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\DOI{10.5802/crgeos.256}
\datereceived{2023-06-20}
\daterevised{2024-03-07}
\dateaccepted{2024-03-12}
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\begin{document}

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\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Dynamic regimes in planetary cores: $\tau$--$\ell$ diagrams}

\alttitle{R\'egimes dynamiques dans les noyaux plan\'etaires : 
diagrammes $\tau$--$\ell$}

\author{\firstname{Henri-Claude} \lastname{Nataf}\CDRorcid{0000-0001-6737-1314}\IsCorresp}
\address{Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD,
Univ. Gustave Eiffel, ISTerre, 38000 Grenoble, France}
\email[H.-C. Nataf]{Henri-Claude.Nataf@univ-grenoble-alpes.fr}

\author{\firstname{Nathana\"el} \lastname{Schaeffer}\CDRorcid{0000-0001-5206-3394}}
\addressSameAs{1}{Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS,
IRD, Univ. Gustave Eiffel, ISTerre, 38000 Grenoble, France}
\email[N. Schaeffer]{Nathanael.Schaeffer@univ-grenoble-alpes.fr}

\begin{abstract}
Planetary cores are the seat of rich and complex fluid dynamics, in
which the effects of rotation and magnetic field combine. The
equilibria governing the strength of the magnetic field produced by the
dynamo effect, the organisation and amplitude of the flow, and those of
the density field, remain debated despite remarkable progress made in
their numerical simulation. This paper describes an approach based on
the explicit consideration of the variation of time scales $\tau$ with
spatial scales $\ell$ for the different physical phenomena involved. 
The \tauell diagrams thus constructed constitute a very complete
graphic summary of the dynamics of the object under study. We highlight
the role of the available convective power in controlling this
dynamics, together with the relevant force balance, for which we derive
a very telling \tauell translation. Several scenarios are constructed
and discussed for the Earth's core, shedding new light on the width of
convective columns, and on the force equilibria to be considered. A
QG-MAC scenario adapted from \cite{aubert2019} gives a good account of
the observations. A diversion to Venus reveals the subtlety and
relativity of the notion of ``fast rotator''. We discuss scaling laws and
their validity domain, and illustrate ``path strategies''. A complete
toolbox is provided, allowing everyone to construct a \tauell diagram
of a numerical simulation, a laboratory experiment, a theory, or a
natural object.{\vspace*{-2pt}}
\end{abstract}

\begin{altabstract}
Les noyaux plan\'etaires sont le si\`ege d'une dynamique des fluides
riche et complexe o\`u se combinent les effets de la rotation et du
champ magn\'etique. Les \'equilibres gouvernant l'intensit\'e du champ
magn\'etique produit par effet dynamo, l'organisation et l'amplitude de
l'\'ecoulement, et celles du champ de densit\'e, demeurent d\'ebattus,
malgr\'e les progr\`es remarquables de leur simulation num\'erique. Cet
article d\'etaille une approche qui repose sur la prise en compte
explicite de la variation des \'echelles de temps $\tau$ avec les
\'echelles spatiales $\ell$ pour les diff\'erents ph\'enom\`enes
physiques impliqu\'es. Les diagrammes \tauell ainsi construits
constituent un r\'esum\'e graphique tr\`es complet de la dynamique de
l'objet \'etudi\'e. Nous soulignons le r\^ole que joue la puissance
convective disponible dans l'\'etablissement de cette dynamique, ainsi
que l'\'equilibre des forces pour lequel nous d\'erivons une traduction
\tauell qui se r\'ev\`ele tr\`es parlante. Plusieurs scenarios sont
ainsi construits et discut\'es pour le noyau terrestre, apportant un
nouvel \'eclairage sur la largeur des colonnes convectives, et sur les
\'equilibres de force \`a consid\'erer. Un scenario QG-MAC adapt\'e de
\cite{aubert2019} rend bien compte des observations. Un d\'etour par
V\'enus r\'ev\`ele la subtilit\'e et la relativit\'e de la notion de
``rotateur rapide''. Nous discutons les lois d'\'echelle et leur
domaine de validit\'e, et illustrons le concept de ``path strategy''.
Une bo\^ite \`a outils compl\`ete est fournie, permettant \`a chacun de
construire le diagramme \tauell d'une simulation num\'erique, d'une
exp\'erience de laboratoire, d'une th\'eorie, ou d'un objet naturel.
\end{altabstract}

\keywords{\kwd{Turbulence}\kwd{tau--ell}\kwd{Dynamo}\kwd{Core}\kwd{Convection}}

\altkeywords{\kwd{Turbulence}\kwd{tau--ell}\kwd{Dynamo}\kwd{Noyau}\kwd{Convection}}

\thanks{French Academy of Science, Electricit\'e de France, EPSRC
(grant no EP/R014604/1), Labex OSUG@2020 (ANR10 LABX56).}

\maketitle

\twocolumngrid

\end{noXML}

\section{Introduction}\label{sec:introduction}

\vspace*{-2pt}

Enormous progress has been achieved in the modeling and understanding
of the magnetic dynamo at work in the core of the Earth and other
planets since the first 3D numerical simulations of
\citet{glatzmaier1995} and \citet{kageyama1995}. It rapidly appeared
that the magnetic fields produced by such numerical simulations met the
main characteristics of the long-term magnetic field observed on Earth,
such as its dipolarity, the presence of high-flux patches at high
latitudes, and symmetry properties
\citep{christensen1999,olson2006,christensen2010a}. Magnetic intensity
scaling laws for planetary and stellar dynamos were obtained by
combining an analysis of the dominant terms of the governing equations
with results of an extensive survey of numerical simulations
\citep{christensen2006,christensen2010b}.

In the meantime, shorter timescale manifestations of the geodynamo were
unveiled, such as a large-scale off-centered anticyclone
\citep[{e.g.},][]{pais2008}, and torsional waves (geostrophic
Alfv\'en waves) with periods of a few years \citep{gillet2010}. These
new observations prompted efforts to run numerical simulations at more
extreme parameter values \citep{schaeffer2017,aubert2017,aubert2023},
increasing the role of rotation by decreasing the Ekman number down to
$\Ek=10^{-7}$, and {increasing} the convective forcing up to
$\Ra/\Ra_c=6300$, where $\Ra$ is the Rayleigh number, and $\Ra_c$ its
critical value. These extreme simulations of the geodynamo successfully
account for fast dynamics retrieved from observations.\looseness=-1

In view of this remarkable progress, it might seem that most problems
are solved. In fact, hot debates are still roaming on several crucial
issues. One of them concerns the dominant length-scale of convective
structures in Earth's core. Column widths of 100~m are suggested by
\citet{yan2022} while \citet{guervilly2019} advocate 30~km.
Extrapolating force-balances from numerical simulations and laboratory
experiments to natural systems is another issue
\citep{aurnou2017,schwaiger2019,teed2023}. The relevance of scenarios
with weak and strong magnetic field branches is also hotly debated
\citep{dormy2016}. One extreme viewpoint being expressed by
\citet{cattaneo2022} who claim that Earth would not have been able to
produce a magnetic field as strong as today without Moon's help.
\looseness=-1

There is room for such diverging views because the distance from
numerically accessible parameters to expected planetary values remains
vertiginous. Laboratory experiments somewhat enlarge the accessible
range but are limited to non-dynamo regimes, making the link with
numerics and observations difficult.

This is the motivation for exploring a different route: instead of
extrapolating available simulations to core conditions, start from the
actual expected properties of the core, and patch scenarios of
turbulence that correspond to different regimes encountered at
different scales. This leads to the construction of \tauell regime
diagrams of turbulence, as briefly introduced by \citet{nataf2015}.

Our experience is that this approach is an excellent intuition-booster.
It provides a simple graphical support that can greatly help
deciphering and testing more mathematically-motivated approaches.
However, we observe that it has not yet received an audience, perhaps
because it clearly advocates a ``fuzzy physics'' method, and also because
it was originally published in a limited-access collection.

In this article, we present a largely renewed and extended version of
the \tauell regime diagrams we originally proposed. We detail the steps
for constructing such diagrams, providing examples of application to
numerical simulations and laboratory experiments. Key properties of
\tauell diagrams are highlighted and illustrated by simple examples.

The central part of the article is devoted to an application to the
Earth's core. We propose scenarios for a non-magnetic rotating
convective core, and for a dynamo-generating rotating convective core.
The resulting diagrams are compared with the predictions of several
scaling analyses \citep{christensen2006, christensen2010b,
davidson2013, aubert2017}.

\citet{nataf2015} were building scenarios from the observed large-scale
flow and magnetic field, and testing how they were compatible with the
expected available power. In this article, our \tauell diagrams are
constructed to satisfy a given constraint on the convective or
dissipated power, a key property of turbulent flows. Comparison with
the observed flow and magnetic field (when available) is used as a
validation test. This is a more challenging exercise, which leads us to
consider the various force balances that could govern the dynamics of
the object. We derive the \tauell translation of the main relevant
force balances (CIA, QG-CIA, MAC, QG-MAC, IMAC). It turns out that
these translations and their graphical representations are very
telling.

This should facilitate the construction of \tauell diagrams for
planets, exoplanets and stars for which no direct observation of the
large-scale flow velocity and magnetic field is available. Indeed,
planets are thermal machines and their thermal evolution is probably
what we can estimate best. Liquid cores of planets cool down on
geological timescales, generating convective motions. Convective power,
which can be estimated from the planet's thermal history
\citep[{e.g.,}][]{stevenson1983,lister2003,nimmo2015,landeau2022,driscoll2023}, sustains fluid flow and magnetic field.
Dissipation of this power by either momentum or magnetic diffusion, or
both, controls the regimes of turbulence the system experiences.
\looseness=-1

We present and illustrate the construction rules and key properties of
\tauell regime diagrams of turbulence in
Section~\ref{sec:construction}.  Section~\ref{sec:phenomena} introduces
the physical phenomena at work in planetary cores, and relates \tauell
diagrams to classical dimensionless numbers. 
Section~\ref{sec:non-magnetic} presents \tauell regime diagrams for a
non-magnetic rotating core. \tauell diagrams of the present-day
geodynamo are built in Section~\ref{sec:Earth}. Both sections emphasize
the crucial role of the available convective power and force balances.
The discussion Section~\ref{sec:discussion} illustrates how \tauell
diagrams bring a new light on several ongoing debates. Limitations and
perspectives are outlined in Section~\ref{sec:limitations}, and we
conclude in Section~\ref{sec:conclusion}. Appendix~\ref{sec:spectra}
provides rules for converting spectra into \tauell language. Simple
Python programs used to build \tauell diagrams are given as
supplementary material, together with examples from numerical
simulations and laboratory experiments.\vspace*{-12pt}

\section{Construction rules and key properties of \tauell diagrams}
\label{sec:construction}
\vspace*{-6pt}

This section presents the rules used to construct \tauell 
diagrams.\vadjust{\vfill\eject} Turbulent systems display a wide range
of length-scales and timescales.
Timescales of physical phenomena such as diffusion or wave propagation
depend upon the length-scale at which they operate. For example,
timescale $\tau_{\nu}$ of momentum diffusion at length-scale $\ell$ can
be written as $\tau_{\nu}(\ell) = \ell^2/\nu$, where $\nu$ is the 
kinematic viscosity. Similarly, turnover time
$\tau_u$ of a vortex of radius $\ell$ is given by $\tau_u(\ell) =
\ell/u(\ell)$, where $u(\ell)$ is the vortex fluid velocity. We build
\tauell regime diagrams by plotting timescales $\tau_x$ as a function
of length-scale $\ell$ in a log--log plot, for all the different
physical phenomena $x$ that govern the fluid flow in a given system.

\onecolumngrid
\vspace*{1pc}
\bigskip
\smallskip
\mbox{}\unskip\noindent\begin{frtextbox}{\textbf{Construction rules of} \tauell \textbf{regime diagrams}}
\enlabel{box1}
\mbox{}

	\tauell regime diagrams are ``object-oriented''. They are built following these steps:
\begin{itemize}
\item Identify physical phenomena that play an important role in the object under study.
\item Document relevant physical properties (viscosity, thermal diffusivity, rotation rate, etc).
\item Build and draw lines $\tau(\ell)$ that control dissipative and wave propagation phenomena.
\item Identify different turbulence regimes the object might experience.
\item Construct and draw lines $\tau(\ell)$ of fields  (velocity, buoyancy, magnetic field) that describe the object's turbulent behaviour, given a dissipated power $\mathcal{P}_{\mathrm{diss}}$.
\item Compare predictions with observables such as large-scale flow velocity and magnetic field, when available.
\end{itemize}

\end{frtextbox}

\vspace*{10pt}

\Smallskip
\vfill\twocolumngrid

\subsection{A simple example: Kolmogorov's universal turbulence}
\label{sec:K41}

We first illustrate the construction rules of \tauell diagrams with the
simple example of Kolmogorov's universal turbulence
\citep{kolmogorov1941}. Although this is not the kind of turbulence we
expect in planetary cores, we pick a range of length-scales and
timescales typical of the Earth core. Figure~\ref{fig:K41} is a log--log
plot of timescales spanning a range from 10~s to 32,000 years versus
length-scales from 1~cm to $R_o=3480$ km, the radius of the core.

\subsubsection{$\tau_{\nu}(\ell)$ line}

We pick a viscosity value $\nu =
10^{-6}{\cdot}\mathrm{m}^2{\cdot}\mathrm{s}^{-1}$ and draw the 
$\tau_{\nu}(\ell)$ viscous dissipation line:
{\begin{equation}\label{eq:nu}
\tau_{\nu}(\ell) = \ell^2/\nu.
\end{equation}}\unskip

\subsubsection{$\tau_u(\ell)$ line}

In Kolmogorov's universal turbulence, kinetic energy cascades down from
large length-scales to small scales, from the energy injection scale
down to the viscous dissipation scale. The range in between is called
the inertial range. The kinetic energy density spectrum $E(k)$ in the
inertial range obeys Kolmogorov's law:
{\begin{equation}\label{eq:K41}
E(k) = C_K \epsilon^{2/3} k^{-5/3},
\end{equation}}\unskip
where $k$ is the wavenumber, $\epsilon$ is the energy injection (and
dissipation) rate per unit mass, and $C_K$ is Kolmogorov's
dimensionless constant, of order~1.

To build line $\tau_u(\ell)$, we need to convert kinetic energy density
into velocity. It is common to define an ``eddy turnover time'' as
$\tau_u(\ell) = \ell/u(\ell)$, with $u^2(\ell) \sim E(k) k$ and $\ell
\sim 1/k$ (see Appendix~\ref{sec:conversion}). This translates into:
{\begin{equation}\label{eq:taufromE}
\tau_u(\ell) \simeq \ell^{3/2} [E(\ell^{-1})]^{-1/2}.
\end{equation}}\unskip
Dropping prefactor $C_K$, Kolmogorov's law yields:
{\begin{equation}\label{eq:tauK41}
\tau_u(\ell) \simeq  \epsilon^{-1/3} \ell^{2/3} .
\end{equation}}\unskip

We draw this $\tau_u(\ell)$ line in Figure~\ref{fig:K41}, assuming that
the energy injection length-scale is $R_o$, and choosing an injection
timescale, which will be discussed later.

\subsubsection{$\ell$-scale Reynolds number}

We terminate line $\tau_u(\ell)$ where it hits viscous line
$\tau_{\nu}(\ell)$. This intersection yields Kolmogorov microscales
$(\ell_K,\tau_K)$, for which $ \tau_K \equiv\tau_u(\ell_K) = \ell_K^2/\nu$,
{i.e.}, $u(\ell_K)\ell_K / \nu = 1$. Defining an $\ell$-scale
Reynolds number $\Re(\ell) = u(\ell) \ell / \nu$, we note that the
intersection of the eddy turnover time line $\tau_u(\ell)$ with the
viscous line $\tau_{\nu}(\ell)$ occurs at $\Re(\ell)=1$. It marks the
transition from the inertial cascade at large scale to the viscous
dissipation regime at small scale.

\subsubsection{Power dissipation markers}

In Kolmogorov's theory, the energy injected at large scale cascades
down with no loss to small scales at which viscous dissipation takes
place. This dissipation range starts at the intersection of lines
$\tau_u$ and $\tau_{\nu}$, where $Re(\ell) \sim 1$, which defines
Kolmogorov microscales $(\ell_K,\tau_K)$. From Equations (\ref{eq:nu})
and (\ref{eq:tauK41}), we deduce:
{\begin{equation}\label{eq:epsilon}
\epsilon = \dfrac{\ell_K^2}{\tau_K^3} = \dfrac{\nu}{\tau_K^2} = 
\dfrac{u^2(\ell_K)}{\tau_K},
\end{equation}}\unskip
where the last expression shows that dissipation rate per unit mass
$\epsilon$ equals kinetic energy at the microscale divided by eddy
turnover time at that scale. We also see that we can attribute to each
$\tau_{\nu}$ value a dissipation rate per unit mass. Multiplying by the
mass of the system, we obtain the total dissipated power. In
Figure~\ref{fig:K41}, we thus draw power markers along the viscous
dissipation line, using the mass of the outer core $M_o = 1.835\times 
10^{24}$ kg. Markers are a factor of one thousand apart and are
labeled. In Figure~\ref{fig:K41}, the timescale for energy injection at
length-scale $R_o$ was chosen to yield a dissipated power
$\mathcal{P}_{\mathrm{diss}} = 3$ TW, marked by the blue arrow. 

