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\begin{document}


{\Large \bf Effects of shear and rotation on the mean electromotive force}\\

\bigskip
\bigskip

\section{}

\subsection{}
\label{sec11}

Assume homogeneous isotropic background turbulence.
Influences of mean shear and rotation on mean electromotive force $\bscE$ weak,
i.e., $\bscE$ linear in shear and angular velocity.

By symmetry arguments
\begin{eqnarray}
\bscE &=& - \beta^{(0)} \bnab \x \bmB  - \beta^{(D)} \bD \cdot (\bnab \x \bmB)
   - (\delta^{(W)} \bW + \delta^{(\Omega)} \bOmega) \x (\bnab \x \bmB)
\nonumber\\
&& - (\kappa^{(W)} \bW + \kappa^{(\Omega)} \bOmega) \cdot (\bnab \bmB)^{(s)}
   - \kappa^{(D)} \hat{\bkappa} (\bD) \cdot (\bnab \bmB)^{(s)} \, ,
\label{eq01}
\end{eqnarray}
with coefficients $\beta^{(0)}$, $\beta^{(D)}$, $\delta^{(\Omega)}$, $\delta^{(W)}$,
$\kappa^{(\Omega)}$, $\kappa^{(W)}$ and $\kappa^{(D)}$ independent of $\bD$, $\bW$ and $\bOmega$,\\
$(\bnab \bmB)^{(s)}$ symmetric part of the gradient tensor of $\bmB$
(i.e., $(\bnab \bmB)^{(s)}_{ij} = (1/2) (\p {\mB}_i / \p x_j + \p {\mB}_j / \p x_j)$), \\
$\hat{\kappa}_{ijk} = \epsilon_{ijl} D_{lk} + \epsilon_{ikl} D_{lj}$, \\
$\bD$ deformation tensor (i.e., $D_{ij} = (1/2) (\p {\mU}_i / \p x_j + \p {\mU}_j / \p x_j)$),
$\bW = \bnab \x \bmU$,\\
$\bOmega$ angular velocity which determines Coriolis force.\\
See RS06.

\subsection{}
\label{sec12}

Assume
\begin{equation}
\bmU = (0, S x, 0) \, , \quad \bOmega = (0, 0, \Omega)
\label{eq11}
\end{equation}
with constants $S$ and $\Omega$.\\
Then
\begin{equation}
\bW = (0, 0, S) \, , \quad
   D_{ij} = \frac{1}{2} S \,( \delta_{i1} \delta_{j2} + \delta_{i2} \delta_{j1}) \, .
\label{eq13}
\end{equation}

\subsection{}
\label{sec13}

Assume further that $\bmB$ is independent of $x$ and $y$,
\begin{equation}
\bmB = (\mB_x (z), \mB_y (z), \mB_z) \, .
\label{eq21}
\end{equation}
$\mB_z$ must be independent of $z$, too.
Then
\begin{equation}
\bnab \x \bmB = \mu \bJ \, , \quad \bJ = (J_x, J_y, 0) \, , \quad
    (\bnab \bmB)^{(s)}_{ij} = - (\mu/2) ( J_x (\delta_{i2} \delta_{j3} + \delta_{i3} \delta_{j2})
    - J_y (\delta_{i1} \delta_{j3} + \delta_{i3} \delta_{j1}))
\label{eq23}
\end{equation}
where
\begin{equation}
\mu J_x = - \frac{\p \mB_y}{\p z} \, , \quad \mu J_y = \frac{\p \mB_x}{\p z} \, .
\label{eq25}
\end{equation}


\subsection{}
\label{sec14}

Under the assumptions introduced so far equation (\ref{eq01}) can be written in the form
\begin{equation}
{\cE}_i  = - \mu \, \eta_{ij} J_j  \, .
\label{eq31}
\end{equation}
We have then
\begin{eqnarray}
\eta_{xx} &=& \eta_{yy} = \eta{zz} = \beta^{(0)}
\nonumber\\
\eta_{xy} &=& (\zeta^{(D)} - \zeta^{(W)}) S - \zeta^{(\Omega)} \Omega
\nonumber\\
\eta_{yx} &=& (\zeta^{(D)} + \zeta^{(W)}) S + \zeta^{(\Omega)} \Omega
\label{eq33}\\
\eta_{xz} &=& \eta_{yz} = \eta_{zx} = \eta_{zy} = 0 \, ,
\nonumber
\end{eqnarray}
where
\begin{equation}
\zeta^{(D)} = \frac{1}{2} (\beta^{(D)} - \kappa^{(D)}) \, , \quad
    \zeta^{(W)} = \delta^{(W)} - \frac{1}{2} \kappa^{(W)} \, , \quad
    \zeta^{(\Omega)} = \delta^{(\Omega)} - \frac{1}{2} \kappa^{(\Omega)} \, .
\label{eq35}
\end{equation}
(In the case of rotation ($S = 0$) we may also write
${\cE}_i = - \mu \beta^{(0)} \, J_i  + \zeta^{(\Omega)} \Omega \, \dd \mB_i / \dd z$.)


