2.1.1 Compressible MHD equations

Let us start from the equations of compressible, viscous, resistive magnetohydrodynamics. First, we have the continuity (mass conservation) equation

(2.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{U})=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is the gas density and $\boldsymbol{U}$ is the fluid velocity field, and the momentum equation

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D70C}\left(\frac{\unicode[STIX]{x2202}\boldsymbol{U}}{\unicode[STIX]{x2202}t}+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U}\right)=-\unicode[STIX]{x1D735}P+\frac{\boldsymbol{J}\times \boldsymbol{B}}{c}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D749}+\boldsymbol{F}(\boldsymbol{x},t),\end{eqnarray}$$

where $P$ is the gas pressure, $\unicode[STIX]{x1D70F}_{ij}=\unicode[STIX]{x1D707}(\unicode[STIX]{x1D735}_{i}U_{j}+\unicode[STIX]{x1D735}_{j}U_{i}-(2/3)\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U})$ is the viscous stress tensor ( $\unicode[STIX]{x1D707}$ is the dynamical viscosity and $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}$ is the kinematic viscosity), $\boldsymbol{F}$ is a force per unit volume representing any kind of external stirring mechanism (impellers, gravity, spoon, supernovae, meteors etc.), $\boldsymbol{B}$ is the magnetic field, $\boldsymbol{J}=(c/4\unicode[STIX]{x03C0})\unicode[STIX]{x1D735}\times \boldsymbol{B}$ is the current density and $\boldsymbol{J}\times \boldsymbol{B}/c$ , the Lorentz force, describes the dynamical feedback exerted by the magnetic field on fluid motions. The evolution of $\boldsymbol{B}$ is governed by the induction equation

(2.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{U}\times \boldsymbol{B})-\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D702}\unicode[STIX]{x1D735}\times \boldsymbol{B}),\end{eqnarray}$$

supplemented with the solenoidality condition

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0.\end{eqnarray}$$

Equation (2.3) is derived from the Maxwell–Faraday equation and a simple, isotropic Ohm’s law for collisional electrons,

(2.5) $$\begin{eqnarray}\boldsymbol{J}=\unicode[STIX]{x1D70E}\left(\boldsymbol{E}+\frac{\boldsymbol{U}\times \boldsymbol{B}}{c}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}$ is the electrical conductivity of the fluid. The first term $\boldsymbol{{\mathcal{E}}}=\boldsymbol{U}\times \boldsymbol{B}$ on the right-hand side of (2.3) is called the electromotive force (EMF) and describes the induction of magnetic field by the flow of conducting fluid from an Eulerian perspective. The second term describes the diffusion of magnetic field in a non-ideal fluid of magnetic diffusivity $\unicode[STIX]{x1D702}=c^{2}/(4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E})$ . Both the Lorentz force and EMF terms in (2.2)–(2.3) play a very important role in the dynamo problem, but so do viscous and resistive dissipation. Finally, we have the internal energy, or entropy equation

(2.6) $$\begin{eqnarray}\unicode[STIX]{x1D70C}T\left(\frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}t}+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}S\right)=D_{\unicode[STIX]{x1D707}}+D_{\unicode[STIX]{x1D702}}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(K\unicode[STIX]{x1D735}T),\end{eqnarray}$$

where $T$ is the gas temperature, $S\propto P/\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6FE}}$ is the entropy ( $\unicode[STIX]{x1D6FE}$ is the adiabatic index), $D_{\unicode[STIX]{x1D707}}$ and $D_{\unicode[STIX]{x1D702}}$ stand for the viscous and resistive dissipation, $K$ is the thermal conductivity and the last term on the right-hand side stands for thermal diffusion (we could also have added an inhomogeneous heat source, or explicit radiative transfer). An equation of state for the thermodynamic variables, like the ideal gas law $P=\unicode[STIX]{x1D70C}{\mathcal{R}}T$ , is also required in order to close this system ( ${\mathcal{R}}$ here denotes the gas constant).

The compressible MHD equations describe the dynamics of waves, instabilities, turbulence and shocks in all kinds of astrophysical fluid systems, including stratified and/or (differentially) rotating fluids, and accommodate a large range of dynamical magnetic phenomena including dynamos and (fluid) reconnection. The ideal MHD limit corresponds to $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D702}=K=0$ . The reader is referred to the astrophysical fluid dynamics lecture notes of Ogilvie (Reference Ogilvie2016), published in this journal, for a very tidy derivation and presentation of ideal MHD.

2.1.2 Important conservation laws in ideal MHD

There are two particularly important conservation laws in the ideal MHD limit that involve the magnetic field and are of primary importance in the context of the dynamo problem. To obtain the first one we combine the continuity and ideal induction equations into

(2.7) $$\begin{eqnarray}\frac{\text{D}}{\text{D}t}\left(\frac{\boldsymbol{B}}{\unicode[STIX]{x1D70C}}\right)=\frac{\boldsymbol{B}}{\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U},\end{eqnarray}$$

where $\text{D}/\text{D}t=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ is the Lagrangian derivative. Equation (2.7) for $\boldsymbol{B}/\unicode[STIX]{x1D70C}$ has the same form as the equation describing the evolution of the Lagrangian separation vector $\unicode[STIX]{x1D6FF}\boldsymbol{r}$ between two fluid particles,

(2.8) $$\begin{eqnarray}\frac{\text{D}\unicode[STIX]{x1D6FF}\boldsymbol{r}}{\text{D}t}=\unicode[STIX]{x1D6FF}\boldsymbol{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U}.\end{eqnarray}$$

Hence, magnetic-field lines in ideal MHD can be thought of as being ‘frozen into’ the fluid just as material lines joining fluid particles. This is called Alfvén’s theorem. Using this equation and (2.4), it is also possible to show that the magnetic flux through material surfaces $\unicode[STIX]{x1D6FF}\boldsymbol{S}$ (deformable surfaces moving with the fluid) is conserved in ideal MHD,

(2.9) $$\begin{eqnarray}\frac{\text{D}}{\text{D}t}(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{S})=0.\end{eqnarray}$$

If a material surface $\unicode[STIX]{x1D6FF}\boldsymbol{S}$ is deformed under the effect of either shearing or compressive/expanding motions, the magnetic field threading it must change accordingly so that $\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{S}$ remains the same. Alfvén’s theorem enables us to appreciate the kinematics of the magnetic field in a flow in a more intuitive geometrical way than by just staring at equations, as it is relatively easy to visualise magnetic-field lines advected and stretched by the flow. This will prove very helpful to develop an intuition of how small- and large-scale dynamo processes work.

A second important conservation law in ideal MHD in the context of dynamo theory is the conservation of magnetic helicity ${\mathcal{H}}_{m}=\int \boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}\,\text{d}^{3}\boldsymbol{r}$ , where $\boldsymbol{A}$ is the magnetic vector potential. To derive it, we first write the Maxwell–Faraday equation for $\boldsymbol{A}$ ,

(2.10) $$\begin{eqnarray}\frac{1}{c}\frac{\unicode[STIX]{x2202}\boldsymbol{A}}{\unicode[STIX]{x2202}t}=-\boldsymbol{E}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D711},\end{eqnarray}$$

where $\unicode[STIX]{x1D711}$ is the electrostatic potential. Combining (2.10) with (2.3) gives

(2.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{F}_{{\mathcal{H}}_{m}}=-2\unicode[STIX]{x1D702}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{B},\end{eqnarray}$$

where

(2.12) $$\begin{eqnarray}\boldsymbol{F}_{{\mathcal{H}}_{m}}=c(\unicode[STIX]{x1D711}\boldsymbol{B}+\boldsymbol{E}\times \boldsymbol{A})\end{eqnarray}$$

is the total magnetic-helicity flux. In the ideal case, we see that (2.11) reduces to an explicitly conservative local evolution equation for $\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}$ ,

(2.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }[c\unicode[STIX]{x1D711}\boldsymbol{B}+\boldsymbol{A}\times (\boldsymbol{U}\times \boldsymbol{B})]=0,\end{eqnarray}$$

where (2.5) with $\unicode[STIX]{x1D702}=0$ has been used to express the magnetic-helicity flux. Note that both ${\mathcal{H}}_{m}$ and $\boldsymbol{F}_{{\mathcal{H}}_{m}}$ depend on the choice of electromagnetic gauge and are therefore not uniquely defined. Qualitatively, magnetic helicity provides a measure of the linkage/knottedness of the magnetic field within the domain considered and the conservation of magnetic helicity in ideal MHD is therefore generally understood as a conservation of magnetic linkages in the absence of magnetic diffusion or reconnection (see e.g. Hubbard & Brandenburg (Reference Hubbard and Brandenburg2011), Miesch (Reference Miesch2012), Blackman (Reference Blackman2015) or Bodo et al. (Reference Bodo, Cattaneo, Mignone and Rossi2017) for discussions of magnetic-helicity dynamics in different astrophysical dynamo contexts).

2.1.3 Magnetic-field energetics

What about the driving and energetics of the magnetic field? An enlightening equation in that respect is that describing the local Lagrangian evolution of the magnetic-field strength $B$ associated with a fluid particle in ideal MHD ( $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D708}=0$ ),

(2.14) $$\begin{eqnarray}\frac{1}{B}\frac{\text{D}B}{\text{D}t}=\hat{\boldsymbol{B}}\hat{\boldsymbol{B}}:\unicode[STIX]{x1D735}\boldsymbol{U}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U},\end{eqnarray}$$

where $\hat{\boldsymbol{B}}=\boldsymbol{B}/B$ is the unit vector defining the orientation of the magnetic field attached to the fluid particle, and we have used the double dot-product notation $\hat{\boldsymbol{B}}\hat{\boldsymbol{B}}:\unicode[STIX]{x1D735}\boldsymbol{U}=\hat{B}_{i}\hat{B}_{j}\unicode[STIX]{x1D6FB}_{i}U_{j}$ . Equation (2.14) follows directly from (2.1) to (2.7), and shows that any increase of $B$ results from either a stretching of the magnetic field along itself by a flow, or from a compression, and that the rate at which $\ln B$ changes is proportional to the local shearing or compression rate of the flow. Note that incompressible motions with no component parallel to the local original/initial background field do not affect the field strength at linear order, and only generate magnetic curvature perturbations (these are shear Alfvén waves). Going back to full resistive MHD, the global evolution equation for the total magnetic energy within a fixed volume, derived for instance in the classic textbook of Roberts (Reference Roberts1967), is

(2.15) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\int \frac{|\boldsymbol{B}|^{2}}{8\unicode[STIX]{x03C0}}\,\text{d}V=-\int \boldsymbol{U}\boldsymbol{\cdot }\frac{(\boldsymbol{J}\times \boldsymbol{B})}{c}\,\text{d}V-\frac{c}{4\unicode[STIX]{x03C0}}\int (\boldsymbol{E}\times \boldsymbol{B})\boldsymbol{\cdot }\text{d}\boldsymbol{S}-\int \frac{|\boldsymbol{J}|^{2}}{\unicode[STIX]{x1D70E}}\,\text{d}V,\end{eqnarray}$$

where the surface integral is taken over the boundary of the volume, oriented by an outward normal vector. The first term on the right-hand side is a volumetric term equal to the opposite of the work done by the Lorentz force on the flow, the second surface term is the Poynting flux of electromagnetic energy through the boundaries of the domain under consideration, and the last term quadratic in $\boldsymbol{J}$ corresponds to ohmic dissipation of electrical currents into heat. In the absence of a Poynting term (for instance in a periodic domain), we see that magnetic energy can only be generated at the expense of kinetic (mechanical) energy. In other words, we must put in mechanical energy in order to drive a dynamo.

2.1.4 Incompressible MHD equations for dynamo theory

Starting from compressible MHD enabled us to show that compressive motions, which are relevant to a variety of astrophysical situations, can formally contribute to the dynamics and amplification of magnetic fields. However, much of the essence of the dynamo problem can be captured in the much simpler framework of incompressible, viscous, resistive MHD, which we will therefore mostly use henceforth (further assuming constant kinematic viscosity and magnetic diffusivity). In the incompressible limit, $\unicode[STIX]{x1D70C}$ is uniform and constant, and the distinction between thermal and magnetic pressure disappears. The magnetic tension part of the Lorentz force provides the only relevant dynamical magnetic feedback on the flow in that case.Footnote ³ Rescaling $\boldsymbol{F}$ and $P$ by $\unicode[STIX]{x1D70C}$ and $\boldsymbol{B}$ by $(4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C})^{1/2}$ , so that $\boldsymbol{B}$ now stands for the Alfvén velocity

(2.16) $$\begin{eqnarray}\boldsymbol{U}_{A}=\frac{\boldsymbol{B}}{\sqrt{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}}},\end{eqnarray}$$

and introducing the total pressure $\unicode[STIX]{x1D6F1}=P+B^{2}/2$ , we can write the incompressible momentum equation as

(2.17) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{U}}{\unicode[STIX]{x2202}t}+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F1}+\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{B}+\unicode[STIX]{x1D708}\unicode[STIX]{x0394}\boldsymbol{U}+\boldsymbol{F}(\boldsymbol{x},t).\end{eqnarray}$$

The induction equation is rewritten as

(2.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{B}=\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U}+\unicode[STIX]{x1D702}\unicode[STIX]{x0394}\boldsymbol{B}.\end{eqnarray}$$

This form separates the physical effects of the electromotive force into two parts: advection/mixing represented by $\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{B}$ on the left, and induction/stretching represented by $\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U}$ on the right. Magnetic stretching by shearing motions is the only way to amplify magnetic fields in an incompressible flow of conducting fluid. In order to formulate the problem completely, equations (2.17)–(2.18) must be supplemented with

(2.19a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}=0,\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0,\end{eqnarray}$$

and paired with an appropriate set of initial conditions, and boundary conditions in space. The latter can be a particularly tricky business in the dynamo context. Periodic boundary conditions, for instance, are a popular choice among theoreticians but may be problematic in the context of the saturation of large-scale dynamos (§ 4.6). Certain types of magnetic boundary conditions are also problematic for the definition of magnetic helicity. The choice of boundary conditions and global configuration of dynamo problems is not just a problem for theoreticians either: as mentioned earlier, the choice of soft-iron versus steel propellers has a drastic effect on the excitation of a dynamo effect in the VKS experiment (Monchaux et al. Reference Monchaux, Berhanu, Bourgoin, Moulin, Odier, Pinton, Volk, Fauve, Mordant and Pétrélis2007), raising the question of whether this dynamo is a pure fluid effect or a fluid–structure interaction effect (see e.g. Gissinger et al. Reference Gissinger, Iskakov, Fauve and Dormy2008b ; Giesecke et al. Reference Giesecke, Nore, Stefani, Gerbeth, Léorat, Herreman, Luddens and Guermond2012; Kreuzahler et al. Reference Kreuzahler, Ponty, Plihon, Homann and Grauer2017; Nore et al. Reference Nore, Castanon Quiroz, Cappanera and Guermond2018).

2.1.5 Shearing sheet model of differential rotation

Differential rotation is present in many systems that sustain dynamos, but can take many different forms depending on the geometry and internal dynamics of the system at hand. As we will discover in § 5.1, working in global cylindrical or spherical geometry is particularly valuable if we seek to understand how large-scale dynamos like the solar or geo-dynamo operate at a global level, because these systems happen to have fairly complex differential rotation laws and internal shear layers. On the other hand, we do not in general need all this geometric complexity to understand how rotation and shear affect dynamo processes at a fundamental physical level. In fact, any possible simplification is most welcome in this context, as many of the basic statistical dynamical processes that we are interested in are usually difficult enough to understand at a basic level. In what follows, we will therefore make intensive use of a local Cartesian representation of differential rotation, known as the shearing sheet model (Goldreich & Lynden-Bell Reference Goldreich and Lynden-Bell1965), that will make it possible to study some essential effects of shear and rotation on dynamos in a very simple and systematic way.

Consider a simple cylindrical differential rotation law $\unicode[STIX]{x1D734}=\unicode[STIX]{x1D6FA}(R)\boldsymbol{e}_{z}$ in polar coordinates $(R,\unicode[STIX]{x1D711},z)$ (think of an accretion disc or a galaxy). To study the dynamics around a particular cylindrical radius $R_{0}$ , we can move to a frame of reference rotating at the local angular velocity, $\unicode[STIX]{x1D6FA}\equiv \unicode[STIX]{x1D6FA}(R_{0})$ , and solve the equations of rotating MHD locally (including Coriolis and centrifugal accelerations) in a Cartesian coordinate system ( $x,y,z$ ) centred on $R_{0}$ , neglecting curvature effects (all of this can be derived rigorously). Here, $x$ corresponds to the direction of the local angular velocity gradient (the radial direction in an accretion disc), and $y$ corresponds to the azimuthal direction. In the rotating frame, the differential rotation around $R_{0}$ reduces to a simple a linear shear flow $\boldsymbol{U}_{S}=-Sx\boldsymbol{e}_{y}$ , where $S\equiv -R_{0}\,\text{d}\unicode[STIX]{x1D6FA}/\text{d}R|_{R_{0}}$ is the local shearing rate (figure 5).

Figure 5. The Cartesian shearing sheet model of differentially rotating flows.

This model enables us to probe a variety of differential rotation regimes by studying the individual or combined effects of a pure rotation, parametrised by $\unicode[STIX]{x1D6FA}$ , and of a pure shear, parametrised by $S$ , on dynamos. For instance, we can study dynamos in non-rotating shear flows by setting $\unicode[STIX]{x1D6FA}=0$ and varying the shearing rate $S$ with respect to the other time scales of the problem, or we can study the effects of rigid rotation on a dynamo-driving flow (and the ensuing dynamo) by varying $\unicode[STIX]{x1D6FA}$ while setting $S=0$ . Cyclonic rotation regimes, for which the vorticity of the shear flow is aligned with the rotation vector, have negative $\unicode[STIX]{x1D6FA}/S$ in the shearing sheet with our convention, while anticyclonic rotation regimes correspond to positive $\unicode[STIX]{x1D6FA}/S$ . In particular, anticyclonic Keplerian rotation typical of accretion discs orbiting around a central mass $M_{\ast }$ , $\unicode[STIX]{x1D6FA}(R)=\sqrt{{\mathcal{G}}M_{\ast }}/R^{3/2}$ , is characterised by $\unicode[STIX]{x1D6FA}=(2/3)S$ in this model.

The numerical implementation of the local shearing sheet approximation in finite domains is usually referred to as the ‘shearing box’, as it amounts to solving the equations in a Cartesian box of dimensions $(L_{x},L_{y},L_{z})$ much smaller than the typical radius of curvature of the system. In order to accommodate the linear shear in this numerical problem, the $x$ coordinate is usually taken as shear periodic,Footnote ⁴ the $y$ coordinate is taken as periodic and the choice of the boundary conditions in $z$ depends on whether some stratification is incorporated in the modelling (if not, periodicity in $z$ is usually assumed).

2.2.1 Reynolds numbers

Let us now consider some important scales and dimensionless numbers in the dynamo problem based on (2.17)–(2.18). First, we define the scale of the system under consideration as $L$ , and the integral scale of the turbulence, or the scale at which energy is injected into the flow, as $\ell _{0}$ . Depending on the problem under consideration, we will have either $L\sim \ell _{0}$ , or $L\gg \ell _{0}$ . Turbulent velocity field fluctuations at scale $\ell _{0}$ are denoted by $u_{0}$ . The kinematic Reynolds number

(2.20) $$\begin{eqnarray}Re=\frac{u_{0}\ell _{0}}{\unicode[STIX]{x1D708}}\end{eqnarray}$$

measures the relative magnitude of inertial effects compared to viscous effects on the flow. The Kolmogorov scale $\ell _{\unicode[STIX]{x1D708}}\sim Re^{-3/4}\ell _{0}$ is the scale at which kinetic energy is dissipated in Kolmogorov turbulence, with $u_{\unicode[STIX]{x1D708}}\sim Re^{-1/4}u_{0}$ the corresponding typical velocity at that scale. The magnetic Reynolds number

(2.21) $$\begin{eqnarray}Rm=\frac{u_{0}\ell _{0}}{\unicode[STIX]{x1D702}}\end{eqnarray}$$

measures the relative magnitude of inductive (and mixing) effects compared to resistive effects in (2.18), and is therefore a key number in dynamo theory.

2.2.2 The magnetic Prandtl number landscape

The ratio of the kinematic viscosity to the magnetic diffusivity, the magnetic Prandtl number

(2.22) $$\begin{eqnarray}Pm=\frac{\unicode[STIX]{x1D708}}{\unicode[STIX]{x1D702}}=\frac{Rm}{Re},\end{eqnarray}$$

is also a key quantity in dynamo theory. Unlike $Re$ and $Rm$ , $Pm$ in a collisional fluid is an intrinsic property of the fluid itself, not of the flow. Figure 6 shows that conducting fluids and plasmas found in nature and in the laboratory have a wide range of $Pm$ . One reason for this wide distribution is that $Pm$ is very strongly dependent on both temperature and density. For instance, in a pure, collisional hydrogen plasma with equal ion and electron temperature,

(2.23) $$\begin{eqnarray}Pm\simeq 2.5\times 10^{3}\frac{T^{4}}{n(\ln \unicode[STIX]{x1D6EC})^{2}},\end{eqnarray}$$

where $T$ is in Kelvin, $\ln \unicode[STIX]{x1D6EC}$ is the Coulomb logarithm and $n$ is the particle number density in $\text{m}^{-3}$ . This collisional formula gives $Pm\sim 10^{25}$ or larger for the very hot ICM of galaxy clusters (although it is probably not very accurate in this context given the weakly collisional nature of the ICM). The much denser and cooler plasmas in stellar interiors have much lower $Pm$ , for instance $Pm$ ranges approximately from $10^{-2}$ at the base of the solar convection zone to $10^{-6}$ below the photosphere. Accretion-disc plasmas can have all kinds of $Pm$ , depending on the nature of the accreting system, closeness to the central accreting object and location with respect to the disc midplane.

Figure 6. A qualitative representation of the magnetic Prandtl number landscape. The grey area depicts the range of $Re$ and $Rm$ (based on root-mean-square velocities) thought to be accessible in the foreseeable future through either numerical simulations or plasma experiments.

Liquid metals like liquid iron in the Earth’s core or liquid sodium in dynamo experiments have very low $Pm$ , typically $Pm\sim 10^{-5}$ or smaller. This has proven a major inconvenience for dynamo experiments, as achieving even moderate $Rm$ in a very low $Pm$ fluid requires a very large $Re$ and therefore necessitates a lot of mechanical input power, which in turns implies a lot of heating. To add to the inconvenience, the turbulence generated at large $Re$ enhances the effective diffusion of the magnetic field, which makes it even harder to excite interesting magnetic dynamics. As a result, the experimental community has started to shift attention to plasma experiments in which $Pm$ can in principle be controlled and varied in the range $0.1<Pm<100$ by changing either the temperature or density of the plasma, as illustrated by (2.23). Finally, due to computing power limitations implying finite numerical resolutions, most virtual MHD fluids of computer simulations have $0.1<Pm<10$ (with a few exceptions at large $Pm$ ). Hence, it is and will remain impossible in a foreseeable future to simulate magnetic-field amplification in any kind of regime found in nature. The best we can hope for is that simulations of largish or smallish $Pm$ regimes can provide glimpses of the asymptotic dynamics.

The large and small $Pm$ MHD regimes are seemingly very different. To see this, consider first the ordering of the resistive scale $\ell _{\unicode[STIX]{x1D702}}$ , i.e. the typical scale at which the magnetic field gets dissipated in MHD, with respect to the viscous scale $\ell _{\unicode[STIX]{x1D708}}$ .

Large magnetic Prandtl numbers. For $Pm>1$ , the resistive cutoff scale $\ell _{\unicode[STIX]{x1D702}}$ is smaller than the viscous scale. This suggests that a lot of the magnetic energy resides at scales well below any turbulent scale in the flow. The situation is best illustrated in figure 7 by taking a spectral point of view of the dynamics in wavenumber $k\sim 1/\ell$ space, introducing the kinetic and magnetic-energy spectra associated with the Fourier transforms in space of the velocity and magnetic field, and the viscous and resistive wavenumbers $k_{\unicode[STIX]{x1D708}}\sim 1/\ell _{\unicode[STIX]{x1D708}}$ and $k_{\unicode[STIX]{x1D702}}\sim 1/\ell _{\unicode[STIX]{x1D702}}$ (this kind of representation will be frequently encountered in the rest of the review). To estimate $\ell _{\unicode[STIX]{x1D702}}$ more precisely in this regime, let us consider the case of Kolmogorov turbulence for which the rate of strain of eddies of size $\ell$ goes as $u_{\ell }/\ell \sim \ell ^{-2/3}$ . For this kind of turbulence, the smallest viscous eddies are therefore also the fastest at stretching the magnetic field. To estimate the resistive scale $\ell _{\unicode[STIX]{x1D702}}$ , we balance the stretching rate of these eddies $u_{\unicode[STIX]{x1D708}}/\ell _{\unicode[STIX]{x1D708}}\sim Re^{1/2}u_{0}/\ell _{0}$ with the ohmic diffusion rate at the resistive scale $\unicode[STIX]{x1D702}/\ell _{\unicode[STIX]{x1D702}}^{2}$ . This gives

(2.24) $$\begin{eqnarray}\ell _{\unicode[STIX]{x1D702}}\sim Pm^{-1/2}\ell _{\unicode[STIX]{x1D708}},\quad Pm\gg 1.\end{eqnarray}$$

Figure 7. Ordering of scales and qualitative representation of the kinetic and magnetic-energy spectra in $k$ (wavenumber) space at large $Pm$ .

