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# Rapidly Rotating Magnetohydrodynamics and the Geodynamo

Emmanuel Dormy

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Emmanuel Dormy. Rapidly Rotating Magnetohydrodynamics and the Geodynamo. Annual Review of Fluid Mechanics, 2025, 57 (1), pp.335-362. 10.1146/annurev-fluid-031224-121649. hal-05365151

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*Annual Review of Fluid Mechanics*

Rapidly Rotating
Magnetohydrodynamics
and the Geodynamo

Emmanuel Dormy

Department of Mathematics and Its Applications, Ecole Normale Supérieure and CNRS, Paris, France; email: Emmanuel.Dormy@ens.fr

ANNUAL
REVIEWS **CONNECT**

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Annu. Rev. Fluid Mech. 2025. 57:335–62

First published as a Review in Advance on
October 1, 2024

The *Annual Review of Fluid Mechanics* is online at
fluid.annualreviews.org

https://doi.org/10.1146/annurev-fluid-031224-121649

Copyright © 2025 by the author(s). This work is
licensed under a Creative Commons Attribution 4.0
International License, which permits unrestricted
use, distribution, and reproduction in any medium,
provided the original author and source are credited.
See credit lines of images or other third-party
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Keywords

dynamo action, magnetohydrodynamics, rotating fluid, Earth's magnetic field

Abstract

The problem of the geodynamo is simple to formulate (Why does the Earth possess a magnetic field?), yet it proves surprisingly hard to address. As with most geophysical flows, the fluid flow of molten iron in the Earth's core is strongly influenced by the Coriolis effect. Because the liquid is electrically conducting, it is also strongly influenced by the Lorentz force. The balance is unusual in that, whereas each of these effects considered separately tends to impede the flow, the magnetic field in the Earth's core relaxes the effect of the rapid rotation and allows the development of a large-scale flow in the core that in turn regenerates the field. This review covers some recent developments regarding the interplay between rotation and magnetic fields and how it affects the flow in the Earth's core.
# 1. INTRODUCTION

Every single one of us has, at least once, played with a compass and experienced some surprise as the compass needle consistently demonstrated a preferred orientation. Such surprise at a physical phenomenon is at the root of scientific research. Even the celebrated Albert Einstein (quoted in Schilpp 1949, p. 9) remembers in his autobiographical notes

> a wonder of this kind I experienced as a child of four or five years when my father showed me a compass. That this needle behaved in such a determined way did not at all fit into the nature of events which could find a place in the unconscious world of concepts. I can still remember—or at least believe I can remember—that this experience made a deep and lasting impression upon me. Something deeply hidden had to be behind things....

While any scientist now knows that the needle simply orientates itself in the ambient magnetic field, the experience of an invisible magnetic field acting on the needle with no apparent contact remains a fascinating one. (Much like gravity, in fact; we just experience gravity much younger than magnetism in our lives, which is probably why we are less surprised.)

The main contribution to this ambient field acting on the needle originates from deep inside the Earth. The largely dipolar nature of this internal magnetic field, when observed at the Earth's surface, has for centuries enabled the use of compasses for navigation at seas.

The Earth is a differentiated planet (Figure 1): When it formed, the heavy elements sank to the center and the lighter silicate floated. It follows that some $3 \times 10^6$ m below our feet, under the mantle, lies a giant sphere of almost pure iron (together with traces of lighter elements). The Earth's magnetic field originates within this core (as can be assessed by a downward potential continuation of the surface field down to the core surface). Because of the geothermal gradient, the temperature at these depths (between 3,000 and 8,000 K) is such that this iron is in a liquid state, despite the huge pressure at these depths. Deeper down, however, the increase in pressure does more than compensate for the increase in temperature, and a solid inner core with a radius of some $1.2 \times 10^6$ m occupies the center of our planet (see Table 1 for a summary of the core properties). As the Earth cools down on geological timescales, the inner core slowly grows in size,

<figure><img src="image_7.png" /><figcaption>Figure 1</figcaption></figure>

Internal structure of the Earth.
<table><caption>Table 1 Some physical properties of the Earth's core</caption><thead><tr><td>Quantity</td><td>Symbol</td><td>Typical value</td><td>Unit</td></tr></thead><tbody><tr><td>Radius of the solid inner core</td><td>r<sub>i</sub></td><td>1.22 × 10<sup>6</sup></td><td>m</td></tr><tr><td>Radius of the core</td><td>r<sub>o</sub></td><td>3.48 × 10<sup>6</sup></td><td>m</td></tr><tr><td>Pressure at r = r<sub>i</sub></td><td>ρ<sub>i</sub></td><td>3.3 × 10<sup>11</sup></td><td>Pa</td></tr><tr><td>Pressure at r = r<sub>o</sub></td><td>ρ<sub>o</sub></td><td>1.3 × 10<sup>11</sup></td><td>Pa</td></tr><tr><td>Core density at r = r<sub>i</sub></td><td>ρ<sub>i</sub></td><td>1.3 × 10<sup>4</sup></td><td>kg m<sup>-3</sup></td></tr><tr><td>Core density at r = r<sub>o</sub></td><td>ρ<sub>o</sub></td><td>1.0 × 10<sup>4</sup></td><td>kg m<sup>-3</sup></td></tr><tr><td>Gravitational acceleration at r = r<sub>i</sub></td><td>g<sub>i</sub></td><td>4.4</td><td>m s<sup>-2</sup></td></tr><tr><td>Gravitational acceleration at r = r<sub>o</sub></td><td>g<sub>o</sub></td><td>10.7</td><td>m s<sup>-2</sup></td></tr><tr><td>Magnetic diffusivity</td><td>η</td><td>1</td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Kinematic viscosity<sup>a</sup></td><td>ν</td><td>10<sup>-6</sup></td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Thermal diffusivity</td><td>κ</td><td>10<sup>-5</sup></td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Solutal diffusivity</td><td>χ</td><td>10<sup>-8</sup></td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Angular velocity of Earth</td><td>Ω</td><td>7.3 × 10<sup>-5</sup></td><td>s<sup>-1</sup></td></tr><tr><td>Typical velocity of the flow</td><td>U</td><td>≈10<sup>-4</sup>–10<sup>-3</sup></td><td>m s<sup>-1</sup></td></tr><tr><td>Coefficient of thermal expansion</td><td>α</td><td>10<sup>-5</sup></td><td>K<sup>-1</sup></td></tr><tr><td>Typical magnetic field strength</td><td>B</td><td>≈4 × 10<sup>-3</sup></td><td>T</td></tr><tr><td>Magnetic permeability</td><td>μ = μ<sub>0</sub></td><td>4π × 10<sup>-7</sup></td><td>N A<sup>-2</sup></td></tr></tbody></table>

releasing light elements at the base of the liquid core (the mixture of iron and lighter elements not being at the eutectic).

The Earth's core is too warm to allow any permanent magnetism (it is well above the Curie point for iron). As originally proposed by André-Marie Ampère (1820), electrical currents within the Earth can account for the geomagnetic field. The Earth is, however, a small planet, and currents within its core would dissipate within some 10⁴ years via ohmic dissipation unless some electromagnetic force maintained them. Paleomagnetic observations indicate that our planet has had a magnetic field for some 3 × 10⁹ years (Tarduno et al. 2007). The theory that these currents are produced by a self-exciting fluid dynamo dates back to Joseph Larmor (1919) over a century ago.

Perhaps surprisingly, this mechanism, while widely accepted, still raises many open questions. This article is an attempt to explain why the problem is so challenging and to review recent progress on this matter.

## 2. DYNAMO ACTION

### 2.1. The Induction Equation

The Maxwell equations in matter govern the coupled evolution of the magnetic induction **B** (often referred to as the magnetic field) and the electric field **E**. They can be written as

$$ \nabla \wedge \mathbf{E} = -\partial_t \mathbf{B}, \qquad 1a. $$

$$ \nabla \wedge \mathbf{B} = \mu \mathbf{j} + \varepsilon \mu \partial_t \mathbf{E}, \qquad 1b. $$

$$ \nabla \cdot \mathbf{E} = \rho_c / \varepsilon, \qquad 1c. $$

$$ \nabla \cdot \mathbf{B} = 0, \qquad 1d. $$

<footnote>ªIt is striking that liquid iron under the conditions of the Earth's core has a similar kinematic viscosity as water at ambient temperature and pressure. Its absolute (dynamic) viscosity is larger but is compensated by a larger density. Molecular values (as opposed to turbulent values) are reported here.</footnote>
**Slow manifold:** for a dynamical system, the manifold on which the dynamics is slow compared to the dynamics off this manifold (here characterized by rapid oscillations)

**Solenoidal:** a solenoidal vector field is a divergence-free vector field

**Laplacian:** also called the Laplace operator; can be noted as either $\Delta$ or $\nabla^2$

where j is the electric current density, $\rho_c$ is the charge density, $\mu$ is the magnetic permeability, and $\varepsilon$ the permittivity.

This system has solutions in the form of electromagnetic waves. In magnetohydrodynamics (MHD), we neglect these rapidly propagating waves and focus on the slowly evolving electromagnetic field (an approach known as evolution on a slow manifold that we meet later in this review). Assuming the free-space value for the magnetic permeability, i.e., $\mu \simeq \mu_0 = 4\pi \times 10^{-7}NA^{-2}$ and $\varepsilon \simeq \varepsilon_0 = (\mu_0c^2)^{-1}Fm^{-1}$, the last term in Equation 1b thus takes the form $c^{-2}\partial_t E$ and can safely be neglected provided the typical velocity of the phenomena is small compared to the speed of light $c$. This is the essence of MHD. We thus simplify Equation 1b to

$$ \nabla \wedge \mathbf{B} = \mu \mathbf{j}. \qquad 2. $$

Ohm's law is then needed to relate electric currents to the electric field via the electrical conductivity, $\sigma$, of the fluid. The reader is probably familiar with its expression for a medium at rest, $\mathbf{j} = \sigma \mathbf{E}$. However, the fluids that we consider are usually not at rest, so it is necessary to introduce some modification for a conductor moving with a velocity $\mathbf{u}$,

$$ \mathbf{j} = \sigma (\mathbf{E} + \mathbf{u} \wedge \mathbf{B}). \qquad 3. $$

Equation 3 is also only valid if $|\mathbf{u}| \ll c$ (see Dormy & Soward 2007 for a more detailed derivation). Equations 1a, 2, and 3 can then be combined to form the so-called induction equation governing the evolution of the magnetic field,

$$ \partial_t \mathbf{B} = \nabla \wedge (\mathbf{u} \wedge \mathbf{B} - \eta \nabla \wedge \mathbf{B}), \qquad 4. $$

with $\eta = (\sigma\mu)^{-1}$ the magnetic diffusivity. This equation is formally completed with Equation 1d, which excludes magnetic monopoles, though strictly speaking the latter could be viewed as a property of the initial conditions, since the solenoidal property of the magnetic field will be preserved under Equation 4.

