2.1. Set-up and assumptions

The mean magnetic field, $\boldsymbol {B}$, evolves according to (e.g. Moffatt Reference Moffatt1978, (7.7))

(2.1)\begin{equation} \frac{\partial\boldsymbol{B} }{\partial t} = \boldsymbol{\nabla}\times\left( \boldsymbol{V}\times\boldsymbol{B} + \boldsymbol{\mathcal{E}} \right) + \eta_m \nabla^2 \boldsymbol{B} , \end{equation}

where $\boldsymbol {V}$ is the mean velocity; $\eta _m$ is the microscopic magnetic diffusivity; and $\boldsymbol {\mathcal {E}}$, the turbulent electromotive force (EMF), is related to the correlation between the fluctuating velocity and magnetic fields. Since the magnetohydrodynamic equations are nonlinear, the evolution equations for moments of a particular order depend on moments of higher orders. For example, the EMF in (2.1) is a double correlation of the fluctuating fields. To keep the system of equations manageable, one has to truncate this hierarchy by applying a closure. To avoid solving for the fluctuating magnetic field, one requires an expression for the EMF in terms of the mean magnetic field itself. If the mean magnetic field is weak and varies slowly, one typically assumes that the EMF depends only on the mean magnetic field and its first derivatives, obtaining a general expression of the form $\mathcal {E}_i = \alpha _{ij} B_j + \eta _{ijk} \partial _j B_k$. Here, and in what follows, repeated indices are summed over. The tensors $\alpha _{ij}$ and $\eta _{ijk}$ may depend on the statistical properties of the velocity field. The expressions for these tensors depend on the closure used.

One of the most widely used closures in dynamo theory is the quasilinear approximation (also called the first-order smoothing approximation (FOSA); or the second-order correlation approximation (SOCA)) (e.g. Moffatt Reference Moffatt1978, § 7.5; Krause & Rädler Reference Krause and Rädler1980, § 4.3). The quasilinear approximation is rigorously valid only when either the magnetic Reynolds number (the ratio of the diffusive to the advective time scale) or the Strouhal number (the ratio of the velocity correlation time to its turnover time) is small (Krause & Rädler Reference Krause and Rädler1980, p. 49). The former is never small in the astrophysical systems of interest, while it is unclear if the latter is small. Nevertheless, in the context of mean-field dynamo theory, the quasilinear approximation often remains qualitatively correct well outside its domain of formal validity. More complicated closures such as the eddy-damped quasinormal Markovian (EDQNM) closure (e.g. Pouquet, Frisch & Léorat Reference Pouquet, Frisch and Léorat1976) and the direct interaction approximation (DIA) (Kraichnan Reference Kraichnan1977) are extremely difficult to work with.

For weakly inhomogeneous non-helical turbulence, the EMF is given in the quasilinear approximation by (Roberts & Soward Reference Roberts and Soward1975, (3.11))

(2.2)\begin{equation} \boldsymbol{\mathcal{E}} ={-} \tfrac{1}{2} \boldsymbol{\nabla} \eta \times \boldsymbol{B} - \eta \boldsymbol{\nabla}\times\boldsymbol{B} , \end{equation}

where $\eta$ is the turbulent diffusivity (proportional to the turbulent kinetic energy). We note that Silant'ev (Reference Silant'ev1999, Reference Silant'ev2000) did not consider the first term above (Appendix B describes how the absence of this term qualitatively changes the behaviour of the system). Comparing this term with the $\boldsymbol {V}\times \boldsymbol {B}$ term in (2.1), we see that the former can be thought of as describing an effective velocity $-\boldsymbol {\nabla }\eta /2$ that acts on the mean magnetic field. This transports the magnetic field in the direction in which the turbulent kinetic energy decreases. By analogy with the reduction of the magnetic field in diamagnetic materials, this is usually referred to as ‘turbulent diamagnetism’ or ‘diamagnetic pumping’. For stratified turbulence, or in the presence of small-scale magnetic fields, additional terms arise (Vainshtein & Kichatinov Reference Vainshtein and Kichatinov1983), but we ignore those effects in this work. As mentioned in the introduction, the effect of helical turbulence ($\alpha$) has been extensively studied, so we restrict ourselves to turbulence that is non-helical pointwise. The first term of (2.2) may be considered one of many contributions to the off-diagonal components of $\alpha _{ij}$. In the literature, such contributions are sometimes described in terms of a vector $\boldsymbol {\gamma }$; see Rädler, Kleeorin & Rogachevskii (Reference Rädler, Kleeorin and Rogachevskii2003, (42)). While the calculations we present can be carried out using the $\alpha _{ij}$ and $\eta _{ijk}$ tensors in their full glory, we use a simpler expression in order to keep the results interpretable.