Simulations under the assumptions of sections \ref{sec12} and \ref{sec13} do not deliver us
the seven coefficients $\beta^{(0)}$, $\beta^{(D)}$, $\delta^{(\Omega)}$, $\delta^{(W)}$,
$\kappa^{(\Omega)}$, $\kappa^{(W)}$ and $\kappa^{(D)}$
but only $\beta^{(0)}$ and the three combinations $\zeta^{(D)}$, $\zeta^{(W)}$ and $\zeta^{(\Omega)}$.
We have, e.g.,
\begin{equation}
\beta^{(0)} = \frac{1}{2} \, (\eta_{xx} + \eta_{yy}) \, ,
\label{eq31}
\end{equation}
in the case of shear ($\Omega = 0$)
\begin{equation}
\zeta^{(D)} = \frac{1}{2 S} \, (\eta_{xy} + \eta_{yx}) \, , \quad
    \zeta^{(W)} = - \frac{1}{2 S} \, (\eta_{xy} - \eta_{yx}) \, ,
\label{eq31}
\end{equation}
and in the case of rotation ($S = 0$)
\begin{equation}
\zeta^{(\Omega)} = - \frac{1}{2 \Omega} \, (\eta_{xy} - \eta_{yx}) \, .
\label{eq31}
\end{equation}


\subsection{}
\label{sec15}

Numerical results can contribute to answers to the following questions.

\bigskip

(a) Applies $\beta^{(0)} = \frac{1}{3} u_\mathrm{rms}^2 \tau_{cor}$ (for $R_m \gg 1$) also
with $St =  O (1)$?\\
Has $\beta^{(0)}$ really no ``magnetic part" (cf., e.g., RR07)?

\bigskip

(b) Is a ``$\bW \x \bJ$ dynamo" possible? \\
Dynamos of that kind are impossible with an incompressible fluid and purely hydrodynamic background turbulence
as long as SOCA applies (RS06, RK06).\\
Assume mean magnetic field $\sim \exp (\iu k z)$ and $\Omega = 0$.\\
A necessary condition for a growing magnetic field is $\eta_{yx} (S + \eta_{xy} k^2) > 0$ (Brandenburg et al. ...)\\
or, equivalently,  $\delta (1 - \delta' k^2) > 0$ with $\delta = \zeta^{(D)} + \zeta^{(W)}$
and $\delta' = - \zeta^{(D)} + \zeta^{(W)}$ (RS06).\\
Excluding cases in which the neglect of contributions to $\bscE$ with higher--order derivatives of $\bmB$
is questionable we may reduce this conditions to $\eta_{yx} S > 0$ or $\delta > 0$.

\bigskip

(c) Under which conditions has the $\bOmega \x \bJ$--effect a reasonable magnitude?\\
It vanishes in SOCA for an incompressible fluid with purely hydrodynamic background turbulence
both in the high--conductivity and in the low--conductivity limit.

\section{}

\subsection{}
\label{sec21}

Consider for a moment SOCA results.
For purely hydrodynamic background turbulence (kinetic forcing) RS06 delivers
\begin{equation}
\beta^{(0)} = \frac{1}{9} \, \frac{u^2_\mathrm{rms} \lambda_c^2}{\eta} \, \hat{\beta}^{(0)} (q)
\label{eq51}
\end{equation}
and
\begin{eqnarray}
\beta^{(D)} &=& \frac{7}{90} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \, \hat{\beta}^{(D)} (P_m, q) \, , \quad
    \kappa^{(D)} = \frac{13}{90} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \, \hat{\kappa}^{(D)} (P_m, q)
\nonumber\\
\delta^{(W)} &=& \frac{1}{36} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \, \hat{\delta}^{(W)} (q) \, , \quad
    \kappa^{(W)} = - \frac{1}{90} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \, \hat{\kappa}^{(W)} (P_m, q)
\label{eq53}\\
\delta^{(\Omega)} &=& - \frac{\sqrt{\pi}}{36 \sqrt{2}} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \sqrt{q}
    \, \hat{\delta}^{(\Omega)} (P_m, q) \, , \quad
    \kappa^{(\Omega)} = \frac{\sqrt{\pi}}{18 \sqrt{2}} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \sqrt{q}
    \, \hat{\kappa}^{(\Omega)} (P_m, q)\, ,
\nonumber
\end{eqnarray}
The $\hat{\beta}^{(0)}$, $\hat{\beta}^{(D)}$, $\ldots$
$\hat{\kappa}^{(\Omega)}$ are dimensionless functions of the magnetic Prandtl number $P_m$
and the parameter $q$, or $q$ only,
\begin{equation}
P_m = \frac{\nu}{\eta} \, , \quad  q = \frac{\lambda_c^2}{\eta \tau_c} \, .
\label{eq55}
\end{equation}
They approach unity for $P_m = 1$ and $q \to 0$.
Further $\lambda_c$ means a correlation length and $\tau_c$ a correlation time,
defined, e.g., by
$\int_0^\infty \langle \bu (\bx, t) \cdot \bu (\bx + \xi \be_x, t) \rangle \dd \xi
= \sqrt{\frac{\pi}{2}} \, u_\mathrm{rms}^2 \lambda_c$
and
$\int_0^\infty \langle \bu (\bx, t) \cdot \bu (\bx, t - \tau) \rangle \dd \tau
= u_\mathrm{rms}^2 \tau_c$.
These definitions agree with the ansatz for correlation function used in RS06,
see relation (RS06 51).
Is this ansatz in agreement with the results of the numerical simulations?


