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\title{Decay of helical magnetic turbulence in unbounded domains}
\author{}
\date{\today,~ $ $Revision: 1.11 $ $}
\begin{document}
\maketitle

Motivation: experiments

\section{Introduction}

Magnetic helicity is a well conserved quantity in highly conducting
magnetohydrodynamic (MHD) turbulence.
On time scales short compared with the microphysical or Spitzer resistive
time, it changes only because of magnetic helicity fluxes out of the
domain.
\cite{BM99} have shown that the conservation of magnetic helicity implies
a slow-down of the turbulent decay law of the energy $E$ from
$E\propto t^{-10/7}$
in hydrodynamic turbulence or $t^{-1}$ in non-helical MHD turbulence to
$t^{-2/3}$ in helical MHD turbulence.
At the same time, the correlation length $\xiM$ increases with time more
rapidly: like $\xiM\propto t^{2/3}$ in the presence of helicity.
This has important consequences for understanding the decay of primordial
magnetic fields in the early universe \citep{BEO96,FC00,CHB01,BJ04},
which may well be helical \citep{Vac01,BKMRPTV17}.

In many astrophysical and laboratory applications of MHD turbulence,
the flow is not confined by walls and may expand in space.
This dilutes both the magnetic energy and the magnetic helicity.
It is then unclear whether the decay laws of magnetic energy and
correlation length are still valid, or how they would be changed.
Investigating this will be one of the goals of this paper.

A related question concerns the case where the local magnetic helicity
changes its sign in space.
This is particularly important in all cases where a helical
magnetic field is generated by a dynamo or a battery mechanism.
This is again because of magnetic helicity conservation, so there can be
no net magnetic helicity if there was none to begin with.
This is particularly obvious in all dynamo and battery scenarios, in
which the magnetic energy is either vanishing initially or it is a weak
seed magnetic field.
With zero initial magnetic energy, there cannot be any magnetic helicity either.

%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{pq_short1152eq_kf60b3_noforce}
%\end{center}\caption[]{
%With equator, first evolution near $\beta=0$ line toward
%origin, but later up again.
%}\label{pns}\end{figure}

%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{pns_short1152eq_kf60b4_noforce}
%\end{center}\caption[]{
%Magnetic helicity density in the northern and southern hemispheres
%for \url{pns_short1152eq_kf60b4_noforce}.
%}\label{pns}\end{figure}

%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{pq_box1152_kf60a_noforce}
%\end{center}\caption[]{
%Small box, helical
%}\label{pq_box1152_kf60a_noforce}\end{figure}

%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{pq_box1152_kf60a_sig0_noforce}
%\end{center}\caption[]{
%Small box, nonhelical
%}\label{pq_box1152_kf60a_sig0_noforce}\end{figure}

\section{Description of the models}

To make contact with earlier work, we restrict ourselves to direct
numerical simulations (DNS) of the magnetohydrodynamic (MHD) equations,
i.e., we assume positive and negative charge carriers to be well coupled
and ignore kinetic effects.
This may not be a good approximation for the actual plasma experiments,
so full kinetic simulations will at some point be unavoidable.
Nevertheless, to guide our intuition for the relatively unexplored
regime of unbounded plasma flows and the interaction between regions of
opposite side of magnetic helicity, it is important to establish some
solid results in MHD that can later serve as a benchmark.

