NUMERICAL STUDIES OF DYNAMO ACTION IN A TURBULENT SHEAR FLOW. I.

Various transport phenomena have traditionally been studied in the framework of mean-field theory (Moffatt 1978; Krause & Rädler 1980; Brandenburg & Subramanian 2005). Applying Reynolds averaging to the induction Equation (3) we find that the mean magnetic field, ${\boldsymbol{B}} ({\boldsymbol{x}} ,t)$, obeys the following (mean-field induction) equation:

$$\begin{eqnarray}&&\left( \frac{\partial }{\partial t}+S{{x}_{1}}\frac{\partial }{\partial {{x}_{2}}} \right){\boldsymbol{B}} -S{{B}_{1}}{{{\boldsymbol{e}} }_{2}}=\nabla \times {\bf{\cal{E} } } +\eta {{\nabla }^{2}}{\boldsymbol{B}} \end{eqnarray} \tag{ 4 }$$

where η is the microscopic resistivity, and ${\bf\cal{E}}$ is the mean electromotive force (EMF), ${\bf\cal{E} } =\langle {\boldsymbol{v}} ^{\prime} \times {\boldsymbol{b}} ^{\prime} \rangle$, where ${\boldsymbol{v}} ^{\prime}$ and ${\boldsymbol{b}} ^{\prime}$ are the fluctuations in the velocity and magnetic fields, respectively. We perform numerical simulations in a cubic domain with a size of $L\times L\times L$, where the mean-field Q of some quantity ${{Q}^{{\rm tot}}}$ is defined by

$$\begin{eqnarray}&&Q({{x}_{3}},t)=\frac{1}{{{L}^{2}}}\int _{-L/2}^{L/2}\int _{-L/2}^{L/2}\;{{Q}^{{\rm tot}}}({{x}_{1}},{{x}_{2}},{{x}_{3}},t)d{{x}_{1}}\;d{{x}_{2}}.\end{eqnarray} \tag{ 5 }$$

Thus, the mean-field quantities discussed here are functions of x₃ and time t. The mean EMF is generally a functional of the mean magnetic field, B_l. For a slowly varying mean magnetic field, the mean EMF can approximately be written as a function of B_l and B_lm (see Brandenburg et al. 2008; Singh & Sridhar 2011):

$$\begin{eqnarray}&&{{\mathcal{E}}_{i}}={{\alpha }_{il}}(t){{B}_{l}}({\boldsymbol{x}} ,t)-{{\eta }_{iml}}(t)\frac{\partial {{B}_{l}}({\boldsymbol{x}} ,t)}{\partial {{x}_{m}}},\end{eqnarray} \tag{ 6 }$$

where ${{\alpha }_{il}}(t)$ and ${{\eta }_{iml}}(t)$ are the transport coefficients which evolve in time at the beginning and saturate at late times.

Previous studies have shown that the mean value of ${{\alpha }_{il}}$ is zero so long as the stirring is non-helical (Brandenburg et al. 2008; Sridhar & Subramanian 2009a, 2009b; Sridhar & Singh 2010; Singh & Sridhar 2011), but it shows zero-mean temporal fluctuations in simulations. Using the definition of the mean-field given in Equation (5), we note that the mean magnetic field ${\boldsymbol{B}} ={\boldsymbol{B}} ({{x}_{3}},t)$. The condition $\nabla \cdot \;{\boldsymbol{B}} =0$ implies that B₃ is uniform in space and that it can be set to zero without loss of generality; hence, we have ${\boldsymbol{B}} =({{B}_{1}},{{B}_{2}},0)$. Thus, Equation (6) for the mean EMF gives ${\bf\cal{E} } =({{\mathcal{E}}_{1}},{{\mathcal{E}}_{2}},0)$ , with

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{\mathcal{E}}_{i}} & = & {{\alpha }_{ij}}{{B}_{j}}-\;{{\eta }_{ij}}{{J}_{j}};\;{\boldsymbol{J}} =\nabla \times {\boldsymbol{B}} =\left( -\frac{\partial {{B}_{2}}}{\partial {{x}_{3}}},\;\frac{\partial {{B}_{1}}}{\partial {{x}_{3}}},\;0 \right), \\ \end{array}\end{eqnarray} \tag{ 7 }$$

where all components of ${{\alpha }_{ij}}$ show zero-mean temporal fluctuations as the forcing is non-helical and the two indexed magnetic diffusivity tensor ${{\eta }_{ij}}$ has four components, $({{\eta }_{11}},{{\eta }_{12}},{{\eta }_{21}},{{\eta }_{22}})$, which are defined in terms of the three indexed object ${{\eta }_{iml}}$ by

$$\begin{eqnarray}&&{{\eta }_{ij}}={{\epsilon }_{lj3}}\;{{\eta }_{i3l}};{\rm which}\;{\rm implies,}\;{{\eta }_{i1}}=-{{\eta }_{i32}};{{\eta }_{i2}}={{\eta }_{i31}}.\end{eqnarray} \tag{ 8 }$$

