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\DOI{10.5802/crgeos.324}
\datereceived{2025-06-17}
\daterevised{2025-12-08}
\dateaccepted{2026-01-05}
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\dateposted{2026-01-28}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\TopicFR{G\'eophysique, g\'eod\'esie}
\TopicEN{Geophysics, geodesy}

\title{Did lunar tides sustain the early Earth's dynamo?}

\alttitle{Les mar\'ees lunaires ont-elles engendr\'e le champ
magn\'etique dynamo de la Terre primitive ?}

\author{\firstname{J\'er\'emie} \lastname{Vidal}\CDRorcid{0000-0002-3654-6633}\IsCorresp}
\address{CNRS, ENS de Lyon, Univ. Lyon 1, LGL-TPE, France}
\email[J. Vidal]{jeremie.vidal@ens-lyon.fr}

\author{\firstname{David} \lastname{C\'ebron}\CDRorcid{0000-0002-3579-8281}}
\address{Universit\'e Grenoble Alpes, CNRS, ISTerre, 38000 Grenoble, France}

\keywords{\kwd{Geodynamo}\kwd{Tides}\kwd{Early
Earth}\kwd{Waves}\kwd{Turbulence}\kwd{Elliptical instability}}

\altkeywords{\kwd{G\'eodynamo}\kwd{Mar\'ees}\kwd{Terre
primitive}\kwd{Ondes}\kwd{Turbulence}\kwd{Instabilit\'e elliptique}}

\thanks{\textsc{ens} de Lyon under the programme ``Terre \&
Plan\`etes'', European Research Council (\textsc{erc}) under the
European Union's Horizon 2020 research and innovation programme (grant
agreement No 847433, \textsc{theia} project), French Academy of
Sciences \& Electricit\'e de France.}

\begin{abstract} 
Geological data show that, early in its history, the Earth had a
large-scale magnetic field with an amplitude comparable to the one of
the present geomagnetic field. However, its origin remains enigmatic
and various mechanisms have been proposed to explain the Earth's field
over geological time scales. Here, we critically evaluate whether tidal
forcing could explain the ancient geodynamo, by combining constraints
from geophysical models of the Earth--Moon system and predictions from
turbulence studies. Our analysis shows that lunar tidal forcing could
have been sufficiently strong before ${-}$3.25 Gy to trigger turbulence
within the Earth's core, and potentially to sustain dynamo action
during that interval. Then, we propose new scaling laws for the
magnetic field amplitude $B$. We expect the latter to scale as $B
\propto \beta^{4/3}$, where $\beta$ is the equatorial ellipticity of
the liquid core, if the turbulence involves weak interactions of
three-dimensional inertial waves. Alternatively, in the regime of
strong tidal forcing, the expected scaling becomes $B \propto \beta$.
When extrapolated to the Earth's core, it suggests that tidal forcing
alone was too weak to possibly explain the ancient geomagnetic field.
Therefore, our study indirectly favours another origin for the early
Earth's dynamo on long time scales (e.g. exsolution of light elements
atop the core, or thermal convection due to secular cooling). 
\end{abstract}

\begin{altabstract} 
Les donn\'ees g\'eologiques indiquent que la Terre poss\'edait, au
d\'ebut de son histoire, un champ magn\'etique \`a grande \'echelle et
dont l'amplitude \'etait comparable \`a celle du champ magn\'etique
actuel. Son origine reste toutefois \'enigmatique, et divers
m\'ecanismes ont \'et\'e propos\'es pour l'expliquer. Dans cet article,
nous \'etudions la possibilit\'e que le for\c{c}age des mar\'ees
lunaires ait pu engendrer la g\'eodynamo ancienne, en combinant les
contraintes des mod\`eles g\'eophysiques du syst\`eme Terre-Lune et les
pr\'edictions relatives \`a la turbulence des \'ecoulements forc\'es
par les mar\'ees. Notre analyse montre que les mar\'ees lunaires
auraient pu \^etre suffisamment intenses avant ${-}$3,25 milliards
d'ann\'ees pour d\'evelopper de la turbulence dans le noyau terrestre,
puis potentiellement maintenir l'effet dynamo durant cette p\'eriode.
Nous proposons ensuite de nouvelles lois d'\'echelle pour l'amplitude
du champ magn\'etique $B$. Nous nous attendons \`a ce que $B$ soit
proportionnel \`a $\beta^{4/3}$, o\`u $\beta$ est l'ellipticit\'e
\'equatoriale du noyau liquide, si la turbulence implique de faibles
interactions non lin\'eaires d'ondes inertielles tridimensionnelles. En
revanche, la loi d'\'echelle attendue serait $B \propto \beta$ dans le
r\'egime des mar\'ees fortes. L'extrapolation g\'eophysique au noyau
terrestre sugg\`ere que les seules mar\'ees lunaires \'etaient trop
faibles pour expliquer le champ magn\'etique ancien de la Terre. Par
cons\'equent, notre \'etude sugg\`ere une autre origine pour la
g\'eodynamo ancienne, comme l'exsolution d'\'el\'ements l\'egers au
sommet du noyau ou la convection thermique due au refroidissement
s\'eculaire de la Terre.
\end{altabstract}

\maketitle
{\vspace*{.5pc}}

\twocolumngrid

\end{noXML}

\defcitealias{farhat2022resonant}{ibid.}

\section{Introduction}

This research article concludes a series published in the
\textit{Comptes Rendus} \citep{deguen2024fluid, nataf2024dynamic,
vidal2024geophysical, plunian2025three}, sparked by the Amp\`ere Prize
awarded in 2021 by the French Academy of Sciences to the
\textsc{g\'eodynamo} research team\footnote{ISTerre, Universit\'e
Grenoble Alpes, France,
\url{https://www.isterre.fr/english/research/research-teams/geodynamo/}.}.
The latter was originally formed to study convective flows within the
Earth's core and the geomagnetic field. Indeed, more than a century
after the seminal idea of \citet{larmor1919could}, it is well accepted
that the geomagnetic field is driven by dynamo action due to turbulent
motions in the Earth's liquid core \citep[e.g.][]{roberts2013genesis,
dormy2025rapidly}. Moreover, paleomagnetism indubitably shows that the
Earth has hosted a large-scale magnetic field for billions of years
\citep[e.g.][]{macouin2004long}. Therefore, the dynamics of the liquid
core over geological time scales can be indirectly probed by studying
the geomagnetic field. For example, geomagnetic data allows
reconstructing surface core flows \citep[e.g.][]{istas2023transient,
rogers2025effects}, which may enhance future predictions of geomagnetic
variations \citep[e.g.][]{madsen2025modelling}, or anchoring models
aimed at understanding the origin of magnetic reversals
\citep[e.g.][]{driscoll2009effects, frasson2025geomagnetic}. In
addition, understanding the emergence of the geodynamo in the ancient
Earth is also of paramount importance, since it would provide
invaluable insights into the long-term evolution of the Earth since its
accretion \citep{halliday2023accretion}.\looseness=1

\subsection{Geologic constraints for dynamo models} \label{sec1.1}

Paleomagnetism indicates that the geomagnetic field is at least
3.4--3.5~Gy old \citep{tarduno2010geodynamo,
biggin2011palaeomagnetism}, with a paleo-amplitude that is roughly
comparable with the present field until ${-}$3.25~Gy. Such observations
agree well with more indirect estimates from geochemistry. Indeed, the
${}^{15}N/{}^{14}N$ isotopic composition of the Archean atmosphere
3.5~Gy ago was found to be rather similar to the one of the present-day
atmosphere \citep{marty2013nitrogen}, which would require a
paleomagnetic amplitude of at least 50\% of the current intensity to
avoid $N_2$ loss in the upper atmosphere
\citep{lichtenegger2010aeronomical}. However, planetary scientists
currently disagree on whether the Earth's magnetic field could have
appeared earlier or not.\ It is indeed very challenging to go further
back in time, as ancient rocks have experienced multiple geological
events throughout their history. As shown in Figure~\ref{fig:paleomag},
some paleomagnetic studies suggest that the geomagnetic field probably
existed during the Eoarchean era with a surface amplitude
${>}$15~$\upmu$T \citep{nichols2024possible}, and possibly 4.2~Gy ago as
inferred from Hadean silicate minerals \citep{tarduno2015hadean,
tarduno2020paleomagnetism, tarduno2023hadaean}. However, the quality of
such ancient paleomagnetic data is strongly disputed. The magnetic
carriers may have a secondary origin, such that the magnetisation could
post-date the formation of the minerals by millions to billions of
years \citep[e.g.][]{weiss2015pervasive, weiss2018secondary,
borlina2020reevaluating, taylor2023direct}.\looseness=1

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:paleomag}Paleointensity at the Earth's surface
during the Hadean and Archean periods. Measurements have been performed
on either single-silicate crystals (e.g.\ zircons) or whole rocks (e.g.\ 
Banded Iron Formations). Data from the \textsc{pint} database
\citep{bono2022pint} and \citet{nichols2024possible}. Geological eons
are also shown (H:~Hadean, A:~Archean).}
\end{figure}

Finding a convincing scenario to explain the ancient Earth's magnetic
field is a long-term goal in geophysical modelling
\citep{landeau2022sustaining}. To this end, dynamo action is believed
to predominantly operate in the Earth's liquid core
\citep[e.g.][]{braginsky1995equations}, as the latter is surrounded by
a silicate mantle having a weaker electrical conductivity (compared to
that of liquid iron at core conditions) in both its upper and lower
regions \citep[e.g.][for the Earth]{yoshino2010laboratory,
jault2015illuminating}. As such, the mantle is often considered as an
electrical insulator on long time scales for dynamo modelling. Note
that dynamo action may be possible in a (basal) magma ocean if the
electrical conductivity of molten silicate rocks is high enough
\citep[e.g.][]{ziegler2013implications, scheinberg2018basal,
stixrude2020silicate, dragulet2025electrical}, but this dynamical
scenario may be energetically expansive \citep{schaeffer2025BMO}. 
However, there is currently no consensus within the community regarding
the physical mechanisms that may have driven core flows in the ancient
core.\looseness=-1

\subsection{Non-consensual dynamo scenarios} \label{sec1.2}

To assess the viability of a candidate dynamo scenario, we should
strive to reproduce the main characteristics of the recorded field over
geological time scales. In particular, we can consider the typical
amplitude of the large-scale field in the dynamo region, which can be
extrapolated from the data. Indeed, for an electrically insulating
mantle, the amplitude $B_{\mathrm{cmb}}$ of the largest-scale dipolar
field at the core-mantle boundary (\textsc{cmb}) is related to that at
the planet's surface $B_s$ by \citep[e.g.][]{moffatt2019self}
{\begin{equation}\label{eq:Bcmbfromdata}
B_{\mathrm{cmb}}\simeq B_s\left(\frac{R_s}{R_{\mathrm{cmb}}}\right)^3,
\end{equation}}\unskip
where $R_{\mathrm{cmb}}$ is a mean core radius, and $R_s$ is the mean
surface radius. With the typical value of $R_{\mathrm{cmb}} \approx
3480$~km for the Earth's core, Figure~\ref{fig:paleomag} leads to the
estimate $B_{\mathrm{cmb}} \sim 10^{-2}\ndash 10^{-1}$~mT for the
early Earth's field before ${-}$3.25~Gy. Deep in the core, the strength
of the dynamo magnetic field could even be larger than at the
\textsc{cmb}, as for instance reported for the current Earth
\citep[with a hidden toroidal field of a few mT,
e.g.][]{gillet2010fast}. Note that it is unclear whether these ancient
dynamos have operated in a strong-field regime (as currently in the
core) or not, which is a regime with a magnetic energy dominating over
the kinetic energy \citep[e.g.][]{moffatt2019self}. If so, a
strong-field behaviour would put a strong constraint to assess the
viability of candidate dynamo{\break} models.

\looseness=-1
Now, do we have some ancient dynamo scenarios meeting the above
constraints? Currently, the main driver of the Earth's core flows is
inner-core crystallisation \citep[e.g.][]{buffett1996thermal}. Indeed,
this scenario has proven successful in reproducing the main
characteristics of the current geomagnetic field using numerical
simulations \citep[e.g.][]{schaeffer2017turbulent, aubert2023state}.
However, inner-core crystallisation was missing in the distant past. 
The exact chronology remains disputed, but a nucleation starting $1 \pm
0.5$~Gy ago seems reasonable from recent thermal-evolution models
\citep[e.g.][]{labrosse2015thermal} or paleomagnetism
\citep{biggin2015palaeomagnetic, bono2019young, zhou2022early,
li2023late}. Prior to inner-core growth, it remains unclear which
mechanism could have sustained a large-scale magnetic field. In
addition to thermal convection alone
\citep[e.g.][]{aubert2009modelling, burmann2025rapidly,
lin2025invariance}, various scenarios have been invoked to trigger
\mbox{turbulence} in the core \citep[e.g.][]{landeau2022sustaining},
such as flows driven by double-diffusive convection in the core
\citep[e.g.][]{monville2019rotating} or by tidal forcing
\citep[e.g.][]{le2015flows}. Indeed, the recent estimates of the
thermal conductivity of liquid iron at core conditions, which do not
show a consensus yet between experimental and computational values
\citep[e.g.][]{williams2018thermal, hsieh2025moderate,
andrault2025long}, might suggest that secular cooling was energetically
less efficient than initially thought to sustain dynamo action{\break}
in the Earth's core.

\subsection{Outline of the manuscript} \label{sec1.3}

Motivated by the paleomagnetic data shown in Figure~\ref{fig:paleomag},
we want to thoroughly assess whether tidal forcing could have sustained
the early Earth's magnetic field. \citet{landeau2022sustaining}
provided preliminary estimates but, as shown below, applying this
scenario to the Earth is still intricate. The extrapolation is
underpinned by some arguments that must be quantitatively revisited in
the light of recent geophysical models and fluid-dynamics studies. 
Thus, this manuscript also intends to explain how the fluid-dynamic
community models tidally driven flows and turbulence in planetary
cores.

