2.1.1. The induction and momentum equation

Dynamo theory typically (minimally) involves the construction of solutions to the pair of coupled partial differential equations for the evolution of the velocity field, ${\boldsymbol u}$ and the magnetic field ${\boldsymbol B}$ in an electrically conducting fluid under the magnetohydrodynamic (MHD) formalism. A complete derivation of the equations and a discussion of their applicability in any given physical situation is beyond the scope of this perspective; the interested reader is directed to Moffatt & Dormy (Reference Moffatt and Dormy2019) and Jones (Reference Jones2008) for details of the derivation. Briefly, under the MHD approximation, which combines the non-relativistic Maxwell equations of electrodynamics with Ohm's law for a moving conductor, these take the form

(2.1)\begin{gather} \frac{\partial {\boldsymbol u}}{\partial t} + {\boldsymbol u} \boldsymbol{\cdot} \boldsymbol{\nabla} {\boldsymbol u} = - \frac{1}{\rho}\boldsymbol{\nabla} p + \frac{1}{\rho}\, {\boldsymbol j} \times {\boldsymbol B}+\nu \nabla^2 {\boldsymbol u} + \frac{\boldsymbol F}{\rho}, \end{gather}

(2.2)\begin{gather}\frac{\partial {\boldsymbol B}}{\partial t} = \boldsymbol{\nabla} \times ( {\boldsymbol u} \times {\boldsymbol B} ) -\boldsymbol{\nabla} \times (\eta \boldsymbol{\nabla} \times {\boldsymbol B} ), \end{gather}

(2.3)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol B} = 0. \end{gather}

Here, the standard fluid parameters are $\rho$, which is the density of the fluid, $p$ which is the pressure and $\nu$ is the kinematic viscosity of the fluid, whilst ${\boldsymbol F}$ represents all the body forces (such as buoyancy or mechanical driving) acting to drive the fluid motion. The additional force term in the Navier–Stokes equations is termed the Lorentz force ${{\boldsymbol F}}_{Lor} = {\boldsymbol j} \times {\boldsymbol B}$ and arises owing to the interaction of the magnetic field with the current ${\boldsymbol j}$ flowing through the conductor. The current is itself given by the pre-Maxwell version of Ampère's law, i.e.

(2.4)\begin{equation} {\boldsymbol j} = \frac{1}{\mu} \boldsymbol{\nabla} \times {\boldsymbol B},\end{equation}

and hence

(2.5)\begin{equation} {{\boldsymbol F}}_{Lor} = \frac{1}{\mu \rho} (\boldsymbol{\nabla} \times {\boldsymbol B}) \times {\boldsymbol B}.\end{equation}

Here, and henceforth in this paper, $\mu$ is the permeability. The evolution equation for the magnetic field given by (2.2) is termed the induction equation. Here, $\eta = (\mu \sigma )^{-1}$ is the magnetic diffusivity which is a property of the conducting fluid, with electrical conductivity $\sigma$. Magnetic diffusivity is large for poor conductors. In circumstances in which the magnetic diffusivity is constant, (2.2) simplifies (utilising (2.3)) to

(2.6)\begin{equation} \frac{\partial {\boldsymbol B}}{\partial t} = \boldsymbol{\nabla} \times ( {\boldsymbol u} \times {\boldsymbol B} ) + \eta \nabla^2 {\boldsymbol B}.\end{equation}

2.1.2. Magnetic boundary conditions

The induction and momentum equations are usually (although not always) solved in a finite domain subject to the imposition of boundary conditions on the fluid velocity and magnetic field. The boundary conditions for the fluid velocity are straightforward and standard; usually one considers impenetrable boundaries with either stress-free or no-slip conditions on the tangential component of the velocity.

The magnetic boundary conditions are more problematic and usually involve some degree of simplification. Typically this involves considering the fluid in a domain $V$ with an exterior bounding surface $S$, external to which is either an insulator or a perfect conductor. If the region outside of the evolution of the dynamo fluid is perfectly conducting, surface charges, $\rho _S$, and surface currents, ${\boldsymbol j}_S$, are allowed on the boundary. If we define $[.]$ as the jump across the surface $S$ and ${\boldsymbol n}$ as an outward pointing normal to that surface, then integrating the pre-Maxwell equations across the surface gives

(2.7a–d)\begin{equation} [ {\boldsymbol n} \boldsymbol{\cdot} {\boldsymbol E} ] = \frac{\rho_S}{\epsilon}, \quad [ {\boldsymbol n} \boldsymbol{\cdot} {\boldsymbol B} ] = 0, \quad [ {\boldsymbol n} \times {\boldsymbol B} ] = \mu {\boldsymbol j}_S , \quad [ {\boldsymbol n} \times {\boldsymbol E} ] = 0, \end{equation}

where $\epsilon$ is the permittivity. If there are no surface currents or charges (i.e. for non-perfectly conducting boundaries) then ${\boldsymbol B}$ is continuous, provided $\mu$ is constant. If the region external to the solution domain does not allow currents (i.e. if this region is an insulator) then this continuity of ${\boldsymbol B}$ defines the problem. If the outside region allows currents then the normal derivative of ${\boldsymbol n} \boldsymbol {\cdot } {\boldsymbol B}$ is also continuous, although the normal derivatives of the tangential components of ${\boldsymbol B}$ are not necessarily continuous (see the detailed discussion in Jones Reference Jones2008).

However, if the outside of the domain is a static perfect conductor it is normal to assume that there is no trapped magnetic field there and so (for no normal flow conditions)

(2.8a,b)\begin{equation} {\boldsymbol n} \boldsymbol{\cdot} {\boldsymbol B} =0, \quad {\boldsymbol n} \times {\boldsymbol j} =0. \end{equation}

This gives

(2.9)\begin{equation} B_z = \frac{\partial B_x}{\partial z} = \frac{\partial B_y}{\partial z} = 0, \end{equation}

at a Cartesian boundary $z=\textrm {const.}$, and

(2.10)\begin{equation} B_r = \frac{\partial (rB_{\theta})}{\partial r} = \frac{\partial(rB_{\phi})}{\partial r} = 0, \end{equation}

at a spherical boundary, $r=\textrm {const.}$

2.3.1. Conservation of magnetic helicity

Because $\boldsymbol {\nabla } \boldsymbol {\cdot } {\boldsymbol B} = 0$, it is often useful to write ${\boldsymbol B} = \boldsymbol {\nabla } \times {\boldsymbol A}$. Here ${\boldsymbol A}$ is termed the vector potential for the magnetic field. Clearly, ${\boldsymbol A}$ is only defined up to a choice of gauge, so that transforming ${\boldsymbol A} \rightarrow {\boldsymbol A} + \boldsymbol {\nabla } \psi$ for any scalar $\psi$ leaves the magnetic field unchanged. It is also then convenient to ‘uncurl’ (2.2) to give

(2.16)\begin{equation} \frac{\partial {{\boldsymbol A}}}{\partial t} =({{\boldsymbol u}}\times {{\boldsymbol B}})-\eta \,\boldsymbol{\nabla} \times {{\boldsymbol B}} + \boldsymbol{\nabla} \phi, \end{equation}

where $\phi$ is related to the choice of gauge.

Various choices of gauge are possible; the Coulomb gauge has

(2.16a,b)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol A} = 0; \quad \nabla^2 \phi = -\boldsymbol{\nabla} \boldsymbol{\cdot} ({{\boldsymbol u}}\times {{\boldsymbol B}}). \end{equation}

The winding gauge (suitable for calculations in Cartesian geometries) has

(2.17a,b)\begin{equation} \nabla_H \boldsymbol{\cdot} {\boldsymbol A} = 0; \quad \nabla_H^2 \phi^\prime = -\boldsymbol{\nabla}_H \boldsymbol{\cdot} ({{\boldsymbol u}}\times {{\boldsymbol B}}), \end{equation}

where $\nabla _H = (\partial _x,\partial _y,0)$ and $\phi ^\prime = \phi - \eta ({\partial A_z}/{\partial z})$ (Prior & Yeates Reference Prior and Yeates2014). A numerically convenient gauge involves setting

(2.17)\begin{equation} \phi = \frac{\partial \psi}{\partial t}, \end{equation}

as described in Brandenburg & Subramanian (Reference Brandenburg and Subramanian2005)

Now, magnetic helicity is defined as

(2.18)\begin{equation} H = \int_V {\boldsymbol A} \boldsymbol{\cdot} {\boldsymbol B} \, \textrm{d} V ,\end{equation}

and its evolution can be easily shown to be given by

(2.19)\begin{equation} \frac{\textrm{d} H}{\textrm{d} t}=-2 \eta \mu \int_V {\boldsymbol j} \boldsymbol{\cdot} {\boldsymbol B} \, \textrm{d} V + F_s,\end{equation}

where $F_s$ represents the surface flux of magnetic helicity. Hence, for a perfectly conducting fluids, with no loss or gain of magnetic helicity through the boundaries (i.e. $F_s = 0$), magnetic helicity is conserved. This has implications for dynamo action, which requires the generation or destruction of magnetic helicity, as we shall see. Magnetic helicity is a measure of the topological complexity and linkage of field lines, and in the absence of diffusive processes (by which the field can reconnect) this complexity is maintained. Furthermore magnetic helicity appears to be a more robust invariant than say total energy in the presence of small diffusion. The dissipative term for total energy, $-\eta \mu \int _V \,{{\boldsymbol j}^2}\, \textrm {d} V$, may remain finite as $\eta \rightarrow 0$, because ${\boldsymbol j}^2$ gets large in this limit. However, the dissipation of magnetic helicity, $-2 \eta \mu \int _V {\boldsymbol j} \boldsymbol {\cdot } {\boldsymbol B} \, \textrm {d} V$, appears to tend to zero as $\eta \rightarrow 0$, so magnetic helicity is well conserved. Of course the situation changes if magnetic helicity is allowed to enter or leave the domain of interest; it may do so via either advective or diffusive processes, as we shall see in § 6.5.2.

In hydrodynamic turbulence the presence of quadratic invariants has consequences for the nature of the cascades. Briefly, the same reasoning applies to MHD. As argued above, for small dissipation energy decays faster than magnetic helicity and cross-helicity (Biskamp Reference Biskamp2003). Therefore, in a turbulent state, energy cascades towards small scales (analogous to the energy cascade in three-dimensional (3-D) hydrodynamics). However, the magnetic helicity cascades toward large scales (analogous to the energy cascade in 2-D hydrodynamics). The inverse cascade of magnetic helicity may lead to the formation of large-scale magnetic fields – this fact may prove to be important for the generation of systematic fields by nonlinear dynamos.

2.4.1. Importance of the magnetic Reynolds number $Rm$

Clearly from the above discussion, magnetic field will decay away unless the advective term in the induction equation is large enough to overcome diffusive effects. The relative importance of the two terms $\boldsymbol {\nabla } \times ({\boldsymbol u} \times {\boldsymbol B})$ and $\eta \nabla ^2 {\boldsymbol B}$ can be established by non-dimensionalising. We choose a typical length scale $L$ which is the size of the object or region under consideration and a typical fluid velocity $U$. On introducing scaled variables $t = ({L}/{U}) \tilde {t}$, ${\boldsymbol x} = L {\tilde {\boldsymbol x}}$, ${\boldsymbol u} = U \tilde {\boldsymbol u}$, and dropping tildes the induction equation becomes

(2.23)\begin{equation} \frac{\partial {\boldsymbol B}}{\partial t} = {\boldsymbol \nabla} \times ({\boldsymbol u} \times {\boldsymbol B}) + Rm^{-1} \nabla^2 {\boldsymbol B},\end{equation}

where $Rm = {U L}/{\eta }$ is the dimensionless magnetic Reynolds number.

In general, large $Rm$ means induction dominates over diffusion, whilst small $Rm$ means diffusion wins out over induction, and as we shall see minimum values of $Rm$ for dynamo action (so-called dynamo bounds) can sometimes be found.

It is worth noting at this point, however, that $Rm$ should only be used as a guide to determine the relative importance of advection and diffusion. In defining $Rm$ it has explicitly been assumed that $L$ is a typical length scale for both the magnetic field and the velocity (i.e. that $\ell _B = \ell _U = L$). This makes sense if $L$ is the size of the astrophysical object (i.e. the largest length scale available). Of course this may not be the case, with the potential for $\ell _B$ to be very different from $\ell _U$. For example if $\ell _B \gg \ell _U$ then the relative importance of advection and diffusion is given by $Rm_T = U \ell _B^2 / \eta \ell _U \sim \omega \ell _B^2/\eta$, where $\omega$ is a typical vorticity amplitude. This may be large even if $\ell _U$ and $U$ are small. Such a basic misunderstanding of the limitations of the information encoded in the magnetic Reynolds number may lead to the incorrect dismissal of certain small-scale flows as the possible origin of large-scale fields. Large-scale fields, as they are weakly diffusive, require very little induction for their maintenance.

2.4.2. A useful technique: axisymmetric field decomposition

It is clear from the form of the induction equation that a non-axisymmetric flow immediately creates a non-axisymmetric field, however, the converse is not true. If both the flow and field are axisymmetric then one can decompose the flow and field by setting (in cylindrical polars $(s,\phi ,z)$)

(2.24)\begin{gather} {\boldsymbol u}=s \varOmega \hat{{\boldsymbol \phi}} + {\boldsymbol u}_P = s \varOmega \hat{{\boldsymbol \phi}} + \boldsymbol{\nabla} \times \left(\frac{\psi}{s}\right) \hat{{\boldsymbol \phi}}, \end{gather}

(2.25)\begin{gather}{\boldsymbol B} = B \hat{{\boldsymbol \phi}} + {\boldsymbol B}_P =B \hat{{\boldsymbol \phi}} + \boldsymbol{\nabla} \times (A \hat{{\boldsymbol \phi}}). \end{gather}

Here, $s = r \sin \theta$ (where $r$ and $\theta$ relate to spherical polars $(r,\theta ,\phi )$). This defines the differential rotation $\varOmega$ the streamfunction $\psi$, the zonal field $B$ (sometimes termed toroidal field in an axisymmetric setting) and the scalar potential $A$. The induction equation then simplifies to

(2.26)\begin{gather} \frac{\partial A}{\partial t} + \frac{1}{s} ({\boldsymbol u}_P \boldsymbol{\cdot} \boldsymbol{\nabla} ) (sA) = \eta \left(\nabla^2 - \frac{1}{s^2}\right) A, \end{gather}

(2.27)\begin{gather}\frac{\partial B}{\partial t} + s ({\boldsymbol u}_P \boldsymbol{\cdot} \boldsymbol{\nabla} ) \left(\frac{B}{s}\right) = \eta \left(\nabla^2 - \frac{1}{s^2}\right) B + s{\boldsymbol B}_P \boldsymbol{\cdot} \boldsymbol{\nabla} \varOmega. \end{gather}

The form of these equations will be exploited in the next section to prove so-called anti-dynamo theorems. Note that, although both the $A$ and $B$ equations have advective and diffusive terms, the field stretching term, $s{\boldsymbol B}_P \boldsymbol {\cdot } \boldsymbol {\nabla } \varOmega$, only appears in (2.27) if gradients in angular velocity are present. (2.26) shows that there is no corresponding source term in the equation for $A$.

2.4.3. Anti-dynamo theorems

It is fair to say that the psychology of the dynamo practitioner has been strongly shaped by the early results of dynamo theory – results that showed the complexity and difficulty of achieving dynamo-generated fields. Key are the so-called anti-dynamo theorems that show the impossibility of dynamo action for large classes of magnetic fields and velocity fields with certain symmetries. We shall not reproduce all of the demonstrations and proofs here, since they are readily available from many sources (for example Dormy & Soward Reference Dormy and Soward2007; Jones Reference Jones2008; Moffatt & Dormy Reference Moffatt and Dormy2019), but the importance of these for the subsequent direction of the development of the field cannot be overstated.