\subsection{Basic equations}

We adopt an isothermal equation of state, so the pressure $p$ is
proportional to the density $\rho$ with $p=\rho\cs^2$, where $\cs=\const$
is the isothermal sound speed.
We solve the induction equation in terms of the magnetic vector potential
$\AAA$, so the magnetic field is given by $\BB=\nab\times\AAA$.
The uncurled induction equation is then given by
\begin{equation}
{\partial\AAA\over\partial t}=\uu\times\BB+\emf_0-\eta\mu_0\JJ,
\end{equation}
where $\uu$ is the velocity, $\JJ=\nab\times\BB/\mu_0$ is the current
density, $\mu_0$ is the vacuum permeability, $\eta$ is the magnetic
diffusivity, and $\emf_0$ is an external electromotive force
that is only used in some cases to produce initial conditions,
so $\emf_0=\bm{0}$ in all of the cases.
The momentum and continuity equations are given by
\begin{equation}
\rho{\DD\UU\over\DD t}=-\cs^2\nab\rho+\JJ\times\BB+\nab\cdot(2\rho\nu\SSSS),
\end{equation}
\begin{equation}
{\DD\ln\rho\over\DD t}=-\nab\cdot\uu,
\end{equation}
where ${\sf S}_{ij}=\half(\partial_i u_j+\partial_j u_i)-\onethird
\delta_{ij}\nab\cdot\uu$ are the components of the rate-of-strain tensor
and $\nu$ is the viscosity.

\subsection{Unbounded domains}

To simulate to simulate the decay of MHD turbulence in an unbounded
domain, we embed a turbulent flow in a larger domain.
It is convenient to scale the wavenumber to the largest wavenumber within
the initially turbulent region, which is chosen to be half the size of
the full domain.

A convenient way to produce a turbulent magnetic field is by driving
it with an electromotive force $\emf_0$ during some time interval
$0\leq t\leq t_\ast$.
We then choose random helical planar waves consisting of the
eigenfunctions of the curl operator with negative eigenvalues;
see \cite{Bra01} for early examples and \cite{HBD04} for more
specific details regarding our implementation.
They are spatially confined to the inner box, i.e., $|\xx|<L_{\rm in}$.

The forcing function can be written as
\EQ
\ff(\xx,t)={\cal N}\,\Rey\left\{\tildeff(\kk,t)
\exp[\ii\kk(t)\cdot\xx+\ii\phi(t)]\right\},
\EN
where ${\cal N}=f_0\cs\,(\cs k/\delta t)^{1/2}$ is a normalization
factor, with $f_0$ determining the strength of the forcing,
$\kk(t)$ is randomly selected at each time step, and
\EQ
\tilde{f}_i(\kk,t)={\sf R}_{ij} \tilde{f}_j^{(0)}(\kk,t),
\EN
where ${\sf R}_{ij}=(\delta_{ij}+\ii\sigma\epsilon_{ijk}\hat{k}_k)/
\sqrt{1+\sigma^2}$ and $-1\leq\sigma\leq1$ determines the helicity
of the forcing function.
Our embedding domain of size $L_{\rm out}$ is periodic and we determine
power spectra for the entire domain.

We monitor the magnetic helicity separately in the upper and lower
parts of the domain.

\subsection{Opposite signs of magnetic helicity}

Opposite signs of magnetic helicity can be modeled by driving MHD
turbulence for some time using an electromotive force  whose
local helicity is modulated in space by a sinusoidal
profile function, i.e.,
\EQ
\sigma(z)=\sin k_1z.
\EN
In these experiments, we adopt a domain of side length $L=2\pi$, where
the current helicity is negative in the upper disc plane and positive
in the lower.
We choose our domain to be fully periodic in all three directions, so,
in addition to $z=0$, the locations $z=\pm\pi$ corresponds to another
place where the local helicity changes sign.

\subsection{Numerical considerations}

Use the {\sc Pencil Code}, where all the relevant tools are already
implemented.

