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## Abstract

We explore the growth of large–scale magnetic fields in a shear flow, due to helicity fluctuations with a finite correlation time, through a study of the Kraichnan–Moffatt model of zero–mean stochastic fluctuations of the parameter of dynamo theory. We derive a linear integro–differential equation for the evolution of large–scale magnetic field, using the first–order smoothing approximation and the Galilean invariance of the –statistics. This enables construction of a model that is non–perturbative in the shearing rate and the –correlation time . After a brief review of the salient features of the exactly solvable white–noise limit, we consider the case of small but non–zero . When the large–scale magnetic field varies slowly, the evolution is governed by a partial differential equation. We present modal solutions and conditions for the exponential growth rate of the large–scale magnetic field, whose drivers are the Kraichnan diffusivity, Moffatt drift, Shear and a non–zero correlation time. Of particular interest is dynamo action when the –fluctuations are weak; i.e. when the Kraichnan diffusivity is positive. We show that in the absence of Moffatt drift shear does not give rise to growing solutions. But shear and Moffatt drift acting together can drive large scale dynamo action with growth rate .

Mean field dynamo due to stochastic in shear flows]Generation of large-scale magnetic fields due to fluctuating in shearing systems Jingade, Singh & Sridhar]N\lsA\lsV\lsE\lsE\lsN\nsJ\lsI\lsN\lsG\lsA\lsD\lsE 1, N\lsI\lsS\lsH\lsA\lsN\lsT\nsK.\nsS\lsI\lsN\lsG\lsH and S.\nsS\lsR\lsI\lsD\lsH\lsA\lsR

## 1 Introduction

Magnetic fields are observed over a wide range of scales in various astrophysical objects. Their origins could be the result of turbulent dynamo processes which can lead to field generation on scales that are larger as well as smaller than the outer scale of underlying turbulence (see, e.g., Moffatt, 1978; Parker, 1979; Krause & Rädler, 1980; Zeldovich et al., 1983; Ruzmaikin et al., 1988; Brandenburg & Subramanian, 2005). Of particular interest here is the subject of large-scale dynamo (LSD) which may be studied in the framework of mean-field theory (Steenbeck et al., 1966; Moffatt, 1978; Krause & Rädler, 1980). The standard paradigm for LSD involves an -effect which arises when the background turbulence possesses mean kinetic helicity, thus breaking the mirror symmetry of turbulence (see, e.g., Brandenburg & Subramanian, 2005). The problem becomes more interesting and complicated when the usual -effect is either absent or subcritical for dynamo growth. Mean velocity shear appears to play a vital role for LSD in such regimes of zero/subcritical . As most astrophysical bodies also possess mean differential rotation, it is natural to ask if large-scale magnetic fields could grow in the presence of a background shear flow when is a purely fluctuating quantity.

Early ideas of stochastically varying with zero mean suggested that it causes a decrement in turbulent diffusion (Kraichnan, 1976; Moffatt, 1978). A number of subsequent studies then considered fluctuating as an important ingredient for the evolution of magnetic fields in objects, such as, the Sun (Silant’ev, 2000; Proctor, 2007), the accretion disks (Vishniac & Brandenburg, 1997), galaxies (Sokolov, 1997; Sur & Subramanian, 2009). Numerical demonstration of the shear dynamo problem (Yousef et al., 2008a, b; Brandenburg et al., 2008; Singh & Jingade, 2015) where large–scale magnetic fields were generated due to non-helically forced turbulence in shear flows, and failure to understand these in terms of simple ideas involving shear–current effect (Kleeorin & Rogachevskii, 2008; Rogachevskii & Kleeorin, 2008; Sridhar & Subramanian, 2009a, b; Sridhar & Singh, 2010; Singh & Sridhar, 2011; Kolekar et al., 2012), brought the focus on stochastic which could potentially lead to the dynamo action generically in shearing systems (Heinemann et al., 2011; McWilliams, 2012; Mitra & Brandenburg, 2012; Proctor, 2012; Richardson & Proctor, 2012; Sridhar & Singh, 2014). There is still a need to verify the model predictions for the growth of first moment of the mean magnetic field in such systems by performing more simulations.

