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\begin{document}

\preprint{NORDITA-2019-044}

\title{Efficient quasi-kinematic large-scale dynamo 
as the small-scale dynamo saturates}% 

\author{Pallavi Bhat}
\thanks{pallavi.bhat@icts.res.in}
\affiliation{% 
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, 
Bangalore 560089, India
}%
\affiliation{% 
Plasma Science and Fusion Center,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
}%
\affiliation{% 
Department of Applied Mathematics, 
University of Leeds, Leeds, LS2 9JT, UK 
}%

\author{Kandaswamy Subramanian}
\affiliation{IUCAA, Post Bag 4, Ganeshkhind, Pune 411007, India}

\author{Axel Brandenburg}
\affiliation{Nordita, KTH Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden}
\affiliation{The Oskar Klein Centre, Department of Astronomy, Stockholm University, SE-10691 Stockholm, Sweden}
\affiliation{McWilliams Center for Cosmology \& Department of Physics, Carnegie Mellon University, PA 15213, USA}
\affiliation{School of Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia}

%\date{\today}% It is always \today, today,
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\date{\today, $ $Revision: 1.150 $ $}

\begin{abstract}
Large-scale magnetic fields in stars and galaxies are thought to arise by mean-field dynamo action due to
the combined influence of both helical turbulence and shear. 
Those systems are also highly conducting and the turbulence therein leads to 
a fluctuation (or small-scale) dynamo which more rapidly amplifies magnetic field fluctuations on the eddy scales and smaller. Will this then
interfere with and suppress the mean (or large-scale) field growth?
Using direct numerical simulations of helical turbulence
(with and without shear),
we identify a novel quasi-kinematic large-scale dynamo which operates 
as the small-scale dynamo saturates.
Thus both dynamos operate efficiently, one after the other, and lead to the generation of significant large-scale fields. 
\end{abstract}

\maketitle

%\tableofcontents
Magnetic fields coherent on large-scales, larger than
the scales of turbulent motions in the system, are prevalent in stars and 
disk galaxies. Their origin is thought to lie in
mean-field or large-scale dynamo (LSD) 
action due to helical turbulence often combined with shear. 
Turbulence in stars and galaxies also has a very high magnetic Reynolds number $\Rm$.
This generically leads to a fluctuation or small-scale dynamo (SSD) which
grows random and small-scale magnetic fields more rapidly \cite{Kaz68,KA92}.
Here, small scales correspond to scales smaller than the outer scale of the
turbulence.

In large $\Rm$ helical turbulence, previous work has shown that, 
while the kinematic phase is dominated by rapidly growing small-scale fields \citep{VC92},
there are hints of large-scale field growth \citep{SB14}. 
A unified dynamo 
grows magnetic fields in the kinematic regime, of both large and small scales, with a shape-invariant 
eigenfunction \citep{SB14,BSB16}.
The saturation of the unified dynamo occurs first at small scales,
and then at progressively larger scales \citep{BSB16,rincon2019}.
Significant large-scale fields tend to arise, but on long resistive time scales due to nonlinear growth 
governed by the slow resistive decay of small-scale helicity \citep{B01}.  
A matter of outstanding concern is whether large-scale fields can be grown unhindered by 
rapidly growing small-scale fields via a fast dynamo action 
\citep{S98,tobias21}.

In this work, we use helically forced turbulence with uniform shear
(and sometimes without)
to identify a novel intermediate second stage of fast growth 
(an exponential one, possibly modulated by some other function) growing large-scale field, 
different from that of the kinematic unified dynamo, 
due to a quasi-kinematic LSD (QKLSD). 
The term `quasi-kinematic' is to signify that this LSD action arises 
as the SSD saturates (indicating that the system has become nonlinear) while the large-scale field is possibly still unaffected 
by the Lorentz force. 
This arises in a previously unidentified parameter regime, but one that is in fact expected to be generic
in astrophysical systems.
This result, that a quasi-kinematic LSD can 
operate in large $\Rm$ systems,
alleviates the long-standing concern that large-scale fields are overwhelmed by rapidly growing 
small-scale fields in such systems. 
Note that this work is limited to the `quasi-kinematic' regime and does 
not deal with the nonlinear issue of catastrophic quenching.

