Dynamical Quenching of the $\alpha^2$ Dynamo
q3astroph.tex
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\title{Dynamical Quenching of the $\alpha^2$ Dynamo}

\centerline{George B. Field$^1$ and Eric G. Blackman$^2$}
\bigskip

\centerline{1. Harvard-Smithsonian Center for Astrophysics, 60 Garden St., 
Cambridge MA, 02138}

\centerline{2. Department of Physics \& Astronomy, and Laboratory}
\centerline{for Laser Energetics, University of Rochester, Rochester NY, 14627}


\centerline{(submitted to ApJ)}


\def\be#1{\begin{equation} \label{#1}}
\def\pref#1{(\ref{#1})}
\def\ee{\end{equation}}
\def\beq{\begin{eqnarray}}
\def\eeq{\end{eqnarray}}
\def\nn{\nonumber \\ }
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\def\lb{\langle}
\def\rb{\rangle}
\def\ts{\times}
\def\curl{\nabla\ts}
\def\kms{\, {\rm km\, s}}
\def\pc{\, {\rm pc}}
\def\cms{\, {\rm cm\, s}}
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\def\tfrac#1#2{{\textstyle{#1\over #2}}}
\def\half{\tfrac{1}{2}}

\def\O{{\cal O}}

\def\OE{\overline{\bf E}}
\def\obeta{\overline{\beta}}
\def\OA{\overline{\bf A}}
\def\oa{\overline{A}}
\def\ovar{\overline{\varphi}}
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%\maketitle
 \baselineskip=24pt
 %\textheight=8.5in
% \hsize=6.5in
%\hoffset=-.5in
% \voffset-1in

\centerline{\bf Abstract}

We present a two-scale approximation for the dynamics of a nonlinear
$\alpha^2$ dynamo. Solutions of the resulting nonlinear equations
agree with the numerical simulations of Brandenburg (2001),
and show that $\alpha$ is quenched by the buildup of magnetic
helicity at the forcing scale $1/k_2$ as the $\alpha$ effect transfers it
from the large scale $1/k_1$. For times $t > (k_1/k_2)R_{M,2}$
in eddy turnover units  (where $R_{M,2}$ is the magnetic Reynolds
number of the forcing scale), $\alpha$  is resistively limited in the form
predicted for the steady-state case. However, for $t << R_{M,2}$,
$\alpha$ takes  on its kinematic value, independent of $R_{M,2}$,
allowing the production of large-scale magnetic energy equal to $k_1/k_2$
times equipartition. Thus the dynamic theory of $\alpha$
predicts substantial "fast" growth of large-scale field despite being
"slow" at large times.

\bigskip
{{\bf Subject Headings}:  MHD; turbulence; 
ISM: magnetic fields;  galaxies: magnetic fields; 
stars: magnetic fields; methods: numerical}





%We use a two-scale mean field formalism and the results  of 
%Pouquet, Frisch and L\'eorat (1976), 
%to derive the time dependent dynamical quenching of a 
%mean field $\alpha^2$ dynamo in a closed system. Our results agree
%with numerical simulations of Brandenburg (2001).
%Dynamical quenching ultimately occurs because the growth of the large
%scale magnetic helicity also implies the growth of small scale
%magnetic helicity of the opposite sign.  This, in turn, implies
%the growth of small-scale current helicity, which  
%decreases the $\alpha$ coefficient and makes the dynamo
%resistively limited (slow) at large times. 
%However, our results also reveal the importance of a 
%fast kinematic phase in which the the mean field grows up to a value 
%$k_1/k_2$ times equipartition with the kinetic turbulence, independently of 
%$R_M$,
%where $k_1$ and $k_2$ are the large scale and small scale wave-numbers respectively.
%This is important because it shows that applying $R_M$ quenching
%formulae at early times is invalid and can lead to a significant
%underestimate in the saturated field strength.




%Gruzinov and Diamond (hereafter GD, 1994, 1995, 1996) 
%and Bhattacharjee and Yuan (hereafter BY, 1995) also used the
%conservation of magnetic helicity to argue that $\alpha$ in
%a closed steady-state system is quenched for strongly 
%sub-equipartition mean field strengths.  

%A second, $R_M$ dependent phase lasts until 
%$t=R_M$, where $R_M$ here is 
% defined using the rms velocity and $k=k_1$ scale.
%For $t>R_M$, our solutions match the numerical simulation quenching curves
%of Brandenburg (2001).




  
%Taken at face value, we find that the GD and BY results 
%need to be reinterpreted when time dependence and turbulent diffusivity 
%are included; then, quenching occurs only when the mean field reaches near 
%equipartition.  In the special case 
%where the scale of the mean field is infinite, the 
%lack of turbulent diffusion  
%leads to resistive quenching of $\alpha$ at asymptotically large times.
%%as in GD and BY. 
%%The result of $\alpha$ quenching at equipartion agrees qualitatively
%Field Blackman \& Chou (hereafter FBC, 1999) 
%employed a very different approach to $\alpha$ quenching than
%that herein and so we also compare both GD and our present work
%to the technique and implications of FBC.  


% an effect not
%anticipated by FBC.  


\section{Introduction}
Large-scale magnetic fields are often interpreted in terms of the equations
of Mean-Field Magnetohydrodynamics (Krause \& R\"adler 1980):
\be{n1}
\part_t\OB=\alpha\nt \OB+\lrp{\beta+\la} \nab^2 \OB\; ,
\ee
where $\OB$ is the mean (or large-scale) magnetic field,
\be{n2}
\la = {\eta c^2\over 4\pi}
\ee
is the magnetic diffusivity in terms of the resistivity $\eta$, and $\alpha$
and $\beta$ are parameters of the
underlying MHD turbulence.  Steenbeck \etal (1966) showed that if the
turbulence is isotropic and incompressible, and
the back reaction of $\OB$ is neglected,
\be{n3}
\alpha = \tfrac{1}{3} \tau\lra{\v\cdot\nt\v} \ee
and
\be{n4}\beta=\tfrac{1}{3} \tau \lra{v^2}\; .
\ee
Here $\tau$ is a typical correlation time of the flow $\v$, and
$\tfrac{1}{2}\lra{\v\cnt\v}$ is its kinetic
helicity, a measure of the net handedness of cyclonic motions (Parker 1955,1979).
$\beta$ represents turbulent diffusion of $\OB$.  
In this paper, the brackets and overbar represent spatial averages.

As $\OB$ grows, 
it exerts a backreaction on the turbulent flow,
%is large, back reaction cannot be neglected, 
and equations \pref{n3} and \pref{n4} must be modified to account for this.
A number of attempts to describe the corresponding saturation
of $\alpha$, or ``$\alpha$ quenching", have been
made.  As part of a general study of homogeneous, isotropic, helical
MHD turbulence, Pouquet, Frisch and L\'eorat (1976; hereafter PFL) used
the Eddy Damped Quasi-Normal Markov (EDQNM) approximation to derive
evolution equations for the spectra of kinetic
energy, magnetic energy, kinetic helicity, and magnetic helicity (defined as
$\lra{\A\cnt \A}/2$, with $\A$ the
vector potential).  They then solved a number of initial-value problems for
these spectra, and found an
$\alpha$ effect like that predicted by Steenbeck \etal (1966). 
By expanding in
terms of
a small quantity  $a$, they found that
$\alpha$ appropriate for a field having a large scale $k^{-1}$ is
\be{n5}
\alpha(k) = -\tfrac{2}{3}\int^\infty_{k/a} \theta_{kqq}
\lrp{H^V_q-H^C_q}dq\;, \ee
where $H^V_q$ is the spectrum of the small-scale kinetic helicity
\be{n6}
H^V=\half\lra{\v\cnt\v}\;,\ee
$H^C_q$  is the spectrum of the small-scale current helicity
\be{n7}
H^C = \half \lra{\b\cnt\b}\;,\ee
(where the small-scale field $\b$, like other magnetic fields in this paper,
 is in velocity units), and $\theta_{kqq}$
is the relaxation time for the interaction
of two wave numbers $q$ and $q'\sim q$ to excite $k\ll q$.  Equation
\pref{n5} is appropriate for the case that the
lower limit of $q$, $k/a$, is much larger than $k$, the wave number of the
large-scale field.
If one replaces $\theta_{kqq}$ by  $\tau$, the first term in
\pref{n5} agrees with \pref{n3}.  However,
the second term in \pref{n5} is new, and its physical significance was
discussed in PFL. It will play an
important role in what follows.

Gruzinov and Diamond (1994, hereafter GD, 1995, 1996) 
and Bhattacharjee and Yuan (1995, hereafter BY) 
recognized that the current 
helicity term in \pref{n5} and \pref{n7} is
related to magnetic helicity, a conserved quantity in ideal MHD, and
exploited that 
fact to find how $\alpha$ is quenched for a closed system 
when it has reached  a steady state.
In this paper, we also link  the current helicity contribution
to $\alpha$ with the equation for magnetic helicity evolution,
but in addition to considering  a steady state, we solve the 
time-dependent problem. As will be discussed,
the results ultimately lead to different conclusions
than those of GD and BY.

There is an important assumption built
into our approach: we assume that the PFL 
current helicity contribution to $\alpha$  
represents the current helicity to all orders 
in the mean field, not the zeroth-order quantities that appear
in the formalism of Field, Blackman and Chou (1999, hereafter FBC).
who expanded turbulent quantities about an isotropic state with $\ob=0$.
However, we have been unable to prove that this assumption is correct.
This issue is addressed in Blackman \& Field (1999), 
where it is shown that the current helicity that formally appears
as a correction in GD and BY should really be the 
zeroth-order contribution as in FBC.  It turns out, however, that the success
of the dynamical theory described below depends crucially
on ignoring this ordering ambiguity.  


In section 2 we discuss the model of
PFL and produce a two-scale simplification of their equations.
(This is supplemented by Appendix A).
In section 3 we solve the resulting time-dependent equations 
for large-scale field growth and show that the results
agree well the numerical simulations of Brandenburg (2001, Hereafter B01).
In section 4 we compare our results to the implications
of previous $\alpha$ quenching models (and supplement this by Appendix B).
We conclude in section 5. 



\setcounter{section}{1}


\section{Using PFL in a Two-Scale Approximation}
 
%Here we  study the $\alpha$ quenching embodied in PFL's
%formulation of $\alpha$.
PFL studied the spectra of kinetic energy,
\be{n10}
E^V = \half \lra{v^2}\; , \ee
magnetic energy
\be{n11}
E^M = \half \lra{B^2}\; , \ee
kinetic helicity $H^V$ (eq.\ \ref{n6}), and magnetic helicity
\be{n12}
H^M = \half \lra{\A\cnt\A} = \half\lra{\A\cdot\B}\; . \ee
(Note the factor of $1/2$ in \pref{n12}.)
It is easy to show that the spectrum of current helicity (eq.\ \ref{n7}) is
related to that of $H^M$ by
\be{n13}
H^C_k = k^2H^M_k\; . \ee
Therefore the evolution of $H^C_k$, needed for evaluating $\alpha$
according to \pref{n5}, is tied to that of
$H^M_k$.

Magnetic helicity conservation will play a role in what follows.  Following
Moffatt (1978), we can use the induction
equation $(c=1)$ in the form
\be{n14}
\part_t \B = -\nt\E = \nt \lrp{\v\times \B-\la\nt \B} \ee
 to show that
\be{n15}
\part_t \lrp{\half \A\cdot\B} = -\half \nabla\cdot \lrb{\B(\phi-\v\cdot
\A)+\v(\A\cdot \B)} - \la \B\cnt \B\; . \ee
With appropriate boundary conditions on $\part V$, the average of the
divergence over the volume $V$ vanishes, and so
\be{n16}
\part_t H^M = -2\la H^C\; , \ee
showing that if $\la=0$, $H^M$ is conserved.  As shown by Moffatt (1978),
this expresses the fact that the linkage
between magnetic lines of force cannot change if they are frozen in the
fluid.  As we shall see, even though the total
magnetic helicity is conserved if $\lambda=0$, $\alpha$ causes it to flow from small
scales to large
scales.  

