\makeatletter
\@ifundefined{HCode}
{
\documentclass[CRPHYS,Unicode,screen,Thematic]{cedram}
\newenvironment{noXML}{}{}
\newenvironment{Table}{\begin{table}}{\end{table}}
\def\thead{\noalign{\relax}\hline}
\def\endthead{\noalign{\relax}\hline}
\def\tabnote#1{\vskip4pt\parbox{.8\linewidth}{#1}}
\def\tsup#1{$^{{#1}}$}
\def\tsub#1{$_{{#1}}$}
\def\sfrac#1#2{{#1}/{#2}}
\def\sofrac#1#2{({#1}/{#2})}
\usepackage{amssymb}
\RequirePackage{etoolbox}
\usepackage{amsmath}
\csdef{Seqnsplit}{\\}
\def\xxboldsymbol#1{\boldsymbol{#1}}
\def\tbody{\noalign{\relax}\hline}
\def\jobid{crphys20240291}
%\graphicspath{{/tmp/\jobid_figs/web/}}
\graphicspath{{./figures/}}
\newcounter{runlevel}
\let\MakeYrStrItalic\relax
\def\refinput#1{}
\def\0{\phantom{0}}
\def\sfrac#1#2{{#1}/{#2}}
\def\stfrac#1#2{({#1})/{#2}}
\def\stofrac#1#2{(({#1})/{#2})}
\def\back#1{}
\def\rmGamma{\Gamma}
\def\rmDelta{\Delta}
\def\rmPi{\Pi}
\def\rmPhi{\Phi}
\DOI{10.5802/crphys.216}
\datereceived{2024-04-15}
\daterevised{2024-10-10}
\dateaccepted{2024-10-11}
\ItHasTeXPublished
\def\ndash{\text{--}}
\def\mathbi#1{\text{\textit{\textbf{#1}}}}
}
{\documentclass[crphys]{article}
\def\CDRdoi{10.5802/crphys.216}
\let\newline\break
\def\sfrac#1#2{{#1}/{#2}}
\def\sofrac#1#2{({#1}/{#2})}
\makeatletter
\def\selectlanguage#1{} 
\def\href#1#2{\url[#1]{#2}}
}
\makeatother

\usepackage{upgreek}

\begin{document}

\begin{noXML}

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

\CDRsetmeta{articletype}{review}

\title{Solid--liquid phase change in planetary cores}

\alttitle{Changement de phase dans les noyaux plan\'etaires}

\author{\firstname{Ludovic} \lastname{Huguet}\CDRorcid{0000-0002-5532-6302}\IsCorresp}
\address{ISTerre, Universit\'e Grenoble Alpes, 
Universit\'e Savoie Mont Blanc, CNRS, IRD, 
Universit\'e Gustave Eiffel, 38000 Grenoble, France}
\curraddr[L. Huguet]{School of Earth and Environment, 
University of Leeds, Leeds LS2 9JT, UK}
\email[L. Huguet]{l.g.huguet@leeds.ac.uk}

\author{\firstname{Quentin} \lastname{Kriaa}\CDRorcid{0000-0002-3518-1660}}
\address{Physics of Fluids Group, 
Max Planck Center for Complex Fluid Dynamics, 
and J. M. Burgers Centre for Fluid Dynamics, 
University of Twente, P.O. Box 217, 7500AE Enschede, 
The Netherlands}
\email[Q. Kriaa]{quentinkriaa@gmail.com}

\author{\firstname{Thierry} \lastname{Alboussi\`ere}\CDRorcid{0000-0002-3692-899X}}
\address{Universit\'e de Lyon, ENSL, UCBL, UJM, CNRS, 
Laboratoire LGL-TPE, France}
\email[T. Alboussi\`ere]{thierry.alboussiere@ens-lyon.fr}

\author{\firstname{Michael} \lastname{~Le Bars}\CDRorcid{0000-0002-4884-6190}}
\address{CNRS, Aix Marseille Univ, 
Centrale Marseille, IRPHE, Marseille, France}
\email[M. Le Bars]{michael.le-bars@univ-amu.fr}

\shortrunauthors

\keywords{\kwd{Crystallization}\kwd{Phase change}\kwd{Two-phase
flows}\kwd{Planetary cores}\kwd{Magnetic field}}

\altkeywords{\kwd{Cristallisation}\kwd{Changement de
phase}\kwd{\'Ecoulements diphasiques}\kwd{Noyaux
plan\'etaires}\kwd{Champ magn\'etique}}

\thanks{European Research Council (ERC)  (grant number 716429), Labex
OSUG@2020 (ANR10 LABX56), Programme National de Plan\'etologie (PNP) of
CNRS-INSU co-funded by CNES}

\begin{abstract}
The ubiquitous phenomena of crystallization and melting occur in
various geophysical contexts across many spatial and temporal scales.
In particular, they take place in the iron core of terrestrial planets
and moons, profoundly influencing their dynamics and magnetic field
generation. Crystallization and melting entail intricate multiphase
flows, buoyancy effects, and out-of-equilibrium thermodynamics, posing
challenges for theoretical modeling and numerical simulations. Besides,
due to the inaccessible nature of the planetary deep interior, our
understanding relies on indirect data from seismology, mineral physics,
geochemistry, and magnetism. Consequently, phase-change-driven flows in
planetary cores constitute a compelling yet challenging area of
research. This paper provides an overview of the role of laboratory
fluid dynamics experiments in elucidating the solid--liquid phase change
phenomena occurring thousands of kilometers beneath our feet and within
other planetary depths, along with their dynamic repercussions. Drawing
parallel with metallurgy, it navigates through all scales of phase
change dynamics, from microscopic processes (nucleation and crystal
growth) to macroscopic consequences (solid--liquid segregation and
large-scale flows). The review delves into the two primary planetary
solidification regimes, top-down and bottom-up, and elucidates the
formation of mushy and/or slurry layers in the various relevant
configurations. Additionally, it outlines remaining challenges,
including insights from ongoing space missions poised to unveil the
diverse planetary regimes.
\end{abstract}

\begin{altabstract}
Les ph\'enom\`enes de cristallisation et de fusion se produisent dans
divers contextes g\'eophysiques \`a diverses \'echelles spatiales et
temporelles. Ils prennent notamment place dans le noyau en fer des
plan\`etes telluriques et des lunes, o\`u ils jouent un r\^ole primordial
dans leur dynamique et dans la g\'en\'eration de leur champ
magn\'etique. La cristallisation et la fusion se caract\'erisent par
des \'ecoulements multiphasiques complexes, des effets de
flottabilit\'e et une thermodynamique hors \'equilibre, difficilement
accessibles \`a la mod\'elisation th\'eorique et aux simulations
num\'eriques. De plus, les profondeurs plan\'etaires demeurent
inaccessibles aux observations directes, et notre compr\'ehension
repose sur des donn\'ees indirectes provenant de la sismologie, de la
physique min\'erale, de la g\'eochimie et du magn\'etisme. Par
cons\'equent, les \'ecoulements induits par un changement de phase dans
les noyaux plan\'etaires constituent un domaine de recherche \`a la fois
passionnant et stimulant. Cet article donne un aper\c{c}u du r\^ole qu'ont
les exp\'eriences de laboratoire en dynamique des fluides pour
\'elucider des ph\'enom\`enes de changement de phase solide-liquide se
produisant \`a des milliers de kilom\`etres sous nos pieds ou dans
d'autres plan\`etes, ainsi que de leurs cons\'equences dynamiques. En
\'etablissant des parall\`eles avec la m\'etallurgie, cet article
navigue \`a travers toutes les \'echelles de la dynamique du changement
de phase, des processus microscopiques (nucl\'eation et
cristallisation) \`a leurs cons\'equences macroscopiques
(s\'egr\'egation solide-liquide et \'ecoulements de grande \'echelle).
Nous d\'ecrivons en d\'etail les deux principaux r\'egimes de
solidification plan\'etaire, descendant et ascendant, ainsi que la
formation de couches de solides interconnect\'ees (``mush'') ou
dispers\'ees (``slurry'') dans les diverses configurations pertinentes.
Enfin, cette revue d\'ecrit les d\'efis restants, notamment dans le
cadre de missions spatiales en cours qui d\'evoileront sans aucun doute
la diversit\'e des r\'egimes\break plan\'etaires.
\end{altabstract}

\maketitle

\end{noXML}

\let\rmmu\upmu

{\vspace*{6pt}}

\section{Introduction}\label{sec1}

{\vspace*{2pt}}

Crystallization and melting are ubiquitous in various geophysical
settings across large ranges of spatial and temporal scales. In most
instances, solidification and melting are associated with subsequent
buoyancy effects with profound dynamical consequences, leading to
stable stratification or intense convective motions. Indeed, beyond
latent heat and heat transfers, solidification and melting in
geophysical settings are often intricately linked with species
segregation due to the heterogeneity of the considered fluids. For
instance, on our planet, the crystallization of ice at the poles
increases the concentration of salt into the ocean water, hence playing
a pivotal role in climate by driving part of the global ocean
circulation. Diving towards the planetary deep, the significance of the
local solid--liquid phase change of the mantle is revealed through
volcanic activity. Partial melting within the lithosphere also shapes
the composition of the crust and contributes to the formation of stable
continents. In the early epochs of Earth's history, more substantial
melting occurred within the mantle, driven by the energetic processes
of accretion and the presence of short-lived radiogenic elements,
resulting in a partial or global magma ocean setting the initial
thermochemical state of today's solid mantle. Still going deeper, the
inner part of the Earth's core has been solidifying over the last
billion years or so, sustaining the Earth's magnetic field generation.
Similar phase changes involving water, silicate, and iron are actually
ubiquitous on all terrestrial planets and icy moons.

\subsection{Phase change and planetary magnetic fields}\label{sec1.1}
The planet of reference for magnetic field production is the Earth,
whose magnetic field is most likely produced by {dynamo} in its liquid
core. This process describes how the motions of an electrically
conducting fluid produce a magnetic field. The coupling between
magnetism and fluid motions leads to the presence of electrical
currents that feed the magnetic field but are constantly subject to
ohmic dissipation. As a result, dynamo action can be sustained only as
long as the electromotive action of the flow is sufficiently large
compared to ohmic dissipation, which translates as a condition on the
magnetic Reynolds number (e.g.,~\cite{cog1999})
{\begin{equation}\label{eq:Rem_intro}
Re_m = \frac{UL}{\eta_m} \gg 1,
\end{equation}}\unskip
where $U$ and $L$ are respectively the characteristic velocity scale
and length scale of the flow, and $\eta_m$ is the magnetic diffusivity.
This necessary yet insufficient requirement places constraints on the
vigor of the flow. Then, a burning question when observing a
dynamo-generated magnetic field is: how do planetary cores sustain
fluid motions that drive a dynamo?

Three contributions are usually mentioned that participate in
sustaining the Earth's magnetic field: \textit{thermal convection} due
to the secular cooling of the planet, and possibly to internal heating
through radioactive decay; \textit{crystallization} of the core at the
center that leads to chemical segregation between pure iron (that
solidifies on the solid inner core) and lighter elements that rise in
the liquid outer core, feeding compositional convection; \textit{latent
heat} release during crystallization that warms the fluid near the
inner core, hence boosting the thermal convection. These three elements
have proven important in driving the Earth's dynamo, and the last two
of them are directly related to core crystallization. Earth's energy
budget makes it very unlikely to sustain a dynamo without contribution
from crystallization, hence questioning the model of a convective Earth
dynamo before the start of the inner core growth
(e.g.,~\cite{lfncs2022}). In addition to these three sources,
mechanical driving processes (libration, precession, tidal deformation)
can provide an alternative source of energy for generating magnetic
fields in planetary\unskip\break
cores~\cite{km1998,lwkcl2011,dsn2011,lcl2015,lbblnst2022}.

Some small Earth-like planets show evidence of past (Mars, Moon) or
present (Ganymede, Mercury) global magnetic field (e.g.,~\cite{brs2015}
or~\cite{bsvvsr2022} for interior structure). These observations have
questioned the scientific community, as long-term, global magnetic
fields seem difficult to sustain in such small cores. In contrast to
the Earth's bottom-up core crystallization, a regime of top-down
crystallization might take place on small planets. Associated flows may
sustain a dynamo, but their complexity has been up-to-now only remotely
tackled.

Planetary cores are very particular geophysical objects, as no direct
observation can be made. Our knowledge of the deepest part of planets
relies on indirect observations from geochemistry (for composition),
geodesy and gravity measurements (for internal structure, mass, moment
of inertia, libration), seismology (for internal structure, density,
velocity, and state), and magnetism. Since the early 20th century, the
image of the Earth's core seen by the seismology shows stratified
layers (E$^\prime$ layer and F-layer)~\cite{icl2018,wipt2023} acting
against the well-mixed state of the liquid core. Seismological studies
have also revealed a complex structure inside the Earth's inner core,
including east-west hemispheric dichotomy, innermost inner core, radial
or lateral variation of elastic anisotropy, and
attenuation~\cite{wipt2023,d2014}. Planetary seismology on the
Moon~\cite{wlgwl2011,ggcl2011,garcia2019lunar} and on
Mars~\cite{skblgcddgho2021,sdrxhgliblo2023,ilddkrksabo2023} have also
demonstrated the presence of a liquid core and some partially solid
part at the top of the core or the bottom. The advance of satellite
observations and planetary space missions has enabled us to better
constrain the structure, intensity, and secular variation of planetary
magnetic fields. The precision of geomagnetic measurements has even
allowed us to infer the dynamics at the top of Earth's core, which
provides key information on the state and the velocity of the fluid
iron motions. However, in contrast with atmospheric and oceanic
observations, the lack of direct measurements challenges our knowledge
of the state and dynamics of planetary cores and makes it necessary to
challenge the output of phase-change experiments and models with the
available data. 

