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\title{Numerical study of an axisymmetric anisotropic conductivity dynamo}
\author{}
\date{\today,~ $ $Revision: 1.12 $ $}
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\maketitle

\begin{abstract}
\cite{PA20} have recently presented an analytic solution to an
axisymmetric dynamo for an anisotropically conducting medium.
The velocity field corresponds to that of a rigidly rotating cylinder
embedded in a conductor at rest.
Here we present numerical simulations for smooth angular velocity profiles.
We also consider the case where the anisotropy is confined to be within
the rotating cylinder only.
\end{abstract}

\section{Introduction}

A major obstacle in dynamo theory has been the fact that the magnetic
field must always be nonaxisymmetric and therefore three-dimensional.
But this is not strictly true if an allows for the conductivity to
be anisotropic.
For that case, \cite{PA20} found recently a solution which is extremely
remarkable in many respects.
(i) Using cylindrical coordinates, $(r,\phi,z)$, the three components of the
magnetic field depend only on radius $r$ and height $z$, but not on the
azimuthal angle $\phi$.
(ii) The velocity field is very simple, corresponding to a rigidly rotating
cylinder embedded in a conducting solid at rest.
(iii) The growth rate increases with increasing conductivity without limit
and does not go to zero as for slow dynamos.
(iv) The solution is given in closed form.

Understanding the properties of such a dynamo could be of interest not
only for the realization of laboratory dynamos, but also for dynamos
in nearly collisionless plasmas, where the vastness of the full
six-dimensional phase space constitutes a significant technical challenge.
It should be kept in mind, however, that the experimental realization
of such an anisotropically conducting medium may, in practice, involve
a nonaxisymmetric design.
This was already discussed by \cite{PA20}, who suggested a composite of
conducting layers arranged in a logarithmic spiral interlaced with
poorly contacting or insulating layers.

The connection between dynamos in a medium of anisotropic conductivity
and the homopolar disk dynamo was already pointed out by \cite{RR84}.
They considered a two-dimensional dynamo in Cartesian coordinates,
for which \cite{ADP20} found an analytic solution.
A feasible design of the homopolar disk dynamo was presented by \cite{PA13}.

\cite{PA20} emphasize that the rotation must be retrograde.
Dynamo action requires then $\alpha>0$.
For $\alpha<0$, dynamo action occurs for prograde rotation.

\section{The model}

We adopt rigid rotation with angular velocity $\Omega_0$ in the inner
cylinder of radius $R$, with the material at rest outside, i.e.,
\begin{equation}
\Omega(r)=\Omega_0\,f(r)
\end{equation}
where $\Omega_0$ is a constant and $f(r)$ is a radial profile modelling
the change from finite to vanishing angular velocity in a continuous
fashion, given by
\begin{equation}
f(r)=1-\Theta(r,d).
\end{equation}
We use the same profile function to also allow for a radial
truncation of the anisotropy using
\begin{equation}
\eta_{ij}=\eta_0\,\delta_{ij}+\eta_1(r)\, q_i q_j
\end{equation}
where $\eta_1=\eta_{10} f(r)$.
We solve the uncurled induction equation,
\begin{equation}
\frac{\partial A_i}{\partial t}=(\uu\times\BB)_i-\eta_{ij} J_j
\end{equation}

We define a magnetic Reynolds number as
\begin{equation}
R_\Omega=\Omega_0 R^2/\eta_0,
\end{equation}
and the degree of anisotropy as
\begin{equation}
s=\eta_{10}/\eta_0.
\end{equation}
\cite{PA20} found dynamo action for $|R_\Omega|>14.61$
for $s\to\infty$ using the optimal angle $\alpha=0.16\pi\approx29\degr$
and wavenumber $kR=1.1$, corresponding to a minimal height of $L_z/R=5.7$.

%\begin{figure}[h!]\begin{center}
%\includegraphics[width=\columnwidth]{psav_I64x32e}
%\end{center}\caption[]{
%\url{psav_I64x32e}
%}\label{psav_I64x32e}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{psav_64x32r1}
\end{center}\caption[]{
\url{psav_64x32r1}
Anisotropic conductor both in the interior and the exterior.
Here, $\eta_0=0.0440$, $L_r=3$, and $L_z=3.9$ %64x32r1
}\label{psav_64x32r1}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{psav_I64x32k4}
\end{center}\caption[]{
\url{psav_I64x32k4}
Isotropic conductor in the exterior, outside the cylinder.
Here, $\eta_0=0.0218$, $L_r=3$, and $L_z=3.6$ %I64x32k4
}\label{psav_I64x32k4}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{p}
\end{center}\caption[]{
\url{p}
Growth rate and k.
}\label{p}\end{figure}

