3.1 White-noise ${\it\alpha}$ fluctuations

Before studying finite-memory effects, we consider the exactly solvable limit of white-noise (i.e. delta-correlated in time) ${\it\alpha}$ fluctuations, for which the normalized correlation function is ${\mathcal{D}}_{WN}(t)={\it\delta}(t)$ , the Dirac delta-function. This gives ${\it\tau}_{{\it\alpha}}=0$ from (2.5), implying that the memory effects are ignored. The focus of this section is to understand the dynamo behaviour due to white-noise ${\it\alpha}$ fluctuations. From (3.11) the mean EMF for white noise is

(3.14) $$\begin{eqnarray}\displaystyle \widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}(\boldsymbol{k},t)=\widetilde{\boldsymbol{U}}(\boldsymbol{k},0)\times \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t), & & \displaystyle\end{eqnarray}$$

which, as expected, depends on the large-scale field at the present time only. The quantity $\widetilde{\boldsymbol{U}}(\boldsymbol{k},0)$ may be simplified as

(3.15) $$\begin{eqnarray}\displaystyle \widetilde{\boldsymbol{U}}(\boldsymbol{k},0) & = & \displaystyle \text{i}\boldsymbol{k}\int \frac{\text{d}^{3}k^{\prime }}{(2{\rm\pi})^{3}}\,\widetilde{{\mathcal{A}}}(\boldsymbol{k}^{\prime })-\int \frac{\text{d}^{3}k^{\prime }}{(2{\rm\pi})^{3}}\,\text{i}\boldsymbol{k}^{\prime }\widetilde{{\mathcal{A}}}(\boldsymbol{k}^{\prime })\nonumber\\ \displaystyle & = & \displaystyle \left.\text{i}\boldsymbol{k}{\mathcal{A}}(\mathbf{0})-\frac{\partial {\mathcal{A}}({\bf\xi})}{\partial {\bf\xi}}\right|_{{\bf\xi}=\mathbf{0}}=\text{i}\boldsymbol{k}{\it\eta}_{{\it\alpha}}+\boldsymbol{V}_{M}.\end{eqnarray}$$

Here, ${\it\eta}_{{\it\alpha}}$ is the ${\it\alpha}$ diffusivity defined in (2.4), and $\boldsymbol{V}_{M}$ is the Moffatt drift velocity (Moffatt Reference Moffatt1978; Sridhar & Singh Reference Sridhar and Singh2014). Both ${\it\eta}_{{\it\alpha}}$ and $\boldsymbol{V}_{M}$ are constants by definition. Substituting (3.14) into (3.13) and using (3.15), we get the following PDE for the large-scale magnetic field:

(3.16a,b ) $$\begin{eqnarray}\displaystyle \frac{\partial \widetilde{\overline{\boldsymbol{B}}}}{\partial t}+[{\it\eta}_{K}k^{2}+\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M}]\widetilde{\overline{\boldsymbol{B}}}=\mathbf{0},\quad \boldsymbol{k}\boldsymbol{\cdot }\widetilde{\overline{\boldsymbol{B}}}=0. & & \displaystyle\end{eqnarray}$$

Here

(3.17a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\eta}_{K}={\it\eta}_{T}-{\it\eta}_{{\it\alpha}}=\text{Kraichnan diffusivity}, & \displaystyle\end{eqnarray}$$

(3.17b ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{V}_{M}=-\left(\frac{\partial {\mathcal{A}}({\bf\xi})}{\partial {\bf\xi}}\right)_{{\bf\xi}=\mathbf{0}}=\int _{0}^{\infty }\overline{{\it\alpha}(\boldsymbol{x},t)\boldsymbol{{\rm\nabla}}{\it\alpha}(\boldsymbol{x},0)}\,\text{d}t=\text{Moffatt drift velocity},\qquad & \displaystyle\end{eqnarray}$$

