Mean Field Dynamo Saturation: Toward Understanding Conflicting Results
Eric G. Blackman
Mean field dynamos may explain the origin of large scale magnetic fields of galaxies, but controversy arises over the extent of dynamo quenching by the growing field. Here we explain how apparently conflicting results may be mutually consistent, by showing the role of magnetic helicity conservation and boundary terms usually neglected. We estimate the associated magnetic energy flowing out of the Galaxy but emphasize that the mechanism of field escape needs to be addressed.
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\begin{document}
\title{Mean Field Dynamo Saturation: Toward Understanding Conflicting
Results}


\author{Eric G. Blackman}
\affil{Department of Physics \& Astronomy, 
University of Rochester, Rochester, NY 91125}

 \author{George B. Field}
\affil{Harvard-Smithsonian Center for Astrophysics,
Cambridge, MA 02138}



\begin{abstract}
Mean field dynamos may explain the origin of 
large scale magnetic fields of galaxies, 
but controversy arises over the extent of dynamo quenching by 
the growing field. Here we explain how apparently conflicting results may  
be mutually consistent, by showing the role of  
magnetic helicity conservation and boundary terms usually neglected.
We estimate the associated magnetic energy flowing out of 
the Galaxy but emphasize that the mechanism of field escape needs 
to be addressed. 
\end{abstract}

\section{Field Growth and Constraining the Turbulent EMF}

Unlike the  turbulent amplification of small scale 
magnetic energy to near equipartition with the the kinetic energy spectrum, 
the mean field dynamo (MFD) 
field generation (c.f. Parker 1979, Kulsrud 1999) 
on scales $>$ turbulent input scale is controversial
(c.f. Field et al. 1999). 
The MFD equation is $\partial_t\bbb=\curl \lb\bfv \ts \bfb \rb+\lambda\nabla^2 \bbb + \curl (\bbV\ts \bbb)$, with the turbulent EMF 
$\lb\bfv\ts\bfb\rb
=\alpha\bbb-\beta\curl\bbb$, 
and pseudo-scalar   $\alpha$ and scalar $\beta$.  
How well MFD growth applies when the when the dynamic magnetic  
backreaction is included depends on the survival of
$\lb\bfv\ts\bfb\rb$.
Blackman \& Field (2000a) used Ohm's law and mean field theory
(e.g.${\bf B}={\bfb}+{\overline {\bf B}}$; \ $\langle\bfb\rangle$ =0)
to constrain the dynamic value of $\lb \bfv\ts \bfb\rb$ analytically.
Deriving  
$\lb\bfv\ts\bfb\rb\cdot{\bbB}
/c = -\eta\lb\bfj\cdot\bfb\rb +\lb\bfe\cdot\bfb\rb$
and then expanding the fluctuating electric field $\bfe$ into its
potentials, gives 
$\lb\bfe\cdot\bfb\rb = -\partial_t\lb\bfa\cdot\bfb\rb/2c+\nabla\cdot
\lb\bfa\ts \bfe -\phi\bfb\rb$.
Thus, for $\lb\bfv\ts\bfb\rb$ not to be resistively 
limited, there must be time variation of 
$\lb \bfa\cdot\bfb\rb$, or non-vanishing boundary terms.  When 
such terms vanish, helical turbulence without mean field gradients 
gives $\alpha \le (b/{\bar B})^2 \alpha_0/R_m$, where
$\alpha_0$ is the kinematic value of the psuedoscalar  
coefficient $\alpha$, and $R_m$ is the magnetic Reynolds number.
There is thus an ambiguity in interpreting all existing
numerical experiments suggesting $\alpha$ quenching (e.g. Cattaneo \& Hughes 1996); the quenching
might not be dynamical, but may be due to boundary conditions.


