2 The model

Let us begin with the standard dynamo equation in the presence of a background linear shear flow, $\boldsymbol{V}=SX_{1}\boldsymbol{e}_{2}$ , where meso-scale magnetic field $\boldsymbol{B}$ evolves according to (see, Moffatt Reference Moffatt1978; Krause & Rädler Reference Krause and Rädler1980; Brandenburg & Subramanian Reference Brandenburg and Subramanian2005; Sridhar & Singh Reference Sridhar and Singh2014):

(2.1) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+SX_{1}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X_{2}}\right)\boldsymbol{B}-SB_{1}\boldsymbol{e}_{2}=\unicode[STIX]{x1D735}\times [\unicode[STIX]{x1D6FC}(\boldsymbol{X},\unicode[STIX]{x1D70F})\boldsymbol{B}]+\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{B};\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0.\end{eqnarray}$$

Here we follow the same notation as in Sridhar & Singh (Reference Sridhar and Singh2014) where the position vector is denoted by $\boldsymbol{X}=(X_{1},X_{2},X_{3})$ with components given in a fixed orthonormal frame $(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3})$ , and $\unicode[STIX]{x1D70F}$ is the time variable. The shear rate, $S$ , and total diffusivity, $\unicode[STIX]{x1D702}_{T}$ , are treated as constant parameters, whereas $\unicode[STIX]{x1D6FC}(\boldsymbol{X},\unicode[STIX]{x1D70F})$ provides a measure of meso-scale kinetic helicity of turbulence. We recall that (2.1) governing the dynamics of a meso-scale magnetic field is obtained by averaging over an ensemble of random velocity fields, $\{\boldsymbol{v}(\boldsymbol{X},\unicode[STIX]{x1D70F})\}$ , which are assumed to have zero-mean isotropic fluctuations, uniform and constant kinetic energy density per unit mass and slow helicity fluctuations.

We employ here the double-averaging scheme (Kraichnan Reference Kraichnan1976; Moffatt Reference Moffatt1983; Sokolov Reference Sokolov1997) under which $\unicode[STIX]{x1D6FC}(\boldsymbol{X},\unicode[STIX]{x1D70F})$ itself is a random variable of space and time, thus making (2.1) a stochastic partial differential equation. It is drawn from a superensemble with zero mean, $\overline{\unicode[STIX]{x1D6FC}(\boldsymbol{X},\unicode[STIX]{x1D70F})}=0$ . It’s statistical properties are given below in (2.11). Next, we separate the meso-scale field, $\boldsymbol{B}=\overline{\boldsymbol{B}}+\boldsymbol{b}$ , into large-scale, $\overline{\boldsymbol{B}}$ , and fluctuating, $\boldsymbol{b}$ , components, where the superensemble average of $\boldsymbol{b}$ vanishes, i.e.  $\overline{\boldsymbol{b}}=\mathbf{0}$ . The governing equation for the large-scale magnetic field $\overline{\boldsymbol{B}}$ can thus be obtained by Reynolds averaging the (2.1) over the superensemble:

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+SX_{1}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X_{2}}\right)\overline{\boldsymbol{B}}-S\overline{B_{1}}\boldsymbol{e}_{2}=\unicode[STIX]{x1D735}\times \overline{\boldsymbol{{\mathcal{E}}}}+\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D6FB}^{2}\overline{\boldsymbol{B}},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\overline{\boldsymbol{B}}=0, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \text{where }\overline{\boldsymbol{{\mathcal{E}}}}=\overline{\unicode[STIX]{x1D6FC}(\boldsymbol{X},\unicode[STIX]{x1D70F})\boldsymbol{b}(\boldsymbol{X},\unicode[STIX]{x1D70F})}. & \displaystyle\end{eqnarray}$$

In order to determine the mean electromotive force (EMF), we must solve for the fluctuating field $\boldsymbol{b}$ , which evolves as:

(2.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+SX_{1}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X_{2}}\right)\boldsymbol{b}-Sb_{1}\boldsymbol{e}_{2}=\unicode[STIX]{x1D735}\times [\unicode[STIX]{x1D6FC}\overline{\boldsymbol{B}}]+\unicode[STIX]{x1D735}\times [\unicode[STIX]{x1D6FC}\boldsymbol{b}-\overline{\unicode[STIX]{x1D6FC}\boldsymbol{b}}]+\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{b},\\ \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{b}=0,\quad \text{with initial condition }\boldsymbol{b}(\boldsymbol{X},0)=\mathbf{0}.\end{array}\right\}\end{eqnarray}$$

As (2.2) and (2.4) involve inhomogeneous terms, it is convenient to solve these in the shearing coordinates $(\boldsymbol{x},t)$ which are expressed in terms of the laboratory coordinates $(\boldsymbol{X},\unicode[STIX]{x1D70F})$ as (see, Sridhar & Subramanian Reference Sridhar and Subramanian2009a ; Sridhar & Singh Reference Sridhar and Singh2010):

(2.5a-d ) $$\begin{eqnarray}x_{1}=X_{1};\quad x_{2}=X_{2}-S\,\unicode[STIX]{x1D70F}\,X_{1};\quad x_{3}=X_{3};\quad t=\unicode[STIX]{x1D70F}.\end{eqnarray}$$

The inverse transformation is:

(2.6a-d ) $$\begin{eqnarray}X_{1}=x_{1};\quad X_{2}=x_{2}+S\,t\,x_{1};\quad X_{3}=x_{3};\quad \unicode[STIX]{x1D70F}=t.\end{eqnarray}$$

Now we can write equations (2.2)–(2.4) in terms of new fields that are functions of $\boldsymbol{x}$ and $t$ : $\overline{\boldsymbol{H}}(\boldsymbol{x},t)=\overline{\boldsymbol{B}}(\boldsymbol{X},\unicode[STIX]{x1D70F})$ ; $\boldsymbol{h}(\boldsymbol{x},t)=\boldsymbol{b}(\boldsymbol{X},\unicode[STIX]{x1D70F})$ ; $a(\boldsymbol{x},t)=\unicode[STIX]{x1D6FC}(\boldsymbol{X},\unicode[STIX]{x1D70F})$ ; and $\overline{\boldsymbol{E}}(\boldsymbol{x},t)=\overline{\boldsymbol{{\mathcal{E}}}}(\boldsymbol{X},\unicode[STIX]{x1D70F})$ . Equations (2.2)–(2.4) then take the form (Sridhar & Singh Reference Sridhar and Singh2014):

(2.7a-c ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{\boldsymbol{H}}}{\unicode[STIX]{x2202}t}-S\overline{H_{1}}\boldsymbol{e}_{2}=\unicode[STIX]{x1D735}\times \overline{\boldsymbol{E}}+\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D6FB}^{2}\overline{\boldsymbol{H}},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\overline{\boldsymbol{H}}=0,\quad \overline{\boldsymbol{E}}=\overline{a\boldsymbol{h}};\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{h}}{\unicode[STIX]{x2202}t}-Sh_{1}\boldsymbol{e}_{2}=\unicode[STIX]{x1D735}\times [a\overline{\boldsymbol{H}}]+\unicode[STIX]{x1D735}\times [a\boldsymbol{h}-\overline{a\boldsymbol{h}}]+\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{h},\\ \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{h}=0,\quad \text{with initial condition }\boldsymbol{h}(\boldsymbol{x},0)=\mathbf{0};\end{array}\right\}\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\text{where }\unicode[STIX]{x1D735}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{x}}-\boldsymbol{e}_{1}St\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{2}}\quad \text{is a time-dependent operator.}\end{eqnarray}$$

We complete defining our model by specifying the statistics of the $\unicode[STIX]{x1D6FC}$ fluctuations. We follow the exact same approach as given in detail in Sridhar & Singh (Reference Sridhar and Singh2014) and recall here only some key relevant points:

1. (i) Shear flows possess a natural symmetry known as Galilean invariance, relating the measurements of correlation functions made by comoving observers whose origins with respect to the laboratory frame translate with the same speed as that of the linear shear flow (Sridhar & Subramanian Reference Sridhar and Subramanian2009a ,Reference Sridhar and Subramanian b ).

