cornerstone to understand rotating flows}

Rotating flows are often shaped by the action of inertial waves, whose
restoring effect is global rotation~\cite{greenspan1968theory}. In a
bounded geometry, these waves can admit counterparts known as inertial
modes.  We assume that the fluid is neutrally buoyant (i.e.\ $N=0$), and
co-rotating with the rigid surrounding envelope at the steady angular
velocity $\Omega_s \boldsymbol{1}_\Omega$ (where
$\boldsymbol{1}_\Omega$ is a unit vector). In the rotating frame, the
fluid boundary is an ellipsoid given in Cartesian coordinates $(x,y,z)$
by $(x/a)^2 + (y/b)^2 + (z/c)^2 = 1$, where $(a,b,c)$ are the
semi-major axes of the ellipsoid.  In this rotating frame, an inertial
mode is an inviscid solution of the eigenvalue problem given by
\changeEqNo{5a\ndash c}
{\begin{subequations}
\label{eq:inertialPB}
\begin{equation}
 \mathrm{i} \omega_i \boldsymbol{u}_i + 2\Omega_s (\boldsymbol{1}_\Omega \times \boldsymbol{u}_i) = - \nabla \Phi_i, \quad \nabla \boldsymbol{\cdot} \boldsymbol{u}_i = 0, \quad \left . \boldsymbol{u}_i \boldsymbol{\cdot} \boldsymbol{1}_n \right |_{\partial V} = 0,
\end{equation}
\end{subequations}}\unskip
\resetEqNo{5}\unskip
\noindent
where $0 \leq |\omega_i| &lt; 2 \Omega_s$ is the eigenvalue of the
$i${th} normal mode, $\boldsymbol{u}_i$ is the eigenvector for
the velocity (respectively $\Phi_i$ for the pressure), and where
$\boldsymbol{1}_n$ is the unit vector normal to the boundary $\partial
V$. Note that the degenerate solutions with $\omega_i=0$ are referred
to geostrophic modes.  They satisfy the Taylor--Proudman theorem 
{\begin{equation}
  {\bOmega} \boldsymbol{\cdot} \nabla \boldsymbol{u}_i = \boldsymbol{0},
  \label{eq:TPtheo}
\end{equation}}\unskip
showing that they are invariant along the rotation axis. Inertial modes
were first evidenced by the pioneering experiment of Aldridge and
Toomre~\cite{aldridge1969axisymmetric}, who considered a full sphere
subject to longitudinal librations (as in
Figure~\ref{fig:manipellip}c). Indeed, some inertial modes
can be excited by librations when the libration frequency $\omega_l$
is close to some eigenvalues $\omega_i$
(Figure~\ref{fig:inertialmodes}a). Theoretical analysis later showed
that such inertial modes can be excited because of viscous effects in
the Ekman boundary
layer~\cite{zhang2013non,cebron2021mean,lin2023resonant}. Other
mechanisms can also sustain inertial waves or modes (after multiple
reflections), such as boundary-layer
instabilities~\cite{sauret2013spontaneous} or \rev{bulk
forcings} \mbox{\cite{morize2010experimental,zhang2017theory,lin2021triadic}.}

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig:inertialmodes}Inertial modes in a rapidly rotating sphere or ellipsoid.
{(a)} Time average of the normalised pressure difference $C_p$ between
the centre and the pole of a water-filled sphere subject to
longitudinal librations (as in Figure~\ref{fig:manipellip}c), where
$\epsilon = \tilde{\epsilon} \omega_l/\Omega_s$ is the libration
amplitude with $\tilde{\epsilon} = 8\text{\textdegree}$, for the Reynolds number
$\mathit{Re}_\omega = \omega_l a^2/\nu = 6.2 \times 10^{4}$ (with the radius $a
= 10$~cm of the sphere). Comparison between experiments, \textsc{dns},
and theoretical predictions. Adapted from figure 3
in~\cite{aldridge1969axisymmetric} and figure 2
in~\cite{sauret2010experimental}. {(b)} Asymptotic distribution
of the normalised eigenvalues $\omega_i/(2\Omega_s)$ in the interval
$[-1,1]$, in a spheroid with $a=b=1$ rotating along the $z-$axis, as a
function of $1/c^2$. Adapted from figure 2 in~\cite{CdV2023spectrum}.}
\end{figure}

Next, we may wonder how inertial modes are modified when the sphere is
deformed into an ellipsoid. Insight into this question can be gained by
looking at the two simplest inertial modes, called the spin-over modes.
They are associated with a uniform vorticity along the equatorial\break $x$-
or $y$-axes. Their eigenvalues $\omega_{\text{so}}$ are given
by~\cite{vantieghem2014inertial}
{\begin{equation}
 \omega_{\text{so}} (a,b,c) = \pm 2 \Omega_s \frac{ab}{\sqrt{a^2+c^2}\sqrt{b^2+c^2}}.
 \label{eq:spinover}
\end{equation}}\unskip
The ellipsoidal deformation is thus responsible for a shift in
frequency of the spherical eigenvalue $\omega_{\text{so}} (1,1,1)\, =\,
\Omega_s$. Similar effects exist for higher-order inertial modes, which
are more or less pronounced depending on the modes' complexity. 
Therefore, detecting some inertial modes could be a way to probe the
ellipticity of a fluid envelope (which is linked to Hopkins' original
idea~\cite{hopkins1839nutations}). Figure~\ref{fig:inertialmodes}(b)
illustrates a more subtle effect of the large-scale ellipsoidal
topography. As proved in~\cite{CdV2023spectrum}, the frequency
distribution of the eigenvalues is different in a sphere and an
ellipsoid.  The probability density function is uniform in a sphere,
but non-uniform in an ellipsoid.  For instance, low-frequency modes
with $|\omega_i| \to 0$ are favoured when the ellipsoid is strongly
elongated with $c/a\gg 1$, whereas high-frequency modes with
$|\omega_i| \to 2 \Omega_s$ are enhanced in flattened ellipsoids with
$c/a\,\ll\, 1$. Consequently, this effect likely weakens the physical
relevance of the oldest works, which employed strongly deformed
ellipsoids to overcome diffusive effects, for modelling the wave-driven
dynamics of geophysical flows in weakly deformed\break spheres.

Finally, a key property of inertial modes in an ellipsoid is that any
divergenceless and square-integrable\footnote{i.e.\ any incompressible
flow, either viscous or not, but with a finite value of the
volume-averaged kinetic energy.} velocity $\boldsymbol{v}$ can be
sought as a combination of the inviscid eigenvectors (including the
geostrophic ones) as~\cite{backus2017completeness,CdV2023spectrum}
{\begin{equation}
\boldsymbol{v}(\boldsymbol{r},t) = \sum_{i=1}^{\infty} \gamma_i(t) \boldsymbol{u}_i(\boldsymbol{r}),
\label{eq:IMexpansion}
\end{equation}}\unskip
where $\boldsymbol{r}$ is the position vector and $\{\gamma_i (t)\}_i$
are arbitrary time-dependent coefficients. Therefore, many experimental
results have been interpreted by using theories based upon inertial
modes.  So far, several phenomena have been experimentally observed,
and the interplay with theory has allowed us to understand a somehow
generic flow response in an ellipsoid (Figures~\ref{fig:poincare}--\ref{fig:turbellip}).

\begin{figure}
{\vspace*{-2pt}}
\includegraphics{fig07}
{\vspace*{-2pt}}
\caption{\label{fig:poincare}Laminar uniform vorticity flows, as obtained in experiments
and predicted by theory. {(a)} Direct resonance for latitudinal
librations. Triaxial ellipsoid with $a=0.078$~m, $b=0.125$~m and
$c=0.1042$~m, subject to librations at $E=1.57 \times 10^{-5}$.
Vertical dashed line shows the value of $\omega_{\text{so}}$ given by
Equation (\ref{eq:spinover}) for this geometry. Experimental values
obtained using Ultrasonic-Doppler-Velocimetry (\textsc{udv}) 
measurements, and the theory is given by a uniform-vorticity
model~\cite{vantieghem2015latitudinal}. See further details
in~\cite{charles2018flows}. {(b)} Vertical component of the
fluid rotation vector $\omega_z/\Omega_s$, as a function of
$\Omega_p/\Omega_s$, in a spheroid with $c/a \simeq 0.847$. Precession
forcing with $E = 2.3 \times 10^{-5}$ and a precession angle of
15\textdegree. Experimental values obtained using \textsc{piv}
measurements. Hysteresis is found by either gradually decreasing
($\circ$) or increasing ($\times$) the value of \textit{Po}. Theory obtained
from~\cite{busse1968steady}. Adapted from figure 4
in~\cite{nobili2021hysteresis}.}
\end{figure}

\begin{figure}
\includegraphics{fig08}
{\vspace*{-2pt}}
\caption{\label{fig:geosCylinderPrecTide}Similarities of mean (laminar) flows observed in ellipsoids
subject to mechanical forcings. Visualisations in a meridional plane
using Kalliroscope flakes. {(a)} Precession in a spheroid with
$c/a = 0.96$. Experiment performed at $E=2.24 \times 10^{-6}$ and
$\Omega_p/\Omega_s=-0.01$, for a precession angle of 20\textdegree.
${\bOmega}_f$ is the axis of rotation of the fluid, which is
tilted from the mantle spin axis along $\Omega_s$. Adapted
from~\cite{noir2000ecoulements}. {(b)} Tidal flow in an
ellipsoid. Experiment performed at $E = 10^{-5}$. Here, the orbital
angular velocity along $\Omega_{\text{orb}}$ (enforced by the two
rollers) is inclined by an angle of 5\textdegree\ from the spin axis along
$\Omega_s$ (which is aligned with ${\bOmega}_f$). Adapted
from~\cite{morize2010experimental}.}
\end{figure}

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig:resellip}Schematic flow response in ellipsoids subject to mechanical
forcings. {Top}: Front visualisations of libration-driven flows in
an ellipsoid at $E\simeq2.7 \times 10^{-5}$. Adapted from figure~3
in~\cite{grannan2017tidally}. {Bottom}: Equatorial visualisations
(using reflective flakes) of precession-driven flows in a sphere (with
orthogonal spin and precession axes) at $E \simeq 1.25 \times 10^{-5}$.
Adapted from Figures 5 and 13 in~\cite{goto2014turbulence}.}
\end{figure}

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig:turbellip}{(a)} Different turbulent regimes of libration-driven
flows in an ellipsoid at $E\simeq2 \times 10^{-5}$. Vertical axis shows
the volume-averaged velocity (measured from equatorial \textsc{piv} in the
experiment), which is also averaged in time using four points per
libration cycle to only retain the long-term time scales of the flow.
Adapted from figure 6 in~\cite{grannan2014experimental}. {(b)}
$3$-D (wave) turbulence versus $2$-D (geostrophic) turbulence.
Equatorial visualisations (using reflective flakes) of
precession-driven turbulence in a sphere (with orthogonal spin and
precession axes) at $E \simeq 1.25 \times 10^{-5}$. Adapted from figure
5 in~\cite{goto2014turbulence}.}
\end{figure}

\subsubsection{Forced (laminar) flows, the first brick}

\rev{Beyond inertial modes or waves}, mechanical forcings often drive a
(laminar) basic flow. It can involve in Equation (\ref{eq:IMexpansion})
either a single modal component (in the case of a perfect resonance, as
in Figure~\ref{fig:inertialmodes}a) or a superposition of few
large-scale modes.  In the particular case of precession, the forced
basic flow is close to a uniform-vorticity flow (even with
viscosity~\cite{busse1968steady,kida2020steady}), which is essentially
a tilted rotation with an irrotational strain to satisfy the
non-penetration at the
boundary~\cite{roberts2011flows,noir2013precession}.  Such
uniform-vorticity flows, which are often referred to as Poincar\'e
flows, are the only ones to carry angular momentum in an
ellipsoid~\cite{ivers2017enumeration,vidal2023precession}. As such,
they are often expected to play a particular role in the rotational
dynamics of the whole planet~\cite{rekier2022earth}.  Depending on the
frequency of the forcing, the forced flow can exhibit direct resonances
(Figure~\ref{fig:poincare}a), whose amplitude is smoothed out by
viscosity in
experiments~\cite{noir2013precession,vantieghem2015latitudinal,vidal2023precession}. 

Despite their apparent simplicity, laminar basic flows in ellipsoids
can have a rather complicated evolution in the parameter space. For
instance, the uniform-vorticity flow in a precessing cavity can be
non-unique for a broad range of precession rates
(Figure~\ref{fig:poincare}b), as found in experiments
~\cite{nobili2021hysteresis,burmann2022experimental} and in agreement
with prior theory~\cite{busse1968steady,cebron2015bistable}. Some
viscously driven flows can also be important ingredients to build a
complete model, such as the oblique shear layers generated by the
breakdown of the oscillatory Ekman layer at the critical
latitudes~\cite{noir2001experimental,kida2011steady}, or laminar
mean flows resulting from nonlinear interactions in the
Ekman boundary layer (as reported in experiments for all three
forcings~\cite{suess1971viscous,sauret2010experimental,noir2012experimental,grannan2017tidally}) 
\rev{or from self-interactions of wave motions 
(as~predicted~\cite{tilgner2007zonal,le2020reflection,lin2021libration} and
experimentally observed~\cite{morize2010experimental}).}  It is
interesting to note the similarity of the mean (zonal) flows obtained
in both tidally-driven and precession-driven flows
(Figure~\ref{fig:geosCylinderPrecTide}). In both cases, the tilted
rotation of the fluid (directly forced for precession, but resulting
from a triadic instability for tidal forcing, see below) generates an
almost identical pattern of zonal flows, which have yet to be fully
understood.  

\subsubsection{The long route to rotating turbulence}

The forced laminar flow is usually linearly unstable to small-amplitude
perturbations.  As outlined in Figure~\ref{fig:resellip}, we often
observe the following route to turbulence in experiments, from the
onset of hydrodynamic instabilities (when the forcing amplitude is
strong enough to overcome diffusive effects) to \rev{anisotropic}
turbulence.

\paragraph*{(1) Instabilities}
Linear stability theory suggests that triadic (parametric)
instabilities are often triggered by mechanical forcings.  In
diffusionless fluids, these triadic (parametric) instabilities obey the
generic resonance condition given by 
{\begin{equation}
 \omega_0 = \omega_1 \pm\, \omega_2,
\end{equation}}\unskip
where $\omega_0$ is the angular frequency of the forced Poincar\'e flow
viewed in the frame rotating with the fluid (e.g.\ $\omega_l$ for
longitudinal librations as in Figure~\ref{fig:manipellip}c), and
$\omega_{1,2}$ are the angular frequencies of two (free) inertial modes
of the inviscid fluid.  A triadic (parametric) instability can only
grow upon the basic flow if its diffusionless growth rate $\sigma$ is
larger than the diffusive damping effects in the system.  For an
ellipsoidal boundary without roughness, the leading-order dissipative
mechanism \rev{in a hydrodynamic experiment} is provided by the
friction within the Ekman layer~\cite{greenspan1968theory}.  For a
laminar Ekman layer, the onset criterion for triadic instabilities is
thus given by
{\begin{equation}
\sigma &gt; \mathcal{O}(E^{1/2}).
\label{eq:growthvsdamping}
\end{equation}}\unskip
\rev{For the mechanical forcings listed above, the diffusionless growth
rate $\sigma$ of the flow instabilities is often proportional to the
ellipsoidal deformation (at the leading order). If so, criterion
(\ref{eq:growthvsdamping}) means that the ellipsoidal deformation must
be greater than the thickness of the Ekman layer (typically 
${\sim}$0.1--1~m in core conditions for a laminar Ekman layer) for the onset of
instabilities.} Other damping mechanisms, such as an enhanced spin-up
because of meso-scale topographies or roughness (see
Section~\ref{sec:smallscale}), could thus naturally affect the
viability of such instability mechanisms.  The most famous example of
such instabilities is the elliptical
instability~\cite{kerswell2002elliptical}, which can be driven by tidal
forcing or librations in
experiments~\cite{lacaze2006magnetic,herreman2009effects,le2010tidal,grannan2014experimental,grannan2017tidally,le2019experimental,lacaze2004elliptical}. 
For precession, various triadic instabilities are expected in inviscid
ellipsoids~\cite{kerswell1993instability}, but their clear
identification remains difficult given the accessible range of parameters in 
experiments~\cite{nobili2021hysteresis,burmann2022experimental,burmann2024precessing}. 
Indeed, current experiments using
water-filled ellipsoids are usually limited to $E \sim
\mathcal{O}(10^{-6})$ (see Tables 1 and 2 in~\cite{le2022fluid}).
Another parametric instability of interest for precession is the
so-called conical shear instability (CSI). It results from triadic
interactions between two inertial modes and the viscous conical shear
layer, \rev{such that this instability can be observed in spherical
geometries~\cite{lin2015shear,cebron2019precessing}.}  The CSI was
probably present in the seminal experiments of
Malkus~\cite{malkus1968precession} and
Vanyo~\cite{vanyo1995experiments}, but the unambiguous experimental
observation only came later \rev{using a
spheroid}~\cite{horimoto2020conical}.  Note that the above parametric
instabilities can also coexist with other hydrodynamic instabilities,
such as the viscous shear-driven instability for
tides~\cite{sauret2014tide}, or boundary-layer instabilities for
precession~\cite{lorenzani2001fluid,cebron2019precessing} and
librations~\cite{noir2009experimental,sauret2013spontaneous}.