We recall
\begin{equation}
R_m = \frac{u_\mathrm{rms} \lambda_c}{\eta} \, , \quad
    St = \frac{u_\mathrm{rms} \tau_c}{\lambda_c} \, .
\label{eq57}
\end{equation}
and note that $R_m / St = q$.

With (\ref{eq35}) and (\ref{eq53}) we obtain
\begin{equation}
\zeta^{(D)}  = - \frac{1}{30} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \, \hat{\zeta}^{(D)} \, , \quad
    \zeta^{(W)} = \frac{1}{30} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \, \hat{\zeta}^{(W)} \, , \quad
    \zeta^{(\Omega)} =  - \frac{\sqrt{\pi}}{18 \sqrt{2}} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \sqrt{q} \,
    \hat{\zeta}^{(\Omega)}
\label{eq59}
\end{equation}
with
\begin{equation}
\hat{\zeta}^{(D)} = \frac{- 7 \hat{\beta}^{(D)} + 13 \hat{\kappa}^{(D)}}{6} \, , \quad
    \hat{\zeta}^{(W)} = \frac{5 \hat{\delta}^{(W)} + \hat{\kappa}^{(W)}}{6} \, , \quad
    \hat{\zeta}^{(\Omega)} = \frac{\hat{\delta}^{(\Omega)} + \hat{\kappa}^{(\Omega)}}{2} \, .
\label{eq61}
\end{equation}
The $\hat{\zeta}^{(D)}$, $\hat{\zeta}^{(W)}$ and $\hat{\zeta}^{(\Omega)}$ are dimensionless
and depend on $P_m$ and $q$.
For $P_m = 1$ and $q \to 0$ they approach unity.
This implies that in this limit $\delta = \zeta^{(D)} + \zeta^{(W)} = 0$.


\bigskip

It is very important to extend the results to the case of mhd background turbulence (magnetic forcing)!!!

\subsection{}
\label{sec22}

Abolishing the restriction to SOCA we represent the results for $\beta^{(0)}$, $\zeta^{(D)}$,
$\zeta^{(W)}$ and $\zeta^{(\Omega)}$ again in the form of (\ref{eq51}) and (\ref{eq59})
but with reinterpreted $\hat{\beta}^{(0)}$, $\hat{\zeta}^{(D)}$, $\hat{\zeta}^{(W)}$
and $\hat{\zeta}^{(\Omega)}$.
We consider them again as functions of $P_m$, further of $R_m$ (possibly also $St$)
and of the Mach number $M = u_\mathrm{rms} / c_s$, where $c_s$ means the sound speed.
In the case of mhd background turbulence (magnetic forcing) they depend in general
on the Alfven number $A = b_\mathrm{rms} / \sqrt{\mu \varrho} \, u_\mathrm{rms}$, too.

\bigskip

As soon as SOCA results for mhd background turbulence are available this concept could be slightly modified.


\subsection{}
\label{sec23}

In the numerical simulation instead of the original quantities $\bscE$, $\bmB$, $\eta_{ij}$,
$\beta^{(0)}$, $\zeta^{(..)}$, $\delta$, $\nu$, $\eta$, $S$, $\Omega$, $u_\mathrm{rms}$,
$b_\mathrm{rms}$, $\lambda_c$, $\tau_c$, etc.
dimensionless ones occur, which we denote by $\tilde{\bscE}$, $\tilde{\bmB}$, $\tilde{\eta}_{ij}$,
$\tilde{\beta}^{(0)}$, $\tilde{\zeta}^{(..)}$, $\hat{\delta}$, $\tilde{\nu}$,
$\tilde{\eta}$, $\tilde{S}$, $\tilde{\Omega}$, $\tilde{u}_\mathrm{rms}$, $\tilde{b}_\mathrm{rms}$,
$\tilde{\lambda}_c$, $\tilde{\tau}_c$, etc.
They are defined by
\begin{eqnarray}
\tilde{\bscE} &=& \bscE / \sqrt{\mu \varrho} c_s^2 \, , \quad
    \tilde{\bmB} = \bmB / \sqrt{\mu \varrho} c_s
\nonumber\\
\tilde{\eta}_{ij} &=& \eta_{ij} / c_s l \, , \quad
    \tilde{\beta}^{(0)} = \beta^{(0)} / c_s l \, , \quad
    \tilde{\zeta}^{(..)} = \zeta^{(..)} / l^2 \, , \quad
    \tilde{\delta} = \delta / l^2
\nonumber\\
\tilde{\nu} &=& \nu / c_s l \, , \quad
    \tilde{\eta} = \eta / c_s l \, , \quad
    \tilde{S} = S l / c_s \, , \quad
    \tilde{\Omega} = \Omega l / c_s
\label{eq71}\\
\tilde{u}_\mathrm{rms} &=& u_\mathrm{rms} / c_s \, , \quad
    \tilde{b}_\mathrm{rms} = b_\mathrm{rms} / c_s \sqrt{\mu \varrho} \, , \quad
    \tilde{\lambda}_c = \lambda_c / l \, , \quad
    \tilde{\tau}_c = \tau_c \, l / c_s \, ,
\nonumber
\end{eqnarray}
where $c_s$ is again the sound speed and $l$ a typical length of the box ($k_1 l = 2 \pi$).
In the simulations the time is measured in units of $l / c_s$ (``sound crossing time").
[How to understand $\varrho$?]
Then we have in particular
\begin{equation}
R_m = \frac{\tilde{u}_\mathrm{rms} \tilde{\lambda}_c}{\tilde{\eta}} \, , \quad
    St = \frac{\tilde{u}_\mathrm{rms} \tilde{\tau}_c}{\tilde{\lambda}_c} \, .
\label{eq73}
\end{equation}