\section{Magnetic Helicity Escape, Dynamo Action, \& Coronal Activity}
The above result highlights the role of total magnetic helicity 
$H^M = \int_V \A\cdot \B\, d^3x$ (Els\"asser 1956), 
where $V$ is a volume of integration,  
and $\A$ is the vector potential.
That MFD growth involves a magnetic helicity inverse cascade 
was demonstrated by Pouquet \etal (1976).  
The $\alpha$ effect conserves $H^M$
by pumping a positive (negative) amount to scales $>L$ (the outer
turbulent scale) and a negative (positive) amount to
scales $\ll L$.  Brandenburg's (2000) simulations confirm this inverse cascade
and the role of $H^M$ conservation.

A large-scale field can be generated only as fast 
as $H_M$ can be removed or dissipated. Presently, simulations have invoked
boundary conditions for which the growth of large scale field is 
resisitvely limited.  Large $R_m$ systems
must rely on open boundary conditions. 
To see this, note that $H^M$ satisfies 
$
{\partial_t} (\A\cdot\B)+c \nabla\cdot (\E\times \A+A_0\B) =
-2c\E\cdot \B
$ 
where $\E = -{\BV\over c}\times \B $.
Consider two cases.
Case (1): The mean scale = universal scale, or 
the integration is over periodic boundaries.  Then  boundary terms 
vanish, so $\partial_t \lb{\bf A}\cdot {\bf B} \rb=
-2c \lb{\bf E}\cdot {\bf B}\rb=
-2c {\bbE}\cdot {\bbB}-2c \lb{\bfe}\cdot {\bfb}\rb=\eta \lb {\bf J}\cdot{\bf B}\rb$
and
$\partial_t ({\bbA}\cdot \bbB) =-2c {\overline {\bf E}}\cdot {\bbB}; \ \  \partial_t \lb{\bfa}\cdot {\bfb} \rb=-2c \lb {\bfe}\cdot {\bfb}\rb$.
This is the case of section 1. Dynamo action is resistively limited. Case (2): The system (e.g. Galaxy or Sun) mean volume V $<<$  universal volume.
Here we must use the gauge invariant
relative helicity $H^M_R$ inside and outside
of the spherical or disk rotator (Berger \& Field 1984).  
The integral over the the universal volume then satisfies
$\partial_t\int_{U}{\bf A} \cdot {\bf B}d^3x=\partial_t{H}^M_{R,in}+\partial_t{H}^M_{R,out}=-2c\int_U{\bf E}\cdot{\bf B}d^3x\simeq 0$.  The formulae for the $H^M_R$ of the mean and fluctuating quantities inside the rotator 
are $\partial_t {{H}}_{R,in}(\bbB)=
-2c\int_{in}{\bbE\cdot\bbB} d^3x
+2c\int_{S_{in}}({\bbA}_p\times 
{\bbE})\cdot d{\bf S}
$
and 
$
\partial_t{{H}}_{R,in}({\bfb})=
-2c\int_{in}{\lb\bfe\cdot\bfb\rb} d^3x
+2c\int_{S_{in}}\lb{\bf a}_p\times 
{\bfe}\rb\cdot d{\bf S}.
\label{r4aae}
$
In a steady state,  $\partial_tH_{R,in}=0=\partial_tH_{R,out}$.
This and ${\bf E}\cdot{\bf B}\simeq 0$ imply that the above surface
terms above must be equal and opposite.  
Moreover, the surface term balances the $\bbE\cdot \bbB$ 
term in the $\partial_tH^M_{R,in}$ equation.
The boundary term can thus allow for a significant turbulent EMF
because the latter is contained in $\bbE\cdot \bbB$.
Dynamo action unrestricted by 
resisitivity is possible only  in case (2).
This is consistent with Pouquet et al. (1976) and 
Brandenburg (2000).  