It is sometimes useful to rewrite Equation 4, using Equation 1d and assuming $\eta$ is constant, in the form

$$ \partial_t \mathbf{B} = \nabla \wedge (\mathbf{u} \wedge \mathbf{B}) + \eta \Delta \mathbf{B}, \quad \nabla \cdot \mathbf{B} = 0, \qquad 5. $$

where $\Delta$ denotes the Laplacian. Assuming that the velocity field $\mathbf{u}$ is solenoidal as well (an incompressible flow), this can be rewritten as

$$ \partial_t \mathbf{B} + (\mathbf{u} \cdot \nabla)\mathbf{B} = (\mathbf{B} \cdot \nabla)\mathbf{u} + \eta \Delta \mathbf{B}, \quad \nabla \cdot \mathbf{B} = 0. \qquad 6. $$

The magnetic field $\mathbf{B}$ is thus governed by an advection diffusion equation, with an extra stretching term (the first term on the right-hand side of Equation 6). The similarity between Equation 5 and the vorticity equation in fluid mechanics (for a barotropic flow or a uniform density fluid) deserves to be highlighted. Numerous analogies follow (e.g., Batchelor 1950). The key distinction, however, is that the vorticity equation is intrinsically nonlinear, whereas the induction equation is linear as long as the field is not strong enough to alter the flow. If $\eta = 0$, the magnetic field evolves as a material line; i.e., it is carried, but it is also stretched by the flow. If $\eta > 0$, the convected field also diffuses relative to the fluid.

## 2.2. Kinematic Dynamos

Equation 4, being linear in $\mathbf{B}$, admits exponentially growing or decaying eigenmodes. Finding exponentially growing solutions is known as the kinematic dynamo problem.
**2.2.1. Growing fields and energy bounds.** An energy evolution equation can be formed by taking the scalar product of Equation 4 with **B**/µ and integrating over space. An integration by part and standard vector manipulations reveals

$$
\frac{d}{dt} \int_{\Omega} \mathbf{B}^2 / (2\mu) \, \mathrm{d}\mathbf{x} = - \int_{\Omega} \mathbf{j} \wedge \mathbf{B} \cdot \mathbf{u} \, \mathrm{d}\mathbf{x} - \int_{\Omega} \mathbf{j}^2 / \sigma \, \mathrm{d}\mathbf{x}. \qquad 7.
$$

(Integration is here performed for simplicity over the whole space, assuming a field decaying suf-
ficiently rapidly at infinity; the same approach can be used in a finite conducting domain with the
addition of suitable boundary terms.) The last term being always negative, it follows that, subject
to sensible boundary conditions, the field necessarily decays, through ohmic diffusion, in the ab-
sence of flow (i.e., for **u** ≡ 0). Evidently a sufficiently large amplitude for the flow, **U**, needs to be
reached for kinematic dynamo action. This is usually measured, introducing a typical length scale
*L*, by the magnetic Reynolds number *R*<sub>m</sub> = *UL*/η, which measures the relative importance of the
two terms on the right-hand side of Equation 4.

The magnetic Reynolds number directly follows from an estimate of the ratio of the order of magnitude (identified here with a {·}) of the first to the second term on the right-hand side of Equation 4:

$$
\mathcal{R}_m = \frac{\{\nabla \wedge (\mathbf{u} \wedge \mathbf{B})\}}{\{\eta \nabla \wedge \nabla \wedge \mathbf{B}\}} = \frac{UB/L}{\eta B/L^2} = \frac{UL}{\eta}, \quad \text{where } U \equiv |\mathbf{u}| \text{ and } B \equiv |\mathbf{B}|. \qquad 8.
$$

Bounds on the minimal values of R_m to be reached for an exponentially growing field can be
obtained from Equation 7. For the spherical geometry we are concerned with in the case of the
geodynamo, i.e., a spherical conductor of radius L surrounded by vacuum, Backus (1958) showed
that dynamo action could not be sustained if R_m < π² (see Busse & Proctor 2007 for a review
of other available bounds for dynamos). [Note that Backus (1958) defined his magnetic Reynolds
number based on the maximum rate of strain e, i.e., as R_m = eL²/η.]

**2.2.2. Kinematic dynamos and optimal flows.**

**2.2.2. Kinematic dynamos and optimal flows.** Some flows were, very early on, identified as kinematic dynamos. Some of these are amenable to analytical treatment, such as the Ponomarenko (Ponomarenko 1973, Gilbert 1988) and the Roberts flows (Roberts 1970, 1972), but most of them require numerical integration. In a spherical geometry, noticeable examples of dynamo flows were introduced by Backus (1958), Gubbins (1973), Kumar & Roberts (1975), Pekeris et al. (1975), and Dudley & James (1989).

A natural question then arises: Can the π² lower bound of Backus (1958) be achieved? Or in other words, What is the optimal flow for dynamo action in the spherical geometry? The problem of optimal flow for dynamo action involves a variational optimization method, which requires the construction of the adjoint problem to Equation 4 (as first considered in Willis 2012).

Proctor (2015) proved that an optimal solution does not exist when the mean square velocity is
fixed; i.e., there is no minimum energy bound on the flow, as bounds require higher-order norms
of **u** (e.g., the maximum velocity) without placing limitations on the rate of strain. It is then natural
to ask what the optimal solution in the sphere is.

Chen et al. (2018) investigated this problem numerically and found an optimal dynamo flow with a threshold as low as $R_m = 64.45$ (their $R_m$ being based on the mean square vorticity, $\omega = |\nabla \wedge \mathbf{u}|$, i.e., $R_m = \omega L^2 / \eta$). The magnetic Reynolds number relevant to the Earth's core can be estimated between 300 and 3,000 (see **Table 1**).

**2.2.3. Transient growth.** Even though the kinematic dynamo problem is purely linear, it remains a challenging one numerically. The reason is that, as noted above, the operator on the right-hand side of Equation 4 is linear but not self-adjoint. As a result the magnetic energy exhibits transient growth. Eigenmodes not being orthogonal, algebraic amplification results from

Norms: notation |·| amounts to a norm yet to be specified; most often the L²-norm, also known as the energy norm (square root of the integrated square), is considered
<figure><img src="image_8.png" /><figcaption>Figure 2</figcaption></figure>

(a) Illustration of transient amplification on a planar field under a uniform shear. In the absence of diffusion, the field grows linearly in time under the effect of the shear. With diffusion being restored, transient algebraic growth is achieved. (b) Azimuthal shear by differential rotation in the sphere, the Ω-effect.

the superposition of modes all decaying over long enough times. From a physical point of view, a shear is enough to amplify the field but only transiently, as the component of the field being sheared decays if it is not being reinforced by the flow. It follows that long integrations of the three-dimensional problem in Equation 4 are needed to assess the dynamo properties of a given flow. Unfortunately, no monotonic exotic energy has so far been identified for this problem (see, e.g., Dauchot & Manneville 1997).

To illustrate this transient amplification, consider a two-dimensional flow $\mathbf{u} = (u_x(y, t), 0, 0)^t$ characterized by a uniform and constant shear $\partial_y u_x(y, t) \equiv S$ [see Figure 2 in the (x, y)-plane]. If we further assume an initial magnetic field of the form $\mathbf{B}_0 = (B_x \cos(z), B_y \cos(z), 0)^t$ (see Figure 2a), the field depends only on the space coordinate $z$, which provides a typical length scale $L$. The evolution of $B_x$ and $B_y$ in time under Equation 4 then simplifies to

$$ \partial_t \begin{pmatrix} B_x \\ B_y \end{pmatrix} = \begin{pmatrix} -\eta & S \\ 0 & -\eta \end{pmatrix} \begin{pmatrix} B_x \\ B_y \end{pmatrix} = \mathcal{L} \begin{pmatrix} B_x \\ B_y \end{pmatrix}. \qquad 9. $$

The time variation of the field strength can be measured by

$$ \|\mathbf{B}\|^2 = \frac{1}{2\pi} \int_0^{2\pi} B_x^2 + B_y^2 \, dz = \frac{1}{2} e^{-2\eta t} \left[ (B_x(0) + S t B_y(0))^2 + B_y(0)^2 \right], \qquad 10. $$

which clearly highlights a transient algebraic amplification for an initial seed field along $B_y$, but no exponential growth. The magnetic field ultimately decays exponentially.

Again the similarity between the induction equation and the vorticity equation is enlightening. This non-self-adjoint nature of the equation points to the existence of subcritical dynamos, which can only be triggered by finite amplitude perturbations. Mannix et al. (2022) recently have shown how nonlinear optimization, usually used to identify minimal disturbances in shear flows, allows systematical probing of such subcritical dynamos.

### 2.2.4. Antidynamo theorems.
Interestingly, a set of negative results exists for kinematic dynamos. Very early on, Cowling (1934) demonstrated that an axisymmetric magnetic field vanishing at infinity cannot be sustained by dynamo action. For the field to be axisymmetric, the field and the flow have to be axisymmetric. The demonstration then amounts to decomposing the field into its component in the meridional plane and the azimuthal component normal to it (or the toroidal component). The azimuthal component can be regenerated by shear [a process
referred to as the Ω-effect (see **Figure 2b**)], but the component in the meridional plane is gov-
erned by an advection–diffusion equation, leading to decay. Once this component has disappeared,
the azimuthal one must also decay (for Braginsky’s detailed proof, see Moffatt & Dormy 2019,
section 6.4). An axisymmetric flow can, however, sustain a nonaxisymmetric field (e.g.,
Ponomarenko 1973).