Cowling's theorem: an axisymmetric field vanishing at infinity cannot be maintained by dynamo action (Cowling Reference Cowling1933).

Proof. As noted above, a non-axisymmetric velocity field immediately generates non-axisymmetric magnetic field and so it is necessary to consider only axisymmetric flows and fields. The evolution of the field is therefore given by (2.26)–(2.27). Multiplying (2.26) by $s^2 A$, using the divergence theorem, and integrating over all space gives

(2.28)\begin{equation} \frac{\textrm{d}}{\textrm{d} t} \int \frac{1}{2} s^2 A^2 \, \textrm{d} V = - \eta \int | \boldsymbol{\nabla} (sA) |^2 \, \textrm{d} V , \end{equation}

assuming that surface terms vanish at infinity. This equation clearly shows that $sA$ decays to zero as $t \rightarrow \infty$; note $sA$ can not decay to a constant since this would imply $A\rightarrow \infty$ as $s\rightarrow 0$. Once $A$ has decayed there is no source term in the toroidal field equation (i.e. 2.27). Similar arguments then ensure the subsequent decay of $B/s$ and therefore the impossibility of the creation of an axisymmetric magnetic field by dynamo action. It is important to note that it is the absence of source terms in (2.26) that causes the problems for dynamo action. Relaxation of the constraint of axisymmetry allows for the re-inclusion of such a source term.

The effect of Cowling's theorem ruling out such simple symmetric solutions made the search for any dynamo solutions (which were necessarily three dimensional!) seem a formidable task. Indeed, for a long time it was not clear that any such dynamo solutions existed. The situation is encapsulated in the following story taken from Krause (Reference Krause1993) (which attributes the source as Paul Roberts) ‘Walter Elsasser and Einstein were friends in Germany before they both emigrated to the US in the 1930s. Several years after Elsasser had settled there (in the late 1930s in fact), he became interested in the origin of the geomagnetic field. Einstein paid him a visit, and (as people do) asked ‘What are you working on these days?’. Elsasser told him, and Einstein invited him to explain dynamo theory to him. Elsasser set-up the problem and then told Einstein about Cowling's theorem. Einstein's response was, ‘If such simple solutions are impossible, self-excited fluid dynamos cannot exist’. For once, the great man's craving for simplicity seems to have misled him’.

Other antidynamo theorems: there are many extensions of Cowling's antidynamo theorem to other geometries and to slightly different set-ups. For example, in Cartesian coordinates $(x,y,z)$, no magnetic field that vanishes at infinity and is independent of $z$ can be generated by dynamo action; the proof proceeds along similar lines to that given above (see e.g. Jones Reference Jones2008). Note again that this is a restriction on the form of a dynamo-generated magnetic field.

Anti-dynamo theorems placing constraints on the form of the fluid velocities that can lead to dynamo action have been proven by Bullard & Gellman (Reference Bullard and Gellman1954), Backus (Reference Backus1958) and in Cartesian coordinates by Zel'dovich (Reference Zel'dovich1957). These results essentially show that a velocity field must have all three components in order to be capable of acting as a dynamo (see the long discussion and derivation in Moffatt Reference Moffatt1978).

In conclusion, both the field and the flow must be sufficiently complicated for dynamo action to occur. A minimal requirement is that the field must be three-dimensional and the flow must not be purely poloidal (i.e. cannot be planar).

2.4.4. Bounds on dynamo action

Even for flows and fields that are not ruled out as dynamos on symmetry grounds, there are bounds that constrain dynamo action in finite domains; a crude measure of the expected efficiency of a dynamo is given by the magnetic Reynolds number $Rm$, which gives the ratio of advection to diffusion at a particular scale in the flow (usually taken to be the integral or system scale). The importance of $Rm$ can be formalised in bounds derived by Backus (Reference Backus1958) and Childress (Reference Childress1969), both of which use the evolution equation for the magnetic energy $E_M$, given by (see (2.13))

(2.29)\begin{equation} \mu \frac{\partial E_M}{\partial t} = - \eta \int | \boldsymbol{\nabla} \times {\boldsymbol B}|^2 \, \textrm{d} V + \int (\boldsymbol{\nabla} \times {\boldsymbol B}) \boldsymbol{\cdot} ({\boldsymbol u} \times {\boldsymbol B}) \, \textrm{d} V . \end{equation}

For field generation in a sphere of radius $a$, matching to a decaying potential dynamo action requires

(2.30)\begin{equation} Rm = \frac{a^2 e_{max}}{\eta} \ge {\rm \pi}^2, \end{equation}

where $e_{max}$ is the maximum of the rate of strain tensor (Backus Reference Backus1958). Note here that $Rm$ is defined in terms of the maximum strain (and not a typical velocity amplitude), which seems natural. A slightly different bound from (2.29) (Childress Reference Childress1969) requires

(2.31)\begin{equation} {Rm = \frac{a u_{max}}{\eta} \ge {\rm \pi}}. \end{equation}

Other bounds on dynamo action are also possible. However, it can be shown that a steady or periodic dynamo can exist in a bounded conductor with an arbitrarily small value of the kinetic energy. Hence there is no lower bound on dynamo action when $Rm$ is defined using the root mean square (rather than maximum) velocity, without placing limitations on the rate of strain (Proctor Reference Proctor2015).

Motivated by the desire to construct experimental dynamos in the laboratory, there is currently an effort to optimise the efficiency of dynamo flows given certain constraints. This involves taking the machinery developed for understanding the transition to turbulence, such as adjoint optimisation methods and using them to maximise the efficiency of dynamo velocities (Willis Reference Willis2012).

2.5.1. Kinematic dynamos

Having convinced ourselves that overly symmetric fields and flows are not good for dynamo action and that sufficient stretching is required, it is time to discuss some flows that do actually work as dynamos. These are not presented in chronological order, as we shall start with the simpler case of flows in an infinite domain, before moving onto flows in spheres and spherical shells.

The Ponomarenko flow (Ponomarenko Reference Ponomarenko1973):

Here, we consider the simplest possible flow that leads to dynamo action. It takes the form of a localised discontinuous ‘screw’ or vortex flow that in cylindrical coordinates $(s, \phi , z)$ is given by

(2.32)\begin{equation} {\boldsymbol u} = \begin{cases} s \varOmega {\hat{{\boldsymbol \phi}}} + U \hat{\boldsymbol z}, \quad s < a,\\ 0, \quad s > a, \end{cases} \end{equation}

where $\varOmega$ and $U$ are constants. Note that this flow is not planar, owing to the presence of the throughflow $U \hat {\boldsymbol z}$, and has kinetic helicity, $H = {\boldsymbol u} \boldsymbol {\cdot } {\boldsymbol \omega } = 2 U \varOmega .$ We shall see the importance of kinetic helicity for large-scale dynamos in § 5. Strong shear naturally occurs at (the physically unrealistic) discontinuity at $s=a$. In principle there are two independent parameters defining the flow $U$ and $\varOmega$. These can be re-expressed in terms of an overall amplitude of the flow given by $U_{amp} = (U^2 + a^2 \varOmega ^2)^{1/2}$ and the pitch angle $\chi = U / a \varOmega$.

The trick for elegant solution for the kinematic dynamo modes in such a configuration is to note first that the flow is steady and, together with the linearity of the induction equation, this implies that magnetic field will either grow or decay exponentially in time, with potentially a complex growth-rate $\lambda = \sigma + {\rm i} \omega$. Secondly, the flow is independent of $z$ and $\phi$ and therefore monochromatic magnetic fields can be sought in those directions. It is therefore advantageous to seek solutions of the form ${\boldsymbol B} = {\boldsymbol b}(s) \exp ( \lambda t + {\rm i} m \phi + {\rm i}kz ),$ where $m$ and $k$ are the azimuthal and vertical wavenumbers, which must be non-zero to avoid the anti-dynamo theorems discussed earlier.

This model is extremely illuminating as it can be solved pseudo-analytically (see Jones Reference Jones2008) for details; marginally stable solutions can be found by setting $\mbox {Re}({\lambda }) = 0$ (for a given $Rm = a U_{amp}/\eta$, $\chi$, $ka$ and $m$). The Ponomarenko flow is indeed a dynamo! It can be thought of as a prototype dynamo that is a model of vortical plume. Dynamo action sets in at $Rm_{crit}=17.72$, for $ka_{crit} = -0.3875$, $m=1$, $a^2 \omega / \eta = -0.41$ and $\chi = 1.31$, so the optimal pitch is $ {O}(1)$. The magnetic field is strongest near $s=a$, where it is generated by (the unphysical) shear. Although it is important to examine how dynamo action onsets, it is also, as we shall see, of great interest to determine the behaviour at large $Rm$. Asymptotic solutions of the Ponomarenko dynamo (in the form of Bessel functions) show that (Gilbert Reference Gilbert1988) the fastest growing modes are given by

(2.35a,b)\begin{equation} |m| = (6(1 + \chi^{-2}))^{-3/4} \left( \frac{a^2 \varOmega}{2 \eta} \right)^{1/2},\quad \sigma = 6^{-3/2} \varOmega (1 + \chi^{-2})^{-1/2}. \end{equation}

In the presence of viscosity, velocities with discontinuities are not realistic and so the Ponomarenko dynamo has been extended to the continuous case where ${\boldsymbol u} = s \varOmega (s) \hat {{\boldsymbol \phi }} + U(s) \hat {\boldsymbol z}$, where $\varOmega (s)$ and $U(s)$ are smooth functions (Gilbert Reference Gilbert1988).

The Roberts flow (Roberts Reference Roberts1972a):

Perhaps the most illuminating, kinematic dynamo flow is the so-called G.O. Roberts flow. This flow, like the Ponomarenko flow, is specially crafted to circumvent both the Cartesian version of Cowling's anti-dynamo theorem and the planar velocity anti-dynamo theorem, whilst still remaining fairly tractable.

The G.O. Roberts flow is the special case of the well-studied ABC (Arnol'd, Beltrami and Childress) flow in Cartesian coordinates $(x,y,z)$ in an infinite domain given by

(2.36)\begin{equation} {\boldsymbol u} = (C \sin z + B \cos y, A \sin x + C \cos z, B \sin y + A \cos x). \end{equation}

For the Roberts flow $A=B=1$, $C=0$, so that ${\boldsymbol u}(x,y) = (\cos y, \sin x, \sin y + \cos x).$ This flow is two-dimensional (in the sense that it only depends on two coordinates), but it has all three components (thus not falling foul of the planar flow anti-dynamo theorem).

Before discussing the dynamo properties of the Roberts flow, it is useful to describe some of its basic hydrodynamic properties. The Roberts flow is integrable in the sense that it can be written in terms of a single steady streamfunction $\psi (x,y)$, so that

(2.37)\begin{equation} {\boldsymbol u} = \left( \frac{\partial \psi}{\partial y}, -\frac{\partial \psi}{\partial x}, \psi \right), \end{equation}

where $\psi = \sin y + \cos x$. The flow takes the form of an array of helical cells with throughflow, see figure 3(a). It has a typical horizontal spatial scale of $2 {\rm \pi}$ and an infinite vertical scale. Moreover the winding sense of each helix in the array is the same. This ensures that the normalised relative kinetic helicity, defined to be

(2.38)\begin{equation} \mathcal{H}_{rel} = \int \frac{| {\boldsymbol u}\boldsymbol{\cdot} {\boldsymbol \omega}| }{| {\boldsymbol u}\boldsymbol{\cdot} {\boldsymbol u} |^{1/2}| {\boldsymbol \omega}\boldsymbol{\cdot} {\boldsymbol \omega}|^{1/2}}\,\textrm{d} V , \end{equation}

is unity (a so-called maximally helical flow); the importance of helicity for the large-scale dynamo properties of the flow will be discussed at some length in § 5.3.1.

Figure 3.

(a) Contours of the streamfunction $\psi$ for the G.O. Roberts flow. Positive (and zero) contours are solid and negative contours are dashed. (b) Growth rate $\sigma$ as a function of wavenumber, $k_z$ for various $Rm$ (after Roberts Reference Roberts1972a). (c,d) Scaled magnetic energy in the plane $z=0$ for two different $Rm=16$ and $512$. As $Rm$ is increased the field is expelled into magnetic boundary layers of width $ {O}(Rm^{-1/2})$. Note only the domain between $0$ and $2 {\rm \pi}$ is shown. The magnetic energy is scaled between $0$ and $1$. Figure courtesy of A. Clarke.

Roberts utilised the same considerations as Ponomarenko (although a year previous) to search for magnetic field solutions to the induction equation of the form

(2.39)\begin{equation} {\boldsymbol B} = {\boldsymbol b}(x,y) \exp ( \lambda t + {\rm i}k_z z), \end{equation}

where ${\boldsymbol b}(x,y)$ is periodic in $x$ and $y$ (including the possibility that ${\boldsymbol b}(x,y)$ has a mean part). The solution of the 2-D problem requires spectral methods yielding a matrix eigenvalue problem for the growth rate $\sigma = \mbox {Re}(\lambda )$ as a function of the wavenumber $k_z$ and $Rm$. For each value of $Rm$ there is an optimal value of $k_z$ that maximises the growth rate, as shown in figure 3(b).

The form of the magnetic field for this flow is very illuminating. Figure 3(c,d) shows the magnetic energy in the plane $z=0$. For this steady flow the field is generated by the flow between the stagnation points. As $Rm$ is increased the field is expelled into magnetic boundary layers of width $ {O}(Rm^{-1/2})$ enhancing diffusion (although the length of the filamentary field structures is determined by the geometry of the flow). This is therefore an example of a small-scale dynamo – the field generated is dominated by structure at a scale smaller than a typical scale in the velocity; such a dynamo relies on the stretching overcoming the dissipative effects of diffusion.

In a tour de force paper, Soward (Reference Soward1987) analysed the behaviour of the Roberts dynamo at high $Rm$ utilising asymptotic methods. He showed that the maximal growth rate for this dynamo $\sigma \rightarrow 0$ as $Rm \rightarrow \infty$ (although very slowly – indeed

(2.40)\begin{equation} \sigma \sim \frac{\log \log Rm}{\log Rm}, \end{equation}

for large $Rm$). In the parlance of dynamo theory this makes the Roberts flow a slow dynamo – we shall discuss slow, fast and quick dynamos later.

Finally for this section we stress again that the Roberts flow is a small-scale dynamo that generates field via stretching. Although the flow is helical, this is not its defining characteristic here. Indeed, Roberts also considered a flow with no net helicity, viz. ${\boldsymbol u}=(\sin 2y, \sin 2x, \sin (x+y))$. Although this is a less efficient dynamo than the helical Roberts flow, it is nonetheless a (slow) dynamo. As noted by H.K. Moffatt ‘Helicity is not essential for dynamo action, but it helps’.

2.5.2. Kinematic dynamos in a spherical domain

The examples above are instructive (and similar types of flows will be utilised later to illustrate other dynamo properties). A natural question to pose, however, is whether similar types of flow can lead to dynamos in a bounded domain – the most natural examples of which are spheres or spherical shells.

In spherical geometry one may decompose an incompressible velocity field in terms of two scalar fields (Bullard & Gellman Reference Bullard and Gellman1954), i.e.

