2.1. Base model: incompressible resistive MHD

As a base model, we assume incompressible resistive MHD with homogeneous mass density $\rho =\text{const}$ ,

(2.1a) \begin{align} \partial _t \boldsymbol{v} + (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} ) \boldsymbol{v} = (\boldsymbol{b} \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{b} - \boldsymbol{\nabla} P + \nu {\nabla}^2 \boldsymbol{v},\end{align}

(2.1b) \begin{align} \partial _t \boldsymbol{b} = \boldsymbol{\nabla} \times (\boldsymbol{v} \times \boldsymbol{b}) + \eta {\nabla}^2 \boldsymbol{b}, \end{align}

(2.1c) \begin{align} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{v} = 0, \qquad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{b} = 0.\end{align}

Here, $\boldsymbol{v}$ represents the fluid velocity, $\boldsymbol{b} \doteq \boldsymbol{B}/\sqrt {4\pi \rho }$ is the local Alfvén velocity, and the normalised total pressure is defined as $P \doteq (P_{\text{kin}} + B^2/8\pi ) / \rho$ , with $P_{\text{kin}}$ being the kinetic pressure (the symbol $\doteq$ denotes definitions). The viscosity, $\nu$ , and resistivity, $\eta$ , are assumed to be constant. To express (2.1) in a more symmetric form, let us rewrite them in terms of the two Elsässer fields $\boldsymbol{z}^\pm$ (Elsasser Reference Elsasser1950), which are also solenoidal,

(2.2) \begin{align} \boldsymbol{z}^\pm \doteq \boldsymbol{v} \pm \boldsymbol{b}, \qquad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{z}^\pm = 0. \end{align}

This leads to two coupled equations for $\boldsymbol{z}^\pm$ ,

(2.3) \begin{align} \partial _t \boldsymbol{z}^\pm =-(\boldsymbol{z}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{z}^\pm -\boldsymbol{\nabla} P +\nu _+ {\nabla}^2 \boldsymbol{z}^\pm +\nu _- {\nabla}^2 \boldsymbol{z}^\mp , \end{align}

where $\nu _+ \doteq (\nu + \eta )/2$ and $\nu _- \doteq (\nu - \eta )/2$ . The normalised total pressure $P$ can be found as follows. By taking the divergence of (2.1a ) and using (2.2), one obtains

(2.4) \begin{align} {\nabla}^2 P = -\boldsymbol{\nabla} \boldsymbol{\cdot} [(\boldsymbol{z}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{z}^\pm ]. \end{align}

Let us introduce the wavevector operator $\hat {\boldsymbol{k}} \doteq - \mathrm{i} \boldsymbol{\nabla}$ , so $\hat {k}^2 \doteq {\hat {\boldsymbol{k}}}^2 = - {\nabla}^2$ . Let us also assume some appropriate (say, periodic) boundary conditions. Then, (2.4) yields

(2.5) \begin{align} P = -\hat {k}^{-2}\hat {\boldsymbol{k}}\boldsymbol{\cdot} [(\boldsymbol{z}^\mp \boldsymbol{\cdot} \hat {\boldsymbol{k}})\boldsymbol{z}^\pm ] + \text{const}, \end{align}

whence $\boldsymbol{\nabla} P$ in (2.3) can be expressed through $\boldsymbol{z}^\pm$ .

Alternatively, $\boldsymbol{\nabla} P$ can be eliminated from (2.3) by taking the curl of this equation,

(2.6) \begin{align} \partial _t \boldsymbol{w}^\pm =-\boldsymbol{\nabla} \times [(\boldsymbol{z}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{z}^\pm ] +\nu _+ {\nabla}^2 \boldsymbol{w}^\pm +\nu _- {\nabla}^2 \boldsymbol{w}^\mp , \end{align}

where the Elsässer vorticities $\boldsymbol{w}^\pm$ are defined as

(2.7) \begin{align} \boldsymbol{w}^\pm \doteq \boldsymbol{\nabla} \times \boldsymbol{z}^\pm . \end{align}

Indeed, due to (2.2), one can express $\boldsymbol{z}^\pm$ using a vector potential $\boldsymbol{a}^\pm$ such that $\boldsymbol{z}^\pm = \boldsymbol{\nabla} \times \boldsymbol{a}^\pm$ . Let us assume the gauge such that $\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{a}^\pm = 0$ . Then,

(2.8) \begin{align} \boldsymbol{w}^\pm = \boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \boldsymbol{a}^\pm ) = - {\nabla}^2 \boldsymbol{a}^\pm \equiv \hat {k}^2 \boldsymbol{a}^\pm , \end{align}

whence

(2.9) \begin{align} \boldsymbol{z}^\pm = \mathrm{i} \hat {k}^{-2} (\hat {\boldsymbol{k}} \times \boldsymbol{w}^\pm ). \end{align}

(Here, we assume that $\boldsymbol{z}^\pm$ and their first spatial derivatives have zero spatial average, so $\hat {k}^2$ is invertible.) Then, (2.3) can be expressed through $\boldsymbol{w}^\pm$ alone,

(2.10) \begin{align} \partial _t \boldsymbol{w}^\pm = -\hat {\boldsymbol{k}} \times \{ [(\hat {\boldsymbol{k}} \times \hat {k}^{-2}\boldsymbol{w}^\mp )\boldsymbol{\cdot} \hat {\boldsymbol{k}}] (\hat {\boldsymbol{k}}\times \hat {k}^{-2}\boldsymbol{w}^\pm ) \} -\hat {k}^2(\nu _+ \boldsymbol{w}^\pm + \nu _- \boldsymbol{w}^\mp ). \end{align}

Writing the Elsässer equations in this form has the benefit that the nonlinear term is now expressed as a product of inverse operators that act only on the Elsässer vorticities themselves, rather than an inverse operator that acts on the product of the Elsässer fields as in (2.5). As we will see shortly, this property makes (2.10) more convenient than (2.3) for the eventual formulation of MFWK. Note also that (2.10) is equivalent to (2.6) with the assumption that $\boldsymbol{z}^\pm$ and their first spatial derivatives have zero spatial average.

2.2. Mean fields versus fluctuations

As in the usual mean-field approach (Krause & Rädler Reference Krause and Rädler1980; Moffatt Reference Moffatt1978), we decompose our Elsässer fields into mean and fluctuating parts,

(2.11) \begin{equation} \boldsymbol{z}^\pm =\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\pm +\widetilde {\boldsymbol{z}}^\pm , \qquad \boldsymbol{w}^\pm =\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm +\widetilde {\boldsymbol{w}}^\pm . \end{equation}

Inserting (2.11) into (2.6) and averaging yields the following equation for the mean fields:

(2.12) \begin{equation} \begin{aligned} \partial _t \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm =& -\boldsymbol{\nabla} \times [(\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\pm ] +\nu _+ {\nabla}^2 \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm +\nu _- {\nabla}^2 \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\mp -\langle \boldsymbol{\nabla} \times [(\widetilde {\boldsymbol{z}}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\widetilde {\boldsymbol{z}}^\pm ] \rangle . \end{aligned} \end{equation}

(Note that this is similar to (2.6), but has an additional source term due to the fluctuating fields.) Subtracting (2.12) from (2.6) yields the following equation for the fluctuations:

(2.13) \begin{equation} \begin{aligned} \partial _t \widetilde {\boldsymbol{w}}^\pm =& -\boldsymbol{\nabla} \times [(\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\widetilde {\boldsymbol{z}}^\pm +(\widetilde {\boldsymbol{z}}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\pm ] +\nu _+ {\nabla}^2 \widetilde {\boldsymbol{w}}^\pm +\nu _- {\nabla}^2 \widetilde {\boldsymbol{w}}^\mp +\boldsymbol{F}^\pm _{\text{NL}}, \end{aligned} \end{equation}

where the nonlinear term

(2.14) \begin{equation} \boldsymbol{F}^\pm _{\text{NL}}\doteq \langle \boldsymbol{\nabla} \times [(\widetilde {\boldsymbol{z}}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\widetilde {\boldsymbol{z}}^\pm ] \rangle -\boldsymbol{\nabla} \times [(\widetilde {\boldsymbol{z}}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\widetilde {\boldsymbol{z}}^\pm ] \end{equation}

corresponds to eddy–eddy interactions.

Within the quasilinear approximation (QLA) (i.e. the collisionless-wave approximation of § 1.2), $\boldsymbol{F}^\pm _{\text{NL}}$ is neglected. Although nonlinear effects are, of course, important for understanding the full picture, our present focus is on understanding mean-field formation, many aspects of which can be studied within the QLA (Tsiolis et al. Reference Tsiolis, Zhou and Dodin2020; Zhu & Dodin Reference Zhu and Dodin2021). As discussed in the previous section, this can also be understood as retaining collective effects while neglecting pairwise interactions of the turbulent fluctuations. Deviations from the QLA are discussed extensively by Jin & Dodin (Reference Jin and Dodin2025).

Although we will proceed with the derivation of MFWK under the QLA, in § 2.5 we will also consider a modified version of the final equations that includes a simple minimal tau approximation (MTA)-like damping term to model the effect of eddy–eddy interactions, $\boldsymbol{F}^\pm _{\text{NL}}$ .Footnote 3

2.3. Wigner–Moyal equation for the fluctuations

2.3.1. Basic notation

Let us first introduce the state ket vectors,

(2.15) \begin{equation} {|{\widetilde {\boldsymbol{z}}}\rangle }\doteq \begin{pmatrix}{|{\widetilde {\boldsymbol{z}}^+}\rangle }\\[4pt]{|{\widetilde {\boldsymbol{z}}^-}\rangle } \end{pmatrix},\qquad {|{\widetilde {\boldsymbol{w}}}\rangle }\doteq \begin{pmatrix}{|{\widetilde {\boldsymbol{w}}^+}\rangle }\\[4pt]{|{\widetilde {\boldsymbol{w}}^-}\rangle } \end{pmatrix}. \end{equation}

The usual fields over spacetime can be understood as the spatial projections of these kets, i.e. $\langle {\boldsymbol{x}|\widetilde {\boldsymbol{z}}^\pm }\rangle =\widetilde {\boldsymbol{z}}^\pm (\boldsymbol{x})$ . The fluctuation equation can then be written as a vector Schrödinger equation for ${|{\widetilde {\boldsymbol{w}}}\rangle }$ ,

(2.16) \begin{equation} \begin{gathered} i\partial _t{|{\widetilde {\boldsymbol{w}}}\rangle }=\,\hat {\boldsymbol{\!H}}' {|{\widetilde {\boldsymbol{w}}}\rangle }, \end{gathered} \end{equation}

with the (generally non-Hermitian) Hamiltonian,

(2.17) \begin{equation} {\boldsymbol {\hat {H}}}'\doteq \begin{pmatrix} {\boldsymbol {\hat{H}}}'^{++}& \quad {\boldsymbol {\hat{H}}}'^{+-}\\[5pt] {\boldsymbol {\hat {H}}}'^{-+}& \quad {\boldsymbol {\hat {H}}}'^{{-}-} \end{pmatrix}, \end{equation}

where we have introduced

(2.18a) \begin{align} \hat {H}_{ij}^{'\pm \pm} & =\delta _{ij}\Bigg (\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_l^\mp \hat {k}_l-i\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,m}^\mp \frac {\hat {k}_l\hat {k}_m}{\hat {k}^2}-i\nu _+\hat {k}^2\Bigg )+i\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,m}^\mp \frac {\hat {k}_l\hat {k}_m}{\hat {k}^2},\\[-6pt]\nonumber\end{align}

(2.18b) \begin{align} \hat {H}_{ij}^{'\pm \mp }&=\delta _{ij}\Bigg (i\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,m}^\pm \frac {\hat {k}_l\hat {k}_m}{\hat {k}^2}+\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{l,mm}\frac {\hat {k}_l}{\hat {k}^2}-i\nu _-\hat {k}^2\Bigg )\nonumber\\[6pt] &\quad +i\Bigg(\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{j,i}-\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{j,l}\frac {\hat {k}_i\hat {k}_l}{\hat {k}^2}\Bigg )+\left(\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{j,il}^\pm -\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{l,ij}\right)\frac {\hat {k}_l}{\hat {k}^2}-\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{j,ll}\frac {\hat {k}_i}{\hat {k}^2}.\end{align}

From (2.9) and (2.7), we have

(2.19) \begin{equation} {|{\widetilde {\boldsymbol{z}}}\rangle }=\mathrm{i} \hat {k}^{-2}\hat {\boldsymbol{{k}}}_{\wedge }{|{\widetilde {\boldsymbol{w}}}\rangle },\qquad {|{\widetilde {\boldsymbol{w}}}\rangle }=\mathrm{i} \hat {\boldsymbol{{k}}}_{\wedge }{|{\widetilde {\boldsymbol{z}}}\rangle }, \end{equation}

where

(2.20) \begin{equation} \hat {\boldsymbol{{k}}}_{\wedge }\doteq \begin{pmatrix} \hat {\boldsymbol{k}}_{\wedge } & \quad 0\\[5pt]0&\quad \hat {\boldsymbol{k}}_{\wedge } \end{pmatrix}, \qquad \hat {\boldsymbol{k}}_{\wedge }\doteq \begin{pmatrix} 0&\quad -\hat {k}_z&\quad \hat {k}_y\\[5pt] \hat {k}_z&\quad 0&\quad -\hat {k}_x\\[5pt] -\hat {k}_y&\quad \hat {k}_x&\quad 0 \end{pmatrix}. \end{equation}

Using (2.19) and (2.13), (2.16) can then be put in a simpler form for the state vector ${|{\boldsymbol{z}}\rangle }$ , with Hamiltonian $\,\hat {\boldsymbol{\!H}}\doteq \mathrm{i}\hat {k}^{-2}\hat {\boldsymbol{{k}}}_{\wedge }\,\hat {\boldsymbol{\!H}}'\hat {\boldsymbol{{k}}}_{\wedge }$ . Specifically, (2.16) becomes

(2.21) \begin{equation} \mathrm{i}\partial _t{|{\widetilde {\boldsymbol{z}}}\rangle }=\,\hat {\boldsymbol{\!H}}{|{\widetilde {\boldsymbol{z}}}\rangle }, \end{equation}

where $\,\hat {\boldsymbol{\!H}}$ is a matrix operator given by

(2.22) \begin{equation}\,\hat{\boldsymbol{\!H}}=\begin{pmatrix} \,\hat{\boldsymbol{\!H}}^{++}& \quad\,\hat{\boldsymbol{\!H}}^{+-}\\[5pt] \,\hat{\boldsymbol{\!H}}^{-+}& \quad\,\hat{\boldsymbol{\!H}}^{{-}{-}} \end{pmatrix}, \end{equation}

and we have also introduced

(2.23a) \begin{align} \hat {H}^{\pm \pm }_{ij}=\delta _{ij}\big (\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\mp _l\hat {k}_l-\mathrm{i}\nu _+\hat {k}^2\big )+\mathrm{i}\frac {\hat {k}_i}{\hat {k}^2}\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,j}^\mp \hat {k}_l, \\[-35pt] \nonumber \end{align}

(2.23b) \begin{align} \hat {H}_{ij}^{\pm \mp }=-\delta _{ij}\mathrm{i}\nu _-\hat {k}^2-\mathrm{i}\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{i,j}+\mathrm{i}\frac {\hat {k}_i}{\hat {k}^2}\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{l,j}\hat {k}_l. \end{align}

Note that the Hamiltonian $\,\hat {\boldsymbol{\!H}}$ is generally not Hermitian (even in the ideal-MHD limit) and is highly non-trivial. Also, because ${|{\boldsymbol{z}}\rangle }$ is a multicomponent vector, one can think of the quasiparticles as quantum-like particles with spin, and the ‘spin-up’ and ‘spin-down’ components are strongly coupled in the presence of inhomogeneous mean fields.

For homogeneous $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\pm$ and vanishing $\nu _\pm$ , the Hamiltonian (2.22) is diagonal, so $|{\tilde {\boldsymbol{z}}^\pm }\rangle$ decouple. In principle, it can also be diagonalised for two-dimensional (2-D) dynamics when $|\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{b}\mkern -1.5mu}\mkern 1.5mu|\gg |\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu|$ (Appendix D), such that the corresponding dynamics can be understood in terms of the resulting phase-space trajectories available to quasiparticles. A more detailed investigation of the properties of the Hamiltonian (2.22) may be of interest for future work and may reveal ways to make greater use of the quasiparticle analogy.Footnote 4 However, such an exploration is beyond the scope of this paper. Also, for three-dimensional dynamics that is of interest in the context of the dynamo problem, a diagonalisation of $\,\hat {\boldsymbol{\!H}}$ does not seem possible.