By substituting Equation (7) for ${\bf\cal{E} }$ in Equation (4), we obtain the evolution equation for the mean magnetic field. The diagonal components, ${{\eta }_{11}}$ and ${{\eta }_{22}}$, augment the microscopic resistivity, η, whereas the off-diagonal components, ${{\eta }_{12}}$ and ${{\eta }_{21}}$, lead to the cross-coupling of B₁ and B₂. It was shown in Singh & Sridhar (2011) that each component of ${{\eta }_{ij}}$ starts from zero at time t = 0 and saturates at some constant value ($\eta _{ij}^{\infty }$) at late times. Here, we aim to measure these saturated quantities.

We use the test-field method to determine the transport coefficients ${{\alpha }_{ij}}$ and ${{\eta }_{ij}}$. This procedure has been described in detail in Brandenburg et al. (2008 and references therein). A brief description of the method is as follows. Assume that ${\boldsymbol{B}}$^q is a set of test-fields and ${{{\bf\cal{E} } }^{q}}$ is the EMF corresponding to the test-field ${\boldsymbol{B}}$^q. By subtracting Equation (4) from Equation (3), we get the evolution equation for the fluctuating field ${\boldsymbol{b}} ^{\prime}$. With properly chosen ${\boldsymbol{B}}$^q and flow ${\boldsymbol{v}} ^{\prime}$, we can numerically solve for the fluctuating field ${{{\boldsymbol{b}} }^{q}}$. This enables us to determine ${{{\bf\cal{E} } }^{q}}$, which can then be used to find ${{\alpha }_{ij}}$ and ${{\eta }_{ij}}$ using $\mathcal{E}_{i}^{q}={{\alpha }_{ij}}B_{j}^{q}-\;{{\eta }_{ij}}J_{j}^{q}$, where ${{{\boldsymbol{J}} }^{q}}=\nabla \times {{{\boldsymbol{B}} }^{q}}$.

There could be various choices for the number and form of the test fields which essentially depend on the problem being solved. For our purposes, we have chosen the test fields, denoted as ${\boldsymbol{B}}$^qc, defined by

$$\begin{eqnarray}&&{{{\boldsymbol{B}} }^{1c}}=B\left( {\rm cos} [k{{x}_{3}}],0,0 \right);\quad {{{\boldsymbol{B}} }^{2c}}=B\left( 0,{\rm cos} [k{{x}_{3}}],0 \right),\end{eqnarray} \tag{ 9 }$$

where B and k are assumed to be constant. Using Equation (9) in the expression $\mathcal{E}_{i}^{q}={{\alpha }_{ij}}B_{j}^{q}\;-\;{{\eta }_{ij}}J_{j}^{q}$, we find the corresponding mean EMF denoted by ${{{\bf\cal{E} } }^{qc}}$ as

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} \mathcal{E}_{i}^{1c} & = & {{\alpha }_{i1}}B\;{\rm cos} [k{{x}_{3}}]+{{\eta }_{i2}}\;Bk\;{\rm sin} [k{{x}_{3}}] \\ \mathcal{E}_{i}^{2c} & = & {{\alpha }_{i2}}B\;{\rm cos} [k{{x}_{3}}]-{{\eta }_{i1}}\;Bk\;{\rm sin} [k{{x}_{3}}];\;i=1,2. \\ \end{array}\end{eqnarray} \tag{ 10 }$$

Here, we have four equations but eight unknowns $({{\eta }_{11}},\ldots {{\eta }_{22}};{{\alpha }_{11}},\ldots {{\alpha }_{22}})$. Therefore, we further consider the following set of test fields, denoted as ${\boldsymbol{B}}$^qs, defined by

$$\begin{eqnarray}&&{{{\boldsymbol{B}} }^{1s}}=B\left( {\rm sin} [k{{x}_{3}}],0,0 \right);\;{{{\boldsymbol{B}} }^{2s}}=B\left( 0,{\rm sin} [k{{x}_{3}}],0 \right),\end{eqnarray} \tag{ 11 }$$

where B and k are assumed to be constant as before. Using Equation (11) in the expression $\mathcal{E}_{i}^{q}={{\alpha }_{ij}}B_{j}^{q}\;-\;{{\eta }_{ij}}J_{j}^{q}$, we find the corresponding mean EMF denoted by ${{{\bf\cal{E} } }^{qs}}$ as