\section{Results}

In \Fig{psav_64x32r1} we show the result for the case where $\eta_1=\const$.
The critical maximal magnetic diffusivity is $\eta_0=0.0440$, corresponding
to $R_\Omega=22.7$, and the vertical wavenumber is $kR=1.6$.
In \Fig{psav_I64x32k4} we show the case where $\eta_1\to0$ for $r>R$.
The critical maximal magnetic diffusivity is $\eta_0=0.0218$, corresponding
to $R_\Omega=45.9$, and the vertical wavenumber is $kR=1.7$.

\begin{table}[htb]\caption{
Models with $\eta_1/\eta_0=5$, $\alpha=28.8\degr$, and $kR=1.1$.
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
$\eta_0$ & $\eta_1$ & $\lambda$ & $k$ \\
\hline
0.0445 & 0.0035 \\%64x32s1
0.04607&-0.0004 \\%64x32s2
0.0467 &-0.0014 \\%64x32s2a
\label{Ttimescale}\end{tabular}}\end{table}

\begin{table}[htb]\caption{
Models with $\eta_1/\eta_0=5$, $\alpha=30\degr$, and $kR=1.1$.
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
$\eta_0$ & $L_z$ & $\lambda$ \\
\hline
$1\times10^{-2}$ & 1 & 0.3221 \\ %128x64e2
$1\times10^{-3}$ &0.2& 1.816  \\ %512x64a1
$2\times10^{-4}$ &0.06& 2.9404 \\ %1024x64a1
$5\times10^{-5}$ &0.03& 3.6706 \\ %2024x64a1
$1\times10^{-5}$ &0.03& 3.6706 \\ %2024x64a1
1e-5 & 0.010 & 3.3899 \\%2048x64b3
1e-5 & 0.012 & 3.9734 \\%2048x64b3
1e-5 & 0.014 & 4.155  \\%2048x64b4
1e-5 & 0.016 & 4.1450 \\%2048x64b4
1e-5 & 0.018 & 4.0539 \\%2048x64b4
5e-6 &0.010 &4.130 \\%2048x64d1
5e-6 &0.008 &3.883 \\%2048x64d1
\label{Ttimescale}\end{tabular}}\end{table}

\begin{table}[htb]\caption{
Models with $\eta_1=0$ for $r\gg1$ and $\eta_{10}/\eta_0=5$.
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
$\eta_0$ & $L_r$ & $L_z$ & $\lambda$ \\
\hline
0.010 & 2 & 2.0 &  0.0886 \\ %I64x32c
0.014 & 2 & 2.0 &  0.0441 \\ %I64x32e
0.016 & 2 & 2.0 &  0.0208 \\ %I64x32f
0.018 & 2 & 2.0 &$-0.0030$\\ %I64x32g
0.018 & 2 & 2.1 &$+0.0024$\\ %I64x32g2
0.018 & 2 & 2.2 &$+0.0066$\\ %I64x32g3
0.018 & 2 & 2.4 &$+0.0123$\\ %I64x32g4
0.018 & 2 & 2.6 &$+0.0155$\\ %I64x32g5
0.018 & 2 & 2.8 &$+0.0171$\\ %I64x32g6
0.018 & 2 & 3.0 &$+0.0177$\\ %I64x32g7
0.018 & 2 & 3.2 &$+0.0176$\\ %I64x32g8
0.020 & 2 & 3.2 &$+0.0071$\\ %I64x32h1
0.022 & 2 & 3.2 &$-0.0035$\\ %I64x32i1
0.022 & 2 & 3.0 &$-0.0059$\\ %I64x32i2
0.022 & 2 & 3.3 &$-0.0027$\\ %I64x32i3
0.022 & 2 & 3.4 &$-0.0021$\\ %I64x32i4
0.022 & 2 & 3.5 &$-0.0016$\\ %I64x32i5
0.022 & 2 & 3.6 &$-0.0012$\\ %I64x32i6
0.022 & 2 & 3.7 &$-0.0009$\\ %I64x32i7
0.022 & 2 & 3.8 &$-0.0007$\\ %I64x32i8
0.022 & 2 & 3.9 &$-0.0006$\\ %I64x32i9
0.022 & 2 & 4.0 &$-0.0006$\\ %I64x32i10
0.022 & 3 & 4.0 &$-0.0002$\\ %I64x32j1
0.022 & 4 & 4.0 &$-0.0011$\\ %I64x32j2
0.022 & 3 & 4.2 &$-0.0007$\\ %I64x32j3
0.022 & 3 & 3.8 &$-0.0002$\\ %I64x32j4
0.0216& 3 & 3.8 &$+0.0014$\\ %I64x32k1
0.0216& 3 & 3.7 &$+0.0014$\\ %I64x32k2
0.0216& 3 & 3.6 &$+0.0013$\\ %I64x32k3
0.0218& 3 & 3.6 &$+0.0001$\\ %I64x32k4
\label{Ttimescale}\end{tabular}}\end{table}