Squire & Bhattacharjee (2015a, b) recently proposed a new mechanism, called the magnetic shear current effect, which leads to the generation of a large scale magnetic field due to the combined action of shear and small scale magnetic fluctuations, if these are sufficiently strong and are near equipartition levels of turbulent motions. Such strong magnetic fluctuations are expected to be naturally present due to small scale dynamo (SSD) action in astrophysical plasmas, which typically have large magnetic Reynolds number (). This new effect thus raises the interesting possibility of the excitation of LSD due to SSD in presence of shear, and it challenges an understanding where SSD in high- systems is thought to weaken the LSD, which could survive only when SSD is suppressed due to shear (Tobias & Cattaneo, 2013; Pongkitiwanichakul et al., 2016; Nigro et al., 2017); but see also Kolokolov et al. (2011); Singh et al. (2017) where it is found that the shear supports and even enhances the growth rate of SSD. However, we are here more concerned with the excitation of a large–scale shear dynamo, quite independent of any small–scale dynamo or strong magnetic fluctuations, which are both absent in most numerical simulations that are relevant. These simulations typically had which were subcritical for SSD and the only source of magnetic fluctuations was due to the tangling of large–scale magnetic fields (Rogachevskii & Kleeorin, 2007), and therefore these fluctuations could never be too strong in the kinematic regime of LSD.

In the present paper we explore the possibility of large-scale dynamo action in presence of background shear flow, due an that varies stochastically in space and time, with vanishing mean. Here we generalize the earlier work by Sridhar & Singh (2014), hereafter SS14, by including the full resistive term in determining the turbulent electromotive force (EMF). Such an extension in the absence of shear was done in Singh (2016). In Section 2 we define our model by writing dynamo equations in shearing coordinates. Integro–differential equation governing the evolution of the large-scale magnetic field is derived under FOSA in Section 3. This is non-perturbative in shearing rate and the correlation time . Here we briefly review the exactly solvable limit of white–noise fluctuations. In Section 4 we reduce the evolution equation into a partial differential equation (PDE) for axisymmetric mean magnetic fields, by assuming small but non-zero . Dispersion relation giving the growth rate is then determined in Section 5 where we present our results in different parameter regimes. We then discuss our findings and conclude in Section 6.

## 2 The model

Let us begin with the standard dynamo equation in the presence of a background linear shear flow, (see, Moffatt, 1978; Krause & Rädler, 1980; Brandenburg & Subramanian, 2005; Sridhar & Singh, 2014):

 (∂∂τ+SX1∂∂X2)\boldmathB−SB1{\boldmathe}2=% \boldmath∇\boldmath×[α(\boldmathX,τ)\boldmathB]+ηT\boldmath∇2\boldmathB;\boldmath∇\boldmath⋅\boldmathB=0. (1)

Here we follow the same notation as in Sridhar & Singh (2014) where the position vector is denoted by with components given in a fixed orthonormal frame , and is the time variable. The shear rate, , and total diffusivity, , are treated as constant parameters. We recall that Eq. (1) governing the dynamics of meso-scale magnetic field is obtained by averaging over an ensemble of random velocity fields, , 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, 1976; Moffatt, 1983; Sokolov, 1997) under which itself is a random variable of space and time, thus making Eq. (1) a stochastic partial differential equation. It is drawn from a superensemble with zero mean, . It’s statistical properties are given below in Eq. (11). Next, we separate the meso-scale field, , into large-scale, , and fluctuating, , components, where the superensemble average of vanishes, i.e., . Governing equation for the large-scale magnetic field can thus be obtained by Reynolds averaging the Eq. (1) over the superensemble:

 (∂∂τ+SX1∂∂X2)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathB−S¯¯¯¯¯¯B1{\boldmathe}2 = \boldmath∇\boldmath×¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯% \boldmathE+ηT\boldmath∇2¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯% \boldmathB,\boldmath∇\boldmath⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathB=0, (2) where¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathE = ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α(\boldmathX,τ)\boldmathb(\boldmathX,τ). (3)

In order to determine the mean electromotive force (EMF), we must solve for the fluctuating field , which evolves as:

 (∂∂τ+SX1∂∂X2)\boldmathb−Sb1{\boldmathe}2 = \boldmath∇\boldmath×[α¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathB]+\boldmath∇% \boldmath×[α\boldmathb−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α% \boldmathb]+ηT\boldmath∇2\boldmathb, \boldmath∇\boldmath⋅\boldmathb=0, with initial condition\boldmathb(% \boldmathX,0)=0. (4)

As Eqs. (2) and (4) involve inhomogeneous terms, it is convenient to solve these in shearing frame where shearing coordinates are expressed in terms of the lab coordinates as (see, Sridhar & Subramanian, 2009a; Sridhar & Singh, 2010):

 x1=X1;x2=X2−SτX1;x3=X3;t=τ. (5)

The inverse transformation is:

 X1=x1;X2=x2+Stx1;X3=x3;τ=t. (6)

Now we can write Eqs. (2)–(4) in terms of new fields that are functions of and : ; ; ; and . Equations (2)–(4) then take the form (Sridhar & Singh, 2014):

 ∂¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH∂t−S¯¯¯¯¯¯¯H1{\boldmathe}2=\boldmath∇\boldmath×% ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathE+ηT\boldmath∇2¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH,\boldmath∇\boldmath% ⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH=0,¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯% \boldmathE=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯a\boldmathh; (7)
 ∂\boldmathh∂t−Sh1{% \boldmathe}2 = \boldmath∇\boldmath×[a¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH]+\boldmath∇% \boldmath×[a\boldmathh−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯a\boldmathh]+ηT\boldmath∇2\boldmathh, ∇⋅h = 0,with initial condition\boldmathh(\boldmathx,0)=0; (8) where\boldmath∇ = ∂∂\boldmathx−{\boldmath% e}1St∂∂x2is a time-dependent % operator. (9)

We complete defining our model by specifying the statistics of fluctuations. We follow the exact same approach as given in detail in Sridhar & Singh (2014) and recall here only some key relevant points:

• Shear flows possess a natural symmetry known as Galilean invariance, relating the measurements of correlation functions made by comoving observers whose origins with resepct to the lab frame translate with the same speed as that of the linear shear flow (Sridhar & Subramanian, 2009a, b).

• Here we are more interested in time–stationary Galilean–invariant statistics, which can be expressed in the shearing frame as (see Sridhar & Singh (2014) for a derivation):

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯a(\boldmathx,t)a(\boldmathx′,t′) = 2A(\boldmathx−\boldmathx′+St′(x1−x′1){\boldmathe}2)D(t−t′),with (10) 2∫∞0D(t)dt = 1,A(0)=ηα≥0. (11)

The correlation time for the fluctuations is defined as,

 τα=2∫∞0dttD(t). (12)

It may be verified that in the absence of shear Eq. (11) reduces to:

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α(\boldmathX,τ)α(\boldmathX′,τ′)=2A(\boldmathX−\boldmathX′)D(τ−τ′). (13)

## 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 by working with its conjugate Fourier variable . Let be the Fourier transform of any quantity , with similar definition in terms of lab-frame coordinates, where denotes the conjugate variable to . Note that the lab-frame wavevector is time-dependent and can be expressed in terms of sheared wavevectors as, ; see Eq. (9). We first solve for as a functional of and , under the scheme of first-order smoothing approximation (FOSA) where we ignore the term in Eq. (8). Thus the fluctuating magnetic field evolves as:

 (∂∂t−ηT\boldmath∇2)\boldmathh−Sh1{\boldmathe}2=\boldmath∇\boldmath×\boldmathM, (14)

where is a source term for fluctuating magnetic field, and is the time-dependent operator defined in Eq. (9). The FOSA solution for the fluctuating magnetic field in the Fourier space is given by (see Appendix A for a derivation),

 ˜\boldmathh(\boldmathk,t)=∫t0dt′˜GηT(\boldmathk,t,t′){i\boldmathK(\boldmathk,t′)\boldmath×˜\boldmathM(\boldmathk,t′)+% {\boldmathe}2S(t−t′)[i\boldmathK(% \boldmathk,t′)\boldmath×˜\boldmathM% (\boldmathk,t′)]1}, (15)

where the sheared Green’s function in Fourier space:

 ˜GηT(\boldmathk,t,t′)=exp[−ηT(k2(t−t′)−Sk1k2(t2−t′2)+S23k22(t3−t′3))] (16)

This is derived in Sridhar & Singh (2010). It may be readily verified that the Eq. (15) satisfies both constraints, and . By making use of Eq. (15), and time-stationary Galilean-Invariant statistics for the 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):

 ˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathE(\boldmathk,t)=2∫t0dt′D(t−t′){˜% \boldmathU(\boldmathk,t,t′)\boldmath×˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(\boldmathk,t′)+{\boldmathe}2S(t−t′)[˜\boldmathU(\boldmathk,t,t′)\boldmath×˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(\boldmathk,t′)]1}, (17)

where

 ˜\boldmathU(\boldmathk,t,t′)=∫d3k′(2π)3˜GηT(% \boldmathk−\boldmathk′,t,t′)i% \boldmathK(\boldmathk−\boldmathk′,t′)˜A(\boldmathK(\boldmathk′,t′)). (18)

is a complex velocity field.

Fourier transforming Eq. (7), the equation governing the large-scale field is:

 ∂˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH∂t−S˜¯¯¯¯¯H1{\boldmathe}2=i% \boldmathK(\boldmathk,t)\boldmath×˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathE−ηTK2(\boldmathk,t)˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH,\boldmathK(% \boldmathk,t)\boldmath⋅˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH=0. (19)

Thus the set of Eqs. (17)–(19) describe the evolution of the large-scale magnetic field, in terms of closed, linear integro-differential equation, where both shear strength, , and the -correlation time, , 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 α fluctuations

It is useful to recall basic properties of an exactly solvable limit of delta-correlated-in-time fluctuations when the normalized correlation function , the Dirac delta-function, giving from Eq. (12). Using this in Eqs. (17) and (18), and noting that from Eq. (16), we find the mean EMF:

 ˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathEWN(\boldmathk,t)=˜\boldmathUWN(\boldmathk,t)% \boldmath×˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(\boldmathk,t)with˜\boldmathUWN(% \boldmathk,t)=i\boldmathK(\boldmathk,t)ηα+\boldmathVM, (20)

where the -diffusivity from the definition given in Eq. (11) and the Moffatt drift velocity is defined as:

 \boldmathVM=−(∂A(\boldmathξ)∂\boldmathξ)\boldmathξ=0=∫∞0¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α(\boldmathX,τ)\boldmath∇α(\boldmathX,0)dτ (21)

The Kraichnan diffusivity, , is defined as, . Using these in Eq. (19) leads to the solution for the large-scale magnetic field (we refer the reader to Sridhar & Singh (2014) for more details):

 ˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(\boldmathk,t)=˜G(\boldmathk,t)[˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(\boldmathk,0)+{\boldmathe}2St˜¯¯¯¯¯H1(\boldmathk,0)],\boldmathk\boldmath⋅˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯% \boldmathH(\boldmathk,0)=0. (22)

where

 ˜G(\boldmathk,t) = exp{−∫t0dt′[ηKK2(\boldmathk,t′)+i\boldmathVM\boldmath⋅\boldmathK(\boldmathk,t′)]} = exp{−ηK[k2t−Sk1k2t2+(S2/3)k22t3]−i[(\boldmathVM\boldmath⋅\boldmathk)t−(S/2)VM1k2t2]},

This solution is identical to the one obtained in Sridhar & Singh (2014). 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, 1976; Moffatt, 1978; Sridhar & Singh, 2014), we see from above that the -diffusivity causes a reduction in the turbulent diffusion of the fields, and if it is sufficiently strong, i.e., when , this can lead to an unstable growth of 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 τα

We now turn to the principal aim of this work where we are more interested in exploring the possibility of large-scale dynamo even when the fluctuations are weak, i.e., when , by taking the memory effects into account. Assuming small but finite correlation time for fluctuations, , we reduce the general set of Equations (17)–(19) into a partial differential equation governing the dynamics of large-scale magnetic field which evolves over times much larger than . In this case, the normalized time correlation function, , is significant only for times and it becomes negligible for larger times. The generalized mean EMF as given in Eq. (17) involves a time integral which can be solved under the small approximation.