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} \mathcal{E}_{i}^{1s} & = & {{\alpha }_{i1}}B\;{\rm sin} [k{{x}_{3}}]-{{\eta }_{i2}}\;Bk\;{\rm cos} [k{{x}_{3}}] \\ \mathcal{E}_{i}^{2s} & = & {{\alpha }_{i2}}B\;{\rm sin} [k{{x}_{3}}]+{{\eta }_{i1}}\;Bk\;{\rm cos} [k{{x}_{3}}];\;i=1,2. \\ \end{array}\end{eqnarray} \tag{ 12 }$$

Using Equations (10) and (12), we can write

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{\alpha }_{i1}} & = & \frac{1}{B}\left( \mathcal{E}_{i}^{1c}{\rm cos} [k{{x}_{3}}]+\mathcal{E}_{i}^{1s}{\rm sin} [k{{x}_{3}}] \right) \\ {{\alpha }_{i2}} & = & \frac{1}{B}\left( \mathcal{E}_{i}^{2c}{\rm cos} [k{{x}_{3}}]+\mathcal{E}_{i}^{2s}{\rm sin} [k{{x}_{3}}] \right);\;i=1,2, \\ \end{array}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{\eta }_{i1}} & = & -\frac{1}{Bk}\left( \mathcal{E}_{i}^{2c}{\rm sin} [k{{x}_{3}}]-\mathcal{E}_{i}^{2s}{\rm cos} [k{{x}_{3}}] \right) \\ {{\eta }_{i2}} & = & \frac{1}{Bk}\left( \mathcal{E}_{i}^{1c}{\rm sin} [k{{x}_{3}}]-\mathcal{E}_{i}^{1s}{\rm cos} [k{{x}_{3}}] \right);\;i=1,2. \\ \end{array}\end{eqnarray} \tag{ 14 }$$

Thus, from Equations (13) and (14), we can determine the unknown quantities ${{\alpha }_{ij}}$ and ${{\eta }_{ij}}$. For the homogeneous turbulence being considered here, the transport coefficients need to be independent of x₃, and therefore the apparent dependence on x₃ through the terms ${\rm sin} [k{{x}_{3}}]$ and ${\rm cos} [k{{x}_{3}}]$ in Equations (13) and (14) has to be compensated for by the ${{x}_{3}}$-dependent $\mathcal{E}_{i}^{\prime }$s given by Equations (10) and (12).

We use the "shear-periodic" boundary conditions to solve Equations (1)–(3) in the same manner as given in Brandenburg et al. (2008). Shear-periodic boundary conditions have been widely used in numerical simulations in a variety of contexts: simulations of local patches of planetary rings (Wisdom & Tremaine 1988), the local dynamics of differentially rotating disks in astrophysical systems (Balbus & Hawley 1998; Binney & Tremaine 2008), the nonlinear evolution of perturbed shear flow in two-dimensions with the ultimate goal of understanding the dynamics of accretion disks (Lithwick 2007), the shear dynamo (Brandenburg et al. 2008; Yousef et al. 2008a, 2008b; Käpylä et al. 2008), etc, are a few examples.

The random forcing function ${\boldsymbol{f}}$ in Equation (1) is assumed to be non-helical, homogeneous, isotropic, and delta-correlated in time. Furthermore, we assume that the vector function ${\boldsymbol{f}}$ is solenoidal and that the forcing is confined to a spherical shell of magnitude $|{{{\boldsymbol{k}} }_{f}}|={{k}_{f}}$, where the wavevector ${{{\boldsymbol{k}} }_{f}}$ signifies the energy-injection scale (${{l}_{f}}=2\pi /{{k}_{f}}$) of the turbulence. This can be approximately achieved by following the method described in Brandenburg et al. (2008). We note that although the random forcing ${\boldsymbol{f}}$ is delta-correlated in time, the resulting fluctuating velocity field ${\boldsymbol{v}}$ will not be delta-correlated in time (this is due to the inertia, as pointed out in Brandenburg et al. 2008). This has been rigorously proved in Singh & Sridhar (2011) in the limit of a small fluid Reynolds number, the limit which we aim to explore in the present manuscript. Another important fact to note is that in the limit of small $\operatorname{Re}$, the non-helical forcing has been shown to give rise to a non-helical velocity field in Singh & Sridhar (2011); whether or not this is true even in the limit of high $\operatorname{Re}$ has not yet been proven. Thus, performing the simulation in the limit of $\operatorname{Re}\lt 1$ with non-helical forcing guarantees that the fluctuating velocity field is also non-helical.