Low magnetic Prandtl numbers. For $Pm<1$ , we instead expect the resistive scale $\ell _{\unicode[STIX]{x1D702}}$ to fall in the inertial range of the turbulence. This is illustrated in spectral space in figure 8. To estimate $\ell _{\unicode[STIX]{x1D702}}$ in this regime, we simply balance the turnover/stretching rate $u_{\unicode[STIX]{x1D702}}/\ell _{\unicode[STIX]{x1D702}}$ of the eddies at scale $\ell _{\unicode[STIX]{x1D702}}$ with the magnetic-diffusion rate $\unicode[STIX]{x1D702}/\ell _{\unicode[STIX]{x1D702}}^{2}$ . Equivalently, this can be formulated as $Rm(\ell _{\unicode[STIX]{x1D702}})=u_{\ell _{\unicode[STIX]{x1D702}}}\ell _{\unicode[STIX]{x1D702}}/\unicode[STIX]{x1D702}\sim 1$ . The result is

(2.25) $$\begin{eqnarray}\ell _{\unicode[STIX]{x1D702}}\sim Pm^{-3/4}\ell _{\unicode[STIX]{x1D708}},\quad Pm\ll 1.\end{eqnarray}$$

Figure 8. Ordering of scales and qualitative representation of kinetic and magnetic-energy spectra at low $Pm$ .

Intuitively, the large- $Pm$ regime seems much more favourable to dynamos. In particular, the fact that the magnetic field ‘sees’ a lot of turbulent activity at low $Pm$ could create many complications. However, and contrary to what was for instance argued in the early days of dynamo theory by Batchelor (Reference Batchelor1950), we will see in the next sections that dynamo action is possible at low $Pm$ . Besides, the large- $Pm$ regime has a lot of non-trivial dynamics on display despite its seemingly simpler ordering of scales.

2.2.3 Strouhal number

Another important dimensionless quantity arising in dynamo theory is the Strouhal number

(2.26) $$\begin{eqnarray}St=\frac{\unicode[STIX]{x1D70F}_{c}}{\unicode[STIX]{x1D70F}_{\text{NL}}}.\end{eqnarray}$$

This number measures the ratio between the correlation time $\unicode[STIX]{x1D70F}_{c}$ and the nonlinear turnover time $\unicode[STIX]{x1D70F}_{\text{NL}}\sim \ell _{u}/u$ of an eddy with a typical velocity $u$ at scale $\ell _{u}$ . A similar number appears in all dynamical fluid and plasma problems involving closures and, despite being of order one in many physical systems worthy of interest (including fluid turbulence), is usually used as a small parameter to derive perturbative closures such as those described in the next two sections. Krommes (Reference Krommes2002) offers an illuminating discussion of the potential problems of perturbation theory applied to non-perturbative systems, many of which are directly relevant to dynamo theory.

2.3.1 Kinematic versus dynamical regimes

The question of the amplification and further sustainment of magnetic fields in MHD is fundamentally an instability problem with both linear and nonlinear aspects. The first thing that we usually need to assess is whether the stretching of the magnetic field by fluid motions can overcome its diffusion. The magnetic Reynolds number $Rm$ provides a direct measure of how these two processes compare, and is therefore the key parameter of the problem. Most, albeit not all, dynamo flows have a well-defined, analytically calculable or at least computable $Rm_{c}$ above which magnetic-field generation becomes possible.

In the presence of an externally prescribed velocity field (independent of $\boldsymbol{B}$ ), the induction equation (2.18) is linear in $\boldsymbol{B}$ . The kinematic dynamo problem therefore consists in establishing what flows, or classes of flows, can lead to exponential growth of magnetic energy starting from an initially infinitesimal seed magnetic field, and in computing $Rm_{c}$ of the bifurcation and growing eigenmodes of (2.18). The velocity field in the kinematic dynamo problem can be computed numerically from the forced Navier–Stokes equation with negligible Lorentz force,Footnote ⁵ or using simplified numerical flow models, or prescribed analytically. This linear problem is relevant to the early stages of magnetic-field amplification during which the magnetic energy is small compared to the kinetic energy of the flow.

The dynamical, or nonlinear dynamo problem, on the other hand, consists in solving the full nonlinear MHD system consisting of (2.1)–(2.6) (or the simpler equations (2.17)–(2.18) in the incompressible case), including the magnetic back reaction of the Lorentz force on the flow. This problem is obviously directly relevant to the saturation of dynamos, but it is more general than that. For instance, some systems with linear dynamo bifurcations exhibit subcritical bistability, i.e. they have pairs of nonlinear dynamo modes involving a magnetically distorted version of the flow at $Rm$ smaller than the kinematic $Rm_{c}$ . There is also an important class of dynamical magnetic-field-sustaining MHD processes, referred to as instability-driven dynamos, which do not originate in a linear bifurcation at all, and have no well-defined $Rm_{c}$ . These different mechanisms will be discussed in § 5.3.

2.3.2 Anti-dynamo theorems

Are all flows of conducting fluids dynamos? Despite the seemingly simple nature of induction illustrated by (2.14), there are actually many generic cases in which magnetic fields cannot be sustained by fluid motions in the limit of infinite times, even at large $Rm$ . Two of them are particularly important (and annoying) for the development of theoretical models and experiments: axisymmetric magnetic fields cannot be sustained by dynamo action (Cowling’s (Reference Cowling1933) theorem), and planar, two-dimensional motions cannot excite a dynamo (Zel’dovich’s (Reference Zel’dovich1956) theorem).

In order to give a general feel of the constraints that dynamos face, let us sketch qualitatively how Cowling’s theorem originates in an axisymmetric system in polar (cylindrical) geometry $(R,\unicode[STIX]{x1D711},z)$ . Assume that $\boldsymbol{B}$ is an axisymmetric vector field

(2.27) $$\begin{eqnarray}\boldsymbol{B}=\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D712}\boldsymbol{e}_{\unicode[STIX]{x1D711}}/R)+R\unicode[STIX]{x1D713}\boldsymbol{e}_{\unicode[STIX]{x1D711}},\end{eqnarray}$$

where $\unicode[STIX]{x1D712}(R,z,t)$ is a poloidal flux function and $R\unicode[STIX]{x1D713}$ is the toroidal magnetic field. Similarly, assume that $\boldsymbol{U}$ is axisymmetric with respect to the same axis of symmetry as $\boldsymbol{B}$ , i.e.

(2.28) $$\begin{eqnarray}\boldsymbol{U}=\boldsymbol{U}_{\text{pol}}+R\unicode[STIX]{x1D6FA}\boldsymbol{e}_{\unicode[STIX]{x1D711}},\end{eqnarray}$$

where $\boldsymbol{U}_{\text{pol}}(R,z,t),$ is an axisymmetric poloidal velocity field in the $(R,z)$ plane and $\unicode[STIX]{x1D6FA}(R,z,t)\boldsymbol{e}_{\unicode[STIX]{x1D711}}$ is an axisymmetric toroidal differential rotation. The poloidal and toroidal components of (2.18) respectively read

(2.29) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}}{\unicode[STIX]{x2202}t}+\boldsymbol{U}_{\text{pol}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D712}=\unicode[STIX]{x1D702}\left(\unicode[STIX]{x1D6E5}-\frac{2}{R}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}R}\right)\unicode[STIX]{x1D712}\end{eqnarray}$$

and

(2.30) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}t}+\boldsymbol{U}_{\text{pol}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}=\boldsymbol{B}_{\text{pol}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FA}+\unicode[STIX]{x1D702}\left(\unicode[STIX]{x1D6E5}+\frac{2}{R}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}R}\right)\unicode[STIX]{x1D713}.\end{eqnarray}$$

Equation (2.30) has a source term, $\boldsymbol{B}_{\text{pol}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FA}$ , which describes the stretching of poloidal field into toroidal field by the differential rotation, and is commonly referred to as the $\unicode[STIX]{x1D6FA}$ effect in the astrophysical dynamo community (more on this in § 4.2.1). However, there is no similar back coupling between $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D712}$ in (2.29), and therefore there is no converse way to generate poloidal field out of toroidal field in such a system. The problem is that there is no perennial source of poloidal flux $\unicode[STIX]{x1D712}$ in (2.29). The advection term on the left-hand side describes the redistribution/mixing of the flux by the axisymmetric poloidal flow in the $(R,z)$ plane and can only amplify the field locally and transiently. The presence of resistivity on the right-hand side then implies that $\unicode[STIX]{x1D712}$ must ultimately decay, and therefore so must the $\boldsymbol{B}_{\text{pol}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FA}$ source term in (2.30), and $\unicode[STIX]{x1D713}$ . Overall, the constrained geometry of this system therefore makes it impossible for the magnetic field to be sustained.Footnote ⁶

Cowling’s theorem is one of the main reasons why the solar and geo- dynamo problems are so complicated, as it notably shows that a magnetic dipole strictly aligned with the rotation axis cannot be sustained by a simple combination of axisymmetric differential rotation and meridional circulation. Note however that axisymmetric flows like the Dudley & James (Reference Dudley and James1989) flow or von Kármán flows (Marié et al. Reference Marié, Burguete, Daviaud and Léorat2003), on which the designs of several dynamo experiments are based, can excite non-axisymmetric dynamo fields with dominant equatorial dipole geometry ( $m=1$ modes with respect to the axis of symmetry of the flow). There are also mechanisms by which nearly axisymmetric magnetic fields can be generated in fluid flows with a strong axisymmetric mean component (Gissinger, Dormy & Fauve Reference Gissinger, Dormy and Fauve2008a ). We will find out in § 4 how the relaxation of the assumption of flow axisymmetry gives us the freedom to generate large-scale dynamos, albeit generally at the cost of a much-enhanced dynamical complexity.

Many other anti-dynamo theorems have been proven using similar reasonings. As mentioned above, the most significant one, apart from Cowling’s theorem, is Zel’dovich’s theorem that a two-dimensional planar flow (i.e. with only two components), $\boldsymbol{U}_{\text{2D}}(x,y,t)$ , cannot excite a dynamo. A purely toroidal flow cannot excite a dynamo either, and a magnetic field of the form $\boldsymbol{B}(x,y,t)$ alone cannot be a dynamo field. All these theorems are a consequence of the particular structure of the vector induction equation, and imply that a minimal geometric complexity is required for dynamos to work. But what does ‘minimal’ mean? As computer simulations were still in their infancy, a large number of applied mathematician brain hours were devoted to tailoring flows with enough dynamical and geometrical complexity to be dynamos, yet simple-enough mathematically to remain tractable analytically or with a limited computing capacity. Flows of two and a half dimensions (or two-dimensional with three components, 2D-3C) of the form $\boldsymbol{U}(x,y,t)$ with three non-vanishing components (i.e. including a $z$ component) are popular configurations of this kind, that make it possible to overcome anti-dynamo theorems at a minimal cost. Some well-known examples are 2D-3C versions of the Roberts flow (Roberts Reference Roberts1972), and the Galloway–Proctor flow (GP, Galloway & Proctor Reference Galloway and Proctor1992). The original version of the latter is periodic in time and therefore has relatively simple kinematic dynamo eigenmodes of the form $\boldsymbol{B}(x,y,z,t)={\mathcal{R}}\{\boldsymbol{B}_{\text{2D}-\text{3C}}(x,y,t)\exp (st+\text{i}k_{z}z)\}$ , where $k_{z}$ , the wavenumber of the magnetic perturbation along the $z$ direction, is a parameter of the problem, $\boldsymbol{B}_{\text{2D}-\text{3C}}(x,y,t)$ has the same time periodicity as the flow and $s$ is the (a priori complex) growth rate of the dynamo for a given $k_{z}$ . While such flows are very peculiar in many respects, they have been instrumental in the development of theoretical and experimental dynamo research and have taught us a lot on the dynamo problem in general. They retain some popularity nowadays because they can be used to probe kinematic dynamos in higher- $Rm$ regimes than in the fully three-dimensional (3-D) problem by concentrating all the numerical resolution and computing power into just two spatial dimensions. A contemporary example of this kind of approach will be given in § 5.4.

Figure 9. The famous stretch-twist-fold dynamo cartoon, adapted from Vainshtein & Zel’dovich (Reference Vainshtein and Zel’dovich1972) and many others.

2.3.3 Slow versus fast dynamos

Dynamos can be either slow or fast. Slow dynamos are dynamos whose existence hinges on the spatial diffusion of the magnetic field to couple different field components. These dynamos therefore typically evolve on a large, system-scale ohmic diffusion time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702},0}=\ell _{0}^{2}/\unicode[STIX]{x1D702}$ and their growth rate tends to zero as $Rm\rightarrow \infty$ , for instance (but not necessarily) as some inverse power of $Rm$ . For this reason, they are probably not relevant to astrophysical systems with very large $Rm$ and dynamical magnetic time scales much shorter than $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702},0}$ . A classic example is the Roberts (Reference Roberts1970, Reference Roberts1972) dynamo, which notably served as an inspiration for one of the first experimental demonstrations of the dynamo effect in Karlsruhe (Stieglitz & Müller Reference Stieglitz and Müller2001; Gailitis et al. Reference Gailitis, Lielausis, Platacis, Gerbeth and Stefani2002). Fast dynamos, on the other hand, are dynamos whose growth rate remains finite and becomes independent of $Rm$ as $Rm\rightarrow \infty$ . Although it is usually very hard to formally prove that a dynamo is fast, most dynamo processes discussed in the next sections are thought to be fast, and so is for instance the previously mentioned Galloway & Proctor (Reference Galloway and Proctor1992) dynamo. The difficult analysis of the Ponomarenko dynamo case illustrates the general trickiness of this question (Gilbert Reference Gilbert1988). A more detailed comparative discussion of the characteristic properties of classic examples of slow and fast dynamo flows is given by Proctor at p. 186 of the Encyclopedia of Geomagnetism and Paleomagnetism edited by Gubbins & Herrero-Bervera (Reference Gubbins and Herrero-Bervera2007).

The standard physical paradigm of fast dynamos, originally due to Vainshtein & Zel’dovich (Reference Vainshtein and Zel’dovich1972), is the stretch, twist, fold mechanism pictured in numerous texts, including here in figure 9. In this picture, a loop of magnetic field is stretched by shearing fluid motions so that the field strength increases by a typical factor two over a turnover time through magnetic flux conservation. If the field is (subsequently or simultaneously) further twisted and folded by out-of-plane motions, we obtain a fundamentally 3-D ‘double tube’ similar to that shown at the bottom of figure 9. In that configuration, the magnetic field in each flux tube has the same orientation as in the neighbouring tube. The initial geometric configuration can then be recovered by diffusive merging of two loops, but with almost double magnetic field compared to the original situation. If we think of this cycle as being a single iteration of a repetitive discrete process (a discrete map), with each iteration corresponding to a typical fluid eddy turnover, then we have all the ingredients of a self-exciting process, whose growth rate in the ideal limit of infinite $Rm$ is $\unicode[STIX]{x1D6FE}_{\infty }=\ln 2$ (inverse turnover times). Only a tiny magnetic diffusivity is required for the merging, as the latter can take place at arbitrary small scale. The overall process is therefore not diffusion limited.

One of the fundamental ingredients here is the stretching of the magnetic field by the flow. More generally, it can be shown that an essential requirement for fast dynamo action is that the flow exhibits Lagrangian chaos (Finn & Ott Reference Finn and Ott1988), i.e. trajectories of initially close fluid particles must diverge exponentially, at least in some flow regions. This key aspect of the problem, and its implications for the structure of dynamo magnetic fields, will become more explicit in the discussion of the small-scale dynamo phenomenology in the next section. Many fundamental mathematical aspects of fast kinematic dynamos have been studied in detail using the original induction equation or simpler idealised discrete maps that capture the essence of this dynamics. We will not dive into this subject any further here, as it quickly becomes very technical, and has already extensively been covered in dedicated reviews and books, including the monograph of Childress & Gilbert (Reference Childress and Gilbert1995) and a chapter by Andrew Soward in a collective book of lectures on dynamos edited by Proctor & Gilbert (Reference Proctor and Gilbert1994).

3.3.1 Small-scale dynamo fields at $Pm>1$

A linear shear flow has a spatially uniform gradient and is therefore the ultimate example of a large-scale shear flow. The magnetic mode that results from this kind of dynamo, on the other hand, has typical reversals at the resistive scale $\ell _{\unicode[STIX]{x1D702}}$ , which of course becomes very small at large $Rm$ . The problem described above is therefore implicitly typical of the large- $Pm$ MHD regime introduced in § 2.2.2. The fastest shearing eddies at large $Pm$ in Kolmogorov turbulence are spatially smooth, yet chaotic viscous eddies, and take on the role of the random linear shear flow in the Zel’dovich model. Interestingly, because this dynamo only requires a smooth, chaotic flow to work, there should be no problem with exciting it down to $Re=O(1)$ (random Stokes flow). On the other hand, $Rm$ must be large enough for stretching to win over diffusion. There is therefore always a minimal requirement to resolve resistive-scale reversals in numerical simulations (typically the 64 Fourier modes per spatial dimension of the Meneguzzi et al. simulation).

Many three-dimensional direct numerical simulations (DNS) of the kind conducted by Meneguzzi et al. have now been performed, that essentially confirm the Zel’dovich phenomenology and folded field structure of the small-scale dynamo in the $Pm>1$ regime. Snapshots of the smooth velocity field and particularly clean folded magnetic field structures in the relatively asymptotic large- $Pm$ regime $Re=1$ , $Pm=1250$ , are shown in figure 14. The Fourier spectra of these two images (not shown) are obviously very different, which is of course reminiscent of the Meneguzzi et al. results. In fact, all simulations down to $Pm=O(1)$ , including the Meneguzzi et al. experiment, essentially produce a dynamo of the kind described above. Figure 15 provides a map in the $(Re,Rm)$ plane of the dynamo growth rate $\unicode[STIX]{x1D6FE}$ of the small-scale dynamo, and a plot of the critical magnetic Reynolds number $Rm_{c,\text{ssd}}$ above which the dynamo is excited ( $Rm$ here and in figures 14 and 16 is defined as $Rm=u_{\text{rms}}\ell _{0}/(2\unicode[STIX]{x03C0}\unicode[STIX]{x1D702})$ , where $u_{\text{rms}}$ is the root mean square (r.m.s.) velocity). $Rm_{c,\text{ssd}}$ is found to be almost independent of $Pm$ and approximately equal to 60 for $Pm>1$ . As $Pm$ decreases from large values to unity, the folded field structure gradually becomes more intricate, but for instance we can always spot very fast field reversals perpendicular to the field itself. This gradual change can be seen on the two left-most 2-D snapshots of figure 16 representing $|\boldsymbol{B}|$ in simulations at $Pm=1250$ and $Pm=1$ .

Figure 14. (a) Two-dimensional snapshot of $|\boldsymbol{u}|$ in a 3-D simulation of non-helical, homogeneous, isotropic smooth random flow forced at the box scale $\ell _{0}$ for $Re=1$ ( $\ell _{\unicode[STIX]{x1D708}}=\ell _{0}$ ). (b) Snapshot of the strength $|\boldsymbol{B}|$ of the kinematic dynamo magnetic field generated by this flow for $Rm=Pm=1250$ , and corresponding magnetic field directions (arrows). The field in this large- $Pm$ regime has a strongly folded geometric structure: it is almost uniform along itself, but reverses on the very fine scale $\ell _{\unicode[STIX]{x1D702}}/\ell _{0}\sim Pm^{-1/2}$ , ${\sim}0.03$ in that example. The brighter regions correspond to large field strengths (adapted from Schekochihin et al. Reference Schekochihin, Cowley, Taylor, Maron and McWilliams2004c ).

Figure 15. Critical magnetic Reynolds number $Rm_{c,\text{ssd}}$ (black solid line with full circles) and growth rates (colour squares) of the kinematic small-scale dynamo excited by non-helical, homogeneous, isotropic turbulence forced at the box scale, as a function of $Re$ . The parameter range of the plot corresponds approximately to the grey box in figure 6. $Rm_{c,\text{ssd}}$ increases by a factor almost four for $Pm<1$ (adapted from Schekochihin et al. Reference Schekochihin, Iskakov, Cowley, McWilliams, Proctor and Yousef2007).

Figure 16. Two-dimensional snapshots of the strength $|\boldsymbol{B}|$ of the kinematic dynamo magnetic field for 3-D simulations of non-helical, homogeneous, isotropic turbulence forced at the box scale. (a)  $Pm=Rm=1250$ , $Re=1$ . (b)  $Pm=1$ , $Re=Rm=440$ . (c)  $Pm=0.07$ , $Rm=430$ , $Re=6200$ (because this particular simulation uses hyperviscous dissipation only, the kinetic Reynolds number in this case is defined using an effective viscosity determined statistically from the simulation data). Note the very different magnetic-field structures between the $Pm=1$ and $Pm=0.07$ cases, despite $Rm$ being essentially the same in both simulations (adapted from Schekochihin et al. Reference Schekochihin, Iskakov, Cowley, McWilliams, Proctor and Yousef2007).

3.3.2 Small-scale dynamo fields at $Pm<1$

What about the $Pm<1$ case? Batchelor (Reference Batchelor1950) argued based on an analogy between the induction equation and the vorticity equation, that there could be no dynamo at all for $Pm<1$ (a concise account of Batchelor’s arguments on the small-scale dynamo can be found in Davidson (Reference Davidson2013), § 18.3). As explained in § 2.2.2, the magnetic field sees a very different, and much more irregular velocity field in the low- $Pm$ regime, and we would naturally expect this to have a negative impact on the dynamo. Whether the dynamo survives in this regime remained an open and somewhat controversial theoretical and numerical question for many years (Vainshtein Reference Vainshtein1982; Novikov, Ruzmaikin & Sokolov Reference Novikov, Ruzmaikin and Sokolov1983; Vainshtein & Kichatinov Reference Vainshtein and Kichatinov1986; Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin1997; Vincenzi Reference Vincenzi2002; Haugen, Brandenburg & Dobler Reference Haugen, Brandenburg and Dobler2004; Schekochihin et al. Reference Schekochihin, Cowley, Maron and McWilliams2004a ; Boldyrev & Cattaneo Reference Boldyrev and Cattaneo2004; Schekochihin et al. Reference Schekochihin, Haugen, Brandenburg, Cowley, Maron and McWilliams2005b ).

The first conclusive numerical demonstrations of kinematic dynamos at low $Pm$ in non-helical isotropic homogeneous turbulence were only performed a few years ago by Iskakov et al. (Reference Iskakov, Schekochihin, Cowley, McWilliams and Proctor2007). While the question of the optimal numerical configuration to reach the low- $Pm$ dynamo regime is not entirely settled, the main results of these Meneguzzi-like simulations of homogeneous, isotropic turbulence are that the critical $Rm$ of the dynamo increases significantly as $Pm$ becomes smaller than one (see figure 15), and that the nature of the low- $Pm$ dynamo is quite different from its large- $Pm$ counterpart. The right-most snapshot of figure 16 shows for instance that the structure of the magnetic field at $Pm<1$ is radically altered in comparison to even the $Pm=1$ case. The disappearance of the folded field structure is perhaps not that surprising, given that we are completely outside of the domain of applicability of Zel’dovich’s smooth flow phenomenology for $Pm<1$ . Unfortunately, a clear physical understanding of the low- $Pm$ small-scale kinematic dynamo process comparable to that of the large $Pm$ case is still lacking. As we will see in the next paragraph, though, the increase in $Rm_{c,\text{ssd}}$ at low $Pm$ can be directly tied to the roughness of the velocity field at the resistive scale, within the framework of the mathematical Kazantsev model.

Numerically, the problem with the low- $Pm$ regime is that one must simultaneously ensure that $Rm$ is large enough to trigger the dynamo ( $Rm_{c,\text{ssd}}$ at low $Pm$ appears to be at least a factor two larger than at large $Pm$ depending on how the problem is set up), and that $Re$ is significantly larger than $Rm$ ! In practical terms, a resolution of $512^{3}$ is required to simulate such high $Re$ turbulence in pseudo-spectral numerical simulations with explicit Laplacian dissipation. Only now is this kind of MHD simulation becoming routine in computational fluid dynamics. Note finally that the excitation of small-scale dynamos at both $Pm>1$ and $Pm<1$ appears to be quite independent of the hydrodynamic turbulent forcing mechanism, and even of the details of the turbulent flow. For instance, results similar to figure 15 have been obtained using hyperviscosity in DNS, MHD shell models (Stepanov & Plunian Reference Stepanov and Plunian2006; Buchlin Reference Buchlin2011) and DNS and large-eddy simulations of the (turbulent) Taylor–Green flow (Ponty et al. Reference Ponty, Mininni, Montgomery, Pinton, Politano and Pouquet2005). Of importance to astrophysics, small-scale fluctuation dynamo action is also known to be effective in $Pm>1$ simulations of turbulent thermal convection, Boussinesq and stratified alike (Nordlund et al. Reference Nordlund, Brandenburg, Jennings, Rieutord, Ruokolainen, Stein and Tuominen1992; Cattaneo Reference Cattaneo1999; Vögler & Schüssler Reference Vögler and Schüssler2007; Pietarila Graham, Cameron & Schüssler Reference Pietarila Graham, Cameron and Schüssler2010; Moll et al. Reference Moll, Pietarila Graham, Pratt, Cameron, Müller and Schüssler2011; Bushby & Favier Reference Bushby and Favier2014). Low- $Pm$ turbulent convection is also widely thought to be the main driver of small-scale solar-surface magnetic fields, although clean, conclusive DNS simulations of a fluctuation dynamo driven by turbulent convection at $Pm$ significantly smaller than one have still not been conducted.