Although the mean-field approach does not formally require scale separation, we associate averages with length/time scales for clarity of exposition. Let us assume that $\eta$ fluctuates at length/time scales (henceforth referred to as the mesoscales) much larger than the scales at which the turbulent velocity fluctuates. We employ a double-averaging approach (Kraichnan Reference Kraichnan1976; Singh Reference Singh2016), in which we treat $\eta$ (at the mesoscale) as a stochastic scalar field which is a function of both position and time (i.e. $\eta = \eta (\boldsymbol {x},t)$). For any mesoscale quantity $\Box$, we use $\langle \Box \rangle$ and $\bar {\Box }$ to denote its averages at the larger scale. We assume this average satisfies Reynolds’ rules (e.g. Monin & Yaglom Reference Monin and Yaglom1971, § 3.1).

If we set the mean velocity to zero, ignore the microscopic diffusivity (which is usually much smaller than the turbulent diffusivity in the systems of interest) and use (2.2), we can write (2.1), the evolution equation for the mesoscale magnetic field, as

(2.3)\begin{equation} \frac{\partial\boldsymbol{B}}{\partial t} = \boldsymbol{\nabla}\times\left( - \frac{1}{2} \boldsymbol{\nabla} \eta \times \boldsymbol{B} - \eta \boldsymbol{\nabla}\times\boldsymbol{B} \right) . \end{equation}

In § 2.3, we assume the fluctuations of $\eta$ are statistically homogeneous, stationary and separable in order to obtain an integro-differential equation for the large-scale magnetic field. In § 2.4, we simplify this equation by assuming the fluctuations of $\eta$ are white noise, while in § 2.5, we also keep terms linear in the correlation time of $\eta$.

2.2. Evolution equation in Fourier space

We now move to Fourier space with

(2.4)\begin{equation} \tilde{f}(\boldsymbol{k},t) \equiv \int \frac{\mathrm{d}\boldsymbol{x}}{(2{\rm \pi})^3}\, {\rm e}^{{\rm i}\boldsymbol{k}\,\boldsymbol{\cdot}\,\boldsymbol{x}} f(\boldsymbol{x},t), \end{equation}

where we have used a tilde to denote the spatial Fourier transform of a quantity. The convolution theorem takes the form

(2.5)\begin{equation} \int\frac{\mathrm{d}\boldsymbol{x}}{(2{\rm \pi})^3}\, {\rm e}^{{\rm i}\boldsymbol{k}\,\boldsymbol{\cdot}\,\boldsymbol{x}} f(\boldsymbol{x}) g(\boldsymbol{x}) = \int\mathrm{d}\boldsymbol{p} \ \tilde{f}(\boldsymbol{p}) \tilde{g}(\boldsymbol{k}-\boldsymbol{p}). \end{equation}

Equation (2.3) then becomes (omitting the temporal arguments whenever there is no ambiguity)

(2.6)\begin{equation} \frac{\partial \tilde{\boldsymbol{B}}(\boldsymbol{k}) }{\partial t} =\boldsymbol{k} \times \int\mathrm{d}\boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2}\right) \times \tilde{\boldsymbol{B}}(\boldsymbol{p}) \, \tilde{\eta}(\boldsymbol{k}-\boldsymbol{p}). \end{equation}

Taking the average of the above, we obtain

(2.7)\begin{equation} \frac{\partial}{\partial t} \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle = \boldsymbol{k} \times \int\mathrm{d}\boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2}\right) \times ( \langle \tilde{\eta}(\boldsymbol{k}-\boldsymbol{p}) \rangle\langle \tilde{\boldsymbol{B}}(\boldsymbol{p}) \rangle + \langle \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p}) \,\tilde{\boldsymbol{b}}(\boldsymbol{p}) \rangle ), \end{equation}

where we have split the mesoscale fields into their mean and fluctuating parts, i.e. $\tilde {\boldsymbol {B}} = \langle \tilde {\boldsymbol {B}}\rangle + \tilde {\boldsymbol {b}}$ and $\tilde {\eta } = \langle \tilde {\eta }\rangle + \tilde {\mu }$. We write the equation for $\tilde {\boldsymbol {b}}$ as