2. (ii) Here we are more interested in time-stationary Galilean-invariant $\unicode[STIX]{x1D6FC}$ statistics, which can be expressed in the shearing frame as (see Sridhar & Singh (Reference Sridhar and Singh2014) for a derivation):

   (2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{a(\boldsymbol{x},t)a(\boldsymbol{x}^{\prime },t^{\prime })}=2{\mathcal{A}}\left(\boldsymbol{x}-\boldsymbol{x}^{\prime }+St^{\prime }(x_{1}-x_{1}^{\prime })\boldsymbol{e}_{2}\right){\mathcal{D}}(t-t^{\prime }),\quad \text{with} & \displaystyle\end{eqnarray}$$
   (2.11) $$\begin{eqnarray}\displaystyle & \displaystyle 2\int _{0}^{\infty }{\mathcal{D}}(t)\,\text{d}t=1,\quad {\mathcal{A}}(\mathbf{0})=\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}\geqslant 0. & \displaystyle\end{eqnarray}$$
   The correlation time for the $\unicode[STIX]{x1D6FC}$ fluctuations is defined as,
   (2.12) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}=2\int _{0}^{\infty }\text{d}t\,t\,{\mathcal{D}}(t).\end{eqnarray}$$
   The intrinsic anisotropy of the $\unicode[STIX]{x1D6FC}$ fluctuations is measured by the Moffatt drift velocity,
   (2.13) $$\begin{eqnarray}\boldsymbol{V}_{M}=-\left(\frac{\unicode[STIX]{x2202}{\mathcal{A}}(\unicode[STIX]{x1D743})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D743}}\right)_{\unicode[STIX]{x1D743}=\mathbf{0}}=\int _{0}^{\infty }\overline{\unicode[STIX]{x1D6FC}(\boldsymbol{X},\unicode[STIX]{x1D70F})\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}(\boldsymbol{X},0)}\,\text{d}\unicode[STIX]{x1D70F}.\end{eqnarray}$$

In the above, we noted two properties of the spatial correlation function, ${\mathcal{A}}$ , namely its value $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ and gradient $\boldsymbol{V}_{M}$ at zero separation. But we can associate one or more length scales relating to its variation in $\unicode[STIX]{x1D743}$ -space. In the estimates made below we use a single scale $\ell$ to denote this correlation length. The temporal correlation function, ${\mathcal{D}}$ is characterized by a single correlation time, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ . Hence the basic constant parameters of our model are $(\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}},\boldsymbol{V}_{M},\ell ,\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}})$ .

3 Evolution equation for the large-scale magnetic field

Here we derive a closed equation for the large-scale magnetic field by exploiting the homogeneity of the problem in the sheared coordinates $\boldsymbol{x}$ by working with its conjugate Fourier variable $\boldsymbol{k}$ . Let $\widetilde{Q}(\boldsymbol{k},t)=\int \text{d}^{3}x\exp (-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x})Q(\boldsymbol{x},t)$ be the Fourier transform of any quantity $Q(\boldsymbol{x},t)$ , with similar definition in terms of laboratory-frame coordinates, where $\boldsymbol{K}$ denotes the conjugate variable to $\boldsymbol{X}$ . Note that the laboratory-frame wavevector $\boldsymbol{K}$ is time dependent and can be expressed in terms of sheared wavevectors $\boldsymbol{k}$ as, $\boldsymbol{K}(\boldsymbol{k},t)=(k_{1}-Stk_{2},k_{2},k_{3})$ ; see (2.9). We need to first solve for $\widetilde{\boldsymbol{h}}(\boldsymbol{k},t)$ as a functional of $\widetilde{a}(\boldsymbol{k},t)$ and $\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t)$ . This is in general a complicated problem by itself, so for a first attempt we use the standard approach of the first-order smoothing approximation (FOSA) wherein the term, $\unicode[STIX]{x1D735}\times [a\boldsymbol{h}-\overline{a\boldsymbol{h}}]$ , is dropped in (2.8). Analogous to (7.124) of Moffatt (Reference Moffatt1978), the condition for FOSA to be valid is:

(3.1a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}}{\ell ^{2}}\ll 1,\quad \text{OR}\quad \frac{\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x1D702}_{T}}\ll \frac{\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}}{\ell ^{2}},\end{eqnarray}$$

where we recall that $\ell$ is the correlation length of the $\unicode[STIX]{x1D6FC}$ fluctuations. The first term of these condition comes from the short correlation assumption by comparing $\unicode[STIX]{x2202}\boldsymbol{h}/\unicode[STIX]{x2202}t$ with $\unicode[STIX]{x1D735}\times [a\boldsymbol{h}-\overline{a\boldsymbol{h}}]$ ; $\unicode[STIX]{x2202}\boldsymbol{h}/\unicode[STIX]{x2202}t$ is of the order $O(h_{0}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}})$ and $\unicode[STIX]{x1D735}\times [a\boldsymbol{h}-\overline{a\boldsymbol{h}}]$ is of the order $O(\unicode[STIX]{x1D6FC}_{0}h_{0}/\ell )$ . For FOSA to be valid, $\unicode[STIX]{x1D6FC}_{0}h_{0}/\ell \ll h_{0}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ . Using $\unicode[STIX]{x1D6FC}_{0}^{2}\sim \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ (from (2.11)), we can write the first condition in (3.1). Similarly, the second condition comes from comparing $\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{h}$ with $\unicode[STIX]{x1D735}\times [a\boldsymbol{h}-\overline{a\boldsymbol{h}}]$ in (2.8); $\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{h}$ is of the order $O(\unicode[STIX]{x1D702}_{T}h_{0}/\ell ^{2})$ , and so, for FOSA to be valid, $\unicode[STIX]{x1D702}_{T}h_{0}/\ell ^{2}\gg \unicode[STIX]{x1D6FC}_{0}h_{0}/\ell$ , which yields the second condition in (3.1) after rearranging and squaring the terms.

Then the fluctuating magnetic field evolves as:

(3.2) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D6FB}^{2}\right)\boldsymbol{h}-Sh_{1}\boldsymbol{e}_{2}=\unicode[STIX]{x1D735}\times \boldsymbol{M},\end{eqnarray}$$

where $\boldsymbol{M}(\boldsymbol{x},t)=a(\boldsymbol{x},t)\overline{\boldsymbol{H}}(\boldsymbol{x},t)$ is a source term for the fluctuating magnetic field, and $\unicode[STIX]{x1D735}$ is the time-dependent operator defined in (2.9). The FOSA solution for the fluctuating magnetic field in the Fourier space is given by (see appendix A for a derivation),

(3.3) $$\begin{eqnarray}\widetilde{\boldsymbol{h}}(\boldsymbol{k},t)=\int _{0}^{t}\text{d}t^{\prime }\,\widetilde{G}_{\unicode[STIX]{x1D702}_{T}}(\boldsymbol{k},t,t^{\prime })\!\left\{\text{i}\boldsymbol{K}(\boldsymbol{k},t^{\prime })\times \widetilde{\boldsymbol{M}}(\boldsymbol{k},t^{\prime })+\boldsymbol{e}_{2}S(t-t^{\prime })[\text{i}\boldsymbol{K}(\boldsymbol{k},t^{\prime })\times \widetilde{\boldsymbol{M}}(\boldsymbol{k},t^{\prime })]_{1}\right\}\!,\end{eqnarray}$$

where the sheared Green’s function in Fourier space:

(3.4) $$\begin{eqnarray}\widetilde{G}_{\unicode[STIX]{x1D702}_{T}}(\boldsymbol{k},t,t^{\prime })=\exp \left[-\unicode[STIX]{x1D702}_{T}\left(k^{2}(t-t^{\prime })-Sk_{1}k_{2}(t^{2}-t^{\prime 2})+\frac{S^{2}}{3}k_{2}^{2}(t^{3}-t^{\prime 3})\right)\right]\!.\end{eqnarray}$$

This is derived in Sridhar & Singh (Reference Sridhar and Singh2010). It may be readily verified that (3.3) satisfies both constraints, $\boldsymbol{K}\boldsymbol{\cdot }\widetilde{\boldsymbol{h}}=0$ and $\widetilde{\boldsymbol{h}}(\boldsymbol{k},0)=\mathbf{0}$ . By making use of (3.3), and time-stationary Galilean-invariant statistics for the $\unicode[STIX]{x1D6FC}$ fluctuations in Fourier space (see appendix B), we obtain the following expression for the mean EMF in Fourier space, after some straightforward algebra (see appendix C for a derivation):