For reasons of computational efficiency, we solve the equations for a
compressible gas; see \cite{LSBH19} for a more detailed motivation.
We employ a periodic or shearing-periodic domain of size $L^3$
with $L=2\pi$, so the smallest wavenumber is $k_1=2\pi/L=1$.
We follow a setup similar to that of Ref.~\cite{BSB16},
except now we also have cases with uniform linear shear.
All simulations (except one) were performed with the \textsc{Pencil Code}
\footnote{DOI:10.5281/zenodo.2315093, github.com/pencil-code}, and
have a resolution of $512^3$ and a magnetic Prandtl number of $\Pm=10$.
The ability of numerical codes to conserve magnetic helicity
has been recognized and was verified on earlier occasions \cite{BS20}.
We have specified the relevant parameters for each run in Table~\ref{runs}.
Turbulence is driven at the forcing wave number of $\kf/k_1=4$ (or $8$ in one case).
The unit of velocity is the sound speed $\cs$, and that of time is $(\cs k_1)^{-1}$. 
Also, we have included a run from the paper \cite{bhat2021} (referred to as Run GP600Pm1a).
This simulation was performed using \textsc{Dedalus} \cite{dedalus2020}. 
The runs leading to LSD action -- with shear (like Run~A) and without shear (like Run~B) --
have helical forcing.
In the following, we discuss results from these runs and compare
with two similar runs (C and D) with
non-helical forcing and thus only SSD action. 

\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*}

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) operates without much interference from the SSD 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.

\begin{figure}[h]
\includegraphics[width=0.48\textwidth]{spectime512Pm10withshear.eps}
\caption{Evolution of $M_k(t)$ for $k=1$, $4$, $10$, $50$, and $100$ for Run A.
The bottom panel shows $\gamma$, the growth rate of $M_1$.
The curve in  dashed red is from the toy model.
The inset compares $\gamma$ between Runs~A and C.}
\label{spectimeRunA}
\end{figure}

In the top panel of \Fig{spectimeRunA}, 
we show the evolution of the magnetic energy spectrum $M_k$ at certain wave numbers $k$.
We characterize large- and small-scale fields through the magnetic energy at $k<\kf$
and $k>\kf$, respectively. The quantity $M_1$ is seen to grow exponentially 
at the same kinematic rate as others until about $t=100$. 
However, there is a novel second phase between $t=100$ to $t\sim250$--$270$, 
where $M_1$ grows fast (exponentially, possibly modulated by another function) 
at a different, albeit slower rate compared to the first stage. 
Meanwhile all modes with $k\ge\kf$ have slowed down towards saturation.
The third phase, after $t\sim 300$, involves the well-known resistively limited nonlinear growth of $M_1$ \citep{B01,CHT2002}.
For Run A, we have also calculated the large-scale field $\meanBB_{\rm rms}$ from 
horizontal or $xy$ averaging and find that 
its energy density
closely follows the $M_1$ curve. 
In the top panel of \Fig{spectimeRunB}, we show the evolution of $\meanBB_{\rm rms}^2$ 
from Run~GP600Pm1a with a completely different code using a different forcing function based
on the Galloway-Proctor flow with no 
large-scale shear. We find that the second phase also shows up in 
this Dedalus run \cite{bhat2021}, as well as 
a \textsc{Pencil Code} run, Run~B 
where there is no shear.  
  
\begin{figure}[h!]
\includegraphics[width=0.48\textwidth]{lsfieldDedalusrun.eps}
\caption{Evolution of energy in $xz$ averaged field $\meanBB_{\rm rms}^2$
and $\gamma$ for Run~GP600Pm1a using a different code, \textsc{Dedalus} \cite{bhat2021}.
}
\label{spectimeRunB}
\end{figure}

The bottom panels of \Figs{spectimeRunA}{spectimeRunB} show the growth rate of the large-scale magnetic energy,
defined as $\gamma=d \ln M_1/dt$ or $\gamma= d \ln\meanBB_{\rm rms}^2/dt$. 
Two successive stages of fast growth can clearly be identified. 
The inset in \Fig{spectimeRunA} compares the
growth rate of $M_1$ between the helical and nonhelical cases to illustrate that in the latter,
where no LSD action occurs, the curve in the second phase simply fluctuates around $\gamma=0$, 
as opposed to the (noisy) plateau in the former case.