\looseness=-1
For instance, to the best of our knowledge, there is no self-consistent
numerical code to efficiently explore the dynamo capability of
orbitally driven flows in realistic core geometries (i.e.\ with a weakly
non-spherical \textsc{cmb}). This is a noticeable difference with
convection-driven dynamos, for which efficient numerical strategies
have been developed for decades
\citep[e.g.][]{christensen2001numerical, schaeffer2013efficient}.
Therefore, we must currently rely on scaling arguments to extrapolate
the few available results to the early Earth. However, planetary
extrapolation is not an easy task because it requires a strong
understanding of the turbulence properties, which are still debated in
the fluid-dynamics community \citep[e.g.\ as recently reviewed
in][]{vidal2024geophysical}. Similarly, a universal scaling law that
could be applied to any tidally driven flow is unlikely to exist,
because the scaling theory should be tailored to each physical
mechanism. 

With the aforementioned goals in mind, the manuscript is divided as
follows. We introduce the basic fluid-dynamics ingredients for core
flows driven by orbital forcings (e.g.\ tides) in
Section~\ref{sec:model}, as they may not be familiar to non-expert
readers. We then focus on the tidal forcing in Section~\ref{sec:tidal},
combining constraints from geophysical models, hydrodynamic studies and
new dynamo scaling laws, to extrapolate our findings to the early
Earth. We discuss the results in Section~\ref{sec:discussion}, and we
end the manuscript{\break} in Section~\ref{sec:conclusion}.

\section{Modelling of orbitally driven flows} \label{sec:model}

In this section, we will present the minimal fluid-dynamics ingredients
needed to model the dynamics of planetary liquid cores subject to
orbital mechanical forcings (such as tides in the early Earth). First,
we introduce in Section~\ref{subsec:modelEQ} an idealised common
description of orbitally driven core flows. Then, we briefly outline
in Section~\ref{subsec:modelROAD} the different steps of the (expected)
flow response of a planetary liquid core to an orbital mechanical
forcing, which will be revisited in Section~\ref{sec:tidal} to carry
out the planetary extrapolation to the early Earth.

\subsection{Equations of fluid dynamics} \label{subsec:modelEQ}

\looseness=-1
We want to model the dynamics of a planetary core prior to the
nucleation of a solid inner core. Thus, we consider a liquid core of
volume $V$ with no inner core, which is surrounded by a rigid and
electrically insulating mantle. The \textsc{cmb} is denoted by
$\partial V$ below. The core geometry is generally assumed to be
spherical in most geodynamo models, as it is sufficient for
convection-driven studies and allows developing very efficient
numerical methods \citep[e.g.][]{christensen2001numerical,
schaeffer2013efficient}. However, it is important to take the small
departures from a spherical \textsc{cmb} into account for tidal and
precession forcings \citep[e.g.][]{le2015flows}. For simplicity, we
assume below that $V$ is an ellipsoid, which agrees with the
mathematical theory of equilibrium figures for a rotating fluid mass
with an orbital partner \citep[e.g.][]{chandrasekhar1987ellipsoidal}.
Then, it is customary to work in the frame rotating with the
ellipsoidal distortion, which is rotating at the angular velocity
$\bOmega_{\epsilon}$. Note that the latter generally differs from the
rotation of the fluid, which will be denoted by $\bOmega_s$ below.
Working in the frame rotating at $\bOmega_{\epsilon}$ will ease the
computations, since the \textsc{cmb} will be steady in that frame and
can be written as $({x}/{a})^2 + ({y}/{b})^2 + ({z}/{c})^2 = 1$, where
$[a,b,c]$ are the ellipsoidal semi-axes and $(x,y,z)$ are the Cartesian
coordinates. Note that dynamical pressure (due to core flows) is
expected to be much weaker than hydrostatic pressure on long time
scales. Consequently, in practice, the flow dynamics is usually
explored in rigid ellipsoids by considering prescribed values of
$[a,b,c]$. Planetary values can be estimated from tidal theory
\citep[e.g.][]{farhat2022resonant} and the theory of planetary figures
\citep[e.g.][]{chambat2010flattening}. Note that the $z$-axis is
usually chosen along the rotation of the planet, such that it is
customary to introduce the two parameters given by
\changeEqNo{2a,b}
{\begin{equation}
\beta = \frac{|a^2-b^2|}{a^2+b^2},\quad f= \frac{a-c}{a},
\end{equation}}\unskip
\resetEqNo{2}\unskip
\noindent
where $ 0 < \beta \ll 1$ is the (equatorial) ellipticity and $0 < f \ll
1$ is the (polar) flattening.

The core is filled with an electrically conducting liquid, which is
assumed to have a constant kinematic viscosity $\nu$ and magnetic
diffusivity~$\eta$. The ratio of these two quantities yields the
magnetic Prandtl number $\mathit{Pm} = \nu/\eta$, whose typical 
value is $\mathit{Pm}
\sim 10^{-6}$ in a planetary liquid core when adopting the typical core
values $\nu \simeq 10^{-6}~\mathrm{m}^2{\cdot}\mathrm{s}^{-1}$
\citep[e.g.][]{de1998viscosity} and $\eta \simeq 1~\mathrm{m}^2{\cdot}
\mathrm{s}^{-1}$ \citep[e.g.][]{nataf2024dynamic}. As a consequence, we
expect the magnetic dissipation to be much larger than the viscous one
in a dynamo regime. Note that we neglect density effects below, to
focus on incompressible fluids with constant density. Indeed, most
fluid-dynamics studies consider orbitally driven flows without buoyancy
effects in the incompressible regime, assuming that the fluid has a
constant density~$\rho_f$. In the frame rotating at
$\bOmega_{\epsilon}$, the fluid velocity $\mathbi{v}$ is then
governed by the incompressible momentum equations given by
{\begin{subequations}
\begin{eqnarray}
\mathrm{d}_t \mathbi{v} + 2 \bOmega_{\epsilon} \times
\mathbi{v} &=& -\nabla \Pi + \nu \nabla^2 \mathbi{v} +
\mathbi{f}_L + \mathbi{f}_P,\Seqnsplit
\nabla \bdot \mathbi{v} &=& 0,
\end{eqnarray}
\end{subequations}}\unskip
where $\mathrm{d}_t = \partial_t + \mathbi{v} \bdot \nabla$ is the
material derivative, $\Pi$ is a reduced pressure term (for
incompressible flows), and $[\mathbi{f}_L,\mathbi{f}_P]$ are
the Lorentz and Poincar\'e forces given by
\changeEqNo{4a{\ndash}b}
{\begin{equation}
\mathbi{f}_L = \frac{1}{\rho_f \mu} (\nabla \times \mathbi{B})
\times \mathbi{B}, \quad \mathbi{f}_P = \mathbi{r} \times
\dot{\bOmega}_{\epsilon},
\end{equation}}\unskip
\resetEqNo{4}\unskip
\noindent
where $\mu$ is the magnetic permeability of the fluid\footnote{We have
$\mu \approx \mu_0$ for core conditions in practice, where $\mu_0$ is
the magnetic permeability of vacuum.} and $\mathbi{r}$ is the
position vector. The momentum equation is coupled, through the Lorentz
force, to the magnetic field equations given by
\citep[e.g.][]{moffatt2019self}
{\begin{subequations} \label{eq:induction}
\begin{eqnarray}
\label{eq:induction-a} \partial_t \mathbi{B} &=& \nabla \times
(\mathbi{v} \times \mathbi{B}) + \eta \nabla^2
\mathbi{B},\Seqnsplit
\nabla \bdot \mathbi{B} &=& 0.
\end{eqnarray}
\end{subequations}}\unskip
Finally, the above equations must be supplemented with appropriate
boundary conditions (BCs) at the \textsc{cmb}. The mantle being
supposed to be an electrical insulator, the magnetic field in the core
must match at the \textsc{cmb} the field in the mantle, which
is given by
\changeEqNo{6a,b}
{\begin{equation}\label{eq:BCmag}
\mathbi{B} = \nabla \Phi,\quad \Phi \to 0\quad\text{when }
|\mathbi{r}| \to +\infty.
\end{equation}}\unskip
\resetEqNo{6}\unskip
\noindent
Next, because the mantle is supposed to be rigid, the velocity must
satisfy the no-penetration BC given by $\mathbi{v} \bdot
\mathbf{1}_n|_{\partial V} = 0$ for any forcing, where $\mathbf{1}_{n}$
is the (outward) unit vector normal to the boundary. Some BCs must also
be enforced on the tangential velocity components, but they are
forcing-dependent.

Finally, it is customary to write down the mathematical problem using
dimensionless variables. In particular, we introduce the Ekman number
$E$ and the Rossby number $\mathit{Ro}$ given by
\changeEqNo{7a,b}
{\begin{equation}
\label{eq:EkRonumbers}
E = \frac{\nu}{\Omega_s R_{\mathrm{cmb}}^2},\quad \mathit{Ro} =
\frac{\mathcal{U}}{\Omega_s R_{\mathrm{cmb}}},
\end{equation}}\unskip
\resetEqNo{7}\unskip
\noindent
where $\Omega_s = |\bOmega_s|$ is the mean angular velocity of the
fluid (e.g.\ $\Omega_s \simeq 7.3 \times 10^{-5}~\mathrm{rad}{\cdot}
\mathrm{s}^{-1}$ currently in the Earth's core), $R_{\mathrm{cmb}}$ is
the mean core radius, and $\mathcal{U}$ is a typical amplitude of the
core flows. We will show in Section~\ref{sec:tidal} that both $E$ and
$\mathit{Ro}$ will be key for the planetary extrapolation of the results.
Indeed, despite $E$ is a very small quantity in planetary cores (e.g.\ 
$E\sim 10^{-15}$ in the Earth and $E\sim 10^{-12}$ in the Moon), it
will mainly control the growth of turbulent flows. Similarly, the
strength of the rotating turbulence (and hence that of the dynamo
magnetic field) will be influenced by the value of $\mathit{Ro}$.

\subsection{Towards dynamo magnetic fields} \label{subsec:modelROAD}

Even with the strong physical assumptions we have employed, the model
presented in Section~\ref{subsec:modelEQ} is extremely difficult to
solve. One of the reasons is that, currently, there is no numerical
code that can efficiently account for magnetic BC (\Custref{6}{eq:BCmag}) in
an ellipsoidal geometry. Nonetheless, previous numerical and
experimental studies have allowed the identification of a rather
general pattern for the flow response to orbital forcings
\citep[e.g.][for a recent review]{vidal2024geophysical}. We briefly
describe it below, as it will guide us for the extrapolation of tidal
forcing in{\break} Section~\ref{sec:tidal}.

To start with, we consider a non-dynamo regime characterised by
negligible magnetic effects, which occurs prior to the establishment of
a planetary magnetic field. At the leading order, an orbital forcing
drives a large-scale flow $\mathbi{U}_0$ in an ellipsoid, which is
governed by
{\begin{subequations} \label{eq:forcedflowU0}
\begin{eqnarray}
\mathrm{d}_t \mathbi{U}_0 + 2 \bOmega_{\epsilon} \times
\mathbi{U}_0 &=& - \nabla \Pi_0 + \nu \nabla^2 \mathbi{U}_0 +
\mathbi{f}_P,\Seqnsplit
\nabla \bdot \mathbi{U}_0 &=& 0.
\end{eqnarray}
\end{subequations}}\unskip
This is a laminar flow, sometimes referred to as a Poincar\'e flow for
precession \citep[e.g.][]{roberts2011flows}, which has a nearly uniform
vorticity and can be written in the bulk as
\citep{noir2013precession}\par\vspace*{-1pc}
\changeEqNo{9a,b}
{\begin{equation}
\label{eq:formGP1}
\mathbi{U}_0 \simeq \bomega_{\epsilon} \times \mathbi{r} +
\nabla \Psi_{\epsilon}, \quad \mathbi{U}_0 \bdot
\mathbf{1}_n |_{\partial V} = 0,
\end{equation}}\unskip
\resetEqNo{9}\unskip
\noindent
where $\bomega_{\epsilon}$ is the fluid rotation vector in the
$\bOmega_{\epsilon}$-frame, and $\nabla \Psi_{\epsilon}$ is a shear
flow that is non-vanishing only when $a \neq b$, $a \neq c$ or $b \neq
c$ (i.e.\ when the \textsc{cmb} is not spherical). Note that orbital
forcings also sustain other laminar flows, such as mean flows due to
weak nonlinear interactions in the boundary layer below the
\textsc{cmb} \citep[e.g.][]{busse1968steady, cebron2021mean}.

It turns out that the forced laminar flows are unlikely to sustain
dynamo action, because their spatial structures are generally too
simple \citep[e.g.][for the precession-driven forced
flow]{tilgner1998models}. However, the flow components departing from
$\mathbi{U}_0$ could be viable candidates for dynamo action.
Indeed, there are often unstable flow perturbations $\mathbi{u}$
that can grow upon $\mathbi{U}_0$ in the bulk of the core with an
amplitude $|\mathbi{u}|\propto\mathrm{exp}(\sigma t)$ in the initial
stage (i.e.\ when $\mathbi{u} \bdot \nabla \mathbi{u}$ remains
negligible in the momentum equation for $\mathbi{u}$), where
$\sigma>0$ is the growth rate of the unstable flows. Physically, such
hydrodynamic instabilities can result from couplings between the shear
component $\nabla \Psi_{\epsilon}$ in Equation~(\Custref{9a}{eq:formGP1}) and
free waves existing in the bulk of the core (e.g.\ inertial waves, whose
restoring force is the Coriolis force associated to the global rotation
of the liquid core). The growth rate $\sigma \geq 0$ of these unstable
bulk flows can be written as
{\begin{equation} \label{eq:growthrateTH}
\sigma \simeq \sigma^i - \sigma^d,
\end{equation}}\unskip
where $\sigma^i>0$ is the diffusionless growth rate (i.e.\ when
$\nu=\eta=0$), and $\sigma^d>0$ is the diffusive damping term. Prior to
the existence of a strong planetary magnetic field, the main
dissipation mechanism is due to viscous dissipation in the Ekman
boundary layer below the \textsc{cmb}, such that $\sigma^d/\Omega_s
\sim \mathcal{O}(E^{1/2})$ with a numerical pre-factor that depends on
the spatial complexity of $\mathbi{u}$
\citep[e.g.][]{greenspan1969theory}. A first condition to reach a
dynamo regime is thus given by
{\begin{equation} \label{eq:sigmaivsdamping}
\frac{\sigma^i}{\Omega_s} > \mathcal{O}(E^{1/2}).
\end{equation}}\unskip
Condition (\ref{eq:sigmaivsdamping}), which gives a condition for bulk
instability in the core, will allow us to estimate when unstable bulk
flows could have been triggered by tidal forcing in the early Earth's
core. Note that other hydrodynamic instabilities may be triggered by
tidal forcing, either in the bulk \citep[e.g.][]{sauret2014tide} or
below the \textsc{cmb} \citep[e.g.\ see a discussion
in][]{cebron2021mean}. However, they are not expected to play a key
role for planetary dynamos (as their existence mostly rely on viscous
effects).