\paragraph*{(2) Rotating turbulence}
Next, \rev{anisotropic} turbulent flows are usually observed in
experiments (Figure~\ref{fig:resellip}). The different mechanisms of
linear hydrodynamic instability are rather well understood now.  On the
contrary, our understanding of the route to turbulence is still
incomplete.  Theoretical works suggest that such a transition could
occur through nonlinear saturation of the primary flows, or because of
various mechanisms involving inertial waves or
modes~\cite{kerswell1999secondary,le2017inertial,le2019experimental}. 
Determining the relevant turbulent regimes, which may be
forcing-dependent, is still an open question. As illustrated in
Figure~\ref{fig:turbellip}(a), experiments have shown that the
mechanically driven turbulence can be either
intermittent~\cite{noir2012experimental,grannan2014experimental,grannan2017tidally} 
(e.g.\ with successive chaotic cycles made of resonant growth,
saturation, collapse and relaminarisation), or
sustained~\cite{goto2014turbulence,horimoto2017sustaining,horimoto2018impact,le2019experimental}. 
Moreover, the developed \rev{anisotropic} turbulence could be either
$3$-D or quasi $2$-D (Figure~\ref{fig:turbellip}b).  These two regimes
have been observed in experiments, such as for
precession~\cite{goto2014turbulence} or
librations~\cite{le2019experimental}. Finally, strong turbulence can be
obtained if the forcing amplitude is large enough, which may be
relevant for engineering
applications~\cite{meunier2020geoinspired,goto2023precessing}.

\subsection{Unanswered questions for future experiments}
\label{subsec:extrapolation}

Experimental (and theoretical) works have allowed us to get
satisfactory physical insight into the flow response of liquid-filled
ellipsoids to mechanical forcings, from the generation of laminar flows
to the transition to turbulence (Figures~\ref{fig:poincare}--\ref{fig:turbellip}).  
However, the story is not over and there is room
for developing new experiments to attack some unsolved problems. 
\rev{As outlined below}, obtaining a better fundamental understanding
of the various turbulent states is crucial, \rev{in particular to
extrapolate the results to geophysical conditions safely.}

\subsubsection{Insight from turbulence studies}

\rev{The ability of mechanically driven flows to exhibit different
states of rotating anisotropic turbulence (Figure~\ref{fig:turbellip}b)
is really fascinating.}  Indeed, it echoes the fundamental problem of
rotating turbulence in low-\textit{Ro} flows~\cite{godeferd2015structure}.
\rev{As sketched in Figure~\ref{fig:WT}(a), different regimes of
rotating turbulence can be obtained as a function of \textit{Ro} and \textit{Re}.}
For instance, rotating turbulence can be in a quasi $2$-D regime in
which energy is concentrated in the form of quasi-geostrophic flows
(i.e.\ with flows almost invariant along the rotation axis).  \rev{Since
rapidly rotating convection is known to sustain low-frequency and quasi
$2$-D flows when $\mathit{Ro} \ll 1$
(e.g.~\cite{guervilly2019turbulent,barrois2022comparison} in spherical
geometries), the regime of quasi-geostrophic turbulence is widely
studied in the geophysical community~\cite{nataf2024dynamic}.  However,
when \textit{Ro} is not vanishingly small, $3$-D inertial waves can also take
part in the anisotropic turbulence and transfer energy to smaller
scales through a direct energy cascade.  Some experiments have indeed
shown the key role played by inertial
waves~\cite{yarom2014experimental,campagne2015disentangling,yarom2017experimental,salhov2019measurements,monsalve2020quantitative}, 
which can still coexist with quasi $2$-D columnar motions and an
inverse
cascade~\cite{duran2013turbulence,campagne2014direct,shaltiel2024direct}.}
\rev{Note that different theories or scaling laws have been proposed
for rotating turbulence, for instance using phenomenological
arguments~\cite{zhou1995phenomenological,nazarenko2011critical,baqui2015phenomenological}, 
various closure
models~\cite{galtier2003weak,cambon2004advances,bellet2006wave} or from
\textsc{dns}~\cite{smith1999transfer,thiele2009structure}.  They often
yield different predictions, against which experimental results can be
tested. Currently, most rotating experiments are designed in the light
of weak Inertial Wave Turbulence (IWT).} \rev{The latter is indeed of
great interest for turbulence modelling because it provides, in the
asymptotic regime $\mathit{Ro} \ll 1$, a natural closure of the turbulent\break
equations~\cite{galtier2023multiple}.} 

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig:WT}{(a)} Schematic regime diagram for rotating turbulence
as a function of \textit{Re} and \textit{Ro}. Adapted from figure 1
in~\cite{godeferd2015structure}. GT: Geostrophic Turbulence. IWT:
Inertial Wave Turbulence. {(b)} Experimental observation of
inertial wave turbulence, adapted and modified
from~\cite{yarom2014experimental}. The experimental set-up is a
rotating cylinder of $90$~cm in height and $80$~cm in diameter, which
is rotating at $\Omega_s = 4 \uppi$~rad${\cdot}$s${}^{-1}$. Colour bar shows the
normalised density energy (in logarithmic scale), measured for the
entire wave number range in the experiment, as a function of the
frequency $\omega$ and the angle $\theta$ between the rotation axis and
the wave vector (measured from the rotation axis). White dashes curves
show dispersion relation (\ref{eq:dispIW}) of plane inertial waves,
along which the energy is concentrated. The strongest energy is near
$\theta \approx \uppi/2$, which evidences a strong geostrophic component.}
\end{figure}

\rev{Weak IWT considers weakly nonlinear interactions (when $\mathit{Ro} \ll 1$)
of plane inertial waves, which satisfy the plane-wave dispersion
relation given by
{\begin{equation}
 \omega(\boldsymbol{k}) = \pm 2 \Omega_s \frac{k_\parallel}{|\boldsymbol{k}|}
\label{eq:dispIW}
\end{equation}}\unskip
from Equations (\ref{eq:inertialPB}a,b), where $\boldsymbol{k}$ is the
wave vector with $k_\parallel = \boldsymbol{k} \boldsymbol{\cdot}
\boldsymbol{1}_\Omega$ and the perpendicular wave vector
$\boldsymbol{k}_\perp = \boldsymbol{k} \times \boldsymbol{1}_\Omega$ of
norm $k_\perp$.} \rev{In the theory, the weak nonlinearities take the
form of three-wave interactions. For wave-vector triads
$[\boldsymbol{k},\boldsymbol{p},\boldsymbol{q}]$ satisfying the
resonance conditions given by
\changeEqNo{12a,b}
{\begin{subequations}
\label{eq:disprelWKE}
\begin{equation}
 \boldsymbol{k} + \boldsymbol{p} + \boldsymbol{q} = \boldsymbol{0}, \quad \omega(\boldsymbol{k}) + \omega(\boldsymbol{p}) + \omega(\boldsymbol{q}) = 0,
\end{equation}
\end{subequations}}\unskip
\resetEqNo{12}\unskip
\noindent
(weak) IWT is governed by a Wave Kinetic Equation (WKE) written in
symbolic form as~\cite{galtier2022physics}
{\begin{equation}
 \partial_t \mathcal{N}(\boldsymbol{k}) = Ro^2 \mathcal{T}(\boldsymbol{k}),
 \label{eq:WKE}
\end{equation}}\unskip
where $\mathcal{N}(\boldsymbol{k}) = \mathcal{E}_a(k_\perp,
k_\parallel)/\omega(\boldsymbol{k})$ is the wave action with the
axisymmetric kinetic energy spectrum\footnote{It is defined as
$\mathcal{E}_a(k_\perp,k_\parallel) = 2 \uppi k_\perp
\mathcal{E}(\boldsymbol{k})$ with the kinetic energy spectrum
$\mathcal{E}(\boldsymbol{k})$, such that the total kinetic energy is
$\propto \int^\infty \mathcal{E}_a (k_\perp, k_\parallel) \,
\mathrm{d}k_\perp \mathrm{d} k_\parallel$.}
$\mathcal{E}_a(k_\perp,k_\parallel)$, and where
$\mathcal{T}(\boldsymbol{k})$ is a collision integral (defined over the
wave vectors satisfying the above triadic conditions).  The solution of
the WKE with a steady non-zero $\mathcal{N}(\boldsymbol{k})$, known as
the Kolmogorov--Zakharov (KZ) spectrum, has received much interest in
the turbulence community.  Finding an explicit KZ solution of the
general WKE is a difficult task but, here, the problem is considerably
simplified in the anisotropic limit $k_\parallel \ll k_\perp$. In this
regime, the WKE admits the anisotropic KZ solution given
by~\cite{galtier2003weak,gelash2017complete}
{\begin{equation}
 \mathcal{E}_a(k_\perp, k_\parallel) \sim \sqrt{\epsilon \Omega_s} k_\perp^{-5/2} k_\parallel^{-1/2}.
 \label{eq:KZspectrum}
\end{equation}}\unskip
The KZ solution predicts a net direct energy cascade within an inertial
range, where the rate of spectral energy transfer is assumed to be
equal to the mean energy dissipation rate per unit of mass $\epsilon$. 
First experimental observations~\cite{yarom2014experimental} did find
that rotating turbulence can exhibit energy along the inertial-wave
dispersion relation (Figure~\ref{fig:WT}b), which was then obtained in
some \textsc{dns}~\cite{clark2015spatio,le2017inertial}. Later on,
dedicated experiments~\cite{monsalve2020quantitative} and
simulations~\cite{le2020evidence,yokoyama2021energy} confirmed the main
spatial properties of anisotropic KZ spectrum (\ref{eq:KZspectrum}).
Moreover, formula (\ref{eq:KZspectrum}) predicts that the
one-dimensional spectrum should scale as $|\boldsymbol{k}|^{-2}$ at
large scales (for which we might assume $k_\perp \sim k_\parallel$),
which is consistent with early experimental~\cite{baroud2002anomalous}
and numerical\break findings~\cite{bellet2006wave,thiele2009structure}.}

\rev{It has been proposed that mechanically driven flows in ellipsoids
could sustain IWT in \mbox{geophysical} conditions (e.g.\ for tidal
flows~\cite{le2017inertial,le2019experimental}).  Despite IWT is
appealing from a \mbox{fundamental} viewpoint, its relevance for mechanically
driven turbulence in ellipsoids is currently not obvious. Indeed, it is
unclear if the underlying assumptions of IWT are satisfied or not in
geophysical conditions.  In particular, we remind the reader that IWT
is a local turbulence theory, which is  valid for homogeneous flows
(i.e.\ unbounded) when $\mathit{Ro} \ll 1$.  First, mechanically driven flows in
ellipsoids likely occur in the low-\textit{Ro} regime, but this regime could
only be valid at large scales (e.g.\ $\mathit{Ro} \sim 10^{-6}$ for planetary
liquid cores with $l\sim L$).  Indeed, at smaller scales, the local
Rossby number is expected to increase as $\ell \to \ell_z$ (assuming
that $u_\ell/\ell$ increases as $\ell$ decreases, e.g.\ $u_\ell/\ell
\sim \epsilon^{1/3} \ell^{-2/3}$ for a Kolmogorov
cascade~\cite{nataf2024dynamic}).  Second, rotating turbulence is often
shaped by spatial inhomogeneities.  The latter notably manifest in the
form of inertial modes at large scales, as discussed for ellipsoids in
Section~\ref{subsec:fullellipsoid}, but which are also found in other
geometries~\cite{bewley2007inertial,lamriben2011excitation,boisson2012inertial}.
Therefore, it would be necessary to extend the WT theory to account for
finite-size geometrical effects to assess the validity of the WT
predictions for the largest scales characterised by $\mathit{Ro} \ll 1$.  The
continuous dispersion relation given by Equation (\ref{eq:dispIW})
should then be replaced by a discrete one in bounded geometries, which
often prevents the occurrence of many triadic interactions. For IWT,
\textsc{dns} in periodic domains ~\cite{bourouiba2008discreteness}
showed that discretisation effects become non-negligible when $\mathit{Ro}
\lesssim 10^{-3}$.  Consequently, it is still unclear whether the KZ
spectrum can properly describe the low-\textit{Ro} regime at rather large
scales in bounded geophysical systems.  Note that a first extension of
the IWT theory has been done by considering a rotating channel that is
infinite in the horizontal directions perpendicular to the rotation
axis~\cite{scott2014wave}.  Yet, the fully bounded case remains to be
considered theoretically and, then, compared to new\break experiments.}

{\vspace*{3pt}}
\subsubsection{$2$-D or not $2$-D? That is another question}
{\vspace*{3pt}}

\rev{Another weakness of IWT is that it does not account for
geostrophic flows,which are associated with $k_\parallel = 0$ in
Equation (\ref{eq:dispIW}). Indeed, there is no direct resonant triad
satisfying Equations (\ref{eq:disprelWKE}a,b) that can transfer energy from $3$-D
waves to the geostrophic component at the leading asymptotic order in
$\mathit{Ro} \ll 1$~\cite{smith1999transfer}. Actually, this results is also
valid for any inviscid flow in a bounded geometry in the low-\textit{Ro}
regime~\cite{greenspan1969non}.  If energy were only supplied to the
$3$-D waves, then geostrophic structures would not be excited and no
inverse cascade of energy could occur in the dynamics.  However, the
preferred regime of rapidly rotating turbulence could be that of
quasi-geostrophic turbulence when $\mathit{Ro} \to 0$.  This may result from the
anisotropic properties of the inertial-wave dispersion relation
~\cite{davidson2006evolution,staplehurst2008structure}, but also
because the geostrophic manifold could be a global attractor of the
rotating flows when $\mathit{Ro} \to 0$~\cite{gallet2015exact}.  If true, this
could thus challenge the validity of the WT theory for geophysical
flows with $\mathit{Ro} \ll 1$. Moreover, most experiments and \textsc{dns} of
rotating turbulence are only performed for moderately small values of
\textit{Ro} (Figure~\ref{fig:parameters}). For such values, nearly geostrophic
components are often observed
(e.g.~\cite{le2017inertial,le2019experimental} for tidal flows).
Therefore, it is essential to understand the mechanisms feeding and
sustaining geostrophic flows from a fundamental viewpoint, but also for
geophysical applications.}

\rev{So far, the transition between these $2$-D and $3$-D regimes is
not fully understood.  Indeed, the dominant mechanisms sustaining
geostrophic flows may be forcing-dependent and \textit{Ro}-dependent.
Moreover, they may also be distinct in forced and decaying turbulence.
Nonetheless, two canonical nonlinear mechanisms are usually put forward
to explain the growth of geostrophic
components~\cite{smith1999transfer}. The first one involve
near-resonant triadic interactions~\cite{bretherton1964resonant}, for
which triads $[\boldsymbol{k},\boldsymbol{p},\boldsymbol{q}]$ satisfy
the detuned resonance conditions given by
{\vspace*{3pt}}
\setcounter{equation}{13}
\changeEqNo{14a,b}
{\begin{subequations}
\begin{equation}
 \omega(\boldsymbol{k}) + \omega(\boldsymbol{p}) + \omega(\boldsymbol{q}) = \mathcal{O}(Ro),
\end{equation}
\end{subequations}}\unskip
\resetEqNo{14}\unskip
\noindent
together with $k_\parallel = 0$ and $p_\parallel = - q_\parallel$. This
is the scenario investigated
in~\cite{smith2005near,clarkdileoni2016quantifying,le2020near}, which
can lead to the onset of geostrophic flows with a growth rate 
${\propto} (k \mathit{Ro})^2$ when $k \mathit{Ro} \ll 1$  or ${\propto} k \mathit{Ro}$ when $k \mathit{Ro} \gtrsim
1$~\cite{le2020near}. The second mechanism involves quartetic
interactions (or four-wave interactions)~\cite{smith1999transfer}. For
primary triads $[\boldsymbol{k},\boldsymbol{p},\boldsymbol{q}]$
satisfying conditions (\ref{eq:disprelWKE}), we can find quaterts
$[\boldsymbol{p},\boldsymbol{q},\boldsymbol{r},\boldsymbol{s}]$
satisfying the quatertic resonance conditions given by
\changeEqNo{15a,b}
{\begin{subequations}
\label{eq:dispquatertic}
\begin{equation}
 \underbrace{\boldsymbol{p} + \boldsymbol{q}}_{=-\boldsymbol{k}} + \boldsymbol{r} + \boldsymbol{s}  = \boldsymbol{0}, \quad \underbrace{\omega(\boldsymbol{p}) + \omega(\boldsymbol{q})}_{=-\omega(\boldsymbol{k})} + \omega(\boldsymbol{r}) + \omega(\boldsymbol{s}) = 0,
\end{equation}
\end{subequations}}\unskip
\resetEqNo{15}\unskip
\noindent
with $s_\parallel = 0$.  We refer the reader to Appendix B
in~\cite{smith1999transfer} for a detailed example. This mechanism,
which also predicts a growth rate $\propto (k \mathit{Ro})^2$, was observed
numerically~\cite{kerswell1999secondary} and
experimentally~\cite{brunet2020shortcut}. Note that other mechanisms
have also been invoked such as various interactions with
eddies~\cite{mininni2009scale,bourouiba2012non,buzzicotti2018energy,lam2023supply}, 
and the shape of the fluid domain could also play a\break
role~\cite{van2020critical}.}