(3.5) $$\begin{eqnarray}\widetilde{\overline{\boldsymbol{E}}}(\boldsymbol{k},t)=2\int _{0}^{t}\text{d}t^{\prime }{\mathcal{D}}(t-t^{\prime })\!\left\{\widetilde{\boldsymbol{U}}(\boldsymbol{k},t,t^{\prime })\times \widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t^{\prime })+\boldsymbol{e}_{2}S(t-t^{\prime })[\widetilde{\boldsymbol{U}}(\boldsymbol{k},t,t^{\prime })\times \widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t^{\prime })]_{1}\right\}\!,\end{eqnarray}$$

where

(3.6) $$\begin{eqnarray}\widetilde{\boldsymbol{U}}(\boldsymbol{k},t,t^{\prime })=\int \frac{\text{d}^{3}k^{\prime }}{(2\unicode[STIX]{x03C0})^{3}}\widetilde{G}_{\unicode[STIX]{x1D702}_{T}}(\boldsymbol{k}-\boldsymbol{k}^{\prime },t,t^{\prime })\text{i}\boldsymbol{K}(\boldsymbol{k}-\boldsymbol{k}^{\prime },t^{\prime })\widetilde{{\mathcal{A}}}(\boldsymbol{K}(\boldsymbol{k}^{\prime },t^{\prime })),\end{eqnarray}$$

is a complex velocity field.

Fourier transforming (2.7), the equation governing the large-scale field is:

(3.7a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\widetilde{\overline{\boldsymbol{H}}}}{\unicode[STIX]{x2202}t}-S\widetilde{\overline{H}_{1}}\boldsymbol{e}_{2}=\text{i}\boldsymbol{K}(\boldsymbol{k},t)\times \widetilde{\overline{\boldsymbol{E}}}-\unicode[STIX]{x1D702}_{T}K^{2}(\boldsymbol{k},t)\widetilde{\overline{\boldsymbol{H}}},\quad \boldsymbol{K}(\boldsymbol{k},t)\boldsymbol{\cdot }\widetilde{\overline{\boldsymbol{H}}}=0.\end{eqnarray}$$

Thus the set of (3.5)–(3.7) describes the evolution of the large-scale magnetic field, $\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t)$ in terms of a closed, linear integro-differential equation, where both shear strength, $S$ , and the $\unicode[STIX]{x1D6FC}$ -correlation time, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ , are treated non-perturbatively. This is the principal general result of this paper, but solving these in full generality is beyond the scope of the present work, and we next pursue these equations analytically by making useful approximations.

3.1 White-noise $\unicode[STIX]{x1D6FC}$ fluctuations

It is useful to recall basic properties of an exactly solvable limit of $\unicode[STIX]{x1D6FF}$ -correlated-in-time $\unicode[STIX]{x1D6FC}$ fluctuations when the normalized correlation function ${\mathcal{D}}_{\text{WN}}(t)=\unicode[STIX]{x1D6FF}(t)$ which is the Dirac delta function, giving $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}=0$ from (2.12). Using this in (3.5) and (3.6), and noting that $\widetilde{G}_{\unicode[STIX]{x1D702}_{T}}(\boldsymbol{k}-\boldsymbol{k}^{\prime },t,t)=1$ from (3.4), we find the mean EMF:

(3.8) $$\begin{eqnarray}\widetilde{\overline{\boldsymbol{E}}}_{\text{WN}}(\boldsymbol{k},t)=\widetilde{\boldsymbol{U}}_{\text{WN}}(\boldsymbol{k},t)\times \widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t)\quad \text{with }\widetilde{\boldsymbol{U}}_{\text{WN}}(\boldsymbol{k},t)=\text{i}\boldsymbol{K}(\boldsymbol{k},t)\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}+\boldsymbol{V}_{M},\end{eqnarray}$$

where the $\unicode[STIX]{x1D6FC}$ -diffusivity, $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}={\mathcal{A}}(\mathbf{0})$ , is given in (2.11) and the Moffatt drift velocity $\boldsymbol{V}_{M}$ is defined in (2.13). The Kraichnan diffusivity, $\unicode[STIX]{x1D702}_{K}$ , is defined as, $\unicode[STIX]{x1D702}_{K}=\unicode[STIX]{x1D702}_{T}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ . Using these in (3.7) leads to the solution for the large-scale magnetic field (see Sridhar & Singh (Reference Sridhar and Singh2014) for more details):

(3.9a,b ) $$\begin{eqnarray}\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t)=\widetilde{{\mathcal{G}}}(\boldsymbol{k},t)[\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},0)+\boldsymbol{e}_{2}St\,\widetilde{\overline{H}}_{1}(\boldsymbol{k},0)],\quad \boldsymbol{k}\boldsymbol{\cdot }\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},0)=0,\end{eqnarray}$$

where

(3.10) $$\begin{eqnarray}\displaystyle \widetilde{{\mathcal{G}}}(\boldsymbol{k},t) & = & \displaystyle \exp \left\{-\int _{0}^{t}\,\text{d}t^{\prime }[\unicode[STIX]{x1D702}_{K}K^{2}(\boldsymbol{k},t^{\prime })+\text{i}\boldsymbol{V}_{M}\boldsymbol{\cdot }\boldsymbol{K}(\boldsymbol{k},t^{\prime })]\right\}\nonumber\\ \displaystyle & = & \displaystyle \exp \{-\unicode[STIX]{x1D702}_{K}[k^{2}t-Sk_{1}k_{2}t^{2}+(S^{2}/3)k_{2}^{2}t^{3}]-\text{i}[(\boldsymbol{V}_{M}\boldsymbol{\cdot }\boldsymbol{k})t-(S/2)V_{M1}k_{2}t^{2}]\}.\qquad\end{eqnarray}$$

This solution is identical to the one obtained in Sridhar & Singh (Reference Sridhar and Singh2014). Thus we find that the inclusion of the turbulent diffusion term in determining the mean EMF makes no difference for the dynamo solution in the white-noise limit. In agreement with earlier findings (Kraichnan Reference Kraichnan1976; Moffatt Reference Moffatt1978; Sridhar & Singh Reference Sridhar and Singh2014), we see from above that the $\unicode[STIX]{x1D6FC}$ -diffusivity causes a reduction in the turbulent diffusion of the fields, and if it is sufficiently strong, i.e. when $\unicode[STIX]{x1D702}_{K}<0$ , this can lead to an instability giving growth of the large-scale magnetic field. Also, the Moffatt drift does not couple to the dynamo growth/decay and contributes only to the phase.

4 Axisymmetric large-scale dynamo equation with finite $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$

We now turn to the principal aim of this work where we are more interested in exploring the possibility of a large-scale dynamo even when the $\unicode[STIX]{x1D6FC}$ fluctuations are weak, i.e. when $\unicode[STIX]{x1D702}_{K}>0$ , by taking the memory effects into account. Assuming a small but finite correlation time for $\unicode[STIX]{x1D6FC}$ fluctuations, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}\neq 0$ , we reduce the general set of (3.5)–(3.7) into a partial differential equation governing the dynamics of the large-scale magnetic field which evolves over times much larger than $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ . In this case, the normalized time correlation function, ${\mathcal{D}}(t)$ , is significant only for times $t\leqslant \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ and it becomes negligible for larger times. The generalized mean EMF as given in (3.5) involves a time integral which can be solved under the small $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ approximation.