If $H_M$ flows through the boundary, then so does magnetic magnetic energy.  
Blackman \& Field (2000b) showed that 
%for field geometries with separable time dependences,  
a typical minimum power leaving the system
when a MFD is operating is given by
 $\dot E^M 
\ge 
{k_{\rm min}\over
8\pi} |\dot H^M|\nn
= {k_{\rm min}\over 6} \left| \lra{\alpha\ob^2}\right|V
$, where $V$ is the system volume.
Dynamos operating 
in the Sun, accretion disks, and the Galaxy would then 
lead to a net escape of magnetic 
energy and small and large scale magnetic helicity. 
Coronal activity from the emergence and
dissipation of helical magnetic flux is thus a prediction
of the MFD in all of these cases, and is observed directly in the Sun 
(c.f. Pevtsov et al. 1999). 
For the Galaxy, \noindent ${\dot E}^M \gsim ({\pi R^2})\alpha \ob^2
\sim10^{40} ({R / 12{\rm kpc}})^2 ({\alpha / 10^5{\rm cm/s}})
({\ob/ 5\ts 10^{-6}{\rm G}})^2 {\rm erg/s}$, in each hemisphere.
Blackman \& Field (2000b) discuss how 
this relation may be 
consistent with coronal energy input rates required by Savage (1995)
and Reynolds et al. (1999).


\section{Open Questions}

An MFD unlimited by resitivivity 
requires the helicity to flow through the boundary
AND that there be some mechanism that enables this flow. Thus there
are two separate issues. Even if the boundary conditions allow it, 
does it actually happen? One may have to include the dynamics of buoyancy 
or winds to fully demonstrate the non-resistive MFD.
Note that turbulent diffusion of the mean
magnetic field (not necessarily the actual field) across the boundary 
is required to maintain a quadrupole field in the Galaxy with a net
flux inside the disk. Similarly, for the Sun, the solar cycle requires
net diffusion through the boundary. The flow of helicity 
would appeal to the same dynamics needed by these constraints.

The analytic and numerical studies that we have seen which 
show catastrophic suppression of the dynamo coefficients, or
resistively limited dynamo action, 
either (1) invoke periodic boundary conditions, and/or (2) 
are 2-D, or (3) do not distinguish between zeroth order isotropic
components of the turbulence and the higher order anistropic
perturbations for a weak mean field (Blackman \& Field 1999). 
This means that there always seems to be an alternative explanation.
and the observed suppression is then ambiguous 
as we have described.  This does not mean that some of the physical concepts
found in the strong suppression results are invalid, but just that they 
may be valid only for the restricted cases considered.  
For example, the observation that the Lagrangian chaos properties of the flow
are changed in the presence of a weak mean field for turbulence in a 
periodic box (e.g. Cattaneo et al. 1996)
needs to be understood in the relation to 
the imposed boundary conditions, and the
shape of the magnetic energy spectrum (e.g. dominated at small or large scales?).  Along these lines, note that the helicity constraint
is global, but also becomes a constraint for any sub-volume
of a periodic box once the system is fully mixed.  





\section{References}
%\begin{enumerate}
\noindent
Blackman, E. G. \& Field, G. B. 1999, ApJ {\bf 521} 597.

\noindent
Blackman, E. G. \& Field, G. B. 2000a, ApJ {\bf 534} 984.

\noindent
Blackman, E. G. \& Field, G. B. 2000b, MNRAS, in press,
astro-ph/9912459.

\noindent Berger M.C. \& Field G.B., 1984, JFM, {\bf 147} 133.

\noindent Brandenburg A., 2000, submitted to ApJ, astro-ph/0006186.

\noindent
Cattaneo, F., \& Hughes, D.W. 1996, Phys.\ Rev.\ E.\ {\bf 54}, 4532.

\noindent
Cattaneo, F., \& Hughes, D.W., Kim E.-J, 1996, PRL {\bf 76}, 2057.


\noindent
Els\"asser, W. M. 1956, Rev.\ Mod.\ Phys.\ {\bf 23}, 135.

\noindent
Field, G. 1986, 
%Magnetic Helicity in Astrophysics, 
in {\it
Magnetospheric Phenomena in Astrophysics}, R. Epstein \& W. Feldman, eds.,
AIP Conf. Proc. 144 (Los Alamos: Los Alamos Sci. Lab.), 324.

\noindent
Field, G. B., Blackman, E. G., \& Chou, H. 1999, ApJ {\bf 513}, 638.

