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\section*{Helicity flux in a shear flow turbulence}

Run similar to Run~A of \cite{Bra18}, but the density changes
from 1 at $z=0$ to about 0.002 at $z=\pi/2$.
Butterfly diagram is similar to before; see \Fig{ppbutter_Strat288a}.
The frequency is now $\omega\approx6.6\,\etatz k_1^2$ instead of
$1.16\,\etatz k_1^2$ in the unstratified case.
This is very surprising.

%\begin{equation}
%{\cal H}(\xx,\yy)=\int \xx\curl^{-1}\yy\,\dd V
%\end{equation}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{ppbutter_S288e}
\end{center}\caption[]{
\url{ppbutter_Strat288a}
}\label{ppbutter_S288e}\end{figure}

The small-scale magnetic helicty has now the same sign
as the large-scale magnetic helicty; see the blue line
in \Fig{pphelflux_Strat288a} compared with $\meanAA\cdot\meanBB$
(red line).
The total $\overline{\AAA\cdot\BB}$ is shown in red.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pphelflux_S288e}
\end{center}\caption[]{
\url{pphelflux_Strat288a}
}\label{pphelflux_S288e}\end{figure}

The second panel of \Fig{pphelflux_Strat288a} shows
at $z>\pi/4$, $\overline{\oo\cdot\rho\uu}$ (green) is almost
as large as $\overline{\jj\cdot\bb}$.
The upper part is fully quenched, it seems, but the
cycle still proceeds, with an even higher frequency.
The small-scale magnetic helicity flux is also of the wrong sign
(positive); see panels 3 and 4.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pphelflux_Om2kf60a}
\end{center}\caption[]{
\url{pphelflux_Om2kf60a}, $\kf=60$.
}\label{pphelflux_Om2kf60a}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pphelflux_hel_b0_Lz2b2}
\end{center}\caption[]{
\url{pphelflux_hel_b0_Lz2b2}
}\label{pphelflux_hel_b0_Lz2b2}\end{figure}

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\bibitem[Brandenburg(2018)]{Bra18}
Brandenburg, A.\yan{2018}{339}{631}
{640}{Magnetic helicity and fluxes in an inhomogeneous $\alpha^2$ dynamo}

\end{thebibliography}

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