It is necessary to compare the numerical results obtained in this parameter regime with the earlier analytical work in which the general functional form for the saturated values of magnetic diffusivities, ${{\eta }_{ij}}$, was predicted (see Equation (60) and the related discussion in Singh & Sridhar 2011). It is useful to recall the following expression for the growth rate of the mean magnetic field, obtained from the mean-field theory (see, e.g., Brandenburg et al. 2008; Singh & Sridhar 2011):

$$\begin{eqnarray}&&\frac{{{\lambda }_{\pm }}}{{{\eta }_{T}}\;{{K}^{2}}}=-1\;\pm \;\frac{1}{{{\eta }_{T}}}\sqrt{\eta _{21}^{\infty }\left( \frac{S}{{{K}^{2}}}+\eta _{12}^{\infty } \right)+{{\epsilon }^{2}}},\end{eqnarray} \tag{ 16 }$$

where,

$$\begin{eqnarray}&&\epsilon =\frac{1}{2}(\eta _{11}^{\infty }-\eta _{22}^{\infty });\quad {\rm and}\quad S\;\lt \;0.\end{eqnarray} \tag{ 17 }$$

Figures 1–3 display the plots of ${{\eta }_{t}}$, $\eta _{12}^{\infty }$, and $\eta _{21}^{\infty }$, versus the dimensionless parameter $\left( -{{S}_{{\rm h}}}\operatorname{Re} \right)$, which demonstrate the comparison of the results from a direct numerical simulation with 64³ mesh points with the theoretical results obtained in Singh & Sridhar (2011). The scalings of the ordinates have been chosen for compatibility with the functional form of Equation (60) in Singh & Sridhar (2011). However, it should be noted that we have performed simulations for values of $(-{{S}_{{\rm h}}}\operatorname{Re})$ up to about 0.7, whereas Singh & Sridhar (2011) have been able to explore larger values of $(-{{S}_{{\rm h}}}\operatorname{Re})$. The plots in Figures 1(a)–(c) are for ${\rm Pr} =1$, but for two sets of values of the Reynolds numbers $\operatorname{Re}={\rm Rm}\approx 0.16$ (the "bold" lines represent the theory and the symbols "◦" represent the simulations) and $\operatorname{Re}={\rm Rm}\approx 0.46$ (the "dashed" lines represent the theory and the symbols "×" represent the simulations). Figures 2(a)–(c) are for $\operatorname{Re}\approx 0.13$ and ${\rm Rm}\approx 0.64$, corresponding to ${\rm Pr} \approx 5$ (the "bold" lines represent the theory and the symbols "◦" represent the simulations). Figures 3(a)–(c) are for $\operatorname{Re}\approx 0.13$ and ${\rm Rm}\approx 0.025$, corresponding to ${\rm Pr} \approx 0.2$ (the "bold" lines represent the theory and the symbols "◦" represent the simulations). Some noteworthy properties are as follows:

• (i)  
  As may be seen from Figure 1, the symbols "◦" and "×" (also the bold and dashed lines) lie very nearly on top of each other. This implies that ${{\eta }_{t}}/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$, $\eta _{12}^{\infty }/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$, and $\eta _{21}^{\infty }/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$ are (approximately) functions of $\left( -{{S}_{{\rm h}}}\operatorname{Re} \right)$ and ${\rm Pr}$. Therefore, the magnitude of χ in Equation (60) of Singh & Sridhar (2011) should be much smaller than unity. This was predicted in Singh & Sridhar (2011), and thus our numerical findings are in good agreement with the theoretical investigations of Singh & Sridhar (2011).
• (ii)  
  We see that ${{\eta }_{t}}$ is always positive. For a fixed value of $(-{{S}_{{\rm h}}}\operatorname{Re})$ the quantity ${{\eta }_{t}}/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$ increases with ${\rm Pr}$, and for a fixed value of ${\rm Pr}$ it slowly increases with $(-{{S}_{{\rm h}}}\operatorname{Re})$ (which is consistent with Brandenburg et al. 2008). An excellent agreement between our numerical findings and the theory presented in Singh & Sridhar (2011) may be seen from the top panels of Figures 1–3.
• (iii)  
  The quantity $\eta _{12}^{\infty }$ approaches zero in the limit where $(-{{S}_{{\rm h}}}\operatorname{Re})$ is nearly zero. In the numerical simulation, it is seen to increase with $(-{{S}_{{\rm h}}}\operatorname{Re})$ for a fixed value of ${\rm Pr}$, and for a fixed value of $(-{{S}_{{\rm h}}}\operatorname{Re})$ it increases with ${\rm Pr}$. $\eta _{12}^{\infty }$ is expected to behave in a more complicated manner. Different signs of $\eta _{12}^{\infty }$ are reported in Brandenburg et al. (2008) and Rüdiger & Kitchatinov (2006), whereas both signs have been predicted in calculations of Singh & Sridhar (2011). The differences between the theory and simulations may be inferred from panel (b) in Figures 1–3.
• (iv)  
  As may be seen from the bottom panels of Figures 1–3, $\eta _{21}^{\infty }$ is always positive. This agrees with the results obtained in earlier works (Rädler & Stepanov 2006; Rüdiger & Kitchatinov 2006; Brandenburg et al. 2008). Once again, the agreement between our numerical findings and the theoretical investigations of Singh & Sridhar (2011) for this crucial component of the diffusivity tensor is remarkably good.⁷

