2. Modelling an internal dynamo region

In this section, we introduce a simplified model of the interior regions beneath the surface of a planet with volume $ S \subset \mathbb{R}^3$ . Specifically, we model the dynamo-active region $ T$ – the region where the electric current density responsible for dynamo action is non-vanishing – as a toroidal volume $ T \subset S$ .

The choice of a toroidal volume is motivated by the nature of axially symmetric dipole magnetic fields generated by spontaneous dynamo action. In such cases, the corresponding electric current density is toroidal and can be expressed as $ \boldsymbol{J} = \alpha (r,z)\nabla \varphi$ , where $ (r,\varphi ,z)$ are cylindrical coordinates, and $ \alpha (r,z)$ is a function of $ r$ and $ z$ . Due to the singularity of $ |\nabla \varphi | = 1/r$ at $ r=0$ , and the fact that the direction of $\nabla \varphi$ is not uniquely defined at $r=0$ , the current density $ \boldsymbol{J}$ must vanish along the planetary axis ( $ r=0$ ). Consequently, a dynamo-active region sustaining such an an axially symmetric magnetic field cannot have a spherical topology. Instead, it can be appropriately modelled as a toroidal volume $ T \subset S$ embedded within the planetary volume $ S \subset \mathbb{R}^3$ .

This observation is closely related to the Poincaré–Hopf and hairy ball theorems (Eisenberg & Guy Reference Eisenberg and Guy1979), which state that a non-vanishing continuous vector field cannot exist tangent to a spherical surface. For a sphere to accommodate a toroidal current, there must be regions where the current vanishes. Practically, this implies the absence of significant macroscopic currents near the planet’s poles, supporting the choice of a topologically toroidal dynamo-active region.

We further assume that the magnetic axis aligns approximately with the planet’s rotational axis and that $ T$ exhibits rotational symmetry around the vertical axis connecting the poles. The planetary structure is modelled with a solid core $ C$ , surrounded by a fluid shell $ L$ , which lies beneath the mantle $ M$ . Here, $ C$ is a spherical volume, while $ L$ and $ M$ are hollow spherical shells. Both the solid core and fluid shell may be electrically conducting; however, the solid nature of the core suggests that convective currents sustaining the planet’s magnetic field are confined to the fluid shell. Hence, $ T \subset L$ .

Additionally, we assume that the toroidal volume $ T$ nearly fills the fluid shell $ L$ , resulting in a ‘nearly spherical’ yet topologically toroidal region (see figure 1). For the purposes of this study, we focus on $ T$ and the exterior region $ O = \mathbb{R}^3 \setminus \bar {T}$ , where $ \bar {T}$ denotes the closure of $ T$ .

A schematic representation of this model is shown in figure 1. In this figure, $ \boldsymbol{J}_{\infty }(\boldsymbol{x}) = \mu _0^{-1} \unicode{x1D6FB} \times \boldsymbol{B}_{\infty }(\boldsymbol{x})$ and $ \boldsymbol{B}_{\infty }(\boldsymbol{x})$ denote the long-term current density and magnetic field, respectively, representing the final equilibrium fields resulting from dynamo action. Here, $ \mu _0$ is the vacuum permeability.

3. Modified Ohm’s law incorporating electron collisions

In the following, we adopt a simplified model of the electrically conducting fluid responsible for dynamo action, described as an ion–electron two-fluid system. This simplified approach is not intended for making quantitative predictions about the magnetic properties of specific planets or stars. Instead, it provides a conceptual framework that highlights the core physical principle: if an additional force is incorporated into Ohm’s law, symmetric dynamos become possible.

Ohm’s law relates the electric field $\boldsymbol{E}({\boldsymbol{x},t})$ to the velocity field $\boldsymbol{u}({\boldsymbol{x},t})$ , the magnetic field $\boldsymbol{B}({\boldsymbol{x},t})$ and the current density $\boldsymbol{J}({\boldsymbol{x},t})$ through the resistivity $\eta \gt 0$ , modelled as a constant, according to

(3.1) \begin{equation} \begin{split} \boldsymbol{E}=\boldsymbol{B}\times \boldsymbol{u}+\eta \boldsymbol{J}. \end{split} \end{equation}

Equation (3.1) originates from the electron fluid momentum equation

(3.2) \begin{equation} \begin{split} m_en_e\frac {{\rm d}\boldsymbol{u}_e}{{\rm d}t}=-en_e({\boldsymbol{E}+\boldsymbol{u}_e\times \boldsymbol{B}})-\nabla P_e+\boldsymbol{R}_e, \end{split} \end{equation}

where $m_e$ , $-e$ , $n_e({\boldsymbol{x},t})$ , $\boldsymbol{u}_e({\boldsymbol{x},t})$ , $P_e({\boldsymbol{x},t})$ and $\boldsymbol{R}_e({\boldsymbol{x},t})$ denote the electron mass, charge, number density, fluid velocity, pressure and resistive force resulting from ion–electron collisions, respectively (Freidberg Reference Freidberg2014). Recalling that the electron fluid velocity $\boldsymbol{u}_e$ is related to the current density $\boldsymbol{J}$ , the ion fluid velocity $\boldsymbol{u}_i({\boldsymbol{x},t})$ , the ion number density $n_i({\boldsymbol{x},t})$ and the ion charge $Ze$ according to $ \boldsymbol{J}=e({Zn_i\boldsymbol{u}_i-n_e\boldsymbol{u}_e})$ , (3.2) can be solved for $\boldsymbol{E}$ . Due to the smallness of the electron to ion mass ratio $\delta =m_e/m_i$ , the velocity of the centre of mass of the ion–electron two-fluid system effectively corresponds to the ion fluid velocity, i.e. $\boldsymbol{u}= ({\boldsymbol{u}_i+\delta \boldsymbol{u}_e})/({1+\delta })\approx \boldsymbol{u}_i$ . Furthermore, assuming quasi-neutrality, we have $n({\boldsymbol{x},t}) =n_i\approx n_e/Z$ , leading to