3.4.1 Kazantsev–Kraichnan assumptions on the velocity field

We consider a three-dimensional, statistically steady and homogeneous fluctuating incompressible velocity field with two-point, two-time correlation function

(3.16) $$\begin{eqnarray}\langle u^{i}(\boldsymbol{x},t)u^{j}(\boldsymbol{x}^{\prime },t^{\prime })\rangle =R^{ij}(\boldsymbol{x}^{\prime }-\boldsymbol{x},t^{\prime }-t).\end{eqnarray}$$

We assume that $\boldsymbol{u}$ has Gaussian statistics,

(3.17) $$\begin{eqnarray}P[\boldsymbol{u}]=C\exp \left[-\frac{1}{2}\int \text{d}t\int \text{d}t^{\prime }\int \text{d}^{3}\boldsymbol{x}\int \text{d}^{3}\boldsymbol{x}^{\prime }\unicode[STIX]{x1D60B}_{ij}(t^{\prime }-t,\boldsymbol{x}^{\prime }-\boldsymbol{x})u^{i}(\boldsymbol{x},t)u^{j}(\boldsymbol{x}^{\prime },t^{\prime })\right],\end{eqnarray}$$

where $C$ is a normalisation factor and the covariance matrix $\unicode[STIX]{x1D60B}_{ij}$ is related to $R^{ij}$ by

(3.18) $$\begin{eqnarray}\int \text{d}\unicode[STIX]{x1D70F}\int \text{d}^{3}\boldsymbol{y}\unicode[STIX]{x1D60B}_{ik}(t^{\prime }-\unicode[STIX]{x1D70F},\boldsymbol{x}^{\prime }-\boldsymbol{y})R^{kj}(\unicode[STIX]{x1D70F}-t,\boldsymbol{y}-\boldsymbol{x})=\unicode[STIX]{x1D6FF}_{i}^{j}\unicode[STIX]{x1D6FF}(t^{\prime }-t)\unicode[STIX]{x1D6FF}(\boldsymbol{x}^{\prime }-\boldsymbol{x}).\end{eqnarray}$$

We further assume that $\boldsymbol{u}$ is $\unicode[STIX]{x1D6FF}$ -correlated in time,

(3.19) $$\begin{eqnarray}\langle u^{i}(\boldsymbol{x},t)u^{j}(\boldsymbol{x}^{\prime },t^{\prime })\rangle =R^{ij}(\boldsymbol{x}^{\prime }-\boldsymbol{x},t^{\prime }-t)=\unicode[STIX]{x1D705}^{ij}(\boldsymbol{r})\unicode[STIX]{x1D6FF}(t^{\prime }-t),\end{eqnarray}$$

where $\boldsymbol{r}=\boldsymbol{x}^{\prime }-\boldsymbol{x}$ is the spatial correlation vector. We restrict the calculation to the isotropic, non-helical case for the time being (the helical case is also interesting in the context of large-scale dynamo theory and will be discussed in § 4.5.2). In the absence of a particular axis of symmetry in the system and of helicity, we are only allowed to use $\unicode[STIX]{x1D6FF}^{ij}$ and $\boldsymbol{r}$ to construct $\unicode[STIX]{x1D705}^{ij}(\boldsymbol{r})$ ,Footnote ⁸ and the most general expression that we can form is

(3.20) $$\begin{eqnarray}\unicode[STIX]{x1D705}^{ij}(\boldsymbol{r})=\unicode[STIX]{x1D705}_{N}(r)\left(\unicode[STIX]{x1D6FF}^{ij}-\frac{r^{i}r^{j}}{r^{2}}\right)+\unicode[STIX]{x1D705}_{L}(r)\frac{r^{i}r^{j}}{r^{2}},\end{eqnarray}$$

where $r=|\boldsymbol{r}|$ and $\unicode[STIX]{x1D705}_{N}$ and $\unicode[STIX]{x1D705}_{L}$ are the tangential and longitudinal velocity correlation functions. For an incompressible/solenoidal vector field, we have

(3.21) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{N}=\unicode[STIX]{x1D705}_{L}+(r\unicode[STIX]{x1D705}_{L}^{\prime })/2.\end{eqnarray}$$

3.4.2 Equation for the magnetic-field correlator

Our goal is to derive a closed equation for the two-point, single-time magnetic correlation function (or, equivalently, for the magnetic-energy spectrum, as the two are related by a Fourier transform)

(3.22) $$\begin{eqnarray}\langle B^{i}(\boldsymbol{x},t)B^{j}(\boldsymbol{x}^{\prime },t)\rangle =H^{ij}(\boldsymbol{r},t).\end{eqnarray}$$

For the same reasons as above, we can write

(3.23) $$\begin{eqnarray}\displaystyle & \displaystyle H^{ij}(\boldsymbol{r},t)=H_{N}(r,t)\left(\unicode[STIX]{x1D6FF}^{ij}-\frac{r^{i}r^{j}}{r^{2}}\right)+H_{L}(r,t)\frac{r^{i}r^{j}}{r^{2}}, & \displaystyle\end{eqnarray}$$

(3.24) $$\begin{eqnarray}\displaystyle & \displaystyle H_{N}=H_{L}+(rH_{L}^{\prime })/2. & \displaystyle\end{eqnarray}$$

Taking the $i$ th component of the induction equation at point $(\boldsymbol{x},t)$ and multiplying it by $\boldsymbol{B}^{j}(\boldsymbol{x}^{\prime },t)$ , then taking the $j$ th component at point $(\boldsymbol{x}^{\prime },t)$ and multiplying it by $\boldsymbol{B}^{i}(\boldsymbol{x},t)$ we find, after adding the two results, the evolution equation for $H^{ij}$ ,

(3.25) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}H^{ij}}{\unicode[STIX]{x2202}t} & = & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}{x^{\prime }}^{k}}(\langle B^{i}(\boldsymbol{x},t)B^{k}(\boldsymbol{x}^{\prime },t)u^{j}(\boldsymbol{x}^{\prime },t)\rangle -\langle B^{i}(\boldsymbol{x},t)B^{j}(\boldsymbol{x}^{\prime },t)u^{k}(\boldsymbol{x}^{\prime },t)\rangle )\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{k}}(\langle B^{k}(\boldsymbol{x},t)B^{j}(\boldsymbol{x}^{\prime },t)u^{i}(\boldsymbol{x},t)\rangle -\langle B^{i}(\boldsymbol{x},t)B^{j}(\boldsymbol{x}^{\prime },t)u^{k}(\boldsymbol{x},t)\rangle )\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D702}\left(\frac{\unicode[STIX]{x2202}^{2}}{{\unicode[STIX]{x2202}x^{k}}^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{{\unicode[STIX]{x2202}{x^{\prime }}^{k}}^{2}}\right)H^{ij},\end{eqnarray}$$

where $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}{x^{\prime }}^{k}\langle \cdot \rangle =-\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x^{k}\langle \cdot \rangle =\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r^{k}\langle \cdot \rangle$ because of statistical homogeneity. Equation (3.25) is exact, but we are now faced with an important difficulty: the time derivative of the second-order magnetic correlation function depends on mixed third-order correlation functions, and we do not have explicit expressions for these correlators. We could write down evolution equations for them too, but their right-hand side would then involve fourth-order correlation functions, and so on. This is a familiar closure problem.

3.4.3 Closure procedure in a nutshell*

In order to make further progress, we have to find a (hopefully physically) way to truncate the system of equations by replacing the higher-order correlation functions with lower-order ones. This is where the Kazantsev–Kraichnan assumptions of a random, $\unicode[STIX]{x1D6FF}$ -correlated-in-time Gaussian velocity field come into play. The assumption of Gaussian statistics implies that $n$ th-order mixed correlation functions involving $\boldsymbol{u}$ can be expressed in terms of $(n-1)$ th-order correlation functions using the Furutsu–Novikov (Gaussian integration) formula,

(3.26) $$\begin{eqnarray}\langle u^{i}(\boldsymbol{x},t)F[\boldsymbol{u}]\rangle =\int \text{d}t^{\prime \prime }\int \text{d}^{3}\boldsymbol{x}^{\prime \prime }\langle u^{i}(\boldsymbol{x},t)u^{l}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime })\rangle \left\langle \frac{\unicode[STIX]{x1D6FF}F[\boldsymbol{u}]}{\unicode[STIX]{x1D6FF}u^{l}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime })}\right\rangle ,\end{eqnarray}$$

where $F[\boldsymbol{u}]$ stands for any functional of $\boldsymbol{u}$ , and the $\unicode[STIX]{x1D6FF}F/\unicode[STIX]{x1D6FF}u^{i}$ are functional derivatives. We can use this formula in (3.25) to replace all the third-order moments appearing in the right-hand side by integrals of products of second-order moments. To illustrate how the closure procedure proceeds, let us isolate just one of these terms,

(3.27) $$\begin{eqnarray}\displaystyle \langle u^{i}(\boldsymbol{x},t)B^{k}(\boldsymbol{x},t)B^{j}(\boldsymbol{x}^{\prime },t)\rangle =\int _{0}^{t}\text{d}t^{\prime \prime }\int \text{d}^{3}\boldsymbol{x}^{\prime \prime }\langle u^{i}(\boldsymbol{x},t)u^{l}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime })\rangle \left\langle \frac{\unicode[STIX]{x1D6FF}[B^{k}(\boldsymbol{x},t)B^{j}(\boldsymbol{x}^{\prime },t)]}{\unicode[STIX]{x1D6FF}u^{l}(\boldsymbol{x}^{\prime \prime },t^{\prime \prime })}\right\rangle . & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Applying the Gaussian integration formula is a critical first step, but more work is needed. In particular, equation (3.27) involves a time integral encapsulating the effects of flow memory. For a generic turbulent flow for which the correlation time is not small compared the relevant dynamical time scales of the problem, the problem is non-perturbative and there is no known method to calculate such an integral exactly. However, as a first step we could still assume that it is small, and perform the integral perturbatively (the expansion parameter will be the Strouhal number). The Kazantsev–Kraichnan assumption of zero correlation time corresponds to the lowest-order calculation. Using (3.19) in (3.27) removes the time integral and leaves us with the task of calculating the equal-time functional derivative $\langle \unicode[STIX]{x1D6FF}[B^{k}(\boldsymbol{x},t)B^{j}(\boldsymbol{x}^{\prime },t)]/\unicode[STIX]{x1D6FF}u^{l}(\boldsymbol{x}^{\prime \prime },t)\rangle$ . This expression can be explicitly calculated using the expression of the formal solution of the induction equation,

(3.28) $$\begin{eqnarray}B^{k}(\boldsymbol{x},t)=\int ^{t}\text{d}t^{\prime }\left[B^{m}(\boldsymbol{x},t^{\prime })\frac{\unicode[STIX]{x2202}u^{k}(\boldsymbol{x},t^{\prime })}{\unicode[STIX]{x2202}x^{m}}-u^{m}(\boldsymbol{x},t^{\prime })\frac{\unicode[STIX]{x2202}B^{k}(\boldsymbol{x},t^{\prime })}{\unicode[STIX]{x2202}x^{m}}+\unicode[STIX]{x1D702}\unicode[STIX]{x0394}B^{k}(\boldsymbol{x},t^{\prime })\right].\end{eqnarray}$$

Functionally differentiating this equation (and that for $B^{j}(\boldsymbol{x}^{\prime },t)$ ) with respect to $\unicode[STIX]{x1D6FF}u^{l}(\boldsymbol{x}^{\prime \prime },t)$ introduces $\unicode[STIX]{x1D6FF}(\boldsymbol{x}^{\prime }-\boldsymbol{x}^{\prime \prime })$ and $\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}^{\prime \prime })$ , which makes the space integral in (3.27) trivial and completes the closure procedure.

The end result of the full calculation outlined above are expressions of all the mixed third-order correlation functions in terms of a combination of products of the two-point second-order correlation function of the magnetic field with the (prescribed) second-order correlation function of the velocity field (and their spatial derivatives).

3.4.4 Closed equation for the magnetic correlator

From there, it can be shown easily using appropriate projection operators and isotropy that the original, complicated unclosed equation (3.25) reduces to the much simpler closed scalar equation for $H_{L}(r,t)$ ,

(3.29) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}H_{L}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D705}H_{L}^{\prime \prime }+\left(\frac{4}{r}\unicode[STIX]{x1D705}+\unicode[STIX]{x1D705}^{\prime }\right)H_{L}^{\prime }+\left(\unicode[STIX]{x1D705}^{\prime \prime }+\frac{4}{r}\unicode[STIX]{x1D705}^{\prime }\right)H_{L},\end{eqnarray}$$

where

(3.30) $$\begin{eqnarray}\unicode[STIX]{x1D705}(r)=2\unicode[STIX]{x1D702}+\unicode[STIX]{x1D705}_{L}(0)-\unicode[STIX]{x1D705}_{L}(r)\end{eqnarray}$$

can be interpreted as (twice) the ‘turbulent diffusivity’. If we now perform the change of variables

(3.31) $$\begin{eqnarray}H_{L}(r,t)=\frac{\unicode[STIX]{x1D713}(r,t)}{r^{2}\unicode[STIX]{x1D705}(r)^{1/2}},\end{eqnarray}$$

we find that (3.29) reduces to a Schrödinger equation with imaginary time

(3.32) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D705}(r)\unicode[STIX]{x1D713}^{\prime \prime }-V(r)\unicode[STIX]{x1D713},\end{eqnarray}$$

which describes the evolution of the wave function of a quantum particle of variable mass

(3.33) $$\begin{eqnarray}m(r)=\frac{1}{2\unicode[STIX]{x1D705}(r)}\end{eqnarray}$$

in the potential

(3.34) $$\begin{eqnarray}V(r)=\frac{2}{r^{2}}\unicode[STIX]{x1D705}(r)-\frac{1}{2}\unicode[STIX]{x1D705}^{\prime \prime }(r)-\frac{2}{r}\unicode[STIX]{x1D705}^{\prime }(r)-\frac{\unicode[STIX]{x1D705}^{\prime }(r)^{2}}{4\unicode[STIX]{x1D705}(r)}.\end{eqnarray}$$

3.4.5 Solutions

We can then look for solutions of (3.32) of the form

(3.35) $$\begin{eqnarray}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}_{E}(r)\text{e}^{-Et}.\end{eqnarray}$$

Keeping in mind that $\unicode[STIX]{x1D713}$ stands for the magnetic correlation function, we see that exponentially growing dynamo modes correspond to discrete $E<0$ bound states. The existence of such modes depends on the shape of the Kazantsev potential, which is entirely determined by the statistical properties of the velocity field encapsulated in the function $\unicode[STIX]{x1D705}(r)$ . The variational result for $E$ is

(3.36) $$\begin{eqnarray}E=\frac{\displaystyle \int 2mV{\unicode[STIX]{x1D713}_{E}}^{2}\,\text{d}r+\int {\unicode[STIX]{x1D713}_{E}^{\prime }}^{2}\,\text{d}r}{\displaystyle \int 2m{\unicode[STIX]{x1D713}_{E}}^{2}\,\text{d}r}.\end{eqnarray}$$

If we are just interested in the question of whether a dynamo is possible, we can equivalently solve the equation

(3.37) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{E}^{\prime \prime }+[E-V_{\text{eff}}(r)]\unicode[STIX]{x1D713}_{E}=0,\end{eqnarray}$$

where

(3.38) $$\begin{eqnarray}V_{\text{eff}}(r)=\frac{V(r)}{\unicode[STIX]{x1D705}(r)}.\end{eqnarray}$$

The ground state describes the fastest growing mode, and therefore the long-time asymptotics in the kinematic regime of the dynamo.

3.4.6 Different regimes

After all this hard work, we are now almost in a position to answer much more specific questions. For instance, we would like to predict whether dynamo action is possible in a smooth flow (something we already know from § 3.2), and in a turbulent flow with Kolmogorov scalings (something the Zel’dovich et al. model does not tell us). Let us assume that the velocity field has power-law scaling,

(3.39) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{L}(0)-\unicode[STIX]{x1D705}_{L}(r)\sim r^{\unicode[STIX]{x1D709}},\end{eqnarray}$$

where $\unicode[STIX]{x1D709}$ is called the roughness exponent. Introducing the resistive scale $\ell _{\unicode[STIX]{x1D702}}=(2\unicode[STIX]{x1D702})^{1/\unicode[STIX]{x1D709}}$ , the effective Kazantsev potential as a function of $\unicode[STIX]{x1D709}$ is

(3.40) $$\begin{eqnarray}\displaystyle V_{\text{eff}}(r) & = & \displaystyle \frac{2}{r^{2}}\quad \text{for }r\ll \ell _{\unicode[STIX]{x1D702}},\end{eqnarray}$$

(3.41) $$\begin{eqnarray}\displaystyle & = & \displaystyle \frac{2-3\unicode[STIX]{x1D709}/2-3\unicode[STIX]{x1D709}^{2}/4}{r^{2}}\quad \text{for }r\gg \ell _{\unicode[STIX]{x1D702}}.\end{eqnarray}$$

The overall shape of the potential is illustrated in figure 17 for different values of $\unicode[STIX]{x1D709}$ . The potential becomes attractive at $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}_{c}=\sqrt{11/3}-1$ , but growing bound modes only exist for $\unicode[STIX]{x1D709}>1$ .

Figure 17. Kazantsev potential as a function of $r$ for different roughness exponents $\unicode[STIX]{x1D709}$ . An attractive potential forms at $\unicode[STIX]{x1D709}_{c}$ . Bound (growing) dynamo modes require $\unicode[STIX]{x1D709}>1$ .

In order to find out which kind of flows are dynamos according to the Kazantsev model, we have to make a small handwaving argument in order to relate the scaling properties of flows with finite correlation times to those of Kazantsev–Kraichnan velocity fields. Recalling that $\langle u^{i}(\boldsymbol{x},t)u^{j}(\boldsymbol{x}^{\prime },t^{\prime })\rangle =\unicode[STIX]{x1D705}^{ij}(\boldsymbol{r})\unicode[STIX]{x1D6FF}(t^{\prime }-t)$ , and that a $\unicode[STIX]{x1D6FF}$ function is dimensionally the inverse of a time, we write

(3.42) $$\begin{eqnarray}\unicode[STIX]{x1D705}(r)\sim \unicode[STIX]{x1D6FF}u(r)^{2}\unicode[STIX]{x1D70F}(r)\sim r\unicode[STIX]{x1D6FF}u(r),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}u(r)$ is the typical velocity difference between two points separated by $r$ , and we have assumed that the relevant time scale in the dimensional analysis is the scale-dependent eddy turnover time $\unicode[STIX]{x1D70F}(r)\sim r/\unicode[STIX]{x1D6FF}u(r)$ (this twiddle-algebra analysis also clarifies the association of $\unicode[STIX]{x1D705}$ with a turbulent diffusivity). Using (3.42), we find that $\unicode[STIX]{x1D709}=2$ for a smooth flow whose velocity increments scale as $\unicode[STIX]{x1D6FF}u\sim r$ (the linear shears of the Zel’dovich phenomenology). Qualitatively, it is tempting to associate this $\unicode[STIX]{x1D709}=2$ case with a large- $Pm$ regime because all the magnetic field sees at large $Pm$ is a large-scale, smooth viscous flow. On the other hand, for a flow with Kolmogorov scaling, $\unicode[STIX]{x1D6FF}u\sim r^{1/3}$ and $\unicode[STIX]{x1D709}=1+1/3=4/3$ . This case would instead correspond to a low- $Pm$ regime in which the magnetic field sits in the middle of the inertial range.

According to the Kazantsev model, a necessary condition for small-scale dynamo action to be possible is that $\unicode[STIX]{x1D709}>1$ . This condition is satisfied in both smooth flows ( $\unicode[STIX]{x1D709}=2$ , $Pm\gg 1$ regime) and rough flows, including Kolmogorov-like turbulence ( $\unicode[STIX]{x1D709}=4/3$ , $Pm\ll 1$ regime). The study of the low- $Pm$ -like case was not part of the original Kazantsev (Reference Kazantsev1967) article, and only appeared in later work by various authors (Vainshtein (Reference Vainshtein1982), Vainshtein & Kichatinov (Reference Vainshtein and Kichatinov1986), Vincenzi (Reference Vincenzi2002) and Boldyrev & Cattaneo (Reference Boldyrev and Cattaneo2004), see also Eyink (Reference Eyink2010), Eyink & Neto (Reference Eyink and Neto2010) for a Lagrangian perspective on the problem). In this respect, we should also mention the work of Kraichnan & Nagarajan (Reference Kraichnan and Nagarajan1967) contemporary to that of Kazantsev. Kraichnan & Nagarajan arrived at a closed equation for the evolution of the magnetic-energy spectrum similar to that derived by Kazantsev using a different closure procedure, and found exponential growth of magnetic energy by solving this equation numerically as an initial value problem for a prescribed Kolmogorov velocity spectrum.

3.4.7 Critical $Rm$

Another interesting result that can be derived from the Kazantsev model is the existence of a critical $Rm$ (Ruzmaikin & Sokolov Reference Ruzmaikin and Sokolov1981). A finite scale separation $\ell _{0}/\ell _{\unicode[STIX]{x1D702}}$ is introduced in the problem by assuming the existence of an integral scale $\ell _{0}$ beyond which the velocity field decorrelates. The shape of the Kazantsev potential in that case is shown in figure 18, and the asymptotic form is

(3.43) $$\begin{eqnarray}\displaystyle V_{\text{eff}}(r) & = & \displaystyle \frac{2}{r^{2}}\quad \text{for }r\ll \ell _{\unicode[STIX]{x1D702}},\end{eqnarray}$$

(3.44) $$\begin{eqnarray}\displaystyle & = & \displaystyle \frac{2-3\unicode[STIX]{x1D709}/2-3\unicode[STIX]{x1D709}^{2}/4}{r^{2}}\quad \text{for }\ell _{\unicode[STIX]{x1D702}}\ll r\ll \ell _{0},\end{eqnarray}$$

(3.45) $$\begin{eqnarray}\displaystyle & = & \displaystyle \frac{2}{r^{2}}\quad \text{for }\ell _{0}\ll r.\end{eqnarray}$$

The potential is now repulsive at both small and large scales. As a result, the existence of an attractive range of scales and bound modes now depends on the existence of a large-enough scale separation $\ell _{0}/\ell _{\unicode[STIX]{x1D702}}$ , or large enough $Rm$ . If the scale separation is too low, magnetic diffusion, whose stabilising effects translate as a repulsive potential at small scales, always wins over stretching, and no dynamo action is possible. This argument is independent of $\unicode[STIX]{x1D709}$ , and therefore predicts the existence of a critical $Rm$ in both large- and low- $Pm$ regimes.

Figure 18. Critical $Rm$ effect in the Kazantsev model. The existence of an attractive potential (growing dynamo modes) requires a large-enough scale separation between the integral scale $\ell _{0}$ and the resistive scale $\ell _{\unicode[STIX]{x1D702}}$ .

3.4.8 Selected results in the large- $Pm$ regime*

The Kazantsev model can be used to derive a variety of quantitative results when the magnetic field is excited at scales much smaller than the spatial correlation scale of the flow. One particularly interesting such case is the large- $Pm$ regime encountered earlier, for which the dynamo is driven by a smooth viscous-scale flow. This is called the Batchelor regime, in reference to Batchelor, Howells & Townsend’s (Reference Batchelor, Howells and Townsend1959) work on the transport of passive scalars. Note that this problem is not of mere applied mathematics interest, and has notably been at the core of important developments on the theory of astrophysical galactic dynamos (Kulsrud & Anderson Reference Kulsrud and Anderson1992).

In the Batchelor regime, the velocity field correlator (3.20)–(3.21) can be expanded in Taylor series by writing $\unicode[STIX]{x1D705}_{L}=\unicode[STIX]{x1D705}_{0}-\unicode[STIX]{x1D705}_{2}r^{2}/4+\cdots \,$ , $\unicode[STIX]{x1D705}_{N}=\unicode[STIX]{x1D705}_{0}-\unicode[STIX]{x1D705}_{2}r^{2}/2+\cdots \,$ , resulting in

(3.46) $$\begin{eqnarray}\unicode[STIX]{x1D705}^{ij}(r)=\unicode[STIX]{x1D705}_{0}\unicode[STIX]{x1D6FF}^{ij}-\unicode[STIX]{x1D705}_{2}\frac{r^{2}}{2}\left(\unicode[STIX]{x1D6FF}^{ij}-\frac{1}{2}\frac{r^{i}r^{j}}{r^{2}}\right)+\cdots \,.\end{eqnarray}$$

A technical digression is in order: while it is possible to make quantitative calculations in physical space starting from (3.46), the easiest and most popular route to calculate magnetic eigenfunctions and the magnetic-energy spectrum goes through Fourier space. This was in fact the primary method used by Kazantsev (Reference Kazantsev1967). We will take that route in the next paragraphs, and find that the results can be expressed in terms of the coefficients of the Taylor expansion (3.46). The correspondence between the two descriptions can be found in Schekochihin et al. (Reference Schekochihin, Boldyrev and Kulsrud2002a ), appendix A.

Fokker–Planck equation. At scales much smaller than the viscous scale (small-scale approximation), the spectral equivalent of the Schrödinger equation (3.32) for the magnetic correlation function is a Fokker–Planck equation for the one-dimensional magnetic energy spectrum

(3.47) $$\begin{eqnarray}M(k,t)={\textstyle \frac{1}{2}}\int k^{2}|\hat{\boldsymbol{B}}_{\boldsymbol{ k}}(t)|^{2}\,\text{d}\unicode[STIX]{x1D6FA}_{\boldsymbol{ k}},\end{eqnarray}$$

where $\hat{\boldsymbol{B}}_{\boldsymbol{k}}(t)$ denotes the Fourier transform in space of $\boldsymbol{B}$ (once again not to be confused in this context with the unit vector in the direction of $\boldsymbol{B}$ ). Namely,

(3.48) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}M}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x1D6FE}}{5}\left(k^{2}\frac{\unicode[STIX]{x2202}^{2}M}{\unicode[STIX]{x2202}k^{2}}-2k\frac{\unicode[STIX]{x2202}M}{\unicode[STIX]{x2202}k}+6M\right)-2\unicode[STIX]{x1D702}k^{2}M,\end{eqnarray}$$

with

(3.49) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}={\textstyle \frac{5}{4}}\unicode[STIX]{x1D705}_{2}={\textstyle \frac{5}{2}}|\unicode[STIX]{x1D705}_{L}^{\prime \prime }(0)|\sim \unicode[STIX]{x1D6FF}u_{\unicode[STIX]{x1D708}}/\ell _{\unicode[STIX]{x1D708}},\end{eqnarray}$$

the typical growth rate of the magnetic energy, of the order of the viscous shearing rate represented by the $\unicode[STIX]{x1D705}_{2}$ parameter in the smooth flow expansion (3.46). These results are only quantitatively valid in the incompressible 3-D case.