After their exponential growth, these flows will saturate in amplitude
due to a non-vanishing nonlinear term $\mathbi{u} \bdot \nabla
\mathbi{u}$, and become turbulent. As outlined in
\citet{vidal2024geophysical}, various regimes of turbulent flows can be
expected, leading to different scaling laws for the velocity amplitude
$\mathcal{U} \sim |\mathbi{u}|$ as a function of the control
parameters (e.g.\ $E$ and~$\mathit{Ro}$). Therefore, we need to determine the
correct scaling law for the amplitude of turbulence.

Once we can estimate the amplitude of turbulent flows in planetary
conditions, we can start assessing their ability to generate a
self-sustained magnetic field. In induction equation
(\Custref{5a}{eq:induction-a}), the magnetic field can growth over time if the
production term \mbox{$\nabla \times (\mathbi{u} \times
\mathbi{B})$} is much larger than the dissipation term $\eta
\nabla^2 \mathbi{B}$. Using orders of magnitude, this yields the
condition that the magnetic Reynolds number $\mathit{Rm}$, given~by
{\begin{equation}\label{eq:Rmnumber}
\mathit{Rm} = \frac{\mathcal{U} R_{\mathrm{cmb}}}{\eta}
\end{equation}}\unskip
where $\eta \simeq 1~\mathrm{m}^2{\cdot}\mathrm{s}^{-1}$ is the typical
value in the Earth's core, must be larger than some threshold value
$\mathit{Rm}_c$ to have a growth of the magnetic energy. In practice, a typical
condition 
{\begin{equation}\label{eq:Rmcdynamo}
\mathit{Rm}_c \sim 40\ndash 100
\end{equation}}\unskip
is often assumed from convection-driven dynamos in spherical geometries
\citep{christensen2006scaling}, in good agreement with theoretical
studies \citep[e.g.][]{chen2018optimal, holdenried2019trio,
vidal2021kinematic}. Yet, larger values $\mathit{Rm}_c \sim 200$ might be
required for orbitally driven dynamos according to prior simulations
\citep{reddy2018turbulent, cebron2019precessing}.

Finally, the nonlinear regime of orbitally driven dynamos remains
unknown.\ Indeed, direct numerical simulations (\textsc{dns}) of
dynamos in ellipsoids have only be performed using either unrealistic
\mbox{(ad-hoc)} numerical approximations that strongly hamper the
planetary extrapolation \citep{cebron2014tidally, vidal2018magnetic},
or were limited to exponential growth regimes
\citep{ivers2017kinematic, reddy2018turbulent}. Therefore, we must rely
on scaling-law arguments to estimate the possible strength of dynamo
fields in planetary interiors. However, there is no consensual dynamo
scaling law in the literature for tidal forcing \citep{le2011impact,
barker2014non, vidal2018magnetic} or precession
\citep{cebron2019precessing}. 

\looseness=-1
Finally, it is worth noting that a dynamo scenario can fail at any
stage. For instance, condition (\ref{eq:sigmaivsdamping}) may not be
fulfilled on long enough time scales to yield turbulent flows
$\mathbi{u}$, or the resulting turbulence may have not remained
vigorous enough to maintain a $\mathit{Rm}$ value above the dynamo onset over
time. Similarly, the associated dynamo magnetic field may be too weak
to match the paleomagnetic estimates given in
Figure~\ref{fig:paleomag}. Therefore, to assess the dynamo capability
of tidally driven flows in the early Earth, we have first to estimate
if conditions (\ref{eq:sigmaivsdamping}) and (\ref{eq:Rmcdynamo}) are
satisfied in the distant past, and then to estimate a typical magnetic
field amplitude using appropriate scaling laws.

\section{Results for tidal forcing} \label{sec:tidal}

As customary in long-term evolution scenarios for the Earth--Moon
system \citep[e.g.][]{farhat2022resonant}, we consider a simplified
model neglecting the effects of Earth's obliquity, of the Moon's
orbital eccentricity and inclination, and of the small phase lag
between the tidal bulge of the Earth and the Moon. Hence, the Earth's
core is supposed to be instantaneously deformed by the tidal potential
into an ellipsoid whose equatorial semi-axes can be written as
\changeEqNo{14a,b}
{\begin{equation}
\frac{a}{R_{\mathrm{cmb}}} = \sqrt{1 + \beta},\quad
\frac{b}{R_{\mathrm{cmb}}} = \sqrt{1 - \beta},
\end{equation}}\unskip
\resetEqNo{14}\unskip
\noindent
where is $R_{\mathrm{cmb}}$ the mean radius of the liquid core. We work
in the frame rotating with the Moon at the angular velocity
$\bOmega_{\epsilon} = \Omega_{\mathrm{orb}} \mathbf{1}_z$, and assume
that the liquid core is co-rotating with the planet at the angular
velocity $\bOmega_s = \Omega_s \mathbf{1}_z$ with respect to the
inertial frame. In fluid-dynamics studies, it is tacitly assumed that
the orbital parameters evolve on a time scale (denoted by $\tau$ below)
that is much longer than that of the turbulent flows. Consequently,
$[\Omega_{\mathrm{orb}}, \Omega_s]$ and the ellipsoidal geometry are
always supposed to be constant for the flow dynamics. This allows
performing parametric studies as a function of the different parameters
over geological time scales, and obtaining scaling laws for
extrapolation to the Earth.

Given the above assumptions, tidal forcing first drives a laminar flow
in an ellipsoidal liquid core of the form (\Custref{9}{eq:formGP1}) with
\citep[e.g.][]{le2015flows}
\changeEqNo{15a,b}
{\begin{equation}
\label{eq:GP1tides}
\bomega_{\epsilon} = \Delta \Omega \, \mathbf{1}_z, \quad
\Psi_{\epsilon} = -\beta \, \Delta \Omega xy,
\end{equation}}\unskip
\resetEqNo{15}\unskip
\noindent
where $\Delta \Omega = \Omega_s - \Omega_{\mathrm{orb}}$ is the
differential rotation between the fluid and the orbit. To assess the
validity of the tidal scenario, we first estimate the orbital
parameters in Section~\ref{subsec:resultsorbit} by using geophysical
models for the early Earth--Moon system. Then, we investigate whether
$\mathbi{U}_0$ can sustain flow instabilities and turbulence over
geological time scales by using constraints from hydrodynamics studies
in Section~\ref{subsec:resultshydro}. Finally, we discuss possible
dynamo effects in Sections~\ref{subsec:resultsdynamo}
and~\ref{subsec:extrapolation} using scaling-law arguments.

\subsection{Estimates from geophysical models}
\label{subsec:resultsorbit}

\looseness=-1
We need to estimate how the geometry of the liquid core has changed
over geological time scales, as well as the spin $\Omega_s$ and the
orbital angular velocity $\Omega_{\mathrm{orb}}$. It is known that the
polar flattening is mainly due to centrifugal effects such that $f
\propto \Omega_s^2$ according to equilibrium hydrostatic models
\citep[e.g.][]{chambat2010flattening}. Moreover, tidal theory for an
incompressible fluid predicts that the equatorial ellipticity of the
core should evolve as $\beta \propto (1/a_M)^3$
\citep[e.g.][]{cebron2012elliptical}, where $a_M$ is the Earth--Moon
distance. Therefore, assuming that the mantle's properties have not
significantly changed over time, the polar flattening $f(\tau)$ and
equatorial ellipticity $\beta(\tau)$ of the core at age $\tau$ before
present can be estimated from the present-day values $[f(0),\beta(0)]$
as
\changeEqNo{16a,b}
{\begin{equation}
\label{eq:betaftime}
\frac{f(\tau)}{f(0)}=\left(\frac{\Omega_s(t)}{\Omega_s(0)}\right)^2,
\quad
\frac{\beta(\tau)}{\beta(0)}=\left(\frac{a_M(0)}{a_M(\tau)}\right)^3,
\end{equation}}\unskip
\resetEqNo{16}\unskip
\noindent
with the present-day values $a_M(0)/R_s \simeq 60.14$ and $\Omega_s(0)
\simeq 7.27 \times 10^{-3}~\mathrm{rad}{\cdot}\mathrm{s}^{-1}$
\citep{farhat2022resonant}.

We have to estimate the present-day values $[f(0),\beta(0)]$, before
extrapolating $[f(\tau),\beta(\tau)]$ back in time. Following
\citet{chambat2010flattening}, the current flattening $f(0)$ can be
obtained using hydrostatic equilibrium theory, with 
{\begin{equation}
f(0) \approx 2.54 \times 10^{-3}
\end{equation}}\unskip
at the \textsc{cmb}. Note that the Earth's nutations provide an
estimate of $f(0)$ with an error margin of only a few percents
\citep[e.g.][]{dehant2017understanding}, such that its value is already
well constrained. For $\beta(0)$, we need to relate the Earth's surface
maximum displacement due to Moon's tides, which is roughly
$s_{\mathrm{max}}(R_s) \approx22$~cm for the solid tides
\citep{AGNEW2015151}, to that at the \textsc{cmb} (see
Figure~\ref{fig:tidesmodel1}). To do so, we have used the open-source
code \textsc{TidalPy} \citep{renaud2023tidalpy} to solve the standard
elastic-gravitational equations for a layered planet subject to a
diurnal tidal potential of degree~2
\citep[e.g.][]{alterman1959oscillations}, using \textsc{prem} as the
reference Earth's internal structure \citep{dziewonski1981preliminary}.
The results are shown in Figure~\ref{fig:tidesmodel2}. We observe that
the maximum tidal displacement at the \textsc{cmb} is nearly equal to
that at the Earth's surface. By writing the equatorial semi-axes as $a
= R_{\mathrm{cmb}} + s_{\mathrm{max}}(R_{\mathrm{cmb}})$ and $b =
R_{\mathrm{cmb}} - s_{\mathrm{min}}(R_{\mathrm{cmb}})$, with
$s_{\mathrm{min}} = -s_{\mathrm{max}}/2$ for a tidal potential of
degree~2 \citep[e.g.][]{AGNEW2015151}, the \textsc{cmb} ellipticity at
present time $\beta(0)$ is thus given by (at leading order in the small
displacement)
{\begin{equation}\label{eq:betaradius}
\beta (0) \simeq \frac{3}{2} \frac{s_{\mathrm{max}}(R_{\mathrm{cmb}})}
{R_{\mathrm{cmb}}} \approx 9.8 \times 10^{-8}.
\end{equation}}\unskip
In the following, we will use the above present-day values to constrain
the extrapolation back in time. Note that such values are subject to
uncertainties, whose influence will be touched upon below in
Section~\ref{sec:discussion}. To do so, we use the orbital model
provided in \citet{farhat2022resonant}. This is a semi-analytical model
that fits the most accurate constraints in the Earth--Moon evolution
(i.e.\ the present tidal dissipation rate and the age of the Moon). As
discussed below in Section~\ref{sec:discussion}, this is not the case
for other orbital models. As illustrated in Figure~\ref{fig:forcing1},
the physical model by \citetalias{farhat2022resonant} also agrees very well
with most available geological data (e.g.\ tidal rhythmites and
cyclostratigraphic records).

