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\title{Magnetic helicity flux in the Roberts flow}
\author{}
\date{\today,~ $ $Revision: 1.9 $ $}
\begin{document}
\maketitle
\section{Introduction}
Magnetic helicity fluxes are believed to play a crucial role in
astrophysical dynamos, when the magnetic Reynolds numbers are very large.
Without magnetic helicity fluxes, the large-scale dynamo would evolve
on a catastrophically slow, resistive timescale, as was demonstrated
using simulations in periodic domains \citep{Bra01}.
This behavior can be understood as a consequence of magnetic helicity
\citep{FB02}, which can only change through microphysical resistivity.
The hope was therefore that nonperiodic boundaries would suffice to
allow for the dynamo to evolve on a faster timescale.
This idea goes back to the work of \cite{BF00}, who showed that
the $\alpha$ effect in mean-field electrodynamics \citep{KR80} can
evade catastrophic quenching only when the dynamo is allowed to shed
preferentially small-scale magnetic helicity.
This was first formulated in the work of \cite{KR82}, using an earlier
finding of \cite{PFL76} that the total $\alpha$ effect is the sum of a
contribution proportional to the kinetic helicity and another proportional
to the current helicity.
The latter quenches the former if the system attains a sufficient amount
of small-scale magnetic helicity.
Subsequent simulations of dynamos with boundary conditions that permit
a magnetic helicity flux have demonstrated that there is indeed a
certain amount of magnetic helicity flux, but most of it is carried
by the large-scale magnetic field and not, as was hoped, by the
small-scale magnetic field.
To understand the driving of magnetic helicity fluxes, we adopt a simple
kinematic flow pattern and study the spreading of an initially localized
magnetic field.
\section{The model}
We adopt a flow geometry where we can control separately the vertical
and horizontal components.
The initial magnetic field can be advected or even amplified by the flow.
Having in mind the transport of magnetic helicity through some surface,
we assume the magnetic field to be helical and confined to a layer of
finite thickness.
Specifically, we adopt an Arnold--Beltrami--Childress (ABC) field with
an envelope in the $z$ direction, i.e.,
\begin{equation}
\BB(\xx,0)=f(z)\left(\begin{matrix}
A\sin kz+C\cos ky\cr
B\sin kx+A\cos kz\cr
C\sin ky+B\cos kx
\end{matrix}\right),
\end{equation}
where $f(z)$ is a smoothed version of a tophat function with
\begin{equation}
f=\left\{
\begin{aligned}
&1\quad\mbox{if $|kz|<\pi$},\\
&0\quad\mbox{otherwise}.
\end{aligned}
\right.
\end{equation}
The velocity is given by a modified Roberts flow of the form
\begin{equation}
\UU=\frac{U_0}{2\kf}\left(\cos\chi\nab\times\psi\zzz
+\kf\sin\chi\tilde{\psi}\zzz\right).
\end{equation}
where $\kf=\sqrt{2}k$ is the effective wavenumber of the flow.
We choose
\begin{equation}
\psi(x,y;\phi)=\cos(kx+\phi)\cos(ky+\phi),
\end{equation}
so the flow is two-dimensional and depends only on $x$ and $y$.
This flow consists of a component with a purely circular horizontal flow
pattern proportional to $\nab\times\psi\zzz$ and a perpendicular component
along the $z$ direction with the same horizontal pattern, which determines
the locations of positive and negative values of $u_z$; see Figure~1 for
a sketch.
The mixing angle $\chi$ controls the relative importance of the vertical
and horizontal components.
For $\chi=45\degr$, we have equally strong horizontal and vertical flows.
For $\chi=0\degr$, the flow only has a horizontal circular component, while
for $\chi=90\degr$, the flow is purely vertical.
The parameters of the model thus include $A$, $B$, $C$, $\chi$, and $\phi$.
%\begin{equation}
%\end{equation}
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pexamz3b_256a}
\includegraphics[width=\columnwidth]{pexamz3b_256b}
\end{center}\caption[]{
Left (\url{pexamz3b_256a}, $\chi=45\degr$):
Upper panel: $t=0.3$, 0.9, 1.4, 1.8;
lower panel: $t=2.3$, 4.6, 5.7, 6.9, 8.0, 9.1, and 10.3.
Right (\url{pexamz3b_256b}, $\chi=90\degr$,
i.e., advection only in the $z$ direction):
Upper panel: $t=0.2$, 0.6, 0.9, 1.2;
lower panel: $t=1.6$, 3.1, 3.9, 4.7, 5.5, 6.2, and 7.0.
}\label{pexamz3b_256a}\end{figure*}
In \Fig{pexamz3b_256a} we compare cases with $\chi=45\degr$ and $90\degr$.
We see that the oppositely signed magnetic helicity flux in the inner parts
are the result dynamo action by the helical Roberts flow.