This is a major setback as it prevents any attempt to seek an axisymmetric solution to the geodynamo problem, despite the dominant axisymmetric nature of the geomagnetic field at the Earth’s surface. Other antidynamo theorems were produced, involving necessary conditions on the flow. For example, Zel’dovich (1957) has shown that planar flows cannot sustain dynamo action for a field vanishing at infinity. In the spherical geometry, Ivers & James (1988) have shown that a flow with no radial component in a spherical conductor fails as well (see Gilbert 2003 and Tobias 2021 for a more detailed overview). Dynamo action is intrinsically three-dimensional.

It is, however, possible to construct nearly axisymmetric dynamos, either in the sense that both the field and the flow are dominantly axisymmetric and deviations from axial symmetry are small (Braginsky 1964) or in the sense that the large-scale flow and field are axisymmetric and small-scale fluctuations are not (Steenbeck et al. 1966). We return to this mean-field approach below.

It is tempting to think that the flow at a small scale will necessarily be inefficient at generat-
ing a field. Indeed, the magnetic Reynolds number being moderate in the Earth's core, the local
magnetic Reynolds number based on a length scale ℓ, i.e., Uℓ/η, will rapidly become too small to
sustain a magnetic field as ℓ decreases.

This reasoning, however, neglects the fact that a small-scale flow can sustain a large-scale field. In this case, neither the small-scale Reynolds number (based on the scale of the flow) nor the large-scale one (based on the scale of the field) is relevant.

Introducing a small flow **u**(**x**, **t**) and a magnetic field separated into mean and small-scale parts **B**₀(**x**, **t**) + **b**(**x**, **t**), with ⟨**b**⟩ = 0, where the notation ⟨·⟩ amounts here to an averaging over the small scale, the induction equation then yields

$$
\partial_t \mathbf{B}_0 = \nabla \wedge (\mathbf{u} \wedge \mathbf{b}) + \eta \Delta \mathbf{B}_0, \qquad 11a.
$$

$$
\partial_t \mathbf{b} = \nabla \wedge (\mathbf{u} \wedge \mathbf{B}_0) + \nabla \wedge (\mathbf{u} \wedge \mathbf{b} - (\mathbf{u} \wedge \mathbf{b})) + \eta \Delta \mathbf{b}. \quad 11b.
$$

For simplicity, let us assume that the field is dominated by the large-scale *L* component (and thus **u** ∧ **b** − ⟨**u** ∧ **b**⟩ ≪ **u** ∧ **B**₀); the above system becomes

$$
\partial_t \mathbf{B}_0 = \nabla \wedge (\mathbf{u} \wedge \mathbf{b}) + \eta \Delta \mathbf{B}_0, \qquad 12a.
$$

$$
\partial_t \mathbf{b} = \nabla \wedge (\mathbf{u} \wedge \mathbf{B}_0) + \eta \Delta \mathbf{b}. \qquad 12b.
$$

Assuming a saturated dynamo is in statistical equilibrium and now considering time-averaged
quantities, we get

$$
\{ \nabla \wedge (\mathbf{u} \wedge \mathbf{b}) \} \sim \{ \eta \Delta \mathbf{B}_0 \}, \qquad 13a.
$$

$$
\{ \nabla \wedge (\mathbf{u} \wedge \mathbf{B}_0) \} \sim \{ \eta \Delta \mathbf{b} \}, \qquad 13b.
$$

and thus the relevant magnetic Reynolds number takes the form $R_{mL\ell} = U\sqrt{L\ell}/\eta$; i.e., it in-
volves the geometrical mean of the two length scales. This property involves only scale separation
and no mean field [though this holds, for example, for $\alpha^2$ dynamos (see Moffatt & Dormy 2019,
section 9.2)]. Thus a small-scale flow can sustain a large-scale field provided $R_{mL\epsilon}$ is larger than an order-one critical value.

Scale separation, as used above, amounts to assuming a gap in the energy spectrum; a clean-cut separation of scale is of course not justified in general and it is to be expected that the system has energy present at all scales. Such simplifying assumptions nonetheless provide useful analytical insight. For more on scale separation and mean field, the reader can refer to Krause & Rädler (1980), Moffatt & Dormy (2019), or Mizerski (2023).

### 2.2.5. Time-dependent flows.
Perhaps surprisingly, all examples of dynamo flows mentioned so far are steady. Interestingly, time dependence of the flow can turn a flow that is not a dynamo at any instant into a dynamo. This was recognized very early by Backus (1958) (the stasis dynamo) and was later used by Bayly & Childress (1988, 1989) to construct dynamos that way and show that they were fast dynamos. Such a flow can also be made smoothly varying in time (Galloway & Proctor 1992, Otani 1993, Skene & Tobias 2023).

As explained above, a simple shear can yield transient amplification but not exponential growth. Alternating the direction of shear can thus be an efficient way to achieve dynamo action, even though the frozen picture of the flow at each instant may not be a dynamo. For example, a succession of shears with different directions in time can act as a kinematic dynamo.

Dormy & Gérard-Varet (2008) considered the simplified model for a shear flow in an infinite domain, as described in Equation 9, but made the direction of the shear time dependent (see also Moffatt & Dormy 2019, section 3.9, for further discussion). When the shear direction varies in time as $\theta(t)$ (i.e., the shear rotates about the axis Oz), the spatial operator acting on the magnetic field thus takes the form $\mathcal{L}'(t) = \mathcal{R}(\theta(t)) \mathcal{L}(\theta(t))^{-1}$, where $\mathcal{L}$ corresponds to the steady shear introduced in Equation 9 and $\mathcal{R}$ is the $2 \times 2$ rotation matrix of angle $\theta$.

In the case of a simple rotating shear $\theta(t) = \alpha_0 t$, the system is completely integrable and is found to admit exponentially growing solutions, i.e., dynamos, for $-1 < \alpha_0 < 0$, with growth rate $\sigma = -R_m^{-1} + \sqrt{-\alpha_0(1+\alpha_0)}$ (expressed in units of the turnover timescale, $L/U$). (This dynamo flow, albeit planar, does not contradict Zel'dovich's theorem, as the field does not decay at infinity.)

Physically, the shear at a given instant yields an amplification of the seed field perpendicular to it, as well as a change in direction of that field. If the shear rotates with the field, the situation can be such that the field remains orthogonal to the shear but with a larger seed. This results in exponential growth (which is why the sign of $\alpha_0$ matters in the above expression). This simple model highlights that in a dynamical configuration, a snapshot of the time-dependent flow does not necessarily need to be a kinematic dynamo.

Perhaps more surprisingly, in the above example, in the case of an oscillating shear $\theta(t) = \alpha \sin(\beta t)$, a Floquet analysis reveals that exponential growth, in the form of a parametric instability, is possible in the limit of a small $\alpha$ (but with $\alpha > R_m^{-1}$).

## 2.3. Magnetohydrodynamic Flows

We can now turn to the coupling of the field and the flow through the Lorentz force. Formally, three equations of motion should be written, one for each: neutrals, ions, and electrons. However, we are dealing with a liquid metal and not a plasma. It is reasonable to assume that collisions occur frequently enough to mechanically couple all three components. We thus consider single-fluid incompressible MHD,

$$ \rho [\partial_t \mathbf{u} + (\mathbf{u} \cdot \nabla) \mathbf{u}] = -\nabla p + \nu \rho \Delta \mathbf{u} + \mathbf{f} + \mathbf{j} \wedge \mathbf{B}, \quad \nabla \cdot \mathbf{u} = 0, \quad 14a. $$

$$ \partial_t \mathbf{B} = \nabla \wedge (\mathbf{u} \wedge \mathbf{B}) + \eta \Delta \mathbf{B}, \quad \nabla \cdot \mathbf{B} = 0, \quad 14b. $$

where $\mathbf{f}$ is an arbitrary body force.
Dotting the first equation with **u** and integrating over space to form an evolution equation for the kinetic energy, one immediately notices that the last term in the first part of Equation 14a yields an expression that is exactly equal and opposite to the first term on the right-hand side of Equation 7.

This highlights that through the combined action of stretching of field lines and the Lorentz force, the system can conservatively convert kinetic energy to magnetic energy and vice versa. Understanding how it achieves this is the essence of dynamo theory.

Assuming that **f** is such that the flow **u** obtained in the absence of a field is a kinematic dynamo, the field will then grow exponentially at first, until the flow is affected by the Lorentz force. In simpler cases, the dynamo saturates to a steady state or statistically steady state (for which the magnetic energy fluctuates around some mean value). Nonlinear dynamos can, however, exhibit complex behaviors. This simple picture should be contrasted with cases in which the effect of the magnetic field on the flow is more drastic and the dynamo collapses. These are sometimes known as self-killing dynamos (Brummell et al. 2001, Fuchs et al. 2001). It can result in a loss of dynamo action if the nonmagnetic system admits several flows as solutions, or it can lead to a cyclic behavior in the form of relaxation oscillations otherwise. Some cases are even metastable; i.e., they can be sustained for several magnetic diffusion times before they die away (see Morin & Dormy 2009). In other cases, the Lorentz force can modify the flow in such a way as to maintain a magnetic field despite the fact that the hydrodynamic flow does not act as a dynamo. In other words, a seed magnetic field of finite size is required to trigger the instability. These are sometimes referred to as essentially nonlinear dynamos (see Brummell et al. 1998). From a bifurcation point of view, these behaviors correspond to supercritical bifurcations, subcritical bifurcations, or detached branches of solutions such as isola.

The intricate relation between the flow and the field is probably best illustrated by the fact that Cowling's theorem can be bypassed via the resulting time-dependent dynamics. In the dynamical regime, Gissinger et al. (2008) considered a spherical dynamo driven by a steadily and axisymmetrically forced flow. They observed time-periodic oscillations between an axial and an equatorial dipole. The equatorial mode is generated first by the axisymmetric flow and saturates, but an axial mode then grows through the nonaxisymmetric part of the velocity field driven by the Lorentz force. The axial dipole then inhibits the equatorial one, which decays almost to zero. As a result, the flow is no longer driven away from axisymmetry by the Lorentz force. The axial dipole then decays and the process repeats in the form of relaxation oscillations. This dynamics could be understood in terms of a simple dynamical model for the amplitude of the relevant modes.