Since the limit $\lim _{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}\rightarrow 0}\widetilde{\overline{\boldsymbol{E}}}(\boldsymbol{k},t)=\widetilde{\overline{\boldsymbol{E}}}_{\text{WN}}(\boldsymbol{k},t)$ , given by (3.8), is non-singular, we proceed by making the following ansatz where, for small $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ , the mean EMF can be expanded in a power series in $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ as:

(4.1) $$\begin{eqnarray}\widetilde{\overline{\boldsymbol{E}}}(\boldsymbol{k},t)=\widetilde{\overline{\boldsymbol{E}}}_{\text{WN}}(\boldsymbol{k},t)+\widetilde{\overline{\boldsymbol{E}}}^{(1)}(\boldsymbol{k},t)+\widetilde{\overline{\boldsymbol{E}}}^{(2)}(\boldsymbol{k},t)+\ldots\end{eqnarray}$$

where $\widetilde{\overline{\boldsymbol{E}}}_{\text{WN}}(\boldsymbol{k},t)\sim O(1)$ and $\widetilde{\overline{\boldsymbol{E}}}^{(n)}(\boldsymbol{k},t)\sim O(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}^{n})$ for $n\geqslant 1$ . Below we verify this ansatz up to $n=1$ , for slowly varying magnetic fields. From (3.5) we determine $\widetilde{\overline{\boldsymbol{E}}}(\boldsymbol{k},t)$ to first order in $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ , for $t\gg \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ , by (i) changing the integration variable from $t^{\prime }$ to $s=t-t^{\prime }$ ; (ii) setting the upper limit of the time integral to $+\infty$ , since ${\mathcal{D}}(s)$ is significant only for times $s\leqslant \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ as mentioned above, suggesting that only short times $0\leqslant s<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ contribute appreciably to the integral in (3.5); and (iii) keeping the terms inside the $\{\;\}$ in the integrand of (3.5) up to only first order in $s$ . To be able to expand in $s$ , we need to first express the (3.6) in the laboratory-frame wavevector $\boldsymbol{K}=\boldsymbol{K}(\boldsymbol{k},t^{\prime })=\boldsymbol{K}(\boldsymbol{k},t-s)$ , so that the Green’s function in (3.4) and therefore the complex velocity field $\widetilde{\boldsymbol{U}}$ in (3.6) becomes time-translational symmetric.Footnote ¹

We first rewrite the mean EMF, given in (3.5), as

(4.2) $$\begin{eqnarray}\widetilde{\overline{\boldsymbol{E}}}(\boldsymbol{k},t)=2\int _{0}^{\infty }\text{d}s\,{\mathcal{D}}(s)\!\left\{\widetilde{\boldsymbol{U}}(\boldsymbol{K},s)\times \widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t-s)+\boldsymbol{e}_{2}Ss[\widetilde{\boldsymbol{U}}(\boldsymbol{K},s)\times \widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t-s)]_{1}\right\}\!,\end{eqnarray}$$

where the complex velocity field, $\widetilde{\boldsymbol{U}}$ , is

(4.3) $$\begin{eqnarray}\widetilde{\boldsymbol{U}}(\boldsymbol{K},s)=\int \frac{\text{d}^{3}K^{\prime }}{(2\unicode[STIX]{x03C0})^{3}}\,\widetilde{G}_{\unicode[STIX]{x1D702}_{T}}(\boldsymbol{K}-\boldsymbol{K}^{\prime },s)\text{i}(\boldsymbol{K}-\boldsymbol{K}^{\prime })\widetilde{{\mathcal{A}}}(\boldsymbol{K}^{\prime }).\end{eqnarray}$$

Equation (4.3) is obtained by changing the integration variable in (3.6) to $\boldsymbol{K}^{\prime }=\boldsymbol{K}(\boldsymbol{k}^{\prime },t^{\prime })=(k_{1}^{\prime }-S(t-s)k_{2}^{\prime },k_{2}^{\prime },k_{3}^{\prime })$ – which has unit Jacobian, giving $\text{d}^{3}k^{\prime }=\text{d}^{3}K^{\prime }$ .

We make further simplification by considering only axisymmetric modes for which $k_{2}=0$ . Note that for the non-axisymmetric modes, $\boldsymbol{K}(\boldsymbol{k},t)=\boldsymbol{e}_{1}(k_{1}-Stk_{2})+\boldsymbol{e}_{2}k_{2}+\boldsymbol{e}_{3}k_{3}$ increases monotonically with time, increasing the wavenumber, which would eventually decay by turbulent diffusivity. Therefore we focus our attention only on axisymmetric modes, for which $\boldsymbol{K}(\boldsymbol{k},t-s)=\boldsymbol{K}(\boldsymbol{k},t)=\boldsymbol{k}=(k_{1},0,k_{3})$ .

Let us first work out $\widetilde{\boldsymbol{U}}(\boldsymbol{k},s)$ and $\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t-s)$ correct up to $O(s)$ .

$(1)\;\;\text{}\underline{\widetilde{\boldsymbol{U}}(\boldsymbol{k},s)\;\text{to}\;O(s):}$ Taylor expanding $\widetilde{\boldsymbol{U}}(\boldsymbol{k},s)$ gives,

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\boldsymbol{U}}(\boldsymbol{k},s)=\widetilde{\boldsymbol{U}}(\boldsymbol{k},0)+s\left.\frac{\unicode[STIX]{x2202}\widetilde{\boldsymbol{U}}}{\unicode[STIX]{x2202}s}\right|_{s=0}+O(s^{2}). & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle \text{where }\widetilde{\boldsymbol{U}}(\boldsymbol{k},0)=\text{i}\boldsymbol{k}\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}+\boldsymbol{V}_{M}\quad \text{(from (3.8))}, & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle \text{and}\quad \left.\frac{\unicode[STIX]{x2202}\widetilde{\boldsymbol{U}}}{\unicode[STIX]{x2202}s}\right|_{s=0}=-\text{i}\unicode[STIX]{x1D702}_{T}\int \frac{\text{d}^{3}K^{\prime }}{(2\unicode[STIX]{x03C0})^{3}}(\boldsymbol{K}-\boldsymbol{K}^{\prime })^{2}(\boldsymbol{K}-\boldsymbol{K}^{\prime })\widetilde{{\mathcal{A}}}(\boldsymbol{K}^{\prime }). & \displaystyle\end{eqnarray}$$

Equation (4.6) is obtained by differentiating equation (4.3) with respect to $s$ and taking the limit $s\rightarrow 0$ ; note that $\boldsymbol{K}=\boldsymbol{k}$ , since $k_{2}=0$ . Using the Fourier transform for $\widetilde{{\mathcal{A}}}$ , together with the properties of the $\unicode[STIX]{x1D6FF}$ -function, we get

(4.7) $$\begin{eqnarray}\displaystyle \left.\frac{\unicode[STIX]{x2202}\widetilde{\boldsymbol{U}}}{\unicode[STIX]{x2202}s}\right|_{s=0} & = & \displaystyle -\text{i}\unicode[STIX]{x1D702}_{T}\!\left\{k^{2}(\boldsymbol{k}{\mathcal{A}}(\unicode[STIX]{x1D743})+\text{i}[\unicode[STIX]{x1D735}{\mathcal{A}}(\unicode[STIX]{x1D743})])+2\text{i}\boldsymbol{k}(\boldsymbol{k}\boldsymbol{\cdot }[\unicode[STIX]{x1D735}{\mathcal{A}}(\unicode[STIX]{x1D743})])\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\boldsymbol{k}[\unicode[STIX]{x1D6FB}^{2}{\mathcal{A}}(\unicode[STIX]{x1D743})]-\text{i}[\unicode[STIX]{x1D6FB}^{2}\{\unicode[STIX]{x1D735}{\mathcal{A}}(\unicode[STIX]{x1D743})\}]-2(\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\{\unicode[STIX]{x1D735}{\mathcal{A}}(\unicode[STIX]{x1D743})\}\right\}_{\unicode[STIX]{x1D743}=0}.\end{eqnarray}$$

Equation (4.7) can be evaluated once we know the functional form for spatial correlator ${\mathcal{A}}(\unicode[STIX]{x1D743})$ . Neglecting derivatives of ${\mathcal{A}}$ that are higher than the first order – see Singh (Reference Singh2016) for detail – we have:

(4.8) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}\widetilde{\boldsymbol{U}}}{\unicode[STIX]{x2202}s}\right|_{s=0}=-\unicode[STIX]{x1D702}_{T}k^{2}(\text{i}\boldsymbol{k}\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}+\boldsymbol{V}_{M})-2\unicode[STIX]{x1D702}_{T}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})\boldsymbol{k}.\end{eqnarray}$$

Equation (4.5) and (4.8) together thus provide the function $\widetilde{\boldsymbol{U}}(\boldsymbol{k},s)$ correct up to $O(s)$ .