Kazantsev spectrum and growth rate. Equation (3.48) has a diffusion term in wavenumber space. If we start with magnetic perturbations at scales much larger than the resistive scale, the magnetic spectrum will both grow in time and spread in wavenumber space until it hits the resistive scale. This early evolution during which all $k\ll k_{\unicode[STIX]{x1D702}}$ is called the diffusion-free regime and is illustrated in figure 19. Assuming that we initially excite a spectrum of magnetic modes $M_{0}(k)$ , the evolution of the spectrum can be computed exactly using Green’s function of (3.48) with negligible magnetic diffusion,

(3.50) $$\begin{eqnarray}M(k,t)=\text{e}^{3/4\unicode[STIX]{x1D6FE}t}\int _{0}^{\infty }\frac{\text{d}k^{\prime }}{k^{\prime }}M_{0}(k^{\prime })\left(\frac{k}{k^{\prime }}\right)^{3/2}\sqrt{\frac{5}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}t}}\exp \left[-\frac{5\,\ln ^{2}\left(k/k^{\prime }\right)}{4\unicode[STIX]{x1D6FE}t}\right].\end{eqnarray}$$

On the large-scale side, the magnetic-energy spectrum grows as $k^{3/2}$ . This result was first derived in Kazantsev’s (Reference Kazantsev1967) paper and is therefore called the Kazantsev spectrum. Interestingly, the energy in each wavenumber grows at the rate $3\unicode[STIX]{x1D6FE}/4$ , but the total energy integrated over wavenumber space grows at the rate $2\unicode[STIX]{x1D6FE}$ because the number of excited Fourier modes also grows in time.

Figure 19. Evolution of the magnetic-energy spectrum in the kinematic, diffusion-free large- $Pm$ regime, starting from an initial magnetic spectrum $M_{0}(k)$ . The magnetic energy in each wavenumber increases, and so does the peak wavenumber as the spectrum spreads.

Figure 20. Evolution of the magnetic-energy spectrum in the kinematic, diffusive large- $Pm$ regime. The shape of the magnetic spectrum is now fixed in time and peaks at the resistive scale $\ell _{\unicode[STIX]{x1D702}}$ , but the magnetic energy continues to grow exponentially.

At the end of the diffusion-free regime, the spectrum hits $k\sim k_{\unicode[STIX]{x1D702}}$ and we enter the diffusive regime, for which the long-time asymptotic form is

(3.51) $$\begin{eqnarray}M(k,t)\propto \left(\frac{k}{k_{\unicode[STIX]{x1D702}}}\right)^{3/2}K_{0}\left(\frac{k}{k_{\unicode[STIX]{x1D702}}}\right)\text{e}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FE}t},\end{eqnarray}$$

where $K_{0}$ is a Macdonald function, $k_{\unicode[STIX]{x1D702}}=\sqrt{\unicode[STIX]{x1D6FE}/(10\unicode[STIX]{x1D702})}\sim Pm^{1/2}k_{\unicode[STIX]{x1D708}}$ , and $\unicode[STIX]{x1D706}$ here is a non-dimensional growth-rate prefactor. The spectrum at large scales is still $k^{3/2}$ , but now peaks at the resistive scale and falls off exponentially at even smaller scales. While the exact value of $\unicode[STIX]{x1D706}$ depends weakly on $Pm$ and on the boundary condition imposed on the viscous side at low $k$ , the asymptotic total energy growth rate is now essentially $3\unicode[STIX]{x1D6FE}/4$ , as the number of excited modes remains constant. The evolution of the spectrum in this regime is illustrated in figure 20.

Interestingly, this general spectral shape is reminiscent of the numerical results shown in figure 10. In fact, and despite all the assumptions behind the Kazantsev derivation, all small-scale dynamo simulations at $Pm>1$ exhibit a spectrum with a positive slope at wavenumbers larger than but close to $k_{\unicode[STIX]{x1D708}}$ . Observing a clean $k^{3/2}$ scaling, though, requires to go to fairly large $Pm$ to reach a proper scale separation between $k_{\unicode[STIX]{x1D708}}$ and $k_{\unicode[STIX]{x1D702}}$ .

Magnetic probability density function and moments. Probability density functions (p.d.f.) and statistical moments of different orders can diagnose subtle dynamical features in both experiments and numerical simulations and as such provide important statistical tools to tackle many turbulence problems. To complete our overview of the Kazantsev theory, we will therefore finally outline a mathematical procedure to calculate the magnetic p.d.f. and moments in the Kazantsev description in the diffusion-free regime. The spatial dependence of the field is ignored for simplicity in this particular derivation: this removes turbulent mixing effects from the model, but not inductive ones. The key point of the derivation (see, e.g. Schekochihin & Kulsrud Reference Schekochihin and Kulsrud2001; Boldyrev Reference Boldyrev2001) is to introduce the characteristic function,

(3.52) $$\begin{eqnarray}Z(\unicode[STIX]{x1D741},t)=\langle \exp [\text{i}\unicode[STIX]{x1D707}_{i}B^{i}(t)]\rangle =\langle \tilde{Z}\rangle ,\end{eqnarray}$$

where $\tilde{Z}=\exp [\text{i}\unicode[STIX]{x1D707}_{i}B^{i}(t)]$ and the bracket average is over all the realisations of the velocity field. This function is interesting because it is the Fourier transform in $\boldsymbol{B}$ of the p.d.f. of $\boldsymbol{B}$ ,

(3.53) $$\begin{eqnarray}Z(\unicode[STIX]{x1D741},t)=\int P[\boldsymbol{B}]\exp [\text{i}\unicode[STIX]{x1D707}_{i}B^{i}(t)]\,\text{d}^{3}\boldsymbol{B}.\end{eqnarray}$$

Using the simplified induction equation

(3.54) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}B^{i}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D70E}_{k}^{i}B^{k},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{k}^{i}=\unicode[STIX]{x2202}u^{i}/\unicode[STIX]{x2202}x^{k}$ is the velocity strain tensor, we obtain an evolution equation for  $Z$ ,

(3.55) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}Z}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D707}_{i}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}_{k}}\langle \unicode[STIX]{x1D70E}_{k}^{i}\tilde{Z}\rangle .\end{eqnarray}$$

This equation can be closed in the Kazantsev model using the same Gaussian integration trick as in § 3.4.3. The result is

(3.56) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}Z}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x1D705}_{2}}{2}T_{kl}^{ij}\unicode[STIX]{x1D707}_{i}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}_{k}}\unicode[STIX]{x1D707}_{j}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}_{l}}Z,\end{eqnarray}$$

where

(3.57) $$\begin{eqnarray}T_{kl}^{ij}=-\frac{1}{\unicode[STIX]{x1D705}_{2}}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D705}^{ij}}{\unicode[STIX]{x2202}r^{k}\unicode[STIX]{x2202}r^{l}}=\unicode[STIX]{x1D6FF}^{ij}\unicode[STIX]{x1D6FF}_{kl}-\frac{1}{4}(\unicode[STIX]{x1D6FF}_{k}^{i}\unicode[STIX]{x1D6FF}_{l}^{j}+\unicode[STIX]{x1D6FF}_{l}^{i}\unicode[STIX]{x1D6FF}_{k}^{j})\end{eqnarray}$$

is the strain correlator for a 3-D, incompressible, isotropic velocity field. Performing an inverse Fourier transform from $\unicode[STIX]{x1D707}$ to $\boldsymbol{B}$ variables and using the transformation

(3.58) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{i}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}_{k}}()\rightarrow \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}B^{i}}[B^{k}()],\end{eqnarray}$$

we obtain a Fokker–Planck equation for the p.d.f.

(3.59) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}P[\boldsymbol{B}]=\frac{\unicode[STIX]{x1D705}_{2}}{2}T_{kl}^{ij}B^{k}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}B^{i}}B^{l}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}B^{j}}P[\boldsymbol{B}].\end{eqnarray}$$

Thanks to the isotropy assumption, this equation simplifies further as a 1-D diffusion equation with a drift for the p.d.f. $P_{B}[B](t)$ of the field strength $B$ ,

(3.60) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}P_{B}[B]=\frac{\unicode[STIX]{x1D705}_{2}}{4}\frac{1}{B^{2}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}B}B^{4}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}B}P_{B}[B],\end{eqnarray}$$

which has the log-normal solution

(3.61) $$\begin{eqnarray}P_{B}[B](t)=\frac{1}{\sqrt{\unicode[STIX]{x03C0}\unicode[STIX]{x1D705}_{2}t}}\int _{0}^{\infty }\frac{\text{d}B^{\prime }}{B^{\prime }}P_{B}[B^{\prime }](t=0)\exp \left(-\frac{[\ln (B/B^{\prime })+(3/4)\unicode[STIX]{x1D705}_{2}t]^{2}}{\unicode[STIX]{x1D705}_{2}t}\right),\end{eqnarray}$$

i.e. the statistics of the logarithm of the field strength are Gaussian. Therefore, the magnetic-field structure is strongly intermittent, despite the fact that the velocity field itself is Gaussian. This is actually expected from the central limit theorem on a general basis, not just in the Kazantsev formalism, as the induction equation is linear in $\boldsymbol{B}$ and involves the multiplicative random variable $\unicode[STIX]{x1D735}\boldsymbol{u}$ on the right-hand side. The magnetic moments of different order $n$ , defined here as

(3.62) $$\begin{eqnarray}\langle B^{n}(t)\rangle =4\unicode[STIX]{x03C0}\int _{0}^{\infty }\text{d}BB^{2+n}P_{B}[B],\end{eqnarray}$$

grow as

(3.63) $$\begin{eqnarray}\langle B^{n}(t)\rangle \propto \exp [{\textstyle \frac{1}{4}}n(n+3)\unicode[STIX]{x1D705}_{2}t].\end{eqnarray}$$

There is currently no similar general result for the magnetic-field p.d.f. in the diffusive regime, although expressions for the $n>2$ moments of the field have been derived in this regime by Chertkov et al. (Reference Chertkov, Falkovich, Kolokolov and Vergassola1999) using a different mathematical technique. Their results show that the structure of the field remains strongly intermittent after the folds’ reversal scale hits the magnetic-diffusion scale.

3.4.9 Miscellaneous observations

Instability threshold subtleties. Equation (3.63) indicates that magnetic moments of different order have different ideal growth rates. Accordingly, we would find that each moment becomes unstable at a different $Rm$ in the resistive case. This behaviour can be attributed to the multiplicative stochastic nature of the inductive term in the linear induction equation. Which moment, then, provides the correct prediction for the overall dynamo threshold? It has been argued that a proper instability threshold in this context is only well defined after taking into account nonlinear saturation terms that tend to suppress the large-deviation events responsible for the field intermittency. The dynamo onset of the full nonlinear problem then appears to be given by the linear threshold derived from the statistical average of the logarithm of the magnetic field, not of the magnetic energy (second-order moment) as calculated in the Kazantsev model (Seshasayanan & Pétrélis Reference Seshasayanan and Pétrélis2018).

Connection with finite-correlation-time turbulence. Statistics of the magnetic field in the kinematic, large- $Pm$ regime have been extensively studied in numerical simulations of the MHD equations. While the log-normal shape of the magnetic p.d.f. appears to be a robust feature of the simulations, intermittency corrections imprinted in magnetic moments, such as predicted by (3.63), cannot be easily diagnosed due to subtle finite-size effects. An in-depth discussion of this thorny aspect of the problem can be found in Schekochihin et al. (Reference Schekochihin, Cowley, Taylor, Maron and McWilliams2004c ). More generally, and despite its crude closure assumptions, the predictions of the Kazantsev model (dynamo growth rate and statistics) are in good overall agreement with the results of DNS at $Pm>1$ . Non-zero correlation-time effects appear to be of relatively minor qualitative importance (see e.g. Vainshtein (Reference Vainshtein1980), Chandran (Reference Chandran1997), Bhat & Subramanian (Reference Bhat and Subramanian2015) for theoretical arguments, and Mason et al. (Reference Mason, Malyshkin, Boldyrev and Cattaneo2011) for a detailed numerical comparison with turbulence DNS), and intermittent statistics of the fluctuation dynamo fields are predicted by the model despite the Gaussianity of the velocity field. This suggests that the kinematic small-scale fluctuation dynamo process is relatively insensitive to the details of the flow driving the dynamo, as long as the former is chaotic.

3.5.1 General phenomenology

The first and most natural question to ask about saturation is arguably that of the dynamo efficiency: how much magnetic energy should we expect relative to kinetic energy in the statistically steady saturated state of the dynamo? Two answers were historically given to this question, one by Batchelor (Reference Batchelor1950) and the other by Schlüter & Biermann (Reference Schlüter and Biermann1950). Batchelor observed that the induction equation for the magnetic field has the same form as the evolution equation for hydrodynamic vorticity. As the vorticity peaks at the viscous scale in hydrodynamic turbulence, he then argued that the dynamo should saturate when $\langle |\boldsymbol{B}|^{2}\rangle \sim u_{\unicode[STIX]{x1D708}}^{2}$ . This argument predicts that very weak fields well below equipartition are sufficient to saturate the dynamo in Kolmogorov turbulence ( $u_{\unicode[STIX]{x1D708}}\sim Re^{-1/4}u_{0}$ ) with $Re\gg 1$ ,

(3.64) $$\begin{eqnarray}\langle |\boldsymbol{B}|^{2}\rangle \sim Re^{-1/2}\langle |\boldsymbol{u}|^{2}\rangle ,\end{eqnarray}$$

admittedly not a very interesting astrophysical prospect! Schlüter & Biermann, on the other hand, argued that the outcome of saturation should be global equipartition between kinetic and magnetic energy,

(3.65) $$\begin{eqnarray}\langle |\boldsymbol{B}|^{2}\rangle \sim \langle |\boldsymbol{u}|^{2}\rangle ,\end{eqnarray}$$

as a result of a gradual build-up of scale-by-scale equipartition from the smallest, least energetic scales. Which one (if either) is correct? Batchelor’s analogy between vorticity and magnetic field was shown to be flawed a long time ago, and numerical simulations exhibit much higher saturation levels than predicted by his model. However, scale-by-scale equipartition does not appear to hold either, as can be seen for instance by inspecting the saturated spectra in figure 10.

In order to get a better handle on the issue, we must have a detailed look at how magnetic growth is quenched physically. Geometry, not just energetics, is very important in this problem, particularly in the large- $Pm$ case. First, not all velocity fields are created equal for the magnetic field, as (2.14) clearly illustrates. Only certain types of motions relative to the magnetic field can make the field change, and it is only by quenching such motions that the magnetic field can saturate. Second, the magnetic tension force $\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{B}$ mediating the dynamical feedback (in incompressible flows) is proportional to the variation of the magnetic field along itself, i.e. to magnetic curvature. A quick look at the folded fields structure in figure 14 reveals the geometric complexity of the problem. Magnetic curvature is very small along the regions of straight magnetic field, and very large in regions where the magnetic field bends to reverse, and therefore so is the Lorentz force back reaction.

These two observations are not independent though because the energy gained by the field through the anisotropic induction term must always be equal and opposite to that lost by the velocity field through the effects of the Lorentz force (§ 2.1.3). In practice, the Lorentz force effectively reduces Lagrangian chaos in the flow in comparison to the kinematic regime, thereby quenching the inductive stretching of the magnetic field. This effect is illustrated in figure 21 by two maps of the stretching Lyapunov exponent of the GP flow in the kinematic and saturated regimes for $Pm=4$ . The Lyapunov exponent reduction is strongly correlated spatially with the structure of the particular realisation of the dynamo magnetic field being saturated. The velocity field saturated by a given small-scale dynamo field realisation therefore remains a kinematic dynamo flow for other independent field realisations/dummy magnetic fields governed by an induction equation (Tilgner & Brandenburg Reference Tilgner and Brandenburg2008; Cattaneo & Tobias Reference Cattaneo and Tobias2009, see also Brandenburg (Reference Brandenburg2018) for a more detailed discussion). A numerical illustration of this effect is provided in figure 22.

Figure 21. Spatial distribution of the finite-time Lyapunov stretching exponent in a dynamo simulation of the GP flow. Light shades correspond to integrable regions with little or no exponential stretching, dark shades to chaotic regions with strong stretching. (a) Kinematic regime map exhibiting fractal regions of chaotic dynamics and stability islands. (b) Dynamical regime. Strongly chaotic regions have almost disappeared (adapted from Cattaneo, Hughes & Kim Reference Cattaneo, Hughes and Kim1996).

Figure 22. Magnetic energy in a simulation of small-scale dynamo action driven by turbulent thermal convection. Dashed line: energy of the saturated magnetic field in the dynamical regime. Dash-dotted line: energy of an independent dummy magnetic field seeded in the course of the saturated phase. The dummy magnetic field grows exponentially, even though the true magnetic field has already saturated the velocity field (adapted from Cattaneo & Tobias Reference Cattaneo and Tobias2009).

3.5.2 Nonlinear growth

Now that we have a better appreciation of the effects of the Lorentz force on the flow, let us come back to the dynamical history of saturation. Various numerical simulations (e.g. Schekochihin et al. Reference Schekochihin, Cowley, Taylor, Maron and McWilliams2004c ; Ryu et al. Reference Ryu, Kang, Cho and Das2008; Beresnyak Reference Beresnyak2012), suggest that the kinematic stage is followed by a nonlinear growth phase during which magnetic energy grows linearly in time instead of exponentially. A possible scenario for such nonlinear growth at large $Pm$ is as follows (Schekochihin et al. Reference Schekochihin, Cowley, Hammett, Maron and McWilliams2002b ). In this parameter regime, the end of the kinematic phase occurs when the stretching by the most dynamo-efficient viscous scales gets suppressed,

(3.66) $$\begin{eqnarray}\displaystyle \boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{B} & {\sim} & \displaystyle \boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\sim u_{\unicode[STIX]{x1D708}}^{2}/\ell _{\unicode[STIX]{x1D708}},\end{eqnarray}$$

(3.67) $$\begin{eqnarray}\displaystyle & {\sim} & \displaystyle B^{2}/\ell _{\unicode[STIX]{x1D708}}\quad \text{(magnetic folds at viscous scale)}.\end{eqnarray}$$

This corresponds to the magnetic ‘Batchelor level’ $\langle \boldsymbol{B}|^{2}\rangle \sim Re^{-1/2}\langle |\boldsymbol{u}|^{2}\rangle$ . However, this need not be the final saturation level. Once the fastest viscous motions are quenched, the baton of magnetic amplification can be passed to increasingly slower but more energetic field-stretching motions at increasingly larger scales. To see how this process can lead to nonlinear growth, let us denote the scale of the smallest, weakest, but fastest as yet unsaturated motions at time $t$ by $\ell ^{\ast }(t)$ , and their stretching rate by $\unicode[STIX]{x1D6FE}^{\ast }(t)=u_{\ell ^{\ast }}/\ell ^{\ast }$ . These motions are the most efficient at amplifying the field at this particular time, but by definition their kinetic energy is also such that $\langle |\boldsymbol{B}|^{2}\rangle (t)\sim u_{\ell ^{\ast }}^{2}$ . From the induction equation, we can then estimate that

(3.68) $$\begin{eqnarray}\displaystyle \frac{\text{d}\langle |\boldsymbol{B}|^{2}\rangle }{\text{d}t} & {\sim} & \displaystyle \unicode[STIX]{x1D6FE}^{\ast }(t)\langle |\boldsymbol{B}|^{2}\rangle ,\end{eqnarray}$$

(3.69) $$\begin{eqnarray}\displaystyle & {\sim} & \displaystyle u_{\ell ^{\ast }}^{3}/\ell ^{\ast }.\end{eqnarray}$$

Now, $u_{\ell ^{\ast }}^{3}/\ell ^{\ast }$ is proportional to the rate of turbulent transfer of kinetic energy $\unicode[STIX]{x1D700}$ at scale $\ell ^{\ast }$ , but note that this quantity is actually time and scale independent in Kolmogorov turbulence. Equation (3.69) therefore shows that for this kind of turbulence, magnetic energy should grow linearly in time at a rate corresponding to a fraction $\unicode[STIX]{x1D701}$ of $\unicode[STIX]{x1D700}$ ,

(3.70) $$\begin{eqnarray}\langle |\boldsymbol{B}|^{2}\rangle \propto \unicode[STIX]{x1D701}\unicode[STIX]{x1D700}t.\end{eqnarray}$$

In this regime, the magnetic field quenches stretching motions at increasingly large $\ell ^{\ast }(t)$ until its energy becomes comparable to the kinetic energy of the largest, most energetic shearing motions $u_{0}$ at the integral scale of the flow $\ell _{0}$ , i.e. the dynamo saturates in a state of global equipartition

(3.71) $$\begin{eqnarray}\langle |\boldsymbol{B}|^{2}\rangle \sim \langle |\boldsymbol{u}|^{2}\rangle .\end{eqnarray}$$

As the scale $\ell ^{\ast }(t)$ of the motions driving the dynamo slowly increases, there should be a corresponding slow increase in the length of magnetic folds. Also, note that the gradual decrease of the maximum instantaneous stretching rate $\unicode[STIX]{x1D6FE}^{\ast }(t)$ leaves the diffusion of the magnetic field unbalanced at increasingly larger scales in the course of the process. This secular increase of magnetic energy should therefore be accompanied by a corresponding slow increase of the resistive scale as $\ell _{\unicode[STIX]{x1D702}}\sim (\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FE}^{\ast }(t))^{1/2}\sim (\unicode[STIX]{x1D702}t)^{1/2}$ for Kolmogorov turbulence. Perpendicular magnetic reversals associated with the spectral peak at $\ell _{\unicode[STIX]{x1D702}}$ do not play an obvious active role in the problem though, as they do not contribute to the Lorentz force. Overall, we therefore expect that the folded structure of the field and general shape of the large- $Pm$ magnetic spectrum should be preserved in this scenario. The particular form of the Lorentz force implies that saturation is possible without the need for scale-by-scale equipartition. Spectra derived from numerical simulations of the saturated regime appear to support the latter conclusion qualitatively (Schekochihin et al. Reference Schekochihin, Cowley, Taylor, Maron and McWilliams2004c ).

The previous argument is non-local in the sense that the magnetic field, which has reversals at the resistive scale, appears to strongly interact with the velocity field at scales much larger than that (that saturated isotropic MHD turbulence has a non-local character remains controversial though, see e.g. Aluie & Eyink Reference Aluie and Eyink2010). A different nonlinear dynamo-growth scenario can be constructed if one instead assumes that nonlinear interactions between the velocity and magnetic field are predominantly local (Beresnyak Reference Beresnyak2012), as could for instance (but not necessarily exclusively) be the case for $Pm$ of order one or smaller. Imagine a situation in which the magnetic spectrum at time $t$ now peaks at a scale (also denoted by $\ell ^{\ast }(t)$ for simplicity) where there is a local equipartition between kinetic and magnetic energy. The dynamics at scales larger than $\ell ^{\ast }(t)$ is hydrodynamic and the turbulent rate of energy transfer from these scales to $\ell ^{\ast }(t)$ is therefore the standard hydrodynamic transfer rate $\unicode[STIX]{x1D700}$ . At scales below $\ell ^{\ast }(t)$ , the magnetic field becomes dynamically significant and the turbulent cascade turns into a joint MHD cascade of magnetic and kinetic energy characterised by a different constant turbulent transfer rate $\unicode[STIX]{x1D700}_{\text{MHD}}$ . Locality of interactions and energy conservation in scale space then lead to the prediction that the magnetic energy should grow as $(\unicode[STIX]{x1D700}-\unicode[STIX]{x1D700}_{\text{MHD}})t$ . For Kolmogorov turbulence, $u_{\ell }\sim \ell ^{1/3}$ , and therefore a linear in time increase of the magnetic energy is associated with an increase of the equipartition scale as $\ell ^{\ast }(t)\sim t^{3/2}$ . Numerical results obtained by Beresnyak (Reference Beresnyak2012) for $Pm=1$ suggest that nonlinear magnetic-energy growth in this scenario may be relatively inefficient, $(\unicode[STIX]{x1D700}-\unicode[STIX]{x1D700}_{\text{MHD}})/\unicode[STIX]{x1D700}\sim 0.05$ . It has been argued recently that this nonlinear growth regime may be relevant to galaxy clusters accreting through cosmic time (Miniati & Beresnyak Reference Miniati and Beresnyak2015). As the turbulence injection scale $\ell _{0}$ also grows in such systems, the relative inefficiency of the dynamo in this nonlinear regime would suggest that the ratio between $\ell ^{\ast }$ and $\ell _{0}$ remains small as both scales grow, so that a statistically steady saturated state is never achieved.

3.5.3 Saturation at large and low $Pm$

It remains unclear which of the previous scenarios, if any, applies to the large- $Pm$ limit and, more generally, whether a unique, universal nonlinear growth scenario applies to different $Pm$ regimes. Numerical simulations of the $Pm>1$ regime (e.g. Haugen et al. Reference Haugen, Brandenburg and Dobler2004; Schekochihin et al. Reference Schekochihin, Cowley, Taylor, Maron and McWilliams2004c ; Beresnyak Reference Beresnyak2012; Bhat & Subramanian Reference Bhat and Subramanian2013) do not as yet provide a definitive answer to these questions. As far as the $Pm\gg 1$ regime is concerned, it has so far proven impossible to explore numerically the astrophysically relevant asymptotic limit $Rm\gg Re\gg 1$ in which the viscous and integral scales of the dynamo flow are properly separated.