\noindent
Frisch, U., Pouquet, A., L\'eorat, J. \& Mazure, A. 1975, JFM {\bf
68}, 769.

%\noindent
%Gruzinov, A. \& Diamond, P. H. 1994, PRL {\bf 72}, 1651.


\noindent Krause F. \& R\"adler K.-H., 1980, {\it 
Mean-field magnetohydrodynamics and dynamo theory}, (New York: Pergamon).

\noindent Kulsrud R., 1999, ARAA, {\bf 37} 37.

\noindent
Parker, E. N., {\it Cosmical Magnetic Fields} (Oxford: Clarendon
Press).

%\noindent
%Parker, E. N. 1955, ApJ {\bf 122}, 293.

\noindent
Pevtsov, A., Canfield, R. C., \& Brown, M. R. eds.\ 1999, {\it Magnetic
Helicity in Space and Laboratory Plasmas}, (American Geophysysical Union).

\noindent
Pouquet, A., Frisch, U., \& Leorat, J. 1976, JFM {\bf 77}, 321.


\noindent
Reynolds R.J., Haffner L.M., Tufte S.L., 1999, ApJ {\bf 525} 21.



\noindent Savage B.D., 1995, in {\it The Physics of the Interstellar Medium
and Intergalactic Medium},  A. Ferrara, C.F. McKee, C. Heiles, \& 
P.R. Shapiro eds. ASP conf ser vol 60. (San Francisco:  PASP) p233.

%\noindent
%Seehafer, N. 1994, Europhys.\ Lett.\ {\bf 27}, 353.

%\noindent
%Shibata, K. 1999, in {\it Magnetic Helicity in Space and
%Laboratory Plasmas}, A. Pevtsov, R. C. Canfield, and M. R. Brown eds.\ (American
%Geophysical Union, 1999).

%\noindent
%Steenbeck, M., Krause, F., \& R\"adler, K. H. 1966, Z. Naturforsch.
%{\bf 21a}, 369.

%\noindent
%Withbroe, G. L. \& Noyes, R. W. 1977, Ann.\ Rev.\ Astron.\
%Astrophys.\ {\bf 15}, 363.
%\end{enumerate}
\end{document}






\subsection{Abbreviations for Journals}

There are macros for many of the oft-referenced journals so that authors
may use the \LaTeX\ names rather than having to look up a particular
journal's specific abbreviation.
Any stylistic requirements of the editors are taken care of by the macros,
so authors need not be concerned about such editorial preferences.

\begin{center}
\begin{tabular}{ll}
\verb"\aj" & Astronomical Journal (\aj )\\
\verb"\araa" & Annual Review of Astronomy and Astrophysics(\araa )\\
\verb"\apj" & Astrophysical Journal (\apj )\\
\verb"\apjs" & \rule[.5ex]{2em}{.4pt}, Supplement Series (\apjs )\\
\verb"\ao" & Applied Optics (\ao )\\
\verb"\apss" & Astrophysics and Space Science(\apss )\\
\verb"\aap" & Astronomy and Astrophysics (\aap )\\
\verb"\aaps" & \rule[.5ex]{2em}{.4pt}, Supplement Series (\aaps )\\
\verb"\azh" & Astronomicheskii Zhurnal (\azh )\\
\verb"\baas" & Bulletin of the AAS (\baas )\\
\verb"\jrasc" & Journal of the RAS of Canada (\jrasc )\\
\verb"\memras" & Memoirs of the RAS (\memras )\\
\verb"\mnras" & Monthly Notices of the RAS (\mnras )\\
\verb"\pra" & Physical Review A: General Physics (\pra )\\
\verb"\prb" & Physical Review B: Solid State (\prb )\\
\verb"\prc" & Physical Review C (\prc )\\
\verb"\prd" & Physical Review D (\prd )\\
\verb"\prl" & Physical Review Letters (\prl )\\
\verb"\pasp" & Publications of the ASP (\pasp )\\
\verb"\pasj" & Publications of the ASJ (\pasj )\\
\verb"\qjras" & Quarterly Journal of the RAS (\qjras )\\
\verb"\skytel" & Sky and Telescope (\skytel )\\
\verb"\sovast" & Soviet Astronomy (\sovast )\\
\verb"\ssr" & Space Science Reviews (\ssr )\\
\verb"\zap" & Zeitschrift f\"ur Astrophysik (\zap )\\
\end{tabular}
\end{center}