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:tidesmodel1}Sketch (not to scale) of the elliptical
geometry of the tidally deformed Earth's core, as seen the orbital
plane of the Moon. $R_s$ is the mean surface radius, and
$R_{\mathrm{cmb}}$ is the mean core radius. The radial displacement
along the major axis in the equatorial plane is $s_{\mathrm{max}}$, and
that along the minor axis is given by $s_{\mathrm{min}} =
-s_{\mathrm{max}}/2$ for a tidal potential of degree~2.}
{\vspace*{.4pc}}
\end{figure}

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:tidesmodel2}Maximum radial displacement
$s_{\mathrm{max}}$ and inverse polar flattening $f^{-1}$ at present
day, as a function of normalised mean radius $r/R_s$, as computed for
tidal theory and hydrostatic equilibrium theory. In both cases, the
same Earth's reference model is chosen
\citep{dziewonski1981preliminary}. Red region shows the solid inner
core, and grey one the liquid outer core.}
{\vspace*{-.4pc}}
\end{figure}

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig:forcing1}Comparison between geologic data and
models for the evolution of the normalised Earth--Moon distance
$a_M/R_s$ in~(a), and of the length of day in~(b).\ Insets show the age
$\tau$ between ${-}$1 and ${-}$0.1~Gy.\ $R_s \simeq 6378$~km is the mean
value of the Earth's radius. Orbital model: \citet{farhat2022resonant}.
Paleontological data: \citet{williams2000geological} and references
therein.\ Cyclostratigraphic data: \citet{zhou2024earth} and references
therein. Tidal rhythmites data: \citet{farhat2022resonant,
eulenfeld2023constraints} and references therein.}
\end{figure}

The corresponding values of $\beta(\tau)$ and $f(\tau)$, computed as a
function of $\tau$ from Equations~(\Custref{16a,b}{eq:betaftime}), are shown in
Figure~\ref{fig:betaftime}. We see that $\beta$ only varies from one
order of magnitude over the Earth's history, that is from $\beta
\approx 10^{-7}$ nowadays to $\beta \approx 10^{-6}$ at ${-}$4.25~Gy.
This narrow range of values will put severe constraints for the
viability of the tidal scenario (as explained below in
Section~\ref{subsec:resultshydro}). Finally, the Moon's orbital
frequency is reconstructed using Kepler's third law as
\changeEqNo{19a,b}
{\begin{equation}
\label{eq:kepler3law}
\frac{\Omega_{\mathrm{orb}}(\tau)}{\Omega_{\mathrm{orb}}(0)} =
\left(\frac{a_M(0)}{a_M(\tau)}\right)^{3/2}, \quad
\Omega_{\mathrm{orb}}(\tau)=\frac{2\uppi}{T_{\mathrm{orb}}(\tau)},
\end{equation}}\unskip
\resetEqNo{19}\unskip
\noindent
where $T_{\mathrm{orb}}(\tau)$ is the Moon's orbital period (whose
current value is $T_{\mathrm{orb}}(0) \simeq 27.3217$~days). As shown
in Figure~\ref{fig:betaftime}, we find that
$\Omega_{\mathrm{orb}}/\Omega_s$ only varied weakly in the distant
past, a typical value being $\Omega_{\mathrm{orb}}/\Omega_s \sim 0.04
\pm 0.01$ during the Hadean and Archean\break eons.

\begin{figure}
\includegraphics{fig05}
{\vspace*{-.2pc}}
\caption{\label{fig:betaftime}Evolution of the Earth's core ellipticity
$\beta$, polar flattening~$f$, and of $\Omega_{\mathrm{orb}}/\Omega_s$,
as a function of age. Dashed line shows the frequency value associated
with the least-damped mode in resonance condition
(\ref{eq:resonancecondition}). Geological eons are also shown
(H:~Hadean, A:~Archean, P:~Proterozoic).}
{\vspace*{-.3pc}}
\end{figure}

\subsection{Hydrodynamic constraints} \label{subsec:resultshydro}

The geophysical models discussed above have allowed us to estimate the
parameters that need to be prescribed in the fluid-dynamics models.
Hence, we can move on the hydrodynamic constraints we have about tidal
flows. Before we can estimate the strength of dynamo action in
Section~\ref{subsec:resultsdynamo}, we need to estimate whether
turbulent tidal flows could have been triggered in the early Earth's
core in Section~\ref{subsec:resultssigma} and estimate their amplitude
in Section~\ref{subsec:resultsturbulence}.

\subsubsection{Onset of turbulent flows} \label{subsec:resultssigma}

As outlined in Section~\ref{subsec:modelROAD}, unstable tidal flows can
possibly grow upon the flow $\mathbi{U}_0$ with an exponentially
increasing amplitude $\propto \exp(\sigma t)$ in the initial stage. The
underlying mechanism is that of a sub-harmonic (parametric)
instability, known as the elliptical instability
\citep{kerswell2002elliptical, le2015flows}. This instability results
from couplings between some normal modes $\mathbi{u}_i$ sustained
by the global rotation of the core, called inertial modes and
oscillating with an angular frequency~$\omega_i$, and the shear
component of the forced flow $\mathbi{U}_0$ through the linearised
term $(\mathbi{u}_i \bdot \nabla) \mathbi{U}_0 +
(\mathbi{U}_0 \bdot \nabla) \mathbi{u}_i$ in the momentum\break
equation.

For this instability to exist, the modes and the tidal forcing must
satisfy a resonance condition in time given by
\citep[e.g.][]{vidal2017inviscid}
{\begin{equation}\label{eq:resonancecondition}
\omega_i \simeq \pm |\Omega_s-\Omega_{\mathrm{orb}}|.
\end{equation}}\unskip
Imperfect resonances can only occur in condition
(\ref{eq:resonancecondition}) for finite values of the ellipticity
$\beta$ or the \mbox{Ekman} number~$E$. Hence, the latter should be
negligible in the early Earth's core with $\beta \ll 1$ and $E \to 0$.
Moreover, we emphasise that Equation~(\ref{eq:resonancecondition}) is
only a necessary condition, because some resonance conditions must also
be fulfilled in space to have a non-zero growth rate $\sigma$ (not
shown here). Then, theory predicts that $\sigma$ is of the form
(\ref{eq:growthrateTH}), in which the diffusionless part $\sigma^i$ is
given by \citep[e.g.][]{vidal2019fossil}
{\begin{equation}\label{eq:sigmatdei}
\frac{\sigma^i}{\Omega_s} \lesssim \frac{(2 \tilde{\Omega} + 3)^2}
{16(1+\tilde{\Omega})^2} |1 - \Omega_0| \beta
\end{equation}}\unskip
with $\Omega_0 = \Omega_{\mathrm{orb}}/\Omega_s$ and $\tilde{\Omega} =
\Omega_0/(1-\Omega_0)$. Note that the upper bound in
Equation~(\ref{eq:sigmatdei}) is reached when the unstable flows have a
large enough spatial complexity \citep{vidal2017inviscid}. Moreover, in
the absence of strong magnetic fields, we can estimate the damping term
$\sigma^d$ from the viscous damping $\sigma^{\nu}$ of the inertial
modes involved in resonance condition (\ref{eq:resonancecondition}).
Boundary-layer theory shows that the viscous damping of inertial modes
in an ellipsoid is given by 
{\begin{equation}\label{eq:dampingBLT}
\sigma^{\nu}\simeq \Omega_s (\sigma^{\nu}_{1/2}\,E^{1/2} +
\sigma^{\nu}_{1}\,E),
\end{equation}}\unskip
where $\sigma^{\nu}_{1/2}>0$ results from the viscous friction in the
Ekman layer below the \textsc{cmb} \citep[e.g.][]{greenspan1969theory},
and $\sigma^{\nu}_{1}\geq 0$ is a bulk contribution
\citep[e.g.][]{liao2001viscous, lemasquerier2017libration}. 

\begin{figure*}
\vspace*{-2pt}
\includegraphics{fig06}
\vspace*{-2pt}
\caption{\label{fig:tdei}(a)~Surface and bulk contributions to the
viscous damping $\sigma^{\nu}$, as a function of the polynomial degree
$n$ of inertial modes with $\beta = 10^{-6}$ and $f = 10^{-2}$. Only
the modes with $0.94 \leq |\omega_i/\Omega_s| \leq 0.98$ that could
satisfy resonance condition (\ref{eq:resonancecondition}) are shown.
(b)~Diffusionless growth rate $\sigma^i/\Omega_s$ of unstable tidal
flows as a function of time, given by formula (\ref{eq:sigmatdei}), and
leading-order damping term $\sigma^d \gtrsim \sigma^{\nu}$ as a
function of the typical length scale of expected unstable flows. We
have used the standard value $\nu =
10^{-6}~\mathrm{m}^2{\cdot}\mathrm{s}^{-1}$ of the core viscosity
\citep[e.g.][]{de1998viscosity}. We have included the prior lower bound
obtained with $\sigma^{\nu}_{1/2}/\Omega_s =2.62$ (blue region).
Geological eons are also shown (H:~Hadean, A:~Archean,
P:~Proterozoic).}
\vspace*{-2pt}
\end{figure*}

Equation~(\ref{eq:dampingBLT}) is illustrated in
Figure~\ref{fig:tdei}(a). Note that the inertial modes can be expressed
in terms of polynomial functions of degree $\leq n$ in rotating
ellipsoids \citep[e.g.][]{backus2017completeness, CdV2025spectrum},
such that they can be computed using dedicated numerical methods
\citep[e.g.][]{vidal2024inertia}. We have only shown the modes for
which we could expect resonances from condition
(\ref{eq:resonancecondition}), that is with $|\omega_i| \sim 0.96 \pm
0.02$ for the early Earth according to Figure~\ref{fig:betaftime}.
Interestingly, we deduce that no modes of degree $n<7$ could be
triggered in the early Earth's core. Indeed, the first modes that
possibly satisfy resonance condition (\ref{eq:resonancecondition}) in
the early Earth are the two $n=7$ modes whose angular frequencies are
{\begin{equation}
\frac{\omega_i}{\Omega_s} \approx \pm 0.965.
\end{equation}}\unskip
Among all modes satisfying the resonance condition, the latter $n=7$
modes also have the lowest surface and bulk damping contributions given
by 
{\begin{equation}
\frac{\sigma^{\nu}_{1/2}}{\Omega_s} \approx 5.73, \quad
\frac{\sigma^{\nu}_{1}}{\Omega_s} \approx 261.
\end{equation}}\unskip
Hence, we can conclude that the damping term had a lower bound in the
early Earth given by
{\begin{equation}\label{eq:dampingTDEIEarlyEarth}
{\sigma^d}/{\Omega_s} \geq 5.73\,E^{1/2} + 261\,E.
\end{equation}}\unskip
This is an improvement with respect to the prior lower bound
${\sigma^d}/{\Omega_s} \geq 2.62\,E^{1/2}$
\citep[e.g.][]{landeau2022sustaining}, which corresponds to the damping
term of the $n=1$ spin-over modes with $\omega_i/\Omega_s \approx \pm
1$ \citep{greenspan1969theory}. However, theory shows that such modes
cannot yield an elliptical instability for the \textsc{cmb} geometry
with $a \geq b \geq c$ \citep{cebron2010systematic}. The damping term
in Equation~(\ref{eq:growthrateTH}) is thus at least two times larger
than previously thought for the early Earth. Moreover, given the
observed scalings of $\sigma^{\nu}_{1/2}$ and $\sigma^{\nu}_{1}$ with
$n$ in Figure~\ref{fig:tdei}(a), perturbations with $n \gg 1000$ would
be required to have a bulk contribution larger than the surface one in
Equation~(\ref{eq:dampingTDEIEarlyEarth}).\looseness=-1

Next, we compare $\sigma^i$ given by Equation~(\ref{eq:sigmatdei}) and
$\sigma^{\nu}$ in Figure~\ref{fig:tdei}(b). We have estimated
$\sigma^{\nu}$ from Figure~\ref{fig:tdei}(a) for unstable flows with
the typical length scale $\ell(n)$ estimated as
\citep{nataf2024dynamic}
{\begin{equation}\label{eq:lton}
\frac{\ell}{R_{\mathrm{cmb}}} \simeq\frac{1}{2}\frac{\uppi}{n+1/2},
\end{equation}}\unskip
where $n$ is the degree of the corresponding modes in the resonance
condition. First, we see that small-scale flows (e.g.\ with $n > 100$)
were certainly entirely damped by viscosity over the entire Earth's
history. Therefore, only some large-scale flows may have become
unstable in the early Earth's core. Moreover, we see that the
largest-scale modes with $n=7$ can only be triggered before ${-}$3.25~Gy.
These results provides a strong constraint on the age of a possible
tidally driven dynamo in the core. Indeed, we can assert that
paleomagnetic measurements younger than ${-}$3.25~Gy, which had a strong
magnetic field amplitude (Figure~\ref{fig:paleomag}a), did record an
ancient large-scale magnetic field driven by another mechanism than
tidal forcing. Yet, it remains possible so far that older points could
evidence an ancient tidally driven geodynamo because tides may have
injected energy into some large-scale unstable flows with $7 \leq n \ll
100$ between ${-}$4.25 and ${-}$3.25~Gy.

\subsubsection{Turbulent flows} \label{subsec:resultsturbulence}

Linear analysis yields predictions for the time window where turbulent
flows may be triggered by elliptical instabilities (sustained by tidal
forcing). Now, we have to estimate the typical velocity
amplitude~$\mathcal{U}$, as~this quantity will play an important role
in the planetary extrapolation. To do so, we can have a look at
simulations of tidally driven flows without magnetic fields. Actually,
several turbulence regimes can be expected. Weakly nonlinear analysis
shows that the normal form of the elliptical instability is that of a
supercritical Hopf bifurcation \citep{knobloch1994normal,
kerswell2002elliptical}. Hence, the saturation amplitude should scale
as $\mathcal{U}/(|\Delta \Omega| a) \propto \sqrt{\beta - \beta_c}$
near the onset, where $\beta_c$ is the critical ellipticity (i.e.\ such
that $\sigma = 0$). On the contrary, we expect $\mathcal{U}/(|\Delta
\Omega| a) \propto \beta - \beta_c \sim \beta$ far enough from the
onset according to phenomenological arguments \citep{barker2014non}.

We estimate $\mathcal{U}$ below from the volume average of the axial
velocity component $u_z$, which has been reported in prior numerical
and experimental studies. Since we have $\mathbi{U}_0 \bdot
\mathbf{1}_z = 0$, any departure from zero will be associated to
tidally driven turbulence and mixing. Moreover, it is often assumed to
be a good proxy for the mixing in the core driven by tides, which is
key dynamo action driven by tides \citep[e.g.][]{vidal2018magnetic}. We
show in Figure~\ref{fig:turbulence} the normalised velocity amplitude
$\mathcal{U}$, obtained from simulations in ellipsoids with a rigid
boundary at $E=1.5 \times 10^{-5}$ and
$|1-\Omega_{\mathrm{orb}}/\Omega_s| \approx 1.98$
\citep{grannan2017tidally}. We do recover the two expected regimes in
the simulations. The transition is believed to occur when $\beta -
\beta_c \sim \mathcal{O}(E)$ from theory
\citep{kerswell2002elliptical}. This is again in broad agreement with
the simulations, but we report here a quite large numerical pre-factor
since it occurs at $\beta - \beta_c \sim 600\,E$. The second regime
with $\beta \gg \beta_c$ can be more efficiently probed by relaxing the
no-penetration BC in the model. This can be achieved by using a
free-surface condition in an ellipsoid \citep{barker2016non}, or by
performing simulations of turbulent flows growing upon
$\mathbi{U}_0$ in Cartesian periodic boxes \citep{barker2014non}.
Such simulations, performed for different values of $E$ and forcing
frequencies $|1-\Omega_{\mathrm{orb}}/\Omega_s|$, are gathered in the
inset of Figure~\ref{fig:turbulence}. Although they have been performed
for different parameters, almost all simulations are well reproduced by
the linear scaling law $u_z/(|\Delta \Omega| a) \propto (0.25 \pm
0.15)\,\beta$ in the second regime (see the inset in
Figure~\ref{fig:turbulence}).

