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\begin{document}

\section*{Small-scale dynamo action at low Prandtl numbers}
Consider nonhelical isotropic turbulence with random large scale
forcing. In the absence of helicity and anisotropies mean-field dynamo
action is not possible, but small-scale magnetic fields
(scale smaller than the forcing scale) might still
grow. Crucial parameters for small-scale dynamo action are the
magnetic Reynolds and Prandtl numbers
\EQA%
\Rm = \frac{\urms}{\eta\kf}\;,\quad
\Pm = \frac{\nu}{\eta}\;,
\EEA%
where $\nu$ and $\eta$ are the viscosity and magnetic
diffusivity. Numerically the case $\Pm=1$ is the easiest case, but
this is also physically the most unrealistic one. For example, in the
Sun $Pm \approx 10^{-4} \ldots 10^{-6}$. Early numerical studies
indicated that the critical magnetic Reynolds number, $\Rmcrit$, at
which the small-scale dynamo is excited increases approximately
proportional to $\Pm^{-1/2}$ at least down to $\Pm=0.1$. This seemed
to indicate that in realistic parameter regimes the small-scale dynamo
is quenched. More recent simulations Iskakov et al.\ (2007),
however, seem to indicate that
below $\Pm \approx 0.05$, $\Rmcrit$ begins to decrease again. 


%\section*{Bottleneck-effect in turbulence}



\section*{Relation between the bottleneck and small-scale dynamo}

A possible explanation for the easier excitation of the small-scale
dynamo is related to the roughness of the velocity field and thus to
the bottleneck effect.

If $\Pm$ is of the order, or larger than unity, the magnetic field
spectrum peaks at the dissipation range of the velocity field. In this
range the velocity is smooth and the magnetic field finds it easy to
grow.

If $\Pm$ is in the intermediate range between the dissipation and
inertial ranges, the maximum of the magnetic energy spectrum coincides
with the bottleneck. This is also the regime where the velocity field
is at its roughest and dynamo action difficult. If, however, $\Pm$ is
decreased even further, the maximum of the magnetic spectrum shifts to
the inertial range where dynamo action is still more difficult than in
the $\Pm \gg 1$ regime but easier than in the bottleneck regime.

In order to test the aforementioned hypothesis we need to determine
whether there is a relation between the slope of the spectrum and the
measured growth rate of the magnetic field.


\begin{table}[h]
\begin{center}
\begin{tabular}{ccccccccc} \hline 
Model & grid     & $\nu$            & $\eta$            &$\Pm$&$\urms$& Re  & Rm & $\sigma$  \\ \hline 
256a  & $256^3$  & $5\cdot 10^{-4}$ & $10^{-3}$         & 0.5 & 0.122 & 158 & 79 & 0.0014 \\ 
512a  & $512^3$  & $2\cdot 10^{-4}$ & $10^{-3}$         & 0.2 & 0.131 & 435 & 87 & -0.007 \\ 
512c  & $512^3$  & $10^{-4}$        & $10^{-3}$         & 0.1 & 0.145 & 970 & 97 & -0.005 \\ 
512d  & $512^3$  & $4\cdot10^{-5}$  & $2\cdot10^{-4}$   & 0.2 & 0.080 &1400 &270 & +0.013 \\
1024a & $1024^3$ & $10^{-4}$        & $10^{-3}$         & 0.1 & 0.146 & 946 & 95 & -0.005? \\ 
1024c & $1024^3$ & $5\cdot 10^{-5}$ & $10^{-3}$ & 0.05 & 0.125-& 1700-& 82- &  +? (too short) \\ 
1024d & $1024^3$ & $2\cdot 10^{-5}$ & $2 \cdot 10^{-4}$ & 0.1 & 0.078 &2600 &260 &  ??? \\ 
512b  & $512^3$  & $10^{-4}$        & $2 \cdot 10^{-3}$ & 0.05& 0.148 & 990 & 49 & -0.012 \\ 
1024b & $1024^3$ & $5\cdot 10^{-5}$ & $10^{-3}$         & 0.05& 0.149 & 1930& 96 & -0.012 \\ 
%
%eta   nu    Pm   lam     urms   Re    Rm  run
%1e-3  5e-4  0.50 0.0014  0.122   158  79  256a
%1e-3  5e-4  0.50 0.0004  0.104   135  67  256b
%1e-3  5e-4  0.50 -0.0000 0.080   104  52  256c
%1e-3  2e-4  0.20 -0.0009 0.131   435  87  512a
%5e-4  1e-4  0.20 -0.0001 0.082   533 107  256d
%2e-4  4e-5  0.20  0.0003 0.079  1286 257  512d
%1e-3  1e-4  0.10 -0.0004 0.145   970  97  512c
%1e-3  1e-4  0.10 -0.0005 0.145   943  94  1024a (need to run more); AB: I get lambda=-0.012
%2e-4  2e-5  0.10 -0.0004 0.074  2388 239  1024d (still running)
%2e-3  1e-4  0.05 -0.0023 0.148   990  49  512b
%1e-3  5e-5  0.05 -0.0019 0.146  1898  95  1024b AB: I get lambda=-0.012
%
\end{tabular}
\caption{Small-scale dynamo runs made so far.
A minus sign behind a number indicates a persisting negative trend.
}\label{tab:fracheli}
\end{center}
\end{table}

