Eric G. Blackman \altaffilmark{1} and George B. Field\altaffilmark{2}
\noindent We develop a new nonlinear mean field dynamo theory that couples field growth to the time evolution of the magnetic helicity and the turbulent electromotive force, $\emfb$. We show that the difference between kinetic and current helicities emerges naturally as the growth driver when the time derivative of $\emfb$ is coupled into the theory. The solutions predict significant field growth in a kinematic phase and a saturation rate/strength that is magnetic Reynolds number dependent/independent in agreement with numerical simulations. The amplitude of early time oscillations provides a diagnostic for the closure.
dynemfprl2.tex
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\begin{document}

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\centerline{\large\bf A New Dynamical Mean-Field Dynamo Theory and 
Closure Approach}
%\centerline{\large\bf on Turbulent Magnetic Dynamo Spectra}

%\bf The Effect of Fractional Kinetic Helicity on Small-Scale Dynamos:}
\medskip
%\centerline{\bf Application to the Galactic Magnetic Field} 

\author{Eric G. Blackman \altaffilmark{1} and George B. Field\altaffilmark{2}}
\affil{1. Department of Physics \&Astronomy and Laboratory for
Laser Energetics, University of Rochester, Rochester NY 14627}
\affil{2. Harvard-Smithsonian Center for Astrophysics, 60 Garden St.
Cambrdige MA, 02138}



\centerline {(in press, Phys. Rev. Lett.)} 
%\altaffiltext{1}{maron@tapir.caltech.edu}

\begin{abstract} 

\noindent We develop a new nonlinear mean field dynamo theory 
that couples field growth to the time evolution of the magnetic helicity
and the turbulent electromotive force, $\emfb$.
We show that the difference between kinetic and current helicities 
emerges naturally as the growth driver when the time derivative of $\emfb$ is 
coupled into the theory. The solutions predict 
significant field growth in a kinematic phase and a   
saturation rate/strength that is magnetic Reynolds number 
dependent/independent in agreement with numerical simulations.
The amplitude of early time oscillations provides a diagnostic for the closure.



\end{abstract}

\centerline{PACS codes: 95.30.Qd;  98.38.Am; 52.55.Ip,
%47.27.Eq; 
52.30.Cv; 98.35.Eg; 
96.60.Hv}
\bigskip
%\tableofcontents
%\listoffigures


%\section{New stuff}

%\section{Introduction}

\ni {\it Introduction-}
Mean field dynamo (MFD) theory has been a useful  framework for modeling
the in situ origin of large-scale magnetic field 
growth in planets, stars, and galaxies 
%[\cite{moffatt,parker,krause,zeldovich83}]
[1-4], and has also been invoked 
to explain the sustenance of fields in  fusion devices
[\cite{ji,bellan}].  
However, whether the backreaction of the magnetic field itself prematurely
quenches the MFD has been debated [7-24].  Recent progress has emerged from 
incorporating  magnetic helicity evolution into the theory.  


%If the MFD growth driver is the residual helicity [\cite{pfl},\cite{kleeorin82}]
%(the difference between current and kinetic helicities),
%rather than just the kinetic helicity of textbook 
%kinematic theory [\cite{moffatt}], then the coupled time evolution of 
%large scale field and total magnetic helicity  
%determines the nonlinear field growth. 
%A simple two-scale approach revealed [\cite{fb02},\cite{bb02}]
%that the predicted field growth and nonlinear saturation are 
%consistent with numerical experiments [\cite{b2001}].  
%As we will show however,  
%these approaches leave an  unresolved question: 
%how does the residual helicity emerge as the growth driver?
%We solve this problem by developing a conceptually new MFD theory
%that couples the field growth to the time evolution of  $both$ the 
%turbulent electromotive force and the magnetic helicity. 

%reveals that mean field dynamos 
%and new approaches to incorporate magnetic helicity evolution.
%[\cite{piddington,vc,ch,cattaneo2002,kulsrud,b2001,fbc,bf2000,fb02,paris01}].
%[\cite{piddington  paris01}]. Ref. [7
%In particular, it is now know that a time-dependent 

To make explicit the problem to be solved, we first average the magnetic 
induction equation to obtain the basic MFD equation
%derived from Maxwell's equations and Ohm's law 
[\cite{moffatt,krause}]:
\beq
\partial_t\OB= \curl \emfb +\curl(\bbV\ts \bbB) -\lambda \nabla^2\OB,
\label{1}
\ee
%\beq
%\partial_t\OB=\alpha \nt \OB+({\beta+\lambda}) \nabla^2 \OB\; ,
%\label{n1}
%\ee
where $\OB$ is the mean (large-scale) magnetic field in Alfv\'en speed units,
$ 
\lambda = {\eta c^2\over 4\pi}
$
is the magnetic diffusivity in terms of the resistivity $\eta$, $\bbV$ is
the mean velocity which we set $=0$, and 
$\emfb=\lb\bfv\ts\bfb\rb$
is the turbulent electromotive force, a correlation between
fluctuating velocity $\bfv$ and magnetic field $\bfb$ in Alfv\'en units.  
%To make (\ref{1}) tractable
%and useful for modeling   mean field growth,  
Textbook treatments [\cite{moffatt,krause}] invoke 
$\emfb=\alpha {\bbB}-\beta \curl \bbB$, 
where $\alpha$ and $\beta$ are pseudoscalar and scalar
correlations of turbulent quantities respectively. 
In the kinematic theory [\cite{moffatt}] $\alpha = -(\tau_c/3)\lb\bfv\cdot\curl\bfv\rb$, where $\tau_c$ is a correlation time, 
and (\ref{1}) is solved with $\alpha$ and $\beta$ as input parameters.
%Determining these correlations requires the time evolution 
%equations for  $\bfv$ and $\bfb$.
%subject to various 
%approximations. The extent to which 
%these approximations are invalid, and for which 
%non-linearities and the backreaction from the growing magnetic field 
%may quench $\emfb$ prematurely has been debated 

%If $
%as is often presumed [\cite{moffatt}], dynamo quenching 
%becomes linked to $\alpha$ quenching.
But the kinematic theory is incomplete.
In a study of
% homogeneous, isotropic, 
helical MHD turbulence, 
Ref. \cite{pfl} 
%used
%the Eddy Damped Quasi-Normal Markov (EDQNM) approximation to 
derived approximate evolution equations for the spectra of kinetic
energy, magnetic energy, kinetic helicity, and magnetic helicity
($\equiv \lb{\bfA\cdot\curl\bfA}\rb$, where  $\bfB=\curl \bfA$).  
%They solved initial-value problems for
%these spectra, 
These calculations suggested that 
$\alpha \simeq (\tau_c/3)\lb\bfb\cdot\curl\bfb-\bfv\cdot\curl\bfv\rb$,
the residual helicity, where $\lb \rb$ indicates spatial or ensemble average.
This form has been employed in attempts to understand  
nonlinear dynamo quenching by coupling  magnetic helicity conservation
into the dynamo through the $\lb\bfb\cdot\curl\bfb\rb$ term 
[\cite{zeldovich83},19-23]
%kleeorin82,kleeorinruz,gd1,gd2,by}].
Although these studies [e.g. \cite{zeldovich83,kleeorin82,kleeorinruz}],
wrote down an equation for the time evolution of $\alpha$,  
they derived quenching  formulae for $\alpha$  
only for the steady-state.  Only after a coupled nonlinear system of 
time-dependent large and small scale magnetic helicity 
equations were solved [\cite{fb02}], was it apparent that 
a dynamical quenching model based on residual helicity 
reveals both a kinematic growth phase and an asymptotic resistively limited 
phase as seen in numerical experiments [\cite{b2001}]. The dynamical approach
has also been applied to dynamos with shear [\cite{bb02}].

%Despite Ref. \cite{pfl}, 
But even in these dynamical approaches,  
the $\emfb$ was assumed to be proportional to the residual helicity.
Here we show that the required residual helicity emerges 
not from $\emfb$, but from $\partial_t\emfb$, and that including the 
$\partial_t\emfb$ equation in addition to the MFD and total
magnetic helicity equations is essential for a complete MFD theory.
%This facilitates coupling the Navier-Stokes and 
%magnetic helicity equations into the MFD.
We first derive $\partial_t\emfb$ and then derive the triplet of 
equations to be solved for the simple
shear free helical dynamo whose solutions can be compared
with existing numerical simulations. We discuss these solutions, physical
implications, and the relation to previous work.



%\section{Deriving $\partial_t \emfb$}




%\subsection{The $\emfb$ Puzzle}

{\it Deriving $\partial_t \emfb$-}
A two-scale nonlinear quenching approach 
invoking the residual helicity in $\emfb$  [\cite{fb02}], 
captures the nonlinear dynamo saturation seen in 
simulations [\cite{b2001}], but 
the derivation of the appropriate $\emfb$  
has been elusive.  To couple  $\lb\bfb\cdot\curl\bfb\rb$ 
of $\emfb$ to the magnetic helicity conservation equation,
the full small-scale field $\bfb$ must enter this 
correlation not a low order approximation. 
The essence of the puzzle [\cite{paris01}] 
is that 
\beq
\emfb(t)=
\lb\bfv(t)\ts\bfb(t)\rb = \int_0^t \lb \bfv(t )\ts 
\partial_{t'}\bfb(t')\rb dt'= -\int_0^t \lb \bfb(t)\ts \partial_{t'}\bfv(t')\rb dt',
\label{c1}
\ee
assuming that $b(0)=0$ and that $t>>0$ so that 
$\lb\bfv(0)\ts\bfb(t)\rb=0$.
The last two terms 
are both exact expressions for  $\emfb$,
however neither leads naturally the residual 
helicity entering $\alpha$: upon using the 
induction equation for $\bfb$, 
the second term on the right leads to a term $\propto
\int\lb\bfv(t)\curl\bfv(t)\rb dt'$. Using the Navier-Stokes 
equation for $\bfv$, the last term contributes a term $\propto
\int\lb\bfb(t)\curl\bfb(t)\rb dt'$.
One emerges with a $choice$
rather than the $difference$ between the two helicities. 
So how does the residual helicity emerge?