In both \Figs{spectimeRunA}{spectimeRunB} we show, as red-dashed 
lines, solutions from the following toy model:
\EQ
\dd{\EM}/\dd t=\left[{\gamma_1-\gamma_2\over1+{\EM}^2/{\EM}^2_1}+{\gamma_2\over1+{\EM}^2/{\EM}^2_2}\right]{\EM}.
\EN
When ${\EM} \ll {\EM}_1 < {\EM}_2$), 
the growth rate of the curve ${\EM}$ asymptotes to $\gamma_1$. 
Later, when ${\EM}_1 < {\EM} \ll {\EM}_2$, the growth 
rate asymptotes to $\gamma_2$. 
The overlay from this toy model is found to match quite well both the 
evolution and growth where the parameters $\gamma_1$, $\gamma_2$ 
are the same as the $\gamma$ calculated from the data.
 
We propose that the LSD action in the second distinctive phase of growth of the
large-scale field is similar to a standard LSD as predicted
by mean-field dynamo theory.
In the first stage, which is entirely kinematic, the faster SSD 
is the main driver and thus governs the growth rate of magnetic energy over all scales. 
Once the growth at smaller scales has slowed down,
LSD action becomes prominent.
Note that the the magnetic evolution curve in the second stage (QKLSD regime) is not a clean exponential but 
has a certain concavity. This is attributed to the fact that it is influenced by the transition 
to or from the other two regimes. 

For the following discussion, it is helpful to
refer to the standard mean-field equations
obtained by splitting the induction equation into one
for the mean or large-scale field $\meanBB$
and the fluctuating or small-scale field $\bb$ \citep{Mof78},
\begin{equation}
\label{meaninduction}
\frac{\partial \meanBB}{\partial t} = \nab \times \left( \meanUU \times \meanBB
+ \EMF- \eta \bf \nab \times \meanBB \right),
\end{equation}
\begin{equation}
\label{residual}
\frac{\partial \bb}{\partial t}={\bf \nab}\times\left(\meanUU\times \bb+
\uu\times \meanBB-\eta \bf \nab \times \bb+\bm{G}\right).
\end{equation}
Here $\EMF = \overline{\uu \times\bb}$ and
$\bm{G} = \uu\times \bb-\EMF$ is a term nonlinear in the fluctuations. 
The mean velocity accounts for the linear shear, $\meanUU=(0,Sx,0)$, 
where $S=\mbox{const}$. 
We also find that helical forcing combined with 
shear induces 
a large-scale flow $\meanUU = (\mean{U}_x(z), \mean{U}_y(z), 0)$ on $xy$ averaging but this 
does not appear to affect the LSD (see below).
This phenomenon is the vorticity dynamo \citep{EKR03,KMB09},
which is known to be suppressed by the magnetic field \citep{KB09,GHJ17}.

\begin{figure}[t!]
\includegraphics[width=0.48\textwidth]{m1compare.eps}
\caption{Comparison of $M_1$ curves between Runs~A and C in the panel (a) and between Runs~B and D in the panel (b).} 
\label{m1comp}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=0.48\textwidth]{sscompare.eps}
\caption{Comparison of $M_4(t)$ shown for Run A and Run~C.
Also comparison of curves from Run A -- $M_1$ growing exponentially while $M_4$ grows linearly.}
\label{sscompare}
\end{figure}

During the kinematic stage, the small-scale field is mainly driven by
$\nab \times \left(\uu \times \bb \right)$, which leads to SSD action. 
All the terms that depend on averaged or mean quantities 
are not significant initially.
In our simulations,
the shearing timescale is much larger than the eddy turn over timescale and thus,
its effect on SSD growth is unimportant \cite{Nishant}.
The small-scale field then grows exponentially as 
${\bb}={\bb}_0 \exp{(\gamma_{\rm SSD} t)}$, 
where $\gamma_{\rm SSD}$ is the SSD growth rate.
In \Eq{meaninduction}, the time evolution of $\EMF$, which drives $\meanBB$,
is then controlled by 
the exponentially growing low wave number tail of the small-scale field, given that 
the velocity field ${\uu}$ is in a statistical steady state. 
The shear term $\nab \times \left( \meanUU \times \meanBB \right)$
is subdominant as $\meanBB$ is at this stage much smaller than $\bb$.  
The rate of change of $\meanBB$ is therefore
expected to be nearly the same as $\gamma_{\rm SSD}$.
This can be seen also in \Fig{m1comp}, where in the left panel we show that the time 
evolution of $M_1$ in both the helical shear dynamo simulations
(Run A) and non-helical shear dynamo (Run C) coincide in the kinematic stage. 
Similarly, in the right panel of \Fig{m1comp}, the $M_1$ curves 
from the helical dynamo (Run B) and the non-helical dynamo (Run D) coincide.
Thus, the kinematic stage is primarily driven by the SSD with $\meanBB$ being
enslaved by $\bb$. 
 