\subsection{From the casting industry to planetary cores}\label{sec1.2}
This review focuses on solid--liquid phase change within planetary cores
and its dynamic consequences. In this context, the phase change is
envisaged through its planetary, large-scale significance. But its
intricate nature, encompassing both macroscopic and microscopic scales,
is always present. At the atomic or molecular level, energy barriers
must be overcome to induce a change in atomic arrangement. Diffusion
processes operating at microscale influence the morphology of the
liquid--solid interface, resulting in the formation of isolated crystals
or dendritic structures, possibly modified by convection. The thickness
of the region undergoing crystallization then represents the mesoscale,
potentially involving compaction and other associated effects. While
most studies focus on specific scales, it remains challenging to
integrate all of them into a single numerical model. Consequently,
experimental approaches hold significant value as they naturally
capture all the complexity of phase change, albeit with the challenge
of disentangling the individual contributions.

Many of our ideas on crystallization in planetary cores originate from
the study of metallurgy (e.g.,~\cite{f1974,kfr2023}). Yet the
conditions are very different: the expected growth rate of
crystallization of the inner core---$V \sim
10^{-10}~\mathrm{m}{\cdot}\mathrm{s}^{-1}$---is unseen in the iron
casting where the  $\mathrm{mm}{\cdot}\mathrm{s}^{-1}$ is more typical.
Similarly, the temperature gradient in the core---$G \sim
10^{-3}~\mathrm{K}{\cdot}\mathrm{m}^{-1}$---is much smaller than usual
values in metallurgy. A consequence is that heuristic laws valid in
metallurgy may not apply to the core. Fortunately, we can still rely on
laws based on physical arguments. For instance, the primary dendrite
spacing is found to scale proportionally to $V^{-1/4} G^{-1/2}$, based
on diffusion length scales and the radius of curvature of the tips of
the dendrites. This kind of scaling has been applied to obtain
estimates of dendrite spacing and grain size in the inner core,
reaching orders of magnitude of tens of meters~\cite{dab2007}, well
beyond day-to-day experience. 

Many factors in the planetary interiors do not have their counterparts
in metallurgy. Compaction due to the deformation of crystallizing
material under its weight or other stresses is a key factor in cores
but is hardly studied in material processing. The effects of rotation
and a background magnetic field are also potentially very important in
planetary interiors. Magnetic fields have been investigated in the
framework of steel casting, but not in the same view. They have been
used to stir the fluid (traveling magnetic fields) or to brake fluid
motion (DC magnetic fields), depending on the configuration and the
desired final properties. At the scale of a solidifying zone, in the
presence of a temperature gradient, thermoelectric currents can develop
at the liquid--solid interface, acting as a short-circuited
thermocouple. These currents have been measured using small electric
potential probes~\cite{lielausis84}. When a constant magnetic field is
applied, the thermoelectric currents produce Lorentz forces and drive a
flow that can compete and reverse the flow induced by compositional
buoyancy~\cite{alboussiere91,lehmann98}. All the ingredients for
similar effects exist in metallic cores but have received little
attention so far in this context.

\subsection{Outline}\label{sec1.3}
This review is organized as follows. In  Section~\ref{sec:cry_planet},
we first describe the state-of-the-art of our thermodynamical
understanding of planetary cores' phase change and the variety of
solidification regimes. This leads us to define two main scenarios,
whose dynamics are described in depth in the following  sections: the
formation of a mushy layer detailed in  Section~\ref{sec:mushylayer},
and the formation of a slurry layer detailed in 
Section~\ref{sec:slurryLayer}, respectively. Mushy layers correspond to
situations where the solid forms a continuous, yet possibly porous
matrix attached to a boundary. Slurry layers correspond to mixtures of
fluid and detached grains in suspension, no matter what their
concentration. The final  Section~\ref{sec:conclusion} details the main
remaining open questions and the upcoming challenges related to ongoing
space missions. It also briefly describes phase change applications in
other planetary settings, which could be each the subject of a long
review.

\section{Crystallization in planetary cores}\label{sec:cry_planet}

\subsection{\label{subsec:stateOfCore}Composition and state}
The solidification scenario of a planetary core depends on its liquidus
temperature (also referred to as the solidification temperature or the
melting temperature) and on its effective temperature profile
(geotherm), which both depend on the pressure profile and core
composition.

Planetary cores are primarily composed of iron\footnote{More
specifically, of an iron-nickel alloy.}, plus some light elements which
could include sulfur, oxygen, silicon, hydrogen, and
carbon~\cite{brs2015,hlh2013}. Mineralogical studies of these various
iron alloys under the pressure and temperature conditions of planetary
cores allow for the determination of their density, melting
temperature, thermal conductivity, and compressibility, depending on
their composition and whether they are in a solid or liquid state. The
melting temperature and thermal conductivity help determine when the
first solid will form due to the thermal evolution of the core. The
liquid compressibility and the slope of the liquidus determine whether
crystallization will begin at the core's center or surface. The density
difference between the liquid and the solid (positive or negative)
determines the dynamics of liquid/solid segregation. All these
parameters have been measured in various experimental studies for
different alloys, such as Fe--Si~\cite{kv2018},
Fe--S~\cite{msfmrpb2007,bw2011}, Fe--S--O~\cite{evswdh2019,pldf2018},
Fe--Si--H~\cite{hthoouo2022}, Fe--Ni--S~\cite{ll2020,pl2022},
Fe--Ni--Si~\cite{ioh2021,kpskbsgkm2019}, and
Fe--Ni--H~\cite{stotnfh2014}.

In the canonical framework, the core is assumed to be well-mixed during
its accretion, leading to a uniform bulk composition before the
crystallization starts~\cite{n2015}. The liquid mixture is then
described by a Fe--X alloy where the element X stands for a light
element. In addition, a partition coefficient equal to 0 is usually
considered, meaning that the iron solid phase is pure. In that case,
the state of the core is described by the typical binary phase diagram
sketched in  Figure~\ref{fig:binaryPhaseDiagramFeX}. Such a diagram is
given for a constant pressure, hence for a fixed depth in the core.
Mineral physics studies provide the eutectic composition and
temperature, and the Clayperon slope for a wide pressure range.
Parameterized equations of state are built to determine which phase
exists for all compositions, temperatures, and pressures. Over time,
the core cools down, so the state of the core is generally expected to
go from fully liquid to partially solid and, ultimately, fully solid.
It is apparent in  Figure~\ref{fig:binaryPhaseDiagramFeX} that the bulk
X mass fraction in the fully liquid state controls the composition of
the crystals that form. Cores on the iron-rich side of the eutectic
(hypo-eutectic) $c_{\mathrm{X}}<c_{\mathrm{X,e}}$ form pure iron
crystals; on the opposite, the X-rich cores (hyper-eutectic)
$c_{\mathrm{X}}>c_{\mathrm{X,e}}$ form
$\mathrm{Fe}_{a}\mathrm{X}_{b}$ crystals, which are
generally considered positively buoyant. 

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:binaryPhaseDiagramFeX}Simplified sketch of the
binary phase diagram of iron (Fe) and a light element denoted ``X'',
showing the state of the core as a function of the X mass fraction
$c_{\mathrm{X}}$ and of temperature. The solid blue line corresponds to
the liquidus, and the solid dark line is the solidus. The composition
of the eutectic is denoted $c_{\mathrm{X,e}}$. The integers $a$ and $b
\neq 0$ in subscript parameterize the composition of the solid forming
on the X-rich side of the eutectic. For simplicity, we focus in 
Section~\ref{subsec:scenariosSolidification} on a core lying on the
iron-rich side of the eutectic (hypo-eutectic). In that case, the
dashed red line shows a typical evolution of the state of the liquid
core assuming thermodynamic equilibrium, for an initial mass fraction
$c_{\mathrm{X}}(t=0)<c_{\mathrm{X,e}}$ and an initial temperature above
the liquidus.}
\end{figure}

\subsection{Dynamical scenarios of
solidification}\label{subsec:scenariosSolidification}
Figure~\ref{fig:binaryPhaseDiagramFeX} offers a simple, local picture
of core crystallization solely based on thermodynamics. But in
practice, core crystallization is a dynamic process due to the presence
of buoyancy effects. Since the liquid Fe--X mixture contains a light
element, pure iron crystals are denser than the liquid, and are
therefore negatively buoyant, while 
$\mathrm{Fe}_{\mathrm{a}}\mathrm{X}_{\mathrm{b}}$ crystals are
positively buoyant. The segregation of the newly solid phase and the
remaining liquid phase determines the dynamical evolution of the cores,
including the growth of a solid layer. Moreover, this segregation may
lead to the formation of convective or stably stratified fluid zones.
Fluid motions driven by buoyant parcels of fluid or by buoyant crystals
are thought to generate enough energy to sustain a dynamo
action~\cite{b1963,lb1995,dp2018}.

The pathways of crystallization and segregation depend on two aspects.
First, it depends on how the density of the crystals compares with that
of the ambient. Second, crystal migration depends on exactly where
solidification happens. For illustration purposes, we focus here on an
iron-sulfur core on the iron-rich side of the eutectic, i.e.\ with a
sulfur concentration $c_{\mathrm{S}}<c_{\mathrm{S,e}}$ and we describe
two scenarios corresponding respectively to the evolution of the Earth
and of a small Earth-like planet (or moons) like Ganymede. More
crystallization scenarios can be found in~\cite{brs2015}. In all
scenarios, the ultimate state of the core is to be entirely solid (for
small cores such as asteroids) or partially solid (for larger cores
such as those of the Moon, Mercury, and Earth).

Figure~\ref{fig:sampleBreuer2015}a shows the typical bottom-up regime
of solidification of the Earth~\cite{j1953}. As the core cools down,
its temperature eventually crosses the liquidus curve at its center
because the slope of the core temperature is steeper than the liquidus.
Consequently, solid crystals first appear at the planet center, as
sketched by red circles. At the bottom of the crystallizing region, an
iron solid layer grows by the formation of a solid matrix or by the
accumulation of free iron crystals (see 
Section~\ref{subsec:mushyslurry}). In the meantime, the residual liquid
depleted in iron is lighter than the bulk Fe--S mixture, so it rises in
the core and nourishes compositional convection above the solid layer
(gray arrows in  Figure~\ref{fig:sampleBreuer2015}a).

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:sampleBreuer2015}Two dynamical scenarios of
solidification of an iron-rich, sulfur hypo-eutectic core: 
(a)~bottom-up solidification in an Earth-like planet; (b)~top-down
solidification in a smaller Earth-like planet. Figure adapted with
permission from Breuer et~al.~\cite{brs2015}, under the
\href{http://creativecommons.org/licenses/by/4.0/}{CC BY 4.0 license}.}
\end{figure}

Consider now the top-down freezing of small Earth-like planets like
Ganymede~\cite{had2006}, as sketched in 
Figure~\ref{fig:sampleBreuer2015}b. In such planets,
studies~\cite{bw2011,fbf1997,cb2007,rvvmd2011,dr2015} suggest that the
slope of the liquidus as a function of depth is either steeper than the
core temperature or even negative~\cite{rvvmd2011,dr2015}. Therefore,
as the core cools down its temperature profile first crosses the
liquidus at the core periphery, where the first crystals
appear~\cite{brs2015,rbs2015,rbs2018} (see the red circles in 
Figure~\ref{fig:sampleBreuer2015}b). Iron crystals might grow from the
core-mantle boundary or spontaneously nucleate freely in the bulk. The
formation of the first crystals on the surface or in the bulk will
depend on physical properties (wettability, roughness, nucleation
rate~\cite{hvhw2018}), and dynamical processes (growth rate,
constitutional supercooling layer~\cite{dab2007}, collisional
breeding~\cite{so1994}). We will deepen the conditions of
crystallization in the  Sections~\ref{subsec:mushyslurry}
and~\ref{subsec:nucleation}. As they are denser than the ambient
liquid, both cases end up with unstable crystals that settle due to
gravity. The remaining light-element-rich liquid is lighter than the
ambient, so it rises. The crystallization region extends from the
core-mantle boundary to a lower limit where the crystals remelt because
of the increasing ambient temperature. As a first-order estimate, this
location is given by the core temperature matching the
liquidus~\cite{brs2015,rbs2015,rbs2018}. This region of solidification
is called the {snow zone,} and the whole phenomenon of
crystallization-settling-remelting is often referred to as {iron snow}.
This snow zone is often thought to be at thermodynamic equilibrium,
leading to the assumptions of fast crystallization and remelting
compared to the crystal dynamics~\cite{dp2018}. For the sake of
simplicity, the departure from equilibrium in the snow zone is assumed
to be negligible, even though supercooling can modify the snow dynamics
in some experiments~\cite{hld2023} (see 
Section~\ref{subsec:bulkCrystallisationXP}). However, quantifying the
departure from equilibrium requires releasing the previous hypotheses,
making the theoretical model complex to handle~\cite{l1992,wdj2023}. At
the lower edge of the snow zone, the molten snowflakes form a liquid
mixture that is richer in iron than the ambient liquid, all the more as
it is concentrated in molten snowflakes, all the less as it dilutes
with the ambient. This {molten snow} is therefore negatively buoyant,
so it sinks towards the center of the core and nourishes compositional
convection---see~\cite{c2015} for a numerical study. Eventually, the
core temperature becomes lower than the liquidus everywhere, so that
snowflakes accumulate at the core center and pile up as a growing inner
core. 