Using the relation (\ref{eq51}) for $\beta^{(0)}$ we find
\begin{equation}
\hat{\beta}^{(0)} = \frac{9}{2} \,
    \frac{\tilde{\eta}_{xx} + \tilde{\eta}_{yy}}{\tilde{\eta} \, R_m^2} \,
    = 9 \, \frac{\tilde{\eta}_t}{\tilde{\eta} R_m^2}  \, .
\label{eq75}
\end{equation}

In the case of shear ($\Omega = 0$) we obtain in an analogous way
\begin{equation}
\hat{\zeta}^{(D)} =  - 15 \, \frac{\tilde{\eta}_{xy} + \tilde{\eta}_{yx}}{R_m^2
     \tilde{\lambda}_c^2 \tilde{S}} \, ,\quad
\hat{\zeta}^{(W)} = - 15 \, \frac{\tilde{\eta}_{xy} - \tilde{\eta}_{yx}}{R_m^2
     \tilde{\lambda}_c^2 \tilde{S}} \, .
\label{eq77}
\end{equation}
Putting
\begin{equation}
\delta = \zeta^{(D)} + \zeta^{(W)} = \frac{1}{30} \, \frac{u^2_\mathrm{rms} \lambda_c^4}{\eta^2} \, \hat{\delta}
\label{eq79}
\end{equation}
we have
\begin{equation}
\hat{\delta} = - (\hat{\zeta}^{(D)} - \hat{\zeta}^{(W)}) \, .
\label{eq81}
\end{equation}

In the case of rotation ($S = 0$) we find
\begin{equation}
\hat{\zeta}^{(\Omega)} =  \frac{9 \sqrt{2}}{\sqrt{\pi}} \,
    \frac{(\tilde{\eta}_{xy} - \tilde{\eta}_{yx}) St^{1/2}}{R_m^{5/2} \tilde{\lambda}_c^2 \,
    \tilde{\Omega}} \,
    = \frac{9 \sqrt{2}}{\sqrt{\pi}} \, \frac{\tilde{v} \, St^{1/2}}{R_m^{5/2} \tilde{\lambda}_c^2 \,
    \tilde{\Omega}} \, .
\label{eq83}
\end{equation}
where $\tilde{v}$ stands for the values given under ``2del-kap".

Finally we specify $\tilde{\lambda}_c$ by putting
\begin{equation}
\tilde{\lambda}_c = \frac{\xi k_1}{k_f} \, ,
\label{eq85}
\end{equation}
where $\xi$ means a numerical factor of the order of unity.

The determination of $St$ requires the calculation of $\tau_c$
according to its definition given under (\ref{eq55}).
It would also be desirable to calculate $\lambda_c$ and so to determine $\xi$.


\section{Numerical results}

Comments/questions concerning Tables 1 and 2\\
Coefficients $\hat{\beta}^{(0)}$, $\hat{\zeta}^{(D)}$ and $\hat{\zeta}^{(W)}$
should be independent of $\tilde{S}$
(as long as $\bscE$ is linear in $S$).\\
The original data of kinshear16a(2) and kinshear16a(3) (in ``notes") are in conflict with each other.
Only one set of data can be correct.\\
NOSOCkinshear16h ``dynamoverd\"achtig".\\
NOSOCmagShear32c, magnohel64F and magnohel64F ``dynamoverd\"achtig".\\
The runs kinshear16d and NOSOCAkinshear16a should provide us with results close to those of SOCA
in the low--conductivity limit (unless $St$ is very small).
In that sense the numerical values obtained for $\hat{\zeta}^{(W)}$ are hardly understandable.\\
The run kinohel32D shows that this problem does not disappear with higher resolution.\\
A distinct dependence of $\beta^{(0)}$ on $A$? \\