Eventually, the growth of the small-scale field slows down from an exponential 
to a more linear 
form as the SSD begins to saturate. 
We find this coincides with the QKLSD regime. 
As seen in the upper panel of \Fig{spectimeRunA}, the
evolution of $M_4$ 
slows down at around $t=100$, when $M_1$ switches to a different rate of exponential growth. 
\FFig{sscompare} shows explicitly the exponentially growing $M_1$ versus linearly growing $M_4$, from Run A.

We can understand the QKLSD regime in the following manner.
In \Eq{residual}, besides the shear and SSD terms, now there are contributions due to those containing mean quantities. 
In particular, the term $\nab \times (\uu\times \meanBB)$, interpreted as the tangling of the large-scale 
field, is expected to be responsible for additional growth of small-scale fields
over and above that when there is no LSD. 
We show that this is indeed the case in \Fig{sscompare}, where 
the linear growth rate of $M_4$ in the helical dynamo in Run A
is larger than that of the non-helical dynamo (Run~C) by a factor of about 8.

\begin{table}
\caption{
Summary of all the runs.}
\vspace{10pt}
{\begin{tabular}{lccccccc}
\hline
\hline
Run & $\kf$ & $u_{\rm rms}$  & $S\tau$ & forcing & $\Rm$ & $\gamma_{\rm theo}$ & $\gamma_{\rm meas}$\\
\hline
A & 4 & 0.13 & 0.38 & helical & 812 & 0.11 & 0.036\\  %Pm10shearhel512a
B & 4 & 0.18 & 0    & helical & 1062 & 0.12 & 0.032\\ %Pm10hel512b
C & 4 & 0.13 & 0.38 & non-hel & 812 & -- & -- \\
D & 4 & 0.19 & 0    & non-hel & 1187 & -- & -- \\
E & 4 & 0.10 & 0.25 & helical & 1000 & 0.066 & 0.026\\ %Pm10shearhel512c
F & 4 & 0.09 & 0.11 & helical & 812  & 0.034 & 0.013\\ %Pm10shearhel512d
G & 8 & 0.09 & 0.27 & helical & 281 & 0.1& 0.032\\    %Pm10shearhel512f 
H & 4 & 0.18 & 0  & helical & 531 & 0.12 & 0.034\\
GP600Pm1a & 8 & 1.0 & 0 & helical & 600 & 0.22 & 0.045\\
\hline
\label{runs}\end{tabular}}
\end{table}

At this stage, as the tangling of large-scale field by the turbulent velocity $\uu$
becomes the more dominant mechanism for growth of small-scale fields $\bb$,
this leads to a correlation between $\bb$ and $\uu$, 
proportional to $\meanBB$. The emf $\EMF = \overline{\uu\times\bb}$,
which then depends on $\meanBB$, can be estimated in the usual fashion
as $\EMF = \alpha \meanBB - \eta_{\rm t}\nab \times \meanBB$ \citep{Mof78}.
Here $\alpha$ and $\eta_{\rm t}$ are the 
turbulent transport coefficients determining the 
effect of small-scale turbulence on the large-scale magnetic field.
Thus, \Eq{meaninduction} for the large-scale field transforms to,
\EQ
{\partial\meanBB \over \partial t}=
\nab \times \left(\meanUU\times\meanBB + \alpha\mean\BB\right)
+\eta_{\rm T}\nabla^2\meanBB,
\quad\nab\cdot\meanBB=0,
\label{mfd}
\EN
with $\eta_{\rm T}=\eta+\eta_{\rm t}$.
This is the standard mean-field dynamo equation,
which has solutions in a periodic box of the form
$\meanBB(\xx,t)=
\mbox{Re}\left[\hat{\BB}(\kk)\,\exp(\ii\kk\cdot\xx+\lambda t)\right]$
at the kinematic stage.
For simplicity, assume that the large-scale field varies only along $z$,
so $k_y=k_x=0$. Then the eigenvalue 
$\lambda$ is given by \citep{BS05},
\EQ
\lambda_\pm=-\eta_{\rm T} k_z^2\pm(\alpha^2 k_z^2-\ii\alpha Sk_z)^{1/2}.
\EN
We note that the $\meanUU$ from the vorticity dynamo does not affect the dispersion
relation as $\nab\times(\meanUU\times\meanBB)=0$ when
$\mean{B}_z=\mean{U}_z=0$ and the fields 
depend only on $z$.
For Run A, the vorticity dynamo becomes suppressed as the magnetic
field continues to saturate \footnote{See Supplemental Material,
which includes Refs.~[10-11], regarding the suppression of the
vorticity dynamo by the magnetic field.}.