(3.3) \begin{equation} \begin{split} \boldsymbol{E}=\boldsymbol{B}\times \boldsymbol{u}+\frac {1}{Ze n}\boldsymbol{J}\times \boldsymbol{B}-\frac {1}{Zen}\nabla P_e+\frac {\boldsymbol{R}_e}{Zen}-\frac {m_e}{e}\frac {{\rm d}\boldsymbol{u}_e}{{\rm d}t}. \end{split} \end{equation}

The second term on the right-hand side is the Hall effect. In the regime of MHD, both the Hall and electron pressure terms (3.3) scale with the ion-gyroradius, and are therefore neglected on the larger spatial scale of the system. The last term on the right-hand side of (3.3) is similarly neglected under the standard MHD assumption (Freidberg Reference Freidberg2014) that the small electron mass makes it smaller than all the other terms, including the resistive term $\boldsymbol{R}_e/Zen$ . If we further assume a linearity relationship between $\boldsymbol{R}_e$ and the current density $\boldsymbol{J}$ , i.e.

(3.4) \begin{equation} \begin{split} \frac {\boldsymbol{R}_{e}}{Zen}=\eta \boldsymbol{J}, \end{split} \end{equation}

we arrive at Ohm’s law (3.1). For completeness, it is important to note that the derivation of Ohm’s law presented previously aligns well with the classical Drude model (Ashcroft & Mermin Reference Ashcroft and Mermin1976), which describes electrons as following random trajectories due to collisions with stationary ions. The emergence of a steady-state characterised by a linear response to the applied electric field is intrinsically tied to the small mass of the electron. This small mass enables an exceptionally fast response time relative to the MHD time scale, thereby ensuring that the system quickly reaches equilibrium where the current density is proportional to the electric field.

The standard form of Ohm’s law (3.1) is valid only within the parameter regime of MHD. However, it is well established that Ohm’s law must be modified when the assumptions underlying this regime no longer hold. For a detailed discussion on the applicability of MHD, we refer to Freidberg (Reference Freidberg2014) and Fitzpatrick (Reference Fitzpatrick2015). Generalised forms of Ohm’s law, such as those in Hall MHD (Acheritogaray et al. Reference Acheritogaray, Degond, Frouvelle and Liu2011) and extended MHD (Abdelhamid, Kawazura & Yoshida Reference Abdelhamid, Kawazura and Yoshida2015), have been employed in applications to dynamo theory (Minnini, Gómez & Mahajan Reference Minnini, Gómez and Mahajan2003; Lingam & Mahajan Reference Lingam and Mahajan2015). The central idea of this study is that the dissipative force arising from electron–electron collisions, which is typically neglected in standard MHD, may play a significant role in geophysical and astrophysical dynamos. This necessitates a corresponding modification of Ohm’s law (3.1). In the remainder of this section, we develop a model for this modified form of Ohm’s law.

Now suppose that we move to a rotating reference frame with speed $\boldsymbol{v}_{\!R}= ({1}/{2})\varOmega r^2\nabla \varphi$ , where $\varOmega \gt 0$ is a constant and $({r,\varphi ,z})$ is a cylindrical coordinate system. In the rotating frame, the ion and electron fluid velocities become $\boldsymbol{u}'=\boldsymbol{u}-\boldsymbol{v}_{\!R}$ and $\boldsymbol{u}_e'=\boldsymbol{u}_e-\boldsymbol{v}_{\!R}$ . It follows that

(3.5) \begin{equation} \begin{split} \boldsymbol{J}=Ze n\left ({\boldsymbol{u}-\boldsymbol{u}_e}\right )=Zen({\boldsymbol{u}'-\boldsymbol{u}_e'})=\boldsymbol{J}', \end{split} \end{equation}

where the prime means that the quantity is evaluated in the rotating frame. Equations (3.4) and (3.5) show that the resistive force $\boldsymbol{R}_e$ cannot take into account dissipative forces that cannot be expressed through the relative velocity $\boldsymbol{u}-\boldsymbol{u}_e$ . Indeed, the force $\boldsymbol{R}_e$ does not change in the presence of rotation $\boldsymbol{v}_{\!R}$ due to the invariance of $\boldsymbol{J}$ described by (3.5). However, the fluid shell surrounding the solid core cannot slide: it is subject to friction forces that tend to make uniform the speed of the fluid with the rotation speed $\boldsymbol{v}_{\!R}$ of the inner core and the overlying mantle. Now imagine sliding a condensed viscous fluid on a rough surface. It is then unlikely that these friction forces can be neglected in the modelling of the fluid shell. The simplest model for this type of friction is given by a restoring force $\boldsymbol{F}_{\!\gamma }$ acting on the electron fluid

(3.6) \begin{equation} \begin{split} \boldsymbol{F}_{\!\gamma }=-Ze\gamma n({\boldsymbol{u}_e-\boldsymbol{v}_{\!R}})=-Ze\gamma n({\boldsymbol{u}-\boldsymbol{v}_{\!R}})+\gamma \boldsymbol{J}, \end{split} \end{equation}

where $\gamma \gt 0$ is a physical constant. The result is a modified Ohm’s law,

(3.7) \begin{equation} \begin{split} \boldsymbol{E}= \boldsymbol{B}\times \boldsymbol{u}+\eta _{\gamma }\boldsymbol{J}-\gamma ({\boldsymbol{u}-\boldsymbol{v}_{\!R}}),\,\,\,\,\eta _\gamma =\eta + ({\gamma }/{Zen}). \end{split} \end{equation}