(2.41)\begin{equation} {\boldsymbol u} = \boldsymbol{\nabla} \times ( T(r,\theta,\phi,t)\,\hat{\boldsymbol r} ) + \boldsymbol{\nabla} \times \boldsymbol{\nabla} \times ( S(r,\theta,\phi,t)\, \hat{\boldsymbol r} ).\end{equation}

Here, $T$ and $S$ are the toroidal and poloidal components. Alternatively we write

(2.42)\begin{equation} {\boldsymbol u} = \sum_{l,m} {\boldsymbol t}_l^m + {\boldsymbol s}_l^m, \end{equation}

where ${\boldsymbol t}_l^m$ and ${\boldsymbol s}_l^m$ are given by

(2.43)\begin{gather} {\boldsymbol t}_l^m = \boldsymbol{\nabla} \times ( t_l^m (r,t) Y_l^m (\theta, \phi)\,\hat{\boldsymbol r}), \end{gather}

(2.44)\begin{gather}{\boldsymbol s}_l^m = \boldsymbol{\nabla} \times \boldsymbol{\nabla} \times (s_l^m (r,t) Y_l^m (\theta, \phi)\, \hat{\boldsymbol r}) \end{gather}

where $-l \le m \le l$, and $Y_l^m$ is the spherical harmonic.

The flow is defined by choosing the values of $l$ and $m$ and the corresponding radial functional form for the scalar fields $t_l^m(r,t)$ and $s_l^m(r,t)$. In a landmark paper, Bullard and Gellman, nearly twenty years before the flows discussed in the last section, chose ${\boldsymbol u} = \epsilon {\boldsymbol t}_1^0 + {\boldsymbol s}_2^2$ and set $t_1^0 = r^2 (1-r)$, and $s_2^2 = r^3 (1-r)^2$. Having made a similar expansion to (2.39) for the magnetic field, they used spectral interaction rules to determine the growth rate of the field. They reported dynamo action for this flow for high enough $Rm$, but unfortunately the dynamo growth was spurious (as shown by subsequent higher resolution calculations). We shall return to this unfortunate property of dynamo calculations in § 2.5.3. Although the reported dynamo action was incorrect, the work of Bullard and Gellman revitalised the field after the depressing wilderness years of anti-dynamo theorems, giving hope that self-excited dynamos were indeed possible. Generalisations and extensions of the Bullard and Gellman type dynamos have been studied by Kumar & Roberts (Reference Kumar and Roberts1975) and Dudley & James (Reference Dudley and James1989).

If the non-axisymmetric components of such flows are small ($ {O}(\epsilon )$) then it is possible to perform an asymptotic expansion with the axisymmetric components of flow and field dominating over the non-axisymmetric parts. This is the nearly axisymmetric dynamo of Braginskii (Reference Braginskii1975), which is an example of a self-consistent mean field model (see § 5).

2.5.3. A note of caution

The evolution equation for the magnetic field as described by the induction equation takes the form of a competition between magnetic field stretching (advection) and diffusion. Often these are both exponential processes; whether magnetic field grows or decays is then determined by the small difference between the efficiency of these processes. It is therefore of vital importance that both of these processes are accurately represented by the numerical scheme utilised to solve the induction equation. Failure to do so will inevitably lead to the incorrect determination of the dynamo properties of the flow. This was first demonstrated by the spurious solutions to the Bullard-Gellman dynamo (Bullard & Gellman Reference Bullard and Gellman1954). Owing to computational limitations, the resolution chosen for the spectral scheme was not sufficient to resolve the dissipative structures (current sheets) in the magnetic field. Hence the dissipation was underestimated and dynamo action was claimed when no such sustained magnetic field generation was possible. It is always this way round. Insufficient resolution in a dynamo calculation will lead to the system appearing to be a dynamo when in fact it is not. This is troubling to computational dynamo theorists, although not as troubling to some as it should be.

It should be clear from the above discussion that any misrepresentation of the magnetic diffusion in a dynamo calculation is to be avoided at all costs. This includes not only misrepresentations that arise owing to lack of resolution, but also those that arise owing to the numerical scheme. It is often the case in fluids calculations that sub-grid processes are modelled (say by the inclusion of a hyperdiffusion, a numerical fix or by nominally solving the diffusionless problem with a stable scheme and using numerical errors to take the form of the dissipative processes (see e.g. Miesch Reference Miesch2015). In hydrodynamics, where there may be many competing terms in the nonlinear evolution equation for the velocity and dissipation may be small, such schemes may not be too damaging – but for dynamo calculations extreme care must be taken in interpreting the results from such schemes. Let me stress again that if one is examining the competition between two processes in the linear induction equation, one of which is known to be incorrectly represented, there is a strong chance that the results are incorrect.

4.2.1. Random dynamos - the Kraichnan–Kazantsev formulation

The discussion developed in this section summarises that of Tobias, Cattaneo & Boldyrev (Reference Tobias, Cattaneo and Boldyrev2012). Turbulent velocity fields have a coherent and random component, both of which may contribute to the dynamo properties. The simplest model of (kinematic) dynamo action driven by a purely random flow on a range of scales is the so-called Kazantsev model (Kazantsev Reference Kazantsev1968). It is an example of a solvable model for the statistics of the magnetic field based on those for a prescribed velocity, and as such may be viewed as an early example of direct statistical simulation (see § 10).

The prescribed velocity takes the form of a Gaussian, $\delta$-correlated (in time) velocity field, with zero mean and a covariance given by $\langle u_i({\boldsymbol x+r}, t)u_j({\boldsymbol x}, \tau ) \rangle =\kappa _{ij}({\boldsymbol x},{\boldsymbol r})\delta (t-\tau )$. Geophysical and astrophysical flows that lead to dynamo action do have means, are in general anisotropic and inhomogeneous (and this should be a feature of any description of dynamo action). However, analytic progress can be made for the dynamo problem by assuming that the underlying flow is isotropic and homogeneous, in which case the velocity correlation function has the form

(4.4)\begin{equation} \kappa_{ij}({\boldsymbol r})=\kappa_N(r)\left(\delta_{ij}-\frac{r_ir_j}{r^2} \right)+\kappa_L(r)\frac{r_ir_j}{r^2},\end{equation}

where $r = |{\boldsymbol r}|$. In addition, for incompressible velocity fields we have that $\kappa _N=\kappa _L+(r\kappa _L^\prime )/2$, and so the velocity statistics can be characterised by the single scalar function $\kappa _L(r)$. Progress is made by defining a corresponding expression for the magnetic covariance $\langle B_i({\boldsymbol x+r}, t)B_j({\boldsymbol x}, t) \rangle =H_{ij}({\boldsymbol x},{\boldsymbol r}, t),$ where for similar reasons to above

(4.5)\begin{equation} H_{ij}=H_N(r,t)\left(\delta_{ij}-\frac{r_ir_j}{r^2} \right)+H_L(r,t)\frac{r_ir_j}{r^2}.\end{equation}

Similarly, $\boldsymbol {\nabla } \boldsymbol {\cdot } {\boldsymbol B}=0$ gives $H_N=H_L+(rH_L^\prime )/2$, and so the correlator is completely determined by $H_L(r,t)$. The evolution equation for $H_L$, in terms of the input function $\kappa _L(r)$, follows from deriving the equation for the magnetic correlator

(4.6)\begin{equation} {\partial_t H}_L=\kappa H_L^{\prime \prime} + \left(\frac{4}{r}\kappa+\kappa' \right)H_L^{\prime} +\left( \kappa''+\frac{4}{r}\kappa' \right)H_L, \end{equation}

where $\kappa (r)=2\eta +\kappa _L(0)-\kappa _L(r)$ is the ‘renormalised’ velocity correlation function. Remarkably, changing variable via $H_L=\psi (r,t) r^{-2}\kappa (r)^{-1/2}$ leads to a related equation that formally coincides with the Schrödinger equation in imaginary time, i.e.

(4.7)\begin{equation} \partial_t \psi=\kappa(r)\psi''-V(r)\psi.\end{equation}

Here, $\psi$ describes the wave function of a quantum particle with variable mass, $m(r)=1/[2\kappa (r)]$, in a 1-D potential ($r>0$) given by

(4.8)\begin{equation} V(r)=\frac{2}{r^2}\kappa(r)-\frac{1}{2}\kappa''(r)-\frac{2}{r}\kappa'(r)- \frac{(\kappa'(r))^2}{4\kappa(r)}. \end{equation}

Equation (4.7) has been investigated in detail for various choices of prescribed $\kappa (r)$ (see Ruzmaikin, Sokoloff & Zeldovich Reference Ruzmaikin, Sokoloff and Zeldovich1990; Chertkov et al. Reference Chertkov, Falkovich, Kolokolov and Vergassola1999; Arponen & Horvai Reference Arponen and Horvai2007). For a more thorough review see Tobias et al. (Reference Tobias, Cattaneo and Boldyrev2012) and Rincon (Reference Rincon2019), although the main results are summarised below.

A homogeneous, isotropic turbulent flow with a wide range of scales, and a well-defined inertial range can be characterised by the second-order structure function $\varDelta _2(r) = \langle |({\boldsymbol u}({\boldsymbol x},t) - {\boldsymbol u}({\boldsymbol x}+{\boldsymbol r},t))\boldsymbol{\cdot}{\boldsymbol e}_r|^2 \rangle$, where ${\boldsymbol e}_r={\boldsymbol r}/r$. The inertial and dissipative ranges are then described by the scaling exponents of $\varDelta _2(r)$ with $\varDelta _2(r) \sim r^{2 \alpha }$, where $\alpha$ is termed the roughness exponent of the flow. In the dissipative sub-range we expect $\alpha =1$ as the velocity is a smooth function of position, whilst for the inertial range the velocity is rough and $\alpha < 1$ – for Kolmogorov turbulence $\alpha = 1/3$. If the slope of the energy spectrum in the inertial range is given by $E_k \sim k^{-p}$ then $p$ is related to the roughness exponent by $p = 2 \alpha +1$.

We shall briefly describe dynamo action in the $Pm \gg 1$ case, where the the velocity is smooth on the dissipative scale of the field, and the $Pm \ll 1$ case where the velocity is rough there. For the smooth case, exponentially growing solutions of (4.6) can be found with the magnetic energy spectrum $E_M$ peaked at the magnetic dissipation scale, just as for the single-scale flows considered earlier. The spectrum for the magnetic energy in the range $1/l_\eta < k < 1/l_\nu$ has $E_M \sim k^{3/2}$, independent of the spectral index for the velocity in the inertial range (Kulsrud & Anderson Reference Kulsrud and Anderson1992). This regime for a smooth velocity is also referred to as the Batchelor regime, as it corresponds to that of Batchelor (Reference Batchelor1959) for passive-scalar advection.

The low $Pm$ case is more complicated, and we shall not go into much detail here. Briefly, when $Pm \ll 1$, the magnetic field dissipates in the inertial range, where $\kappa (r) \sim r^{1+\alpha }$ (see e.g. Tobias et al. Reference Tobias, Cattaneo and Boldyrev2012). Hence in the inertial range the Schrödinger (4.7) has an effective potential with the following properties. At small scales the effective potential is regularised by magnetic diffusion; growing dynamo solutions correspond to bound states for the wave function $\psi$. These are guaranteed to exist when $0 < \alpha < 1$. Hence, dynamo action is always possible even in the case of a rough velocity at low $Pm$ (Boldyrev & Cattaneo Reference Boldyrev and Cattaneo2004). However, it is important to note that the effective potential always remains $\propto 1/r^2$ in the inertial range; its depth decreases as $\alpha \rightarrow 0$. Hence, it is harder to drive dynamo action the rougher the velocity – this has serious consequences for our ability to generate magnetic fields in liquid metal experiments, as discussed in § 9.

Equation (4.7) can be solved asymptotically and solutions demonstrate that the growth time of the dynamo is of the order of the turnover time of the eddies at the diffusive scale ($\ell _\eta$). Moreover, it shows (Boldyrev & Cattaneo Reference Boldyrev and Cattaneo2004) that there is a dramatic increase in the effort (computational or experimental) as the velocity becomes rougher (1+$\alpha$ gets smaller). Hence the effort required to drive a dynamo in a rough velocity also increases.

For these random flows we may now describe the behaviour of the critical magnetic Reynolds number $Rm_c$ as one moves from large $Pm$ to small $Pm$. At high $Pm$ the effort necessary to drive a dynamo is modest. As $Pm$ decreases through unity the viscous scale becomes smaller than the resistive scale and the dynamo begins to operate in the inertial range. There is then an increase in the effort required to sustain dynamo action. However, once the dynamo is in the inertial range, further decreases in $Pm$ do not make any difference to the effort required (as measured by $Rm_c$). We also note here that the Kazantsev model relates to the addition of multiplicative noise in the induction equation. Mathematically this case was considered by, for example, Laval et al. (Reference Laval, Blaineau, Leprovost, Dubrulle and Daviaud2006) who were interested in the effects of turbulence on the dynamo threshold. They showed that there are two regimes – an intermittent regime, with a power law distribution for $B$ and at higher $Rm$, a second threshold where the field saturates towards a stable non-zero value.

This picture is largely consistent with numerical models of dynamo action in random flows (Yousef, Brandenburg & Rdiger Reference Yousef, Brandenburg and Rdiger2003; Schekochihin et al. Reference Schekochihin, Cowley, Maron and McWilliams2004, Reference Schekochihin, Haugen, Brandenburg, Cowley, Maron and McWilliams2005) as we shall see in the next section.

The Kazantsev model as proposed is extremely restrictive, although it can easily be extended to cases where the correlation time of the turbulence is finite (Vainshtein & Kichatinov Reference Vainshtein and Kichatinov1986; Kleeorin, Rogachevskii & Sokoloff Reference Kleeorin, Rogachevskii and Sokoloff2002), provided the growth time of the dynamo is long compared with this correlation time. Anisotropic versions of the Kazantsev model have also been constructed by Schekochihin et al. (Reference Schekochihin, Cowley, Maron and McWilliams2004b). These solvable models will also prove useful in understanding the generation of organised field (as discussed in § 5.3.4).

4.2.2. Numerical solutions of random dynamos – the low $Pm$ problem

Owing to the importance of understanding how dynamos onset in turbulent flows at low $Pm$ for experiments (see § 9), there has been much numerical effort in this direction. These numerical calculations are fraught with difficulty as extremely large calculations are required to capture the separation of spatial scales. Tobias et al. (Reference Tobias, Cattaneo and Boldyrev2012) calculate that the resolution of a numerical model required to answer the question of the behaviour of the critical value of $Rm$ for dynamo action at low $Pm$ is beyond the reach of current computational resources (requiring a calculation of size $10\ 000^3$ points). However, numerical calculations are beginning to yield some indication of the role of low $Pm$.

In order to relate the Kazantsev models to numerical kinematic numerical simulations, where the Navier–Stokes equations are solved (with no Lorentz force) to provide the input to the induction equation, the Kazantsev models have been extended take into account departures from Gaussianity in the flow, with similar conclusions being drawn as for Gaussian flows. The results are summarised in figure 7, which shows $Rm_c$ as a function of $Re$ for a collection of such calculations (Schekochihin et al. Reference Schekochihin, Iskakov, Cowley, McWilliams, Proctor and Yousef2007). At large and moderate $Pm$ these results are consistent with the predictions of the Kazantsev model described above, as shown in figure 7. As noted above, calculations rapidly becomes prohibitive at small $Pm$, and so this regime is not really accessible to direct numerical simulation (DNS), unless large-eddy simulations (LES) are utilised (Ponty et al. Reference Ponty, Mininni, Pinton, Politano and Pouquet2007). However, care must be taken here – in this regime the dynamo growth is controlled by the roughness exponent of the flow and so one would require an LES scheme with exceptional representation of the roughness.

Figure 7.