For the case without shear ($\alpha^2$ dynamo), the growing mode
has $\lambda = |\alpha|k_z - \eta_{\rm T} k_z^2$. 
When shear dominates (standard $\alpha\Omega$ dynamo), such that
$\alpha k_z/S\ll1$,
\EQ
\lambda_\pm \approx -\eta_{\rm T} k_z^2 \pm |\half\alpha Sk_z|^{1/2} (1-i),
\label{lam_approx}
\EN

\begin{figure}[t!]
\includegraphics[width=0.4\textwidth]{varshear.eps}
\caption{Comparison between
theoretical $\gamma_{\rm theo}=-2\eta_{\rm T} k^2
\pm 2|\half\alpha Sk_z|^{1/2}$ and
measured $\gamma_{\rm meas}=d\ln{M_1}/dt$, for Runs~A, E and F (with varying shear, $S$).}
\label{varshear}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=0.4\textwidth]{butterPm10shearhel512a.eps}
\caption{Space-time diagram where the field components have been $xy$ averaged.}
\label{butter}
\end{figure}

We now ask, can the QKLSD growth in \Fig{spectimeRunA},
be understood in terms of the above standard mean-field dynamo properties?
For homogeneous, isotropic and fully helical turbulence forced
at a wave number $\kf$, we estimate
$\alpha \sim u_{\rm rms}/3$ and $\eta_t \sim u_{\rm rms}/(3\kf)$
\citep{SBS08}. 
The real part of the
second term (with shear) in \Eq{lam_approx} governs 
the growth rate, while the first term would be smaller in the supercritical case. 
The magnetic energy density growth rate 
$\gamma_{\rm theo} = 2\mbox{Re}\lambda_+$. 
In our Run A, $\urms\sim0.13$, $\kf=4$ and $S\tau\sim0.38$, which leads to
$\gamma_{\rm theo}=2\mbox{Re}\lambda_+\sim0.11$, 
which is larger than the measured value of $\gamma_{\rm meas}\sim0.036$.
This yields an `efficiency factor' $c_{\rm eff}\sim 0.32$.
Note that in the runs with shear, 
we estimate the $\urms$ after subtracting out $\meanUU$.

From \Eq{lam_approx}, we observe that 
the growth rate is not expected to change much as we change $\kf$. 
We have run a case with $\kf=8$, where $\urms\sim 0.09$ and $S\tau\sim 0.27$,
yielding $\gamma_{\rm theo}\sim 0.1$. This theoretical estimate is 
similar to the $\gamma_{\rm theo}$ of Run A.
This is also confirmed by the measurement of the growth rate of $\sim0.032$
(similar to the measured value of $0.036$ in Run A). 
In the no-shear case, the theoretical estimate of the growth rate for the large scale magnetic 
energy is given by $\urms\kf/6$ \cite{SB14}.
For Run B, where $\urms\sim0.18$ this leads to $\gamma_{\rm theo}\sim0.12$. 
Here, with $\gamma_{\rm meas}\sim0.032$, we have $c_{\rm eff}\approx 0.26$.
For the run from Dedalus, GP600Pm1a, $\urms\sim1.$ and $\kf \sim8.5$. 
We have to account for a factor of $2\pi$ considering the the non-dimensionalization of the equations in Dedalus. 
Thus we find $\gamma_{\rm theo}\sim0.22$.
And with $\gamma_{\rm meas}\sim0.045$, we have $c_{\rm eff}\approx 0.20$, which is  
similar to that in runs from \textsc{Pencil Code}, assuring us that the QKLSD regime is robust.