\rev{Back to geophysical flows}, nearly $2$-D turbulence is
unambiguously the preferred regime for rapidly rotating convective
flows~\cite{nataf2024dynamic}. On the contrary, it is unclear which
regime is preferred for mechanically-driven turbulence in geophysical
(i.e.\ $\mathit{Ro} \ll 1$ and $E \to 0$) or experimental conditions (with finite
values of \textit{Ro} and $E$). \rev{Fortunately}, future experiments could
shed new light on this controversial question. For a new-generation
experiment with a metre-size ellipsoid filled with water at room
temperature, it seems very difficult to lower the Ekman number below $E
\sim 10^{-7}$ with a typical value $\Omega_s\sim 10$~rad${\cdot}$s${}^{-1}$ for
the angular velocity of currently used rotating tables.  Nonetheless,
this may be sufficient to start disentangling the various mechanisms
and turbulent regimes in an ellipsoid.  \rev{Finally, it would also be
worth extending the theory to account for ellipsoidal (or spherical)
boundaries for a better quantitative comparison with the experiments.
Indeed, since geostrophic flows have very different properties in
unbounded and bounded systems (e.g.\ due to the beta
effect~\cite{greenspan1968theory}), the finite-size geometry could
quantitatively affect the predictions previously obtained in either
unbounded or cylindrical\break geometries.}  

\subsubsection{Towards geophysical extrapolation}

In addition to the fundamental \rev{and enthralling} question \rev{of
the properties of rotating turbulence}, we also have to obtain scaling
laws to correctly extrapolate the experimental results to geophysical
conditions (which are still out of reach).  Some scaling laws,
\rev{which have been proposed for linear and nonlinear features of
mechanically driven flows}, are gathered in Table~\ref{table:instab}.
\rev{For instance}, scaling laws are crucial \rev{if we want} to
estimate the contribution of mechanically driven flows to tidal
dissipation in the subsurface oceans of some icy
moons~\cite{lemasquerier2017libration,wilson2018can}, or to \rev{assess
their viability to sustain} planetary magnetic fields (to go beyond
prior numerical
studies~\cite{reddy2018turbulent,vidal2018magnetic,cebron2019precessing}).
\rev{In the latter case, dimensional analysis tells us that the typical
magnetic field strength $B$ driven by dynamo action could be given at
the fluid surface by~\cite{davidson2013scaling}
{\begin{equation}
 B \propto \sqrt{\mu \rho_m} \, (L \mathcal{P})^{1/3},
 \label{eq:scalingB}
\end{equation}}\unskip
where $\mu$ is the fluid magnetic permeability, $\rho_m$ is the mean
fluid density, and $\mathcal{P}$ is the mean energy production rate per
unit of mass. Moreover, on the long time scales characterising dynamo
action, we may also expect the dynamics to be close to a steady state
yielding $\mathcal{P} \sim \epsilon$ from the energy balance, where
$\epsilon$ is the mean energy dissipation rate per unit of mass. Hence,
finding a quantitative estimate of $\epsilon$ is often a cornerstone in
turbulence theory and for geophysical extrapolation.  Direct
measurements of $\epsilon$ are difficult because they require a
well-resolved velocity field.  Since we have $\epsilon = \epsilon
(u_\ell,\ell)$ by dimensional analysis, a first and accessible step
towards geophysical extrapolation is therefore to estimate the typical
strength $u_\ell$ of mechanically driven turbulent flows  (e.g.\  
Table~\ref{table:instab}).} \rev{Yet, not all the predictions have been
quantitatively validated with experiments. Moreover, another pitfall is
that prior laws were obtained without a clear \mbox{understanding} of the
underlying rotating turbulence regime. As an example, the energy
dissipation law is expected to be given from formula
(\ref{eq:KZspectrum}) by~\cite{galtier2003weak}
\vspace*{-2pt}
{\begin{equation}
 \epsilon \sim \frac{u_\ell^4 l_\parallel}{\Omega_s l_\perp^3}
\end{equation}}\unskip
with $\ell_\perp \sim 1/k_\perp$ and $\ell_\parallel \sim
1/k_\parallel$, which differs from the usual scaling $\epsilon \sim
u_\ell^3/\ell$ in isotropic Kolmogorov turbulence. Since these
dissipation scalings would yield different estimates from Equation
(\ref{eq:scalingB})}, new experiments are required to validate the
scaling laws for the different regimes.

\begin{table}
\caption{\label{table:instab}Proposed scaling laws for the diffusionless growth rate
$\sigma$ and the flow amplitude $u_\ell$ for precession and tidal
forcings in ellipsoids. The proposed numerical prefactors can be found
in the original works. $\eta_e = |a^2-b^2|/(a^2+b^2)$ is the equatorial ellipticity
of the ellipsoid with semi-major axes $(a,b,c)$, and $\eta_p$ its polar
flattening. $\Delta \omega$ is the differential rotation between the
fluid and the mantle due to precession. EI: elliptical instability,
VSDI: Viscous shear-driven instability, CSI: conical shear instability,
SI: Shear instability{\vspace*{-2pt}}}
\fontsize{9.8}{12}\selectfont
\begin{tabular}{lcccc}
\thead
{Forcing} &amp; \multicolumn{2}{c}{Tides} &amp; \multicolumn{2}{c}{Precession} \\
\cline{2-3}\cline{4-5}
{mechanism} &amp; EI &amp; VSDI &amp; CSI &amp; KSI\\
\endthead
Growth rate $\sigma$ &amp; $\mathcal{O}(\eta_e |\Omega_s-\Omega_{\mathrm{orb}}|)$ &amp; - &amp; $\mathcal{O}(|\Delta \omega| E^{1/5})$ &amp; $\mathcal{O}(|\Delta \omega| \eta_p)$  \\
Turbulence amplitude $u_\ell$ &amp; $\mathcal{O}(\eta_e |\Omega_s-\Omega_{\mathrm{orb}}| a)$ &amp; $\mathcal{O}(\eta_e^2 \Omega_s a E^{-\chi})$ &amp; $\mathcal{O}(|\Delta \omega| E^{2/5} a)$ &amp; $\mathcal{O}(|\Delta \omega| \eta_p a)$ \\
References &amp;~\cite{barker2013non,grannan2017tidally,vidal2019fossil} &amp; \cite{sauret2014tide} &amp; \cite{lin2015shear,vidal2023precession} &amp;~\cite{kerswell1993instability,barker2016turbulence,vidal2023precession}
\botline
\end{tabular}
\end{table}

Additional physical ingredients would also be worth including in new
experiments, such as a shell geometry (i.e.\ with an inner core).  Prior
experimental
works~\cite{lacaze2005elliptical,lemasquerier2017libration} showed that
the flow response to mechanical forcings in a shell is qualitatively
similar to that in a full ellipsoid, even if the mathematical theory of
inertial modes is very different in a shell
geometry~\cite{rieutord2000wave}.  This similarity should thus be
further investigated using future experiments.  Including density
stratification (with $N^2 &gt; 0$) would also be of great importance for
other geophysical systems.  For example, there are (nearly) isolated
and lenticular vortices in the
oceans~\cite{bashmachnikov2015properties} or Jovian
planets~\cite{marcus1993jupiter}, whose dynamics (from birth to death)
could affect mixing processes. Dedicated experiments could be useful to
model the dynamics inside~\cite{labarbe2021diffusive,vidal2024igw} and
outside~\cite{le2021numerical} such geophysical eddies, which can be
modelled by ellipsoidal
vortices~\cite{aubert2012universal,de2017laboratory,lemasquerier2020remote}. 
Investigating the flow response in an ellipsoid with coupled global
rotation and density stratification is currently very challenging (see
Section~\ref{sec:stratification}), but it may be amenable to
experimental work in the future.  Finally, the ongoing (and future)
space missions will likely shed new light on turbulent flows in Jovian
planets.  Therefore, the flow compressibility may be worth considering
in next-generation experiments to model the dynamics of gas giants in a
global geometry (e.g.\ following a pioneering experimental
work~\cite{su2020acoustic}). 

\vspace*{-2pt}
\section{Slowly varying flows over meso-scale and rough topographies}
\label{sec:smallscale}
\vspace*{-2pt}

We now consider how small-wavelength topographies (i.e.\ meso-scale and
roughness) can affect rotating flows.  Several phenomena well studied
in non-rotating but possibly stratified fluids, such as the drag due to
an obstacle or wall-driven turbulence, remain poorly quantified for
rotating flows. Moreover, small-wavelength topography could also
inhibit the growth of bulk (turbulent) flows driven by orbital
mechanical forcings (as discussed in Section~\ref{sec:ellipsoid}), by
enhancing the dissipation in Equation (\ref{eq:growthvsdamping}). For
all these reasons, it is key to assess the effects of small-wavelength
topographies on rotating flows.  We discuss in
Section~\ref{subsec:meso} how meso-scale topographies and roughness
affect bulk rotating flows in thick-layer systems, and we investigate
how a turbulent boundary layer (BL) is modified by roughness in
Section~\ref{subsec:turbulentEkman}.

\subsection{Bulk flows over meso-scale topographies and roughness}
\label{subsec:meso}

\subsubsection{Meso-scale topography}

Flows interacting with topographical features have garnered interest
within the oceanographic and atmospheric research communities,
benefiting from the abundance of data available to study both the
dynamics of the flows and the characteristics of the topography.  For
atmospheric and oceanic applications (i.e.\ thin-layer systems), it is
customary to solve the dynamics using the $f$-plane approximation, that
is to only retain the component of the planet's rotation vector
${\bOmega}$ onto the local direction of gravity by
introducing the Coriolis parameter $f= 2 \Omega_s \cos \theta$, where
$\theta$ is the colatitude in spherical coordinates. In the context of
stratified fluids with $0&lt;|f|&lt;N$, where $N$ is the BV frequency as
defined above in Equation (\ref{eq:BVN}), energy transfer and
dissipation is well captured by the propagation and breaking of Lee
waves (i.e.\ internal gravity waves emitted over mountains), thereby
promoting efficient vertical mixing processes~\cite{legg2021mixing}. 
Experimental investigations have revealed the existence of various flow
regimes, ranging from linear wave radiation to blocked flows.  They
depend on the relationship between the potential energy acquired by a
fluid parcel as it traverses from the base of a valley to the summit of
a topographic feature, and the kinetic energy available within the
background flow (as quantified by a Froude number).  In scenarios
characterised by low values of the Froude number, linear and weakly
nonlinear Lee waves propagate in the ocean, radiating energy away from
the topography.  Conversely, when the Froude number is large, the
available kinetic energy is not sufficient to transport parcels over
the topography. This leads to the emergence of blocked flows and regimes
of wave emission, but with an effective wavelength larger than that of
the topographical features. A more detailed description of flows over
topography in stratified fluids can be found in~\cite{legg2021mixing}.

In thick-layer fluid systems, the departures from a strictly adiabatic
density gradient (i.e.\ when $N^2 \neq 0$) are expected but
undocumented. Hence, the flow dynamics with weak or without
stratification (i.e.\ either weakly stable or neutral) also deserves
consideration for such geophysical applications.  However, such
dynamical regimes have gained much less attention relative to strongly
stratified regimes. Let us delve into the simplest scenario of an
incompressible, neutrally buoyant fluid in rapid rotation steadily
passing a single obstacle.  As described below, the observed flows are
contingent upon the obstacle's aspect ratio and the fluid velocity. 

\paragraph*{Single bump}
For isolated bumps, defined as objects with a comparable horizontal
extent $L$ in both the stream-wise and cross-stream directions, Taylor
columns (i.e.\ flow structures that are almost invariant along the
rotation axis) dominate the flow when $H \mathit{Ro}/L \ll 1$ (where $H$ is the
total fluid column height).  Physically, it means that the vertical
travel time of inertial waves over the whole fluid column above the
bump, which is given by $H/(\Omega_s L)$ for inertial waves with the
same wavelength as the topography, must be much shorter than the
typical time it takes to horizontally translate the obstacle by a
distance $L$.  Conversely, a wake of inertial waves predominates when
$H \mathit{Ro} / L \gg 1$.  These phenomena have been validated in experiments
employing various shapes of obstacles, such as elevated straight
cylinders/bumps~\cite{taylor1923experiments,hide1966experimental,hide1968slow} 
or cylindrical depressions~\cite{boyer1984rotating}.

\paragraph*{Single ridge}
For elongated obstacles like ridges, the dynamics additionally hinges on
the cross-sectional shape.  It has been demonstrated that the formation
of Taylor columns is inhibited for elongated and non-slender objects of
vertical ($h$) and horizontal stream-wise ($L_{\parallel}$) dimensions
with $h\gtrsim L_{\parallel}$, and with significant cross-stream
dimensions (i.e.\ $L_{\perp} \gg {L_{\parallel}}$, see
Figure~\ref{fig:slendernonslender}a). In such cases, the flow manifests
purely as a wake of inertial
waves~\cite{heikes1982observations,johnson1982effects} (even at
vanishingly small values of \textit{Ro}). This was recently confirmed by
laboratory experiments~\cite{machicoane2018wake}, where an elongated
cylinder is steadily towed in a rotating square tank for small to
moderate values of \textit{Ro}. As shown in Figure~\ref{fig:machiacoane}, bulk
flows dominated by $3$-D inertial waves are observed in such
experiments.  Conversely, experiments conducted with slender elongated
obstacles (i.e.\ $h \ll L_{\parallel} \ll L_{\perp}$, see
Figure~\ref{fig:slendernonslender}b) showed geostrophic dominated
dynamics of blocked flows at low values of \textit{Ro}, and distorted zonal
circulations at larger values of \textit{Ro}~\cite{weeks1997transitions}. 