${\it\alpha}$

(3.18) $$\begin{eqnarray}\displaystyle \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)=\widetilde{{\mathcal{G}}}(\boldsymbol{k},t)\,\widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},0),\quad \boldsymbol{k}\boldsymbol{\cdot }\widetilde{\overline{\boldsymbol{B}}}=0, & & \displaystyle\end{eqnarray}$$

where

(3.19) $$\begin{eqnarray}\displaystyle \widetilde{{\mathcal{G}}}(\boldsymbol{k},t)=\exp \{-{\it\eta}_{K}k^{2}t-\text{i}(\boldsymbol{V}_{M}\boldsymbol{\cdot }\boldsymbol{k})t\}. & & \displaystyle\end{eqnarray}$$

Equations (3.18) and (3.19) provide complete solution to the problem of white-noise ${\it\alpha}$ fluctuations, where the growth or decay of the mean magnetic field $\widetilde{\overline{\boldsymbol{B}}}$ is determined by the Green’s function $\widetilde{{\mathcal{G}}}(\boldsymbol{k},t)$ . We note some general properties below.

1. (i) Weak ${\it\alpha}$ fluctuations have ${\it\eta}_{{\it\alpha}}<{\it\eta}_{T}$ so that ${\it\eta}_{K}>0$ . In this case, modes of all wavenumbers $\boldsymbol{k}$ decay.

2. (ii) Strong ${\it\alpha}$ fluctuations have ${\it\eta}_{{\it\alpha}}>{\it\eta}_{T}$ so that ${\it\eta}_{K}<0$ . This belongs to the case when the ${\it\alpha}$ diffusivity compensates and overcomes the total (turbulent plus microscopic) diffusivity ${\it\eta}_{T}$ . This leads to the growth of modes of all wavenumbers $\boldsymbol{k}$ by the process of negative diffusion which was first obtained by Kraichnan (Reference Kraichnan1976). It may be noted that the process of negative diffusion is, in a sense, self-limiting, as it would increasingly create smaller-scale structures, where the necessary assumption of scale separation cannot continue to be valid indefinitely. This would eventually lead to breakdown of the two-scale framework.

3. (iii) The Moffatt drift velocity $\boldsymbol{V}_{M}$ contributes only to the phase and does not determine the growth or decay of the large-scale magnetic field.

Therefore, the necessary condition for dynamo action for white-noise ${\it\alpha}$ fluctuations is that they must be strong, i.e. ${\it\eta}_{K}<0$ . It is necessary to consider ${\mathcal{D}}(t)\neq {\it\delta}(t)$ to explore memory effects, which will have ${\it\tau}_{{\it\alpha}}\neq 0$ . This is studied in the next section.

4.1 Derivation of the governing equation

We note that the normalized time correlation function, ${\mathcal{D}}(t)$ , has a singular limit: i.e. $\lim _{{\it\tau}_{{\it\alpha}}\rightarrow 0}{\mathcal{D}}(t)={\mathcal{D}}_{WN}(t)={\it\delta}(t)$ . Here we wish to consider non-zero but small ${\it\tau}_{{\it\alpha}}$ , which implies that the function ${\mathcal{D}}(t)$ is significant only for times $t\leqslant {\it\tau}_{{\it\alpha}}$ and becomes negligible for larger times. We simplify the mean EMF given by (3.11), together with (3.12), by solving the time integral for small ${\it\tau}_{{\it\alpha}}$ . Since the limit $\lim _{{\it\tau}_{{\it\alpha}}\rightarrow 0}\widetilde{\overline{\mathbf{\mathscr{E}}}}(\boldsymbol{k},t)=\widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}(\boldsymbol{k},t)=\widetilde{\boldsymbol{U}}(\boldsymbol{k},0)\times \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)$ is evidently non-singular, we make the ansatz that, for small ${\it\tau}_{{\it\alpha}}$ , the EMF can be expanded in a power series in ${\it\tau}_{{\it\alpha}}$ :

(4.1) $$\begin{eqnarray}\displaystyle \widetilde{\overline{\mathbf{\mathscr{E}}}}(\boldsymbol{k},t)=\widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}(\boldsymbol{k},t)+\widetilde{\overline{\mathbf{\mathscr{E}}}}^{(1)}(\boldsymbol{k},t)+\widetilde{\overline{\mathbf{\mathscr{E}}}}^{(2)}(\boldsymbol{k},t)+\cdots \,, & & \displaystyle\end{eqnarray}$$

where $\widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}(\boldsymbol{k},t)\sim O(1)$ and $\widetilde{\overline{\mathbf{\mathscr{E}}}}^{(n)}(\boldsymbol{k},t)\sim O({\it\tau}_{{\it\alpha}}^{n})$ for $n\geqslant 1$ . Below, we verify this ansatz up to $n=1$ , for slowly varying magnetic fields.