Onset of dynamo action at moderate $Pm$, from Schekochihin et al. (Reference Schekochihin, Iskakov, Cowley, McWilliams, Proctor and Yousef2007). (a) Growth rate of magnetic energy as a function of $Pm$ for five values of $Rm$. (b) Growth/decay rates in the parameter space $(Re, Rm)$. Also shown are the interpolated stability curves $Rm_c$ as a function of $Re$ based on the Laplacian. Hyperviscous runs are shown separately.

Turbulence is, however, significantly more complicated than random flows that are $\delta$-correlated in time, limiting the applicability of the Kazantsev model. For these flows it should be the case that characteristics other than the spectral slope of the velocity (and hence the roughness exponent) control the dynamo growth. We discuss this case in the next section.

4.2.3. Flows with coherence

In this section we consider the case more relevant to geophysical and astrophysical flows, where the flow has two components, a random component as described above and a systematic component whose coherence time is long compared with its turnover time. Such coherent structures, such as long-lived vortices, are ubiquitous in flows where rotation and stratification are important. The structures also exist on a wide range of spatial scales and so it is natural to ask what determines the kinematic dynamo growth rate in a multi-scale flow with coherent structures.

The theory for such flows (which do not now have short correlation times) was developed by Tobias & Cattaneo (Reference Tobias and Cattaneo2008a,Reference Tobias and Cattaneob). This was achieved in two stages, the first step involved demonstrating that for such flows, knowledge of the form of the spectrum is not enough to determine the dynamo properties. They considered a flow with both long-lived vortices and a random component with a well-defined spectrum. They took advantage of the G.O. Roberts trick for dynamos with $2 \frac {1}{2}$-D velocities, by synthesising the flows from solutions to the active scalar equations. They also generated a second (random) flow with the same spectrum as the first by randomising the phases of the spectral components. They found that the flow with coherent long-lived structures is a much more efficient dynamo than the flow that is purely random. Here, by efficient we mean that at the same $Rm$ the dynamo growth rate is higher for the coherent flow. The sustained systematic stretching from the long-lived coherent structures is pivotal in generating field efficiently. In particular, helical vortices are able to generate magnetic field structures – in this case the field generated takes the form of a helix reminiscent of that generated by a Ponomarenko dynamo.

It is now reasonable to assume that a multi-scale kinematic dynamo will be dominated by the coherent parts of the flow – the long-lived structures (rather than the temporally $\delta$-correlated random components) will determine the dynamo growth rate and the form of the field. So, if a dynamo consists of a superposition of such coherent eddies with a range of spatial scales and turnover times (all shorter than their correlation time), what factors determine the small-scale dynamo growth rate?

Progress can be made by assuming that each eddy acts as dynamo largely independently of the eddies at very different scales. This assumption was validated in a model problem by Cattaneo & Tobias (Reference Cattaneo and Tobias2005). A further step is to assume that each of the dynamo eddies are ‘quick dynamos.’ As defined by Tobias & Cattaneo (Reference Tobias and Cattaneo2008a), a ‘quick dynamo’ is one that reaches a neighbourhood of its maximal growth rate quickly as a function of $Rm$. Here, ‘quickly’ is a somewhat imprecise term – although a rule of thumb would be for the growth rate to come within $10\,\%$ of its maximum for $Rm \sim 10 Rm_c$; by this definition most dynamos are quick dynamos. If this is the case, then the dynamo is driven by the coherent eddy which has the shortest turnover time $\tau$ and has $Rm \geq { {O}}(1)$. Both the local $Rm$ and $\tau$ are functions of the spatial scale and so the location in the spectrum of the eddy responsible for dynamo action depends on the spectral slope as well as the magnetic Reynolds number on the integral scale.

The difference between dynamos that are dominated by random components and those that have long-lived coherent structures has recently been confirmed by Seshasayanan, Dallas & Alexakis (Reference Seshasayanan, Dallas and Alexakis2017a). They considered the onset of dynamo action in a randomly forced flow subject to the effects of rotation. As the rotation is increased, the flow develops more spatial and temporal coherence with the coherent vortices eventually winning out over the random element of the flow. This has the effect of reducing the critical $Rm$ at which dynamo action occurs, even at low $Pm$ – the coherence engendered by the rotation turns a low $Pm$ dynamo into a high $Pm$ dynamo as predicted by Tobias & Cattaneo (Reference Tobias and Cattaneo2008a).

5.1.1. Spatial and temporal averaging

Perhaps the most utilised form of averaging assumes that the fluctuations are characterised by a length scale $\ell _0$ (perhaps the scale of the energy containing eddies or fluctuating magnetic field). This is assumed small compared with the larger scale $L$ of the variation in the averaged quantities. For a system-scale dynamo $L$ will typically be ${ {O}}(R)$ where $R$ is the length scale of the domain. It is then possible to define an intermediate scale $a$ that satisfies $\ell _0 \ll a \ll L$. The spatial average can then be defined as (Moffatt Reference Moffatt1978)

(5.7)\begin{equation} {\overline{\boldsymbol B}} \equiv \langle {\boldsymbol B} \rangle_a \equiv \frac{3}{4 {\rm \pi}a^3}\int_V {\boldsymbol B}({\boldsymbol x}+{\boldsymbol \xi},t){\rm d}^3 {\boldsymbol \xi}, \end{equation}

where $V$ is a sphere of radius $a$. A fairly drastic, although perhaps also very natural, example of spatial averaging is to average over one spatial coordinate. For example the butterfly diagram of figure 2 is constructed by averaging the surface sunspot data over longitude and plotting the averaged field as a function of latitude and time. For this type of averaging there is a natural separation of spatial scales, with the axisymmetric component of field formally being separated in azimuthal wavenumber space from the non-zero wavenumbers. This type of ‘zonal’ averaging is often utilised in planetary and stellar physics where zonal symmetry of the underlying turbulent statistics is a natural assumption (see e.g. Braginskii Reference Braginskii1975). In local Cartesian numerical models, we shall see that it is sometimes natural to average over horizontal coordinates.

A similar separation and averaging procedure is available if the quantity to be averaged varies on two time scales. For example if the fluctuations have a characteristic short time scale $t_0$ (perhaps as defined by the correlation time) and the averaged quantities vary on a long time scale $T$ then one can average over an intermediate time scale $\tau$ with $t_0 \ll \tau \ll T$ by defining

(5.8)\begin{equation} {\overline{\boldsymbol B}} \equiv \langle {\boldsymbol B} \rangle_\tau \equiv \frac{1}{2 \tau}\int_{-\tau}^{\tau} {\boldsymbol B}({\boldsymbol x},t+\tau')\, \textrm{d} \tau'. \end{equation}

5.1.2. Ensemble averaging

All of the averaging processes described above rely on some separation of scales (either spatial or temporal) between the average quantities and the fluctuations (this occurs naturally in the case of coordinate averaging). In most turbulent systems, this separation is difficult to achieve, and there is energy at all scales. Whether this really is a difficulty for the theory is open to debate (see e.g. Brandenburg & Subramanian Reference Brandenburg and Subramanian2005). A more natural averaging procedure, from a statistical viewpoint, is to take an average over realisations of the turbulence – a so-called ensemble average. If the flow is ergodic then this method of averaging should (in combination with the underlying symmetries of the turbulence) yield the same answers as temporal or spatial averaging in the relevant asymptotic limits (Moffatt Reference Moffatt1978).

5.1.3. Averaging by filtering and general quasilinear (GQL) averaging

A discussion of averaging procedures would not be complete without including filtering, either spectral or Gaussian (Germano Reference Germano1992). These filters are chosen to smooth small-scale spatial features and are often representative of the action of observations at finite spatial resolution. It is worth noting that such procedures typically do not satisfy the Reynolds rules of averaging; for example Gaussian filtering does not satisfy (5.3) whilst spectral filtering typically does not satisfy (5.6). However, Gaussian filtering can be used in dynamo calculations (see e.g. Hollins et al. Reference Hollins, Sarson, Shukurov, Fletcher and Gent2017) whilst spectral filtering can be made to satisfy the Reynolds averaging rules by restricting the interactions to remove triad decimation in pairs (Kraichnan Reference Kraichnan1985). This filtering forms the basis of the generalised quasilinear approximation introduced by Marston, Chini & Tobias (Reference Marston, Chini and Tobias2016).

5.3.1. Deductions from linearity

Clearly, calculation of the EMF is possible from evaluation of ${\boldsymbol b}'$ (recalling that the velocity field is prescribed in the kinematic regime), so the solution of (5.13) is desirable. However, as noted previously, the solution of this equation together with that of the mean equation is equivalent to solving the full problem. It is, therefore, useful to see what can be learned from the structure of (5.13). This equation is formally a linear equation for ${\boldsymbol b}'$, with a forcing term given by $\boldsymbol {\nabla } \times ({\boldsymbol u}' \times {\overline{\boldsymbol B}})$. Solutions to this equation for ${\boldsymbol b}'$ are linearly (although not homogeneously) related to $\overline{\boldsymbol B}$; that is

(5.15)\begin{equation} {\boldsymbol b}' = {\boldsymbol b}'_{ss}+{\mathcal{L}}(\overline{\boldsymbol B}). \end{equation}

Here, ${\mathcal {L}}$ is a linear operator and ${\boldsymbol b}'_{ss}$ is that fluctuation field that exists in the absence of a mean field $\overline{\boldsymbol B}$. At high $Rm$ this ‘small-scale dynamo field’ is inescapable as we have discussed at length above. However, it is not clear that the presence of this field poses any difficulty for the calculation of the EMF. Clearly ${\boldsymbol b}'_{ss}$ is, by definition, that field that exists in the absence of any large-scale field $\overline{\boldsymbol B}$. It seems unlikely therefore that, except in extremely contrived situations, it can contribute to an EMF at high $Rm$; since its contribution to the EMF would inevitably drive a large-scale field, which is supposed absent. In the nonlinear regime, this argument clearly does not hold (as the small-scale dynamo subspace may saturate and become unstable to large-scale perturbations as we shall discuss later). Indeed some mechanisms for large-scale field generation rely on interactions arising nonlinearly from a saturated small-scale dynamo (Squire & Bhattacharjee Reference Squire and Bhattacharjee2016).

Hence, it seems that, although the fluctuation dynamo field in the kinematic regime is not linearly and homogeneously related to $\overline{\boldsymbol B}$, the EMF is. This is not to say that the presence of small-scale dynamo action is irrelevant; indeed, as we shall see in § 5.4.2, in the kinematic regime at high $Rm$ the growth rate of the dynamo (at both large and small scales) is completely controlled by the presence of a small-scale dynamo (Nigro et al. Reference Nigro, Pongkitiwanichakul, Cattaneo and Tobias2017).

The linear dependence of the EMF on the averaged field suggests an integral representation of the form

(5.16)\begin{equation} {\mathcal{E}}_i({\boldsymbol x},t) = \int \int K_{ij}({\boldsymbol x},t;{\boldsymbol \xi},\tau) \overline{{B}_j}({\boldsymbol x}+{\boldsymbol \xi},t+\tau) \, {\rm d}^3 {\boldsymbol \xi}\, {\rm d} \tau, \end{equation}

for some kernel $K_{ij}$.

If a separation of spatial scales (for example) also pertains then it is possible to expand $\overline{\boldsymbol B}$ in a Taylor series, i.e.

(5.17)\begin{equation} \overline{{B_i}} ({\boldsymbol x+{\boldsymbol \xi}}) = {\overline{B_i}} ({\boldsymbol x}) + \xi_j \frac{\partial {\overline{B_i}}}{\partial x_j}({\boldsymbol x}) + \frac{1}{2} \xi_j \xi_k \,\frac{\partial^2 {\overline{B_i}} }{\partial x_j \partial x_k}({\boldsymbol x}) + \cdots. \end{equation}

For scale separation, $|{{\boldsymbol \xi }}|$ is small compared with a typical scale of $\overline{\boldsymbol B}$, and higher-order terms may be neglected. As noted by Moffatt (Reference Moffatt1978) terms in the Taylor expansion involving time derivatives of the mean field can be written in terms of the mean field and spatial derivatives through back substitution into (5.10).

It is therefore natural, having made this approximation, to postulate a series expansion of the EMF in terms of spatial gradients of the average field of the form

(5.18)\begin{equation} \mathcal{E}_i = \alpha_{ij} {\overline{B_j}} + \beta_{ijk} \frac{\partial {\overline{B_j}}}{\partial x_k}+\cdots ,\end{equation}

where the coefficients $\alpha _{ij}$ and $\beta _{ijk}$ are known as turbulent transport coefficients.

In general, $\alpha _{ij}$ and $\beta _{ijk}$ need to be calculated from the turbulent flow statistics (usually this requires some approximations as we shall see). However, some general statements about their form can be made immediately from (5.18).

We note that $\boldsymbol {\mathcal {E}}$ is a polar vector, whereas $\overline{\boldsymbol B}$ is an axial vector. From this we can immediately deduce that $\alpha _{ij}$ is a pseudo-tensor, which can be decomposed into symmetric and anti-symmetric parts as

(5.19)\begin{equation} \alpha_{ij} = \alpha_{ij}^s - \epsilon_{ijk} \gamma_k. \end{equation}

Hence, the ‘$\alpha$-term’ in (5.18) takes the form $\alpha _{ij}^s{}{\overline{B_j}} + ({\boldsymbol \gamma } \times \overline{{\boldsymbol B}})_i$. The anti-symmetric part of the $\alpha$-tensor therefore contributes an extra mean velocity to the mean-field equations. The symmetric part of the $\alpha$-tensor obviously depends on the flow. In a system with preferred directions, given by say gravity ${\boldsymbol g}$ and rotation ${\boldsymbol \varOmega }$, it is given by Krause & Raedler (Reference Krause and Raedler1980)

(5.20)\begin{equation} \alpha_{ij} = \alpha_1 \delta_{ij} \hat{\boldsymbol g} \boldsymbol{\cdot} \hat{{\boldsymbol \varOmega}} + \alpha_2 \hat{g}_i \hat{\varOmega}_j + \alpha_3 \hat{g}_j \hat{\varOmega}_i. \end{equation}

This form is useful; however, more enlightening is to consider the case where the turbulence is statistically isotropic (although still not reflectionally symmetric). In this case

(5.21)\begin{equation} \alpha_{ij} = \alpha \delta_{ij}, \end{equation}

with ${\boldsymbol \gamma }=0$. The constant $\alpha$ is a pseudo-scalar, which can only be non-zero if the turbulence lacks reflexional symmetry.

Similar considerations can lead to progress in understanding the second term in the expansion. For the simple case of isotropic turbulence, the $\beta _{ijk}$-tensor is also isotropic and so

(5.22)\begin{equation} \beta_{ijk} = \beta \epsilon_{ijk}. \end{equation}

Here, $\beta$ is a pure scalar (i.e. it may be non-zero even for turbulence with no broken symmetry).

I stress here that, in general, these tensors (even if deemed to be useful) will not be homogeneous or isotropic. The presence of rotation, stratification and mean flows tend to give a preferred direction to the turbulence and the tensors will not take a particularly simple or enlightening form. There is now a significant body of work ascribing importance to the generation of an EMF via particular parts of the tensors arising from different physical interactions. I shall argue later that what really matters are the low-order statistics of the MHD turbulence (in this case the EMF itself). Little progress can be made by ascribing and naming effects.

However, it is still worth following the theory through to its logical conclusion; it remains to determine the transport coefficients $\alpha _{ij}$ and $\beta _{ijk}$, ideally in terms of the prescribed flow. There have been many attempts at this that we shall summarise in § 5.3.3. Before doing this, we shall briefly examine why this theory has proven so popular by demonstrating solutions to the kinematic mean-field equations.