%\subsection{The Time Derivative}

%{\it Time Derivative of $\emfb$-}
Rather than impose the form of the $\emfb$, we solve for it dynamically 
using
\beq
\partial_t\emfb=\lb\partial_t\bfv\ts\bfb\rb +\lb\bfv\ts\partial_t\bfb\rb.
\label{timed}
\ee
To proceed, we need equations for $\partial_t\bfb$ and $\partial_t\bfv$.  
Assuming $\div\bfv=0$ we have
%$\lb\bfv\ts\partial_t\bfb\rb$
%proceed, we need the equations for $\bfb$ and $\bfv$, 
%If we assume that $\nabla\cdot\bfv=0$, these are 
\begin{equation}
\begin{array}{r}\partial_{t} \bfb = \OB\cdot\nabla\bfv - \bfv\cdot\nabla\OB + 
%\bfb\cdot\nabla\bfv -\bfv\cdot\nabla\bfb 
\curl(\bfv\ts\bfb) 
-\curl\lb\bfv\ts\bfb\rb +
\lambda\nabla^{2}\bfb,
\end{array}
\label{c3}
\ee
and
%\begin{equation}
%\begin{array}{r}
%\partial_{t} {v_i}=(\delta_{ij}-{\nabla_i\nabla_j\over \nabla^2}) 
%(-\bfv\cdot\nabla v_j-\lb\bfv\cdot\nabla v_j\rb
%+{\OB}\cdot\nabla{b_j} + {\bfb}\cdot\nabla{\bB_j} + 
%{\bf b}\cdot\nabla{b_j}
%\\-\lb {\bf b}\cdot\nabla{b_j}\rb)
%+ \nu\nabla^{2}{v_i} 
%+{f_i}({\bf x}, t),
%\end{array}
%\label{c2a}
%\end{equation}
%\beq
%p_{eff}=p_{th}+\bfb^2/2 - \lb\bfb^2\rb/2 + \bbB\cdot\bfb
%\ee
%and thus includes the thermal and magnetic contributions to the 
\begin{equation}
\begin{array}{r}
\partial_{t} {v}_q=P_{qi}({\OB}\cdot\nabla{b_i} + {\bfb}\cdot\nabla{\bB_i} 
-\bfv\cdot\nabla v_i+\lb\bfv\cdot\nabla v_i\rb
+{\bf b}\cdot\nabla{b}_i-\lb {\bf b}\cdot\nabla{b_i}\rb)
+ \nu\nabla^{2}{v_q} 
+{f_q},
\end{array}
\label{c2}
\end{equation}
where $\bf f$ is a divergence-free 
forcing function uncorrelated with $\bfb$, $\nu$ is the viscosity, 
and $P_{qi}\equiv (\delta_{qi}-\nabla^{-2}\nabla_q\nabla_i)$
is the projection operator that arises after taking the divergence
of the incompressible Navier-Stokes equation to eliminate the total
fluctuating pressure (magnetic + thermal).
Using Reynolds rules [\cite{rad}] to interchange  brackets with time
and spatial derivatives, the 5th term of 
(\ref{c3}) and the 4th and 6th terms in the parentheses of 
(\ref{c2}) do not contribute when put into the averages so we 
ignore them.   

The contribution to $\partial_t\emfb$ 
from the 3rd term in (\ref{timed}) can be derived 
by direct use of (\ref{c3}) in configuration space.
We assume isotropy of the resulting velocity
and magnetic field correlations for terms linear in $\bbB$. We  also 
retain the triple correlations.  The contribution 
to $\partial_t\emfb$ from the 2nd
term in (\ref{timed}) also contributes terms linear in $\bbB$, 
and triple correlations.  Here the terms linear in $\bbB$ are best
derived in Fourier space. For this, 
we follow the technique in the appendix of Ref. \cite{gd2}, 
which invokes the Fourier transform of the terms linear in $\bbB$
 contributing to $\lb\partial_t\bfv\ts\bfb\rb$,  
supplemented by a linear expansion of the projection operator in $k_1<<k_2$, 
where $k_1$ is the characteristic wavenumber of the 
bracketed or mean quantities and 
$k_2$ is the characteristic wavenumber of the fluctuating quantities
$\bfb$ and $\bfv$. Collecting all surviving 
terms, we then have for (\ref{timed})
\beq
\begin{array}{r}
\partial_t{\emfb}=
{1\over 3}(\lb\bfb\cdot\curl\bfb\rb
-\lb\bfv\cdot\curl\bfv\rb)\bbB 
- {1\over 3}\lb
\bfv^2\rb\curl\bbB
%-\zeta
+\nu\lb\nabla^2\bfv\ts\bfb\rb +\lambda\lb\bfv\ts\nabla^2\bfb\rb
+ {\bf T}^V+{\bf T}^M,
%-{\tilde\zeta}\lb\bfv\ts\bfb\rb,
\end{array}
\label{timed33}
\ee
%where $\tilde\zeta$ includes the microphysical dissipation
%terms and the last term of (\ref{timedp}).
%(\ref{timed33}).
where ${{\bf T}^M}=\lb \bfv\ts\curl(\bfv\ts\bfb)\rb$ 
and ${T_{j}^V}=\lb(\ep_{jqn}\lb P_{qi}(\bfb\cdot\nabla b_i
-\bfv\cdot\nabla v_i) b_n\rb$
are  the triple correlations. Note that 
the 3rd, 4th, 6th and 8th  
terms in (\ref{timed33})
 come from  
the $\lb\bfv\ts\partial_t\bfb\rb$ term of (\ref{timed})
and the 2nd 5th and 7th terms come from the 
% which in turn arise from the 2nd, 3rd and 6th terms of (\ref{c3})
%after using Reynolds rules 
$\lb\partial_t\bfv\ts\bfb\rb$ term of (\ref{timed}).





%In deriving the contribution from 
%$\lb\partial_t\bfv\ts\bfb\rb$ we used the technique in 
%the appendix of [\cite{gd2}].
%This requires Fourier transforming (\ref{c2}), then 


%6th term of (\ref{timed3}) plus a term 
%${1\over k_2^2} \lb\nabla[\nabla\cdot(\bfv\cdot\nabla\bfv)] \ts \bfb\rb$
%arising from $-\lb \nabla p\ts \bfb\rb$. 
%Both vanish (i.e. $\bf T$=0)
%in a two-scale approach where $\nabla\cdot\bfv=\nabla\cdot\bfb=
%-i{\bf k_2}\cdot\bfv=-i{\bf k_2}\cdot\bfb=0$, but 
%we will explore $\bf T\ne 0$ later.
%$\nabla \bfv=-i{\bf k}_2\bfv$
%and $\nabla \cdot \bfv=-i{\bf k_2}\cdot \bfv=0$ which we assume.
%An alternative is to take these terms $\propto
%{\tilde \zeta}\lb\bfv\ts\bfb\rb$ 
%where $\tilde\zeta$ is described below.


We are primarily interested in the component of $\emfb$ parallel to 
$\bbB$. For this we have
\beq
\partial_t\emf_{||}=(\lb\partial_t\bfv\ts\bfb\rb
 +\lb\bfv\ts\partial_t\bfb\rb)\cdot\bbB/|\bbB|+
\lb\bfv\ts\bfb\rb\cdot\partial_t(\bbB/|\bbB|).
\label{timedp}
\ee
Substituting (\ref{timed33}) into 
(\ref{timedp}) gives 
\beq
\partial_t\emf_{||}= {\tilde\alpha}{\bbB^2/|\bbB|}
-{\tilde\beta}{\bbB\cdot\curl\bbB}/|\bbB|-{\tilde \zeta}\emf_{||}
%-\lb\bfv\ts\bfb\rb\cdot\partial_t({\bbB/|\bbB|}),
\label{2}
\ee
where 
${\tilde\alpha}
=(1/3)(\lb\bfb\cdot\curl\bfb\rb-\lb\bfv\cdot\curl\bfv\rb)$,  ${\tilde\beta} = (1/3)\lb \bfv^2\rb$, and  
%$\tilde\zeta>0$ 
$\tilde \zeta$ 
accounts for microphysical dissipation
terms, the last term of (\ref{timedp}), and
any additional contribution  arising from
${\bf T}^M + {\bf T}^V\ne 0$. 
Note that $\tilde \alpha$ and $\tilde\beta$ appear similar
to the usual $\alpha$ and $\beta$ dynamo coefficients in $\emfb$, 
but they are fundamentally different because they
are coefficients in $\partial_t\emfb$ (and thus 
have different units) and do not involve $\tau_c$.
Note also that if isotropy of like correlations were strongly violated, 
$\tilde\alpha$ and $\tilde\beta$
would be anisotropic tensors in analogy to tensor generalizations 
of $\alpha$ and $\beta$ [\cite{kr,rk}]. We have not considered
that here.  
%Even if this offset is a factor of 2, 
%only a weak suppression is predicted.

Following [\cite{fb02}] we 
% these equations
%where $k_2$ is that of the small-scale, taken to be at the 
%forcing wavenumber and $k_1$ is that of the large scale.
define large and small-scale magnetic
helicities as $H_1^M\equiv \lb\bbA\cdot\bbB\rb_{vol}$
and $H_2^M\equiv\lb\bfa\cdot\bfb\rb_{vol}$,
where $\lb\rb_{vol}$ indicates a global spatial average.
Then 
$\lb\bfb\cdot\curl\bfb\rb_{vol} = k_2^2H_2^M$ and 
$\lb\bbB\cdot\curl\bbB\rb_{vol} = k_1^2H_1^M$.
%We include $\lb \rb$ in the definition of $H_1^M$ 
%to allow for a $k_1=1$ mode in which $\bbB$ changes sign 
%in the volume. 
We define the small-scale kinetic helicity
$H_2^V=\lb\bfv\cdot\curl\bfv\rb$.
%, and the small-scale kinetic helicity
%$H_2^V=\lb\bfv\cdot\curl\bfv\rb$.
We assume $\bbV=0$, and a force-free large-scale field 
for which the $\partial_tH_1^M$ equation 
becomes degenerate [\cite{fb02}] with that of $\partial_t\bbB^2$. 
Then $|\bbB|=k_1^{1/2}|H_1^{1/2}|$.
We can thus rewrite (\ref{timed}) as
\beq
\partial_t\emf_{||}=
k_1^{1/2}|H_1^M|^{1/2}(k_2^2H_2^M-H_2^V)/3-k_1^{3/2}(H_1^M/(|H_1^M|^{1/2})
{\tilde\chi}-{\tilde \zeta}\emf_{||}.
\label{4time}
\ee