(2.8)\begin{align} \frac{\partial \tilde{\boldsymbol{b}}(\boldsymbol{k}) }{\partial t} &= \boldsymbol{k} \times \int\mathrm{d}\boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2}\right) \times ( \langle \tilde{\eta}(\boldsymbol{k}-\boldsymbol{p}) \rangle \tilde{\boldsymbol{b}}(\boldsymbol{p}) + \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p}) \langle \tilde{\boldsymbol{B}}(\boldsymbol{p}) \rangle\nonumber\\ &\quad +\, \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p}) \,\tilde{\boldsymbol{b}}(\boldsymbol{p}) - \langle \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p}) \,\tilde{\boldsymbol{b}}(\boldsymbol{p}) \rangle ). \end{align}

We now apply the quasilinear approximation, where the equations for the fluctuating fields are truncated by keeping only terms which are at most linear in the fluctuating fields. We then obtain

(2.9)\begin{equation} \frac{\partial \tilde{\boldsymbol{b}}(\boldsymbol{k}) }{\partial t} = \boldsymbol{k} \times \int\mathrm{d}\boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2}\right) \times ( \langle \tilde{\eta}(\boldsymbol{k}-\boldsymbol{p}) \rangle \tilde{\boldsymbol{b}}(\boldsymbol{p}) + \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p}) \langle \tilde{\boldsymbol{B}}(\boldsymbol{p}) \rangle ). \end{equation}

2.3. Homogeneity and separability

To simplify the preceding expression, we assume that the moments of $\eta (\boldsymbol {x},t)$ are statistically homogeneous and stationary. We can then write, say, $\langle \mu (\boldsymbol {x},\tau _1) \mu (\boldsymbol {y},\tau _2)\rangle = C(\boldsymbol {x} - \boldsymbol {y}, \tau _1 - \tau _2)$. Further, we assume that $\langle \mu (\boldsymbol {x},\tau _1) \mu (\boldsymbol {y},\tau _2)\rangle$ can be written as the product of a temporal correlation function and a spatial correlation function, i.e. $C(\boldsymbol {x} - \boldsymbol {y}, \tau _1 - \tau _2) = Q(\boldsymbol {x} - \boldsymbol {y}) \, S(\tau _1 - \tau _2)$. In Fourier space, these assumptions can be expressed as

(2.10a)$$\begin{gather} \langle \tilde{\eta}(\boldsymbol{k},t)\rangle = \bar{\eta} \operatorname{\delta}(\boldsymbol{k}), \end{gather}$$

(2.10b)$$\begin{gather}\langle \tilde{\mu}(\boldsymbol{p},\tau_1) \tilde{\mu}(\boldsymbol{q},\tau_2)\rangle = \tilde{Q}\left(\boldsymbol{p}\right) S(\tau_1-\tau_2) \operatorname{\delta}(\boldsymbol{p}+\boldsymbol{q}). \end{gather}$$

For $S$, we require

(2.11)\begin{equation} 2 \int_0^\infty S(t) \, \mathrm{d} t = 1, \end{equation}

and define the correlation time of $\eta$ as

(2.12)\begin{equation} \tau_\eta \equiv 2 \int_0^\infty t S(t) \, \mathrm{d} t. \end{equation}

We can then write

(2.13)\begin{equation} \boldsymbol{k} \times \int\mathrm{d}\boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2}\right) \times \tilde{\boldsymbol{b}}(\boldsymbol{p}) \langle \tilde{\eta}(\boldsymbol{k}-\boldsymbol{p}) \rangle ={-} \bar{\eta} k^2 \tilde{\boldsymbol{b}}(\boldsymbol{k}), \end{equation}

and

(2.14)\begin{equation} \boldsymbol{k} \times \int\mathrm{d}\boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2}\right) \times \langle \tilde{\boldsymbol{B}}(\boldsymbol{p}) \rangle \langle \tilde{\eta}(\boldsymbol{k}-\boldsymbol{p}) \rangle ={-} \bar{\eta} k^2 \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle . \end{equation}