2.3.2. Wigner–Moyal equation

Right-multiplying (2.16) by ${\langle {\widetilde {\boldsymbol{z}}}|}$ and subtracting the adjoint of the resulting equation yields the von Neumann equation for the density operator of Alfvénic fluctuations, $\hat {\boldsymbol{W}}\doteq {|{\widetilde {\boldsymbol{z}}}\rangle }\!{\langle {\widetilde {\boldsymbol{z}}}|}$ ,

(2.24) \begin{equation} \mathrm{i}\partial _t\hat {\boldsymbol{W}}=\,\hat {\boldsymbol{\!H}}\hat {\boldsymbol{W}}-\hat {\boldsymbol{W}}\,\hat {\boldsymbol{\!H}}^\dagger . \end{equation}

Applying the Wigner–Weyl transform (Appendix B) to (2.24) yields the Wigner–Moyal equation (WME), which governs the dynamics of Alfvénic fluctuations in the phase space $(\boldsymbol{x},\boldsymbol{k})$ ,

(2.25) \begin{align} \mathrm{i}\partial _t\boldsymbol{W}=\boldsymbol{H}\star \boldsymbol{W}-\boldsymbol{W}\star \boldsymbol{H}^\dagger . \end{align}

Here, $\boldsymbol{H}$ is the symbol of the Hamiltonian $\,\hat {\boldsymbol{\!H}}$ ,

(2.26) \begin{align} {\boldsymbol{H}}=\begin{pmatrix} {\boldsymbol{H}}^{++}&\quad {\boldsymbol{H}}^{+-}\\[5pt] {\boldsymbol{H}}^{-+}&\quad {\boldsymbol{H}}^{{-}{-}} \end{pmatrix}, \end{align}

and the individual components are given by

(2.27a) \begin{align} H^{\pm \pm }_{ij}=\delta _{ij}\big (\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\mp _l\star k_l-\mathrm{i}\nu _+k^2\big )+\mathrm{i}\frac {k_i}{k^2}\star \mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,j}^\mp \star k_l, \\[-30pt]\nonumber \end{align}

(2.27b) \begin{align} H_{ij}^{\pm \mp }=-\delta _{ij}\mathrm{i}\nu _-k^2-\mathrm{i}\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{i,j}+\mathrm{i}\frac {k_i}{k^2}\star \mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,j}^\mp \star k_l, \end{align}

the Moyal star product $\star$ (B.8) is defined in Appendix B, and $\boldsymbol{W}$ is the symbol of $\hat {\boldsymbol{W}}$ , also known as the Wigner matrix. The latter can be expressed as

(2.28) \begin{equation} \boldsymbol{W}=\begin{pmatrix} \boldsymbol{W}^{++}& \quad \boldsymbol{W}^{+-}\\[5pt] \boldsymbol{W}^{-+}&\quad \boldsymbol{W}^{{-}{-}} \end{pmatrix}, \end{equation}

where

(2.29) \begin{equation} \begin{gathered} W^{\sigma _1\sigma _2}_{ij}=\int \mathrm{d} \boldsymbol{s}\,\mathrm{e}^{-\mathrm{i}\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{s}}\widetilde {z}^{\sigma _1}_i(\boldsymbol{x}+\boldsymbol{s}/2)\widetilde {z}^{\sigma _2}_j(\boldsymbol{x}-\boldsymbol{s}/2). \end{gathered} \end{equation}

Note that the Wigner matrix $\boldsymbol{W}$ is Hermitian, i.e.

(2.30) \begin{equation} W_{ij}^{\sigma _1\sigma _2}(t,\boldsymbol{x},\boldsymbol{k})=W_{ji}^{\sigma _2\sigma _1*}(t,\boldsymbol{x},\boldsymbol{k}). \end{equation}

Additionally, since the Elsässer fields are real, we also have that

(2.31) \begin{equation} W_{ij}^{\sigma _1\sigma _2}(t,\boldsymbol{x},\boldsymbol{k})=W_{ji}^{\sigma _2\sigma _1}(t,\boldsymbol{x},-\boldsymbol{k}). \end{equation}

2.3.3. Average Wigner matrix and the GO expansion

By averaging (2.25) with the same averaging operation used to define the mean fields, we obtain the averaged WME,

(2.32) \begin{align} \mathrm{i}\partial _t\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{W}\mkern -1.5mu}\mkern 1.5mu=\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{H}\mkern -1.5mu}\mkern 1.5mu\star \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{W}\mkern -1.5mu}\mkern 1.5mu-\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{W}\mkern -1.5mu}\mkern 1.5mu\star \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{H}\mkern -1.5mu}\mkern 1.5mu^\dagger . \end{align}

Here, $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{H}\mkern -1.5mu}\mkern 1.5mu=\boldsymbol{H}$ , since $\boldsymbol{H}$ is independent of the fluctuating fields, and the averaged Wigner matrix is the average of $\boldsymbol{W}$ ,

(2.33) \begin{equation} \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{W}\mkern -1.5mu}\mkern 1.5mu\equiv \langle \boldsymbol{W} \rangle =\begin{pmatrix} \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{W}\mkern -1.5mu}\mkern 1.5mu^{++}& \quad \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{W}\mkern -1.5mu}\mkern 1.5mu^{+-}\\[5pt] \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{W}\mkern -1.5mu}\mkern 1.5mu^{-+}&\quad \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{W}\mkern -1.5mu}\mkern 1.5mu^{{-}{-}} \end{pmatrix}, \end{equation}

where

(2.34) \begin{equation} \mkern 1.5mu\overline {\mkern -1.5muW\mkern -1.5mu}\mkern 1.5mu^{\sigma _1\sigma _2}_{ij}=\int \mathrm{d} \boldsymbol{s}\,\mathrm{e}^{-\mathrm{i}\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{s}}\big\langle \widetilde {z}^{\sigma _1}_i(\boldsymbol{x}+\boldsymbol{s}/2)\widetilde {z}^{\sigma _2}_j(\boldsymbol{x}-\boldsymbol{s}/2) \big\rangle . \end{equation}

Notice that the average Wigner matrix (2.33) can be understood as the Fourier transform of the symmetrised two-point correlation tensor of the Elsässer fields.

Below, the original Wigner matrix (2.28) will not be needed, so we will call (2.33) ‘the’ Wigner matrix and omit the bar in $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{W}\mkern -1.5mu}\mkern 1.5mu$ to simplify notation. The properties (2.30) and (2.31) hold for this averaged matrix as well. Also, the trace of $\boldsymbol{W}$ , to some extent,Footnote 5 can be interpreted as the phase-space density of turbulent quasiparticles. In this sense, the WME can be considered as a generalisation of the Liouville equation. Its interpretation as a quantum extension of kinetic theory becomes clearer when we consider the series expansion of the Moyal star,

(2.35) \begin{equation} \begin{aligned} \mathrm{i}\partial _t \boldsymbol{W} &=\boldsymbol{H}\mathrm{e}^{\mathrm{i}\hat {\mathcal{L}}/2}\boldsymbol{W}-\boldsymbol{W}\mathrm{e}^{\mathrm{i}\hat {\mathcal{L}}/2}\boldsymbol{H}^\dagger \\ &=\boldsymbol{H}(1+\mathrm{i}\hat {\mathcal{L}}/2+\cdots )\boldsymbol{W}-\boldsymbol{W}(1+\mathrm{i}\hat {\mathcal{L}}/2+\cdots )\boldsymbol{H}^\dagger , \end{aligned} \end{equation}

where the Janus operator $\hat {\mathcal{L}}$ (B.9) is defined in Appendix B, and, basically, stands for the canonical Poisson bracket in the $(\boldsymbol{x},\boldsymbol{k})$ space, $A\hat {\mathcal{L}}B = \{A, B\}$ . Note that $\hat {\mathcal{L}}$ effectively scales as the GO parameter $l/L$ , so the higher-order terms (denoted ‘…’ in (2.35)) can be omitted. Furthermore, this equation can be further simplified when $W$ and $H$ are scalar functions (as can be the case, for example, in drift-wave turbulence (Zhu & Dodin Reference Zhu and Dodin2021)). In this case, (2.35) becomes the familiar ‘classical’ wave kinetic equation when $l/L \to 0$ ,

(2.36) \begin{equation} \partial _t W\approx \{H_{\mathrm{H}},W\}+2 H_{\mathrm{A}} W, \end{equation}

where $H_{\mathrm{H}}$ and $H_{\mathrm{A }}$ are the real (Hermitian) and the imaginary (anti-Hermitian) parts of the Hamiltonian. However, keep in mind that, for MHD turbulence, $\boldsymbol{W}$ and $\boldsymbol{H}$ are non-commuting matrices. Furthermore, when $L$ is determined by MI, it is often the case that $l\sim L$ , so the GO approximation of the WME is inapplicable.

2.4. Turbulent source term for the mean field

As with any quadratic functional of the fluctuating field (Dodin Reference Dodin2022), the turbulent source term for the mean field,

(2.37) \begin{equation} \boldsymbol{S}^\pm \doteq -\langle \boldsymbol{\nabla} \times [(\widetilde {\boldsymbol{z}}^\mp \boldsymbol{\cdot} \boldsymbol{\nabla} )\widetilde {\boldsymbol{z}}^\pm ] \rangle , \end{equation}

can be expressed through the Wigner matrix $\boldsymbol{W}$ ,

(2.38) \begin{align} S_i^\pm =&-\big\langle \epsilon _{ijk}\partial _{j}\big(\widetilde {z}^\mp _l\partial _l\widetilde {z}^\pm _k\big) \big\rangle \nonumber\\[5pt] =&-\epsilon _{ijk}\big\langle \big(\big\langle {\boldsymbol{x}|\mathrm{i}\hat {k}_j|\widetilde {z}^\mp _l}\big\rangle \big\langle {\boldsymbol{x}|\mathrm{i}\hat {k}_l|\widetilde {z}^\pm _k}\big\rangle -\big\langle {\boldsymbol{x}|\widetilde {z}^\mp _l}\big\rangle \big\langle {\boldsymbol{x}|\hat {k}_j\hat {k}_l|\widetilde {z}^\pm _k}\big\rangle \big) \big\rangle \nonumber\\[5pt] =&-\epsilon _{ijk}\big\langle \big(\big\langle {\boldsymbol{x}|\hat {k}_j|\widetilde {z}^\mp _l}\big\rangle \big\langle {\widetilde {z}^\pm _k|\hat {k}_l|\boldsymbol{x}}\big\rangle -\big\langle {\boldsymbol{x}|\widetilde {z}^\mp _l}\big\rangle \big\langle {\widetilde {z}^\pm _k|\hat {k}_j\hat {k}_l|\boldsymbol{x}}\big\rangle \big) \big\rangle \nonumber\\[5pt] =&\,\epsilon _{ijk}\int \frac {\mathrm{d}\boldsymbol{k}}{(2\pi )^3}\big (k_lk_j\star W_{kl}^{\pm \mp }-k_l\star W_{kl}^{\pm \mp }\star k_j\big ), \end{align}

where $\epsilon _{ijk}$ is the Levi–Civita symbol and the transition from the third line to the fourth line uses (B.4).

Note that the source term involves only the off-diagonal blocks of the Wigner matrix, $\boldsymbol{W}^{\pm \mp }$ , i.e. mean fields are generated only by correlations between $\widetilde {\boldsymbol{z}}^+$ and $\widetilde {\boldsymbol{z}}^-$ . This can also be expected from the original Elsässer equations (2.3), where the nonlinear term vanishes if either $\boldsymbol{z}^+$ or $\boldsymbol{z}^-$ is zero.

2.5. Summary of the main equations

In summary, our mean-field wave-kinetics model is as follows:

(2.39a) \begin{align} \partial _t \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm = -\hat {\boldsymbol{k}} \times \left \{ \left[\left(\frac {\hat {\boldsymbol{k}}}{\hat {k}^2} \times \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\mp \right)\boldsymbol{\cdot} \hat {\boldsymbol{k}}\right] \left(\frac {\hat {\boldsymbol{k}}}{\hat {k}^2}\times \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm \right ) \right\} -\hat {k}^2(\nu _+ \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm + \nu _- \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\mp )+\boldsymbol{S}^\pm , \end{align}

(2.39b) \begin{align} \mathrm{i}\partial _t\boldsymbol{W}=\boldsymbol{H}\star \boldsymbol{W}-\boldsymbol{W}\star \boldsymbol{H}^\dagger , \end{align}

where

(2.40) \begin{equation} S_i^\pm =\epsilon _{ijk}\int \frac {\mathrm{d}\boldsymbol{k}}{(2\pi )^3}\big (k_lk_j\star W_{kl}^{\pm \mp }-k_l\star W_{kl}^{\pm \mp }\star k_j\big ), \end{equation}

and the matrix

(2.41) \begin{equation} {\boldsymbol{H}}=\begin{pmatrix} {\boldsymbol{H}}^{++}&\quad {\boldsymbol{H}}^{+-}\\[5pt] {\boldsymbol{H}}^{-+}&\quad {\boldsymbol{H}}^{{-}{-}} \end{pmatrix} \end{equation}

consists of

(2.42a) \begin{align} H^{\pm \pm }_{ij}=\delta _{ij}\big (\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\mp _l\star k_l-\mathrm{i}\nu _+k^2\big )+\mathrm{i}\frac {k_i}{k^2}\star \mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,j}^\mp \star k_l, \end{align}

(2.42b) \begin{align} H_{ij}^{\pm \mp }=-\delta _{ij}\mathrm{i}\nu _-k^2-\mathrm{i}\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{i,j}+\mathrm{i}\frac {k_i}{k^2}\star \mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,j}^\pm \star k_l. \end{align}

Note that (2.39) are equivalent to the original MHD system within the QLA and do not assume scale separation between the fluctuations and mean fields, which is typically done in the literature (Brandenburg Reference Brandenburg2018; Hughes Reference Hughes2018). As we show in Appendix C, (2.39) conserve the total energy and cross-helicity, while allowing for the transfer of these invariants between the mean and fluctuating fields.

To aid comparisons with existing theories, we will also often work with the following ad hoc modification of (2.39b):

(2.43) \begin{equation} \mathrm{i}\partial _t\boldsymbol{W}=\boldsymbol{H}\star \boldsymbol{W}-\boldsymbol{W}\star \boldsymbol{H}^\dagger -\mathrm{i}\tau _c^{-1}\boldsymbol{W}+\boldsymbol{T}, \end{equation}

where $\tau _c$ is the correlation time that determines the damping of fluctuations through wave–wave collisions. Such a damping term is used in the popular MTA closure (Blackman & Field Reference Blackman and Field2002; Brandenburg & Subramanian Reference Brandenburg and Subramanian2005). The forcing term $\boldsymbol{T}$ , which is yet to specified, is added to allow for turbulent equilibria at non-zero dissipation (§ 3).

3.1. Properties of statistically homogeneous turbulent equilibria

3.1.1. Equilibrium condition

In the absence of zeroth-order mean fields, (2.39) give the following equations for the equilibrium:

(3.3a) \begin{align} 0 =\boldsymbol{S}_0^\pm ,\\[-30pt] \nonumber \end{align}

(3.3b) \begin{align} 0=(2\pi )^3\big(\boldsymbol{H}_0\boldsymbol{F}-\boldsymbol{F}\boldsymbol{H}_0^\dagger -\mathrm{i}\tau _{c}^{-1}\boldsymbol{F}\big)+\boldsymbol{T}, \end{align}

(3.3c) \begin{align} S_{0i}^\pm =\epsilon _{ijk}\int \mathrm{d}\boldsymbol{k}\big(k_lk_j\star F_{kl}^{\pm \mp }-k_l\star F_{kl}^{\pm \mp }\star k_j\big), \\[-30pt] \nonumber \end{align}

(3.3d) \begin{align} \begin{gathered} \boldsymbol{H}_{0}^{\pm \pm }=-\mathrm{i}\nu _+k^2\boldsymbol{1}_3,\qquad \boldsymbol{H}_{0}^{\pm \mp }=-\mathrm{i}\nu _-k^2\boldsymbol{1}_3, \end{gathered} \\ \nonumber \end{align}

where $\boldsymbol{1}_3$ is the $3\times 3$ identity matrix. Thus, for a given $\boldsymbol{F}$ , the matrix $\boldsymbol{T}$ must satisfy

(3.4) \begin{equation} \boldsymbol{T}^{\sigma _1\sigma _2}=\mathrm{i}(2\pi )^3\big[\big(2\nu _+ k^2+\tau _c^{-1}\big)\boldsymbol{F}^{\sigma _1\sigma _2}+\nu _-k^2\big(\boldsymbol{F}^{\sigma _1\mkern 1.5mu\overline {\mkern -1.5mu\sigma _2\mkern -1.5mu}\mkern 1.5mu}+\boldsymbol{F}^{\mkern 1.5mu\overline {\mkern -1.5mu\sigma _1\mkern -1.5mu}\mkern 1.5mu\sigma _2}\big)\big], \end{equation}

where $\sigma _{1/2}=+/-$ and $\mkern 1.5mu\overline {\mkern -1.5mu\sigma \mkern -1.5mu}\mkern 1.5mu\doteq -\sigma$ .