\bigskip

Comments/questions concerning Tables 3 and 4\\
Coefficients $\hat{\beta}^{(0)}$ and $\hat{\zeta}^{(\Omega)}$ should be independent of $\tilde{\Omega}$
(as long as $\bscE$ is linear in $\Omega$).\\

\bigskip

Comments/questions concerning Tables 5 and 6\\
In the case $S = \Omega = 0$ the $\hat{\zeta}^{(D)}$, $\hat{\zeta}^{(W)}$ and $\hat{\zeta}^{(\Omega)}$
are meaningless.
Here, e.g., $\tilde{\eta}_{xy}$ and $\tilde{\eta}_{yx}$ should be given.
They should be sufficiently close to zero.\\

\bigskip

General\\
Results obtained with the SOCA procedure are questionable unless $R_m$ or $St$ are sufficiently small.\\
Realization of homogeneous isotropic turbulence??\\
$\delta$--correlated forcing??\\
$St$ not of the order of unity??\\

\newpage

\begin{table}
\begin{center}
\caption{Shear but no rotation ($\Omega = 0$), kinetic forcing ($A = 0$),  $k_f/k_1 = 5$, $\xi = 1$.
See also Tab.~\ref{tab7}.
\label{tab1}}
\begin{tabular}{cccccccccc}
\hline
$\tilde{\eta}$ & $P_m$  &  $R_m$  &  $M$  &  $\hat{\beta}^{(0)}$  &  $\hat{\zeta}^{(D)}$
&  $\hat{\zeta}^{(W)}$  &   $\hat{\delta}$  &  $\tilde{S}$  &  run  \cr
\hline
0.005 & ? &   6.00 &   0.15 &   0.71000 &   0.79687 &   0.65104 &  -0.14583 & -0.2 & kinshear32b (SOCA!!)  \cr
0.001 & ? &  40.00 &   0.20 &   0.00045 &   0.00680 &   0.01617 &   0.00937 & -0.2 & kinshear16a(1) (SOCA!!) \cr
0.100 & 1 & 0.38 & 0.038 & 1.24 & 1.85 & -0.22 & -2.07  & -0.2 & kinshear16a(2) (SOCA) \cr
0.050 &   1 &   0.21 &   0.05 &   1.42289 &   2.31861 &   0.10835 &  -2.21027 & -0.2 & kinshear16d (SOCA) \cr
0.020 &   1 &   0.83 &   0.08 &   1.65917 &   2.81427 &   0.74575 &  -2.06852 & -0.2 & kinshear16b (SOCA) \cr
0.200 &   0.5 &   0.038 &   0.038 &   2.48684 &   7.38833 &  -0.89595 &  -8.28428 & -0.2 & kinshear16a(3) (SOCA) \cr
0.200 &   0.5 &   0.038 &   0.038 &   1.64855 &   3.20204 &  -0.95308 &  -4.15513 & -0.2 & NOSOCkinshear16f \cr
0.100 &   1 &   0.076 &   0.038 &   1.24342 &   1.95745 &  -0.17854 &  -2.13599 & -0.2 & NOSOCkinshear16a \cr
0.050 &   2 &   0.152 &   0.038 &   0.83362 &   0.74419 &   0.13715 &  -0.60704 & -0.2 & NOSOCkinshear16g \cr
0.020 &   5 &   0.38 &   0.038 &   0.41447 &   0.17581 &   0.09427 &  -0.08154 & -0.2 & NOSOCkinshear16e \cr
0.010 &  10 &   0.76 &   0.038 &   0.22749 &   0.05171 &   0.03723 &  -0.01448 & -0.2 & NOSOCkinshear16b \cr
0.005 &  20 &   1.52 &   0.038 &   0.11842 &   0.01350 &   0.01118 &  -0.00232 & -0.2 & NOSOCkinshear16c \cr
0.002 &  50 &   3.80 &   0.038 &   0.04893 &   0.00262 &   0.00234 &  -0.00029 & -0.2 & NOSOCkinshear16d \cr
0.001 & 100 &   7.60 &   0.038 &   0.02633 &   0.00049 &   0.00069 &   0.00020 & -0.2 & NOSOCkinshear16h \cr
0.100 &   1 &  0.072 &   0.036 &   1.48090 &   2.11588 &  -0.068721 & -2.18461 & -0.2 & kinohel32D \cr
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Shear but no rotation ($\Omega = 0$), magnetic forcing, $k_f/k_1 = 5$, $\xi = 1$.
\label{tab2}}
\begin{tabular}{ccccccccccc}
\hline
$\tilde{\eta}$ & $P_m$  &  $R_m$  &  $M$  &  $A$  &  $\hat{\beta}^{(0)}$  &  $\hat{\zeta}^{(D)}$
