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\begin{document}

\preprint{}

\title{Supplemental Material \--- Efficient quasi-kinematic large-scale dynamo\\
as the small-scale dynamo saturates
}% Force line breaks with \\


\author{Pallavi Bhat}
%\email{P.Bhat@leeds.ac.uk}
%AB: replaced your email
%PB: thanks, missed this!
\email{pallavi.bhat@icts.res.in}
\author{Kandaswamy Subramanian}

\author{Axel Brandenburg}

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\maketitle

\subsection{Shear helical dynamo: large-scale flows and late time behavior}

\begin{figure}[b]
\includegraphics[width=\columnwidth]{psat.eps}
\caption{Comparison of $\meanu_y$, $\meanB_y$, and three
representations of $\meanBB_{\rm rms}$ for Run~A.
The dotted line is scaled by $\times20$.}
\label{psat}
\end{figure}

\begin{figure}[t]
\includegraphics[width=\columnwidth]{pcomp2.eps}
\caption{Comparison of the rms values of the $z$ dependent ($xy$-averaged,
black lines) and $x$ dependent ($yz$-averaged, red lines) mean fields,
$\meanBB_{\rm rms}^{(z)}$ and $\meanBB_{\rm rms}^{(x)}$, respectively,
for Run~A (solid line) and a similar one at $\Rm=300$ (dotted lines).}
\label{pcomp2}
\end{figure}

We mentioned in the main paper that a large-scale component arises
in the velocity field.
This is shown in \Fig{psat}(a).
The large-scale flow is seen to arise early on, just after the end of
phase~I, at $t=100$.
This is due to a vorticity dynamo \cite{EKR03} and arises also without
magnetic field \cite{KMB09}.
Note that, because the large-scale flow varies only along the $z$ direction,
the relevant terms in the mean-field dynamo equation basically go to
zero and thus this large-scale flow is not responsible for any magnetic
field generation.
At $t=400$, the large-scale flow weakens, changes its form, and
disappears by $t=1000$.
This suppression is due to the magnetic field \cite{KB09,GHJ17}.
\FFig{psat}(b) shows that a dynamo wave begins to emerge at $t=100$,
but it has initially a larger phase speed than at later times.
In \Fig{psat}(c), we see that there is exponential growth in phases
I and II with different growth rates.
\FFig{psat}(d) shows that this growth of $\meanBB_{\rm rms}^2$ does not
grow linearly in time.
There is also evidence for another period of exponential growth 
at later times after phase II; see \Fig{psat}(e).
The growth rate however is very slow
and this perhaps arises due to a temporary fluctuation 
during the resistive phase.

\begin{figure*}[tp]
  \centering
\includegraphics[width=0.32\textwidth]{bbyt50-red2.png}
\includegraphics[width=0.32\textwidth]{bbyt200-red2.png}
\includegraphics[width=0.32\textwidth]{bbyt400-red2.png}
  \caption{
$B_y$ for Run~A at $t=50$, $200$ and $400$.
   } 
  \label{byboxes}
\end{figure*}

As mentioned in the main paper, a fourth stage involves 
fratricide of this $\alpha\Omega$ dynamo by its $\alpha^2$ sibling \cite{HRB11}.
This can be seen by plotting the $x$ dependent ($yz$-averaged) mean field,
which we denote by $\meanBB_{\rm rms}^{(x)}$ with superscript ${(x)}$.
The usual $z$ dependent ($xy$-averaged) mean field is now denoted by
$\meanBB_{\rm rms}^{(z)}$ with superscript ${(z)}$.
This fratricide happens at much later times during $\urms\kf t=1000$--$2000$; see \Fig{pcomp2},
where the black solid curve of $\meanBB_{\rm rms}^{(z)}$ shows decay
and simultaneously, $\meanBB_{\rm rms}^{(x)}$ in solid red is growing. 
Importantly, it appears that the phase II is not $\Rm$ dependent
as seen by comparing the near parallel solid and dashed black lines in \Fig{pcomp2} between $t=100$ to $400$.
Thus, there is no evidence for catastrophic quenching of this phase in this range of $\Rm$ explored.
\subsection{Spatial organization of the field}

