%MJK This subsection for Matthias to re-write
%\subsubsection{Kinematics}
%
%%MJK I took this out now, as we do not have any other runs with varying kf. But, can be put back if you see necessary...
%%\begin{table*}[t!]\caption{
%%Summary of the runs with  only $\kf$ being varied. 
%%}
%%%\centerline{
%%\begin{tabular}{ccccc} \hline \hline
%% $\kf/k_1$ & $\eta_{11}/\eta$ & $\eta_{22}/\eta$ & $\eta_{12}/\eta$ & $\eta_{21}/\eta$ \\
%% \hline
%% 5 &   6.092(0.4461) & 4.855(0.3023)  &  2.103(0.8647)  &  1.884(0.08792)  \\
%%20 & 3.851(0.1451)  &  3.704(0.0537) &   0.076(0.02659)  &  0.120(0.08777) \\
%%40 & 
%%\end{tabular}\\[2mm]
%%Notes: $S=-0.1$, $\Omega=0$, $A_{zx}=1$. Uncertainties in brackets.
%%\end{table*}
%
%Noteworthy things to mention. The overall behavior of all the eta components follows
%closely the measured values from the decimated forcing cases. Disagreement with the 
%monochromatic forcing function occurs for the $\eta_{xy}$ component, which shows a 
%reversed sign in between the kinematic cases and monochromatic magnetic forcing. 
%We interpret this being related to the presence of a strong mean magnetic field nearly
%at all times with the monochromatic forcing, while the decimated forcing cases have
%weaker fields; hence agreement in the latter case with kinematics, while not in the
%former.
%
%\begin{table*}[t!]
%\begin{center}
%\caption{
%Summary of the 
%%MJK Guessing and Adding
%kinematic magnetically forced
%%MJK
%runs with  
%%only $A_{zx}$ 
%varying vertical aspect ratio.
%%being varied. 
%}
%%\centerline{
%\begin{tabular}{ccccc} \hline \hline
%%MJK Provide run name, could be something like SM1k, SM4k, SM16k, where 'k' would refer to kinematic, and number to the aspect ratio, and then M would indicate only magnetic forcing.
%%MJK Please consider also providing Rm, and normalizing the same way as in other tables.
%  $A_{zx}$ & $\eta_{xx}/\eta$ & $\eta_{yy}/\eta$ & $\eta_{xy}/\eta$ & $\eta_{yx}/\eta$ \\
% \hline
%1 & 6.092(0.4461) & 4.855(0.3023)  &  2.103(0.8647e)  &  1.884(0.08.792) \\
%4 & 6.949(0.5745) &   4.756(0.09928)  & 1.258(0.09545)  & 1.546(0.1328) \\
%16 & 6.656(0.3665)  &  4.826(0.02134)   & 1.236(0.1447)  & 1.567(0.1034) \\
%\end{tabular}\\[2mm]
%Notes: $S=-0.1$, $\Omega=0$, $\kf=5$. 
%%MJK Should be in $\pm$ format to be consistent with other tables.
%Uncertainties in brackets.
%\end{center}
%\end{table*}

%\subsection{Kinematic versus full NLTFM results}
%%Matthias' results on the kinematic runs and comparison with selected results from Javier
%\begin{figure*}\begin{center}
%\hspace*{-4mm}\includegraphics[width=0.34\textwidth]{b11rms_comp.eps}
%\hspace*{-5.9mm}\includegraphics[width=0.34\textwidth]{b11rms_ddt_comp.eps}
%\hspace*{-5.9mm}\includegraphics[width=0.34\textwidth]{b11rms_ddt2_comp.eps}\\
%\hspace*{-4mm}\includegraphics[width=0.34\textwidth]{eta12_comp.eps}
%\hspace*{-5.9mm}\includegraphics[width=0.34\textwidth]{eta12_ddt_comp.eps}
%\hspace*{-5.9mm}\includegraphics[width=0.34\textwidth]{eta12_ddt2_comp.eps}\\
%\hspace*{-4mm}\includegraphics[width=0.34\textwidth]{eta21_comp.eps}
%\hspace*{-5.9mm}\includegraphics[width=0.34\textwidth]{eta21_ddt_comp.eps}
%\hspace*{-5.9mm}\includegraphics[width=0.34\textwidth]{eta21_ddt2_comp.eps}
%\end{center}\caption{
%Comparison of the kinematic (broken lines)  and full (solid lines) versions of the NLTFM.
%Left: constant timestep $10^{-4}$, middle: adapted timestep, {\tt cdt=cdtv=0.2}, right: {\tt cdt=cdtv=0.7}.
%The evolving mean field reaches at $t\approx15$ a level $\approx 0.1 B_{\rm rms}$
%($\meanB_x$ roughly constant, $\meanB_y$ roughly linearly time-dependent).
%At this time, first deviations between kinematic and full can be seen in $\eta_{12}$.
%For {\tt cdt=cdtv=0.2}, the adapted timestep falls below $10^{-4}$ at $t\approx 80$,
%remaining $> 10^{-4}$ for  {\tt cdt=cdtv=0.7}.
%$\Rm=10.2$, $\Lu=16.8$, $A_{zx}=1$, $\kf=5$, $S=-0.1$.
%}\label{fig:eta_aspect}
%\vspace{8mm}
%\end{figure*}