\newcommand{\yprl}[5]{:~#1, ``#5,'' {\em Phys.\ Rev.\ Lett. }{\bf #2}, #3--#4}
\newcommand{\yprlN}[4]{:~#1, ``#4,'' {\em Phys.\ Rev.\ Lett. }{\bf #2}, #3}

\subsection*{References}

\begin{list}{}{\leftmargin 3em \itemindent -3em\listparindent \itemindent
\itemsep 0pt \parsep 1pt}\item[]

%\noindent Rogachevskii, I. \& Kleeorin, N. (2003), PhRvE, 68, 036301

Iskakov, A. B., Schekochihin, A. A., Cowley, S. C., McWilliams, J. C.,
Proctor, M. R. E.\yprlN{2007}{98}{208501}
{Numerical demonstration of fluctuation dynamo at low magnetic Prandtl numbers}

\end{list}

\vspace{1cm} \noindent {\small \emph{$ $Id: notes.tex,v 1.4 2008/07/09 03:22:26 brandenb Exp $ $}

\newpage

\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{pspec_1024d}
\caption{
Spectra of kinetic and magnetic energies,
compensated by $\epsilon^{-2/3}k^{5/3}$,
for Run 1024d with $\Rey=2600$, $\Rm=260$, so $\Pm=0.1$.
The horizontal dash-dotted line indicates the Kolmogorov constant at 1.5.
Note that the compensated magnetic energy spectrum peaks at $k/k_1\approx40$.
}\label{pspec_1024d}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{pbrms_1024d}
\caption{
Evolution of normalized rms magnetic field,
for Run 1024d with $\Rey=2600$, $\Rm=260$, so $\Pm=0.1$.
Note the absence of a clear trend for either increasing or decreasing
magnetic fields even after 12 turnover times.
[Should compare with lower $\Rm$ runs to check whether this
run is just marginal!]
}\label{pbrms_1024d}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{pspec_1024a}
\caption{
Spectra of kinetic and magnetic energies,
compensated by $\epsilon^{-2/3}k^{5/3}$,
for Run 1024a with $\Rey=950$, $\Rm=95$, so $\Pm=0.1$.
The horizontal dash-dotted line indicates the Kolmogorov constant at 1.5.
Note that the compensated magnetic energy spectrum peaks at $k/k_1\approx40$.
}\label{pspec_1024a}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{pbrms_1024a}
\caption{
Evolution of normalized rms magnetic field,
for Run 1024a with $\Rey=950$, $\Rm=95$, so $\Pm=0.1$.
Even here no clear trend can be seen.
}\label{pbrms_1024a}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{pspec_1024b}
\caption{
Spectra of kinetic and magnetic energies,
compensated by $\epsilon^{-2/3}k^{5/3}$,
for Run 1024b with $\Rey=1930$, $\Rm=96$, so $\Pm=0.05$.
The dash-dotted line indicates the Kolmogorov constant at 1.5.
}\label{pspec_1024b}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{pbrms_1024b}
\caption{
Evolution of normalized rms magnetic field,
for Run 1024b with $\Rey=1930$, $\Rm=96$, so $\Pm=0.05$.
}\label{pbrms_1024b}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{pspec_512d}
\caption{
Spectra of kinetic and magnetic energies,
compensated by $\epsilon^{-2/3}k^{5/3}$,
for Run 512d with $\Rey=1400$, $\Rm=270$, so $\Pm=0.2$.
The dash-dotted line indicates the Kolmogorov constant at 1.5.
}\label{pspec_512d}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{pbrms_512d}
\caption{
Evolution of normalized rms magnetic field,
for Run 512d with $\Rey=1400$, $\Rm=270$, so $\Pm=0.2$.
}\label{pbrms_512d}
\end{figure}


\end{document}