5.3.2. Solution of the mean-field (filtered) equations

With the assumption that the transport coefficients are isotropic, so that $\alpha _{ij} = \alpha \delta _{ij}$ and $\beta _{ijk} = \beta \epsilon _{ijk}$ the mean-field dynamo equations take the form

(5.23)\begin{equation} \frac{\partial \overline{\boldsymbol B}}{\partial t} = \boldsymbol{\nabla} \times (\overline{\boldsymbol U} \times \overline{\boldsymbol B}) +\boldsymbol{\nabla} \times (\alpha \overline{\boldsymbol B}) - \boldsymbol{\nabla} \times (\beta \boldsymbol{\nabla} \times \overline{\boldsymbol B}) +\eta \nabla^2 {\overline{\boldsymbol B}}. \end{equation}

If $\beta$ is constant, $\boldsymbol {\nabla } \times (\beta \boldsymbol {\nabla } \times \overline{\boldsymbol B}) = - \beta \nabla ^2 {\overline{\boldsymbol B}}$ so the equation becomes

(5.24)\begin{equation} \frac{\partial \overline{\boldsymbol B}}{\partial t} = \boldsymbol{\nabla} \times (\overline{\boldsymbol U} \times \overline{\boldsymbol B}) +\boldsymbol{\nabla} \times (\alpha \overline{\boldsymbol B}) +(\eta+\beta) \nabla^2 {\overline{\boldsymbol B}}. \end{equation}

We can now see the role of the transport coefficients clearly for this choice of isotropic turbulence. The $\boldsymbol {\nabla } \times (\alpha \overline{\boldsymbol B})$ term is an inductive term leading to the generation of magnetic field (we shall spell this out in the next section) whilst for this simple configuration the $\beta$ term acts like an enhanced diffusivity. We note that other components of the $\beta$ tensor can act so as to generate field for non-isotropic flows. Given that there are 27 of these coefficients, it is fair to say that the role of them all has yet to be established. It is not even clear that it is desirable so to do, as argued above.

The adoption of the mean-field dynamo equations circumvents Cowling's theorem! All the non-axisymmetric parts of the dynamo are encoded in the small scales, which are not solved for. Hence axisymmetric solutions for the mean fields are allowed (and informative). Combining (5.24) with the axisymmetric formalism equations given in (2.26)–(2.27) one derives the axisymmetric mean-field dynamo equations

(5.25)\begin{gather} \frac{\partial A}{\partial t} + \frac{1}{s} ({\boldsymbol u}_P \boldsymbol{\cdot} \boldsymbol{\nabla} ) (sA) = \alpha B + \eta_T \left(\nabla^2 - \frac{1}{s^2}\right) A, \end{gather}

(5.26)\begin{gather}\frac{\partial B}{\partial t} + s ({\boldsymbol u}_P \boldsymbol{\cdot} \boldsymbol{\nabla} ) \left(\frac{B}{s}\right) = \boldsymbol{\nabla} \times (\alpha{\boldsymbol B}_P) \boldsymbol{\cdot} \hat{{\boldsymbol \phi}} + \eta_T \left(\nabla^2 - \frac{1}{s^2}\right) B + s{{\boldsymbol B}_P} \boldsymbol{\cdot} \boldsymbol{\nabla} \varOmega, \end{gather}

where now it is understood that $A$ and $B$ represent the poloidal and toroidal parts of the mean magnetic field fields and $\eta _T = \eta + \beta$.

Cowling's antidynamo theorem therefore is defeated by the presence of a source term $\alpha B$ in the equation for the poloidal field. This source term, which is now called the $\alpha$-effect had been previously derived by Parker (Reference Parker1955). In the equation for the evolution of the azimuthal field both the shear (the $\omega$-effect) and the $\alpha$-effect can operate to generate this component of the field from the meridional field. Depending on the relative importance of the $\alpha$ and $\omega$ effects, mean-field dynamos may be termed $\alpha ^2$, $\alpha \omega$ or $\alpha ^2 \omega$ dynamos. There are two ways of generating azimuthal field $B$ from poloidal field ${\boldsymbol B}_P$: the $\alpha$-effect or the $\omega$-effect. If the $\alpha$-effect dominates, the dynamo is called an $\alpha ^2$-dynamo. If the $\omega$-effect dominates it is an $\alpha \omega$ dynamo. We can also have $\alpha ^2 \omega$ dynamos where both mechanisms operate equally.

The importance and utility of the mean-field ansatz can be seen by analysing the kinematic mean-field dynamo solutions in a 2-D local Cartesian geometry. We seek solutions independent of $y$, with no meridional flow and constant shear, $U'$, in the $\alpha \omega$ limit (i.e. we set $\boldsymbol {\nabla } \times (\alpha {\boldsymbol B}_P) \boldsymbol {\cdot } \hat {{\boldsymbol \phi }} = 0$), so that

(5.27a,b)\begin{gather} {\boldsymbol B} = \left(- \frac{\partial A}{ \partial z}, B, \frac{\partial A}{ \partial x} \right) , \quad {\boldsymbol u} = (0, U' z, 0), \end{gather}

(5.28a,b)\begin{gather}\frac{\partial A}{\partial t} = \alpha B + \eta_T \nabla^2 A, \quad \frac{\partial B}{\partial t} = J(A,U'z) + \eta_T \nabla^2 B, \end{gather}

where $J(\,f,g) = \,f_x g_z - g_x \,f_z$. We seek wave-like solutions for $A$ and $B$ proportional $\exp (\sigma t + i {\boldsymbol k} \boldsymbol {\cdot } {\boldsymbol x})$, with ${\boldsymbol k}= (k_x,0,k_z)$ to yield a dispersion relation of the form

(5.29)\begin{equation} (\sigma + \eta_T k^2)^2 = {\rm i} k_x \alpha U', \end{equation}

with $k^2 = k_x^2+k_z^2$, and so

(5.30)\begin{equation} \sigma = \frac{1+{\rm i}}{\sqrt{2}} (\alpha U' k_x)^{1/2} - \eta_T k^2. \end{equation}

In a finite Cartesian domain of size $d$ in the $z$-direction, one sets $k_z= {\rm \pi}/ d$ and large-scale dynamo action will occur if the dimensionless dynamo number $D = \alpha U' d^3/ \eta _T^2$ exceeds a threshold. The bifurcation to large-scale dynamo action occurs in a Hopf bifurcation to travelling waves. If $D > 0$ the waves travel in the negative $x$-direction, whilst if $D < 0$ they travel in the positive $x$-direction.

At this point it is immediately clear why mean-field theory has proved so popular. For an axisymmetric (indeed 1-D) model the theory can yield travelling wave solutions reminiscent of the waves of sunspot activity seen in the solar observations. The theory can easily be extended to bounded Cartesian domains – although care must be taken in carrying out the instability calculations owing to the difference between convective and absolute instabilities (Bassom & Gilbert Reference Bassom and Gilbert1997; Tobias, Proctor & Knobloch Reference Tobias, Proctor and Knobloch1997), and to spherical domains (Roberts Reference Roberts1972b).

5.3.3. Calculations of kinematic transport coefficients: theory and computation

We described above how solution of the mean-field equations can lead to large-scale magnetic fields for simple choices of the transport coefficients. We also described in § 5.3.1 that basic properties of the transport coefficients ($\alpha _{ij}$ and $\beta _{ijk}$) can be determined via consideration of symmetries (and on making the assumptions of homogeneity).

However, it is important to calculate these transport coefficients for solvable and realistic models of turbulence (both in the kinematic and dynamic regime). We begin by describing the kinematic evaluation of these coefficients – this is where the theory gets controversial, as all the current analytical calculations of transport coefficients necessarily involve some form of approximation.

First-order smoothing:

The simplest ansatz in which to calculate the transport coefficients in the kinematic regime invokes the quasilinear approximation. This involves the discard of the fluctuation/fluctuation $\rightarrow$ fluctuation interactions in the equation for the fluctuating field; that is we set $\boldsymbol {\mathcal {G}}=0$ in (5.13) to yield

(5.31)\begin{equation} \frac{\partial {\boldsymbol b}'}{\partial t}=\boldsymbol{\nabla} \times (\overline{{\boldsymbol U}} \times {\boldsymbol b}')+\boldsymbol{\nabla} \times ({\boldsymbol u}' \times {\overline{\boldsymbol B}})+\eta \nabla^2 {\boldsymbol b}'. \end{equation}

The discard of the nonlinear term $\boldsymbol {\mathcal {G}}$ can be understood as assuming that the Kubo number (to be defined below) of the MHD turbulence is small (in the sense that the fluctuation/fluctuation interaction is small compared with the mean/fluctuation interaction). This may occur in many circumstances of interest, although the conditions for which the approximation is valid is difficult to determine a priori. Formally, however, this is certainly the case when the fluctuating field is small compared with the mean field i.e. $|{\boldsymbol b}'| \ll |{\overline{\boldsymbol B}}|$. It is also applicable when $Rm \ll 1$ in which case one may also neglect the $\partial {\boldsymbol b}'/\partial t$ term to leave

(5.32)\begin{equation} \eta \nabla^2 {\boldsymbol b}'=-\boldsymbol{\nabla} \times (\overline{{\boldsymbol U}} \times {\boldsymbol b}')-\boldsymbol{\nabla} \times ({\boldsymbol u}' \times {\overline{\boldsymbol B}}), \end{equation}

which can then easily be solved using standard linear transform techniques (see Moffatt (Reference Moffatt1978) for a detailed description) to yield ${\boldsymbol b}'$ and hence, as ${\boldsymbol u}'$ is known, $\boldsymbol {\mathcal {E}}$.

The nonlinear term $\boldsymbol {\mathcal {G}}$ can also be formally neglected a priori if the correlation time $\tau _c$ of the turbulence (which is given in the kinematic regime) is sufficiently short. In particular if the Kubo number $u_{rms} \tau _c/\ell \ll 1$ then (5.31) is justified. If in addition $Rm \gg 1$ then

(5.33)\begin{equation} \frac{\partial {\boldsymbol b}'}{\partial t}\approx\boldsymbol{\nabla} \times (\overline{{\boldsymbol U}} \times {\boldsymbol b}')+\boldsymbol{\nabla} \times ({\boldsymbol u}' \times {\overline{\boldsymbol B}}). \end{equation}

Again, this linear system may be solved by transform methods. In the simplified case with no local mean flow one may write

(5.34)\begin{equation} {\boldsymbol{\mathcal{E}}}(t) = \overline{{\boldsymbol u}(t) \times \int_0^t \boldsymbol{\nabla} \times ( {\boldsymbol u}(t') \times \overline{{\boldsymbol B}(t')} ) \,\textrm{d} t'}. \end{equation}

In a statistically steady state and for isotropic turbulence this simplifies to give

(5.35)\begin{equation} {\boldsymbol{\mathcal{E}}}(t) = \int_0^t (\hat{\alpha}(t-t') \overline{{\boldsymbol B}(t')} - \hat{\eta}_T(t-t') \overline{\boldsymbol{\nabla} \times {\boldsymbol B}(t')})\, \textrm{d} t'. \end{equation}

where $\hat {\alpha }(t-t') = -\frac {1}{3}\overline {{\boldsymbol u}'(t) \boldsymbol {\cdot } {\boldsymbol \omega }'(t')}$ and $\hat {\eta }_T(t-t')= \frac {1}{3}\overline {{\boldsymbol u}'(t) \boldsymbol {\cdot } {\boldsymbol u}'(t')}$.

For a mean field that is sufficiently slowly varying compared with the turbulence the integral kernels can be approximated by $\delta$-functions and so

(5.36)\begin{equation} {\boldsymbol{\mathcal{E}}}(t) = \alpha \overline{{\boldsymbol B}} - \hat{\eta}_T \overline{\boldsymbol{\nabla} \times {\boldsymbol B}}, \end{equation}

with

(5.37)\begin{gather} \alpha = - \tfrac{1}{3} \int_0^t \overline{{\boldsymbol u}'(t) \boldsymbol{\cdot} {\boldsymbol \omega}'(t')} \, \textrm{d} t' \approx - \tfrac{1}{3} \tau_c \overline{{\boldsymbol u}' \boldsymbol{\cdot} {\boldsymbol \omega}'}, \end{gather}

(5.38)\begin{gather}\eta_T = \tfrac{1}{3} \int_0^t \overline{{\boldsymbol u}'(t) \boldsymbol{\cdot} {\boldsymbol u}'(t')} \, \textrm{d} t' \approx \tfrac{1}{3} \tau_c \overline{{\boldsymbol u}' \boldsymbol{\cdot} {\boldsymbol u}'}. \end{gather}

First-order smoothing, which is applied within the kinematic framework, is designed so that the EMF may be related to the mean magnetic field through properties of the prescribed velocity field. The smoothing is such that the fluctuating magnetic field is ‘slaved’ to the velocity field and the mean magnetic field and so the EMF can be readily calculated. At high $Rm$, for correlation times that are not short, this slaving is not maintained and the magnetic fluctuations rapidly become decorrelated with the velocity field. This has two major effects. The first is that the EMF cannot be evaluated explicitly and one must resort to numerical procedures for its calculation (as described in the next section). The second, and more severe, effect is that the loss of correlation can lead to a diminution of the EMF relative to the fluctuations. This may have serious consequences for large-scale dynamo theory in the kinematic regime at high $Rm$ as described below – although, as we shall see, in section there will be some effects that can mitigate the loss of correlation and to some extent rescue the theory.

Numerical evaluation of the transport coefficients:

With the growth of computing power, it is now possible to evaluate the transport coefficients numerically, albeit at moderate $Rm$ for 3-D flows. This is, of course, possible both when the magnetic field is weak and does not act back on the flow and therefore the transport coefficients are in the kinematic regime, or in the fully nonlinear regime so that the effects of the Lorentz force on transport coefficients can be calculated – an important topic discussed in § 6.5.

Constructing a reliable method for the calculation of the transport coefficients is not trivial (especially when $Rm$ is not small) since it involves the calculation of averages and care must be taken to ensure that these averages are converged. Calculation of the $\alpha$-coefficient is conceptually straightforward. Consider (5.12) and (5.18). These are two separate expressions for the EMF. In the case where the averages are spatial, the flow and magnetic field can be decomposed into a spatial mean (in this case a uniform field) and spatial fluctuations. Now a uniform field has no spatial derivatives and so (5.18) simplifies to

(5.39)\begin{equation} \mathcal{E}_i = \alpha_{ij} {\overline{B_j}}. \end{equation}

Hence, it is possible to impose constant mean fields with different orientations and calculate $\mathcal {E}_i = \overline {{\boldsymbol u'} \times {\boldsymbol b'}}$ numerically to identify the different components of the $\alpha$-tensor. For a very turbulent flow at high(ish) $Rm$ it turns out that $\overline {{\boldsymbol u'} \times {\boldsymbol b'}}(t)$ in the kinematic regime may be a rapidly varying timeseries which takes a broad distribution of values with a large variance and a relatively small mean (see e.g. Cattaneo & Hughes (Reference Cattaneo and Hughes2006) who computed the alpha coefficient in a plane layer of rotating Boussinesq convection). This is an indication that the systematic behaviour in the kinematic regime may be in competition with (and perhaps dominated by) the disorganised or fluctuating dynamo behaviour (see § 5.4 for a more complete discussion of this); however, it is fair to say that the relative effectiveness of the fluctuating and organised magnetic fields may depend on both the form of the spectrum of the velocity and the value of $Rm$ as demonstrated by Cattaneo & Tobias (Reference Cattaneo and Tobias2014) – at very high $Rm$ the dynamo is controlled by smaller eddies from further down the spectrum with shorter correlation times, which means that averaging is a more effective procedure and the distribution of the EMF is narrowed. Thus, in addition to spatial averaging, temporal (or ensemble) averaging should also be utilised to ensure well converged statistics.