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig:turbulence}Normalised velocity
$\mathcal{U}/(|\Delta \Omega| a)$ with $\Delta \Omega = \Omega_s -
\Omega_{\mathrm{orb}}$, as a function of $\beta - \beta_c$ (with
$\beta_c\approx10^{-2}$) in \textsc{dns} of tidally driven flows in
rigid ellipsoids \citep{grannan2017tidally}. Inset also shows the
velocity but as a function of $\beta$ when $\beta \gg \beta_c$ for
additional simulations in ellipsoids with a free-surface condition
\citep{barker2016non}, and in Cartesian boxes \citep{barker2014non}. In
the latter case, we have defined $a = \sqrt{1+\beta}$ for the
normalisation. Coloured region shows scaling law $\mathcal{U}/(|\Delta
\Omega| a) = (0.25 \pm 0.15)\,\beta$, which broadly agrees with
simulations.}
{\vspace*{-.5pc}}
\end{figure}

To apply these results to the early Earth, we need to estimate how
supercritical the early Earth's core was before ${-}$3.25~Gy. Going back
to Figure~\ref{fig:tdei}(b), we see that super-criticality is never
very large (i.e.\ $\sigma^i/\sigma^{\nu} \ll 10$). We can estimate a
lower bound for the critical value of the ellipticity at the onset from
Equations~(\ref{eq:sigmatdei}) and~(\ref{eq:dampingTDEIEarlyEarth}).
This yields
{\begin{equation}
\beta_c \gtrsim \frac{5.73\,E^{1/2}}{|1 - \Omega_{\mathrm{orb}}/
\Omega_s|}\frac{16(1+\tilde{\Omega})^2}{(2\tilde{\Omega}+3)^2},
\end{equation}}\unskip
from which we obtain the typical value $\beta_c \geq 2.5 \times
10^{-7}$ before ${-}$3.25~Gy. Hence, we estimate that we had at most
$\beta/\beta_c \lesssim 2$ after ${-}$4~Gy from
Figure~\ref{fig:betaftime}.

Assuming that the transition between the two regimes still occurs at
$\beta - \beta_c \sim 600\,E$ when $E\ll1$, the early Earth's core
would have been in the second regime for most of the Hadean and Archean
eons (since $\beta - \beta_c \gg 600\,E$). Therefore, we can consider
that the tidally driven velocity amplitude (in planetary cores) is
given by
\changeEqNo{28a,b}
{\begin{equation}
\label{eq:scalinglawUtides}
\mathcal{U} \simeq \alpha_1 \beta |\Delta \Omega| R_{\mathrm{cmb}},
\quad \alpha_1 = 0.25 \pm 0.15,
\end{equation}}\unskip
\resetEqNo{28}\unskip
\noindent
as deduced from the simulations above. It is worth noting here that
\citet{landeau2022sustaining} assumed that $\alpha_1=1$ in their
velocity scaling law, which is at odds with the numerical results
gathered in \mbox{Figure}~\ref{fig:turbulence}.

Finally, we point out that scaling law (\Custref{28}{eq:scalinglawUtides}) says
almost nothing about the characteristics of the underlying turbulence.
Indeed, the properties of tidally driven turbulence remains largely
disputed for planetary conditions \citep[e.g.][]{vidal2024geophysical}.
Scaling law (\Custref{28}{eq:scalinglawUtides}) may apply for tidally driven
flows characterised by weakly nonlinear interactions of
three-dimensional waves when $\mathit{Ro} \ll 1$ \citep{le2017inertial,
le2019experimental, le2021evidence}, as well as with strong geostrophic
flows when $\mathit{Ro} \gtrsim 10^{-2}$ \citep[e.g.][]{barker2014non,
barker2016non, vidal2018magnetic}.

\subsection{A dynamo scaling law} \label{subsec:resultsdynamo}

Given the scaling law for the velocity amplitude, we evaluate in
Figure~\ref{fig:Rmevolution} the value of $\mathit{Rm}$, defined in
Equation~(\ref{eq:Rmnumber}), as a function of time. We have chosen a
critical value $\mathit{Rm}_c = 100$ for the onset of dynamo action, which is
standard in dynamo studies. Our results show a more pessimistic view
than the one presented in \citet{landeau2022sustaining}, in which the
$\mathit{Rm}$ values seem overestimated due to the chosen upper bound value
$\alpha_1 = 1$ in the velocity scaling law. Indeed, taking all
uncertainties into account (e.g.\ on $\alpha_1$ in the scaling law), we
see that the $\mathit{Rm}$ value is rather loosely constrained during most of
the Hadean and Archean eons. The uncertainties in the Earth--Moon model
yield $\mathit{Rm}$ values that can vary by a factor~2, and those in scaling law
(\Custref{28}{eq:scalinglawUtides}) even yield more larger variations. As such,
tidally driven dynamo action might have well never existed or ceased
${-}$4.25~Gy ago, or even have operated until the flow turbulence ceased
near ${-}$3.25~Gy. Therefore, a putative tidally driven dynamo was likely
less super-critical than previously thought. Smaller $\mathit{Rm}$ values not
only narrow the time window for a tidally driven dynamo, but also
weaken the magnetic field possibly sustained by such a mechanism. In
particular, since the amplitude of the tidal forcing decreases over
time (see Figure~\ref{fig:tdei}b), $\mathit{Rm}$--$\mathit{Rm}_c$ decreases during the
Hadean and Archean eons. Hence, we expect the magnetic field driven by
tidal forcing to weaken over time. This may be at odds with the
paleomagnetic measurements shown in Figure~\ref{fig:paleomag}(a), which
may suggest that the (maximum) amplitude of the geomagnetic field did
not vary much between ${-}$3.5 and ${-}$2.5~Gy. In the following, we focus
on dynamo action during the late Hadean and Archean eras (where more
powerful tidally driven dynamos may be expected).

\begin{figure}
\includegraphics{fig08}
{\vspace*{.3pc}}
\caption{\label{fig:Rmevolution}Magnetic Reynolds number $\mathit{Rm}$ as a
function of age~$\tau$.\ Second $y$-axis shows the \mbox{amplitude}
$\mathcal{U}$ of the expected turbulence according to formula
(\protect\Custref{28}{eq:scalinglawUtides}) when \mbox{$\tau \leq -3.25$}~Gy. Gray zone
shows the non-dynamo region $\mathit{Rm} \leq \mathit{Rm}_c$ with \mbox{$\mathit{Rm}_c = 100$}.
Predictions younger than ${-}$3.25~Gy are hidden, since the elliptical
instability was not triggered (see Figure~\ref{fig:tdei}b). Prior bound
from \citet{landeau2022sustaining} has been included for comparison.
Geological eons are also shown (H: Hadean, A: Archean).\looseness=-1}
\end{figure}

Since state-of-the-art \textsc{dns} cannot properly investigate dynamo
action for realistic \textsc{cmb} geometries and in geophysical
conditions (even for convective flows in spherical geometries),
appropriate scaling laws must be developed to establish a connection
between dynamo modelling and geophysical parameters. We assume that
$\mathit{Rm}$--$\mathit{Rm}_c$ was large enough at that time, to render the proposed
scaling laws for dynamo action in the vicinity of the onset invalid
\citep[e.g.][]{fauve2007scaling}. A fruitful approach in dynamo theory
is to consider power-based scaling laws \citep{christensen2006scaling,
christensen2010dynamo, davidson2013scaling}. For convection-driven
dynamos, such laws are often tested against numerical results
regardless of the parameters \citep[e.g.][]{oruba2014predictive,
schwaiger2019force}, and have provided useful insight into planetary
extrapolation. In such scaling theories, the saturated magnetic energy
density per unit of mass, which is given by
{\begin{equation}\label{eq:MagNRJ}
\mathcal{E}(\mathbi{B}) = \frac{1}{\rho_f V} \int_V
\frac{\mathbi{B}^2}{2 \mu}\,\mathrm{d}V \sim \frac{1}{2}
\frac{B^2}{\rho_f \mu}
\end{equation}}\unskip
where $\rho_f$ is the mean fluid density and $B$ is a typical magnetic
field strength in the dynamo region, should be somehow related to the
available power per unit of mass for dynamo action $\mathcal{P}_M =
\mathcal{P}/(\rho_f V)$, where $\mathcal{P}$ is the mean energy
production rate (in~W), and to the Joule dissipation per unit of mass
given by
{\begin{equation}\label{eq:epsilon_eta}
\epsilon_{\eta} = \frac{1}{\rho_f V} \int_V \frac{\eta}{\mu} |\nabla
\times \mathbi{B}|^2\,\mathrm{d}V \sim \frac{\eta}{\rho_f \mu}
\frac{B^2}{\ell_B^2},
\end{equation}}\unskip
where $\ell_B \sim \sqrt{2\eta\,\mathcal{E}(\mathbi{B})/
\epsilon_{\eta}}$ is a magnetic dissipation length scale. Note that
$\epsilon_{\eta}$ is expected to dominate over the viscous dissipation
$\epsilon_{\nu}$ for a turbulent dynamo when $\mathit{Pm} \ll 1$. Here, we
follow \citet{davidson2013scaling} to assume that, in the low-$\mathit{Ro}$
regime characterising the Earth's core, the magnetic field solely
scales with $\mathcal{P}_M$ and $R_{\mathrm{cmb}}$. If so, we obtain
from dimensional analysis that
{\begin{equation}\label{eq:lawBdavidson}
B \propto \sqrt{\rho_f \mu}\, (R_{\mathrm{cmb}} \mathcal{P}_M)^{1/3}.
\end{equation}}\unskip
Plugging typical values for the current Earth's liquid core (i.e.\ 
$\rho_f \sim 10^4~\mathrm{kg}{\cdot}\mathrm{m}^{-3}$, $\mathcal{P} \sim
1$--10~TW) into Equation~(\ref{eq:lawBdavidson}) yields $B \sim
1$--3~mT, which is a satisfactory upper bound for the current magnetic
field amplitude atop the core \citep[e.g.][]{gillet2010fast}. This law
generalises previous laws for buoyancy-driven dynamos 
\citep[e.g.][]{christensen2009energy}, which assume that $B$ varies
with the advected energy flux to the power $1/3$. As such, scaling law
(\ref{eq:lawBdavidson}) is often a cornerstone for extrapolation to
natural dynamos. However, in practice, other laws may also be relevant
(e.g.\ see in Appendix~\ref{sec:appendix} for weakly turbulent dynamos).
Finally, for the comparison with paleomagnetic data, note that only
upper bounds are generally obtained from scaling law
(\ref{eq:lawBdavidson}). Indeed, the surface field atop the dynamo
region is usually only a fraction of typical field amplitude $B$
\citep[e.g.][]{aubert2017spherical}. Moreover, the surface field is not
always purely dipolar. This is measured by introducing the dipolar
fraction $0 \leq f_{\mathrm{dip}} \leq 1$ as a pre-factor in the dynamo
scaling law \citep{christensen2006scaling}. Numerical simulations show
that self-sustained dynamos can have very different values of
$f_{\mathrm{dip}}$ over the parameter space, as reported for
convection-driven \citep[e.g.][]{oruba2014predictive,
schwaiger2019force} or precession-driven dynamos
\citep{cebron2019precessing}. Therefore, only a fraction of the
produced dynamo field has a dipolar morphology at the \textsc{cmb}.
Without further knowledge, we will discard such prefactors for tidally
driven dynamos below, to focus on upper-bound estimates.

Next, could we faithfully use scaling law (\ref{eq:lawBdavidson}) for
tidally driven dynamos? Since there are no dynamo simulations of tidal
flows in ellipsoids against which we can compare theoretical
predictions, we cannot be assertive. However, we expect the above
power-based arguments to remain largely valid for orbitally driven
flows. We have re-analysed in Figure~\ref{fig:scalingB2}
magnetohydrodynamics simulations of tidally driven flows performed in
Cartesian periodic boxes at $\mathit{Ro} \gtrsim 10^{-2}$ \citep{barker2014non}.
Note that it was not possible to separate $\epsilon_{\nu}$ and
$\epsilon_{\eta}$ in the re-analysis of the published
magnetohydrodynamic simulations. Yet, a good agreement is found with
scaling law (\ref{eq:lawBdavidson}), assuming that $\mathcal{P}_M \sim
\epsilon_{\nu} + \epsilon_{\eta}$ in a statistically steady state. Such
observations are very promising, but quantitative applications to
planets remain somehow speculative at present for tidal forcing.
Indeed, it is difficult to safely estimate $\mathcal{P}_M$ for
geophysical conditions.\vspace*{-.15pc}

\begin{figure}
\includegraphics{fig09}
{\vspace*{-.3pc}}
\caption{\label{fig:scalingB2}Simulations of tidally driven flows at
moderate values of $\mathit{Ro}$, performed in Cartesian periodic boxes
of unit length $L$ at $\mathit{Pm}=1$. Data from \citet{barker2014non}.
Typical magnetic field amplitude $B$ defined from
Equation~(\ref{eq:MagNRJ}), as a function of total dissipation
$\epsilon = \epsilon_{\nu} + \epsilon_{\eta}$.}
{\vspace*{-.6pc}}
\end{figure}

\subsection{Towards planetary conditions} \label{subsec:extrapolation}

We have found that dynamo scaling law (\ref{eq:lawBdavidson}) is likely
valid for tidally driven flows. However, it remains difficult to apply
this law in practice, because tidally driven turbulence is still poorly
understood at planetary core conditions. To make progress in this
direction, we first present in Section~\ref{subsubsec:rationale} the
arguments that underpin our planetary extrapolation in
Section~\ref{subsubsec:cartoon}.