%\section{The Dynamo Equations-}

%\section{The Dynamo Equations-}

{\it Dynamo Equations-}
We couple (\ref{4time})
to the equations for small and large-scale magnetic helicity 
evolution for a dynamo in which the kinetic energy is
externally forced and $\bbV=0$. 
We interpret $\bbB$, $\bbA$ and $\emfb$ 
as the $k_1$ ($0 < k_1 < k_2$) component of $\bfB, \bfA$ and $(\bfv\ts\bfb)$ 
of a closed system to facilitate  comparison with simulations of Ref. 12.  The total magnetic helicity, $H^M=\lb\bfA\cdot\bfB\rb_{vol}$, 
then satisfies [\cite{moffatt}]
$
\partial_t\lb\bfA\cdot\bfB\rb_{vol}=-2\lb{\bfE}\cdot\bfB\rb_{vol},
$
where $\bf E= -\partial_t {\bf A} -\nabla \phi$,
and  $\phi$ is the scalar potential.
%the induction equation  
%$\partial_t {\bf B}= -\curl {\bf E}$, and Ohm's law  
%${\bf E} = - \bfv \ts {\bf B} +\eta \bfJ$, 
%imply
The large and small scale 
integrated magnetic helicity equations are then [\cite{b2001,fb02,bf00}]
\beq
\partial_t\lb\bbA\cdot \bbB\rb_{vol}=2\lb{\emfb}\cdot\bbB\rb_{vol}-2\lambda \lb{\bbB\cdot \curl\bbB}\rb_{vol},
\label{h1}
\ee
and
\beq
\partial_t\lb\bfa\cdot \bfb\rb_{vol}=-2\lb{\emfb}\cdot\bbB\rb_{vol} -2\lambda 
\lb{\bfb\cdot\curl\bfb}\rb_{vol}.
\label{h2}
\ee
%Following [\cite{fb}] we invoke
%our two scale assumption for these equations
%where $k_2$ is that of the small-scale, taken to be at the 
%forcing wavenumber and $k_1$ is that of the large scale.
%We define $H_1^M\equiv \lb\bbA\cdot\bbB\rb$
%and $H_2^M\equiv\lb\bfa\cdot\bfb\rb$, so that
%$\lb\bfj\cdot\bfb\rb = k_2^2H_2^M$ and 
%$\lb\bbJ\cdot\bbB\rb = k_1^2H_1^M$.
When $\bbB$ is force-free, the two-scale
approximation allows us to write 
(\ref{h1}) and (\ref{h2}) as 
\beq
\partial_t H_1^M=2\emf_{||}k_1^{1/2}|H_1^M|^{1/2}
-2\lambda k_1^2H_1^M
\label{h3}
\ee
and 
\beq
\partial_t H_2^M=-2\emf_{||}k_1^{1/2}|H_1^M|^{1/2}-2\lambda k_2^2H_2^M.
\label{h4}
\ee
%Then  
%\beq
%-\partial_t\emf_{||}=k_1^{1/2}{H_1^M}^{1/2}(k_2^2H_2^M+H_2^V)-k_1^{3/2}
%({H_1^M}^{1/2})\sigma.
%\label{5}
%\ee
We need to solve (\ref{h3}), (\ref{h4}), and (\ref{4time})
after  converting them into dimensionless form.
We define the dimensionless quantities
$h_1\equiv H_1^M (k_2/v_2^2)$ and $h_2\equiv H_2^M (k_2/v_2^2)$,
$\ren\equiv v_2/\lambda k_2$, $Pr_M\equiv \nu/\lambda$, 
$\tau\equiv tv_2k_2$, $Q=-\emf_{||}/v_2^2$, 
$\chi={\tilde\beta}/v_2^2$, $\zeta = {\tilde \zeta}/v_2 k_2=(1+Pr_M)/\ren$, and 
use $H_2^V=-k_2v_2^2$.
For (\ref{h3}), (\ref{h4}) and (\ref{4time}) respectively, 
this gives
\beq
\partial_\tau h_1=-2Qh_1^{1/2}(k_1/k_2)^{1/2}-2h_1(k_1/k_2)^2/\ren,
\label{7}
\ee
\beq
\partial_\tau h_2=2Qh_1^{1/2}(k_1/k_2)^{1/2}-2h_2/\ren,
\label{8}
\ee
and
\beq
\partial_\tau Q= -\left(k_1/k_2 \right)^{1/2}h_1^{1/2}(1+h_2)/3
+(k_1/k_2)^{3/2}h_1^{1/2}\chi-\zeta Q.
\label{6}
\ee


%\section

{\it Solutions-}
Since $H_2^V < 0$, $H_1^M>0$ and $H_2^M<0$ for
a growing solution.  The solutions of (\ref{7}), (\ref{8}), and (\ref{6}) for
two different $\ren$ are shown
in Figs. 1 \& 2 over different time ranges 
for both $\zeta=2/R_M$ and $\zeta=1$
we use ${\tilde \alpha}\propto {\tilde \beta}$ in Fig. 1,
but the resulting solutions are only weakly sensitive to 
the form of $\tilde \beta$ as shown in Fig. 2.
In Figs.1\&2 we also plotted the empirical fit formula
to numerical simulations [\cite{b2001}]
(equation (54) of Ref \cite{b2001}) using our dimensionless parameters,
to demonstrate the good agreement.

In Figs. 1\&2 we  also compare the triplet
solution of $h_1$ with the doublet solution [\cite{fb02}]
that results from solving (\ref{7}) and (\ref{8}) but 
with $imposing$ $\emfb=\alpha\bbB-\beta\curl\bbB$ such that  
$Q=Q_{d}\equiv -(k_1/k_2)^{1/2}h_1^{1/2}(1+h_2)\tau_c/3+
(k_1/k_2)^{3/2}h_1^{1/2}\chi\tau_c$,
%
%
%$\emf_{||}=
%(-k_1^{1/2}|H_1^M|^{1/2}(k_2^2H_2^M-H_2^V)+k_1^{3/2}(H_1^M/(|H_1^M|^{1/2})\sigma)\tau_c$
%, 
where the correlation time $\tau_c$ is a
free parameter taken to be $\sim 1$.
Note that the present triplet solution does not involve
$\tau_c$ in the dynamo coefficients $\tilde\alpha$ and $\tilde\beta$.
But a remarkable result emerges: 
Fig. 1 shows that the triplet solution  matches the doublet solution
at early times for $\zeta=1$, which corresponds to 
a damping time $\zeta^{-1}\sim \tau_c$. This arises from a closure
in which the triple correlations ${\bf T}^M$ and  ${\bf T}^V$
lead to a damping with time constant $\sim \tau_c$,
and the damping suppresses the oscillations.
Fig. 2 also shows that 
the  $\zeta=2/\ren$ and $\zeta=1$ cases are indistinguishable at late times.
%A $\zeta\sim 1$ could arise if the large-scale field were
%changing direction rapidly, if $Pr\sim \ren$, 
%or if ${\bf T} \ne 0$.
We can also compare the kinematic regimes of the triplet and
doublet solutions.
The rise to the first peak of $h_1$ in Fig. 1a
is independent of $\ren$ and there the two $\ren$ triplet solutions
overlap.  This is the  kinematic regime.  
The end of the doublet kinematic regime
occurs at $h_1\simeq 1$, as
seen in Fig. 1.  Again, the doublet and triplet match when $\zeta\sim 1$.
In sum:
{\it The doublet solution emerges as the limit of the triplet
solution when the triple correlations act as a damping term.}
This closure can be tested with future simulations.


%High
%resolution simulations
% than Ref. \cite{b2001} 
%are required to test for the subsequent oscillations.

The maximum kinematic growth rate for $h_1$ is a function of $\zeta$
because it occurs where 
% can be 
%seen from semi-analytic considerations of 
%(\ref{7}), (\ref{8}) and (\ref{6}). 
%The maximum growth rate occurs 
$Q$ is a minimum. 
If we ignore resistive terms so that $h_1 = -h_2$,
and assume that $\zeta<<1$ and $\chi=1/3$, 
then Eq. (\ref{6}) implies that 
%and (\ref{8})
the maximum growth rate occurs when $h_1\simeq 1 - k_1/k_2$.  From Fig 1a, 
the minimum of $Q$ during the first oscillation is 
$\sim -1/3$ (found to be independent of $k_1/k_2$).  Setting 
$\partial_\tau h_1 \sim n h_1$, the maximum kinematic growth rate  from
(\ref{7}) is then $n \sim  (2/3)(k_1/k_2)^{1/2}(1 - k_1/k_2)^{-1/2}
\sim 0.33$, for $k_2=5k_1$.
However, when $\zeta=1$, the minimum of $Q$ from (\ref{6}) 
occurs where $Q=Q_d$. In this case, $n\sim (2/3)(k_1/k_2)(1-k_1/k_2)\sim 0.11$
for $k_2=5k_1$. 
This demonstrates that the kinematic growth rate for
$\zeta<<1$ is $\sim 3$ times that for $\zeta=1$, and 
why the triplet kinematic growth rate with $\zeta=1$ matches that of the 
doublet. Both results are seen in Fig 1.

Inspection of (\ref{7}), (\ref{8}), and (\ref{6}) 
% (\ref{gr})  
reveals why there are oscillations for a positive seed $h_1$
and $\zeta<<1$ (and independent of whether $\emfb(t=0)=0$ or $\emfb(t=0)\ne 0$). 
As long as $-1< h_2 <0$, $Q$ grows more negative and 
$h_1$ and  $h_2$  grow with mutually opposite signs.
As $h_2$ passes through $-1$ from above,   
$\partial_\tau Q$  changes sign immediately but 
$h_1$ continues to grow positive, albeit more slowly, 
until $Q$ changes sign.  Then, $\partial_\tau h_1$ 
changes sign and $h_1$ decreases. But $\partial_\tau h_2$  
changes sign when $\partial_\tau h_1$  does, 
so when $h_2$ eventually passes back through $-1$
from below,  $\partial_\tau Q$ reverses sign again, and 
eventually $Q$ becomes negative and $h_1$ again grows.  
Large $\ren$ terms only weakly damp the oscillations.
This describes what happens in Fig 1a.   
If instead, $\zeta \sim 1$, 
%$\zeta^2 \gsim (4/3) (k_1/k_2)$, 
once $\partial_\tau Q$ is depleted 
by the growth of $h_2$, the $\zeta$ term of
(\ref{6}) takes over and $Q$ decays without oscillating.  
Then $h_1$ grows without oscillations (Fig 1b).
%For $\zeta\sim 1$ the growth rate 
%in (\ref{gr})
%is significantly reduced from the $\zeta=0$ case.
%and the early-time
%growth rates for the doublet and triplet solutions nearly match, as seen
%in Fig. 1b.



%Roughly speaking, without resistive terms, 
%$h_2= -h_1$, and the oscillation from (\ref{gr}) proceeds with
%a frequency that depends on $h_1$, namely  
%$\omega \sim h_1(2/3)^{1/2}(k_1/k_2)^{1/2}$.
%The pulses thus have narrow peaks
%and wide bases. 

%\section{Discussion}

{\it Discussion-}
Textbook kinematic MFD theories solve only
the MFD equation itself [\cite{moffatt}]. 
Recent nonlinear approaches incorporating magnetic helicity evolution 
dynamically [\cite{fb02,bb02}] solve a doublet: 
the MFD equation (or the $\partial_t H_1^M$ equation)
and the total magnetic helicity evolution equation
(the $\partial_t H_1^M + \partial_t H_2^M$ equation).  
The present paper solves a triplet: the MFD equation, the total 
magnetic helicity evolution equation, $and$ the $\partial_t \emfb$ equation. 
Only the present approach shows how the difference between 
kinetic and current helicities (the residual helicity)
emerges as the MFD driver in a time dependent theory.  
The residual helicity in turn couples to the total  
magnetic helicity evolution equation.
The physical interpretation of the solutions for a closed  system 
is that as the large scale
helical field grows from MFD action, the small scale magnetic helicity 
grows of the opposite sign. At early times, kinematic growth is 
unimpeded, and the large scale field energy grows to 
$\bB^2 \gsim (k_1/k_2)v_2^2$.  Eventually, the 
small-scale magnetic helicity backreacts on the kinetic
helicity, suppressing the growth rate to an $R_M$ dependent
value. Ultimately $\bB^2 \simeq (k_2/k_1)v_2^2 $ at saturation.
This picture also arises in the imposed $\emfb$  doublet approach
[\cite{fb02}]. The approaches agree in the  
asymptotic $R_m$ dependent growth phase, matching 
simulations [\cite{b2001}]. However, oscillations  
are possible at early times only in the triplet approach.
The amplitude of such oscillations 
serves as a direct diagnostic for the MHD closure scheme, which can be 
tested with future numerical experiments.
For $\zeta=1$, the agreement between the two approaches becomes
exact for all times. This  corresponds to the triple correlations
contributing a simple damping term with characteristic time scale 
$\sim \tau_c$.






\ni 
We thank A. Brandenburg for discussions and for
pointing out a proper derivation of (\ref{timed33}).
EB acknowledges DOE grant DE-FG02-00ER54600, 
and the Aspen Center for Physics and NORDITA for hospitality.