Equations (3.2) and (3.4) of PFL are
\be{n17}
\lrp{\part_t +2\la k^2} E^M_k = k\Gamma_k \lrp{E^V_k-E^M_k} + 2\alpha(k)
H^C_k - 2\beta(k) k^2E^M_k
\ee
and
\be{n18}
\lrp{\part_t +2\la k^2} H^M_k = {\Gamma_k\over k} \lrp{ H^V_k-H^C_k} +
2\alpha(k)E^M_k-2\beta(k)H^C_k\; . \ee
We have restored the Ohmic dissipation term $2\la k^2$ which appears in PFL
Table 1, but which is omitted in PFL
(3.2) and (3.4).  We also have omitted $\hat{\Gamma}_k$, which PFL state is
a first-order correction to $\Gamma_k$.

%These equations are to be used with caution, because they apply in different
%form depending on the value of $k$.  
The
numerical results of PFL show that if helical MHD turbulence is excited predominately at a single wave number $k_2$ (the outer scale of the
turbulence), a pulse of excitation at $k_1$, say, moves toward smaller
values (larger scales) in what is
sometimes called an inverse cascade.  B01 has shown
that this inverse cascade is non-local in the sense that the excitation
jumps from $k_2$ to $k_1$.

We are interested in applying \pref{n18} to the case $k=k_1$.  Because according to (5)
$\alpha (k_1)$
is based on helicity at $k\gg k_1$, and
$E^M(k_1)$ can be significant, the $\alpha$-effect pumps magnetic
helicity from $k_2$ to $k_1$.  As we will see later, if
$\la$ is small magnetic helicity conservation requires that an equal and
opposite amount of helicity must be established at
$k_2$.  Because of \pref{n13}, the last term of \pref{n18}, 
$(2\beta(k) H^C_k)$ has the
same qualitative effect as $2\la k^2H^M_k$.
The first term on the right is
proportional to
\be{n19}
\Gamma_k = \tfrac{4}{3}k \int^{ak}_0 \theta_{kkq}E^M_qdq\; , \ee
which depends on the magnetic energy at $k\ll k_1$. In the
numerical results of PFL  there is a
peak at $k_1$, with little energy at $k<k_1$.  Hence we assume that the 
$\Gamma_k$ term is negligible.
Note that $\alpha(k)=\half \alpha^R$ and $\beta(k)=\nu^V_k$ of PFL.

We may integrate 
of \pref{n17} and \pref{n18} over $k$, and approximate the results by
\be{n20}
\part_tE^M_1 = 2\alpha k^2_1H^M_1 - 2(\la+\beta) k^2_1 E^M_1 \ee
and
\be{n21}
\part_t H^M_1 = 2\alpha E^M_1-2(\la+\beta) k^2_1 H^M_1\; , \ee
where
\be{n22}
E^M_1 = \int_{q\sim k_1} E^M_q dq\ee
etc., in effect setting $\alpha(k_1)$ and $\beta(k_1)$ equal to their values 
derived from contrubutions at $k=k_2$.  (Note that $\alpha$ is dimensionally
a speed and $\beta$ a diffusivity. 
For magnetic and kinetic spectrum approximately Kolmogorov,
the dominant contribution to  both $\alpha$ and
$\beta$ comes from the forcing scale.)
This  two-scale approach, wherein the forcing scale is equal to
the scale at which the small scale field is peaked, is justified 
only when the forcing is sufficiently helical (Maron \& Blackman 2002).

It is reassuring that \pref{n20} and \pref{n21} are exactly the equations
one gets from two-scale theory applied to
the small scale $k_2$ and the large scale $k_1$ (Appendix A).  In what
follows we will often use $E^M_1=B^2_1/2$, where
$B_1$ is the field at scale $k^{-1}_1$.  Note that $B_1$ can be used
interchangeably with $\ob$.

In the same spirit, we can replace $\theta_{kqq}$ in \pref{n5} by a typical
value $\tau$ related to the peak at $k_2$, to
obtain
\be{n23}
\alpha = -\tfrac{2}{3}\tau \lrp{H^V_2-H^C_2}\; .\ee

\section{Dynamical quenching and comparison to numerical simulations}


Here we show that the solutions of the equations in the  
two-scale formalism of the previous section 
agree well with the numerical results of \cite{b2001}.
B01 studied the dynamo effect in a nearly
incompressible conducting fluid with periodic
boundary conditions.   $B_1$ is   allowed to grow at various wave numbers
$k$, consistent with the boundary conditions,
as a result of the $\alpha$ effect, thus  simulating  a non-linear 
$\alpha^2$ dynamo.
The results of B01 are qualitatively similar to the 
numerical results of PFL, in
that a pulse of excitation propagates to large
scales.  As B01 kept $H^V_2$ approximately constant by driving the MHD
turbulence with a helical force, in the light of \pref{n23},
  \pref{n20} and \pref{n21} become
\be{n37}
\part_t E^M_1 = 2 \lrp{\alpha_0 +\tfrac{2}{3}\tau k^2_2H^M_2} k^2_1H^M_1 -
2(\la+\beta) k^2_1 E^M_1 \ee
and
\be{n38}
\part_t H^M_1 = 2 \lrp{\alpha_0 +{\tfrac{2}{3}}\tau k^2_2
H^M_2}E^M_1-2(\la+\beta)k^2_1H^M_1\;,\ee
where
\be{38p}
\alpha_0 = -\tfrac23\tau H^V_2 = {\it const.} \ee
Remarkably, the nonlinear differential equations \pref{n37} and \pref{n38}
have a force-free solution for $B_1$ in which
\be{n39}
E^M_1=k_1H^M_1\; . \ee
Thus, \pref{n38} and \pref{38p} reduce to one
equation that $H_1$ must satisfy, namely 
\be{n40}
\part_t H^M_1 = 2k_1 \lrp{\alpha_0 + \tfrac{2}{3}\tau k^2_2
H^M_2}H^M_1-2(\la+\beta)k^2_1 H^M_1 \; .\ee
To solve \pref{n40} and \pref{40p}, we 
express $H^M_2$ in terms of $H^M_1$.  To
do this, we use the conservation of magnetic helicity, \pref{n16}, 
in the form
\be{40p}
\part_t H^M_1+\part_t H^M_2 = -2\la \lrp{k^2_1 H^M_1 +k^2_2 H^M_2}\; .
\ee
%\subsection{Discussion and Comparison to Previous Work}Case 3: Time-dependent dynamo action and dynamical quenching in a periodic box}
Equations 
\pref{n40} and \pref{40p} are the coupled equations 
in $H^M_1$ and $H_2^M$ which need to be
solved. 
If $H_2^M$ is small, $H_1^M$, and hence the large-scale field $B_1$
grows exponetially, driven by the first term on the right hand
side of \pref{n40}. The magnetic helicity conservation equation \pref{40p} 
shows that, for small $\lambda$, 
growth of $H_1^M$ is not free, but comes at the
expense of growing $H_2^M$ with the opposite sign.  
This decreases the value of $\alpha$ in \pref{n40}.
This ``$\alpha$-quenching'' slows the growth of 
$H_1^M$, leading to a steady state when the right-hand side
of $\pref{n40}$ vanishes.

To solve \pref{n40} and \pref{40p}, we rewrite them in dimensionless form.
We define the dimensionless magnetic helicities
 $h_1\equiv  2 H^M_1k_2/v_2^2$ and $h_2 \equiv 2 H^M_2 k_2/v_2^2$ and 
write time in units of $1/k_2v_2$. We also define  
$R_M\equiv (v_2/k_1)/\lambda$.
(Note that this definition of $R_M$ is based on the forcing-scale
RMS velocity but on the large scale, $k_1^{-1}$.  We will later employ
a second magnetic Reynolds number $R_{M,2}\equiv R_M (k_1/k_2)$.)
We also need a prescription for $\alpha_0$ and for $\beta$.
We assume that the kinetic helicity is forced maximally, and
we take $\tau=2/k_2v_2$ implying that $\alpha_0= 2v_2/3$. 
%We also take $\beta_0=v_2/k_2$.
Unfortunately, a rigorous prescription for $\beta$ in 3-D is lacking, but 
as in B01, we will consider two cases, 
$\beta=\beta_0\alpha/\alpha_0$
and $\beta=\beta_0\equiv v_2/k_2$.
%Ref. \cite{vainshtein} found both $\alpha$ and $\beta$ 
%to be suppressed similarly, but this was for waves.


Using the above scalings we can replace 
(\ref{n40}) and (\ref{40p})
with dimensionless equations given by
\beq
\partial_t h_1 ={4\over 3}\left({k_1\over k_2}\right)h_1(1+h_2)-2h_1
\left[{k_1\over k_2 R_M}+{k_1^2\over k_2^2}(1+q_2h_2)\right]
\label{40b}
\eeq
and
\beq
\nn
\partial_t h_2 =-{4\over 3}\left({k_1\over k_2}\right)h_1(1+h_2)+2h_1
{k_1^2\over k_2^2}(1+q_2 h_2)
-{2\over R_M}{h_2k_2\over k_1},
\label{41b}
\eeq
where $q_2=0$ in the above equations corresponds to $\beta(t)=\beta_0=$constant.
and $q_2=1$ corresponds to $\beta(t)=\alpha(t)\beta_0/\alpha_0$.  
Solutions of these coupled equations are shown
in Figs. 1-4. The key parameters 
are $k_2/k_1$, $R_M$, and $q_2$. 
In the figures, we have also compared
these results to the empirical fits of numerical simulations 
in \cite{b2001}. We have taken $h_1(t=0)=10^{-3}$,
but the sensitivity to $h_1(0)$ is only  
logarithmic (see \pref{tkin} below).
In Fig. 1, we have used $k_2/k_1=5$, following B01, 
and in Fig. 2 we have used $k_2/k_1=20$.


In the figures, the solid lines represent our numerical 
solutions to \pref{40b} and \pref{41b}, whereas 
the dotted lines represent the formula given 
in \cite{b2001}, which is an 
empirical fit to simulation 
data assuming that $\alpha$ and $\beta$ are prescribed according
to \pref{48} and \pref{49} below.
More explicitly, B01 found that the growth of $\OB$ was
well described by the formula 
\beq
{B_1^2/B_{1,0}^2\over (1-B_1^2/B^2_{1,sat})^{1+{\alpha_0k_1-k_1^2\beta_0\over 
\lambda k_1^2}}}=
e^{2(\alpha_0k_1-k_1^2\beta_0)t}, 
\label{axel1}
\eeq
where $B_{1,0}= B_1(t=0)$.
This can be rewritten using the notation above as 
a dimensionless equation for $t$ in units of $(k_2v_2)^{-1}$, namely 
\beq
t={k_2\over 2k_1}{ {\rm Ln}[(h_1/h_0)(1 - h_1 k_1^2/k_2^2)^{R_M(k_1/k_2-2/3)-1}]
\over {2 / 3} - {k_1/k_2}}.
\label{axel2}
\eeq
Note that \pref{axel1} and \pref{axel2} 
correspond to $\alpha$ and $\beta$ quenching of the form 
\beq
\alpha={\alpha_0\over  1+s_B B_1^2/v_2^2}
\label{48}
\eeq
and 
\beq
\beta={\beta_0\over  1+s_B B_1^2/v_2^2},
\label{49}
\eeq
where 
$s_B\sim R_M (k_1/k_2)(2/3-k_1/k_2)=R_{M,2} (2/3-k_1/k_2)$,
and $R_{M,2}\equiv v_2/k_2\lambda$.
%After making a table of $t(h_1)$ from \pref{axel2}
%we can plot the points together with the solutions
%of  \pref{40b} and \pref{41b}
%% for the cases Rm=100, 1000 and k2/k1=5.
Eqns. \pref{48} and \pref{49} are derived from those in B01 by
re-scaling Eq. (55) of B01 with our notation.  
It can also be shown directly that, 
up to terms of order $1/R_M$,  
 \pref{axel2} is consistent 
with that derived by substituting \pref{48} and \pref{49} 
into \pref{40b} and solving for $t$.
Note that in contrast to the suggestion of B01, it is 
actually the forcing-scale magnetic Reynolds number, $R_{M,2}$, that plays 
a prominent role in these 
formulae. 