\subsection{Mathematical Symbols}

The following  macros produce various troublesome or laborious mathematical symbols.
\begin{center}
\begin{tabular}{llll}
\verb"\deg" & (\deg)&\verb"\farcs"&(\farcs)\\
\verb"\sun"& (\sun)&\verb"\fp"&(\fp)\\
\verb"\earth"&(\earth)&\verb"\micron"&(\micron)\\
\verb"\la"&({$\la$})&\verb"\onehalf"	&(\onehalf)\\
\verb"\ga"&({$\ga$})&\verb"\onethird"&(\onethird)\\
\verb"\arcmin"&(\arcmin)&\verb"\twothirds"&(\twothirds)\\
\verb"\arcsec"&(\arcsec)&\verb"\onequarter"&(\onequarter)\\
\verb"\fd"&(\fd)&\verb"&\threequarters"&(\threequarters)\\
\verb"\fh"&(\fh)&\verb"\ubvr "	&(\ubvr)\\
\verb"\fm"&(\fm)&\verb"\ub"	&(\ub)\\
\verb"\fs"&(\fs)&\verb"\bv"	&(\bv)\\
\verb"\fdg"&(\fdg)&\verb"\vr"	&(\vr)\\
\verb"\farcm"&(\farcm)&\verb"\ur"	&(\ur)\\
\end{tabular}
\end{center}
\pagebreak
%\section{Examples}

%These instructions give an overview of the basic markup commands
%that need to be entered in a paper.
%Authors are encouraged to examine
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%these examples are named \verb"psample1.tex"
%and \verb"psample2.tex".  The file \verb"psample1.tex" is a paper prepared
%with the PASP macros utilizing a \emphasize{minimal} amount of markup.
%A more ``complete'' paper requiring most of the capabilities
%of the package is provided as \verb"psample2.tex";
%this file is annotated with comments that describe
%the purpose of the markup.

\section{Instructions to Editors}
Please observe the following guidelines.

 All contributions must carry the eyebrow slug (giving the title of the
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\medskip

\noindent \verb"\def\@jourvol{**VOLUME**}"

\noindent \verb"\def\cpr@year{**PUBLICATION YEAR**}"

\noindent \verb"\def\vol@title{**TITLE**}"

\noindent \verb"\def\vol@author{**EDITORS**}"

\medskip

\noindent and replace ``**VOLUME**'', etc., by the relevant information.

\section{Persons to Contact}

\begin{description}
\item For details concerning the volume number allocated to your proceedings
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\end{document}



\begin{itemize}
\item[1)] Current helicity $\lra{\B\cnt \B}$ can be directly measured for
flux tubes penetrating the surface into the solar corona. This quantity
measures the parallel current, and hence, the twist of the lines of
force. Generally, twist implies linkage of lines of force, hence
magnetic helicity. With radii in the
range $10^3$-$10^4$ km, such tubes represent relatively small-scale fields, as
$R_\odot = 7\times 10^5$km.  
\item[2)] There is a correlation between the sign of $\lra{\B\cnt\B}$ and
the sign of the observed twist of the field-aligned features in the
photosphere, implying that the parallel current responsibile for the twist
originates below the photosphere and continues into the corona.
\item[3)] In laboratory experiments, twisted flux tubes are subject to kink
instability, leading to reconnection and magnetic energy release.
\item[4)] The Yohkoh satellite detected evidence of magnetic reconnection
in flares of various sizes.  Shibata (1999) put forward a model in which
clouds of plasma (``plasmoids") are confined in twisted flux tubes which
reconnect with nearby flux, ejecting the plasmoid together with its
twisted flux, hence magnetic helicity.   Ejection of helicity is an important
part of the model.
\item[5)] The ejection of large plasmoids are observed as Coronal Mass
Ejections (CMEs).  If an interplanetary spacecraft happens to pass through a
CME, its magnetic field can be observed {\it in situ}.  At least some CMEs
appear to carry  magnetic helicity.
\end{itemize}

