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\title{Dynamos in ideal magnetohydrodynamics?}
\author{}
\date{\today,~ $ $Revision: 1.10 $ $}
\begin{document}
\maketitle
\section{Introduction}
Ideal magnetohydrodynamics (MHD) describes the magnetic field evolution
in a perfectly conducting fluid and is given by the induction equation
of the form
\EQ
{\partial\BB\over\partial t}=\nab\times(\uu\times\BB),
\label{IndEqnIdeal}
\EN
where $\BB$ is the magnetic field, $\uu$ is the velocity, and $t$ is time.
This equation is routinely stated in the astrophysical literature,
especially in the context of numerical simulations of the MHD
equations, where some sort of magnetic diffusion and the corresponding
dissipation are usually implied by the particular numerical scheme
under consideration \citep{SN92,Pen03,Derigs16}.
In many cases, the corresponding operator cannot easily be
expressed in terms of explicit mathematical operations.
This is one reason for omitting the diffusion and dissipation terms
in the astrophysical literature.
Another reason is the fact that the magnetic diffusivity is very small
in most astrophysical settings.
This does not generally imply that the magnetic dissipation is small
\citep{Hendrix,GN96}, although it can be small if the magnetic Prandtl
number is large, i.e., the kinematic viscosity is much larger than the
magnetic diffusivity (Bra14,BR19).
We now know that magnetic field generation by dynamo action is a generic
process in most astrophysical flows, because they tend to be turbulent,
so a certain fraction of the kinetic energy will be diverted into
magnetic energy (BL13).
Thus, the process of dynamo action is a typical ingredient of
astrophysical turbulence simulations.
This raises the question whether dynamo action of any type is possible
at all when \Eq{IndEqnIdeal} is solved and the magnetic diffusivity is
thus strictly zero.
Early work of \cite{MP85} demonstrated that, in the special case of
steady flows, the eigenvalue of \Eq{IndEqnIdeal} must be zero in the
strictly ideal case.
The result of \cite{MP85} is somewhat counterintuitive, because one
expects the magnetic field always to grow when there is no magnetic
diffusivity.
What happens, however, at least for a steady flow, is that the magnetic
field develops progressively smaller structures, so the magnetic field
increases---potentially even exponentially---by concentrating on itself
into ever tinier instructions.
Because the field structure changes all the time, this field does not
correspond to an eigenfunction.
The situation may be different in a turbulent and thus time-dependent
flow, where the velocity is constantly changing before the field has a
chance to concentrate itself too much.
This may lead to a field that has statistically always the same
typical size.
Whether or not this really happens is unclear and needs to be
investigated.
Studying this in more detail is the main purpose of the present work.
\section{Models}
\subsection{Analysis tools}
A and EP methods,
resetting $\BB$ to $\nab\alpha\times\nab\beta$,
compare $\Brms^{\rm EP}/\Brms^{\rm A}$ versus time,
compare $\Jrms/\Brms$ versus time (smaller structures),
kurtosis and pdfs of $B_i$,
magnetic helicity spectra $\HM(k,t)$.
The present experiment also allows us to address the question what
happens with magnetic helicity.
Even when writing the magnetic vector potential in the more general
form as
\EQ
\AAA=q\alpha\nab\beta-(1-q)\beta\nab\alpha,
\EN
where $0\leq q\leq1$ is some weighting factor, the magnetic field is still
always independent of $q$, i.e., the local magnetic helicity density is
always zero.
On the other hand, it is still possible to have a non-vanishing magnetic
helicity spectrum, which is defined as \citep[cf.][]{BN11}
\begin{eqnarray}
H_{\rm M}(k)&=&\;\half\!\!\!\!\!\!\!\!\sum_{k_- < |{\kk}|\leq k_+} \!\!\!\!\!\!
(\tilde{\AAA}\cdot\tilde{\BB}^\ast+\tilde{\AAA}^\ast\cdot\tilde{\BB}),
\label{EHvec}
\end{eqnarray}
where $k_\pm=k\pm\delta k/2$ and $\delta k=2\pi/L$ is the wavenumber
increment and also the smallest wavenumber in the domain $L^3$.
\subsection{ABC flows}
To understand the nature of dynamo action in the steady ABC flow
(with $A=B=C=1$), we compare solutions with the A and EP methods
at progressively smaller diffusivity.
The initial magnetic field is expressed in terms of Euler potentials
and is given by $\alpha=\cos y$ and $\beta=\cos z$.
\FFig{sABC256d} shows that there is an initial windup phase during
which the magnetic field increases approximately exponentially or
even superexponentially.