\subsubsection{Heuristic rationale} \label{subsubsec:rationale}

\looseness=-1
We can further analyse the dynamo simulations presented in
Figure~\ref{fig:scalingB2}, as they can guide us for the planetary
extrapolation below. Indeed, these simulations were performed for
moderate values of the Rossby number $\mathit{Ro} \gtrsim 10^{-2}$, for which
scaling \mbox{arguments} have been proposed in rotating turbulence. In
such a regime, rotating turbulence usually exhibits strong nearly
two-dimensional (geostrophic) flows \citep[e.g.][]{le2019experimental,
le2020near}. Two different scaling theories have been proposed when $\mathit{Ro}
\lesssim 1$, such that that the mean viscous dissipation could either
scale as \citep{nazarenko2011critical, baqui2015phenomenological}
\changeEqNo{32a,b}
{\begin{equation}
\epsilon_{\nu} \sim \frac{u^3_{\ell_{\parallel},\ell_{\perp}}}
{\ell_{\perp}} \quad\text{or}\quad
\epsilon_{\nu} \sim \frac{u^3_{\ell_{\parallel},\ell_{\perp}}}
{\ell_{\parallel}},
\end{equation}}\unskip
\resetEqNo{32}\unskip
\noindent
where $u_{\ell_{\parallel},\ell_{\perp}}$ is the velocity amplitude at
the length scales $\ell_{\parallel}$ and $\ell_{\perp}$ (where
$\ell_{\parallel}$ is the length scale parallel to the rotation axis,
and $\ell_{\perp}$ is the one perpendicular to it). We see that we
recover from the two formulas the usual Kolmogorov prediction
$\epsilon_{\nu} \sim u^3_{\ell}/\ell$ for homogeneous isotropic
turbulence when $l_{\perp} = l_{\parallel} = \ell$. Assuming that
$u_{\ell_{\parallel}, \ell_{\perp}} \lesssim \mathcal{U}$ and
$\ell_{\perp} \sim \ell_{\parallel} \sim \ell$ at large scales, the two
laws should reduce at large scales to
{\begin{equation}\label{eq:epsU3/ltides}
\epsilon_{\nu} \sim \frac{\mathcal{U}^3}{\ell} \propto \beta^3
\end{equation}}\unskip
where $\ell$ is some length scale and $\mathcal{U}$ is the flow
amplitude given by Equations~(\Custref{28a,b}{eq:scalinglawUtides}). As
illustrated in Figure~\ref{fig:heuristicBarker}, it turns out that the
total dissipation $\epsilon = \epsilon_{\nu} + \epsilon_{\eta}$ in the
dynamo simulations is in very good quantitative agreement with the
above scaling law (i.e.\ when $\epsilon_{\nu}$ is replaced
by~$\epsilon$). Moreover, dynamo action operates in a weak-field regime
(not shown) with
$\mathcal{E}(\mathbi{B})/\mathcal{E}(\mathbi{u}) < 1$, where
$\mathcal{E}(\mathbi{u})$ is the kinetic energy.

\begin{figure}
\includegraphics{fig10}
\vspace*{-.2pc}
\caption{\label{fig:heuristicBarker}Simulations of tidally driven flows
at moderate values of $\mathit{Ro}$, performed in Cartesian periodic boxes of
unit length $L$ at $\mathit{Pm}=1$. Data from \citet{barker2014non}. Total
dissipation $\epsilon_{\nu} + \epsilon_{\eta}$ as a function of
equatorial ellipticity~$\beta$. Inset shows the magnetic field
amplitude as a function of~$\beta$.}
\vspace*{-.2pc}
\end{figure}

These results show that, for planetary extrapolation, the injected
power $\mathcal{P}_M$ for dynamo action can be estimated from the bulk
dissipation sustained by the turbulent flows in a statistically steady
state. Moreover, the theoretical prediction laws could hold in a dynamo
regime by considering the total dissipation (instead of the
viscous one). Such heuristic findings are consistent with the fact
that, in the nonlinear regime, the elliptical-instability mechanism
involves inertial-wave motions. Indeed, inertial waves are barely
affected by magnetic effects in realistic core conditions, and are such
that \mbox{$\mathcal{E}(\mathbi{B})/\mathcal{E}(\mathbi{u})<1$}.
This results from the dispersion relation of magnetohydrodynamic waves
in unbounded fluids \citep[e.g.][]{moffatt2019self}, and have also been
obtained numerically in an ellipsoid \citep{vidal2019fossil,
gerick2020pressure}. It is also worth noting that mean-field dynamo
theory shows that turbulent interactions of inertial waves could
sustain weak-field dynamo action \citep{moffatt1970dynamo,
moffatt1970turbulent}. Therefore, we assume below that inertial wave
motions play a key dynamical role in sustaining a weak-field dynamo
regime driven by tidal forcing, which will allow extrapolating our
theoretical laws to the Earth.

\subsubsection{A plausible extrapolation} \label{subsubsec:cartoon}

The regime $\mathit{Ro} \lesssim 1$ described above could apply to short-period
Hot Jupiters \citep[e.g.][]{barker2014non, barker2016non} or binary
systems \mbox{\citep[e.g.][]{vidal2019fossil},} in which tidal forcing can be
much stronger such that large \mbox{values} $\beta \to 10^{-2}$
could be obtained. On the contrary, the Earth's core is characterised
by much smaller values $\mathit{Ro} \sim 10^{-7}\ndash 10^{-6}$. For such
small values $\mathit{Ro} \ll 1$, tidal forcing is believed to sustain a regime
of inertial-wave turbulence \citep{le2017inertial, le2019experimental,
le2021evidence}. This is a regime of weak turbulence, characterised by
weakly nonlinear interactions of three-dimensional inertial waves. Like
in Kolmogorov turbulence, inertial-wave turbulence involves energy
being injected at some large scale, denoted by $\ell$ below. This
energy is then transmitted to smaller scales via a direct cascade in an
inertial range, in which the input power is balanced by dissipation at
every scale, until energy is finally dissipated at sufficiently small
scales at a rate~$\epsilon$. However, contrary to isotropic homogeneous
turbulence, inertial-wave turbulence is described by an anisotropic
energy spectrum \citep{galtier2003weak, galtier2023multiple}, depending
on the two length scales $\ell_{\parallel} \geq \ell_{\perp}$
introduced above.

By analogy with the regime $\mathit{Ro} \lesssim 1$ described in
Section~\ref{subsubsec:rationale}, we assume that the injected power
$\mathcal{P}_M$ available for dynamo action can be estimated in a
statistically steady state from the dissipation $\epsilon$ of turbulent
flows given by wave-turbulence theory when $\mathit{Ro} \ll 1$. This rests on
the fact that inertial waves are barely modified by magnetic effects at
core conditions, such nonlinear interactions of almost pure inertial
waves will still be triggered in a weak-field dynamo regime. The main
difference with the pure hydrodynamic regime would be that, for small
values $\mathit{Pm} \ll 1$, the dissipation would occur on a diffusive magnetic
length scale larger than the viscous one, such that the width of the
inertial range (in the wavenumber space) would be shortened compared to
the hydrodynamic case. Hence, we estimate the effective dissipation as
\citep{galtier2003weak, galtier2023multiple}
{\begin{equation}\label{eq:IWT}
\epsilon_{\mathit{Ro} \ll 1} \sim \frac{\ell_{\parallel}}{\ell_{\perp}}
\frac{u_{\ell_{\parallel},\ell_{\perp}}^4}{\Omega_s\ell_{\perp}^2}
\end{equation}}\unskip
with an expected $\mathcal{O}(1)$ pre-factor from theory
\citep{zeman1994note, zhou1995phenomenological}, where
$u_{\ell_{\parallel},\ell_{\perp}}$ is the velocity amplitude at the
length scales $\ell_{\parallel} \geq \ell_{\perp}$. Since the
dissipation is a constant in the theory, it can be estimated from the
knowledge of $u_{\ell_{\parallel},\ell_{\perp}}$ at some length scales.

Figure~\ref{fig:tdei}(b) shows that tidal forcing can only inject
energy at rather large scales $\ell$ (i.e.\ at fraction of the core
radius denoted by $\alpha_2$ below), for which we may assume
$\ell_{\perp} \sim \ell_{\parallel} \sim \ell$. Moreover, scaling law
(\Custref{28}{eq:scalinglawUtides}) shows that $u_{\ell_{\parallel},
\ell_{\perp}} \lesssim \mathcal{U}$ at large scales. Altogether, this
allows us to estimate the mean dissipation in a regime of inertial-wave
turbulence as
{\begin{equation}\label{eq:IWT2}
\epsilon_{\mathit{Ro} \ll 1} \lesssim \frac{\mathcal{U}^4}{\Omega_s \ell^2}
\end{equation}}\unskip
for tidal forcing, with $\ell = \alpha_2 R_{\mathrm{cmb}}$ and
$\alpha_2 \simeq 0.01\ndash1$. Formula (\ref{eq:IWT2}) is compatible
with an energy spectrum scaling as $\ell^2$ at large scales
\citep{galtier2003weak}, which is consistent with prior studies in
rotating turbulence \citep[e.g.][]{baroud2002anomalous,
thiele2009structure, li2025energy}. For completeness, we remind the
reader that another predictive law ought to be used for tidally driven
turbulence with moderately small values $\mathit{Ro} \lesssim 1$. This should be
given by dissipation law (\ref{eq:epsU3/ltides}) as explained above,
together with an energy injection at the large scale $\ell = \alpha_2
R_{\mathrm{cmb}}$ with $\alpha_2 = 0.01\ndash1$ that is consistent
with Figure~\ref{fig:heuristicBarker}.

Finally, we can combine Equations
(\ref{eq:epsU3/ltides})--(\ref{eq:IWT2}) with dynamo scaling law
(\ref{eq:lawBdavidson}) to obtain
{\begin{equation}\label{eq:BIWTtides}
B \propto \sqrt{\rho_f \mu} \begin{cases}
\alpha_2^{-1/3}\,\mathcal{U} & \text{if } \mathit{Ro} \lesssim 1\\
\alpha_2^{-2/3}\,\mathit{Ro}^{1/3} \mathcal{U} & \text{if } \mathit{Ro} \ll 1
\end{cases}
\end{equation}}\unskip
for tidal forcing in the early core, which yields
{\begin{equation}
B \propto \begin{cases}
\beta & \text{if } \mathit{Ro} \lesssim 1\\
\beta^{4/3} & \text{if } \mathit{Ro} \ll 1
\end{cases}
\end{equation}}\unskip
by using scaling law (\Custref{28}{eq:scalinglawUtides}) for the amplitude of
turbulence. Note that the magnetic field amplitude is expected to be
$\mathit{Ro}^{1/3}$ smaller when $\mathit{Ro} \ll 1$ than in a nearly two-dimensional
(geostrophic) regime at $\mathit{Ro} \lesssim 1$. We can also calculate from law
(\ref{eq:BIWTtides}) the ratio of the magnetic energy to the kinetic
energy per unit of mass $\mathcal{E}(\mathbi{u})$ as
{\begin{equation}
\mathcal{E}(\mathbi{B})/\mathcal{E}(\mathbi{u}) \propto
\begin{cases}
\alpha_2^{-2/3} & \text{if } \mathit{Ro} \lesssim 1\\
\alpha_2^{-4/3}\,\mathit{Ro}^{2/3} & \text{if } \mathit{Ro} \ll 1.
\end{cases}
\end{equation}}\unskip
Hence, only weak-field dynamos with $\mathcal{E}(\mathbi{B})/
\mathcal{E}(\mathbi{u}) \ll 1$ are expected for a wave-turbulence
when $\mathit{Ro} \ll 1$.

The predictions from scaling law (\ref{eq:BIWTtides}) as a function of
the age are illustrated in Figure~\ref{fig:takehomeB}. We obtain as a
typical estimate $B \sim 10^{-5}\ndash10^{-3}$~mT during the Hadean
period 4.25--4~Gy ago (i.e.\ when tidal forcing was maximal), and the
field amplitude would then ultimately decrease until ${-}$3.25~Gy (since
tidal forcing had a decreasing amplitude during the Archean era).\ The
predicted amplitudes are thus at least ten times smaller than the
\textsc{cmb} field~$B_{\mathrm{cmb}}$, which is estimated from surface
measurements with Equation~(\ref{eq:Bcmbfromdata}).\ Therefore, it
seems unlikely that the ancient geodynamo was solely sustained by a
wave-turbulence regime at $\mathit{Ro} \ll 1$ driven by tidal
forcing.

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig:takehomeB}Paleointensity at the \textsc{cmb}
$B_{\mathrm{cmb}}$, reconstructed Figure~\ref{fig:paleomag} using
Equation~(\ref{eq:Bcmbfromdata}) with $R_s = 6378$~km and
$R_{\mathrm{cmb}} = 3480$~km, and upper bounds for $B$ (dotted and
dashdot lines) from scaling law (\ref{eq:BIWTtides}) when $\mathit{Ro} \ll 1$.
Numerical estimates using $\rho_f \simeq 1.1 \times 10^4~\mathrm{kg}
{\cdot}\mathrm{m}^{-3}$ for a mean density (accounting for the mass of
the outer and inner cores), $\eta =
1~\mathrm{m}^{2}{\cdot}\mathrm{s}^{-1}$, and $\mu = 4 \uppi \times
10^{-7}~\mathrm{H}{\cdot}\mathrm{m}^{-1}$. Geological eons are also shown
(H:~Hadean, A:~Archean).}
\end{figure}

\section{Geophysical discussion} \label{sec:discussion}

Our extrapolation suggests that tidal forcing may have been too weak to
generate a dynamo magnetic field with an amplitude matching the
(scarce) Hadean and Archean paleomagnetic measurements, at least for
flows in a wave-turbulence regime. Since several assumptions were made
to arrive at this conclusion, we discuss below if our main results
could be modified or not by adopting other modelling
choices.