The physics of saturation at low $Pm$ also very much remains terra incognita at the time of writing of these notes. The only nonlinear results available in this regime are due to Sahoo, Perlekar & Pandit (Reference Sahoo, Perlekar and Pandit2011) and Brandenburg (Reference Brandenburg2011). One of the positive conclusions of the latter study is that dynamical simulations at low $Pm$ appear to be slightly less demanding in terms of numerical resolution than their kinematic counterpart because the saturated magnetic field diverts a significant fraction of the cascaded turbulent energy before it reaches the viscous scale. Brandenburg’s simulations at largest $Rm$ also suggest that the efficiency of the dynamo should be quite large even at low $Pm$ , with magnetic energy of the order of at least $30\,\%$ of the kinetic energy.

3.5.4 Reconnecting dynamo fields

Another potentially very important problem in the nonlinear regime is that of the stability of small-scale dynamo fields (in particular magnetic folds at large $Pm$ ) to fast MHD reconnection for Lundquist numbers $Lu=U_{A}L/\unicode[STIX]{x1D702}$ of the order of a few thousand (or equivalently $Rm$ of a few thousand in an equipartitioned nonlinear dynamo regime characterised by a typical Alfvén velocity $U_{A}$ of the order of the r.m.s. flow velocity). This regime has only become accessible to numerical simulations in the last few years, but plasmoid chains typical of this process (Loureiro, Schekochihin & Cowley Reference Loureiro, Schekochihin and Cowley2007; Loureiro et al. Reference Loureiro, Samtaney, Schekochihin and Uzdensky2012) have now been observed in different high-resolution simulations of small-scale dynamos by Andrey Beresnyak and by Alexei Iskakov and Alexander Schekochihin (figure 23), as well as in simulations of MHD turbulence driven by the magnetorotational instability (Kadowaki, De Gouveia Dal Pino & Stone Reference Kadowaki, De Gouveia Dal Pino and Stone2018). The relevance and implications of fast, stochastic reconnection processes for the saturation of dynamos in general, and the small-scale dynamo in particular, are currently not well understood, although there is some nascent theoretical activity on the problem (Eyink Reference Eyink2011; Eyink, Lazarian & Vishniac Reference Eyink, Lazarian and Vishniac2011). A tentative phenomenological model of the reconnecting small-scale dynamo, inspired by similar results on MHD turbulence in a guide field, is due in an upcoming review by Schekochihin (Reference Schekochihin2019).

Figure 23. Two-dimensional snapshots of $|\boldsymbol{u}|$  (a) and $|\boldsymbol{B}|$  (b) in a nonlinear simulation of small-scale dynamo driven by turbulence forced at the box scale at $Re=290$ , $Rm=2900$ , $Pm=10$ (the magnetic-energy spectra for this $512^{3}$ spectral simulation suggest that it is reasonably well resolved). At such large $Rm$ , the dynamo field becomes weakly supercritical to a secondary fast-reconnection instability in regions of reversing field polarities associated with strong electrical currents. The instability generates magnetic plasmoids and outflows, leaving a small-scale dynamical imprint on the velocity field (unpublished figure courtesy of Iskakov and Schekochihin).

3.5.5 Nonlinear extensions of the Kazantsev model*

Let us finally quickly review a few nonlinear extensions of the Kazantsev model that readers begging for more quantitative mathematical derivations may enjoy exploring further. The general idea of these models is to solve (3.25) for the magnetic correlator for a dynamical velocity field consisting of the original stochastic kinematic Kazantsev–Kraichnan field, plus a nonlinear magnetic-field-dependent dynamical correction $\boldsymbol{u}_{\text{NL}}$ accounting for the effect of the Lorentz force on the flow. By massaging the new unclosed mixed correlators associated with $\boldsymbol{u}_{\text{NL}}$ in (3.25), one picks up new quasilinear or nonlinear terms in Kazantsev’s equation (3.29). Stationary solutions of this modified equation for the saturated magnetic correlator are then sought. Of course, in the absence of a good analytical procedure to solve the Navier–Stokes equation with a magnetic back reaction, the exact form of $\boldsymbol{u}_{\text{NL}}$ must be postulated on phenomenological grounds. Subramanian (Reference Subramanian1998), for instance, introduced an ambipolar drift proportional to the Lorentz force, $\boldsymbol{u}_{\text{NL}}=a\,\boldsymbol{J}\times \boldsymbol{B}$ , where $a$ is a dimensional constant. The closure procedure leads to a simple dynamical renormalisation of the magnetic diffusivity $\unicode[STIX]{x1D702}\rightarrow \unicode[STIX]{x1D702}_{\text{NL}}=\unicode[STIX]{x1D702}+2aH_{L}(0,t)$ , and therefore to a renormalisation of the magnetic Reynolds number, i.e.  $Rm_{\text{NL}}=u_{0}\ell _{0}/\unicode[STIX]{x1D702}_{\text{NL}}$ ( $H_{L}$ is the longitudinal magnetic correlator introduced in (3.23)). The saturated state of the model is the neutral eigenmode of the marginally stable linear problem, scaled by an amplitude fixed by the condition $Rm_{\text{NL}}=Rm_{c}$ . This model predicts a total saturated magnetic energy much smaller than the kinetic energy (by a factor $Rm_{c}$ ) with locally strong equipartition fields organised into non-space-filling narrow rope structures.

Another model, put forward by Boldyrev (Reference Boldyrev2001), postulates a velocity strain tensor of the form

(3.72) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}u_{\text{NL}}^{i}}{\unicode[STIX]{x2202}x^{k}}=-\frac{1}{\unicode[STIX]{x1D708}}\left(B^{i}B^{k}-\frac{1}{3}\unicode[STIX]{x1D6FF}^{ik}B^{2}\right),\end{eqnarray}$$

motivated by the dynamical feedback of the Lorentz force on viscous eddies expected in the early phases of saturation in the large $Pm$ regime. This prescription neatly leads to a modified version of the Fokker–Planck equation (3.59) for the p.d.f. of magnetic-field strength, whose stationary solution is a Gaussian. Simulations suggest that the magnetic p.d.f. in the saturated regime of the dynamo is exponential, but the determination of the exact shape of the p.d.f. appears to bear some subtle dependence on rare but intense stretching events. Phenomenological considerations can be used to fine tune a very similar model to generate exponential p.d.f.s (Schekochihin et al. Reference Schekochihin, Maron, Cowley and McWilliams2002c ). Yet another possibility, explored by Schekochihin et al. (Reference Schekochihin, Cowley, Taylor, Hammett, Maron and McWilliams2004b ), is to postulate a local magnetic-field-orientation-dependent anisotropic correction $\unicode[STIX]{x1D705}^{a}$ to the correlation tensor $\unicode[STIX]{x1D705}^{ij}$ of the Kazantsev velocity field to model the effects of magnetic tension on the flow. The generalised correlator can be expressed in spectral space as

(3.73) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}^{ij}(\boldsymbol{k}) & = & \displaystyle \unicode[STIX]{x1D705}^{(\text{i})}(k,|\unicode[STIX]{x1D707}|)(\unicode[STIX]{x1D6FF}^{ij}-\hat{k}_{i}\hat{k}_{j})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D705}^{a}(k,|\unicode[STIX]{x1D707}|)(\hat{B}^{i}\hat{B}^{j}+\unicode[STIX]{x1D707}^{2}\hat{k}_{i}\hat{k}_{j}-\unicode[STIX]{x1D707}\hat{B}^{i}\hat{k}_{j}-\unicode[STIX]{x1D707}\hat{k}_{i}\hat{B}^{j}),\end{eqnarray}$$

where $\unicode[STIX]{x1D707}=\hat{\boldsymbol{k}}\boldsymbol{\cdot }\hat{\boldsymbol{B}}$ and $\hat{\boldsymbol{k}}=\boldsymbol{k}/k$ . The Kazantsev equation derived from this model is more complicated and must be solved numerically. The results appear to reproduce the evolution of the magnetic spectra of simulations of the saturated large- $Pm$ regime of the small-scale dynamo reasonably well.

Overall, we see that multiple nonlinear extensions of the Kazantsev formalism are possible. The potential of this kind of semi-phenomenological approach to saturation has probably not yet been fully explored.

4.2.1 Coherent large-scale shearing: the $\unicode[STIX]{x1D6FA}$ effect

The winding up of a weak magnetic-field component parallel to the direction of the velocity gradient into a stronger magnetic-field component parallel to the direction of the velocity field itself, the $\unicode[STIX]{x1D6FA}$ effect is undoubtedly a very important inductive process in any shearing or differentially rotating system (Cowling Reference Cowling and Kuiper1953). This effect is illustrated in figure 26 in spherical geometry. As discussed in § 2.3.2, however, this effect is only good at producing toroidal field out of poloidal field, and cannot by itself sustain a dynamo.Footnote ⁹

Figure 26. A poor perspective drawing of the $\unicode[STIX]{x1D6FA}$ effect in a spherical fluid system with latitudinal differential rotation $\unicode[STIX]{x1D6FA}(\unicode[STIX]{x1D703})\boldsymbol{e}_{z}$ (maximum at the equator in this example): the latitudinal shear winds up an initial axisymmetric poloidal field into a stronger axisymmetric toroidal field in the regions of fastest rotation. In the absence of resistivity and any other dynamical effect, the growth of the toroidal field is linear, not exponential, in time. In the resistive case, and in the absence of further three-dimensional dynamical effects, the field as a whole is ultimately bound to decay (Cowling’s theorem).

4.2.2 Helical turbulence: Parker’s mechanism and the $\unicode[STIX]{x1D6FC}$ effect

After Cowling’s (Reference Cowling1933) anti-dynamo work, it became clear that more complex, three-dimensional, non-axisymmetric physical mechanisms were required to sustain the poloidal field against resistive decay. One such mechanism was first identified in a landmark publication by Parker (Reference Parker1955a ), and is commonly referred to as the $\unicode[STIX]{x1D6FC}$ effect after the theoretical work of Steenbeck et al. (Reference Steenbeck, Krause and Rädler1966), Steenbeck & Krause (Reference Steenbeck and Krause1966) that we will introduce in § 4.3. In his work, Parker considered the effects of helical fluid motions, typical of rotating convection in the Earth’s core or in the solar convection zone, on an initially straight magnetic field perpendicular to the axis of rotation (figure 27 a). Field lines rising with the hot convecting fluid also get twisted in the process, and thereby acquire a component perpendicular to the original field. A statistical version of this mechanism involving an ensemble of localised small-scale swirls, all with the same sign of helicity, would effectively couple the large-scale toroidal and poloidal field components, which could then lead to effective large-scale dynamo action. Parker’s mechanism can equally turn toroidal field into poloidal field, and poloidal field into toroidal field (the initial horizontal orientation of the magnetic field drawn in figure 27 a is arbitrary). It should therefore be sufficient to excite a dynamo even in the absence of large-scale shearing.

Figure 27. (a) Sketch of the dynamics of a magnetic flux tube in Parker’s mechanism for a right-handed helical velocity fluctuation $\boldsymbol{u}$ , showing a left-handed large-scale magnetic writhe associated with a large-scale current $\overline{\boldsymbol{J}}$ , and a right-handed internal twist associated with a small-scale current $\boldsymbol{j}$ . This particular configuration is generally thought to be representative of the dynamics in the southern hemisphere of rotating stars with a strongly stratified convection zone, where motions have a net cyclonic bias (§ 4.4.1). (b,c) Computation of the Cauchy solution of an initially straight magnetic flux tube in a cyclonic velocity field (b), and corresponding magnetic-helicity spectrum (c) (adapted from Yousef & Brandenburg Reference Yousef and Brandenburg2003).

In practice, there are several issues with this simple picture, the most important of which will be discussed in § 4.3.6 and in § 4.6. Besides, as we will discover later in this section and in § 5, the $\unicode[STIX]{x1D6FC}$ effect is not the only statistical effect capable of exciting a large-scale dynamo. Parker’s idea, however, was seminal in the development of large-scale dynamo theory and remains one of its central pillars, not least because it is directly connected to rotating dynamics, and rotation and large-scale magnetism in the Universe always seem to go hand in hand.

4.2.3 Writhe, twist and magnetic helicity

An important aspect of the kinematic evolution of a magnetic field in a helical velocity field is the associated dynamics of magnetic helicity ${\mathcal{H}}_{m}$ subject to the constraint of total magnetic-helicity conservation introduced in § 2.1.2. To illustrate this, consider again the simple three-dimensional evolution of an initially straight magnetic flux tube in a steady right-handed helical swirl depicted in figure 27(a). As the swirl acts upon the flux tube and makes it rise, a ‘Parker loop’, or large-scale writhe, is created. The current associated with this loop is anti-parallel to the large-scale field direction (and so is the vector potential in the Coulomb gauge), i.e. the process generates negative magnetic helicity on scales larger than that of the loop. But the same motion simultaneously twists the magnetic-field lines around the flux tube, thereby generating a local current within the tube with a positive projection along the local field, i.e. positive local magnetic helicity. If we ignore resistive effects and assume that there are no helicity losses out of the domain under consideration, equation (2.13) shows that the total magnetic helicity of the system is conserved, so that the negative helicity/left-handed writhe generated on large scales must be exactly balanced by the positive helicity/right-hand twist generated on small scales if ${\mathcal{H}}_{m}=0$ initially (for a left-handed helical swirl, the sign of large-scale and small-scale magnetic helicity is opposite). Overall, Parker’s kinematic mechanism therefore tends to generate a partition of large- and small-scale magnetic helicities of opposite signs.

This effect can be illustrated in a more quantitative way by computing for instance the magnetic-helicity spectrum of a twisted magnetic flux tube (figure 27 b,c). Similar computations for fully turbulent helically driven systems lead to the same results and conclusions (e.g. Mininni Reference Mininni2011). In the period preceding the very first simulations of helical MHD turbulence, it was suggested that such dynamics may be a consequence of the existence of an inverse ‘cascade’ of magnetic helicity associated with the magnetic helicity conservation constraint (Frisch et al. Reference Frisch, Pouquet, Leorat and Mazure1975). This interpretation can be broadly justified in the framework of a three-scale interaction model, however numerical simulations suggest that the magnetic helicity transfer process in helical MHD turbulence is very different in nature from hydrodynamic cascades, as it does not proceed on turbulent dynamical time scales and is non-local in spectral space (Brandenburg, Dobler & Subramanian Reference Brandenburg, Dobler and Subramanian2002; Brandenburg & Subramanian Reference Brandenburg and Subramanian2005a ).

Finally, let us point out that the magnetic tension associated with the small-scale curvature of twisted magnetic-field lines should be an important source of back reaction of the field on the flow in the dynamical stages of the dynamo. Accordingly, we will discover in § 4.6 that the dynamics of magnetic helicity is a key ingredient of nonlinear theories of large-scale helical dynamos driven by an $\unicode[STIX]{x1D6FC}$ effect.

4.3.1 Two-scale approach

The general idea behind mean-field theory is to split the magnetic field, velocity field and the corresponding MHD equations into mean and fluctuating parts in order to isolate the net r.m.s. effect of the fluctuations on the mean (‘large-scale’) fields. Depending on the problem, the mean can be defined as an average over a statistical ensemble of realisations of the flow, as an average over one dimension (e.g. the toroidal direction for problems in spherical or cylindrical geometry), over two dimensions, or, if there is a proper spatial and temporal dynamical scale separation in the problem, as an average over the small-scale dynamics. In what follows, we will assume that any such averaging procedure satisfies the so-called Reynolds rules (for a more detailed discussion of averaging in the large-scale dynamo problem, see Brandenburg & Subramanian (Reference Brandenburg and Subramanian2005a ), chap. 6.2, and Moffatt (Reference Moffatt1978), chap. 7.1), and will essentially consider a formulation based on a two-scale decomposition of the dynamics. Namely, the magnetic and velocity fields are split into large-scale, mean-field parts ( $\ell \gg \ell _{0}$ , denoted by an overline) and small-scale, fluctuating parts ( $\ell \leqslant \ell _{0}$ , denoted by $\boldsymbol{b}$ and $\boldsymbol{u}$ ),

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{U}=\overline{\boldsymbol{U}}+\boldsymbol{u}, & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{B}=\overline{\boldsymbol{B}}+\boldsymbol{b}. & \displaystyle\end{eqnarray}$$

We will also occasionally interpret (4.1) as a decomposition into axisymmetric and non-axisymmetric field components in systems with a rotation axis.Footnote ¹⁰ In order to minimise the physical complexity, we also restrict the analysis to the incompressible formulation of the kinematic dynamo problem with a uniform magnetic diffusivity, equation (2.18).

Let us start with the induction equation averaged over scales larger than $\ell _{0}$ ,

(4.3) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\overline{\boldsymbol{B}}}{\unicode[STIX]{x2202}t}+\overline{\boldsymbol{U}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\boldsymbol{B}}=\overline{\boldsymbol{B}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\boldsymbol{U}}+\unicode[STIX]{x1D735}\times (\overline{\boldsymbol{u}\times \boldsymbol{b}})+\unicode[STIX]{x1D702}\unicode[STIX]{x0394}\overline{\boldsymbol{B}}. & & \displaystyle\end{eqnarray}$$

To predict the evolution of $\overline{\boldsymbol{B}}$ , we must determine the mean electromotive force (EMF) $\overline{\boldsymbol{{\mathcal{E}}}}=\overline{\boldsymbol{u}\times \boldsymbol{b}}$ driving the dynamo. This term involves the statistical cross-correlations between small-scale magnetic and velocity fluctuations of the kind that we encountered in Parker’s phenomenology. The big question, of course, is how do we calculate it? Ideally, we would like to find an expression for $\overline{\boldsymbol{{\mathcal{E}}}}$ in terms of just $\overline{\boldsymbol{B}}$ and the statistical properties of the fluctuating velocity field $\boldsymbol{u}$ . The logical next step is therefore to attempt to solve the induction equation for $\boldsymbol{b}$ in terms of $\boldsymbol{u}$ and $\overline{\boldsymbol{B}}$ , and to substitute the solution into the above expression for $\overline{\boldsymbol{{\mathcal{E}}}}$ . This equation is obtained by subtracting (4.3) from the full induction equation (2.18),

(4.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{b}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times [(\boldsymbol{u}\times \overline{\boldsymbol{B}})+(\overline{\boldsymbol{U}}\times \boldsymbol{b})+(\boldsymbol{u}\times \boldsymbol{b}-\overline{\boldsymbol{u}\times \boldsymbol{b}})]+\unicode[STIX]{x1D702}\unicode[STIX]{x0394}\boldsymbol{b}.\end{eqnarray}$$

The term in the first pair of parentheses on the right-hand side describes the induction of small-scale magnetic fluctuations, and mixing of the field, due to the tangling and shearing of the mean field by small-scale velocity fluctuations. This is the term that we are most interested in order to compute a mean-field dynamo effect. The term in the second pair of parentheses describes the effect of a large-scale velocity field on small-scale magnetic fluctuations. Such a velocity field usually has a much slower turnover time and amplitude compared to the fluctuations, and we will therefore consider that it has a subdominant role in the inductive dynamics of small-scale fluctuations for the purpose of this discussion. Finally, we have a combination of two terms in the third pair of parentheses, which will be henceforth referred to as the ‘tricky term’ (also often referred to as the ‘pain in the neck’ term). This term is quadratic in fluctuations, and there is no obvious way to simplify it without making further assumptions: so we meet again, old closure foe! We will come back to this problem in § 4.3.6.

4.3.2 Mean-field ansatz

Equation (4.4) is a linear relationship between $\boldsymbol{b}$ and $\overline{\boldsymbol{B}}$ if $\boldsymbol{U}$ is independent of $\boldsymbol{B}$ (as is the case in the kinematic regime), i.e.

(4.5) $$\begin{eqnarray}{\mathcal{L}}_{\boldsymbol{U}}(\boldsymbol{b})=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \overline{\boldsymbol{B}}),\end{eqnarray}$$

where ${\mathcal{L}}_{\boldsymbol{U}}$ is a linear operator functionally dependent on $\boldsymbol{U}$ (Hughes Reference Hughes, Pogorelov, Audit and Zank2010). Despite its seemingly innocuous nature, there is actually quite a lot of complexity hiding in this equation due to the presence of the tricky term on the left-hand side, but for the time being we are going to ignore the presence of this term and simply postulate that (4.5) is indicative of a straightforward linear relationship between the fluctuations $\boldsymbol{b}$ and the mean field $\overline{\boldsymbol{B}}$ . If this holds, then we may expand the mean EMF as

(4.6) $$\begin{eqnarray}(\overline{\boldsymbol{u}\times \boldsymbol{b}})_{i}=a_{ij}\overline{B}_{j}+b_{ijk}\unicode[STIX]{x1D6FB}_{k}\overline{B}_{j}+\cdots \,,\end{eqnarray}$$

where the spatial derivative is with respect to the slow spatial variables over which $\overline{\boldsymbol{B}}$ is non-uniform. This is called the mean-field expansion. Terms involving higher-order derivatives are discarded because the spatial derivative is slow. In the general case of inhomogeneous, stratified, anisotropic, differentially rotating flows, this expansion is usually recast in terms of a broader combination of Greek-letter tensors and vectors that neatly isolate different symmetries (Krause & Rädler Reference Krause and Rädler1980; Rädler & Stepanov Reference Rädler and Stepanov2006),

(4.7) $$\begin{eqnarray}\overline{{\mathcal{E}}}_{i}=\unicode[STIX]{x1D6FC}_{ij}\overline{B}_{j}+(\unicode[STIX]{x1D738}\times \overline{\boldsymbol{B}})_{i}-\unicode[STIX]{x1D6FD}_{ij}(\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}})_{j}-[\unicode[STIX]{x1D739}\times (\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}})]_{i}-\frac{\unicode[STIX]{x1D705}_{ijk}}{2}(\unicode[STIX]{x1D6FB}_{j}\overline{B}_{k}+\unicode[STIX]{x1D6FB}_{k}\overline{B}_{j}).\end{eqnarray}$$

Here, $\unicode[STIX]{x1D736}$ and $\unicode[STIX]{x1D737}$ are symmetric second-order tensors, $\unicode[STIX]{x1D738}$ and $\unicode[STIX]{x1D739}$ are vectors and $\unicode[STIX]{x1D73F}$ is a third-order tensor, namely

(4.8a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{ij}={\textstyle \frac{1}{2}}(a_{ij}+a_{ji}),\quad \unicode[STIX]{x1D6FD}_{ij}={\textstyle \frac{1}{4}}(\unicode[STIX]{x1D700}_{ikl}b_{jkl}+\unicode[STIX]{x1D700}_{jkl}b_{ikl}),\end{eqnarray}$$

(4.9a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{i}=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D700}_{ijk}a_{jk},\quad \unicode[STIX]{x1D6FF}_{i}={\textstyle \frac{1}{4}}(b_{jji}-b_{jij}),\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{ijk}=-{\textstyle \frac{1}{2}}(b_{ijk}+b_{ikj}).\end{eqnarray}$$

All these quantities formally depend on $Rm$ , $Pm$ and the statistics of the velocity field (and magnetic field, in the dynamical regime). The two most emblematic mean-field effects are those deriving from $\unicode[STIX]{x1D736}$ and $\unicode[STIX]{x1D737}$ . The former is related to Parker’s mechanism and can drive a large-scale dynamo, while the latter is easily interpreted as a turbulent magnetic diffusion (albeit not necessarily a simple one).

4.3.3 Symmetry considerations

Symmetry considerations can be used to determine which of the previous mean-field quantities are in principle non-vanishing for a given problem. In order to illustrate at a basic level how this is generally done, and what kind of mean-field effects we might expect, we will essentially discuss three particular symmetries in the rest of this section: isotropy, parity and homogeneity.

As mentioned in § 3.4, there is no preferred direction or axis of symmetry in an isotropic three-dimensional system. Tensorial dynamical quantities can only be constructed from $\unicode[STIX]{x1D6FF}_{ij}$ and the antisymmetric Levi-Civita tensor $\unicode[STIX]{x1D700}_{ijk}$ in this case. On the other hand, if a fluid system has one or several particular directions, as is the case in the presence of rotation, stratification, a non-uniform large-scale flow $\overline{\boldsymbol{U}}$ or a strong, dynamical large-scale magnetic field, then we are in principle also allowed to use a combination of quantities such as the rotation vector $\unicode[STIX]{x1D734}$ , the direction of gravity (or inhomogeneity) $\boldsymbol{g}$ , the mean-flow deformation tensor $\overline{\unicode[STIX]{x1D63F}}=[\unicode[STIX]{x1D735}\overline{\boldsymbol{U}}+(\unicode[STIX]{x1D735}\overline{\boldsymbol{U}})^{\text{T}}]/2$ , the mean-flow vorticity $\overline{\boldsymbol{W}}=\unicode[STIX]{x1D735}\times \overline{\boldsymbol{U}}$ or even $\overline{\boldsymbol{B}}$ in the dynamical case to construct mean-field tensors.