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%\addcontentsline{toc}{section}

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%\end{thebibliography}
\bigskip
\bigskip
\bigskip
\bigskip


%---------------------------------------------------------------------
\vspace{-.1cm} \hbox to \hsize{ \hfill \epsfxsize7.5cm
\epsffile{emff1a.ps} \epsfxsize8cm \epsffile{emff3.ps} \hfill }  

\noindent {Figure 1: (a) Plot for $k_1=1$, $k_2=5$ with $\zeta=2/\ren$. 
The top and middle oscillating curves are for
 $h_1$ with $\ren = 200$  and $\ren = 1000$, and the 
bottom oscillating curve is  $Q$ for $\ren=1000$.
The thin lines are the doublet solutions 
for $h_1$ from Ref. \cite{fb02} which used an imposed $\emfb$. 
Here ${\chi} \propto (1+h_2)$.
(b) Same as (a) but with $\zeta\sim 1$.}
%The larger damping factor  suppresses the oscillations
%and makes the growth rate at early times match
%the doublet solution.}
%------------------------------------------------------------------------------
\vspace{-.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{emff2a.ps} \epsfxsize8cm \epsffile{emff2b.ps} \hfill }

\noindent { Figure 2:
%\vspace{-.1cm} \hbox to \hsize{ \hfill  
(a) Same as Fig. 1a but for broader time range.  
(b) Same as (a)  but for $\zeta=1$.  
For this time range, the doublet and triplet solutions are indistinguishable.
The dotted curves are fits to simulations [\cite{b2001}].
The lines slightly below each of the thick lines are for 
$\chi\propto 1/(1+k_1 h_1/k_2)$ demonstrating 
the weak dependence on $\tilde \beta$.

 





%------------------------------------------------------------------------------
%\vspace{-.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
%\epsffile{emff6.ps} \hfill } \noindent 
%{Figure 6:
%Plot of $h_1(t)$ for $k_1=1$, $k_2=5$ without damping
%for $\ren = 50$. The top curve is the triplet,
%the middle is the doublet and the dotted curve is
%the simulation fitting formula [\cite{b2001}]. 
%Note that for low $\ren$ the oscillations are largely
%suppressed, though the triplet solution is slightly above the other two.}
%------------------------------------------------------------------------------









\end{document}

\vfill
\eject
\section*{Calculation of $\lb\partial_t\bfv \ts \bfb\rb$}

Ignoring viscous terms and bracketed terms  we have for nav stokes: 
\beq
\partial_t\bfv=-v_i\partial_iv_j-\partial_j p -\partial_j b^2/2
-\bB_i\partial_jb_i-b_i\partial_j\bB_i+b_i\partial_ib_j
+\bB_i\partial_ib_j+b_i\partial_i\bB_j
\label{21}
\ee
and thus 
\beq
\begin{array}{r}
\lb\partial_t\bfv\ts\bfb\rb
=-\lb\ep_{ijk} b_k v_i\partial_iv_j\rb
-\lb\ep_{ijk} b_k \partial_j p\rb
-\lb\ep_{ijk} b_k \partial_j b_i\rb\bB_i
-\lb\ep_{ijk} b_k  b_i\rb\partial_j\bB_i\\
+\lb\ep_{ijk} b_k  \partial_ib_j\rb\partial_j\bB_i
+\lb\ep_{ijk} b_k  b_i\rb\partial_i\bB_j
\end{array}
\ee
where in deriving the latter equation I ignored 
the terms with 3 factors of $\bfb$ and 
the first term on the right of (\ref{21})
which are discarded due to 
our closure arguments. 
The pressure term survives and will be dealt with
below. Collecting the terms with $\bbB$ in the above
equation and perfomring the 
usual manipulations, we have using isotropy
\beq
\begin{array}{r}
\lb\partial_t\bfv\ts\bfb\rb 
= -\lb\ep_{ijk} b_k \partial_j p\rb
+ (1/3)\lb \bfb\cdot\curl\bfb\rb\bB_m
- (1/3)\lb \bfb^2\rb(\curl\bbB)_m
+ (1/3)\lb \bfb\cdot\curl\bfb\rb\bB_m \\
- (1/3)\lb \bfb^2\rb\curl\bB_m=
-\lb\ep_{ijk} b_k \partial_j p\rb+
(2/3)\lb\bfb\cdot\curl\bfb\rb\bB_m
-(2/3)\lb\bfb^2\rb\bB_m
\end{array}
\label{bfv}
\ee
Now we must deal with the pressure.  Take divergence
of nav stokes, and use $\nabla^2 p= -k_2^2p$, then 
we have
\beq
\begin{array}{r}
-k_2^2p = -\partial_j(v_i\partial_iv_j)-\partial_j^2 b^2/2
-\partial_j\bB_i\partial_jb_i-\bB_i\partial_j^2b_i
-\partial_jb_i\partial_j\bB_i\\
-b_i\partial_j^2\bB_i+\partial_jb_i\partial_ib_j
+\partial_j\bB_i\partial_ib_j+\partial_jb_i\partial_i\bB_j.
\end{array}
\ee
Now we  ignore terms without $\bbB$ as they are 
terms which eventually don't contribute in this closure scheme.
Also ignore $\partial_j^2$ term on $\bbB$ compared to
$\partial_j^2$ term on $\bfb$. Replace the latter with  
with multiplication by $-k_2^2$.  
Then divide overall by $-k_2^2$ and collect
like terms. The result is then
\beq
p = (2/k_2^2)\partial_q\bB_i\partial_q b_i-\bB_ib_i
-(2/k_2^2)\partial_ib_q\partial_q\bB_i
\ee
Now take the gradient, and place into the relevant
term of $\lb\partial_t\bfv\ts\bfb\rb$.
In what follows I ignore terms which have two derivatives
on $\bbB$  as they turn into terms which are $(k_1/k_2)^2$ down
from other terms. We have then
\beq
\begin{array}{r}
\lb\ep_{mjk}b_k\partial_j p\rb
=(2/k_2^2)\lb\ep_{mjk}b_k\partial_j\partial_q b_i\rb\partial_q\bB_i
-\lb\ep_{mjk}b_kb_i\rb\partial_j\bB_i-\lb\ep_{mjk}b_k\partial_j b_i\rb\bB_i\\
-(2/k_2^2)\lb\ep_{mjk}b_k\partial_j\partial_i b_q\rb\partial_q\bB_i.
\end{array}
\ee
Isotropizing the 3rd and 4th terms and rewriting we have
\beq
\begin{array}{r}
\lb\ep_{mjk}b_k\partial_j p\rb
=-(1/3)\lb b^2\rb(\curl \bbB)_m + (1/3)\lb\bfb\cdot\curl\bfb\rb\bB_m
+(2/k_2^2)\lb\ep_{mjk}b_k\partial_j\partial_q b_i\rb\partial_q\bB_i\\
-(2/k_2^2)\lb\ep_{mjk}b_k\partial_j\partial_i b_q\rb\partial_q\bB_i
\end{array}
\label{last}
\ee
Now we must work on the last two terms. Consider the penultimate term
first. The procedure to isotropize the correlation is to take 
$\lb\ep_{mjk}b_k\partial_j\partial_q b_i\rb =
\ep_{miq} Q$ where $Q$ is isotropic. Then mulitply both sides
by $\ep_{miq}$. Then we get $Q=(1/6)(\lb b_q \partial_j^2 b_q\rb$
But $\partial_j^2 = -k_2^2$. 
Thus $\ep_{miq} Q=(-k_2^2/6)\lb \bfb^2 \rb \ep_{miq}$ 
Thus upon plugging this into the penultimate term of (\ref{last})
this term contributes $(-1/3)\lb\bfb^2\rb(\curl \bbB)_m$.
A similar procedure for the last term in (\ref{last}) produces
an identical contribution.  Thus  collecting terms in 
(\ref{last}) produces  
\beq
\lb\ep_{mjk}b_k\partial_j p\rb
=-\lb \bfb^2\rb(\curl \bbB)_m + (1/3)\lb\bfb\cdot\curl\bfb\rb\bB_m
\label{last}
\ee
This contribution comes into $\lb\partial_t\bfv\ts\bfb\rb$
with an overall minus sign, thus 
\beq
-\lb\ep_{mjk}b_k\partial_j p\rb
=\lb \bfb^2\rb
(\curl \bbB)_m 
- (1/3)\lb\bfb\cdot\curl\bfb\rb\bB_m.
\label{last}
\ee
Note the fact that 3 terms have combined to give an overall
coefficient of 1 on the second term.
Now (\ref{last}) must be added to  the last 2 terms in (\ref{bfv})
so we get overall,
\beq
\lb\partial_t\bfv\ts\bfb\rb=
(1/3)\lb\bfb^2\rb (\curl \bbB)_m 
+ (1/3)\lb\bfb\cdot\curl\bfb\rb \bB_m 
\label{final}
\ee
which explains the signs of the 
contributions to the $\bfb$ term  coefficients in equation (9).





%\end{document}



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\def\bbA{\bar{\bf A}}
\def\bA{\bar {A}}

\def\bff{{\bf f}}
\def\bfx{{\bf x}}
\def\BS{{\bf S}}
\def\bV{\bar V}
\def\bfa{{\bf a}}
\def\bfh{{\bf h}}
\def\bfS{{\bf S}}

\def\a{{\alpha}}
\def\be{{\beta}}
\def\ga{\gamma}
\def\de{\delta}
\def\ep{\epsilon}
\def\ve{\varepsilon}
\def\ze{\zeta}
\def\et{\eta}
\def\th{\theta}
\def\vt{\vartheta}
\def\io{\iota}
\def\ka{\kappa}
\def\la{\lambda}
\def\vpi{\varpi}
\def\rh{\rho}
\def\vr{\varrho}
\def\si{\sigma}
\def\vs{\varsigma}
\def\ta{\tau}
\def\up{\upsilon}
\def\ph{\phi}
\def\vp{\varphi}
\def\ch{\chi}
\def\ps{\psi}
\def\om{\omega}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\Th{\Theta}
\def\La{\Lambda}
\def\Si{\Sigma}
\def\Up{\Upsilon}
\def\Ph{\Phi}
\def\Ps{\Psi}
\def\Om{\Omega}
\def\lb{\langle}
\def\rb{\rangle}
%





\def\b0{b^{(0)}}
\def\v0{v^{(0)}}
\def\w0{\omega^{(0)}}
\def\bb0{\bfb^{(0)}}
\def\bv0{\bfv^{(0)}}
\def\bw0{\bfw^{(0)}}
\def\bj0{\bfj^{(0)}}
\def\bbj{\bar {\bf j}}
\def\bbm{\bar {\bf m}}




\subsection{Folded field line structure} \label{structure}

The magnetic field, in the course of amplification and entanglement, develops
structure not identifiable in the power spectrum. Define

\begin{equation} \label{eq:sa}
k_\perp^2 = <(\bb \times \nabla \times \bb)^2> / <b^2>^2 \hspace{7mm}
k_\parallel^2 = <(\bb \cdot \nabla \bb)^2> / <b^2>^2
\end{equation}
\begin{equation} \label{eq:sb}
k_P^2 = <\left( \nabla \frac{b^2}{2} \right)^2> / <b^2>^2 \hspace{7mm}
k_\circ^2 = <(\bb \cdot \nabla \times \bb)^2> / <b^2>^2
\end{equation}
%\begin{equation} \label{eq:sc}
%C_S = <( \bb \cdot (\nabla \vv) \cdot \bb)^2 >
%\end{equation}

%\begin{equation} f^{-1} = \frac{\lambda_\perp}{\lambda_\parallel} \sim
%\left( \frac{c_\parallel}{c_\perp} \right)^{1/2} \end{equation}

Define a field line folding factor $f = k_\perp/k_\parallel$,
which parameterizes the magnetic tension per energy and the unwinding
time of field lines.