\subsubsection{Energy}

In Kolmogorov's universal turbulence, kinetic energy is dominated by
large length-scales. In \tauell diagrams, we retrieve kinetic energy
$\mathcal{E}_k$ from the square of the inverse of $\tau_u(R_o)$ since:
{\begin{equation}\label{eq:E}
\mathcal{E}_k = \frac{1}{2} \int_{V_o} {\rho [u(\mathbf{r})]^2
\,\mathrm{d}V} \sim M_o [u(R_o)]^2 = \frac{M_o R_o^2} {\tau_u^2(R_o)},
\end{equation}}\unskip
where $V_o$ is the volume of the liquid core. We use this property to
compare the amplitudes of the various energy reservoirs when dealing
with planetary cores. Note that this applies as long as the slope of
the $E(k)$ kinetic energy density spectrum is negative ({i.e.},
slope of $\tau_u(\ell)$ is less than $3/2$), in order for large-scale
energy to dominate.

\onecolumngrid
\bigskip
\smallskip
\mbox{}\unskip\noindent\begin{frtextbox}{\textbf{Key properties of} \tauell \textbf{regime diagrams}}
\enlabel{box1}
\mbox{}
\begin{itemize}
\item \tauell regime diagrams gather in a simple graphical
representation many of the ingredients that control the dynamics of a
turbulent fluid system.
\item In \tauell regime diagrams, intersections of lines $\tau_x(\ell)$
and $\tau_y(\ell)$ of physical phenomena $x$ and $y$ occur where
$\ell$-scale dimensionless number $Z(\ell) =
{\tau_y(\ell)}/{\tau_x(\ell)}$ equals 1. They mark a change in the
system's dynamical regime.
\item Usual integral-scale values of dimensionless numbers are obtained
from the ratio of relevant $\tau_x(\ell)$ and $\tau_y(\ell)$ times at
integral scale $\ell=R_o$.
\item Total dissipated power can be marked along $\tau(\ell)$ lines of
dissipative phenomena.
\item Energies of different types (kinetic, gravitational, magnetic)
are represented by the inverse square of corresponding $\tau(R_o)$.
\item \tauell regime diagrams are a useful tool to infer or test
turbulence scenarios. They are \textit{not} a theory of turbulence.
\end{itemize}
\end{frtextbox}

\vspace*{10pt}

\Smallskip
\vfill\twocolumngrid

\section{Physical phenomena in planetary cores and dimensionless
numbers}\label{sec:phenomena}

We now turn our attention to planetary cores. Flow within planetary
cores are mostly powered by thermal or thermo-compositional convection.
They often produce a magnetic field. Most importantly, these flows
occur in a rotating spherical system.

In this section, we introduce the \tauell lines these physical
phenomena contribute. We illustrate the resulting \tauell regime
diagram template, using values pertaining to Earth's core, and relate
the diagram to various dimensionless numbers used to characterize
planetary core dynamics. 

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:K41}\tauell regime diagram for Kolmogorov's
universal turbulence \citep{kolmogorov1941}. Teal line labeled $\nu$ is
viscous dissipation line $\tau_{\nu}(\ell)=\ell^2/\nu$. The thick blue
line is eddy turnover time $\tau_u(\ell)=\epsilon^{-1/3} \ell^{2/3}$
inferred from Kolmogorov's law, assuming that energy is injected at
core radius length-scale $\ell=R_o$. These two lines intersect where
the $\ell$-scale Reynolds number $\Re(\ell) \sim 1$, as marked by a
circle. Total energy dissipation rate $\mathcal{P}_{\mathrm{diss}}$ can be read
at this intersection (blue arrow), using square markers drawn and
labelled along the viscous line. Markers are a factor of $10^3$ apart,
the 1 TW marker being filled.}
\end{figure}

\subsection{Physical phenomena and their \tauell expressions}
\label{sec:scales}

Table~\ref{tab:scales} gives the expressions of major $\tau(\ell)$
times pertaining to planetary cores. $\tau_{\nu}(\ell)$ and
$\tau_u(\ell)$ times have been introduced in Section~\ref{sec:K41}.
Convection adds thermal diffusion and buoyancy scales. Rotation,
spherical boundaries, and magnetic field contribute key additional
timescales.

%tab1
\begin{table}
\caption{\label{tab:scales}Notation and expression of $\tau(\ell)$
times of relevant physical phenomena for planetary cores}
\begin{tabular}{ccc}
\thead
Time & Expression & Phenomenon \\
\endthead
$\tau_{\nu}(\ell)$ & $\ell^2/\nu$ & Viscous dissipation\\
$\tau_{\kappa}(\ell)$ & $\ell^2/\kappa$ & Thermal diffusion\\
$\tau_{\chi}(\ell)$ & $\ell^2/\chi$ & Compositional diffusion\\
$\tau_{\eta}(\ell)$ & $\ell^2/\eta$ & Magnetic dissipation\\
$t_{\Omega}$ & $1/\Omega$ & Rotation\\
$\tau_{\mathrm{Rossby}}(\ell)$ & $R_o/\Omega \ell$ & Rossby wave propagation\\
$\tau_{\mathrm{Alfven}}(\ell)$ & $\ell \sqrt{\rho\mu}/B_0$ & Alfv\'en wave propagation{\vspace*{3pt}}\\
$\tau_{\rho}(\ell)$ & $\sqrt{\frac{\ell}{g}\frac{\rho}{|\Delta\rho(\ell)|}}$ & Buoyancy (or free-fall)\\
$\tau_u(\ell)$ & $\ell / u(\ell)$ & Eddy turnover\\
$\tau_b(\ell)$ & $\ell \sqrt{\rho\mu}/b(\ell)$ & Alfv\'en wave collision
\botline
\end{tabular}
\tabnote{\textit{Fluid properties:} density $\rho$; kinematic viscosity
$\nu$; thermal and compositional diffusivities $\kappa$ and $\chi$,
respectively; magnetic diffusivity $\eta$;  magnetic permeability
$\mu$. \textit{System properties:} radius $R_o$; gravity $g$; rotation
rate $\Omega$; large-scale magnetic field $B_0$. \textit{Turbulent flow
properties:} $\Delta \rho(\ell)$, $u(\ell)$ and $b(\ell)$ are
$\ell$-scale density anomaly, flow velocity, and magnetic field
intensity, respectively. We also write $\tau_{\eta}(R_o)$ as $T_{\eta}$
for short. Adapted from Table 1 of Chapter 8.06 of \textit{Treatise on
Geophysics} \citep{nataf2015} with permission.}
{\vspace*{-12pt}}
\end{table}

We discuss the origin and meaning of these various $\tau(\ell)$ scales,
and illustrate the \tauell template they provide in Earth's core
example (Figure~\ref{fig:template}), using its properties presented in
Section~\ref{sec:core}. We first ignore the magnetic field and build
the template of Figure~\ref{fig:template}a.

\begin{figure*}
\includegraphics{fig02}
{\vspace*{1pt}}
\caption{\label{fig:template}\tauell regime diagram templates for the
Earth's core. The time-length relationships of relevant physical
phenomena, as given in Table~\ref{tab:scales}, are drawn in a log--log
plot of timescale versus length-scale, using Earth core values from
Table~\ref{tab:properties} presented in Section~\ref{sec:core}. (a)
\textit{Template ignoring the magnetic field.} The steep solid lines
labeled $\chi$, $\nu$ and $\kappa$ are diffusion times
$\tau_{\chi}(\ell)$, $\tau_{\nu}(\ell)$, and $\tau_{\kappa}(\ell)$,
respectively. The dash--dot horizontal line is the rotation time
$t_{\Omega}$. The Rossby line $\tau_{\mathrm{Rossby}}(\ell)$ is drawn
as a wavy line pinned to $t_{\Omega}$. Markers along lines
$\tau_{\nu}(\ell)$ and $\tau_{\mathrm{Rossby}}(\ell)$ indicate viscous
power dissipation. Markers are a factor of 1000 apart, the 1 TW marker
being filled. The circle labeled $\Ek$ at the intersection of the
viscous $\tau_{\nu}(\ell)$ and $t_{\Omega}$ lines mark the length-scale
at which the $\ell$-scale Ekman number equals 1. Same thing for the
thermal Ekman number $\Ek_{\kappa}(\ell)$.	 (b) \textit{Template
including the magnetic field.} Same as (a) with additional lines and
labels brought by the magnetic field. The steep solid red line labeled
$\eta$ is the magnetic diffusion line $\tau_{\eta}(\ell)$. Markers
along that line indicate Ohmic power dissipation. Markers are a factor
of 1000 apart, the 1 TW marker being filled. The Alfv\'en line
$\tau_{\mathrm{Alfven}}(\ell)$ is drawn as a wavy line pinned to the
observed large-scale magnetic field Alfv\'en time $t_b^{\mathrm{obs}}$.
Circles labeled $\Ek_{\eta}$, $\Lu$ and $\lambda$ at line intersections
mark scales at which the corresponding $\ell$-scale dimensionless
number (see Table~\ref{tab:dimensionless}) equals 1. See
Sections~\ref{sec:onset_note} and~\ref{sec:core} for more information.}
{\vspace*{1pt}}
\end{figure*}

\subsubsection{Diffusion}

Lines labeled  $\nu$, $\kappa$ and  $\chi$ are diffusion \tauell lines.
Timescales of diffusive phenomena all share the same form $\tau(\ell) =
\ell^2/D$, where diffusivity $D$ is $\nu$, $\kappa$ or  $\chi$
depending upon which field diffuses: momentum, temperature, or
composition, respectively.

\subsubsection{Convection}

We introduce a ``buoyancy'' or ``free-fall'' timescale $\tau_{\rho}(\ell) =
\sqrt{({\ell}/{g}) ({\rho}/{|\Delta\rho(\ell)|})}$, which is the
time it takes for a parcel of fluid with density anomaly
$\Delta\rho(\ell)$ to rise or sink a distance $\ell$ in the absence of
diffusion. $\tau_{\rho}(\ell)$ relates to density anomaly $\Delta\rho$
at length-scale $\ell$. Density anomaly, flow velocity and magnetic
field constitute the three fields for which we seek an adequate
turbulent description.

We have shown in Section~\ref{sec:K41} that the value of $\tau_u$ at
integral scale $R_o$ measures the kinetic energy of the flow.
Similarly, gravitational energy $\mathcal{E}_g$ is measured by
$\tau_{\rho}(R_o)$ (as long as the slope of $\tau_{\rho}(\ell)$ is less
than $3/2$) since:
{\begin{equation}\label{eq:E_g}
\mathcal{E}_g = \frac{1}{2} \int_{V_o} { g r \Delta\rho(\mathbf{r})
\,\mathrm{d}V} \sim M_o g R_o \frac{\Delta\rho(R_o)}{\rho} = \frac{M_o
R_o^2} {\tau_{\rho}^2(R_o)}.
\end{equation}}\unskip

{\vspace*{-1pt}}

\subsubsection{Rotation}

{\vspace*{-1pt}}

Rotation is a crucial ingredient of planetary core dynamics. It adds
one important time in our \tauell regime diagram: the rotation time
$t_{\Omega} = \Omega^{-1}$, {i.e.,} one day divided by $2 \uppi$.
Physical phenomena operating at timescales smaller than $t_{\Omega}$
are not influenced by planet's spin, while those with longer timescales
feel the effect of rotation. We thus draw a horizontal line at
$t_{\Omega}$ in the diagram of Figure~\ref{fig:template}a. The
intersection of this line with the viscous line yields Ekman layer's
thickness $\ell_E = \sqrt{\nu/\Omega}$. Viscous forces are balanced by
Coriolis acceleration in these thin Ekman layers. $\ell_E$ is the only
length-scale one can build from $\nu$ and $\Omega$ alone, and it
controls friction, hence viscous dissipation, that takes place at
boundaries. It turns out that boundaries bring up new important
dynamical constraints and scales.

\subsubsection{Rotation and spherical boundaries} \label{sec:QG}

We will not review here the vast literature on rotating fluids in
containers. The book of \citet{greenspan1968} remains amazingly
central. At this stage, let us simply recall that Navier--Stokes
equation reduces to \textit{geostrophic equilibrium} when Coriolis
acceleration dominates:
{\begin{equation}\label{eq:geostrophic}
2 \rho \boldsymbol{\Omega} \times \mathbf{u} = -\nabla P,
\end{equation}}\unskip
where $\mathbf{u}$ and $P$ are fluid velocity and pressure,
respectively. Taking the curl of this equation yields
\textit{Proudman--Taylor constraint}:
{\begin{equation}\label{eq:PT}
\frac{\partial \mathbf{u}}{\partial z} = 0,
\end{equation}}\unskip
where $z$ coordinate is parallel to vector $\boldsymbol{\Omega}$.
Reintroducing acceleration term $\rho \partial_t \mathbf{u}$ allows for
the propagation of \textit{inertial waves}.

Proudman--Taylor constraint would inhibit all fluid motions in a
rotating fluid bounded by a solid container. Accounting for the
presence of a thin Ekman layer that accommodates a velocity jump
between the fluid bulk and the boundary, \textit{geostrophic} flows are
allowed, which follow contours of equal fluid column-height (measured
in the $z$-direction), {i.e.,} azimuthal flows in a spherical
container. In most situations, \textit{Quasi-geostrophic} (QG) fluid
motions are also observed, which approximately satisfy Proudman--Taylor
constraint ({i.e.,} $z$-invariance) in the bulk (at least for one
component, typically the azimuthal velocity).

It is important to note that Proudman--Taylor constraint is established
by the propagation of inertial waves in the fluid, and is effective
only when they had time to reach a boundary. Thus, a localized eddy of
radius $\ell$ grows into a columnar vortex at a speed equal to $\Omega
\ell$ \citep{davidson2006}. This means that large eddies rapidly form
quasi-geostrophic columns, while it takes more time for small eddies to
form core-size columns. Time for reaching quasi-geostrophy is thus
given by:
{\begin{equation}\label{eq:Rossby}
\tau_{\mathrm{Rossby}}(\ell) = \frac{R_o}{\Omega \ell},
\end{equation}}\unskip
as written in Table~\ref{tab:scales}. This line is drawn as a wavy line
in Figure~\ref{fig:template}a. It is pinned to time
$t_{\Omega}$ at $\ell=R_o$, and we extend it until it reaches viscous
line $\tau_{\nu}(\ell)$.

Note that $\tau_{\mathrm{Rossby}}(\ell)$ also equals the time it takes
for a Rossby wave of wavelength $\ell$ to propagate one wavelength
(hence its name) \citep{nataf2015}. The intersection of the
$\tau_u(\ell)$ line with the Rossby line has $\ell =\sqrt{u(\ell) R_o/
\Omega}$, which defines a \textit{Rhines scale} (originally more
precisely defined as $\ell =\sqrt{u/\beta}$ in a thin shell, where
$\beta = 2 \Omega \sin\theta/R_o$ is the northward gradient of Coriolis
frequency at colatitude $\theta$ \citep{rhines1975}, here extended to a
wide gap \citep{busse1970,schopp1997}).

Flow is quasi-geostrophic for scales above line
$\tau_{\mathrm{Rossby}}(\ell)$. In the triangle formed by the Rossby
line, the viscous line and line $t_{\Omega}$, flow structures are
elongated parallel to the spin axis but not enough to reach both
boundaries. Flow is 3D beneath line $t_{\Omega}$.

\subsubsection{Quasi-geostrophic dissipation}

Quasi-geostrophic vortices dissipate kinetic energy by Ekman friction
at no-slip boundaries of the liquid core. We approximate energy loss
rate $p_{\ell}$ of a single QG vortex of radius $\ell$ by:
{\begin{equation}
p_{\ell} = \rho \nu \frac{u^2(\ell)}{\ell_E^2} \ell_E \ell^2,
\end{equation}}\unskip
with $\ell_E$ the Ekman layer thickness. Summing contributions of all
$\ell$-scale QG vortices filling the entire core, we obtain the total
power dissipated by Ekman friction $\mathcal{P}_{QG}$. Dividing by the
mass of the core, we obtain the QG viscous dissipation rate per unit
mass $\epsilon_{QG}$ at scale $\ell$ as:
{\begin{equation}\label{eq:epsilonQG}
\epsilon_{QG} = \nu \frac{u^2(\ell)}{R_o \ell_E}
\end{equation}}\unskip
Ekman friction matters for QG vortices. We therefore draw the
corresponding power markers on the Rossby line, above which flow is
quasi-geostrophic. Time $\tau_{QG}$ for which total viscous dissipation
by \mbox{Ekman} friction equals $\mathcal{P}_{QG}$ is then obtained from
Equation (\ref{eq:epsilonQG}) as:
{\begin{equation}\label{eq:tauQG}
\tau_{QG} = \left[ \frac{M_o R_o}{\mathcal{P}_{QG}}
\sqrt{\frac{\nu}{\Omega^3}} \right]^{1/4}.
\end{equation}}\unskip
This provides markers drawn in Figure~\ref{fig:template}a as diamonds
along the Rossby line, a factor of one thousand apart, the TW marker
being filled.