Note that we introduced an effective resistivity $\eta _{\gamma }$ . Remarkably, the restoring force contains a toroidal component $({\boldsymbol{u}-\boldsymbol{v}_{\!R}})\cdot \nabla \varphi$ that can potentially balance a toroidal current density $\boldsymbol{J}\cdot \nabla \varphi \neq 0$ . Indeed, the difference $\boldsymbol{u}-\boldsymbol{u}_e$ between ion fluid velocity and electron fluid velocity can have the same orientation as the difference $\boldsymbol{u}_e-\boldsymbol{v}_{\!R}$ between electron fluid velocity and rotation velocity, implying that the resistive force $\eta \boldsymbol{J}$ can be balanced by the restoring force $-\gamma ({\boldsymbol{u}_e-\boldsymbol{v}_{\!R}})$ . Physically, the fast moving ions attract the electron fluid in one direction, while rotation of the planet effectively drags them in the other.

It is worth noting that the resistive force, which is proportional to $\boldsymbol{u} - \boldsymbol{u}_e$ , and the frictional force, which is proportional to $\boldsymbol{u} - \boldsymbol{v}_{\!R}$ , share the same mathematical structure. In the case of the resistive force, this structure represents the linear response to electron–ion collisions. Conversely, the frictional force models the linear response to electron–electron collisions, which interact with the rotating inner core and outer mantle as they collide with the boundary of the toroidal domain, propagating this effect through subsequent collisions. We also remark that the friction force $\boldsymbol{F}_{\!\gamma }$ is consistent with Galilean invariance, as the velocity differences $\boldsymbol{u} - \boldsymbol{v}_{\!R}$ and $\boldsymbol{u} - \boldsymbol{u}_e$ remain invariant across inertial frames under Galilean relativity.

We emphasise that (3.6) represents a simplified description of the actual friction force at play in the fluid shell, and is chosen to simplify the later analysis. This friction force is ultimately rooted in the momentum transfer associated with electron collisions. An accurate derivation of the friction force should take into account the individual condition of the planet, e.g. the graduality of the transition from liquid to solid phase in Earth’s interior, and would require kinetic modelling of particle collisions in a rotating setting, leading to a corresponding electron fluid momentum equation subject to boundary conditions for the velocity field that take into account the planet’s rotation. If viscous resistive MHD equations are chosen to model the electron momentum equation in the rotating frame, the resulting modification of Ohm’s law in the non-rotating rest frame is

(3.8) \begin{equation} \begin{split} \boldsymbol{E}=\boldsymbol{B}\times \boldsymbol{u}+\eta \boldsymbol{J}+ (({m_e\nu _e})/{e})\Delta \boldsymbol{u}_e\approx \boldsymbol{B}\times \boldsymbol{u}+\eta \boldsymbol{J}+ (({m_e\nu _e})/{e})\Delta \boldsymbol{u}, \end{split} \end{equation}

where $\nu _e$ is the electron fluid kinematic viscosity, $\Delta$ the Laplacian operator, we used the fact that $\Delta \boldsymbol{v}_{\!R}=\boldsymbol{0}$ and neglected the term proportional to $\Delta ({\boldsymbol{J}/Zen})$ , which is expected to represent a higher order correction in the plasma regime under consideration. Note that, although $\boldsymbol{v}_{\!R}$ does not appear explicitly in $\Delta \boldsymbol{u}$ , the presence of the operator $\Delta$ implies that planetary rotation is felt throughout the toroidal domain via the no-slip boundary conditions on $\boldsymbol{u}$ .

It is important to note that since the difference $\boldsymbol{u} - \boldsymbol{u}_e$ can be small relative to $\boldsymbol{u}$ , the viscous term may be comparable to the resistive term. This implies that neglecting viscous effects in resistive MHD may not always be a valid approximation, as clear from the numerical examples given below. In particular, note that when approaching the boundary of the fluid shell, the current density progressively drops towards zero, leaving the friction/viscous forces the dominant toroidal component in Ohm’s law. In fact, the condition $\eta \boldsymbol{J}\sim m_e\nu _e\Delta \boldsymbol{u}/e$ can be used to estimate the characteristic spatial scale $L$ of the relative velocity $\boldsymbol{u}-\boldsymbol{v}_{\!R}$ in the fluid shell needed to sustain a given current density $\boldsymbol{J}$ . We have

(3.9) \begin{equation} \boldsymbol{u}-\boldsymbol{v}_{\!R}\sim \frac {eL^2\eta }{m_e\nu _e}\boldsymbol{J}\sim \frac {e\mu _0L^2}{m_e\mathrm{P_m}}\boldsymbol{J} \implies L^2\sim \frac {m_e\mathrm{P_m}\lvert {\boldsymbol{u}-\boldsymbol{v}_{\!R}}\rvert }{e\mu _0\lvert {\boldsymbol{J}}\rvert }, \end{equation}