Both scenarios described here are based on sketches and idealized
hypotheses: they now need to be confronted with the reality of lab
experiments.

\subsection{Mushy or slurry layers}\label{subsec:mushyslurry}
During the crystallization of a binary mixture, the equilibrium
temperature at the interface between the solid and liquid is equal to
the liquidus temperature. However, the growth of the solid by
crystallization requires a slight departure from this equilibrium,
leading to the formation of \mbox{supercooled} regions ahead of the
solidification front (i.e.\ the liquid temperature is below the freezing
point). In addition, the composition of the liquid and the solid differ
in terms of a light element composition depending on its solubility
(generally very small for light elements (S, C, H, O), except for Si).
Near the solidification front, the liquid is enriched in light
elements, leading to the formation of a compositional gradient.
Therefore, the fluid ahead of the front of solidification is often in a
constitutionally supercooled state~\cite{rc1953}. The morphological
stability of the interface, which depends on both the temperature and
composition gradients, has been extensively examined theoretically and
experimentally~\cite{kfr2023,ms1964}. In the conditions of planetary
crystallization, the pressure gradient also plays a major role in the
front stability, as the core and liquidus temperatures are
pressure-dependent~\cite{dab2007,splm2005} 
(Figure~\ref{fig:sampleBreuer2015}). If the interface is
morphologically unstable, the front of solidification will encounter
cellular growth, dendritic growth, or equiaxed crystal growth when
considering an increasingly large zone of the constitutional
supercooled layer~\cite{kfr2023,c2002} 
(Figure~\ref{fig:frontsolidification}). In planetary cores, the
stability of the front of solidification depends on the liquidus and
Clapeyron slope  (Figure~\ref{fig:frontsolidification}). The liquidus
slope $\sfrac{\partial T_m}{\partial c}|_P$ is negative or positive
depending on whether the liquid composition is hypo-eutectic or
hyper-eutectic  (Figure~\ref{fig:binaryPhaseDiagramFeX}). The Clapeyron
slope $\sfrac{\partial T_m}{\partial P}|_c$ is mainly positive for iron
alloys at high pressure, yet, for Fe--S at an intermediate pressure,
the Clapeyron slope is negative~\cite{pldf2018,pl2022,clh2008}. In
addition, the solidification rate and advective transport of the solute
in the liquid play a key role in controlling the structure of the
thermal and chemical fields near the solidification front. While
crystallization is very slow and is of the order of one millimeter a
year~\cite{n2015,lpl2001}, advective transport in the liquid can be
several orders of magnitude larger~\cite{habdll2016}. For the Earth,
the theoretical analysis of the stability of the solidification front,
at the conditions of the inner core boundary, suggests an unstable
front of solidification~\cite{dab2007,splm2005,flr1981}.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:frontsolidification}Schematic view (for the Earth's
core) of the core temperature profile (very close to the adiabatic
profile), liquidus profile, and Clayperon slope, associated with a
stable (A)~or unstable (B--D)~solidification front. An unstable front
of solidification will encounter cellular (B), dendritic (C), or
equiaxed growth (D). The thickness of the constitutional supercooled
region (black dashed lines) depends on the core temperature and the
liquidus profile, which both depend on the growth rate and
composition.}
\vspace*{1pc}
\end{figure}

Figure~\ref{fig:dendritevsequiaxed} shows two experiments of equiaxed
and dendritic crystallization using analog material, like metallurgy
studies. Equiaxed crystal growth leads to the formation of a two-phase
region in which free crystals nucleate and grow, and the surrounding
liquid can be convectively stable or unstable: this is a so-called
\textit{slurry layer}  (Figure~\ref{fig:dendritevsequiaxed}a).
Alternatively, dendritic growth leads to the formation of a two-phase
region in which dendrites coexist with an interstitial liquid that can
also be stable or unstable: a so-called \textit{mushy layer} 
(Figure~\ref{fig:dendritevsequiaxed}b). In a mushy layer, primary
dendrites grow mainly in the same direction. Both regimes can co-exist
for the same conditions; mushy layers preferentially grow from a
substrate, and slurry layers exist if enough nucleation sites are
present. Hence, by analogy with metallurgy, the crystallization
conditions at the inner core boundary may lead to the formation of a
mushy layer~\cite{dab2007,splm2005,habdll2016,flr1981} or of a slurry
layer~\cite{splm2005,wdj2021}. The presence of a slurry layer depends
on the thickness of the constitutional supercooled
layer~\cite{splm2005}, and the ability to produce nucleation sites in
the bulk of this layer (see 
Section~\ref{subsec:nucleation})~\cite{splm2005,rl1983}. The stability
of the front of solidification in other planetary cores remains poorly
known even though the presence of dendrite crystallization has been
suggested in asteroids based on observations in metallic
meteorites~\cite{hs1992,ch2006}, and the presence of an iron snow
regime (slurry layer) has been suggested at the top of the core of
small moons or planets~\cite{had2006,rbs2015} (see 
Figure~\ref{fig:sampleBreuer2015}b).

\begin{figure}
\vspace*{10pt}
\includegraphics{fig04}
\caption{\label{fig:dendritevsequiaxed}(a)~A slurry layer in a tank
cooled from above. Equiaxed crystals form near the top boundary,
settle, and accumulate at the bottom of the tank. (b)~Mushy layer with
dendrites growing from the bottom of a tank cooled from
below~\cite{habdll2016,h2014}. Both cases used ammonium chloride
solutions as an analog to the Fe--X mixture~\cite{jhus1966}.}
\end{figure}

\subsection{Nucleation paradox}\label{subsec:nucleation}
Crystallization has been extensively studied in the literature
(e.g.,~\cite{kfr2023}), given its importance in industrial processes
(metallurgy, casting,\,\ldots). It is divided into two steps: nucleation
followed by crystal growth. While the second step has also been widely
studied in the context of planetary cores, the nucleation step has been
largely ignored~\cite{n2015,splm2005}. For liquid metals, classical
nucleation theory~\cite{kfr2023,c2002} requires either a very large
supercooling in a homogeneous environment or a {wettable} surface
reducing the supercooling. Experiments in various pure metals or
alloys~\cite{c2002,t1950,gsw1965,smm1987} have shown that the critical
supercooling is quite large compared to the melting temperature, i.e.\ 
about 20\% of $T_m$. To explain the stability of the supercooled
metals, Frank suggested that local structures exist in supercooled
liquids~\cite{f1952}. This hypothesis was later confirmed by
identifying atomic structures with a degree of icosahedral short-range
order in experiments in alloys~\cite{klghrrrr2003} or in
iron~\cite{shsbh2002,gw2008,imkn2009}. These icosahedral short-range
order structures may lead to the formation of stable or metastable
quasi-crystals~\cite{sbgc1984}, which would preferentially grow instead
of crystalline phase~\cite{klghrrrr2003}. The undercoolability (meaning
the ability to remain stable for a supercooled liquid) of metal alloys
then depends on the stability of these local structures in the
supercooled liquid, which would ultimately depend on the presence of
other phases than the iron~\cite{hssbchh2004}. These short-range order
structures have been suspected to play a major role in determining the
melting temperature and the viscosity of high-pressure
liquids~\cite{br2015}. However, the experimental detection of such
structure in a supercooled liquid at high pressure and temperature is
extremely difficult. Still, the detection of footprints, such as the
twining of dendrites in Al alloys~\cite{kjr2013}, might be plausible.
One may think that the advance of electromagnetic levitation
containerless experiments~\cite{kglhhkrrr2002} in various environments
including microgravity~\cite{gsbhvkgphk2021,mbggghkkvvo2023} along with
laser shock experiments~\cite{lwlxtzzxgs2020,khabbbbccgco2022}, which
already provide physical properties of iron at high-pressure, may be
able to determine the degree of supercooling for iron alloys at the
high pressure and high temperature, which up-to-now remains unknown.

Recently, Huguet et~al.~\cite{hvhw2018} extrapolated the theory and
results from the classical nucleation theory to planetary core
conditions. They also investigated plausible causes of deviation from
the canonical theory. They revealed that the nucleation barrier is
likely to be very high even at the center of the Earth's core. The
Earth's inner core nucleation paradox arises from the incompatibility
between the slow cooling rate of the Earth's core and the high
energetic barrier for homogeneous nucleation. While heterogeneous
nucleation seems to be unlikely at the center of the Earth's
core~\cite{hvhw2018} (it is located thousands of kilometers away from
any solid surface), the presence of the solid surface at the
core-mantle boundary should ease the nucleation in a top-down scenario
of crystallization~\cite{hhvj2018}. 

Motivated by this work, several studies have attempted to resolve the
paradox using molecular dynamics
simulations~\cite{wawd2023,wwad2021,szmwh2022,dpa2019}. The presence of
light elements can either reduce or increase the nucleation
barrier~\cite{wawd2023}. In a two-step scenario of nucleation, Sun 
et~al.~\cite{szmwh2022} found that the body-centered cubic (bcc) phase
of iron has a lower nucleation barrier than the stable hexagonal
close-packed (hcp) phase. However, the nucleation barrier remains
significantly high even considering light elements or complex pathways
of nucleation. This energetic barrier might be particularly significant
for the existence of a slurry layer since the latter requires
nucleation sites in the bulk: the origin of these nucleation sites
remains a matter of speculation in planetary cores~\cite{rl1983},
especially for bottom-up solidification in which no solid phase is
initially present~\cite{hvhw2018}. The recent molecular
dynamic~\cite{wawd2023} and geodynamical studies~\cite{lkc2020} further
highlight the importance of the nucleation process in the
crystallization of planetary cores, which remains a subject of active
research.

We will now describe in more detail the dynamics of a mushy, and then
of a slurry layer in the context of planetary cores. We will highlight
fluid dynamics or crystallization experiments that unravel the dynamics
of such layers in cores.

\section{Mushy layer}\label{sec:mushylayer}
Here we present the growth of a mushy layer and its dynamics in the
context of planetary cores. Mushy layers have been considerably studied
through theoretical, numerical, and experimental approaches because of
their importance in industrial casting and sea ice dynamics (see
e.g.,~\cite{bw1995,w1997,fuww2006}). Experiments have used metallic
alloys or salt
solutions~\cite{habdll2016,cgjh1970,sh1988,cc1991,tj1992,wk1994,ww1997}
(see  Figures~\ref{fig:dendritevsequiaxed}b and~\ref{fig:mush_height}).
Ammonium chloride and sodium chloride solutions have proven good
analogs for studying the solidification of metals~\cite{jhus1966}.
Following these works, dedicated experimental studies have been
performed to unravel the effect of
convection~\cite{habdll2016,bmshca2005} and magnetic
field~\cite{bfbs1997} on the structure of the Earth's inner core. Note
that a core mushy layer differs in terms of structure and evolution
compared to the classical casting case: while the dendrites are
canonically seen as a steady solid matrix growing from a surface, large
planetary dendrites might flow under their weight, melt due to external
forces, and sink into the liquid core.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig:mush_height}(a)~Top view of a mushy layer growing
under high gravity level (136~g)~\cite{habdll2016,h2014}. Red circles
denote the chimneys, which are evidence of mushy layer convection. 
(b)~Side view of the mushy layer consisting of ammonium chloride
dendrites. Dendrites formed grains with different preferential
orientations~\cite{h2014}. (c)~The thickness of the mushy layer as a
function of time for different apparent gravities. Squares are the
experimental results and dashed lines are predictions from the
evolution model~\cite{habdll2016}. (d)~Same as (c) for different
viscosities, which correspond to different gravity levels given as an
indication~\cite{tj1992}. Panels (a,b) and (c,d) adapted with
permission from Huguet~\cite{h2014} and Huguet 
et~al.~\cite{habdll2016}, \mbox{respectively}.}
\end{figure}