We wish to determine $\widetilde{\overline{\mathbf{\mathscr{E}}}}(\boldsymbol{k},t)$ , which is correct to first order in ${\it\tau}_{{\it\alpha}}$ , for $t\gg {\it\tau}_{{\it\alpha}}$ . Since ${\mathcal{D}}(s)$ is strongly peaked for times $s\leqslant {\it\tau}_{{\it\alpha}}$ and becomes negligible for larger $s$ , as mentioned above, most of the contribution to the integral in (3.11) comes only from short times $0\leqslant s<{\it\tau}_{{\it\alpha}}$ . Hence in (3.11) we can (i) set the upper limit of the $s$ integral to $+\infty$ , and (ii) keep the terms inside the $\{~\}$ in the integrand up to only first order in $s$ . Below, we first work out $\widetilde{\boldsymbol{U}}(\boldsymbol{k},s)$ and $\widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t-s)$ correct up to $O(s)$ .

1. (i) $\widetilde{\boldsymbol{U}}(\boldsymbol{k},s)$ to $O(s)$ : Taylor expansion of the function $\widetilde{\boldsymbol{U}}(\boldsymbol{k},s)$ gives

   (4.2) $$\begin{eqnarray}\displaystyle \widetilde{\boldsymbol{U}}(\boldsymbol{k},s)=\widetilde{\boldsymbol{U}}(\boldsymbol{k},0)+s\left.\frac{\partial \widetilde{\boldsymbol{U}}}{\partial s}\right|_{s=0}+O(s^{2}). & & \displaystyle\end{eqnarray}$$
   Let us rewrite $\widetilde{\boldsymbol{U}}(\boldsymbol{k},s)$ , correct up to $O(s)$ , as
   (4.3) $$\begin{eqnarray}\displaystyle \widetilde{\boldsymbol{U}}(\boldsymbol{k},s)=\boldsymbol{P}(\boldsymbol{k})+s\,\boldsymbol{Q}(\boldsymbol{k}), & & \displaystyle\end{eqnarray}$$
   where
   (4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{P}(\boldsymbol{k})=\widetilde{\boldsymbol{U}}(\boldsymbol{k},0)=\text{i}\boldsymbol{k}{\it\eta}_{{\it\alpha}}+\boldsymbol{V}_{M}\quad \text{(from equation (3.15))}, & \displaystyle\end{eqnarray}$$
   (4.5) $$\begin{eqnarray}\displaystyle & \displaystyle \text{and}\quad \left.\boldsymbol{Q}(\boldsymbol{k})=\frac{\partial \widetilde{\boldsymbol{U}}}{\partial s}\right|_{s=0}=-\text{i}{\it\eta}_{T}\int \frac{\text{d}^{3}k^{\prime }}{(2{\rm\pi})^{3}}\,(\boldsymbol{k}-\boldsymbol{k}^{\prime })^{2}(\boldsymbol{k}-\boldsymbol{k}^{\prime })\widetilde{{\mathcal{A}}}(\boldsymbol{k}^{\prime }). & \displaystyle\end{eqnarray}$$
   The integrand in (4.5) can be expanded to obtain the following expression for $\boldsymbol{Q}(\boldsymbol{k})$ (see appendix A):