Using (2.13), (2.9) can be written as

(2.15)\begin{equation} \frac{\partial \tilde{\boldsymbol{b}}(\boldsymbol{k}) }{\partial t} ={-} \bar{\eta} k^2 \tilde{\boldsymbol{b}}(\boldsymbol{k}) + \boldsymbol{k} \times \int\mathrm{d}\boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2}\right) \times \langle \tilde{\boldsymbol{B}}(\boldsymbol{p}) \rangle \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p}), \end{equation}

which gives us

(2.16)\begin{align} \tilde{\boldsymbol{b}}(\boldsymbol{k},t) &= \boldsymbol{k} \times \int_0^t \mathrm{d}\tau \int\mathrm{d}\boldsymbol{p} \exp({- \bar{\eta} k^2 \left(t-\tau \right)}) \left( \frac{\boldsymbol{k} + \boldsymbol{p}}{2} \right) \times \langle \tilde{\boldsymbol{B}}(\boldsymbol{p}, \tau) \rangle \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p}, \tau)\nonumber\\ &\quad + \tilde{\boldsymbol{b}}(\boldsymbol{k},0). \end{align}

We assume that the initial fluctuations of the mesoscale magnetic field are uncorrelated with $\mu$. Using the above along with (2.10b), we can write

(2.17)\begin{align} \langle \tilde{\mu}(\boldsymbol{q},t) \, \tilde{\boldsymbol{b}}(\boldsymbol{k},t) \rangle &=\boldsymbol{k} \times \int_0^t \mathrm{d}\tau \exp({- \bar{\eta} k^2 \left(t-\tau \right)}) \left( \boldsymbol{k} + \frac{\boldsymbol{q}}{2} \right)\nonumber\\&\quad \times \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}+\boldsymbol{q}, \tau) \rangle \tilde{Q}\left(\boldsymbol{q}\right) S(t-\tau). \end{align}

Putting the above in (2.7) gives us an equation for $\langle \tilde {\boldsymbol {B}}(\boldsymbol {k}+\boldsymbol {q}, \tau ) \rangle$. However, this is an integro-differential equation which is difficult to solve in general. The resulting equation can be simplified by assuming $\tau _\eta$ is small. In § 2.4, we assume $\tau _\eta =0$ and simplify the evolution equation for the large-scale magnetic field. In § 2.5, we simplify the evolution equation neglecting $O(\tau _\eta ^2)$ terms.

2.4. Evolution equation with white-noise fluctuations

Assuming $S(t) = \operatorname {\delta }(t)$, we write (2.17) as

(2.18)\begin{equation} \langle \tilde{\mu}(\boldsymbol{q},t) \, \tilde{\boldsymbol{b}}(\boldsymbol{k},t) \rangle =\frac{1}{2} \, \boldsymbol{k} \times \left[ \left( \boldsymbol{k} + \frac{\boldsymbol{q}}{2} \right) \times \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}+\boldsymbol{q}, t) \rangle \right] \tilde{Q}\!\left(\boldsymbol{q}\right). \end{equation}

Recalling that $\boldsymbol {k}\boldsymbol {\cdot }\langle \tilde {\boldsymbol {B}}(\boldsymbol {k}, t)\rangle = 0$, we can use the above to write

(2.19)\begin{align} &\boldsymbol{k} \times \int \mathrm{d}\boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2}\right)\times \langle \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p},t) \, \tilde{\boldsymbol{b}}(\boldsymbol{p},t) \rangle \nonumber\\ &\quad = \int\mathrm{d}\boldsymbol{s} \, \frac{1}{8} \, \tilde{Q}\!\left(\boldsymbol{s}\right) ( 4 k^4 - 8k^2 \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{s} + 3 \left( \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{s} \right)^2 + 2 k^2 s^2 - s^2 \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{s} ) \langle \tilde{\boldsymbol{B}}(\boldsymbol{k},t)\rangle , \end{align}

where $\boldsymbol {s} \equiv \boldsymbol {k}-\boldsymbol {p}$. Defining

(2.20a–d) \begin{equation} A^{(0)} \equiv Q(\boldsymbol{0}) ,\quad A^{(1)}_i \equiv \left. \frac{\partial Q(\boldsymbol{\xi}) }{\partial \xi_i} \right|_{\boldsymbol{\xi} = \boldsymbol{0}} ,\quad {\mathsf{A}}^{(2)}_{ij} \equiv \left. \frac{\partial^2 Q(\boldsymbol{\xi}) }{\partial \xi_i \partial \xi_j} \right|_{\boldsymbol{\xi} = \boldsymbol{0}} ,\quad A^{(3)}_{i} \equiv \left. \frac{\partial^3 Q(\boldsymbol{\xi}) }{\partial \xi_i \partial \xi_j \partial \xi_j} \right|_{\boldsymbol{\xi} = \boldsymbol{0}}, \end{equation}