The solutions of \pref{40b} and \pref{41b} 
are subtle and interesting. Some insight
can be gained by their sum
\beq
\nn
\partial_t h_1 + \partial_t h_2 
= -{2\over R_M}\left({h_1k_1\over k_2} +{h_2k_2\over k_1}\right),
\label{41bb}
\eeq
which corresponds to 
\pref{40p}, the conservation of total magnetic helicity. 
If we make the 
astrophysically relevant assumption that $R_M >> 1$, the right hand side of
\pref{40p} is small for all $h_1$ and $h_2$. It follows that 
$\partial_t (h_1 + h_2)=0$ 
and for $h(t=0)=0$,  this implies $h_2 = - h_1$. 
In this period, we can self-consistently ignore $1/R_M$ in
\pref{40b}. If $q_2 = 1$, this phase ends 
when $h_2= -1$, so that $h_1 = 1$. 
This is manifested in figure 3.
    
This kinematic phase precedes the asymptotic saturation of the dynamo 
investigated by other authors, in which all time derivatives
vanish exactly. For this to happen, the right hand side of \pref{41bb} must
vanish, which is equivalent to demanding that $h_2 = -(k_1/k_2)^2 h_1$. 
Since the right hand sides of \pref{40b} and \pref{41b} 
are proportional to $1 + h_2$ when terms
of order $1/R_M$ are neglected, their vanishing requires that $h_2 = -1$, 
and therefore, that $h_1 = (k_2/k_1)^2$. This is observed in figures 1 and 2.
The asymptotic saturation  (when the field growth ceases)
takes a time of order $t_{sat} \sim  R_M k_2/k_1$, 
which in astrophysics is often huge. Thus, although in principle it is correct
that $\alpha$ is resistively limited 
(as seen from our solutions in figures 4 \& 5)
as suggested by BY, GD, Vainshtein \& Cattaneo (1992) and 
Cattaneo \& Hughes (1994), this is 
less important than the fact that for a time $t_{kin} < R_M$ 
the kinematic value of $\alpha$ applies.
The time scale $t_{kin}$ here is given by
a few kinematic growth time scales for the $\alpha^2$ dynamo,
more specifically, 
\beq
t_{kin}\sim {\rm Ln}[1/h_1(0)] (k_2/k_1)/(4/3-2k_1/k_2).
\label{tkin}
\eeq
For $h_1(0)=0.001$, $k_2/k_1=5$, $t_{kin}\sim 37$, as seen in Fig 3.
 
Note that $t_{kin}$ is sensitive to $k_2/k_1$
and independent of $R_M$.  Figure 3 shows that there is significant
disagreement in this regime with \pref{48},
but this formula was used in B01 only to model the  regime $t>R_M$, 
so the result is not unexpected. 
We can see from the solution  for $\alpha$ itself
that indeed our solutions do match \pref{48} 
for  $t > R_M$ (figures 4 and 5).  
Figure 4 shows the difference in the $\alpha$ along with \pref{49}
for the two values $R_M=10^2$ and $R_M=10^3$. Notice again
the disagreement with the  formula \pref{48}
until $t=R_M$, and agreement afterward.  
This marks the time  at which the resistive term
on the right of \pref{40b} becomes competitive with the  
terms involving $(1+h_2)$. Asymptotic saturation 
does not occur until $t\sim t_{sat}=R_M k_2/k_1$ as described above.




%For $t>t_{kin}$, the current helicity
%term in $\alpha$, resulting from a growing $|h_2|$, begins to 
%significantly deplete the $\alpha$ effect.


Finally, note that $q_2$ corresponds to 
$\beta=\beta_0$.  
In general, this leads to a lower value of $h_1$ in the asymptotic 
saturation phase because 
this enforces zero saturation of $\beta$, whereas there is
still some saturation of $\alpha$ in this limit.
(Note that $q_2=0$ corresponds to the case of GD discussed further
in appendix B.)
For large $k_2/k_1$ the solutions of \pref{40b}
and \pref{41b} are insensitive to $q_2=0$ or $q_2=1$. This is because
the larger $k_2/k_1$, the smaller the influence of the
$q_2$ terms in \pref{40b} and \pref{41b}.  This is highlighted 
in figure 6 where the result for $q_2=0$ is plotted with the B01
fit. This suggests that for large-scale separation, the magnetic energy
saturation is  insensitive to the form of $\beta$ quenching. 
However, in real dynamos, magnetic flux and not just magnetic energy
may be needed, so the insensitivity can be misleading because
$\beta$ is needed to remove flux of the opposite sign.
From the low $k_2/k_1$ cases, it is clear that that $q_2=1$
is a better fit to the simulations of \cite{b2001}.






\vspace{.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{mhd01_blackman_f3.ps} 
\hfill } \noindent 
{Figure 1: Solution for $h_1(t)$, $f_h=1$,$q_2=1$. Here $k_2/k_1 =5$ and
the three curves from left to right have $R_M=100,250,500$
respectively. The dotted lines are plotted from the formula used to 
quasi-empirically fit the simulations in B01 as described
in the text.




\vspace{.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{mhd01_blackman_f4.ps} 
\hfill } \noindent 
{Figure 2: Solution for $h_1(t)$, $f_h=1$, $q_2=1$. Here $k_2/k_1 =20$ and
the three curves from left to right have $R_M=10^7,10^8,10^9$
respectively. The dotted lines are plotted from the formula for used to quasi-empirically fit simulations of B01.





\vspace{.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
%\epsffile{/home/blackman/inprogress/paris01/mhd01_blackman_f5.ps} 
\epsffile{H12345brane.ps} 
\hfill } \noindent 
{Figure 3: The early-time solution for $h_1(t)$, $f_h=1$, $q_2=1$. Here  
for $k_2/k_1=5$, and $R_M=10^2,10^3, 10^4$ from left to right respectively.
Notice the significant departure from the 
formula of B01 at these early times. For $t< t_{kin}$ there
is no dependence on $R_M$ and the growth proceeds kinematically.}



\vspace{.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{alpha235brane.ps} 
\hfill } \noindent 
{Figure 4: Solution of $\alpha/\alpha_0(t)$ 
for $h_1(t)$, $f_h=1$, $q_2=0$. Here $k_2/k_1 =5$ and
the solid lines are our solutions for $R_M=10^2$ (top curve)
and $R_M=10^3$ (bottom curve) respectively.
The top and bottom dotted curves are  from \pref{48}, interpreted
from Ref. \cite{b2001}}. Notice the longer kinematic
phase for our solutions, the overshoot, and the  convergence
of the solution for $R_M=10^2$ with that of \pref{48} at $t=R_M$.


\vspace{.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{alpha235bran.ps} 
\hfill } \noindent 
{Figure 5: This is the extension of  Figure 5 for later times. Notice 
the convergence of the $R_M=10^3$ solution to \pref{48} near $t=R_M$.

\bigskip
\bigskip

%\vspace{.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
%\epsffile{mhd01_blackman_f6.ps} 
%\hfill } \noindent 
%{
Figure 6 (separate file) 
: Solution for $h_1(t)$, $f_h=1$, $q_2=0$. Here $k_2/k_1 =50$ and
the three curves from left to right have $R_M=10^2,10^3,10^4$
respectively. The dotted lines are plotted from the formula 
used to quasi-empirically fit simulations of Ref. \cite{b2001}}. For such 
large $k_2/k_1$ the fit to the data is only weakly sensitive to 
whether $q_2=1$ or $q_2=0$. 





\bigskip
\bigskip






\section {Implications  and Comparison to Previous Work}


The physical picture of the quenching process just described
is this: helical turbulence is forced at $k_2$ ($=5$ in
B01), and kept approximately constant by forcing.  
Hence $\alpha_0 = -2\tau H^V_2/3 = {\it const}$.  
If $H_1^M$, the magnetic 
helicity at $k_1$ (which reaches $1$ here as a result of boundary
conditions), is initially small --- so that $|2k^2_1H^M_1/3|\ll |\alpha_0|$, \pref{n40} (or \pref{40b}) 
shows that it will be exponentially
amplified provided that the damping due to
$\beta +\la$ does not overcome the $\alpha$ effect.  
%It is important to
%note that turbulent diffusion can add to the
%effect of molecular diffusion by breaking up the field at $k_1$ and sending
%its helicity back to $k_2$.  
Initially, $\alpha=\alpha_0$, 
acting like a pump that  moves magnetic helicity from $k_2$ to $k_1$ 
and driving the dynamo.  This kinematic phase lasts
until $t_{kin}$ as given by \pref{tkin}. 
Eventually, the growing 
%From
%\pref{n44} and \pref{n53}, 
$H^M_1$
%, because of 
%current helicity conservation (see \pref{40p} or \pref{41b}), 
results in a  growing $H^M_2$ of opposite sign,
which reduces $\alpha$ through $H^C_2$.  
$R_M$-dependent quenching kicks in at $t=t_{kin}$, 
but it is not until $t=R_M$ that the asymptotic formulae 
\pref{48} and \pref{49} 
are appropriate.  Asymptotic saturation, defined by the time 
at which $B_1$ approaches 
its maximum possible value of $(k_2/k_1)^{1/2} v_2$,  
occurs at $t=t_{sat}=R_Mk_2/k_1$.
For $t\ge R_M$ our numerical solution, like the full numerical
simulations of B01,
is  well fit by the $\alpha$ in \pref{48} with a corresponding $\beta$
of \pref{49}. Our two-scale approach is also consistent with 
B01 in that magnetic helicity jumps from $k_2$ to $k_1$ without filling
in the intermediate wavenumbers.  


The emergence of the time scale $t_{kin}$ is 
interesting because it shows how 
one can misinterpret the implications of the
asymptotic quenching formula \pref{48} and \pref{49}.
These formulae are appropriate only for $t> R_M$. 
The large-scale field actually grows kinematically
up to a value $B_1=(k_1/k_2)^{1/2} v_2$  by $t=t_{kin}$
and ultimately up to $B_1\sim (k_2/k_1)^{1/2}v_2$ by $t=t_{sat}$.
For large $R_{M,2}$, 
these values of $B_1$ are  both 
much larger than the quantity $v_2/R_{M,2}^{1/2}$, which 
would have been inferred to be the saturation value if 
one assumed 
\pref{48} and \pref{49} were valid at all times.

%For example, if we cavalierly apply these results for the Galaxy
%(by ignoring the shear and assuming an $\alpha^2$ dynamo
%that produces force free large scale fields),
%and use $k_2/k_1=20$, $B_1(0)=10^{-9}$, 
%$v_2=10$km/s,  and $k_2^{-1} \sim 100pc$, 
%we would find the end of the kinematic
%regime to be at $t=140$, or about $1.4\ts 10^9$yr. 
%After this stage the field growth would proceed
%very slow because of the large $R_M$, but the saturation field strength at this
%time is  $B_1 \sim v_2/4.5$.  This is within a factor of
%3 of the present value(!)  Even though 
%ultimately the quenching is resistively limited, 
%the suggestion of FBC, that quenching should occur only after 
%$B_1\sim v_2$, is nearly true when $k_1/k_2$ is not too small.
%However the approach of FBC was completely different, 
%assuming a uniform and steady $B_1$ and a steady value of
%kinetic and current helicity.
%Magnetic helicity conservation was not used as a dynamical constraint.