%which has the
%following dynamical significance. When the resistive scale is
%$\lambda_\eta = \lambda_\perp$, the magnetic force term $b \cdot \nabla b$
%responsible for the unfolding of field lines scales as
%$b_{\lambda_\perp}^2 \lambda_\parallel^{-1}$.


The equations of magnetohydrodynamics (MHD) are
\begin{equation} \label{eq:momentum}
\rho\left(\partial_t \vv + \vv \cdot \bnabla \vv\right)
= - \bnabla\left(P + \Pi + \frac{B^2}{8 \pi}\right)
+ \frac{1}{4 \pi} \B \cdot \bnabla \B + \rho \nu \nabla^2 \vv
\end{equation}
\begin{equation} \label{eq:induction}
\partial_t \B = \bnabla \times ( \vv \times \B) + \lambda\nabla^2 \B
\end{equation}
\begin{equation} \label{eq:continuity}
\partial_t \rho+\bnabla \cdot (\rho\vv) = 0,
\end{equation}
%\begin{equation} \label{eq:passive}
%\partial_t c + \bnabla \cdot (c \vv) = \nu_c \nabla^2 c
%\end{equation}
\begin{equation} \label{eq:divB}
\bnabla \cdot \B = 0
\end{equation}
\begin{equation} \label{eq:braginski}
\Pi = - 3 \nu_{ii} (\hat{B}\hat{B} - \frac{1}{3}I)
(\hat{B}\hat{B} - \frac{1}{3}I) : \nabla v
\end{equation}
\vspace{0.2 cm}

%+ A \nabla \left[ \left((\nabla \times \B) \times \B \right) \times \B \right]

%The concentration of a passive scalar advected by the fluid evolves
%according to

\begin{center}
\begin{tabular}{llll}
\hline
$\vv$     & fluid velocity        & $\B$  & magnetic field            \\
$\rho$    & fluid density         & $P$    & fluid pressure            \\
$\nu$     & momentum diffusivity  & $\eta  $& magnetic diffusivity      \\
$c$       & passive scalar density& $\nu_c$& passive scalar diffusivity\\
$\Pi$     & Braginsky pressure    & $\hat{B}$& Magnetic unit vector \\
%$\nu_{ii}$& ion   diffusivity     & $A$    & ambipolar parameter       \\
%$n_i$     & Ion number density    & $n_n$  & Neutral number density    \\
%$\sigma_{ni}$& Neutral-ion cross section \hspace{2mm}&$v_{th}$& Thermal velocity\\
\hline
\end{tabular} \end{center}

%$\sigma$ & electrical conductivity & & \\
% = 1/(n_n n_i m_i \sigma_{ni} v_{th})$&


\subsection{Index of simulations}

\begin{center} {\it Table \tablesim: Index of simulations} \end{center}
\small
\begin{center}
\begin{tabular}{rcccccccl}
\hline
Sim  & Grid & $f_h$ & $s_f$ & $\nu_v$ & $\nu_b$ & Pr & $\epsilon_f$ & Notes\\
\hline
 A0&$64^3$&0.0&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
 A1&$64^3$&0.1&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
 A2&$64^3$&0.2&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
 A3&$64^3$&0.3&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
 A4&$64^3$&0.4&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
 A5&$64^3$&0.5&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
 A6&$64^3$&0.6&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
 A7&$64^3$&0.7&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
 A8&$64^3$&0.8&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
 A9&$64^3$&0.9&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
A10&$64^3$&1.0&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
A11&$64^3$&0.4&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&Start with large-sacle field\\
\hline
\end{tabular}
\end{center}
\vspace{2mm}
\normalsize


\subsection{Computational Pitfalls}

A few subtle dangers threaten the validity of turbulent dynamo simulations.

1. For saturated dynamics, magnetic unwinding causes kinetic motion and
viscous dissipation at scales below $\lambda_{\nu_v}$ (section
Z). Therefore, the use of a hyperviscous operator in the Navier-Stokes
equation of the form $\nu_n \nabla^{2n} v$, where $n>1$, is not valid.
If used, it would anomalously destroy small-scale magnetic fields,
giving the false impression that the saturated magnetic spectrum has
a larger-scale character for the absence of a large dynamical range.

2. In hydrodynamic turbulence, inertial range dynamics are insensitive to
whether energy is removed at the inner scale by physical diffusivity
or by dealiasing. In MHD turbulence, dealiasing destroys folded
field line structure, increasing the unwinding and viscous dissipation
by a factor of $(\lambda_\parallel/\lambda_\perp)^2$, which can be as
large as $200$ in our simulations. Therefore, the resistivity must be
large enough to prevent magnetic energy from reaching the aliasing
scale.

3. Our choice of periodic 



\appendix

\section{Code} \label{code}

We evolve the incompressible MHD equations in Fourier space where they
take the form (Lesieur 1990)
\begin{eqnarray}
&
\partial_t \tilde{v}_\alpha
= -i k_\gamma \left( \delta_{\alpha\beta} - \frac{k_\alpha k_\beta}{k^2}
\right)
  \left( \widetilde{v_\beta v}_\gamma - \widetilde{b_\beta b}_\gamma \right) -
\nu_n k^{2n} \tilde{v}_\alpha& \label{eq:Fmomentum}, \\
&\partial_t \tilde{b}_\alpha = -i k_\beta
(\widetilde{v_\beta b}_\alpha - \widetilde{b_\beta v}_\alpha) - \nu_n k^{2n}
\tilde{b}_\alpha, &
\label{eq:Finduction}\\
& k_\alpha\tilde{v}_\alpha=0, \hspace{15mm}
  k_\alpha\tilde{b}_\alpha=0, &\label{eq:Fdivb} \\
&\partial_t \tilde{c} = -i k_\beta \widetilde{v_\beta c} - \nu_n
k^{2n} \tilde{c}. \label{eq:Fpassive}&
\end{eqnarray}

%& k_\alpha\tilde{v}_\alpha=0, &\label{eq:Fdivv}\\
%& k_\alpha\tilde{b}_\alpha=0, &\label{eq:Fdivb}\\

Equations \ref{eq:Fmomentum}, \ref{eq:Finduction}, and
\ref{eq:Fpassive} constitute a system of ordinary differential
equations with time as the dependent variable and the Fourier
coefficients $\{\tilde{v}_\alpha,\, \tilde{b}_\alpha,\, \tilde{c}\}$
as the independent variables. We employ a modified version of the
second order Runge-Kutta algorithm (RK2) to advance the variables in
time.

We make one departure from RK2 and treat diffusive terms with an
integrating factor. Consider an equation of the form
\beq
\partial_t \tilde{q}(k) = A - \nu_n k^{2n} \tilde{q}(k),
\label{eq:modeleq}
\ee
where $A$ comprises the non-diffusive terms.  Its solution, with $A$
constant throughout the interval $\Delta t$, is
\beq
\tilde{q}(\Delta t) = \left[ \tilde{q}(0) + \frac{A}{\nu_n k^{2n}}
(e^{\nu_n k^{2n}\Delta t} - 1) \right] e^{-\nu_n k^{2n}\Delta t}
\label{eq:modifiedE1}
\ee
We use this expression in place of E1 in each stage of RK2. To
lowest order in $\nu_n k^{2n}\Delta t$, equation \ref{eq:modifiedE1}
reduces to E1. However, it has the advantage that it yields stable solutions
to equation \ref{eq:modeleq} with constant $A$ for arbitrary values of
$\nu_n k^{2n}\Delta t$ whereas E1 yields unstable solutions for
$\nu_n k^{2n}\Delta t>2$

We employ standard dealiasing according to the $2/3$ rule (Canuto 1988)
except with the Braginsky and ambipolar terms, which require special
treatment (section \ref{braginsky}).



%Energy loss due to resistivity and dealiasing.

Define the one-dimensional kinetic and magnetic energy spectra as
\begin{equation}
E_v = \int E_v(k) d\!k \hspace{15mm} E_b = \int E_b(k) d\!k
\end{equation}





It is useful show graphically the distinction between the various scales 
of in situ amplification, to better emphasize this point.
The relevant scales for the Galaxy, in decreasing magnitude, are the
large scale coherence scale ($\lambda_g$), the forcing scale
($\lambda_f$), the viscous scale ($\lambda_\nu$), and the resistive
scale ($\lambda_\eta$).  Figure \figstages\, summarizes the kinetic
and magnetic spectra at each stage of the dynamo. 
%Table \tablegalaxy\,
%contains a list of sample parameters for the galactic and
%protogalactic environments.



The small-scale dynamo proceeds by field line stretching in 
linear and nonlinear stages. In the linear stage the magnetic field is very
weak and unable to affect the kinetics. Kulsrud and Anderson (1992)
predict exponential magnetic growth with a $k^{3/2}$ magnetic
spectrum terminating at the resistive scale.
This is confirmed by previous non-helical numerical simulations.

Simulations in the nonlinear regime 
differ from what might be expected by extension
of the Kulsrud-Anderson picture.
The nonlinear stage commences when the magnetic energy
approaches the viscous-scale kinetic energy.   
Instead of the field saturating on progressively
larger and larger scales, building up to near equipartition
on the input scale, simulations show that the field piles up on the smallest
scales when the forcing is non-helical. 
Although the total magnetic energy saturates to a value in near
equipartition with the kinetic energy, 
the fact that the small-scale magnetic field
of the Galaxy is observed to occur on the input scale of the turbulence
means that the non-helical simulation results contradict
the observations. In addition, this also has implications for protogalactic
models of the present large scale galactic field which thrive on 
a small-scale dynamo.

In this paper we study the 
role kinetic helicity plays determining the spectrum 
of small-scale magnetic fields. 
We show how a source of kinetic helicity in the turbulence
allows the magnetic field 
to build up on the input scales.  The results are found 
to depend on the fraction of input kinetic helicity.
%if the helicity fraction
%is too low, then there is no significant buildup on the input
%scale. If the helicity fraction is too high, then the Kolmogorov
%spectrum is not maintained. But for an intermediate range, the
%helicity alone can solve the problem.  The reason for these effects
%is that the fractional kinetic helicity partially inhibits the cascade
%of kinetic energy, leaving some kinetic energy at large scales long enough
%to stretch the magnetic field there before draining.
%Our study revelas that the
%kinetic helicity plays a fundamental role in the 
%generation of the magnetic field on the input scale of the
%turbulence, in addition to the role kinetic helicity plays
%for the generation of the large scale field.



\section{Nature of the Turbulent ISM}

There are four primary  phases of the ISM
(Spitzer 1978, Heiles 2000). Below, $n_p$ and $n_e$ are the
hydrogen and free-electron densities in $cm^{-3}$ $T$ is the temperature in $K$,
$b_{rms}$ is the RMS magnetic field in $Gauss$, and $v_s$, $v_A$, and $V_>$ 
are the acoustic, Alfv\'en, and dominant kinetic turbulent speeds in cm/s.
The last column shows the ratio of the dominant kinetic to magnetic
energy scales of the turbulence.