Since the limit , given by Eq. (20), is non-singular, we proceed by making the following ansatz where, for small , the mean EMF can be expanded in a power series in as:

 ˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathE(\boldmathk,t)=˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathEWN(\boldmathk,t)+˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathE(1)(\boldmathk,t)+˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathE(2)(\boldmathk,t)+… (24)

where and for . Below we verify this ansatz up to , for slowly varying magnetic fields. From Eq. (17) we determine to first order in , for , by (i) changing the integration variable from to ; (ii) setting the upper limit of the time integral to , since is significant only for times as mentioned above, suggesting that only short times contribute appreciably to the integral in Eq. (17); and (iii) keeping the terms inside the in the integrand of Eq. (17) up to only first order in . To be able to expand in , we need to first express the Eq. (18) in lab frame wave vector , so that the green’s function in Eq. (16) and therefore the complex velocity field in Eq. (18) becomes time-translational symmetric 2.

We first rewrite the mean EMF, given in Eq. (17), as

 ˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathE(\boldmathk,t)=2∫∞0dsD(s){˜\boldmathU(\boldmathK,s)\boldmath×˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯% \boldmathH(\boldmathk,t−s)+{\boldmathe}2Ss[˜\boldmathU(\boldmathK,s)\boldmath×˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(\boldmathk,t−s)]1}, (25)

where the complex velocity field, , is

 ˜\boldmathU(\boldmathK,s)=∫d3K′(2π)3˜GηT(\boldmathK−\boldmathK′,s)i(\boldmathK−% \boldmathK′)˜A(\boldmathK′). (26)

Equation (26) is obtained by changing the integration variable in (18) to  — which has unit Jacobian giving .

We make further simplification by considering only axisymmetric modes for which . Note that for the non-axisymmetric modes, 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 .

Let us first work out and correct up to .

Taylor expanding gives,

 ˜\boldmathU(\boldmathk,s) = ˜\boldmathU(\boldmathk,0)+s∂˜\boldmathU∂s∣∣∣s=0+O(s2). (27) where˜\boldmathU(\boldmath% k,0) = i\boldmathkηα+\boldmathVM(from Eq.~{}(???)), (28) and∂˜\boldmathU∂s∣∣∣s=0 = −iηT∫d3K′(2π)3(\boldmathK−\boldmathK′)2(\boldmathK% −\boldmathK′)˜A(\boldmathK′). (29)

Eq. (29) is obtained by differentiating Eq. (26) w.r.t and taking the limit , remember here that , since . Using the Fourier transform for , together with the properties of delta-function, we get

 =−iηT{k2(\boldmathkA(\boldmathξ)+i[\boldmath∇A(\boldmathξ)])+2i\boldmathk(\boldmathk⋅[\boldmath∇A(\boldmath% ξ)]) (30) −\boldmathk[\boldmath∇2A(% \boldmathξ)]−i[\boldmath∇2{\boldmath∇A(\boldmathξ)}]−2(\boldmathk⋅% \boldmath∇){\boldmath∇A(\boldmathξ)}}\boldmathξ=0

Equation (30) can be evaluated once we know the functional form for spatial correlator . We follow Singh (2016), by focusing on the limit when the correlation length of is greater than the scale of variation of the mean magnetic field. In this case derivatives of that are higher than the first order can be neglected, as justified in the Appendix A in Singh (2016). Hence

 ∂˜\boldmathU∂s∣∣∣s=0=−ηTk2(i\boldmathkηα+% \boldmathVM)−2ηT(\boldmathk% \boldmath⋅\boldmathVM)\boldmathk. (31)

Eqs. (28) and (31) together, thus provides the function correct up to .