Dynamical quenching or time-dependent approaches recognizing the current
helicity as a contributor to $\alpha$ 
have been discussed elsewhere (\cite{zeldovich,kleeorinetal,kleeorinruz,kr}
see also Ji 1999; Ji and Prager 2001), 
but here we have specifically linked the  PFL $\alpha$ correction to the 
helicity conservation in a simple two-scale approach. 
Other quenching studies for closed systems such as
%such as the work of Vainshtein \& Cattaneo (1992),  
Cattaneo \& Hughes (1994)  and \cite{by} advocated
values of $\alpha$ which are resistively limited
and of a form in agreement with \pref{48} 
but with the assumption of a steady  $B_1$.
Assuming \pref{n23}, and using \pref{n40} and \pref{40p} in the steady
state, their formulae can be easily derived. 
However, one must also have a prescription for $\beta$.  
If $\beta$ is proportional to $\alpha$, then formulae like
\pref{48} and \pref{49} emerge.  If  $\beta(t)=\beta_0$, 
as in GD, then a formula for $\alpha$
$without$ resistively limited quenching emerges (this 
requires a re-interpretation of their formulae--see appendix B).  
On the other hand, we have shown that for large $k_2/k_1$, the 
dynamo quenching is largely insensitive to $\beta$.

An important point to re-emphasize  is
that even when resistive quenching formulae are found 
from steady-state analyses, this does not 
necessarily  reflect the saturation value of $B_1$.
The fact that there exists a kinematic regime up until $t_{kin}$ means that by
the time formulae like \pref{48} and \pref{49} 
are valid, the field may have already grown substantially, as we have shown.
That being said, all of our analysis here is for the growth of 
magnetic energy to saturation for the simple $\alpha^2$ dynamo  in a box
as in B01.
In such  a system, 
the magnetic energy can saturate at super-equipartition values 
because it is force free. The extent to which these 
idealized studies apply to 
real astrophysical systems with boundaries,
shear, and stratification, or to dynamo cycle periods, 
still remains to be seen.


\section{Conclusion}

We have shown that the evolution equations of PFL, together with their 
formula for $\alpha$, leads to dynamical $\alpha$
quenching from the $\alpha$-induced flow of magnetic helicity
from small to large scales; the associated buildup of small-scale 
current helicity of the opposite sign eventually 
suppresses $\alpha$. This simple $\alpha^2$ dynamo process 
can be modeled using a two-scale formalism. We have identified
a time scale $t_{kin}$ up to which the dynamo
in a periodic box operates independently of $R_M$ and grows
to large values, of order  $\sim (k_1/k_2)^{1/2} v_2$.
At later times, the dynamo becomes slow.  
The dynamo coefficients become 
resistively limited, depending strongly on $R_M$.  
Our solutions agree with the numerical simulations of B01
for the regime of $t>R_M$, where B01
showed that \pref{48} fits the data.




{\bf Acknowledgments}

\noindent We thank the ITP at UC Santa Barbara 
and its participants for stimulating interactions during the
Spring 2000 workshop on astrophysical turbulence. In particular, we thank 
A. Brandenburg, S. Cowley, R. Kulsrud, J. Maron, and P. Diamond  
for discussions.  We also warmly thank the participants
of the memorable Virgin Gorda MHD turbulence Workshop of Dec 2001, namely
A. Brandenburg, R. Kulsrud, J. Maron, B. Mattheaus, and A. Pouquet 
for the intense and extended discussions which have directly influenced
the revised version of this paper.
EB acknowledges support from DOE grant DE-FG02-00ER54600. 






%The details of 
%this process depend upon the turbulent viscosity, which can drive magnetic
%helicity back to small scales.  We have
%shown that when applied to peaked spectra, the PFL equations reduce to the
%familiar two-scale approximation.

\def\cno#1{\eqno{({\rm C.#1})}}


\section*{Appendix A: The Equivalence of PFL
and Two-Scale Theory}
\appendix  Here we show that \pref{n20} and \pref{n21} also follow from
two-scale theory.
Multiplying (1)  by $\OB$ gives
$$
\begin{array}{rcl}\part_t \lrp{\half \ob^2} &=& \alpha\OB\cdot\nt \OB -
(\la+\beta)\OB\cdot\nab^2\OB \\ [6pt]
&=& \alpha \OB\cdot\nt \OB+(\la+\beta) \OB\cdot \nt \nt \OB \\ [6pt]
&\doteq&\alpha\OB\cdot\nt \OB - (\la+\beta) (\nt \OB)^2\; , \end{array}
\ano{1}
$$
where
$\doteq$ means equal to within a divergence; from \pref{n15}, it is the true
equality for the indicated boundary conditions
.  Now let $k^{-1}_1$ be the scale of $\ob$.  Then
\beq
\OB\cnt \OB &=& \OB_1\cnt\OB_1\, \nn
&=& 2H^C_1 = 2k^2_1H^M_1\; , \nonumber
\eeq
and 
$$\lrp{\nt\OB}^2  \;  = \; 2 k^2_1 E^M_1 \; . \hskip.6in\ano{2}
$$
Thus (A.1)  becomes
$$
\part_t E^M_1 = 2\alpha k^2_1H^M_1-2(\la+\bet)k^2_1E^M_1 \ano{3}
$$
in agreement with \pref{n20}.

From \pref{n15}
$$
\part_t \lrp{\half\OA \cdot\OB} \doteq - \OE\cdot\OB\; . \ano{4}
$$
Since from \pref{n1} and \pref{n14}
$$\hskip.6in \OE = -\alpha\OB+(\la+\beta)\nt \OB\; ,\ano{5}$$
$$ \part_t \lrp{\half\OA\cdot\OB}  \doteq  \alpha \ob^2 -
(\la+\beta)\OB\cnt\OB \ano{6}$$
  or 
$$
\hskip.4in \part_t H^M_1  \doteq  2\alpha E^M_1 - 2(\la+\beta) k^2_1 H^M_1\;
, \ano{7}
$$
in agreement with \pref{n21}.  We conclude that if the Alfv\'en effect is
omitted, and the field is concentrated at $k_1$ and $k_2$, equations (3.2)
and (3.4) of PFL are equivalent to the two-scale
approximation.














%(2) An approach employing perturbative field theoretic
%techniques is invoked by Verma (2001) to
%demonstrate the inverse cascade of magnetic
%helicity and accompanying energy transfer to larges scales (though
%under the assumption of zero mean field). Application to the initial
%stages of large scale field growth of the Galaxy was considered there.



\def\dno#1{\eqno{({\rm B.#1})}}
\section*{Appendix B\\ Reinterpretation of GD Quenching Formula}










%Reynolds number of the turbulence.  A distinctive
%feature of \pref{10p} is the appearance of $R_M$, which is often very large
%in astronomy.  



GD (1994) were the first to use the conservation of magnetic helicity to
obtain a formula for $\alpha$ in a closed system in a
steady state.  Their conclusion that $\alpha$ saturates when $B_1$ is of the
order of $R^{-1/2}_m$ $v_2$ stimulated the
present investigation, for if correct, it would imply that the $\alpha$
effect in the Galaxy would be useless in explaining any
fields larger than $10^{-16}$ Gauss, as $R_m\cong 10^{20}$ in the
interstellar medium.
Taking their formula for $\alpha$, 
we show that their result was misinterpreted.

GD did not assume that $\B_1$ is constant in space, so we can
use \pref{n21} with $k_1\ne 0$.  Because they
assumed a steady state, we put $\part_t=0$, so that
$$
\alpha E^M_1 = \la H^C_1 +\beta H^C_1\; , \dno{1}
$$
which agrees with eq.\ (9) in GD (1994) (except for a sign error in the
latter which is not propagated in the rest of their
paper.)

From \pref{n23} and \pref{38p} we have
$$
\alpha= \alpha_0 + \tfrac23 \tau H^C_2\; , \dno{2}
$$
so
$$
H^C_2 = \frac{3}{2\tau} (\alpha-\alpha_0)\; . \dno{3}
$$
If  $t>t_{sat}$
$$
H^C_1 = -H^C_2 = -{3\over 2\tau} (\alpha-\alpha_0) \; . \dno{4}
$$
If we substitute (B.4) into the first term on the right-hand side of (B.1),
we get
$$
\alpha = {\alpha_0 +\beta_0 \R\cnt \R\over 1+R^2}\; , \dno{5}
$$
where
$$
\R = \left({\tau\over 3\la}\right)^{1/2} \B_1 = {{\sqrt 2}R^{1/2}_{M,2}\over v_2}\B_1\; , \dno{6}
$$
and we have used $\tau = 2/k_2 v_2$ and $\la = \beta_0 /R_{M,2}=v_2 /3k_2 
{R_{M,2}}$.
Following GD (1994) we have put $\beta=\beta_0$, where $\beta_0$ is a constant.

(B.5) is the same as equation (4) of GD (1994), so their work is consistent
with this paper for $t>t_{sat}$.  However, GD
(1994) went on to conclude that $\alpha $ saturates 
when $B_1\sim R^{-1/2}_{M,2}
v_2$, apparently assuming from (B.5) that
the criterion for saturation is $R\sim 1$.

However, one must be careful about the second term in (B.5).  Recall that it
is proportional to $H^C_1$, which is constrained
by (B.4).  When one substitutes $H^C_1$ from (B.4) into the second term in
(B.1), one finds that
$$
\alpha = {\alpha_0\over 1+{R^2\over 1+R_{M,2}}} = {\alpha_0 \over 1+{R_{M,2}\over
1+R_{M,2}} \lrp{B_1\over v_2}^2}\; , \dno{7}
$$
so that $\alpha$ saturates not at $B_1\sim R^{-1/2}_Mv_2$, but at
$$
B_{1,\widetilde{\rm sat}} \lrp{1+R_{M,2}\over R_{M,2}}^{1/2} v_2\; , \dno{8}
$$
or, as $R_{M,2}$ gets large
$$
B_{1,\widetilde{\rm sat}} v_2\; , \dno{9}
$$
rather than being resistively limited as  GD suggest.


However, if instead of $\beta=\beta_0$ 
we employ $\beta=\alpha\beta_0/\alpha_0$ (or $q_2=1$ in \pref{40b} and
\pref{41b}), 
then it can be shown analytically 
that the resistively limited 
asymptotic forms $\pref{48}$ and $\pref{49}$ are correct.




%We attribute the difference between
%this result and Appendix C to the fact that with
%$k_1\ne 0$, turbulent diffusion is effective in moving $H^m$ from $k_1$ back
%to $k_2$, thus allowing the $\alpha$ effect to
%continue.  

%It is interesting to compare (B.8) with the result of PFL in \pref{54p},
%taking account of \pref{n59}, which gives the saturated
%value of $B_1$ if it is allowed to grow in an $\alpha^2$ dynamo.  The two
%formulas  agree when $B_1$ is large, with
%$$
%\alpha_{\rm sat} \sim {k_1\over k_2}v_2 {1+R_M\over R_{M,2}}\; .\dno{10}
%$$





%\noindent  (NOTE: Having just said all the above, something occurred
%to me:  the time derivatives of isotropic quantities can depend on $\OB$
% in the following sense: The time evolution of  $\lb\b^{(0)}.\b^{(0)}\rb$
%certainly does depend on $\OB$ {\bf if} $\b(t=0)=0$.)