Computing the turbulent diffusivity $\beta _{ijk}$ is somewhat more problematic since that relates the EMF to gradients of the organised magnetic field. This causes problems as there is no simple method of imposing a magnetic field with a spatial gradient that is not prone to grow exponentially via dynamo action at high $Rm$ in the kinematic regime. Three methods for evaluating the turbulent diffusivity have been suggested; the ‘test field’ method, the ‘turbulent Ångström’ method and the ‘Lagrangian statistics’ method.

The test field method, introduced by Schrinner et al. (Reference Schrinner, Rädler, Schmitt, Rheinhardt and Christensen2005) involves the solution of an auxiliary equation and is a simple extension of the procedure outlined above for the calculation of the $\alpha$-effect. Here the aim is to calculate the EMF for a variety of applied test magnetic fields with different orientations and field gradients. If the test fields are chosen to be constants then the $\alpha$-tensor can be recovered in a similar manner to that described above. In principle the $\beta$-coefficients can be backed out by imposing a series of spatially varying test fields and solving for the fluctuations, and hence the EMF. The test field method has been criticised (Cattaneo & Hughes Reference Cattaneo and Hughes2009) in that it requires a prohibitively large computational cost to evaluate the transport coefficients at high $Rm$. Nonetheless, at the moderate $Rm$ values that are currently achievable for current numerical computations convergence of results does appear achievable if ensemble averages are used in addition to spatial averages.

The turbulent Ångström method (Tobias & Cattaneo Reference Tobias and Cattaneo2013a) is based on that used to measure the thermal conductivity of solids. The method consists of applying an oscillatory source of magnetic field (usually via an oscillating current) with a given frequency larger than the largest natural frequency of the turbulence. Here the turbulent diffusivity can be calculated from the response measured at the frequency of the imposed oscillating field. This method is attractive as the turbulent diffusivity can be calculated to any required precision, simply by analysing a long enough timeseries for the magnetic field, although this may be computationally expensive.

The final method, which involves calculating Lagrangian statistics, is formally valid in the infinite $Rm$ limit and is outlined in Moffatt (Reference Moffatt1978). Briefly, both the $\alpha$ and $\beta$ tensors may be evaluated numerically by calculating averages over Lagrangian trajectories. This method can (and has) been extended to the finite $Rm$ case by adding stochastic fluctuations to the Lagrangian trajectories (Drummond & Horgan Reference Drummond and Horgan1986). This elegant method appears to merit further investigation.

5.3.4. Organised Kazantsev dynamos

We have described above how organised magnetic fields may be expected to emerge for flows that lack reflectional symmetry, and how, under certain assumptions, the evolution of the large-scale field may be determined in the kinematic regime by turbulent transport coefficients that encode the eddy/eddy $\rightarrow$ mean interactions. Even earlier, in § 4.2.1, we also discussed how the Kazantsev model for random (short correlation time) flows could yield insight into the onset of small-scale dynamos.

It turns out that the Kazantsev model can be extended to random flows that lack reflectional symmetry; in this case the velocity and magnetic correlators are defined using two functions – one, as before, related to the energy density (either kinetic or magnetic), the other to the helicity (either kinetic or magnetic) (Vainshtein & Kichatinov Reference Vainshtein and Kichatinov1986). The two parts of the magnetic correlator are then described by a pair of coupled Schrödinger-like equations (Vainshtein & Kichatinov Reference Vainshtein and Kichatinov1986; Berger & Rosner Reference Berger and Rosner1995). The analysis is now more complicated, yet it is possible to demonstrate that the system remains self-adjoint (Boldyrev, Cattaneo & Rosner Reference Boldyrev, Cattaneo and Rosner2005). These authors demonstrated that, for sufficient kinetic helicity, extended states are found that do not decay exponentially at infinity. These are the ‘large-scale dynamo solutions’, analogous to the mean-field solutions. It can, however, be shown that at large $Rm$ the largest growth rates remain associated with the localised bound states, so that the overall dynamo growth rate remains controlled by small-scale dynamo action (Malyshkin & Boldyrev Reference Malyshkin and Boldyrev2008a,Reference Malyshkin and Boldyrevb). This competition between large and small-scale dynamos in the kinematic regime will be the theme of our discussion in the next few sections.

5.4.1. The role of shear: suppressing the small-scale dynamo

Most efforts to promote the large-scale dynamo over the small-scale dynamo involve the addition of a large-scale shear flow. As we have described above, in the absence of shear, the kinematic growth rate is determined by the stretching of the small-scale dynamo. This effect also manifests itself in the distribution of the EMF, which in certain circumstances can be calculated to have a small mean and a large variance caused by the fluctuations. If one is to promote the generation of organised magnetic fields over that of small-scale fields then two strategies immediately suggest themselves; either utilise the shear to boost the mean EMF by introducing new effects (i.e. new terms in the transport coefficients) or utilise the shear to suppress the small-scale dynamo.

Much attention has focussed on the first of these strategies (see e.g. Yousef et al. Reference Yousef, Heinemann, Schekochihin, Kleeorin, Rogachevskii, Iskakov, Cowley and McWilliams2008; Käpylä & Brandenburg Reference Käpylä and Brandenburg2009; Sridhar & Singh Reference Sridhar and Singh2010 among others). Indeed it can be shown that the presence of a prescribed large-scale shear flow can lead to extra terms in the transport coefficients and hence in the EMF. Moreover a shear flow can in principle interact with a weak and highly fluctuating EMF to produce large-scale field (Richardson & Proctor Reference Richardson and Proctor2010). All of these mechanisms can be shown to boost the generation of a mean EMF, in certain circumstances. However, at high $Rm$, if this large-scale field is ever to be seen in the kinematic regime, there must be some mechanism present to suppress the small scales which have a tendency to dominate. It would also help to differentiate the two dynamos if the large-scale dynamo were to exhibit a different dynamics than that of the small scales.

A rather different, and potentially more promising, way to proceed is to recognise that, as indicated by fast dynamo theory, the small-scale dynamo relies on the presence of chaotic stretching to survive at high $Rm$. In order to give the large-scale dynamo a chance (Courvoisier & Kim Reference Courvoisier and Kim2009), it is imperative to suppress the small-scale dynamo by reducing the chaotic stretching in the flow. Such a strategy was proposed by Tobias & Cattaneo (Reference Tobias and Cattaneo2013b). They argued that the addition of a shear flow to a chaotic (or turbulent) flow will increase the regions of integrability in the flow, thus reducing chaos and hence the effectiveness of the small-scale dynamo at high $Rm$. If such a dynamo also has broken reflectional symmetry and is capable of generating an EMF and hence large-scale fields then it is possible that the suppression of the small-scale dynamo will lead to the emergence of large-scale fields. Tobias & Cattaneo (Reference Tobias and Cattaneo2013b) and Cattaneo & Tobias (Reference Cattaneo and Tobias2014) investigated the properties of just such a dynamo. They considered a velocity field in a Cartesian domain that was comprised of a combination of generalised Galloway–Proctor flows imposed at a range of scales together with a large-scale shear flow. In particular they considered a multi-scale generalisation of the cellular Galloway–Proctor flow that is maximally helical and takes the form of an infinite array of clockwise and anti-clockwise rotating helices. The pattern of helices itself rotates in a circle with a scale-dependent frequency and can be decorrelated on a time scale $\tau _d$. The decorrelation time of the flow is also scale-dependent and scales in the same manner as the turnover time. Hence the magnetic Reynolds number is also a function of scale and was chosen to decrease as the spatial scale decreases (see Tobias & Cattaneo Reference Tobias and Cattaneo2013b for details).

To this time-dependent helical multi-scale flow, a steady unidirectional large-scale shear of the form

(5.40)\begin{equation} {\boldsymbol u}_s = (V_0 \sin y, 0, 0),\end{equation}

was added and dynamo solutions were sought (in a similar manner to the calculations for the Galloway–Proctor flow of § 4.1) at high $Rm$.

In the absence of shear, the flow is an excellent small-scale dynamo. For $\chi = Rm/Rm_c \gg 1$ all the magnetic energy is, as expected, concentrated at small scales and generated on a typical time scale of a Galloway–Proctor eddy. There is no sign of a large-scale organised component of the magnetic field. The situation changes significantly when shear is added. Although the small-scale dynamo is still operational, its growth-rate is reduced (see Tobias & Cattaneo Reference Tobias and Cattaneo2013b). At high $Rm$ in the kinematic regime the role of an imposed shear is always to lower the growth rate of a small-scale dynamo. As explained above, this is due to the introduction of regions of integrability to the otherwise chaotic flow. This statement has caused some controversy in the literature, but is correct at high enough $Rm$ since the operation of such a dynamo relies on the presence of chaos in the flow. Dynamos for which the shear increases the dynamo growth rate are either poor small-scale dynamos to start with, or are simply not being investigated at high enough $\chi = Rm/Rm_c$.

Moreover, once the small-scale dynamo is suppressed, it is reasonable to require that the large-scale dynamo can be detected. Figure 8(a), which shows $\overline {B_x}(\,y,t)$ the average (in $x$) of the toroidal (shear-aligned) field at a given $z$-level, demonstrates that this expectation is met. The average field can be seen to take the form of systematic dynamo waves propagating in the regions of strong shear (of the type envisaged by Parker). In the absence of helicity of the small-scale flow, the shear still amplifies fields in bands but there is no systematic generation of spatio-temporally coherent field. That the shear reduces the fluctuations is clear on examination of the probability distribution (as shown by the p.d.f.) for the EMF in figure 8(b). This shows the distribution of EMF as measured by imposing a mean field. In the absence of shear the distribution is dominated by the fluctuations with a large variance and a small mean (as shown by the black curve). As the strength of the shear is increased the variance of the distribution decreases markedly, although the mean shifts barely at all. This demonstrates (for this flow at least) that the crucial role of shear at high $Rm$ in the kinematic regime is to suppress the fluctuations, rather than modify the mean.

Figure 8.

(a) Space–time $(t,y)$ plot of average magnetic field $\overline {B_x}(\,y,t)$ at fixed $z=0$ for dynamo with shear $(V_0=5)$ and small-scale flow with net helicity (after Cattaneo & Tobias Reference Cattaneo and Tobias2014); here $0 \le y < 2 {\rm \pi}$ whilst $0 \le t \le 40$. (b) Probability density function of EMF for a fixed small-scale helical flow and a range of strength of shear flow (defined in (5.40)) $V_0 = 0$ (black), $V_0 = 1$ (red), $V_0 = 2$ (yellow), $V_0 = 5$ (green), $V_0 = 10$ (cyan), and $V_0 = 20$ (blue). As the shear is increased the distribution narrows significantly with a small change in the mean EMF (after Cattaneo & Tobias Reference Cattaneo and Tobias2014) © AAS. Reproduced with permission.

These results led Cattaneo & Tobias (Reference Cattaneo and Tobias2014) to propose a general ‘suppression principle’, namely ‘At high $Rm$ large-scale dynamo action can only be observed if there is a mechanism that suppresses the small-scale fluctuations.’ As discussed earlier, the kinematic framework dynamo theory is linear and the solutions are superposable – large-scale dynamo action can therefore only be observed if the small-scale dynamo is suppressed. In the cases discussed above the shear acted so as to suppress the fluctuations. It is possible, however, to envisage other mechanisms that may act in this way. In most circumstances, however, dynamo action proceeds in the nonlinear regime (see § 6). There one can envisage a nonlinear mechanism that leads to either the suppression of the small-scale dynamo or the saturation of the fluctuating field at a reasonable amplitude. In the saturated regime, whether the large-scale field can be observed depends on the ratio of the saturation amplitude for the large and small scales, which is itself a contentious issue in dynamo theory, as we shall see.

5.4.2. What does mean-field theory get right in the kinematic regime?

Mean-field theory describes the evolution of filtered or averaged quantities. In order for this theory to have utility in the kinematic regime, one would expect that application of the filter to the eigenfunctions of the full equations should correspond to the eigenfunctions of the filtered equations. Moreover, the growth rate of the large-scale structure of the solutions should coincide with the growth rate predicted by the filtered equations. As we have argued above, this second expectation is never met at high $Rm$ – the growth rate is determined by the stretching of the small scales (Tobias & Cattaneo Reference Tobias and Cattaneo2008a).

However, it is possible to construct more complicated cases, for example with the addition of shear, where the filtered (mean-field) equations have a growth rate and a frequency (Parker Reference Parker1955; Tobias & Cattaneo Reference Tobias and Cattaneo2013b). The shear breaks the isotropy of the filtered equations and the isotropy of the statistics of the velocity in the unfiltered system and therefore one expects a breaking of symmetry of the statistics of the solutions. In that case one might hope that the frequency, as manifested by the speed of propagation of the dynamo waves to be the same both in the solutions of the filtered equations and those of the unfiltered equations in a statistical sense. Here, symmetry breaking acts to perturb the frequency away from zero and is controlled by a change in the symmetry of the large scales. Nigro et al. (Reference Nigro, Pongkitiwanichakul, Cattaneo and Tobias2017) considered just such a case based on the Tobias–Cattaneo dynamo of § 5.4.1. They determined a range of solutions with a travelling helical (dynamo) wave, that remains coherent for long periods of time and whose frequency is determined by mean-field effects. The wave could only be identified from the rest of the structure by a persistent phase-coherent signal with the rest of the solution being incoherent in time. They argued from this that it is better to consider a definition of large-scale dynamo action that considers the time scale of evolution of the pattern, rather than one that relies on spatial scales alone, i.e. spatio-temporal averaging is the natural choice for determining organisation. Further, as kinematic large-scale dynamo action consists of a long-lived coherent pattern embedded in a sea of incoherent fluctuations; the filtered equations simply average over the fluctuations. Encouragingly, they predicted that mean-field electrodynamics is capable of predicting the frequency of this coherent signal.

This gives hope for the construction of a sensible framework for deriving a filtered equation in the nonlinear regime. As argued by Nigro et al. (Reference Nigro, Pongkitiwanichakul, Cattaneo and Tobias2017), as nonlinearity becomes important different scales may continue to grow at different rates and saturate at relative amplitudes completely different from those of the kinematic phase, controlling the fluctuations relative to the mean. If this were the case then they argue that the filtering technique used to identify spatio-temporally coherent large-scale fields in the kinematic regime may continue to be utilised in the nonlinear regime. We shall therefore finally (and none too soon for the referee!) move into the dynamic regime, and examine solutions to the coupled Navier–Stokes/induction system.

6.2.1. High $Pm$ dynamos: general considerations and random flows

Just as for the kinematic regime, there is a large difference between high and low $Pm$ regimes with regard to the saturation of dynamos. In the high $Pm$ regime the dynamo is operating at scales in the sub-inertial range of the velocity for which the kinetic Reynolds number is small and the inertial term in the momentum equation can be neglected. The velocity can then formally be decomposed into two parts ${\boldsymbol u} = {\boldsymbol u}_{F} + {\boldsymbol u}_M$ where ${\boldsymbol u}_{F}$ is the original velocity driven by the forcing (either mechanical or buoyancy) i.e. it is the velocity of the kinematic dynamo problem and has a characteristic scale $\gg l_{\eta }$, the other ${\boldsymbol u}_M$ is the magnetically driven velocity. These two components then evolve independently (Brummell et al. Reference Brummell, Cattaneo and Tobias2001), as the inertial terms has been neglected.