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig:slendernonslender}Illustration of the concept of {(a)} a non-slender
obstacle with $h \gtrsim L_\parallel$ and {(b)}~a slender
obstacle with $h \ll L_\parallel$, where $h$ is the obstacle's
thickness, $L_\parallel$ is its stream-wise dimension \rev{(i.e.\ along
the direction of the velocity $\boldsymbol{U}$)}, and $L_\perp$ is its
cross-stream dimension. In the two cases, the obstacle is elongated
with $L_\perp \gg L_\parallel$. }
\end{figure}

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig:machiacoane}Inertial waves radiated by an elongated (non-slender) cylinder
steadily towed in a rotating tank. Adapted
from~\cite{machicoane2018wake}. {(a)} Sketch of the experimental
setup. {(b)} Cross-section showing the vertical velocity from
\textsc{piv} normalised by the towing speed. In the experiment, the
Rossby number is small and the obstacle Reynolds number is less than
$10^3$.}
\end{figure}

\paragraph*{2-D topography}
Considerably less attention has been devoted to spatially more
intricate topographies in neutrally buoyant fluid (e.g.\ arrangements of
multiple hills), particularly concerning angular momentum transfer and
energy dissipation within the fluid. This issue was recently tackled
experimentally~\cite{burmann2018effects}, by investigating the impact
of meso-scale bottom topography on the spin-up time, that is\ the time
required for the fluid's rotation to realign with that of the container
after a sudden increase. This time is a proxy for the rate of angular
momentum exchange.  In the experiment, the bottom of a straight
circular cylinder was paved with a chessboard-like array of square
blocks with varying horizontal extents and fixed height
(Figure~\ref{fig:spin_up}). The decay rate of the mean kinetic energy
was monitored in horizontal cross-sections, using \textsc{piv} velocity
fields.  The evolution of the mean energy is exponential, and the
spin-up time is defined as the $e$-folding time.  Topographic effects
yield a significantly shorter spin-up time, up to 10 times for a
specific horizontal length scale of the blocks, meaning that in this
experiment the axial torque can be up to 10 times larger with a
topography than for a flat cylinder. In the canonical flat end walls
case, the spin-up time follows a scaling law $\tau \propto
E^{-1/2}$\cite{greenspan1968theory}.  With topography, this scaling is
altered with an exponent decreasing with the wavelength.  At the
smallest wavelength considered in this study, it is almost independent
of $E$. In addition, the authors observed a vertical propagation of
inertial waves generated near the bottom topography, with a typical
wavelength comparable to (or smaller than) the topographic blocks. 
These observations suggest that the mechanism by which the BL
communicates with the bulk has evolved from a large-scale Ekman
circulation for a flat cylinder to a more efficient inertial wave
radiation-dominated regime with topography.  This process is similar to
the dissipation of tidal energy by the emission of Lee waves in the
bottom ocean~\cite{legg2021mixing}.  A spin-up time independent of
Ekman at the smaller wavelength suggests that the initial transfer of
angular momentum from the surface to the fluid is governed by strong
nonlinear dynamics near the bottom topography, which ultimately
radiates energy in the bulk via inertial waves.  This scenario for
small-wavelength topography at small (but finite) values of \textit{Ro}
requires further investigation, including a more detailed flow
visualisation near the topography. 

\begin{figure}
\includegraphics{fig14}
\caption{\label{fig:spin_up}{(a)} Meso-scale topography at the bottom of a cylinder
(of radius $R$) in spin-up experiments~\cite{burmann2018effects}. ROI:
Region of Interest considered in the original paper. {(b)}
Roughness in spin-up experiments with rough end walls conducted at
\textsc{eth} Z\"urich. {(c)} Spin-up time $\tau$ (scaled by that
of a flat end wall cylinder $\tau_{\text{flat}}$) as a function of the
horizontal length of topography $\delta$ (scaled by the thickness
$\delta_\nu$ of the Ekman layer). Red shaded and blue shaded areas
indicate roughness and meso-scale topography, respectively.}
\end{figure}

While such experiments represented an initial step to get insights into
the effects of complex topography patterns on fluid dynamics (and its
interaction with the surrounding solid), several unresolved questions
remain. In particular, a detailed characterisation of the flow near the
topography, and the development of a model to forecast torque and
energy dissipation, deserve further consideration.

\subsubsection{Rough surfaces, a challenge for experimental approach}


Roughness, that is topography embedded in the BL, is also likely to be
present in geophysical systems (e.g.\ at the Earth's
\textsc{cmb}~\cite{narteau2001smallscale,le2006dissipation}, Earth's
surface and ocean floor).  Its presence will affect the dissipation
processes and transfer of angular momentum. Accounting for roughness in
numerical models is also difficult, and it often entails solving the
large-scale flows using \textsc{dns} in which the effects of the
roughness are parameterised using models of boundary layers dynamics
(see for instance Section~\ref{subsec:turbulentEkman} below for
turbulent boundary layers). Controlled experiments do not suffer from
the limitations of \textsc{dns} and can be used to validate the
idealised models of rough end-walls.  In a typical laboratory
experiment, employing water at room temperature in a tank of typical
size $L\sim50$~cm rotating at the angular velocity $\Omega_s\sim 2\uppi
$~rad${\cdot}$s${}^{-1}$, the Ekman number is of order $E \sim 10^{-6}$. This
results in a viscous BL thickness of approximately $\delta_{\nu} \sim$
50--100~$\rmmu$m. This~entails employing textured walls, with a typical
height and wavelength on the order of $\delta\sim 1$--100~$\rmmu$m.  The
primary challenge in such experiments lies in controlling the
roughness.  Indeed, at such sub-millimetre scales, constructing
consistently rough walls that remain flat and statistically homogeneous
on the small scales of the roughness ($\delta\sim 1$--100~$\rmmu$m) is
non-trivial. We present in Figure~\ref{fig:spin_up} previously
unpublished experimental data from a preliminary experiment on spin-up
with rough end walls, conducted at \textsc{eth}
Z\"urich\footnote{Conducted during the graduate internship of 
Y.~Charles.}. Here, sandpaper of varying grain sizes is affixed to the end
walls of a cylinder, and the azimuthal velocity within the fluid bulk
is measured using Laser Doppler Velocimetry (\textsc{ldv}).  The
spin-up time is determined as the $e$-folding time of the decay in the
amplitude of the azimuthal circulation.  Not surprisingly, these
initial findings suggest that the spin-up time is shortened when the
roughness is embedded within the Ekman layer. Once the grain size
exceeds the BL thickness, a shift in dynamics occurs, still associated
with an enhanced coupling but not as significant. In all cases, the
increase of the torque by the roughness with respect to a flat wall is
merely of the order of 20\%, much less than that of a meso-scale
topography in the same range of Rossby and Ekman numbers. A speculative
scenario at low values of \textit{Ro} posits that the roughness enhances
angular momentum exchange and energy dissipation within the BL itself,
while the spin-up is still communicated to the bulk through large-scale
Ekman pumping, less efficient than the wave radiation with meso-scale
topography. This preliminary data suggest that further experiments are
worth considering, in particular to characterise the large scale
circulation.

\subsubsection{Experimental perspectives}

Extensive research has been devoted to exploring the impact of
meso-scale topography on the dynamics of geophysical fluids within the
oceanic and atmospheric communities.  However, these findings
often lack direct applicability to thick-layer fluid systems.  Due to
the inherent challenges of flows over topography for \textsc{dns}
(particularly in nonlinear regimes), experiments are indispensable to
fill the gap between the numerically accessible regimes and the
inaccessible regimes of planets and moons.  Meanwhile, the observed
pivotal role of inertial waves suggests the potential development of 
linear and weakly nonlinear wave drag models to estimate the torque and
energy dissipation (alike that of Lee waves in the Earth's oceans and
atmosphere~\cite{legg2021mixing}), or reduced quasi-geostrophic models
as in the recent works on flows over rough
seafloors~\cite{radko2023generalized,radko2023sandpaper}.   Experiments
over a broad range of parameters and topography shapes will be
necessary to test those models, and extend them beyond the weakly
nonlinear limit. This would entail using challenging velocity
measurements near the topography, together with measures of global
quantities such as the torque exerted by the fluid on the solid
boundaries. 

Finally, the interaction between the large-scale flows driven by an
oscillatory mechanical forcing (as the ones discussed in
Section~\ref{sec:ellipsoid}) and small-wavelength topography remains
unexplored. Ever since it was proposed by
Hopkins~\cite{hopkins1839nutations} in the middle of the nineteenth
century that one could probe the internal structure of planets from
their rotation, the role of the topographic torque between cores or
subsurface oceans and their surrounding shells has been debated (e.g.\ at
the Earth's
\textsc{cmb}~\cite{dehant2017understanding,rekier2022earth}).  In
contrast to the steady flows discussed above, the mechanically driven
flows are oscillatory with periods close to the rotation period.  The
interaction of oscillating flows with topography in neutrally buoyant
fluids remains inadequately documented, especially within thick-layer
fluid systems. While canonical problems in ellipsoidal geometries have
received extensive attention, the modification of inertial waves or
modes, and their nonlinear couplings with topography, remains largely
unexplored even in the simplest scenarios involving single bumps or
ridges.  Further experimental work is thus needed to elucidate the
mechanisms of energy dissipation and angular momentum transfer for such
geophysical applications.

\subsection{Turbulent boundary layers with flat and rough end-walls}
\label{subsec:turbulentEkman}

Like in the Earth's atmosphere~\cite{coleman1999similarity}, the Ekman
boundary layer (BL) could be turbulent in thick fluid envelopes. For
that, the local Reynolds number must be much larger than the critical
values 55--150 for the onset of boundary-layer
instabilities~\cite{lilly1966instability,caldwell1970characteristics,aelbrecht1999experimental}. 
For instance, the mechanical (orbital) forcings discussed in
Section~\ref{sec:ellipsoid} can sustain some boundary-layer
instabilities in experimental conditions (e.g.\ for
librations~\cite{noir2009experimental}, see
Figure~\ref{fig:BLinstab}a), but also in some geophysical conditions
(e.g.\ at the Moon's \textsc{cmb}, where the local Reynolds number is
${\sim}10^4$--10$^5$ because of precession-driven
flows~\cite{cebron2019precessing}).  Similarly, a transition to
turbulent BLs is also commonly observed in laboratory experiments
during spin-down of a rotating tank (Figure~\ref{fig:BLinstab}b). A
description of turbulent Ekman BLs is interesting for geophysical
modelling. For instance, wall friction is enhanced by a turbulent BL
(compared to the classical $E^{1/2}$ scaling for a laminar Ekman BL),
such that wall-driven turbulence could weaken or inhibit the growth of
the mechanically driven bulk flows discussed in
Section~\ref{sec:ellipsoid}. However, \rev{obtaining a turbulent BL
dynamics is challenging} using global \textsc{dns}, in which the
boundary layers are \rev{often} laminar \rev{because of the numerical
difficulty to resolve for both the largest scales of the flows and the
smaller ones associated with turbulent BLs}. Therefore, it is worth
obtaining mathematical parametrisations for turbulent Ekman BLs, which
could be later implemented in global numerical models (or used for
geophysical extrapolation).

\begin{figure}
\includegraphics{fig15}
\caption{\label{fig:BLinstab}Transition from laminar to turbulent Ekman boundary layers
(BLs). {(a)} BL instabilities driven by longitudinal librations
in a sphere at $\mathit{Re}_\delta \sim 100$ and $E \simeq 5 \times 10^{-5}$.
Flow visualisation using Kalliroscope flakes (which make the volume of
fluid opaque, except for the first mm below the boundary). Adapted
from~\cite{noir2009experimental}. {(b)} Ekman BL instabilities
(in the form of rolling waves~\cite{caldwell1970characteristics}, here
oriented with an angle of about 20\textdegree\  from the geostrophic
flow), and turbulence triggered during the spin-down of a rotating
tank. Visualisation using particle-streak photographs. Adapted from
figure 3 in~\cite{sous2013friction}.}
\end{figure}

In practice, the ultimate goal is often to obtain an estimate of the
drag coefficient ${\tau_\star}/{(\rho U^2)}$ in geophysical conditions,
where $\tau_\star$ is the surface shear stress on the wall, $\rho$ is
the density (which is constant for an incompressible fluid), and $U$ is
the amplitude of the geostrophic velocity far enough from the wall.
Initially motivated by atmospheric applications, the turbulent Ekman BL
forced by a steady geostrophic flow on a plane has been studied using
experiments, \textsc{dns} and theory, complemented by more than a
hundred years of meteorological 
data~\cite{hess2002evaluating,hess2002evaluating2}.  Classical similarity
theory provides asymptotic laws for the velocity in a turbulent Ekman
BL~\cite{csanady1967resistance,blackadar1968asymptotic,spalart1989theoretical}, 
which is often in good agreement with measurements.  First, we remind
the readers the basics of the similarity theory for a highly turbulent
Ekman BL with roughness, and we discuss its usual simplified version
for smooth boundaries.  Constraints obtained from experimental works
are later discussed and, finally, we outline some remaining open
questions for geophysical modelling and future experimental works.

\subsubsection{Theoretical considerations for turbulent Ekman BLs}

We consider the mean (free-stream) geostrophic velocity
$\boldsymbol{U}=U \boldsymbol{1}_x$ in the bulk, which flows over a
smooth or rough planar boundary.  We introduce the local Reynolds
number as
\changeEqNo{18a,b}
{\begin{subequations}
\label{eq:Relocal}
\begin{equation}
\mathit{Re}_\delta = \frac{U \delta_{\nu}}{\nu} \quad \text{with} \ \delta_{\nu}=\sqrt{\frac{2 \nu}{f}},
\end{equation}
\end{subequations}}\unskip
\resetEqNo{18}\unskip
\noindent
where $\delta_\nu$ is the laminar BL thickness, and $f=2 \Omega_s$ is
the Coriolis parameter in experimental conditions. If $\mathit{Re}_\delta \geq$
200--300, a turbulent Ekman BL can be established above the boundary.
When the Reynolds number is large enough, turbulent BLs are often
assumed to be described by universal laws, such as the law-of-the-wall
that is expected to be valid for any wall-driven turbulent flows (e.g.\ 
for channel flows).    Such descriptions are thus cornerstones in the
theory of turbulence~\cite{pope2000turbulent}, as many turbulent models
are calibrated to reproduce them for simple flows.  However, there is a
noticeable difference between non-rotating and rotating flows.  
The wall shear stress is expected to be along the direction of the
background flow for non-rotating flows, whereas the stress is tilted
from the geostrophic flow $\boldsymbol{U}$ with rotation
(Figure~\ref{fig:EkmanBL}a).  Indeed, laminar and turbulent Ekman BLs
display a spiral structure as a function of the distance $|z|$ from the
wall and, close to the wall, the boundary shear stress is tilted from
the free-stream velocity with the veering angle $\beta$. As $\mathit{Re}_\delta$
increases, turbulence tends to reduce the veering angle from its
laminar value of 45\textdegree.  Hence, the theory has to be slightly
modified to work with coordinates defined from the stress direction.

\begin{figure}
\includegraphics{fig16}
\caption{\label{fig:EkmanBL}Neutral turbulent Ekman BL over a smooth plane boundary, for a
geostrophic (bulk) flow $U$ in the $x$ direction. Adapted from figure 2
in~\cite{ansorge2014global}. {(a)} Hodograph showing the
modification of the Ekman spiral (obtained for a laminar BL) as a
function of $\mathit{Re}_\delta$ for a turbulent Ekman BL. $(v_x,v_y)$ are the
Cartesian components of the mean velocity in the coordinate system
aligned with the geostrophic flow $\boldsymbol{U}$. The surface veering
angle $\beta$ is given by the slope near the origin. {(b)}
Normalised mean velocity $u/u_\star$ as a function of the normalised
distance $z/z_0$ to the wall. In the \textsc{dns}, $u_\star$ is
estimated from $u_\star^2 = \nu \partial_z Q$ with $Q =
(u^2+v^2)^{1/2}$. The linear law $u/u_\star = z/z_0$ (in the viscous
sub-layer), and law-of-the-wall (\ref{eq:loglaw1}a) with $\kappa =
0.41$ and $C=5$ (i.e.\ the $y-$intercept at $z/z_0=1$), are also shown.}
\end{figure}

To do so, we denote by $u$ the mean shear-wise velocity (i.e.\ in the
wall-stress direction) departing from the bulk flow $\boldsymbol{U}$,
and by $v$ the (mean) cross-shear velocity. The veering angle is thus
given by $\beta = \arctan(v/u)$ at the wall. It is then postulated
that, away from the wall, the velocity decrease from the geostrophic
bulk flow is independent of viscosity and is represented by the
so-called velocity-defect-law.  The latter is given from dimensional
analysis by~\cite{csanady1967resistance,blackadar1968asymptotic}
\changeEqNo{19a,b}
{\begin{subequations}
\label{eq:defectsimilar}
\begin{equation}
 \frac{u-U_g}{u_\star} = G_1 \left ( \frac{f z}{u_\star}  \right ), \quad \frac{v - V_g}{u_\star} = G_2 \left (\frac{f z}{u_\star} \right ),
\end{equation}
\end{subequations}}\unskip
\resetEqNo{19}\unskip
\noindent
where $U_g$ is the component of the geostrophic bulk flow
$\boldsymbol{U}$ in the shear-wise direction (respectively $V_g$ in the
cross-shear direction), $u_\star$ is the friction velocity estimated
from the mean wall-stress $\tau_\star = \rho u_\star^2$, and
$(G_1,G_2)$ are some universal functions. On the contrary, near the
wall, the average velocity is expected to be given by the
law-of-the-wall as
\changeEqNo{20a,b}
{\begin{subequations}
\label{eq:wallsimilar}
\begin{equation}
 \frac{u}{u_\star} = F_1 \left ( \frac{z}{z_r} \right ), \quad  \frac{v}{u_\star} = 0,
\end{equation}
\end{subequations}}\unskip
\resetEqNo{20}\unskip
\noindent
where $z_r$ is an adjustable constant, and $F_1$ is a universal
function.  Finally, it is assumed that there is an overlap region in
which the velocity law-of-the-wall can be matched with the
velocity-defect law (as originally proposed for turbulent flows in
channels and circular tubes~\cite{millikan1938critical}).  Admissible
solutions are obtained by similarity
analysis~\cite{csanady1967resistance,blackadar1968asymptotic}, which
shows that $(F_1,G_1)$ are logarithmic functions and $G_2$ is constant.
Notably, we obtain the usual form of the law-of-the-wall as
\changeEqNo{21a,b}
{\begin{subequations}
\label{eq:lawofthewallrough}
\begin{equation}
\frac{u}{u_\star}=\frac{1}{\kappa} \ln \left ( \frac{z}{z_r} \right ), \quad  \frac{v}{u_\star} = 0,
\end{equation}
\end{subequations}}\unskip
\resetEqNo{21}\unskip
\noindent
where $\kappa$ is the von K\'arm\'an constant, and $z_r$ is a scale
height (sometimes called the roughness length).  Finally, the matching
between Equations (\ref{eq:defectsimilar}b) and (\ref{eq:wallsimilar}b)
in the overlap region provides the wall stress through\footnote{Note
the retraction~\cite{spalart2009retraction} of the more recently
proposed logarithmic law with shifted origin~\cite{spalart2008direct}.}
\changeEqNo{22a\ndash c}
{\begin{subequations}
\label{eq:t1t3}
\begin{equation}
 \frac{U \cos(\vartheta)}{u_\star} = \frac{1}{\kappa} \ln \left ( \frac{\delta_\star}{z_r} \right ) + B, \quad  \sin(\vartheta) = A \frac{u_\star}{U}, \quad 
  \vartheta = \beta+ C_5 \frac{z_r}{\delta_\star},
\end{equation}
\end{subequations}}\unskip
\resetEqNo{22}\unskip
\noindent
with $\delta_\star=u_\star/f$, and where $\beta$ is the veering angle
between the boundary (shear) stress vector and $\boldsymbol{U}$ is the
free-stream geostrophic velocity. Note that the basic theory states
that $C_5=0$~\cite{csanady1967resistance,blackadar1968asymptotic}, but
the $C_5$-correction has been later proposed to have a better agreement
with \textsc{dns} when $\mathit{Re}_\delta &lt; 5000$ for smooth
boundaries~\cite{spalart1989theoretical}.