To understand how the magnetic field changes in different stages of
growth, we show in \Fig{byboxes} the magnetic field component $B_y$
of Run~A at times $t=50$, $200$ and $400$.
At $t=50$, the system is in kinematic stage and thus the fields are of small-scale nature. 
At $t=200$, the large-scale dynamo (LSD) is active and one finds that
the fields have started ordering themselves on larger and larger scales. 
In the third stage of resistive decay of helicity, the fields become
increasingly organized and at $t=400$, there is a coherent field
on the largest scale in the box.

\subsection{Magnetic helicity evolution in the dynamo with 
shear and helical forcing}

In the main paper we mentioned that 
the build up of small-scale helicity during the second stage is expected
to eventually quench the LSD.
Initially, the LSD due to $\alpha$-effect results in a helical 
polarization of the field, which is 
of opposite signs on small and large scales. The
Lorentz force associated with this small scale helical field
can back react to then quench the $\alpha$-effect 
\cite{B01,BBS02,BB02,BS05}.
Moreover, the large-scale field growth, which as discussed in the
main paper is similar in helical
and non-helical runs during the first stage when the SSD dominates,
can however be differentiated at later stages by its helicity properties. 
Thus, it is of interest to examine the magnetic helicity power spectrum
$H(k)$ for this standard signature of the LSD and 
study how it evolves in the first two stages of large-scale field growth discussed in the main paper.

In the kinematic stage, when the SSD is dominant, there is no clear
separation between the positively helical and the negatively helical
fields as shown by $H(k)$ in \Fig{helspec} for Run~A.
However, towards the end of the second stage (the curve at $t=250$), a
clear separation in scales based on helicity develops, i.e., the helicity
on smaller $k$, $k<\kf$ is one sign represented by blue diamonds and the
helicity on larger $k$ is the opposite sign represented by red squares.
It is this accumulation of small-scale helicity that could induce
a magnetic back reaction to the initial kinetic $\alpha$-effect
and quench the LSD such that the exponential growth of the large-scale
field transits to a resistively limited growth in the third stage.

\begin{figure}[h]
\includegraphics[width=0.48\textwidth]{helspecevol.eps}
\caption{The magnetic helicity power spectrum is shown at three times,
$t=100,150$ \& $250$ for Run~A. The blue diamonds represent negative 
helicity and red squares represent positive helicity.}
\label{helspec}
\end{figure}
 
\begin{figure}[h]
\includegraphics[width=0.48\textwidth]{pchposneg.eps}
\caption{Evolution of (a) $H_\pm(t)$ and (b) $C_\pm(t)$ for Run~A
and a similar run with $\Rm=300$.
The inset highlights the early growth of $H_+(t)$.
}\label{pchposneg}
\end{figure}

At late times, when the $\alpha^2$ dynamo is operating, magnetic helicity
continues to build up at larger scales; see \Fig{pchposneg}(a).
Here, $H_\pm(t)=\int_\pm H(k,t)\,dk$, where $\int_\pm$ denote the
integrals separately for positive and negative arguments, respectively.
The gradual build-up of $H_-(t)$ happens by dissipating magnetic helicity
of positive sign at small scales.
The dissipation of $H_+(t)$ is proportional to the corresponding
current helicity, $C_+(t)$, where $C_\pm(t)=\int_\pm k^2 H(k,t)\,dk$
are the contributions from positive and negative current helicity,
respectively.
Eventually, $C_+$ and $C_-$, begin to cancel each other;
see \Fig{pchposneg}(b).
This leads to the asymptotic steady state
where the total current helicity goes to zero and the
large-scale field goes to the box scale as in \cite{B01}.
Note this late time behavior only occurs on the long 
resistive time scales.

\bibliography{reftwody}
\end{document}
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