\subsection{State of a mushy layer growing at the Earth's inner
core}\label{subsec:ICmushy} 
Crystallization of the inner core is a major source of energy for
convection in the outer core, which generates the Earth's magnetic
field. Reciprocally, convection alters the transfer of light elements
to the inner core's surface, significantly impacting crystallization.
Fearn et~al.~\cite{flr1981} showed that the inner core boundary
conditions lead to the formation of dendrites~\cite{lr1981}. Based on
thermodynamic arguments, they demonstrated that this dendritic mushy
layer might extend down to the center of the Earth's
core~\cite{flr1981}. The mushy zone can be subject to convective
instabilities~\cite{w1992}: the boundary layer mode and the mushy layer
mode. The first mode would be unstable in the inner core since the
solute released upon solidification is lighter than the
ambient~\cite{habdll2016}. The mushy layer mode of convection is
controlled by the mushy layer Rayleigh number
$R_m$~\cite{wwo2010,wwo2011,wwo2013,rw2013} written as:
{\begin{equation}
R_m=\frac{\beta^\star \rmDelta C g \rmPi(\rmPhi) h }{\nu \kappa},
\label{eq:Rayleigh_R}
\end{equation}}\unskip
where $\beta^\star = \beta - \rmGamma \alpha$ is the expansion
coefficient due to temperature and concentration changes ($\beta$ is
the solute expansion coefficient, $\alpha$ is the thermal expansion
coefficient, $\rmGamma$ is the liquidus slope), $\rmDelta C$ is the
difference of composition across the height of the mushy layer, $\nu$
and $\kappa$ are the viscosity and thermal diffusivity of the
interstitial liquid. In contrast with the classical Rayleigh number
definition, the mushy layer Rayleigh number is proportional to the
thickness $h$ of the layer. The permeability $\rmPi(\rmPhi)$ depends on the
solid fraction $\rmPhi$ and is described by the Kozeny-Carman
equation~\cite{c1956}
{\begin{equation}
\rmPi(\rmPhi)=\frac{\lambda_1^2}{4}\frac{(1-\rmPhi)^3}{k_0 \rmPhi},
\label{eq:permea}
\end{equation}}\unskip
where $\lambda_1$ is the primary interdendritic spacing which is set by
the morphological instability of the front of
solidification~\cite{tj1992}. The parameter $k_0$ is a constant
(usually taken equal to 5 for directional solidification) but depends
on the mean flow direction~\cite{tsosa2019}. The mushy layer mode is
unstable when the critical Rayleigh number is above $25$~\cite{tj1992}.
The effective Rayleigh number is limited by permeability, as vigorous
convection induces more crystallization which increases the solid
fraction and so decreases the permeability~\cite{habdll2016}. This
second mode of convection would usually lead to the formation of
chimneys as the iron-depleted liquid may dissolve iron dendrites in
regions of upwelling~\cite{w1992}. The formation of chimneys is well
documented by theoretical~\cite{w1997,wwo2013,rw2013,w1991} and
experimental works in
metallurgy~\cite{cgjh1970,sh1988,bfbs1997,mh1970}. Based on
observations in experiments and evidence from meteorites~\cite{eb1982},
Bergman et~al.~\cite{bfbs1997} suggest that the chimneys could be
several hundreds of meters large at the inner core boundary. But in the
presence of a magnetic field, the convection may take a different form
than the chimneys-mode~\cite{bf1994}. A strong enough magnetic field
might suppress chimney convection~\cite{bfb1999}, even if the mushy
layer is convectively unstable.

The particular conditions at the inner core boundary are different from
those in the canonical metallurgy literature, due to the simultaneous
presence of vigorous convection in the outer core and a very slow inner
core growth. Most of the experimental
studies~\cite{habdll2016,bmshca2005} on the mushy layer at the inner
core boundary have explored the effect of the convection in and above
the mushy layer on its internal structure (solid fraction, size of the
dendrites, or presence of chimneys). To investigate the effect of the
vigor of this convection, several experimental studies have been
performed by increasing or decreasing the Rayleigh number $R_m$ (see
Equation~(\ref{eq:Rayleigh_R}))~\cite{habdll2016,tj1992}. Tait and 
Jaupart~\cite{tj1992} used a viscosifier to decrease the Rayleigh
number. Huguet  et~al.~\cite{habdll2016} increased the apparent gravity
$g$ to increase the Rayleigh number $R_m$. 
Figure~\ref{fig:mush_height} shows the time evolution of the thickness
of a growing mushy layer. Since all other controlling parameters
(composition and heat flux) are similar between the experiments, the
final thickness of the mushy layer is a proxy of the solid fraction as
a function of the Rayleigh number. The solid fraction is significantly
increased by the vigorous convection of the liquid in and above it.
Extrapolated to the inner core conditions, Huguet 
et~al.~\cite{habdll2016} suggested that the inner core mushy layer has a
solid fraction very close to one, meaning that the remaining liquid is
very sparse.

At the inner core boundary, the heat flux extracted by the convection
induces a directional solidification of the mushy layer, which produces
a preferred orientation of the crystals~\cite{kfr2023,hh1962,c1964}.
Their orientation is determined by the crystal lattice, which depends
on pressure, temperature, and crystal composition. For example, iron
has a body-centered cubic lattice at ambient pressure (1 bar), but at
high pressure, the most stable lattice for iron is still debated,
between a hexagonal-closed package lattice~\cite{thot2010} and
body-centered cubic~\cite{vwgbdpa2008,blfzds2017}. Analog materials
commonly used in mushy layer studies, such as ammonium chloride have a
body-centered cubic lattice, and ice growing from salty water has a
hexagonal closed package lattice. However, Bergman 
et~al.~\cite{bcj2002} showed that in directional solidification
experiments with hexagonal close-packed crystals, the fluid motions
above the mushy layer force the crystals to have a preferred
orientation parallel to the flow  (Figure~\ref{fig:bergman_mush}). With
a laboratory model of the inner core growth, Bergman 
et~al.~\cite{bmshca2005} showed that the convective patterns in the
liquid induce a texturing of the mushy layer, which might explain the
structure of the inner core as seen by seismological
studies~\cite{wipt2023,d2014}. 

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig:bergman_mush}Transverse thin section of sea ice
crystals with a mean flow parallel to the front of solidification (a)
and with no mean flow (b). The white arrow in (a) shows the direction
of convective flow near the interface. Panels (a and b) adapted from
Bergman et~al.~\cite{bcj2002} with permission of John Wiley and Sons.}
\end{figure}

The typical length scale of the iron crystals (dendrites or equiaxed
crystals) is controlled by crystallization at the inner core boundary.
The primary spacing between dendrites (related to the grain width) is
proportional to the growth rate and the chemical and temperature
gradients~\cite{kfr2023}. Based on morphological
analysis~\cite{dab2007} and experimental studies~\cite{b1997,b1998},
the typical size of the dendrites in the Earth's core is thought to be
between $10^{-1}$ and $10^2$~m. Experiments with high apparent gravity
have also shown that the interdendritic spacing and grain size decrease
with the vigor of convection~\cite{habdll2016}. Besides, at the length
and time scales of laboratory
experiments~\cite{habdll2016,bmshca2005,bfbs1997}, the network of
dendrites is seen as a fixed matrix in the mushy layer. But in
planetary cores, dendrites can also compact under their weight on a
very long timescale~\cite{sykh1996}. Theoretical and numerical
works~\cite{lkc2020,sykh1996} show that the compaction of a mushy layer
will depend on the viscosity of the solid, the mush permeability
(related to the solid fraction), and the growth rate of the solid. At
the inner core conditions, Shimizu et~al.~\cite{sykh1996} predict that
the mushy layer will compact on the length scale of a few kilometers. 

In conclusion, the Earth's inner core mushy layer, if it exists at all,
is likely very thin and very depleted in liquid, with strong lateral
heterogeneity of crystal orientation as fluid flow and magnetic field
induce localized preferred orientation. The presence of an initial
supercooled layer, as suggested by the nucleation
paradox~\cite{hvhw2018}, may imply a phase of rapid growth that changes
the structure of the mushy layer. Therefore, the history of the
crystallization of the mushy layer is hidden in the layering of the
Earth's inner core~\cite{lkc2020}.

\subsection{Mushy zone at the top of planetary cores}\label{sec3.2}
In  Section~\ref{subsec:ICmushy}, we discussed the state of the mush
growing from the center of the Earth's core, but as we saw in 
Section~\ref{subsec:scenariosSolidification}, crystallization can occur
from the core-mantle boundary. While many studies argue in favor of the
formation of a slurry layer (snow layer) in this regime of
crystallization~\cite{had2006,rbs2015} (see 
Section~\ref{sec:slurryLayer}), other studies for small planetary cores
suggest that iron crystals grow from the core-mantle
boundary~\cite{ch2006,hhvj2018,sesb2016,nbn2019} 
(Figure~\ref{fig:topmush}).

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig:topmush}(a,b)~Schematic view of the destabilization
of a growing mushy layer (with a growth rate $V$) at the top of the
core of a small planet~\cite{hhvj2018,nbn2019}. (a)~Rayleigh--Taylor
instability of the iron layer. The wavelength of the instability is
about the thickness of the layer $h_c$~\cite{nbn2019}.  (b)~Necking of
independent dendrites. The typical dendrite size depends on the iron's
rheology~\cite{hhvj2018}. In (a) and (b) iron crystals fall into the
liquid core (with a falling velocity $w$) and remelt or accumulate at
the bottom. (c) Photos of an experiment using ammonium chloride aqueous
solution cooled from above~\cite{c2001}. A large chunk of the mushy
layer falls into the liquid and produces the crystallization of tiny
equiaxed crystals. A temperature probe is inserted into the middle of
the tank. Panel (c) adapted with permission from Chen~\cite{c2001}.} 
\end{figure}

Based on the observation of metallic meteorite samples, Haack and
Scott~\cite{hs1992} suggested that dendrite can inwardly grow attached
to the core-mantle boundary. Such a mushy layer may differ from the one
at the inner core boundary as composition, temperature, and pressure
conditions are quite different. The existence of a mushy layer at the
top of a core depends on the morphological stability of the interface.
Deguen~\cite{d2009} suggests that conditions at the core-mantle
boundary of the asteroid might be compatible with an unstable interface
and, therefore the formation of a mushy layer.

In a mushy layer composed of iron solid dendrite and a hypo-eutectic
liquid, growing from the top will lead to a stably stratified
interstitial liquid and unstable dendrites (as they are denser than the
surrounding liquid~\cite{habdll2016,nbn2019}). The absence of
convection and the slow growth rate in the mush induce a highly porous
mushy layer, in contrast to the highly compact mushy layer at the inner
core boundary~\cite{habdll2016}. Therefore, the mushy layer can be seen
as a solid interconnected matrix layer with viscosity depending on the
porosity~\cite{nbn2019}  (Figure~\ref{fig:topmush}a), or as independent
dendrites growing towards the center~\cite{hhvj2018} 
(Figure~\ref{fig:topmush}b). A mushy layer will be subjected to
Rayleigh--Taylor instability~\cite{nbn2019}. The length and time scales
of the delamination of the mushy layer depend on the rheology and the
solid fraction, which are controlled by the strain, rate, temperature,
and composition.

Experimental studies of unstable mushy layers~\cite{c2001,ante2020} 
have shown that large chunks of a mushy layer, with a typical size
ranging between the interdendritic spacing and the mush thickness, can
fall into the liquid  (Figure~\ref{fig:topmush}c). However, the
rheology of salt or ice dendrites is in the brittle regime. The
fragmentation of dendrites is often observed in experiments due to the
vigorous convection and strong strain rate applied to secondary or
tertiary arms of dendrites~\cite{jhus1966,ccr2004,skwl2018}. In
planetary cores, the rheology of iron at high pressure and temperature
is still poorly known~\cite{rt2020} and the strain rate is likely to be
small (for example, the typical convective velocity for the Earth's
core is of the order of mm/s).

On another end, independent dendrites attached to the core-mantle
boundary can be seen as slab necking under gravity during the
subduction process in the Earth's
mantle~\cite{rfbm2008,dsg2012,bsr2015}. The necking and delamination of
dendrites could produce a large range of crystal sizes falling into the
liquid core~\cite{hhvj2018,nbn2019}. Moreover, the fragmentation of
dendrites induces the formation of equiaxed crystals through the
production of secondary nucleation sites~\cite{jhus1966,ccr2004}. It is
also well-known that stronger convection will produce more fragments
resulting in a larger amount of equiaxed crystals in the
bulk~\cite{skwl2018}. 

Laboratory experiments and theoretical studies applied to the growth of
the top-down mushy layer are missing to determine which destabilization
mode dominates. One might suggest that the existence of an iron snow
might result from the growth of a thin mushy layer producing a high
quantity of small crystals falling into the liquid cores, as further
described in  Section~\ref{sec:slurryLayer}. The existence of such a
regime would solve the paradoxical existence of a large nucleation
barrier for homogeneous nucleation since primary dendrites would grow
from the core-mantle boundary and secondary nucleation would provide
sites for the iron snow~\cite{d1984}.