As illustrated in Figure~\ref{fig:EkmanBL}(b), turbulent Ekman BLs are
characterised by different sub-layers in agreement with the
aforementioned analysis. First, there is a viscous sub-layer close to
the wall in which $u_\star/U$ varies almost linearly with the distance
from the wall~\cite{pope2000turbulent}.  Far enough from the wall,
there is the overlap region in which $u/u_\star$ approximately obeys
the law-of-the-wall, before reaching the outer layer (i.e.\ the last
sub-layer before the bulk free-stream flow). Between the viscous and
overlap sub-layers, there is a transition zone (the buffer layer) in
which the mean velocity is neither linear nor logarithmic.

\subsubsection{Adjustable constants and boundary stress}

The above analysis is often assumed to be a universal feature of
wall-driven turbulence~\cite{pope2000turbulent}, but there is a kind of
black magic to estimate which results are independent of the turbulence
mechanism. For instance, there are no universally accepted values for
$\kappa$. Proposed values for non-rotating BLs are $\kappa = 0.36$--0.45~\cite{george2007there}, and fitted values range between $0.36$
and $0.59$ for turbulent Ekman BLs~\cite{caldwell1972laboratory}.
However, the standard value is often chosen to be $\kappa \simeq 0.41$.
Without roughness, it is generally assumed that $z_r \propto
\nu/u_\star$~\cite{coleman1999similarity,shingai2004study}, such that
the law-of-the-wall is generally rewritten as 
\changeEqNo{23a,b}
{\begin{subequations}
\label{eq:loglaw1}
\begin{equation}
\frac{u}{u_\star} = \frac{1}{\kappa} \ln \left ( \frac{z}{z_0} \right ) + C, \quad z_0 = \frac{\nu}{u_\star}.
\end{equation}
\end{subequations}}\unskip
\resetEqNo{23}\unskip
\noindent
In this case, the value of $C$ is also not consensual.  Traditional
estimates for non-rotating BLs put it between $4.9$ and $5.1$, but
values from $4$ to $10$ may also be acceptable~\cite{george2007there}. 
For turbulent Ekman BLs, very different values $C = 0.2$--0.4~\cite{caldwell1972laboratory} and 
$C=5$--5.5~\cite{spalart1989theoretical,sous2013friction} have been
proposed in the literature. The rough case has received much less
consideration for Ekman layers~\cite{howroyd1975characteristics}, but
has been investigated without global rotation with various
roughnesses~\cite{raupach1991rough,kadivar2021review}.  For roughness
of height $h_r$, such works have shown that deviations from the smooth
regime occur when $u_\star h_r/\nu&gt;2$--15, and a fully rough regime is
reached when $u_\star h_r/\nu&gt;25$--90. In the fully rough state, data
for sand roughness suggested that $z_r \approx h_r/30$.  Note that the
higher-order $C_5-$term has not been tested yet with roughness.

In addition to the von K\'arm\'an constant $\kappa$ and $z_r$, there
are also three unknown constants $(A,B,C_5)$ that appear in Equations
(\ref{eq:t1t3}a--c) to complete the theory. It is natural to try
estimating their values using \textsc{dns}, as it is possible to design
numerical models that are very close to the theoretical assumptions of
the turbulence model. These adjustable parameters are also assumed to
have the same values for rough and smooth
walls~\cite{spalart1989theoretical}.  Hence, some works have attempted
to obtain their values using smooth boundaries with \textsc{dns}. To do
so, it is customary~\cite{braun2020turbulence} to introduce the local
Reynolds number given by Equation (\ref{eq:Relocal}a) such that
$\delta_\star/z_0=\mathit{Re}_\delta^2
(u_\star/U)^2/2$~\cite{coleman1999similarity}. Next, Equations
(\ref{eq:t1t3}a--c) can be rewritten
as~\cite{shingai2004study,shih2023turbulent} 
\changeEqNo{24a\ndash c}
{\begin{subequations}
\begin{equation}
 \frac{U}{u_\star} \cos(\vartheta) + \frac{2}{\kappa} \ln \left (\frac{U}{u_\star} \right ) = \frac{2}{\kappa} \ln(\mathit{Re}_\delta) - \frac{\ln(2)}{\kappa} + B, \quad  \sin(\vartheta)  = A \frac{u_\star}{U}, \quad 
  \vartheta = \beta+ \frac{2 C_5}{\mathit{Re}_\delta^2} \left( \frac{U}{u_\star} \right)^2.
\end{equation}
\end{subequations}}\unskip
\resetEqNo{24}\unskip
\noindent
For consistency, note that there is an inconsequential typo in the sign
of $\kappa^{-1 }\ln (2)$ in~\cite{deusebio2014numerical}. Typically, it
is found that $C_5 \approx
-52$~\cite{coleman1990numerical,coleman1999similarity,shingai2004study,deusebio2014numerical} 
when $\mathit{Re}_\delta&lt; 5000$, with for instance the mean values $(5.91,1.35)$
when $2000 \leq \mathit{Re}_\delta \leq 4000$~\cite{braun2020turbulence}. Note
that different values are obtained when $C_5$ is set to zero or not.
For instance, it has been proposed that $(6.9,{-}0.35)$ when $C_5=0$ or
$(5.4,0.26)$ when $C_5\approx-52$~\cite{shingai2004study}, but the
others values $(7.1,{-}0.73)$ with $C_5=0$ or $(5.1,0)$ with $C_5 \approx
-52$ could also be acceptable~\cite{coleman1999similarity}. The
similarity theory has also been compared with measurements of the
atmospheric boundary
layer~\cite{hess2002evaluating,hess2002evaluating2}, showing a broad
agreement, but with variations of $(A,B)$ as a function of the
latitude.  This strongly suggests that, for turbulent Ekman BLs, the
different coefficients do not have universal values in the similarity
theory. 

Finally, we can use the above theory to estimate the drag coefficient
for geophysical applications.  The wall stress being $\rho u_\star^2
\cos (\beta)$ in the $\boldsymbol{U}$ direction, the friction
coefficient between the fluid and a rigid wall can be estimated as
$k_f=(u_\star/U)^2 \cos(\beta)$. For precession-driven flows, the Ekman
BL theory with either
$C_5=0$~\cite{cebron2019precessing,nobili2021hysteresis} or $C_5
\approx -52$~\cite{shih2023turbulent} has been used. Note that the
simpler non-rotating limit with $\vartheta=\beta=0$ has also been
employed (e.g.\ for Venus~\cite{yoder1995venus}, sometimes together with
$B=\kappa^{-1} \ln (2 \kappa)$ as in~\cite{williams2001lunar}), which
simplifies further to a constant $U/u_\star$ if $\delta_\star/z_0$ is
imposed. For instance, the constant value $(u_\star/U)^2=k_f=0.002$ was
used to model the friction at the Moon's
\textsc{cmb}~\cite{yoder1981free}.

\subsubsection{Experimental investigations}

Only a few experimental works have explored the Ekman BL turbulence.
Two kinds of experiments have been designed, using either annular
rotating wind
tunnels~\cite{caldwell1972laboratory,kreider1973ekman,howroyd1975characteristics} 
or water-filled tanks mounted on a turntable~\cite{ferrero2005physical,sous2012tsai,sous2013friction}.
Contrary to \textsc{dns}, the comparison between experimental results
and the similarity theory is more complicated. First, there is an
influence of the vortical nature of the geostrophic flow in the
interior, whereas the imposed flow is rectilinear in the theory. This
effect can be quantified by introducing the apparatus Rossby number
$\mathit{Ro}_a = U/(rf)$, where $r$ is the radius of the observation point.
Typical experimental values are $\mathit{Ro}_a \sim 0.1$--1
in~\cite{caldwell1972laboratory,kreider1973ekman} and 
$\mathit{Ro}_a \sim 0.48$--2.36~\cite{sous2013friction}, whereas we have $\mathit{Ro}_a \to 0$ in the
theory. A large enough turntable should thus be employed to approach
the theoretical regime. For instance, as stated
in~\cite{howroyd1975characteristics}, a wind tunnel with a radius of
approximately $11$~m should be used to avoid significant curvature
effects.  To weaken such effects, recent
experiments~\cite{ferrero2005physical,sous2012tsai,sous2013friction}
have been carried out on the \textsc{coriolis} table, which is the
largest turntable worldwide with a tank whose diameter is $13$~m. 
Another difficulty comes from the experimental measurements. The
friction velocity $u_\star$ has been obtained using wall-stress
sensors~\cite{caldwell1972laboratory} (requiring difficult calibration
and alignment procedures), by using linear regression of
law-of-the-wall
(\ref{eq:lawofthewallrough})~\cite{kreider1973ekman,howroyd1975characteristics}, 
or from the rate of change of the angular momentum during spin-up or spin-down processes~\cite{sous2013friction}.
The comparison suggests that the two latter techniques are the more
accurate ones~\cite{caldwell1972laboratory,sous2013friction}. For the
velocity profiles, the oldest studies used hot-wire
anemometry~\cite{caldwell1972laboratory,kreider1973ekman,howroyd1975characteristics}, 
whereas the most recent ones employed \textsc{piv} techniques~\cite{ferrero2005physical,sous2012tsai,sous2013friction}. 

Given the intrinsic experimental difficulties, it is a tour de force
that experiments have managed to confirm the similarity theory. The
first experimental results, obtained at $\mathit{Re}_\delta \sim
600$--2000~\cite{caldwell1972laboratory}, indeed showed a broadly good
agreement with the similarity theory when $z/z_r &gt; 10^2$. Yet, all the
theoretical predictions were not confirmed at that time. For instance,
the shear-wise velocity-defect-law (\ref{eq:defectsimilar}a) was not
fully obtained in~\cite{caldwell1972laboratory}, as well as a vanishing
cross-shear velocity as expected from law-of-the-wall
(\ref{eq:lawofthewallrough}b) in the overlap region.  The experimental
confirmation of such predictions was later given by
Kreider~\cite{kreider1973ekman}, as illustrated in
Figure~\ref{figu:kreiderexp} where we have digitised some of his
results. In particular, we observe that the measurements of the
cross-shear velocity roughly vanish when $z/z_r \lesssim 20$, as
expected from law-of-the-wall (\ref{eq:lawofthewallrough}b) in the
overlap region. Another striking observation is that these old
measurements are already in good agreement with the similarity theory,
even for the moderate values $\mathit{Re}_\delta \sim 150$--400. More recent
experiments~\cite{ferrero2005physical,sous2013friction} did further
confirm the robustness of the similarity theory for smooth boundaries
with cleaner measurements, but, quantitatively, the different studies
slightly disagree.  For instance, the constants $(A,B)$ were found to
be (7.32, ${-}$2.67) in~\cite{sous2013friction} and $(6.01,0.018)$
in~\cite{caldwell1972laboratory}, which typically gives a friction
about 15--20\% higher in~\cite{caldwell1972laboratory} than reported
in~\cite{sous2013friction}. Such differences have been attributed to
roughness by~\cite{sous2013friction}, which was not well controlled
in~\cite{caldwell1972laboratory}.

\begin{figure}
\includegraphics{fig17}
\caption{\label{figu:kreiderexp}Turbulent Ekman BLs in the pioneering experiments of
Kreider~\cite{kreider1973ekman}. (a)~Normalised shear-wise
velocity $u/u_\star$ as a function of $z/z_r$. Dash-dot line shows
law-of-the-wall (\ref{eq:lawofthewallrough}a) with $\kappa \approx
0.41$. {(b)} Normalised cross-shear velocity $v/u_\star$ as a
function of $z/z_r$. Dashed line shows law-of-the-wall
(\ref{eq:lawofthewallrough}b) for the cross-shear velocity.}
\end{figure}

So far, there is only one study that investigated roughness effects on
turbulent Ekman BLs~\cite{howroyd1975characteristics}. The experimental
setup is illustrated in Figure~\ref{fig:BLexp}(a). Various kinds of
bottom roughness were employed, namely a coarse grained sandpaper, a
textured paint, bare duct floor coated with clear varnish, and an
artificial celanese turf (usually used for indoor-outdoor carpeting).
However, note that these experiments only allowed exploring the regime
with $0.2 \leq u_\star z_r /\nu \leq 4.6$.  A simple way to explore the
effects of roughness on the Ekman layer is to look at the veering angle
$\beta$ and the drag coefficient. As shown in
Figure~\ref{fig:BLexp}(b), we observe that the rough points are very
close to the smooth points (as expected in this regime).  However, the
roughness height $z_r$ is found to have more influence on the friction
velocity (Figure~\ref{fig:dragcoeff}a). Indeed, $u_\star/U$ increases
when the dimensionless roughness height $z_r/H$ increases (where $H$ is
defined as the geostrophic height above which the angular deflection of
the velocity vector no longer changes in magnitude). Moreover, all the
data collapse on a master curve, suggesting that roughness has somehow
universal effects. As shown in the inset, the rough points are well
predicted by law (\ref{eq:t1t3}a) with $\kappa \approx 0.41$ and $B
\approx -5.87$.  Finally, to compare the results
with~\cite{howroyd1975characteristics} and
without~\cite{caldwell1972laboratory,kreider1973ekman} roughness on a
single plot, we show in Figure~\ref{fig:dragcoeff}(b) the evolution of
the geostrophic drag coefficient $u_\star^2/U^2$ as a function of the
surface Rossby number $\mathit{Ro}_s = U/(f z_r)$.  We find that all the rough
points seem to collapse on a power law of the form $u_\star^2/U^2
\propto \mathit{Ro}_s^{-0.4}$, whose exponent is larger than that \mbox{predicted} by a
prior eddy-viscosity model yielding $u_\star^2/U^2 \propto
\mathit{Ro}_s^{-0.238}$~\cite{prandtl1924windverteilung}.  Note that the smooth
and rough points also almost collapse on a single curve, which seems to
be well described by a nonlinear eddy-viscosity
model~\cite{kung1968momentum,hess2002evaluating}.  Moreover, we see
that the geostrophic drag coefficient estimated
from~\cite{kreider1973ekman} is larger than that estimated with
roughness~\cite{howroyd1975characteristics}. This is somehow
unexpected, but this may suggest that roughness was present but not
reported in the experiment~\cite{kreider1973ekman}. Future experimental
studies could thus start reproducing these points to elucidate the
origin of this puzzling observation, before exploring other rough
regimes with larger values of $u_\star z_r/\nu$.

