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\section{Comparison with analytic SOCA calculations}
\label{SOCA}

In R\"adler \& Stepanov (2006, referred to as RS06 in the following)
the mean electromotive force has been calculated in the second--order correlation approximation
for a generally inhomogeneous turbulence in a incompressible rotating fluid
showing a position--dependent mean motion.
In this context the second--order correlation approximation was understood as neglect of terms
in the induction equation as well as in the momentum balance.
Both the Coriolis force and the derivatives of the mean velocity were assumed to be small enough
so that the mean electromotive force is linear in the angular velocity $\OO$
and the gradient tensor of $\meanUU$.
Detailed results were obtained for a special correlation function for the background turbulence.

Let us apply the results to the situations considered in this paper.
In the case of rotation without shear we obtain
\EQ
\frac{\delta}{\eta_{\rm t}} = \frac{\sqrt{\pi}}{4 \sqrt{2}} \Co \, \Rm (\lambda_c k_{\rm f})^2
    \sqrt{q} \, \delta^o (q, \Pm)
\label{B01}
\EN
with $\Co$, $\Rm$ and $\Pm$ as defined above, $q = \lambda_{\rm c}^2 / \eta \tau_{\rm c}$,
and $\lambda_c$ and $\tau_{\rm c}$ being correlation length and time.
When introducing the Strouhal number $\mbox{St} = u_{\rm rms} k_{\rm f} \tau_{\rm c}$,
we have $q = \Rm / \mbox{St}$.
The function $\delta^o$ approaches unity if $\Pm = 1$ and $q \to 0$.
It can be calculated according to
\EQ
\delta^o (q, \Pm) =
\frac{\delta^{o (\Omega)}(q, \Pm) + \kappa^{o (\Omega)}(q, \Pm)}{ 2 \beta^{o (0)}(q)} \, ,
\label{B03}
\EN
from the functions $\delta^{o (\Omega)}$, $\kappa^{o (\Omega)}$ and $\beta^{o (0)}$
defined and plotted in RS06.

Proceeding to the case of shear without rotation we note first
that due the above--mentioned assumption on the linearity in the mean--velocity gradient,
that is in $S$, both $\kappa_{11}$ and $\kappa_{22}$ are equal to zero.
Further we have
\EQ
\frac{\eta_{12}}{\eta_{\rm t}} = - \frac{19}{20} \mbox{Sh} \, \Rm (\lambda_{\rm c} k_{\rm f})^2
    \eta_{12}^o (q, \Pm) \, , \quad
    \frac{\eta_{21}}{\eta_{\rm t}} = - \frac{7}{20} \mbox{Sh} \, \Rm (\lambda_{\rm c} k_{\rm f})^2
    \eta_{21}^o (q, \Pm)
\label{B05}
\EN
with functions $\eta^o_{12}$ and $\eta^o_{21}$ which approach unity if $\Pm = 1$ and $q \to 0$.
They are given by
\EQA
\eta_{12}^o (q, \Pm) = \frac{13 \kappa^{o (D)}(q, \Pm) + 5 \delta^{o (W)}(q, \Pm)
    + \kappa^{o (W)}(q, \Pm)}{19 \beta^{o (0)}(q)} \, , \quad
%AB: added line break if not part of ApJ environment...
\\
    \eta_{21}^o (q, \Pm) = \frac{13 \kappa^{o (D)}(q, \Pm) - 5 \delta^{o (W)}(q, \Pm)
    - \kappa^{o (W)}(q, \Pm)}{7 \beta^{o (0)}(q)}
\label{B07}
\ENA
with the functions $\kappa^{o (D)}$, $\delta^{o (W)}$, $\kappa^{o (W)}$ and $\beta^{o (0)}$ of RS06.

[From (\ref{B01}) we conclude
that $\ln (\delta / \eta_{\rm t}) = \ln \Co + const$
if all relevant parameters except $\Co$ are constant.
The data with $\Co \leq 1.15$ given in Fig.1 correspond
to $\ln (\delta / \eta_{\rm t}) \approx 0.86 \ln \Co + const$.\\
We further conclude that
$\ln (\delta / \eta_{\rm t}) = \ln \Rm + const$
if all relevant parameters (including $q$) except $\Rm$ are constant.
The data of Fig.2 with $0.65 \leq \Rm \leq 5.0$ lead
to $\ln (\delta / \eta_{\rm t}) \approx 0.76 \ln \Rm + const$. \\
It is hard to evaluate Fig.3 in that sense.\\
From (\ref{B05}) we conclude
that $\ln (\eta_{12} / \eta_{\rm t}) = \ln \Rm + const$
and $\ln (\eta_{21} / \eta_{\rm t}) = \ln \Rm + const$
if all relevant parameters except $\Rm$ are constant.
From the data for $0.7 \leq \Rm \leq 10$ in Fig.4 we conclude
that $\ln (\eta_{12} / \eta_{\rm t}) \approx 0.87 \ln \Rm + const$,
from the data for $0.7 \leq \Rm \leq 7$ there
$\ln (\eta_{21} / \eta_{\rm t}) \approx  0.61 \ln \Rm + const$.\\
All evaluations of the figures are very crude, maybe erroneous.
Moreover I am not sure to which extent the assumptions on the constancy
of the other parameters are justified.\\
These considerations say, of course, nothing about, e.g., the dependencies on $\lambda_{\rm c} k_{\rm f}$,
$q$ or $\Pm$.]


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