\subsection{Melting of a mushy layer at the inner core
boundary}\label{sec3.3}
Finally, Earth's seismology has highlighted the existence of the
transition region between the outer core and the inner
core~\cite{sp1991,ok2015}. This region, the so-called F-layer, is
characterized by the observation of slower P-waves than predicted by
the Preliminary Reference Earth Model (PREM)~\cite{da1981}, which
assumes a well-mixed outer core. Therefore, the F-layer is expected to
be a denser and stably stratified fluid layer. Its origin is still
debated but one explanation comes from the melting of the inner core.
Melting of the inner core would release a denser liquid enriched in
iron, sitting at the base of the outer core. Several studies have
examined mechanisms for such melting including a translation convection
mode of the inner core~\cite{adm2010,mcms2010} and a heterogeneous heat
flux at the inner core boundary~\cite{gsmr2011}. A model experiment was
set up in Alboussi\`ere et~al.~\cite{adm2010} to mimic the phase changes
induced by the translation of the inner core 
(Figure~\ref{fig:FlayerConv}). One hemisphere crystallizes and releases
light elements, while the other melts and releases dense liquid iron
with few light elements. This was modeled in a volume of salted water,
injecting fresh (light) water on the right side of the bottom surface
through a porous surface, while more salted water was injected on the
left side. Overall, the buoyancy flux of light elements exceeded
slightly that of heavy elements to account for the slow net
crystallization of the inner core. Then, a dense layer started to grow
from the bottom, providing a mechanism for the existence of the
F-layer. Light plumes could still cross that dense layer and drive a
convective flow throughout the whole volume. The corresponding flow
necessarily exists in the Earth's core, at least to drive the
geodynamo. 

\begin{figure}
\vspace*{2pt}
\includegraphics{fig08}
\vspace*{4pt}
\caption{\label{fig:FlayerConv}Convection experiment with light and
dense fluid released from the bottom and formation of an
F-layer~\cite{adm2010}. The shadowgraph on the left shows the dense
fluid (with purple dye) and light fluid rising through. Using a
synthetic Schlieren method, the vertical gradient (in the middle) and
the horizontal gradient (on the right) of the refractive index reveal
the dense layer and the convective plume, respectively. Figure adapted
with permission from Alboussi\`ere et~al.~\cite{adm2010}.}
\vspace*{6pt}
\end{figure}

Several experiments using a reactive porous medium or a mushy layer
have shown that the melting implies a change in the structure of the
matrix and the liquid
composition~\cite{habdll2016,hhw2004,hhw2005,ybha2015}. Moreover,
melting induces an enhanced crystallization in a deeper part of the
mushy layer~\cite{habdll2016,ybha2015} as heavy melt drives stronger
compositional convection and recrystallization 
(Figure~\ref{fig:melting_mush}). This leads to a higher solid fraction
inside the mush~\cite{habdll2016,ybha2015}, the formation of a dense
layer above the mush, and the disappearance of cones of
chimneys~\cite{habdll2016}  (Figure~\ref{fig:melting_mush}).

\begin{figure}
\includegraphics{fig09}
\vspace*{4pt}
\caption{\label{fig:melting_mush}Heating and melting from above of a
growing mushy layer~\cite{habdll2016}. The crystallization and
convection inside the mushy layer induce the formation of chimneys and
meandering plumes ($t= 260$~min). Note that double-diffusive convection
occurs in the liquid layer as seen in other experiments~\cite{c1997}.
Once heating and melting start, chimneys melt and mushy layer
convection becomes stronger, as evidenced by the turbulent plumes
($t=268$, 294, 320~min). Figure adapted with permission from Huguet
et~al.~\cite{habdll2016}.}
\vspace*{6pt}
\end{figure}

\section{Slurry layer}\label{sec:slurryLayer}
In  Section~\ref{sec:cry_planet}, we show that the crystallization of a
planetary core can lead to the formation of free crystals in a slurry
layer. Here, we examine the fluid dynamics and the phase change of
crystals in such a particle-laden environment. A key ingredient of the
slurry dynamics is the difference of density between the crystals and
the interstitial fluid. When crystals are sufficiently concentrated,
this density difference manifests as an effective buoyancy on the
ambient fluid, through which crystals may collectively force a
large-scale flow beyond the scale of a single crystal. Despite the
existence of studies on such flows in other geophysical contexts
(landslides~\cite{fmy2009}, turbidity currents~\cite{nhkm2002,oms2019},
snow avalanches~\cite{kms2018}), the planetary situation of crystals
settling in a bulk liquid is much different and fewer experiments have
been performed to model this phenomenon. In addition, crystal formation
and remelting are two complex ingredients that must be included to
model core evolution. With an incremental approach, the next 
subsections first describe the collective dynamics of inert particles 
(Section~\ref{subsec:collective}) before adding the additional
complexity of phase change  (Section~\ref{subsec:phasechange_slurry}).

\subsection{Collective motion of inert
particles}\label{subsec:collective}

\subsubsection{What numbers control the motion of
particles?}\label{sec4.1.1}
Suppose a suspension of particles settles in an infinite domain. The
motion of a single particle depends on its radius $r_p$, the local
gravity $g$, the particle density $\rho_p$, and finally the fluid
density $\rho_a$ and kinematic viscosity $\nu_a$. The motion of
multiple particles depends on their local volume fraction $\phi$, and
if the ambient fluid is moving, it also depends on the ambient fluid
velocity $U_a$. According to the Vaschy--Buckingham theorem, four
dimensionless numbers can be used to characterize the motion of a
suspension in a moving ambient: the density ratio $\rmPi_\rho \equiv
\rho_p/\rho_a$ quantifies the particles' inertia; the volume fraction
$\phi$ controls the collective behavior of particles and their forcing
on the fluid; the particle Reynolds number $Re_p \equiv 2 w_s r_p /
\nu_a$ characterizes the regime of the flow past a single particle. It
depends on the terminal settling velocity $w_s$ of a single particle,
which is usually implicitly defined as a function of $Re_p$. A
classical expression for $Re_p<800$ is~\cite{csst2011}:
{\begin{equation}\label{eq:wsSchillerNaumann}
w_s = \frac{w_s^{\mathrm{Stokes}}}{1+0.15Re_p^{0.687}},
\end{equation}}\unskip
where the Stokes velocity is defined as
{\begin{equation}\label{eq:wsStokes}
w_s^{\mathrm{Stokes}} = \frac{2gr_p^2(\rho_p-\rho_a)}{9\nu_a \rho_a}.
\end{equation}}\unskip
Finally, the Rouse number $\mathcal{R} \equiv w_s/U_a$ quantifies how
fast particles gravitationally drift with respect to ambient fluid
motions. This can be shown from the momentum equation of a single small
particle settling in a moving fluid: for a sufficiently small particle
with low inertia (see~\cite{kfl2023} for details), the buoyancy of this
particle balances its drag that is proportional to the difference
between the local fluid velocity $\mathbi{v}_a$ and the local particle
velocity $\mathbi{v}_p$. This readily gives $\mathbi{v}_p = \mathbi{v}_a - w_s
\mathbi{e}_r$ where $\mathbi{e}_r$ is the radial unit vector pointing outward
from the core center. Such a small particle follows the ambient flow
and constantly drifts in the direction of gravity with a velocity
$w_s$; the Rouse number quantifies the relative gravitational drift
between the particle and the local fluid motions with $|\mathbi{v}_p -
\mathbi{v}_a|/|\mathbi{v}_a| = w_s/|\mathbi{v}_a|=\mathcal{R}$. This 4D parameter
space is a reference to characterise the dynamics of inert monodisperse
particles with a locally uniform volume volume fraction $\phi$.
Additional ingredients of relevance for planetary interiors (e.g.\ 
heterogeneities of concentration, phase change, polydispersity\,\ldots),
that would largely expand the size of this parameter space, are
incrementally considered in the next  sections.

The difficulty in resolving the particulate dynamics in large-scale
turbulent geophysical flows has led many experimentalists to model
slurries as a fluid whose effective density (and viscosity) differs
from the ambient---for example using salt water instead of a dense
suspension of particles. Such an approximation corresponds to the
limits $Re_p \rightarrow 0$ and $\mathcal{R} \rightarrow 0$. We show
below some experimental works that revealed the need to refine the
description of the particles'\unskip\break
dynamics.

To gain understanding on the dynamics of settling-driven convection,
two different approaches can be found: some authors focus on the
fundamental building blocks that nourish this convection at a local
scale---continuous releases of buoyancy in \textit{particle-laden
plumes}, and instantaneous releases of buoyancy in \textit{particle
clouds} or particle-laden thermals---while other authors investigate
the slurry dynamics directly as a whole with a global approach.
Numerical simulations and models of core evolution usually adopt this
second vision and consider slurry layers like the snow zone as a
uniform region with quasi-steady dynamics (e.g.,~\cite{rbs2015}). Yet, 
Section~\ref{subsec:nucleation} suggests that the slurry dynamics is
neither steady nor uniform in the snow zone. Therefore, the motions
produced in the snow zone might be described as a sum of localized
releases of snowflakes in different locations and at different times.
How should these releases be parameterized in terms of crystal sizes,
concentrations, distance in space, and delay in time between two bursts
of crystals? Should these releases be instantaneous or have a finite
duration? In this regard, studies of local convective structures bring
some light on the dynamics that might be at play in slurry layers. Both
the local and global approaches are addressed respectively in 
Sections~\ref{subsubsec:localConvection}
and~\ref{subsubsec:globalConvection}. Since we are interested in
large-scale flows in liquid iron that has a low viscosity 
$10^{-6}~\mathrm{m}^2{\cdot}\mathrm{s}^{-1}$, these flows are likely
turbulent, which is why we restrict the description to such flows.

\subsubsection{Modelling convection with a local
approach}\label{subsubsec:localConvection}
Particle-laden plumes are often studied in a steady state with the
``filling box'' technique (see the illustration with a one-phase plume in
Figure~\ref{fig:particleLadenPlumes}a): from an inlet at the center of
the top wall of the tank, a first source feeds a negatively buoyant
plume with a volume flux $Q_p$; in the meantime, a second source feeds
a volume flux $Q_{\mathrm{in}}$ that nourishes a downward displacement
flow in the whole tank; finally, a sink lets a volume flux
$Q_{\mathrm{out}}$ spill from the tank, thus maintaining a constant
volume of fluid in the tank. This setup is particularly relevant to
quantify the plume entrainment which characterizes its coupling with
the ambient fluid. McConnochie et~al.~\cite{mcm2021} recently used this
technique with particle plumes. They revealed that particles modify the
entrainment rate of such steady plumes when they settle in opposite
directions to the initial volume flux, e.g., when downward-settling
particles rain out from an upward-oriented plume. Other experiments by
Zurner et~al.~\cite{ztdmm2023} focused more precisely on the coupling
between the particulate motion and the flow. In their experiments,
particles were continuously sieved from the top of an injection column
and formed a steady settling-driven plume---see
Figure~\ref{fig:particleLadenPlumes}b. By measuring both the local
fluid velocities through PIV and the velocity of each particle through
PTV, the authors quantified the energy transfer from the particles to
the fluid. Their results show that particles of low particle Reynolds
number are more efficient at transferring energy to the flow than
particles of large particle Reynolds number. This was interpreted as a
consequence of the difference in near-field flow around particles.

\begin{figure}
\vspace*{2pt}
\includegraphics{fig10}
\vspace*{4pt}
\caption{\label{fig:particleLadenPlumes}(a)~Steady-state reached in a
``filling box'' experiment with a dyed salty plume generated in fresh
water. (b)~Experimental setup used by Z\"urner 
et~al.~\cite{ztdmm2023} to generate a central settling-driven plume
from the injection column into the water tank. Panels (a) adapted from
Dadonau et~al.~\cite{dpl2020}, (b) adapted from  Z\"urner 
et~al.~\cite{ztdmm2023}, with permission of the Cambridge
University Press and the American Physical Society,\unskip\break 
respectively.}
\end{figure}

In both experiments~\cite{mcm2021,ztdmm2023}, the particulate nature of
the buoyancy forcing directly affects the particle plume, modifying its
coupling with the ambient and, if multiple plumes are driving
large-scale convection, with other particle plumes. It points to the
importance of the density and size of crystals in solidifying cores.