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Furthermore, we show the time dependence of the root-mean-squared value of the total magnetic field (${{B}_{{\rm rms}}}$) in Figure (4), which explicitly demonstrates the decay of ${{B}_{{\rm rms}}}$ following three sets of values of control parameters: (i) $\operatorname{Re}\approx 0.128$, ${\rm Rm}\approx 0.643$ (corresponding to ${\rm Pr} \approx 5.0$; shown by the bold line), ${{S}_{{\rm h}}}\approx -1.545$; (ii) $\operatorname{Re}\approx 0.16$, ${\rm Rm}\approx 0.16$ (corresponding to ${\rm Pr} \approx 1.0$; shown by the dashed line), ${{S}_{{\rm h}}}\approx -1.237$; and (iii) $\operatorname{Re}\approx 0.127$, ${\rm Rm}\approx 0.025$ (corresponding to ${\rm Pr} \approx 0.25$; shown by the dashed-dotted line), ${{S}_{{\rm h}}}\approx -1.560$. Results shown in Figure 4 are from a direct numerical simulation with 64³ mesh points and ${{k}_{f}}/K=10.03$.

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We explored this parameter regime for completeness in order to investigate the dynamo action when ${\rm Rm}\lt 1$ and $\operatorname{Re}\gt 1$. The kinematic theory of the shear-dynamo problem was developed in Sridhar & Singh (2010), which is valid for a low magnetic Reynolds number but places no restriction on the fluid Reynolds number. We computed all of the components of ${{\alpha }_{ij}}$ and ${{\eta }_{ij}}$ using the test-field method and investigated the possibility of a dynamo action. We find that all of the components of ${{\alpha }_{ij}}$ show fluctuations in time with mean zero, and therefore we do not expect the generation of any net helicity in the flow in these parameter regimes. We summarize all of our results for $\operatorname{Re}\gt 1$ and ${\rm Rm}\lt 1$ in detail in Table 1.

Table 1.  Summary of the Simulations for $\operatorname{Re}\gt 1$ and ${\rm Rm}\lt 1$

Run	$\operatorname{Re}$	${\rm Rm}$	${{k}_{f}}/K$	$-{{S}_{{\rm h}}}$	${\rm Ma}$ a	Grid	${{\eta }_{t}}/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$	${{\eta }_{12}}/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$	${{\eta }_{21}}/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$	Comments
A	5.50	0.14	10.03	0.136	0.110	643	0.000366	0.000044	0.0000279	No dynamo
B	4.63	0.70	10.03	0.014	0.139	643	0.006845	0.000174	0.0000764	No dynamo
C	4.69	0.70	10.03	0.057	0.141	643	0.007000	0.000628	0.0003048	No dynamo
D	4.83	0.73	10.03	0.103	0.145	643	0.007247	0.001224	0.0005103	No dynamo
E	5.63	0.84	10.03	0.141	0.169	643	0.006654	0.002330	0.0006008	No dynamo
F	41.14	0.82	3.13	0.186	0.258	643	0.000110	0.000017	0.0000092	No dynamo
G	48.40	0.41	3.13	0.186	0.258	643	0.000025	0.000003	0.0000021	No dynamo

^aMach Number.