   (4.6) $$\begin{eqnarray}\displaystyle \boldsymbol{Q}(\boldsymbol{k})=-{\it\eta}_{T}k^{2}(\text{i}\boldsymbol{k}{\it\eta}_{{\it\alpha}}+\boldsymbol{V}_{M})-2{\it\eta}_{T}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})\boldsymbol{k}+\text{i}\boldsymbol{k}{\it\eta}_{T}C_{1}+2\text{i}{\it\eta}_{T}\boldsymbol{k}\boldsymbol{\cdot }\overleftrightarrow{\boldsymbol{C}_{2}}-{\it\eta}_{T}\boldsymbol{C}_{3},\qquad & & \displaystyle\end{eqnarray}$$
   where ${\it\eta}_{{\it\alpha}}$ and $\boldsymbol{V}_{M}$ are the constants defined in (2.4) and (3.17), respectively, which shows that they depend on the spatial correlation function ${\mathcal{A}}$ and its first spatial derivative. Both these constants have appeared already in Sridhar & Singh (Reference Sridhar and Singh2014). But $C_{1}$ (scalar), $\overleftrightarrow{\boldsymbol{C}_{2}}$ (dyad) and $\boldsymbol{C}_{3}$ (vector) are three new constants given by (see appendix A)
   (4.7a-c ) $$\begin{eqnarray}\displaystyle C_{1}=\left[{\rm\nabla}^{2}{\mathcal{A}}({\bf\xi})\right]_{{\bf\xi}=\mathbf{0}},\quad \overleftrightarrow{\boldsymbol{C}_{2}}=\left[\boldsymbol{{\rm\nabla}}\boldsymbol{{\rm\nabla}}{\mathcal{A}}({\bf\xi})\right]_{{\bf\xi}=\mathbf{0}},\quad \boldsymbol{C}_{3}=\left[{\rm\nabla}^{2}\{\boldsymbol{{\rm\nabla}}{\mathcal{A}}({\bf\xi})\}\right]_{{\bf\xi}=\mathbf{0}}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
   which depend on second- or third-order spatial derivatives of ${\mathcal{A}}$ . Thus, in general, the mean EMF will be determined by five constants $({\it\eta}_{K},\boldsymbol{V}_{M},C_{1},\overleftrightarrow{\boldsymbol{C}_{2}},\boldsymbol{C}_{3})$ , which can all be explicitly found once the form of the spatial correlation function ${\mathcal{A}}({\bf\xi})$ is chosen. To keep the analysis simple, we assume that the function ${\mathcal{A}}({\bf\xi})$ is such that its second- or higher-order spatial derivatives are negligible at ${\bf\xi}=\mathbf{0}$ , and therefore we ignore the constants $C_{1}$ , $\overleftrightarrow{\boldsymbol{C}_{2}}$ and $\boldsymbol{C}_{3}$ in the present work. Thus we write
   (4.8) $$\begin{eqnarray}\displaystyle \boldsymbol{Q}(\boldsymbol{k})=-{\it\eta}_{T}k^{2}\boldsymbol{P}(\boldsymbol{k})-2{\it\eta}_{T}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})\boldsymbol{k}. & & \displaystyle\end{eqnarray}$$

2. (ii) $\widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t-s)$ to $O(s)$ : Taylor expansion of the function $\widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t-s)$ gives