{\center
\begin{tabular}{lcccccccc}
\hline
ISM Phase&$n_p$&$n_e$&$T$&$b_{\rm rms}$&$v_S$ & $v_A$ & ${L_>\over L_{>,b}}$\\
\hline
Cloud&$>1000$&$>10^{-2}$&$30$  &$>3\cdot10^{-4}$&$>5\cdot10^4$&$10^6\sim V_>$&$\sim 1$\\
Cold &$1$    &$10^{-5}$ &$80$  & $3\cdot10^{-6}$&$10^5      $&$10^6\sim V_>$&$\sim1$\\
Warm &$0.01$ &0.01      &$8000$& $3\cdot10^{-7}$&$10^6      $&$10^6\sim V_>$&$\sim1$\\
Hot  &$0.003$&$0.003$   &$10^6$&$       10^{-7}$&$10^7      $&$10^6\sim V_>$&$\sim 1$
\\
\hline
\end{tabular} \\
}

\begin{center} {\it Table \tableism: ISM parameters} \end{center}

The warm phase, alternatively known as the diffuse ionized gas (DIG)
or the warm ionized medium (WIM), is generally associated with
ionizing stars.  
The cloud phase consists of
gravitationally condensed matter and is host to star forming activity.
The hot phase is associated with supernova remnants
and superbubbles, and tends to dominate in the galactic corona more
than 100 pc from the disk midplane. 














(2) A plasma with a signifigant fraction of neutrals
which then isotropize the viscosity. 
(3) A sufficiently magnetized quasi-netural plasma 
such that ambipolar diffusion arises from 
the relative ion-neutral motion (section \ref{sec:ambipolar}).
(4) The somewhat artifical but simple case 
where the plasma is fully ionized but the viscosity and 
resistivity (section \ref{sec:pure}) are isotropic. 



\subsection{Scales} \label{Simulation} \label{scales}

We state our assumptions for the parameters of galactic turbulence in
table \tablegalaxy, followed by a discussion of our inability to
directly simulate them.

\begin{center}
\begin{tabular}{llll}
\hline
Quantity          &Symbol           & Galactic & Protogalactic \\
\hline
Magnetic coherence scale&$\lambda_g$&$10^{22}$ cm   & - \\
Forcing scale     &$\lambda_f$    &$10^{21}$ cm    &$10^{23}$ cm \\
Forcing velocity  &$v_f$          &$10^{6}$ cm/s   &$10^{7}$ cm/s \\
Viscous scale     &$\lambda_\nu$  &$10^{17}$ cm    &$10^{22}$ cm\\
Resistive scale   &$\lambda_\eta$ &$10^{12}$ cm    &$10^{12}$ cm\\
Viscosity         &$\nu$          &$10^{20}$ ${\rm cm}^2/s$ &$10^{29}$
                                                              ${\rm cm}^2/s$ \\
Magnetic diffusivity&$\eta$       &$10^{14}$ ${\rm cm}^2/s$ &$10^{10}$
                                                              ${\rm cm}^2/s$ \\
Prandtl number    &$P$            &$10^6$          &$10^{19}$\\
Neutral free path &$\lambda_{nn}$ &$10^{15}$ cm    & - \\
Ion free path     &$\lambda_{ii}$ &$10^{13}$ cm    &$10^{22}$ cm \\
Temperature       &$T$            &$10^4$ K        &$10^7$ K \\
Proton density    &$n$            &1${\rm cm}^{-3}$&$10^{-2}{\rm cm}^{-3}$ \\
\hline
\end{tabular}
\end{center}
\begin{center} {\it Table \tablegalaxy: Galactic turbulence parameters}
\end{center}


Figure \figfinalstate\, illustrates the nonlinearly saturated spectra
for a sequence of viscosities, all at fixed resistivity. The magnetic
spectra all fall in the same place, with the exception of the one
associated with the highest viscosity. 
%This suggests that the 
%kinematics at any scale other than the forcing scale are irrelevant to
%the cascade. The kinetic spectra also have greatly reduced magnitudes
%compared to the magnetic spectra.





\begin{center} {\it Table \tablesim: Index of simulations} \end{center}
\small
\begin{center}
\begin{tabular}{rcccccccl}
\hline
Simulation & Grid & $f_h$ & $s_f$ & $\nu_v$ & $\nu_b$ & Pr & $\epsilon_f$ 
& notes
\\
\hline
 A0-A3\ ;\ A4-A10
 &$64^3$&0.0,0.05,0.1,0.2\ ; \ 0.4-1.0&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A4-A10&$64^3$&
%0.4-1.0&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A1&$64^3$&0.05&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A2&$64^3$&0.1&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A3&$64^3$&0.2&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A4&$64^3$&0.4&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A5&$64^3$&0.5&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A6&$64^3$&0.6&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A7&$64^3$&0.7&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A8&$64^3$&0.8&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A9&$64^3$&0.9&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A10&$64^3$&1.0&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&\\
% A11&$64^3$&0.0&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&
%                                                 Start from large-scale field\\
% A12&$64^3$&0.4&4.5&$3\cdot10^{-3}$&$1\cdot10^{-3}$&   3&5&
%                                                 Start from large-scale field\\
%B0&$128^3$&0.0&4.5&$3\cdot10^{-3}$&$3.3\cdot10^{-4}$&   9&5&\\
%B4&$128^3$&0.4&4.5&$3\cdot10^{-3}$&$3.3\cdot10^{-4}$&   9&5&\\
%B5&$128^3$&0.5&4.5&$3\cdot10^{-3}$&$3.3\cdot10^{-4}$&   9&5&\\
%B6&$128^3$&0.6&4.5&$3\cdot10^{-3}$&$3.3\cdot10^{-4}$&   9&5&\\
%B7&$128^3$&0.7&4.5&$3\cdot10^{-3}$&$3.3\cdot10^{-4}$&   9&5&\\
%C0&$128^3$&0.0&2.5&$1\cdot10^{-3}$&$2.5\cdot10^{-4}$&   4&2&\\
%C4&$128^3$&0.4&2.5&$1\cdot10^{-3}$&$2.5\cdot10^{-4}$&   4&2&\\
%C5&$128^3$&0.5&2.5&$1\cdot10^{-3}$&$2.5\cdot10^{-4}$&   4&2&\\
%C6&$128^3$&0.6&2.5&$1\cdot10^{-3}$&$2.5\cdot10^{-4}$&   4&2&\\
%C7&$128^3$&0.7&2.5&$1\cdot10^{-3}$&$2.5\cdot10^{-4}$&   4&2&\\
%C9&$128^3$&0.9&2.5&$1\cdot10^{-3}$&$2.5\cdot10^{-4}$&   4&2&\\
\hline
\end{tabular}
\end{center}
\vspace{2mm}
\normalsize


\begin{center} {\it Table \tablescales: Computational scales} \end{center}
\begin{center}
\begin{tabular}{lllll}
\hline
$v_\lambda$&Velocity at scale $\lambda$&\hspace{1cm}&$b_\lambda$&
Magnetic field at scale $\lambda$\\
$\nu_v$ &Kinematic diffusivity &\hspace{1cm}& $\nu_b$& Magnetic diffusivity\\
$\lambda_\nu$&Viscous scale&\hspace{1cm}&$\lambda_\eta$& Resistive scale\\
$\lambda_f$& Forcing scale &&$\vv_f$&Outer scale RMS velocity\\
%$\lambda_s$&Shear scale&\hspace{1cm}&$v_s$&Shear velocity\\
%$t_f=\lambda_f/v_f$&Forcing timescale&\hspace{1cm}&$t_s=\lambda_s/v_s$&Shear
%  timescale\\
%$\lambda_\perp$&Transverse folding scale&\hspace{1cm}&$\lambda_\parallel$&
%Longitudinal folding scale\\
%$f=\lambda_\parallel/\lambda_\perp$&Fieldline folding factor&\hspace{1cm}&
%$s=k/(2\pi)=1/\lambda$& Wavenumber\\
%$\lambda_a=3/N$&Aliasing scale&\hspace{1cm}&$N^3$& Grid size\\
%l_g
\hline
\end{tabular} \vspace{0.2 cm}
\end{center}


In the framework of mean field dynamo theory, the resistively
limited rate  can be briefly understood as follows
Splitting $\bf B,E,J$ into mean
and fluctuating components i.e. ${\bf B}= \bfb+{\bbB}$ 
and assuming no mean velocity we have
\beq 
\partial_t{\bbB}=\curl \lb\bfv\ts\bfb\rb 
%= \curl( \alpha{\bbB}-\curl\beta\curl\bbB)
\label{d1}
\end{equation}
We can write the turbulent EMF $\lb\bfv\ts\bfb\rb/c$
as
\beq
\lb\bfv\ts\bfb\rb\cdot\bbB/c =-\lb{\bf E}\cdot{\bf B}\rb+\eta{\overline {\bf J}}\cdot{\bar {\bf B}} 
=-\eta\lb\bfj\cdot\bfb\rb-\lb{\bf e}\cdot\bfb\rb
\ee
% $$\lb{\bf E}\cdot{\bf B}\rb-\eta{\bar {\bf J}}\cdot
%{\bar {\bf B}}=
%
%\beq
%\lb\bfv\ts\bfb\rb\cdot\bbB/c = -\eta\lb\bfj\cdot\bfb\rb -
%\lb\bfe\cdot\bfb\rb
%\ee
Now from conservation equation of small scale magnetic
helicity, the last term is the sum of time and spatial divergence:
\beq
\lb{\bf e}
\cdot\bfb\rb = -\partial_t\lb\bfa\cdot\bfb\rb/2c+\nabla\cdot
\lb\bfa\ts \bfe -\phi\bfb\rb
\ee
But if this vanishes 
if $\lb \ \rb$ is periodic \& stationary.
Thus 
\beq
\lb \bfv \ts \bfb \rb\cdot\bbB/c 
= \alpha\bbB^2 -\beta\bfB\cdot\curl \bbB
=-\eta\lb\bfj\cdot\bfb\rb.
\ee
Moreover, the equation for helicity evolution for the
large scale field is given by 
\beq
\partial_t ({\overline {\bf A}}\cdot{\overline {\bf B}})
=-2c\bbE\cdot\bbB= 2\lb \bfv \ts \bfb \rb\cdot\bbB 
\label{abar}
\ee

Thus one can see the direct dependence of the turbulent
EMF (which determines the mean field growth) on the resistivity
when the boundary conditions are periodic.


%\beq
%\alpha \le \alpha_{max} R_m^{-1} {(\bbB/\bfb})^{-2}
%\ee

%$\rightarrow$  ``quenching'' from boundary conditions

The difficulty is to understand which approximations 
are sufficient to properly test mean field theory.
It is likely that a more realistic global
disk simulation with open boundaries  is necessary.


What limits the spectral shape of the small scale field











The free electron density $n_e$ constitutes a
spatially-varying index of refraction responsible for the
scintillation of pulsar radio sources.  Assuming that 
fluctuations in entropy, and hence $n_e$, occur in pressure
equilibrium and evolve as a passive scalar, the turbulent dynamics
determines the observed spectrum of $n_e$ fluctuations.
More specifically, 
the scattering measure is defined ${\rm SM} = \int C_n^2(z) dz$,
where the electron density spectrum is given by $E_{n_e}(k,z) =
C_n^2(z) k^{-\alpha}$ and $z$ is the path from the source to the
observer. Assuming a Kolmogorov form and an outer scale $\lambda_>$,
we have the relation $C_n^2 \sim n_e^2 \lambda_>^{-2/3}$.