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We find no evidence of dynamo action in this particular parameter regime. This is shown clearly in Figure 5, in which we plot the time dependence of root-mean-squared value of the total magnetic field (${{B}_{{\rm rms}}}$) and demonstrate the absence of the dynamo action in this parameter regime. Figure 5 shows results from the direct simulation with 64³ mesh points for the following four sets of parameter values: (i) $\operatorname{Re}\approx 24.57$, ${\rm Rm}\approx 0.614$, ${{k}_{f}}/K=5.09$, ${{S}_{{\rm h}}}\approx -0.118$ (shown by the bold line); (ii) $\operatorname{Re}\approx 22.40$, ${\rm Rm}\approx 0.448$, ${{k}_{f}}/K=5.09$, ${{S}_{{\rm h}}}\approx -0.128$ (shown by the dashed line); (iii) $\operatorname{Re}\approx 43.17$, ${\rm Rm}\approx 0.863$, ${{k}_{f}}/K=3.13$, ${{S}_{{\rm h}}}\approx -0.177$ (shown by the dashed dotted line); and (iv) $\operatorname{Re}\approx 36.54$, ${\rm Rm}\approx 0.365$, ${{k}_{f}}/K=3.13$, ${{S}_{{\rm h}}}\approx -0.209$ (shown by the triple-dot–dashed line).

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We now report our analysis concerning the growth of the mean magnetic field in a background linear shear flow with non-helical forcing at small scales for the case where $\operatorname{Re}\lt 1$ and ${\rm Rm}\gt 1$. This is a particularly interesting regime for the following reasons: (i) it is an important fact to note that in the limit of small $\operatorname{Re}$, the non-helical forcing has been shown to give rise to a non-helical velocity field (see the discussion below Equation (46) of Singh & Sridhar 2011); (ii) for low $\operatorname{Re}$ the Navier–Stokes Equation (1) can be linearized, and thus it becomes an analytically more tractable problem compared to the case of high $\operatorname{Re}$. Such solutions have been rigorously obtained without the Lorentz forces and were presented in Singh & Sridhar (2011). Therefore, it appears more reasonable to develop a theoretical framework in the limit $\operatorname{Re}\lt 1$ and ${\rm Rm}\gt 1$ before one aims to develop a theory which is valid for both ($\operatorname{Re}$, ${\rm Rm}$) $\gt 1$. Such thoughts motivated us to perform numerical experiments in this limit to look for the dynamo action. Figures 6–8 display the time dependence of the root-mean-squared value of the total magnetic field and the spacetime diagrams of mean fields ${{B}_{1}}({{x}_{3}},t)$ and ${{B}_{2}}({{x}_{3}},t)$ for three different combinations of $\operatorname{Re}$ and ${\rm Rm}$. These simulations were performed with 128³ mesh points. We have scaled the magnetic fields in Figures 6–8 with respect to ${{B}_{{\rm eq}}}$, where ${{B}_{{\rm eq}}}={{({{\mu }_{0}}\langle \rho v_{{\rm rms}}^{2}\rangle )}^{1/2}}$. Scalings in these figures have been chosen for compatibility with Figures 7 and 8 of Brandenburg et al. (2008). Below, we list a few useful points related to the dynamo action when $\operatorname{Re}\lt 1$ and ${\rm Rm}\gt 1$ based on careful investigation of Figures 6–8.

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• (i)  
  The top panels of Figures 6–8 clearly show the growth of ${{B}_{{\rm rms}}}$, demonstrating the shear dynamo due to non-helical forcing ($B_{{\rm rms}}^{2}=\langle {{B}^{2}}\rangle +\langle {{b}^{2}}\rangle$, where B and b are the magnitudes of the mean and fluctuating magnetic fields, respectively). Thus, the ${{B}_{{\rm rms}}}$ field may grow either due to B or b, or due to both B and b.
• (ii)  
  Denoting the magnetic diffusion timescale as ${{\tau }_{\eta }}={{(\eta k_{f}^{2})}^{-1}}$ and the eddy turn over timescale as ${{\tau }_{{\rm edd}}}={{({{v}_{{\rm rms}}}{{k}_{f}})}^{-1}}$, we write ${{\tau }_{\eta }}={\rm Rm}\;{{\tau }_{{\rm edd}}}$. The magnetic fields in these simulations survive for times, say $t=640\;{{\tau }_{{\rm edd}}}$, which for ${\rm Rm}\approx 32$ (corresponding to Figure 8) implies $t\approx 20\;{{\tau }_{\eta }}$, i.e., 20 times the diffusion timescale. This is a clear indication of the dynamo action as the magnetic fields survive much longer than the magnetic diffusion timescale.
• (iii)  
  Spacetime diagrams in Figures 6–8 reveal that the mean magnetic fields start developing only after times which are few times the magnetic diffusion timescale (${{\tau }_{\eta }}$).
• (iv)  
  Although the mean magnetic field starts developing at much later times, ${{B}_{{\rm rms}}}$ starts growing at earlier times. The possibility of the growth of a mean-squared field with no net mean magnetic field at these early times cannot be ruled out.