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\end{thebibliography}

\bye


 








\begin{figure}[]
\begin{center}
\includegraphics[width=.8\textwidth]{
/home/blackman/inprogress/paris01/H178920bran.ps}
\end{center}
\caption[]
{Figure 2:Solution for $h_1(t)$, $f_h=1$,$q_2=1$. Here $k_2/k_1 =5$ and
the three curves from left to right have $R_M=100,250,500$
respectively. The dots are plotted from the formula used to 
quasi-empirically fit the simulations in Ref. \cite{b2001}}
\label{/home/blackman/inprogress/paris01/H11002505005bran.ps}
\end{figure}




\begin{figure}[]
\begin{center}
\includegraphics[width=.8\textwidth]{H178920bran.ps}
\end{center}
\caption[]
{Solution for $h_1(t)$, $f_h=1$, $q_2=1$. Here $k_2/k_1 =20$ and
the three curves from left to right have $R_M=10^7,10^8,10^9$
respectively. The dots are plotted from the formula for used to quasi-empirically fit simulations of Ref. \cite{b2001}}
\label{/home/blackman/inprogress/paris01/H178920bran.ps}
\end{figure}








\begin{figure}[]
\begin{center}
\includegraphics[width=.8\textwidth]
{/home/blackman/inprogress/paris01/
h1R239k5e.ps}
\end{center}
\caption[]{
The early time solution for $h_1(t)$, $f_h=1$, $q_2=1$. Here  
for $k_2/k_1=5$, and $R_M=10^2,10^3, 10^9$ from left to right respectively.}
\label{h1R239k5e.ps}
\end{figure}

\begin{figure}[]
\begin{center}
\includegraphics[width=.8\textwidth]{/home/blackman/inprogress/paris01/
H123450branq20.ps}
\end{center}
\caption[]
Solution for $h_1(t)$, $f_h=1$, $q_2=0$. Here $k_2/k_1 =50$ and
the three curves from left to right have $R_M=10^2,10^3,10^4$
respectively. The dots are plotted from the formula for used to quasi-empirically fit simulations of Ref. \cite{b2001}}. For such large $k_2/k_1$ the
fit to the data is not strongly sensitive to whether $q_2=1$ 
or $q_2=0$. 
\label{H123450branq20.ps}
\end{figure}


For the question of the actual field amplification and
time scales associated with cycle periods, 
a time-dependent, non-linear dynamical theory is required.
(Indeed, note also that in laboratory plasmas, time
dependent effects are also important when it comes to helicity 
magnetic injection in Tokomaks \cite{ji}. 
%to helicity conservation. 
%%helicity can be changed 
%injected  through the surface terms [26]. 
Here the time derivative term of $(\ref{03a2})$ important.

An important step forward was the work of Ref. \cite{pfl},  
based on a Eddy Damped Quasi-Normal Markovian spectral 
closure scheme. They produced an apporoximate set
of equations describing the evolution of magnetic and kinetic energy
and helicity. Refs. \cite{b2001,maronblackman} performed 
numerical simulatiions of the process, and Ref. \cite{fb}
further simplified the equations of \cite{pfl}
 by considering a two scale approach for fully helically
forced turbulence,  Ref. \cite{fb} 
assumes that the large-scale field grows primarily 
on scale $k_1$ and the small scale turbulent field is
primiarly at $k_2 >> k_1$. This is consistent with
the simulations.

The basic concept of the semi-analytic model is that the growth of
the large scale field is the result of a segregation of magnetic helicity.
Magnetic helicity of one sign grows on the large scale, while the opposite
sign grows on the small scale, up to resistively limited conservation.
Moreover, the growth of the small scale magnetic helicity also
grows current helicity which suppresses $\alpha$ as in \cite{pfl}.
There are two essential equations that we need to work with.
The equation for total magnetic helicity growth (\ref{03a2})  
and
the equation for large scale magnetic helicity evolution (\ref{15}).

We ignore the boundary terms in equation (\ref{15}). 
Using subscript 1 and 2 to indicate mean and fluctuating scales,
and $M$ to indicate ``magnetic'' 
we then have $H^M_1 \equiv H^0(\OB)$.
Thus (\ref{15}) implies
\beq
\partial_t H^M_1  =  2\alpha E^M_1 - 2(\la+\beta) {\overline {\bf J}}\cdot \OB
, 
\eeq
where where $E_1^M = \OB^2$.
Now we replace spatial derivatives of the mean scale  with $k_1$,
and then note that the current helicty term on the right is related
to the magnetic helcity 
${\overline {\bf J}}\cdot \OB=k_1^2H^M_1$.
We then have  
%\doteq
\beq
\partial_t H^M_1  =  2\alpha E^M_1 - 2(\la+\beta) k^2_1 H^M_1\;
, 
\label{40}
\eeq
Ref. \cite{fb} describes the conditions for which 
this is consistent with the analagous equation that arises
from the spectral treatment of \cite{pfl}.

The second equation we need is a rewrite of (\ref{03a2}).
Since the left-most side is the sum of contribution
from large and small scales, as is the right most side, we have
then in the two scale approximation
\beq
\partial_t H^M_1  + \partial_t H^M_2  
=  - 2\la k^2_1 H^M_1- 2\la k^2_2 H^M_2.
\label{41}
\eeq

To solve these equations,  
we need a prescription for $\alpha$ and a prescription for
$\beta$. A configuration space approach and its pitfalls
are described in the next section, but for now,
let us extract the dynamical form derived from \cite{pfl}.
For $\alpha$ they find essentially
\beq
\alpha=-(2/3)\tau (\lb\bfv\cdot\curl \bfv \rb - \lb\bfb\cdot\curl\bfb\rb),
\label{PFLalpha}
\eeq
where $\tau$ is the correlation time of the turbulence
at $k_2$.








In
Appendix B we study this equation and find that its solution depends on
the age of the system $t$.  If $t>(\la k^2_1)^{-1}$ as is the case for
Brandenburg, the appropriate solution is
\be{n41}
k^2_2 H^M_2 = -k^2_1 H^M_1\; , \ee
so that the total current helicity remains constant.  Brandenburg
(2001a)
proposed that this is the case, and observed
it to be so in his simulations.  Substituting \pref{n41} into \pref{n40}
yields a nonlinear equation in $H^M_1$ alone:
\be{n42}
\part_t H^M_1 =2k_1 \lrp{\alpha_0 -\tfrac{2}{3}\tau k^2_1 H^M_1}H^M_1 -
2(\la+\beta) k^2_1 H^M_1\; . \ee
 Brandenburg's simulations suggest that $\beta$ is proportional to
$\alpha$, so 
\be{n43}
\beta=\beta_0 {\alpha\over \alpha_0}\;.
\ee
We shall adopt \pref{n43} in what follows, and comment later on what happens
if $\beta={\it const.}$, as advocated by GD (1995).  If we let
\be{n44}
x={2\tau k^2_1H^M_1\over 3\alpha_0}=-{2\tau k^2_2 H^M_2\over 3\alpha_0} \ee
and
\be{n45}
n_0=2k_1 \lrp{\alpha_0 -\beta_0k_1}\; ,\ee
we can write \pref{n42} in the form
\be{n46}
{dx\over bx-x^2}=n_0dt\; ,\ee
where
\be{n47}
b=1-{2\la k^2_1\over n_0}\;.\ee
The solution to \pref{n46} is
\be{n48}
{x\over b} = \lrc{ 1+ \lrp{ {b\over x_0}-1} \exp-2k_1 \lrb{ \alpha_0 -k_1
(\beta_0+\la)}t}^{-1}\;,\ee
where
\be{n49}
x_0=x(0)\;.\ee
If $x_0\ll 1$, the second term in the braces dominates, and
\be{n50}
x=x_0\, {\rm exp} 2k_1 \lrb{\alpha_0-k_1(\beta_0+\la)} t\;.\ee
This is the usual unstable mode of the linear $\alpha^2$ dynamo (Moffatt
1978), which grows if
\be{n51}
\alpha_0>k_1(\beta_0+\la)\;.\ee
However, whatever the value of $x_0$, as $t\to\infty$,
\be{n52}
x\to x(\infty)=b\;.\ee

Now from \pref{38p} and \pref{n44}
\be{n53}
\alpha(t)=\alpha_0 \lrb{1-x(t)}\; . \ee
In particular, 
\beq
\alpha(\infty) &= &\alpha_0 (1-b) = {2\la k^2_1\alpha_0\over n_0} \nn
&=& k_1\la {\alpha_0\over \alpha_0-k_1\beta_0}\; . \label{n54} \eeq
If $\alpha_0\gg k_1\beta_0$, as is required for a robust instability,
$\alpha(\infty) = k_1\la$ is resistively
limited.  If instead we assume with Gruzinov
and Diamond (1995) that $\beta=\beta_0$
is constant, independent of $B_1$, we find that
\be{54p}
\alpha(\infty) = k_1 (\beta_0+\la)\;,
\ee 
of which the magnitude greatly
exceeds $\alpha(\infty)$ in the previous case if ${ R}_M = \beta_0/\la\gg
1$.  Thus, the theoretical prediction of
$\alpha(\infty)$ depends heavily on the value of $\beta$.  Brandenburg found
that $\alpha(\infty)$ is strongly
quenched, inconsistent with the result we predict if $\beta=\beta_0$.










In an externally forced system, the kinetic helicity correlations 
are enforced at a fixed value. At early times,
the second term on the right is small.  Thus we 
explore the idea that the second term on the right represents
the main effect of the backreaction on the first term.
The idea that the second term represents a backreaction was
investigated by \cite{gd1,by} for a stationary system 
and a correction to $\alpha$ was derived. 
Here we pursue a fully dynamical correction.
(The stationary limit of the dynamical treatment of $\alpha$ 
does agree with \cite{gd1,by} but see section 2.4).
%kleeorin



To understand the quenching in this formalism, 
we need to solve the equations 



In this case we can write
\beq
\alpha=\alpha_0+ 2\tau k_2 H^M_2 /3.
\eeq
Unfortunately, a prescription for $\beta$ in 3-D is lacking, but 
we will consider two cases, 
$\beta=\beta_0\alpha/\alpha_0$ 
and $\beta=\beta_0$ and show that at large $k_2/k_1$
the difference is not essential.
Ref. \cite{vainshtein} found both $\alpha$ and $\beta$ 
to be suppressed similarly, but this was for waves.
The subject of dynamical quenching of $\beta$
and $\alpha$ can be investigated separately and further work
on $\beta$ quenching is needed.
If the helicity is forced maximally, then $\alpha_0\sim 2v_2/3$,
where $v_2$ is the speed associated with the rms kinetic
spectrum ( dominated by the $k_2$ forcing scale).
We also take $\beta_0=v_2/k_2$.
Using these prescriptions, setting 
$\tau=1/k_2 v_2$, writing time in units of
$\tau$, magnetic helicities in units of $v_2^2/k_2$,
and using $R_M=(v_2/k_1)/\lambda$
we can replace 
(\ref{40}) and (\ref{41})
with dimensionless equations, namely 
\beq
\partial_t h_1 ={4\over 3}\left({k_1\over k_2}\right)h_1(1+h_2)-2h_1
\left[{k_1\over k_2 R_M}+{k_1^2\over k_2^2}(1+q_2h_2)\right]
\label{40b}
\eeq
and
\beq
\nn
\partial_t h_2 =-{4\over 3}\left({k_1\over k_2}\right)h_1(1+h_2)+2h_1
\left[{k_1\over k_2 R_M}+{k_1^2\over k_2^2}(1+q_2 h_2)\right]\\
\hskip0.4in
-{2\over R_M}\left({h_1k_1\over k_2} +{h_2k_2\over k_1}\right).
\label{41b}
\eeq
where $q_2=0$ in the above equations corresponds to $\beta(t)=\beta_0$
and $q_2=1$ corresponds to $\beta(t)=\alpha(t)\beta_0/\alpha_0$

Examples of the numerical solution of these equations are shown
in Figs. \pref{H11002505005bran.ps}, \pref{H178920bran.ps}
and \pref{H123450branq20.ps}. The key parameters which we
investigate are $k_2/k_1$, $R_M$, and $q_2$. We also compare
these results to the empirical fits of numerical simulations 
in \cite{b2001}.