A second important class of symmetry is parity invariance. In three dimensions, a parity transformation, or point reflection, is a combination of a reflection through a plane (an improper rotation) and a proper rotation of $\unicode[STIX]{x03C0}$ around an axis perpendicular to that plane. Parity symmetry is therefore connected to mirror symmetry, although the two must be distinguished in principle in three dimensions (see discussion in Moffatt Reference Moffatt, Balian and Peube1977, chap. 7, footnote 4). We essentially ignore this distinction in what follows and will talk indiscriminately of mirror/parity/reflection symmetry breaking. Under a point reflection, the position vector transforms as $\boldsymbol{r}\rightarrow -\boldsymbol{r}$ , and the velocity field transforms as $\boldsymbol{U}\rightarrow -\boldsymbol{U}$ . Now, if we look at some local, homogeneous, isotropic hydrodynamic solution of the Navier–Stokes equation driven by a non-helical force and with no rotation, it is clear that the image of the velocity field under a parity transformation is itself a solution of the equations. In the MHD and/or rotating case, on the other hand, the image of a solution is only a physical solution itself if we keep $\boldsymbol{B}$ and/or $\unicode[STIX]{x1D734}$ the same, i.e.  $\boldsymbol{B}\rightarrow \boldsymbol{B}$ , and $\unicode[STIX]{x1D734}\rightarrow \unicode[STIX]{x1D734}$ . Accordingly, we say that the velocity field is a true vector, while the magnetic field and rotation vectors are pseudo-vectors.Footnote ¹¹ Scalars and higher-order tensors can also be divided into pseudo and true quantities, depending on how they transform under reflections. Pseudo-scalars simply change sign under reflection. Let us finally introduce a few simple rules for the manipulation of vectors: vector products of pairs of true vectors or pairs of pseudo-vectors produce a pseudo-vector, while mixed vector products involving a true and a pseudo-vector produce a true vector. Curl operators turn a true vector into a pseudo-vector, and vice-versa (as illustrated by Ampere’s law, or the Biot–Savart law of magnetostatics between the electric current, a true vector field constructed from the velocities of charged particles and the magnetic field). Let us now come back to (4.7). The mean EMF on the left-hand side, being the vector product of a true vector and a pseudo-vector, is a true vector, while the mean magnetic field on the right-hand side is a pseudo-vector. Thanks to the particular decomposition used in the equation, it is then straightforward to see that $\unicode[STIX]{x1D736}$ and $\unicode[STIX]{x1D73F}$ must be pseudo-tensors, while $\unicode[STIX]{x1D737}$ is a true tensor. Similarly, $\unicode[STIX]{x1D739}$ must be a pseudo-vector, and $\unicode[STIX]{x1D738}$ is a true vector.

Why is this important? If a flow is parity invariant, as is for instance the case of standard homogeneous, isotropic non-helical turbulence, then its image under a parity transformation has the exact same statistical properties, and should produce the exact same physical statistical effects (with the same sign). However, we have identified several pseudo-quantities, most importantly $\unicode[STIX]{x1D736}$ , which by construction must change sign under a parity transformation. These quantities must therefore vanish for a parity-invariant flow. In order for them to be non-zero, parity invariance must be broken one way or the other, and a particular way to achieve this is by making the flow helical. Parker’s big idea was to suggest that this kind of effect can in turn lead to large-scale dynamo action. We will find in the next paragraphs that this can indeed be proven rigorously in some particular regimes. Of course, it is also this argument that motivated the set-up of the dynamo simulations of Meneguzzi et al. (Reference Meneguzzi, Frisch and Pouquet1981) discussed in the introduction of this section.

More generally, the previous discussion highlights that the different large-scale statistical effects encapsulated by the Greek-letter tensors introduced in (4.7) are only non-zero when some particular symmetries are broken. For pedagogical purposes, we will essentially concentrate on kinematic and dynamical problems involving simple isotropic, homogeneous $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ effects in this section. Important variants of these effects, as well as additional effects arising in stratified, rotating and shearing flows will however also be touched upon in § 4.4.

4.3.4 Mean-field equation for pseudo-isotropic homogeneous flows

In what follows, we will more specifically be concerned with the Cartesian version of the problem of large-scale dynamo action driven by isotropic, homogeneous, statistically steady, but non-parity-invariant flows (also referred to as pseudo-isotropic flows), the archetype of which is our now familiar helically forced isotropic turbulence. Although the helical nature of many flows in nature is a consequence of rotation, we will not explicitly try to capture this effect here, and instead assume that helical motions are forced externally (a discussion of the $\unicode[STIX]{x1D6FC}$ effect in rotating, stratified flow will be provided in § 4.4.1). On the other hand, we retain a large-scale shear flow typical of differentially rotating systems to accommodate the possibility of an $\unicode[STIX]{x1D6FA}$ effect, but neglect possible anisotropic statistical effects associated with this shear for simplicity (the latter will be discussed in § 4.4.3). Under all these assumptions, $\unicode[STIX]{x1D6FC}_{ij}=\unicode[STIX]{x1D6FC}\,\unicode[STIX]{x1D6FF}_{ij}$ and $\unicode[STIX]{x1D6FD}_{ij}=\unicode[STIX]{x1D6FD}\,\unicode[STIX]{x1D6FF}_{ij}$ (or $b_{ijk}=\unicode[STIX]{x1D6FD}\,\unicode[STIX]{x1D716}_{ijk}$ ) are both finite and independent of space and time, and all the other mean-field effects in (4.7) vanish. Substituting (4.6) into (4.3), we obtain a closed evolution equation for the mean field,

(4.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{\boldsymbol{B}}}{\unicode[STIX]{x2202}t}+\overline{\boldsymbol{U}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\boldsymbol{B}}=\overline{\boldsymbol{B}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\boldsymbol{U}}+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}+(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})\unicode[STIX]{x0394}\overline{\boldsymbol{B}}.\end{eqnarray}$$

The term on the right-hand side proportional to the curl of $\overline{\boldsymbol{B}}$ , the $\unicode[STIX]{x1D6FC}$ effect, is the only one, apart from the shear, that can couple the different components of the mean field for the simple system under consideration. The last term on the right-hand side, the $\unicode[STIX]{x1D6FD}$ effect, acts as an effective turbulent magnetic diffusivity operating on the mean field.

The presence of the $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ terms in the linear equation (4.11) implies that we can now in principle obtain exponentially growing mean-field dynamo solutions if the statistical properties of the flow allow it.Footnote ¹² This conclusion may seem puzzling at first in view of Cowling’s theorem. If we interpret large-scale averages as azimuthal averages, how is it possible that (4.11), which only involves axisymmetric quantities, has growing solutions? The key here is to realise that we have massaged the original problem quite a bit to arrive at this result. Equation (4.11) is not the pristine axisymmetric induction equation but a simpler model equation. The exact evolution equation for $\overline{\boldsymbol{B}}$ , equation (4.3), has a dynamo-driving term $\overline{\boldsymbol{{\mathcal{E}}}}$ , which is quadratic in non-axisymmetric (or small-scale) fluctuations. In order to arrive at (4.11), we have simply expressed this term using a simple closure equation (4.6) motivated by the linear nature of (4.4). In other words, the three-dimensional nature of the dynamo is now hidden in the mean-field coefficients $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ . We will demonstrate in § 4.3.6 that these coefficients are indeed functions of the statistics of small-scale fluctuations.

4.3.5 The $\unicode[STIX]{x1D6FC}^{2}$ , $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D6FA}$ dynamo solutions

In a Cartesian domain with periodic boundary conditions, we may seek simple solutions of (4.11) in the form

(4.12) $$\begin{eqnarray}\overline{\boldsymbol{B}}=\overline{\boldsymbol{B}}_{\boldsymbol{k}}\exp [st+\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}]+\text{c.c.},\end{eqnarray}$$

subject to $\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{B}}_{\boldsymbol{k}}=0$ . We first consider the case with no mean flow or shear. Substituting (4.12) into (4.11) with $\overline{\boldsymbol{U}}=\mathbf{0}$ , and solving the corresponding eigenvalue problem, we find a branch of growing dynamo modes with a purely real eigenvalue corresponding to a dynamo growth rate

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=|\unicode[STIX]{x1D6FC}|k-(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})k^{2}.\end{eqnarray}$$

For $\unicode[STIX]{x1D6FD}>0$ , the maximum growth rate and optimal wavenumber are

(4.14a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\text{max}}=\frac{\unicode[STIX]{x1D6FC}^{2}}{4(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})},\quad k_{\text{max}}=\frac{|\unicode[STIX]{x1D6FC}|}{2(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})}\end{eqnarray}$$

(for consistency, we should have $k_{\text{max}}\ell _{0}<1$ , but to check this we first need to learn how to calculate $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ ). The dynamo is possible here because the $\unicode[STIX]{x1D6FC}$ effect couples the two independent components of $\overline{\boldsymbol{B}}$ in both ways through the curl operator (in particular, it also couples the toroidal field component to the poloidal field component in cylindrical or spherical geometry). For this reason, this mean-field dynamo model is usually called the $\unicode[STIX]{x1D6FC}^{2}$ dynamo. It is this kind of coupling, illustrated in figure 28(a), that drives the helical large-scale kinematic dynamo effect in the simulations shown in figures 24 and 25. A discussion of magnetic helicity and linkage dynamics in the $\unicode[STIX]{x1D6FC}^{2}$ dynamo, complementary to the Parker mechanism picture shown earlier in figure 27, can be found in Blackman & Hubbard (Reference Blackman and Hubbard2014).

Figure 28. The $\unicode[STIX]{x1D6FC}^{2}$  (a),  $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D6FA}$  (b) mean-field dynamo loops.

What about the astrophysically relevant situation involving a mean shear flow or differential rotation, on top of some smaller-scale turbulence? While this case is of course most directly connected to problems in global spherical and cylindrical geometries, it is most easily analysed in the Cartesian shearing sheet model introduced in § 2.1.5. In the presence of linear shear, $\overline{\boldsymbol{U}}=\boldsymbol{U}_{S}=-Sx\boldsymbol{e}_{y}$ , the mean-field equation (4.11) reads

(4.15) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{\boldsymbol{B}}}{\unicode[STIX]{x2202}t}=-SB_{x}\boldsymbol{e}_{y}+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}+(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})\unicode[STIX]{x0394}\overline{\boldsymbol{B}}.\end{eqnarray}$$

The $\overline{\boldsymbol{U}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\boldsymbol{B}}$ term on the left-hand side of (4.11) vanishes for a purely azimuthal mean flow because $\overline{\boldsymbol{B}}$ is axisymmetricFootnote ¹³ (independent of $y$ in our notation). The couplings in (4.15) allow different forms of large-scale dynamo action illustrated in figure 28(b). In the most standard case, commonly referred to as the $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ dynamo, toroidal/azimuthal ( $y$ ) field is predominantly generated out of the poloidal field (of which $B_{x}$ is one component) via the $\unicode[STIX]{x1D6FA}$ effect (the first term on the right-hand side), while the $\unicode[STIX]{x1D6FC}$ effect associated with the small-scale helical turbulence turns the toroidal field back into poloidal field. In $\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D6FA}$ dynamos, the $\unicode[STIX]{x1D6FC}$ effect also significantly contributes to the conversion of poloidal field into toroidal field.

Let us briefly look at axisymmetric mean-field solutionsFootnote ¹⁴   $\overline{\boldsymbol{B}}(x,z,t)$ of (4.15) in the $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ case. We again seek plane-wave solutions in the form of (4.12), with a wavenumber $\boldsymbol{k}=(k_{x},0,k_{z})$ . In the $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ limit $S\gg \unicode[STIX]{x1D6FC}k_{z}$ , the solution of the linear dispersion relation leads to a branch of growing dynamo solutions if the so-called dynamo number $D\equiv \unicode[STIX]{x1D6FC}Sk_{z}/[2(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})^{2}k^{4}]$ is larger than unity (Brandenburg & Subramanian Reference Brandenburg and Subramanian2005a ). The unstable eigenvalue in this case has an imaginary part,

(4.16) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{R}}(s)\equiv \unicode[STIX]{x1D6FE}={\textstyle \frac{1}{2}}|S\unicode[STIX]{x1D6FC}k_{z}|^{1/2}-(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})k^{2}, & \displaystyle\end{eqnarray}$$

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{I}}(s)\equiv \unicode[STIX]{x1D714}={\textstyle \frac{1}{2}}|S\unicode[STIX]{x1D6FC}k_{z}|^{1/2}, & \displaystyle\end{eqnarray}$$

i.e. the solutions take the form of growing oscillations. The interplay between the $\unicode[STIX]{x1D6FC}$ effect and the $\unicode[STIX]{x1D6FA}$ effect can therefore produce dynamo waves involving periodic field reversals. This conclusion, first obtained by Parker (Reference Parker1955a ), was key to the realisation that differentially rotating turbulent MHD systems can host large-scale oscillatory dynamos, and sparked an enormous amount of research on planetary, solar, stellar and galactic dynamo modelling.

4.3.6 Calculation of mean-field coefficients: first-order smoothing

The mean-field formalism appears to be a very convenient and intuitive framework to model large-scale turbulent dynamos and to simplify their underlying nonlinear MHD complexity. However, we have so far ignored several difficult key questions: under which conditions is the mean-field ansatz justified? And are there systematic ways to derive mean-field coefficients such as $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ from first principles, given a velocity field with prescribed statistical properties?

The former question largely conditions the answer to the latter. To illustrate the nature of the problem, recall that we have carefully avoided to discuss how to handle the tricky term in (4.4) in our earlier discussion of the mean-field ansatz. However, if we want to calculate $\unicode[STIX]{x1D6FC}$ , we need to do something about it! The radical closure option is to simply neglect this term. This approximation is often called the first-order smoothing approximation (FOSA), second-order correlation approximation (SOCA), Born approximation or quasilinear approximation. It has the merit of simplicity, but can only be rigorously justified in two limiting cases:

1. (i) small velocity correlation times (either small Strouhal numbers $St=\unicode[STIX]{x1D70F}_{c}/\unicode[STIX]{x1D70F}_{\text{NL}}\ll 1$ , or random wave fields);

2. (ii) low magnetic Reynolds numbers $Rm=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/\unicode[STIX]{x1D70F}_{\text{NL}}\ll 1$ ,

where $\unicode[STIX]{x1D70F}_{\text{NL}}=\ell _{0}/u_{\text{rms}}$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}=\ell _{0}^{2}/\unicode[STIX]{x1D702}$ in the two-scale model (assuming $Pm=O(1)$ ). To see this, we borrow directly from Moffatt (Reference Moffatt1978) and consider the ordering of (4.4),

(4.18) $$\begin{eqnarray}\underbrace{\frac{\unicode[STIX]{x2202}\boldsymbol{b}}{\unicode[STIX]{x2202}t}}_{O(b_{\text{rms}}/\unicode[STIX]{x1D70F}_{c})}=\underbrace{\unicode[STIX]{x1D735}\times [(\boldsymbol{u}\times \overline{\boldsymbol{B}})]}_{O(\overline{B}/\unicode[STIX]{x1D70F}_{\text{NL}})}+\underbrace{\unicode[STIX]{x1D735}\times [(\boldsymbol{u}\times \boldsymbol{b})-(\overline{\boldsymbol{u}\times \boldsymbol{b}})]}_{\text{Tricky}\;\text{term}:\;O(b_{\text{rms}}/\unicode[STIX]{x1D70F}_{\text{NL}})}+\underbrace{\unicode[STIX]{x1D702}\unicode[STIX]{x0394}\boldsymbol{b}}_{O(b_{\text{rms}}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})},\end{eqnarray}$$

where $\overline{\boldsymbol{U}}=\mathbf{0}$ has been assumed for simplicity. The first important thing to notice here is that the typical rate of change of $\boldsymbol{b}$ on the left-hand side is ordered as the inverse correlation time of the velocity field, because significant field amplification requires coherent flow stretching episodes whose typical duration is $O(\unicode[STIX]{x1D70F}_{c})$ . Another handwaving way to look at this is to formally integrate (4.18) up to a time $t$ , and to approximate the integral of the tricky term by $\unicode[STIX]{x1D70F}_{c}$ (the times during which $\boldsymbol{u}$ remains self-correlated) times the integrand. Either way, we see that the tricky term is negligible compared to the left-hand side in the limit $St\ll 1$ . The problem, of course, is that $St$ is usually $O(1)$ for standard fluid turbulence such as Kolmogorov turbulence. In this case, the only limit in which the tricky term is formally negligible is the diffusion-dominated regime $Rm\ll 1$ in which the resistive term dominates. Unfortunately, this regime is not very interesting from an astrophysical perspective either.

4.3.7 FOSA derivations of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ for homogeneous helical turbulence

The calculation of mean-field electrodynamics coefficients under FOSA can be done with a variety of mathematical methods, most if not all of which have already been skilfully laid out in textbooks and reviews by the inventors and main practitioners of the theory (e.g. Moffatt Reference Moffatt1978; Krause & Rädler Reference Krause and Rädler1980; Brandenburg & Subramanian Reference Brandenburg and Subramanian2005a ; Rädler & Rheinhardt Reference Rädler and Rheinhardt2007). Many of these presentations are very detailed though, and can feel a bit intimidating or overwhelming to newcomers in the field. The aim of this paragraph is therefore to provide concise, low-algebra versions of the calculations of the $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ effects distilling the essence of these methods.

In order to illustrate in the most simple possible way how this kind of derivation works, we start with a physically intuitive calculation in the $St\ll 1$ , $Rm\gg 1$ regime. In this limit, both the tricky term and the resistive term can be neglected in (4.18), but we cannot formally neglect the first term on the right-hand side of the equation, because there is no particular reason to assume that $\boldsymbol{b}$ and $\overline{\boldsymbol{B}}$ should be of the same order. In fact, they are not and, as mentioned in § 4.3.2, it is precisely this term that makes a large-scale dynamo possible in the first place. In order to calculate $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ , we simply substitute the formal integral solution $\boldsymbol{b}(\boldsymbol{x},t)$ of (4.18) into the expression of the mean EMF,

(4.19) $$\begin{eqnarray}\overline{\boldsymbol{{\mathcal{E}}}}(\boldsymbol{x},t)\equiv \overline{\boldsymbol{u}(\boldsymbol{x},t)\times \boldsymbol{b}(\boldsymbol{x},t)}=\overline{\boldsymbol{u}(t)\times \int _{t_{0}}^{t}\unicode[STIX]{x1D735}\times [\boldsymbol{u}(\boldsymbol{x},t^{\prime })\times \overline{\boldsymbol{B}}(\boldsymbol{x},t^{\prime })]\,\text{d}t^{\prime }}.\end{eqnarray}$$

In writing this expression, we have implicitly assumed that $t-t_{0}$ is much larger than the typical flow correlation and resistive time scales, so that the correlation $\overline{\boldsymbol{u}(t)\times \boldsymbol{b}(t_{0})}$ can be neglected, and the integral is understood to be independent of the lower bound $t_{0}$ . The spatial dependence of $\overline{\boldsymbol{{\mathcal{E}}}}$ is to be understood as a large-scale dependence on the scale of the mean field itself. The right-hand side of (4.19) only contains correlations that are quadratic in fluctuations, as a result of the neglect of the tricky term in (4.18), hence the name second-order correlation approximation (or first-order smoothing approximation, if we take the equivalent view that it is the quadratic term in (4.18) which has been neglected). Now, a bit of tensor algebra shows that (4.19) in the isotropic case reduces to

(4.20) $$\begin{eqnarray}\overline{\boldsymbol{{\mathcal{E}}}}(\boldsymbol{x},t)=\int _{0}^{t}[\hat{\unicode[STIX]{x1D6FC}}(t-t^{\prime })\overline{\boldsymbol{B}}(\boldsymbol{x},t^{\prime })-\hat{\unicode[STIX]{x1D6FD}}(t-t^{\prime })\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}(\boldsymbol{x},t^{\prime })]\,\text{d}t^{\prime },\end{eqnarray}$$

where

(4.21a,b ) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FC}}=-{\textstyle \frac{1}{3}}\overline{\boldsymbol{u}(t)\boldsymbol{\cdot }\unicode[STIX]{x1D74E}(t^{\prime })},\quad \hat{\unicode[STIX]{x1D6FD}}={\textstyle \frac{1}{3}}\overline{\boldsymbol{u}(t)\boldsymbol{\cdot }\boldsymbol{u}(t^{\prime })},\end{eqnarray}$$

and $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$ . The coefficients $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are uniform in this derivation as a result of the assumption of statistical homogeneity. Now, since the correlation time of the flow is assumed small compared to its typical turnover time, we can approximate the integrals in (4.20) by $\unicode[STIX]{x1D70F}_{c}$ times the integrand at $t^{\prime }=t$ , i.e.

(4.22) $$\begin{eqnarray}\overline{\boldsymbol{{\mathcal{E}}}}=\unicode[STIX]{x1D6FC}\overline{\boldsymbol{B}}-\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}},\end{eqnarray}$$

with

(4.23a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=-{\textstyle \frac{1}{3}}\unicode[STIX]{x1D70F}_{c}\overline{(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D74E})},\quad \unicode[STIX]{x1D6FD}={\textstyle \frac{1}{3}}\unicode[STIX]{x1D70F}_{c}\overline{|\boldsymbol{u}|^{2}}.\end{eqnarray}$$

As expected by virtue of the FOSA, equation (4.22) has the form of the isotropic version of the mean-field expansion (4.6) postulated in § 4.3.2, but we have now also found explicit expressions for $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ .

The previous calculation relies on a description of the statistical properties of the flow in configuration (correlation) space, and is representative of the general method used by Krause, Steenbeck and Rädler in the 1960s to develop a much broader theory of mean-field electrodynamical effects in differentially rotating stratified flows (their most important papers, originally published in German, have been translated in English by Roberts & Stix (Reference Roberts and Stix1971), and form the core of the textbook by Krause & Rädler Reference Krause and Rädler1980). However, mean-field coefficients can also be derived by means of a spectral representation of the turbulent dynamics. This formalism is perhaps slightly less intuitive physically than the procedure outlined above, but it makes the calculations a bit more compact and straightforward in the general case, and is also frequently encountered in the literature (the reference textbook here is Moffatt Reference Moffatt1978). In order to introduce this alternative in a nutshell, we concentrate uniquely on the problem of an $\unicode[STIX]{x1D6FC}$ effect generated by helical homogeneous incompressible turbulence acting upon a uniform mean magnetic field. However, we do not a priori restrict the calculation to one of the two possible FOSA limits in this case and, in anticipation of a further discussion of rotational effects in § 4.4.3, we do not assume isotropy from the start either (a more detailed version of this derivation, also including a determination of the $\unicode[STIX]{x1D737}$ tensor for a non-uniform field, can be found in Moffatt & Proctor Reference Moffatt and Proctor1982). Denoting the space–time Fourier transforms of $\boldsymbol{u}$ and $\boldsymbol{b}$ over the small-scale, fast variables by $\hat{\boldsymbol{u}}(\boldsymbol{k},\unicode[STIX]{x1D714})$ and $\hat{\boldsymbol{b}}(\boldsymbol{k},\unicode[STIX]{x1D714})$ , we first solve (4.18) in spectral space subject to FOSA as

(4.24) $$\begin{eqnarray}\hat{\boldsymbol{b}}(\boldsymbol{k},\unicode[STIX]{x1D714})=\frac{\text{i}(\overline{\boldsymbol{B}}\boldsymbol{\cdot }\boldsymbol{k})}{(-\text{i}\unicode[STIX]{x1D714}+\unicode[STIX]{x1D702}k^{2})}\hat{\boldsymbol{u}}(\boldsymbol{k},\unicode[STIX]{x1D714}).\end{eqnarray}$$

Substituting this expression into the general expression for the mean EMF,

(4.25) $$\begin{eqnarray}\overline{{\mathcal{E}}}_{i}=\iiiint \unicode[STIX]{x1D700}_{ilm}\overline{\hat{u}_{l}^{\ast }(\boldsymbol{k}^{\prime },\unicode[STIX]{x1D714}^{\prime })\hat{b}_{m}(\boldsymbol{k},\unicode[STIX]{x1D714})}\text{e}^{\text{i}[(\boldsymbol{k}-\boldsymbol{k}^{\prime })\boldsymbol{\cdot }\boldsymbol{x}-(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}^{\prime })t]}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\,\text{d}\boldsymbol{k}^{\prime }\,\text{d}\unicode[STIX]{x1D714}^{\prime },\end{eqnarray}$$

and introducing the spectrum tensor of the turbulence $\unicode[STIX]{x1D731}_{lm}(\boldsymbol{k},\unicode[STIX]{x1D714})$ , defined by

(4.26) $$\begin{eqnarray}\overline{\hat{u}_{l}^{\ast }(\boldsymbol{k},\unicode[STIX]{x1D714})\hat{u}_{m}(\boldsymbol{k}^{\prime },\unicode[STIX]{x1D714}^{\prime })}=\unicode[STIX]{x1D731}_{lm}(\boldsymbol{k},\unicode[STIX]{x1D714})\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{k}^{\prime })\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}^{\prime }),\end{eqnarray}$$

we find after integration over $\boldsymbol{k}^{\prime }$ and $\unicode[STIX]{x1D714}^{\prime }$ that

(4.27) $$\begin{eqnarray}\overline{{\mathcal{E}}}_{i}=\left(\iint \frac{\text{i}\unicode[STIX]{x1D700}_{ilm}k_{j}}{(-\text{i}\unicode[STIX]{x1D714}+\unicode[STIX]{x1D702}k^{2})}\unicode[STIX]{x1D731}_{lm}(\boldsymbol{k},\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\right)\overline{B}_{j}.\end{eqnarray}$$

Recalling the mean-field ansatz for a uniform mean field, $\overline{{\mathcal{E}}}_{i}=\unicode[STIX]{x1D6FC}_{ij}\overline{B}_{j}$ , the term in parenthesis is easily identified with $\unicode[STIX]{x1D6FC}_{ij}$ . In order to simplify this expression further, we introduce the helicity spectrum function