In the case with $\chi=90\degr$, we see that the spreading
of the magnetic helicity flux to the right (left) is
symmetric at early times.
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pexamz3b_256ABC110chi30del0}
\includegraphics[width=\columnwidth]{pexamz3b_256ABC110chim30del0}
\end{center}\caption[]{
p/m-run with $\chi=\pm30\degr$;
upper panel: $t=0.2$, 0.6, 0.9, 1.2;
lower panel: $t=1.6$, 3.1, 3.9, 4.7, 5.5, 6.2, and 7.0.
}\label{pexamz3b_256ABC110chi30del0}\end{figure*}
In \Fig{pexamz3b_256ABC110chi30del0} we see that the sign
of $\chi$, which determines the sign of the helicity of the flow,
changes the overall sense of propagation of the helical field
into the exterior.
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pexamz3b_256ABC110chi30del90}
\includegraphics[width=\columnwidth]{pexamz3b_256ABC110chim30del90}
\end{center}\caption[]{
p/m-run with $\chi=\pm30\degr$ and $\delta=90\degr$;
upper panel: $t=0.2$, 0.6, 0.9, 1.2;
lower panel: $t=1.6$, 3.1, 3.9, 4.7, 5.5, 6.2, and 7.0.
}\label{pexamz3b_256ABC110chi30del90}\end{figure*}
In \label{pexamz3b_256ABC110chi30del90} we compare cases with
$\delta=90\degr$, corresponding to Roberts flow-II with zero
pointwise helicity, but opposite signs of the pumping direction.
For positive (negative) values of $\chi$, there is a positive (negative)
magnetic helicity flux for $z>0$.
to the right
%\begin{table}[htb]\caption{
%}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
%\hline
%\label{Ttimescale}\end{tabular}}\end{table}
The induction equation for the magnetic field, $\BB=\nab\times\AAA$, is
solved in terms of the magnetic vector potential $\AAA$ and obeys
\begin{equation}
\frac{\partial\AAA}{\partial t}=\UU\times\BB+\eta\nabla^2\AAA
\end{equation}
where $\eta$ is the magnetic diffusivity.
Its value is given in terms of the magnetic Reynolds number,
\begin{equation}
\Rey=U_0/k\eta.
\end{equation}
We choose the side length of the domain to be $L=2\pi$, so that
$k=2\pi/L=1$.
We define mean fields through $xy$ averaging, e.g.,
\begin{equation}
\meanAA(z,t)=\int\AAA(x,y,z,t)\,\dd x\,\dd y/L^2.
\end{equation}
We determine the evolution of the mean magnetic helicity density
$h=h_{\rm m}+h_{\rm f}$, where $h_{\rm m}=\meanAA\cdot\meanBB$ and
$h_{\rm f}=\overline{\aaaa\cdot\bb}$ are the contributions from the mean
and fluctuating parts.
Likewise, the magnetic helicity flux is
\begin{verbatim}
ToDo:
bihelical fields, k=1,5.
\end{verbatim}
%r e f
\begin{thebibliography}{}
\bibitem[Blackman \& Field(2000)]{BF00}
Blackman, E. G., \& Field, G. B.\ymn{2000}{318}{724}
%{724}{Coronal activity from dynamos in astrophysical rotators}
%Blackman, E. G., \& Field, G. B.\yapj{2000}{534}{984}
%{988}{Constraints on the magnitude of $\alpha$ in dynamo theory}
\bibitem[Brandenburg(2001)]{Bra01}
Brandenburg, A.\yapj{2001}{550}{824}
%{840}{The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical hydromagnetic turbulence}
\bibitem[Field \& Blackman(2002)]{FB02}
Field, G. B., \& Blackman, E. G.\yapj{2002}{572}{685}
%{692}{Dynamical quenching of the $\alpha^2$ dynamo}
\bibitem[Kleeorin \& Ruzmaikin(1982)]{KR82}
Kleeorin, N. I., \& Ruzmaikin, A. A.\yjour{1982}{Magne\-to\-hydro\-dynamics}{18}{116}
%{122}{Dynamics of the average turbulent helicity in a magnetic field}
Translation from Magnitnaya Gidrodinamika, 2, pp.\ 17--24 (1982)
\bibitem[Krause \& R\"adler(1980)]{KR80}
Krause, F., \& R\"adler, K.-H.\ybook{1980}
{Mean-field Magneto\-hydro\-dy\-na\-mics and Dynamo Theory}{Oxford: Pergamon Press}
\bibitem[Pouquet et al.(1976)]{PFL76}
Pouquet, A., Frisch, U., \& L\'eorat, J.\yjfm{1976}{77}{321}
%{354}{Strong MHD helical turbulence and the nonlinear dynamo effect}
\end{thebibliography}
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