It is perhaps useful to stress at this stage that the equations are invariant under the **B** → −**B** symmetry. The Lorentz force (the last term in the first part of Equation 14a) is quadratic in **B** and the induction equation, Equation 14b, is linear in **B**.

It may also be useful to recall that, the flow governed by Equation 14a being incompressible and in a rigid container, the pressure plays the role of a Lagrange multiplier to enforce the solenoidal condition on the flow. This equation can conveniently be rewritten in the equivalent form

$$ \partial_t \mathbf{u} = \mathbf{P} \left[ -(\mathbf{u} \cdot \nabla) \mathbf{u} + \nu \Delta \mathbf{u} + \mathbf{f} / \rho + (\mu \rho)^{-1} (\nabla \wedge \mathbf{B}) \wedge \mathbf{B} \right], \quad 15. $$

where $\mathbf{P}$ is the Leray projector onto a divergence-free vector field (e.g., Temam 1984), defined as

$$ \mathbf{P}(\mathbf{v}) = \mathbf{v} - \nabla\phi, \quad \text{with } \Delta\phi = \nabla \cdot \mathbf{v}. \quad 16. $$

**Magnetic diffusion time:** usually defined as $\tau_{\eta} = L^2/\eta$, where $L$ is the domain size; in a sphere, because the slowest decaying mode is the dipole with eigenvalue $-\pi^2$, a better estimate is the dipole diffusion time $\tau_d = L^2/\pi^2\eta$.
### 3. RAPIDLY ROTATING FLOWS

#### 3.1. Geostrophy

The system in Equations 14a and 14b is valid in an inertial frame of reference. However, when considering a rotating frame, we need to introduce fictitious forces to account for the acceleration due to the rotation of the reference frame. Two terms arise, the centrifugal force, which takes the form of a gradient term $-\frac{1}{2}\nabla(\Omega^2 s^2)$ (where $s$ denotes the distance from the rotation axis) and as such does not affect the flow bounded by two solid spheres (in practice it is responsible for the ellipsoidal shape of the Earth), and the Coriolis term $-2\Omega \wedge \mathbf{u}$, which does affect the flow. This apparent force is always perpendicular to $\mathbf{u}$ and therefore does not contribute to the energy budget of the flow. It does, however, very significantly modify the dynamics of the flow.

The incompressible Navier-Stokes equation in a rotating frame can be written in nondimensional form based both on a typical length scale $L$ and on an initial amplitude of the flow $U$ as

$$R_0 [\partial_t \mathbf{u} + (\mathbf{u} \cdot \nabla) \mathbf{u}] = -\nabla\pi + \mathcal{E}\Delta\mathbf{u} - 2\mathbf{e}_z \wedge \mathbf{u}, \quad \nabla \cdot \mathbf{u} = 0, \qquad 17.$$

where $R_0 = U/\Omega L$ is the Rossby number and $\mathcal{E} = \nu/\Omega L^2$ is the Ekman number. A fundamental result for the above equation, the Taylor-Proudman theorem, stems from the limit of vanishing Ekman and Rossby numbers. Neglecting inertial and viscous forces, the dominant force balance is

$$2\mathbf{e}_z \wedge \mathbf{u} = -\nabla\pi. \qquad 18.$$

It is then natural to take the curl of this balance, which yields

$$(\mathbf{e}_z \cdot \nabla) \mathbf{u} = \partial_z \mathbf{u} = 0. \qquad 19.$$

In other words, the flow is, in this limit, bidimensional and independent of the direction of the axis of rotation.

Contours satisfying this constraint are known as geostrophic contours. In a spherical shell, such as the liquid outer core, these are concentric cylinders, $C(s)$, as illustrated in Figure 3a. The geostrophic flow consists of an azimuthal flow that is a function of the distance to the axis of rotation only, i.e., $\mathbf{u}_G = u_\phi(s) \mathbf{e}_\phi$. The geostrophic cylinders $C(s)$ can thus only rotate as solid bodies. The particular cylinder tangent to the equator of the inner sphere (i.e., at a distance $s = r_i \approx 0.35r_0$

<figure><img src="image_9.png" /><figcaption>Figure 3</figcaption></figure>

<footnote>(a) Geostrophic cylinders in a spherical shell. (b) Original sketch of the cartridge belt geometry of the marginally unstable convective motions in an internally heated rotating sphere. Panel adapted with permission from Busse (1970). (c) Rendering of isosurfaces of the axial vorticity at the onset of convection for $\mathcal{E} = 2 \times 10^{-6}$. Panel reproduced from Dormy (1997).</footnote>
away from the axis) is known as the tangent cylinder. It plays a crucial role as it separates an outer region—in which the geostrophic contours have the outer sphere as their only boundary—from two distinct regions, above and below the solid inner core.

Naturally, the geostrophic balance highlighted in Equation 19 is broken either at short timescales (inertial waves) or at small length scales. If $\ell$ is such that $U/\Omega\ell \sim O(1)$ or $\ell = U/\Omega$, then inertia cannot be neglected at that scale. Inertial effects kick in and the flow becomes three-dimensional. Likewise, at a scale such that $\ell \sim \varepsilon^{1/2}L$, the viscous effect will kick in [this is the Ekman boundary layer scale, as pressure is assumed not to vary across the boundary layer (e.g., Vallis 2017)]. Considering this equation at a large scale, however, the approximation seems to hold.

Such reasoning regarding the full equation can, however, be misleading. Consider now the curl of Equation 17,

$$ \mathcal{R}_0 [\partial_t \omega + \nabla \wedge (\omega \wedge \mathbf{u})] = \varepsilon \Delta \omega + 2 \partial_z \mathbf{u}. \qquad 20. $$

This reveals that, no matter how small $\mathcal{R}_0$ is, if the vertical scale of the flow is large enough, $\partial_z \mathbf{u}$ can be made as small as the left-hand side. (This reasoning can be made a little more rigorous by going to Fourier space, rewriting the Coriolis term as $ik_z \hat{\mathbf{u}}$, and considering the limit of a small $k_z$.) The flow can then become three-dimensional. For example, an instability at a large vertical size was identified by Billant (2021), who considered a pair of counterrotating z-invariant vortices and considered a computational domain with a wide vertical extent. A three-dimensional instability develops with a vertical length scale $L_z \simeq \mathcal{R}_0^{-1}$ and the vortices cease to be bidimensional (see Figure 4). The Billant instability can only be understood once the curl of Equation 17 has been written. This result should also be related to the earlier observation by van Kan & Alexakis (2020) that in rapidly rotating turbulence within elongated domains, the transition from an inverse to a forward cascade depends critically on the vertical extent of the domain (see also Seshasayanan & Gallet 2020).

If, on the other hand, one balances the two terms on the right-hand side of Equation 20, one gets the relevant small horizontal length scale at which viscosity becomes significant for nearly

<figure><img src="image_10.png" /><figcaption>Figure 4</figcaption></figure>

Time evolution of the axial vorticity in a pair of vortices prone to a three-dimensional large-scale instability, as obtained by Billant (2021). The initial condition corresponds to a quasi-steady vortex pair computed with the two-dimensional Navier-Stokes equations. Yellow and blue contours indicate positive and negative isovalues, respectively. Figure adapted with permission from Graphical Abstract for Billant (2021).
z-independent vortices, which is $\ell \sim \mathcal{E}^{1/3} L$. It also reveals near the equator of a spherical boundary, with the geometric constraint $(\ell_{\perp}/L) \sim (\ell_{\parallel}/L)^2$, the $\ell_{\perp} \sim \mathcal{E}^{2/5} L$ and $\ell_{\parallel} \sim \mathcal{E}^{1/5} L$ scalings of the equatorial singularity of the Ekman layer (see Marcotte et al. 2016), with $\ell_{\perp}$ and $\ell_{\parallel}$ denoting, respectively, the length scale perpendicular and parallel to the direction of the rotation axis.

## 3.2. Thermal Convection

Dynamo action involves the conversion of kinetic energy into magnetic energy. As energy will be lost, largely through ohmic dissipation and to a very minor extent through weak viscous effects, some energy source is needed. In the Earth's liquid core, the source of kinetic energy is likely convective motions, whereby lighter fluid elements ascend through the core, thus converting potential energy into kinetic energy. The light elements can be light either because they are locally warmer (thermal convection) or because they contain lighter elements (compositional convection).

Estimates for the thermal conductivity in the Earth's core vary over quite a wide range (for a review, see Landeau et al. 2022), yet the conductivity may be large enough (as large as 250 W m⁻¹ K⁻¹) to sustain significant heat transfer through conduction alone. Solutal convection caused by the slow cooling of the Earth and the exsolution of light elements provides an important additional source of buoyancy. Such convection driven by compositional density differences interestingly raises the possibility of a stable layer of lighter fluid accumulating at the top of the core (Braginsky 1993, Buffett et al. 2016, Philidet et al. 2020). This leaves aside the delicate question of the generation of a magnetic field on Earth before the formation of the inner core, for which thermal convection and exsolution, however, still appear to be the most likely driving mechanisms (for a discussion, see Landeau et al. 2022).

We focus here on the so-called Boussinesq approximation, which is arguably the simplest model for convection within the Earth's core. Utilizing dimensionless variables defined with respect to a characteristic length scale $L = r_o - r_i$ (the liquid core depth), a viscous timescale $L^2/v$, and $T_i - T_o$ (the temperature drop across the core), the dimensionless forms of the momentum and buoyancy equations are

$$ \mathcal{E}[\partial_t \mathbf{u} + (\mathbf{u} \cdot \nabla) \mathbf{u}] = -\nabla \pi + \mathcal{E} \Delta \mathbf{u} - 2 \mathbf{e}_z \wedge \mathbf{u} + \mathcal{R} a T \mathbf{r}, \quad \nabla \cdot \mathbf{u} = 0, \qquad 21a. $$

$$ \partial_t T + (\mathbf{u} \cdot \nabla) T = \mathcal{P} r^{-1} \Delta T. \qquad 21b. $$

This system admits three control (or input) parameters, here the Ekman, Rayleigh, and Prandtl numbers defined by $\mathcal{E} = v/\Omega L^2$, $\mathcal{R}a = \alpha g (T_i - T_o)L/\Omega\kappa$, and $\mathcal{P}r = v/\kappa$, where $\alpha$ is the coefficient of thermal expansion, $g$ is the gravitational acceleration at $r = r_o$ (for a self-gravitating body with nearly constant density, the internal gravity increases linearly with $r$), and $\mathbf{r} = r \mathbf{e}_r$. The classical, nonrotation-based Rayleigh number can be recovered as $\mathcal{R} = \mathcal{R}a/\mathcal{E}$.