We write as,

 ˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(\boldmathk,t−s)=˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(\boldmathk,t)−s∂˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(\boldmathk,t)∂t+…. (32)

where it is assumed that . In Eq. (32), we need only up to to find up to . We write this by substituting Eq. (20) in Eq. (19) and using :

 ∂˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH∂t∣∣∣O(1)=S˜¯¯¯¯¯H1{\boldmathe}2+i\boldmathk\boldmath×˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathEWN−ηTk2˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯% \boldmathH=S˜¯¯¯¯¯H1{\boldmathe}2−(ηKk2+i\boldmathk⋅\boldmathVM)˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH (33)

Time-integral in Eq. (25) is then solved by using definitions provided in Eqs. (11) and (12) when we substitute the expressions derived just above for the terms in in Eq. (25). We get after straightforward algebra the following expression for the mean EMF which is correct upto :

 ˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathE(\boldmathk,t) = ˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathEWN(% \boldmathk,t)+τα{(i\boldmathk⋅˜\boldmathUWN)˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯% \boldmathEWN−2ηT(\boldmathk⋅% \boldmathVM)\boldmathk\boldmath×˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH}+ (34) +Sτα{˜¯¯¯¯¯¯¯H1% {\boldmathe}2\boldmath×˜\boldmathUWN+{\boldmathe}2[˜\boldmathUWN\boldmath×˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH]1}

This verifies the ansatz of Eq. (24) up to , as claimed. It is important to note that the Eq. (34) is valid only for slowly varying large–scale magnetic fields. To lowest order this condition can be explicitly stated as: . To obtain the sufficient condition for the validity of Eq. (34), use Eq. (33) for to get the following conditions for three dimensionless quantities which need to be small:

 |Sτα|≪1,|ηKk2τα|≪1,|kVMτα|≪1. (35)

Using Eq. (34) in Eq. (19) we obtain:

 ∂˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH∂t = [S˜¯¯¯¯¯H1{\boldmathe}2+ηαk2˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH−i(\boldmathk\boldmath⋅\boldmathVM)˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH][1+i(\boldmathk\boldmath⋅\boldmathVM)τα−ηαk2τα]−ηTk2˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH+ (36) +2iηTk2τα(\boldmathk⋅\boldmathVM)˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH+Sτα[VM2˜¯¯¯¯¯H3−VM3˜¯¯¯¯¯H2−iηαk3˜¯¯¯¯¯H2][−ik3{\boldmathe}1+ik1{\boldmathe}3], with\boldmathk=(k1,0,k3),andk1˜¯¯¯¯¯H1+k3˜¯¯¯¯¯H3=0.

Equation (36) is the linear partial differential equation obtained by reducing the linear integro-differential equation (see Eqs. (17)–(19)) under the condition of (35). Nonetheless, it describes the evolution of an axisymmetric, large–scale magnetic field over times that are much larger than . It depends on (i) the diffusivity ; (ii) properties of alpha-correlation in terms of , and ; (iii) shear . These parameters must satisfy the three conditions given in Eq. (35) for the validity of the Eq. (36). We note here again that the set of Eqs. (17)–(19) are non-perturbative in both and , whereas Eq. (36) is valid only when .

## 5 Growth rate of modes when τα is non-zero

As usual in numerical works on the related subject (see, e.g., Brandenburg et al., 2008; Singh & Jingade, 2015) where “horizontal” (plane of shear; in this case the plane) averages are performed to define the large–scale magnetic fields, it is therefore useful to consider one–dimensional propagating modes. This is equivalent to setting and equal to zero. Here we only need to set in Eq. (36). In this case the wavevector points along the “vertical” () direction, thus resulting in a uniform which is of no interest for dynamo action. Hence we set , and take . Making these substitutions in Eq. (36) we find:

 ∂˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH∂t = [S˜¯¯¯¯¯H1{\boldmathe}2+ηαk2˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH−ikVM3˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH][1+ikVM3τα−ηαk2τα]−ηTk2˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH+ (37) +2iηTτα(k3VM3)˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH+S[ikVM3τα−ηαk2τα]˜¯¯¯¯¯H2%\boldmath$e$1

Seeking modal solutions of the form,

 ˜¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\boldmathH(k,t)=[˜¯H01(k){\boldmathe}1+˜¯¯¯¯¯H02(k){\boldmathe}2]exp(λt), (38)

and substituting this in Eq. (37) we get the following dispersion relation:

 λ± = −ηKk2−η2αk4τα+(kVM3)2τα+ikV