Note that in the absence of dissipation ( $\tau _c^{-1}=0$ and $\nu _\pm =0$ ), (3.4) gives $\boldsymbol{T}=0$ ; i.e. any such homogeneous turbulent background is quasilinearly self-consistent without external driving. Also note that the consistency of the mean-field equation, $\boldsymbol{S}_0=0$ , follows directly from the background homogeneity, $\boldsymbol{F}(\boldsymbol{x},\boldsymbol{k})=\boldsymbol{F}(\boldsymbol{k})$ ,

(3.5) \begin{align} S_{0i}^\pm &=\epsilon _{ijk}\int \mathrm{d}\boldsymbol{ k}\,\big(k_lk_j\star F_{kl}^{\pm \mp }-k_l\star F_{kl}^{\pm \mp }\star k_j\big)\nonumber\\ &=\epsilon _{ijk}\int \mathrm{d} \boldsymbol{k}\,(k_lk_j-k_l k_j) F_{kl}^{\pm \mp }\nonumber\\ &=0. \end{align}

3.1.2. Wigner matrix of isotropic MHD turbulence

For a homogeneous turbulent background, we have

(3.6) \begin{align} F_{ij}^{\sigma _1\sigma _2}(\boldsymbol{k})&=\frac {1}{(2\pi )^3}\int \mathrm{d}\boldsymbol{ s}\,\mathrm{e}^{-\mathrm{i}\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{s}}\left\langle \widetilde {z}_i^{\sigma _1}\left(\boldsymbol{x}+\frac {\boldsymbol{s}}{2}\right)\widetilde {z}_j^{\sigma _2}\left(\boldsymbol{x}-\frac {\boldsymbol{s}}{2}\right)\right\rangle \nonumber\\[4pt] &=R_{ij}^{\sigma _1\sigma _2}(-\boldsymbol{k}), \end{align}

where $\boldsymbol{R}^{\sigma _1\sigma _2}(\boldsymbol{k})$ is the Fourier transform of the two-point correlation tensor,

(3.7) \begin{equation} R_{ij}^{\sigma _1\sigma _2}(\boldsymbol{r})=\big\langle \widetilde {z}^{\sigma _1}_i(\boldsymbol{x}-\boldsymbol{r}/2)\widetilde {z}^{\sigma _2}_j(\boldsymbol{x}+\boldsymbol{r}/2) \big\rangle . \end{equation}

If we further assume that the turbulence is isotropic (Oughton, Rädler & Matthaeus Reference Oughton, Rädler and Matthaeus1997), then

(3.8) \begin{equation} R_{ij}^{\sigma _1\sigma _2}(\boldsymbol{k})=\left(\delta _{ij}-\frac {k_i k_j}{k^2}\right)\frac {E^{\sigma _1\sigma _2}(k)}{4\pi k^2}+\mathrm{i} \epsilon _{ijk}\frac {k_k}{k^2}\frac {H^{\sigma _1\sigma _2}(k)}{8\pi k^2}, \end{equation}

where $\boldsymbol{R}^{\sigma _1\sigma _2}=\boldsymbol{R}^{\sigma _2\sigma _1}$ , $k\doteq |\boldsymbol{k}|$ and the $H^{\sigma _1\sigma _2}$ are zero for non-helical turbulence. Note that $E^{\sigma _1\sigma _2}(k)$ and $H^{\sigma _1\sigma _2}(k)$ are the spectra of energy-like and helicity-like quantities, respectively, in the following sense:

(3.9) \begin{equation} \begin{gathered} \big\langle \boldsymbol{z}^\sigma _1\boldsymbol{\cdot} \boldsymbol{z}^\sigma _2 \big\rangle =2\int \mathrm{d} k \,E^{\sigma _1\sigma _2}(k),\qquad \big\langle \boldsymbol{z}^\sigma _1\boldsymbol{\cdot} \big(\boldsymbol{\nabla} \times \boldsymbol{z}^\sigma _2\big) \big\rangle =\int \mathrm{d} k \,H^{\sigma _1\sigma _2}(k). \end{gathered} \end{equation}

In other words, isotropic MHD turbulence corresponds to

(3.10) \begin{align} \boldsymbol{F} & =\begin{pmatrix} \boldsymbol{F}^+&\quad \boldsymbol{F}^C\\[3pt] \boldsymbol{F}^C&\quad \boldsymbol{F}^- \end{pmatrix},\nonumber\\[3pt] F^\chi _{ij}(\boldsymbol{k}) & =\left(\delta _{ij}-\frac {k_i k_j}{k^2}\right)\frac {E^\chi (k)}{4\pi k^2}-\mathrm{i} \epsilon _{ijk}\frac {k_k}{k^2}\frac {H^\chi (k)}{8\pi k^2} \end{align}

with $\chi =+,\,-,\,C$ . It will also be convenient to work with the symmetric and antisymmetric combinations $\boldsymbol{F}^S\doteq (\boldsymbol{F}^+ +\boldsymbol{F}^-)/2$ and $\boldsymbol{F}^A\doteq (\boldsymbol{F}^+ -\boldsymbol{F}^-)/2$ , and we also extend this notation to $E$ and $H$ . Then,

(3.11) \begin{align} \int _0^\infty \mathrm{d} k \,E^{S}(k)&=\frac {1}{2}\langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} {\widetilde{\boldsymbol{v}}}+{\widetilde{\boldsymbol{b}}}\boldsymbol{\cdot} {\widetilde{\boldsymbol{b}}} \rangle ,\nonumber \\ \int _0^\infty \mathrm{d} k \,E^{C}(k)&=\frac {1}{2}\langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} {\widetilde{\boldsymbol{v}}}-{\widetilde{\boldsymbol{b}}}\boldsymbol{\cdot} {\widetilde{\boldsymbol{b}}} \rangle ,\nonumber \\ \int _0^\infty \mathrm{d} k \,E^{A}(k)&=\langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} {\widetilde{\boldsymbol{b}}} \rangle , \nonumber\\ \int _0^\infty \mathrm{d} k \,H^{S}(k)&=\langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} \widetilde {\boldsymbol{w}}+{\widetilde{\boldsymbol{b}}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle ,\nonumber \\ \int _0^\infty \mathrm{d} k \,H^{C}(k)&=\langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} \widetilde {\boldsymbol{w}}-{\widetilde{\boldsymbol{b}}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle ,\nonumber \\ \int _0^\infty \mathrm{d} k \,H^{A}(k)&=2\langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle =2\langle \widetilde {\boldsymbol{w}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle , \end{align}

where $\widetilde {\boldsymbol{w}}\doteq \boldsymbol{\nabla} \times \widetilde {\boldsymbol{v}}$ and $\widetilde {\boldsymbol{j}}\doteq \boldsymbol{\nabla} \times \widetilde {\boldsymbol{b}}$ .

3.2. Linear modulational dynamics

3.2.1. Linearised equations

The perturbed quantities are governed by the following linearised MFWK equations:

(3.12a) \begin{align} \partial _t \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm = -\hat {k}^2(\nu _+ \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm + \nu _- \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\mp )+\boldsymbol{s}^\pm , \end{align}

(3.12b) \begin{align} \mathrm{i}\partial _t\boldsymbol{f}=\boldsymbol{H_0}\star \boldsymbol{f}-\boldsymbol{f}\star \boldsymbol{H_0}^\dagger +\boldsymbol{h}\star \boldsymbol{F}-\boldsymbol{F}\star \boldsymbol{h}^\dagger -\mathrm{i}\tau _c^{-1}\boldsymbol{f},\end{align}

(3.12c) \begin{align} s_i^\pm =\epsilon _{ijk}\int \mathrm{d}\boldsymbol{k}\,\big(k_lk_j\star f_{kl}^{\pm \mp }-k_l\star f_{kl}^{\pm \mp }\star k_j\big), \end{align}

(3.12d) \begin{align} h^{\pm \pm }_{ij} & =\delta _{ij}\big (\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\mp _l\star k_l\big )+\mathrm{i}\,\frac {k_i}{k^2}\star \mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,j}^\mp \star k_l,\nonumber\\[3pt] h_{ij}^{\pm \mp } & =-\mathrm{i}\mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu^\pm _{i,j}+\mathrm{i}\,\frac {k_i}{k^2}\star \mkern 1.5mu\overline {\mkern -1.5muz\mkern -1.5mu}\mkern 1.5mu_{l,j}^\pm \star k_l. \end{align}

Note that this model is fundamentally different from the commonly used kinematic approximation (Krause & Rädler Reference Krause and Rädler1980). The kinematic approximation assumes that all magnetic fields (that is, both turbulent and mean components) are sufficiently weak such that the flow (again, both mean and turbulent parts) can be taken as prescribed. The momentum equation is then never invoked, and the resulting mean-field theory is built on the induction equation alone. It has already been pointed out in the literature (Courvoisier et al. Reference Courvoisier, Hughes and Proctor2010a ,Reference Courvoisier, Hughes and Proctor b ) that such an approach, which artificially privileges the magnetic field over the flow, is fundamentally inconsistent in the presence of substantial magnetic field fluctuations (which are to be expected in MHD turbulence), but the general theory for this case has been lacking. Our approach fixes this problem in that it is formulated in terms of the Elsässer fields and thus treats velocity and magnetic fields on the same footing.

3.2.2. Eikonal perturbations

Let us now look for the linear eigenmodes of (3.12). For that, let us consider perturbed quantities of the following form:

(3.13a) \begin{align} \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\pm =\textrm {Re} \left (\boldsymbol{{z}}^\pm \mathrm{e}^{\mathrm{i}\Theta }\right )\!, \end{align}

(3.13b) \begin{align} \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm =\textrm {Re} \left (\boldsymbol{{w}}^\pm \mathrm{e}^{\mathrm{i}\Theta }\right )\!, \end{align}

(3.13c) \begin{align} \boldsymbol{f}=\left (\boldsymbol{{f}}(\boldsymbol{k})\mathrm{e}^{\mathrm{i}\Theta }\right )_{\mathrm{H}}\!, \end{align}

(3.13d) \begin{align} \boldsymbol{h}=\frac {1}{2}\big(\boldsymbol{{h}}_{(+)}(\boldsymbol{k})\mathrm{e}^{i\Theta }+\boldsymbol{{h}}_{(-)}^{\dagger }(\boldsymbol{k})\mathrm{e}^{-i\Theta ^*}\big), \end{align}

with

(3.14) \begin{equation} \Theta =-\varOmega t+\boldsymbol{K}\boldsymbol{\cdot} \boldsymbol{x}, \end{equation}

where $\varOmega$ and $\boldsymbol{K}$ are the modulational frequency and wavevector, respectively, and $\boldsymbol{{w}}^\pm =\mathrm{i}\boldsymbol{K}\times \boldsymbol{{z}}^\pm$ . (The index H denotes the Hermitian part.) We will use the convention that the modulational wavevector $\boldsymbol{K}$ is real, while the modulational frequency $\varOmega$ may be complex. Note that the mean-field polarisations, $\boldsymbol{{z}}^\pm$ and $\boldsymbol{{w}}^\pm$ , are constants, while $\boldsymbol{{h}}_{(\pm )}$ and $\boldsymbol{{f}}$ are functions of $\boldsymbol{k}$ . We also use the convention that the argument of a function will only be explicitly written when it is first defined, not obvious from the context, or judged to provide helpful information.

3.2.3. Equations for the polarisations

Using (B.11), we can write the following equations for the polarisations $\boldsymbol{{z}}^\pm$ and $\boldsymbol{{f}}$ :

(3.15) \begin{align} \varOmega \boldsymbol{{z}}^\pm =-\mathrm{i} K^2(\nu _+ \boldsymbol{{z}}^\pm +\nu _-\boldsymbol{{z}}^\mp )-\frac {\boldsymbol{K}}{K^2}\times \boldsymbol{s}^\pm , \\[-33pt] \nonumber \end{align}

(3.16) \begin{align} \begin{aligned} \varOmega '\boldsymbol{{f}}-\boldsymbol{H}_0(\boldsymbol{k}_+)\boldsymbol{{f}}+\boldsymbol{{f}}\boldsymbol{H}_0^{\dagger }(\boldsymbol{k}_-)=\boldsymbol{{h}}_{(+)}\boldsymbol{F}(\boldsymbol{k}_-)-\boldsymbol{F}(\boldsymbol{k}_+)\boldsymbol{{h}}_{(-)}, \end{aligned} \end{align}

where

(3.17) \begin{align} -\left(\frac {\boldsymbol{K}}{K^2}\times \boldsymbol{s}^\pm \right)_i=\int \mathrm{d}\boldsymbol{k}\left(K_n\mathrm{f}_{in}^{\pm \mp }-\frac {K_iK_mK_n}{K^2}\mathrm{f}_{mn}^{\pm \mp }\right), \end{align}

and

(3.18) \begin{align} \boldsymbol{{h}}_{(\pm )}=\begin{pmatrix} \boldsymbol{{h}}_{(\pm )}^{++}&\quad \boldsymbol{{h}}_{(\pm )}^{+-}\\[5pt] \boldsymbol{{h}}_{(\pm )}^{-+}&\quad \boldsymbol{{h}}_{(\pm )}^{{-}{-}} \end{pmatrix}, \end{align}

where

(3.19a) \begin{align} \mathrm{h}^{\pm \pm }_{(+)ij}=(\boldsymbol{{z}}^\mp \boldsymbol{\cdot} \boldsymbol{k})\left(\delta _{ij}-\frac {k_{+i}K_j}{k_+^2}\right), \end{align}

(3.19b) \begin{align} \mathrm{h}^{\pm \mp }_{(+)ij}=\,\mathrm{z}_i^\mp K_j-(\boldsymbol{{z}}^\mp \boldsymbol{\cdot} \boldsymbol{k})\frac {k_{+i}K_j}{k_+^2}, \end{align}

(3.19c) \begin{align} \,\mathrm{h}^{\pm \pm }_{(-)ij}=(\boldsymbol{{z}}^\mp \boldsymbol{\cdot} \boldsymbol{k})\left(\delta _{ij}+\frac {K_ik_{-j}}{k_-^2}\right), \end{align}

(3.19d) \begin{align} \mathrm{h}^{\pm \mp }_{(-)ij}= -\mathrm{z}_j^\pm K_i+(\boldsymbol{{z}}^\pm \boldsymbol{\cdot} \boldsymbol{k})\frac {K_i k_{-j}}{k_-^2}.\end{align}

Also, $\varOmega '\doteq \varOmega +\mathrm{i}\tau _c^{-1}$ , $\boldsymbol{k}_\pm \doteq \boldsymbol{k}\pm \boldsymbol{K}/2$ and $k_{\pm i}$ refers to the $i$ th element of $\boldsymbol{k}_\pm$ . We study applications of these equations in § 4 and § 5.

4.1. Derivation

The generic non-local linear response of the turbulent EMF $\boldsymbol{\mathcal{E}}$ to the mean fields $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{b}\mkern -1.5mu}\mkern 1.5mu$ and $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu$ can be written as follows:

(4.5) \begin{equation} \boldsymbol{\mathcal{E}}(t,\boldsymbol{x})=\int \mathrm{d} t'\int \mathrm{d}\boldsymbol{x}'\big(\boldsymbol{G}^{(\boldsymbol{b})}(t,\boldsymbol{x};t',\boldsymbol{x}')\,\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{b}\mkern -1.5mu}\mkern 1.5mu(t',\boldsymbol{x}')+\boldsymbol{G}^{(\boldsymbol{v})}(t,\boldsymbol{x};t',\boldsymbol{x}')\,\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu(t',\boldsymbol{x}')\big). \end{equation}

Note that we are allowing for a possible dependence on the mean flow, which is often neglected. (Since the mean flow is generally inhomogeneous, for example, as in (3.13a ), it cannot be removed by a Galilean transformation.) For the remainder of our calculation of the turbulent EMF, we will also work with the traditional velocity and magnetic fields, as opposed to the Elsässer fields, in order to aid comparisons with existing theories.