we write

(2.21)\begin{align} & \boldsymbol{k} \times \int \mathrm{d} \boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2} \right) \times \langle \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p},t) \, \tilde{\boldsymbol{b}}(\boldsymbol{p},t) \rangle\nonumber\\ &\quad = \frac{1}{8} (4 A^{(0)} k^4 - 8 i A^{(1)}_i k^2 k_i - 3 {\mathsf{A}}^{(2)}_{ij} k_ik_j - 2 {\mathsf{A}}^{(2)}_{ii} k^2 + {\rm i} A^{(3)}_i k_i ) \langle \tilde{\boldsymbol{B}}(\boldsymbol{k},t) \rangle.\end{align}

Note that the functions defined in (2.20a–d) depend only on the value of $Q(\boldsymbol {\xi })$ and its spatial derivatives at the origin. Putting this in (2.7) and using (2.14), we obtain

(2.22)\begin{equation} \frac{\partial}{\partial t} \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle =( - \bar{\eta} k^2 + g(\boldsymbol{k}) ) \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle + {\rm i} h(\boldsymbol{k}) \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle, \end{equation}

where

(2.23a)$$\begin{gather} g(\boldsymbol{k}) \equiv{}- \tfrac{3}{8} {\mathsf{A}}^{(2)}_{ij} k_i k_j - \tfrac{1}{4} {\mathsf{A}}^{(2)}_{ii} k^2 + \tfrac{1}{2} A^{(0)} k^4, \end{gather}$$

(2.23b)$$\begin{gather}h(\boldsymbol{k})\equiv{}\tfrac{1}{8} A^{(3)}_i k_i - A^{(1)}_i k^2 k_i. \end{gather}$$

We see that $g(\boldsymbol {k})$ describes corrections to the turbulent diffusivity (along with a $k^4$ hyperdiffusive term), while the term involving $h(\boldsymbol {k})$ describes advection of the large-scale magnetic field with an effective velocity $A^{(3)}_i /8 - A^{(1)}_i k^2$.

To aid the interpretation of (2.22), we note that, if the spatial correlation function of the fluctuations of $\eta$ is an isotropic Gaussian (see Appendix A), we can write

(2.24)\begin{equation} \frac{\partial}{\partial t} \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle ={-} \left( \bar{\eta} - \frac{9}{8} \, \beta \right) k^2 \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle + O(k^4), \end{equation}

where $\beta \equiv A^{(0)}/l_c^2 > 0$ represents the diffusivity arising from fluctuations of $\eta$ with a correlation length $l_c$. Thus, we find that fluctuations of $\eta$ reduce the turbulent diffusion of the large-scale magnetic field.

2.5. Evolution equation with non-zero correlation time

We expand

(2.25)\begin{equation} \langle \tilde{\boldsymbol{B}}(\boldsymbol{p}, \tau) \rangle = \langle \tilde{\boldsymbol{B}}(\boldsymbol{p}, t) \rangle - \left( t - \tau \right) \frac{ \partial }{\partial t}\!\langle \tilde{\boldsymbol{B}}(\boldsymbol{p}, t) \rangle + O( (t-\tau)^2 ). \end{equation}

The idea is that when we substitute this into (2.17), assume $t\gg \tau _\eta$ and perform the time integral, the powers of $(t-\tau )$ become powers of $\tau _\eta$. The convergence of this series requires that the large-scale magnetic field vary on a time scale much larger than $\tau _\eta$. Note that on the right-hand side of the above, we can neglect $O(t-\tau )$ contributions to ${ \partial }/{\partial t}\!\langle \tilde {\boldsymbol {B}}(\boldsymbol {p}, t) \rangle$ and use (2.22). Similarly, we can expand

(2.26)\begin{equation} \exp{(- \bar{\eta} k^2 (t-\tau ) )} = 1 - (t-\tau ) \bar{\eta} k^2 + O( (t-\tau)^2 ). \end{equation}