Going back to the viscous line $\tau_{\nu}(\ell)$, we terminate it at
the spin-up time $R_o/\sqrt{\nu \Omega} \sim 13 \times 10^3$ years,
which is the time it takes for a change in the outer boundary spinning
rate to be transmitted to the entire volume of the core.

\subsubsection{A note on convection onset}
\label{sec:onset_note}

We can already illustrate an interesting insight provided by this
simple \tauell template, by adding the scales appearing at the onset of
convection in this rotating spherical system. This topic has a long
history, starting with the pioneer studies of \citet{roberts1968} and
\citet{busse1970}, followed by \citet{jones2000}, \citet{dormy2004},
\citet{zhang2007}. It is found that convection sets in as a travelling
thermal Rossby wave, forming columns aligned with the spin axis, whose
width is controlled by viscosity in the bulk of the spherical shell. A
\tauell translation of the length-scale, period, and Rayleigh number at
convection onset is given in Appendix~\ref{sec:onset}.

In Figure~\ref{fig:template}a, a black triangle marks the length-scale
and period at convection onset. It lies at the intersection of the
Rossby and thermal diffusion lines, close to the viscous diffusion
line, as expected for a viscously-controlled quasi-geostrophic thermal
Rossby wave. We read a column-width of about 100 m. Is this the typical
convective length-scale in the Earth's core? We will get back to this
question in Section~\ref{sec:length-scale}. The orange triangle at
$\ell=R_o$ marks the critical free-fall time $T_{\rho}^c$ deduced from
the critical Rayleigh number (see Appendix~\ref{sec:onset}).

\subsubsection{Magnetic field, magnetic dissipation, magnetic energy}

Magnetic fields are often produced and sustained by dynamo action
within planetary cores. We now add the magnetic field to build the
template of Figure~\ref{fig:template}b. The red line labeled $\eta$ is
the magnetic diffusion line $\tau_{\eta}(\ell)$. Magnetic dissipation
markers are labeled along that line, following the same rule as in
Equation~(\ref{eq:epsilon}).

The presence of a magnetic field allows the propagation of
magnetohydrodynamic waves called Alfv\'en waves \citep{alfven1942}. In
a uniform magnetic field $B_0$, these waves propagate at speed $V_A =
B_0 / \sqrt{\rho \mu}$, where $\mu$ is fluid's magnetic permeability.
Assuming a large-scale magnetic field $B_0$, we construct line
$\tau_{\mathrm{Alfven}}(\ell) = \ell \sqrt{\rho \mu}/B_0$, the time it
takes for an Alfv\'en wave to propagate over a distance $\ell$. It is
drawn as a red wavy line in Figure~\ref{fig:template}b.

To describe the magnetic field in the system, we define a similar
timescale, replacing $B_0$ by $\ell$-scale magnetic field $b(\ell)$.
Note that $\tau_b(R_o) = \tau_{\mathrm{Alfven}}(R_o)$ provides the
magnitude of magnetic energy $\mathcal{E}_m$ (as long as the slope of
$\tau_b(\ell)$ is less than $3/2$), since:
{\begin{equation}\label{eq:E_m}
\mathcal{E}_m = \frac{1}{2 \mu} \int_{V_o} {[b(\mathbf{r})]^2\,\mathrm{d}V}
\sim \frac{M_o}{\rho \mu} [b(R_o)]^2 = \frac{M_o R_o^2}{\tau_b^2(R_o)}.
\end{equation}}\unskip

\subsection{Dimensionless numbers}
\label{sec:dimensionless}

In order to connect to the huge literature pertaining to geophysical
and astrophysical fluid dynamics, it is important to relate our \tauell
regime diagrams to widely used dimensionless numbers. Dimensionless
numbers provide the minimum number of parameters needed to describe a
physical system. They permit a comparison of widely different systems
that yield the same dimensionless numbers. These numbers are
dimensionless combinations of properties and field variables that
appear when the equations governing the dynamics of the system under
study are made dimensionless by normalizing their various terms by
``typical scales''.

For example, Reynolds number for a system of size $L$ will be written:
$\Re = U L / \nu$, where $U$ is a typical fluid velocity, and $\nu$
kinematic viscosity. Usually, it is when this dimensionless number is
of order 1 that a change of regime occurs. In this example: a change
between a regime where momentum diffusion dominates over advection when
$\Re<1$ to one where advection dominates for $\Re>1$.

Most dimensionless numbers can be written as the ratio of two times. In
our approach, we define length-scale dependent dimensionless numbers,
constructed as the ratios of the timescales of the relevant physical
phenomena at that length-scale. We thus define $\ell$-scale Reynolds
number as: $\Re(\ell) = \tau_{\nu}(\ell) / \tau_u(\ell)$, where
$\tau_{\nu}(\ell)$ is momentum diffusion timescale at length-scale
$\ell$, while $\tau_u(\ell)$ is the overturn time of a vortex of radius
$\ell$. Table~\ref{tab:dimensionless} gives the expressions of
$\ell$-scale dimensionless numbers pertaining to planetary liquid core
dynamics.

%tab2
\begin{table*}
\caption{\label{tab:dimensionless}Expressions of $\ell$-scale
dimensionless numbers}
\tabcolsep8pt
\begin{tabular}{cccc}
\thead
Number & Expression & Time ratio & Name \\
\endthead
$\Re(\ell)$ & { $\dfrac{u(\ell)\ell}{\nu}$} & {
$\dfrac{\tau_{\nu}(\ell)}{\tau_u(\ell)}$} & Reynolds \vspace*{3pt}\\
$\Ra(\ell)$ & { $\dfrac{g \ell^3 |\Delta\rho(\ell)|/\rho}{\kappa \nu}$}
& { $\dfrac{\tau_{\kappa}(\ell) \,
\tau_{\nu}(\ell)}{\tau_{\rho}^2(\ell)}$} & Rayleigh \vspace*{3pt}\\
$\Ek(\ell)$ & { $\dfrac{\nu}{\Omega \ell^2}$} & {
$\dfrac{t_{\Omega}}{\tau_{\nu}(\ell)}$} & Ekman \vspace*{3pt}\\
$\Ek_{\kappa}(\ell)$ & { $\dfrac{\kappa}{\Omega \ell^2}$} & {
$\dfrac{t_{\Omega}}{\tau_{\kappa}(\ell)}$} & Thermal Ekman \vspace*{3pt}\\
$\Ek_{\eta}(\ell)$ & { $\dfrac{\eta}{\Omega \ell^2}$} & {
$\dfrac{t_{\Omega}}{\tau_{\eta}(\ell)}$} & Magnetic Ekman \vspace*{3pt}\\
$\Ro(\ell)$ & { $\dfrac{u(\ell)}{\Omega \ell}$} & {
$\dfrac{t_{\Omega}}{\tau_u(\ell)}$} & Rossby \vspace*{3pt}\\
$\Roff(\ell)$ & { $\dfrac{\sqrt{g \ell |\Delta \rho(\ell)|/\rho}}{\Omega
\ell}$} & { $\dfrac{t_{\Omega}}{\tau_{\rho}(\ell)}$} & Free-fall Rossby \vspace*{3pt}\\
$\Rm(\ell)$ & { $\dfrac{u(\ell)\ell}{\eta}$} & {
$\dfrac{\tau_{\eta}(\ell)}{\tau_u(\ell)}$} & Magnetic Reynolds \vspace*{3pt}\\
$\Lu(\ell)$ & { $\dfrac{\ell B_0}{\eta \sqrt{\rho \mu}}$} & {
$\dfrac{\tau_{\eta}(\ell)}{\tau_{\mathrm{Alfven}}(\ell)}$} & Lundquist \vspace*{3pt}\\
$\Lambda_d(\ell)$ & { $\dfrac{b(\ell) B_0}{\rho \mu u(\ell) \Omega
\ell}$} & { $\dfrac{\tau_u(\ell) \, t_{\Omega}}{\tau_b(\ell) \,
\tau_{\mathrm{Alfven}}(\ell)}$} & Dynamical Elsasser \vspace*{3pt}\\
$\lambda(\ell)$ & { $\dfrac{B_0}{\sqrt{\rho \mu} \Omega \ell}$} & {
$\dfrac{t_{\Omega}}{\tau_{\mathrm{Alfven}}(\ell)}$} & Lehnert 
\botline
\end{tabular}
\tabnote{These numbers are also expressed as ratios of characteristic
$\ell$-scale times, which are defined in Table~\ref{tab:scales}. One
recovers the classical expression of these numbers at integral scale by
setting $\ell = R_o$. Adapted from Table 2 of Chapter 8.06 of
\textit{Treatise on Geophysics} \citep{nataf2015} with permission.}
\end{table*}

In \tauell regime diagrams, the intersection of the $\tau_x(\ell)$ and
$\tau_y(\ell)$ lines of physical phenomena $x$ and $y$ occurs where
$\ell$-scale dimensionless number $Z(\ell) = \tau_y(\ell) /
\tau_x(\ell)$ equals 1. Each such intersection marks a change in the
system's dynamic regime.

In Figures~\ref{fig:template}a and~\ref{fig:template}b, we have labeled
several line intersections, where specific dimensionless numbers equal
1. The intersections of line $t_{\Omega}$ and the $\nu$, $\kappa$ and
$\eta$ lines indicate where corresponding $\ell$-scale Ekman numbers
equal 1 in the \tauell plane. The intersection of the Alfv\'en and
magnetic diffusion lines defines where the $\ell$-scale Lundquist
number equals 1, marking a change from propagating Alfv\'en waves at
larger scales to damped waves at smaller scales. Similarly, the
intersection of the Alfv\'en and $t_{\Omega}$ lines yields
$\lambda(\ell) \sim 1$, where $\lambda$ is the Lehnert number
\citep{lehnert1954,jault2008}. System rotation favors quasi-geostrophic
Alfv\'en waves at timescales above this intersection.

More dimensionless numbers, such as $\Re$, $\Rm$, $\Ra$, $\Ro$,
$\Roff$, and $\Lambda_d$, will appear when we plot lines
$\tau_{\rho}(\ell)$, $\tau_u(\ell)$ and $\tau_b(\ell)$ of the system's
density, velocity and magnetic fields for the different turbulence
scenarios we will explore.\vspace*{-12pt}

\subsection{A word on Earth's core properties}
\label{sec:core}

Core properties used to build the \tauell diagrams of
Figure~\ref{fig:template} are listed in Table~\ref{tab:properties}.
Most are taken from Peter Olson's review in  \textit{Treatise on
Geophysics} \citep{olson2015}. Some of them are known with great
precision (to about 1$\permil$ for core radius $R_o$ and liquid core
mass $M_o$), but others are poorly constrained (to about 1 or 2 orders
of magnitude for viscosity $\nu$ and compositional diffusivity $\chi$).
In addition, most physical properties are expected to vary with radius.
None of these (important) subtleties are taken into account in the
``fuzzy'' approach we advocate for building \tauell regime diagrams.
Note that we systematically drop all numerical prefactors, including~$2
\uppi$.

%tab3
\begin{table*}
\caption{\label{tab:properties}Properties of Earth's core}
\tabcolsep8pt
\begin{tabular}{cccc}
\thead
Symbol & Value & Unit & Property \\
\endthead
$\nu$    & $10^{-6}$              & m$^2{\cdot}$s$^{-1}$ & Kinematic viscosity \\
$\kappa$ & $5 \times 10^{-6}$     & m$^2{\cdot}$s$^{-1}$ & Thermal diffusivity \\
$\chi$   & $10^{-9}$              & m$^2{\cdot}$s$^{-1}$ & Compositional diffusivity \\
$\eta$   & $1$                    & m$^2{\cdot}$s$^{-1}$ & Magnetic diffusivity \\
$\rho$   & $10.9 \times 10^3$     & kg${\cdot}$m$^{-3}$  & Density \\
$\alpha$ & $1.2 \times 10^{-5}$   & K$^{-1}$ & Thermal expansion coefficient \\
$C_P$    & 850                    & J${\cdot}$kg$^{-1}{\cdot}$K$^{-1}$ & Specific heat capacity \\
$R_o$    & $3.48 \times 10^{6}$   & m & Core radius \\
$R_i$    & $1.22 \times 10^{6}$   & m & Inner core radius \\
$M_o$    & $1.835 \times 10^{24}$ & kg & Outer core mass \\
$g$      & $8$                    & m${\cdot}$s$^{-2}$ & Gravity \\
$t_{\Omega}$ & $1.38 \times 10^{4}$ & s & Earth's rotation time ({i.e.} $1/2\uppi$ day) \\
$\mathcal{P}_{\mathrm{diss}}$ & $3 \times 10^{12}$ & W & Available convective power \\
$t_u^{\mathrm{obs}}$ & $9 \times 10^{9}$ & s & $R_o$-scale core flow time ({i.e.} ${\simeq}300$ years) \\
$t_b^{\mathrm{obs}}$ & $1.4 \times 10^{8}$ & s & $R_o$-scale Alfv\'en wave time ({i.e.} ${\simeq}4$ years)
\botline
\end{tabular}
\tabnote{See \citet{olson2015} for details and uncertainties. Adapted
from Table 3 of Chapter 8.06 of \textit{Treatise on Geophysics}
\citep{nataf2015} with permission.}
{\vspace*{-6pt}}
\end{table*}

As noted in Section~\ref{sec:introduction}, the power available to
drive the dynamics of the system under study is a key ingredient. It
largely controls the different turbulence regimes the system will
experience. Thermal evolution of the Earth has received considerable
attention (see \citet{nimmo2015, landeau2022,driscoll2023} for
reviews). It is now well established that the dynamics of Earth's core
today is powered by its slow cooling, enhanced by the resulting growth
of the solid inner core. As iron-nickel alloy crystallizes at its
surface, it releases latent heat and light elements that drive
convection and power the geodynamo.

Despite uncertainties on isentropic heat flux, the available convective
power is found to be in the range 0.8--$5 \times 10^{12}$~W for
present-day core \citep{nimmo2015,landeau2022}. We adopt value
$\mathcal{P}_{\mathrm{diss}} = 3$ TW. This value is pointed by a blue
arrow on the Rossby line in Figure~\ref{fig:template}a, and by a red
arrow on the magnetic dissipation line in Figure~\ref{fig:template}b.

The last two rows in Table~\ref{tab:properties} are not used to build
\tauell diagrams, but instead to test their relevance. Large-scale
vortex turnover time $t_u^{\mathrm{obs}} \simeq 300$ years is retrieved
from core flow inversions of magnetic secular variation
\citep[{e.g.,}][]{pais2008}. Large-scale Alfv\'en wave propagation time
$t_b^{\mathrm{obs}} \simeq 4$ years is deduced from the discovery and
analysis of ``torsional oscillations'' in Earth's core
\citep{gillet2010}.

It is also observed that the Lowes--Mauersberger spectrum of magnetic
energy is flat at the core-mantle boundary up to harmonic degree 10
\citep[{e.g.},][]{langlais2014}. This means that energy is independent
of length-scale in this scale-range, which translates into a
$\tau_b(\ell) \propto \ell^{3/2}$ trend at large-scale (see
Appendix~\ref{sec:spectra_LM}). Similarly, core flow inversions favor
an almost flat harmonic spectrum of kinetic energy up to degree 10
\citep{roberts2013,aubert2013,gillet2015,baerenzung2016}. These trends
are sketched by colored disks labeled $t_u^{\mathrm{obs}}$ and
$t_b^{\mathrm{obs}}$ in Figure~\ref{fig:template}.

\section{\tauell regime diagrams for a non-magnetic 
\mbox{rotating} convective core} \label{sec:non-magnetic}

Building a turbulence scenario for a given system, starting from the
power it dissipates, requires estimating the balance of forces at
different length-scales. We thus start by expressing the \tauell
translation of expected relevant force balances. We then focus on a
quasi-geostrophic regime, which we illustrate with the \tauell diagram
of an actual numerical simulation. We then propose an idealized
scenario of turbulent convection in Earth's core in the absence of a
magnetic field.

\subsection{\tauell expression of force balances in rotating
convection} \label{sec:force_balance}

Let us start from the Navier--Stokes equation for deviations from
hydrostatic equilibrium in an incompressible fluid under the Boussinesq
approximation:
{\begin{equation}\label{eq:NS}
\rho (\partial_t \mathbf{u} + \mathbf{u} \cdot \nabla  \mathbf{u} + 2
\bOmega \times \mathbf{u}) =  {-}\nabla P + \Delta \rho
\mathbf{g} + \rho \nu \nabla^2 \mathbf{u},
\end{equation}}\unskip
where the symbols have their usual meaning. The acceleration term on
the left-hand side includes advection and Coriolis, while the
right-hand side figures pressure gradient, buoyancy, and viscous
forces.