where $\mathrm{P_m}=\mu _0\nu _e/\eta$ is the (electron) magnetic Prandtl number. For given $\boldsymbol{u}-\boldsymbol{v}_{\!R}$ , we see that when $\mathrm{P_m}$ is smaller, the current density and the turbulence scale $L$ tend to be very small. Conversely, in areas where $\mathrm{P_m}$ is higher, substantial currents can flow and larger scales $L$ are allowed. As an example, consider Earth’s liquid iron core for a relative velocity of $\boldsymbol{u} - \boldsymbol{v}_{\!R} \approx 10^{-4} \, \text {m s}^{-1}$ . Using estimated values for resistivity, $\eta \approx 40 \, \unicode{x03BC} \varOmega \,\text{cm}$ (Ohta et al. Reference Ohta, Kuwayama, Hirose, Shimizu and Ohishi2016), and kinematic viscosity, $\nu \approx 1.5\times 10^{-6} \, \text{m}^2\,\rm s^{-1}$ (de Wijs et al. Reference de Wijs, Kresse, Vocadlo, Dobson, Alfè, Gillan and Price1998), noting that $\nu _e\approx \sqrt {{m_i}/{m_e}}\nu$ (Freidberg Reference Freidberg2014), one finds a turbulence scale of $L \approx 10^{-3}\,\text{m}$ for the geomagnetic field $\boldsymbol{B} \approx 10^{-6} \, \text{T}$ such that $\unicode{x1D6FB} \times \boldsymbol{B} \approx 10^{-13} \, \rm T\, m^{-1}$ . A similar estimate for Saturn’s metallic hydrogen, with $\nu \approx 0.4\times 10^{-6}\mathrm{m}^2\,\rm s^{-1}$ , $\eta \approx 0.2\times 10^{-3} \unicode{x03BC}\Omega \,\text{cm}$ (Preising et al. Reference Preising, French, Mankovich, Soubiran and Redmer2023) and $\unicode{x1D6FB} \times \boldsymbol{B}\approx 10^{-14}\,\rm T\, m^{-1}$ , gives $L\approx 10\,\text{m}$ when $\boldsymbol{u}-\boldsymbol{v}_{\!R}\approx 10^{-2}\,\text{ m s}^{-1}$ . In those plasma regimes, by contrast, the ion gyroradii $r_{i}$ are found to be approximately $ 10^{-5} \, \text{m}$ and $10^{-4}\,\text{m}$ for the Earth and Saturn, respectively. The MHD approximation with the modified Ohm’s law is hence valid here (see table 1). We also note that these estimates on $L$ likely represent lower bounds as $\nu _e\approx \sqrt {m_i/m_e}\nu$ only takes into account electron–electron collisions.

Finally, we remark that similar considerations apply to the ratio of the friction force $\boldsymbol{F}_{\!\gamma }$ and the resistive term $\eta \boldsymbol{J}$ , as $\boldsymbol{F}_{\!\gamma }$ and $\Delta \boldsymbol{u}$ model the same dissipative force.

Table 1. Comparison of parameters for Earth’s liquid iron core and Saturn’s metallic hydrogen. $L$ is the spatial turbulence scale at which the resistive term $\eta \boldsymbol{J}$ is comparable to the viscous term $m_e\nu _e\Delta \boldsymbol{u}/e$ .

4. Modified Cowling’s theorem

Let us examine how Cowling’s theorem changes under the modified Ohm’s law (3.7). It will be sufficient to consider an axially symmetric poloidal magnetic field

(4.1) \begin{equation} \begin{split} \boldsymbol{B}=\nabla \varPsi \times \nabla \varphi , \end{split} \end{equation}

where $\varPsi ({r,z,t})$ denotes the flux function, and an axially symmetric velocity field

(4.2) \begin{equation} \begin{split} \boldsymbol{u}=\nabla \varTheta \times \nabla \varphi +g\nabla \varphi , \end{split} \end{equation}

with $\varTheta ({r,z,t})$ and $g({r,z,t})$ single-valued functions. For simplicity, we hereafter assume $\eta _{\gamma }\gt 0$ to be constant. Using (3.7), the induction equation for the magnetic field $\boldsymbol{B}$ reads as

(4.3) \begin{equation} \begin{split} \frac {\partial \boldsymbol{B}}{\partial t}=\unicode{x1D6FB} \times \left ({\boldsymbol{u}\times \boldsymbol{B}-\eta _{\gamma }\boldsymbol{J}+\gamma \boldsymbol{u}}\right )-\gamma \varOmega \nabla z. \end{split} \end{equation}

Substituting (4.1), we have

(4.4) \begin{align} \unicode{x1D6FB} \times \left ({\frac {\partial \varPsi }{\partial t}\nabla \varphi }\right )=&\unicode{x1D6FB} \times \left \{{-\left ({\nabla \varPsi \cdot \boldsymbol{u}}\right )\nabla \varphi +\frac {g}{r^2}\nabla \varPsi +\frac {\eta _{\gamma }}{\mu _0}\left [{r\frac {\partial }{\partial r}\left ({\frac {1}{r}\frac {\partial \varPsi }{\partial r}}\right )+\frac {\partial ^2\varPsi }{\partial z^2}}\right ]\nabla \varphi }\right \}\nonumber\\ &+\gamma \unicode{x1D6FB} \times \left ({\nabla \varTheta \times \nabla \varphi +g\nabla \varphi -\frac {1}{2}\varOmega r^2\nabla \varphi }\right ). \end{align}

Considering a toroidal volume $T\subset \mathbb{R}^3$ as domain for the induction equation, we can remove the curl operator by introducing a potential $\varPhi ({r,z,t})$ such that

(4.5) \begin{align} \frac {\partial \varPsi }{\partial t}\nabla \varphi & = \nabla \varPhi +\left [{\gamma g-\frac {\gamma }{2}\varOmega r^2-\left ({\nabla \varPsi \cdot \boldsymbol{u}}\right )+\frac {\eta _{\gamma }}{\mu _0}\left ({ \Delta \varPsi -2\nabla \varPsi \cdot \nabla \log r }\right )}\right ]\nabla \varphi \nonumber\\& \quad +\frac {g}{r^2}\nabla \varPsi +\gamma \nabla \varTheta \times \nabla \varphi . \end{align}

The toroidal component of this equation reads as

(4.6) \begin{equation} \begin{split} \frac {\partial \varPsi }{\partial t}= \gamma g-\frac {\gamma }{2}\varOmega r^2-{\nabla \varPsi \cdot \boldsymbol{u}}+\frac {\eta _{\gamma }}{\mu _0} \left ({\Delta \varPsi -2\nabla \varPsi \cdot \nabla \log r}\right ). \end{split} \end{equation}

A key observation is that this governing equation allows for a zero magnetic field solution whenever the toroidal velocity equals the planet’s rotational speed, specifically when $ g = {\varOmega r^2}/{2}$ . Hence, the system is not frictionally driven, allowing for spontaneous dynamo action.