$(2)\;\;\text{}\underline{\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t-s)\;\text{to}\;O(s):}$ We write as,

(4.9) $$\begin{eqnarray}\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t-s)=\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t)-s\frac{\unicode[STIX]{x2202}\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t)}{\unicode[STIX]{x2202}t}+\cdots \,.\end{eqnarray}$$

where it is assumed that $|\widetilde{\overline{\boldsymbol{H}}}|\gg s|\unicode[STIX]{x2202}\widetilde{\overline{\boldsymbol{H}}}/\unicode[STIX]{x2202}t|\gg s^{2}|\unicode[STIX]{x2202}^{2}\widetilde{\overline{\boldsymbol{H}}}/\unicode[STIX]{x2202}t^{2}|,\text{etc}$ . In (4.9), we need $\unicode[STIX]{x2202}\widetilde{\overline{\boldsymbol{H}}}/\unicode[STIX]{x2202}t$ only up to $O(1)$ to find $\widetilde{\overline{\boldsymbol{H}}}(\boldsymbol{k},t-s)$ up to $O(s)$ . We write this by substituting (3.8) in (3.7) and using $\widetilde{\boldsymbol{U}}_{\text{WN}}(\boldsymbol{k},t)=\widetilde{\boldsymbol{U}}_{\text{WN}}(\boldsymbol{k})=\text{i}\boldsymbol{k}\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}+\boldsymbol{V}_{M}$ :

(4.10) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}\widetilde{\overline{\boldsymbol{H}}}}{\unicode[STIX]{x2202}t}\right|_{O(1)}=S\widetilde{\overline{H}_{1}}\boldsymbol{e}_{2}+\text{i}\boldsymbol{k}\times \widetilde{\overline{\boldsymbol{E}}}_{\text{WN}}-\unicode[STIX]{x1D702}_{T}k^{2}\,\widetilde{\overline{\boldsymbol{H}}}=S\widetilde{\overline{H}_{1}}\boldsymbol{e}_{2}-(\unicode[STIX]{x1D702}_{K}k^{2}+\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{\text{M}})\widetilde{\overline{\boldsymbol{H}}}.\end{eqnarray}$$

The time integral in (4.2) is then solved by using the definitions provided in (2.11) and (2.12) when we substitute the expressions derived above for the terms in $\{\;\}$ in (4.2). We get, after straightforward algebra, the following expression for the mean EMF, which is correct up to $O(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}})$ :

(4.11) $$\begin{eqnarray}\displaystyle \widetilde{\overline{\boldsymbol{E}}}(\boldsymbol{k},t) & = & \displaystyle \widetilde{\overline{\boldsymbol{E}}}_{\text{WN}}(\boldsymbol{k},t)+\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}\{(\text{i}\boldsymbol{k}\boldsymbol{\cdot }\widetilde{\boldsymbol{U}}_{\text{WN}})\widetilde{\overline{\boldsymbol{E}}}_{\text{WN}}-2\unicode[STIX]{x1D702}_{\text{T}}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{\text{M}})\boldsymbol{k}\times \widetilde{\overline{\boldsymbol{H}}}\}\nonumber\\ \displaystyle & & \displaystyle +\,S\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}\{\widetilde{\overline{H_{1}}}\boldsymbol{e}_{2}\times \widetilde{\boldsymbol{U}}_{\text{WN}}+\boldsymbol{e}_{2}[\widetilde{\boldsymbol{U}}_{\text{WN}}\times \widetilde{\overline{\boldsymbol{H}}}]_{1}\}.\end{eqnarray}$$

This verifies the ansatz of (4.1) up to $n=1$ , as claimed. It is important to note that the (4.11) is valid only for slowly varying large-scale magnetic fields. To lowest order this condition can be explicitly stated as: $|\widetilde{\overline{\boldsymbol{H}}}|\gg \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}|\unicode[STIX]{x2202}\widetilde{\overline{\boldsymbol{H}}}/\unicode[STIX]{x2202}t|$ . To obtain the sufficient condition for the validity of (4.11), use (4.10) for $\unicode[STIX]{x2202}\widetilde{\overline{\boldsymbol{H}}}/\unicode[STIX]{x2202}t$ to get the following conditions for three dimensionless quantities which need to be small:

(4.12a-c ) $$\begin{eqnarray}|S\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}|\ll 1,\quad |\unicode[STIX]{x1D702}_{K}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}|\ll 1,\quad |kV_{M}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}|\ll 1.\end{eqnarray}$$

Since we have expanded EMF in small $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ , it is only the first of the two FOSA conditions in (3.1) that becomes relevant. This must be added to the above three conditions for (4.11) to be valid. Using (4.11) in (3.7) we obtain:

(4.13) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{\overline{\boldsymbol{H}}}}{\unicode[STIX]{x2202}t} & = & \displaystyle [S\widetilde{\overline{H}}_{1}\boldsymbol{e}_{2}+\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\widetilde{\overline{\boldsymbol{H}}}-\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})\widetilde{\overline{\boldsymbol{H}}}][1+\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}]-\unicode[STIX]{x1D702}_{T}k^{2}\widetilde{\overline{\boldsymbol{H}}}\nonumber\\ \displaystyle & & \displaystyle +\,2\text{i}\unicode[STIX]{x1D702}_{T}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}_{M})\widetilde{\overline{\boldsymbol{H}}}+S\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}[V_{M2}\widetilde{\overline{H}}_{3}-V_{M3}\widetilde{\overline{H}}_{2}-\text{i}\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k_{3}\widetilde{\overline{H}}_{2}][-\text{i}k_{3}\boldsymbol{e}_{1}+\text{i}k_{1}\boldsymbol{e}_{3}],\nonumber\\ \displaystyle & & \displaystyle \text{with }\boldsymbol{k}=(k_{1},0,k_{3}),\;\text{and}\;k_{1}\widetilde{\overline{H}}_{1}+k_{3}\widetilde{\overline{H}}_{3}=0.\end{eqnarray}$$

Equation (4.13) is the linear partial differential equation obtained by reducing the linear integro-differential equation (see (3.5)–(3.7)) under the condition of (4.12). It describes the evolution of an axisymmetric, large-scale magnetic field over times that are much larger than $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ . It depends on (i) the diffusivity $\unicode[STIX]{x1D702}_{T}$ ; (ii) properties of the $\unicode[STIX]{x1D6FC}$ correlation in terms of $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ , $\boldsymbol{V}_{M}$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ ; (iii) shear $S$ . These must satisfy the three conditions given in (4.12) and the first condition in (3.1) for the validity of (4.13). We note here again that the set of (3.5)–(3.7) is non-perturbative in both $S$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ , whereas (4.13) is valid only when $|S\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}|\ll 1$ .

5 Growth rate of modes when $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ is non-zero

As usual in numerical works on the related subject (see, e.g. Brandenburg et al. Reference Brandenburg, Rädler, Rheinhardt and Käpylä2008; Singh & Jingade Reference Singh and Jingade2015) where ‘horizontal’ (plane of shear; in this case the $X_{1}-X_{2}$ plane) averages are performed to define the large-scale magnetic fields, it is useful to consider one-dimensional propagating modes. This is equivalent to setting $K_{1}$ and $K_{2}$ equal to zero. Here we only need to set $k_{1}=0$ in (4.13). In this case the wavevector $\boldsymbol{k}=(0,0,k)$ points along the ‘vertical’ ( $\pm \boldsymbol{e}_{3}$ ) direction, thus resulting in a uniform $\widetilde{\overline{H}}_{3}$ which is of no interest for dynamo action. Hence we set $\widetilde{\overline{H}}_{3}=0$ , and take $\widetilde{\overline{\boldsymbol{H}}}(k,t)=\widetilde{\overline{H}}_{1}(k,t)\boldsymbol{e}_{1}+\widetilde{\overline{H}}_{2}(k,t)\boldsymbol{e}_{2}$ . Making these substitutions in (4.13) we find:

(5.1) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{\overline{\boldsymbol{H}}}}{\unicode[STIX]{x2202}t} & = & \displaystyle [S\widetilde{\overline{H}}_{1}\boldsymbol{e}_{2}+\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\widetilde{\overline{\boldsymbol{H}}}-\text{i}kV_{M3}\widetilde{\overline{\boldsymbol{H}}}][1+\text{i}kV_{M3}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}]-\unicode[STIX]{x1D702}_{T}k^{2}\widetilde{\overline{\boldsymbol{H}}}\nonumber\\ \displaystyle & & \displaystyle +\,2\text{i}\unicode[STIX]{x1D702}_{T}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}(k^{3}V_{M3})\widetilde{\overline{\boldsymbol{H}}}+S[\text{i}kV_{M3}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}]\widetilde{\overline{H}}_{2}\boldsymbol{e}_{1}.\end{eqnarray}$$

