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\begin{document}


\section*{Representation of the results}

\bigskip

$\bullet$\\
Start with SOCA relations for the low--conductivity limit, e.g.,
\begin{equation}
\eta_t = \frac{u_{rms}^2 \lambda_c^2}{9 \eta} \, ,
\label{eq01}
\end{equation}
where $\lambda_c$ means a correlation length.
Denote dimensionless quantities by a tilde.
We have
\begin{equation}
\tilde{\eta}_t = \frac{\eta}{c_s l} \, , \quad
    \tilde{u}_{rms} = \frac{u_{rms}}{c_s} \, ,
\label{eq03}
\end{equation}
with the normalization length $l$ such that $k_1 l = 2 \pi$.
Define the magnetic Reynolds number $R_m$ by
\begin{equation}
R_m = \frac{u_{rms} \lambda_c}{\eta}
\label{eq05}
\end{equation}
and put
\begin{equation}
\lambda_c = l \, \frac{k_1}{k_f}.
\label{eq07}
\end{equation}
Then (\ref{eq01}) is equivalent to
\begin{equation}
\tilde{\eta}_t = \frac{\tilde{u}_{rms}^2}{9 \tilde{\eta}} \Big( \frac{k_1}{k_f} \Big)^2 \, .
\label{eq09}
\end{equation}
Represent the numerical results for $\tilde{\eta}_t$ in the form
\begin{equation}
\tilde{\eta}_t = \frac{\tilde{u}_{rms}^2}{9 \tilde{\eta}} \Big( \frac{k_1}{k_f} \Big)^2 \, f (R_m, \cdots )
\label{eq11}
\end{equation}
and plot $f (R_m , \cdots )$.

\bigskip

$\bullet$\\
Consider a situation with shear.
Then SOCA delivers for the low--conductivity limit
\begin{equation}
\eta_{xy} = - \frac{u_{rms}^2 \lambda_c^4}{15 \eta^2} S \, .
\label{eq13}
\end{equation}
We have
\begin{equation}
\tilde{S} = \frac{S l}{c_s} \, .
\label{eq15}
\end{equation}
Represent the numerical results for $\tilde{\eta}_{xy}$ in the form
\begin{equation}
\tilde{\eta}_{xy} = - \frac{\tilde{u}_{rms}^2 \tilde{S}}{15 \tilde{\eta}^2} \Big( \frac{k_1}{k_f} \Big)^4
    \, f (R_m, \cdots)  \, .
\label{eq17}
\end{equation}
At the moment the SOCA result for $\eta_{yx}$ is not sure.

\bigskip

$\bullet$\\
Consider rotation.
SOCA delivers for the low--conductivity limit
\begin{equation}
\eta_{xy} = - \eta_{yx} = - \frac{\sqrt{\pi} \, u_{rms}^2 \lambda_c^4}{18 \sqrt{2} \, \eta^2} \, \Omega \,
    \sqrt{q} \, , \quad q = \frac{\lambda_c^2}{\eta \tau_c} \, \to \, 0 \, .
\label{eq19}
\end{equation}
We have
\begin{equation}
\tilde{\Omega} = \frac{\Omega l}{c_s} \, .
\label{eq21}
\end{equation}
Represent the numerical results, e.g., in the form
\begin{equation}
\tilde{\eta}_{xy} = - \frac{\sqrt{\pi} \, \tilde{u}_{rms}^2 \tilde{\Omega}}{18 \sqrt{2} \tilde{\eta}^2}
    \Big( \frac{k_1}{k_f} \Big)^4 \, f (R_m, \cdots) \quad \mbox{???}  \, .
\label{eq23}
\end{equation}



\end{document}