\subsection{Variants of the forcing function} \label{forcing}

%MJK Moved to the appendix
In most of our simulations we use stochastic monochromatic non-helical
%forcing, that is applied to velocity or/and induction equations
%AB: that -> which, added "the"
forcing, which is applied to the velocity or/and induction equations
through the terms $\FFK$ and $\FFM$, respectively, and their fluctuating
parts to the test problems through the terms $\ffK$ and $\ffM$.
The 
wavevectors
%of the forcing are selected such that no mean field 
%AB: of -> in, better say "forcing function"
in the forcing function are selected such that no mean field 
%or flow
%AB: better repeat "mean"
or mean flow
%is directly
%AB: is -> are
are directly
sustained, that is, the case 
$k_y=0$
is excluded.\footnote{Without shear, only those with $k_x=k_y=0$
had to be excluded, but due to shear-periodicity,
$2\pi/k_x$ is no longer an integer fraction of $L_x$.}
However, due to roundoff errors, it is unavoidable that averages over harmonic functions
deviate slightly from zero. We call this effect ``leakage of the forcing into the mean fields".
Being normally negligible, 
strong shear could produce a linearly growing $\meanB_y$ out of a small $\meanB_x$ due to such leakage.
This is why we checked  
its effect in purely magnetic runs 
and found the growth of $\meanB_y$ to be limited and both component to stay within
margins close to numerical precision.
Nevertheless,
%as will be discussed at more length in 
%AB: omit
as will be discussed in 
Sect.~\ref{overall},
with this magnetic forcing setup,
the mean magnetic fields very quickly (in a few tens of time units)
reach dynamically effective strengths
without showing a clear exponential stage.

Hence, another forcing setup was designed, referred to as ``decimated forcing".
%MR: the following unclear:
%Speaking about the modulus of the wavevector, there is hardly a low-k end, as all k's lie in a narrow shell.
%What is  k_{\rm min} for, say, kf=5?
%All our k_i's are 3 or 4 (for the isotropic box), so not $k_i \leq k_{\rm min}=2$
%In addition to ensuring that the case $k_x=k_y=0$ in excluded, we cut out even more 
%AB: in -> is
In addition to ensuring that the case $k_x=k_y=0$ is excluded, we cut out even more 
wavenumbers from the low-$k$ end of the spectrum, so that $k_i \leq k_{\rm min}$, where 
$i=x,y,z$ and $k_{\rm min}+1$ is the lowest wavenumber included in the forcing. 
For all the cases investigated here, we use $k_{\rm min}$=2.
As will be discussed at length in the results section, the decimated
forcing has the advantage of reducing the amplitude of the mean fields
%generated during the initial stages, and allowing us, e.g., to determine
%AB: can just omit ", e.g.,"
generated during the initial stages, and allowing us to determine
the growth rate of the exponentially growing dynamo instability.
%MR: I'd like to say the following:
%MJK Why do we need to be so lengthy and sound like we want to shoot ourselves on the foot?
%The standard forcing implemented in the {\sc Pencil Code}\footnote{
%\url{http://github.com/pencil-code}} employs all wavevectors from a thin shell in $k$ space
%with a chosen center radius $\kf$ which fit into the computational box.
%For reasons described in the results section, we also employed
%a decimated set of only 32 wavevectors which are parallel or almost parallel to the spatial diagonals of the box.
%MJK I would say simply this instead.
We note that in this case we have a set of only 32 wavevectors which are parallel or almost
parallel to the spatial diagonals of the box.
While the standard choice is expected to provide a good approximation to homogeneous isotropic 
velocity turbulence, isotropy 
%MJK not according to Nishant recent findings, hence 'could'
%AB: ok
%is lost 
could be lost
in the decimated case. 
%MJK Already said, and sufficient.
%As a side effect, the level of leakage is reduced
%and the growth of the mean fields is slowed down, allowing the identification of an exponential stage. 
%MJK Adding, and Nishant should check this.
However, as 
%is shown in Fig.~\ref{fig:pdfs_uu_decimated}, 
is discussed in Appendix~\ref{sect:forcings},
the generated turbulence does not
markedly vary in its properties from the standard forcing function. 
%MJK added 
%AB: ok
Also, repeating the 
kinetically forced runs with decimated forcing do not significantly alter the 
%MJK more specific.
%results.
dynamo solutions.