In Fig. \pref{H11002505005bran.ps}, we have used $k_2/k_1=5$
and in Fig. \pref{H178920bran.ps} we have used $k_2/k_1=20$.
In both figures, the dots represent the formula given in \cite{b2001}
which is an empirical fit to the simulation 
data of the non-linear helical dynamo. 
Ref. \cite{b2001} found that the growth of $\OB$ was
well described by the formula 
\beq
{B_1^2/B_{1,0}^2\over (1-B_1^2/B^2_{1,sat})^{1+{\alpha_0k_1-k_1^2\beta_0\over 
\lambda k_1^2}}}=
e^{2(\alpha_0k_1-k_1^2\beta_0)t} 
\label{axel1}
\eeq
which  can be rewritten
dimensionally in the notation above as dimensionless equation
for $t$ in units of $k_2v_2$
\beq
t={k_2\over 2k_1}{ {\rm Ln}[(h_1/h_0)(1 - h_1 k_1^2/k_2^2)^{R_M(k_1/k_2-2/3)-1}]
\over {2 / 3} - {k_1/k_2}}.
\label{axel2}
\eeq
Note that these formulae correspond to $\alpha$ and $\beta$ quenching
of the form 
\beq
\alpha={\alpha_0\over  1+s_B B_1^2/v_2^2}
\eeq
and 
\beq
\beta={\beta_0\over  1+s_B B_1^2/v_2^2}
\eeq
where 
$s_B\sim R_M (k_1/k_2)(2/3-k_1/k_2)=R_{M,2} (2/3-k_1/k_2)$,
and $R_{M,2}\equiv v_2/k_2\lambda$.

After making a table of $t(h_1)$ from \pref{axel2}
we then  plot these points together with the solutions
of  \pref{40b} and \pref{41}
% for the cases Rm=100, 1000 and k2/k1=5.
The dots represent the results of this procedure and thus
represent \pref{axel2}.
The thin lines are our analytic solutions.


\vspace{.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{/home/blackman/inprogress/paris01/H11002505005bran.ps} 
\hfill } \noindent 
{Figure 1: Solution for $h_1(t)$, $f_h=1$, $q_2=1$. Here $k_2/k_1 =5$ and
the three curves from left to right have $R_M=100,250,500$
respectively. The dots are plotted from the formula used to 
quasi-empirically fit the simulations in Ref. \cite{b2001}}


\vspace{.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{/home/blackman/inprogress/paris01/H178920bran.ps}
\hfill } 
{ Figure 2:
Solution for $h_1(t)$, $f_h=1$, $q_2=1$. Here $k_2/k_1 =20$ and
the three curves from left to right have $R_M=10^7,10^8,10^9$
respectively. The dots are plotted from the formula for used to quasi-empirically fit simulations of Ref. \cite{b2001}}





\vspace{-.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{/home/blackman/inprogress/paris01/h1R239k5e.ps}
\hfill } \noindent {\it Figure 3: The early time solution for $h_1(t)$, $f_h=1$, $q_2=1$. Here  
for $k_2/k_1=5$, and $R_M=10^2,10^3, 10^9$ from left to right respectively.}


%Third, note that at very early times the dynamo proceeds kinematically,
%and independent of $R_M$. This regime occurs 
%because at early times, $h_2 << 1$, and 
%therefore the $\alpha$ effect is dominated by the kinetic helicity
%term. In this regime, the resisitve terms are not important
%and $h_2 \sim h_1$ (assuming infinitessimal small initial values) 
%as can be seen from ignoring the resistive terms  in \pref{40b}.
%Thus, this stage ends as $h_1$ approaches 1. 




Finally, note that if we set $q_2=0$, this corresponds to 
$\beta=\beta_0$.  In general, this makes the dynamo saturate
at values of 
For large $k_2/k_1$ the solutions of \pref{40b}
and \pref{41b} are insensitive to $q_2=0$ or $q_2=1$. This is because
the large $k_2/k_1$, the smaller the influence of the
$q_2$ terms in the equations.  This is shown
in figure 6 where the result for $q_2=0$ is plotted.

This suggests that for large-scale separation, the magnetic energy
saturation is not sensitive to the form of $\beta$ quenching. 
However, in real dynamos, magnetic flux and not just magnetic energy
is needed, so this can be misleading.
It is clear however, from the low $k_2/k_1$ cases, that $q_2=1$
($\beta$ quenching the same as $\alpha$ quenching)
is a better fit to the simulations of Brandeburg 2001a.






\vspace{-.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{/home/blackman/inprogress/paris01/H123450branq20.ps}
\hfill } \noindent {\it Figure 4: Solution for $h_1(t)$, $f_h=1$, $q_2=0$. Here $k_2/k_1 =50$ and
the three curves from left to right have $R_M=10^2,10^3,10^4$
respectively. The dots are plotted from the formula for used to quasi-empirically fit simulations of B01. 
{\it For such large $k_2/k_1$ the
fit to the data is not strongly sensitive to whether $q_2=1$ 
or $q_2=0$.}}

\bigskip



This semi-analytic model agrees very well with
the simulations of \cite{b2001}.
The suppression of the growth of the large 
scale field is the result of the growth of the small
scale current helicity, which in turn is associated with
the buildup of the small scale magnetic helicity.
In the earliest phase of growth, magnetic helicity is exactly
conserved and for any values of $R_M >>1$, the large scale helicity
and magnetic energy grow independent of $R_M$ at first. This
happens until $h_1\sim 1$ which corresponds to $2H_1^M\sim v_2^2/k_2$.
Since $H_1^M=H_2^M$ at this stage, $H_2^C$, the current helicity associated
with $H_2^M$ approaches $H_2^V$, the kinetic helicity, so that the
difference in (\ref{PFLalpha}) begins to drop.










Remarkably now we are at the stage where for a periodic box,
the above results from Refs. \cite{pfl,fb}, and the simulations 
by \cite{b2001} all show convincingly that mean field dynamo action
in a periodic box is slow.  
This ultimately means that those who have suggested that
meanfield dynamo action is slow are correct as applied to a periodic
box.  The work in Refs. \cite{ch,gd1,by} all advocated
values of $\alpha$ which are resistively limited,
and of the form $\alpha=\alpha_0/(1 + R_M b^2/{\bar B}^2)$.
The value of 
However, these values were all derived 





%Ref. \cite{b2001} showed numerically that in a periodic box, forcing with kinetic helicity.
%does indeed lead to growth of magnetic energy on the $k=1$ scale.
%The growth proceeds in two phases. 
%In the first phase, the field on the large scale and atfig. the forcing
%scale both grow at the same rate.  In this phase,
%the, resistivity 




******
In  externally forced simulations, the 
first term on the right of \pref{PFLalpha}
(the kinetic helicity)
is typically maintained at a fixed value. 
At early times, the second term on the right is small.  Thus 
the second term on the right (current helicity)
can be thought of as a  backreaction on the first term.
It arises from inclusion of the Navier-Stokes equation.
The idea that the second term represents a backreaction was
investigated by \cite{gd1,by} for a stationary system 
and a correction to $\alpha$ was derived. However, 
here we derive the fully dynamical correction \cite{fb}.
%kleeorin

We re-write \pref{PFLalpha} as
\beq
\alpha=\alpha_0+ 2\tau k_2 H^M_2 /3.
\eeq




We can estimate the value of $B_1$ at saturation as follows. From \pref{n42}
and \pref{n44}
\be{n55}
\left| H^M_{1,{\rm sat}}\right| \sim {|H^V_2|\over k^2_1} \;.\ee
Because $\B_1$ is force free
\be{n56}
E^M_{1,{\rm sat}}={|H^V_2|\over k_1}\; . \ee
If $\v_2$ is maximally helical
\be{n57}
|H^V_2| \sim k_2 E^V_2 \ee
so
\be{n58}
E^M_{1,{\rm sat}} \sim {k_2\over k_1} E^V_2 \ee
whence
\be{n59}
B_{1,{\rm sat}}\sim \lrp{ {k_2\over k_1}}^{1/2} v_2 \; ,\ee
a result found by Brandenburg (2001a).
Recall that FBC implies that
\be{n60}
B_{1,{\rm sat}}\sim v_2\;.\ee
The ratio of \pref{n59} to \pref{n60} is $(k_2/k_1)^{1/2}=2.3$ in the
simulation of Brandenburg (2001a).  Thus, the predictions for
$B_{1,{\rm sat}}$ do not differ very much, although the underlying physics
is quite different (Appendix C).



\section*{Appendix B\\
Conservation of $H^M$ vs $H^C$}

When $\la$ is small, $H^M$ is conserved, so in \pref{n37} we can write
$H^M_2
= -H^M_1$.  In general, however, this is not
true, and one must use \pref{40p}:
$$
\part_t \lrp{H^M_1+H^M_2}=-2\la \lrp{k^2_1H^M_1+k^2_2H^M_2}\; . \bno{1}
$$
Here we show that this has a general solution, and show, as Brandenburg
(2001a) conjectured, that if $\la$ is large,
$k^2_2 H^M_2 =-k^2_1H^M_1$.

If we let
$$
\hat{H}^M_{1,2}=e^{2\la k^2_1 t}H^M_{1,2} \bno{2}
$$
(B1) becomes
$$
d \hat{H}^M_1 = e^{-\la (k^2_2 -k^2_1)t} d\lrp{e^{2\la
(k^2_2-k^2_1)t}\hat{H}^M_2}\; . \bno{3}
$$
If we integrate by parts, the solution to (B3) is
$$
H^M_1+H^M_2 = \lrb{H^M_1(0)+ H^M_2(0)} e^{-2\la k^2_1 t} -2\la
\lrp{k^2_2-k^2_1} \int^t_0 e^{- 2\la k^2_1(t-t')}H^M_2(t')
dt'\; . \bno{4}
$$
There are two limiting cases: $\la k^2_2 t\ll 1$ (so $\la k^2_1 t \ll$ 1
also) 
and $\la k^2_1 t\gg 1$.  In
the first case the integral is $\sim
\overline{H}^M_2 t$, where $\overline{H}^M_2$ is a representative value of
$H^M_2$.  Then the ratio of the second term
on the right to the second term on the left is
$$
\sim {2\la (k^2_2-k^2_1)\overline{H}^M_2 t\over H^M_2} \sim \la k^2_2 t\ll 1
\bno{5}
$$
since $k_2>k_1$.  Hence the second term on the right can be neglected.  In
the special case that $H^M(0)=
H^M_1(0)+H^M_2(0)=0$, (B4) then implies that
$$
H^M_2=-H^M_1\; . \bno{6}
$$

In the second case, $\la k^2_1 t\gg 1$.  Then the contribution  to the
integral comes only from $\Delta t'\sim 1/2\la
k^2_1$, so the second term on the right is
$$
\sim - \lrp{ {k^2_2\over k^2_1}-1} H^M_2 (t)\; . \bno{7}
$$
Hence in the special case that $H^M(0)=0$,
$$
H^M_1 = -H^M_2 - \lrp{ {k^2_2\over k^2_1}-1} H^M_2 \bno{8}
$$
as conjectured by Brandenburg (2001a), so that
$$
 H^C_1 = k^2_1 H^M_1 = -k^2_2 H^M_2 = -H^C_2 \; . \bno{9}
$$

\def\cno#1{\eqno{({\rm C.#1})}}


\section*{Appendix E: 
Deriving PFL's $\alpha$ Correction in Configuration Space: 
GD vs FBC and the Ordering Ambiguity}

Our above two-scale framework for $\alpha$-quenching is 
appealing because it nicely couples the equations and concepts 
of magnetic helicity evolution  
to the current helicity contribution in $\alpha$.  The current
helicity contribution was interpreted as a correction to
the kinetic helicity contribution of kinematic theory.
However, the approach above,  and that of GD and BY, relies 
crucially on the fact that the current 
helicity contribution to $\alpha$ as presented in PFL
and thus in \pref{n23}, should really be the
total current helicity associated with $k_2$.  To
see what we mean by {\it total} and to show the complication, we consider 
the derivation of the turbulent EMF in configuration space.