We now assume weakly inhomogeneous turbulence, such that $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\pm$ and $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{w}\mkern -1.5mu}\mkern 1.5mu^\pm$ are perturbations to an otherwise homogeneous turbulent background. In this case, we can take $\boldsymbol{G}^{(\boldsymbol{b},\boldsymbol{v})}(t,\boldsymbol{x};t',\boldsymbol{x}')=\boldsymbol{G}^{(\boldsymbol{b},\boldsymbol{v})}(t-t',\boldsymbol{x}-\boldsymbol{x}')$ such that in Fourier space we have

(4.6) \begin{equation} \boldsymbol{\mathcal{E}}(\varOmega ,\boldsymbol{K})=\boldsymbol{G}^{(\boldsymbol{b})}(\varOmega ,\boldsymbol{K})\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{b}\mkern -1.5mu}\mkern 1.5mu(\varOmega ,\boldsymbol{K})+\boldsymbol{G}^{(\boldsymbol{v})}(\varOmega ,\boldsymbol{K})\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu(\varOmega ,\boldsymbol{K}). \end{equation}

(For the remainder of our calculation of the turbulent EMF, we will also work with the traditional velocity and magnetic fields, as opposed to the Elsässer fields, in order to aid comparisons with existing theories.) An expression relating the Fourier coefficients of $\boldsymbol{\mathcal{E}}$ , $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{b}\mkern -1.5mu}\mkern 1.5mu$ and $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu$ at frequency $\varOmega$ and wavenumber $\boldsymbol{K}$ therefore yields the non-local response kernel $\boldsymbol{G}(\varOmega ,\boldsymbol{K})$ .

To do this, we first note that the EMF can be directly written in terms of our Wigner matrix as follows:

(4.7) \begin{align} \mathcal{E}_i&=-\frac {1}{2}\langle \widetilde {\boldsymbol{z}}^+\times \widetilde {\boldsymbol{z}}^- \rangle _i \nonumber \\ &=-\frac {1}{2}\int \frac {\mathrm{d}\boldsymbol{k}}{(2\pi )^3}\, \epsilon _{ijk}W^{+-}_{jk}. \end{align}

Curiously, (4.7) can be expressed even more concisely as

(4.8) \begin{equation} \begin{gathered} \boldsymbol{\mathcal{E}}=\bigstar \int \frac {\mathrm{d}\boldsymbol{k}}{(2\pi )^3}\,\boldsymbol{W}^{-+}, \end{gathered} \end{equation}

where $\bigstar$ is the Hodge star, $(\bigstar \boldsymbol{A})_i\doteq \epsilon _{ijk}A_{jk}/2$ . The desired expression can therefore be obtained with the modulational mode analysis described in § 3, by solving the WME to obtain the Wigner matrix in terms of the mean fields (which enter the WME through $\boldsymbol{H}$ ).

We can immediately see that there may, in principle, be some contribution to the turbulent EMF from $\boldsymbol{F}$ , which has nothing to do with the mean fields,

(4.9) \begin{align} \mathcal{E}_{0i}=&-\frac {\epsilon _{ijk}}{2}\int \mathrm{d}\boldsymbol{k}\,F_{jk}^{+-}\nonumber \\[3pt] =&-\frac {\epsilon _{ijk}}{4}\int \mathrm{d}\boldsymbol{ k}\,\big(F_{jk}^{+-}+F_{kj}^{-+}\big)\nonumber \\[3pt] =&-\frac {\epsilon _{ijk}}{4}\int \mathrm{d}\boldsymbol{ k}\,\big(F_{jk}^{+-}-F_{jk}^{-+}\big). \end{align}

It can be easily shown that such a background EMF $\boldsymbol{\mathcal{E}}_0$ vanishes for a broad class of turbulent backgrounds. For example, isotropy is a sufficient but not necessary condition for $\boldsymbol{F}^{+-}=\boldsymbol{F}^{-+}$ . We henceforth understand $\boldsymbol{\mathcal{E}}$ to refer to the mean-field contribution alone and will not discuss $\boldsymbol{\mathcal{E}}_0$ further.

The mean-field contribution to the turbulent EMF is captured by the perturbed Wigner matrix,

(4.10) \begin{equation} \mathcal{E}_i=-\epsilon _{ijk}\int \mathrm{d}\boldsymbol{k} f_{jk}^{+-}. \end{equation}

As in § 3, we now take our perturbed quantities to be of the following form:

(4.11) \begin{align} \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu^\pm =\textrm {Re} \big(\boldsymbol{{v}}^\pm \mathrm{e}^{\mathrm{i}\Theta }\big)\!,\qquad \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{b}\mkern -1.5mu}\mkern 1.5mu^\pm =\textrm {Re} \big(\boldsymbol{{b}}^\pm \mathrm{e}^{\mathrm{i}\Theta }\big)\!,\qquad \boldsymbol{\mathcal{E}}=\textrm {Re}\big(\boldsymbol{\mathfrak{E}}\mathrm{e}^{\mathrm{i}\Theta }\big)\!, \nonumber \\ \boldsymbol{f}=\big(\boldsymbol{{f}}(\boldsymbol{k})\mathrm{e}^{\mathrm{i}\Theta }\big)_{\mathrm{H}}\!,\qquad \boldsymbol{h}=\frac {1}{2}\big(\boldsymbol{{h}}_{+}(\boldsymbol{k})\mathrm{e}^{i\Theta }+\boldsymbol{{h}}_-^{\dagger }(\boldsymbol{k})\mathrm{e}^{-i\Theta ^*}\big), \end{align}

with $\Theta =-\varOmega t+\boldsymbol{K}\boldsymbol{\cdot} \boldsymbol{x}$ , where $\varOmega$ and $\boldsymbol{K}$ are the modulational frequency and wavevector, respectively. In terms of the notation of (4.6), one has

(4.12) \begin{equation} \boldsymbol{\mathcal{E}}(\varOmega ,\boldsymbol{K})=\boldsymbol{\mathfrak{E}}, \qquad \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{b}\mkern -1.5mu}\mkern 1.5mu(\varOmega ,\boldsymbol{K})=\boldsymbol{{b}}, \qquad \mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu(\varOmega ,\boldsymbol{K})=\boldsymbol{{v}}. \end{equation}

Therefore, our relevant equations are

(4.13) \begin{align} \mathfrak{E}_i=-\frac {1}{2}\int \mathrm{d}\boldsymbol{k} \,\epsilon _{ijk}\mathrm{f}_{jk}^{+-}, \end{align}

along with (3.16), $\boldsymbol{{v}}=(\boldsymbol{{z}}^++\boldsymbol{{z}}^-)/2$ and $\boldsymbol{{b}}=(\boldsymbol{{z}}^+-\boldsymbol{{z}}^-)/2$ .

Solving (3.16) for the off-diagonal blocks of the perturbed Wigner matrix yields

(4.14) \begin{equation} \begin{aligned} \boldsymbol{{f}}^{\pm \mp }&=\frac {C_1(\boldsymbol{k})}{C(\boldsymbol{k})}\big [\boldsymbol{{h}}^{\pm +}_{(+)}\boldsymbol{F}^{+\mp }(\boldsymbol{k}_-)+\boldsymbol{{h}}^{\pm -}_{(+)}\boldsymbol{F}^{-\mp }(\boldsymbol{k}_-)-\boldsymbol{F}^{\pm +}(\boldsymbol{k}_+)\boldsymbol{{h}}^{+\mp }_{(-)}-\boldsymbol{F}^{\pm -}(\boldsymbol{k}_+)\boldsymbol{{h}}^{-\mp }_{(-)}\big ]\\[3pt] &\quad +\frac {C_2(\boldsymbol{k})}{C(\boldsymbol{k})}\big [\boldsymbol{{h}}^{\mp +}_{(+)}\boldsymbol{F}^{+\pm }(\boldsymbol{k}_-)+\boldsymbol{{h}}^{\mp -}_{(+)}\boldsymbol{F}^{-\pm }(\boldsymbol{k}_-)-\boldsymbol{F}^{\mp +}(\boldsymbol{k}_+)\boldsymbol{{h}}^{+\pm }_{(-)}-\boldsymbol{F}^{\mp -}(\boldsymbol{k}_+)\boldsymbol{{h}}^{-\pm }_{(-)}\big ]\\[3pt] &\quad +\frac {C_3(\boldsymbol{k})}{C(\boldsymbol{k})}\big [\boldsymbol{{h}}^{\mp +}_{(+)}\boldsymbol{F}^{+\mp }(\boldsymbol{k}_-)+\boldsymbol{{h}}^{\mp -}_{(+)}\boldsymbol{F}^{-\mp }(\boldsymbol{k}_-)-\boldsymbol{F}^{\mp +}(\boldsymbol{k}_+)\boldsymbol{{h}}^{+\mp }_{(-)}-\boldsymbol{F}^{\mp -}(\boldsymbol{k}_+)\boldsymbol{{h}}^{-\mp }_{(-)}\big ]\\[3pt] &\quad +\frac {C_4(\boldsymbol{k})}{C(\boldsymbol{k})}\big [\boldsymbol{{h}}^{\pm +}_{(+)}\boldsymbol{F}^{+\pm }(\boldsymbol{k}_-)+\boldsymbol{{h}}^{\pm -}_{(+)}\boldsymbol{F}^{-\pm }(\boldsymbol{k}_-)-\boldsymbol{F}^{\pm +}(\boldsymbol{k}_+)\boldsymbol{{h}}^{+\pm }_{(-)}-\boldsymbol{F}^{\pm -}(\boldsymbol{k}_+)\boldsymbol{{h}}^{-\pm }_{(-)}\big ], \end{aligned} \end{equation}

where

(4.15) \begin{align} C(\boldsymbol{k})&=[\mathcal{W}^2(\varOmega ,\boldsymbol{K},\boldsymbol{k})+\nu _-^2( k_+^4+k_-^4)]^2-4\nu _-^4k_+^4k_-^4,\nonumber \\[3pt] C_1(\boldsymbol{k})&=\mathcal{W}(\varOmega ,\boldsymbol{K},\boldsymbol{k}) [\mathcal{W}^2(\varOmega ,\boldsymbol{K},\boldsymbol{k})+\nu _-^2(k_+^4+k_-^4) ],\nonumber \\[3pt] C_2(\boldsymbol{k})&=-2\nu _-^2k_+^2k_-^2\mathcal{W}(\varOmega ,\boldsymbol{K},\boldsymbol{k}),\nonumber \\[3pt] C_3(\boldsymbol{k})&=-\mathrm{i}\nu _-k_+^2 [\mathcal{W}^2(\varOmega ,\boldsymbol{K},\boldsymbol{k})+\nu _-^2( k_+^4+k_-^4) ]+2\mathrm{i}\nu _-^3k_-^4k_+^2,\nonumber \\[3pt] C_4(\boldsymbol{k})&=-\mathrm{i}\nu _-k_-^2 [\mathcal{W}^2(\varOmega ,\boldsymbol{K},\boldsymbol{k})+\nu _-^2( k_+^4+k_-^4) ]+2\mathrm{i}\nu _-^3k_+^4k_-^2,\nonumber \\[3pt] \mathcal{W}(\varOmega,&\boldsymbol{K},\boldsymbol{k})\doteq \varOmega '+\mathrm{i}\nu _+ \big(k_+^2+k_-^2 \big), \end{align}

and, as before, $\boldsymbol{k}_\pm =\boldsymbol{k}\pm \boldsymbol{K}/2$ . Noting that

(4.16) \begin{align} D_1(\boldsymbol{k})\doteq \frac {C_1(\boldsymbol{k}_+)}{C(\boldsymbol{k}_+)} & =\frac {C_1(-\boldsymbol{k}_+)}{C(-\boldsymbol{k}_+)},\qquad D_2(\boldsymbol{k})\doteq \frac {C_2(\boldsymbol{k}_+)}{C(\boldsymbol{k}_+)}=\frac {C_2(-\boldsymbol{k}_+)}{C(-\boldsymbol{k}_+)},\nonumber\\ D_3(\boldsymbol{k})\doteq \frac {C_3(\boldsymbol{k}_+)}{C(\boldsymbol{k}_+)} & =\frac {C_4(-\boldsymbol{k}_+)}{C(-\boldsymbol{k}_+)},\qquad D_4(\boldsymbol{k})\doteq \frac {C_4(\boldsymbol{k}_+)}{C(\boldsymbol{k}_+)}=\frac {C_3(-\boldsymbol{k}_+)}{C(-\boldsymbol{k}_+)}, \end{align}

let us define

(4.17) \begin{equation} \boldsymbol{{h}}(\boldsymbol{k})\doteq \boldsymbol{{h}}_{(+)}(\boldsymbol{k}_+)=-\boldsymbol{{h}}_{(-)}^\intercal (-\boldsymbol{k}_+), \end{equation}

where $^\intercal$ denotes transposition and

(4.18) \begin{align} \mathrm{h}_{ij}^{\pm \pm }(\boldsymbol{k})=(\boldsymbol{{z}}^\mp \boldsymbol{\cdot} \boldsymbol{k})\left[\delta _{ij}-\frac {(k_i+K_i)K_j}{(\boldsymbol{k}+\boldsymbol{K})^2}\right],\nonumber\\[3pt] \mathrm{h}_{ij}^{\pm \mp }(\boldsymbol{k})=\mathrm{z}_i^\mp K_j-(\boldsymbol{{z}}^\mp \boldsymbol{\cdot} \boldsymbol{k})\frac {(k_i+K_i)K_j}{(\boldsymbol{k}+\boldsymbol{K})^2}. \end{align}

Substituting (4.14) into (4.13) yields

(4.19) \begin{align} \mathfrak{E}_i=-\frac {1}{2}\int \boldsymbol{{d}}\boldsymbol{k}\,\epsilon _{ijk}\big \{&-\big [\varDelta _1(\boldsymbol{k})\mathrm{h}^{-+}_{jl}(\boldsymbol{k})+\varDelta _2(\boldsymbol{k})\mathrm{h}^{++}_{jl}(\boldsymbol{k})\big ]F^{++}_{lk}(\boldsymbol{k})\nonumber\\ &+\big [\varDelta _1(\boldsymbol{k})\mathrm{h}^{+-}_{jl}(\boldsymbol{k})+\varDelta _2(\boldsymbol{k})\mathrm{h}^{{-}{-}}_{jl}(\boldsymbol{k})\big ]F^{{-}{-}}_{lk}(\boldsymbol{k})\nonumber\\[3pt] &+\big [\varDelta _1(\boldsymbol{k})\mathrm{h}^{++}_{jl}(\boldsymbol{k})+\varDelta _2(\boldsymbol{k})\mathrm{h}^{-+}_{jl}(\boldsymbol{k})\big ]F^{+-}_{lk}(\boldsymbol{k})\nonumber\\[3pt] &-\big [\varDelta _1(\boldsymbol{k})\mathrm{h}^{{-}{-}}_{jl}(\boldsymbol{k})+\varDelta _2(\boldsymbol{k})\mathrm{h}^{+-}_{jl}(\boldsymbol{k})\big ]F^{-+}_{lk}(\boldsymbol{k})\big \}, \end{align}

where

(4.20) \begin{align} \varDelta _1(\boldsymbol{k}) & =D_1-D_2=\frac {\mathcal{V}(\varOmega ,\boldsymbol{K},\boldsymbol{k})}{\mathcal{V}^2(\varOmega ,\boldsymbol{K},\boldsymbol{k})+\nu _-^2\kappa ^4(\boldsymbol{K},\boldsymbol{k})},\nonumber\\[3pt] \varDelta _2(\boldsymbol{k})=D_3-D_4 & =-\frac {\mathrm{i}\nu _-\kappa ^2}{\mathcal{V}^2(\varOmega ,\boldsymbol{K},\boldsymbol{k})+\nu _-^2\kappa ^4(\boldsymbol{K},\boldsymbol{k})},\nonumber\\[3pt] \mathcal{V}(\varOmega ,\boldsymbol{K},\boldsymbol{k}) & \doteq \varOmega '+\mathrm{i}\nu _+[k^2+(\boldsymbol{k}+\boldsymbol{K})^2],\nonumber\\[3pt] \kappa ^2(\boldsymbol{K},\boldsymbol{k}) & \doteq (\boldsymbol{k}+\boldsymbol{K})^2-k^2. \end{align}

Finally, noting that

(4.21) \begin{align} \mathrm{h}^{\pm \pm }_{ij}(\boldsymbol{k})=M_{ijk}(\boldsymbol{k})\mathrm{z}^\mp _k,\qquad \mathrm{h}_{ij}^{\pm \mp }(\boldsymbol{k})=N_{ijk}(\boldsymbol{k})\mathrm{z}_k^\mp , \end{align}

where

(4.22) \begin{align} M_{ijk}=\left(\delta _{ij}-\frac {(k_i+K_i)K_j}{(\boldsymbol{k}+\boldsymbol{K})^2}\right)k_k, \qquad N_{ijk}=\delta _{ik}K_j-\frac {(k_i+K_i)K_j}{(\boldsymbol{k}+\boldsymbol{K})^2}k_k, \end{align}

we have

(4.23) \begin{align} G_{ij}^{(\boldsymbol{v})}(\varOmega ,\boldsymbol{K}) & =\frac {\epsilon _{imn}}{2}\int \mathrm{d}\boldsymbol{k}\big \{M_{mlj} \big[\varDelta _1\big(F_{ln}^{-+}-F_{ln}^{+-}\big)+\varDelta _2\big(F_{ln}^{++}-F_{ln}^{{-}{-}}\big)\big]\nonumber\\[4pt] & \quad +N_{mlj}\big[\varDelta _1\big(F_{ln}^{++}-F_{ln}^{{-}{-}}\big)+\varDelta _2\big(F_{ln}^{-+}-F_{ln}^{+-}\big)\big]\big \}, \end{align}

(4.24) \begin{align} G_{ij}^{(\boldsymbol{b})}(\varOmega ,{\boldsymbol{K}}) & =\frac {\epsilon _{imn}}{2}\int \mathrm{d}{\boldsymbol{k}}\big \{M_{mlj} \big[\varDelta _1\big(F_{ln}^{-+}+F_{ln}^{+-}\big)-\varDelta _2\big(F_{ln}^{++}+F_{ln}^{{-}{-}}\big)\big]\nonumber\\[4pt] & \quad +N_{mlj} \big[\varDelta _1\big(F_{ln}^{++}+F_{ln}^{{-}{-}}\big)-\varDelta _2\big(F_{ln}^{-+}+F_{ln}^{+-}\big)\big]\big \}. \end{align}

Equations (4.23) and (4.24) are the exact (within the QLA) forms of the linear non-local response kernel of the turbulent EMF to mean velocity and magnetic fields, respectively. We emphasise that, in deriving them, we have made no additional approximations, such as scale separation or any particular symmetries of the turbulence (e.g. isotropy). It is immediately evident from (4.23) that the contribution to the EMF from the mean flow vanishes if $\boldsymbol{F}^{++}=\boldsymbol{F}^{{-}{-}}$ and $\boldsymbol{F}^{+-}=\boldsymbol{F}^{-+}$ , that is, if $\widetilde {\boldsymbol{z}}^+$ and $\widetilde {\boldsymbol{z}}^-$ have the same two-point correlations. This condition is somewhat trivially satisfied by hydrodynamic turbulence (in which $\widetilde {\boldsymbol{b}}=0$ such that $\widetilde {\boldsymbol{z}}^+=\widetilde {\boldsymbol{z}}^-$ ) but can also be satisfied by more general MHD turbulence, as will be discussed shortly.