&  $\hat{\zeta}^{(W)}$  &   $\hat{\delta}$  &  $\tilde{S}$  &  run \cr
\hline
0.200 &   1 &   0.007 &   0.007 & ? &   1.001 &   6.494 &   2.767 &  -3.727 & -0.2 & NOSOCmagshear16a(1) \cr
0.100 &   1 &   0.032 &   0.016 & ? &   1.310 &   6.389 &   3.682 &  -2.706 & -0.2 & NOSOCmagshear16a(2) \cr
0.050 &   2 &   0.140 &   0.035 & ? &   1.240 &   4.200 &   3.013 &  -1.186 & -0.2 & NOSOCmagshear16a(3) \cr
0.020 &   5 &   0.010 &   0.001 & ? &   0.273 &   0.092 &   0.057 &  -0.036 & -0.2 & NOSOCmagShear32f \cr
0.010 &  10 &   0.620 &   0.031 & ? &   1.267 &   1.233 &   1.206 &  -0.027 & -0.2 & NOSOCmagShear32b \cr
0.005 &  20 &   1.92  &   0.048 & ? &   0.728 &   0.439 &   0.464 &   0.026 & -0.2 & NOSOCmagShear32c \cr
0.10 &   1 &  0.03 &  0.015 & 12.00 &  2.59 &  9.58 &  4.92 & -4.67 & -0.2 & magnohel32A   \cr
0.10 &   1 &  0.11 &  0.056 &  6.43 &  2.41 &  6.70 &  3.47 & -3.23 & -0.2 & magnohel32B   \cr
0.10 &   1 &  0.39 &  0.20 &  3.52 &  1.76 &  2.23 &  1.79 & -0.45 & -0.2 & magnohel32C  \cr
0.10 &   1 &  1.16 &  0.58 &  2.17 &  1.17 &  1.02 &  0.54 & -0.48 & -0.2 & magnohel64D  \cr
0.10 &   1 &  2.50 &  1.25 &  1.61 &  0.57 &  1.12 &  0.12 & -1.00 & -0.2 & magnohel64E  \cr
0.01 & 10 & 10.0 & 0.50 & 2.44 & 2.24 e-1 & -4.26 e-3 & 1.97 e-2 & 2.40 e-2 & -0.2 & magnohel64H \cr
0.0010 &  100 &   108.86 &  0.54 & 2.64 & 2.24 e-2 & -5.93 e-5 &  1.65 e-5 &  7.59 e-5 & -0.2 & magnohel64F \cr
0.0001 & 1000 &  1095.00 &  0.55 & 2.76 & 2.21 e-3 & -2.6 e-7  & -4.6 e-7 & -1.9 e-7 & -0.2 & magnohel64G \cr
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Rotation but no shear ($S = 0$), kinetic forcing ($A = 0$), $k_f/k_1 = 5$, $\xi = 1$.
See also Tab.~\ref{tab8}.
\label{tab3}}
\begin{tabular}{cccccccc}
\hline
$\tilde{\eta}$ & $P_m$  &  $R_m$  &  $M$  &  $\hat{\beta}^{(0)}$
&  $\hat{\zeta}^{(\Omega)}$  & $\tilde{\Omega}$  &  run  \cr
\hline
0.005 & ? &   5.96 &   0.15 &   1.16042 &   0.16 & 0.01 & rot32f  \cr
0.005 & ? &   5.96 &   0.15 &   1.15535 &   0.14 & 0.02 & rot32g  \cr
0.005 & ? &   5.96 &   0.15 &   1.11481 &   0.051 & 0.1 & rot32h  \cr
0.005 & ? &   5.96 &   0.15 &   1.08441 &   0.073 & 0.1 & rot32c  \cr
0.005 & ? &   6.00 &   0.15 &   1.07850 &   0.048 & 0.2 & rot32a  \cr
0.005 & ? &   6.00 &   0.15 &   0.87500 &   0.016 & 0.5 & rot32b  \cr
0.005 & ? &   6.08 &   0.15 &   0.68949 &   0.035 & 1.0 & rot32d  \cr
0.005 & ? &   6.04 &   0.15 &   0.47564 &   0.00046 & 2.0 & rot32e  \cr
0.005 & ? &   5.96 &   0.15 &   0.81331 &   0.036 & 0.1 & rotNOSOCA32c  \cr
0.005 & ? &   5.96 &   0.15 &   0.84523 &   0.038 & 0.2 & rotNOSOCA32a  \cr
0.005 & ? &   6.00 &   0.15 &   0.73500 &   0.031 & 0.5 & rotNOSOCA32b  \cr
0.005 & ? &   6.08 &   0.15 &   0.66709 &   0.0079 & 1.0 & rotNOSOCA32d  \cr
0.005 & ? &   6.00 &   0.15 &   0.48500 &   0.0010 & 2.0 & rotNOSOCA32e  \cr
0.002 & ? &  20.20 &   0.20 &   0.33526 &   0.00064 & 0.5 & rot64a   \cr
0.002 & ? &  19.50 &   0.20 &   0.31953 &   0.00024 & 0.2 & rotNOSOCA64b  \cr
0.002 & ? &  20.20 &   0.20 &   0.36063 &   0.0024 & 0.5 & rotNOSOCA64c  \cr