We then write (2.17) as

(2.27)\begin{equation} \langle \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p},t) \, \tilde{\boldsymbol{b}}(\boldsymbol{p},t) \rangle = \tilde{Q}\left(\boldsymbol{k}-\boldsymbol{p}\right) \boldsymbol{p} \times \left[ \left( \frac{\boldsymbol{k} + \boldsymbol{p}}{2} \right) \times \boldsymbol{\mathcal{B}}(\boldsymbol{k},t) \right], \end{equation}

where

(2.28)\begin{equation} \boldsymbol{\mathcal{B}}(\boldsymbol{k},t) \equiv \frac{1}{2}\langle \tilde{\boldsymbol{B}}(\boldsymbol{k}, t) \rangle - \frac{\tau_\eta}{2} \frac{\partial}{\partial t}\!\langle \tilde{\boldsymbol{B}}(\boldsymbol{k}, t) \rangle - \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}, t) \rangle \frac{\tau_\eta}{2} \, \bar{\eta} k^2. \end{equation}

Using (2.22), we write

(2.29)\begin{equation} \boldsymbol{\mathcal{B}}(\boldsymbol{k},t) = \frac{1}{2}\langle \tilde{\boldsymbol{B}}(\boldsymbol{k}, t) \rangle - \frac{\tau_\eta}{2} \, g(\boldsymbol{k}) \langle \tilde{\boldsymbol{B}}(\boldsymbol{k},t) \rangle - \frac{{\rm i} \tau_\eta}{2} \, h(\boldsymbol{k}) \langle \tilde{\boldsymbol{B}}(\boldsymbol{k},t) \rangle . \end{equation}

We note that $\boldsymbol {k}\boldsymbol{\cdot }\boldsymbol {\mathcal {B}}(\boldsymbol {k},t) = 0$ (since $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {B} = 0$) and use (2.27) and (2.29) to write

(2.30)\begin{align} & \boldsymbol{k} \times \int \mathrm{d}\boldsymbol{p} \left(\frac{\boldsymbol{k}+\boldsymbol{p}}{2}\right)\times \langle \tilde{\mu}(\boldsymbol{k}-\boldsymbol{p},t) \, \tilde{\boldsymbol{b}}(\boldsymbol{p},t) \rangle\nonumber\\ &\quad = \left[ g(\boldsymbol{k}) + {\rm i} h(\boldsymbol{k}) \right] \left[ 1 - \tau_\eta g(\boldsymbol{k}) - {\rm i} \tau_\eta h(\boldsymbol{k}) \right] \langle \tilde{\boldsymbol{B}}(\boldsymbol{k},t) \rangle, \end{align}

where $h$ and $g$ are defined in (2.23). Putting this in (2.7) and using (2.14), we write

(2.31)\begin{equation} \frac{\partial}{\partial t} \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle = \left[ g(\boldsymbol{k}) + i h(\boldsymbol{k}) \right] \left[ 1 - \tau_\eta g(\boldsymbol{k}) - {\rm i} \tau_\eta h(\boldsymbol{k}) \right] \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle - \bar{\eta} k^2 \langle \tilde{\boldsymbol{B}}(\boldsymbol{k}) \rangle + O(\tau_\eta^2) . \end{equation}

2.6. Growth rate of the large-scale magnetic field

Let us now focus on the problem of whether a particular Fourier mode of the large-scale magnetic field grows or decays. We assume $\langle \tilde {\boldsymbol {B}}(\boldsymbol {k},t) \rangle \propto \exp (\lambda t)$. Plugging this into (2.31) and taking its real part, we find

(2.32)\begin{equation} \operatorname{Re}(\lambda) ={-} \bar{\eta} k^2 + g(\boldsymbol{k}) - \tau_\eta [ g(\boldsymbol{k}) ]^2 + \tau_\eta \left[ h(\boldsymbol{k}) \right]^2 + O(\tau_\eta^2) , \end{equation}

where $h$ and $g$ are defined in (2.23). From the fact that, above, only $[ g(\boldsymbol {k}) ]^2$ contains a $k^8$ term, we can see that the growth rate always becomes negative for large-enough $k$ (small-enough scales) as long as $\tau _\eta \ne 0$. Note that while in the white-noise case, $h(\boldsymbol {k})$ only contributed a drift term, it now affects the growth rate as well.

Since we assumed the large-scale magnetic field varies on time scales much larger than $\tau _\eta$, our derivation is self-consistent only when $\left |\tau _\eta \lambda \right | \ll 1$.