In Table 2, we present the results from test-field simulations performed in the regime $\operatorname{Re}\lt 1$ and ${\rm Rm}\gt 1$. We have performed runs for two sets of values of the fluid Reynolds number, $\operatorname{Re}$: runs R1 and R2 with $\operatorname{Re}\approx 0.4$, and runs S1–S4 with $\operatorname{Re}\approx 0.6$. We confirm that all of the components of the magnetic diffusivity tensor, ${{\eta }_{t}}$, ${{\eta }_{12}}$, and ${{\eta }_{21}}$, increase with increasing magnetic Reynolds number, ${\rm Rm}$, which is in agreement with Brandenburg et al. (2008). Comparing the numerical values of the η tensor in Table 2 (with ${\rm Rm}\gt 1$) with those in Table 1 (with ${\rm Rm}\lt 1$), we see that each component of ${{\eta }_{ij}}$ increases with ${\rm Rm}$ for the range of values considered in this work. For larger values of ${\rm Rm}$, we refer the reader to Brandenburg et al. (2008) where the possibility of the relevant component, ${{\eta }_{21}}$, becoming negative at much larger ${\rm Rm}(\gt 100)$ was discussed; however, the error bars were quite large, and therefore no conclusion could be drawn regarding the mean-field dynamo action. As mentioned earlier, our interest is in the intermediate ${\rm Rm}$ values (5–40), and we show our findings with maximum of ${\rm Rm}$ being below 38. We note that the component ${{\eta }_{21}}$ remains positive even in this parameter regime, thus confirming earlier claims that the shear-current effect cannot be responsible for the observed large-scale dynamo action.

Table 2.  Summary of the Simulations for $\operatorname{Re}\lt 1$ and ${\rm Rm}\gt 1$

Run	$\operatorname{Re}$	${\rm Rm}$	$-{{S}_{{\rm h}}}$	${\rm Ma}$	Grid	${{\eta }_{t}}/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$	${{\eta }_{12}}/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$	${{\eta }_{21}}/({{\eta }_{T}}{{\operatorname{Re}}^{2}})$
R1	0.393	9.837	0.391	0.10	1283	3.657	0.644	0.011
R2	0.396	19.796	0.583	0.10	1443	4.735	1.695	0.057
S1	0.592	5.920	0.325	0.15	1443	1.508	0.340	0.059
S2	0.598	14.938	0.515	0.15	1443	2.110	0.704	0.084
S3	0.598	29.875	0.515	0.15	1443	2.400	0.786	0.140
S4	0.603	37.692	0.638	0.15	1443	2.435	1.196	0.189

Note. All runs have ${{k}_{f}}/K=5.10$.

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It is instructive to note the magnitude of the magnetic power at different length scales in the simulations and study its evolution in time. Although the forcing is done at a single length scale, a typical kinetic energy spectrum has a peak at the stirring scale with significantly less power at other length scales (e.g., see the dashed lines in the various panels of Figure 9). In Figure 9, we display the energy spectra obtained in one of the three simulations (for different combinations of the control parameters, all with $\operatorname{Re}\lt 1$), corresponding to that shown in Figure 8. Thus, Figures 8 and 9 show results obtained from one particular simulation with 128³ mesh points, $\operatorname{Re}\approx 0.641$, ${\rm Rm}\approx 32.039$, ${{k}_{f}}/K=5.09$, and ${{S}_{{\rm h}}}\approx -0.60$. A few noteworthy points are discussed below in detail.