The transient settling of particle clouds reveals some additional
physics that depends on the cloud Reynolds number based on its radius
and bulk velocity. Several studies have focused on clouds of large
cloud Reynolds number
(e.g.,~\cite{rahimipourDynamicBehaviourParticle1992,ruggaberDynamicsSedimentClouds2000,
bushParticleCloudsHomogeneous2003,zhaoEffectAirRelease2012,laiTwophaseModelingSediment2013,
landeauExperimentsFragmentationBuoyant2014,penasBubbleladenThermalsSupersaturated2021,ksfl2022}).
They are typically generated in the lab by releasing some buoyant
material from the inlet of an injector that is located above
water~\cite{zhaoEffectAirRelease2012}, exactly at the free
surface~\cite{ruggaberDynamicsSedimentClouds2000,bushParticleCloudsHomogeneous2003,
landeauExperimentsFragmentationBuoyant2014} 
or
submerged~\cite{ruggaberDynamicsSedimentClouds2000,laiTwophaseModelingSediment2013,
landeauExperimentsFragmentationBuoyant2014,ksfl2022}.
In most experiments, the injector is initially sealed by a membrane
that keeps the buoyant material until it is ruptured (see 
Figure~\ref{fig:MontagesRhoda}a). After rupturing the membrane, the
buoyant mixture rolls up and a large-scale turbulent cloud forms in
which the particles swirl as long as the cloud falls faster than the
settling speed of an individual crystal (see snapshots one and two in 
Figure~\ref{fig:MontagesRhoda}b); as the cloud dilutes, it decelerates
and particles rain out of the cloud when the latter is as slow as their
settling speed~\cite{deguenExperimentsTurbulentMetalsilicate2011} (see
the third snapshot in  Figure~\ref{fig:MontagesRhoda}b); after this
separation, particles settle almost vertically with a constant velocity
(see the fourth snapshot in  Figure~\ref{fig:MontagesRhoda}b). The
larger the Rouse number, the earlier the separation (compare 
Figures~\ref{fig:MontagesRhoda}b and~\ref{fig:MontagesRhoda}c). It
means that a given buoyancy anomaly released from a fixed initial cloud
size gradually loses its ability to produce large-scale fluid motions
as the particle size is increased, due to the decoupling between eddies
and the particles that gravitationally drift with respect to the 
flow---in other words, the flow experiences transitions that are bound to
the particulate nature of the buoyant material. The entrainment of
particle clouds is also affected by the particles' properties:
in~\cite{ksfl2022} the authors showed that particle clouds have a
maximum growth rate and hence dilute the fastest for a finite particle
size, 75\% faster than first-order estimates that disregard the
particulate nature of the buoyancy\unskip\break 
forcing.

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig:MontagesRhoda}(a)~Experimental setup used in Kriaa
et~al.~\cite{ksfl2022}. A cylinder partly immersed below the free
surface is initially sealed by a latex membrane; after rupturing the
membrane with a needle, glass spheres settle in fresh water, reflect
the green light from the vertical laser sheet, and their motion is
recorded by the camera with the green filter. The orange coloring
(rhodamine B) is initially present with particles in the cylinder; as
particles fall, this dye is tracked by the camera with an orange
filter. The influence of background rotation can be studied since the
setup is entirely mounted on a rotating table. (b,c)~Snapshots show, in
the absence of background rotation, the gradual separation between
particles (in white) and the released dye (in orange), all the faster
as the Rouse number $\mathcal{R}$ increases from (b) to (c). Time
intervals $\rmDelta t$ between snapshots are (b) $\rmDelta t=3.0s$ and (c)
$\rmDelta t=1.2s$. Panels (a) and (b) adapted from Kriaa 
et~al.~\cite{ksfl2022} with the permission of the American 
Physical\unskip\break Society.}
\end{figure}

To sum up, these experiments with plumes and clouds show that the
particle size, through $Re_p$, $\mathcal{R}$ and $\phi$, affects the
efficiency of the collective drag, the decoupling of particles from
fluid motions, and the entrainment capacity of these local flows.
Therefore, it controls the ability of the slurry to generate a
large-scale flow, with consequences for dynamo generation. The next 
section confirms this conclusion and addresses another key question:
when no length scale nor velocity scale is imposed by plumes and
particle clouds, how does a slurry layer evolve and what is its spatial
structure?

\subsubsection{A global approach of convection}\label{subsubsec:globalConvection}
Besides the local analysis of convection through its building blocks,
some experiments directly analyze the behavior of a whole layer of
particles settling in an initially quiescent fluid. The simplest
approach is to consider this layer uniform, settling above another
layer of clear lighter fluid. This configuration has been investigated
in experiments to analyze the onset of the fluid motions at the
interface between the two layers through a Rayleigh--Taylor-like
instability~\cite{boffettaIncompressibleRayleighTaylor2017,
careyInfluenceConvectiveSedimentation1997,
friesInfluenceParticleConcentration2021}, 
typically after sliding a gate open. For example, Fries 
et~al.~\cite{friesInfluenceParticleConcentration2021} let a mixture
of water and particles settle above slightly denser sugary water, with
a motivation to study the settling and deposition of ash from warm
volcanic clouds. Other authors let dense particles settle in the air
down a settling column until they penetrate in a tank of
constant~\cite{careyInfluenceConvectiveSedimentation1997,
mizukamiParticlegeneratedTurbulenceHomogeneous1992} or piecewise
linear~\cite{manvilleVerticalDensityCurrents2004} density profile, and
they analyze the particle instabilities that develop near the free
surface. Some studies in a Hele-Shaw cell (see 
Figure~\ref{fig:harada} extracted from Harada 
et~al.~\cite{haradaParticlelikeFluidlikeSettling2012}) or in numerical
simulations~\cite{yamamotoNumericalSimulationConcentration2015} have
shown that particles behave collectively when the interparticle
distance of a uniform suspension, proportional to $r_p \phi^{-1/3}$, is
sufficiently small for the hydrodynamic perturbations of neighboring
particles to overlap, enabling them to collectively drag the
interstitial fluid. A paramount consequence is that the collective
settling of particles considerably accelerates their fall compared to
the individual settling velocity of isolated particles.

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig:harada}Snapshots of experiments where a suspension
of white particles settles above the clear fluid in a narrow cell. From
left to right the inter-particle distance $\lambda_p$ increases (it is
respectively equal to 97~$\rmmu$m, 1200~$\rmmu$m and  3700~$\rmmu$m),
leading to a transition from a collective fluid-like behavior
(particles undergo a Rayleigh--Taylor-like fluid instability) to
individual particle-like behavior (particles essentially settle
vertically). This transition to individual behavior is reflected by the
ratio of the inter-particle distance $\lambda_p$ over the expected
width of the Rayleigh--Taylor mushrooms $l_{\mathrm{RT}}$, which equals
$5.24\times 10^{-3}$, $6.42\times 10^{-2}$ and $1.34\times 10^{-1}$
from left to right. Figure extracted from Harada 
et~al.~\cite{haradaParticlelikeFluidlikeSettling2012} with permission
from Springer Nature.}
\end{figure}

These previous studies mainly focus on the transient formation of the
Rayleigh--Taylor-like instability and essentially analyze the
propagation of the front of particles without analyzing the fluid
motions that ensue at larger times. If crystals continuously settle in
a snow zone, what would be the dynamics of this layer over timescales
larger than necessary for crystals to cross through the whole layer?
Insight can be gained from experiments of bubble-driven convection:
some bubbles are continuously injected at the bottom of a water tank,
over the entire cross-section of the tank. Several of these studies
focused on the formation of rolls in 2D tanks having a narrow gap
between two of their side walls, as observed in experiments of
Kimura~\cite{kimuraCellFormationConvective1988} 
(Figure~\ref{fig:bubbleDrivenFlows}a) and in numerical simulations by
Climent and
Magnaudet~\cite{climentLargeScaleSimulationsBubbleInduced1999} 
(Figure~\ref{fig:bubbleDrivenFlows}b). The experiments showed that the
rate of injection of bubbles affects the aspect ratio of these rolls,
while the size of bubbles controls their ability to couple with fluid
motions and cross through the cells when they are sufficiently
large~\cite{kimuraCellFormationConvective1988}. However, such
two-dimensional studies have evidenced very different behaviors
compared to observations in 3-D flows. Different regimes exist
depending on the bubble injection
rate~\cite{mezuiBuoyancydrivenBubblyFlows2022}, and while 2D bubble
columns tend to show well-structured stacks of rolls, experiments in 3D
columns are more chaotic~\cite{muddeGravityDrivenBubblyFlows2005}.
Experiments of Iga and Kimura~\cite{igaConvectionDrivenCollective2007}
with a uniform flux of bubbles injected at the bottom of a water tank
with a $24~\mathrm{cm} \times 24~\mathrm{cm}$ cross-section have shown
that the bubble-driven convection leads to irregular 3D patterns that
do not reach a steady state, as shown in 
Figure~\ref{fig:bubbleDrivenFlows}c.

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig:bubbleDrivenFlows}(a)~Steady convective cells
produced in a narrow 2D water tank by a constant injection of bubbles.
The whiter the regions are the more concentrated in bubbles. The
injection rate of bubbles increases from the first to the third row.
(b)~Similar observations in numerical simulations: black dots denote
the bubble on the first row, while the second row shows the velocity
field. Results were obtained for an aspect ratio $L/H=4$ and a Rayleigh
number of $2.07\times 10^5$. (c)~Sketches of the evolution of a 3D
bubble-laden as seen from above, with the time origin corresponding to
the start of bubble injection. The dark regions are laden with bubbles.
Figure (a) and Figure (c) are respectively adapted from
Kimura~\cite{kimuraCellFormationConvective1988} and Iga and
Kimura~\cite{igaConvectionDrivenCollective2007}
\tsup{\text{\textcopyright}}The Japan Society of Fluid Mechanics and
IOP Publishing Ltd and reproduced by permission of IOP Publishing. All
rights reserved. Figure (b) is adapted from Climent 
et~al.~\cite{climentLargeScaleSimulationsBubbleInduced1999} with the
permission of the American Physical Society.}
\end{figure}

Although bubbles differ from solid crystals by their ability to deform
and the possible existence of a slip velocity at their interface, such
bubble-driven flows provide valuable information. They support the idea
that the motion of crystals in the slurry layer has little chance of
being uniform and steady, even when disregarding issues due to bulk
crystallization.

These studies confirm the order-of-magnitude impact that collectivity
can have on the timescale of crystal segregation in the core and on the
estimates of convective velocities that may or may not drive a dynamo.
Unfortunately, collective effects are still poorly understood,
difficult to investigate experimentally, and challenging to model
numerically. Further experiments are required to gain more
understanding of these aspects, especially in the context of a global
settling-driven convection of solid particles over large timescales,
for which the parameter space ($\rmPi_\rho$, $\phi$, $Re_p$ and
$\mathcal{R}$) remains largely unexplored.

\subsection{Influence of phase change in slurry
layers}\label{subsec:phasechange_slurry}

\subsubsection{Crystallization}\label{subsec:bulkCrystallisationXP}
Slurry layers involve the crystallization of free crystals when rising
or falling. In metallurgy, the formation of equiaxed crystals above the
front of solidification is associated with the convective motion of the
fluid~\cite{bw1996}. Experiments of equiaxed crystal solidification
(with analog material) have shown the complexity of the multiphase flow
associated with crystal growth, as a slurry layer encounters processes
of nucleation, fragmentation, sedimentation, and convective fluid
motion~\cite{bw1995,bw1996,kslw2013}. These experiments~\cite{bw1996}
and early numerical simulations~\cite{bw1995,rdb2003,wl2009} have shown
that the microscale dynamics (nucleation, grain growth, sedimentation)
cannot be disregarded for an understanding of the macroscale
segregation in industrial casting~\cite{b2002b}. These dynamics can be
modeled only at a small scale or with a few crystals. In the most
recent numerical studies, the growth of equiaxed crystals with fluid
motions can be modeled using the phase-field method and lattice
Boltzmann methods in small computational domains, but with the presence
of up to a few hundreds of crystals~\cite{stosa2020,yst2021,t2023}.
Other experiments with equiaxed crystals have investigated the growth
rate of individual falling crystals as a function of the surrounding
constitutional supercooling~\cite{aacl1999,bcb2007}, or the production
rate of crystals through fragmentation of dendrites via convective
motion~\cite{ccr2004,hll1997,lskw2017}. However, despite the existence
of recent techniques to track particles in multiphase
flows~\cite{ante2020,anse2019}, the fluid motions that naturally emerge
in slurry layers make it challenging to perform quantitative
measurements, especially in large fluid domains. In planetary cores,
the formation of a slurry layer (the Earth's F-layer or a snow layer in
small cores) is thought to be accompanied by a stably stratified layer
due to the release of light elements upon crystal growth and the
sinking of dense crystal~\cite{had2006,wdj2023,wdj2018}. Following the
seminal studies done by Loper and Roberts~\cite{lr1977} and
Loper~\cite{l1992}, theoretical one-dimensional models have been
developed~\cite{wdj2023,wdj2018} for the Earth's F-layer. They showed
that the bulk crystallization of iron crystals and their sedimentation
are compatible with the existence of a stratified layer at the bottom
of the Earth's core. Numerical models of the thermal evolution of
Ganymede or Mars core in a regime of iron snow led to the same
conclusion~\cite{dp2018,rbs2018}. However, even with several
assumptions made (one-dimensional and two-phase flow, Boussinesq
approximation), the most advanced model for such a slurry layer
requires at least 8 equations and 14 input parameters~\cite{wdj2023}.