\begin{figure}
\includegraphics{fig18}
\caption{\label{fig:BLexp}{(a)} Experimental setup to investigate a turbulent
Ekman BL with roughness~\cite{howroyd1975characteristics}. Annular
rotating wind tunnel with a duct in wood, whose floor is inclined from
an angle of 45\textdegree\  to the horizontal. The duct has a width of
$45.72$~cm and a height of $30.48$~cm. {(b)} Veering angle
$\beta$ (in degrees) as a function of the modified Rossby number $\mathit{Ro}_f
= u_\star/\sqrt{f' \nu}$, where $f'\approx f+U/r$ is the modified
Coriolis parameter~\cite{sous2013friction}. Measurements for
smooth~\cite{kreider1973ekman,sous2013friction} rough
boundaries~\cite{howroyd1975characteristics} are shown. The transition
from a laminar to a turbulent Ekman BL occurs at $\mathit{Ro}_f \simeq
9$~\cite{sous2013friction}, corresponding to a BL instability at a
local (boundary layer) Reynolds number of $\mathit{Re}_\delta \approx
150$~\cite{caldwell1970characteristics,sous2013friction}.}
\end{figure}

\begin{figure}
\includegraphics{fig19}
\caption{\label{fig:dragcoeff}Effects of roughness of turbulent Ekman BLs
from~\cite{howroyd1975characteristics}. Common symbols are used in the
two panels. {(a)} Dimensionless friction velocity $u_\star/U$ as
a function of $z_r/H$. $H$ is the geostrophic height above which the
angular deflection of the velocity vector no longer changes in
magnitude. Inset shows the comparison between the rough points and law
(\ref{eq:t1t3}a) with the fitted parameters $\kappa \approx 0.41$ and
$B \approx -5.87$ assuming $C_5=0$. {(b)}~Geostrophic drag
coefficient $(u_\star/U)^2$ as a function of the surface Rossby number
$\mathit{Ro}_s = U/(fz_r)$. Experimental points for smooth boundaries
from~\cite{caldwell1970characteristics,kreider1973ekman}. Dashed (blue)
line shows the empirical power law $u_\star/U \propto \mathit{Ro}_s^{-0.2}$,
which is close from the nonlinear law $u_\star/U =
\alpha_1/(\log_{10}(\mathit{Ro}_s) - \alpha_2)$ obtained from a nonlinear
eddy-viscosity model~\cite{kung1968momentum,hess2002evaluating} with
the fitted parameter $\alpha_1 = 0.2238$ and $\alpha_2 \approx 1.542$
(dotted pink line). Inset shows, using the same axes, the comparison
between the rough points and law (\ref{eq:t1t3}a) with the fitted
parameters $\kappa \approx 0.41$ and $B \approx -5.87$ assuming
$C_5=0$.}
\end{figure}

\subsubsection{Remaining open questions and experimental perspectives}

The similarity theory outlined above was derived for high-$\mathit{Re}_\delta$
turbulent Ekman BLs above planar boundaries, but experiments and
\textsc{dns} showed that it is also quite robust for moderately large
values $\mathit{Re}_\delta \sim 200$--1000. Yet, all these works show that the
constants in the theory are not as universal as initially expected, as
they can significantly vary in the different configurations. Therefore,
this calls for dedicated studies for each application. Another
challenge is to properly assess the validity of the similarity theory
with roughness, to go beyond the pioneering experiments presented
in~\cite{howroyd1975characteristics}. Indeed, roughness is expected in
many geophysical systems (e.g.\ at a planet's
\textsc{cmb}~\cite{narteau2001smallscale,le2006dissipation}).  In
particular, next-generation experiments should investigate the
different regimes as a function of the roughness height $z_r$, to
properly quantify the enhancement of the friction by roughness. Another
limitation is that, in the similarity theory, the turbulent Ekman BL is
forced by a steady geostrophic flow.  However, the orbital forcings
discussed in Section~\ref{sec:ellipsoid} can sustain oscillatory and
non-geostrophic large-scale flows (e.g.\ equatorial rotation with a
nearly diurnal frequency in the rotating frame for
precession~\cite{noir2013precession}), which may lead to turbulent
Ekman BLs~\cite{cebron2019precessing}. So far, a numerical study
proposed the fitted values $(A,B,C_5)=(5.74,1.55,{-}25)$ for the
precession-driven flow oscillating at $f/2$ over a smooth
boundary~\cite{shih2023turbulent}, which lead to smaller values of
$u_\star$ and $\beta$ compared to those obtained with a steady flow. 
Therefore, it would be worth pursuing in this direction by considering
the effects of oscillatory flows on turbulent Ekman BLs in future works
(e.g.\ to assess the effects of the flow frequency).  In particular,
previous experimental works on oscillatory turbulent
BLs~\cite{jonsson1980new,sleath1987turbulent}, which used
oscillatory-flow water tunnels, could be revisited by including global
rotation.

For geophysical applications to thick-layer fluid systems, we may also
question the validity of the similarity theory with curvature effects. 
It is well-known that the laminar BL solution fails to be valid at the
so-called critical latitudes~\cite{greenspan1968theory}, where the
laminar BL thickness diverges. Hence, we may also anticipate some
issues for the similarity theory at these critical latitudes. Indeed, a
prior numerical work~\cite{shih2023turbulent} showed that it is not
straightforward to properly extrapolate such local laws for turbulent
Ekman BLs to a global near-spherical geometry.  Moreover, the Ekman BL
could be turbulent at some latitudes and laminar at others (e.g.\ near
the pole and the critical latitudes, respectively).  The proper
extrapolation of the similarity theory to this case, where the mass
fluxes at each latitude are interconnected (e.g.\ by the Ekman pumping),
remains to be found.  Moreover, the turbulent theory may not be valid
in the presence of meso-scale topographies, which deserves further
investigation. If the meso-cale topography has slender or non-slender
bumps, we may preserve or disrupt the viscous sub-layer.  It is also
known that wall-driven turbulence is affected by the convexity of
meso-scale topography~\cite{mendez2018boundary}.  Another limitation of
the above turbulent theory is that some thick-layer systems may only be
at the margin of turbulence.  For instance, the expected values of the
BL Reynolds number $\mathit{Re}_\delta \sim 100$--500 due to precession-driven
flows at the Earth's
\textsc{cmb}~\cite{buffett2021conditions,sikdar2023differential} might
be large enough to sustain wall-driven instabilities, but not to obtain
a fully turbulent BL. Since experimental works clearly show some
deviations from the similarity theory when $\mathit{Re}_\delta$ is not large
enough, the similarity theory may not give accurate enough predictions
for such applications. 

\begin{figure}
\includegraphics{fig20}
\caption{\label{fig:BLS}Turbulent Ekman BL with stratification in \textsc{dns}.
{(a)} Instantaneous visualisation of wall shear stress. Colour
coded from black (low) to white (high). White arrows indicate the
orientation of bulk geostrophic flow $\boldsymbol{U}$ and that of
$u_\star$. Tilted white line indicates a large-scale structure in the
flow. Inset shows a vertical cross-section of $u/|\boldsymbol{U}|$
along the red arrow, with colour coded from black to white in the range
$[0.12, 1.11]$. \textsc{dns} for a turbulent BL at $\mathit{Re}_\delta \simeq
500$ in the intermediate stratified regime, adapted from Figures 9 and
11 in~\cite{ansorge2014global}. (b)~Deviations from
law-of-the-wall (\ref{eq:loglaw1}) measured by the quantity $\Phi =
(\kappa z/u_\star) \ \partial_z Q$ with $\kappa = 0.41$, as a function
of the ratio $N/\Omega_s$. \textsc{dns} (using a large-eddy-simulation
approximation) at $\mathit{Re}_\delta \simeq 5 \times 10^{7}$ and $Pr \sim
5$--10, adapted from figure 9 in~\cite{taylor2008stratification}.}
\vspace*{-5pt}
\end{figure}

Note that turbulent Ekman BLs can also be affected by density
variations.  In particular, buoyancy effects deserve consideration to
\rev{understand} the interplay between turbulent BLs and
\rev{convective flows}.  Temperature should be governed by a similar
law-of-the-wall~\cite{huang1995law}, but the validity of a universal
law-of-the-wall for temperature \rev{was also
questioned}~\cite{bradshaw1995law}. The interaction between rotating
convection and turbulent Ekman BLs remains to be studied, but the
non-rotating case has already been \rev{considered}. It could indeed be
key for \rev{elucidating the transition} between the different regimes
of turbulent convection (see in Section~\ref{sec:convection}), or for
estimating basal melting of ice shelves~\cite{rosevear2021role}.  The
stably stratified regime is also of paramount importance for some
geophysical applications.  As illustrated in Figure~\ref{fig:BLS}(a),
stratified Ekman BLs have some similarities with unstratified Ekman BLs
(e.g.\ a friction velocity $u_\star$ tilted from the bulk geostrophic
flow).  However, strong departures from the law-of-the-wall can be
obtained depending on the value of $N/\Omega_s$ (even for
high-$\mathit{Re}_\delta$ BLs, see Figure~\ref{fig:BLS}b). Different turbulent
regimes can be identified as a function of the strength of
stratification~\cite{ansorge2014global}, from weakly stratified Ekman
BLs (similar to neutral ones) to strongly stratified Ekman BLs (which
can exhibit global intermittency and turbulent collapse, as for
non-rotating stratified BLs~\cite{mahrt2014stably}).  Except in the
weakly stratified regime, the standard law-of-the-wall fails to
describe the observed turbulence with
stratification~\cite{taylor2008stratification,ansorge2014global}. A
quantitative understanding of stratified turbulent Ekman BLs is thus
considerably more difficult than without stratification.  Other
similarities theories have been proposed to go beyond the classical
log-law theory (e.g.\ 
Monin-Obukhov~\cite{monin1970atmospheric,obukhov1971turbulence} or
Perlin~\cite{perlin2005modified,taylor2008stratification} theories),
but their range of validity remains to be carefully determined. 
Finally, wave motions can also be emitted by the turbulent dynamics in
stratified Ekman
BLs~\cite{taylor2007internal,taylor2008stratification,deusebio2014numerical}, and be radiated away in the bulk. 
The vertical energy flux associated with such waves is likely
negligible compared to the turbulent friction in the BL, but these
waves could affect entrainment and mixing processes in the near-wall
region (e.g.\ without global\break
rotation~\cite{linden1975deepening,xuequan1986mixing,munroe2014internal}).

To finish with, we have pointed out that turbulent Ekman BLs are far from being fully understood. 
As such, it calls for new experimental (and numerical) studies before
conducting geophysical extrapolation of the results.  Such works should
explore the BL dynamics for low to medium values of $\mathit{Re}_\delta$ to
understand the route to wall-driven turbulence, as well as
high-$\mathit{Re}_\delta$ BLs to characterise the properties of the fully
turbulent regimes and to assess the validity of the turbulence
theories.

\section{Convection-driven flows with topography}
\label{sec:convection}

Many geophysical systems are subject to thermal convection (e.g.\ the
Earth's liquid core~\cite{jones2015tog}, subsurface
oceans~\cite{soderlund2024physical}, or the
Sun~\cite{hanasoge2016seismic}).  It is thus natural to seek physical
insight into turbulent convection using experiments. Experimental
setups have mainly considered variations around the classical
configuration of Rayleigh--B\'enard convection (RBC), for which
convection is driven by the temperature difference $\Delta T$ between a
bottom plate and a top one at the distance $L$. As illustrated in
Figure~\ref{fig:sketchRB}, RBC is well suited to experimentally model
the local properties of turbulent convection of a thick-layer fluid
system (even with rotation, as suggested by a recent numerical
work~\cite{gastine2023latitudinal}). 

\begin{figure}
\includegraphics{fig21}
\caption{\label{fig:sketchRB}Thermal convection in a thick-layer envelope (e.g.\ with radial
gravity $\boldsymbol{g}$ in a spherical shell), as modelled by
Rayleigh--B\'enard convection (RBC) in experiments (with axial gravity).
Visualization of temperature variations with multiple-scale topography
on top and bottom planes is also illustrated (adapted from figure 2
in~\cite{zhu2019scaling}).}
\end{figure}

In the following, we assume that convection can be described using the
Oberbeck--Boussinesq (OB) approximation (i.e.\ with constant physical
properties, except its density that varies with temperature in the
buoyancy force). Despite the restricted range of validity of the OB
approximation~\cite{spiegel1960boussinesq,gray1976validity}, it is
often employed in theoretical works and usually offers a good
(qualitative) understanding of most experimental results. RBC is then
controlled by the Rayleigh number $Ra$, the Prandtl number $Pr$, and
the convective Rossby number $\mathit{Ro}_c$ (which is a proxy of the Rossby
number based on $L$). They are defined by
\changeEqNo{25a\ndash c}
{\begin{subequations}
\begin{equation}
Ra = \frac{\alpha g \Delta T L^3}{\nu \kappa}, \quad Pr = \frac{\nu}{\kappa}, \quad \mathit{Ro}_c = E \left ( \frac{Ra}{Pr} \right )^{1/2},
\end{equation}
\end{subequations}}\unskip
\resetEqNo{25}\unskip
\noindent
where $\alpha$ is the thermal expansion coefficient of the fluid, $g$
is the acceleration of ambient gravity, $\nu$ is the kinematic
viscosity of the fluid, and $\kappa$ is the thermal diffusivity.
Convection is affected by global rotation when $\mathit{Ro}_c \ll 1$ and, in
contrast, is barely modified by rotation when $\mathit{Ro}_c \geq 1$. 

One of the difficulties in convection modelling is to reach the extreme
parameters characterising geophysical systems 
(Table~\ref{table:convection}). For a water-filled tank with $\Delta T \sim
100$~K (at maximum), it is difficult to obtain high-\textit{Ra} convection
given the thermal expansion coefficient of water $\alpha \sim
10^{-4}$~K${}^{-1}$  (which does not efficiently convert temperature
differences into density variations to explore high-\textit{Ra} convection).
This effect can be partly balanced by increasing the height $L$ of the
tank (e.g.\ with $L\sim4$~m in the \textsc{tropoconvex}
experiment~\cite{cheng2020laboratory}), but there are then some
mechanical and thermal constraints (e.g.\ it is difficult to ensure
thermal insulation on the lateral walls). Therefore, water-filled
experiments are typically limited to $Ra \sim 10^{12}$--$10^{13}$, and
most experiments probing more vigorous convection use other fluids with
$0.6 \leq Pr \leq 7$~\cite{roche2020ultimate} (e.g.\ He or SF${}_6$).

\begin{table}
\caption{\label{table:convection}Typical values of the Rayleigh number \textit{Ra}, the Prandtl
number \textit{Pr}, the Ekman number $E$, and the convective Rossby
number \textit{Ro}$_c$ in natural (turbulent) systems and in laboratory
experiments}
 \begin{tabular}{lcccc}
\thead
 {} &amp; \textit{Ra} &amp; \textit{Pr} &amp; $E$ &amp; \textit{Ro}$_c$ \\
\endthead
Earth's core~\cite{jones2015tog} &amp; $10^{20}$--$10^{30}$ &amp; $10^{-2}$--$1$ &amp; $10^{-15}$ &amp; $\ll$1 \\
Fully molten silicate layers~\cite{solomatov2015tog,zhang2022internal} &amp; $10^{25}$--$10^{30}$ &amp; 1--$10^4$ &amp; $10^{-13}$ &amp; $10^{-2}$--$10^2$ \\
Sun~\cite{hanasoge2016seismic,schumacher2020colloquium} &amp; $10^{19}$--$10^{24}$ &amp; $10^{-7}$--$10^{-4}$ &amp; ${\leq}10^{-14}$ &amp; $\geq$1 \\
Experiments~\cite{roche2020ultimate,ecke2023turbulent} &amp; ${\leq}10^{15}$ &amp; $10^{-2}$--$10^3$ &amp; $5 \times 10^{-9}$--$\infty$ &amp; $10^{-4}$--$\infty$
\botline
\end{tabular}
\end{table}

\rev{A detailed account of the fundamental aspects of convective flows
is available elsewhere for
non-rotating~\cite{roche2020ultimate,lohse2024ultimate} and
rotating~\cite{ecke2023turbulent} convection, so we will not discuss
them here. Instead, we will briefly discuss experimental (and
numerical) works dedicated to the interplay between turbulent
convection and small-wavelength topography.  The effects of a
topographic $\upbeta$-plane, as well as of global spherical-like
geometries, are also beyond the scope of the following review (although
they are directly relevant for modelling thick-layer systems).  We
refer the interested reader to~\cite{cardin2015tog,potherat2024seven}
for geophysically motivated descriptions of such convection
experiments.}  We first discuss the interplay between small-wavelength
topography and RBC (both without and with rotation), and then we
outline some experimental perspectives.