In the remainder of this section, we examine (4.23) and (4.24) for specific turbulent backgrounds. We also compare them with traditional mean-field theories and the numerically inferred form of the non-local response kernel $\boldsymbol{G}^{(\boldsymbol{b})}$ used in the literature.

4.2. Hydrodynamic turbulence

Consider homogeneous hydrodynamic turbulence, where $\widetilde {\boldsymbol{z}}^+=\widetilde {\boldsymbol{z}}^-$ and thus $\boldsymbol{F}\doteq \boldsymbol{F}^{\pm \pm }=\boldsymbol{F}^{\pm \mp }$ . Then (4.19) becomes simply

(4.25) \begin{equation} \begin{gathered} \mathfrak{E}_i=\epsilon _{ijk}\int \mathrm{d}\boldsymbol{k}\,\Delta (\boldsymbol{k})[(\boldsymbol{{b}}\boldsymbol{\cdot} \boldsymbol{k})\delta _{jl}-\mathrm{b}_j K_l]F_{lk}, \end{gathered} \end{equation}

where

(4.26) \begin{equation} \Delta (\boldsymbol{k})\doteq \varDelta _1(\boldsymbol{k})-\varDelta _2(\boldsymbol{k})=\frac {1}{\varOmega '+\mathrm{i}\nu k^2+\mathrm{i}\eta (\boldsymbol{k}+\boldsymbol{K})^2}. \end{equation}

This is the generic answer for hydrodynamic turbulence for general $\boldsymbol{F}(\boldsymbol{k})$ . Notably, the EMF is entirely independent of $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu$ in this case.

If we further assume that the turbulent flow is isotropic (3.10), we have

(4.27) \begin{equation} F_{lp}(\boldsymbol{k})=\left(\delta _{lp}-\frac {k_lk_p}{k^2}\right)\frac {E(k)}{4\pi k^2}-\mathrm{i}\epsilon _{lpq}\frac {k_q}{k^2}\frac {H(k)}{8\pi k^2}, \end{equation}

where $E(k)$ and $H(k)$ are the energy and helicity spectra, respectively,

(4.28) \begin{equation} \frac {1}{2}\langle v^2 \rangle =\int _0^\infty \mathrm{d} k \,E(k),\qquad \langle \boldsymbol{v}\boldsymbol{\cdot} (\boldsymbol{\nabla} \times \boldsymbol{v}) \rangle =\int _0^\infty \mathrm{d} k \,H(k). \end{equation}

We now assume sufficiently weak dissipation, such that $\tau _c\nu _\pm k^2\ll 1$ and $\tau _c\nu _\pm k K \ll 1$ over the range where $E(k)$ and $H(k)$ substantially contribute to the integrals over $\boldsymbol{k}$ . In this case, we can use the following approximation for $\varDelta$ :

(4.29) \begin{equation} \begin{gathered}\Delta (\boldsymbol{k})\approx \frac {1}{\varOmega '+\mathrm{i}\eta K^2}\Bigg (1-\frac {2\mathrm{i}(\nu _+k^2+\eta \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{K})}{\varOmega '+\mathrm{i}\eta K^2}\Bigg ). \end{gathered} \end{equation}

Note that we have assumed dissipation to be weak, so that the terms $\nu _+ k^2$ and $\eta \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{K}$ could be taken out of the denominator of $\varDelta$ . (The $\eta K^2$ terms are retained for reasons that will become evident shortly.) With (4.29), the integration over $\boldsymbol{k}$ in (4.25) can be performed without assuming any particular form of the spectra $E(k)$ and $H(k)$ , and one obtains the relatively familiar form

(4.30a) \begin{align} \boldsymbol{\mathfrak{E}}=\mathcal{A}(\varOmega ,\boldsymbol{K})\boldsymbol{{b}}-\mathcal{B}(\varOmega ,\boldsymbol{K})(\mathrm{i}\boldsymbol{K}\times \boldsymbol{{b}}),\\[-30pt]\nonumber \end{align}

(4.30b) \begin{align} \mathcal{A}(\varOmega ,\boldsymbol{K})=\frac {\alpha }{N(\varOmega ,\boldsymbol{K})}+\frac {2}{3}\frac {\nu _+\tau _c^2\langle \widetilde {\boldsymbol{w}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{w}} \rangle }{N^2(\varOmega ,\boldsymbol{K})},\\[-30pt]\nonumber \end{align}

(4.30c) \begin{align} \mathcal{B}(\varOmega ,\boldsymbol{K})=\frac {\beta }{N(\varOmega ,\boldsymbol{K})}-\frac {2}{3}\frac {\nu _+\tau _c^2\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{w}} \rangle }{N^2(\varOmega ,\boldsymbol{K})}, \end{align}

where

(4.31) \begin{equation} N(\varOmega ,\boldsymbol{K})\doteq 1-\mathrm{i}\tau _c\varOmega +\tau _c\eta K^2, \end{equation}

$\alpha$ and $\beta$ are the local transport coefficients defined in (4.4),

(4.32) \begin{align} \langle {\widetilde{\boldsymbol{w}}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times {\widetilde{\boldsymbol{w}}} \rangle =\int _0^\infty \mathrm{d} k\, k^2 H(k), \qquad \langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times {\widetilde{\boldsymbol{w}}} \rangle =\int _0^\infty \mathrm{d} k\, k^2 E(k), \end{align}

and the corresponding response kernel is given by

(4.33) \begin{equation} G_{ij}^{(\boldsymbol{b})}(\varOmega ,\boldsymbol{K})=\mathcal{A}(\varOmega ,\boldsymbol{K})\delta _{ij}+\mathrm{i}\mathcal{B}(\varOmega ,\boldsymbol{K})\epsilon _{ijk}K_k. \end{equation}

The first terms in the expressions for $\mathcal{A}$ and $\mathcal{B}$ are exactly the empirical non-local response kernel (4.3), with $\tau =\tau _c$ and

(4.34) \begin{equation} l^2=\tau _c\eta . \end{equation}

The second terms are viscous and resistive corrections. In particular, note that in the ideal limit ( $\nu _\pm \to 0$ ) and for short correlation times ( $\tau _c\varOmega \ll 1$ ), we recover the original local statement of the $\alpha$ -effect, that is, $\mathcal{A}\to \alpha$ and $\mathcal{B}\to \beta$ .

These results are consistent with reports of the non-locality parameter $\tau$ being of the order of the turnover time across multiple simulations with different values of the magnetic Reynolds number. In contrast, the inferred value of $l$ varies more widely in the literature (Rheinhardt & Brandenburg Reference Rheinhardt and Brandenburg2012; Brandenburg Reference Brandenburg2018), and (4.34) can be considered the first actual calculation of $l$ from first principles (modulo the fact that we rely on the QLA and introduce $\tau _c$ in (2.43) ad hoc, as usual).

4.3. Isotropic MHD turbulence

The non-local turbulent EMF for isotropic MHD turbulence (3.10) can be written as follows:

(4.35) \begin{equation} \boldsymbol{\mathfrak{E}}=\mathcal{A}(\varOmega ,\boldsymbol{K})\boldsymbol{{b}}-\mathcal{B}(\varOmega ,\boldsymbol{K})(\mathrm{i}\boldsymbol{K}\times \boldsymbol{{b}})+\mathcal{C}(\varOmega ,\boldsymbol{K})\boldsymbol{{v}}-\mathcal{D}(\varOmega ,\boldsymbol{K})(\mathrm{i}\boldsymbol{K}\times \boldsymbol{{v}}), \end{equation}

where

(4.36a) \begin{align} \mathcal{A}(\varOmega ,\boldsymbol{K})& =\mathrm{i}\int \mathrm{d} \boldsymbol{k}\,\frac {k_{\boldsymbol{{b}}}^2}{k^2}\left[\frac {\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{K}+K^2}{(\boldsymbol{k}+\boldsymbol{K})^2}(\varDelta _1+\varDelta _2)\left(\frac {H^C(k)}{8\pi k^2}- \frac {H^S(k)}{8\pi k^2}\right)\right.\nonumber\\[5pt] &\quad{} \left. +2\left(\varDelta _2\frac {H^S(k)}{8\pi k^2}-\varDelta _1\frac {H^C(k)}{8\pi k^2}\right)\right], \end{align}

(4.36b) \begin{align} \mathcal{B}(\varOmega ,\boldsymbol{K}) & =\mathrm{i}\int \mathrm{d} \boldsymbol{k}\left[\frac {k_{\boldsymbol{{b}}}^2}{k^2}\frac {k^2+\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{K}}{(\boldsymbol{k}+\boldsymbol{K})^2}(\varDelta _1+\varDelta _2)\left(\frac {E^C(k)}{4\pi k^2}- \frac {E^S(k)}{4\pi k^2}\right)\right.\nonumber\\ &\quad{} \left.+\frac {k^2-k_{\boldsymbol{K}}^2}{k^2}\left(\varDelta _1\frac {E^S(k)}{4\pi k^2}-\varDelta _2\frac {E^C(k)}{4\pi k^2}\right)\right], \end{align}

(4.36c) \begin{align} \mathcal{C}(\varOmega ,\boldsymbol{K})=\mathrm{i}\int \mathrm{d} \boldsymbol{k}\,\frac {k_{\boldsymbol{{v}}}^2}{k^2}\Bigg [\frac {\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{K}+K^2}{(\boldsymbol{k}+\boldsymbol{K})^2}(\varDelta _1+\varDelta _2)-2\varDelta _2\Bigg ]\frac {H^A(k)}{8\pi k^2}, \end{align}

(4.36d) \begin{align} \mathcal{D}(\varOmega ,\boldsymbol{K})=\mathrm{i}\int \mathrm{d} \boldsymbol{k}\Bigg [\frac {k_{\boldsymbol{{v}}}^2}{k^2}\frac {k^2+\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{K}}{(\boldsymbol{k}+\boldsymbol{K})^2}(\varDelta _1-\varDelta _2)+\frac {k_{\boldsymbol{K}}^2-k^2}{k^2}\varDelta _1\Bigg ]\frac {E^A(k)}{4\pi k^2}, \end{align}

$k_{\boldsymbol{\xi }}\doteq (\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{\xi })/\xi$ denotes the component of $\boldsymbol{k}$ along a given vector $\boldsymbol{\xi }$ and the functions $E^\chi (k)$ and $H^\chi (k)$ are defined in (3.11). In particular, for ideal MHD in the commonly assumed GO limit, the above coefficients acquire a more familiar form,

(4.37a) \begin{align} \mathcal{A}(\varOmega ,\boldsymbol{K})\approx -\frac {\mathrm{i}}{3\varOmega '}\langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times {\widetilde{\boldsymbol{v}}}-{\widetilde{\boldsymbol{b}}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times {\widetilde{\boldsymbol{b}}} \rangle ,\\[-30pt]\nonumber \end{align}

(4.37b) \begin{align} \mathcal{B}(\varOmega ,\boldsymbol{K})\approx \frac {\mathrm{i}}{3\varOmega '}\langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} {\widetilde{\boldsymbol{v}}} \rangle ,\\[-30pt]\nonumber \end{align}

(4.37c) \begin{align} \mathcal{C}(\varOmega ,\boldsymbol{K})\approx 0,\\[-30pt]\nonumber \end{align}

(4.37d) \begin{align} \mathcal{D}(\varOmega ,\boldsymbol{K})=\frac {\mathrm{i}}{3\varOmega '}\langle {\widetilde{\boldsymbol{v}}}\boldsymbol{\cdot} {\widetilde{\boldsymbol{b}}} \rangle . \end{align}

Let us now discuss each of these contributions to the non-local EMF and their relationship to the results of local mean-field theories. As established in the previous discussion for hydrodynamic turbulence (§ 4.2), the transport coefficients $\mathcal{A}$ and $\mathcal{B}$ correspond to the usual $\alpha$ -effect driven by kinetic helicity and turbulent diffusivity, respectively. Note that for MHD turbulence, the MFWK framework captures the cancellation of kinetic helicity by current helicity, i.e. the magnetic $\alpha$ -effect, also known as the Pouquet effect (Pouquet, Frisch & Léorat Reference Pouquet, Frisch and Léorat1976). The coefficient $\mathcal{A}$ is therefore qualitatively in agreement with previous theories, up to non-local corrections (the difference between (4.36a ) and (4.37a )). Notably, it is commonly stated that the magnetic $\alpha$ -effect is captured only beyond the QLA, as done in the eddy-damped quasinormal Markovian (Orszag Reference Orszag1970) or MTA (Blackman & Field Reference Blackman and Field2002) closures. However, as can be seen from our calculation, this is not the case. Although we included an MTA-like ad hoc damping term in (2.43), this modification to the QLA is inessential for capturing the magnetic $\alpha$ -effect in (4.37a), and taking $\tau _c\to \infty$ results only in replacing $\varOmega '$ with $\varOmega$ . The coefficient $\mathcal{B}$ also agrees with the vast majority of previous mean-field theories in that it contains no contribution to the turbulent diffusivity from magnetic fluctuations (Vainshtein & Kichatinov Reference Vainshtein and Kichatinov1983; Rädler et al. Reference Rädler, Kleeorin and Rogachevskii2002; Squire & Bhattacharjee Reference Squire and Bhattacharjee2015), although such a contribution was found with the two-scale direct interaction approximation in Yoshizawa (Reference Yoshizawa1990).

The contribution to the turbulent EMF proportional to the mean-flow vorticity, as determined by $\mathcal{D}$ , is known as the Yoshizawa effect (Yoshizawa Reference Yoshizawa1990; Yokoi Reference Yokoi2013) and is related to the cross-helicity $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle$ . As with the $\alpha$ -effect term, the coefficient $\mathcal{D}$ agrees exactly with previous mean-field theories in the GO limit ( $lK\ll 1$ ) and for short correlation times ( $\tau _c\varOmega \ll 1$ ).