Later, a truly exponential growth commences, although the growth
is then less rapid than during the windup phase.
%\section{}
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{sABC256d}
\end{center}\caption[]{
Dynamo action for $\eta=10^{-3}$ at $256^3$ with the A method (solid line)
and magnetic field decay with the EP method (dotted line).
}\label{sABC256d}\end{figure}
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pEPcomp2}
\end{center}\caption[]{
Similar to \Fig{sABC256d} but for $\eta=10^{-3}$ (lowermost lines),
$5\times10^{-4}$ (both at $256^3$), $2\times10^{-4}$ (at $512^3$),
$10^{-4}$, $5\times10^{-5}$, and $2\times10^{-5}$ (at $1024^3$).
Note that the EP method agrees with the A method only during
the initial wind-up phase, and never during the later dynamo phase.
}\label{pEPcomp2}\end{figure}
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pEPcomp3}
\end{center}\caption[]{
The ratio $\Brms^{\rm EP}/\Brms^{\rm A}$ during early times.
}\label{pEPcomp3}\end{figure}
\begin{table}[htb]\caption{
}\vspace{12pt}\centerline{\begin{tabular}{lrrccccc}
Run & $N$ & $\eta\quad$ & $k_{\rm Tay}^{A}$ & $k_{\rm Tay}^{EP}$ \\
\hline
A & 256 & $10^{-3}$ & 20.8 & 20.2 \\%sABC256d
B & 256 & $5\times10^{-4}$ & 29.8 & 28.5 \\%sABC256e
C & 512 & $2\times10^{-4}$ & 46.0 & 45.7 \\%sABC512b
D &1024 & $ 10^{-4}$ & 65.9 & 67.1 \\%sABC1024b
E &1024 & $5\times10^{-5}$ & 95.0 & 103.6 \\%sABC1024c
F &1024 & $2\times10^{-5}$ & 158.6 & 262.4 \\%sABC1024d
\label{Ttimescale}\end{tabular}}\end{table}
\FFig{pEPcomp2} shows that the windup phase is prolonged as one
decreases the magnetic diffusivity.
This phase can also be described with the EP method at finite
magnetic diffusivity.
\FFig{p} shows that at $t=10$, the A and EP methods agree reasonably
well, although both methods suffer from poor resolution at the
lowest magnetic diffusivity.
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{sABC1024b_10}
\includegraphics[width=\columnwidth]{sABC1024c_10}
\includegraphics[width=\columnwidth]{sABC1024d_10}
\end{center}\caption[]{
$B_z(x,y)$ in a given plane $z=\const$ for
$\eta=$1e-4, 5e-5, and 2e-5.
In all cases, the resolution is $1024^3$.
}\label{p}\end{figure}
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{ppdf_sABC1024d_7}
\end{center}\caption[]{
Histograms for the ABC flow at $t=7$
for the run with $\eta=2\times10^{-5}$.
}\label{ppdf7}\end{figure}
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{ppdf_sABC1024d_30}
\end{center}\caption[]{
Histograms for the ABC flow at $t=30$
for the run with $\eta=2\times10^{-5}$.
}\label{ppdf30}\end{figure}
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{sGP512a_1}
\includegraphics[width=\columnwidth]{sGP512a_6}
\includegraphics[width=\columnwidth]{sGP512b_6}
\includegraphics[width=\columnwidth]{sGP512c_6}
\end{center}\caption[]{
$B_x(y,z)$ at $x=0$ for
$\eta=$1e-3, 1e-4, and 1e-5.
In all cases, the resolution is $512^3$.
}\label{sGP}\end{figure}
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pGPcomp2}
\end{center}\caption[]{
Time evolution for GP flow.
}\label{pGP}\end{figure}
At a given resolution, here $1024^3$,
the EP method reproduces the initial windup phase
until a certain time.
At higher resolution, the time over which the
EP method agrees with the A method will be longer.
In the early phase, however, the magnetic field evolution appears
to be similar to the windup in 2-D, when there is no dynamo.
\FFig{ppdf7} shows the pdf of the 3 components of $\BB$ at $t=7$ with the
EP and A methods, respectively, and compares in a log-log plot.
With the EP method, the pdf is more extended, but the magnetic
field decays and at later times, only the pdf obtained with the
A method has extended tails; see \Fig{ppdf30} for $t=30$.
In all cases, the PDF has powerlaw tails proportional to $1/B_i^2$.
\section{Galloway--Proctor flow}
The Galloway--Proctor flow is time dependent.