\subsection{Non-rotating (and rotating) RBC with small-scale
topography}

We first consider the limit case of non-rotating RBC (i.e.\ $\mathit{Ro}_c \to
\infty$), which is appropriate for the dynamics of slowly rotating
convective systems (e.g.\ solar convection, and possibly molten silicate
layers).  In particular, estimating the efficiency of heat transport by
convection is key for understanding the heat budget of these systems
and their long-term thermal evolution over billions of years. 
Convective heat transport is usually characterised by the Nusselt
number $Nu$ given by $Nu = Q L/(k \Delta T)$, where $k$ is the thermal
conductivity (in W${\cdot}$m${}^{-1}{\cdot}$K${}^{-1}$), and $Q$ is the heat flux per
unit of area with convection (in W${\cdot}$m$^{-2}$).   There is still a
long-standing controversy about the asymptotic scaling law $Nu \sim
f(Ra,Pr)$ for turbulent convection in the asymptotic high-$Ra$ regime.
Indeed, classical theory predicts $Nu \sim
Ra^{1/3}$~\cite{malkus1954heat}, whereas the ultimate theory states
that $Nu \sim Ra^{1/2}Pr^{1/2}$ (at least for $Pr \leq 1$) when
convection operates in a diffusion-free
regime~\cite{spiegel1963generalization,kraichnan1962turbulent}. 
Actually, in the theory, the ultimate regime of RBC is expected to be
linked to the existence of turbulent BLs.  \rev{A few studies have thus
investigated the nature of the BLs in high-\textit{Ra} convection.  It was
first conjectured that BL turbulence would occur when the local
friction Reynolds number is greater than ${\sim}$200 in non-rotating
RBC~\cite{schumacher2016transitional}.  Later on, some
experiments~\cite{ahlers2012logarithmic,ahlers2014logarithmic} and
\textsc{dns}~\cite{van2015logarithmic,zhu2018transition,he2024turbulent,scheel2017predicting,blass2021effect} 
did confirm that the viscous and thermal BLs become turbulent and can
be described by similar laws-of-the-wall when the friction Reynolds
number is large enough.  Yet, some local departures from the
law-of-the-wall have also been documented in the presence of
large-scale
plumes~\cite{van2015logarithmic,zhu2018transition,he2024turbulent}.
Moreover, despite the presence of turbulent BLs, contradictory results
were found for the scaling of $Nu$ with
\textit{Ra}~\cite{zhu2018transition,he2024turbulent}.} Whether very high-\textit{Ra}
RBC with smooth plates is governed by the classical theory or falls
into the ultimate regime is still thus unknown in experimental
conditions, \rev{and it remains the subject of active
research~\cite{roche2020ultimate,lohse2024ultimate}.}

This fundamental problem has been revisited by using topographic
effects for a few years.  \rev{Indeed}, small-scale topography can
favour wall-driven flows~\cite{liot2016boundary,liot2017velocity}, such
as thermal plumes and wall-driven turbulence (see
Figure~\ref{fig:convetion}a). \rev{As such, it could} possibly
\rev{favour} the transition to the ultimate
regime~\cite{toppaladoddi2017roughness}.  Different experiments of RBC
with small-scale topography on the walls have thus been conducted,
using topography with different shapes and
heights~\cite{shen1996turbulent,du1998enhanced,ciliberto1999random,du2000turbulent,
tisserand2011comparison,salort2014thermal,wei2014heat,xie2017turbulent,rusaouen2018thermal,zhu2019scaling}. 
Most studies employed either water or another fluid with $Pr \geq 1$,
but a few used an air-filled
experiment~\cite{liot2016boundary,du2024thermal}. These studies did
find that small-scale topography can often enhance heat transport by
turbulent convection, with either an exponent larger than $1/3$ or with
a higher numerical prefactor.   Scaling laws with exponents close to
$1/2$ have even been experimentally reported in some
studies~\cite{roche2001observation,zhu2018wall}.  This suggests that
turbulent convection might reach a diffusion-free regime in the
presence of wall-driven turbulence (at least for the considered values
of \textit{Ra}). However, topography does not always enhance heat transport.
By further increasing the value of \textit{Ra}, the $1/2$ scaling can be
lost~\cite{zhu2017roughness}, which can depend on the chosen\break
topographies~\cite{zhu2019scaling}.

\begin{figure}
{\vspace*{-1pt}}
\includegraphics{fig22}
{\vspace*{-1pt}}
\caption{\label{fig:convetion}{(a)} Observation of topography-driven flows in an
experiment of RBC with rough plates at $Ra \simeq 2.6\times10^9$ and
$Pr \simeq 5.4$. Visualization of the flow (streak image of the fluid,
seeded with thermochromic liquid crystal spheres) near the upper rough
(cold) plate. Wall-driven (cold) eruptions are in brown, and warmer
flow structures are in green and blue. Adapted from figure 4
in~\cite{du2000turbulent}. {(b)} Observation of wall-driven mode
in rotating RBC. Numerical simulation in a cylinder at $Ra=5 \times
10^{10}$, $Pr=5.2$ and $E=10^{-7}$. Colour bar shows the vertical
velocity in the mid-plane of the cylinder. Adapted from figure 5
in~\cite{favier2020robust}.}
{\vspace*{-1pt}}
\end{figure}

Finally, contrary to the non-rotating case, the interaction between
rotating RBC and small-scale topography has only received scant
attention  \cite{joshi2017heat,tripathi2024regimes}.  Further work is
thus required to get insight into the effects of topography on RBC.
Rotating RBC is also often impacted by the presence of (zonal) wall
modes along the lateral boundaries (Figure~\ref{fig:convetion}b). Their
presence is largely indifferent to the bulk
convection~\cite{zhang2020boundary,de2023robust}, and they may have a
non-trivial topological origin~\cite{favier2020robust,zhang2024non}.  From an
experimental viewpoint, these modes can affect heat flux measurements
in the bulk, and reduce our ability to precisely study the regimes of
bulk turbulence~\cite{ecke2023turbulent}.  A recent theoretical
work~\cite{terrien2023suppression} showed that such wall-driven flows
can be strongly weakened by including well-chosen topographic defaults
on the lateral walls, which paves the way for future experimental
applications.

\subsection{Experimental perspectives}

Characterising heat transport of RBC with small-scale topography in the
presence (or absence) of wall-driven turbulence is key \rev{from a
fundamental viewpoint}, but it could also \rev{be important} for
engineering~\cite{garcia2012influence} and geophysical applications.
\rev{For instance, the BL Reynolds number due to convection may have
the typical value 100--1000 at the Earth's
\textsc{cmb}~\cite{jones2015tog}, which could be sufficient for the
onset of BL instabilities and wall-driven turbulence.} Further
experimental work, together with joint theoretical and numerical
studies, may therefore help to make progress on this fundamental
question For rotating RBC, including topographic defaults on lateral
walls seems key to weaken the strength of wall-driven flows in future
experiments~\cite{terrien2023suppression}. Finally, we could naturally
wonder how much the various convection scaling laws remain valid in the
presence of non-Oberbeck--Boussinesq (NOB) effects (i.e.\ non-uniform
material properties of the fluid due to temperature variations) or
compressibility. Such effects can be important in some experimental
conditions (e.g.\ a gas near its saturated vapor
curve~\cite{urban2019elusive} to explore high-\textit{Ra} convection, or a gas
subject to extreme rotation rates~\cite{menaut2019experimental}), but
also in some geophysical systems (e.g.\ for mantle convection in
super-Earth~\cite{ricard2023compressible}, or in gas giants). As for
incompressible RBC~\cite{doering1996variational}, theory shows that the
Nusselt number in non-rotating compressible (anelastic) convection
remains bounded by $Ra^{1/2}$~\cite{alboussiere2024upper}. Hence, the
ultimate regime of non-rotating RBC might also exist with
compressibility. As such, the interplay between small-scale topography
and compressible convection would certainly deserve consideration in
the future. 

Finally, exploring the effects of meso-scale topographies on rapidly
rotating convection in planetary-core geometries would be worth
considering.   Indeed, the low-frequency dynamics of planetary-core
convection is strongly shaped by global rotation, which tends to
sustain columnar motions that are nearly invariant along the rotation
axis of the fluid.  However, prior
\mbox{theoretical}~\cite{bell1996influence,herrmann1998stationary,bassom1996localised} 
and experimental~\cite{westerburg2003centrifugally} works in the
cylindrical annulus geometry showed that this low-frequency dynamics
can be modified in the presence of sinusoidal deformations on top and
bottom walls (e.g.\ in trapping columnar motions near bumps, or in
sustaining low-frequency Rossby waves). Similar effects were found in
\textsc{dns} using a quasi-geostrophic approximation, in a rotating
spherical shell with a non-slender meridional
ridge~\cite{calkins2012effects}. The convective flow was dominated by
zonal circulation when \textit{Ra} is large enough and, at low values of \textit{Ro},
blocked flows were reported (as in~\cite{weeks1997transitions} for
slender obstacles). Moreover, it was observed that topography can
transfer energy from large-scale (convective) flows to smaller-scale
flows in the form of Rossby waves (\rev{whose amplitude} scales as
$\mathit{Ro}^{1/2}$).  Future experiments could build upon such prior works to
explore these effects, such as in the idealised annulus geometry that
is amenable to experimental investigation (where Earth's gravity is
replaced by centrifugal gravity~\cite{potherat2024seven}).

\section{Brief discussion of rotating stratified flows}
\label{sec:stratification}

Beyond neutrally buoyant fluids and convection, understanding the
dynamics of rotating and stratified flows with topography is also
crucial for some geophysical applications (e.g.\ the wave-driven drag
over a topography, wave-driven mixing processes in rotating stratified
ellipsoids, or the turbulence in stratified Ekman BLs).  The effects of
density stratification have not been reviewed in the previous sections,
as they deserve specific review articles. We refer the interested
reader \rev{to the two reviews in this special issue on 
different regimes of stratified turbulence without 
topography~\cite{cortet2024turbulence,lefauve2024cras},} 
and to~\cite{legg2021mixing,baines2022topographic} for theoretical 
descriptions of topographic effects in stratified flows. 
However, rotating experiments with stratification and topography are
not very common~\cite{boyer2000laboratory,stewart2024stratified}.  In
brief, these experimental works have mainly explored the regime
$N/\Omega_s \sim \mathcal{O}(1)$, and often considered the
low-frequency dynamics (e.g.\ to investigate spin-up processes, or the
quasi-geostrophic dynamics). In the following, we explain why it is
still very difficult to study rotating stratified flows in the
laboratory. Indeed, there are intrinsic difficulties that need to be
overcome in future experimental works before answering unsolved
geophysical questions.

\subsection{Accessible experimental regimes}

Since the strength of stratification is subject to strong uncertainties
for geophysical applications, exploring the dynamics for a broad range
of values of $N/\Omega_s$ would be desirable.  We can estimate BV
frequency (\ref{eq:BVN}) for an incompressible fluid from $N^2 \sim
(g/\ell_V) |\Delta \rho|/\rho$ in laboratory conditions, where $\ell_V$
is the height of the container (in metres), $g \simeq
9.81$~m${\cdot}$s${}^{-2}$ is the value of Earth's gravity, and $|\Delta
\rho|/\rho$ is the relative density anomaly. For salty (NaCl) water,
the density anomaly can vary from $0$ to $\sim 0.3$ (at most) by
changing the salt concentration.  As such, the values of $N$ that are
accessible in a water-filled experiment are limited
(Figure~\ref{fig:stratified}a).  Typically, for a metre-size tank
(which is required to lower the value of $E$), we cannot impose an
initial stratification with a BV frequency greater than $N \sim
1.7$~rad${\cdot}$s${}^{-1}$. For a rotating table with a rotation rate
$\Omega_s$ between 0.1--$10$~rad${\cdot}$s${}^{-1}$, the maximum value would be
$N/\Omega_s \sim 10$--20.  Beyond the regime $N/\Omega_s \leq 1$,
stratified regimes with $N/\Omega_s \geq 1$ could also be considered in
experiments, but a first technical difficulty is to impose the initial
stratification profile.  The standard approach (known as the
double-bucket method) does not offer great flexibility to impose
arbitrary stratification, but other strategies have been
proposed~\cite{economidou2009density}.  Another difficulty would also
result from the velocity measurements.  To probe the low-\textit{Ro} dynamics
with $\mathit{Ro} \sim 10^{-3}$--$10^{-2}$ in a metre-size experiment, the velocity
would have a typical amplitude $u_\ell \sim
10^{-4}$--$10^{-1}$~m${\cdot}$s${}^{-1}$ for an angular velocity $\Omega_s \sim
0.1$--10~rad${\cdot}$s${}^{-1}$, which could be below the detection level of
the different velocimetry techniques.   Therefore, it seems difficult
to simultaneously explore low-\textit{Ro} and strongly stratified regimes with
currently available experiments.

\begin{figure}
{\vspace*{3pt}}
\includegraphics{fig23}
{\vspace*{3pt}}
\caption{\label{fig:stratified}{(a)} Typical values (in logarithmic scale) of the BV
frequency $N \sim \sqrt{(g/\ell_V) |\Delta \rho|/\rho}$ (in
rad${\cdot}$s${}^{-1}$) in a water-filled experiment at room temperature, where
$\ell_V$ is the height of the fluid container (in metres), $g \simeq
9.81$~m${\cdot}$s${}^{-2}$ is the acceleration of gravity, and $|\Delta
\rho|/\rho$ is the (relative) density anomaly. The water density can be
changed by varying the salt concentration (from $0$ to
approximately $300$~kg${\cdot}$m${}^{-3}$). Red curves show the values $N \in
[0.1,1,4.2]$~rad${\cdot}$s${}^{-1}$. {(b)} How to keep centrifugal
gravity under control in a rotating experiment. Colour bar shows the
rotation rate $\Omega_s$ (in rad${\cdot}$s${}^{-1}$) required to reach the
Ekman\break number $E$ for a water-filled experiment with equal horizontal
and vertical length scales $\ell_H \sim \ell_V$. Dashed line
(respectively, dotted line) shows the length scale $\ell_H$ given by
Equation (\ref{eq:negligibleCG}) such that $\Delta_g = 10^{-1}$
(respectively, $\Delta_g = 10^{-2}$).}
\end{figure}

Rapidly rotating regimes with $N/\Omega_s \leq 1$ would also be
difficult to probe because of centrifugal effects. Indeed, if global
rotation is strong enough, centrifugal gravity will deform the initial
vertical stratification.  To maintain a vertical stratification with
negligible centrifugal effects, as well as small viscous ones with $E
\ll 1$, a large tank with a relatively moderate rotation is required. 
This can be estimated by the ratio between centrifugal forces to
Earth's gravity given by $\Delta_g =\Omega^2 \ell_H/g$ (where $\ell_H$
is the horizontal length scale of the tank, where centrifugal gravity
is maximum).  Assuming that the tank has a typical aspect ratio
$\ell_H/\ell_V \sim 1$, we can introduce the Ekman number $E$ such that
$\Delta_g = \nu^2/(E^2 \ell_H^3 g)$. For a fixed value of $\Delta_g$,
the maximum value of $\ell_H$ is thus given by
{\begin{equation}
   \ell_H \sim \left( \frac{\nu^2}{g \Delta_g E^2} \right)^{1/3},
   \label{eq:negligibleCG}
\end{equation}}\unskip
where $g\simeq9.81$~m${\cdot}$s$^{-2}$ is the value of Earth's gravity, and
$\nu=10^{-6}$~m$^2{\cdot}$s$^{-1}$ is the kinematic viscosity of water at
ambient temperature. Formula (\ref{eq:negligibleCG}) shows that when
$\Delta_g$ is reduced (i.e.\ the more centrifugal gravity is
negligible), then the horizontal length scale $\ell_H$ increases. As
shown in Figure~\ref{fig:stratified}(b), to reach a value $E=10^{-6}$
with a rotation rate $\Omega_s \sim 1$~rad${\cdot}$s${}^{-1}$, we must use a
tank with $\ell_H \sim 1$~m if $\Delta_g=10^{-1}$ (or $\ell_V \sim 4$~m
if $\delta_g=10^{-2}$).  For a value of the Ekman number $E \sim
10^{-8}$, we would instead need a much larger tank with $\ell_H \sim
20$~m. Therefore, the dynamics of rotating stratified fluids with $E &lt;
10^{-6}$ cannot be probed with a metre-size tank. The only alternative
would be to change the aspect ratio of the tank.  While keeping $\ell_H
\sim1$~m to weaken wall-driven flows, we could increase $\ell_V$ up to
4--5~m (e.g.\ as in the \textsc{tropoconvex} convection experiment) but
at the expense of a smaller value of the BV frequency (according to
Figure~\ref{fig:stratified}a).