At saturation, the magnetic backreaction opposes magnetic shear growth
at time scales shorter than the forcing time scale. Shear at smaller
scales exists, but is best interpreted to be the result of energy transfer
from the magnetic field to the velocity, rather than the reverse. 
Alternatively, 
energy can be exchanged cyclically between $\vv$ and $\bb$, such as
for spandex waves (section \ref{spandex}).  If additionally the
magnetic field loses only a small fraction of its energy to the
velocity during its cascade from the forcing to the resistive scale
(section \ref{unwinding}), then the magnetic cascade will be
independent of the kinematic dynamics. Hence, the magnetic spectrum
will be independent of viscosity.


Armstrong, Rickett, \& Spangler (1995) 
summarize our knowledge of
turbulence obtained from various pulsar lines of sight.  Information
about the smallest scales ($10^{9} {\rm cm}$) comes from diffractive
scintillation, angular broadening, and decorrelation bandwidth
observations. Refractive scintillation and dispersion measure
fluctuations probe structure at scales of $\sim 10^{14}$ cm. Rotation
measure and direct angle-resolved observations cover the outer scale
conditions at $\sim 10^{19}$ cm. By splicing together the information
at each scale, they obtain an averaged spectrum with $C_n^2 \sim 5
\cdot 10^{-17} {\rm cm}^{-20/3}$ and a power law index consistent with
a uniform value of $5/3$ throughout the 11 decades of range. The
uniformity of the index plus the fact that it is smaller than 2
suggest a turbulent process. Assuming a value of $n_e \sim 10^{-2}$ in
the dominant regions of scintillation, we have an outer scale of
$\lambda_> \sim 3\cdot 10^{18}$.  They also place an upper bound on
the inner scale of $10^{10}$ cm and mention that much smaller values
can exist on the most active lines of sight.




To roughly estimate the required  turbulent energy dissipation rate in
the ISM we take an the observed outer scale
velocity of $v_> \sim 10$ km/s, a lengthscale of $L_> \sim 100$ pc, an
average density of $n_p \sim 0.1 {\rm cm}^{-3}$, a hot ISM scale height
of $500$pc, and radius of $10$kpc. This gives 
the turbulent dissipation rate $V_>^3/L_> \sim 
10^{41} {\rm erg s^{-1}}$.  
%Given galactic dimensions of $r_\star \sim 10$ kpc and $h_\star \sim
%200$ pc, we have an energy dissipation rate of $\sim $ erg $cm^{-3}$
%$s^{-1}$.
The primary sources of 
this turbulence in the ISM is supernovae, and stellar winds.
A supernova injects $\sim 10^{51}$ ergs of energy into the ISM, and
event which occurs every $\sim$ 100 years. As the supernova ejecta
expands, some fraction of the energy $(\sim 4 \%)$ (Dyson \& Williams
1987) goes into kinetic motion of the surrounding ISM, with the rest
going into heat. The turbulent energy input associated with supernovae
is therefore $\sim 3 \cdot 10^{41}$ erg/s.  This is the principal
origin of the hot turbulent coronal gas.
Massive O stars contribute energy to the interstellar medium through
winds and the expansion of ionization fronts. The winds tend to have
velocities of 2000 km/s and a mass flux of $10^{-6} M_\odot
yr^{-1}$, implying a mechanical energy output of $10^{36}$ erg/s.
There are $\sim$ 20,000 O stars (Mathis 1986, ApJ301,423),
giving an energy input rate of $2 \cdot 10^{40}$ erg/s.






%Observations of anisotropic scattering (Molnar et al. 1995, Spangler
%\& Cordes 1998) suggest an origin in anisotropic electron density
%fluctuations.

%\subsection{Overall Energetics and the Role of Forced Turbulence}









Our results have direct implications for the 
Galaxy because observations indicate that indeed its
small scale field is in near equipartition with the turbulent kinetic energy
at the forcing scale of 100pc and that the kinetic spectrum is Kolmogorov.
This is not reproducible with simulations that do not include kinetic
helicity. 
Significant kinetic helicity is thus at minimum,
playing a role in maintaining the magnetic 
energy spectrum peaked at the input scale, aside from any
role it may play in generating the large scale field. 
galactic dynamo.  






For the Galaxy, 
Faraday rotation and synchroton polarization observations 
reveal that the Galaxy contains 
both a small scale field and a large scale field.
By small scale we mean ordered on scales less than the turbulent
forcing scale ($\sim 100$pc, from  supernova and stellar winds)
at which the turbulent kinetic energy spectrum peaks.
By large scale we mean ordered on scales $>>$ forcing scale,
$\sim 2-3$ kpc for the Galaxy as determined  by the distances between
radial field reversals. (Zweibel \& Heiles) (Rand \& Kulkani) (Beck et al.).

The magnetic energy density in the small scale field of the ISM dominates that 
of the large scale field by a factor 10 or so, and the energy
density of the small scale field is in near equipartition with the kinetic
turbulent energy on the 100pc input scale. 
The magnetic energy density thus peaks at the same
scale the kinetic energy spectrum peaks. The spectrum of the latter
is determined to be Kolmogorov from the $100$pc input scale down $11$ orders
of magnitude from a range of scintillation and direct
imaging observations.

The large scale field origin has traditionally been a subject of debate.
Is the large scale field mainly produced
in situ, or is it mainly a residual
of flux freezing from amplification in the protogalaxy or primordial gas?
The controversy on this issue has to do with how the backreaction
of the growing small scale field and the growing large scale field 
affects the rate of growth of the large scale field itself.
This question has persisted for decades 
because the problem is inherently very non-linear and
so is difficult to understand without a feedback loop between
numerical simulations and analytic studies. This has been
improving recently.  








The small scale field growth at the forcing scale and the mean
field dynamo growth at the large scale
are not necessarily inconsistent.



%-----------------------------------------------------------------
\hbox to \hsize{ \hfill \epsfxsize7cm \epsffile{final.state.128.ps}
\hfill \epsfxsize7cm \epsffile{final.state.256.ps} \hfill}
\vspace{-4mm} \noindent {\it Figure \figfinalstate: The saturated
spectra for a sequence of viscosities and fixed resistivity. Lines
with points indicate the kinetic spectra, and lines without points
indicate the magnetic spectra. The simulations in the left and right
panels have $128^3$ and $256^3$ resolution, respectively}


%------------------------------------------------------------------------------
\vspace{-3.5cm} \hbox to \hsize{ \hfill \epsfxsize10cm
\epsffile{scales.ps} \hfill } \noindent {\it Figure \figstages: Kinetic and
magnetic energy spectra for successive stages of the galactic dynamo.}


We have studied the role of kinetic helicity 
in determining the saturated shape of the magnetic energy spectrum
in forced incompressible MHD turbulence. 
While our numerical simulations show that a force free large
scale magnetic field grows in a slow dynamo (e.g. at a resistively
limited rate) in our periodic box as seen also in Brandenburg (2000), our 
main focus here is on the saturated state of the
small scale dynamo. The implications for the small scale dynamo 
are likely less sensitive to our choice of periodic boundary conditions, 
the absence of rotational shear, and the absence of density stratification.

We have found that the peak of the spectrum for the small scale dynamo moves
from the resistive scale to the input scale when the forced
kinetic helicity fraction is $\ge 0.5$, while the kinetic energy
spectrum maintains its Kolmogorov shape.    
We have discussed how this basic shift in the magnetic peak occurs because 
of two features (1) the conservation of magnetic helicity and
(2) the fact that the non-linear cascade
term for the velocity in the Navier-Stokes equation decreases 
with increasing kinetic helicity. 


The saturated state is characterized by equal kinetic and
magnetic energy densities, and the magnetic backreaction neutralizes
shear motions having less energy than the magnetic field.  The,
only shear capable of transferring energy to the magnetic field is
that from the forcing scale, and the magnetic cascade proceeds from the
forcing scale to the resistive scale according to one time scale, the
forcing time scale.  Unlike the helicity free case,  for the helical
case, the field is also more effectively  being resupplied at the input scale
and the buildup of the small scale fields
is combatted by the inverse cascade of magnetic helicity.
All of the above shows that at least for a periodic box, the dynamics
are strongly influenced by the resistive scale.

















































turbulence (leading to the $\alpha$ effect), differential rotation 
(the $\Omega$ effect),  
and turbulent diffusion (the $\beta$ effect) to exponentiate an initial seed mean magnetic field. 
Ref. \cite{steenbeck} developed a 
formalism for describing the concept \cite{parker} that helical turbulence  
can twist toroidal ($\phi$)  fields 
into the poloidal ($r,z$) direction, where they can be
acted upon by differential rotation to regenerate a powerful large scale
toroidal magnetic field.  The turbulent diffusion serves to
redistribute the flux so that inside the bounded volume of interest,
a net flux can grow. 
































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%\maketitle
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As $\OB$ grows, 
it exerts a backreaction on the turbulent flow,
%is large, back reaction cannot be neglected, 
and equations (\ref{n3}) and (\ref{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 (
$H^M=\lb{\bfA {\cnt} \bfA}/2\rb$, with $\bfA$ 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 dimensionless expansion parameter $a << 1$, they found that
$\alpha$ appropriate for a field having a large scale $k^{-1}$ is
***




The mean field $\bbB$ satisfies the induction 
equation (moffatt,steenbeck) 
\beq
{\partial\OB\over \partial t} = -c\curl\bbE,
\label{2.4a} 
\ee
where
\beq
\bbE=-\lb\OV \ts \bbB\rb/c  - \lb\bfv\ts\bfb\rb/c+
\nu_M\nt\OB,
\label{2.4aa} 
\ee
\beq
\lb\bfv\ts\bfb\rb_i = \alpha_{ij}{\bB}_j-\beta_{ijk}\partial_j{\bB}_k
\label{2.5a}
\ee
is the turbulent electromotive force (EMF), and $\nu_M=\eta c^2/4\pi$ is the magnetic
diffusivity defined with the resistivity $\eta$.
Here $\alpha_{ij}$ contains Parker's twisting (the $\alpha$ effect)
and $\beta_{ijk}$ contains the turbulent diffusivity. Ref. (steenbeck)
calculated $\OE$ to first order in $\OB$ for isotropic
$\alpha_{ij}$ and $\beta_{ijk}$ and hence the pseudo-scalar 
and scalar dynamo coefficients $\alpha$ and 
$\beta$ respectively 
to zeroth order in $\OB$ from the statistics of the turbulence,
ignoring the Navier-Stokes equation.  
When the Navier-Stokes equation is not used, we speak of
the ``kinematic theory.'' 
Using the equation for the fluctuating field $\partial_t\bfb$,
 plugging it into $\lb\bfv\ts\bfb\rb$, the standard approach
gives a kinematic $\alpha \propto -\int \lb\bfv(t)\cdot\curl\bfv(t')\rb dt'
\sim -\tau_c \lb\bfv\cdot\curl\bfv\rb dt$
and $\beta\propto\int \lb\bfv(t)\cdot\bfv(t')\rb dt'
 \sim \tau_c \lb\bfv\cdot\bfv\rb dt$.  I will come back
to correcting the form for $\alpha$ in section 2.3 and 2.4.
because we really want a fully dynamic theory, that 
accounts for the dynamo coefficients'
dependence on  $\bfb$ and $\OB$.
 Only then can one fully address the fundamental
problem of mean field dynamo theory:  how does the growing
magnetic field affect the rate and saturation level of growth?
%The back-reaction on the dynamo coefficients to first
%order in $\overline {\bf B}$ and $\overline {\bf V}$ was calculated
%in Ref. 9, and Ref. 10 calculates $\alpha$ 
%to all orders in $\ob$ when mean field gradients are small.
%See also Refs. 11, 12, which find a catastrophically quenched
%alpha, but subject to the issues raised in Ref. 13.
%We will come back to the issue of catastrophic suppression is expected.