0.002 & ? &  25.50 &   0.26 &   0.15370 &   0.000087 & 1.0 & rotNOSOCA64d  \cr
0.002 & ? &  24.00 &   0.24 &   0.17531 &   0.000042 & 2.0 & rotNOSOCA64e  \cr
0.001 & ? &  45.79 &   0.23 &   0.16905 &   0.000020 & 0.5 & rotNOSOCA128a2 \cr
0.001 & ? &  45.79 &   0.23 &   0.15188 &   0.000044 & 0.5 & rot128c   \cr
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Rotation but no shear ($S = 0$), magnetic forcing, $k_f/k_1 = 5$, $\xi = 1$.
\label{tab4}}
\begin{tabular}{ccccccccc}
\hline
$\tilde{\eta}$ & $P_m$  &  $R_m$  &  $M$  &  $A$  &  $\hat{\beta}^{(0)}$
&  $\hat{\zeta}^{(\Omega)}$  &  $\tilde{\Omega}$  &  run \cr
\hline
0.005 & ? &   2.65 &   0.07 &   1.98 &   1.52 &   0.42 & 0.01 & brot32h \cr
0.005 & ? &   2.65 &   0.07 &   1.98 &   1.57 &   0.36 & 0.02 & brot32g \cr
0.005 & ? &   2.65 &   0.07 &   1.98 &   1.48 &   0.20 & 0.05 & brot32f \cr
0.005 & ? &   2.64 &   0.07 &   1.98 &   1.39 &   0.11 & 0.10 & brot32a \cr
0.005 & ? &   2.60 &   0.07 &   2.00 &   1.30 &   0.049 & 0.20 & brot32b \cr
0.005 & ? &   2.40 &   0.06 &   2.23 &   1.27 &   0.016 & 0.50 & brot32c \cr
0.005 & ? &   2.10 &   0.05 &   2.65 &   1.57 &   0.0051 & 1.00 & brot32d \cr
0.005 & ? &   1.73 &   0.04 &   3.36 &   2.19 &   0.0014 & 2.00 & brot32e \cr
0.005 & ? & ? & ? & ? & ? & ? & 0.01 & brotNOSOCA32h \cr
0.005 & ? & ? & ? & ? & ? & ? & 0.02 & brotNOSOCA32g \cr
0.005 & ? & ? & ? & ? & ? & ? & 0.05 & brotNOSOCA32f \cr
0.005 & ? &   2.64 &   0.07 &   1.99 &   1.27 &   0.14 & 0.10 & brotNOSOCA32b \cr
0.005 & ? &   2.78 &   0.07 &   1.88 &   0.97 &   0.064 & 0.20 & brotNOSOCA32a \cr
0.005 & ? &   2.40 &   0.06 &   2.23 &   1.19 &   0.021 & 0.50 & brotNOSOCA32c \cr
0.005 & ? &   2.10 &   0.05 &   2.65 &   1.45 &   0.0068 & 1.00 & brotNOSOCA32d \cr
0.005 & ? &   1.73 &   0.04 &   3.36 &   2.05 &   0.0027 & 2.00 & brotNOSOCA32e \cr
0.002 & ? &  10.00 &   0.10 &   1.59 &   0.55 &   0.0027 & 0.10 & brot64a \cr
0.002 & ? &   9.90 &   0.10 &   1.63 &   0.43 &   0.0020 & 0.20 & brot64b \cr
0.002 & ? &   9.80 &   0.10 &   1.63 &   0.41 &   0.0045 & 0.20 & brotNOSOCA64b \cr
0.002 & ? &   7.10 &   0.07 &   2.70 &   0.47 &   0.00043 & 2.00 & brotNOSOCA64e \cr
0.001 & ? &  22.40 &   0.11 &   1.67 &   0.16 &   0.000046 & 0.50 & brot128a \cr
0.001 & ? &  22.40 &   0.11 &   1.67 &   0.14 &   0.00024 & 0.50 & brotNOSOCA128a3 \cr
0.001 & ? &  22.00 &   0.11 &   1.64 &   0.23 &   0.00014 & 0.20 & brot128b \cr
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Neither shear nor rotation ($S = \Omega = 0$), kinetic forcing ($A = 0$), $k_f/k_1 = 5$, $\xi = 1$.
\label{tab5}}
\begin{tabular}{cccccccc}
\hline
$\tilde{\eta}$ & $P_m$  &  $R_m$  &  $M$  &  $\hat{\beta}^{(0)}$ & $\tilde{\eta}_{xy}$ & $\tilde{\eta}_{xy}$
&  run  \cr
\hline
0.005 & ? &   5.96 &   0.15 &   0.93188 & ? & ? &   rotNOSOCA32o  \cr
0.002 & ? &  19.29 &   0.19 &   0.31048 & ? & ? &   rotNOSOCA64o  \cr
0.001 & ? &  43.00 &   0.22 &   0.16180 & ? & ? &   rotNOSOCA128o \cr
0.001 & ? &  43.20 &   0.22 &   0.21841 & ? & ? &   rot128o2  \cr