\subsection{Influence of the orbital scenario}


Obviously, one source of uncertainty arises from the parameters given
by the orbital scenario, which was less constrained in the distant
past. We have here employed the semi-analytical model by
\citet{farhat2022resonant}, as it fits most of the available geological
\mbox{proxies} for the history of the Earth--Moon system and reproduce
the age of the Moon's formation fairly well. However, we should assess
how the results could be affected by adopting other orbital scenarios.
Most models reasonably well agree on the recent Earth--Moon evolution
(i.e.\ after ${-}$1~Gy), as they are constrained by geological data.
However, the models can significantly differ further back in time.

The comparison between different orbital models is shown in
Figure~\ref{fig:orbitaluncertainties}. Note that we have discarded
models that cannot be extrapolated during the Hadean and the Archean
eras \citep[e.g.][]{green2017explicitly, zeeden2023earth,
zhou2024earth}. We find that the presented models fall within the error
bars of the model by \citet{farhat2022resonant}. This is probably due
to the conservation of angular momentum that is enforced in all the
models, which gives good constraints on the Earth--Moon distance for a
given value of the Earth's spin rate. Yet, the evolution curves can
differ in time between the models. In particular, the model by
\citet{tyler2021tidal} does not reproduce the estimated age of the
Moon, overestimating the Earth--Moon distance during the Hadean and
Archean eras. A similar conclusion can be drawn for
\citet{daher2021long}, as shown in Figure~6 of
\citet{eulenfeld2023constraints}. On the contrary, the model by
\citet{touma1994evolution} is found to be quite close to the orbital
model by \citet{farhat2022resonant}. Consequently, our results already
account for most of the uncertainties of the community regarding the
orbital scenario. However, the next generation of orbital models may
change the overall picture. 

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig:orbitaluncertainties}Uncertainties in the orbital
model. Comparison between the predictions in the distant past from
different orbital models \citep{touma1994evolution, tyler2021tidal,
farhat2022resonant} for the Earth--Moon distance $a_M/R_s$ and the
length of day.\ Inset is analogous to Figure~\ref{fig:forcing1}(a), and
to Figure~6 in \citet{eulenfeld2023constraints}. Geological eons are
also shown (H:~Hadean, A:~Archean, P:~Proterozoic).}
\end{figure}

\subsection{Rheological uncertainties}

Another key quantity in the model is the present-day value of the
equatorial ellipticity $\beta(0)$ at the \textsc{cmb}. Seismological
observations of the peak-to-peak amplitude of the \textsc{cmb}
topography \citep[e.g.][]{koper2003constraints, sze2003core} suggest
that the current value of the equatorial ellipticity may be
order-of-magnitude larger than our considered value in
Equation~(\ref{eq:betaradius}). However, as this elliptical deformation
results from mantle dynamics, it is nearly in phase with the spin
frequency of the Earth (i.e.\ $\Delta \Omega \approx 0$). As such, we do
not expect this elliptical deformation to play a role in the
elliptical-instability mechanism. This is only the asynchronous
component of the ellipticity, which is driven by tidal forcing, that is
able to drive an elliptical instability inside the liquid core. The
\mbox{amplitude} of the tidal potential is quite well \mbox{constrained} at the
present time \citet{AGNEW2015151}, such that the main uncertainties on
the value of $\beta(0)$ probably come from the mantle's rheology.

We have employed the \textsc{prem} model
\citep{dziewonski1981preliminary} to account for the Earth's rheology
in Figure~\ref{fig:tidesmodel2}. More recent Earth models could
naturally be used together with the open-source code \textsc{TidalPy}
\citep{renaud2023tidalpy}, but this is beyond the scope of the present
study. More importantly, we have assumed that the mantle remained rigid
over time. This seems to be a reasonable assumption throughout most the
Earth's history, but the Earth's mantle was probably molten after the
giant impact that formed the Moon \citep[at least partially,
e.g.][]{nakajima2015melting}. Then, a crystallising basal magma ocean
(\textsc{bmo}) probably survived in the aftermath for millions of years
to a billion of years \citep[e.g.][]{labrosse2007crystallizing,
boukare2025solidification}. How such a \textsc{bmo} could have
interacted with tidal forcing remains largely unknown
\citep[e.g.][]{korenaga2025tidala, korenaga2025tidalb,
korenaga2025tidalc}, as well as its possible interactions with core
flows. A crude extrapolation of old fluid-dynamics experiments on the
spin-up of rotating immiscible fluids \citep{pedlosky1967spin,
o1992spin} might suggest that \textsc{cmb} dissipation is not
significantly reduced in the presence of a low-viscosity \textsc{bmo}
(due to interfacial friction with the core, and Ekman friction with the
solid mantle above). However, further work is needed to elucidate this
question.\looseness=-1

\subsection{Early core's conditions remain elusive}

In addition to the orbital parameters and mantellic properties, there
are also uncertainties regarding the physical state of the liquid core
in the distant past. In particular, additional dissipation mechanisms
may have operated within the core, hindering the onset of tidally
driven turbulence over geological timescales.\looseness=1

\subsubsection{Hydrodynamic dissipation in the core}

The value of the core's viscosity $\nu$ appears to be another physical
quantity of interest when determining the occurrence of tidally driven
turbulence. Indeed, as shown in Figure~\ref{fig:tdei}, it controls the
leading-order damping inhibiting the growth of unstable flows. We have
chosen the standard value $\nu=10^{-6}~\mathrm{m}^2{\cdot}\mathrm{s}^{-1}$
\citep[e.g.][]{de1998viscosity}, but the core value may be between
$10^{-7}~\mathrm{m}^2{\cdot}\mathrm{s}^{-1}$ and $3 \times
10^{-6}~\mathrm{m}^2{\cdot}\mathrm{s}^{-1}$ \citep{mineev2004viscosity}.
Thus, in Figure~\ref{fig:tdei}, the damping term may be decreased by a
factor of $\sqrt{10} \approx 3$ with the lowest value, or increased by
a factor $\sqrt{3} \approx 2$ with the largest one. Similar effects
could be obtained if the liquid core were rotating in the bulk faster
or slower than the mantle. We have here assumed that the liquid core is
co-rotating with the mantle at the angular velocity~$\Omega_s$, but the
core may be rotating a bit faster than the mantle according to some
tidal models \citep[e.g.][]{wahr1981effect}. Our model would then
remain largely unchanged, except that the value of $E$ would be smaller
as it must be based on the fluid rotation \citep[see
in][]{greenspan1969theory}.

Note that possible interactions between tidal forcing and buoyancy
effects, such as with either convective flows or density
stratification, could also provide additional dissipation mechanisms in
the core. Convection would essentially sustain slowly varying flows in
the core, whereas tidal forcing may mainly \mbox{trigger} high-frequency
motions with inertial-wave turbulence. Because of this separation of
time scales, strong interactions are not expected between both flows
\citep[unless convective flows could locally cancel out the rotation of
the core, e.g.][]{de2023interactions, de2023tidal}. Convection would
then provide an additional damping, but the latter might be rather weak
for fast tidal forcing \citep[e.g.][]{duguid2020convective,
duguid2020tidal, vidal2020efficiency, vidal2020turbulent}. Note that
the early core may have been instead (at least partially) stably
stratified in density, according to thermal evolution modelling
\citep[only for large values of the thermal conductivity of liquid iron
at core conditions, e.g.][]{labrosse2015thermal} or if the early core
had been insufficiently mixed after giant impacts
\citep{landeau2016core}. In this case, density stratification would not
modify the largest growth rate of the elliptical instability $\sigma^i$
\citep{vidal2019fossil}. Similarly, boundary-layer theory suggests that
the value of the damping rate would not change much with a stable
density stratification \citep{friedlander1989asymptotic}, such that the
predictions shown in Figure~\ref{fig:tdei}(b) may remain quantitatively
accurate. However, preliminary simulations show that density
stratification could significantly weaken the strength of radial flows
and mixing \citep{vidal2018magnetic}. We may thus expect dynamo action
to be less favourable with stratification, but further work is needed
to carefully investigate the interplay with density stratification.
\looseness=1

\subsubsection{What about magnetic damping?}

It is unclear whether the early core was subject to strong magnetic
effects during the Hadean era, that is before an undoubted dynamo
action was recorded in paleomagnetic data. However, the Earth's core
had a magnetic field since from at least ${-}$3.5~Gy, whatever its
dynamical origin. Hence, we may wonder if the presence of an ambient
magnetic field (possibly of different origin) could alter the onset of
tidally driven flows within the core during the Archean era.
\looseness=1

Actually, the elliptical-instability mechanism would be largely
unchanged in the presence of a background field. To quantify magnetic
effects, we usually introduce the Lehnert number defined as
{\begin{equation}
\mathit{Le}=\frac{B_{\mathrm{cmb}}}{\Omega_s R_{\mathrm{cmb}}\sqrt{\rho_f\mu}}.
\end{equation}}\unskip
A typical value is $\mathit{Le} \sim 10^{-4}\ndash10^{-3}$ in the early core,
which is similar to the current value in the Earth's core. For such low
values, the (high-frequency) inertial waves responsible for the
elliptical instability are only weakly affected by the magnetic
field{\break}
\citep[only in the absence of an inner core, otherwise see
in][]{lin2018tidal}. Given the expected forcing frequency (in
dimensionless units) $|1-\Omega_{\mathrm{orb}}/\Omega_s| \gg \mathit{Le}$ of
tidal forcing inside the early core, the injection of energy can only
come from resonances with nearly inertial waves in the resonance
condition \citep[e.g.][]{kerswell1994tidal, vidal2019fossil}. Then, the
diffusionless growth rate $\sigma^i$ of the elliptical instability,
given in Equation~(\ref{eq:sigmatdei}), will remained unchanged at the
leading order in $\mathit{Le}$. However, Ohmic diffusion below the \textsc{cmb}
will provide an additional magnetic damping term $\sigma^{\eta}$ in the
growth rate equation. \citet{kerswell1994tidal} showed that the
magnetic damping $\sigma^{\eta}$ and the viscous one $\sigma^{\nu}$ are
probably of comparable order of magnitude for plausible core fields.
Therefore, the effective damping term in Figure~\ref{fig:tdei} may be
increased by a factor of (up to) two. To summarise, if we consider an
ambient magnetic field in the early core, tidal forcing may not have
been strong enough to sustain turbulence during the Hadean and Archean
eras.\vspace*{-.2pc}

\subsection{Scaling-law uncertainties} \vspace*{-.15pc}

The presented tidal scenario heavily relies on different scaling laws,
which must be used to extrapolate the results in the turbulent regime
to core conditions. We have done our best to constrain the various
scaling laws as much as possible, by uniquely combining an Earth--Moon
evolution scenario with theoretical results and re-analyses of the most
up-to-date numerical simulations of tidally driven turbulent flows.
Yet, other modelling uncertainties also call the dynamo extrapolation
into question. In particular, the outcome of turbulent tidally driven
flows and dynamo action remains uncertain as discussed below.

\subsubsection{Comparison with earlier works}

We have shown in Figure~\ref{fig:Rmevolution} that the prior
predictions presented in \citet{landeau2022sustaining} are likely
overestimated, since we have used the same modelling assumptions in our
study. This mainly results from the chosen numerical prefactors in the
\mbox{different} extrapolation laws, which were set to unity for a first
proof-of-concept study.

Notably, we have revisited the value of $\alpha_1$ in velocity scaling
law (\Custref{28}{eq:scalinglawUtides}), showing that $\alpha_1 = 1$ disagrees
with the available numerical results gathered in
\mbox{Figure}~\ref{fig:turbulence}.\ Instead, the numerical simulations
\mbox{suggest} that $\alpha_1 \simeq 0.25 \pm 0.15$ for planetary
extrapolation. Note that we have estimated the value of $\mathcal{U}$
directly from~$u_z$.\ Actually, we found that $|u_z| \sim
0.7\ndash0.8\,|\mathbi{u}|$ for most of the dynamo simulations
presented in Figures~\ref{fig:scalingB2} and~\ref{fig:heuristicBarker},
where $|\mathbi{u}|$ is estimated from the kinetic energy
$\mathcal{E}(\mathbi{u})$. This is in good agreement with a
preliminary estimate of the radial mixing induced by tidally driven
flows \citep[see Figure~4 in][]{vidal2018magnetic}. However, if the
horizontal mixing were also important for dynamo action, the velocity
prefactor $\alpha_1$ may be increased up to~0.6. Yet, this would only
lead to a small increase of the magnetic field amplitude (see the
dashdot line in Figure~\ref{fig:takehomeB}). Future research work may
thus shed new light on this estimate. For instance, it is unknown
whether the value of $\alpha_1$ could be increased or not when $E$ is
lowered (and similarly~$\beta$) but, heuristically, we always expect
$\alpha_1 < 1$ to sustain tidally driven turbulence in the core over
long time scales. Otherwise, tidally driven turbulent flows would
become of the same amplitude as the shear component of the forced flow
$\mathbi{U}_0$ when $\alpha_1 \to 1$, which would temporarily stop
the injection of energy and hinder the development of a wave-turbulence
regime. This may echo some peculiar regimes in rotating turbulence,
which are sometimes observed in experiments involving
growth-and-collapse phases \citep[e.g.][]{mcewan1970inertial,
malkus1989experimental}.\ We might also obtain during the energy growth
regimes of self-killing dynamos \citep[e.g.][]{reuter2009wave,
fuchs1999self}, which have already been reported in simulations of
precession-driven flows in a sphere \citep{cebron2019precessing}.\ If
such exotic regimes were obtained, our theoretical predictions for the
flow turbulence and its dynamo capability would be\break invalid.

Finally, there are also strong uncertainties associated with dynamo
scaling law (\ref{eq:lawBdavidson}). Obviously, the lack of numerical
methods for simulating self-sustained dynamos in ellipsoids currently
hampers our ability to assess its validity for planetary relevant
regimes. For instance, the inset of Figure~\ref{fig:heuristicBarker}
shows that the simulations are compatible with
{\begin{equation}
B \approx 0.55 \sqrt{\rho_f \mu}\,(R_{\mathrm{cmb}}\epsilon)^{1/3},
\end{equation}}\unskip
where $\epsilon$ is given by scaling law (\ref{eq:epsU3/ltides}) in the
$\mathit{Ro} \lesssim 1$ regime.\ The observed numerical prefactor in the above
law shows that the dynamo scaling laws generally come with
non-unit numerical prefactors. For instance, the dynamo predictions
could be \mbox{overestimated} by a factor of nearly 2 if the observed
prefactor were the same in the $\mathit{Ro} \ll 1$ regime.