We already recalled in Section~\ref{sec:QG} that Equation (\ref{eq:NS})
reduces to geostrophic equilibrium (\ref{eq:geostrophic}) when Coriolis
acceleration strongly dominates, yielding Proudman--Taylor constraint
(\ref{eq:PT}). However, this equation is a diagnostic equation that
can't be used on its own, as gets clear when Equation (\ref{eq:NS}) is
curled to obtain the vorticity equation (see \citet{jones2015} for a
more complete treatment). Reintroducing other accelerations and forces
in the vorticity equation can lead to two different situations: (i)
Proudman--Taylor constraint is broken and we get a three-term balance
between Coriolis, inertia (or viscosity) and buoyancy; (ii) Coriolis
acceleration is still dominant, and flow is quasi-geostrophic at
leading order, with a small velocity gradient along the spin axis
$(\bOmega \cdot \nabla) \mathbf{u}$, scaling as $1/\ell_{\parallel}$,
where $\ell_{\parallel} \sim R_o$ \citep{julien2012}.

\subsubsection{Coriolis-Inertia-Archimedes (CIA)}
\label{sec:CIA}

Let's first consider the first situation, with a three-term balance of
Coriolis, inertia and Archimedean forces. At a given length-scale
$\ell_{\Omega}$, retaining these forces in the curl of Equation
(\ref{eq:NS}) yields:
{\begin{equation}\label{eq:CIA}
\frac{\Omega u}{\ell_{\Omega}} \sim \frac{u^2}{\ell_{\Omega}^2} \sim
\frac{\Delta \rho }{\rho} \frac{g}{\ell_{\Omega}}.
\end{equation}}\unskip
Translating in \tauell language, we get:
{\begin{equation}\label{eq:CIA_tauell}
t_{\Omega} \tau_u(\ell_{\Omega}) \sim \tau_u^2(\ell_{\Omega}) \sim
\tau_{\rho}^2(\ell_{\Omega}),
\end{equation}}\unskip
which implies:
{\begin{equation}\label{eq:CIA_tauell}
\tau_u(\ell_{\Omega}) \sim t_{\Omega}  \sim \tau_{\rho}(\ell_{\Omega}).
\end{equation}}\unskip
This is the regime we expect when the $\tau_u(\ell)$ line reaches the
$t_{\Omega}$ line, where $\Ro(\ell_{\Omega}) \sim 1$.

\subsubsection{Quasi-geostrophic Coriolis-Inertia-Archimedes (QG-CIA)}
\label{sec:QG-CIA}

The second situation is more relevant for the Earth's core in the
absence of a magnetic field. We still consider a three-term balance of
Coriolis, inertia and Archimedean forces, but in which the Coriolis
term is reduced to its ageostrophic part:
{\begin{equation}\label{eq:QG-CIA}
\frac{\Omega u}{\ell_{\parallel}} \sim \frac{u^2}{\ell_{\perp}^2} \sim
\frac{\Delta \rho }{\rho} \frac{g}{\ell_{\perp}}.
\end{equation}}\unskip
The first term corresponds to vortex stretching, the second one to
vorticity advection, and the last one to vortex generation by buoyancy
\citep{cardin1994,aubert2001,jones2015}.

Translating in \tauell language, and assuming $\ell_{\parallel} \sim
R_o$, we get:
{\begin{equation}\label{eq:QG-CIA_tauell}
\frac{R_o}{\ell_{\perp}}t_{\Omega} \tau_u(\ell_{\perp}) \sim
\tau_{\rho}^2(\ell_{\perp}) \sim \tau_u^2(\ell_{\perp}).
\end{equation}}\unskip
Equation (\ref{eq:QG-CIA_tauell}) implies: 
{\begin{equation}
\tau_u(\ell_{\perp}) \simeq \frac{R_o}{\ell_{\perp}}t_{\Omega} \equiv
\tau_{\mathrm{Rossby}}(\ell_{\perp}),
\end{equation}}\unskip
meaning that the $\tau_u(\ell)$ of a flow in QG-CIA balance plots on
the Rossby line $\tau_{\mathrm{Rossby}}(\ell)$ (see
Table~\ref{tab:scales}). Note that where flow is quasi-geostrophic, the
$\ell$ that we use to describe its dynamics in \tauell diagrams is the
length-scale in the equatorial plane ({i.e.,} the width of the
columns).\vspace*{-6pt}

\subsection{\tauell diagram of a remarkable numerical simulation}
\label{sec:Guervilly}

We focus here on rapid rotation regimes (small Rossby number), for
which the leading order balance is quasi-geostrophic. Important results
have been obtained for this regime by \citet{guervilly2019}, who
performed numerical simulations of thermal convection at low Prandtl
number ($\Pr = 10^{-2}$ and $10^{-1}$) in a sphere, at Ekman numbers
down to $\Ek(R_o) = 10^{-8}$ in 3D, and to $\Ek(R_o) = 10^{-11}$ in
quasi-geostrophic 2D. In this section, we build the \tauell diagram of
their most extreme 3D simulation, at $\Ek(R_o)=10^{-8}$and 
$\Pr=10^{-2}$, and discuss its implications. Figure~\ref{fig3}a
displays the \tauell diagram we obtain.

\begin{figure*}
\includegraphics{fig03}
\caption{\label{fig3}\tauell diagrams for non-magnetic rapidly rotating
convection. Refer to Figure~\ref{fig:template}a and
Section~\ref{sec:phenomena} for a complete description of the
background ``template''. (a) \textit{3D numerical simulation from
\citet{guervilly2019}.} Field variables $u$ and $\rho$ of the
simulation are represented by blue $\tau_u(\ell)$ and orange
$\tau_{\rho}(\ell)$ lines, respectively, which are \tauell translations
of their respective volume-averaged order $m$-spectra (see
Appendix~\ref{sec:spectra_m}). Simulation's viscous dissipation in the
bulk can be read on line $\tau_{\nu}(\ell)$ at the blue arrow, while 
viscous dissipation in Ekman layers is marked by a blue arrow on the
Rossby line. Additional pale blue line labeled $u_r$ gives the \tauell
line of radial velocity, which gets much smaller than azimuthal
velocity at large length-scales. (b) \textit{Scenario for the Earth
assuming a QG-CIA force balance.} The available convective power
$\mathcal{P}_{\mathrm{diss}} \simeq 3$ TW sets time $\tau_*$ on the
Rossby line (blue arrow) at which QG-CIA force balance yields the
dominant vortex radius $\ell_*=\ell_{\perp} \simeq 200$ km, such that
$\tau_u(\ell_{\perp}) = \tau_{\rho}(\ell_{\perp}) =
\tau_{\mathrm{Rossby}}(\ell_{\perp})$. The QG-CIA balance governs flow
at length-scales $\ell < \ell_*$ all the way to the intersections with
diffusion lines $\tau_{\kappa}(\ell)$ and $\tau_{\nu}(\ell)$. At
length-scales $\ell > \ell_*$, azimuthal flow velocities dominate over
radial velocities (labeled $u_r$).}
\end{figure*}

We first build its ``template'' as in Figure~\ref{fig:template}a, using
radius $R_o$ and $t_{\Omega}$ (spin rate's inverse) of the actual
Earth's core as length-scale and timescale, respectively, in order to
compare with the Earth. Input dimensionless parameters $\Ek$ and $\Pr$
of the simulation provide values needed to build lines
$\tau_{\nu}(\ell)=\ell^2/\nu$ and $\tau_{\kappa}(\ell)=\ell^2/\kappa$.
As in Figure~\ref{fig:template}a, power dissipation markers are drawn
along the Rossby and viscous lines, using outer core mass $M_o$. The
simulation provides the power dissipated by viscosity in the bulk,
pointed by a blue arrow on the viscous line, and the slightly larger
viscous dissipation in the Ekman boundary layer, pointed by a blue
arrow on the Rossby line.

We now turn to extracting lines $\tau_u(\ell)$ and $\tau_{\rho}(\ell)$
from the simulation. Line $\tau_u(\ell)$, labeled $u$, is obtained from
the conversion of the volumetric average of a snapshot's azimuthal
order $m$-kinetic energy spectrum, following Equation
(\ref{eq:tauell-m_u}) of Appendix~\ref{sec:spectra_m}. Note that line
$\tau_u(\ell)$ stays above the Rossby line at all length-scales,
implying that the flow is quasi-geostrophic, as expected from the high
degree of $z$-invariance observed in this simulation
\citep{guervilly2019}. Remember that, in this case, length-scale $\ell$
has to be understood as the flow's length-scale in the equatorial
plane. Flow becomes anisotropic at large length-scale, as shown by the
additional $\tau_{u_r}(\ell)$ line, labeled $u_r$, of radial
velocities. The azimuthal over radial velocity ratio increases with
$\ell$. Line $\tau_{\rho}(\ell)$, labeled $\rho$, is obtained in a
similar way, using the conversion rule given by Equation
(\ref{eq:tauell-m_rho}) of Appendix~\ref{sec:spectra_m}, with gravity
given by $g = {\Ra(R_o) R_o \, \kappa \nu}/{R_o^4}$ and
$\Ra(R_o)=2.5 \times 10^{10}$.

Both $\tau_u(\ell)$ and $\tau_{\rho}(\ell)$ lines display a sharp
timescale minimum, very close to the Rossby line, defining length-scale
$\ell_{\perp}$. This simulation thus nicely illustrates the QG-CIA
force balance, with  $\tau_{\mathrm{Rossby}}(\ell_{\perp}) \sim
\tau_u(\ell_{\perp}) \sim \tau_{\rho}(\ell_{\perp})$, as advocated by
\citet{guervilly2019}. Note that the same force balance appears to
apply for $\ell < \ell_{\perp}$, as envisioned by \citet{rhines1975}.
Length-scale $\ell_{\perp}$ coincides with the length-scale given by
power dissipation occurring in Ekman layers (blue arrow pinned to the
Rossby line). As expected from Equations (\ref{eq:tauQG}) and
(\ref{eq:epsilon}), viscous \mbox{dissipation} is mostly due to flow at the
minimum $\tau_u(\ell)$ time.

\subsection{\tauell diagram for a non-magnetic Earth core}
\label{sec:non-magnetic-Earth}

We now have all elements to start building a \tauell scenario for
rotating convection in a non-magnetic Earth's core, which we present in
Figure~\ref{fig3}b. The goal is to create and draw realistic
$\tau_u(\ell)$ and $\tau_{\rho}(\ell)$ lines over the background
``template'' of Figure~\ref{fig:template}a. We assume that viscous
dissipation mainly occurs in Ekman layers. Applying Equation
(\ref{eq:tauQG}), we obtain time $\tau_* = [({M_o
R_o}/{\mathcal{P}_{\mathrm{diss}}}) \sqrt{{\nu}/{\Omega^3}}
]^{1/4}$, which yields the dissipated power
$\mathcal{P}_{\mathrm{diss}}$ that we estimate for the Earth's core
(see Section~\ref{sec:core}), marked by a blue arrow in
Figure~\ref{fig3}b. We further formulate the ansatz of a QG-CIA force
balance at the ``optimum'' length-scale $\ell_*$, which thus lies on
the Rossby line. We plot point ($\ell_*$, $\tau_*$) in
Figure~\ref{fig3}b.

Inspired by Figure~\ref{fig3}a, and by Rhines' arguments, we infer that
flow obeys the QG-CIA balance for all length-scales $\ell < \ell_*$. We
thus plot $\tau_u(\ell)$ and $\tau_{\rho}(\ell)$ lines along the Rossby
line all the way to their intersection with dissipation
$\tau_{\nu}(\ell)$ and $\tau_{\kappa}(\ell)$ lines, respectively. It
remains to draw lines $\tau_u(\ell)$ and $\tau_{\rho}(\ell)$ for $\ell$
larger than $\ell_{\perp}$. Flow becomes anisotropic for $\ell >
\ell_{\perp}$, with radial velocities decreasing as $\ell$ increases,
while azimuthal velocities increase with $\ell$. For $\ell >
\ell_{\perp}$ we thus loosely prescribe $\tau_{u_r}(\ell) \propto
\ell^{2}$ (for a spectral energy density $E(k) \propto k$),
$\tau_{u_{az}}(\ell) \simeq \tau_{u}(\ell) \propto \ell$ (for $E(k)
\propto k^{-1}$), and $\tau_{\rho}(\ell) \propto \ell^{1/2}$ (for a
{\unskip\break}$k^{-2}$-spectrum).

Reading the \tauell diagram of Figure~\ref{fig3}b, we see that core
flow in a non-magnetic Earth would be quasi-geostrophic at all scales,
with azimuthal velocities reaching 3~m${\cdot}$s$^{-1}$, much larger than
present-day core flow velocities represented by its
$t_u^{\mathrm{obs}}$ value and trend. The radius of dominant columnar
vortices would be around 200~km. Ekman layer viscous dissipation
would dominate over bulk viscous dissipation by many orders of
magnitude.

\section{\tauell regime diagrams for the Earth's core}
\label{sec:Earth}

In this section, we examine which \tauell regime diagrams to expect for
the Earth's core. Our goal is not to come up with an optimal or
accurate scenario, but rather to illustrate how \tauell diagrams can
help inventing and testing such scenarios. We now consider the presence
of a magnetic field and try to document Earth's core \tauell diagram,
for which we presented a template in Figure~\ref{fig:template}b. Our
starting point is the available convective power, as in
Section~\ref{sec:non-magnetic-Earth}. Building a turbulence scenario
requires again estimating the balance of forces at different
length-scales.\vspace*{-6pt}

\subsection{\tauell expression of force balances in rotating convective
dynamos} \label{sec:force_balance_dynamo}

Following the approach of Section~\ref{sec:force_balance}, we add the
Lorentz force in Navier--Stokes' equation:
{\begin{equation}\label{eq:NS_dynamo}
\rho (\partial_t \mathbf{u} +  \mathbf{u} \cdot \nabla \mathbf{u} + 2
\bOmega \times \mathbf{u} ) = {-}\nabla p + \Delta \rho
\mathbf{g} + \mathbf{j} \times \mathbf{b} + \rho \nu \nabla^2
\mathbf{u},
\end{equation}}\unskip
where $\mathbf{j}$ is the electric current density. Following
\citet{aurnou2017}, \citet{aubert2019}, \citet{schwaiger2019},
\citet{schwaiger2020}, we consider two different situations: (i)
Proudman--Taylor constraint is broken by the magnetic field and we get
a three-term balance between Lorentz, buoyancy, and Coriolis (MAC force
balance); (ii) Coriolis acceleration is dominant, and flow is
quasi-geostrophic at leading order and obeys a QG-MAC force balance.
\vspace*{-3pt}

\subsubsection{Magneto-Archimedean-Coriolis (MAC)} \label{sec:MAC}

When magnetic Lorentz force is strong enough to break quasi-geostrophy
at scale $\ell_*$, one can get a balance between Lorentz, buoyancy and
Coriolis forces, such that:
{\begin{equation}
\frac{\Omega u}{\ell_*} \sim \frac{b^2}{\rho \mu \ell_*^2} \sim
\frac{\Delta \rho }{\rho} \frac{g}{\ell_*},
\end{equation}}\unskip
analogous to the CIA balance with fluid velocity replaced by Alfv\'en
wave velocity in the advection term. Translating in \tauell language,
we get:
{\begin{equation}\label{eq:MAC_tauell}
t_{\Omega} \tau_u(\ell_*) \sim \tau_b^2(\ell_*) \sim
\tau_{\rho}^2(\ell_*),
\end{equation}}\unskip
which implies:
{\begin{equation}
\Lambda_d(\ell_*) = \frac{t_{\Omega} \tau_u(\ell_*)}{\tau_b^2(\ell_*)}
\sim 1.
\end{equation}}\unskip
Graphically, this means that $\tau_b(\ell_*)$ and $\tau_{\rho}(\ell_*)$
both plot at mid-distance between $\tau_u(\ell_*)$ and $t_{\Omega}$.
This is the regime we get when the magnetic field is strong enough 
\citep[see][]{aurnou2017}. 
The dynamical Elsasser number $\Lambda_d$ is of
order one at scale $\ell_*$.\vspace*{-3pt}

\subsubsection{Quasi-geostrophic magneto-Archimedean-{\ubreak}Coriolis
(QG-MAC)} \label{sec:QG-MAC}

When the leading order force balance is quasi-geostrophic, the Coriolis
term should only involve its ageostrophic part, at a length-scale
$\ell_{\parallel} \sim R_o$. QG-MAC balance at lenght-scale
$\ell_{\perp}$ therefore writes:
{\begin{equation}
\frac{\Omega u}{\ell_{\parallel}} \sim \frac{b^2}{\rho \mu
\ell_{\perp}^2} \sim \frac{\Delta \rho }{\rho} \frac{g}{\ell_{\perp}},
\end{equation}}\unskip
analogous to the QG-CIA balance with fluid velocity replaced by
Alfv\'en wave velocity in the advection term. Translating in \tauell
language, we get:
{\begin{equation}\label{eq:QG-MAC_tauell}
\frac{R_o}{\ell_{\perp}}t_{\Omega} \tau_u(\ell_{\perp}) \sim
\tau_{\rho}^2(\ell_{\perp}) \sim \tau_b^2(\ell_{\perp}),
\end{equation}}\unskip
which can also be written:
{\begin{equation}\label{eq:QG-MAC_tauell_alt}
\tau_{\mathrm{Rossby}}(\ell_{\perp}) \tau_u(\ell_{\perp}) \sim
\tau_{\rho}^2(\ell_{\perp}) \sim \tau_b^2(\ell_{\perp}).
\end{equation}}\unskip
Graphically, this means that $\tau_b(\ell_{\perp})$ and
$\tau_{\rho}(\ell_{\perp})$ both plot at mid-distance between
$\tau_u(\ell_{\perp})$ and $\tau_{\mathrm{Rossby}}(\ell_{\perp})$.