(4.28) $$\begin{eqnarray}H(\boldsymbol{k},\unicode[STIX]{x1D714})=-\text{i}k_{n}\unicode[STIX]{x1D700}_{nlm}\unicode[STIX]{x1D731}_{lm}(\boldsymbol{k},\unicode[STIX]{x1D714}),\end{eqnarray}$$

and remark that

(4.29) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{ilm}\unicode[STIX]{x1D731}_{lm}(\boldsymbol{k},\unicode[STIX]{x1D714})=\frac{k_{i}}{k^{2}}H(\boldsymbol{k},\unicode[STIX]{x1D714})\end{eqnarray}$$

for an incompressible flow (i.e. the left-hand side, being a cross-product of two vectors $\hat{\boldsymbol{u}}(\boldsymbol{k},\unicode[STIX]{x1D714})$ and $\hat{\boldsymbol{u}}^{\ast }(\boldsymbol{k},\unicode[STIX]{x1D714})$ both perpendicular to $\boldsymbol{k}$ , is oriented along $\boldsymbol{k}$ ). Substituting (4.29) into (4.27) and using the condition $H(-\boldsymbol{k},-\unicode[STIX]{x1D714})=H(\boldsymbol{k},\unicode[STIX]{x1D714})$ deriving from the reality of $\boldsymbol{u}$ to eliminate the imaginary part of the integral, we arrive at the result

(4.30) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{ij}=-\unicode[STIX]{x1D702}\iint \frac{k_{i}k_{j}}{\unicode[STIX]{x1D714}^{2}+\unicode[STIX]{x1D702}^{2}k^{4}}H(\boldsymbol{k},\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\end{eqnarray}$$

showing that the $\unicode[STIX]{x1D736}$ tensor is a weighted integral of the helicity spectrum of the flow under generic anisotropic conditions. In the isotropic case, $H(\boldsymbol{k},\unicode[STIX]{x1D714})=H(k,\unicode[STIX]{x1D714})$ , and $\unicode[STIX]{x1D6FC}_{ij}=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}_{ij}$ with

(4.31) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\frac{\unicode[STIX]{x1D6FC}_{ii}}{3}=-\frac{\unicode[STIX]{x1D702}}{3}\iint \frac{k^{2}}{(\unicode[STIX]{x1D714}^{2}+\unicode[STIX]{x1D702}^{2}k^{4})}H(k,\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}.\end{eqnarray}$$

One of the main advantages of the spectral formalism, clearly, is that it makes it easy in principle to investigate either analytically or numerically the dynamo properties of flows with different prescribed spectral distributions, from simple ‘monochromatic’ helical wave fields to more complex random flows characterised by a broad range of frequencies and wavenumbers. As stressed by Moffatt (Reference Moffatt1978) though, one must be wary of the application of the formalism at large $Rm$ if the flow under consideration has some significant helicity and energy at frequencies $\unicode[STIX]{x1D714}<u_{0}/\ell _{0}$ , in which case the FOSA approximation cannot be justified. On the other hand, we can safely use (4.31) to derive $\unicode[STIX]{x1D6FC}$ analytically in the $St=O(1)$ , $Rm\ll 1$ limit. In this case, $\unicode[STIX]{x1D714}\ll \unicode[STIX]{x1D702}k^{2}$ and the weight function in the integral in (4.31) can be approximated by $1/(\unicode[STIX]{x1D702}^{2}k^{2})$ . Using the property

(4.32) $$\begin{eqnarray}\overline{(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D74E})}=\iint H(\boldsymbol{k},\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714},\end{eqnarray}$$

we obtain the low- $Rm$ result (Moffatt Reference Moffatt1970a )

(4.33) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=-{\textstyle \frac{1}{3}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{(H)}\overline{(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D74E})},\end{eqnarray}$$

with

(4.34) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{(H)}=\frac{1}{\unicode[STIX]{x1D702}}\frac{\displaystyle \iint k^{-2}H(\boldsymbol{k},\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}}{\displaystyle \iint H(\boldsymbol{k},\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}}.\end{eqnarray}$$

An extension of the calculation to a non-uniform mean field similarly leads to

(4.35) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}={\textstyle \frac{1}{3}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{(E)}\overline{|\boldsymbol{u}|^{2}},\end{eqnarray}$$

with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{(E)}$ similarly defined as a weighted integral of the kinetic energy spectrum $E(k,\unicode[STIX]{x1D714})$ .

The previous results all lead to the comforting conclusion that the $\unicode[STIX]{x1D6FC}$ effect is directly proportional to the net kinetic helicity of the flow, in line with Parker’s original intuition of large-scale dynamo action driven by cyclonic convection. The only difference between (4.23) and (4.33) derived in the low- $St$ and low- $Rm$ FOSA limits respectively is the characteristic time involved. In both cases, the results translate that the twisting of magnetic-field lines into Parker loops occurs on a short time compared to the dynamical turnover time. The amount of coherent twisting by each individual swirl is limited by magnetic diffusion in the $Rm\ll 1$ case, and by the fast decorrelation of the flow in the $St\ll 1$ case. The small but systematic effects of these ‘impulsive’ twists simply add up statistically.

The problem, however, gets significantly more complicated if we consider the more realistic and astrophysically relevant regime $St=O(1)$ , $Rm\gg 1$ for which the field is essentially frozen-in. For instance, when $\unicode[STIX]{x1D70F}_{c}=O(\ell _{0}/u_{0})$ , a Parker loop can do a full $360^{\circ }$ turn before the swirl decorrelates, leaving us with zero net magnetic field in the direction perpendicular to the original field orientation. While this scenario is extreme, the point is that the net effect of a statistical ensemble of generic turbulent swirls with $St=O(1)$ at large $Rm$ is much harder to assess than in the calculations above due to cancellation effects. This kind of complication notably creates some significant difficulties with the determination of the kinematic value of $\unicode[STIX]{x1D6FC}$ at large $Rm$ for some families of chaotic flows with long correlation times (Courvoisier, Hughes & Tobias Reference Courvoisier, Hughes and Tobias2006), although not necessarily for generic isotropically forced helical turbulence with $St=O(1)$ (Sur, Brandenburg & Subramanian Reference Sur, Brandenburg and Subramanian2008). On top of all that, we will also discover later in § 4.5 that the neglect of the tricky term is actually generically problematic at large $Rm$ , even when $St\ll 1$ , as a result of the excitation of small-scale dynamo fields.

Note finally that the FOSA result (4.33), combined with $\unicode[STIX]{x1D6FD}\ll \unicode[STIX]{x1D702}$ , ensures that the theory is consistent with the two-scale assumption in the regime of low $Rm$ . In the $\unicode[STIX]{x1D6FC}^{2}$ dynamo problem, for instance, the wavenumbers $k$ at which growth occurs at low $Rm$ are of the order $k\ell _{0}\sim \unicode[STIX]{x1D6FC}\ell _{0}/\unicode[STIX]{x1D702}\sim Rm^{2}\ll 1$ (assuming $|\unicode[STIX]{x1D74E}|_{\text{rms}}\sim u_{\text{rms}}/\ell _{0}$ ). In the $St\ll 1$ , $Rm\gg 1$ regime, on the other hand, equation (4.23) applies, $\unicode[STIX]{x1D6FD}\gg \unicode[STIX]{x1D702}$ , and $k\ell _{0}\sim \unicode[STIX]{x1D6FC}\ell _{0}/\unicode[STIX]{x1D6FD}\sim \overline{(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D74E})}\ell _{0}/\overline{\boldsymbol{u}^{2}}$ . The theory is therefore formally only self-consistent in this regime if the flow has small fractional helicity, i.e.  $|\unicode[STIX]{x1D74E}|_{\text{rms}}\ll u_{\text{rms}}/\ell _{0}$ .

4.3.8 Third-order moment closures: eddy-damped quasi-normal Markovian and $\unicode[STIX]{x1D70F}$ -approach*

While the FOSA closure has been central to the historical development of mean-field electrodynamics, it’s very limited formal domain of validity implies that its predictive power and practical applicability is also formally very limited. Other closure schemes, such as the eddy-damped quasi-normal Markovian (EDQNM) (Orszag Reference Orszag1970; Pouquet, Frisch & Leorat Reference Pouquet, Frisch and Leorat1976) or minimal $\unicode[STIX]{x1D70F}$ approximation (MTA) closures (Vainshtein & Kichatinov Reference Vainshtein and Kichatinov1983; Kleeorin, Rogachevskii & Ruzmaikin Reference Kleeorin, Rogachevskii and Ruzmaikin1990; Blackman & Field Reference Blackman and Field2002) operating at the next order in the hierarchy of moments, have been implemented in the context of large-scale dynamo theory in order to attempt to remedy this problem. To illustrate the idea behind these closures, consider the time derivative of the mean EMF,

(4.36) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{\boldsymbol{{\mathcal{E}}}}}{\unicode[STIX]{x2202}t}\equiv \overline{\boldsymbol{u}\times \frac{\unicode[STIX]{x2202}\boldsymbol{b}}{\unicode[STIX]{x2202}t}}+\overline{\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}\times \boldsymbol{b}},\end{eqnarray}$$

and substitute $\unicode[STIX]{x2202}\boldsymbol{b}/\unicode[STIX]{x2202}t$ by its expression in (4.18), and $\unicode[STIX]{x2202}\boldsymbol{u}/\unicode[STIX]{x2202}t$ by its expression given by the Navier–Stokes equation for $\boldsymbol{u}$ , assuming that the contribution of the Lorentz force is of the form $\overline{\boldsymbol{B}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{b}$ (this is formally only valid for small $|\overline{\boldsymbol{B}}|\ll b_{rms}$ , see e.g. discussion in Proctor Reference Proctor, Thompson and Christensen-Dalsgaard2003). From this equation, it is easy to see that the tricky term in (4.18) and the inertial term in the Navier–Stokes equation introduce third-order correlations in the resulting evolution equation for the second-order moment $\overline{\boldsymbol{{\mathcal{E}}}}$ . In order to close this equation, one possibility is to model the effects of these triple correlations (fluxes) in terms of a simple relaxation of the EMF, $-\overline{\boldsymbol{{\mathcal{E}}}}/\unicode[STIX]{x1D70F}$ ( $\unicode[STIX]{x1D70F}$ , the relaxation time, is sometimes assumed to be scale dependent). This qualitatively amounts to modelling the effects of the turbulent fluxes as a diffusion-like/damping effect. The result of the simplest version of this kind of calculation after applying this new closure ansatz is (e.g. Blackman & Field Reference Blackman and Field2002; Brandenburg & Subramanian Reference Brandenburg and Subramanian2005a ),

(4.37) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{\boldsymbol{{\mathcal{E}}}}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D6FC}^{\prime }\overline{\boldsymbol{B}}-\unicode[STIX]{x1D6FD}^{\prime }\,\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}-\frac{\overline{\boldsymbol{{\mathcal{E}}}}}{\unicode[STIX]{x1D70F}},\end{eqnarray}$$

where now

(4.38a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{\prime }=-{\textstyle \frac{1}{3}}[\overline{\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D74E}}-\overline{(\unicode[STIX]{x1D735}\times \boldsymbol{b})\boldsymbol{\cdot }\boldsymbol{b}}],\quad \unicode[STIX]{x1D6FD}^{\prime }={\textstyle \frac{1}{3}}\overline{|\boldsymbol{u}|^{2}}.\end{eqnarray}$$

One seeming advantage of this scheme, compared to FOSA, is that $\unicode[STIX]{x1D70F}$ is not formally required to be small in comparison to the typical eddy turnover time. However, it is not a panacea either, because there is no guarantee that the net effect of triple correlations is to relax the mean EMF in the form prescribed by an MTA-like closure. In fact, it is all but certain that this is not strictly true for most turbulence problems, and that even the qualitative validity of this kind of prescription is problem and regime dependent. In the mean-field dynamo context, it has for instance notably been found that analytical predictions that come out of the MTA are not always in agreement with the FOSA predictions in the (admittedly rather peculiar) regimes in which the latter is formally valid (e.g. Rädler & Rheinhardt Reference Rädler and Rheinhardt2007, see § 4.4.3 below for a discussion of a specific example). On the other hand, analyses of numerical simulations of turbulent-transport and large-scale dynamo problems suggest that an MTA closure with $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\text{NL}}=O(1)$ is not entirely unfounded either in some common fluid turbulence regimes (Brandenburg, Käpylä & Mohammed Reference Brandenburg, Käpylä and Mohammed2004; Brandenburg & Subramanian Reference Brandenburg and Subramanian2005b ) (of course, that $\unicode[STIX]{x1D70F}$ in this model should be of the order of $\unicode[STIX]{x1D70F}_{\text{NL}}$ is expected from a simple order of magnitude analysis of the different correlation functions involved in the closure). Overall, it seems like we have no choice at the moment but to accept the theoretical uncertainties and confusion that come with this kind of problem. Closures are notoriously difficult in all areas of turbulence research (Krommes Reference Krommes2002), and dynamo theory is no exception to this.

Keeping these caveats in mind, it is nevertheless interesting to note that the MTA closure applied to the helical dynamo problem gives rise in (4.38) to a magnetic contribution to the $\unicode[STIX]{x1D6FC}$ effect associated with small-scale helical magnetic fluctuations interacting with the turbulent velocity field. The questions of the exact meaning, validity and applicability of this result are beyond the realm of pure kinematic theory, and will be critically discussed in § 4.6 in the context of saturation of large-scale dynamos.

4.4.1 The $\unicode[STIX]{x1D6FC}$ effect in a stratified, rotating flow

As we discovered in § 4.3.3, an $\unicode[STIX]{x1D6FC}$ effect is only possible in flows that are not parity invariant, a particular example of which is helically forced, pseudo-isotropic homogeneous turbulence. The kinematic derivations presented in § 4.3.7 showed that $\unicode[STIX]{x1D6FC}$ in this problem is directly proportional (in the FOSA regimes) to the net average kinetic helicity of the flow. Here, we will see that the $\unicode[STIX]{x1D6FC}$ effect is also non-zero in a rotating, stratified flow involving a gradient of density and/or kinetic energy, such as stratified rotating thermal convection typical of many stellar interiors (see e.g. Brandenburg & Subramanian (Reference Brandenburg and Subramanian2005a ) for a similar discussion). Beyond its obvious astrophysical relevance, this particular problem provides a good illustration of how one can use symmetry arguments to simplify matters. Zooming in on a patch of stratified, rotating turbulence, we can locally identify two preferred directions in the system, that of gravity denoted by the unit vector $\hat{\boldsymbol{g}}=\boldsymbol{g}/g$ , and that of rotation, denoted by the unit vector $\hat{\unicode[STIX]{x1D734}}=\unicode[STIX]{x1D734}/\unicode[STIX]{x1D6FA}$ . The pseudo-tensor $\unicode[STIX]{x1D736}$ can only be constructed from these two vectors in the following way (to lowest order in $g$ and $\unicode[STIX]{x1D6FA}$ ):

(4.39) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{ij}=\unicode[STIX]{x1D6FC}_{0}\hat{\boldsymbol{g}}\boldsymbol{\cdot }\hat{\unicode[STIX]{x1D734}}\,\unicode[STIX]{x1D6FF}_{ij}+\unicode[STIX]{x1D6FC}_{1}{\hat{g}}_{j}\hat{\unicode[STIX]{x1D6FA}}_{i}+\unicode[STIX]{x1D6FC}_{2}{\hat{g}}_{i}\hat{\unicode[STIX]{x1D6FA}}_{j}.\end{eqnarray}$$

Of course, we still have to calculate the three mean-field coefficients. This is difficult to achieve in the general case, but explicit expressions can be obtained for a few analytically prescribed flows. One of the most well-known results in this context is the expression

(4.40a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{0}\,\hat{\boldsymbol{g}}\boldsymbol{\cdot }\hat{\unicode[STIX]{x1D734}}=-\frac{16}{15}\unicode[STIX]{x1D70F}_{c}^{2}\overline{|\boldsymbol{u}|^{2}}\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\ln \left(\unicode[STIX]{x1D70C}\sqrt{\overline{|\boldsymbol{u}|^{2}}}\right),\quad \unicode[STIX]{x1D6FC}_{1}=\unicode[STIX]{x1D6FC}_{2}=-\frac{\unicode[STIX]{x1D6FC}_{0}}{4}\end{eqnarray}$$

derived in the $St\ll 1$ FOSA regime for an anelastic flow model (characterised by $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u})=0$ ) encapsulating the effects of stratification through a simple exponential vertical-dependence parametrisation of the background density and turbulence intensity (Krause Reference Krause1967; Steenbeck & Krause Reference Steenbeck and Krause1969). Equation (4.40) obviously provides a more explicit mathematical validation of Parker’s idea that rotating convection can generate an $\unicode[STIX]{x1D6FC}$ effect than equation (4.23), however it also suggests that some stratification along the rotation axis, not just rotation, is actually required in a rotating flow to obtain a non-zero $\unicode[STIX]{x1D736}$ tensor. The physical reason underlying this result was identified by Steenbeck et al. (Reference Steenbeck, Krause and Rädler1966). In a stratified environment, rising fluid expands horizontally, while sinking fluid is compressed. Thereby, and upon the action of the Coriolis force, upflows in the northern hemisphere are made to rotate clockwise, while downflows are made to rotate counter-clockwise. In both cases, the flow acquires negative helicity. The effect is opposite in the southern hemisphere. The net result, expressed by (4.40), is an antisymmetric distribution of $\unicode[STIX]{x1D6FC}$ with respect to the equator. This argument and (4.40) also predict that $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}\unicode[STIX]{x1D711}}$ (or $\unicode[STIX]{x1D6FC}_{yy}$ in the Cartesian model with a vertical rotation axis), the key coefficient that couples back the toroidal field to the poloidal field in an $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ dynamo, is positive in the northern hemisphere. We will discover later in § 5.1 that this result is actually problematic in the solar dynamo context.

The stratification effect described above formally vanishes in the incompressible Boussinesq regime of convection rotating about a vertical axis, because in this limit all motions are assumed to take place at a vertical scale much smaller than the typical density scale height. Note however that the presence of vertical boundaries in the system also generates converging and diverging motions that get acted upon by the Coriolis force. It is actually this type of horizontal motions, not expansions or compressions in a stratified atmosphere, that Parker apparently had in mind when he introduced the idea of an $\unicode[STIX]{x1D6FC}$ effect generated by cyclonic convection (he used the words ‘influxes’ and ‘effluxes’). In his work, Parker focused on helical motions of a given sign, but in Boussinesq convection the dynamics of overturning rotating convective motions actually leads to the generation of flow helicity of opposite signs at the top and bottom. In plane-parallel Boussinesq convection between two plates and rotating about the vertical axis (the rotating Rayleigh–Bénard problem) in particular, the existence of an ‘up-down’ dynamical mirror symmetry with respect to the mid-plane of the convection layer (Chandrasekhar Reference Chandrasekhar1961) implies that the profile of (horizontally averaged) flow helicity induced by the Coriolis force is antisymmetric in $z$ . Accordingly, the volume-averaged flow helicity of the system is zero in this case.Footnote ¹⁵ In the density-stratified case, this up-down symmetry is broken (e.g. Graham Reference Graham1975; Gough et al. Reference Gough, Moore, Spiegel and Weiss1976; Graham & Moore Reference Graham and Moore1978; Massaguer & Zahn Reference Massaguer and Zahn1980) and the vertical profile of horizontally averaged helicity has no particular symmetry (Käpylä, Korpi & Brandenburg Reference Käpylä, Korpi and Brandenburg2009). The volume-averaged helicity does not therefore in general vanish in this case, illustrating further that it is stratification effects that generate a helicity imbalance in this problem.

4.4.2 Turbulent pumping*

In an inhomogeneous fluid flow, the statistics of the velocity field depend on the coordinate along the inhomogeneous direction $\hat{\boldsymbol{g}}$ . If we thread such a flow with a large-scale magnetic field, then the latter will be expelled from the regions of higher turbulence intensity to the regions of lower turbulent intensity. In a stratified turbulent fluid layer in particular, $u_{\text{rms}}$ usually decreases along $\boldsymbol{g}$ as density increases, and we therefore expect the field to be brought downwards. This is usually referred to as diamagnetic pumping (Zel’dovich Reference Zel’dovich1956; Rädler Reference Rädler1968; Moffatt Reference Moffatt1983). While this effect is not inducing magnetic field on its own, it can contribute to the large-scale transport of the field, and may notably lead to its accumulation deep into stellar interiors.

From a mean-field theory perspective, inhomogeneity implies the presence of a non-vanishing $\unicode[STIX]{x1D738}\times \overline{\boldsymbol{B}}$ term in (4.7), as we now have a preferred direction to construct a true vector $\unicode[STIX]{x1D738}=\unicode[STIX]{x1D6FE}\,\boldsymbol{g}$ . Remark that $\unicode[STIX]{x1D738}\times \overline{\boldsymbol{B}}$ has the same form as $\overline{\boldsymbol{U}}\times \overline{\boldsymbol{B}}$ , so that $\unicode[STIX]{x1D738}$ is easily interpreted as an effective large-scale velocity advecting the field. A detailed FOSA derivation in the $St\ll 1$ limit, for a simple flow model encapsulating the effects of inhomogeneity through an exponential dependence of the turbulence intensity, gives (Krause Reference Krause1967)

(4.41) $$\begin{eqnarray}\unicode[STIX]{x1D738}=-{\textstyle \frac{1}{6}}\unicode[STIX]{x1D70F}_{c}\unicode[STIX]{x1D735}\overline{|\boldsymbol{u}|^{2}},\end{eqnarray}$$

confirming the diamagnetic character of the effect. Note finally that the further presence of rotation or large-scale flows in the problem raises additional contributions to $\unicode[STIX]{x1D738}$ , not all of which are oriented along $\boldsymbol{g}$ (see e.g. Rädler & Stepanov Reference Rädler and Stepanov2006). In spherical geometry, this notably makes for the possibility of large-scale pumping along the azimuthal direction.

4.4.3 Rädler and shear-current effects*

Additional statistical effects distinct from the standard $\unicode[STIX]{x1D6FC}$ effect formally arise if the turbulence is rendered anisotropic by the presence of large-scale rotation or shear, even in the absence of stratification. These effects are not necessarily very intuitive physically but their mathematical form can be captured easily within the framework of mean-field electrodynamics using some of the symmetry arguments introduced in § 4.3.3. Namely, in the presence of rotation, we can now form a non-vanishing $\unicode[STIX]{x1D739}\times (\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}})$ term in the expression (4.7) for the mean EMF by making $\unicode[STIX]{x1D739}$ proportional to the rotation vector, $\unicode[STIX]{x1D739}\equiv \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D734}$ . The associated statistical effect is generally referred to as the Rädler effect after its original derivation by Rädler (Reference Rädler1969a ,Reference Rädler b ) or, more explicitly, as the ‘ $\unicode[STIX]{x1D734}\times \overline{\boldsymbol{J}}$ ’ effect. Similarly, we may expect that $\unicode[STIX]{x1D739}$ is formally non-vanishing if the turbulence is sheared by a large-scale flow $\overline{\boldsymbol{U}}$ with associated vorticity $\overline{\boldsymbol{W}}=\unicode[STIX]{x1D735}\times \overline{\boldsymbol{U}}$ , in which case $\unicode[STIX]{x1D739}\equiv \unicode[STIX]{x1D6FF}_{W}\overline{\boldsymbol{W}}$ . This effect is generally referred to as the shear-current effect, or ‘ $\overline{\boldsymbol{W}}\times \overline{\boldsymbol{J}}$ ’ effect (Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2003).

The Rädler effect has a subtle connection to parity invariance and flow helicity. To see this, let us consider a flow whose statistics are rendered anisotropic (more precisely axisymmetric) by the presence of rotation. In particular, the energy and helicity spectra are such that $E(\boldsymbol{k},\unicode[STIX]{x1D714})=E(k,\unicode[STIX]{x1D707},\unicode[STIX]{x1D714})$ and $H(\boldsymbol{k},\unicode[STIX]{x1D714})=H(k,\unicode[STIX]{x1D707},\unicode[STIX]{x1D714})$ , where $\unicode[STIX]{x1D707}=\hat{\boldsymbol{k}}\boldsymbol{\cdot }\hat{\unicode[STIX]{x1D734}}$ . A spectral FOSA derivation similar to that outlined in § 4.3.7 shows that (Moffatt & Proctor Reference Moffatt and Proctor1982)

(4.42) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FA}=\frac{2\unicode[STIX]{x1D702}^{2}}{5}\iint \frac{k^{3}\unicode[STIX]{x1D707}\unicode[STIX]{x1D714}H(k,\unicode[STIX]{x1D707},\unicode[STIX]{x1D714})}{(\unicode[STIX]{x1D714}^{2}+\unicode[STIX]{x1D702}^{2}k^{4})^{2}}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714},\end{eqnarray}$$

and

(4.43) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{ij}=\unicode[STIX]{x1D6FC}\,\unicode[STIX]{x1D6FF}_{ij}+\unicode[STIX]{x1D6FC}_{1}(\unicode[STIX]{x1D6FF}_{ij}-3\hat{\unicode[STIX]{x1D6FA}}_{i}\hat{\unicode[STIX]{x1D6FA}}_{j}),\end{eqnarray}$$

with

(4.44) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=-\frac{\unicode[STIX]{x1D702}}{3}\iint \frac{k^{2}}{(\unicode[STIX]{x1D714}^{2}+\unicode[STIX]{x1D702}^{2}k^{4})}H(k,\unicode[STIX]{x1D707},\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\end{eqnarray}$$

and

(4.45) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{1}=\frac{\unicode[STIX]{x1D702}}{6}\iint \frac{k^{2}(3\unicode[STIX]{x1D707}^{2}-1)}{(\unicode[STIX]{x1D714}^{2}+\unicode[STIX]{x1D702}^{2}k^{4})}H(k,\unicode[STIX]{x1D707},\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}.\end{eqnarray}$$

These expressions show that both $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}$ and $\unicode[STIX]{x1D6FC}_{ij}$ are weighted integrals of the helicity spectrum of the flow, and therefore vanish if the flow has no helicity at all. However, notice that the weight function in the $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}$ integral is odd in $\unicode[STIX]{x1D707}$ , and even in the $\unicode[STIX]{x1D6FC}_{ij}$ integrals. This implies that $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}$ , unlike the $\unicode[STIX]{x1D6FC}$ effect, can be non-zero even if there is an equal statistical amount of right- and left-handed velocity fluctuations, and the flow as a whole has zero net helicity. In particular, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}\neq 0$ and $\unicode[STIX]{x1D736}=\mathbf{0}$ if $H(k,-\unicode[STIX]{x1D707},\unicode[STIX]{x1D714})=-H(k,\unicode[STIX]{x1D707},\unicode[STIX]{x1D714})$ . This situation corresponds to a flow with an up–down symmetry with respect to planes perpendicular to the rotation axis, such as the incompressible Rayleigh–Bénard convection rotating about the vertical axis discussed in § 4.4.1. A corollary of this result is that the Rädler effect is formally present in unstratified rotating flows. The derivation of Moffatt & Proctor (Reference Moffatt and Proctor1982) also makes it clear that the effect survives even if the forcing of the flow is itself non-helical, and all the flow helicity is induced by the effects of the Coriolis force. In other words, we expect the Rädler effect to be present even in incompressible, homogeneous turbulence forced isotropically and non-helically, as long as the system rotates. In a generic stratified, rotating, turbulent system, both $\unicode[STIX]{x1D6FC}$ and Rädler effects are expected to coexist, although it is sometimes argued that the latter should be smaller due to the presence of an extra slow spatial derivative ( $\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}$ ) in the expression of its EMF. Analyses of simulations of stratified rotating convection in which both effects are present suggest that the Rädler effect is smaller than the $\unicode[STIX]{x1D6FC}$ effect, but not altogether negligible, with $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}$ being comparable in magnitude to the isotropic turbulent diffusivity coefficient $\unicode[STIX]{x1D6FD}$ (Käpylä et al. Reference Käpylä, Korpi and Brandenburg2009).