\subsubsection{Disputed wave-turbulence regime}

\looseness=-1
We have assumed that tidal forcing establishes a regime of
inertial-wave turbulence, as often postulated after
\citet{le2017inertial, le2019experimental, le2021evidence}. However,
inertial-wave turbulence is a research topic that is far from being
well understood, especially because it is very challenging to obtain in
simulations or in laboratory experiments. As such, current
investigations still strive observing predictions of wave-turbulence
theory in set-ups mimicking as much as possible the theoretical model
\citep[e.g.][]{yarom2014experimental, monsalve2020quantitative}.
Therefore, it is still unclear whether the quantitative predictions of
wave turbulence theory could be directly applied to rotating turbulent
flows in bounded geometries at $\mathit{Ro} \ll 1$, such as in the early Earth's
core with $\mathit{Ro} \sim 10^{-7}$. On the contrary,
Figure~\ref{fig:heuristicBarker} shows that simulations at moderate
values $\mathit{Ro} \sim 10^{-2}$ already agree fairly well with the dissipation
law~(\ref{eq:epsU3/ltides}).

\looseness=-1
Moreover, how magnetic effects modify inertial-wave turbulence remains
speculative so far. We conjecture that inertial-wave turbulence can
persist in a weak-field dynamo regime, since high-frequency inertial
waves are barely affected by magnetic fields in an ellipsoid. The main
difference would be that the dissipation would occur on a diffusive
magnetic length scale for small values $\mathit{Pm} \ll 1$. This may agree with
the qualitatively view drawn from magnetohydrodynamics simulations of
tides \citep{barker2014non, vidal2018magnetic} and precession
\citep{barker2016turbulence, kumar2024dynamo}, showing that the
obtained turbulence could be largely unchanged in a weak-field dynamo
regime. Future numerical works, for instance using local simulations
at smaller values of $\mathit{Pm}$, may shed new light on this point.

\subsubsection{A magnetostrophic dynamo regime?}

Apart from wave turbulence, another possibility could be that tidally
driven turbulence at $\mathit{Ro} \ll 1$ could be in a regime of geostrophic
turbulence characterised by the presence nearly two-dimensional
(geostrophic) flows. This would challenge the applicability of
wave-turbulence theory when $\mathit{Ro} \ll 1$ \citep[e.g.][]{gallet2015exact},
since geostrophic flows are filtered out in the wave-turbulence theory.
However, a regime of hydrodynamic geostrophic turbulence is likely to
be modified by magnetic effects. This notably rests on the properties
of low-frequency waves in rotating systems subject to magnetic effects.
Indeed, slow quasi-geostrophic wave motions morph into various
low-frequency waves shaped by the magnetic field, such as torsional
Alfv\'en waves \citep[e.g.][]{luo2022waves1} and magneto-Coriolis waves
\citep[e.g.][]{luo2022waves2,gerick2024interannual}. Consequently, a
magnetostrophic regime is expected to supersede purely geostrophic
turbulence when $\mathit{Ro} \ll 1$ \citep[e.g.][]{hollerbach1996theory}. In
such a regime, the Coriolis force could balance the Lorentz force such
that $B$ would be given by \citep[e.g.][]{christensen2010dynamo}
{\begin{equation}\label{eq:Bmagnetostrophic}
B \propto \sqrt{\rho_f \mu} \sqrt{l_B \Omega_s \mathcal{U}}.
\end{equation}}\unskip
Formula (\ref{eq:Bmagnetostrophic}) would then give the field estimate
$B \lesssim 5$~mT in the early Earth's core at ${-}$4~Gy, using the same
physical parameters as in Figure~\ref{fig:takehomeB} and with
$\ell_B/R_{\mathrm{cmb}} = 0.02$ as in \citep[as
in][]{starchenko2002typical}. A magnetostrophic regime may be a better
candidate than inertial-wave turbulence to reach a strong-field dynamo
regime. Unfortunately, there is no evidence so far that magnetostrophy
could be achieved with tidal forcing.

\section{Concluding remarks} \label{sec:conclusion}
\subsection{Summary}

We have thoroughly explored the capability of tidal forcing to explain
the ancient geodynamo. We have combined geophysical constraints from
Earth--Moon evolution models, theoretical predictions, and re-analyses
of recent results on tidally driven turbulent flows. This has allowed
us to show that tidal forcing was likely strong enough to sustain
turbulence prior to ${-}$3.25~Gy, and possibly dynamo action. However,
the self-sustained magnetic field was certainly too weak to explain
paleomagnetic measurements if tides were only sustaining an
inertial-wave turbulence regime in the ancient core. We hope that our
results will guide future studies of tidally driven flows, to possibly
strive beyond the limits we have identified. For example, we outlined
that magnetostrophic turbulence driven by tidal forcing might produce a
strong-field dynamo consistent with the observations. However, evidence
of such a tidally driven regime remains to be found.

Alternatively, other mechanisms in the core could be invoked to explain
the ancient geodynamo \citep[see in][]{landeau2022sustaining}. For
instance, the \textsc{cmb} heat flow extracted by mantle convection
could power thermal convection in the ancient core
\citep[e.g.][]{al2024coupled}, but this requires low-to-moderate values
of the thermal conductivity of liquid iron at core conditions
\citep[e.g.][]{hsieh2025moderate, andrault2025long}. Alternatively,
exsolution (or precipitation) of light elements below the \textsc{cmb}
could sustain double-diffusive turbulent convection in the core
\citep[e.g.][]{monville2019rotating}. Yet, a difficulty with such
mechanisms may be to obtain vigorous enough turbulence, or to sustain
magnetic fields with a dipolar morphology. Actually, a weak tidally
driven dynamo might have been important in providing an optimal
magnetic seed of finite amplitude to kick-start an efficient
convection-driven geodynamo \citep{cattaneo2022earth}.

\subsection{Towards precession in the Moon?}

Beyond Earth, the Moon is another planetary body for which we have
geological samples that could help to understand planetary dynamos over
long time scales. The analysis of Apollo samples
(Figure~\ref{fig:Moonpaleomag}) has revealed that the Moon had a
planetary dynamo field in the distant past
\citep[e.g.][]{wieczorek2023lunar}, with a (possibly intermittent)
magnetic activity from about ${-}$4.2~Gy until at least ${-}$1.9~Gy. Recent
analyses confirmed that the Moon was first characterised by a
high-field epoch \citep{lepaulard2019survey, jung2024assessing}, which
persisted from ${\sim}$3.9 until ${\sim}$3.5~Gy ago, with measured
surface field intensities of 40--110~$\upmu$T. This high-field period
was then succeeded by an epoch with a declining field
\citep{tikoo2014decline, strauss2021constraining}, whose surface
amplitude fell to below $10~\upmu$T after ${-}$3.5~Gy. Note that it
remains unclear whether a lunar core dynamo was long-lived
\citep[e.g.][]{cai2024reinforced, cai2025persistent}, episodic
\citep[e.g.][]{evans2022episodic}, or instead limited to the first
\mbox{hundred} millions of years of the Moon's life \citep{zhou2024lunar}. In
any case, the magnetic activity certainly ceased between ${-}$1.9 and
${-}$0.8~Gy \citep{tikoo2017two, mighani2020end}.

\begin{figure}
\includegraphics{fig13}
\vspace*{-.35pc}
\caption{\label{fig:Moonpaleomag}Paleointensity at the Moon's surface,
inferred from Apollo rocks. Olive region: high-field dynamo epoch. Old
Apollo values extracted from \citet{lepaulard2019survey}.}
\vspace*{-.25pc}
\end{figure}

Actually, such paleomagnetic data put very tough constraints for dynamo
action inside the ancient Moon. As inferred from
Equation~(\ref{eq:Bcmbfromdata}), any viable \mbox{dynamo} scenario
should be capable of generating a magnetic field that is 10 larger in
the Moon's core than in the Earth's one, despite its core radius being
about 10 times smaller (e.g.\ $B_{\mathrm{cmb}} \sim 10$--70~mT during
the high-field epoch, with $R_{\mathrm{cmb}} \approx 200$--380~km).
However, standard dynamo scenarios currently fail to explain the
observed field values \citep{wieczorek2023lunar}. It has been proposed
that the ancient Moon's dynamo could result from precession-driven
flows \citep[e.g.][]{dwyer2011long}, which have many dynamical
similarities with tidal flows \citep[e.g.][]{vidal2024geophysical}. The
\textsc{g\'eodynamo} team and its collaborators have worked on
precession for a long time, making pioneering contributions to the
hydrodynamics \citep[e.g.][]{noir2001numerical, lin2015shear} and
magnetohydrodynamics \citep{lin2016precession, cebron2019precessing} of
such flows. The present work, at the crossroad of the
\textsc{g\'eodynamo}'s research activities, may thus also guide future
studies of precession-driven flows.

\section*{CRediT authorship contribution statement}

\textbf{J\'er\'emie Vidal:} conceptualisation, formal \mbox{analysis},
methodology, software, visualisation, funding, writing---original
draft, writing---review and \mbox{editing}.

\noindent
\textbf{David C\'ebron:} funding, writing---review and editing. Both
authors gave final approval for submission, and agreed to be held
accountable for the work performed therein.

\section*{Acknowledgements}

The authors warmly acknowledge an anonymous referee for the thorough
revision of the manuscript, which helped to greatly improve its
quality. JV thanks Les Houches School of Physics for the hospitality
and stimulating discussions during the workshop ``Physics of Wave
Turbulence and beyond'', which occurred in September 2024 and where the
main idea of the study first emerged. JV also acknowledges the
organisers of the Advanced Summer School ``Mathematical Fluid
Dynamics'', which was held in Corsica in April 2025 and where part of
the work was \mbox{finalised}.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organisation that could benefit from this article, and have
declared no affiliations other than their research organisations.

\section*{Funding}

 JV received funding from \textsc{ens} de Lyon
under the programme ``Terre \& Plan\`etes''. DC received funding from
the European Research Council (\textsc{erc}) under the European Union's
Horizon 2020 research and innovation programme (grant agreement No
847433, \textsc{theia} project). As part of the \textsc{g\'eodynamo}
team, DC greatly acknowledges the support from the French Academy of
Sciences \& Electricit\'e de France.

\CDRGrant[ERC]{847433}

\section*{Supplementary materials}

%Supporting information for this article is available on the journal's
%website under \printDOI\ or from the author.
%\CDRsupplementaryTwotypes{supplementary-material}{\cdrattach{crgeos-324-suppl.pdf}}

The \textsc{matlab} code used to compute the Earth's flattening in
Figure~\ref{fig:tidesmodel2} is available at
\url{http://frederic.chambat.free.fr/hydrostatic/HYDROSTATIC\_dec2011.zip}.
The code \textsc{TidalPy} used to compute the tidal deformations in
Figure~\ref{fig:tidesmodel2} is available at
\url{https://doi.org/10.5281/zenodo.14867405}.

\back{}

\appendix{}
\def\appendixlabel{Appendix \Alph{section}}

\section{A weakly turbulent dynamo law?}\label{sec:appendix}

Scaling law (\ref{eq:lawBdavidson}) has proven to fairly well reproduce
the dynamo simulations shown in Figure~\ref{fig:scalingB2}. However,
other scaling laws may also be appropriate for orbitally driven
dynamos. For instance, we can assume that the magnetic energy saturates
when the injected power is statistically balanced by Joule heating
(i.e.\ $\mathcal{P}_M \sim \epsilon_{\eta}$). This simple reasoning
yields \citep{christensen2004power}
{\begin{equation}\label{eqA.1}
B\propto \sqrt{\rho_f\mu}\left(\frac{\ell^2_B}{\eta}
\epsilon_{\eta}\right)^{1/2},
\end{equation}}\unskip
where $\ell_B$ is defined as in Equation~(\ref{eq:epsilon_eta}). This
scaling law was found to agree quite well with numerical results
\citep[e.g.][]{christensen2010dynamo, oruba2014predictive}. However,
the above scaling law cannot generally be employed as a predictive
dynamo law for the extrapolation to planets. Indeed, $\ell_B$ is
largely unknown for core conditions, since it could be non-constant and
regime-dependent. Yet, it is sometimes assumed to be some fraction of
the core radius \citep[as found in some simulations,
e.g.][]{starchenko2002typical}. Similarly, relating $\epsilon_{\eta}$
to the input parameters of the problem is often elusive.

We have re-analysed in Figure~\ref{fig14} the dynamo simulations of
precession-driven flows in a sphere reported in
\citet{cebron2019precessing}. The simulations show that $B\propto
\epsilon_{\eta}^{1/2}$ when $B$ is estimated from
Equation~(\ref{eq:MagNRJ}), which agrees with scaling law
(\ref{eqA.1}). Moreover, the magnetic length scale is found to only
weakly vary across the parameter space, which suggests that the dynamos
are not very turbulent.

\begin{figure}
\includegraphics{fig14}
{\vspace*{-.3pc}}
\caption{\label{fig14}Typical magnetic field amplitude $B$ defined from
Equation~(\ref{eq:MagNRJ}), as a function of magnetic dissipation
$\epsilon_{\eta}$ in numerical simulations of precession-driven flows
inside a sphere \citep[data from][]{cebron2019precessing}. The
calculated magnetic dissipation length scale is
$\ell_B/R_{\mathrm{cmb}} = 0.077 \pm 0.046$. Colour bar shows the value
of $\log_{10}(\epsilon_{\eta}/(\epsilon_{\eta} + \epsilon_{\nu}))$ for
each simulation.}
{\vspace*{-.6pc}}
\end{figure}

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