In contrast, the effect of $\mathcal{C}$ predicted by the MFWK framework is qualitatively new. The term in the EMF that is proportional to the mean flow has received limited attention in the literature. Typically, Galilean invariance is invoked to argue that any term proportional to the mean flow should vanish. This is, of course, true if the flow is homogeneous, and, indeed, the contribution from $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu$ vanishes in the limit of negligible $K$ ; i.e. $\mathcal{C} \to 0$ as $K\to 0$ . However, the contribution of $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu$ at non-negligible $K$ is non-zero. To the best of our knowledge, the only other work where this subject is discussed is Rädler & Brandenburg (Reference Rädler and Brandenburg2010). Using our notation, the result of Rädler & Brandenburg (Reference Rädler and Brandenburg2010) can be expressed as follows:

(4.38a) \begin{align} \mathcal{E}=\ldots +C\boldsymbol{\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu},\\[-30pt]\nonumber \end{align}

(4.38b) \begin{align} C=\frac {1}{3}\int \mathrm{d} \tau \int \mathrm{d} \boldsymbol{\xi } \,[\mathcal{G}^{(\nu )}(\tau ,\xi )-\mathcal{G}^{(\eta )}(\tau ,\xi )]\langle \widetilde {\boldsymbol{v}}(t,\boldsymbol{x})\boldsymbol{\cdot} \widetilde {\boldsymbol{j}}(t-\tau ,\boldsymbol{x}-\boldsymbol{\xi }) \rangle ,\\[-30pt]\nonumber \end{align}

(4.38c) \begin{align} \mathcal{G}^{(\gamma )}(\tau ,\xi )\doteq \begin{cases} (4\pi \gamma \tau )^{-3/2}\exp (-\xi ^2/4\gamma \tau ),& \tau \gt 0,\\ 0,&\tau \leqslant 0. \end{cases}\\[-20pt]\nonumber \end{align}

The ellipsis indicates the additional contributions to the EMF that are not relevant to the present discussion.

Although (4.38) and (4.36c ) are in agreement in that the basic effect is driven by flow–current alignment $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ , they differ in several significant ways beyond what can be attributed to non-local corrections. Firstly, (4.38) predicts a finite contribution to the non-local EMF even for constant mean flows. In Rädler & Brandenburg (Reference Rädler and Brandenburg2010), it is argued that this does not violate Galilean invariance because the frame of reference is fixed by the assumption of isotropic turbulence, i.e. that shifting from a frame with a constant mean flow and isotropic turbulence to the frame where the mean flow is zero would render the turbulence anisotropic. However, because the constant shift to the total velocity $\boldsymbol{v}=\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu+\widetilde {\boldsymbol{v}}$ that would be introduced by a change in reference frame would be absorbed by the mean flow $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{v}\mkern -1.5mu}\mkern 1.5mu$ and not contribute to the fluctuating fields $\widetilde {\boldsymbol{v}}$ that comprise the turbulence. In contrast, (4.36c ) has the correct limiting behaviour for constant mean flows, as the transport coefficient $\mathcal{C}$ vanishes as $K\to 0$ .

Although the violation of Galilean invariance alone indicates that (4.38) is unphysical, let us also mention other ways in which it disagrees with our theory. Equation (4.38) predicts that the mean-flow contribution to the EMF vanishes when $\nu =\eta$ , and, in particular, in the ideal limit $\nu _\pm \to 0$ . In contrast, (4.36c ) at $\nu =\eta$ (i.e. $\nu _-=0$ ) predicts non-zero $C$ ,

(4.39) \begin{equation} \mathcal{C}(\varOmega ,\boldsymbol{K})=\mathrm{i}\int \mathrm{d} \boldsymbol{k}\,\frac {k_{\boldsymbol{{v}}}^2}{k^2}\frac {\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{K}+K^2}{(\boldsymbol{k}+\boldsymbol{K})^2[\varOmega '+\mathrm{i}\nu _+(k^2+(\boldsymbol{k}+\boldsymbol{K})^2]}\frac {H^A(k)}{8\pi k^2}, \end{equation}

or, in the ideal limit,

(4.40) \begin{equation} \mathcal{C}(\varOmega ,\boldsymbol{K})=\frac {\mathrm{i}}{\varOmega '}\int \mathrm{d} \boldsymbol{k}\,\frac {k_{\boldsymbol{{v}}}^2}{k^2}\frac {\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{K}+K^2}{(\boldsymbol{k}+\boldsymbol{K})^2}\frac {H^A(k)}{8\pi k^2}, \end{equation}

both of which are generally non-zero. These discrepancies may be related to the treatment of the mean flow as an externally prescribed field, as opposed to a self-consistent field.

Note that flow–current correlations have also been found to produce additional contributions to the $\alpha$ -effect in the case of non-stationary and (or) inhomogeneous turbulence, or when the correlation time scales of the magnetic and velocity fluctuations are not equal (Mizerski, Yokoi & Brandenburg Reference Mizerski, Yokoi and Brandenburg2023; Yokoi Reference Yokoi2023; Hughes, Mason & Proctor Reference Hughes, Mason and Proctor2024). However, that effect is different from the one we discuss here. In our case, the flow–current effect manifests as a contribution to the EMF proportional to the mean flow rather than the mean magnetic field, is found for stationary and homogeneous turbulent backgrounds, and does not depend on the correlation times of the turbulent fields.

In the following section, we will see that this little-known contribution to the EMF couples the mean magnetic fields and flows in such a way that enables a previously unknown $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -driven dynamo effect.

5.1. Derivation

The dispersion relation of modulational modes follows directly from the linearised MFWK equations written in terms of the polarisations of the perturbed quantities (3.15) and (3.16). Let us first rewrite (3.15) in a form that aids the eventual dispersion matrix formulation,

(5.1) \begin{equation} \begin{pmatrix} (\mathrm{i}\varOmega - \nu _+K^2) \boldsymbol{1}_3& \quad -\nu _- K^2 \boldsymbol{1}_3\\[4pt] -\nu _- K^2 \boldsymbol{1}_3&\quad (\mathrm{i}\varOmega - \nu _+K^2) \boldsymbol{1}_3 \end{pmatrix}\boldsymbol{{z}}=\mathrm{i}\begin{pmatrix} \boldsymbol{\mathcal{S}}^+ \\[4pt] \boldsymbol{\mathcal{S}}^- \end{pmatrix}, \end{equation}

where $\boldsymbol{{z}}\doteq (\boldsymbol{{z}}^+,\boldsymbol{{z}}^-)^\intercal$ and

(5.2) \begin{align} \mathcal{S}^\pm _i&=\int \mathrm{d}\boldsymbol{k}\,K_n\left(\mathrm{f}_{in}^{\pm \mp }-\frac {K_iK_m}{K^2}\mathrm{f}_{mn}^{\pm \mp }\right)\nonumber\\[4pt] &=K_n\left(\delta _{im}-\frac {K_iK_m}{K^2}\right)\int \mathrm{d}\boldsymbol{k}\,\mathrm{f}_{mn}^{\pm \mp }. \end{align}

To find the dispersion matrix, we must express $\boldsymbol{\mathcal{S}}$ in terms of the mean fields $\boldsymbol{{z}}^\pm$ . To do this, the integral of the perturbed Wigner matrix (4.14) can be simplified by shifting and flipping integration variables as done in § 4.1,

(5.3) \begin{align} \int \mathrm{d}\boldsymbol{k}\,\mathrm{f}_{mn}^{\pm \mp } & =\bigg [\int \mathrm{d}\boldsymbol{k}\, M_{mpj}\big(D_2F_{pn}^{\mp \pm }+D_3F_{pn}^{\mp \mp }\big)+N_{mpj}\big(D_2F_{pn}^{\pm \pm }+D_3F_{pn}^{\pm \mp }\big)\nonumber\\& \quad +M_{npj}\big(D_1F_{pm}^{\mp \pm }+D_4F_{pm}^{\mp \mp }\big)+N_{npj}\big(D_1F_{pm}^{\pm \pm }+D_4F_{pm}^{\pm \mp }\big) \bigg ]\mathrm{z}_j^\pm \nonumber\\ & \quad +\bigg [\int \mathrm{d}\boldsymbol{k}\, M_{mpj}\big(D_1F_{pn}^{\pm \mp }+D_4F_{pn}^{\pm \pm }\big)+N_{mpj}\big(D_1F_{pn}^{\mp \mp }+D_4F_{pn}^{\mp \pm }\big)\nonumber\\ & \quad +M_{npj}\big(D_2F_{pm}^{\pm \mp }+D_3F_{pm}^{\pm \pm }\big)+N_{npj}\big(D_2F_{pm}^{\mp \mp }+D_3F_{pm}^{\mp \pm }\big)\bigg ]\mathrm{z}_j^\mp , \end{align}

where the matrices $M_{ijk}$ and $N_{ijk}$ are defined in (4.22) and the functions $D_n$ are defined as follows:

(5.4) \begin{align} D_1 & =\frac {\mathcal{W}\big(\mathcal{W}^2+\nu _-^2\kappa _1^4\big)}{\big(\mathcal{W}^2+\nu _-^2\kappa _1^4\big)^2-4\nu _-^4\kappa _2^8},\nonumber\\[8pt]D_2 & =\frac {-2\nu _-^2\kappa _2^4\mathcal{W}}{\big(\mathcal{W}^2+\nu _-^2\kappa _1^4\big)^2-4\nu _-^4\kappa _2^8},\nonumber\\[8pt] D_3 & =\frac {-\mathrm{i}\nu _-(\boldsymbol{k}+\boldsymbol{K})^2\big(\mathcal{W}^2+\nu _-^2\kappa _1^4\big)+2\mathrm{i}\nu _-^3k^2\kappa _2^4}{\big(\mathcal{W}^2+\nu _-^2\kappa _1^4\big)^2-4\nu _-^4\kappa _2^8},\nonumber\\[8pt] D_4 & =\frac {-\mathrm{i}\nu _-k^2\big(\mathcal{W}^2+\nu _-^2\kappa _1^4\big)+2\mathrm{i}\nu _-^3(\boldsymbol{k}+\boldsymbol{K})^2\kappa _2^4}{\big(\mathcal{W}^2+\nu _-^2\kappa _1^4\big)^2-4\nu _-^4\kappa _2^8}, \end{align}

and $\mathcal{W}\doteq \varOmega '+\mathrm{i}\nu _+\kappa _0^2$ , $\kappa _0^2\doteq k^2+(\boldsymbol{k}+\boldsymbol{K})^2$ , $\kappa _1^4\doteq k^4+(\boldsymbol{k}+\boldsymbol{K})^4$ and $\kappa _2^4\doteq k^2(\boldsymbol{k}+\boldsymbol{K})^2$ .

After extensive but straightforward calculation, one can rewrite this as follows:

(5.5) \begin{equation} \begin{gathered} \mathrm{i}\boldsymbol{S}^\pm =\boldsymbol{P}^\pm \boldsymbol{{z}}^\pm +\boldsymbol{Q}^\pm \boldsymbol{{z}}^\mp , \end{gathered} \end{equation}

where the matrices $\boldsymbol{P}^\pm$ and $\boldsymbol{Q}^\pm$ are given by

(5.6a) \begin{align} P_{ij}^\pm & =\mathrm{i} K_n\int \mathrm{d}\boldsymbol{k}\left[\frac {K_p}{K^2}F_{pn}^{\mp \mp }\mathcal{M}_{ij}(D_{12},D_1,D_{34},D_{13})-F^{\mp \mp }_{in}D_3k_j\right.\nonumber\\[4pt] &\quad +\frac {K_p}{K^2}F_{pn}^{\mp \pm }\mathcal{M}_{ij}(D_{34},D_4,D_{12},D_{24})-F^{\mp \pm }_{in}D_2k_j\nonumber\\[4pt] &\quad - F^{\mp \mp }_{ni}\mathcal{N}_j(D_2,D_4,D_{24})-F^{\mp \pm }_{ni}\mathcal{N}_j(D_3,D_1,D_{13})\bigg], \end{align}

(5.6b) \begin{align} Q^\pm _{ij} & =\mathrm{i} K_n\int \mathrm{d}\boldsymbol{k}\left[\frac {K_p}{K^2}F_{pn}^{\pm \pm }\mathcal{M}_{ij}(D_{12},D_2,D_{34},D_{24})-F^{\pm \pm }_{in}D_4k_j\right.\nonumber\\[6pt] & \quad +\frac {K_p}{K^2}F_{pn}^{\pm \mp }\mathcal{M}_{ij}(D_{34},D_3,D_{12},D_{13})-F^{\pm \mp }_{in}D_1k_j\nonumber\\[6pt] & \quad - F^{\pm \pm }_{ni}\mathcal{N}_j(D_1,D_3,D_{13})-F^{\pm \mp }_{ni}\mathcal{N}_j(D_4,D_2,D_{24})\bigg], \end{align}

where

(5.7) \begin{align} \mathcal{M}_{ij}(f_1,f_2,f_3,f_4)&\doteq f_1 K_iK_j-f_2K^2\delta _{ij}+\left(f_3-D\frac {\boldsymbol{K}\boldsymbol{\cdot} (\boldsymbol{k}+\boldsymbol{K})}{(\boldsymbol{k}+\boldsymbol{K})^2}\right)K_ik_j\nonumber\\&\quad{}+f_4\frac {K^2}{(\boldsymbol{k}+\boldsymbol{K})^2}(k_i+K_i)K_j, \end{align}

(5.8) \begin{align} \mathcal{N}_{j}(f_1,f_2,f_3)\doteq f_1K_j+\left(f_2-f_3\frac {\boldsymbol{K}\boldsymbol{\cdot} (\boldsymbol{k}+\boldsymbol{K})}{(\boldsymbol{k}+\boldsymbol{K})^2}\right)k_j, \end{align}

and we use the shorthand $D\doteq D_1+D_2+D_3+D_4$ and $D_{mn}\doteq D_m+D_n$ .

5.2. Dispersion relation

The general dispersion relation and polarisations for the modulational modes of MHD turbulence are given by

(5.9) \begin{align} \det \boldsymbol{\Pi }=0, \qquad \boldsymbol{\Pi }\boldsymbol{{z}}=0, \end{align}

where $\boldsymbol{{z}}\doteq (\boldsymbol{{z}}^+,\boldsymbol{{z}}^-)^\intercal$ , $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\pm =\boldsymbol{{z}}^\pm \exp {(-\mathrm{i}\varOmega t+\mathrm{i}\boldsymbol{K}\boldsymbol{\cdot} \boldsymbol{x})}$ and the dispersion matrix $\boldsymbol{\Pi }$ is given by

(5.10) \begin{equation} \boldsymbol{\Pi }=\begin{pmatrix} \boldsymbol{\Pi }^{++}& \quad \boldsymbol{\Pi }^{+-}\\[4pt] \boldsymbol{\Pi }^{-+} & \quad \boldsymbol{\Pi }^{{-}{-}} \end{pmatrix}, \end{equation}

where

(5.11a) \begin{align} \boldsymbol{\Pi }^{\pm \pm } & = (\mathrm{i}\varOmega -\nu _+K^2)\boldsymbol{1}_3-\boldsymbol{P}^\pm , \end{align}

(5.11b) \begin{align} \boldsymbol{\Pi }^{\pm \mp } & =-\nu _-K^2\boldsymbol{1}_3-\boldsymbol{Q}^\pm , \end{align}

and $\boldsymbol{P}^\pm$ and $\boldsymbol{Q}^\pm$ are defined in (5.6a ) and (5.6b ), respectively. Equation (5.9) is the general dispersion relation of modulational modes of generic incompressible resistive MHD turbulence within the MTA closure (which reduces to the QLA in the limit $\tau _c\to \infty$ ). We emphasise that no particular properties of the turbulence, e.g. isotropy or scale-separation, have been assumed to derive these equations.

5.3. Examples

Although (5.9) is powerful in its generality, the intricate dependence of the integrands in (5.6a ) and (5.6b ) on $\varOmega$ and $\boldsymbol{K}$ means that obtaining the solutions to (5.9) is challenging in its own right. In the remainder of this section, we make a series of simplifying assumptions to obtain explicit solutions of (5.9) that allow us to compare the predictions of MFWK with those of traditional mean-field theories.