We use here the circular polarized version in
the definition of GP92.
\FFig{sGP} shows $B_x$ for $t=10$
(title says {\em incorrectly} $t=1$),
and $t=60$ (second row) for $\eta=10^{-3}$,
and then the same for $\eta=10^{-4}$ and $\eta=10^{-5}$
in rows 3 and 4.
\FFig{pGP} shows the time evolution for all 3 cases.
The EP solution departs from the correct solution for $t\ga4$.
The A solution with $\eta=10^{-5}$ is under-resolved.
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pEP_turb512e}
\end{center}\caption[]{
Time evolution for a delta-correlated fully helical flow.
}\label{pEP_turb512e}\end{figure}
\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{turb512e_2}
\end{center}\caption[]{
$B_x(y,z)$ at $x=0$ for the delta-correlated fully helical flow.
}\label{turb512e_2}\end{figure}
\begin{figure*}[h!]\begin{center}
\includegraphics[width=\textwidth]{ppower_all_turb512e}
\end{center}\caption[]{
Magnetic energy and helicity spectra
for the delta-correlated fully helical flow.
}\label{ppower_all_turb512e}\end{figure*}
\section{Delta-correlated turbulence}
In mean-field electrodynamics \citep{KR80}, it is shown that there is
an $\alpha$ effect even in the absence of magnetic diffusion.
This is done by using the high connectivity limit, in which the magnetic
diffusion operator is neglected.
This approximation is only valid when the Strouhal number $\St=\urms\kf\tau$
is small, i.e., when the correlation time $\tau$ of the flow is zero.
To realize such a flow, we now
XX
Instead of using a delta-correlated forcing function with plane waves
in the momentum equation,
as one usually does, we now use this function for the velocity.
\FFig{pEP_turb512e} shows the time evolution.
\FFig{turb512e_2} compares $B_x(y,z)$ at $x=0$ and $t=200$.
\FFig{ppower_all_turb512e} compares magnetic energy and helicity spectra
for four times.
In \FFig{ppower_all_turb512e}, we compare magnetic energy and helicity
spectra from the EP and A methods at four times.
We see that $\HM(k,t)$ shows positive and negative contributions of
small and large wavenumbers, respectively.
This agrees with early studies on the properties of $\alpha^2$ dynamos
\citep{See96,Ki99} and was also seen in the early phase of resistive
turbulent $\alpha^2$ dynamos \citep{Bra01}.
Interestingly, both the EP and the A methods reproduce this behavior
correctly, and even at late times, the EP method recovers this behavior
qualitatively.
%\begin{verbatim}
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\begin{thebibliography}{}
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%{840}{The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical hydromagnetic turbulence}
\bibitem[Derigs et al.(2016)]{Derigs16}
Derigs, D., Winters, A. R., Gassner, G. J., \& Walch, S.\yjcp{2016}{317}{223}
\bibitem[Galsgaard \& Nordlund(1996)]{GN96}
Galsgaard, K., \& Nordlund, \AA.\yjgr{1996}{101}{13445}
%{13460}{Heating and activity of the solar corona: I. boundary shearing of an initially homogeneous magnetic-field}
\bibitem[Hendrix et al.(1996)]{Hendrix}
Hendrix, D. L., van Hoven, G., Mikic, Z., \& Schnack, D. D.\yapj{1996}{470}{1192}
%{1197}{The viability of ohmic dissipation as a coronal heating source}
\bibitem[Ji(1999)]{Ji99}
Ji, H.\yprl{1999}{83}{3198}
%{3201}{Turbulent dynamos and magnetic helicity}
\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[Moffatt \& Proctor(1985)]{MP85}
Moffatt, H. K., \& Proctor, M. R. E.\yjfm{1985}{154}{493}
%{507}{Topological constraints associated with fast dynamo action}
\bibitem[Pen et al.(2003)]{Pen03}
Pen, U.-L., Arras, P., \& Wong, S.\yapjs{2003}{149}{447}
%{455}{A free, fast, simple, and efficient total variation diminishing magnetohydrodynamic code}
\bibitem[Seehafer(1996)]{See96}
Seehafer, N.\ypre{1996}{53}{1283}
%{1286}{Nature of the $\alpha$ effect in magnetohydrodynamics}
\bibitem[Stone \& Norman(1992)]{SN92}
Stone, J. M., \& Norman, M.\yapjs{1992b}{80}{791}
%{818}{ZEUS-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions: II. The magnetohydrodynamic algorithms and tests}
\end{thebibliography}
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