We have varied the shear parameter $S\tau$ to see its effect
on the QKLSD growth rate. 
In \Fig{varshear}, we compare the theoretical estimate of the growth rate 
for Runs~A, E and F (with different values of the shear parameter) against the measured value.
We find that $c_{\rm eff}$ is roughly the same in all three cases, thus leading 
to the points in \Fig{varshear} falling nearly on a straight line. 

Next we examine the oscillatory behavior of the LSD in the runs with shear. 
In \Fig{butter}, we show $xy$ averaged field $\meanB_y$ in a $zt$ space-time diagram. 
As in earlier work at lower $\Rm$ \citep{KB09},
the oscillations begin only during the QKLSD regime. 
To make an estimate of the time period of this cycle from the mean-field theory, we take 
$\mbox{Re}\lambda_+$ in \Eq{lam_approx}
to be $0$ (which 
approximately holds as the LSD saturates), 
and obtain $\omega_{\rm cyc}=\eta_{\rm T} k_z^2$. 
Thus, the period is $T=2\pi/\omega_{\rm cyc}=6\pi(\kf/k_1)^2$.
For Run A, such an estimate yields $T\sim300$ and from the simulation shown in \Fig{butter}, this is the period of the oscillations in the large-scale field. 
It appears that the mean-field theory is satisfactorily applicable 
to understand these features of the QKLSD.  

A caveat is that we are applying kinematic mean-field theory 
to a system which is already affected by the Lorentz force. 
The zeroth order agreement between the measured growth rate and that estimated from the simple 
$\alpha\Omega$ dynamo theory
is possibly because the nonlinearity has not affected the large scales at this stage yet \cite{S98}.
Note that the quasi-kinematic LSD here arises upon saturation of the SSD
and seems to be in alignment with the `suppression principle' 
put forward by \citet{TC2013}.

An important question is regarding the effects of $\Rm$ on the QKLSD regime.
To check this we have lowered in Run~H the $\Rm$ by a factor of $2$ compared to that of Run~B,
and we find that the growth rate remains the same;
see Supplemental Material for details of the third stage governed by helicity evolution and
also a fourth phase of nonlinear mode switching. 
Our simulations are in a parameter regime (manifest in astrophysical systems) 
which allows for separation in large-scale field growth time scales between the three stages. 
This makes it possible to identify the QKLSD stage distinctively.
An intermediate stage of LSD growth was seen in earlier works;
see Fig.~8 of Ref.~\cite{KKB08} and Fig.~2 of Ref.~\cite{PBMP2012},
but was not investigated in detail.

In conclusion, we have demonstrated via direct numerical simulations the presence of
a novel second stage growth of large-scale field (QKLSD regime), one that occurs 
between the kinematic stage driven by the SSD and the saturation stage driven by magnetic helicity decay.
Interestingly, we find that the QKLSD 
growth sets in when the SSD
slows down.
Our detailed analysis suggests that 
the classical mean-field theory applies reasonably well to understanding the large-scale field characteristics 
such as growth rate and frequency of oscillations.
Moreover, the growth of the large-scale field in the QKLSD regime seems to be independent of $\Rm$.
Our work thus gives the first detailed evidence for how an LSD
operates quasi-kinematically once the small-scale field has saturated and develops an identity even amidst strong small-scale fields. 

%\acknowledgements
We thank Anvar Shukurov and Steve Tobias for helpful suggestions.
PB thanks Nuno F.\ Loureiro for support under NSF-DOE award No.\ DE-SC0016215.
This project was completed using funding from the European Research Council (ERC) under the 
European Union's Horizon 2020 research and innovation programme (grant agreement no.\ D5S-DLV-786780).
This work was supported through the the Swedish Research Council, grant 2019-04234.
The simulations in the paper were performed on the MIT-PSFC partition
of the Engaging cluster at the MGHPCC facility (www.mghpcc.org), which
is funded by DoE grant number DE-FG02-91-ER54109.
Additional simulations were performed using resources provided by
the Swedish National Infrastructure for Computing (SNIC)
at the Royal Institute of Technology in Stockholm.

\begin{figure}[t!]
\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}

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

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.

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}

\section{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]{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}