Seeking modal solutions of the form,

(5.2) $$\begin{eqnarray}\widetilde{\overline{\boldsymbol{H}}}(k,t)=[\widetilde{\overline{H}}_{01}(k)\boldsymbol{e}_{1}+\widetilde{\overline{H}}_{02}(k)\boldsymbol{e}_{2}]\exp (\unicode[STIX]{x1D706}t),\end{eqnarray}$$

and substituting this in (5.1) we get the following dispersion relation:

(5.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{\pm } & = & \displaystyle -\unicode[STIX]{x1D702}_{K}k^{2}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}^{2}k^{4}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}+(kV_{M3})^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}+\text{i}kV_{M3}[2(\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x1D702}_{T})k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}-1]\nonumber\\ \displaystyle & & \displaystyle \pm \,|S|\sqrt{[\text{i}kV_{M3}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}][1+\text{i}kV_{M3}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}]}.\end{eqnarray}$$

We are more interested in the growth rate $\unicode[STIX]{x1D6FE}=\text{Re}\{\unicode[STIX]{x1D706}\}$ , as the dynamo action corresponds to the case when $\unicode[STIX]{x1D6FE}>0$ . From the dispersion relation (5.3) we have:

(5.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6FE}_{\pm }=\text{Re}\{\unicode[STIX]{x1D706}_{\pm }\}=-\unicode[STIX]{x1D702}_{K}k^{2}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}^{2}k^{4}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}+(kV_{M3})^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}\pm |S|[\unicode[STIX]{x1D712}_{R}^{2}+\unicode[STIX]{x1D712}_{I}^{2}]^{1/4}\cos (\unicode[STIX]{x1D713}/2),\\ \displaystyle \text{where }\cos (\unicode[STIX]{x1D713})=\frac{\unicode[STIX]{x1D712}_{R}}{(\unicode[STIX]{x1D712}_{R}^{2}+\unicode[STIX]{x1D712}_{I}^{2})^{1/2}},\quad \cos (\unicode[STIX]{x1D713}/2)=\frac{\sqrt{\unicode[STIX]{x1D712}_{R}+\sqrt{\unicode[STIX]{x1D712}_{R}^{2}+\unicode[STIX]{x1D712}_{I}^{2}}}}{\sqrt{2}(\unicode[STIX]{x1D712}_{R}^{2}+\unicode[STIX]{x1D712}_{I}^{2})^{1/4}},\\ \displaystyle \unicode[STIX]{x1D712}_{R}=\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}-1)-(kV_{M3}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}})^{2},\quad \unicode[STIX]{x1D712}_{I}=-kV_{M3}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}(2\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}-1).\end{array}\right\}\end{eqnarray}$$

Below, we make some comments about the growth rate derived above.

1. (i) The growth rate $\unicode[STIX]{x1D6FE}$ of the large-scale dynamo is linear in the shear rate $|S|$ , assuming that the parameters ( $\unicode[STIX]{x1D702}_{K},\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}},V_{M3},\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ ) are all independent of $S$ . This linear scaling is observed in earlier numerical works (Brandenburg et al. Reference Brandenburg, Rädler, Rheinhardt and Käpylä2008; Yousef et al. Reference Yousef, Heinemann, Schekochihin, Kleeorin, Rogachevskii, Iskakov, Cowley and Mcwilliams2008a ,Reference Yousef, Heinemann, Rincon, Schekochihin, Kleeorin, Rogachevskii, Cowley and Mcwilliams b ; Singh & Jingade Reference Singh and Jingade2015).

2. (ii) For zero shear, the growth rate as given from (5.4) becomes identical to the one derived in Singh (Reference Singh2016), where the generalization to the Kraichnan problem as well as the possibility of Moffatt drift driven dynamos were explored in detail.

3. (iii) The last term involving shear in (5.4) is identical to the corresponding term in the expression for the growth rate derived in Sridhar & Singh (Reference Sridhar and Singh2014), with an important difference being that there the angle $\unicode[STIX]{x1D713}$ was defined using the tangent function, which introduces an error when either of the two, $\unicode[STIX]{x1D712}_{R}$ and $\unicode[STIX]{x1D712}_{I}$ , take negative values. Here we correct this by explicitly writing $\cos (\unicode[STIX]{x1D713}/2)$ in terms of $\unicode[STIX]{x1D712}_{R}$ and $\unicode[STIX]{x1D712}_{I}$ .

5.1 Dimensionless growth rate function

The growth rate function $\unicode[STIX]{x1D6E4}$ is defined using dimensionless quantities,

(5.5a-e ) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{\pm }=\unicode[STIX]{x1D6FE}_{\pm }\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}};\quad \unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}};\quad \unicode[STIX]{x1D700}_{S}=S\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}};\quad \unicode[STIX]{x1D700}_{K}=\unicode[STIX]{x1D702}_{K}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}};\quad \unicode[STIX]{x1D700}_{M}=kV_{M3}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D700}_{K}$ measure the wavenumber of modal mean magnetic field in terms of $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D702}_{K}$ , respectively. With the first condition of (3.1), $\unicode[STIX]{x1D6FD}/k^{2}=\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}\ll \ell ^{2}$ . These parameters can vary as,

(5.6a-e ) $$\begin{eqnarray}0\leqslant \unicode[STIX]{x1D6FD}\ll (k\ell )^{2};\quad \unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D700}_{K}>0;\quad |\unicode[STIX]{x1D700}_{S}|\ll 1;\quad |\unicode[STIX]{x1D700}_{K}|\ll 1;\quad |\unicode[STIX]{x1D700}_{M}|\ll 1.\end{eqnarray}$$

The parameter $\unicode[STIX]{x1D6FD}$ can be larger or smaller than unity depending on whether the mean-field varies over scales smaller or larger than $\ell$ , respectively. The second condition comes from $\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D700}_{K}=\unicode[STIX]{x1D702}_{T}k^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}>0$ , and last three constraints come from (4.12). Multiplying the expression for $\unicode[STIX]{x1D6FE}_{\pm }$ in (5.4) by $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ , and denoting by $\unicode[STIX]{x1D6E4}_{{>}}$ ( $\unicode[STIX]{x1D6E4}_{{<}}$ ) the larger (smaller) of $\unicode[STIX]{x1D6E4}_{+}$ and $\unicode[STIX]{x1D6E4}_{-}$ , we get

(5.7) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{\stackrel{{>}}{{<}}}=-\unicode[STIX]{x1D700}_{K}-\unicode[STIX]{x1D6FD}^{2}+\unicode[STIX]{x1D700}_{M}^{2}\pm \frac{|\unicode[STIX]{x1D700}_{S}|}{\sqrt{2}}\sqrt{\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FD}-1)-\unicode[STIX]{x1D700}_{M}^{2}+\sqrt{[\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FD}-1)+\unicode[STIX]{x1D700}_{M}^{2}]^{2}+\unicode[STIX]{x1D700}_{M}^{2}}}.\end{eqnarray}$$

Note that the radicand in (5.7) is greater than zero. In figure 1 we show the behaviour of $\unicode[STIX]{x1D6E4}$ as a function of $\unicode[STIX]{x1D6FD}$ by keeping other parameters fixed. Below we list some properties of the growth rate function as defined in (5.7).

1. (i) For fixed $\unicode[STIX]{x1D700}_{K}$ , $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D700}_{M}$ , $\unicode[STIX]{x1D6E4}_{{>}}$ ( $\unicode[STIX]{x1D6E4}_{{<}}$ ) increases (decreases) monotonically with shear.

2. (ii) When $\unicode[STIX]{x1D700}_{M}$ is non-zero, then the radicand in (5.7) vanishes at $\unicode[STIX]{x1D6FD}=1/2$ , where the two roots coincide; see green solid and dashed curves in figure 1. Both roots are identical for $0\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$ when $\unicode[STIX]{x1D700}_{M}=0$ and branch out for $\unicode[STIX]{x1D6FD}>1$ ; see red solid and dashed curves.

3. (iii) In the absence of the Moffatt drift, the necessary condition for dynamo action is that the $\unicode[STIX]{x1D6FC}$ fluctuations must be strong, i.e.  $\unicode[STIX]{x1D700}_{K}<0$ , regardless of the strength of the shear parameter $|\unicode[STIX]{x1D700}_{S}|$ which should be kept smaller than unity in the present model. The dynamo is then driven through the $-\unicode[STIX]{x1D700}_{K}$ term in (5.7) by the process of negative diffusion first suggested by Kraichnan (Reference Kraichnan1976).