   (4.9) $$\begin{eqnarray}\displaystyle \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t-s)=\widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)-s\frac{\partial \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)}{\partial t}+\cdots \,, & & \displaystyle\end{eqnarray}$$
   where it is assumed that $|\widetilde{\overline{\boldsymbol{B}}}|\gg s|\partial \widetilde{\overline{\boldsymbol{B}}}/\partial t|\gg s^{2}|\partial ^{2}\widetilde{\overline{\boldsymbol{B}}}/\partial t^{2}|\gg s^{3}|\partial ^{3}\widetilde{\overline{\boldsymbol{B}}}/\partial t^{3}|$ , etc. In order to find $\widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t-s)$ , which is correct up to $O(s)$ , we need $\partial \widetilde{\overline{\boldsymbol{B}}}/\partial t$ in (4.9) only up to $O(1)$ . Using (4.1) together with (3.14) and (3.15) in (3.13) we find
   (4.10) $$\begin{eqnarray}\displaystyle \left.\frac{\partial \widetilde{\overline{\boldsymbol{B}}}}{\partial t}\right|_{O(1)}=\text{i}\boldsymbol{k}\times \widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}(\boldsymbol{k},t)-{\it\eta}_{T}k^{2}\widetilde{\overline{\boldsymbol{B}}}=-[\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{P}+{\it\eta}_{T}k^{2}]\widetilde{\overline{\boldsymbol{B}}}, & & \displaystyle\end{eqnarray}$$
   which is substituted in (4.9) to obtain
   (4.11) $$\begin{eqnarray}\displaystyle \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t-s)=\widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)[1+s(\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{P}+{\it\eta}_{T}k^{2})]+O(s^{2}). & & \displaystyle\end{eqnarray}$$
   Then, from the above analysis, we may write
   (4.12) $$\begin{eqnarray}\displaystyle & & \displaystyle \widetilde{\boldsymbol{U}}(\boldsymbol{k},s)\times \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t-s)=\boldsymbol{P}(\boldsymbol{k})\times \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,s[(\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{P}+{\it\eta}_{T}k^{2})\boldsymbol{P}(\boldsymbol{k})\times \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)+\boldsymbol{Q}(\boldsymbol{k})\times \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)]+O(s^{2}).\end{eqnarray}$$
   Noting that $\widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}(\boldsymbol{k},t)=\boldsymbol{P}(\boldsymbol{k})\times \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)$ and using the definition of $\boldsymbol{Q}(\boldsymbol{k})$ as given in (4.8), we get
   (4.13) $$\begin{eqnarray}\displaystyle \{~\}\text{of (3.11)} & = & \displaystyle \widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}(\boldsymbol{k},t)\nonumber\\ \displaystyle & & \displaystyle +\,s\{(\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{P})\,\widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}-2{\it\eta}_{T}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})\,\boldsymbol{k}\times \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)\}+O(s^{2}).\end{eqnarray}$$
   Using (4.13) and the properties of ${\mathcal{D}}(t)$ , given in (2.4) and (2.5), we solve the integral over $s$ in (3.11) to obtain the mean EMF:
   (4.14) $$\begin{eqnarray}\displaystyle \widetilde{\overline{\mathbf{\mathscr{E}}}}(\boldsymbol{k},t)=\widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}(\boldsymbol{k},t)+{\it\tau}_{{\it\alpha}}\{(\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{P})\,\widetilde{\overline{\mathbf{\mathscr{E}}}}_{WN}-2{\it\eta}_{T}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})\,\boldsymbol{k}\times \widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)\} & & \displaystyle\end{eqnarray}$$
   accurate to $O({\it\tau}_{{\it\alpha}})$ , which verifies the ansatz of (4.1) up to $n=1$ , as claimed. We note that (4.14) for the mean EMF is valid only when the large-scale magnetic field is slowly varying. To lowest order, this condition can be stated as $|\widetilde{\overline{\boldsymbol{B}}}|\gg {\it\tau}_{{\it\alpha}}|\partial \widetilde{\overline{\boldsymbol{B}}}/\partial t|$ . Using (4.10) for $\partial \widetilde{\overline{\boldsymbol{B}}}/\partial t$ , we see that the sufficient condition for (4.14) to be valid is that the following two dimensionless quantities be small:
   (4.15) $$\begin{eqnarray}\displaystyle |{\it\eta}_{K}k^{2}{\it\tau}_{{\it\alpha}}|\ll 1,\quad |kV_{M}{\it\tau}_{{\it\alpha}}|\ll 1. & & \displaystyle\end{eqnarray}$$
   Substituting (4.14) in (3.13) we obtain the following PDE governing the evolution of the large-scale magnetic field:
   (4.16) $$\begin{eqnarray}\displaystyle \frac{\partial \widetilde{\overline{\boldsymbol{B}}}}{\partial t}=\{[-{\it\eta}_{K}k^{2}-{\it\eta}_{{\it\alpha}}^{2}k^{4}{\it\tau}_{{\it\alpha}}+(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})^{2}{\it\tau}_{{\it\alpha}}]+\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})[2({\it\eta}_{{\it\alpha}}+{\it\eta}_{T})k^{2}{\it\tau}_{{\it\alpha}}-1]\}\,\widetilde{\overline{\boldsymbol{B}}}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