\subsection{\tauell diagram of a remarkable numerical simulation}
\label{sec:S2}

Let us start by building and discussing the \tauell regime diagram 
(Figure~\ref{fig:core}a) of one of the most extreme dynamo simulations
available today: the S2 DNS of \citet{schaeffer2017}.

\begin{figure*}
\includegraphics{fig04}
\caption{\label{fig:core}\tauell diagrams for rapidly rotating
convective dynamo. Refer to Figure~\ref{fig:template}b and
Section~\ref{sec:phenomena} for a complete description of the
background ``template''. (a) \textit{3D numerical simulation S2 of
\citet{schaeffer2017},} Field variables $u$, $b$ and $\rho$ of the
simulation are represented by blue $\tau_u(\ell)$, red $\tau_b(\ell)$,
and orange $\tau_{\rho}(\ell)$  lines, respectively, which are \tauell
translations of their respective time- and volume-averaged degree
$n$-spectra (see Appendix~\ref{sec:spectra_n}). Simulation's viscous
dissipation can be read on line $\tau_{\nu}(\ell)$ at the blue arrow,
while Ohmic dissipation is marked by a red arrow on line
$\tau_{\eta}(\ell)$. The black vertical dotted line marks the
length-scale at which a QG-MAC force balance appears to be achieved.
(b) \textit{Scenario for the Earth assuming a MAC force balance.} The
available convective power $\mathcal{P}_{\mathrm{diss}} \simeq 3$ TW
sets time $\tau_*$ on the $\tau_{\eta}(\ell)$ line (red arrow).
Length-scale of maximum dissipation $\ell_* \simeq 7$ km is obtained
by combining $\Rm(\ell_*) = 1$ and a MAC force balance at this same
length-scale. Circles labeled $\Re$ and $\Rm$ at the intersections of
line $\tau_u$ with lines $\tau_{\nu}(\ell)$ and $\tau_{\eta}(\ell)$
mark the scales at which the corresponding $\ell$-scale dimensionless
number (see Table~\ref{tab:dimensionless}) equals 1. Red and blue stars
on the right $y$-axis mark magnetic intensity and velocity amplitude,
respectively, predicted by \citet{christensen2006}'s scaling laws. Blue
cross from \citet{davidson2013}'s velocity scaling law.} 
\end{figure*}

We first build its ``template'' as in Figure~\ref{fig:template}b, using
radius $R_o$ and $t_{\Omega}$ of the actual Earth's core as
length-scale and timescale, respectively, in order to compare with the
Earth. Input dimensionless parameters of the simulation
($\Ek(R_o-R_i)=10^{-7}$, $\Pr=1$, $\Pm=0.1$) provide values needed to
build lines $\tau_{\nu}(\ell)$, $\tau_{\kappa}(\ell)$ and
$\tau_{\eta}(\ell)$. As in Figure~\ref{fig:template}b, power
dissipation markers are drawn and labeled along the magnetic diffusion
line, and along the Rossby and viscous lines, using outer core mass
$M_o$. Ohmic and viscous dissipations $D_{\eta}$ and $D_{\nu}$ of the
simulation are obtained from Table~2 of \citet{schaeffer2017}, and
scaled to Earth's core by: $\mathcal{P} = D \rho \nu^3/(R_o-R_i)$. They
are pointed by a red arrow on the magnetic diffusion line, and a blue
arrow on the viscous line, respectively.

Next, we turn to extracting lines $\tau_u(\ell)$ and $\tau_b(\ell)$
from the simulation's spherical harmonic spectra, applying conversion
rules of Equations (\ref{eq:tauell_u}) and (\ref{eq:tauell_b}) of
Appendix~\ref{sec:spectra_n}, respectively. The simulated acceleration
of gravity $g$ at the top boundary is obtained from:
{\begin{equation}
g = \frac{\Ra^* R_o \, \kappa \nu}{(R_o-R_i)^4},
\end{equation}}\unskip
with $\Ra^* = \Ra /\beta R_o = 2.4 \times 10^{13}$, where $\Ra$ is the
classical large-scale Rayleigh number, and $\beta$ the imposed
codensity gradient at the top boundary. We then obtain line
$\tau_{\rho}(\ell)$ by applying Equation (\ref{eq:tauell_rho}) to the
codensity spectrum multiplied by $\Pr^2$.

Reading the resulting \tauell diagram, we see that: magnetic energy
largely dominates over kinetic energy ($\tau_b(R_o) \ll \tau_u(R_o)$);
Ohmic dissipation dominates over viscous dissipation (compare
dissipation powers indicated by arrows pinned to lines
$\tau_{\eta}(\ell)$ and $\tau_{\nu}(\ell)$, respectively); flow should
be largely quasi-geostrophic, since the $\tau_u(\ell)$ line stays above
the Rossby line down to dissipation length-scales. We also observe that
both $\tau_u(\ell)$ and $\tau_b(\ell)$ lines have slopes steeper than 1
at large length-scale, while the slope of the $\tau_{\rho}(\ell)$ line
is closer to 1/2. Finally, we observe that a QG-MAC force balance seems
approximately satisfied at length-scale $\ell_{\perp} \simeq R_o/18$,
marked by a vertical dotted line, at which we observe
$({\tau_b}/{t_{\Omega}}) ({\tau_b}/{\tau_u}) \sim
{R_o}/{\ell_{\perp}} $ and $\tau_b(\ell_{\perp}) \sim
\tau_{\rho}(\ell_{\perp})$.

\subsection{A simple MAC-balance scenario} \label{sec:MAC_scenario}

Figure~\ref{fig:core}b proposes a first attempt to complete the
template of Figure~\ref{fig:template}b with plausible $\tau_u(\ell)$, 
$\tau_b(\ell)$ and $\tau_{\rho}(\ell)$ lines. As in
Section~\ref{sec:non-magnetic-Earth}, our starting point is the
available convective power $\mathcal{P}_{\mathrm{diss}} \simeq 3$ TW.
We formulate the ansatz that it is dissipated by Joule heating only,
which means that line $\tau_b(\ell)$ should get down to (but not below)
time $\tau_* = \sqrt{M_o \eta / \mathcal{P}_{\mathrm{diss}}}$ that
yields a dissipation equal to $\mathcal{P}_{\mathrm{diss}}$, as pointed
by the red arrow on line $\tau_{\eta}(\ell)$. However, in contrast with
the situation of Section~\ref{sec:non-magnetic-Earth}, we cannot
attach line $\tau_b(\ell)$ to this point, because we anticipate that
the velocity would be too weak at that scale to amplify the magnetic
field.

One of the guidelines of our \tauell approach is that regime changes
should occur where relevant $\ell$-scale dimensionless numbers reach 1,
{i.e.,} at the intersection of corresponding \tauell lines. In a
dynamo, we expect a regime change for $\Rm(\ell) \sim 1$, at the
intersection of lines $\tau_u(\ell)$ and $\tau_{\eta}(\ell)$, below
which magnetic diffusion takes precedence over induction. Furthermore,
if the magnetic field is strong, the intensity of turbulence is
strongly reduced in that low $\Rm$ regime. We thus assume that Ohmic
dissipation is maximum at length-scale $\ell_*$ where $\Rm(\ell_*) =
1$, implying $\tau_b(\ell_*) \simeq \tau_*$.

This first condition links the velocity field to the magnetic field but
is not sufficient to provide $\ell_*$. Another constraint is needed,
which we get from a force balance. As a first guess, we request that
our system obeys a MAC force balance at length-scale $\ell_*$.
Recalling Equation~(\ref{eq:MAC_tauell}), we have:
{\begin{equation*}
t_{\Omega} \tau_u(\ell_*) \sim \tau_b^2(\ell_*) \sim
\tau_{\rho}^2(\ell_*).
\end{equation*}}\unskip
Together with condition $\Rm(\ell_*)=1$, this sets $\ell_*$ at a
position such that $\tau_u(\ell_*) = \tau_{\eta}(\ell_*) =
\tau_*^2/t_{\Omega}$, and implies $\tau_{\rho}(\ell_*) = \tau_*$, as
drawn in Figure~\ref{fig:core}b. Graphically, $\tau_*$ plots at
mid-distance between $\tau_u(\ell_*)$ and $t_{\Omega}$. We can also
write:
{\begin{equation}\label{eq:MAC_l_star}
\left(\frac{\ell_*}{R_o}\right)^2 = 
\frac{\tau_*}{t_{\Omega}} \frac{\tau_*}{T_{\eta}} = 
\frac{1}{t_{\Omega} T_{\eta}^2} 
\frac{M_o R_o^2}{\mathcal{P}_{\mathrm{diss}}},
\end{equation}}\unskip
where $T_{\eta} \equiv \tau_{\eta}(R_o)$.

The next step consists in drawing lines $\tau_u(\ell)$ and
$\tau_b(\ell)$ for $\ell < \ell_*$ and for $\ell > \ell_*$. For $\ell <
\ell_*$, we note that the regime of MHD turbulence in the presence of a
strong magnetic field is characterized by steep energy density spectra:
$E(k) \propto k^{-3}$ and $E_m(k) \propto k^{-5}$ \citep{alemany1979}.
Converting to \tauell language with Equation (\ref{eq:tauell_new}), we
get $\tau_u(\ell) \propto \ell^0$ and $\tau_b(\ell) \propto \ell^{-1}$,
yielding the slopes drawn in Figure~\ref{fig:core}b. Adding rotation
further reduces the intensity of turbulence \citep{nataf2008,
kaplan2018}, but we lack constraints on the resulting energy spectra.

In the dynamo region, for $\ell > \ell_*$, we assume $k^{-1}$ energy
density spectra for both $u$ and $b$ ({i.e.}, $\tau(\ell) \propto
\ell$), implying that line $\tau_b(\ell)$ follows the Alfv\'en wave
line $\tau_{\mathrm{Alfven}}(\ell)$. This choice is mostly for
pedagogical reasons as explained below. Finally, we rather arbitrarily
assume $\tau_{\rho}(\ell) \propto \ell^{1/2}$ at all length-scales.
This completes the \tauell diagram shown in Figure~\ref{fig:core}b.

Reading this \tauell diagram, we see that our scenario yields velocity
and magnetic amplitudes that are not too far from the observed
$t_u^{\mathrm{obs}}$ and $t_b^{\mathrm{obs}}$ values. They translate
into a magnetic to kinetic energy ratio of about $10^4$, according to
Equations (\ref{eq:E}) and (\ref{eq:E_m}).  Bulk and boundary viscous
dissipations have comparable amplitudes, both many orders of magnitude
smaller than Ohmic dissipation, as assumed. The smallest QG vortices
(on the Rossby line) are very sluggish, with a turnover time of about 1
year and a radius of 1 km. Magnetic diffusion is largest at a
length-scale of about 10 km.

It is interesting to observe that our scenario implies that the
Alfv\'en line intersects line $\tau_{\eta}(\ell)$ at time $t_{\Omega}$,
meaning that all three $\ell$-scale dimensionless numbers $\Ek_{\eta}$,
$\Lu$ and $\lambda$ equal 1 at this same scale. This implies that
$\tau_b(R_o)$ can be deduced from the intersection of
$\tau_{\eta}(\ell)$ and $t_{\Omega}$ lines. In other words, even though
our scenario has been built to achieve a given Ohmic dissipation
$\mathcal{P}_{\mathrm{diss}}$, the actual value of $\tau_b(R_o)$ does
not depend on $\mathcal{P}_{\mathrm{diss}}$. Only the kinetic energy
depends on $\mathcal{P}_{\mathrm{diss}}$, as given by these scaling
laws:
{\begin{eqnarray}
\tau_u(R_o) &=& \left(\frac{T_{\eta} \tau_*^2 }{t_{\Omega}}
\right)^{1/2} = \left(\frac{\Omega M_o R_o^2}{\mathcal{P}_{\mathrm{diss}}}
\right)^{1/2} \label{eq:MACu} \Seqnsplit
\tau_b(R_o) &=& (t_{\Omega} T_{\eta})^{1/2} = 
\frac{R_o}{\sqrt{\Omega \eta}}.	\label{eq:MACb}
\end{eqnarray}}\unskip

\subsection{A simple QG-MAC balance scenario}
\label{sec:QG-MAC_scenario}

The simple MAC-balance scenario of the previous section meets a
problem: line $\tau_u(\ell)$ plots above the Rossby line for all
length-scales $\ell > \ell_*$, and all these scales have
$\Lambda_d(\ell) < 1$. This means that flow should be quasi-geostrophic
in this scale range, even though the magnetic field is strong.
Leading-order force balance should therefore be quasi-geostrophic, with
convective motions forming columnar vortices parallel to the spin axis.
Figure~\ref{fig5}a shows a first plausible alternative scenario, built
as follows.

\begin{figure*}
\includegraphics{fig05}
\caption{\label{fig5}\tauell diagrams of two QG-MAC force balance
scenarios for the Earth core. Refer to Figure~\ref{fig:template}b and
Section~\ref{sec:phenomena} for a complete description of the
background ``template'', and to Figure~\ref{fig:core}b for more
details. Both scenarios assume that the available convective power
$\mathcal{P}_{\mathrm{diss}} \simeq 3$ TW is dissipated by Joule
effect, setting time $\tau_*$ marked by a red arrow on the magnetic
diffusion line, and that the flow obeys a QG-MAC balance at a
prescribed length-scale $\ell_{\perp}=R_o/10$ indicated by a blue vertical
dot-dash line. (a) \textit{a simple QG-MAC balance scenario.} It is
assumed that $\tau_u(\ell)$ and $\tau_b(\ell)$ are proportional to
$\ell$ between $\ell_*$ and $R_o$ (see
Section~\ref{sec:QG-MAC_scenario}). (b) \textit{a QG-MAC balance
scenario \`a la Aubert.} This scenario assumes $\tau_u(\ell) \propto
\ell^{1/2}$ for $\ell_* < \ell < \ell_{\perp}$ (invariant vorticity)
and $\tau_u(\ell) \propto \ell^{3/2}$ for $\ell_{\perp} < \ell < R_o$
(following the trend of observations $t_u^{\mathrm{obs}}$). Same trends
for $\tau_b(\ell)$ (see Section~\ref{sec:QG-MAC_JA_scenario}).}
\end{figure*}

As in Section~\ref{sec:MAC_scenario} we first obtain the minimum
magnetic time $\tau_b$, which is set to $\tau_* = \sqrt{M_o \eta /
\mathcal{P}_{\mathrm{diss}}}$, marked by a red arrow on line
$\tau_{\eta}(\ell)$, and the corresponding length-scale of maximum
dissipation $\ell_*$ is such that $\Rm(\ell_*)=1$. This first condition
links the velocity field to the magnetic field but is not sufficient to
provide $\ell_*$. Another constraint is needed, which we get from a
QG-MAC force balance. We recall that such a balance at length-scale
$\ell_{\perp}$ is given by Equation (\ref{eq:QG-MAC_tauell}): 
{\begin{equation*}
\frac{R_o}{\ell_{\perp}}t_{\Omega} \tau_u(\ell_{\perp}) \sim
\tau_b^2(\ell_{\perp}) \sim \tau_{\rho}^2(\ell_{\perp}).
\end{equation*}}\unskip

Graphically, this means that $\tau_b(\ell_{\perp})$ plots at
mid-distance between $\tau_u(\ell_{\perp})$ and
$\tau_{\mathrm{Rossby}}(\ell_{\perp})$. One then needs to guess at
which scale $\ell_{\perp}$ this balance should apply. In contrast with
our simple MAC scenario, one cannot choose $\ell_{\perp} = \ell_*$,
because this would place $\tau_u(\ell_*)$ very far down, below the
$t_{\Omega}$ line, breaking quasi-geostrophy, and yielding
$\mathcal{E}_k \gg \mathcal{E}_m$ in strong disagreement with
observations. And we also need to decide how velocity and magnetic
fields vary between $\ell_*$ and $R_o$. As in
Section~\ref{sec:MAC_scenario} we assume $\tau_u(\ell) \propto \ell$
and $\tau_b(\ell) \propto \ell$. We then obtain:
{\begin{equation}\label{eq:QG-MAC_l_star}
\left(\frac{\ell_*}{R_o} \right)^3 = 
\left(\frac{\ell_{\perp}}{R_o} \right)^2
\frac{\tau_*}{t_{\Omega}} \frac{\tau_*}{T_{\eta}} .
\end{equation}}\unskip

Figure~\ref{fig5}a displays the \tauell diagram of such a QG-MAC
scenario with $\ell_{\perp} = R_o/10$. Comparing with
Figure~\ref{fig:core}b, we see that this scenario predicts a larger
magnetic over kinetic energy ratio, with $\tau_u$ above the Rossby line
down to scales of a few hundred meters. Another difference is the level
of line $\tau_{\rho}(\ell)$.