\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{psav_box1152_kf60b2_sig1}
\end{center}\caption[]{
Slices of $B_z$ (top) and $u_z$ (bottom) in a $(4\pi)^3$ domain
with $\beta\gg1$.
\url{psav_box1152_kf60b2_sig1}
}\label{psav_box1152_kf60b2_sig1}\end{figure}

\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{ptime_box1152_kf60d_sig1}
\end{center}\caption[]{
Evolution of max and rms values of $\vA/\cs$ and $u/\cs$
during the driven and decaying phases ($t\cs k_1<2$)
for a simulation with $k_*/k_1=60$.
\url{ptime_box1152_kf60d_sig1}
}\label{ptime_box1152_kf60d_sig1}\end{figure}

\begin{figure}\begin{center}
\includegraphics[width=.49\columnwidth]{pkt1152_box1152_kf60d_sig1}
\includegraphics[width=.49\columnwidth]{pq_box1152_kf60d_sig1}
\end{center}\caption[]{
Spectra (left) and $pq$ diagram (right) for Run~A.
\url{pkt1152_box1152_kf60d_sig1}
}\label{pkt1152_box1152_kf60d_sig1}\end{figure}

\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{pns_short1152eq_kf180c}
\end{center}\caption[]{
Magnetic helicity density above and below the midplane.
Run~B; \url{pns_short1152eq_kf180c}.
}\label{pns_short1152eq_kf180c}\end{figure}

\begin{figure}\begin{center}
\includegraphics[width=.49\columnwidth]{pkt1152_short1152eq_kf180c}
\includegraphics[width=.49\columnwidth]{pq_short1152eq_kf180c}
\end{center}\caption[]{
Spectra (left) and $pq$ diagram (right) for Run~B.
\url{pkt1152_short1152eq_kf180c}
}\label{pkt1152_short1152eq_kf180c}\end{figure}

\section{Results}

\subsection{Unbounded domain}

We begin with the case of turbulence being embedded in a larger domain.
We drive turbulence until $t=t_\ast$, where $t_\ast$ is given by
$t_\ast\cs k_1=5$, corresponding to five sound travel times.
The forcing function has an amplitude $f_0=10^{-3}$, which results in a
maximum Alfv\'en speed of about unity, i.e., $\vA/\cs\approx1$.
The rms field based on the full domain size, $\vA^{\rm rms}/\cs$,
is only about 0.1.

In \Fig{psav_box1152_kf60b2_sig1} we show
slices of $B_z$ and $u_z$ at the end of the run.

In \Fig{ptime_box1152_kf60d_sig1}, we show the evolution of the
maximum and rms values of $\vA/\cs$ and $u/\cs$ during the driven
and decaying phases ($t\cs k_1<2$) for a simulation with $k_*/k_1=60$.
We see that the maximum velocity and magnetic fields are reached
slightly before $t_\ast$, which is marked by a vertical line.
The magnetic field dominates over the velocity field by a factor
of about 1.7, as is typical for magnetically driven turbulence; see
\cite{BKMRPTV17} for comparison of magnetically and kinetically driven
turbulence in periodic domains with uniform be distributed magnetic
helicity.

\subsection{Runs with an equator}

Runs are driven by an emf during the time interval $0<t<t_\ast$.
The rms Mach number reaches then a value of about 0.1 and decays
after that.
During that time, the magnetic energy peak has moved from the initial
value of $k_0/k_1=60$ to about 8.

\begin{table}\caption{
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
Run \\
\hline
A \\%box1152_kf60a_noforce
B \\%box1152_kf60a_sig0_noforce
C \\%short1152eq_kf60b3_noforce
D & 10 & \url{short1152eq_kf60b4_noforce} \\
\label{Ttimescale}\end{tabular}}\end{table}

\begin{verbatim}
\end{verbatim}

\section{Todo}

Higher Rm, two-scale approach to analysis,
box bc.

With an equator, the total magnetic helicity is zero,
but not in difference hemispheres.

Flux of magnetic helicity across equator is small.

Show xiM(t) and Em(t).

%r e f
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\bibitem[Christensson et al.(2001)]{CHB01}
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{Inverse cascade in decaying 3D magnetohydrodynamic turbulence}

\bibitem[Field \& Carroll(2000)]{FC00}
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\bibitem[Haugen et al.(2004)]{HBD04}
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\bibitem[Vachaspati(2001)]{Vac01}
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\end{thebibliography}

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