Note that various studies use different choices for $L$, which can be set either to the core radius $r_o$ or to the gap width $r_o - r_i$. The core radius is usually preferred when considering, or comparing with, convection in a full sphere. Also the factor of 2 in front of the Coriolis term can sometimes be absorbed in the definitions of nondimensional numbers.

It is interesting to note that the temperature $T$ in the above system could be equivalently replaced by a concentration in lighter elements $\xi$ governed by the same equation, albeit with a different diffusion coefficient (see Table 1) and boundary conditions.

The Boussinesq model is a sensible model for the Earth's core as liquid iron is, to a good approximation, incompressible. A rigorous derivation of the Boussinesq model involves a two-step limiting process (Malkus 1964, Dormy & Soward 2007). First, we consider fluid motions that
are considerably slower than the speed of sound. Second, we investigate a layer that is relatively thin compared to the scale height of the density variations of the fluid at rest (i.e., stratification due to its own weight). The first limit on its own leads to what is known as the anelastic approximation, effectively filtering out sound waves. The second limit allows us to consider a uniform density for the fluid. By filtering out sound waves, we are in effect studying a slow manifold. This approach is similar to neglecting the displacement current in Maxwell's equations discussed in Section 2.1 (effectively filtering out electromagnetic waves).

The Rayleigh number is here defined based on the temperature drop; it could alternatively be based on the heat flux. We could equally use Equation 21b as an advection diffusion equation for light elements in the core, i.e., solutal convection. It is usually assumed (Braginsky & Roberts 1995) that the two mechanisms are present and important. They are modeled by similar equations, though they are characterized by very different diffusion coefficients (see **Table 1**).

In a rotating spherical shell, the convective flow at the onset manifests as prograde drifting, equatorially symmetric, thermal Rossby waves. Using a homogeneous distribution of heat sources in a full sphere, these waves develop at a critical distance from the axis (see Busse 1970, Jones et al. 2000). When a difference of temperature is maintained between both bounding spheres, these waves develop near the tangent cylinder at the onset of convection (Dormy et al. 2004). In both cases, the critical Rayleigh number and azimuthal wave number (i.e., the number of columns) then scale as $Ra_c \sim m_c \sim \mathcal{E}^{-1/3}$ (recall that $\mathcal{E} \simeq 10^{-15}$ in the core). [The radial length scale for the onset of convection is, however, intricate and differs from $\mathcal{E}^{1/3}$ (see Dormy et al. 2004).] The $\mathcal{E}^{1/3}$ scaling of the azimuthal length scale relates to the nearly z-independent length scale derived in the previous section. Very small length scales are needed at the onset of convection because no heat can be carried by a flow along the geostrophic contours. Likewise, the thermal wind resulting from buoyancy entering the dominant balance (e.g., Vallis 2017) cannot carry heat across the core.

It follows that viscosity is essential at the onset of convection and necessarily enters the leading order balance. Rapid rotation impedes convection, pushing it toward smaller and smaller scales and larger and larger forcing as $\mathcal{E}$ decreases.

These convective columns at the onset are often confused with convection rolls. In fact, the pattern illustrated in **Figure 3** drifts extremely rapidly (at a frequency $\omega_c \sim \mathcal{E}^{-2/3}$), so that the fluid particles are not spiraling around the columns but instead oscillate very rapidly (increasingly fast as $\mathcal{E}$ decreases). The transition from this wave structure at the onset to a flow as the Rayleigh number is increased has not yet been described. Soward (1977) showed that the first nonlinear effects would be destabilizing, leading to a subcritical onset of convection. Gastine et al. (2016) numerically investigated changes in the characteristics of the flow for nonlinear rotating convection in the case of an aspect ratio of 0.6 with a central mass distribution (a model suited for the Sun). The force balance close to the onset was identified, involving viscosity, buoyancy, and Coriolis (and pressure); this is often referred to as the VAC balance (A stands for Archimedean). The VAC balance holds close to the onset of convection. As the Rayleigh number is increased, a second regime is identified. An increase of the supercriticality is accompanied by a gradual transition to a turbulent quasi-geostrophic regime. This is the so-called Coriolis–inertia–Archimedean (CIA) balance, which Gastine et al. (2016) identified for $Ra > 6Ra_c$. Viscosity does not seem to affect the leading order dynamics. For much larger forcing, a turbulent convection state hardly influenced by rotation could be reached. The threshold for this extreme limit is found to increase rapidly as the Ekman number is decreased.

**Figure 5** illustrates scale-dependent force balances (boundary layers excluded). Spectra of the various forces are presented for $Ra = 50 \times Ra_c$, well within the CIA balance, with an Ekman number $\mathcal{E} = 10^{-4}$ and $Pr = 1$. The x-axis represents the spherical harmonic degree $l$ (equivalent in the spherical geometry to the Fourier mode $k$), and the various curves correspond to
<figure><img src="image_11.png" /><figcaption>Figure 5</figcaption></figure>

Scale-dependent force balances in thermal convection at $\mathcal{E} = 10^{-4}$, $Pr = 1$, and $Ra = 50 \times Ra_c$. Panel a presents the forces, whereas panel b presents the compensated curls of forces. Figure provided by Rob Teed.

different forces. This representation is very much in use to try and understand force balances in numerical models of the geodynamo (Aubert et al. 2017; Schwaiger et al. 2019, 2021; Aubert 2023).

**Figure 5a** presents forces as they are usually investigated. It reveals a dominant balance between Coriolis and pressure: the geostrophic flow. Following the procedure usually used for dynamo simulation, we then form the ageostrophic Coriolis by subtracting the pressure forces from Coriolis. As highlighted by Teed & Dormy (2023), this is a dangerous thing to do, as the pressure also compensates for the gradient part of all the other forces. This is perhaps best illustrated by the fact that the ageostrophic Coriolis overcomes the Coriolis force in part of the spectrum. In this plot, viscosity is out of the leading order balance at all scales and seems unimportant. A deviation from the CIA balance at a large scale appears to happen for $l > 7$. Buoyancy then seems to become less important as $l$ increases.

Following Teed & Dormy (2023), we then consider spectra of the compensated curls in **Figure 5b**. This is not the only approach to extract the solenoidal component of each force. Leray projected forces can also be considered. They are not strictly identical to the curls for two reasons. First, curl spectra are compensated only for the horizontal scale $l$ in the form $\hat{C}_l = (l(l+1))^{-1/2} C_l$. Second, Equation 16 allows for an arbitrary harmonic potential in the absence of physical boundary conditions on the solenoidal part of each force (see Teed & Dormy 2023).

It is interesting to note in **Figure 5b** that the CIA balance is never strictly achieved and that the role of buoyancy in the balance is in fact increasing with $l$. Finally, viscosity balances inertia at small scales, as expected. This highlights, for a simulation of nonlinear rotating convection, the importance of considering solenoidal force balances.

# 4. RAPIDLY ROTATING DYNAMOS

## 4.1. Governing Equations

Together with the induction equation, the governing equations for convectively driven dynamos in a rotating frame of reference can be written in nondimensional form; with $L = r_o - r_i$, $L^2/\eta$, and $T_i - T_o$ as units of length, time, and temperature, respectively, and $(\rho\mu\eta\Omega)^{1/2}$ as the unit for
<table><caption>**Table 2** Some nondimensional numbers and their orders of magnitude for the geodynamo</caption><thead><tr><th>Name</th><th>Symbol</th><th>Definition</th><th>Approximate value</th></tr></thead><tbody><tr><td colspan="4"><strong>Control parameters used in the article</strong></td></tr><tr><td>Rayleigh number (modified)<sup>a</sup></td><td>R̃<sub>a</sub></td><td>αgΔTL/ηΩ</td><td>≈[10<sup>5</sup>–10<sup>8</sup>]</td></tr><tr><td>Ekman number</td><td>E</td><td>ν/ΩL<sup>2</sup></td><td>≈10<sup>-15</sup></td></tr><tr><td>Magnetic Ekman number<sup>b</sup></td><td>E<sub>m</sub></td><td>η/ΩL<sup>2</sup></td><td>≈10<sup>-9</sup></td></tr><tr><td>Thermal Roberts number</td><td>q<sub>T</sub></td><td>κ/η</td><td>≈10<sup>-5</sup></td></tr><tr><td>Solutal Roberts number</td><td>q<sub>ξ</sub></td><td>χ/η</td><td>≈10<sup>-8</sup></td></tr><tr><td colspan="4"><strong>Alternative control parameters</strong></td></tr><tr><td>Magnetic Prandtl number</td><td>P<sub>m</sub></td><td>ν/η ≡ E/ε<sub>m</sub></td><td>≈10<sup>-6</sup></td></tr><tr><td>Thermal Prandtl number</td><td>P<sub>rT</sub></td><td>ν/κ ≡ E/ε<sub>m</sub>q<sub>T</sub></td><td>≈0.1</td></tr><tr><td>Solutal Prandtl number<sup>c</sup></td><td>P<sub>rξ</sub></td><td>ν/χ ≡ E/ε<sub>m</sub>q<sub>ξ</sub></td><td>≈10<sup>2</sup></td></tr><tr><td>Lewis number</td><td>L<sub>e</sub></td><td>κ/χ ≡ q<sub>T</sub>/q<sub>ξ</sub></td><td>≈10<sup>3</sup></td></tr><tr><td colspan="4"><strong>Diagnostic parameters</strong></td></tr><tr><td>Elsasser number</td><td>Λ</td><td>B<sup>2</sup>/Ωρμη</td><td>≈[1–10]</td></tr><tr><td>Rossby number</td><td>R<sub>o</sub></td><td>U/ΩL</td><td>≈10<sup>-6</sup></td></tr><tr><td>Reynolds number</td><td>R<sub>e</sub></td><td>UL/ν ≡ R<sub>o</sub>/ε</td><td>≈10<sup>9</sup></td></tr><tr><td>Magnetic Reynolds number</td><td>R<sub>m</sub></td><td>UL/η ≡ R<sub>o</sub>/ε<sub>m</sub></td><td>≈10<sup>3</sup></td></tr><tr><td>Lundquist number</td><td>L<sub>u</sub></td><td>BL/η√ρμ ≡ Λ<sup>1/2</sup>ε<sub>m</sub><sup>-1/2</sup></td><td>≈10<sup>5</sup></td></tr><tr><td>Lorentz number<sup>d</sup></td><td>L<sub>o</sub></td><td>B/ΩL√ρμ ≡ Λ<sup>1/2</sup>ε<sub>m</sub><sup>1/2</sup></td><td>≈10<sup>-4</sup></td></tr></tbody></table>

Several others can be constructed as combinations of these, but these are the most commonly used.