\subsection{Specific properties of inertia-gravity waves in
experimental conditions}

Inertial waves and modes have proven useful to characterise the flow
response of rotating fluids with large-scale ellipsoidal topographies
(see Section~\ref{sec:ellipsoid}), but also with small-wavelength
topographies (see Section~\ref{sec:smallscale}). When global rotation
and density stratification are coupled, inertia-gravity waves (IGWs)
could be expected to play a similar role.  IGWs are mixed waves that
are sustained by Coriolis force and gravity.  Let us discuss their
properties in the case of a fine-tuned experiment, in which rotation is
aligned with gravity and centrifugal effects are negligible.   
We consider a fluid that is stratified in density
with a constant BV frequency $N&gt;0$ along Earth's gravity
$\boldsymbol{g} = -g \boldsymbol{1}_z$ (where $g$ is a constant), and
rotating along the same axis at the angular velocity
${\bOmega} = \Omega_s \boldsymbol{1}_z$.  In the rotating
frame, IGWs satisfy the eigenvalue problem given\break by~\cite{vidal2024igw}
\changeEqNo{27a,b}
{\begin{subequations}
\label{eq:IGW1}
\begin{equation}
-\omega_i^2 \boldsymbol{u}_i + \mathrm{i} \omega_i (2 {\bOmega} \times \boldsymbol{u}_i) + N^2 (\boldsymbol{u}_i \boldsymbol{\cdot} \boldsymbol{1}_z) \boldsymbol{1}_z = -\mathrm{i} \omega_i \nabla \Phi_i, \quad \nabla \boldsymbol{\cdot} \boldsymbol{u}_i = 0,
\end{equation}
\end{subequations}}\unskip
\resetEqNo{27}\unskip
\noindent
where $\omega_i$ is the angular frequency of the wave, and
$\boldsymbol{u}_i$ is the eigenvector for the wave velocity
(respectively $\Phi_i$ for the pressure).  It turns out that Equation
(\ref{eq:IGW1}a) is a mixed hyperbolic-elliptic equation, whose nature
changes as a function of $\omega_i$ only (i.e.\ the equation is either
entirely hyperbolic in the fluid volume for a given value of
$\omega_i$, or is entirely elliptic in the volume). As such, it can be
shown (using a plane-wave analysis) that local IGWs can only exist when
$\min(2\Omega_s,N)\leq |\omega_i| \leq \max (2\Omega_s,N)$. In this
interval, $\omega_i$ satisfies the dispersion relation\break given by
{\begin{equation}
 \omega_i^2 \boldsymbol{k}^2 = N^2 |\boldsymbol{1}_z \times \boldsymbol{k}|^2 + 4 \Omega_s^2 (\boldsymbol{1}_z \boldsymbol{\cdot} \boldsymbol{k})^2,
 \label{eq:IGWrelationdisp}
\end{equation}}\unskip
where $\boldsymbol{k}$ is the (local) wave vector of the plane wave. 
As illustrated in Figure~\ref{fig:IGW}, the local analysis agrees very
well with experimental results~\cite{peacock2005effect}.  In the other
interval $0 &lt; |\omega_i| \leq \min(2\Omega_s,N)$, the equation is
elliptic such that there are no wave motions (i.e.\ plane waves are
evanescent, as for high-frequency disturbances without
stratification~\cite{nosan2021evanescent}). However, if the entire
fluid is forced in volume (not like in~\cite{peacock2005effect}, where
the oscillatory forcing was only local in space), then some
low-frequency eigenmodes can exist in this frequency
interval~\cite{friedlander1982internal,vidal2024igw}. Such $3$-D modes,
which appear similar to the low-frequency Kelvin waves in oceanography,
owe their existence to the presence of lateral (vertical)
walls~\cite{cdv2024gravito}. 

\begin{figure}
\includegraphics{fig24}
\caption{\label{fig:IGW}Experimental investigation of inertia-gravity waves (IGWs),
for a rotating stratified fluid with aligned global rotation
${\bOmega}=\Omega_s \boldsymbol{1}_z$ and gravity
$\boldsymbol{g} = - g \boldsymbol{1}_z$ in the regime $N/\Omega_s &gt; 2$.
Adapted from~\cite{peacock2005effect}. {(a)} Sketch of the
experimental setup. A solid sphere, which is embedded in the fluid,
oscillates at the angular frequency $\omega$. The oscillation generates
IGWs, which manifest in the form of wave beams (red regions, in which
the phase and group velocities are orthogonal). {(b)} Angle
$\theta$ (in degrees) between the wave vector $\boldsymbol{k}$ and
gravity in dispersion relation (\ref{eq:IGWrelationdisp}). Excellent
agreement is found between the experimental results and theory. No
waves were reported when $|\omega| \leq 2 \Omega_s &lt; N$.}
\end{figure}

The existence of a frequency gap in experimental conditions has
annoying consequences, as it hampers our ability to safely extrapolate
experimental results to thick-layer systems. Indeed, a frequency gap
implies that an oscillatory forcing (e.g.\ a velocity flowing over a
small-wavelength topography) will not be able to excite (locally)
propagating waves for some frequencies in experimental conditions (but
it may excite evanescent disturbances, as found without
stratification~\cite{nosan2021evanescent}). However, such a frequency
gap is not globally expected in a thick-layer fluid system in which
gravity is radial~\cite{friedlander1982gafd,dintrans1999gravito}.  In
such systems, for a given value of $\omega_i$, the fluid volume is
indeed separated into regions in which waves can propagate, and other
regions in which wave motions are locally evanescent. Therefore, at
different latitudes, the same oscillatory forcing could excite or not
waves in thick-layer fluid systems (contrary to the experimental
setup).  The mathematical properties of IGWs are thus drastically
different in experimental models and in geophysical systems, such that
experimental results should be extrapolated with care to thick fluid
envelopes. 

\section{Concluding remarks}\label{sec:ccl}
\subsection{A common need for new experiments}

This paper enlightens the critical role of experimental studies in
advancing our understanding of topographic effects in geophysical fluid
dynamics.  One key aspect discussed is the need to bridge the gap
between numerically accessible regimes, and the often inaccessible flow
regimes encountered in geophysical fluid envelopes.  Given the inherent
challenge of simulating flows over topography with \textsc{dns}, we
have here emphasised the importance of experiments in complementing
theoretical and numerical works.  These problems have been extensively
studied in the context of thin-layer systems (e.g.\ stratified oceans
and atmospheres) using experiments, \textsc{dns} or theory.  However,
those results are not directly relevant for thick-layer systems, such
as planetary cores and subsurface oceans.  Those have received
comparatively less attention from theoretical and experimental
viewpoints, partly because of the scarcity of direct observations. 
Meanwhile, indirect observations (e.g.\ the accurate measurements of the
Earth's and Moon rotation, or the measurements of the upcoming space
missions to Jovian satellites) require a better understanding of the
role of topography for mixing, angular momentum transfer, or magnetic
field generation. Density stratification is also not well-constrained
in such hidden layers, such that all possible situations should be
considered (including neutral regimes, weak or strong stable
stratifications, or convection). 

Since the most advanced \textsc{dns} are limited in the accessible
range of parameters, but also in the geometries they can simulate,
experiments have thus played a crucial role in making progress in this
field. For instance, they often allow us to validate asymptotic
theoretical models or derive (empirical) scaling laws, which can be
extrapolated to the out-of-reach range of parameters of geophysical
bodies. However, particular attention must be given to the regimes in
which the experiments operate (i.e.\ are they already governed by the
dynamics prevailing at planetary conditions?).  An illustrative example
is the debate about the expected rotating turbulence regime driven by
mechanical (orbital) forcings.  Indeed, experiments conducted at
moderately small values of the Ekman and Rossby numbers likely operate
in a different turbulent regime from that in geophysical conditions.

We have also identified some key points that would help to pave the way
for future experiments.  Firstly, it appears quite clear that we should
consider larger experimental devices. This is required to reduce the
value of the Ekman number \rev{to probe high-\textit{Re} turbulence while
keeping the Rossby number small enough}, but also to allow measurements
in BLs and in the vicinity of topographical features.  Such measurements
are indeed currently too challenging using sub-metre size experiments. 
If we target a typical value $E\sim 10^{-8}$, it would be achievable
with a two-metre size experiment rotating at $60$ \textsc{rpm}, and
using water at ambient temperature as a working fluid.  Few devices of
this class exist, such as the Big Sister three-metre sodium experiment
in \mbox{Maryland}~\cite{zimmerman2011bi,zimmerman2014turbulent} or the
forthcoming two-metre \textsc{dresdyn}
device~\cite{stefani2019dresdyn}. While the targeted low-$E$ values are
accessible with such experiments, they are dedicated to specific
forcing and focus on hydromagnetic flows. Hence, they do not offer
enough flexibility to address the questions discussed here.  It may be
more desirable to think of the next-generation experiments as an
experimental platform that would allow investigating a broader range of
problems, similar to the unique \textsc{coriolis} facility hosted in
the \textsc{legi} laboratory (Figure~\ref{fig:ccl}a). This is an open
tank with a diameter of $13$~m, rotating at $6$~\textsc{rpm} and
embarking a fluid layer with a maximum height of one metre (resulting
in an Ekman number $E\sim10^{-6}$).  This unique instrument is well
suited to model thin-layer dynamics with stratification at the cost of
a moderate Ekman number.  Yet, it is not ideal to investigate the flow
dynamics in closed containers representative of thick-layer systems. 

\begin{figure}
\includegraphics{fig25}
\caption{\label{fig:ccl}{(a)} \textsc{coriolis} experimental facility
in \textsc{legi} laboratory (Grenoble, France). This is the largest turntable
worldwide, with a diametre of $13$~m and a height of $1$~m, rotating at
$6$~\textsc{rpm} at maximum. It has notably been used to investigate
the dynamics of Ekman layers
~\cite{aelbrecht1999experimental,ferrero2005physical,sous2012tsai,sous2013friction}. 
{(b)} \textsc{theia} experimental facility, currently under
construction in \textsc{isterre} laboratory (Grenoble, France). It will be
dedicated to the study of rotating flows with topography. The turntable
will be capable of embarking tanks with a diameter of ${\sim}$1~m and a
height of ${\sim}$2~m, rotating at a maximum rotation frequency of
$10$~\textsc{rpm}.}
\end{figure}

Another benefit of large experiments would be to investigate the
combined effects of stratification and rotation when $N \lesssim
\Omega_s$, in contrast with the existing studies dedicated to oceanic
dynamics that mainly explored the other regimes with $N \gg \Omega_s$.  As
shown in Figure~\ref{fig:stratified}, it would be possible to perform
experiments in the weakly stratified regimes at low Ekman numbers, and
possibly in the more strongly stratified regime $1 \leq N/\Omega_s \leq
10 $.  Finally, a larger device would also allow us to investigate the
same range of Ekman numbers as we currently do with sub-metre size
experiments, but using higher viscosity fluids (10--100 times) in
order to increase the Ekman BL thickness by a factor 3 to 10. 
Detailed investigations of the effect of roughness in such a device
would be achievable with topographic features, ranging from a few
microns to several\break millimetres. 

\subsection{Larger experiments, a realistic prospect?}

\rev{We have argued that substantial advancement in the field of
rapidly rotating geophysical fluid dynamics is contingent upon the
development of new large-scale facilities.} A few laboratories have
already acquired large-capacity turntables capable of embarking
experiments of 1--2 m in size, such as the ones used
in~\cite{machicoane2018wake,lemasquerier2021zonal} or the upcoming
\textsc{theia} experimental facility in Grenoble
(Figure~\ref{fig:ccl}b). Even if they cannot reach low-$E$ regimes with
$E \ll 10^{-8}$, these facilities pave the way for the development of
even more ambitious devices.  However, there will be some unavoidable
difficulties when working with large-scale devices at small Ekman
numbers.  As anticipated from stability analyses, such rapidly rotating
flows will be prone to instability for much smaller-amplitude
perturbations than in the current devices. It is therefore essential to
pay close attention to the potential adverse effects, including small
temperature fluctuations across the tank, the rotation of the Earth
that will prevent from achieving perfect solid-body
rotation~\cite{boisson2012earth,triana2012precessional}, or any
deviation from the intended geometry exceeding some characteristic
length scales (e.g.\ the BL thickness). \rev{Another challenge inherent
to a unique apparatus is the repeatability and validation of the
results.} It has become standard practice in the computational fluid
dynamics community to conduct periodic benchmark studies on canonical
problems, which allow testing and validating the different algorithms
(e.g.~\cite{christensen2001numerical,marti2014full} for the geodynamo).
\rev{The development of a similar approach, based on well-understood
simple problems (e.g.\ the resonances of eigenmodes, or linear
spin-up/down), would be particularly important in the context of unique
devices aimed at achieving unprecedented ranges of parameters.}

\rev{In our opinion, developing such new large-scale facilties should
only be undertaken within the context of an open-access collaborative
project (e.g.\ as in astronomy with the large telescopes, or in
particule physics with the \textsc{cern}).  As outlined below, it is
essential that the collaborative structure in question fulfils some
fundamental conditions.}

\paragraph*{(1) Motivations}
\rev{It should be driven by common interests, identified by a broad
scientific community (e.g.\ the needs for a better understanding of the
rapidly rotating fluid dynamics and related phenomena, which are shared
among all geophysical sciences).}

\paragraph*{(2) Accessibility}
\rev{It should provide \emph{easy access} to the facility for all users
(e.g.\ similar to astronomical observatories or supercomputers).}

\paragraph*{(3) Scientific recognition}
\rev{While open access to publications is becoming a standard practice in 
academia, technical solutions or inconclusive results are rarely acknowledged 
and never rewarded. To maintain a culture of risk-taking and ambitious project
development, it is essential to recognise the significant time, energy
and creativity required. This may not always result in a peer-reviewed
publication, but it is nevertheless a crucial aspect of academic
endeavour. A collaborative platform involving recognised experts should
offer an alternative to the classical publication scheme to share this
unconventional outcomes, for instance in the form of open access
repositories (as done by the \textsc{cern}).}

\paragraph*{(4) Scientific commitment}
\rev{The success of such an enterprise is contingent upon the
dedication of senior scientists who are past the stage of developing
their careers.  This would allow them to assume the risks inherent to
these high-risk and high-gain projects.  Moreover, it would be worth
incorporating early-career and intermediate-career scientists into the
management structure (to stimulate creativity and to ensure the
continuity of the legacy).}

\paragraph*{(5) Long-term funding and legacy}
\rev{It is not feasible to obtain financial support for a collaborative
platform through the conventional grant schemes, which are constrained
by temporal limitations. To achieve this, a host institution must
demonstrate a long-term commitment, while national and international
institutions must pledge sustainable financial support.  Finally, given
the likelihood of more rigorous financial constraints in the near
future, a shared financial support would be the most optimal management
strategy to secure such ambitious research activities.}

For all these reasons, we believe that the experimental geophysical
fluid dynamics community would greatly benefit from large collaborative
projects rather than individual incentives.  While already done in
other fields (e.g.\ in astronomy or particle physics), this seems
however at odds with the current research organisation that tends to
reward individuals rather than consortiums. 

\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.

\CDRGrant[ERC]{847433}
\CDRGrant[ETH]{\#ETH-04 22-1}
\CDRGrant[ERC]{833848}
\CDRGrant[ANR]{ANR10 LABX56}

\section*{Funding}

{JV, RM and DC are supported by the European Research Council
(ERC) under the European Union's Horizon 2020 research and innovation
programme (grant agreement No 847433, \textsc{theia} project). JN and
VG are supported by ETH grant \#ETH-04 22-1. FB received funding from
the European Research Council (ERC) under the European Union's
Horizon 2020 research and innovation program (grant agreement No
833848, \textsc{uemhp} project). DC acknowledges the French Academy of
Sciences, Electricit\'e de France, Labex \textsc{osug}@2020 (ANR10 LABX56).}

\section*{Acknowledgements}

The authors thank S. Fauve and M. Le Bars for the
opportunity to write this review article. DC thanks the French Academy
of Science and Electricit\'e de France for granting their `Amp\'ere
Prize' to the ISTerre `Geodynamo' team. JV acknowledges Y. Colin de
Verdi\`ere for fruitful discussions about waves in rotating and
stratified fluids. DC and RM thank D. Jault for discussions on
topographic effects in planetary cores. DC thanks M. Solazzo for his
valuable help and his engineering work on rotating experiments with
topographic effects. The authors would like to dedicate the paper to
Stijn Vantieghem, who made major contributions in our field but passed
away in 2023 at the age of 39.

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