4. (iv) Moffatt drift always contributes positively to the dynamo growth. Considering the case of zero shear, we see from (5.7) that $\unicode[STIX]{x1D700}_{M}>\unicode[STIX]{x1D700}_{M}^{\text{crit}}$ , with $\unicode[STIX]{x1D700}_{M}^{\text{crit}}=\sqrt{\unicode[STIX]{x1D700}_{K}+\unicode[STIX]{x1D6FD}^{2}}$ , can always facilitate LSD in both the weak and strong $\unicode[STIX]{x1D6FC}$ fluctuation regimes, for sufficiently low values of $\unicode[STIX]{x1D6FD}$ such that $\unicode[STIX]{x1D700}_{M}^{\text{crit}}\ll 1$ .

5. (v) The growth rate is always negative for $\unicode[STIX]{x1D6FD}\gg 1$ due to the $-\unicode[STIX]{x1D6FD}^{2}$ term in (5.7), as is also shown in figure 1.

6. (vi) The growth rate for $\unicode[STIX]{x1D6FD}\approx 0$ i.e. largest scale possible is given for small values of $\unicode[STIX]{x1D700}_{M}\ll 1$ as,

   (5.8) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{{>}}\simeq -\unicode[STIX]{x1D700}_{K}+\frac{|\unicode[STIX]{x1D700}_{S}|}{\sqrt{2}}|\unicode[STIX]{x1D700}_{M}|^{1/2},\end{eqnarray}$$
   which implies Moffatt drift couples strongly with shear and growth is possible for weak $\unicode[STIX]{x1D6FC}$ -fluctuations i.e.  $\unicode[STIX]{x1D700}_{K}>0$ when $|\unicode[STIX]{x1D700}_{S}||\unicode[STIX]{x1D700}_{M}|^{1/2}>\sqrt{2}\unicode[STIX]{x1D700}_{K}$ .

Figure 1. The two roots, $\unicode[STIX]{x1D6E4}_{{>}}$ (solid) and $\unicode[STIX]{x1D6E4}_{{<}}$ (dashed), of the growth rate function defined in (5.7) are shown as a function of $\unicode[STIX]{x1D6FD}$ for $\unicode[STIX]{x1D700}_{M}=0$ (red; thick) and $0.3$ (green; thin) with $|\unicode[STIX]{x1D700}_{S}|=0.5$ , where (a) and (b) correspond to weak ( $\unicode[STIX]{x1D700}_{K}=0.1$ ) and strong ( $\unicode[STIX]{x1D700}_{K}=-0.1$ ) $\unicode[STIX]{x1D6FC}$ fluctuations, respectively.

5.2 Growth rates as functions of the wavenumber

We henceforth consider only the dominant root $\unicode[STIX]{x1D6E4}_{{>}}$ and study its wavenumber dependence. Following SS14, we first identify natural length and time scales whose corresponding wavenumber and frequency are defined as,

(5.9a,b ) $$\begin{eqnarray}k_{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}})^{-1/2}>0;\quad \unicode[STIX]{x1D70E}=|\unicode[STIX]{x1D702}_{K}|k_{\unicode[STIX]{x1D6FC}}^{2},\end{eqnarray}$$

where $k_{\unicode[STIX]{x1D6FC}}$ can be recognized as an inverse diffusion length due to $\unicode[STIX]{x1D6FC}$ -diffusivity $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ . Here $|k|>k_{\unicode[STIX]{x1D6FC}}$ and $|k|<k_{\unicode[STIX]{x1D6FC}}$ are called high and low wavenumbers, respectively. From (5.5) and (5.9) we see that $\unicode[STIX]{x1D6FD}=(k/k_{\unicode[STIX]{x1D6FC}})^{2}$ . Since the parameters $\unicode[STIX]{x1D700}_{K}$ and $\unicode[STIX]{x1D700}_{M}$ involve the wavenumber $k$ in their definitions, we find it better to rewrite an expression for $\unicode[STIX]{x1D6E4}_{{>}}$ using new dimensionless dynamo numbers, which are defined in terms of known constants:

(5.10a,b ) $$\begin{eqnarray}{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x1D702}_{T}},\quad {\mathcal{D}}_{M}=\frac{V_{M3}^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}}.\end{eqnarray}$$

We first make use of (5.4)–(5.10) to express the growth rates as a function of wavenumber and constant dynamo parameters in the weak and strong regimes of $\unicode[STIX]{x1D6FC}$ -fluctuations.

Figure 2. Normalized growth rate $\unicode[STIX]{x1D6FE}_{{>}}/\unicode[STIX]{x1D70E}$ as a function of $|k/k_{\unicode[STIX]{x1D6FC}}|$ for ${\mathcal{D}}_{M}=0.2$ . (a) and (b) correspond to weak ( ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}=0.5$ ) and strong ( ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}=1.5$ ) $\unicode[STIX]{x1D6FC}$ fluctuations respectively. Solid, dashed, dash-dotted and dotted curves correspond to $\unicode[STIX]{x1D700}_{S}=0.6$ , 0.4, 0.2 and 0, respectively.

Weak $\unicode[STIX]{x1D6FC}$ fluctuations: Here, $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}<\unicode[STIX]{x1D702}_{T}$ , i.e.  ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}<1$ and $|\unicode[STIX]{x1D702}_{K}|=+\unicode[STIX]{x1D702}_{K}=\unicode[STIX]{x1D702}_{T}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ , giving

(5.11) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D6FE}_{{>}}}{\unicode[STIX]{x1D70E}}=-\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}+\frac{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}{1-{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}\left[-\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{4}+{\mathcal{D}}_{M}\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}+\frac{|\unicode[STIX]{x1D700}_{S}|}{\sqrt{2}}\sqrt{{\mathcal{R}}_{a}(k)+{\mathcal{R}}_{b}(k)}\right]\qquad & \displaystyle\end{eqnarray}$$

(5.12) $$\begin{eqnarray}\displaystyle & \displaystyle \text{with }{\mathcal{R}}_{a}(k)=\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}\left[\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}-1\right]-{\mathcal{D}}_{M}\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2} & \displaystyle\end{eqnarray}$$

(5.13) $$\begin{eqnarray}\displaystyle & \displaystyle \text{and }{\mathcal{R}}_{b}(k)=\left(\left\{\left[\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}-1\right]+{\mathcal{D}}_{M}\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}\right\}^{2}+{\mathcal{D}}_{M}\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}\right)^{1/2}. & \displaystyle\end{eqnarray}$$

In figure 2(a) we show the wavenumber dependence of the normalized growth rate $\unicode[STIX]{x1D6FE}_{{>}}/\unicode[STIX]{x1D70E}$ for different choices of the shear parameter, at fixed ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ and ${\mathcal{D}}_{M}$ , when the $\unicode[STIX]{x1D6FC}$ fluctuations are weak. Interestingly, the growth rate is positive for fairly small wavenumbers, thus facilitating a truly large-scale dynamo, with a wavenumber cutoff beyond which the growth rate turns negative. At much larger wavenumbers, The growth rate varies as $\unicode[STIX]{x1D6FE}\propto -k^{4}$ due to the $\unicode[STIX]{x1D702}_{T}$ -correction in the present model. Shear boosts the growth rates at all wavenumbers, and thus it can support the dynamo action for sufficiently strong Moffatt drift.

Strong $\unicode[STIX]{x1D6FC}$ fluctuations: In this case, $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}>\unicode[STIX]{x1D702}_{T}$ , i.e.  ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}>1$ and $|\unicode[STIX]{x1D702}_{K}|=-\unicode[STIX]{x1D702}_{K}$ , giving

(5.14) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FE}_{{>}}}{\unicode[STIX]{x1D70E}}=+\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}+\frac{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}-1}\left[-\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{4}+{\mathcal{D}}_{M}\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}+\frac{|\unicode[STIX]{x1D700}_{S}|}{\sqrt{2}}\sqrt{{\mathcal{R}}_{a}(k)+{\mathcal{R}}_{b}(k)}\right].\end{eqnarray}$$

Here the small wavenumbers grow as all the effects, Kraichnan diffusivity, Moffatt drift and shear, contribute positively to the dynamo action; see figure 2(b). Similar to the case of weak $\unicode[STIX]{x1D6FC}$ fluctuations, the growth rate here too is a non-monotonic function of $k$ and it becomes negative for sufficiently large wavenumbers.

