% $Id: notes.tex,v 1.5 2019/04/23 04:39:28 brandenb Exp $
%\documentclass{article}
\documentclass[twocolumn]{article}
\setlength{\textwidth}{160mm}
\setlength{\oddsidemargin}{-0mm}
\setlength{\textheight}{220mm}
\setlength{\topmargin}{-8mm}
\usepackage{graphicx,natbib,bm,url,color}
\graphicspath{{./fig/}{./png/}}
\thispagestyle{empty}
\input apj
\input gafd_journ
\def\red{\textcolor{red}}
\def\blue{\textcolor{blue}}
\title{Turbulent decay with an imposed magnetic field}
\author{}
\date{\today,~ $ $Revision: 1.5 $ $}
\begin{document}
\maketitle

\section{}

In the presence of an imposed magnetic field, turbulent motions produce
magnetic fluctuations by tangling on all length scales.
Here we consider the case where turbulence is being driven at small
length scales ($\kf/k_1=60$ or $180$).
This leads to magnetic energy being produced at all larger scales.
We find that the magnetic field exhibits an approximate $k^2$ subinertial
range spectrum while the kinetic energy spectrum is shallower
(perhaps closer to a linearly increasing one, i.e., proportional to $k$).

In both helical and nonhelical cases, the decay laws follows that of
nonmagnetic and nonhelical turbulence in the sense that the evolution is
close to the $\beta=4$ line, corresponding to the Loitsiansky integral
being conserved; see \cite{BK17} for the corresponding motivation.
This behavior is in many ways rather surprising.
First, given that the magnetic field is strong and plays a dominant role,
one would have expected to have an evolution similar to that of usual
MHD turbulence.
This is not the case.
Second, in the case with magnetic helicity, we might have expected there
to be a difference, especially because the presence of magnetic helicity
is detectable even at late times.
Again, however, this is not the case.

Of course, we know is that with an imposed magnetic field, magnetic
helicity is no longer conserved \citep{Ber97,BM04}.
It remained unclear, however, what this implies for the magnetic
field evolution in the system.
We know now that the decay behavior is indeed the same as far as the
evolution in the $pq$ diagram is concerned, but the magnetic field still
exhibits a non-vanishing magnetic helicity spectrum.
Moreover, the magnetic helicity spectrum is strongly bihelical.
We can now conclude that this is how the system adapted to the constraint
of having magnetic helicity without it affecting the overall dynamics:
the net magnetic helicity is actually close zero!

%\begin{itemize}\itemsep-0.2mm
%\item
%\end{itemize}

%\begin{equation}
%Density contrast
%\end{equation}

%\begin{figure}[h!]\begin{center}
%\includegraphics[width=\columnwidth]{pMukohyama}
%\end{center}\caption[]{
%Black: magnetic energy spectrum, $\nu=\eta=2\times10^{-6}$,
%imposed field $B_0=0.1$, initially $k^{-1}$ spectrum. $\sigma=1$.
%\url{H1152km1c_B01}.
%Red: \url{H1152km1p4b_sig01}
%}\label{pMukohyama}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pkt1152_1152a}
\end{center}\caption[]{
\url{pkt1152_1152a}.
}\label{pkt1152_1152a}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pkt1152_H1152a}
\end{center}\caption[]{
\url{pkt1152_H1152a}.
}\label{pkt1152_H1152a}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pq_1152a}
\end{center}\caption[]{
\url{pq_1152a}.
}\label{pq_1152a}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pq_H1152a}
\end{center}\caption[]{
\url{pq_H1152a}.
}\label{pq_H1152a}\end{figure}

%r e f
\begin{thebibliography}{}

\bibitem[Berger(1997)]{Ber97}
Berger, M. A.\yjgr{1997}{102}{2637}
{2644}{Magnetic helicity in a periodic domain}

\bibitem[Brandenburg \& Kahniashvili(2017)]{BK17}
Brandenburg, A., \& Kahniashvili, T.\yprl{2017}{118}{055102}
{Classes of hydrodynamic and magnetohydrodynamic turbulent decay}

\bibitem[Brandenburg \& Matthaeus(2004)]{BM04}
Brandenburg, A., \& Matthaeus, W. H.\ypreN{2004}{69}{056407}
{Magnetic helicity evolution in a periodic domain with imposed field}

\end{thebibliography}

\vfill\bigskip\noindent\tiny\begin{verbatim}
$Header: /var/cvs/brandenb/tex/tina/Mukohyama/notes.tex,v 1.5 2019/04/23 04:39:28 brandenb Exp $
\end{verbatim}

\end{document}