The importance of boundary conditions can be seen by considering two
cases. In the first case, when size of the 
system  = universal scale, or the average is over periodic
boundary conditions, then there are no boundary terms:
$\partial_t \lb{\bf A}\cdot {\bf B} \rb=
-2c \lb{\bf E}\cdot {\bf B}\rb=
-2c {\bbE}\cdot {\bbB}-2c \lb{\bfe}\cdot {\bfb}\rb\simeq 0$
% c \eta \lb {\bf J}\cdot{\bf B}\rb
and
$\partial_t ({\bbA}\cdot \bbB) =-2c {\overline {\bf E}}\cdot {\bbB}; \ \  \partial_t \lb{\bfa}\cdot \bfb \rb=-2c \lb {\bfe}\cdot {\bfb}\rb$.
In the second case, the system (e.g. Galaxy) scale  $<<$  universal scale.
Then we must invoke gauge invariant
relative helicities inside and outside
the spherical or disk rotator.  The integral of the total magnetic helicity
over the universal volume then satisfies
$
\partial_t\int_{U}{\bf A} \cdot {\bf B}d^3x
= \partial_t{H}_{R,in}+\partial_t{H}_{R,out}=-2c\int_U{\bf E}\cdot{\bf B}d^3x\simeq 0,
%\label{r4aab1}
$
The formulae for the $H_R$ 
associated with the mean and fluctuating quantities inside the rotator 
satisfy 
%$   \partial_t \int{\bbA}\cdot \bbB d^3x=
%-2c \int_{SYS} {\overline {\bf E}}\cdot {\bbB}d^3x$$
%\smallskip
%$$
$
\partial_t {{H}}_{R,in}(\bbB)=
-2c\int_{in}{\bbE\cdot\bbB} d^3x
+2c\int_{S_{in}}({\bbA}_p\times 
{\bbE})\cdot d{\bf S}
%\label{r4aad}
$
and 
$
\partial_t{{H}}_{R,in}({\bfb})=
-2c\int_{in}{\lb\bfe\cdot\bfb\rb} d^3x
+2c\int_{S_{in}}\lb{\bf a}_p\times 
{\bfe}\rb\cdot d{\bf S}.
\label{r4aae}
$

%$\dot H^M\simeq {4\pi \over 3} \lra{\alpha\ob^2}R^3_\odot$
%%\; . \lab{9}
Let $H^M=\int H^M_k\, dk$; keep in mind that both $H^M_k$ and $H^M$ are
signed quantities.  



Blackman \& Field (1999) 
find the magnitude of this quantity in each hemisphere to be
%As (8) changes sign at the 
%equator, we evaluate its integral over each polar hemisphere separately.
%For one hemisphere we get

where the angle
brackets refer to a hemisphere volume average.  According to Blackman \&
Field (1999), both terms in the brackets are of the order $v_0 L
\ob^2/R_\odot$, where $L=3\times 10^8$cm is the outer scale of the
turbulence, $v_0 = 10^4$cm~s$^{-1}$ is the turbulent velcocity at that
scale, and $\ob= 100\,$G in the mean magnetic field (Parker 1979).   



$
\noindent A mode's presence in the lower corona requires $k>k_{\rm
min}=2\pi/R_\odot$, so 
$\dot E^M\ge {\pi\over 3} \left| \lra{\alpha\ob^2}\right|R^2_\odot$
Summing over both polar hemispheres, the total power in ejected magnetic
fields satisfies
%\be
$P \ge {2\pi\over 3} \left| \lra{\alpha\ob^2}\right|R^2_\odot = 4\times
10^{27}{\rm erg~s}^{-1}
%\; , \lab{13}
%\ee
$ using $\alpha\sim 40\,$cm~s$^{-1}$ (Parker 1979).
This energy is available for particle and wind acceleration.
%reconnection, which in turn can heat coronal
%gas, create Alfv\'en waves which accelerate it in a wind, and accelerate
%energetic particles, we should 
This lower limit is in fact consistent with solar observations, as it is 
$\sim 1/6$ the total measured non-thermal energy loss (Withbroe \& Noyes 1977).