4.2 The Kraichnan problem with non-zero ${\it\tau}_{{\it\alpha}}$

First we consider the Kraichnan problem and extend it to include finite ${\it\tau}_{{\it\alpha}}$ in order to understand the combined effect of the ${\it\alpha}$ fluctuations when $\boldsymbol{V}_{M}=\mathbf{0}$ , but ${\it\eta}_{{\it\alpha}}>0$ and ${\it\tau}_{{\it\alpha}}>0$ . SS14 defined a length scale whose corresponding wavenumber was

(4.17) $$\begin{eqnarray}\displaystyle k_{{\it\alpha}}=({\it\eta}_{{\it\alpha}}{\it\tau}_{{\it\alpha}})^{-1/2}>0. & & \displaystyle\end{eqnarray}$$

This is used to define high or low wavenumbers ( $k$ ); $|k|>k_{{\it\alpha}}$ are called high wavenumbers, and $|k|<k_{{\it\alpha}}$ are called low wavenumbers. When $\boldsymbol{V}_{M}=\mathbf{0}$ , one of the two conditions in (4.15) are met trivially, and the other one implies that $|k|$ must be small enough such that $|{\it\eta}_{K}k^{2}{\it\tau}_{{\it\alpha}}|\ll 1$ . Using (4.17) and setting $\boldsymbol{V}_{M}=\mathbf{0}$ in (4.16), we get

(4.18) $$\begin{eqnarray}\displaystyle \frac{\partial \widetilde{\overline{\boldsymbol{B}}}}{\partial t}=-{\it\eta}_{K}k^{2}\left[1+\frac{{\it\eta}_{{\it\alpha}}}{{\it\eta}_{K}}\left(\frac{k}{k_{{\it\alpha}}}\right)^{2}\right]\widetilde{\overline{\boldsymbol{B}}},\quad \text{with}~\boldsymbol{k}\boldsymbol{\cdot }\widetilde{\overline{\boldsymbol{B}}}=0. & & \displaystyle\end{eqnarray}$$

The solutions are of the exponential form, $\widetilde{\overline{\boldsymbol{B}}}(\boldsymbol{k},t)=\widetilde{\overline{\boldsymbol{B}}}_{0}(\boldsymbol{k})\exp ({\it\gamma}t)$ , where $\boldsymbol{k}\boldsymbol{\cdot }\widetilde{\overline{\boldsymbol{B}}}_{0}(\boldsymbol{k})=0$ . Substituting this in (4.18), we get the growth rate as

(4.19) $$\begin{eqnarray}\displaystyle {\it\gamma}=-({\it\eta}_{K}k_{{\it\alpha}}^{2})\left(\frac{k}{k_{{\it\alpha}}}\right)^{2}\left[1+\frac{{\it\eta}_{{\it\alpha}}}{{\it\eta}_{K}}\left(\frac{k}{k_{{\it\alpha}}}\right)^{2}\right],\quad \text{when}~|{\it\eta}_{K}k^{2}{\it\tau}_{{\it\alpha}}|\ll 1. & & \displaystyle\end{eqnarray}$$

Following SS14 we normalize the growth rates by the characteristic frequency $({\it\sigma})$ , which is defined as

(4.20) $$\begin{eqnarray}\displaystyle {\it\sigma}=|{\it\eta}_{K}|k_{{\it\alpha}}^{2}=\left(\frac{|{\it\eta}_{T}-{\it\eta}_{{\it\alpha}}|}{{\it\eta}_{{\it\alpha}}}\right)\frac{1}{{\it\tau}_{{\it\alpha}}}\geqslant 0. & & \displaystyle\end{eqnarray}$$

Below we consider the behaviour of the growth rate as a function of the wavenumber, for weak and strong ${\it\alpha}$ fluctuations, and show this in figure 1 where we also show the corresponding results of SS14 to illustrate some qualitative differences. (We note, however, that the expressions for the growth rates in SS14 were simpler and did not explicitly depend on the factor ${\it\eta}_{{\it\alpha}}/|{\it\eta}_{K}|$ .)