<sup>a</sup>As outlined in the text, there are several possible definitions of the Rayleigh number. We use here the relevant one for rapidly rotating dynamos (referred to as modified). This number is very poorly known in the Earth's core, as to estimate it for a Boussinesq model, only the superadiabatic temperature gradient should be taken into account.

<sup>b</sup>Also called the magnetic Rossby number.

<sup>c</sup>Also known as the Schmidt number.

<sup>d</sup>Sometimes also called the Lehnert number.

the magnetic field, these equations are

$$ \mathcal{E}_m [\partial_t \mathbf{u} + (\mathbf{u} \cdot \nabla)\mathbf{u}] = -\nabla\pi + \mathcal{E} \Delta\mathbf{u} - 2\mathbf{e}_z \wedge \mathbf{u} + \tilde{R}_a T \mathbf{r} + (\nabla \wedge \mathbf{B}) \wedge \mathbf{B}, \quad \nabla \cdot \mathbf{u} = 0, \quad 22a. $$

$$ \partial_t T + (\mathbf{u} \cdot \nabla)T = q \Delta T, \qquad 22b. $$

$$ \partial_t \mathbf{B} = \nabla \wedge (\mathbf{u} \wedge \mathbf{B}) + \Delta \mathbf{B}, \quad \nabla \cdot \mathbf{B} = 0. \qquad 22c. $$

Equations 22a–c involve four independent nondimensional parameters: the Ekman number $\mathcal{E}$, the magnetic Ekman number $\mathcal{E}_m$, the Roberts number $q$, and the modified Rayleigh number $\tilde{R}_a$. Estimates for these parameters are given in **Table 2**. The Roberts number $q$ assumes different values depending on whether thermal or solutal convection is being considered. In both cases, this is a very small number. $\tilde{R}_a \equiv R_{aq}$ measures the energy input, which is large. The three other parameters $\mathcal{E}$, $\mathcal{E}_m$, and $q$ are extremely small, which complicates the numerical work significantly and calls for distinguished limits (Dormy 2016).

For a fixed aspect ratio, four nondimensional control parameters are enough to control this system. The state of the system then involves two vector fields and a scalar field. To simplify, and since we are interested in the dynamo property, we focus on the mean magnetic energy as measured by the Elsasser number (see **Table 2**) as our state variable.
**Hyperdiffusivity:**

consists of replacing
Laplacians in the
equations with a
higher-order operator
to reduce dissipation at
a large scale while
preserving significant
dissipation at smaller
scales; this approach
intends to account for
turbulent effects

4.2. Dynamo Branches

The first successful self-consistent model for the geodynamo was obtained by Glatzmaier & Roberts (1995). This was then a state-of-the-art calculation, aiming for the most extreme parameters achievable, using hyperdiffusivities. Shortly after this, systematic parameter surveys were performed exploring the parameter space with unparameterized equations (Christensen et al. 1999, Kutzner & Christensen 2000, Christensen & Aubert 2006). This resulted in phase diagrams (i.e., regions of the parameter space characterized by dynamos and others not), such as the one presented in **Figure 6**. Two branches were identified very early by Kutzner & Christensen (2000). They are highlighted in **Figure 6**, in which dynamo modes are presented as a function of $\tilde{Ra}'(\equiv \tilde{Ra}/Ra_{q})$ and $\epsilon_m$. We refer to these two types of dynamo as weak dipolar (WD) and fluctuating multipolar (FM). Oruba & Dormy (2014) showed that the transition to FM originates from the inertial term entering the balance between viscosity and Coriolis, which dominates on the WD branch.

A bistability between the WD and FM modes has been observed with stress-free conditions (Goudard & Dormy 2008, Simitev & Busse 2009) and further explored by Petitdemange (2018), who showed that the bistability also exists with no-slip boundary conditions in the limit of a small $\mathcal{E}$.

The limit of a vanishing $\mathcal{E}$ has since then proven remarkably challenging numerically, independently of the choice of $\mathcal{E}_m$. These are extremely demanding simulations (e.g., Takahashi et al. 2005, Aubert et al. 2017, Schaeffer et al. 2017, Sheyko et al. 2018). Numerical models for the geodynamo generally fall within two categories: those aimed at understanding and those aimed at reproducing Earth-like features (obviously with many shades of gray in between).

Morin & Dormy (2009) constructed bifurcation diagrams in the case of $\mathcal{E} = 3 \times 10^{-4}$, as shown in **Figure 6**. They reveal first a complex behavior, with isola, subcritical, and then supercritical bifurcations. Perhaps more importantly, such diagrams reveal at a smaller $\mathcal{E}_m$ the existence of a third dynamo branch (see Dormy 2016, Dormy et al. 2018). For this regime of a small $\mathcal{E}_m$ the dynamo is characterized first by a weak field branch (WD) followed by runaway field growth and then a strong field branch [referred to as strong dipolar (SD)] on which the magnetic field relaxes the constraint of rapid rotation.

Chandrasekhar (1961) recognized that both magnetic fields and rotation, when considered individually, tend to impede convective motions. However, when these effects are combined, the Lorentz force can partly relax the constraint of rapid rotation (see Chandrasekhar 1961, Eltayeb & Roberts 1970, Fearn 1979, Horn & Aurnou 2022, Mason et al. 2022). Roberts (1979) thus conjectured a bifurcation diagram with two branches, a weak and a strong field branch (see also Moffatt & Dormy 2019, chapter 13).

The importance of the strong field branch is better illustrated in **Figure 7**. The top two graphs correspond to a cross section through the parameter space along $q = \mathcal{E}/\mathcal{E}_m$ (i.e., $Pr = 1$, as in **Figure 6**), followed by a projection along the $(\mathcal{E}, \mathcal{E}_m)$-plane, so that the plot only depends on $\tilde{Ra}$. The residual dependence on $\mathcal{E}$ and $\mathcal{E}_m$ is highlighted by the dispersion of the open symbols on these graphs.

Menu et al. (2020) explored this strong field branch further and showed that it does not suffer from a transition to the FM branch as long as the Lorentz and Coriolis forces remain dominant at large length scales.

It is interesting to note that the transition to the two-branch regimes illustrated in **Figure 6** happens through what appears to be a cusp singularity (e.g., Arnold 1992). For a value of $\mathcal{E}_m$ slightly above the transition, the supercritical branch exhibits a sharp step that announces the cusp formation. With more than two control parameters in the system, other types of singularities
<figure><img src="image_12.png" /><figcaption>Figure 6</figcaption></figure>

(Left two columns) Phase diagram from three-dimensional numerical simulations in a rotating spherical shell at six different Ekman numbers; two regimes emerge: a weak dipolar (WD) regime marked with circles and a fluctuating multipolar (FM) regime marked with diamonds. Crosses are failed dynamos, circles are dipolar dynamos (the larger the circle the larger the field), and diamonds are multipolar dynamos. Yellow, orange, red, and purple lines indicate bifurcation diagrams in the case of $\mathcal{E} = 3 \times 10^{-4}$. Left two columns adapted with permission from Christensen & Aubert (2006). (Right column) Bifurcation diagrams for $\mathcal{E} = 3 \times 10^{-4}$; $q = \mathcal{E}/\mathcal{E}_m$; and $\mathcal{E}_m = 2 \times 10^{-4}$, $10^{-4}$, $5 \times 10^{-5}$, and $2 \times 10^{-5}$. Lines are tentative interpretations. Open symbols indicate a WD regime and solid symbols a strong dipolar (SD) regime. Each data point corresponds here to the mean of a fully three-dimensional numerical simulation. Data for right column from Morin & Dormy (2009) and Dormy (2016).
<figure><img src="image_13.png" /><figcaption>Figure 7</figcaption></figure>

(a,b) Field strength as measured by the Elsasser number $Λ$ and the modified Elsasser number $Λ' = ΛL/(R_mℓ_B)$ as a function of $\tilde{Ra}$. A master curve emerges in the strong field regime (solid dots). Panels a and b adapted from Dormy et al. (2018). (c) Anticipated bifurcation diagram restricted to the $(\tilde{Ra}, \mathcal{E}_m, B)$ space at given small values of $\mathcal{E}$ and $q$. The numbers indicated in gray on the bottom plane identify, for each region, the number of stable branches for the signed magnetic field, which is odd when the hydrodynamic flow is stable to a magnetic perturbation and even otherwise (owing to the $B \to -B$ symmetry). The cusp catastrophe is here indicated by an orange square. The blue dot corresponds to a weak field dynamo and the red dot to a strong field dynamo. The dotted green curve indicates a runaway field growth. There exist, however, paths such as the one indicated in dotted red connecting continuously the blue to the red dot by avoiding the singularity. The green square in the figure indicates the crossing of the strong field fold with the onset of the dynamo bifurcation line.

are possible in addition to the cusp outlines in **Figure 7**. The swallowtail (see Arnold 1992), for example, has so far not been reported for this problem.