\subsection{A QG-MAC balance scenario \`a la Aubert}
\label{sec:QG-MAC_JA_scenario}

In the previous scenario, choosing $\ell_{\perp} = R_o/10$ was borrowed
from \citet{aubert2017} and \citet{aubert2019}, who find it in good
agreement with their numerical simulation results. Following
\citet{davidson2013}, \citet{aubert2019} proposes a $\tau_u(\ell)$
scaling for  $\ell_* < \ell < \ell_{\perp} $ that differs from the one
we used in Section~\ref{sec:QG-MAC_scenario}. In that interval,
\citet{davidson2013} infers that vorticity is independent of $\ell$.
This translates into $\tau_u(\ell) \propto \ell^{1/2}$ instead of
$\tau_u(\ell) \propto \ell$. Using the same scaling for $\tau_b(\ell)$,
length-scale $\ell_*$ is then given by:
{\begin{equation}\label{eq:JA_l_star}
\left(\frac{\ell_*}{R_o} \right)^{5/2} =
\left(\frac{\ell_{\perp}}{R_o}\right)^{3/2} 
\frac{\tau_*}{t_{\Omega}} \frac{\tau_*}{T_{\eta}} .
\end{equation}}\unskip

The corresponding \tauell regime diagram is shown in
Figure~\ref{fig5}b, where we chose to follow the observed $\ell^{3/2}$
trend for $\tau_u$ and $\tau_b$ above $\ell_{\perp}$ (blue and red
disks, respectively). This scenario provides an amazing fit to the
observed $t_u^{\mathrm{obs}}$ and $t_b^{\mathrm{obs}}$ values.
Furthermore, we observe that the dynamical Elsasser number
$\Lambda_d(\ell) = ({t_{\Omega}}/{\tau_b(\ell)})
({\tau_u(\ell)}/{\tau_b(\ell)})$ remains below 1 at all length-scales
$\ell$, validating our assumption of leading-order Quasi-Geostrophy.

This scenario yields the following (ugly-looking) scaling laws for the
largest scale velocity and magnetic fields:
{\begin{eqnarray}
\tau_u(R_o) &=& 
\left(\frac{R_o}{\ell_{\perp}}\right)^{1/10}
\left(\frac{T_{\eta}^2 \tau_*^6 }{t_{\Omega}^3} \right)^{1/5}
\label{eq:QG-MAC_JAu} \Seqnsplit
\tau_b(R_o) &=&
\left(\frac{R_o}{\ell_{\perp}}\right)^{13/10}
(t_{\Omega} T_{\eta} \tau_*^3)^{1/5}.
\label{eq:QG-MAC_JAb}
\end{eqnarray}}\unskip

\subsection{The interesting case of Venus} \label{sec:Venus}

The internal structure of Venus is very poorly known, but we know that
it does not generate a detectable magnetic field. This important
difference from its sister planet Earth is classically explained by a
different thermal history, leading to a hot mantle convecting beneath a
rigid lid, preventing core cooling, hence halting the convective engine
of the dynamo \citep{stevenson1983,nimmo2002}.

Venus and Earth also differ by their spinning rate: one turn in 243
days instead of one day. This difference is usually considered as
unimportant since rotation still appears overwhelming, with an Ekman
number $\Ek(R_o) \sim 10^{-13}$ \citep{russell1980}. However, if we
adopt for Venus the same physical properties as for Earth, including
its available convective power $\mathcal{P}_{\mathrm{diss}}$, but
update the spinning rate to the one of Venus, we meet a problem,
illustrated in Figure~\ref{fig:Venus}.

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig:Venus}Devil's advocate \tauell regime diagram for
the core of Venus. All relevant properties are assumed identical to
that of the Earth's core (see Section~\ref{sec:core}), except for the
spin rate (rotation period of 243 days). Venus ``template'' is built as
in Figure~\ref{fig:template}b. Value $t_u^{\mathrm{obs}} \simeq
10^{12}$ s is deduced from an upper bound on Venus' undetected magnetic
field. We speculate that the difference in rotation period has a
dramatic impact on the dynamo, and Section~\ref{sec:Venus} proposes a
tentative IMAC force balance scenario, which results in the displayed
\tauell diagram.}
\end{figure}

We first observe that Ohmic dissipation time $\tau_*= \sqrt{M_o \eta /
\mathcal{P}_{\mathrm{diss}}}$ plots below the $t_{\Omega}$ line. We
also see that the core of Venus would not be able to dissipate such a
power by friction in its Ekman layers, as indicated by the markers
along the Rossby line (compare with Figure~\ref{fig3}b). Venus does not
appear to be a ``fast rotator'', and we should not expect the flow to
be dominantly Quasi-Geostrophic as in Earth's core. It might be more
similar to the solar dynamo.

We therefore propose a very tentative ``devil's advocate'' scenario, in
which we assume that dissipation takes place in the bulk, with
equipartioned viscous and Ohmic dissipations, yielding the blue arrow
on the viscous line, and the red arrow on the magnetic diffusion line.
In this region far below the $t_{\Omega}$ line, Kolmogorov energy
density spectra $E(k) \propto k^{-5/3}$ seem plausible for both kinetic
and magnetic energies, which translate into $\tau_u(\ell) \propto
\ell^{2/3}$ and $\tau_b (\ell) \propto \ell^{2/3}$, as drawn in
Figure~\ref{fig:Venus} from their respective dissipation scales up to
timescale $t_{\Omega}$, which is met at a length-scale $\ell_{\Omega}
\simeq 5$ km.

The force balance we expect there is of type
Inertia-Magneto-Archimedean-Coriolis (IMAC), given by:
{\begin{equation}\label{eq:IMAC_tauell}
t_{\Omega} \tau_u(\ell_{\Omega}) \sim 
\tau_u^2(\ell_{\Omega}) \sim 
\tau_b^2(\ell_{\Omega})\sim
\tau_{\rho}^2(\ell_{\Omega}),
\end{equation}}\unskip
which we use to plot $\tau_{\rho}(\ell_{\Omega})$. At length-scales
larger than $\ell_{\Omega}$ the IMAC scenario drawn in
Figure~\ref{fig:Venus} assumes $\tau_{\rho}(\ell) \propto \ell^{1/2}$,
$\tau_u(\ell) \propto \ell^{2/3}$, and $\tau_b(\ell) \propto \ell^2$.
The unusual very large $\ell$ exponent for the magnetic field,
corresponding to an $E(k) \propto k$ magnetic energy density spectrum,
predicts an undetectable magnetic field at the largest length-scale.
Such a spectrum is indeed observed at the surface of the Sun
\citep{finley2023}, where the magnetic field is much stronger at small
length-scales than at large length-scales. 

Our exercise is very formal since we recall that there are good reasons
to believe that such a $\mathcal{P}_{\mathrm{diss}}$ convective power
is not available for Venus \citep{stevenson1983,nimmo2002}.
Nevertheless, our diversion to Venus questions the notion of ``fast
rotator'' and shows that for a given convective power
$\mathcal{P}_{\mathrm{diss}}$, planet's spin rate might play a more
important role than simply inferred from the low Ekman number it
delivers.

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

In this section, we try to show how \tauell diagrams can help building
a new vision on several \mbox{ongoing} debates, such as the validity domain of
various scaling laws and the controversy on the dominant length-scale
in the core. We also illustrate the link with ``path strategies'' and
propose extensions of this{\unskip\break} approach.

\subsection{Validity domain of dynamo regimes}

One goal of our approach is to better appreciate the validity domain of
the various regimes encountered in planetary cores. For example, no
dynamo will exist if line $\tau_u(\ell)$ plots too high in the diagram,
yielding $\Rm(R_o) \sim 1$. In Earth's core, this would only happen for
$\mathcal{P}_{\mathrm{diss}}$ values several orders of magnitude
smaller than our reference value of $3$ TW. Actually, very low values
(even negative) are not excluded by some thermal history models, before
the birth of the inner core \citep[{e.g.,}][]{landeau2022}.

At the other end, large $\mathcal{P}_{\mathrm{diss}}$ values can pull
the dynamo generation domain ($\Rm(\ell) > 1$) partly beneath the
Rossby line. Complete columnar vortices won't then have time to form at
length-scales below the Rossby line, and we might have a somewhat
different turbulence regime. \citet{davidson2014} proposed an original
dynamo scenario that corresponds to such a situation. Noting that
inertial waves are strongly helical, and that flow helicity is a key
ingredient for dynamo action, he suggests that Earth's dynamo might
operate this way. None of the three scenarios we presented (Figures
\ref{fig:core}b, \ref{fig5}a, \ref{fig5}b) puts $\Rm(\ell) \sim 1$
below the Rossby line. However, it would only take
$\mathcal{P}_{\mathrm{diss}} \simeq 10$ TW for the QG-MAC scenario
\textit{\`a la Aubert} (Figure~\ref{fig5}b) to qualify (remember we
only consider orders of magnitude).

An even more dramatic regime change should occur when $\tau_* <
t_{\Omega}$, as in our devil's advocate scenario for Venus 
(Figure~\ref{fig:Venus}). Then, the object should not be considered as
a ``fast rotator'' anymore, and the dynamo regime is more of a
small-scale type. This occurs when $\mathcal{P}_{\mathrm{diss}} > M_o
\eta \Omega^2$. For the Earth's core, this translates into a power of
$10^4$ TW, which was never reached in Earth's history.

We believe that \tauell diagrams could help inferring relevant dynamo
regimes for planets and stars.

\subsection{Dynamo scaling laws}\label{sec:scaling_laws}

\citet{christensen2010b} nicely reviews a number of scaling laws
proposed to infer the magnetic field intensity of planets (and stars).
Among the nine proposed scaling laws he lists, five relate magnetic
intensity to planetary rotation rate $\Omega$, with no influence of the
available convective power $\mathcal{P}_{\mathrm{diss}}$, three involve
both  $\Omega$ and $\mathcal{P}_{\mathrm{diss}}$, and one only
$\mathcal{P}_{\mathrm{diss}}$. The latter one, first proposed by
\citet{christensen2006}, stems from the analysis of a large corpus of
numerical simulations of the geodynamo, backed by an appraisal of the
dominant terms in the governing equations. It is the law preferred by
\citet{christensen2010b} who shows that it is in good agreement with
the measured magnetic intensity of planets and stars.

Translated in \tauell formulation, Christensen's preferred laws yield:
{\begin{eqnarray}
\tau_u(R_o) &=& \frac{1}{c_u} 
\left(\frac{M_o R_o^2\Omega^{1/2}}{\tilde{F} \mathcal{P}_{\mathrm{diss}}}\right)^{2/5}
\simeq 
\left(\frac{T_{\eta}^2 \tau_*^4}{t_{\Omega}}\right)^{1/5}
\label{eq:CAu} \Seqnsplit
\tau_b(R_o) &=& 
\frac{1}{\sqrt{2 c_b f_{\mathrm{ohm}}}}\left(
\frac{M_o R_o^2}{\tilde{F}\mathcal{P}_{\mathrm{diss}}}\right)^{1/3}
\simeq (T_{\eta} \tau_*^2)^{1/3}, \nonumber \\
\label{eq:CAb}
\end{eqnarray}}\unskip
from Equations (32) and~(15) of \citet{christensen2010b}. Both
large-scale flow velocity $U \equiv R_o/\tau_u(R_o)$ and magnetic field
$B  \equiv (\rho \mu)^{-1/2} R_o/\tau_b(R_o)$ do not depend upon
magnetic diffusivity\footnote{Note that one can define a dissipation
time $T_{\mathrm{diss}} = \sqrt[3]{{M_o
R_o^2}/{\mathcal{P}_{\mathrm{diss}}}}$, which makes it more explicit
that $T_{\eta} \tau_*^2= T_{\mathrm{diss}}^3$ is independent of
magnetic diffusivity $\eta$.}, and $B$ is also independent of spin rate
$\Omega$. These predictions are displayed by blue and red stars on the
right axis in Figure~\ref{fig:core}b. While $B$-prediction agrees well
with observations, $U$-prediction over-estimates flow velocity.

\citet{davidson2013} re-examined this question and noted that inertia
is playing too strong a role in \citet{christensen2006}'s simulations.
He derived a QG-MAC-balance variant and obtained a revised flow
velocity law (his Equation (15)), which translates into:
{\begin{equation}\label{eq:D13u}
\tau_u(R_o) \simeq \Omega^{1/3}
\left(\frac{M_o R_o^2}{\mathcal{P}_{\mathrm{diss}}}\right)^{4/9} = 
\left(\frac{T_{\eta}^4 \tau_*^8}{t_{\Omega}^3}\right)^{1/9},
\end{equation}}\unskip
in which $U$ is independent of magnetic diffusivity $\eta$. Davidson's
prediction is drawn as a blue $\times$ in Figure~\ref{fig:core}b.

Our discussion on scenario validity domain questions the obtention of
universal dynamo scaling laws. However, we have seen that such scaling
laws can be easily derived from the \tauell diagram of various
scenarios. Each scenario produces $\tau_u(R_o)$ and $\tau_b(R_o)$ that
combine $\tau_*$ (or $T_{\mathrm{diss}}$), $T_{\eta}$ and $t_{\Omega}$
with various powers. Equations (\ref{eq:MACu}) and (\ref{eq:MACb}) give
the scaling laws for a simple MAC scenario. Although built to dissipate
a given convective power $\mathcal{P}_{\mathrm{diss}}$ by Ohmic
dissipation, this scenario provides a large-scale magnetic field $B$
that does not depend upon $\mathcal{P}_{\mathrm{diss}}$, while $U$ is
independent of magnetic diffusivity $\eta$. The prediction of our
QG-MAC scenario ``\`a la Aubert'' is given by Equations
(\ref{eq:QG-MAC_JAu}) and (\ref{eq:QG-MAC_JAb}) for $U$ and $B$,
respectively. 

\subsection{Dominant length-scale controversy}
\label{sec:length-scale}

Several recent studies have revived a debate on the dominant
length-scales to be expected in the Earth's core \citep{yan2021,
bouillaut2021,yan2022, cattaneo2022, nicoski2023, abbate2023,
hawkins2023}. Fast rotation inhibits thermal convection, and viscosity
is needed to break the Taylor--Proudman constraint
\citep{chandrasekhar1961}. Studies on the onset of thermal convection
in rapidly rotating plane layers \citep{chandrasekhar1961}, annulus
\citep{busse1970}, and spherical shells or spheres \citep{busse1970,
jones2000, dormy2004} therefore all show that, at the onset of
convection, the flow consists in columnar vortices aligned with the
rotation axis, with a width $\ell_c \sim R_o [\Ek(R_o)]^{1/3}$ (see
Appendix~\ref{sec:onset}). In the Earth's core, such columns would be
30~m in diameter, extending several thousand kilometers in length. At
least two effects could strongly widen such columns: (i) the magnetic
field \citep{chandrasekhar1961, fearn1979, sreenivasan2011,
aujogue2015}; (ii) non-linear flow advection \citep{rhines1975,
ingersoll1982, gillet2007b, jones2015, guervilly2019}.

The effect of the magnetic field can be studied at the onset of
convection, but only for given---often simplistic---geometries of the
magnetic field that never occur in natural systems. These studies show
that convective cells can get as large as the depth of the container
$R_o$ when a magnetic field is present. Concerning planetary dynamos,
one can argue that such an effect can only be invoked when the dynamo
has already produced a strong enough magnetic field. This led
\citet{cattaneo2022} to call on the Moon for help, arguing that in an
unmagnetized Earth, ``convection would consist of hundreds of thousands
of extremely thin columns'' that would be unable to produce a magnetic
field.

This view is clearly challenged by \citet{guervilly2019} who advocate a
dominant convective \mbox{length-scale} around 30 km. Their prediction rests
on the extrapolation of a QG-CIA force balance to Earth's core
conditions, and is backed by numerical simulations such as the one we
present in Figure~\ref{fig3}a. Our QG-CIA scenario, shown in
Figure~\ref{fig3}b obeys the same rules. It predicts an even larger
dominant length-scale (${\sim}200$~km) because it is built on the
available convective power $\mathcal{P}_{\mathrm{diss}}$ rather than on
the observed large-scale velocity, which is reduced by the magnetic
field. Interestingly, Figure~\ref{fig3}a shows that the simulations of
\citet{guervilly2019} remain very close to the convection onset, which
would yield a critical length-scale $\ell_c$ close to the intersection
of the $\tau_{\kappa}(\ell)$ and $\tau_{\mathrm{Rossby}}(\ell)$ lines
(see Appendix~\ref{sec:onset}). Yet, they reach a turbulent QG-CIA
balance because of the low value of the Prandtl number ($\Pr=10^{-2}$)
that allows for large Reynolds numbers near the onset
\citep{aubert2001, kaplan2017}.