Weak ${\it\alpha}$ fluctuations: This case has $0<{\it\eta}_{{\it\alpha}}<{\it\eta}_{T}$ , so that ${\it\eta}_{K}$ is positive. The normalized growth rate ( ${\it\gamma}/{\it\sigma}$ ) may be expressed as

(4.21) $$\begin{eqnarray}\displaystyle \frac{{\it\gamma}}{{\it\sigma}}=-\left(\frac{k}{k_{{\it\alpha}}}\right)^{2}\left[1+\frac{{\it\eta}_{{\it\alpha}}}{|{\it\eta}_{K}|}\left(\frac{k}{k_{{\it\alpha}}}\right)^{2}\right],\quad \text{when}~|k|\ll \frac{k_{{\it\alpha}}}{\sqrt{{\it\sigma}{\it\tau}_{{\it\alpha}}}}. & & \displaystyle\end{eqnarray}$$

The growth rate is negative definite for finite $k$ and it is a monotonically decreasing function of the wavenumber (solid curve in figure 1). This is even qualitatively different from SS14 where the high wavenumbers always grow for weak ${\it\alpha}$ fluctuations (dash-dotted curve). This also highlights the fact that the inclusion of the resistive term in the fluctuating field equation is a non-trivial extension of the SS14 model.

Figure 1.

Growth rate ${\it\gamma}/{\it\sigma}$ as a function of $|k/k_{{\it\alpha}}|$ , when $\boldsymbol{V}_{M}=\mathbf{0}$ . Weak ( ${\it\eta}_{K}>0$ ) and strong ( ${\it\eta}_{K}<0$ )  ${\it\alpha}$ fluctuations correspond to solid and dashed curves, respectively. Results of SS14 are also shown; dash-dotted and triple-dot-dashed curves correspond to weak and strong ${\it\alpha}$ fluctuations, respectively.

Strong ${\it\alpha}$ fluctuations: This case has $0<{\it\eta}_{T}<{\it\eta}_{{\it\alpha}}$ , so that ${\it\eta}_{K}$ is negative. The normalized growth rate ( ${\it\gamma}/{\it\sigma}$ ) may be expressed as

(4.22) $$\begin{eqnarray}\displaystyle \frac{{\it\gamma}}{{\it\sigma}}=+\left(\frac{k}{k_{{\it\alpha}}}\right)^{2}\left[1-\frac{{\it\eta}_{{\it\alpha}}}{|{\it\eta}_{K}|}\left(\frac{k}{k_{{\it\alpha}}}\right)^{2}\right],\quad \text{when}~|k|\ll \frac{k_{{\it\alpha}}}{\sqrt{{\it\sigma}{\it\tau}_{{\it\alpha}}}}. & & \displaystyle\end{eqnarray}$$

In this case the growth rate can be positive for a range of wavenumbers, before becoming negative at larger wavenumbers (dashed curve in figure 1). This is qualitatively similar to the results of SS14 (triple-dot-dashed curve) in this regime and the differences at large wavenumbers arise due to reasons mentioned above.

Thus the necessary condition for dynamo action is that the ${\it\alpha}$ fluctuations must be strong, i.e. ${\it\eta}_{K}<0$ . Recall that this is, in a sense, similar to the white-noise or the original Kraichnan model, but we note the following important difference: in the white-noise case, the growth rate increases monotonically with wavenumber ( $k$ ) for strong ${\it\alpha}$ fluctuations, with largest allowed wavenumbers (smallest allowed length scales) growing the fastest; whereas, here, we find that the growth rate is a non-monotonic function of $k$ , and, as a result, there exists a wavenumber cutoff at some large $k$ beyond which the growth rate turns negative. This makes it a special dynamo as the magnetic power at smallest length scales would be suppressed due to the existence of the wavenumber cutoff (see dashed curve in figure 1), thus enabling a bona fide large-scale dynamo, unlike the white-noise case, where much of the magnetic power lies at the smallest allowed length scales.

It would indeed be interesting if even weak ${\it\alpha}$ fluctuations could lead to large-scale dynamo action. In the next section we explore the combined effect of $\boldsymbol{V}_{M}\neq 0$ and ${\it\tau}_{{\it\alpha}}\neq 0$ , and ask the following question: When both ${\it\tau}_{{\it\alpha}}$ and $\boldsymbol{V}_{M}$ are non-zero, could large-scale magnetic fields grow even when ${\it\alpha}$ fluctuations are weak, i.e. when ${\it\eta}_{K}>0$ ?