\begin{table}[htb]\caption{
Models with $\eta_1=0$ for $r\gg1$ and $\eta_{10}/\eta_0=5$.
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
$d_\Omega$ & $d_\eta$ & $\eta_0$ & $L_r$ & $L_z$ & $\lambda$ \\
\hline
0.02 & 0.02 & 0.0218& 3 & 3.6 &$+0.0001$\\ %I64x32k4
0.05 & 0.02 & 0.0218& 3 & 3.6 &$-0.0013$\\ %I64x32k4d2
0.05 & 0.05 & 0.0218& 3 & 3.6 &$+0.0034$\\ %I64x32k4d3
\label{Ttimescale}\end{tabular}}\end{table}

\begin{table}[htb]\caption{
Models with anisotropy throughout, i.e.,
$\eta_1=\eta_{10}=5\,\eta$.
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
$\eta_0$ & $L_r$ & $L_z$ & $\lambda$ \\
\hline
0.0218& 3 & 1.8 &$+0.1010$\\ %64x32c2
0.0300& 3 & 1.8 &$-0.0082$\\ %64x32d1
0.0300& 3 & 2.0 &$+0.0051$\\ %64x32d2
0.0300& 3 & 2.2 &$+0.0292$\\ %64x32d3
0.0294& 3 & 1.8 &$-0.0080$\\ %64x32e2
0.0294& 3 & 2.2 &$+0.0360$\\ %64x32e4
0.0306& 3 & 2.2 &$+0.0220$\\ %64x32f1
0.0312& 3 & 2.2 &$+0.0146$\\ %64x32g1
0.0324& 3 & 2.2 &$+0.0018$\\ %64x32h1
0.0324& 3 & 2.4 &$+0.0197$\\ %64x32h2
0.0340& 3 & 2.6 &$+0.0197$\\ %64x32i1
0.0360& 3 & 2.8 &$+0.0148$\\ %64x32j1
0.0380& 3 & 3.0 &$+0.0095$\\ %64x32k1
0.0400& 3 & 3.2 &$+0.0042$\\ %64x32l1
0.0410& 3 & 3.2 &$-0.0018$\\ %64x32m1
0.0410& 3 & 3.3 &$+0.0017$\\ %64x32m2
0.0410& 3 & 3.4 &$+0.0046$\\ %64x32m3
0.0420& 3 & 3.5 &$+0.0018$\\ %64x32n1
0.0425& 3 & 3.6 &$+0.0016$\\ %64x32o1
0.0430& 3 & 3.7 &$+0.0013$\\ %64x32p1
0.0435& 3 & 3.8 &$+0.0008$\\ %64x32q1
0.0440& 3 & 3.9 &$+0.0002$\\ %64x32r1
\label{Ttimescale}\end{tabular}}\end{table}

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%r e f
\begin{thebibliography}{}

\bibitem[Alboussi{\`e}re et al.(2020)]{ADP20}
Alboussi\`ere, T., Drif, K., \& Plunian, F.\ypreN{2020}{101}{033107}
{Dynamo action in sliding plates of anisotropic electrcal conductivity}

\bibitem[Plunian \& Alboussi{\`e}re(2020)]{PA20}
Plunian, F., \& Alboussi\`ere, T.\yjourN{2020}{Phys. Rev. Research}{2}{013321}
{Axisymmetric dynamo action is possible with anisotropic conductivity}

\bibitem[Priede \& Avalos-Z\'u\~niga(2020)]{PA13}
Priede, J., \& Avalos-Z\'u\~niga, R.\yjour{2013}{Phys. Lett. A}{377}{2093}
{2096}{Feasible homopolar dynamo with sliding liquid-metal contacts}

\bibitem[Ruderman \& Ruzmaikin(1984)]{RR84}
Ruderman, M. S., \& Ruzmaikin, A. A.\ygafd{1984}{28}{77}
{88}{Magnetic field generation in an anisotropically conducting fluid}

\end{thebibliography}

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