Our \tauell diagrams clearly emphasize the role of
$\mathcal{P}_{\mathrm{diss}}$ in controlling the dominant length-scale,
both with and without a magnetic field. This is particularly clear in
Figure~\ref{fig3}b where the minima of curves $\tau_u(\ell)$ and
$\tau_{\rho}(\ell)$ would ``slide'' along the Rossby line following the
$\mathcal{P}_{\mathrm{diss}}$ blue arrow. This diagram also suggests
that while viscosity in the bulk controls the convective length-scale
at the onset (black triangle), it progressively looses its importance
as viscous dissipation in the Ekman layers takes over. To achieve a
turbulent convection regime where viscosity in the bulk is unimportant,
it is crucial to provide another way of dissipating the convective
power, like Ekman layer friction or Ohmic dissipation, which are both
present in Earth's core. Because you always need to dissipate the power
injected into turbulence, the previous sentence sounds rather obvious.
Yet, this might explain why several recent studies find no evidence of
a diffusion-free scaling, in which viscosity plays no role. In the
numerical simulations of \citet{yan2021, yan2022, nicoski2023},
boundaries are stress-free so that no other energy sink than viscous
dissipation in the bulk can equilibrate the convective power.

\subsection{Path strategy}\label{sec:path}

Our \tauell approach has much in common with the ``path strategy''
devised by Julien Aubert and colleagues \citep{aubert2017, aubert2021,
aubert2022, aubert2023}. Their strategy also targets the dynamical
regime of a given object, the Earth's core, and aims at defining a
``path'' that numerical simulations should follow to reach that regime.
As stressed by \citet{dormy2016}, minimizing $\Pm$ at a given Ekman
number $\Ek(R_o) = \nu / \Omega R_o^2$ is not a good strategy, since it
increases magnetic Ekman number $\Ek_{\eta}(R_o) = \eta / \Omega
R_o^2$, while \citet{christensen2010a} show that $\Ek_{\eta}(R_o) <
10^{-4}$ is needed for obtaining Earth-like geodynamo simulations.
Hence the need for targeting a more elaborate ``distinguished limit''
\citep{dormy2016}.

Aubert and his colleagues stress the importance of respecting the
hierarchy of dynamic times, and propose a unidimensional path based on
scaling laws obtained by \citet{davidson2013} for a QG-MAC force
balance. Position along the path is measured by an $\epsilon$
parameter. As exposed in \citet{aubert2023}, the idea is to keep the
magnetic diffusion time $T_{\eta} = \tau_{\eta}(R_o)$ fixed to the
Earth value, and to  decrease the magnetic Ekman number step by step,
by decreasing the rotation time $t_{\Omega}$ from values at which
appropriate numerical simulations are available (defining $\epsilon=1$)
down to the actual value of the Earth ($\epsilon=10^{-7}$). Long-period
behaviour of the Earth's core is well recovered from $\epsilon=1$,
while decreasing $\epsilon$ to Earth's value allows for the development
of short-term dynamics.\looseness=1

Figure~\ref{fig:path} shows how \tauell diagrams can help visualize and
extend this strategy. We show three diagrams, which correspond to three
different values of $\epsilon$: $1$, $10^{-3}$, and $10^{-7}$. Their
``template'' are built from the values of the magnetic Ekman and
Prandtl numbers listed for ``prior PB'' in Table 1 of
\citet{aubert2023}. The values of $\mathcal{P}_{\mathrm{diss}}$ (red
arrow), $t_u^{\mathrm{obs}}$ (blue disk), and $t_b^{\mathrm{obs}}$ (red
disk), are from the corresponding simulation outputs given in his 
Table~2. We then add the $\tau_{\rho}(\ell)$, $\tau_u(\ell)$ and
$\tau_b(\ell)$ lines that our ``\`a la Aubert'' QG-MAC balance scenario
(see Section~\ref{sec:QG-MAC_JA_scenario}) predicts for the listed
$\mathcal{P}_{\mathrm{diss}}$. An amazing fit of the ``observed'' $t_u$
and $t_b$ is obtained, choosing $\ell_{\perp} = R_o/5$.

\begin{figure*}
{\vspace*{3pt}}
\includegraphics{fig07}
\caption{\label{fig:path}\tauell regime diagrams illustrating the
``path strategy'' devised by Aubert and colleagues. (a) 0\% of path
($\epsilon = 1$). (b) 43\% of path ($\epsilon = 10^{-3}$). (c) 100\%
of path ($\epsilon = 10^{-7}$). See text for explanations.}
{\vspace*{3pt}}
\end{figure*}

A key contribution of Aubert and colleagues was to determine how the
input parameters of numerical simulations should scale with $\epsilon$
in order to follow the path. This is where the QG-MAC scaling laws of
\citet{davidson2013} come in, and this is also where \tauell diagrams
can help. Using the scaling laws obtained for $\tau_u(R_o)$ and
$\tau_b(R_o)$ in Section~\ref{sec:QG-MAC_JA_scenario}, we test two
different paths. Note that parameter $\epsilon$ is defined such that
$\Ek_{\eta}[\epsilon] = \sqrt{\epsilon} \, \Ek_{\eta}[1]$, implying 
$t_{\Omega}[\epsilon] = \sqrt{\epsilon} \, t_{\Omega}[1]$.

Aubert and colleagues chose to keep $\Rm(R_o)$ constant along the path,
{i.e.,} $\tau_u(R_o)[\epsilon] = C^{te}$. Equation
(\ref{eq:QG-MAC_JAu}) then implies $\tau_*[\epsilon] \propto
\sqrt{t_{\Omega}[\epsilon]} \propto \epsilon^{1/4}$, since $T_{\eta}$
does not vary with $\epsilon$, and assuming that $\ell_{\perp}/R_o$ is
constant as well. We thus get $\mathcal{P}_{\mathrm{diss}}[\epsilon]
\propto \epsilon^{-1/2}$ since $\tau_* = \sqrt{M_o \eta /
\mathcal{P}_{\mathrm{diss}}}$. Equation (\ref{eq:QG-MAC_JAb}) then
implies $\tau_b(R_o)[\epsilon] \propto \epsilon^{1/2}$. These scalings
agree with those of \citet{aubert2023}, as can be seen in
Figure~\ref{fig:path}.

Instead of keeping $\Rm(R_o)$ constant along the path, one could choose
to keep the dynamo-generating domain above the Rossby line, as we
obtain for the Earth. Imposing that $\Rm(\ell)=1$ occurs at the
intersection of lines $\tau_{\eta}(\ell)$ and
$\tau_{\mathrm{Rossby}}(\ell)$ yields $\tau_u(R_o)[\epsilon] \propto
\epsilon^{1/4}$, $\tau_b(R_o)[\epsilon] \propto \epsilon^{3/8}$, and
$\mathcal{P}_{\mathrm{diss}}[\epsilon] \propto \epsilon^{-11/12}$. It
would be interesting to run numerical simulations along such a path.

\section{Limitations and perspectives}\label{sec:limitations}

Let us first recall that our \tauell approach is \textit{not} a theory
of turbulence. We try to formulate plausible scenarios by identifying
scales at which a change in turbulence regime should occur, and by
patching scaling laws appropriate for each regime. We thus entirely
depend on the availability of such laws, which can be brought by
experiments, theory, and numerical simulations.

Our approach contains a fair number of assumptions and approximations.
How realistic are the conversion rules we employ to ``translate'' force
balances and turbulent spectra in \tauell language? Does the minimum
dynamic time control dissipation? What controls length-scale
$\ell_{\perp}$ that we had to guess for writing QG-MAC-type force
balances?

For simplicity reasons, we have treated planetary cores as simple full
spheres. The application to actual planets requires to at least
consider spherical shells of various thicknesses instead. An extension
to giant planets and stars also requires taking into account
compressibility and free-slip boundaries.

Our results suggest that dissipation of Quasi-Geostrophic flows in
Ekman layers takes over bulk dissipation in rapidly rotating convection
when it gets strongly super-critical. Can it be tested? How smooth is
the transition? Our devil's advocate scenario for Venus suggests a
transition to a very different regime if dissipation is too large to be
taken up by Ekman friction. Is there evidence for such a transition?
How sharp is it?

We observe that density anomaly spectra from numerical simulations are
rarely displayed, while they convey valuable information. We also note
that spectra from laboratory experiments are scarce (but see
\citet{madonia2023}) and too often given in ``arbitrary units'',
preventing their conversion into \tauell representation. We are lacking
experimental data on turbulence for rotating convection in a sphere in
presence of strong magnetic fields.

\tauell diagrams provide hints on how velocity and magnetic field scale
with length-scale. This might be useful for observers who need such
constraints to tune their magnetic field and core-flow inversions
\citep{gillet2015, baerenzung2016}. As a matter of fact, our analysis
suggests that the flat spherical harmonic spectra observed at low
degrees $n$ for both flow velocity and magnetic field cannot extend
much beyond degree 10 without meeting dissipation problems.

\section{Conclusion}\label{sec:conclusion}

\tauell regime diagrams are a simple graphical tool that proves useful
for inventing or testing dynamic scenarios for planetary cores.
Tradition in fluid dynamics is to characterize systems by dimensionless
numbers, usually based on ``typical'' large-scale \mbox{quantities.} Past
decades have seen large efforts to develop a more detailed description
of phenomena that operate at different scales. This has led to the
apparition of even more dimensionless numbers, in which the various
scales involved do not always figure very clearly, and to the
construction of somewhat unintelligible scaling laws. By defining
$\tau$ timescales that depend on $\ell$ length-scales over their entire
range, we hope to make these choices more explicit. By providing a
simple graphical identity to these scales, we wish to make their
meaning more intuitive. Contrary to spectra in ``arbitrary units'',
\tauell diagrams give insight into regimes and balances which are
paramount to rotating, magnetized and/or stratified fluids, where waves
can be present and significantly alter the dynamics.

Because they put together most key properties of a given object,
\tauell regime diagrams constitute a nice identity card. We think this
applies to numerical simulations and laboratory experiments as well.
Both approaches enable extensive parameter surveys, which are crucial
for exploring and understanding different regimes. Being
object-oriented, \tauell diagrams are not easily applied to such
surveys, but we think they would very valuably complement classical
scaling law plots. The idea would be to draw \tauell diagrams for a few
representative members and end-members of the survey, which would
nicely illustrate their validity range.

Our article thus has two goals. The first goal is to provide all
ingredients for building your own \tauell diagram, be it of a numerical
simulation, a laboratory experiment or theory. To that end, we included
construction rules, examples, technical appendices, and Python scripts
(supplementary material). The second goal is to demonstrate the
potential of \tauell regime diagrams for suggesting and testing various
scenarios for Earth's dynamo.

Convinced that available convective power $\mathcal{P}_{\mathrm{diss}}$
is a key control parameter, and the one that can most readily be
estimated for other planets and exoplanets, we have modified our
original approach \citep{nataf2015} to propose and discuss a few
scenarios built upon this input data. This results in a more
challenging exercise, calling for force balance inspection. We show
that the \tauell translation of relevant force balances is very handy
and telling.

We built several geodynamo scenarios to test MAC (Figure 
\ref{fig:core}b) and QG-MAC (Figure~\ref{fig5}a, \ref{fig5}b) force
balances. The validity domain of these scenarios shows up well in
\tauell diagrams. A QG-MAC scenario ``\`a la Aubert'' looks
particularly appealing, and could have applied to the Earth over its
entire history. We note that in such a scenario, flow in the
dynamo-generating region remains Quasi-Geostrophic, with a dynamical
Elsasser smaller than 1, even though the magnetic to kinetic energy
ratio is of order $10^4$. In contrast, Venus would have a hard time
entering that regime, because of its slow rotation
(Figure~\ref{fig:Venus}). This calls for a re-analysis of what is
called a ``fast rotator''.

\tauell regime diagrams also help us addressing several on-going
debates, such as the the validity of various scaling laws, and the
question of the dominant convective length-scale in the Earth's core.
We speculate that dissipation in Ekman layers drives non-magnetic rapidly 
rotating convection towards a \mbox{QG-CIA}
force balance when the flow is turbulent
enough, promoting dominant length-scales much larger than the iconic
$R_o \, \Ek^{1/3}$ length-scale at convection onset
(Figure~\ref{fig3}b).

We use \tauell diagrams to illustrate the concept of ``path strategy''
developed by Julien Aubert and colleagues (Figure~\ref{fig:path}), and
we propose an interesting alternative to their original path.

\section*{Declaration of interests}

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

\section*{Dedication}

The manuscript was mostly written by HCN. NS ran all presented
numerical simulations. Both authors have given approval to the final
version of the manuscript.

\section*{Acknowledgments}

We thank the French Academy of Science and Electricit\'e de France for
granting their ``Amp\`ere Prize'' to our ``Geodynamo'' team, and
B\'ereng\`ere Dubrulle for early encouragements. HCN thanks Peter
Davidson, Julien Aubert, Thomas Gastine, Franck Plunian, Sacha Brun,
Antoine Strugarek and Quentin Noraz for useful discussions, and
Emmanuel Dormy for his review. The authors would like to thank the
Isaac Newton Institute for Mathematical Sciences, Cambridge, for
support and hospitality during the programme ``Frontiers in dynamo
theory: from the Earth to the stars'' where work on this paper was
undertaken. This work was supported by EPSRC grant no EP/R014604/1.
ISTerre is part of Labex OSUG@2020 (ANR10 LABX56). Our manuscript
greatly benefited from the thorough reviews of three anonymous referees
(including rather harsh but stimulating comments by a ``lengths and
time scales person'').

\CDRGrant[EPSRC]{EP/R014604/1}
\CDRGrant[LABEX]{ANR10 LABX56}

\back{}

\section*{Supplementary data}

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

\CDRsupplementaryTwotypes{supplementary-material}{\cdrattach{crgeos-256-suppl.pdf}}
\CDRsupplementaryTwotypes[application/zip]{supplementary-material}{\cdrattach{tau-ell\_programs.zip}}

All Python scripts and data used to produce the figures of this article
are available in package at article's URL and at
\url{https://gricad-gitlab.univ-grenoble-alpes.fr/natafh/shell\_tau-ell\_programs}.

\appendix{}

\gdef\appendixlabel{Appendix A}
\section{\tauell representation of turbulent spectra}
\label{sec:spectra}

\renewcommand\theequation{A\arabic{equation}}

Our ``fuzzy physics'' approach targets gathering dominant physical
phenomena in a common frame. Some of them are classically defined in
wavenumber or frequency space, rather than in physical space, hence the
need for some conversion. This appendix lists the different types of
energy spectra that are usually obtained from observations, numerical
simulations and experiments, and derives recipes for converting them
into \tauell language.

\subsection{Choosing a conversion rule} \label{sec:conversion}

We consider different expressions of total energy per unit mass
$\mathcal{U}$:
{\begin{eqnarray}
\mathcal{U} &=& \int{E(k)\,\mathrm{d}k} \quad \mbox{or} \quad 
\mathcal{U} = \sum_{k_i}{E}_B(k_i) \quad \mbox{or} \nonumber \\
&&  \quad \mathcal{U}(r) = \frac{1}{\rho}\sum_{n}\mathcal{L}_b(n,r).
\end{eqnarray}}\unskip
$E(k)$ is the classical \textit{spectral energy density} introduced by
\citet{kolmogorov1941}. $E_B(k_i)$ is the discrete equivalent of $E(k)$
for flow ``in a box'' \citep{lesieur2008}. $\mathcal{L}_b(n,r)$ is the
spherical harmonic degree $n$ component at radius $r$ of
Lowes--Mauersberger spectrum widely used in geomagnetism
\citep{lowes1966}. Note that we denote $n$ the spherical harmonic
degree, often denoted $l$ or $\ell$, in order to avoid confusion with
our length-scale $\ell$.

A flow with spectral energy density $E(k) \propto k^{-5/3}$ yields the
same $k$-exponent for its discrete energy spectrum $E_B$
\citep{lesieur2008,stepanov2014}, and a $n^{-5/3}$ Lowes--Mauersberger
spectrum. However, pre-factors may differ. More importantly, the
conversion of energy spectra into \tauell equivalents is questionable.
Indeed, no exact conversion between spectral energy density and eddy
velocity can be drawn, as thoroughly discussed by \citet{davidson2005}.

In \citet{kolmogorov1941}'s universal turbulence, an \textit{eddy
turnover time} is classically derived as:
{\begin{equation}\label{eq:tauell_K41}
\tau_u(\ell) \simeq \ell^{3/2} [E(k)]^{-1/2} \quad \mbox{with }
\ell \simeq 1/k,
\end{equation}}\unskip
where $\ell$ is the ``size'' of the eddy. A similar result is obtained
using \textit{velocity increments} $S_2(\ell) = <[u(x+\ell)-u(x)]^2> =
C_2 (\epsilon \ell)^{2/3}$. This approach is appealing for
Kolmogorov-type self-similar inertial range where no length-unit other
than $\ell$ should appear, and where large eddies are more energetic
than small eddies. This conversion rule was used in \citet{nataf2015}.