The turbulent EMF can be written in three different ways:
\beq
\begin{array}{rcl}
\lb\v\ts\b\rb = \lb\v(0)\ts\b(0)\rb + \int_0^t \lb\partial_{t'} \v(t')\ts\b(t')\rb dt' +
\int_0^t \lb\v(t')\ts\partial_{t'} \b(t')\rb dt'\\ 
\\
= \lb\v(t)\ts\b(0)\rb + \int_0^t \lb \v(t )\ts \partial_{t'}\b(t')\rb dt'\\ 
\\
= \lb\v(0)\ts\b(t)\rb + \int_0^t \lb \partial_{t'}\v(t')\ts\b(t)\rb dt' 
\end{array}
\label{c1}
\eeq
The three lines in \pref{c1} simply correspond to the 3 relevant 
ways of using the formula $f(t)= f(0) + \int^t_0 \partial_{t'}f(t')dt'$, where $g$ is an arbitrary
function of time. If we assume that 
$t>>0$, and that widely
separated turbulent quantities do not correlate, the first terms on the
right of the 2nd and 3rd lines respectively, can be dropped.
We then have
\beq
\begin{array}{rcl}
\lb\v\ts\b\rb 
=\int_0^t \lb \v(t )\ts \partial_{t'}\b(t')\rb dt' 
=\int_0^t \lb \partial_{t'}\v(t')\ts\b(t)\rb dt'. 
\end{array}
\label{c2}
\eeq
The dynamo theorist will recognize the second term in \pref{c2}
as the standard textbook starting point (e.g. Moffatt 1978;
Krause \& R\"adler 1980)
for evaluating the turbulent EMF for a kinematic dynamo, but here we
have not made any assumptions about the backreaction yet.

To illustrate the point, consider the  simple case in which $\nabla\OB=0$.
The equation for the small scale field is then
\begin{equation}
\begin{array}{r}\partial_{t'} \b = \OB\cdot\nabla\v + 
\b\cdot\nabla\v -\v\cdot\nabla\b -\curl\lb\v\ts\b\rb +
\lambda\nabla^{2}\b.
\end{array}
\label{c3}
\end{equation}
The penultimate term goes away when included in \pref{c2}
and we ignore the last term.  The first terms on the right,
upon the assumption that the dominant contributions to correlations
are isotropic in $\v$,  gives the 
``textbook'' expression for $\alpha$ plus extra terms, that is  
\beq
\lb\v\ts\b\rb \simeq -{\OB\over 3}\int_0^t \lb\v(t)\cdot\curl\v(t')\rb dt'
+ Q (v^2 b).
\label{c4}
\eeq
The terms  symbolized by $Q(v^2 b)$ 
are typically ignored
using some version of the first order smoothing approximation.
This is of questionable validity, given that the small
scale field rapidly grows to exceed the mean field. 
We will come back to the relevance of these terms below.

Now if instead we use the last term of \pref{c2}
to expand the EMF, we must then invoke the Navier-Stokes equation for the
time derivative of the turbulent velocity 
\begin{equation}
\partial_{t} {\v}=-\v\cdot\nabla\v-\lb\v\cdot\nabla\v\rb
-\nabla p_{\em{eff}}
 +{\OB}\cdot\nabla{\bf b} + {\bf b}\cdot\nabla{\bf b}
-\lb {\bf b}\cdot\nabla{\bf b}\rb
+ \nu\nabla^{2}{\bf v}
+{\bf f}({\bf x}, t),
\label{c5}
\end{equation}
where $\bf f$ is a forcing function and $p_{eff}$ is the magnetic and
thermal pressure. Upon plugging this into \pref{c2},
the second and sixth terms  on the right vanish.
If we ignore the viscosity, and assume the dominant contribution
to correlations are isotropic, we then have 
\beq
\lb\v\ts\b\rb \simeq {\OB\over 3}\int_0^t \lb\b(t)\cdot\curl\b(t')\rb dt'
+ {\tilde Q} (v^2 b,fb)
\label{c6}
\eeq

Notice two things about \pref{c4} and \pref{c6}:
First they are equal to each other since they were derived from different
choices of the expansion of $\lb\v\ts\b\rb$.
Second, they do not cleanly include
the combination of the total residual helicity, required in
\pref{n23} when placed into \pref{n18} to derive \pref{n38},
which are used in the explicit
derivations of $\alpha$ quenching in section 2.  
The only way that  \pref{c4} and \pref{c6} can 
have the form of the desired relative helicity is if
$Q(v^2b)={\OB\over 3}\int_0^t \lb\b(t)\cdot\curl\b(t')\rb dt'$
and if 
${\tilde Q}(v^2b,fb)=-{\OB\over 3}\int_0^t \lb\v(t)\cdot\curl\v(t')\rb dt'$.
We have been unable  to prove that this is the case for the astrophysically
relevant weak $\bar B$ regime.

There is indeed another approach to calculating the EMF
that does in fact reveal a combination
of relative magnetic helicity entering in $\alpha$, namely 
the approach of FBC which led to the result in \pref{n8}.
Indeed one is tempted by the similarity between \pref{n8} and \pref{n5},
to make an identification between them, 
but they are different because $\v$ and $\b$ enter \pref{n5}
whereas  $\v^{(0)}$ and $\b^{(0)}$, the statistically isotropic
parts of $\v$ and $\b$,  enter \pref{n8}. 
To see this more explicitly, we write 
\beq
\v=\v^{(0)}+\v^{(A)},
\label{c7}
\eeq
 where 
$A$ indicates an anisotropic contribution,
the result of the backreaction from $\OB$. Similarly, 
 \beq
\b=\b^{(0)}+\b^{(A)}.
\label{c8}
\eeq
(Even when $\b$ is the result
of stirring up an initial seed $\OB$, there is still a $\b^{(0)}$ which
is the statistically isotropic part of $\b$.)
We then assume that the statistics of the zeroth-order
turbulent correlations are those of a homogeneous isotropic,
``known'' base state.
The goal is to express turbulent correlations in terms of the 
zeroth-order quantities. It is sufficient to demonstrate the basic idea
invoked to all orders in $\OB$ in FBC with that 
derived to linear order in Blackman \& Field (1999). 
Noting that $\lb\v\ts\b\rb^{(0)}$ vanishes, the lowest
order contribution to the turbulent EMF is
\beq
\begin{array}{rcl}
\lb\v\ts\b\rb^{(1)} = 
\lb\v^{(0)}\ts\b^{(1)}\rb + \lb\v^{(1)}\ts\b^{(0)}\rb\\
= \int^t_0 \lb \v^{(0)}(t)\ts\partial_{t'} \b^{(1)}(t')\rb dt'   + \int^t_0 \lb\partial_{t'} \v^{(1)}(t')\ts\b^{(0)}(t)\rb dt'
\end{array}
\label{c9}
\eeq
To linear order, using the induction equation for $\b^{(1)}$ and
the Navier-Stokes equation for $\v^{(1)}$, it can be
shown that by analogy to the derivations of \pref{c4} and \pref{c6},
(combined with a revised first order smoothing approximation
that assumes $|\b^{(1)}/\OB| < 1$)
\pref{c9} becomes
\beq
\lb\v\ts\b\rb^{(1)} = 
-{\OB\over 3}\left(\int^t_0 \lb\v^{(0)}(t)
\cdot\curl\v^{(0)}(t')\rb dt'-\int^t_0 \lb\b^{(0)}(t)
\cdot\curl\b^{(0)}(t')\rb dt'\right).
\label{c10}
\eeq
FBC showed that in the case of negligible mean field gradients,
it is still the zeroth order kinetic and current helicities which appear
most explicitly in $\alpha$, even to all orders in $\OB$.

It is clear that the zeroth order helicities are not necessarily 
equal to those constructed with the full turbulent quantities
since
%example, that 
% $\lb\v\cdot\curl\v\rb\ne \lb\v^{(0)}\cdot\curl\v^{(0)}\rb$
%results from the fact that  
\beq
\lb\v\cdot\curl\v\rb = 
\lb\v^{(0)}\cdot\curl\v^{(0)}\rb+
\lb\v^{(A)}\cdot\curl\v^{(0)}\rb+\lb\v^{(A)}\cdot\curl\v^{(0)}\rb
\label{c11}
\eeq
and similarly 
\beq
\lb\b\cdot\curl\b\rb = 
\lb\b^{(0)}\cdot\curl\b^{(0)}\rb+
\lb\b^{(A)}\cdot\curl\b^{(0)}\rb+\lb\b^{(A)}\cdot\curl\b^{(0)}\rb,
\label{c12}
\eeq
so the extra terms on the right must be dealt with.
One might ask however, if the first terms on the right of
\pref{c11} and \pref{c12} dominate, why can't we 
simply replace $\lb\b^{(0)}\cdot\curl\b^{(0)}\rb$ by
$\lb\b\cdot\curl\b\rb$ wherever the former occurs?
The reason is that if the appropriate helicity which enters $\alpha$  
is $\lb\b^{(0)}\cdot\curl\b^{(0)}\rb$, 
then the magnetic helicity, $H_2$, entering on the 
right side of \pref{n38} would be of zeroth order.
However the helicities entering on the left of \pref{40p} would
then have to be at least second order. 
Moreover, no such relation as \pref{n41} could apply, relating
$H_2^{M(0)}$ to $H_1^{M(0)}$, since the latter is zero.
Thus the helicity entering $\alpha$ would not be simply
related to the quantity under the time derivative
in \pref{n40}, whereas the derivations in
section 2 require such a relation.

The procedure outlined to derive \pref{c10} 
and the subtlety just described with respect to ordering
is basically the  ``ordering ambiguity'' that was discussed in
Blackman \& Field (1999).  There it was shown that 
e.g. GD effectively derived 
the form \pref{c9} 
(and therefore \pref{n8} rather than \pref{n5})
 by linearizing in terms of $\OB$ 
but did not identify that they had derived the zeroth order
contribution to $\alpha$.  Thus GD and BY, and FBC were actually
using a similar expansion.  The subsequent manipulations
of GD and BY presumed however 
that they had derived $\alpha$ as a function of the full $\v$ and
$\b$.  Their derivation was in configuration space, so this issue
was identifiable, as opposed to the $k$ space derivation of PFL.

In this paper we have assumed that  PFL's helicity correction to $\alpha$ is
indeed that of  the full current helicity. 
We cannot reproduce that this 
correction is the same as that derived in configuration
space. Thus, in our comparisons of the present results 
to GD and BY, and the comparisons of GD, BY, and PFL to FBC, 
we are simply overlooking this  ``ordering ambiguity'' that
we have earlier identified in GD and BY as being 
incorrect (Blackman \& Field 1999). In other words, 
our comparisons of the present work to those previous 
papers must be considered more carefully 
as we continue to ponder over why the 
correction to $\alpha$ that appears in PFL does not
cleanly emerge in configuration space.


A final point: the 
assumption of a stationary and stable isotropic base state represented by
zeroth order quantities such as $H^{M(0)}_2$ as invoked by
FBC may be realistic for a fixed $B_1$, but it 
is not realistic during the growth phase of a dynamo.  
%Technically speaking therefore, the expressions for 
%$\alpha$ given by  FBC and GD and BY, 
%$\alpha$ that depends on the properties of the isotropic part of the
%turbulence, denoted by superscript $(0)$.
For example, in the $\alpha^2$ dynamo, such as that simulated by Brandenburg
(2001a), instability drives the turbulence into a dynamical state
with an ever-changing $B_1$.  This process
drives $H^M_1(t)$, hence $H^M_2(t)$, away from that in the isotropic state
that was the basis of the perturbative analysis of FBC and 
implicitly of GD and BY.  This further highlights
the importance of distinguishing zeroth order correlations
from the full correlations.







Using their expansion in $a$, PFL also described the ``Alfv\'en effect",
%in which a large scale 
%field facilitates the propagation of Alfv\'en waves of smaller
%scale.  
%Since the magnetic energy $E^M$ and
%kinetic energy $E^V$ are in equipartition in such waves, PFL found a term
which  promotes the relaxation of $E^M_k-E^V_k$
to zero at large $k$.  They also found a similar term which promotes
the relaxation of $H^C_k-H^V_k$ to $0$ at large $k$.  
Recent numerical simulations  (Maron \& Blackman 2002)
show that this tendency toward equipartition at each $k$ above the 
forcing scale only occurs
when sufficient kinetic helicity is injected into the system (otherwise
the magnetic field peaks at the resistive scale), 
but this is consistent with the case studied by PFL.
%Alfv\'enic on small scales is correlated with the amount 
%of kinetic helicity forced into the system; 
PFL  suggested, in their discussion of helical turbulence, 
that if the large-scale field grows large,
this tendency toward equipartition
%Alfv\'enic turbulence 
will quench $\alpha$ by
reducing the integral in \pref{n5}.  In principle,
the quenching can occur in  
even if there is not exact equipartition at each $k$ but
if the separate integrals are equal and opposite.