%
%conduction, radiative and solar wind energy flux
%in coronal holes.
%cover $\sim 1/2$ the area of the Sun.  
%is $25 \ts 10^27$ ergs/sec.




%If one wishes to have an acknowledgments section, it should be
%set off simply with the command
%\begin{quote}
%\verb"\acknowledgments"
%\end{quote}

%(scale height $h \sim$ 1kpc) 
%is $\sim 10^{41}$erg/sec, whereas that input into the highly
%ionized coronal gas ($h\gsim {\rm few}$ kpc)
%is $\sim 10^{40}$erg/sec. This compares
%favorably with (\ref{gal}).


%estimate 


explain 
This is also implied by  the study of Reynolds et al (1999), 
who argue that 
spatial variations of [S II]/H-Alpha and [N II]/H-Alpha line intensity 
ratios  in the halos of our and
other galaxies are inconsistent with pure photoionization models.
Instead, a  secondary heating mechanism is required that
i
the required input heating rate of $4.1\ts 10^{40}$erg/s over a 12kpc radius. 
This again compares well with (\ref{gal}).

Both the Pouquet et al. (1976) and the
Brandenburg (2000) results show dynamo action 
proceeding only at the resistive time scale.  These results are both consistent
with their choice of boundary conditions. 
Note that the  Pouquet calculations used a particularly low 
magnetic Reynolds number, $R_M=30$, 
well below that of astrophysical systems like the Galaxy.

-----
The energy spectrum $E^M_k$ must satisfy $E^M_k \ge {1\over 8\pi} k|H^M_k|$.
(e.g. Frisch \etal (1975)) for a  helical magnetic field to be realizable.
If we demand that $\partial_tH_{R,in}$ and 
$\partial_tH_{R,out}$ vanish, then when combined with 
${\bf E}\cdot{\bf B}\simeq 0$, the two surface
terms in these equations above must be equal and opposite.  
Balancing the surface term
with the $\bbE\cdot \bbB$ term in the $\partial_tH_{r,in}$ equation
implies a relative helicity flow out the boundary of order 
$\dot H_R = {4\pi\over 3} R^3_\odot \lra{\alpha\ob^2-\beta\OB\cnt \OB}$ 
Also, the implications of the 2-D Zeldovich constraint 
need to be re-investigated in the context of 




For larger $R_M$, related calculations which do not include boundary/divergence
terms (e.g. Seehafer 1994; Gruzinov \& Diamond 1994; Cattaneo \& Hughes 1996) 
would predict that large-scale dynamo should be severely curtailed, proceeding
at resitive time scales.  

%Two types of MHD magnetic field amplification are 
%(1) turbulent amplification (=small scale
%dynamo) and (2) mean field dynamo theory (MFD).  The former 
%involves random walk field line stretching. In 3-D, 
%near equipartition is reached between the magnetic and 
%kinetic energy spectra. This is not 
%controversial, though the exact level of equipartition may vary.
%depending on forcing function and Prandtl number. 
%The value of $\alpha$ that we have scaled to 
%is from Ruzmaikin et al. (1988).  
%Ferri\`ere (1993) 
%suggests that $\alpha\sim 2 \ts 10^4$cm/sec at maximum,
%which would lower the above estimate of the limit by a factor of 4. 
 
%have shown that 
%dynamo action can occur in large $R_M$ systems only if 
%small-scale helicity can escape to a body's surface, something
%impossible for the boundary conditions of Seehafer (1994) and
%Cattaneo \& Hughes (1996).  
%Blackman and Field (1999) have shown that 
%helicity can and must escape from the surface for a working dynamo in
%systems of finite size.  
%and that the expected
%rate of escape limits dynamo action to a rate approximately $L/R$ times
%the maximum value, where $L$ is
%the outer scale of the turbulence and $R$ is
%the radius of the system.  This suppression by a factor $\sim 20$   is
%sufficiently mild to allow the $\ao$ dynamo to operate in the Sun.