Next, we evaluate the integral

(4.7) \begin{equation} \begin{split} \frac {1}{2}\frac {\rm d}{{\rm d}t}\left \lvert \left \lvert \varPsi \right \rvert \right \rvert _{T}^2=\frac {1}{2}\frac {\rm d}{{\rm d}t} \int _{T}\varPsi ^2\,{\rm d}V, \end{split} \end{equation}

where $\lvert \lvert \cdot \rvert \rvert _{T}$ denotes the standard $L^2({T})$ norm. We have

(4.8) \begin{equation} \begin{split} \frac {1}{2}&\frac {\rm d}{{\rm d}t}\left \lvert \left \lvert \varPsi \right \rvert \right \rvert _{T}^2=\int _{T}\varPsi \left [{\gamma g-\frac {\gamma }{2}\varOmega r^2-\nabla \varPsi \cdot \boldsymbol{u}+\frac {\eta _{\gamma }}{\mu _0}\left ({\Delta \varPsi -2\nabla \varPsi \cdot \nabla \log r}\right )}\right ]\,{\rm d}V. \end{split} \end{equation}

Noting that $\Delta \log r=0$ , and integrating by parts under the boundary conditions

(4.9a) \begin{align} \varPsi &=0\,\,\,\,\mathrm{on}\,\,\partial T, \end{align}

(4.9b) \begin{align} \boldsymbol{u}\cdot \boldsymbol{n}& =0\,\,\,\,\mathrm{on}\,\,\partial T, \end{align}

where $\partial T$ denotes the boundary of $T$ with unit outward normal $\boldsymbol{n}$ , it follows that

(4.10) \begin{equation} \begin{split} \frac {1}{2}\frac {\rm d}{{\rm d}t}\left \lvert \left \lvert \varPsi \right \rvert \right \rvert _{T}^2= \gamma \int _T\varPsi \left ({g-\frac {1}{2}\varOmega r^2}\right )\,{\rm d}V -\frac {\eta _{\gamma }}{\mu _0}\left \lvert \left \lvert \nabla \varPsi \right \rvert \right \rvert ^2_T. \end{split} \end{equation}

For a suitable choice of the toroidal flow $g$ , the integrand of the first term on the right-hand side becomes positive, indicating the possibility of dynamo action. For example, setting $C_0^t=\lvert \lvert \varPsi \rvert \rvert _T^2$ , $C_1^t=\lvert \lvert \nabla \varPsi \rvert \rvert _T^2$ and $g=K\varPsi + ({1}/{2})\varOmega r^2$ , at a given instant $t$ , we have $ ({1}/{2}) (\textrm{d}/{{\rm d}t})\lvert \lvert \varPsi \rvert \rvert _T^2=\gamma K C_0^t- (({\eta _\gamma })/({\mu _0}))C_1^t$ , which is positive for large enough $K\in \mathbb{R}_{\gt 0}$ . If the system achieves an equilibrium, defining $\varPsi _{\infty }=\lim _{t\rightarrow +\infty }\varPsi$ and $g_{\infty }=\lim _{t\rightarrow +\infty }g$ , we find that

(4.11) \begin{equation} \begin{split} \left \lvert \left \lvert \nabla \varPsi _{\infty } \right \rvert \right \rvert ^2_T=\frac {\gamma \mu _0}{\eta _{\gamma }}\int _T\varPsi _{\infty }\left ({g_{\infty }-\frac {1}{2}\varOmega r^2}\right )\,\mathrm{d}V. \end{split} \end{equation}

Evidently, for a suitable equilibrium toroidal flow $g_{\infty }$ in $T$ , the right-hand side becomes positive, and this equality admits non-trivial solutions $\nabla \varPsi _{\infty }\neq \boldsymbol{0}$ in $T$ (see the following for more details). Note that, however, the right-hand side of (4.11) vanishes as soon as the restoring force (3.6) is absent, i.e. $\gamma =0$ , recovering Cowling’s theorem.

In $\mathbb{R}^3$ , the rate of change in magnetic energy is

(4.12) \begin{equation} \begin{split} \frac {1}{2}\frac {\rm d}{{\rm d}t}\left \lvert \left \lvert \boldsymbol{B} \right \rvert \right \rvert _{\mathbb{R}^3}^2=-\mu _0\int _{T}{\boldsymbol{E}\cdot \boldsymbol{J}}\,{\rm d}V=\mu _0\int _{T}\left [{\boldsymbol{u}\times \boldsymbol{B}+\gamma ({\boldsymbol{u}-\boldsymbol{v}_{\!R}})}\right ]\cdot \boldsymbol{J}\,{\rm d}V-\mu _0\eta _{\gamma }\left \lvert \left \lvert \boldsymbol{J} \right \rvert \right \rvert ^2_{T}, \end{split} \end{equation}

where we used Maxwell’s equation $\partial \boldsymbol{B}/\partial t=-\unicode{x1D6FB} \times \boldsymbol{E}$ , the fact that $\boldsymbol{E}$ and $\boldsymbol{B}$ must vanish at infinity, the fact that the current density is confined to $T$ , and (3.7). Now suppose that the flow $\boldsymbol{u}=g\nabla \varphi$ is purely toroidal. Noting that $\mu _0\boldsymbol{J}=-({\Delta \varPsi -2\nabla \varPsi \cdot \nabla \log r})\nabla \varphi$ is purely toroidal, it follows that