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• (i)  
  Initially, the magnetic power is very small compared to the kinetic power and it is mainly concentrated at large k (i.e., small length scales), as may be seen from panel (a) of Figure 9. Also, there is essentially no magnetic power at small k (i.e., large length scales) during the initial stage of the simulation.
• (ii)  
  The strength of the total magnetic field decreases up to a certain time due to dissipation (compare panels (a) and (b) of Figure 9), before it starts building up due to the dynamo action.
• (iii)  
  From the top panel of Figure 8, we see that the root-mean-squared value of the total magnetic field starts growing due to the dynamo action ($B_{{\rm rms}}^{2}=\langle {{B}^{2}}\rangle +\langle {{b}^{2}}\rangle$, where B and b are the magnitudes of the mean and fluctuating magnetic fields, respectively). As the ${{B}_{{\rm rms}}}$ field may grow either due to B or b, or due to both B and b, it seems necessary to understand this in more detail. From Figure 9, it may be seen that the magnetic energy, when it begins to grow, grows at all scales until it saturates at small length scales.
• (iv)  
  The small-scale field grows, which averages out to zero, and hence does not show up in the spacetime diagrams of Figure 8. This is generally referred to as the fluctuation dynamo. The growth rate changes and becomes smaller after the fluctuation dynamo saturates (which happens at $t\;{{v}_{{\rm rms}}}\;{{k}_{f}}\approx 150$ in Figure 8 and the corresponding power spectrum at that time is shown in panel (d) of Figure 9).
• (v)  
  Although there is non-zero magnetic energy at the large scales at time $t\;{{v}_{{\rm rms}}}\;{{k}_{f}}\approx 150$ (see panel (d) of Figure 9), we begin to see some features in the spacetime diagrams of the mean magnetic field (shown in Figure 8) only beyond $t\;{{v}_{{\rm rms}}}\;{{k}_{f}}\approx 150$. Thus, it is possible that ${\boldsymbol{B}} ={\bf 0}$ while $\langle {{B}^{2}}\rangle$ is finite.
• (vi)  
  The mean magnetic field starts developing beyond $t\;{{v}_{{\rm rms}}}\;{{k}_{f}}\approx 150$ (which is about five times the magnetic diffusion timescale) and saturates at $t\;{{v}_{{\rm rms}}}\;{{k}_{f}}\approx 330$ (see Figure 8), after which time the magnetic energy essentially stops evolving at all length scales, as may be seen from Figure 9.
• (vii)  
  When the magnetic energy saturates at some value, we see significant magnetic power at the largest scale.

We recall that in the kinematic stage, the magnetic field at all length scales grows at the same rate, i.e., the magnetic spectrum remains shape invariant (Brandenburg & Subramanian 2005; Subramanian & Brandenburg 2014). From panels (b) to (d) of Figure 9, the magnetic spectrum evolves in a nearly shape invariant manner. During this kinematic stage, much of the magnetic power still lies at small scales, but the power at the largest scales also grows with time. This initial growth of magnetic energy occurs at turbulent (fast) timescales. Toward the end of the kinematic stage, the growth rates of large- and small-scale magnetic fields become different due to the back reaction from Lorentz forces. The small-scale fields saturate, whereas the large-scale fields continue to grow, thus dominating over small-scale fields at much later times; shown in panels (e) and (f) of Figure 9.

It may be seen from the top panels of Figures 6–8 that ${{B}_{{\rm rms}}}$ shows exponential growth. We denote the initial exponential growth rate of ${{B}_{{\rm rms}}}$ as $\gamma$. It is evident from Figure (10) that the dimensionless growth rate (${{\gamma }^{*}}=\gamma /({{v}_{{\rm rms}}}{{k}_{f}})$) appears to scale as ${{\gamma }^{*}}\propto -{{S}_{{\rm h}}}$ in the range of parameters explored in this work. This result is in agreement with (Brandenburg et al. 2008; Yousef et al. 2008b; Heinemann et al. 2011; Richardson & Proctor 2012).

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In Table 3, we summarize the details of various simulations performed in different parameter regimes. We note that larger shear contributes positively to the mean-field dynamo action; compare Runs C1 and C2, where the shear in C2 is five times larger compared to that in C1, with the rest of the parameters being the same.

Table 3.  Summary of Simulations in Different Parameter Regimes

Run	$\operatorname{Re}$	${\rm Rm}$	${{k}_{f}}/K$	$-{{S}_{{\rm h}}}$	${\rm Ma}$	Grid	Comments
A1	0.47	0.47	10.03	1.27	0.0235	643	No dynamo
A2	0.73	0.91	10.03	0.41	0.0727	1283	No dynamo
A3	0.76	0.57	1.54	2.78	0.0701	1283	No dynamo
B1	41.20	0.82	3.13	0.186	0.258	643	No dynamo
B2	4.65	0.69	10.03	0.0285	0.139	643	No dynamo
B3	4.99	0.75	10.03	0.133	0.150	643	No dynamo
C1	0.59	29.47	5.09	0.12	0.03	1283	No dynamo
C2	0.59	29.47	5.09	0.66	0.03	1283	Dynamo
C3	0.85	25.50	5.09	0.226	0.129	1283	Dynamo
C4	0.75	33.60	10.03	0.236	0.0674	1283	Dynamo
D1	1.04	41.66	3.13	0.367	0.13	1283	Dynamo
D2	1.79	89.51	5.09	0.215	0.0911	1283	Dynamo

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