A three-dimensional bifurcation diagram can be speculated near the cusp (i.e., at a small $\mathcal{E}_m$), as illustrated in **Figure 7c**. This diagram illustrates part of the current numerical difficulties. For a small enough $\mathcal{E}_m$ at a given $\mathcal{E}$, the double fold characteristic of the weak (blue dot) and strong
(red dot) field branch exists. The runaway catastrophe can be obtained by increasing $\tilde{Ra}$ from the blue to the red point (dotted green curve). There exist, however, paths in parameter space (such as the one indicated in dotted red) connecting continuously the blue to the red dot by avoiding the singularity. Dynamos along this red path will continuously evolve from weak to strong (note that this path is not a distinguished limit). At a larger $\mathcal{E}_m$ (above the cusp), force balances are not as clearly identified and there is a continuous transition of states more or less influenced by viscous and inertial effects. Some of these models may be strongish, in the sense that they will necessarily share some properties of the strong dipolar state. Studying these systems for values of $\mathcal{E}_m$ below the formation of the cusp offers a guarantee of two separated branches and therefore well-identified force balances.

The green square in the figure indicates the crossing of the strong field fold with the onset of the dynamo bifurcation line. For values of $\mathcal{E}_m$ below this crossing, the strong dynamo branch exists below the linear onset of the dynamo. This crossing, anticipated by theory, is so far speculative, in the sense that it has not yet been produced numerically in the spherical geometry [but it has been observed in Cartesian simulations, for which dynamo action was even achieved below the onset of nonmagnetic convection (see Stellmach & Hansen 2004, Cooper et al. 2020)]. Note that, because $\tilde{Ra}_c \sim \mathcal{E}^{-1/3}$, the onset of convection (needed to reach the weak branch supercritically) increases to larger values of $\tilde{Ra}$ as $\mathcal{E} \to 0$ (for a given $q$).

One of the main issues with numerical dynamos is therefore to assess their force balance, which is often viewed as a way to ensure they have reached the relevant strong field balance without having to perform a bifurcation study. Recent efforts involve computing the scale-dependent balance (Aubert et al. 2017) as well as, in some cases, the variation in space and in time (Schaeffer et al. 2017) of solenoidal forces. There are risks of complications when comparing force spectra. Most forces in this system contain a gradient part that does not influence the dynamics (as highlighted by Equation 15). Teed & Dormy (2023) thus advocated for a solenoidal forces balance, which can be obtained by considering the curls of forces. These are presented in **Figure 8**, which highlights a typical strongish case. While the forces plotted in **Figure 8a** highlight a crossing between the Lorentz force and the buoyancy force near $l = 10$, sometimes argued to be dynamically relevant, the solenoidal force balance does not present such a crossing. The balance appears simpler

<figure><img src="image_14.png" /><figcaption>**Figure 8**</figcaption></figure>

Force balances and solenoidal force balances for $\mathcal{E} = 10^{-5}$, $\mathcal{E}_m = 10^{-5}$, $q = 1$, and $\tilde{Ra}' = 10$. Forces and curl of forces are shown in panels *a* and *b*, respectively, as in **Figure 5**. Figure adapted from Teed & Dormy (2023).
and nearly independent of scale (except that viscosity now is part of the main balance at a small scale). The purely geostrophic flow has been filtered by this process. Interestingly, the Coriolis and Lorentz forces would be even closer to equilibrium at larger scales if the dynamo was strong, i.e., with a smaller $\mathcal{E}_m$.

It should be stressed that the choice of control parameters being used here is neither unique nor the classical one (in a historical sense). Yet it does help us understand what the issues are. Extreme values of $\mathcal{E}$, $\mathcal{E}_m$, and $q$, such as the ones relevant to the Earth's core dynamics, are well out of reach of current computational resources. Varying $\mathcal{E}$ and $\mathcal{E}_m$ may be interpreted physically as considering planetary interiors, existing or not, with different sizes $L$ and/or rotation rates $\Omega$. The situation differs, however, when it comes to the Roberts number $q$. This is a property of the fluid only, not of the flow. Liquid metals, albeit good conductors of electricity, dissipate electrical current much more efficiently than they diffuse heat, let alone chemical concentrations. As a result the Roberts number must be small for any planetary core we can think of. It is currently overestimated in all numerical simulations.

In much the same way, the magnetic Prandtl number $P_m = \mathcal{E}/\mathcal{E}_m$ is a property of the fluid. It is, for very similar reasons, also a very small number. Most of the earlier studies relied on the following set of four control parameters: $\mathcal{E}$, $P_m$, $Pr = P_m/q$, and $\mathcal{R} \equiv \tilde{Ra}/\mathcal{E}q$. They were thus initially presented in a slightly different way than in **Figure 6**.

Since neither of the small parameters can assume its geophysically relevant value, the question arises of the region of the parameter space amenable to numerical treatment that is most relevant to the Earth.

With $\mathcal{E}$ being as small as the computer permits, but significantly larger than $10^{-15}$, what is the best set of parameters to approach the relevant balance of forces? Dormy (2016) suggested that $\mathcal{E}$ and $\mathcal{E}_m$ should be related through a distinguished limit, instead of systematically trying to maximize $\mathcal{E}_m$ at a given $\mathcal{E}$ (i.e., minimize $P_m$) as was previously done. A small parameter $\varepsilon$ can be introduced, and, setting $\mathcal{E} = \mathcal{E}_0\varepsilon$ (with $\mathcal{E}_0$ a coefficient kept constant in the limit), the distinguished limit is set by choosing $\mathcal{E}_m = \mathcal{E}_{m0}\varepsilon^\alpha$ and $q = q_0\varepsilon^\beta$ ($\alpha, \beta > 0$). The parameter $\tilde{Ra}$, controlling the input of energy, should be kept constant in the limiting process. The coefficients $\alpha, \beta, \mathcal{E}_0, \mathcal{E}_{m0}, q_0$, and $\tilde{Ra}$ are to be chosen so as to preserve some relevant properties of the solution, such as force balances, as $\varepsilon$ is varied, and to tend to Earth's core values for a very small $\varepsilon$.

Choosing $\alpha < 1$ ensures $P_m$ vanishes in the limiting process. This would obviously not be the case if $\mathcal{E}_m$ had simply been set to zero instead of being related to $\mathcal{E}$.

This is the essence of the distinguished limit advocated for by Dormy (2016), which was sometimes wrongly interpreted as a large $P_m$ limit. Setting a small value for $\mathcal{E}_m$, at a given $\mathcal{E}$, can result in $P_m > 1$ (for a moderate $\varepsilon$). It is sometimes argued that this has the unphysical consequence that the viscous dissipation will be comparable to the ohmic dissipation or that the magnetic field will be small scale. Both are incorrect (see Dormy 2016, Dormy et al. 2018). However, it has the unfortunate consequence that, until $P_m = \mathcal{E}/\mathcal{E}_m$ is decreased below unity in the limiting process, the dissipative length scale for the field and the flow will be inverted. This has consequences on the resulting energy spectra at a small scale (see Schekochihin 2022).

This idea of a distinguished limit [presented by the author before publication in 2014 at the Institute for Pure and Applied Mathematics (IPAM) and at the Institut de Geophysique du Globe de Paris (IPGP)] quickly caught on. Aubert et al. (2017) adopted it under the name path theory, with $\alpha = \beta = 1/2$, and followed this solution down to very small values of $\varepsilon$. Aubert (2023) recently reached the end of his path with extremely small values of the parameters $\mathcal{E} = 3 \times 10^{-12}$ and $\mathcal{E}_m = 2.37 \times 10^{-9}$, though the integration could only be performed so far for a very short time and using hyperdiffusivities.
### 4.3. The Taylor State

In the limit of vanishing $\mathcal{E}$ and $\mathcal{E}_m$, Equation 22a takes the form

$$2\mathbf{e}_z \wedge \mathbf{u} = -\nabla\pi + \tilde{R}a T \mathbf{r} + (\nabla \wedge \mathbf{B}) \wedge \mathbf{B}, \quad \nabla \cdot \mathbf{u} = 0. \qquad 23.$$

Note that the small parameter $q$ remains in Equation 22b, though the limit of a vanishing $q$ could also be envisaged. This limit equation is referred to as the magnetostrophic approximation because of its similarities with the geostrophic limit discussed earlier. It was first studied by Bryan Taylor (1963). The first question to ask is whether this equation always admits a solution; i.e., can we find a $\mathbf{u}$ satisfying the equation for any $\mathbf{B}$? Taylor (1963) identified a solvability condition. Taking the $\phi$-component of the above equation yields

$$2\mathbf{u} \cdot \mathbf{e}_z = \partial_\phi \pi + [(\nabla \wedge \mathbf{B}) \wedge \mathbf{B}]_\phi. \qquad 24.$$

If we now integrate on one of the geostrophic contours identified in Figure 3a, we note that the left-hand side vanishes because each cylinder can be closed by a polar cap on the bounding sphere. Since there is no radial flow on the spherical boundary and $\nabla \cdot \mathbf{u} = 0$, there can be no net flow through the cylinders. The first term on the right-hand side must also vanish (as the integral along $\phi$ of a $\phi$-derivative). The Taylor constraint follows

$$\int_{C(s)} [(\nabla \wedge \mathbf{B}) \wedge \mathbf{B}]_{\phi} \, dS = 0. \qquad 25.$$

If, and only if, this nonlinear condition is met by the field $\mathbf{B}$, the flow can be reconstructed from $\mathbf{B}$ by considering the curl of Equation 23:

$$2\partial_z \mathbf{u} = \tilde{R}a (\nabla T \wedge \mathbf{r}) + \nabla \wedge [(\nabla \wedge \mathbf{B}) \wedge \mathbf{B}]. \qquad 26.$$

Thanks to the boundary conditions of impenetrability of the spherical bounding spheres, $u_r$ and $u_\phi$ are uniquely recovered. $u_\phi$, however, can only be recovered up to an arbitrary geostrophic flow $u_\phi(s)$.

It is natural to split the flow into three components: Integrating for the first term on the right-hand side of the above equation yields the thermal wind [familiar to meteorologists (e.g., Vallis 2017)], the second term yields a magnetic wind, and an arbitrary geostrophic flow remains to be determined. Interestingly, the thermal wind, on its own, can drive a dynamo (see Sarson 2003).