(4.13) \begin{equation} \begin{split} \frac {1}{2}\frac {\rm d}{{\rm d}t}\left \lvert \left \lvert \boldsymbol{B} \right \rvert \right \rvert _{\mathbb{R}^3}^2= \mu _0\gamma \int _{T}\left ({{g}-\frac {1}{2}\varOmega r^2 }\right )\boldsymbol{J}\cdot \nabla \varphi \,{\rm d}V-\mu _0\eta _{\gamma }\left \lvert \left \lvert \boldsymbol{J} \right \rvert \right \rvert ^2_T. \end{split} \end{equation}

For example, if the current density has the same orientation as the planet’s rotation speed $\boldsymbol{v}_{\!R}$ , and the toroidal flow $g$ is large enough, the first term on the right-hand side may exceed the second one, leading to an increase in magnetic energy. The amplification of the magnetic field is not limited to the toroidal volume $T$ . Indeed, in $O$ , we have

(4.14) \begin{equation} \begin{split} \frac {1}{2}\frac {\rm d}{{\rm d}t}\left \lvert \left \lvert \boldsymbol{B} \right \rvert \right \rvert _O^2= -\int _{ O}\unicode{x1D6FB} \cdot \left ({\boldsymbol{E}\times \boldsymbol{B}}\right )\,{\rm d}V-\mu _0\int _O\boldsymbol{E}\cdot \boldsymbol{J}\,{\rm d}V=\int _{\partial T}\boldsymbol{E}\times \boldsymbol{B}\cdot \boldsymbol{n}\,{\rm d}S, \end{split} \end{equation}

where we used Maxwell’s equation $\partial \boldsymbol{B}/\partial t=-\unicode{x1D6FB} \times \boldsymbol{E}$ , the fact that $\boldsymbol{E}$ and $\boldsymbol{B}$ must vanish at infinity, and the fact that $\boldsymbol{J}=\boldsymbol{0}$ in $O$ (there is no current outside $T$ ). The tangential electric field $\boldsymbol{E}_t=\boldsymbol{n}\times ({\boldsymbol{E}\times \boldsymbol{n}})$ must be continuous across the toroidal boundary $\partial T$ . Furthermore, since the electric current $\boldsymbol{J}$ vanishes in $O$ , from (3.7), we conclude that

(4.15) \begin{equation} \begin{split} \boldsymbol{E}_t= -\gamma ({\boldsymbol{u}-\boldsymbol{v}_{\!R}})\,\,\,\,\mathrm{on}\,\,\partial T, \end{split} \end{equation}

where we used the fact that $\boldsymbol{u}\cdot \boldsymbol{n}=0$ on $\partial T$ (recall (4.9)). Now suppose that at the time $t=0$ , the initial seed electric current $\boldsymbol{J}_0=\boldsymbol{J}({\boldsymbol{x},0})\neq \boldsymbol{0}$ is positively oriented along the toroidal direction $\varphi$ . Then, there is some time interval in which $\nabla \varPsi$ points towards the centre of $T$ . More precisely, the unit outward normal on $\partial T$ is given by $\boldsymbol{n}=-\nabla \varPsi /\lvert {\nabla \varPsi }\rvert$ , leading to

(4.16) \begin{equation} \begin{split} \frac {1}{2}\frac {\rm d}{{\rm d}t}\left \lvert \left \lvert \boldsymbol{B} \right \rvert \right \rvert _O^2=\gamma \int _{\partial T}\left ({\frac {g}{r^2}-\frac {\varOmega }{2}}\right )\left \lvert {\nabla \varPsi }\right \rvert \,{\rm d}S. \end{split} \end{equation}

This quantity becomes positive for sufficiently large $g\gt 0$ and vanishes when $\gamma =0$ . From this equation, we also see that an equilibrium solution is given by $g_{\infty }=r^2\varOmega /2$ on $\partial T$ , i.e.

(4.17) \begin{equation} \begin{split} \left ({\boldsymbol{u}_{\infty }-\boldsymbol{v}_{\!R}}\right )\cdot \nabla \varphi =0\,\,\,\,\mathrm{on}\,\,\partial T, \end{split} \end{equation}

where $\boldsymbol{u}_{\infty }=\lim _{t\rightarrow +\infty }\boldsymbol{u}$ .

For completeness, it is useful to show how Cowling’s theorem is modified if the modified Ohm’s law (3.8) involving viscosity is used in place of (3.7). For simplicity, we consider a purely toroidal flow $\boldsymbol{u}=g\nabla \varphi$ . We have

(4.18) \begin{equation} \begin{split} \frac {1}{2}\frac {\rm d}{{\rm d}t}\left \lvert \left \lvert \boldsymbol{B} \right \rvert \right \rvert ^2_{\mathbb{R}^3}=\frac {\mu _0 m_e\nu _e}{e}\int _{T}\left ({2\nabla g\cdot \nabla \log r-\Delta g}\right )\boldsymbol{J}\cdot \nabla \varphi \,{\rm d}V-\mu _0\eta \left \lvert \left \lvert \boldsymbol{J} \right \rvert \right \rvert ^2_T. \end{split} \end{equation}