To fix ideas, consider the case where the velocity is kinematically driven solely on one scale, say the integral scale of the calculation. As noted in § 4.1. this will lead kinematically to a growing dynamo eigenfunction for the magnetic field that is concentrated at the resistive $Rm^{-1/2}$ scale. If $Rm$ is large (i.e. in the kinematic fast dynamo regime) then this is much smaller than the scale of the velocity. These small-scale filamentary fields now drive flows on scales smaller than the velocity via the Lorentz force, eventually generating strong enough magnetic energy at the velocity scale to modify the original flow. Because of the simplicity of system at low $Re$, estimates for the strength of the generated magnetic field can be obtained where $B^2/U^2 \sim Rm^{1/2} Re^{-1}$, where $B$ is measured in units of the Alfvén velocity. This estimate is borne out at moderate $Rm$, but breaks down at higher $Rm$ (Brummell et al. Reference Brummell, Cattaneo and Tobias2001). At such low $Re$ and $Rm$ the saturation mechanism is simply to drive flows via the Lorentz force that can counteract the advective properties of the driven flow.

For multi-scale velocity fields at moderate and large $Re$ the situation is more complicated and saturation can be investigated using semi-analytical models, phenomenological models and numerical experiments. The semi-analytical models are ultimately based on some closure of the MHD equations. For random flows described by the Kazantsev formalism, the magnetically driven velocity produces a change in the velocity correlator, which leads to the nonlinear saturation (Subramanian Reference Subramanian1999, Reference Subramanian2003). Alternatively a Fokker–Planck equation for the probability distribution function for magnetic fluctuations can be derived. It can be shown that the coefficients of this equation again are determined by the velocity correlation function which can be modified nonlinearly in a similar manner to above (Boldyrev Reference Boldyrev2001). However, all of these results rely on some assumption of the role of the Lorentz force and that the flows remain $\delta$-correlated in time even in the presence of a saturating magnetic field.

At high $Pm$ an interesting phenomenological description for multi-scale dynamos has been proposed (Schekochihin et al. Reference Schekochihin, Cowley, Hammett, Maron and McWilliams2002). The initial kinematic growth of field at the resistive scale (which is much smaller than the viscous scale) is modified when ${\boldsymbol u} \boldsymbol {\cdot } \boldsymbol {\nabla } {\boldsymbol u} \approx {\boldsymbol B} \boldsymbol {\cdot } \boldsymbol {\nabla } {\boldsymbol B}$ at that scale. The left-hand side is easily estimated to be $v^2/l_\nu$. To estimate the right-hand side the authors use the foliated structure of the magnetic field to yield an estimate of $b^2/l_\nu$, where $b$ is a typical field strength at the scale $l_\eta$. The nonlinear saturation therefore begins when the magnetic energy comes into equipartition with the kinetic energy at the viscous scale, and dynamo growth associated with eddies at the viscous scale is halted. This suppression is believed to be the result of a subtle modification of the eddy geometry; the Lorentz force causes the eddies to align with the local magnetic field, thus reducing induction. However, slightly larger eddies can still sustain growth and so growth will continue until the magnetic energy comes into equipartition with the energy of these eddies; this process continues until the magnetic energy reaches equipartition with the kinetic energy at the integral scale. This nonlinear adjustment is characterised by a growth of the magnetic energy on an algebraic (rather than exponential) time with a shift to the generation of magnetic fields on larger scales. It is possible to construct models where the final state only reaches a fraction of global equipartition. This occurs when $Pm$ is large but $Pm \le Re^{1/2}$, and for this regime $B^2/U^2 \approx Pm/Re^{1/2}$ (Schekochihin et al. Reference Schekochihin, Cowley, Hammett, Maron and McWilliams2002). In this scenario, however, the characteristic scale of the saturated magnetic field is still smaller than the viscous scale.

6.2.2. High $Pm$ dynamos: numerical experiments

There have been many simulations of dynamo saturation at moderate to high $Pm$ that seem to confirm the phenomenological picture described above (Maron, Cowley & McWilliams Reference Maron, Cowley and McWilliams2004). In the kinematic phase the magnetic spectrum appears compatible with the $k^{3/2}$ prediction of the Kazantsev model, whilst the magnetic field does indeed appear foliated. Dynamo saturation occurs when the magnetic and kinetic energies are comparable; as the saturation progresses the magnetic spectrum grows and flattens with the creation of magnetic structures at larger scales. The magnetic energy always exceeds the kinetic energy and the field is more intermittent than the velocity, with the p.d.f. for the velocity field remaining close to that of a Gaussian whilst that for the magnetic field is better described by an exponential. The degree of intermittency is, however, reduced in the saturated state as compared with the kinematic state (Cattaneo Reference Cattaneo1999). Interestingly, the high $Pm$ formalism seems to persist for $Pm \sim { {O}}(1)$ (see Tobias et al. (Reference Tobias, Cattaneo and Boldyrev2012) for more details) although quite how well the phenomenological picture is replicated is unclear, owing to numerical limitations. It is clear though that at $Pm \sim { {O}}(1)$ the magnetic fields are more intermittent that the velocity and the local magnetic field introduces statistical anisotropy in the flow in the nonlinear regime.

6.5.1. Catastrophic (Vainshtein–Cattaneo) quenching

Perhaps the greatest challenge to mean-field theory in the dynamic regime arises from arguments proposed in the early nineties by Vainshtein and Cattaneo (Cattaneo & Vainshtein Reference Cattaneo and Vainshtein1991; Vainshtein & Cattaneo Reference Vainshtein and Cattaneo1992) and extended by Gruzinov & Diamond (Reference Gruzinov and Diamond1994). The arguments concern the level of saturation of the organised field that can be generated by an EMF, before the EMF is itself suppressed by the nonlinear effects of Lorentz force in the momentum equation.

One way to proceed would be to assume on energetic grounds that the Lorentz force can only begin to suppress a kinematic effect once the energy in the magnetic field is comparable with that of the turbulence. Traditionally (and naively) it has been assumed that this occurs when the magnetic energy of the mean field is in equipartition with the kinetic energy of the flow, i.e. when

(6.8)\begin{equation} \langle \rho u^2 \rangle \sim B_0^2 / \mu, \end{equation}

where $B_0^2 = \overline{{\boldsymbol B}} \boldsymbol {\cdot } \overline{{\boldsymbol B}}$ is the energy in the mean field. Hence a simple phenomenological model would then be to take the nonlinear dependence of the transport coefficients as

(6.9a,b)\begin{equation} \alpha = \frac{\alpha_0}{1+ B_0^2/ {\mathcal{B}}^2 }, \quad \beta = \frac{\beta_0}{1+ B_0^2/ {\mathcal{B}}^2 },\end{equation}

where $\alpha _0$ and $\beta _0$ are the values of the transport coefficients derived in the kinematic approximation and ${\mathcal {B}}^2$ is the non-dimensional equipartition energy (i.e. normalised with the kinetic energy of the flow). Note that this prescription implicitly assumes that the mean field dominates (or at least is of the same order of magnitude as) the fluctuating field in the saturated regime, and so it is the mean field that leads to saturation of the transport. The expressions in (6.9a,b) are attractive as they lead to a saturation of the mean field for an energy comparable with that of the turbulence. Moreover these expressions rely only on the calculation of the mean or organised fields, which requires little in the way of computational expense.

However, the assumption that the magnetic energy is dominated by the organised component is questionable at high $Rm$ as pointed out clearly by Cattaneo & Vainshtein (Reference Cattaneo and Vainshtein1991) and Vainshtein & Cattaneo (Reference Vainshtein and Cattaneo1992). The distribution of magnetic energy with spatial scale at high $Rm$ is a key issue for understanding nonlinear dynamo saturation. One can proceed on either physical or mathematical grounds to understand this distribution – computational models, although suggestive and informative, are not at the point of unambiguously settling this issue.

On physical grounds, it is argued that if there is a large-scale component of the field generated kinematically in a turbulent dynamo then turbulence will also amplify magnetic fields on the small scale via the action of eddies with short turnover times. Hence one might expect the small-scale fields (kinematically) to be more energetic than those on large scales. If diffusion plays a role in setting the ratio of these energies then one might expect a relationship of the form

(6.10)\begin{equation} \overline{b^2} \sim Rm^p B_0^2 ,\end{equation}

where $b^2 = {\boldsymbol b'}\boldsymbol {\cdot } {\boldsymbol b'}$ and $0 \le p \le 2$ is a flow- and geometry-dependent coefficient.

If this is the case, and assuming that magnetic fields with this energy do distort the eddies with correlations that lead to the generation of organised field (i.e. those with significant broken reflectional symmetry), then this may indicate that the transport coefficients are altered significantly once this strong small-scale field reaches equipartition with the kinetic energy of the turbulence, i.e. when

(6.11)\begin{equation} \langle \rho u^2 \rangle \sim Rm^p B_0^2 / \mu.\end{equation}

One might then expect that the expressions in (6.9a,b) to be replaced by

(6.12a,b)\begin{equation} \alpha = \frac{\alpha_0}{1+Rm^p B_0^2/ {\mathcal{B}}^2 } , \quad \beta = \frac{\beta_0}{1+Rm^p B_0^2/ {\mathcal{B}}^2 } . \end{equation}

At high $Rm$, one can then argue that the efficiency of the transport coefficients is strongly suppressed, even when the organised field $B_0$ is small; a phenomenon known as ‘catastrophic’ quenching (Cattaneo & Vainshtein Reference Cattaneo and Vainshtein1991; Vainshtein & Cattaneo Reference Vainshtein and Cattaneo1992). It certainly seems plausible that the transport coefficients begin to be suppressed when the mean field is weak. However, the generation of a mean EMF relies on correlations in the turbulence between the fluctuating magnetic field and eddies, which is a subtle interplay, so arguments on physical grounds may be misleading. Hence, other approaches to this issue have been employed.

The first of these is to derive analytical expressions for the EMF (and hence the transport coefficients) utilising closure approximations (such as those discussed in § 6.4) and to then apply conservation laws that are valid in the non-dissipative system. This approach can yield significant insight (in fact even understanding the role played by conservation laws is important) but is only as good as the closure approximations that are utilised. For a more extensive discussion see Diamond, Hughes & Kim (Reference Diamond, Hughes and Kim2005) and Brandenburg & Subramanian (Reference Brandenburg and Subramanian2005). We give a brief explanation here.

Let us start by describing a closed domain, with no fluxes in or out of the region. We noted in (2.19) that the total magnetic helicity changes only via dissipation or through surface fluxes from the domain. Hence, for an ideal closed system, magnetic helicity is a quadratic (although not sign-definite) invariant quantity. For a closed dissipative system with no surface fluxes, global magnetic helicity can only be created or destroyed via irreversibility introduced by diffusive processes, i.e.

(6.13)\begin{equation} \frac{\textrm{d}}{\textrm{d} t} \langle {\boldsymbol A}\boldsymbol{\cdot} {\boldsymbol B}\rangle = - 2 \eta \langle\kern1.5pt {\boldsymbol j}\boldsymbol{\cdot} {\boldsymbol B} \rangle. \end{equation}

We recall here that angular brackets refer to volume averages. Note at this point that, for this closed system, this immediately allows some inference about $\langle\kern1.5pt {\boldsymbol j}\boldsymbol {\cdot } {\boldsymbol B}\rangle$ in the steady state, namely that $\langle\kern1.5pt {\boldsymbol j}\boldsymbol {\cdot } {\boldsymbol B}\rangle = 0$.

Furthermore, if one defines an intermediate average (such as coordinate averaging in the mean-field sense as discussed in § 5.1) and denotes it by an overbar then in the steady state

(6.14)\begin{equation} \langle\kern1.5pt {\boldsymbol j}' \boldsymbol{\cdot} {\boldsymbol b}' \rangle + \langle\kern1.5pt \overline{{\boldsymbol j}} \boldsymbol{\cdot} \overline{{\boldsymbol B}} \rangle = 0, \end{equation}

so that the average current helicity from the fluctuations is of the opposite sign to that from the means in the steady state, and if there is no mean current then $\langle\kern1.5pt {\boldsymbol j}' \boldsymbol {\cdot } {\boldsymbol b}' \rangle = 0$.

Indeed, if there is no mean current another exact result emerges from consideration of Ohm's law, giving

(6.15)\begin{equation} {\boldsymbol{\mathcal{E}}} \boldsymbol{\cdot} {\boldsymbol B}_0 = -\frac{1}{\sigma} \langle\kern1.5pt {\boldsymbol j'} \boldsymbol{\cdot} {\boldsymbol b'} \rangle + \langle {\boldsymbol e'} \boldsymbol{\cdot} {\boldsymbol b'} \rangle = \alpha B_0^2,\end{equation}

where ${\boldsymbol B}_0$ is the uniform mean field, with no associated current. This exact result relates the transport coefficient $\alpha$ to the fluctuating current density and the average correlation between the fluctuating magnetic field and EMF.

Finally, for a closed domain one may obtain evolution equations for the magnetic helicity contributions from large and small scales, namely

(6.16)\begin{equation} \frac{\textrm{d}}{\textrm{d} t} \langle \overline{{\boldsymbol A}}\boldsymbol{\cdot} \overline{{\boldsymbol B}}\rangle = 2 \langle \overline{{\boldsymbol {\mathcal{E}}}}\boldsymbol{\cdot} \overline{{\boldsymbol B}} \rangle - 2 \eta \langle\kern1.5pt \overline{{\boldsymbol j}}\boldsymbol{\cdot} \overline{{\boldsymbol B}} \rangle, \end{equation}

and

(6.17)\begin{equation} \frac{\textrm{d}}{\textrm{d} t} \langle {{\boldsymbol a}'}\boldsymbol{\cdot} {{\boldsymbol b}'}\rangle = -2 \langle \overline{{\boldsymbol {\mathcal{E}}}}\boldsymbol{\cdot} \overline{{\boldsymbol B}} \rangle - 2 \eta \langle\kern1.5pt {{\boldsymbol j}'}\boldsymbol{\cdot} {{\boldsymbol b}'} \rangle. \end{equation}

Hence, the $\langle \overline {{\boldsymbol {\mathcal {E}}}}\boldsymbol {\cdot } \overline{{\boldsymbol B}} \rangle$ term acts to transfer magnetic helicity between fluctuating and mean magnetic fields. It is clear then for this closed system that if the large-scale magnetic helicity is to grow it must be at the expense of the small-scale magnetic helicity; recall though that magnetic helicity is not sign definite so it is certainly possible for these to grow together if they are of opposite signs.

We note here that the expressions derived above allow evaluation of an expression for the projection of the average EMF onto the mean field, i.e. ${\boldsymbol {\mathcal {E}} \boldsymbol {\cdot } {\boldsymbol B}_0}$, in terms of correlations between ${\boldsymbol j}'$ and ${\boldsymbol b}'$, instead of correlations between ${\boldsymbol u}'$ and ${\boldsymbol b}'$. However, in the nonlinear regime we know none of ${\boldsymbol u}'$, ${\boldsymbol b}'$ or ${\boldsymbol j}'$, so what has this manipulation gained us?

The answer arises from utilising results from calculations employing the closure approximations described in § 6.4 – specifically calculations employing the EDQNM approximation, and generalisations thereof. These calculations (see e.g. Pouquet et al. Reference Pouquet, Frisch and Leorat1976; Kleeorin et al. Reference Kleeorin, Rogachevskii and Sokoloff2002) yield an approximate relationship between the $\alpha$-coefficient for homogeneous isotropic turbulence and the kinetic and current helicity, namely that

(6.18)\begin{equation} \alpha \sim - \frac{\tau_c}{3} \langle {\boldsymbol u}' \boldsymbol{\cdot} {\boldsymbol \omega}' - {\boldsymbol j}' \boldsymbol{\cdot} {\boldsymbol b}' \rangle. \end{equation}

At this point it is worth commenting again on the status and interpretation of this expression. Unlike the earlier global balance relationships, which are exact, it is to be stressed that this expression can only be derived via approximate closure relationships or via linearisation about a pre-existing turbulent MHD state at low $Rm$ and short $\tau _c$ (Proctor Reference Proctor2003). Equation (6.18) has clearly replaced the calculation of correlations between ${\boldsymbol u}'$ and ${\boldsymbol b}'$ with those between ${\boldsymbol j}'$ and ${\boldsymbol b}'$. This is only possible if some dynamical relationship between ${\boldsymbol j}'$ and ${\boldsymbol u}'$ has been approximated.