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Neither shear nor rotation ($S = \Omega = 0$), magnetic forcing, $k_f/k_1 = 5$, $\xi = 1$.
\label{tab6}}
\begin{tabular}{ccccccccc}
\hline
$\tilde{\eta}$ & $P_m$  &  $R_m$  &  $M$  &  $A$  &  $\hat{\beta}^{(0)}$ & $\tilde{\eta}_{xy}$ & $\tilde{\eta}_{xy}$
&  run \cr
\hline
0.005 & ? &   2.65 &   0.07 &   1.98 &   1.59 & ? & ? &   brot32o \cr
0.002 & ? &  10.00 &   0.10 &   1.59 &   0.45 & ? & ? &   brotNOSOCA64o \cr
0.001 & ? &  23.80 &   0.12 &   1.50 &   0.25 & ? & ? &   brot128o \cr
0.001 & ? &  23.20 &   0.12 &   1.53 &   0.18 & ? & ? &   brotNOSOCA128o \cr
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Shear but no rotation ($\Omega = 0$), kinetic forcing ($A = 0$), $k_f/k_1 = 5$, various $\xi$.
\label{tab7}}
\begin{tabular}{ccccccccccccc}
\hline
$\tilde{\eta}$ & $P_m$ & $\tilde{S}$ & $\xi$ & $R_m$ & $St$ & $q$ & $M$ & $\hat{\beta}^{(0)}$
& $\tilde{\zeta}^{(D)}$ & $\tilde{\zeta}^{(W)}$ & run \cr
\hline
* & * & * & * & * & * & * & * & * & * & * & * \cr
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Rotation but no shear ($S = 0$), kinetic forcing ($A = 0$), $k_f/k_1 = 5$, various $\xi$.
$St$ calculated with $\tilde{\tau}_c = 0.045$.
In brackets the ratios of the results of the simulations and the RS06 results.
\label{tab8}}
\begin{tabular}{ccccccccccc}
\hline
$\tilde{\eta}$ & $P_m$ & $\tilde{\Omega}$ & $\xi$ & $R_m$ & $St$ & $q$ & $M$ & $\hat{\beta}^{(0)}$
& $\hat{\zeta}^{(\Omega)}$  &   run \cr
\hline
0.5 & 1 &  0.05 & 1.5 & 2.22 e-2 & 5.55 e-3 & 4.0  & 0.037 & 0.427 (1.04)& 6.49 e-2 (0.42)&  SOCArot16c \cr
0.5 & 1 &  0.2  & 1.3 & 1.92 e-2 & 6.40 e-3 & 3.0  & 0.037 & 0.571 (1.25)& 9.55 e-2 (0.51)&  SOCArot16d \cr
0.5 & 1 &  0.2  & 1.4 & 2.07 e-2 & 5.95 e-3 & 3.48 & 0.037 & 0.491 (1.14)& 6.58 e-2 (0.38)&  SOCArot16d \cr
0.5 & 1 &  0.2  & 1.5 & 2.22 e-2 & 5.55 e-3 & 4.0  & 0.037 & 0.427 (1.04)& 4.65 e-2 (0.30)&  SOCArot16d \cr
0.5 & 1 &  0.2  & 1.6 & 2.37 e-2 & 5.20 e-3 & 4.56 & 0.037 & 0.375 (0.95)& 3.36 e-2 (0.23)&  SOCArot16d \cr
0.5 & 1 &  1.0  & 1.5 & 2.22 e-2 & 5.55 e-3 & 4.0  & 0.037 & 0.427 (1.04)& 5.26 e-2 (0.34)&  SOCArot16a \cr
0.5 & 1 &  2.0  & 1.5 & 2.22 e-2 & 5.55 e-3 & 4.0  & 0.037 & 0.427 (1.04)& 4.59 e-2 (0.30)&  SOCArot16b \cr
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Some RS06 results for $P_m = 1$. Values taken from diagrams (RS06 Fig 1), rather rough.
\label{tab9}}
\begin{tabular}{ccccc}
\hline
$q$ & $\hat{\beta}^{(0)}$ & $\hat{\delta}^{(\Omega)}$ & $\hat{\kappa}^{(\Omega)}$ & $\hat{\zeta}^{(\Omega)}$ \cr
\hline
0 & 1 & 1 & 1 & 1 \cr
1.0 & 0.66 & 0.562 & 0.126 & 0.344 \cr
3.0 & 0.453 & - & - & 0.187 \cr
3.16 & 0.437 & 0.316 & 0.0316 & 0.174 \cr
3.5 & 0.431 & - & - & 0.171 \cr
4.0 & 0.411 & - & - & 0.155 \cr
4.5 & 0.395 & - & - & 0.143 \cr
10.0 & 0.224 & 0.0316 & 0.00316 & 0.0174 \cr
31.6 & 0.0794 & 0.00891 & $<$ 0 & $<$ 0.0045 \cr
100.0 & 0.0282 & 0.00141 & $<$ 0 & $<$ 0.0007 \cr
\hline
\end{tabular}
\end{center}
\end{table}


\end{document}