5.3.1. Isotropic turbulence

For isotropic turbulent backgrounds (3.10), we can, without loss of generality, let the modulational wavevector $\boldsymbol{K}$ lie along $\boldsymbol{e}_{z}$ ( $\boldsymbol{K}=K\boldsymbol{e}_{z}$ ). By incompressibility, we then have $\mathrm{z}_z^\pm =0$ , so we may consider the following reduced system:

(5.12) \begin{equation} \mkern 1.5mu\underline {\mkern -1.5mu \boldsymbol{\Pi }\mkern -1.5mu}\mkern 1.5mu\mkern 1.5mu\underline {\mkern -1.5mu \boldsymbol{{z}}\mkern -1.5mu}\mkern 1.5mu=0,\qquad \det \mkern 1.5mu\underline {\mkern -1.5mu \boldsymbol{\Pi }\mkern -1.5mu}\mkern 1.5mu=0, \end{equation}

where $\mkern 1.5mu\underline {\mkern -1.5mu \boldsymbol{{z}}\mkern -1.5mu}\mkern 1.5mu\doteq (\mathrm{z}_x^+,\mathrm{z}_y^+,\mathrm{z}_x^-,\mathrm{z}_y^-$ ) and the dispersion matrix $\mkern 1.5mu\underline {\mkern -1.5mu \boldsymbol{\Pi }\mkern -1.5mu}\mkern 1.5mu$ can be written as

(5.13) \begin{align} \mkern 1.5mu\underline {\mkern -1.5mu \boldsymbol{\Pi }\mkern -1.5mu}\mkern 1.5mu\doteq \begin{pmatrix} M_S+M_A&\quad m_S+m_A&\quad N_S+N_A&\quad n_S+n_A\\[2pt] -m_S-m_A&\quad M_S+M_A&\quad -n_S-n_A&\quad N_S+N_A\\[2pt] N_S-N_A&\quad n_S-n_A&\quad M_S-M_A&\quad m_S-m_A\\[2pt] -n_S+n_A&\quad N_S-N_A&\quad -m_S+m_A&\quad M_S-M_A\\ \end{pmatrix}. \end{align}

Here,

(5.14a) \begin{align} M_S \doteq \mathrm{i}\varOmega -\nu _+K^2+\mathrm{i} K\int \mathrm{d}\boldsymbol{k}\frac {k_x^2}{k^2}& \left[\frac {E_S}{4\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{M}\mkern -1.5mu}\mkern 1.5mu(D_1,D_{13},D_{34},D_{24})\right.\nonumber\\& \quad \left.+\frac {E_C}{4\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{M}\mkern -1.5mu}\mkern 1.5mu(D_4,D_{24},D_{12},D_{13})\right ],\\[-25pt]\nonumber \end{align}

(5.14b) \begin{align} N_S \doteq -\nu _-K^2+\mathrm{i} K\int \mathrm{d}\boldsymbol{k}\frac {k_x^2}{k^2}& \left[\frac {E_S}{4\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{M}\mkern -1.5mu}\mkern 1.5mu(D_2,D_{24},D_{34},D_{13})\right.\nonumber\\& \quad \left. +\frac {E_C}{4\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{M}\mkern -1.5mu}\mkern 1.5mu(D_3,D_{13},D_{12},D_{24})\right],\\[-25pt]\nonumber \end{align}

(5.14c) \begin{align} M_A\doteq -\mathrm{i} K\int \mathrm{d}\boldsymbol{k}\frac {E_A}{4\pi k^2}\frac {k_x^2}{k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{M}\mkern -1.5mu}\mkern 1.5mu(D_1,D_{13},D_{34},D_{24}),\\[-30pt]\nonumber \end{align}

(5.14d) \begin{align} N_A\doteq \mathrm{i} K\int \mathrm{d}\boldsymbol{k}\frac {E_A}{4\pi k^2}\frac {k_x^2}{k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{M}\mkern -1.5mu}\mkern 1.5mu(D_2,D_{24},D_{34},D_{13}), \\[-30pt]\nonumber\end{align}

(5.14e) \begin{align} m_S\doteq K\int \mathrm{d} \boldsymbol{k}\frac {k_x^2}{k^2}\left[\frac {H_S}{8\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{N}\mkern -1.5mu}\mkern 1.5mu(D_3,D_4,D_{24})+\frac {H_C}{8\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{N}\mkern -1.5mu}\mkern 1.5mu(D_2,D_1,D_{13})\right],\\[-30pt]\nonumber \end{align}

(5.14f) \begin{align} n_S\doteq K\int \mathrm{d} \boldsymbol{k}\frac {k_x^2}{k^2}\left[\frac {H_S}{8\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{N}\mkern -1.5mu}\mkern 1.5mu(D_4,D_3,D_{13})+\frac {H_C}{8\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{N}\mkern -1.5mu}\mkern 1.5mu(D_1,D_2,D_{24})\right],\\[-30pt]\nonumber \end{align}

(5.14g) \begin{align} m_A\doteq -K\int \mathrm{d} \boldsymbol{k}\frac {k_x^2}{k^2}\frac {H_A}{8\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{N}\mkern -1.5mu}\mkern 1.5mu(D_3,D_4,D_{24}),\\[-30pt]\nonumber \end{align}

(5.14h) \begin{align} n_A \doteq K\int \mathrm{d} \boldsymbol{k}\frac {k_x^2}{k^2}\frac {H_A}{8\pi k^2}\mkern 1.5mu\underline {\mkern -1.5mu \mathcal{N}\mkern -1.5mu}\mkern 1.5mu(D_4,D_3,D_{13}), \\[-12pt]\nonumber\end{align}

where

(5.15) \begin{align} \mkern 1.5mu\underline {\mkern -1.5mu \mathcal{M}\mkern -1.5mu}\mkern 1.5mu(f_1,f_2,f_3,f_4)\doteq 2K\left(-f_1+f_2\frac {k_x^2}{(\boldsymbol{k}+\boldsymbol{K})^2}\right)+k_z\left(f_3-f_4\frac {K(k_z+K)}{(\boldsymbol{k}+\boldsymbol{K})^2}\right), \end{align}

(5.16) \begin{align} \mkern 1.5mu\underline {\mkern -1.5mu \mathcal{N}\mkern -1.5mu}\mkern 1.5mu(f_1,f_2,f_3)\doteq f_1-\left(f_2-f_3\frac {K(k_z+K)}{(\boldsymbol{k}+\boldsymbol{K})^2}\right).\\[-6pt]\nonumber \end{align}

The dispersion relation can then be factored as follows:

(5.17) \begin{align} \det \mkern 1.5mu\underline {\mkern -1.5mu \boldsymbol{\Pi }\mkern -1.5mu}\mkern 1.5mu & =(a+\mathrm{i} b)(a-\mathrm{i} b),\nonumber\\ a& \doteq M_S^2-M_A^2-N_S^2+N_A^2-m_S^2+m_A^2+n_S^2-n_A^2,\nonumber\\ b& \doteq 2(M_sm_s-N_sn_s-M_Am_A+N_An_A). \end{align}

In the remainder of the section, we focus on solving (5.17) for isotropic backgrounds as an example.

5.3.2. Ideal limit

The dispersion matrix (5.13) takes a particularly simple form in the ideal limit $\nu _\pm \to 0$ . In this limit, one has $D_1\to 1/\varOmega '$ and all of the $D_{n\neq 1}$ are zero, so the modulational frequency $\varOmega$ can be pulled outside of the integrals in (5.14a). Then, we have

(5.18a) \begin{align} M_S =\mathrm{i}\varOmega +\frac {\mathrm{i} K}{\varOmega '}\int \mathrm{d}\boldsymbol{k}\left\{\frac {E_S}{4\pi k^2}\frac {k_x^2}{k^2}\right. & \left[2K\left(-1+\frac {k_x^2}{(\boldsymbol{k}+\boldsymbol{K})^2}\right)\right]\nonumber\\[5pt] & \left.-\frac {E_C}{4\pi k^2}\frac {k_x^2}{k^2}\left[k_z\Big (\frac {K (k_z+K)}{(\boldsymbol{k}+\boldsymbol{K})^2}\Big )\right]\right\}, \end{align}

(5.18b) \begin{align} N_S=\frac {\mathrm{i} K}{\varOmega '}\int \mathrm{d}\boldsymbol{k}\left\{\frac {E_S}{4\pi k^2}\frac {k_x^2}{k^2}\right. & \left[k_z\left(-\frac {K(k_z+K)}{(\boldsymbol{k}+\boldsymbol{K})^2}\right)\right]\nonumber\\[5pt]& \left.+\frac {E_C}{4\pi k^2}\frac {k_x^2}{k^2}\left[2K\left(\frac {k_x^2}{(\boldsymbol{k}+\boldsymbol{K})^2}\right)\right]\right\},\end{align}

(5.18c) \begin{align} M_A=-\frac {\mathrm{i} K}{\varOmega '}\int \mathrm{d}\boldsymbol{k}\frac {E_A}{4\pi k^2}\frac {k_x^2}{k^2}\left[2K\left(-1+\frac {k_x^2}{(\boldsymbol{k}+\boldsymbol{K})^2}\right)\right], \end{align}

(5.18d) \begin{align} N_A=\frac {\mathrm{i} K}{\varOmega '}\int \mathrm{d}\boldsymbol{k}\frac {E_A}{4\pi k^2}\frac {k_x^2}{k^2}\left[k_z\left(-\frac {K(k_z+K)}{(\boldsymbol{k}+\boldsymbol{K})^2}\right)\right], \end{align}

(5.18e) \begin{align} m_S= \frac {K}{\varOmega '}\int \mathrm{d} \boldsymbol{k}\frac {k_x^2}{k^2}\left\{\frac {H_C}{8\pi k^2}\left[-1+\frac {K(k_z+K)}{(\boldsymbol{k}+\boldsymbol{K})^2}\right]\right \},\\[-25pt]\nonumber \end{align}

(5.18f) \begin{align} n_S=\frac {K}{\varOmega '}\int \mathrm{d} \boldsymbol{k}\frac {k_x^2}{k^2}\left\{\frac {H_S}{8\pi k^2}\left[\frac {K(k_z+K)}{(\boldsymbol{k}+\boldsymbol{K})^2}\right]+\frac {H_C}{8\pi k^2}\right\},\\[-30pt]\nonumber \end{align}

(5.18g) \begin{align} m_A=0,\\[-30pt]\nonumber \end{align}

(5.18h) \begin{align} n_A =\frac {K}{\varOmega '}\int \mathrm{d} \boldsymbol{k}\frac {k_x^2}{k^2}\frac {H_A}{8\pi k^2}\left[\frac {K(k_z+K)}{(\boldsymbol{k}+\boldsymbol{K})^2}\right]. \end{align}

Hence, the dispersion relation (5.17) becomes the product of two quadratic polynomials in $\varOmega \varOmega '$ . It can then be easily solved, yielding eight total solutions $\varOmega (\boldsymbol{K})$ . The solutions obtained in this manner are what will be presented throughout this section. Also, unless explicitly stated otherwise, we will assume the limit ${St}\to \infty$ , i.e. $\varOmega '\approx \varOmega$ , for clarity.

Although the ${Rm},\,{St}\to \infty$ limit is formally outside of the QLA applicability domain, we believe this to be a worthwhile approach nonetheless, for the following reasons. Firstly, it is often the case that equations derived in a certain asymptotic limit qualitatively hold outside of that same limit. In particular, the accuracy of the QLA for the modulational dynamics of MHD is a rather complicated issue, and the conventional validity criterion min $\{{St},{Rm}\}\ll 1$ can be understood as a sufficient, but not necessary condition (Jin & Dodin Reference Jin and Dodin2025). Secondly, the simplified picture that emerges in the QLA can serve as a stepping stone towards a more complex future theory that will contain quasilinear dynamics as a limiting case.

In addition, we will also supplement qualitatively novel predictions of MFWK with a study of their dependence on ${St}$ within the MTA-like closure used in (2.43), i.e. retaining the $\tau _c^{-1}$ term in $\varOmega '=\varOmega +\mathrm{i}\tau _c^{-1}$ . This is, of course, a highly simplified model of the effect of the higher-order correlations neglected in the QLA, the results of which must therefore be taken with a grain of salt. Nevertheless, establishing the dependence on ${St}$ within this limited approach can serve as a tentative predictor of the robustness of the collective effects found in the ideal limit ( ${St}\gg 1$ ) to eddy–eddy interactions.

5.4. Parametrization of isotropic helical background

As discussed in § 3.1, the Wigner matrix of a generic isotropic turbulent background is fully determined by the six spectra that characterise the total energy $E_S(k)$ , residual energy $E_C(k)$ , cross-helicity $E_A(k)$ , total (kinetic $+$ current) helicity $H_S(k)$ , residual (kinetic $-$ current) helicity $H_C(k)$ and flow–current alignment $H_A(k)$ . Solving (5.17) for different background spectra will therefore allow us to determine how the modulational dynamics of the system depend on the properties of the turbulent equilibrium.

For simplicity, we assume Gaussian test spectra:

(5.19a) \begin{align} \frac {E_S}{4\pi k^2}=\frac {\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{v}} \rangle +\langle \widetilde {\boldsymbol{b}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle }{(\sqrt {2\pi })^3}\frac {l^3}{2}\exp \left(-\frac {l^2 k^2}{2}\right), \end{align}

(5.19b) \begin{align} \frac {E_C}{4\pi k^2}=\frac {\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{v}} \rangle -\langle \widetilde {\boldsymbol{b}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle }{(\sqrt {2\pi })^3}\frac {l^3}{2}\exp \left(-\frac {l^2 k^2}{2}\right), \end{align}

(5.19c) \begin{align} \frac {E_A}{4\pi k^2}=\frac {2\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle }{(\sqrt {2\pi })^3}\frac {l^3}{2}\exp \left(-\frac {l^2 k^2}{2}\right), \end{align}

(5.19d) \begin{align} \frac {H_S}{8\pi k^2}=\frac {\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{v}} \rangle +\langle \widetilde {\boldsymbol{b}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{b}} \rangle }{(\sqrt {2\pi })^3}\frac {l^3}{2}\exp \left(-\frac {l^2 k^2}{2}\right), \end{align}

(5.19e) \begin{align} \frac {H_C}{8\pi k^2}=\frac {\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{v}} \rangle -\langle \widetilde {\boldsymbol{b}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{b}} \rangle }{(\sqrt {2\pi })^3}\frac {l^3}{2}\exp \left(-\frac {l^2 k^2}{2}\right), \end{align}

(5.19f) \begin{align} \frac {H_A}{8\pi k^2}=\frac {\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{b}} \rangle +\langle \widetilde {\boldsymbol{b}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{v}} \rangle }{(\sqrt {2\pi })^3}\frac {l^3}{2}\exp \left(-\frac {l^2 k^2}{2}\right). \end{align}

Such spectra correspond to two-point correlations of the form

(5.20) \begin{equation} \langle \widetilde {\boldsymbol{v}}(\boldsymbol{x}+\boldsymbol{r}/2)\boldsymbol{\cdot} \widetilde {\boldsymbol{v}}(\boldsymbol{x}-\boldsymbol{r}/2) \rangle =\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{v}} \rangle \exp \left(-\frac {r^2}{2l^2}\right) \end{equation}

and are a convenient and popular choice in the mean-field dynamo literature (Rädler & Stepanov Reference Rädler and Stepanov2006; Squire & Bhattacharjee Reference Squire and Bhattacharjee2015). Note that we have assumed the same Gaussian form and same characteristic correlation length $l$ for all spectra. Although this is surely a simplification, it enables a minimal parametrization of the properties of the turbulent background in terms of the single-point statistics. Although six spectra are needed to fully define the isotropic Wigner matrix, we need only five parameters to capture physical properties of the turbulent equilibria that have the potential to qualitatively impact the modulational dynamics, since

(5.21) \begin{equation} \tau \doteq l/v_{\mathrm{rms}} \end{equation}

simply serves to set the characteristic time scale of the problem.Footnote 8 In the following section, we will normalise all modulational frequencies, $\varOmega$ to $\tau ^{-1}$ and all modulational wavevectors $\boldsymbol{K}$ to $l^{-1}$ . Then, our turbulent equilibria (after the Gaussian spectral ansatz) can be fully characterised by the five dimensionless parameters,

(5.22a) \begin{align} m \doteq b_{\mathrm{rms}}^2/v_{\mathrm{rms}}^2,\\[-30pt]\nonumber \end{align}

(5.22b) \begin{align} a\doteq \langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle /v_{\mathrm{rms}}b_{\mathrm{rms}}, \\[-30pt]\nonumber\end{align}

(5.22c) \begin{align} h_v\doteq l\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{v}} \rangle /v_{\mathrm{rms}}^2, \\[-30pt]\nonumber\end{align}

(5.22d) \begin{align} h_b\doteq l\langle \widetilde {\boldsymbol{b}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{b}} \rangle /b_{\mathrm{rms}}^2, \\[-30pt]\nonumber\end{align}

(5.22e) \begin{align} h_c\doteq l\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{b}} \rangle /v_{\mathrm{rms}}b_{\mathrm{rms}},\\[-15pt]\nonumber \end{align}

where $ \langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{v}} \rangle =v_{\mathrm{rms}}^2$ and $\langle \widetilde {\boldsymbol{b}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle =b_{\mathrm{rms}}^2.$ The relative strength of magnetic fluctuations is captured by the parameter $m$ , with $m=0$ corresponding to hydrodynamic turbulence and $m=1$ corresponding to equipartition. The parameter $a$ is the normalised cross-helicity, which is non-zero only in the case of ‘imbalanced’ MHD turbulence, i.e. when energy is unevenly distributed between $\widetilde {\boldsymbol{z}}^+$ and $\widetilde {\boldsymbol{z}}^-$ . (Note that $\langle \widetilde {\boldsymbol{z}}^+\boldsymbol{\cdot} \widetilde {\boldsymbol{z}}^+ \rangle -\langle \widetilde {\boldsymbol{z}}^-\boldsymbol{\cdot} \widetilde {\boldsymbol{z}}^- \rangle =2\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle$ .) The parameters $h_v$ and $h_b$ are the normalised kinetic and current helicities, respectively. The parameter $h_c$ captures flow–current alignment and can be understood as the ‘helical counterpart’ to the cross-helicity $a$ in that it captures the imbalance of the Elsässer-fields helicities ( $\langle \widetilde {\boldsymbol{z}}^+\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{z}}^+ \rangle -\langle \widetilde {\boldsymbol{z}}^+\boldsymbol{\cdot} \boldsymbol{\nabla} \times \widetilde {\boldsymbol{z}}^+ \rangle =2\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ ).