5.2.1 Dynamo action for zero Moffatt drift

This corresponds to the Kraichnan problem, extended to include a non-zero $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ . There are two cases to consider, the one in the absence of shear and the other when shear is present.

1. Shear absent (only $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ non-zero)

Using (5.11)–(5.14) by setting ${\mathcal{D}}_{M}=0$ and $|\unicode[STIX]{x1D700}_{S}|=0$ the normalized growth rate can be expressed as, for

Weak $\unicode[STIX]{x1D6FC}$ fluctuations: when $0<\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}<\unicode[STIX]{x1D702}_{T}$ , i.e.  ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}<1$ ,

(5.15) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D70E}}=-\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}-\frac{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}{1-{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{4}.\end{eqnarray}$$

Here, the growth is negative definite for all values of $k$ and a monotonically decreasing function of $k$ . At large wavenumbers, it varies as $\unicode[STIX]{x1D6FE}\propto -k^{4}$ , a correction due to finite $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ and inclusion of a finite resistive term in the fluctuating field equation. The first term in the (5.15) is due to Kraichnan diffusivity (compare it with (3.10) by setting $S=0$ and $\boldsymbol{V}_{M}=0$ ).

Strong $\unicode[STIX]{x1D6FC}$ fluctuations: when $0<\unicode[STIX]{x1D702}_{T}<\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ , i.e.  ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}>1$ ,

(5.16) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D70E}}=\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}-\frac{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}-1}\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{4}.\end{eqnarray}$$

In this regime, the growth rate is positive for a certain range of wavenumbers and it becomes negative for large wavenumbers as mentioned above. In figure 3 we compare our model (which has non-zero $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ ) with the original Kraichnan model – we see that a non-zero $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ introduces a high wavenumber cutoff in the case of strong $\unicode[STIX]{x1D6FC}$ -fluctuations, which agrees with the conclusions of Singh (Reference Singh2016).

Figure 3. Normalized growth rate when Moffatt drift and shear are zero. Plotted as a function of $|k/k_{\unicode[STIX]{x1D6FC}}|$ . Solid curve shows finite $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ correction and dashed-dotted curve is for the white-noise case.

2. The effect of shear

Using (5.7) we rewrite the growth rate function more explicitly as:

(5.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{{>}}=-\unicode[STIX]{x1D700}_{K}-\unicode[STIX]{x1D6FD}^{2},\quad \text{when }0\leqslant \unicode[STIX]{x1D6FD}\leqslant 1 & \displaystyle\end{eqnarray}$$

(5.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{{>}}=-\unicode[STIX]{x1D700}_{K}-\unicode[STIX]{x1D6FD}^{2}+|\unicode[STIX]{x1D700}_{S}|\sqrt{\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FD}-1)},\quad \text{when }\unicode[STIX]{x1D6FD}>1. & \displaystyle\end{eqnarray}$$

We note that the shear does not couple to the dynamo growth rate when $\unicode[STIX]{x1D6FD}$ is smaller than or equal to unity, or in other words, when $|k|<k_{\unicode[STIX]{x1D6FC}}$ .Footnote ² Since $|\unicode[STIX]{x1D700}_{S}|\ll 1$ , the dominant term in (5.18) is $-\unicode[STIX]{x1D6FD}^{2}$ for $k>k_{\unicode[STIX]{x1D6FC}}$ , which makes the growth rate negative definite. Thus, for weak $\unicode[STIX]{x1D6FC}$ fluctuations which have $\unicode[STIX]{x1D700}_{K}>0$ , shear alone cannot drive a large-scale dynamo at any wavenumber. Therefore, the necessary condition for dynamo action in this case is that the $\unicode[STIX]{x1D6FC}$ fluctuations must be strong. We now look at the properties of growth rate as a function of wavenumber.

Weak $\unicode[STIX]{x1D6FC}$ fluctuations: Here, $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}<\unicode[STIX]{x1D702}_{T}$ , i.e.  ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}<1$ and $|\unicode[STIX]{x1D702}_{K}|=+\unicode[STIX]{x1D702}_{K}=\unicode[STIX]{x1D702}_{T}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ , giving

(5.19) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FE}_{{>}}}{\unicode[STIX]{x1D70E}} & = & \displaystyle -\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}-\frac{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}{1-{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{4},\quad \text{when }0<|k|<k_{\unicode[STIX]{x1D6FC}}\end{eqnarray}$$

(5.20) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}+\frac{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}{1-{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}\left[-\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{4}+|\unicode[STIX]{x1D700}_{S}|\sqrt{\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}\left[\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}-1\right]}\right],\nonumber\\ \displaystyle & & \displaystyle \text{when }|k|>k_{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

We can see from (5.19) that the growth rate is negative definite in the range $0<|k|<k_{\unicode[STIX]{x1D6FC}}$ , as inferred above. Dynamo action is not possible for $|k|>k_{\unicode[STIX]{x1D6FC}}$ for the following reason. When $|k|>k_{\unicode[STIX]{x1D6FC}}$ , shear contributes to the growth rate (see (5.20)). Since the model is valid for $|\unicode[STIX]{x1D700}_{S}|\ll 1$ , in order to increase the strength of that term we can increase ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ (while keeping it less than unity), but this will also strengthen the second term, which is $\propto -k^{4}$ , due to the finite $\unicode[STIX]{x1D702}_{T}$ correction in the fluctuating field equation, thereby killing dynamo action.

Figure 4. Normalized growth rate $\unicode[STIX]{x1D6FE}_{{>}}/\unicode[STIX]{x1D70E}$ as a function of $|k/k_{\unicode[STIX]{x1D6FC}}|$ for $|\unicode[STIX]{x1D700}_{S}|=0.3$ . (a,b) Correspond to weak ( ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}=0.5$ ) and strong ( ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}=1.8$ ) $\unicode[STIX]{x1D6FC}$ fluctuations, respectively. Solid and dashed curves correspond to this work and SS14, respectively.

Strong $\unicode[STIX]{x1D6FC}$ fluctuations: Here, $\unicode[STIX]{x1D702}_{T}<\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$ , i.e.  ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}>1$ and $|\unicode[STIX]{x1D702}_{K}|=-\unicode[STIX]{x1D702}_{K}=\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D702}_{T}$ , giving

(5.21) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FE}_{{>}}}{\unicode[STIX]{x1D70E}} & = & \displaystyle \left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}-\frac{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}-1}\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{4},\quad \text{when }0<|k|<k_{\unicode[STIX]{x1D6FC}}\end{eqnarray}$$

(5.22) $$\begin{eqnarray}\displaystyle & = & \displaystyle \left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}+\frac{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}}{{\mathcal{D}}_{\unicode[STIX]{x1D6FC}}-1}\left[-\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{4}+|\unicode[STIX]{x1D700}_{S}|\sqrt{\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}\left[\left(\frac{k}{k_{\unicode[STIX]{x1D6FC}}}\right)^{2}-1\right]}\right],\nonumber\\ \displaystyle & & \displaystyle \text{when }|k|>|k_{\unicode[STIX]{x1D6FC}}|.\end{eqnarray}$$

The growth rate $\unicode[STIX]{x1D6FE}$ becomes positive for a certain range of wavenumber depending upon the strength of the $\unicode[STIX]{x1D6FC}$ -fluctuations, eventually becoming negative at large wavenumbers due to the $-k^{4}$ term arising due to the finite $\unicode[STIX]{x1D702}_{T}$ correction. This behaviour is compared in figure 4 with Sridhar & Singh (Reference Sridhar and Singh2014); we can see that there is good agreement at low wavenumbers whereas at large wavenumbers there is a difference. The derivation of Sridhar & Singh (Reference Sridhar and Singh2014) had neglected the effect of turbulent resistivity on the fluctuating component of the magnetic field, and they had noted that this would lead to an overestimation of growth rates at large wavenumbers. This is what we find in the present work: retaining this term makes the growth rate negative at large wavenumbers and, for weak $\unicode[STIX]{x1D6FC}$ -fluctuations, the behaviour is indeed qualitatively different. Therefore, including the $\unicode[STIX]{x1D702}_{T}$ term gives a bonafide large-scale dynamo action by predicting the high wavenumber cutoff. Thus, in the absence of Moffatt drift, the necessary condition for the large-scale dynamo when shear is present is the same as the case when it is absent.