%implies a relative helicity flow out the boundary of order 
%$
%\dot H_R = {4\pi\over 3} R^3_\odot \lra{\alpha\ob^2-\beta\OB\cnt \OB}.
%$ 
%Taking the $\alpha$ term as representative,


%In this paper we describe the link between the production of large-scale
%magnetic helicity by the solar dynamo, and the escape of a net amount of
%small-scale magnetic helicity, hence magnetic energy.  


%\section{Magnetic Helicity}



%Steenbeck, Krause, and R\"adler (1966) developed a
%formalism for describing Parker's (1955) concept that helical turbulence  
%can twist toroidal fields  into the poloidal direction, where they can be
%acted upon by differential rotation to regenerate a powerful large-scale
%magnetic field.  Their formalism involved breaking $\B$ into a large-scale
%component $\OB$ and a small-scale component $\b$, and similarly for the
%velocity field $\BV$.  They showed that $\OB$ satisfies the induction
%equation
%\be
%{\partial\OB\over \partial t} = \nt \lrp{\OV\times \OB} - \nt c\OE+
%\lambda\nabla^2\OB\;,\lab{4} \ee
%where the first term describes the effect of differential rotation
%(``$\Omega$-effect"),
%\be
%c\OE = -\alpha\OB +\beta\nt \OB 
%\lab{5}
%\ee
%is the ``turbulent emf," and $\lambda=\eta c^2/4\pi$ is the magnetic
%diffusivity.  Here $\alpha$ represents Parker's twisting (``$\alpha$ effect")
%and $\beta(\gg \lambda)$ is the turbulent diffusivity. Steenbeck \etal
%calculated $\OE$ to first order in $\ob$ and hence $\alpha$ and
%$\beta$ to zero order in $\ob$ from the statistics of the turbulence. 
% Recently Field, Blackman \& Chou (1999) have calculated $\alpha$ to all
%orders in $\ob$.  

%When (\ref{5}) is substituted into (\ref{4}), we have the
%mean-field dynamo equation for the case $\alpha$ and $\beta$ are constant:
%\be
%{\partial\OB\over \partial t} = \nt \lrp{\OV\times \OB} + \alpha \nt
%\OB+(\beta+\lambda) \nabla^2 \OB\; , \lab{6}
%\ee
%which in the approximation that $\alpha$ and $\beta$ are independent of
%$\ob$,s a linear equation for $\OB$ which can be solved as an eigenvalue
%problem for the growing modes in the Sun and other bodies.

%Blackman \& Field (1999) show that in the framework of mean field
%dynamo theory, 
%%From (\ref{2}) and (\ref{5}), we see that in this 
%large-scale magnetic
%helicity is created at the rate
%$
%-2 c \OE\cdot \OB= 2(\alpha\ob^2 - \beta \OB\cnt \OB) 
%$
%However, from above we know that  the total magnetic helicity is
%constant in the absence of Ohmic dissipation.  How is this apparent
%contradiction resolved?




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%\section{Escape of Small-Scale Helicity from the Sun}

%In July 1998, a Chapman Conference entitled ``Magnetic Helicity in Space
%and Laboratory Plasmas" 
%forum for recent work on magnetic helicity in the solar photosphere and
%corona.  We may summarize some of the papers as follows:

%In summary, currents along $\B$ twist emerging flux tubes, endowing them
%with current helicity and therefore magnetic helicity as well.  Instability
%leads to reconnection, allowing magnetic flux to escape in plasmoids carrying
%magnetic helicity.  The net result is that helical fields below the
%photosphere escape the Sun, carrying magnetic helicity with them

%{1\over 8\pi}\int k|\dot H^M_k| \, dk\nn\ge 
%{k_{\rm min}\over 8\pi} \int |\dot H^M_k| \, dk 