Note that the parameters are scaled such that their values lie between zero and one for ‘realistic’ turbulence, which we define to include turbulence that can be feasibly obtained in numerical simulations with the appropriate choice of forcing functions. For example, $h_v=1$ is commonly referred to as ‘maximally helical turbulence’ in the literature, and, understandably, it is a popular scenario for numerical studies of the $\alpha$ -effect (Brandenburg Reference Brandenburg2001; Sur, Brandenburg & Subramanian Reference Sur, Brandenburg and Subramanian2008; Mitra et al. Reference Mitra, Käpylä, Tavakol and Brandenburg2009), although such turbulence is indeed highly stylised and not realistic in the usual sense of the word.

The mean-field effects associated with $h_v$ , $h_b$ and $m$ are relatively well understood, since the kinetic-helicity-driven $\alpha$ -effect and its quenching by current helicity are arguably the central results of traditional mean-field theory for homogeneous MHD turbulence and have been the subject of much numerical investigation. Furthermore, these effects alone can be accommodated within the usual mean-field treatment where only the dynamics of the mean magnetic field are considered, since the mean induction equation decouples from the momentum equation in the absence of cross-helicity or flow–current alignment, as discussed in § 4.

For cases where $a=0$ and $h_c=0$ , we therefore expect MFWK to largely corroborate the results of previous mean-field theories. In contrast, the mean-field dynamics associated with cross-helicity (as parametrised by $a$ ) and flow–current alignment (parametrised by $h_c$ ) have not yet been fully analysed. As discussed in § 4, cross-helicity and flow–current alignment couple the mean induction and momentum equations such that the usual kinematic approach cannot be applied. Determining the mean-field modes associated with these effects has therefore, until now, been intractable in the general case.Footnote 9 In particular, we report a new dynamo effect driven by correlations between the fluctuating velocity and current, $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ .

5.5. Benchmark example: the $\alpha ^2$ -dynamo

We first benchmark the modulational dynamics obtained with the MFWK formalism by reproducing the expected $\alpha$ -effect and associated $\alpha ^2$ -dynamo, i.e. solve (5.17) for $a=h_c=0$ .

Figure 1. The imaginary (a–c) and real (d–f) parts of modulational frequency $\varOmega$ versus modulational wavevector $K$ at $m=a=h_b=h_c=0$ and ${St}=10^5$ for $h_v=0$ (a,d), $h_v=0.5$ (b,e) and $h_v=1$ (c,f). The magnetic-energy fraction $f_b \dot {=} {b}^2/(\textit{v}^2+\textit{b}^2)$ of the corresponding eigenmodes are given in panels (g)–(i). The frequencies are given in units of the inverse turnover time $\tau ^{-1}$ and wavevectors are given in units of the inverse characteristic eddy size $l^{-1}$ .

5.5.1. Hydrodynamic turbulence

Figure 1 shows the modulational modes obtained from MFWK for hydrodynamic helical turbulence. As expected, the kinetic helicity supports a stationary ( $\textrm {Re} \varOmega =0$ ) unstable solution that is helically $\boldsymbol{{b}}$ -polarised ( $\mathrm{b}_x=\mathrm{ib}_y$ , $\mathrm{v}_x=\mathrm{v}_y=0$ ), with a growth rate that is maximised at relatively large modulational wavelengths ( $lK\lt 1$ ). To further confirm that the unstable $\boldsymbol{{b}}$ -polarised mode is indeed the expected $\alpha ^2$ -dynamo, figure 2 shows that the growth rate of the unstable $\boldsymbol{{b}}$ -polarised mode indeed converges to the local $\alpha ^2$ -dynamo growth rate,

(5.23) \begin{equation} \varGamma =\alpha K-(\eta +\beta ) K^2, \end{equation}

in the ${St}\ll 1$ limit, and still qualitatively agrees with (5.23) for larger values of ${St}$ in that it is unstable strictly for $K\lt 1$ and is stabilised by current helicity.

Note that in the ideal limit ( $\nu _\pm \to 0$ ), and unless driven unstable by helicity, the modulational perturbations obtained from the MFWK framework are damped at the rate $\tau _c^{-1}/2$ , irrespective of the modulational wavenumber $K$ . In this sense, the interpretation of $\beta$ in the local expansion of the EMF (also, $\mathcal{B}$ in (4.35)) as a turbulent diffusivity relies on the assumption of short correlation times, and the analogy must therefore be applied with caution. A true dissipation such as viscosity $\nu$ would damp the modulational perturbations at the rate $\nu K^2$ . Figure 1 also shows another (less) unstable stationary mode that is helically $\boldsymbol{{v}}$ -polarised ( $\mathrm{iv}_x=\mathrm{v}_y$ , $\mathrm{b}_x=\mathrm{b}_y=0$ ; note the opposite handedness relative to the $\boldsymbol{{b}}$ -polarised mode) and is unstable for relatively larger values of $K$ . It may seem, at first glance, that this is simply the hydrodynamic $\alpha$ -effect, also known as the H $\alpha$ -effect (Moiseev et al. Reference Moiseev, Sagdeev, Tur, Khomenko and Yanovskii1983); however, it has been argued that the latter vanishes in incompressible turbulence (Khomenko, Moiseev & Tur Reference Khomenko, Moiseev and Tur1991). The incompressible analogue is known as the anisotropic kinetic alpha instability (Frisch, She & Sulem Reference Frisch, She and Sulem1987); however, as the name might suggest, this effect requires anisotropy, while our figure 1 is for an incompressible, isotropic turbulent background. Therefore, this mode remains to be interpreted. Although the generation of mean flows from hydrodynamic turbulence has received considerable attention (Rüdiger Reference Rüdiger1980; Yokoi & Yoshizawa Reference Yokoi and Yoshizawa1993; Elperin, Kleeorin & Rogachevskii Reference Elperin, Kleeorin and Rogachevskii2003; Yokoi & Brandenburg Reference Yokoi and Brandenburg2016), our focus is on the turbulent dynamo problem. We therefore do not discuss this hydrodynamic mode further, except to note that it is stabilised by magnetic fluctuations and vanishes entirely for $m \gtrsim 0.2$ . Instead, we invite future investigations that may resolve this mystery.

Figure 2. The growth rate $\varGamma \dot {=} \mathrm{ Im }\, \varOmega$ versus modulational wavenumber $K$ for various $\textit{St}\,\dot {=}\tau _c/\tau$ . The classic $\alpha ^2$ -dynamo dispersion relation (5.23) is shown in the black dashed line. The frequencies are given in units of the inverse turnover time $\tau ^{-1}$ and wavevectors are given in units of the inverse characteristic eddy size $l^{-1}$ .

5.6. Dynamo driven by flow–current alignment

Let us now examine the effect of flow–current alignment by solving (5.17) for non-zero values of the parameter $h_c$ . As discussed in § 4, flow–current alignment (equivalently, an imbalance in the helicities of the Elsässer fields) couples the mean momentum and induction equations, and the fundamental modulational dynamics associated with this property have not yet been established. Let us therefore first isolate the influence of flow–current alignment by removing all other potential effects ( $a=h_v=h_b=0$ ). The resulting modes are shown in figure 3.

Figure 3. The imaginary (a–c) and real (d–f) parts of modulational frequency $\varOmega$ normalised to $\tau ^{-1}$ at $m=1$ , $a=h_b=h_v=0$ and ${St}=10^5$ for $h_c=0$ (a,d), $h_c=0.5$ (b, e) and $h_c=1$ (c,f). The magnetic energy fraction $f_b\dot {=} {b}^2/({v}^2+{b}^2)$ of the corresponding eigenmodes is presented in panels (g)–(i). The frequencies are given in units of the inverse turnover time $\tau ^{-1}$ and wavevectors are given in units of the inverse characteristic eddy size $l^{-1}$ .

It can be seen that the flow–current alignment $h_c$ indeed drives a strange and novel dynamo effect, the properties of which we shall discuss at length in the remainder of this section. Let us first highlight that this effect is emphatically not one that can be captured under the assumption of scale separation, as the mode is stable for $lK\lessapprox 1$ . Moreover, in the absence of viscosity or resistivity, the growth rate asymptotes to a constant non-zero value as $K\to \infty$ , although we will show in Appendix E that this property is limited to the case $a=0$ .

In contrast to the stationary $\alpha ^2$ -dynamo, the unstable modes driven by flow–current alignment propagate as they grow. The propagation speed of these ‘correlation waves’ is given by

(5.24) \begin{equation} v_{\rm {ph}}^\pm =\sqrt {\langle \widetilde {\boldsymbol{z}}^\mp \boldsymbol{\cdot} \widetilde {\boldsymbol{z}}^\mp \rangle /3} \end{equation}

for $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^\pm$ polarised modes, and they are further discussed in Appendix E. Furthermore, for $a=0$ these modes are polarised such that the associated energy is split evenly between the mean flow and magnetic fields, or equivalently, between $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^+$ and $\mkern 1.5mu\overline {\mkern -1.5mu\boldsymbol{z}\mkern -1.5mu}\mkern 1.5mu^-$ . Note that this means that a $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo drives a net conversion of kinetic energy to magnetic energy as long as the turbulent magnetic energy is below equipartition ( $m\lt 1$ ). As will be discussed shortly, the energy balance for the $a\neq 0$ case is more complicated, but the ultimate consequence of dynamo action is the same.

Figure 4. The maximum growth rate $\varGamma _{\mathrm{max}}$ (colourbars) of the (a) $\alpha ^2$ -dynamo versus flow–current alignment ( $h_c$ ) and current helicity ( $h_b$ ); (b) $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo versus the dimensionless kinetic ( $h_v$ ) and current ( $h_b$ ) helicities. The growth rates are given in units of the inverse turnover time $\tau ^{-1}$ and wavevectors are given in units of the inverse characteristic eddy size $l^{-1}$ .

5.6.1. Interaction with kinetic helicity

Although the EMF for isotropic MHD turbulence can be neatly split into the sum of separate ‘effects’ (for example, the term proportional to the mean flow that arises from $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ ), these additive contributions to the EMF certainly do not guarantee additive contributions to the solutions of (5.17), and the various statistical properties of the turbulence can interact in non-obvious ways. To illustrate this, figure 4 shows the dependence of the maximum growth rate of the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo on the kinetic and current helicities. This figure also compares the said maximum growth rate with that of the $\alpha ^2$ -dynamo, particularly in how it depends on flow–current alignment and the current helicity.

It can be seen that although flow–current alignment has almost no effect on the $\alpha ^2$ -dynamo (as opposed to current helicity, which can exactly cancel the destabilising drive for $h_v=mh_b$ ), kinetic helicity partially suppresses the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo. Interestingly, the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo is not perceptibly affected by current helicity. Let us therefore, in the remainder of this section, take $h_v=mh_b$ , so that the $\alpha ^2$ -dynamo is completely stabilised. This will allow us to focus on the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo and pin down the dimension of the parameter space that is irrelevant for the dynamics of interest.

Beyond a simple suppression of the maximum growth rate, kinetic helicity can qualitatively impact the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -driven mode properties shown in figure 3. As seen in figure 5, kinetic helicity breaks the degeneracy of the modes shown in figure 3, shifting one unstable branch towards higher $K$ and the other towards lower $K$ . For brevity, let us refer to the former mode as $M_1$ and the latter as $M_2$ (each mode label refers to two separate solutions of (5.17) with opposite signs of Re $\varOmega$ ). As shown in figure 5, at sufficiently high values of $h_v$ , $M_2$ splits into two unstable $K$ regions at low and high $K$ . The modes are both circularly polarised in both $\mathrm{z}^+$ and $\mathrm{z}^-$ , but with opposite handedness ( $\mathrm{z}_x^\pm =\mathrm{iz}_y^\pm$ for $M_1$ and $\mathrm{iz}_x^\pm =\mathrm{z}_y^\pm$ for $M_2$ ).

Figure 5. The imaginary (a–c) and real (d–f) parts of modulational frequency $\varOmega$ at $m=h_c=1$ , $a=0$ and ${St}=10^5$ for $h_v=h_b=0.3$ (a,d), $h_v=h_b=0.8$ (b, e) and $h_v=h_b=0.9$ (c,f). The magnetic energy fraction $f_b\dot {=} \mathrm{b}^2/(\mathrm{v}^2+\mathrm{b}^2)$ of the corresponding eigenmodes are given in panels (g–i). The frequencies are given in units of the inverse turnover time $\tau ^{-1}$ and wavevectors are given in units of the inverse characteristic eddy size $l^{-1}$ .

5.6.2. Cross-helicity interaction

Let us now examine the impact of cross-helicity on the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo. As can be seen in figure 6, the overall effect of cross-helicity is to suppress the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo and shift the instability range to smaller $K$ , for both $M_1$ and $M_2$ .

At first, this may seem like bad news for the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo, since cross-helicity is a measure of the imbalance between the Elsässer fields ( $\langle \widetilde {\boldsymbol{z}}^+\boldsymbol{\cdot} \widetilde {\boldsymbol{z}}^+ \rangle -\langle \widetilde {\boldsymbol{z}}^-\boldsymbol{\cdot} \widetilde {\boldsymbol{z}}^- \rangle =2\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle$ ), and flow–current alignment is a measure of the imbalance between the Elsässer helicities ( $\langle \widetilde {\boldsymbol{z}}^+\boldsymbol{\cdot} \widetilde {\boldsymbol{w}}^+ \rangle -\langle \widetilde {\boldsymbol{z}}^-\boldsymbol{\cdot} \widetilde {\boldsymbol{w}}^- \rangle =2\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ ). Although they both capture some form of imbalance between the Elsässer fields, cross-helicity and flow–current alignment are entirely independent properties of MHD turbulence (determined by the spectra $E_A$ and $H_A$ , respectively). Whether such turbulence is realised in astrophysical environments or other environments of interest is a question that we leave to future work.

Figure 6. The imaginary (a,b) and real (c,d) parts of modulational frequency $\varOmega$ versus $K$ and $a$ at $m=h_v=h_b=h_c=1$ and ${St}=10^5$ , for both $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -driven modes $M_1$ (a,c) and $M_2$ (b,d). The frequencies are given in units of the inverse turnover time $\tau ^{-1}$ and wavevectors are given in units of the inverse characteristic eddy size $l^{-1}$ .

5.6.3. Dependence on St

The results presented in this section thus far are truly quasilinear, in that they are obtained in the ${St}\to \infty$ limit of (2.43). For ‘realistic’ large- ${Rm}$ turbulence, ${St}$ is expected to be order-one. Let us therefore explore how strongly damped the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo is by eddy–eddy collisions, at least within the simple MTA-type model used in (2.43). Figure 7(a) shows the maximum growth rate of the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo over the $(m,h_c)$ parameter space, and figure 7(b) shows the critical ${St}$ below which the fastest growing $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -mode is completely stabilised. It can be seen that although the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{j}} \rangle$ -dynamo can still be unstable at ${St}\sim 1$ , it is, in general, far less robust to the damping effect of decorrelations than the $\alpha ^2$ -dynamo, whose growth rate scales linearly with $\tau _c$ in the ${St}\to 0$ limit.

Figure 7. (a) The maximum growth rate $\varGamma _{\mathrm{max}}$ normalised to $\tau ^{-1}$ of $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle$ -dynamo, at ${St}=10^5$ and (b) critical St, ${St}_{\mathrm{crit}}$ , required for instability versus $h_c$ and $m$ , both at $a=h_v=h_b=0$ . Note that ${St}_{\mathrm{crit}}$ can only be defined for parameter values where the $\langle \widetilde {\boldsymbol{v}}\boldsymbol{\cdot} \widetilde {\boldsymbol{b}} \rangle$ -dynamo is unstable; the grey region on (b) indicates values of $(m,h_c)$ for which there is no instability at any value of ${St}$ . The colourbar is limited to the values $(0,15)$ for visibility, although ${St}_{\mathrm{crit}}$ reaches much larger values near the stability boundary.