To understand how the Rädler and shear-current effects may affect large-scale dynamos in rotating shear flows, let us have a look at the mean-field dynamo problem for small-scale homogeneous turbulence forced isotropically and non-helically in the simplest possible unstratified, rotating shearing sheet configuration, $\unicode[STIX]{x1D734}=\unicode[STIX]{x1D6FA}\boldsymbol{e}_{z}$ , $\boldsymbol{U}_{S}=-Sx\boldsymbol{e}_{y}$ , for which $\overline{\boldsymbol{W}}=-S\boldsymbol{e}_{z}$ and the only non-zero components of the deformation tensor are $\overline{\unicode[STIX]{x1D60B}}_{xy}=\overline{\unicode[STIX]{x1D60B}}_{yx}=-S/2$ . Under these assumptions, there are no mean $\unicode[STIX]{x1D736}$ and $\unicode[STIX]{x1D738}$ effects and the kinematic evolution equations for the $x$ and $y$ components of a $z$ -dependent mean magnetic field (defined as the average over $x$ and $y$ of the total magnetic field) can be cast in the simple form

(4.46) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\overline{B}_{x}}{\unicode[STIX]{x2202}t}=-\unicode[STIX]{x1D702}_{yx}\frac{\unicode[STIX]{x2202}^{2}\overline{B}_{y}}{\unicode[STIX]{x2202}z^{2}}+(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D702}_{yy})\frac{\unicode[STIX]{x2202}^{2}\overline{B}_{x}}{\unicode[STIX]{x2202}z^{2}}, & \displaystyle\end{eqnarray}$$

(4.47) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\overline{B}_{y}}{\unicode[STIX]{x2202}t}=-SB_{x}-\unicode[STIX]{x1D702}_{xy}\frac{\unicode[STIX]{x2202}^{2}\overline{B}_{x}}{\unicode[STIX]{x2202}z^{2}}+(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D702}_{xx})\frac{\unicode[STIX]{x2202}^{2}\overline{B}_{y}}{\unicode[STIX]{x2202}z^{2}}, & \displaystyle\end{eqnarray}$$

where we have introduced a contracted generalised anisotropic turbulent diffusion tensor $\unicode[STIX]{x1D702}_{ij}$ appropriate to the configuration of the problem, namely

(4.48) $$\begin{eqnarray}b_{ijz}=\unicode[STIX]{x1D702}_{il}\unicode[STIX]{x1D700}_{jzl}.\end{eqnarray}$$

Using (4.8)–(4.10), it can be shown that

(4.49) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{xx}=\unicode[STIX]{x1D702}_{yy}=\unicode[STIX]{x1D6FD}, & \displaystyle\end{eqnarray}$$

(4.50) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{yx}=\unicode[STIX]{x1D6FA}\left(\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}-\frac{\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6FA}}}{2}\right)-S\left[\unicode[STIX]{x1D6FF}_{W}-\frac{1}{2}(\unicode[STIX]{x1D705}_{W}-\unicode[STIX]{x1D6FD}_{D}+\unicode[STIX]{x1D705}_{D})\right], & \displaystyle\end{eqnarray}$$

(4.51) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{xy}=-\unicode[STIX]{x1D6FA}\left(\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}-\frac{\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6FA}}}{2}\right)+S\left[\unicode[STIX]{x1D6FF}_{W}-\frac{1}{2}(\unicode[STIX]{x1D705}_{W}+\unicode[STIX]{x1D6FD}_{D}-\unicode[STIX]{x1D705}_{D})\right], & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}$ is the usual isotropic turbulent diffusion coefficient, $\unicode[STIX]{x1D6FD}_{D}$ is an anisotropic contribution to the $\unicode[STIX]{x1D737}$ tensor arising from the presence of the large-scale strain $\overline{\unicode[STIX]{x1D63F}}$ associated with the shear flow, and $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6FA}}$ , $\unicode[STIX]{x1D705}_{W}$ and $\unicode[STIX]{x1D705}_{D}$ are contributions to the mean-field $\unicode[STIX]{x1D73F}$ tensor arising (similarly to $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}$ and $\unicode[STIX]{x1D6FF}_{W}$ ) from the presence of rotation, large-scale vorticity, and strain associated with the shear flow (for a detailed derivation, see Rädler & Stepanov Reference Rädler and Stepanov2006; Squire & Bhattacharjee Reference Squire and Bhattacharjee2015a ).

We see that all these effects amount to a special kind of off-diagonal diffusion that couples the components of the magnetic field perpendicular to the new special direction introduced in the problem. Can these couplings, most importantly that associated with the $\unicode[STIX]{x1D702}_{yx}$ coefficient that couples back the toroidal field to the poloidal field, drive a dynamo of their own? To discover this, we can derive from (4.46) to (4.47) the complex growth rate of a horizontal mean-field mode with a simple $\exp (st+\text{i}k_{z}z)$ dependence,

(4.52) $$\begin{eqnarray}s=k_{z}\sqrt{\unicode[STIX]{x1D702}_{yx}(-S+\unicode[STIX]{x1D702}_{xy}k_{z}^{2})}-(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})k_{z}^{2}.\end{eqnarray}$$

Further assuming $|S|\gg \unicode[STIX]{x1D702}_{xy}k_{z}^{2}$ , similarly to what we did when we derived the $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ dispersion relation in § 4.3.5, we find that a necessary condition for dynamo growth is

(4.53) $$\begin{eqnarray}S\unicode[STIX]{x1D702}_{yx}<0,\end{eqnarray}$$

i.e.  $S$ and $\unicode[STIX]{x1D702}_{yx}$ must have opposite sign. If the dynamo is realisable, the maximum growth rate and optimal wavenumber are

(4.54a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\text{max}}=\frac{|S\unicode[STIX]{x1D702}_{yx}|}{4(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})},\quad k_{z,\text{max}}=\frac{(S\unicode[STIX]{x1D702}_{yx})^{1/2}}{2(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})}.\end{eqnarray}$$

But under which circumstance(s), if any, is the condition (4.53) satisfied? To answer this question, we need to calculate $\unicode[STIX]{x1D702}_{yx}$ using a closure assumption such as the FOSA or MTA. At this point, things become tricky. Indeed, FOSA calculations (Krause & Rädler Reference Krause and Rädler1980; Moffatt & Proctor Reference Moffatt and Proctor1982) show that the Rädler effect alone can promote a dynamo ( $\unicode[STIX]{x1D702}_{yx}S<0$ ) when the rotation is anticyclonic (corresponding to positive $S/\unicode[STIX]{x1D6FA}>0$ with our convention), but not the shear-current effect ( $S\unicode[STIX]{x1D702}_{yx}\geqslant 0$ , Rädler & Stepanov Reference Rädler and Stepanov2006; Rüdiger & Kitchatinov Reference Rüdiger and Kitchatinov2006; Sridhar & Subramanian Reference Sridhar and Subramanian2009; Sridhar & Singh Reference Sridhar and Singh2010; Squire & Bhattacharjee Reference Squire and Bhattacharjee2015a ). Calculations based on the MTA closure agree with FOSA calculations as far as the Rädler effect is concerned, but find dynamo growth ( $S\unicode[STIX]{x1D702}_{yx}<0$ ) for the shear-current effect (Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2003, Reference Rogachevskii and Kleeorin2004)! Which one, if any, is right in practice? We know that the results derived with FOSA at least hold rigorously if $Rm\ll 1$ or $St\ll 1$ , which establishes the absence of a shear-current-driven dynamo and the existence of a Rädler-effect-driven dynamo in these limits (at least for the kind of sheared or rotating random flows considered in the FOSA derivations). However, there is no guarantee that these conclusions extend to standard MHD turbulence regimes. As discussed in § 4.3.8, the MTA is an empirical closure model that seems to be in reasonable agreement with numerical results for a few simple turbulent-transport problems with $St=O(1)$ , but it cannot be rigorously shown to be valid in any particular regime either, leaving us in the dark for the time being.

Looking at the bigger picture, we see that the previous discussion of a shear-current-effect dynamo (or lack thereof) raises the more general, very intriguing question of the possibility of large-scale dynamo excitation in non-rotating sheared turbulence with zero mean helicity. This problem, including the sign of $S\unicode[STIX]{x1D702}_{yx}$ in non-rotating sheared turbulence, will be discussed much more extensively in § 5.2 in the light of several recent numerical developments.

4.5.1 The overwhelming growth of small-scale dynamo fields

In the previous paragraphs, we discussed the very limited formal asymptotic mathematical range of parameters under which kinematic mean-field electrodynamics can be rigorously derived, but there is actually an even bigger potential physical problem looming over the theory at large $Rm$ . Indeed, we have learned in § 3 that homogeneous, isotropic turbulence independently generates small-scale magnetic fluctuations $\boldsymbol{b}$ through a small-scale dynamo effect, provided that $Rm$ exceeds a value $Rm_{c,\text{ssd}}$ of a few tens to a few hundreds, depending on $Pm$ (figure 15). This dynamo produces equipartition field strengths over a dynamical (turbulent) turnover time scale, which is usually much smaller than the rotation or shearing time scales over which kinematic mean-field dynamo modes grow. This strongly suggests that the derivation of a meaningful solution of the generic problem of large-scale magnetic-field growth at large $Rm$ should start from a state of saturated small-scale MHD turbulence characterised by $\overline{|\boldsymbol{b}|^{2}}\gg |\overline{\boldsymbol{B}}|^{2}$ and $\overline{\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}}\sim \overline{|\boldsymbol{b}|^{2}}$ , rather than from a state of hydrodynamic turbulence. This was recognised as a major issue for large-scale astrophysical dynamo theory after the work of Kulsrud & Anderson (Reference Kulsrud and Anderson1992) described in § 3.4.8 (see also the review by Kulsrud et al. Reference Kulsrud, Cowley, Gruzinov and Sudan1997), although obviously the tricky questions of the interactions of small-scale MHD turbulence and large-scale dynamos already pervaded several earlier calculations, including the Pouquet et al. (Reference Pouquet, Frisch and Leorat1976) paper mentioned earlier (see also Ponty & Plunian (Reference Ponty and Plunian2011) for a specific numerical example of a classic large-scale helical dynamo being overwhelmed by a small-scale dynamo as $Rm$ increases).

We are unfortunately not in a very good place to address this problem at this stage of exposition of the theory. A key observation is that small-scale dynamo modes were purely and simply discarded in the FOSA treatment as a result of the neglect of the tricky term in (4.4). Mathematically, we can think of these modes as fast-growing homogeneous solutions of (4.5), which have nothing to do with $\overline{\boldsymbol{B}}$ . In other words, our earlier assumption that magnetic fluctuations $\boldsymbol{b}$ are small and uniquely the product of the stretching and tangling of the mean field $\overline{\boldsymbol{B}}$ is incorrect at large $Rm$ . This in turn begs the question of the interpretation and practical applicability of the linear mean-field ansatz (4.6) in this regime (Cattaneo & Hughes Reference Cattaneo and Hughes2009; Hughes Reference Hughes, Pogorelov, Audit and Zank2010; Cameron & Alexakis Reference Cameron and Alexakis2016). In order to make progress on a unified theory of large- and small-scale dynamos, our first priority should therefore be to derive a linear theory that at the very minimum accommodates both types of dynamos. Such a theory is available in the form of a helical extension of the Kazantsev model discussed in § 3.4.

4.5.2 Kazantsev model for helical turbulence*

A generalisation of the Kazantsev model to helical flows was first derived by Vainshtein (Reference Vainshtein1970) shortly after the introduction of mean-field electrodynamics. Vainshtein used a Fourier-space representation (see also Kulsrud & Anderson Reference Kulsrud and Anderson1992; Berger & Rosner Reference Berger and Rosner1995), but in the following we will stay in the correlation-vector space as a matter of continuity with § 3.4. Helicity can be accommodated in the model by adding a new term to the correlation tensor of the flow,

(4.55) $$\begin{eqnarray}\unicode[STIX]{x1D705}^{ij}(\boldsymbol{r})=\unicode[STIX]{x1D705}_{N}(r)\left(\unicode[STIX]{x1D6FF}^{ij}-\frac{r^{i}r^{j}}{r^{2}}\right)+\unicode[STIX]{x1D705}_{L}(r)\frac{r^{i}r^{j}}{r^{2}}+g(r)\unicode[STIX]{x1D700}^{ijk}r^{k},\end{eqnarray}$$

where $g$ is a scalar function that vanishes for parity-invariant flows. A corresponding helical term is added to the magnetic correlation tensor,

(4.56) $$\begin{eqnarray}H^{ij}(\boldsymbol{r},t)=H_{N}(r,t)\left(\unicode[STIX]{x1D6FF}^{ij}-\frac{r^{i}r^{j}}{r^{2}}\right)+H_{L}(r,t)\frac{r^{i}r^{j}}{r^{2}}+K(r)\unicode[STIX]{x1D700}^{ijk}r^{k}.\end{eqnarray}$$

Proceeding as in § 3.4 to close the problem, we obtain a system of two coupled equations for $H_{L}(r)$ and $K(r)$ (Vainshtein & Kichatinov Reference Vainshtein and Kichatinov1986; Subramanian Reference Subramanian1999; Boldyrev, Cattaneo & Rosner Reference Boldyrev, Cattaneo and Rosner2005),

(4.57) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}H_{L}}{\unicode[STIX]{x2202}t}=\frac{1}{r^{4}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r^{4}\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}H_{L}}{\unicode[STIX]{x2202}r}\right)+GH_{L}-4hK, & \displaystyle\end{eqnarray}$$

(4.58) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}K}{\unicode[STIX]{x2202}t}=\frac{1}{r^{4}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left[r^{4}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}(\unicode[STIX]{x1D705}K+hH_{L})\right], & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}(r)$ is given by (3.30), $h(r)=g(0)-g(r)$ , and $G(r)=\unicode[STIX]{x1D705}^{\prime \prime }+4\unicode[STIX]{x1D705}^{\prime }/r$ . When $g=0$ (no helicity in the flow), $H_{L}$ decouples from $K$ , equation (4.57) reduces to (3.29) and (4.58) for the evolution of the magnetic-helicity correlator $K$ reduces to a diffusion equation with no source term. From there, it can be shown (Vainshtein Reference Vainshtein1970; Subramanian Reference Subramanian1999; Boldyrev Reference Boldyrev2001; Boldyrev et al. Reference Boldyrev, Cattaneo and Rosner2005) that taking the $r\rightarrow \infty$ limit of this problem leads to a mean-field $\unicode[STIX]{x1D6FC}^{2}$ equation reminiscent of (4.11),

(4.59) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\langle \boldsymbol{B}\rangle }{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D735}\times \langle \boldsymbol{B}\rangle +(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D6E5}\langle \boldsymbol{B}\rangle ,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}\equiv g(0)$ , $\unicode[STIX]{x1D6FD}\equiv \unicode[STIX]{x1D705}_{L}(0)/2$ and the mean field here is to be interpreted as an average of the full magnetic field over the Kraichnan velocity-field ensemble. Because the velocity is correlated on scales $\ell _{0}$ , this averaging procedure is equivalent to a spatial average over scales much larger than $\ell _{0}$ .

This seems promising, but let us now look at the full solutions of (4.57)–(4.58) to gain a better understanding of the situation. Using the transformation

(4.60a-c ) $$\begin{eqnarray}H_{L}=\frac{\sqrt{2}}{r^{2}}W_{h},\quad K=-\frac{\sqrt{2}}{r^{4}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}(r^{2}W_{k}),\quad \boldsymbol{W}=\left(\begin{array}{@{}c@{}}W_{h}\\ W_{k}\end{array}\right),\end{eqnarray}$$

Boldyrev et al. (Reference Boldyrev, Cattaneo and Rosner2005) showed that (4.57)–(4.58) can be cast into a self-adjoint spinorial form,

(4.61) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{W}}{\unicode[STIX]{x2202}t}=-\tilde{\unicode[STIX]{x1D619}}^{\text{T}}\,\tilde{\unicode[STIX]{x1D611}}\,\tilde{\unicode[STIX]{x1D619}}\boldsymbol{W},\end{eqnarray}$$

where

(4.62) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D619}}=\left(\begin{array}{@{}cc@{}}\sqrt{2}/r & 0\\ 0 & \displaystyle -\frac{1}{r^{2}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}r^{2}\end{array}\right), & \displaystyle\end{eqnarray}$$

(4.63) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D611}}=\left(\begin{array}{@{}cc@{}}\hat{E} & C\\ C & B\end{array}\right), & \displaystyle\end{eqnarray}$$

(4.64) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{E}=-\frac{1}{2}r\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}B\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}r+\frac{1}{\sqrt{2}}(A-rA^{\prime }), & \displaystyle\end{eqnarray}$$

and

(4.65) $$\begin{eqnarray}\displaystyle & \displaystyle A(r)=\sqrt{2}[2\unicode[STIX]{x1D702}+\unicode[STIX]{x1D705}_{N}(0)-\unicode[STIX]{x1D705}_{N}(r)], & \displaystyle\end{eqnarray}$$

(4.66) $$\begin{eqnarray}\displaystyle & \displaystyle B(r)=\sqrt{2}[2\unicode[STIX]{x1D702}+\unicode[STIX]{x1D705}_{L}(0)-\unicode[STIX]{x1D705}_{L}(r)], & \displaystyle\end{eqnarray}$$

(4.67) $$\begin{eqnarray}\displaystyle & \displaystyle C(r)=\sqrt{2}[g(0)-g(r)]r. & \displaystyle\end{eqnarray}$$

Therefore, the generalised helical case too can be diagonalised and has orthogonal eigenfunctions. It turns out that there are now two kinds of growing eigenmodes, discrete bound modes (‘trapped particles’ in the quantum-like description) with growth rates $\unicode[STIX]{x1D6FE}_{n}$ , and a continuous spectrum of free modes (‘travelling particles’) with growth rates $\unicode[STIX]{x1D6FE}_{\text{free}}$ . The growth rates are such that $\unicode[STIX]{x1D6FE}_{n}>\unicode[STIX]{x1D6FE}_{0}>\unicode[STIX]{x1D6FE}_{\text{free}}>0$ , where

(4.68) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{0}=\frac{g^{2}(0)}{2\unicode[STIX]{x1D702}+\unicode[STIX]{x1D705}_{L}(0)}=\frac{2\unicode[STIX]{x1D6FC}^{2}}{4(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})},\end{eqnarray}$$

is twice the maximum standard mean-field $\unicode[STIX]{x1D6FC}^{2}$ dynamo growth rate for the magnetic field (see e.g. Malyshkin & Boldyrev Reference Malyshkin and Boldyrev2007, remember that we are looking at the growth rates of the quadratic magnetic correlator here). In other words, not only do we have trapped growing modes reminiscent of the small-scale dynamo modes derived in § 3.4, there are now also free ‘large-scale’ modes asymptotic to helical mean-field dynamo modes in the limit $r\rightarrow \infty$ . Of course, this convergence is not totally surprising considering that the $\unicode[STIX]{x1D6FC}^{2}$ mean-field theory can be derived rigorously in the asymptotic two-scale approach in the limit of short correlation times. The family of free large-scale growing modes can be envisioned in the quantum-like description as caused by a helicity-induced modification of the potential at large scales. When $r/\ell _{0}\gg 1$ , the effective helical potential is

(4.69) $$\begin{eqnarray}V_{\text{eff}}(r)\simeq \frac{2}{r^{2}}-\frac{\unicode[STIX]{x1D6FC}^{2}}{(\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD})^{2}}\end{eqnarray}$$

and therefore tends to a strictly negative constant value as $r\rightarrow \infty$ , allowing large-scale eigenfunctions with negative energy (positive growth rates). The shape of the full potential is shown in figure 29, to be compared with figure 17 for the non-helical case.

Figure 29. Helical Kazantsev potential as a function of correlation length $r$ .

We see that the smaller the helicity of the flow (as represented by $\unicode[STIX]{x1D6FC}$ ), the smaller the growth rate of the free large-scale modes must be, due to the constraint $\unicode[STIX]{x1D6FE}_{0}>\unicode[STIX]{x1D6FE}_{\text{free}}$ . The growth rate of the bound modes, on the other hand, is not bounded from above by the amount of helicity in the flow. In all cases, the model predicts that the bound modes grow faster than the free modes, and are therefore expected to dynamically saturate the turbulence before the free modes saturate.

Another major conclusion of this analysis, however, is that the kinematic helical dynamo problem at large $Rm$ is not reducible to a simple dichotomy between fast, small-scale dynamo modes and slow, large-scale dynamo ones. Solving (4.57)–(4.58) numerically for a helical velocity field with a Kolmogorov spectrum, Malyshkin & Boldyrev (Reference Malyshkin and Boldyrev2007, Reference Malyshkin and Boldyrev2009) found that the faster-growing bound modes that reduce to ‘small-scale dynamo’ modes when $h=0$ in (4.58) themselves develop correlations on scales $r>\ell _{0}$ in the presence of helicity, and are therefore expected to contribute to the growth of the large-scale magnetic field. A comparison between some of the bound eigenfunctions and the fastest-growing unbound eigenfunction is shown in figure 30 for two different regimes. The results show that the fastest kinematic modes in helical turbulence at large $Rm$ are not pure mean-field or small-scale dynamo modes, but hybrid modes energetically dominated by small-scale fields, with a significant large-scale magnetic tail.

4.5.3 The $Pm$ -dependence of kinematic helical dynamos

It is finally worth pointing out in relation to our discussion in § 3.3.2 of small-scale dynamos at low $Pm$ that the helical Kazantsev model predicts that the presence of flow helicity at the resistive scale $\ell _{\unicode[STIX]{x1D702}}\sim Rm^{-3/4}$ has the effect of decreasing $Rm_{c}$ for bound modes at low $Pm$ in comparison to the non-helical case, down to a value comparable to or even smaller than in the large- $Pm$ case (Malyshkin & Boldyrev Reference Malyshkin and Boldyrev2010). In other words, helicity/rotation facilitates the low- $Pm$ growth of modes identified as small-scale dynamo modes in the non-helical case. While this problem deserves further numerical scrutiny, a rotationally induced decrease of $Rm_{c}$ has for instance been reported in simulations of dynamos driven by turbulent convection at $Pm=1$ (Favier & Bushby Reference Favier and Bushby2012), and in simulations of kinematic dynamos driven by rotating 2.5-D flows, but forced non-helically (Seshasayanan, Dallas & Alexakis Reference Seshasayanan, Dallas and Alexakis2017). More generally, numerical simulations suggest that large-scale dynamos in rotating/helical turbulent flows are much less dependent on $Pm$ than small-scale ones (see e.g. Mininni Reference Mininni2007; Brandenburg Reference Brandenburg2009b ). A handwaving physical explanation for this is that the large-scale field always feels the whole sea of small-scale turbulent velocity and magnetic fluctuations. Pure ‘non-helical’ small-scale dynamo fields, on the other hand, appear to be much more dependent on the details of the flow and dissipation.

Figure 30. Fourier spectra of selected dynamo eigenfunctions in the helical Kazantsev model (black thin lines: bound modes, red spiky solid line: most unstable unbound mode). (a)  $Pm=150$ case. (b)  $Pm=6.7\times 10^{-4}$ case. The kinetic helicity of the flow is maximal in both cases (adapted from Malyshkin & Boldyrev Reference Malyshkin and Boldyrev2009).

4.6.1 Phenomenological considerations

The simplest possible outcome of large-scale dynamo saturation, which is also arguably the most desirable one to explain the characteristics of magnetic fields in astrophysical systems such as our Sun or galaxy, is that the large-scale magnetic field reaches equipartition with the underlying turbulence,

(4.70) $$\begin{eqnarray}|\overline{\boldsymbol{B}}|^{2}\sim \overline{\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}}\equiv B_{\text{eq}}^{2}.\end{eqnarray}$$