In our object-oriented approach, the integral length scale $R_o$ plays
a role, and when translating numerical simulation energy spectra in
\tauell form, we were thus tempted to simply define $\tau_u(\ell)$ as
$\ell(n) \left[\mathcal{S}_u(n) \right] ^{-1/2}$ where
$\mathcal{S}_u(n)$ is the degree $n$ component of the spherical
harmonic spectrum of $u^2$, and $\ell(n)$ is given by:
{\begin{equation}\label{eq:ell_of_n}
\ell(n) \simeq \frac{1}{2} \frac{\uppi R_o}{n+1/2}.
\end{equation}}\unskip
It turns out that this choice is not consistent with our \tauell
representation, in which the $n^{-5}$ spectrum of a system in Rhines'
equilibrium at all scales should translate into $\tau_u(\ell) \propto
\ell^{-1}$ and plot along the Rossby line.

We thus use conversion rules similar to that of Equation
(\ref{eq:tauell_K41}), such as:
{\begin{eqnarray}\label{eq:tauell_new}
\hspace*{-1.6pc}
\tau_u(\ell(n)) \simeq \ell(n) [n \, \mathcal{S}_u(n)]^{-1/2}
\quad\mbox{with } \ell(n) \simeq \frac{1}{2} \frac{\uppi R_o}{n+1/2}.
\hspace*{-2pc}
\nonumber\\ 
\end{eqnarray}}\unskip

\subsection{Application to various relevant spectra}
\label{sec:application}

We now detail the \tauell conversion of spectra commonly obtained from
observations, numerical experiments and laboratory experiments.

\subsubsection{Lowes--Mauersberger spectrum} \label{sec:spectra_LM}

In geomagnetism, the variation of magnetic energy with length-scale is
usually measured by its Lowes--Mauersberger spectrum \citep{lowes1966}.
This spectrum is expressed in terms of Gauss coefficients $g_n^m$ and
$h_n^m$ of scalar magnetic potential $V$, which defines the internal
magnetic field at any radius above the core-mantle boundary when the
mantle is considered as an electrical insulator.

Potential $V(r,\theta,\varphi)$ is then solution of Laplace equation
and can be expressed in terms of spherical harmonics as:
{\begin{eqnarray}
 V(r,\theta,\varphi) &=& R_o \sum_{n=1}^\infty \sum_{m=0}^n \left(
\frac{R_o}{r}\right)^{n+1} P_n^m(\cos \theta) \nonumber \\
&& \times \, (g_n^m \cos m \varphi + h_n^m \sin m
\varphi),
\end{eqnarray}}\unskip
where $P_n^m$ are the Schmidt semi-normalized associated Legendre
functions of degree $n$ and order~$m$.

Following \citet{langlais2014}, the Lowes--Mauersberger spectrum at any
$r>R_o$ is then given by the suite of $\mathcal{L}_b(n)$ defined by:
{\begin{equation}
\mathcal{L}_b(n,r) = (n+1) \sum_{m=0}^n ((g_n^m)^2 + (h_n^m)^2)
\left(\frac{R_o}{r}\right)^{2n+4}.
\end{equation}}\unskip
The total magnetic energy per unit mass at radius $r$ is obtained as:
{\begin{equation}\label{eq:Btot}
\mathcal{U}_m(r) \equiv \frac{B^2(r)}{2 \rho \mu} =
\frac{1}{\rho}\sum_{n=1}^\infty \mathcal{L}_b(n,r).
\end{equation}}\unskip
Spherical harmonic degree $n$ is related to our $\ell$ length-scale by:
{\begin{equation}\label{eq:n2ell}
\ell(n) \simeq \frac{1}{2} \frac{\uppi R_o}{n+1/2}.
\end{equation}}\unskip
We deduce from Equations (\ref{eq:Btot}) and (\ref{eq:n2ell}) that the
magnetic field $b(\ell,r)$ at length-scale $\ell$ and radius $r$ is
given by:
{\begin{equation}
b(\ell(n),r) \simeq \sqrt{2 \mu \mathcal{L}_b(n,r)}.
\end{equation}}\unskip
We finally obtain $\tau_b(\ell)$ from:
{\begin{equation}\label{eq:tauell_LM}
\tau _b(\ell(n)) \simeq \frac{\ell(n)}{\sqrt{2 n \mathcal{L}_b(n)/\rho}}
\quad \mbox{with } \ell(n) = \frac{1}{2} \frac{\uppi R_o}{n+1/2}.
\end{equation}}\unskip
A flat Lowes--Mauersberger spectrum (constant $\mathcal{L}_b(n)$), such
as observed for the Earth's magnetic field at the core-mantle boundary,
thus translates into $\tau_b(\ell) \propto \ell^{3/2}$.

\subsubsection{Degree $n$-spectra from numerical simulations}
\label{sec:spectra_n}

Numerical simulations of planetary dynamos are most often performed
using a pseudo-spectral  expansion in spherical harmonics
$Y_n^m(\theta, \varphi)$ of degree $n$ and order $m$. Degree-$n$ or
order-$m$ spectra are thus readily obtained for both velocity, magnetic
and codensity fields. These spectra are usually for $u^2$, $b^2$ and
$C^2$ in dimensionless units, such that the sum over all $n$ and $m$
yields $2/\rho$ times the energy per unit mass of that dimensionless
field. Given length-scale $\mathrm{L}$ and time-scale $\mathrm{T}$
chosen in the simulation, $u$ and $b$ spectra should be multiplied by
$\mathrm{L}^2/\mathrm{T}^2$ (assuming that $b$ is expressed in Alfv\'en
wave velocity units).

Given these precisions, the procedure is similar to that exposed in
Appendix~\ref{sec:spectra_LM}. $\ell(n)$ is still given by:
{\begin{equation}\label{eq:tauell_ell}
\ell(n) = \frac{1}{2} \frac{\uppi R_o}{n+1/2}.
\end{equation}}\unskip
One should keep in mind that for a given degree $n$, the corresponding
length-scale varies linearly with radius. This is ignored in our
approach. If $\mathcal{S}_x(n)$ is the $n$-element of the dimensionless
$x^2$ spectrum, then the corresponding $\tau_x(\ell)$ lines for $x =
u$, $b$ and $\rho$ are given by:
{\begin{eqnarray}
\tau_u(\ell(n)) &\simeq& \frac{\mathrm{T}}{\mathrm{L}}
\frac{\ell(n)}{\sqrt{n \mathcal{S}_u(n)}}, \label{eq:tauell_u}
\Seqnsplit
\tau_b(\ell(n)) &\simeq& \frac{\mathrm{T}}{\mathrm{L}} 
\frac{\ell(n)}{\sqrt{n \mathcal{S}_b(n)}},
\label{eq:tauell_b} \Seqnsplit
\tau_{\rho}(\ell(n)) &\simeq& \sqrt{\frac{\ell(n)}{g \, 
\sqrt{n \mathcal{S}_{\rho}(n)}}}, \label{eq:tauell_rho}
\end{eqnarray}}\unskip
where gravity $g$ (in dimensional units) is obtained from the input
Rayleigh number.

In the example of Figure~\ref{fig3}a from \citet{guervilly2019}, the
simulated acceleration of gravity $g$ at the top boundary is obtained
from:
{\begin{equation}
g = \frac{\Ra R_o \, \kappa \nu}{R_o^4},
\end{equation}}\unskip
with $\Ra = 2.5 \times 10^{10}$.

In the example of Figure~\ref{fig:core}a from \citet{schaeffer2017},
$g$ is obtained from:
{\begin{equation}
g = \frac{\Ra^* R_o \, \kappa \nu}{(R_o-R_i)^4},
\end{equation}}\unskip
with $\Ra^* = \Ra/\beta R_o = 2.4 \times 10^{13}$, where $\beta$ is the
imposed codensity gradient at the top boundary. We then obtain the
$\tau_{\rho}(\ell)$ line by applying Equation (\ref{eq:tauell_rho}) to
the codensity spectrum multiplied by $\Pr^2$.

\subsubsection{Order $m$-spectra from numerical simulations}
\label{sec:spectra_m}

Quasi-geostrophic vortices are better characterized by their order-$m$
spectra $\mathcal{S}_x(m)$ than by their degree-$n$ spectra, in
particular in 2D QG simulations. Thus \citet{guervilly2019} display
$m$-spectra of their 3D and QG convection simulation results.
Translation into \tauell is obtained as in
Appendix~\ref{sec:spectra_n}, replacing $\mathcal{S}_x(n)$ by
$\mathcal{S}_x(m)$:
{\begin{eqnarray}
\tau_u(\ell(m)) &\simeq& \frac{\mathrm{T}}{\mathrm{L}}
\frac{\ell(m)}{\sqrt{m \mathcal{S}_u(m)}}, \label{eq:tauell-m_u}
\Seqnsplit
\tau_b(\ell(m)) &\simeq& \frac{\mathrm{T}}{\mathrm{L}} 
\frac{\ell(m)}{\sqrt{m \mathcal{S}_b(m)}},
\label{eq:tauell-m_b} \Seqnsplit
\tau_{\rho}(\ell(m)) &\simeq& \sqrt{\frac{\ell(m)}{g \,
\sqrt{m \mathcal{S}_{\rho}(m)}}}, \label{eq:tauell-m_rho}
\end{eqnarray}}\unskip
with:
{\begin{equation}\label{eq:tauell-m_ell}
\ell(m) = \frac{1}{2} \frac{\uppi R_o}{m+1/2}.
\end{equation}}\unskip

For the simulation presented in Figure~\ref{fig3}a, $\mathcal{S}_x(m)$
and $\mathcal{S}_x(n)$ spectra are very similar, apart for even-odd
oscillations in the $n$-spectra due to equatorial symmetry.

\subsubsection{Frequency spectrum}
\label{sec:spectra_exp}

In laboratory experiments, turbulent spectra are more easily obtained
from signal $x(\mathbf{r},t)$ measured in the time-domain ($t=0$ to
$T$) at a given position $\mathbf{r}$. Power spectral density (PSD) is
then computed from its Fourier transform $\hat{x}_T(\mathbf{r},f)$ as:
{\begin{equation}\label{eq:PSD}
\tilde{E}(\mathbf{r},f) = \lim_{T \to \infty} \frac{1}{T} |
\hat{x}_T(\mathbf{r},f) |^2.
\end{equation}}\unskip

When a mean flow $\mathbf{U}(\mathbf{r})$ is present, and when
turbulence is weak enough, a time record of velocity reflects the
advection of the spatial variation of velocity
\citep[e.g.,][]{frisch1995}. Extensions to intense turbulence have been
developed \citep{pinton1994}. Taylor's hypothesis \citep{taylor1938}
then permits to obtain a kinetic energy density wavenumber spectrum
$E(\mathbf{r},k)$ from the velocity frequency power spectrum
$\tilde{E}(\mathbf{r},f)$:
{\begin{equation}\label{eq:PSDtok}
E(\mathbf{r}, k(f)) = \frac{U}{2 \uppi}\tilde{E}(\mathbf{r}, f)
\quad\mbox{with } k(f) = \frac{2 \uppi f}{U},
\end{equation}}\unskip
where $U = \Vert \mathbf{U}(\mathbf{r})\Vert$. Strictly speaking, this is a
$k_{\sslash}$-spectrum, valid for wavenumber $\mathbf{k}$ parallel to
mean velocity vector $\mathbf{U}(\mathbf{r})$. It is the same as a
$k$-spectrum for isotropic turbulence. The \tauell translation is then
obtained from Equation (\ref{eq:tauell_K41}).

Note that Taylor's hypothesis cannot be applied to magnetic spectra
unless magnetic diffusion is small enough for the frozen flux
approximation to apply.

\gdef\appendixlabel{Appendix B}
\section{\tauell diagram for convection onset} \label{sec:onset}

\renewcommand\theequation{B\arabic{equation}}

Although \tauell regime diagrams are built to span a large range of
length- and time-scales, they provide an interesting insight on what
controls single scales appearing at the onset of convection. Here we
compare what would be the convection threshold in Earth's core,
depending on whether it is rotating or not. We display the results in
Figure~\ref{fig:onset}, using properties of the Earth's core listed in
Table~\ref{tab:properties}. We extended the figure to very long
$\tau$-values in order to include the intersection of the viscous line
and the right $y$-axis.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig:onset}\tauell diagram illustrating convection onset
in the Erth's core in the absence of a magnetic field. In the absence
of rotation, convection sets in with a cell size comparable with $R_o$.
Orange disk marks the value of $\tau_{\rho}$ at convection onset. It is
half-way between $\tau_{\kappa}(R_o)$ and $\tau_{\nu}(R_o)$, where
$\Ra(R_o) \sim 1$. Rotation controls the length-scale at which
convection sets in. We compare a  $\Pr < 1$ case with a $\Pr > 1$ case.
Black down-triangle marks column radius $\ell_c$ and period $\tau_c$ of
the quasi-geostrophic vortices that form at convection onset when
considering viscosity and thermal diffusivity ($\Pr<1$). Orange
down-triangle at $\ell=R_o$ indicates the value of $\tau_{\rho}$ at
onset. Up-triangles display the same quantities when considering
viscosity and compositional diffusivity $\chi$ ($\Pr>1$). Dotted lines
help reading their graphical construction.} 
\end{figure}

In the absence of rotation, the threshold of convection is governed by
a balance between buoyancy and the combined action of momentum and
thermal diffusions. It takes place at the largest length-scale $\ell =
R_o$ (or at ``scale height'' $\mathcal{H} = {C_P}/{\alpha g}$ if it
is smaller than $R_o$), and for $\Ra(R_o) \sim 1$ (note that the
critical value $\Ra_c$ is in fact much larger than 1 due to several
powers of $2\uppi$, which we dropped for simplicity). Expressing
$\ell$-scale Rayleigh number as $\Ra(\ell) = {\tau_{\kappa}(\ell)
\tau_{\nu}(\ell)}/{\tau_{\rho}^2(\ell)}$ as in
Table~\ref{tab:dimensionless}, we obtain critical $\tau_{\rho}$ at the
convection onset: $T_{\rho}^c = \sqrt{\tau_{\kappa}(R_o)
\tau_{\nu}(R_o)}$. This value is plotted in Figure~\ref{fig:onset} as
an orange disk on the right $y$-axis, at a time half-way between
$\tau_{\kappa}(R_o)$ and $\tau_{\nu}(R_o)$.

Things get very different when the system is rapidly rotating.
Proudman--Taylor constraint inhibits convective flows, and viscosity is
needed to break this geostrophic constraint \citep{chandrasekhar1961}. 
Convection marginal stability in rapidly rotating spheres has a long
history
\citep{chandrasekhar1961,roberts1968,busse1970,jones2000,zhang2007}.
Convective structures are found to be quasi-geostrophic at onset,
forming columnar vortices with a width small enough to enable viscosity
in the bulk to alleviate Proudman--Taylor constraint, yielding the
famous $\Ek^{1/3}$ law. These structures can also be viewed as thermal
Rossby waves.

It is interesting to examine the graphical \tauell representation of
this situation. We follow the local stability analysis of
\citet{busse1970}. Omitting numerical prefactors, his Equations
(4.11)--(4.13) yield:
{\begin{eqnarray}
\ell_c &\simeq& R_o (\Pr^{-1} +1)^{1/3} \Ek^{1/3} \Seqnsplit
\tau_c &\simeq& t_{\Omega}\, \Pr^{1/3} (1+ \Pr)^{2/3} \Ek^{-1/3} \Seqnsplit
T_{\rho}^c &\simeq& t_{\Omega} (\Pr^{-1} +1)^{2/3} \Pr^{1/2} \Ek^{-1/3},
\end{eqnarray}}\unskip
where $\ell_c$ is column radius, $\tau_c$ time period, $T_{\rho}^c$
free-fall time (for $\ell=R_o$) at onset, and $\Pr = \nu/D$ the Prandtl
number, where diffusivity $D$ is either thermal diffusivity $\kappa$ or
compositional diffusivity $\chi$. Note that $\Ek$ is here the classical
large-scale Ekman number $\Ek(R_o)$.

Figure~\ref{fig:onset} translates these results graphically. We compare
cases with $\Pr<1$ and $\Pr>1$, by selecting diffusivity $D = \kappa$
or $D = \chi$. At convection onset, column width is at the intersection
of line $\tau_{\mathrm{Rossby}}$ and either line $\tau_{\nu}$ or line
$\tau_D$, depending on which it encounters first. Thermal (or
compositional) diffusivity governs period since $\tau_c \simeq
\tau_D(\ell_c)$ in both cases. We also observe that $T_{\rho}^c =
\sqrt{\tau_{\nu}(\ell_c) \tau_D(\ell_c)}$. Graphically, this places
$T_{\rho}^c$ along the right $y$-axis ($\ell=R_o$) at a time half-way
between $\tau_{\nu}(\ell_c)$ and $\tau_D(\ell_c)$ on our log--log
plots. Note that onset parameters from global stability analysis
\citep{jones2000,dormy2004} differ substantially from those of the
local stability analysis of \citet{busse1970} for $\Pr < 1$.

\bibliographystyle{crgeos}
\bibliography{crgeos20230563}
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\end{document}