We thus see that the first term on the right-hand side can be positive for a suitable toroidal flow $g$ , indicating the possibility of dynamo action. For example, noting that $\boldsymbol{J}=-\mu _0^{-1}({\Delta \varPsi -2\nabla \varPsi \cdot \nabla \log r})\nabla \varphi$ and setting $g=K\varPsi /\mu _0$ , one obtains $({1}/{2})({\rm d}/{{\rm d}t})\lvert \lvert \boldsymbol{B} \rvert \rvert ^2_{\mathbb{R}^3}=\mu _0({K (({m_e\nu _e})/{e})-\eta })\lvert \lvert \boldsymbol{J} \rvert \rvert _T^2$ , which is positive for large enough $K\in \mathbb{R}_{\gt 0}$ . A similar calculation to (4.16) shows that magnetic field growth is not limited to $T$ ,

(4.19) \begin{equation} \begin{split}{ \frac {1}{2}\frac {\rm d}{{\rm d}t}\left \lvert \left \lvert \boldsymbol{B} \right \rvert \right \rvert ^2_{O}=\frac {m_e\nu _e}{e}\int _{\partial T}r^{-2}\left ({2\nabla g\cdot \nabla \log r-\Delta g}\right ){\left \lvert {\nabla \varPsi }\right \rvert }\,{\rm d}S. }\end{split} \end{equation}

5. Modified neutral point argument

Cowling’s theorem is related to the so-called neutral point argument. Let us see how the neutral point argument changes in the present setting. Suppose that the equilibrium flux function $\varPsi _{\infty }$ is a well-behaved non-constant function in $\mathbb{R}^3$ . Then, by symmetry, it will attain local maximima or minimima somewhere in $\mathbb{R}^3$ . At these neutral points, $\nabla\Psi_{\infty}$ and $\boldsymbol{B}_{\infty}=\nabla\Psi_{\infty} \times \nabla\varphi$ vanish. Let N denote a neutral point, and consider a circle C of radius $\epsilon\sim \lvert {\boldsymbol{B}_{\infty }({N+\epsilon \boldsymbol{\delta }})}\rvert$ , where $\boldsymbol{\delta }$ is some unit vector. Let $\Sigma _C$ denote a surface in the $({r,z})$ -plane, with normal $\boldsymbol{n}_C=r\nabla \varphi$ , enclosed by $C$ and such that $N\in \Sigma _C$ . Using Ohm’s law (3.1), Stokes’ theorem and noting that, at equilibrium, the electric field $\boldsymbol{E}_{\infty }=-\nabla \varPhi _{\infty }$ is given as the gradient of a single-valued potential $\varPhi _{\infty }({{r,z}})$ , we have

(5.1) \begin{equation} \begin{split}{ 0=\int _{\Sigma _C}\boldsymbol{E}_{\infty }\cdot \boldsymbol{n}_C\,{\rm d}S= \int _{\Sigma _C}\boldsymbol{B}_{\infty }\times \boldsymbol{u}_{\infty }\cdot \boldsymbol{n}_C \,{\rm d}S+\frac {\eta }{\mu _0}\int _C\boldsymbol{B}_{\infty }\cdot {\rm d}\boldsymbol{l}. }\end{split} \end{equation}

However, by construction, $\Sigma _C=\pi \epsilon ^2$ while $C=2\pi \epsilon$ . In proximity of $N$ , (5.1) can thus be satisfied only if $ ({\mu _0}/{\eta })\boldsymbol{u}_{\infty }\sim {\epsilon ^{-1}}$ , implying that $\boldsymbol{u}_{\infty }$ is singular at $N$ for non-vanishing and bounded coefficients $\eta$ and $\mu _0$ . If the same argument is applied to the modified Ohm’s law (3.7), one obtains

(5.2) \begin{equation} \begin{split}{ 0=\int _{\Sigma _C}\left [{\boldsymbol{B}_{\infty }\times \boldsymbol{u}_{\infty }-\gamma ({\boldsymbol{u}-\boldsymbol{v}_{\!R}})}\right ]\cdot \boldsymbol{n}_C \,{\rm d}S+\frac {\eta _{\gamma }}{\mu _0}\int _C\boldsymbol{B}_{\infty }\cdot {\rm d}\boldsymbol{l}. }\end{split} \end{equation}

Now, the second and third terms on the right-hand side are both second-order terms in $\epsilon$ , and the neutral point argument does not apply.

We conclude this section by noting that modifying Ohm’s law only by retaining the Hall effect does not suffice to invalidate the neutral point argument. This is because $ \boldsymbol{J}_{\infty } \times \boldsymbol{B}_{\infty } \cdot \boldsymbol{n}_C = 0$ , given that $ \boldsymbol{J}_{\infty } \times \boldsymbol{n}_C = \boldsymbol{0}$ .

6. Axially symmetric poloidal equilibrium configurations

Let us construct axially symmetric poloidal equilibrium configurations, i.e. steady solutions of the induction equation such that

(6.1a) \begin{align} \boldsymbol{B}_{\infty }&=\nabla \varPsi _{\infty }\times \nabla \varphi ,\\[-10pt]\nonumber \end{align}

(6.1b) \begin{align} \boldsymbol{J}_{\infty }&=-\frac {1}{\mu _0}\left ({\Delta \varPsi _{\infty }-2\nabla \varPsi _{\infty }\cdot \nabla \log r}\right )\nabla \varphi ,\\[-10pt]\nonumber \end{align}

(6.1c) \begin{align} \boldsymbol{E}_{\infty }&=-\nabla \varPhi _{\infty },\\[-10pt]\nonumber \end{align}

(6.1d) \begin{align} \boldsymbol{u}_{\infty }&=\nabla \varTheta _{\infty }\times \nabla \varphi +g_{\infty }\nabla \varphi . \end{align}