Only a few experiments have been developed to investigate the dynamic
of crystallization of a slurry layer in the configuration of iron
snow~\cite{hld2023,ante2020}. Huguet  et~al.~\cite{hld2023} recently
investigated the solidification of free ice crystals, their
sedimentation, and their melting in a hotter layer 
(Figure~\ref{fig:snow}). A cyclic pattern was observed of alternating
bursts of crystallization and quiet periods with no crystallization
(Figure~\ref{fig:snow}a). The bursts of crystallization were due to the
formation of large numbers of crystals in a supercooled layer by
fragmentation of dendrites. The ice crystals then melted as they rose
in hotter regions of the tank. Quiet periods were due to the inhibition
of crystallization as the supercooling was reduced in the tank, caused
by the release of latent heat of solidification, and the convection
induced by the motions of crystals. These experiments revealed the wide
range of crystal sizes formed upon solidification in the slurry layer,
as shown in Figure~\ref{fig:snow}b. Crystallization in slurry is
thought to produce a polydisperse distribution of crystals, which will
have consequences on their fall and melting in the iron snow\unskip\break 
regime. 

\begin{figure}
\includegraphics{fig14}
\caption{\label{fig:snow}Experimental study of the homogeneous
crystallization of a tank of water cooled from below. (a)~Shows a
space-time diagram from a vertical cross-section in the middle of the
tank: darker regions correspond to ice. The dynamics exhibit clear
alternative behaviors with periods of intense upward snow followed by
quiet periods. The black dashed line denotes the maximum height of
rising ice crystal i.e.\ the height of the liquidus. (b)~Shows the
measured probability distribution function of rising snow particle
radius, highlighting the large distribution. Panels (a,b) adapted with
permission from Huguet et~al.~\cite{hld2023}.}
\end{figure}

\subsubsection{Crystals remelting}\label{sec4.2.2}
According to the simplistic vision of iron snow previously introduced
in  Section~\ref{subsec:scenariosSolidification}, the bottom of the
snow zone is delimited by the location where the local temperature is
equal to the liquidus temperature; below this position, the temperature
is larger than the liquidus and snowflakes remelt. Models of the
thermal evolution of small Earth-like planets have time steps of order
$10^6$~yrs~\cite{rbs2015,rbs2018}, which is insufficient to resolve
settling and remelting of snowflakes. Therefore, these processes are
considered instantaneous, and all snowflakes are assumed to remelt
right at the lower edge of the snow zone. In practice, particles have a
finite melt rate, and they travel a finite distance before disappearing
there exists a ``\textit{remelting layer}'' between the snow zone and the
zone of compositional convection.

The remelting layer is usually disregarded. Therefore, models of
compositional convection in the deeper core are fully liquid models in
which the whole buoyancy is of a fluid nature. As shown in 
Figure~\ref{fig:olsonExperiments}a, Olson 
et~al.~\cite{olsonLaboratoryExperimentsRaindriven2017} designed an
experiment where drops of aqueous solution rained out from a shower
head into quiescent air, penetrated through a first layer of silicone
oil in which the drops settled---the analog of the snow zone---, and
finally penetrated through a second layer of pure water---the analog
of the convective zone---where convection developed depending on the
buoyancy flux of raindrops, see  Figure~\ref{fig:olsonExperiments}b. By
relating the Rayleigh number which quantifies the convective forcing,
to the Reynolds number based on the average velocity of plumes
propagating down the second convective layer, the authors showed that
this plume velocity verifies an inertial scaling that is independent of
viscosity and diffusivity. This observation led Olson 
et~al.~\cite{olsonLaboratoryExperimentsRaindriven2017} to use
diffusion-free dynamo scalings to analyze whether a dynamo could be
sustained by precipitation-driven convection on Earth, Mercury, or
Ganymede; the details for each case can be found
in~\cite{olsonLaboratoryExperimentsRaindriven2017}. 

\begin{figure}
\includegraphics{fig15}
\caption{\label{fig:olsonExperiments}(a)~Sketch of the setup used by
Olson  et~al.~\cite{olsonLaboratoryExperimentsRaindriven2017} to model
precipitation-driven dynamos. The light grey top layer contains
silicone oil and typically stands for the snow zone; the white bottom
layer contains fresh water and stands for the convective zone. The dyed
aqueous drops that are continuously injected represent settling
crystals. (b)~Time series of the compositional convection that develops
in the lower convective layer. The width of every snapshot is 18~cm. We
thank Olson et~al.~for providing panels (a) and (b) which correspond to
the experiments used
in~\cite{olsonLaboratoryExperimentsRaindriven2017}.}
\end{figure}

Few studies include phase change and particle settling in a large-scale
flow. Large-scale studies consider either a fully fluid convection or a
settling-driven convection of inert particles. From the physicist and
mathematician's viewpoints, melting and dissolving are
equivalent~\cite{wellsMeltingDissolvingVertical2011,gagliardiThinFilmModeling2018,favierRayleighBenardConvection2019}, 
so information can be found in the literature on individual particles
melting or dissolving as they settle. Most studies have tried to
measure and predict the trajectory and shrinking of a single solid
sphere falling and dissolving in a homogeneous
ambient~\cite{riceCompleteDissolutionSpherical1979,riceTranspirationEffectsSolids1982,elperinMassHeatTransfer2005,
riceDissolutionSolidSphere2006}. 
Recently, Huguet et~al.~\cite{hbl2020} studied a settling sphere of
$Re_p \sim 10^3$--$10^4$. They showed that when the particle melts, its
trajectory has a lower amplitude of oscillation and a drag coefficient
almost twice larger than when it is inert. Aligning with the work
of~\cite{zmm2019}, Huguet et~al.~\cite{hbl2020} conjecture that
is it due to enhanced mixing in the sphere's wake due to the melt's
buoyancy. These observations are important for the timescale of crystal
segregation in the core; future experiments could clarify the role of
the melt liquid by using other materials whose melt liquid is neutrally
buoyant, as done in~\cite{heSedimentationSingleSoluble2023}. The
experiments in~\cite{hbl2020} also revealed that when a melting sphere
falls in a stratified ambient, it excites internal waves that contain 
${\sim}$1\% of the initial potential energy of the sphere for a
Brunt-V\"ais\"al\"a frequency $N\sim 0.1\ndash0.2$~Hz. How the waves of
several crystals might combine in the snow zone, and how they would
couple with the crystals' motion remain open questions that require
further investigation.

Recent experiments were conducted that include the collective aspect of
the particles' dynamics while they dissolve (see Chapter 4
in~\cite{kriaaFlowsInducedSettling2023}). To do so, homemade dyed sugar
grains of controlled size were sieved with a constant mass rate over a
cylindrical water tank. A plume of dissolving sugar grains formed over
a wide surface area, and a vertical laser sheet enabled to visualize
the flow in the midplane. The sugar grains were visible in the laser
sheet because they reflected the laser light. In addition, the sugar
was cooked like caramel and doped with Rhodamine B, hence the wakes of
dissolved sugar were visible in the flow (see 
Figure~\ref{fig:SugarExperiments}a,b). By
aligning with the usual assumptions of models of core thermal 
evolution---a uniform and constant mass flux of monodisperse particles is fed
through the bottom of the snow zone (in experiments, through the
water-free surface)---these experiments revealed the importance of the
remelting layer and how the size of grains hugely affects its dynamics.
First, for an identical mass rate, varying the diameter of sugar grains
from 45$~\rmmu$m to  363$~\rmmu$m controls a flow transition, from a
laminar plume that gains intensity over a long transient when forced by
the rectilinear fall of large grains 
(Figure~\ref{fig:SugarExperiments}a), to a turbulent lazy plume that
emerges faster when forced by fast-dissolving small grains 
(Figure~\ref{fig:SugarExperiments}c). This flow transition is
determined by the enhanced forcing imposed by smaller grains, and their
ability to nourish a Rayleigh--Taylor-like instability at the plume
onset---both because smaller grains behave more collectively and
hardly decouple from fluid motions. Second, the plume velocity
explicitly depends on the size of grains through their settling
velocity which controls how dilute the grains are in the plume. For
these two reasons, the size of grains imprints a lasting signature on
the plume, even after full dissolution/melting. Third, the thickness of
the remelting layer depends both on the mass flux imposed and on the
size of grains (for a mass rate of 0.1~g/s, grains of radius
$r_p=45~\rmmu\mathrm{m}$ dissolve over a few centimeters whereas grains
of radius $r_p=363~\rmmu\mathrm{m}$ dissolve deeper than the tank
height of 38~cm). Since the mass flux of crystals depends on the
dynamics of the snow zone that feeds the melting layer with crystals,
it reinforces the need to accurately model the fluid motions and the
crystal sedimentation in the snow zone.

\begin{figure}
\includegraphics{fig16}
\caption{\label{fig:SugarExperiments}Snapshots of sugar plumes of
different sugar radii and similar mass rates (see above snapshots). The
vertical extent of each snapshot is (a)~38~cm, (b)~23~cm, (c)~38~cm.
Snapshots of the experiments presented in Chapter 4 of 
Kriaa~\cite{kriaaFlowsInducedSettling2023}.}
\end{figure}

If crystals are sufficiently large, they may get deep in the core
before remelting. As long as they remain large enough for their
settling velocity to largely exceed the surrounding flow velocity
($\mathcal{R} \gg 1$), crystals are expected to settle radially inward
irrespective of the ambient fluid motions and to inject their buoyancy
deep in the core. Only when crystals are sufficiently small can they be
entrained in ambient flows? Here again, constraints on the size of
crystals are paramount to better model the core evolution and where
compositional convection might take place. We mentioned in 
Section~\ref{subsec:bulkCrystallisationXP} that polydisperse
distributions are expected; it remains an open question to know how
polydisperse experiments behave compared to these experiments with
monodisperse distributions. Crystals of different sizes settle with
different velocities; therefore, they might collide, form clusters, and
shield one another from the ambient fluid (see such effects
in~\cite{k1995}), thus delaying phase change, as already observed in
the monodisperse experiments. 

Such idealized experiments show that many open questions remain, even
before addressing the geophysically relevant effects of compressibility
and magnetism. Such effects will be difficult to tackle experimentally.
Yet, to account for the very complex physics involved in
phase-change-driven convection, including out-of-equilibrium
thermodynamics and crystal size distribution that are inaccessible to
Direct Numerical Simulations, laboratory experiments are, at the very
least, necessary to test parameterization schemes and to benchmark
numerical models that will then address planetary mysteries.

\section{Implications and limitations}\label{sec:conclusion}

\subsection{Consequences for observations}\label{sec5.1}
A better understanding and modelling of phase-change driven flows in
planetary cores is especially relevant in the current context of new
observations from recent (e.g., Bepi Colombo) and upcoming (e.g.,
Juice, Europa Clipper, Psyche) space missions, which will help to
unravel the evolution of planetary cores. The dynamics of internal
flows will remain inaccessible to observations on many planets, at
least within the foreseeable future. The challenge of lab experiments
on solidification and melting is therefore to provide large-scale
signatures that are closely linked to the microscale physics at play in
the core, so as to constrain the core dynamics from planetary-scale
observations.

\subsubsection{Dynamo and magnetic observations}\label{sec5.1.1}
A better understanding of crystal migration could help guide future
magnetic observations. The crystals' size affects how they force the
flow, decouple from it, and how deep they remelt, thus affecting the
characteristic flow velocity $U$ and the size of the convective zone
$L$. If they are sufficiently large, the magnetic Reynolds number $Re_m
= UL/\eta_m$ can exceed the minimum value required for dynamo action.
The more turbulent the flow, the more high-order multipoles can be
excited and therefore the more multipolar the magnetic field might be.

However, magnetic observations at the surface of a planet do not only
depend on the inner core dynamics. For the same $Re_m$, the magnetic
multipoles produced by a deep convective zone (scenario of top-down
freezing) decay over a large distance before being measured at the
planet's surface, favoring the predominance of the magnetic dipole;
conversely, the magnetic field produced by a convective zone near the
core-periphery decays over a smaller distance and is prone to show
higher-order multipoles~\cite{rbs2018}. In addition, solidification can
lead to stratification, as expected in Ganymede where top-down freezing
leads to the rejection of a sulfur-rich residual liquid rising to the
core's top~\cite{rbs2015,rbs2018}. Such a stratification filters the
magnetic field, reducing higher-order multipoles relative to the
magnetic dipole~\cite{c2015}. So far, the existence of such a
stratification has been deduced from thermodynamical arguments (see
e.g.,~\cite{rbs2018}). Aspects of fluid dynamics remain to be
investigated, like the possible erosion of the stratification by the
underlying convection, the two-way coupling between the stratification
and the flow produced by the settling crystals, and the transport of
crystals by inertio-gravity as observed in other astrophysical contexts
in astrophysical fluid
dynamics~\cite{magnanPhysicalPictureAcoustic2024}.

\subsubsection{Seismic observation of the Earth's inner
core}\label{sec5.1.2}
Seismological studies of the Earth's inner core have revealed large and
small-scale heterogeneities of the seismic properties (attenuation,
anisotropy, or wave velocity), which are thought to be consequences of
the evolution of the inner core and then of its crystallization
pathway~\cite{wipt2023,d2014}. Radial and regional variations of
anisotropy are related to the principal orientation of the crystals and
their intrinsic properties. Several studies modeling traveling waves in
the complex medium have attempted to understand the seismic
observations~\cite{c2007,ca2013,lcdm2016} and their link to the history
of solidification~\cite{ca2013}.