Taylor (1963) argues that the requirement that the Taylor state be satisfied for all times provides an equation for the geostrophic flow. It must be such that the field provided by Equation 22c continues to satisfy Taylor's constraint as time proceeds. The geostrophic flow can thus be viewed as a projector, projecting the magnetic field (via Equation 22c) on the subspace (or manifold) of solutions satisfying the constraint in Equation 25, much like the Leray projector in Equation 15 ensures that the velocity field remains on the divergent-free subspace. The key distinction, however, is that the Leray projector (Equations 16a and 16b) enforces a linear constraint, whereas the Taylor state is characterized by a nonlinear constraint (Equation 25).

The Taylor state is in effect a slow manifold approach, similar in spirit to the ones we have previously discussed. Interestingly, a double limit was taken to obtain Equation 23. It is not surprising that the convergence of the full solution to the Taylor state requires some care in the way this double limit is taken. As stressed above, different distinguished limits can yield different behaviors in the limiting process. The importance of the ratio of two small parameters in a double limit does not come as a surprise and was, for example, highlighted in MHD in the context of astrophysical instabilities (see Fromang et al. 2007).

Walker et al. (1998) pointed to the possible existence of very high-frequency modes in the magnetostrophic limit. These yield at least numerical difficulties, but maybe, although the energy
is decaying for the unforced Taylor state, such high-frequency waves may generate a blow-up of the solution's gradients (a classical example of such a loss of regularity, although in a very different context, would be the formation of discontinuities in the inviscid Burger's equation). The well-posedness of this limit raises complicated mathematical issues and can only be tackled so far for a linearized Cartesian setup (Gallagher & Gérard-Varet 2017). An additional issue is that of the stability of the Taylor state (see, for example, the Billant type of instabilities discussed above). A first set of numerical solutions in the case of axisymmetric configurations has, however, been obtained (Roberts & Wu 2014, Wu & Roberts 2015, Li et al. 2018), possibly opening the way to Taylor state dynamos.

The first deviation from the Taylor state, when the constraint of vanishing torque is not exactly met, is an acceleration of the cylinders (this requires restoring at least the linear part of inertia for axisymmetric motions). The resulting waves, as those cylinders are coupled by the magnetic field, are known as torsional oscillations. They have been identified in the geomagnetic records and constitute a remarkably active area of research (e.g., Buffett et al. 2009; Gillet et al. 2010; Wicht & Christensen 2010; Teed et al. 2014, 2015; Hori et al. 2015, 2023; Gerick et al. 2021).

## 5. REVERSALS AND DISTINGUISHED LIMITS

One cannot conclude an article on the geodynamo without touching on the question of polarity reversals. Of course, as noted earlier, if $[u(x, t), B(x, t), T(x, t)]$ is a solution of Equations 22a-c, then so is $[u(x, t), -B(x, t), T(x, t)]$. So if a flow can sustain a given polarity, the same flow can also sustain the opposite one. The simplest possible approach to model transitions between those two symmetric states is probably to think of a double-well potential (e.g., Hoyng et al. 2002) (see Figure 9a). Some fluctuations are then needed to explore the two stable states.

One route to gain a better understanding of reversals is to argue that fluctuations in the strength of the Earth's dipole field are a consequence of turbulent fluid motions in the liquid metal core and use stochastic models to represent the influence of turbulent flow on the generation of a magnetic field (Buffett et al. 2013, 2022; Holdenried-Chernoff et al. 2023). Understanding the small-scale, highly anisotropic turbulence in the Earth's core is a challenging problem (Moffatt & Loper 1994, Nataf & Schaeffer 2015, Aurnou & King 2017), and stochastic models can offer a practical strategy for dealing with the poorly known small-scale dynamics.

An alternative approach is to argue that the reversals are due to a large-scale process such as a breaking of symmetry. Pétrélis et al. (2009) showed that as the equatorial symmetry of the flow is

a

<figure><img src="image_15.png" /></figure>

b

<figure><img src="image_16.png" /><figcaption>Figure 9</figcaption></figure>

(a) The double-well potential as a model for polarity reversals. (b) The dipole-quadrupole coupling close to the saddle-node bifurcation.
broken, away from the onset of convection, two families of fields become coupled (symmetric and antisymmetric with respect to the equator). An amplitude equation can be derived for the evolution of a field with dipole symmetry *D* and one with quadrupole symmetry *Q*. A dynamical system exhibiting Earth-like reversals, as well as excursions, can be derived from these equations under simplifying assumptions.

Stable ($B_s$) and unstable ($B_u$) mixed modes with both polarities are dynamically coupled. As the symmetry breaking increases, the stable and unstable nodes approach a saddle-node bifurcation (see **Figure 9b**). The reversals obtained from this model could also be successively compared to reversals obtained in the von Kármán sodium (VKS) dynamo experiment (for more detail, see Pétrélis et al. 2009). Some turbulent fluctuations remain necessary to obtain reversals in this model below the saddle-node bifurcation. An interesting property of this model is that, for a given level of fluctuations, the frequency of reversals will depend on the distance to the saddle-node bifurcation (i.e., on the symmetry breaking). Note that these fluctuations can also be replaced by a large-scale mode coupling the symmetries, thus resulting in a deterministic dynamical system (see Gissinger et al. 2010).

In three-dimensional direct numerical models, it was soon observed that the fluctuating multi-
polar FM mode is characterized by fluctuations, including reversals, of its nondominating dipolar
component. Because of this, it has sometimes been argued that the geodynamo operates very near
this transition (Christensen 2010). This appears, however, to require unrealistically small-scale
convection cells (especially once the field has relaxed the constraint of rapid rotation). Besides, it
is difficult to imagine that the Earth has maintained its dynamo close to this transition throughout
its history.

Yet some nonlinearity, driving fluctuations, is needed in the system to trigger reversals. An alternative route to inertia consists of introducing a distinguished limit, in which *q* is very small, despite the fact that *E*<sub>m</sub> is not as small as it should be. Decreasing *q* (which corresponds to a large *Pr* in the classical setup) increases the strength of nonlinearities in the heat equation. [The author and C.A. Jones came to this conclusion independently and had the possibility to develop the idea further through discussions at the Isaac Newton Institute (INI) meeting on dynamos from September to December 2022.] This would result in a modified distinguished limit and may result into geophysically relevant reversals. It is interesting to note that a distinguished limit, by construction, aims at preserving some properties of the limit system, the geodynamo, in parameter regimes that are computationally affordable. Various distinguished limits can thus arise depending on the properties that need to be captured by the models. In costly three-dimensional models, it may also be needed to artificially increase the reversal rate in order to study these.

Majumder et al. (2024) attempted to achieve inertia-less (in the sense of a low ε_m) reversals at a small ε_m and E by setting q = 1 (rather than q = E/ε_m as in most studies). This is the opposite of the low q limit advocated above. They obtained reversals but at very large forcing, when the field ceases to be dominantly dipolar. Further studies are needed to assess the force balances in their regime.

As far as the author is aware, computational studies with a small $\mathcal{E}_m$ (needed to achieve a strong field balance) and a small $q$ (probably needed for reversals) have not been published so far. These are, of course, very challenging studies, as long integration times are needed in order to study polarity reversals.

Kinematically unstable saturated dynamos shed an interesting light on the dynamical nature of the flow and field interactions. To gain a deeper understanding of how dynamo systems reach saturation, Cattaneo & Tobias (2009) have examined the kinematic dynamo characteristics of velocity fields generated by nonlinearly saturated dynamos. They solved a system with two induction equations, one for the actual (active) magnetic field and one for an independent (passive) vector

**Excursions:**

sometimes described
as failed reversals; i.e.,
the initial polarity
recovers after what
started like a reversal
field. They showed that it is possible for the flow of a saturated dynamo to still act as an efficient kinematic dynamo. Interestingly, Schrinner et al. (2010) showed that dynamos on the WD branch are kinematically stable, whereas the tracer field grows exponentially in the multipolar dynamo regime. Reversals in numerical dynamos have thus, so far, only been obtained precisely when the flow becomes kinematically unstable. The transition to a kinematically unstable dynamo is an indicator that several magnetic modes are in competition and highlights the dynamical nature of the flow and field interactions. Whether or not it is relevant to reversals deserves further study. This kinematic instability over a saturated dynamo should be contrasted with magnetic instabilities (e.g., Fearn 1979) in which the realized field may itself become unstable. This is yet another mechanism that could be relevant to polarity reversals.

The problem of the origin of the Earth's magnetic field is fascinating, if only because of the stark contrast between the simplicity of the initial question (Why does the compass point north?) and the mathematical developments required to try and address it. I have tried to present in this review a personal view of the state of affairs with the understanding of the geodynamo. This is an extremely active field of research and I unavoidably have missed some relevant references, yet I hope those presented provide useful entry points to this fascinating problem.

## FUTURE ISSUES

1. How do the rapidly oscillating thermal Rossby waves at the onset of convection interact with the flow as the Rayleigh number is increased?

2. How does this transition from wave to flow affect the onset of dynamo action and the weak field branch?

3. Can the strong field branch, relevant to the geodynamo, be extended below the onset of dynamo action for sufficiently small $\mathcal{E}, \mathcal{E}_m$, and $q$?

4. If so, can dynamo action be sustained below the onset of convection itself for sufficiently small $\mathcal{E}, \mathcal{E}_m$, and $q$?

5. Can geophysically relevant polarity reversals be obtained in a low-inertia regime by decreasing $q$ and thus increasing nonlinearities in the heat equation?

6. If so, will such reversals be related to an equatorial symmetry breaking of the flow and/or to the evolution of the flow to a kinematically unstable dynamo flow?

7. Is the Taylor state (dropping all small parameters) well posed?

8. If so, is the Taylor state stable, and can it be solved for numerically?

## DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.

## ACKNOWLEDGMENTS

I am indebted to Keith Moffatt and Andrew Soward for sharing their understanding of these problems over the years. I would also like to thank Bruce Buffett, Stéphan Fauve, Chris Jones, Rob Teed, and Steve Tobias for most helpful, often enlightening, discussions.
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