%MJK Thanks! Can you also check the text, edit it to your liking and write about the figure below? 
%Nishant, please add the text and spectra here, that we discussed over Zoom! 
%NS: added component-wise PDFs which might better address
%NS: the anisotropy argument
%AB: I'd use plot_io
%NS: thanks, done now. Also, it might be better to move this and
%NS: discussions about this possibly to an appendix; this plot
%NS: hardly has any surprises, and is only a technical detail to address
%NS: concerns related to the anisotropy argument
%AB: yes, appendix could be good.
%MJK OK, I can cook up an appendix, but after the results from the analysis proposed by Axel yesterday become available.
%MJK Now moving into an appendix
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{pdfs_uu_decimated}
\end{center}\caption{
PDFs of all three components of the velocity field from $64^3$ hydro runs
with $\kf=5$. (a): undecimated; (b): $k\leq2$ are decimated from the forcing
function. All pdfs are nearly Gaussians with kurtosis $\sim 3$.
}\label{fig:pdfs_uu_decimated}
\vspace{8mm}
\end{figure}

%NS: added this as Axel's suggested, in whole theta-phi plane
%NS: features already in undecimated case, but note that it is
%NS: a low resolution (64^3) test.
%NS: I will try to get a line plot too at, say, specific phi,
%NS: as function of theta
%AB: what is the percentage in the two cases? A color bar would fix this.
%AB: Also, use xticks=8,yticks=6
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{uuaniso}
\end{center}\caption{
Snapshots of $\sqrt{\bra{(\uu\cdot\nnn)^2}}$ normalized by the instantaneous
$\urms$ obtained from the corresponding snapshot, in the plane of
azimuthal angle $\phi$ and colatitude $90-\theta$, with
$\nnn=(\sin{\theta} \cos{\phi},\, \sin{\theta} \sin{\phi},\, \cos{\theta})$.
Left: undecimated; right: $k\leq2$ are decimated from the forcing
function. These normalized contours are saturated at levels 0.4 (minimum)
and 0.7 (maximum).
}\label{fig:uuaniso}
\vspace{8mm}
\end{figure}

%NS: the line plot with phi at theta=pi/4
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{uu_aniso_phi}
\end{center}\caption{
Variation of $\zeta=\sqrt{\bra{(\uu\cdot\nnn)^2}}/\urms$
with the azimuthal angle $\phi$ at polar angle $\theta=45\degr$
(averaged over 8 snapshots);
$\nnn=(\sin{\theta} \cos{\phi},\, \sin{\theta} \sin{\phi},\, \cos{\theta})$.
Solid/black: undecimated; dashed/red: $k\leq2$ are decimated from the forcing
function.
}\label{fig:uu_aniso_phi}
\vspace{8mm}
\end{figure}

%\cite{SB15b} argued that an incoherent $\alpha$--shear scenario
%should not lead to 
%amplification of a mean magnetic field unless the diagonal components of $\bm\alpha$
%would be markedly larger than the off-diagonals. 
%Although this was not so 
%in the moderate shear case, studied by \cite{BRRK08}, we have now clearly identified
%such a situation, both in kinetically and magnetically forced SMHD cases, see
%\Fig{fig:histo}. Hence the possibility of an incoherent $\alpha$--shear dynamo cannot be
%ruled out for our simulations.
%MJK I am very confused about SB15b appendix, but modifying now to something more neutral.

%MR: Where would we know from that the small-scale magnetic fields due to tangling 
%        are less relevant compared to those by the magnetic forcing directly?
%MJK Because Lu is large from the beginning, and does not change as function of time similarly to the mean fields.
%MJK Added "primarily" above, however. 
%MR: Actually, Lu covers the full magnetic field and the mean fields gain their final strength very early on.
%MJK Not always! The decimated cases behave differently.
%MR: But in Table 2 all runs have standard forcing, right?
%MJK Yes, but this sentence is not referring only to those. The decimated runs are in Table 1, but for them we do not list Lu, but I have of course computed it.
%MR: No, the sentence only refers to those: "As can be seen from the listed $\Lu$ ..."and
%MJK Ooooof OMG. One word in this sentence, namely "Primarily" is information coming from Table 1, from
%MJK the decimated runs. The rest comes from Table 2. This is simply not shown information, as I cannot
%MJK list the Lu in that table, and even that would not help, because it is the time evolution that matters.
%MJK So, it is impossible to describe it in any more detail in the paper. I think we are absolutely fine with this
%MJK statement.
%MR: I'm very sorry, but we can't say "As can be seen from the listed \Lu" that the fluctuations by the forcing are dominating as it cannot be seen.
%       Also, he reader cannot know that "primarily" is somehow meant to involve information from some other place.

\red{Our kinetically forced FMHD runs reproduce the earlier results 
of similar systems (compare our upper leftmost panel 
of \Fig{fig:but} to Figure~7 of \cite{BRRK08})
with rather coherent patches in $\meanB_y$, while the SMHD counterpart
(lower left panel of \Fig{fig:but}) shows somewhat more erratic pattern; here, 
however, we must note that the time series in the case of the SMHD run is much 
longer. These results are in disagreement with the purely kinetically forced,
incompressible, runs of \cite{SB15b} (their Figure~9(a)), which show a 
much more erratic pattern than what we observe either in FMHD 
or SMHD.}