As shown in the  Section~\ref{sec:mushylayer}, the structure of the
mushy layer originates from the dynamic of the crystallization (growth
rate, crystal alignment, chimneys, remelting). Beyond experimental
works of crystallization, several experimental studies have attempted
to correlate seismic observations to crystallization and dynamics of
the Earth's inner core~\cite{habdll2016,bghi2000,beo2002}, using
ultrasonic waves as analog to seismic waves. Solidification texturing
of the mushy layer~\cite{b1997} can cause the apparent elastic
anisotropy~\cite{bghi2000} as crystal elongation aligns with heat flux.
A model has been proposed involving convective translation of the inner
core~\cite{adm2010}: this would produce crystallization in one
hemisphere and melting in the other. Huguet  et~al.~\cite{habdll2016}
have shown that the melting of a mushy layer lowers its attenuation and
have argued that the western hemisphere of the inner core melts as it
has a weak attenuation. The translation, as such, does not explain the
anisotropy nor the innermost inner core. But pre-existing
texture~\cite{beo2002} or post-solidification
deformation~\cite{sykh1996,d2012} of the solid mushy layer might alter
the solidification texturing, which can perhaps explain anisotropy
variation in the inner core or a different structure in the innermost
part of the inner core. With the increase in the number and quality of
seismic observations, the relation between the mushy layer structure
and its seismic properties will deserve more investigation, guided by
experimental results to identify the evolution of the Earth's inner
core. 

\subsection{Additional questions to address}\label{sec5.2}
On a very long timescale, a growing mushy layer might deform under its
weight. Few theoretical and numerical studies with 1D models of
compaction~\cite{lkc2020,sykh1996} have shown that the length-scale of
compaction is relatively small compared to the size of the inner core
and that a significant amount of liquid in a compacted mushy layer can
be trapped. However, the complexity of extending these models to 2D or
3D renders an experimental approach highly valuable. Compaction in
porous medium has been extensively studied through experiments and
numerical simulations~\cite{ggsve2015,dwzz1994,bblsswz2017}, but not in
regard to a mushy layer growing at the inner core boundary. The next
set of experiments may investigate the compaction using ``soft''
material like paraffin or lead and applying external compression force
or high apparent gravity. We have already seen how convection in a mush
increases its solid fraction, without deformation of the solid
structure. It would be interesting to see in experiments the second
phase of compaction, where the solid fraction increases further because
of solid deformation. 

Experiments on the crystallization of a core have been performed in
many configurations and with different materials (ice, analog salt
solution, metals). However, only a few studies have been performed in
the configuration of a snow layer~\cite{hld2023,ante2020} or of a
top-down mushy layer~\cite{c2001,ante2020}. Future experiments would
have to focus on the stability of a top-down mushy layer and its
interaction with the top cooling boundary. In the slurry layer, the
processes of crystal growth and fragmentation can be investigated
experimentally to predict the size distribution of iron crystals in the
snow layer, which is until now unknown. Indeed, a slurry layer has
little chance of being monodisperse: if crystals form anywhere in the
bulk, upward or downward settling enables crystals of many sizes to
meet and interact in the core; if large pieces of solid iron are
delaminated from the core-mantle boundary, they might trigger
nucleation in a supercooled ambient, especially if their detachment
results in the dispersal of smaller fragments. As we have seen, the
crystal size has a drastic influence on convection. Hence, although
experiments with monodisperse particle distributions are a necessary
step to fully comprehend the physics at play, it is unclear how their
results would be affected when moving on to polydisperse distributions.
Disentangling all the effects that are coupled might require keeping
track of the space-and-time evolution of the size distribution in the
flow.  Two types of experimental hurdles must be overcome. First,
surface tension plays a much stronger role in experiments compared to
planetary scales. Secondly, deforming solid materials in experiments is
difficult, as the available stress is limited by the size of the setup.

The dynamical consequences of planetary rotation remain largely
unexplored in experiments. Some studies have considered its influence
on miscible
plumes~\cite{fernandoDevelopmentPointPlume1998,
goodmanHydrothermalPlumeDynamics2004,frankAnticyclonicPrecessionPlume2017,
sutherlandPlumesRotatingFluid2021},
bubble-laden plumes (e.g.,~\cite{frankEffectsBackgroundRotation2021})
or particle
clouds~\cite{ksfl2022,ayotteMotionTurbulentThermal1994,
helfrichThermalsBackgroundRotation1994,
kriaaInfluencePlanetaryRotation2024}, 
always in the configuration where the rotation axis and gravity are
aligned hence corresponding to flows inside the tangent cylinder. While
rotation tends to constrain flow structures to align with and be
invariant along the rotation axis, the buoyancy force tends to
constrain radial motions. How these constraints compete with one
another when gravity and rotation are not aligned, and how the
spherical geometry of the core shapes convection, remain two
challenging questions for experimentalists. 

The influence of magnetic field on the two-phase flow in mushy or
slurry layer has only been sparsely investigated by experiments and
theoretical work~\cite{bfb1999,bm2020}. Experimental studies usually
require metallic phases, which are opaque, making two-phase flow
tracking very difficult. Magnetohydrodynamics can be also
experimentally studied with a transparent material such as a
water-electrolyte solution and an extremely intense magnetic
field~\cite{apbds2016,apsd2018}. One might think the solidification of
such a water-electrolyte solution makes it possible to have direct
visualization of two-phase flows affected by a magnetic field.

\subsection{Similar questions in other fields}\label{sec5.3}
Many more communities share common challenges with core dynamicists,
starting with geoscientists studying the cooling of magma reservoirs.
Crystallizing magmatic flows are prodigiously complex. Crystals can
form in the bulk or on the
walls~\cite{turnerConvectionMixingMagma1986,sparksFluidDynamicsEvolving1997} 
of, for example, a magma chamber---a competition that is challenging
to model and unravel from geological
samples~\cite{holnessCrystalSettlingConvection2017,
holnessMagmaChambersMush2019}.
Since crystal growth injects latent heat, changes the local fluid
composition, and modifies its rheology, the fluid properties and
buoyancy hugely depend on the distribution and dynamics of
crystals~\cite{sparksFluidDynamicsEvolving1997,
holnessMagmaChambersMush2019,namurCrystallizationInterstitialLiquid2014}.
The resulting convection is reflected in the zonation of rocks on the
walls and the alignment of crystals with the
flow~\cite{holnessCrystalSettlingConvection2017,holnessMagmaChambersMush2019},
which depend on the exact convective patterns and chamber
geometry~\cite{turnerConvectionMixingMagma1986}. Yet, even from the
sole point of view of microphysical processes, the interpretation of
geological samples is complex: the porous crystals can temporarily
retain liquid or move in a saturated
environment~\cite{sparksFluidDynamicsEvolving1997,
namurCrystallizationInterstitialLiquid2014}, 
which affects their phase change and possibly their density; with
polydispersity~\cite{kleinInfluenceCrystalSize2017} and
aggregation~\cite{holnessCrystalSettlingConvection2017}, all these
ingredients modify crystal migration, with direct feedback on magma
heterogeneity and\unskip\break 
rheology.

In watercourses, basins, or coasts of cold regions
(e.g.,~\cite{d1984,moralesmaquedaPolynyaDynamicsReview2004,
martinFrazilIceRivers1981}), water can be supercooled, leading to
the formation of sub-millimetric to millimetric ice crystals called
\textit{frazil ice} in turbulent streams. In this context as well, the
large number of crystals forming in the turbulent flow cannot be
resolved numerically, which is why parameterizations
(e.g.,~\cite{matsumuraLagrangianModellingFrazil2015,rw2018}) or
continuum models
(e.g.,~\cite{hermanHighresolutionSimulationsInteractions2020}) are used
to model the crystals. Despite numerous laboratory and field studies on
frazil ice, the exact nucleation mechanism for crystals remains
unknown. Additional questions remain to be solved concerning the
influence on frazil growth and aggregation of, notably, initial
supercooling, thermal evolution of the water column, turbulence
intensity, and double-diffusive mixing in saline waters. In-situ
observations (e.g.,~\cite{d1984}) and laboratory experiments
(e.g.,~\cite{ushioLaboratoryStudySupercooling1993}) have provided
valuable information on the morphology of frazil crystals. Yet, despite
the careful analyses of ice samples in thin 
sections or CAT-SCANS~\cite{dubeInnerStructureAnchor2014}, a lot
remains to be investigated on their aggregation in anchor ice, whether
the latter essentially grow in situ or by collecting frazil crystals,
what controls the ability of frazil ice to stick to pebbles or
vegetation, and how frazil aggregates manage to trap sediments,
pollutants or organisms in grease ice (for a review,
see~\cite{barretteUnderstandingFrazilIce2021}).

A field sharing many challenges with those of the present paper is
cloud dynamics, and how it is coupled to cloud microphysics. Although
hydrometeors are widely modeled as a continuum in analog experiments
(e.g.,~\cite{friesInfluenceParticleConcentration2021,shyTurbulentStratifiedInterfaces1991,krugerDynamicsDowndraughtsCold2020}) 
and simulations
(e.g.,~\cite{oms2019,hermanHighresolutionSimulationsInteractions2020,boffettaEulerianDescriptionDilute2007,realiLayerFormationSedimentary2017,lemusModellingSettlingDrivenGravitational2021}),
the mathematical grounds of the continuum approach are regularly
violated due to excessive particle
dilution~\cite{chouNumericalStudyParticleinduced2016,melladoTwofluidFormulationCloudtop2010}.
Disregarding the particulate nature of the hydrometeors, which imposes
a spatially-discretized coupling between the air and the
particles~\cite{haradaParticlelikeFluidlikeSettling2012,yamamotoNumericalSimulationConcentration2015a}, 
leads to persistent shortcomings to model how the particle-scale
dynamics are coupled to the cloud
macrophysics~\cite{melladoTwofluidFormulationCloudtop2010,melladoCloudTopEntrainmentStratocumulus2017}:
hydrometeors only drag air in their vicinity and they release or absorb
latent heat locally. Since drag and rates of phase change depend
crucially on the size of
hydrometeors~\cite{levichPhysicochemicalHydrodynamics1962,fuRadiativeTransfer2006,pruppacherMicrophysicsCloudsPrecipitation2010}, 
cloud microphysics directly affects latent heating and evaporative
cooling in stratocumuli, altering cloud convection and stability, with
a direct consequence on cloud coverage and hence cloud
feedback~\cite{woodStratocumulusClouds2012}. Through precipitation,
microphysics also hugely affects cloud aggregation which ``remains one
of the largest sources of uncertainty in climate models and thus for
reliable projections of climate
change''~\cite{benistonGrandChallengesClimate2013}: during extreme
precipitation events, the simultaneous fall and evaporation of
hydrometeors can cause violent downbursts that spread on the floor as
cold winds, favoring the ascent of warm updrafts above them, leading to
cloud aggregation near the
downbursts~\cite{malardelAtmosphericBuoyancydrivenFlows2012,mullerSpontaneousAggregationConvective2022}.

These questions are equally relevant on other planets, for example,
concerning helium rains due to H-He demixing in Jupiter
(e.g.,~\cite{ss1977,nettelmannExplorationDoubleDiffusive2015}).
Convective storms are common in the gas giants of our solar
system~\cite{markhamRainyDowndraftsAbyssal2023}. The influence of
microphysics on their atmosphere dynamics has been mentioned to explain
the heterogeneity of ammonia in Jupiter's atmosphere through the
formation of ammonia-rich
hailstones~\cite{huesoConvectiveStormsAtmospheric2020,guillotStormsDepletionAmmonia2020}, 
as expected during some long-lasting evaporatively-driven downdrafts in
the deep Jovian atmosphere~\cite{markhamRainyDowndraftsAbyssal2023}.
More work is also required to model clouds on exoplanets. Taking the
example of tidally-locked hot Jupiters, the huge temperature contrast
between the sub-stellar point and the night side strongly constrains
their atmospheric
dynamics~\cite{parmentier3DMixingHot2013,hellingExoplanetClouds2019}.
This creates a planetary-scale 3D flow that affects the dispersal and
stability of tracers through vertical mixing, with consequences on
cloud coverage if cold traps deplete the atmosphere of some chemicals
through precipitation~\cite{parmentier3DMixingHot2013}. Here again, the
size of the meteors greatly affects the outcome of the scenarios that
have been modeled~\cite{parmentier3DMixingHot2013}, but the huge amount
of chemicals involved further complexify the
picture~\cite{hellingExoplanetClouds2019}, notably due to the possible
change in composition of the particles as they simultaneously grow and
migrate in the
atmosphere~\cite{hellingExoplanetClouds2019,marley2013clouds}.

\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.

\section*{Funding}
LH is supported by the European Research Council (ERC) under the
European Unions Horizon 2020 research and innovation programme (grant
number 716429). ISTerre is part of Labex OSUG@2020 (ANR10 LABX56). MLB
and TA are supported by the Programme National de Plan\'etologie (PNP)
of CNRS-INSU co-funded by CNES.

\CDRGrant[ERC]{716429}
\CDRGrant[ANR]{ANR10 LABX56}

\back{}

\bibliographystyle{crunsrt}
\bibliography{crphys20240291}
\refinput{crphys20240291-reference.tex}

\end{document}