One key limitation in this expression is the assumption that the correlation time of the turbulence, $\tau _c$ remains unaffected by the presence of large-scale magnetic field, which is not necessarily true at high $Rm$, as discussed earlier. Nonetheless, what is clear is that the expression (6.18) does reduce to that given in (5.38) in the kinematic regime; the correction to the kinematic result often being termed a magnetically driven $\alpha$-effect, $\alpha _M$.

Because, in this approximation, the current helicity from the small scales plays a key role determining the correction to the transport coefficient, it is now possible to combine this with the exact results of (6.17) to derive an evolution equation for the transport coefficient $\alpha$. For details of this see Gruzinov & Diamond (Reference Gruzinov and Diamond1994), Kleeorin, Rogachevskii & Ruzmaikin (Reference Kleeorin, Rogachevskii and Ruzmaikin1995) and Blackman & Brandenburg (Reference Blackman and Brandenburg2002); which include the further assumptions/simplifications (such as relating $\langle {\boldsymbol a}' \boldsymbol {\cdot } {\boldsymbol b}'\rangle$ to $\langle\kern1.5pt {\boldsymbol j}' \boldsymbol {\cdot } {\boldsymbol b}'\rangle$ via a spectral relationship) required to derive the evolution equation.

The evolution equation is given in full in Blackman & Brandenburg (Reference Blackman and Brandenburg2002) who also show that, in the saturated steady state for large-scale fields with no associated currents (i.e. uniform fields),

(6.19)\begin{equation} \alpha = \frac{\alpha_0}{1+Rm B_0^2/ {\mathcal{B}}^2 }, \end{equation}

a result first derived by Gruzinov & Diamond (Reference Gruzinov and Diamond1994), which is in agreement with the catastrophic quenching result of Cattaneo & Vainshtein (Reference Cattaneo and Vainshtein1991) in (6.12a,b). There is therefore now little debate as to the nature of the quenching of the $\alpha$-effect in closed systems with no large-scale currents; particularly as we shall see that the quenching formula is consistent with the results of numerical experiments (albeit at low and moderate $Rm$.)

Physically this result can be interpreted as the conservation of magnetic helicity placing a topological constraint on the nature of the magnetic field (as discussed in § 2.3.1). In order to grow the large-scale magnetic field, the knotted small-scale field that is naturally generated kinematically must be untangled, which is impossible if the topological constraint is maintained. If the presence of small magnetic diffusion is the only source of irreversibility allowing the untangling of magnetic field, then this places a severe constraint on the level of the mean field that can be generated – or at least the time scale on which it can be generated.

One interpretation of a formula such as that of (6.19) is that large-scale field generation is completely suppressed for mean fields an order of magnitude smaller than equipartition ($B_0 \sim {O}(Rm^{-1/2} \mathcal {B})$). However, consideration of the full evolution equation given in, say, Blackman & Brandenburg (Reference Blackman and Brandenburg2002) demonstrates that even with this ‘catastrophic’ quenching formula the energy in the large-scale magnetic field continues to grow on a time scale controlled by the irreversible process, in this case magnetic diffusion (an ohmic time). In the long run (for example in the case of the Sun a time scale comparable with the age of the star) reaching a statistically steady state with significant large-scale field may be possible, see also Moffatt & Dormy (Reference Moffatt and Dormy2019). However, ‘in the long run we are all dead’ (Keynes Reference Keynes1923).

6.5.2. Large-scale currents and helicity fluxes: can they help?

‘E pur si muove’ Attributed (perhaps incorrectly) to Galileo.

Catastrophic quenching as envisaged by Vainshtein & Cattaneo (Reference Vainshtein and Cattaneo1992) and Gruzinov & Diamond (Reference Gruzinov and Diamond1994) places a severe constraint on the applicability of mean-field theory at high $Rm$. Large-scale organised fields can only be generated from a weak seed field on an ohmic time scale, this being the time taken for the action of ohmic dissipation to untangle the knotted field. However, it may be that other processes can be identified that break reversibility and hence the topological constraints placed on the field, which we shall discuss below. Another option is that the magnetic field may not have been generated all the way from kinematic values by a turbulent EMF, but may simply be sustained by an EMF that arises from an instability of the field itself. These more laminar ‘essentially nonlinear’ dynamos will be described in § 7.

When a mean current $\overline{\boldsymbol{j}}$ is allowed, this naturally leads to the presence of large-scale current helicity $\langle\kern1.5pt \overline {{\boldsymbol j}} \boldsymbol {\cdot } \overline{{\boldsymbol B}}\rangle$. Care does need to be taken here as the presence of a large-scale current leads to both questions of the nature of the separation of scales and the presence of a turbulent diffusion because of the presence of large-scale field gradients. The inclusion of a mean current may seems natural, since it is difficult to see how a dynamo can generate a magnetic field with no mean current. In that case (6.19) (subject to the same assumptions and approximations) is modified to

(6.20)\begin{equation} \alpha = \frac{\alpha_0}{1+Rm B_0^2/ {\mathcal{B}}^2 } + \frac{Rm \, \beta \langle\kern1.5pt \overline{{\boldsymbol j}} \boldsymbol{\cdot} \overline{{\boldsymbol B}}\rangle}{{{\mathcal{B}}^2 }+Rm \langle \overline{{\boldsymbol B}}^2 \rangle}, \end{equation}

as first derived by Gruzinov & Diamond (Reference Gruzinov and Diamond1994); here, $\beta$ is the magnitude of the turbulent diffusivity. Thus, although the kinematic $\alpha$-effect is catastrophically quenched, if one can (somehow) generate large-scale fields with a non-trivial current helicity then they can be maintained via a magnetically driven EMF, the second term in (6.20) (see also § 7).

However, it is not at all clear how one generates these significant large-scale fields from a weak seed field (e.g. in a galaxy). It may be that shear flows play a major role here in producing a large-scale field strong enough so that nonlinear effects can kick in. However, if one is relying on turbulence (i.e. fluctuation–fluctuation interactions) to do the job then there must be another process breaking reversibility for this to proceed quickly. Blackman & Field (Reference Blackman and Field2000) proposed that catastrophic quenching, or the associated generation of large-scale fields on an ohmic time scale, may be circumvented if irreversibility is introduced to the system by allowing magnetic helicity fluxes into and out of the domain. For example if magnetic helicity associated with small scales is preferentially lost on a time scale $\tau _{flux}$ then the rate of generation of large-scale field may occur on this time scale. Mathematically this can be seen by reinstating the helicity flux term into (6.13) and hence also into (6.16) and (6.17). The presence of this term breaks the reliance on ohmic processes to reconfigure (untangle) the magnetic field, and hence may alleviate the catastrophic quenching or allow the more rapid generation of large-scale field. There are two, physically distinct, contributions to the helicity flux; one diffusive and one ideal. Clearly the diffusive flux should become analogously weak to the ohmic dissipation of magnetic helicity in the limit of high $Rm$, so we are relying on the ideal flux to do the job!

There are a number of issues with this picture that require further investigation. The first is that the ideal flux term takes the form of a surface integral; it is not obvious whether a term that relies on losses through a surface can lead to a significant impact on the generation of large-scale fields throughout a large volume. The second is whether the ideal fluxes manage to avoid scaling with the magnetic Reynolds number (not directly, since the diffusivity does not appear explicitly in these terms, but through the form of the magnetic field). This may be checked using numerical calculations as described in the next subsection, at least at moderate $Rm$; this potential lifebelt for the generation of large-scale fields at high $Rm$ should be investigated more closely in the near future. Finally, it may be that it is enough to separate spatially the region of large-scale field generation from that of large-scale field storage. Faced with the proposal of catastrophic quenching, Parker (Reference Parker1993) introduced the concept of interface dynamos, which made use of precisely this idea. More investigation of dynamos with inhomogeneous turbulence and shear flows may reveal whether this model can function at high $Rm$. Shear-enhanced diffusion and other methods of introducing irreversibility on a fast time scale may also play an important role here.

6.5.3. Numerical investigations of catastrophic quenching

The theoretical arguments proposed above can be somewhat tested by devising numerical experiments. These, in general, take two complementary forms: dynamo experiments examine the nature of the generated magnetic field when it is allowed to evolve freely in a turbulent flow, whereas turbulent transport experiments typically impose a large-scale field (perhaps a uniform field or a large-scale field with a large-scale current) and measure the response of the turbulence in creating an EMF. Of course, the first of these approaches is more natural, with the magnetic field self-consistently evolving with the turbulence. The second approach, though, offers more control and the potential to determine the functional dependence of the EMF on parameters such as the imposed field strength, rotation rate, and magnetic Prandtl number.

However, it must be stressed that such numerical simulations are limited in the range of $Rm$ that are accessible and it is difficult to extrapolate the results of these to the astrophysically relevant high $Rm$ regime. For example, a spectral calculation performed with $10^9$ degrees of freedom with a reasonable separation of scales between large and small scales is able to reach magnetic Reynolds numbers of $ {O}(10^3)$ (Hughes & Tobias Reference Hughes and Tobias2010).

Simulations are usually configured in one of two ways. In the first a flow is driven via a prescribed body force, typically chosen to drive a relatively small-scale flow with significant kinetic helicity (to maximise the chances of driving an EMF and hence a large-scale dynamo) and potentially a shear flow. In the second driving is via buoyancy in thermal convection, where helicity emerges naturally via the interaction with rotation; clearly the second of these is of more relevance astrophysically although there is less control over the form of the flow.

The list of numerical experiments performed over the past twenty or so years is long, and a complete review is well beyond the scope here. The interested reader is directed to Brandenburg & Subramanian (Reference Brandenburg and Subramanian2005) for a thorough review of the early attempts and Brandenburg (Reference Brandenburg2018) for a briefer review of later attempts.

In a closed domain all simulations (albeit at low and moderate $Rm$) are consistent with the picture of catastrophic quenching of the EMF (or diffusive growth of the mean field). For example Cattaneo & Hughes (Reference Cattaneo and Hughes1996) utilised a forcing that, in the absence of magnetic field, was designed to drive a small-scale Galloway–Proctor flow (here small-scale is defined as $k=1$) and measured the response of the turbulent EMF to an imposed uniform ($k=0$) magnetic field in a triply periodic box. They found results consistent with (6.12a,b). Analogous results were found for dynamo calculations driven by helical turbulence by Blackman & Brandenburg (Reference Blackman and Brandenburg2002) (and see references therein), where for closed systems the large-scale field only grew ohmically after an exponential kinematic phase. It seems as though there is now reasonable agreement for homogeneous, confined systems; the magnetic field can only emerge after an ohmic time scale.

For open systems, where boundary conditions that allow magnetic helicity to escape or enter the domain, the situation is more controversial, and more detailed calculations at higher $Rm$ are needed. Briefly, it does appear that the dynamos in these open domains do seem to have a different character to those in closed domains – at least at moderate $Rm$. For example, Hubbard & Brandenburg (Reference Hubbard and Brandenburg2010) found that, as $Rm$ is increased both the diffusive volume term and helicity flux contribution to irreversibility decrease, although intriguingly the diffusive volume term decreases more rapidly as a function of $Rm$ (at least for moderate $Rm$). Linear extrapolation would seem to indicate that the boundary term would dominate at sufficiently high $Rm$. Similar results have also been found by Del Sordo, Guerrero & Brandenburg (Reference Del Sordo, Guerrero and Brandenburg2013). However, it does seem in all current cases as though the ratio of mean field to fluctuating field continues to decrease with $Rm$ (Brandenburg Reference Brandenburg2018; Cattaneo, Bodo & Tobias Reference Cattaneo, Bodo and Tobias2020). Clearly this issue is difficult to address computationally, although the status of mean-field theory in the nonlinear regime may become clearer in the next few years.

8.2.1. Consequences of MAC balance: Taylor's constraint

Enforcing MAC balance imposes a serious constraint on the nature of the possible dynamo solutions. Here, we derive the simplest constraint on the form of the integrated Lorentz force known as Taylor's constraint.

Taking the $\phi$-component (in cylindrical polar coordinates $(s,\phi ,z)$ of (8.10) and integrating over cylinders aligned with the rotation axis one obtains

(8.11)\begin{equation} \int 2 \rho ({\boldsymbol \varOmega} \times {\boldsymbol u})_\phi \,\textrm{d} S \sim - \int (\boldsymbol{\nabla} p)_\phi \,\textrm{d} S + \int (\kern1.5pt {\boldsymbol j} \times {\boldsymbol B})_\phi \, \textrm{d} S + \int (\rho {\boldsymbol g})_\phi \, \textrm{d} S,\end{equation}

where $\textrm {d} S = s \, \textrm {d} \phi \, \textrm {d} z$. As buoyancy is purely radial the final term is zero and the pressure term integrates to zero. Finally the Coriolis term can also be shown to be zero by using the divergence theorem over the cylinder. Hence, the only term to contribute to (8.11) is the Lorentz force and so

(8.12)\begin{equation} \int (\kern1.5pt {\boldsymbol j} \times {\boldsymbol B})_\phi \, \textrm{d} S = \frac{1}{\mu} \int (({\boldsymbol{\nabla} } \times {\boldsymbol B}) \times {\boldsymbol B})_\phi \, \textrm{d} S = 0.\end{equation}

This Taylor constraint ensures that the Lorentz torque on any cylindrical surface parallel to the rotation axis is zero.

It is possible to construct dynamos that solve the induction equation in addition to maintaining Taylor's constraint both in two (Roberts & Wu Reference Roberts and Wu2018) and three (Li, Jackson & Livermore Reference Li, Jackson and Livermore2018) dimensions. This elegant formalism certainly provides a way forward for finding exact solutions that satisfy MAC balance. Numerical computations usually do, however, solve the full momentum equation (albeit in the wrong parameter regime) and utilise the size of the departures from the solution to (8.12) (the degree of ‘Taylorisation’) as a measure of the degree to which they have been successful in obtaining solutions that are in MAC balance.

8.3.1. The search for a distinguished limit

As noted above, geodynamo models are usually integrated in completely the wrong parameter range, but sometimes yield solutions that resemble the geomagnetic field. When and why is this the case, and how can one continue to achieve solutions in MAC balance as computers get faster? It is tempting to try to compute numerical models with all the parameters set to be as close as possible to their correct geophysical values (in particular low $E$ and low $Pm$). Is this the most sensible approach, given current computational limitations?

An alternative, and more considered, approach was pioneered by Dormy (Reference Dormy2016) who studied the properties of a large set of dynamo models at moderate Ekman number (with $E \approx {O}(10^{-4}$)). He argued that, at such moderate values amenable to rapid computation, it was necessary to increase $Pm$ from its order-one value to achieve the correct balance for the generated magnetic field; indeed it is also of interest to examine rapidly rotating dynamos where inertia is completely suppressed (Hughes & Cattaneo Reference Hughes and Cattaneo2016) – in this regime $Pm$ is infinite. Of course, when $E$ is reduced, as computational resources become more lavish, one should move away from these models and $Pm$ should be reduced in tandem with the Ekman number. Dormy argued that a distinguished limit should be found where all parameters, including $Pm$ scale with the Ekman number as $E \rightarrow 0$ to leave one on the balanced dynamo branch. This is also the approach taken by Aubert et al. (Reference Aubert, Gastine and Fournier2017), as described above, although the scalings selected for the variation of the parameters are slightly different.