\medskip

























































Our results
are also in approximate agreement with other
features in his closed box simulation: (i) the field at $k_1$ is force free; (ii) the sign
of $\alpha$ is the same as that of $H_2^C-H_2^V$; (iii) $\alpha$ saturates as $B_1$ grows; (iv) saturation occurs when 
$B_1\sim (k_2/k_1)v_1$, corresponding to superequipartition;
(v) if $\beta=\beta_0(\alpha/\alpha_0)$, as indicated by Brandenburg, 
the saturation value we find for $\alpha$ 
is resistively limited (e.g. slow dynamo), in agreement with Brandenburg.
If  on the other hand, $\beta=\beta_0=constant$, as in 
GD, the dynamo growth is not resistively limited (and thus
a fast dynamo) in disagreement with Brandenburg.



%From \pref{n53} this continues until $x\sim 1$, at
%which point $\alpha$ is quenched and the
%process stops.  Whether $\alpha(\infty)$ is limited by turbulent diffusion
%or molecular diffusion depends on how
%$\beta$  depends on $B_1$.


\section*{Appendix C: Comparison of FBC with the Present Paper}



Field, Blackman and Chou (1999; hereafter FBC) used a different approach
to find the dependence of $\alpha$ on
$\ob$, in which the turbulent $\v$ and $\b$ entering $\alpha$ are expanded
about their values in a state with
$\ob=0$.  If $\la$ is small, their result can be written in the form
\be{n8}
\alpha(0) = -\tfrac{2}{3} \int^\infty_0 (2\gamma_q)^{-1}
\lrp{H^{V(0)}_q-H^{C(0)}_q} g \lrp{q\ob/\gamma_q}dq\; ,\ee
where $\gamma^{-1}_q$ is the relaxation time for modes $q$, $H^{V(0)}_q$ and
$H^{C(0)}_q$ are the helicity spectra in
the state $\ob=0$, and
%\be{n9}
$
g(x) = 3(x-\tan^{-1} x)/ x^3.$
%\; . \ee
Although \pref{n8} superficially resembles \pref{n5}, there are significant
differences: (i) FBC assumed that the
large-scale field has infinite scale ($k=0$) so $q$ takes all values down to $0$; (ii) in
\pref{n8} the helicities pertain to the state
with $\ob=0$, rather than the actual state of arbitrary $\ob$ in \pref{n5};
(iii) \pref{n8} includes the function $g$,
which approaches unity for $q\ob/\gamma_q\ll 1$, and $\sim
(q\ob/\gamma_q)^{-2}$ when it is $\gg 1$.  Thus
$\alpha$ is quenched when $\ob>\gamma_q/q$, which for a typical $q$ in a
peaked
spectrum is just $v$, the turbulent speed. We discuss this further in
Appendix B.










Unlike Brandenburg (2001a) FBC assumed that the large-scale field $\B_1$ is
constant in time and space.  As shown by
Matthaeus and Goldstein (1982), in this case $H^M$ as defined by \pref{n12}
is not conserved, but there is an analogous
quantity $H^M_a$ given by
$$
H^M_a= \A_1\cdot \B_1 +H^M_2 = \A_1\cdot \B_1+\half \lra{\a\cdot\b} \cno{1}
$$
which is conserved, provided that $\A_1$ is defined in terms of the
large-scale electric field $\E_1=-\lra{\v\times \b}$ by
$$
\A_1 = -\int^t \E_1dt\; , \cno{2}
$$
so that 
$$
\part_t \A_1 =  -\E_1 = \lra{\v\times \b} = \alpha \B_1 - \beta\nt \B_1 =
\alpha \B_1\; . \cno{3}
$$
(Note that because $\nt \B_1=0$ no $\A_1$ can be defined in a periodic
system with $\nt \A_1=\B_1$.)

To show that $H^M_a$ is conserved, write
$$
\begin{array}{rl}
\part_t H^M_a &= \B_1 \cdot \part_t \A_1+\half \lra{\part_t
\a\cdot\b+\part_t\b\cdot \a} \\[4pt]
&= -\E_1\cdot \B_1+\half \lra{\part_t \a\cnt \a+\part_t(\nt \a)\cdot \a}
\\[4pt]
&= \lra{\v\times \b} \cdot \B_1+\lra{\lrp{\v\times \B-\la\nt \b}\cdot\b} \; ,
\end{array}\cno{4}
$$
where the gauge potential term vanishes because of the periodic boundary
conditions.  Using 
$$
\lra{\lrp{\v\times \B}\cdot\b} = \lra{(\v\times \B_1)\cdot \b} = -\B_1\cdot
\lra{\v\times \b}\; , \cno{5}
$$
we have
$$
\begin{array}{rl}
\part_t H^M_a &= -\la\lra{\b\cnt \b} \\[4pt]
&= -2\la H^C_2 \;  , \end{array} \cno{6}
$$
which vanishes if $\la=0$, proving conservation of $H^M_a$.

Since we can also write
$$
\part_t H^M_a = \B_1\cdot \lra{\v\times \b}+\part_t H^M_2 = \alpha\B^2_1
+\part_t H^M_2\; , \cno{7}
$$
we have
$$
\begin{array}{rl}
\part_t H^M_2 &= -\alpha B^2_1 - 2\la H^C_2 \\[4pt]
&= -2\alpha E^M_1 - 2\la k^2_2 H^M_2\; , \end{array} \cno{8}
$$
to which the solution with $H^M_2(0)=0$ is
$$
H^C_2 = k^2_2 H^M_2 = -{\alpha E^M_1\over \la} \lrp{1-e^{-2\la k^2_2 t}}\; .
\cno{9}
$$
From \pref{38p} and \pref{n23} we therefore have
$$
\alpha = \alpha_0 - {2\tau \alpha E^M_1\over 3\la} \lrp{1-e^{-2\la k^2_2 t}}
\;  , \cno{10}
$$
so
$$
\alpha = {\alpha_0 \over 1+{2\tau E_1^M\over 3\la} \lrp{1-e^{-2\la k^2_2 t}}}\;
. \cno{11}
$$
If $t\gg 1/2\la k^2_2$,
$$
\alpha= {\alpha_0 \over 1+\frac13 R_m\lrp{B_1\over v_2}^2}\; , \cno{12}
$$
where we have used $\la=v_2 /k_2 R_m$, $\tau=1/k_2v_2$, and $E^M_1=B^2_1/2$.
This resembles the suggestion of GD
(1994) that $B_{1,{\rm sat}}\sim R^{-1/2}_mv_2$, and is different from FBC's
conclusion that $B_{1,{\rm sat}} \sim v_2$.  We
attribute this to the fact that in the saturated state the only $H^M$ that
can be pumped from $k_2$ to $k_1$ must be extracted
by resistive effects, owing to the lack of turbulent diffusion from $k_1$ to
$k_2$.  (However, see Appendix D for further
discussion of GD (1994)).

But 
in the Galaxy  for example, $t\ll 1/2\la k^2_2$ because $\la$ is so small, and in that
case, (C.11) becomes
$$
\alpha = {\alpha_0 \over 1+t/t_\ast}\; , \cno{13}
$$
where
$$
{t_\ast\over \tau} = {3\over 2} \lrp{v_2\over B_1}^2\; . \cno{14}
$$
Thus, with $\tau\sim 10^7\y$, $t\sim 10^{10}\y$, and $v_2$ equivalent to
$3\mu G$ in the Galaxy, $B_1$ can be as large
as $0.1\mu G$ without any approach to saturation.  As will be seen in
Appendix D, if $k_1$ is finite, this problem is overcome,
and the results agree with FBC.  In summary, adoption of a strictly
vanishing $k_1$, following FBC, forces a resistively
quenched $\alpha$.  However, the time to reach that state is so long that
$\alpha$ can be large over the  life of the Galaxy.


While these two concepts are consistent with what
we find, we have explicitly calculated the 
the dynamics and the level saturation level of the mean
field, which differs from previous suggestions:
Refs. \cite{ch,by} considered a uniform field so the
dynamics of the growth could not fully be considered.










(It should be noted that the 
$B_1 \sim \lb v_2 /R_M^{1/2}$ which is equivalent
to the Zeldovich relation \cite{z57} when $v_2\sim b_2$ 
should not influence one's estimate for 
when a mean field dynamo should quench: 
this relation was derived in 2-D, for a fixed $\OB$ as
a limit for how large the small scale field can grow \cite{bfplasma}. 
It has nothing to do with growing the large scale field in an MFD.)
However, here 
we have shown that there is a much larger saturation
value. From the discussion above, recall that the kinematic regime
proceeds until $h_1\sim 1$, or equivalently $B_1 \sim v_2 (k_1/k_2)^{1/2}$.





%There are several key features to notice.
%First, when $q_2=1$, the
%curves saturate at the value of $h_1=k_2^2/k_1^2$.
%This corresponds to a value of $B_1/v_2 = (k_2/k_1)^{1/2}$.
%This is expected  analytically, as can be seen by setting the sum
%of \pref{40} and \pref{41} to zero. Then one has
%$h_1/h_2=25$  in saturation, but $h_2 \sim 1$ in saturation
%since this occurs  when $\alpha\rightarrow 0$.
%Thus $|h_2|\sim v_2^2/k_2$ in saturation, which is dimensionless
%unit of magnetic helicity. Note that this result is independent
%of $R_M$.

%Second, note that increasing $R_M$ increases the time
%at which the ultimate saturation occurs.
%The saturation time is given approximately by  
%\beq
%t_{sat} \sim R_M (k_2/k_1).
%\label{tsat}
%\eeq
%This marks the time at which the resistive terms fully dominate the evolution
%of equations \pref{40b} and \pref{41b}.  




There are several key features to notice.
First, when $q_2=1$, the
curves saturate at the value  $h_1=k_2^2/k_1^2$.
This corresponds to $B_1/v_2 = (k_2/k_1)^{1/2}$.
This is expected  analytically, as can be seen by setting the sum
of \pref{40b} and \pref{41b} to zero. Then one has
$h_1/h_2=k_2^2/k_1^2$  in saturation. But $h_2 \sim 1$ in saturation
since this occurs as  the two terms in $\alpha$ become comparable.
Thus $|h_2|\sim v_2^2/k_2$ in saturation, which is our dimensionless
unit of magnetic helicity. Note that this result is independent
of $R_M$.

Second, note that increasing $R_M$ increases the time
at which the ultimate saturation occurs.
The saturation regime is given approximately by  
\beq
t_{sat} \ge R_M (k_2/k_1).
\label{tsat}
\eeq
This marks the regime 
at which the resistive terms fully balance
the other terms in both 
equations \pref{40b} and \pref{41b}. 

Third, it is important to recognize 
that at very early times the dynamo proceeds kinematically,
and independent of $R_M$. This regime occurs 
because at early times, $h_2 << 1$, and 
therefore the $\alpha$ effect is dominated by the kinetic helicity
term.  In this regime, the resisitve terms are not important
and $h_2 \sim h_1$ as can be seen from ignoring the 
resistive terms  in \pref{40b}. This stage ends as $h_1\rightarrow 1$,
which corresponds to $H_1^M\sim v_2^2/k_2$.
The time scale for this phase to end  is given by
a few kinematic growth time scales for the $\alpha^2$ dynamo, namely
\beq
t_{kin}\sim {\rm Ln}[1/h_1(0)] (k_2/k_1)/(2/3-k_1/k_2).
\label{tkin}
\eeq







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