Recalling the modified Ohm’s law (3.7), the equilibrium equation is

(6.2) \begin{align} -\nabla \varPhi _{\infty } & =\left (\!{\nabla \varTheta _{\infty }\times \nabla \varphi \cdot \nabla \varPsi _{\infty }-\frac {\eta _{\gamma }}{\mu _0}\Delta \varPsi _{\infty }+2\frac {\eta _\gamma }{\mu _0}\nabla \varPsi _{\infty }\cdot \nabla \log r-\gamma g_{\infty }+\frac {1}{2}\gamma \varOmega r^2}\right )\!\nabla \varphi \nonumber\\& \quad -\frac {g_{\infty }}{r^2}\nabla \varPsi _{\infty }-\gamma \nabla \varTheta _{\infty }\times \nabla \varphi . \end{align}

The poloidal component of $\boldsymbol{u}_{\infty }$ can be eliminated by setting $\varTheta _{\infty }=\mathrm{constant}$ . However, the electrostatic potential must be single-valued, leading to $g_{\infty }=r^2 {\rm d}f({\varPsi _{\infty }}){\rm d}\varPsi _{\infty }$ and $\varPhi _{\infty }=f({\varPsi _{\infty }})$ . The equilibium flux function $\varPsi _{\infty }$ is then determined by a Poisson equation. The full set of equilibrium equations is

(6.3a) \begin{align} \Delta \varPsi _{\infty }=2\nabla \varPsi _{\infty }\cdot \nabla \log r-\frac {\gamma \mu _0}{\eta _{\gamma }} r^2\left ({\frac {{\rm d}f({\varPsi _{\infty }})}{{\rm d}\varPsi _{\infty }}-\frac {\varOmega }{2}}\right )\,\,\,\,\mathrm{in}\,\,T, \end{align}

(6.3b) \begin{align} \varPhi _{\infty }=f({\varPsi _{\infty }})\,\,\,\,\mathrm{in}\,\,T,\end{align}

(6.3c) \begin{align} g_{\infty }=r^2\frac {{\rm d}f({\varPsi _{\infty }})}{{\rm d}\varPsi _{\infty }}\,\,\,\,\mathrm{in}\,\,T,\end{align}

(6.3d) \begin{align} \varTheta _{\infty }=\mathrm{constant}\,\,\,\,\mathrm{in}\,\,T, \end{align}

(6.3e) \begin{align} \varPsi _{\infty }=0\,\,\,\,\mathrm{on}\,\,\partial T, \end{align}

(6.3f) \begin{align} \frac {{\rm d}f({\varPsi _{\infty }})}{{\rm d}\varPsi _{\infty }}=\frac {\varOmega }{2}\,\,\,\,\mathrm{on}\,\,\partial T. \end{align}

For given $f({\varPsi _{\infty }})$ , this system admits non-trivial regular solutions in $T$ thanks to the source term proportional to $\gamma$ in the first equation. The result reveals that the electric field $\boldsymbol{E}_{\infty }$ is balanced by the ideal convective term $\boldsymbol{B}_{\infty }\times \boldsymbol{u}_{\infty }$ (see 6.3b ), while the resistive term $\eta _{\gamma }\boldsymbol{J}_{\infty }$ is balanced by the friction term $-\gamma ({\boldsymbol{u}_{\infty }-\boldsymbol{v}_{\!R}})$ as shown in (6.3a ). This type of force balance is consistent with kinetic theory of plasmas, which informs us that equilibria of the Boltzmann equation usually belong to the kernel of the collision operator, i.e. in a steady-state, ideal terms and dissipative terms vanish independently (see e.g. Sato & Morrison (Reference Sato and Morrison2024) and references therein). Indeed, MHD and related two-fluid theories are reduced models originating from kinetic theory. As previously noted, zero magnetic field solutions are permitted when $g_{\infty } = ({r^2 \varOmega })/{2}$ in $T$ . Additionally, it is important to observe that the toroidal component of the fluid velocity, $g_{\infty } \nabla \varphi$ , can deviate significantly from the planet’s rotation speed, $\boldsymbol{v}_{\!R} = ({1}/{2}) \varOmega r^2 \nabla \varphi$ , within $T$ when a non-zero magnetic field is present (see 6.3a ).

We also remark that $\boldsymbol{J}_{\infty }$ vanishes on $\partial T$ due to the boundary condition (6.3f ) coming from (4.17). Hence, for any $\boldsymbol{x}\in \mathbb{R}^3$ , the (continuous) magnetic field vanishing at infinity and corresponding to the current distribution resulting from system (6.3) can be obtained through the Biot–Savart integral,

(6.4) \begin{equation} \begin{split}{ \boldsymbol{B}_{\infty }({\boldsymbol{x}})=\frac {\mu _0}{4\pi }\int _{T}\frac {\boldsymbol{J}_{\infty }({\boldsymbol{x}'})\times \left ({\boldsymbol{x}-\boldsymbol{x}'}\right )}{\left \lvert {\boldsymbol{x}-\boldsymbol{x}'}\right \rvert ^3}\,{\rm d}V'. }\end{split} \end{equation}

The properties of the Biot–Savart integral (6.4), as well as a generalisation to bounded domains, can be found in Enciso, Garcia-Ferrero & Peralta-Salas (Reference Enciso, Garcia-Ferrero and Peralta-Salas2018). We remark that since the current density $\boldsymbol{J}_{\infty }$ is an axially symmetric toroidal flow, it follows that in $O$ , the equilibrium magnetic field $\boldsymbol{B}_{\infty }$ is an axially symmetric dipole-type magnetic field.