Substituting
(\ref{2.4aa}) into (\ref{2.4a}), 
gives the MFD equation
\beq
{\partial\OB\over \partial t} = \nt \lb{\OV \times \OB}\rb 
+\curl(\alpha \bbB)-\curl(\beta +\lambda)\curl \OB. \label{6a}
\ee
The first term on the right is the non-controversial 
$``\Omega-{\rm effect}.''$
If one assumes $\overline {\bf V}$, 
$\alpha$ and $\beta$ to be independent of $\OB$, then 
(\ref{6}) 
can be solved as a linear eigenvalue
problem for the growing modes of $\OB$ 
in the Sun and other bodies.
%Boundary conditions play an important role in allowing net flux growth.
However a rapid growth of the fluctuating field necessarily accompanies the 
MFD in a turbulent medium.  
Its impact upon the growth of the mean
field, and the impact of the mean field itself on its own growth 
have been
controversial.  





Mean field dynamo 

{\it Dynamic Dynamo Equations-}

Define $\emfb$ such that 
\beq
\emfb\cdot\bbB=\emf_{||}|\bbB|=-\lb\bfv\ts\bfb\rb\cdot\bbB
\label{1}
\ee
where $\emf = -\lb\bfv\ts\bfb\rb\cdot\bbB/|\bbB|$.





\beq
\label{}
\ee

\beq
\label{}
\ee

\beq
\label{}
\ee

\beq
\label{}
\eeBelow I will work with the first term only in what follows to show
that it is bounded but the
second one should be similar:
The solution to (\ref{o2}) is 
\beq
b_{i,L}=\int_0^t\{T Exp[\int_{t'}^t M(t'')dt'']\}_{ij}F_j(t')dt'
\ee
where 
\beq
M_{ij}=\partial_iv_j
\ee
and 
\beq
F_j=\bB_m\partial_mv_j-v_m\partial_m\bB_j.
\ee
Let $Q_{ij}(t,t')=[Te^{\int_{t'}^tM(t'')dt''}]_{ij}$,
where $T$ is time ordered product. 
Then
\beq
\lb\bfv\ts\bfb\rb_k=
\int_0^t\lb \epsilon_{kri}v_r(t)Q_{ij}(t,t')\partial_m v_j(t')\rb 
\bB_m(t') dt'
-\int^t_{0}\lb\epsilon_{kri}v_rQ_{ij}(t,t')v_m(t')\rb 
\partial_m\bB_j (t')dt'.
\ee
Thus the exact expressions for the contribution from the
second term in (\ref{o5}) to $\alpha_{km}$ and $\beta_{kmj}$,
assuming we can pull $\OB$ terms out of the time integrals
are 
\beq
\alpha_{km}=\int_{0}^t \lb \epsilon_{kri}v_r(t)Q_{ij}(t,t')\partial_m v_j(t')\rb dt'
\ee
and
\beq
\beta_{kmj}=\int_{0}^t \lb \epsilon_{kri}v_r (t)Q_{ij}(t,t')v_m(t')\rb dt'.
\ee

To estimate the order of magnitude, we have 
\beq
||\alpha_{km}||= ||\int_{0}^t\lb \ep_{kri}v_rQ_{ij}\partial_m v_j\rb dt'||.
\ee
Since $\lb\rb$ and $\int dt$ are integrations, we pull
$||$ inside so that we have 
\beq
||\alpha_{km}||\le \int_{0}^t\lb||\ep_{kri}v_rQ_{ij}\partial v_j||\rb dt'.
\ee
Define $A_{ki}\equiv \ep_{kri}v_r$ and $N_{jm}\equiv M_{mj}=\partial_mv_j$
so that
\beq
||\alpha_{km}||\le\int^{t}_{0}\lb||A_{ki}||||Q_{ij}||||N_{jm}||\rb dt'.
\ee
Similarly, 
\beq
||\beta_{kmj}||\le\int^t_{0}\lb||A_{ki}||||Q_{ij}||||v_m||\rb dt'.
\ee
The offending factor is of course $Q_{ij}$.
For this we have 
\beq
\begin{array}{r}
||Q_{ij}||= ||Te^{\int_{t'}^t M(t'')dt''}||
=||\delta_{ij}+\int_{t'}^tM(t'')dt'' +
\int_{t'}^tM(t'')dt''\int_{t'}^{t'''}M(t''')dt''' +...||
\\
\le {\sqrt 3}  + \int_{t'}^t||M(t'')||dt''+
\int_{t'}^t\int_{t'}^{t'''} ||M(t'')||
||M(t''')||dt'''dt'' +...
\label{schwarz}
\end{array}
\ee
Define $t_{ed}$ such that 
$||M||\le 1/t_{ed}$ for all $t$. Then we  have 
for (\ref{schwarz})
\beq
||Q_{ij}||= ||Te^{\int_{t'}^t M(t'')dt''}||
\le {\sqrt 3} + (t-t')/t_{ed} + (1/2)(t-t')^2/t_{ed}^2 + ...
= {\sqrt 3}-1+ e^{(t-t')/t_{ed}}
\ee
Since this quantity appears inside the correlation
of $\alpha_{ij}$ and $\beta_{ijk}$ we can replace with $t-t'$ with $t_c$ so
we have inside a correlation
 \beq
||Q_{ij}|| \le Exp[(t_c/t_{ed})] +{\sqrt 3} -1.
\ee
That means that 
\beq
||\alpha_{ij}||_{max,tot} -||\alpha_{ij}||_{lowest}
\le (e^{t_c/t_{ed}}-1)\int_0^t\lb||A_{ki}||||N_{jm}||\rb 
\ee
and
\beq
||\beta_{kmj}||_{max,tot} -||\beta_{kmj}||_{lowest}
\le (e^{t_c/t_{ed}}-1)\int_0^t\lb||A_{ki}||||v_{j}||\rb 
\ee
where the subscript ``lowest'' indicates that
norm determined with $Q_{ij}= \delta_{ij}$.

A similar procedure to the above 
would apply for the terms contributing to $\alpha_{ij}$
and $\beta_{ijk}$ from the
second term in (\ref{o5}).

The upshot is that we would be left with our desired
lowest order tensor contributions to $\alpha$ and $\beta$
changed by a small factor for $t_c < t_{ed}$.



It is not sufficien
The reason is that the appropriate helicity which then enters $\alpha$  
is $\lb\bfb^{(0)}\cdot\curl\bfb^{(0)}\rb$ to lowest order. 
Then the $H_1^M$ entering on the left of \pref{n40} would
be second order. But then the magnetic helicity, $H_2^M$, entering 
 \pref{40p} would also be second order. 
Thus the current helicity entering $\alpha$ is zeroth order whereas
that entering the helicity conservation equation would
be second order. There is an ordering mismatch.
This is a problem because the success of the model of 2.3
depends on our being able to circumvent this ordering ambiguity
and presume that the current helicity entering \pref{PFLalpha} is exactly
$k_2^2$ times the $H_2^M$ entering \pref{40p}.



will use is that any surviving mixed correlations are proportional
to $\lb\bfv\ts\bfb\rb$. With a constant that has to be motivated
physically. The 4th term on the right is
of this form. Now we show that for our purposes the pressure term is also
ove this form. 

We will later find that a two-scale approximation
is extremely fruitful, where the small scale
quantites of interest vary on scale $k_2^{-1}$ and 
large scale quantities vary on scale $k_1^{-1}$.
Taking advantage of this here, we can write
\beq
p_{eff}= k_2^{-2} (\nabla\cdot(\bfb\cdot\nabla\bfb)-\nabla\cdot
(\bfv\cdot\nabla\bfv))
\label{peff}
\ee
Upon placing this into (\ref{timed3}) we 
can elimitate the contribution from the first  term on the right 
of (\ref{peff}) by
isotropy.
We then have 
\beq
\begin{array}{r}
\lb\partial_t\bfv\ts\bfb\rb +\lb\bfv\ts\partial_t\bfb\rb=
{1\over 3}(\lb\bfb\cdot\curl\bfb\rb(1-k_1^2/k_2^2)
-\lb\bfv\cdot\curl\bfv\rb)\bbB 
\\- 
{1\over 3}\lb
\bfv^2 + \bfb^2(1-k_1^2/k_2^2)\rb\curl\bbB-{\tilde\zeta}
\lb\bfv\ts\bfb\rb,
\end{array}
\label{timed3}
\ee
where we have placed all of the surviving cross 
correlation terms in ito a single term proportional to $\tilde\zeta$.
This approximation will be further motivated shortly.







only for a value $h_2 < -1$ does $\partial_\tau Q=0$.
When 

Once $h_1=-h_2=1$, then $h_2$ stops growing significantly
and
and 
The solution of the 
(\ref{7})
 in this regime
is given by 
\beq
h_1=h_{1,0}Exp[(2/3)(k_1/k_2)(1-k_1/k_2)t]
\ee
whereas the solution of (\ref{7}) (\ref{9}) in this regime
is given by
\beq
\begin{array}{r}
h_1=h_{1,0}Exp[(2/3)^{1/2}(k_1/k_2)^{1/2}(1-k_1/k_2)^{1/2}\tau]\\
+\tau h_{1,0}\left[
{2\over 3}(k_1/k_2)(1-k_1/k_2)-\left({2\over 3}\right)^{1/2}(k_1/k_2)^{1/2}(1-k_1/k_2)^{1/2}\right]
\end{array}
\ee




In the kinematic regime,




Note that the $\sigma$ term is the one which is ultimately
related to turbulent diffusion, and the $\xi$ term
is the one ultimately related to 
but is not exactly the turbulent diffusion
coefficient because it enters in the equation for the 
time derivative of $\emfb$.
In the isotropic approximation,
the form of $\sigma$ appears not to be suppressed by the growing mean
field but surely when $\bbB$ increases beyond equipartition some suppression
should result and it should not be isotropic.  We are led to believe
that the appropriate quenching of $\sigma$ 
can only be determined by an anisotropic treatment which is
beyond the scope of the current paper. As it turns out however,
the overall solution of our triplet of equations are not 
particularly sensitive to the form
of $\sigma$, just like the results of [\cite{bb02}] and [\cite{fb02}] 
were weakly sensitive to the turbulent diffusion.
In addition, we have also used the isotropic form of $\xi_{ij}$,
the residual helicity tensor, and this should also be 
generalized. It was argued
in [\cite{KR99}] that the dominant contribution 
to the current helicty ternsor was from
the isotropic part. We assume that the forcing justifies taking
the kinetic part as being isotropic. 


 




%\beq
%t={1/2\lambda}[h_1/h_0/(1-k_1^2/k_2^2)
%\label{}
%\ee

\beq
\tau=(k_2/k_1) Ln[(h_1/h_{10})/(1-k_1^2/k_2^2)^{1+R_{m,2}(k_2/k_1-1)/3}]
\label{9}
\ee 





-usual treaments

-pouquet, fb

-but fbc approach

-need pressure argument, 

-[\cite{KR99}] isotropy argument

-need to get

and $\beta$ are parameters of the
underlying MHD turbulence.  Steenbeck et al (1966) showed that if the
turbulence is isotropic and incompressible, and
the back reaction of $\OB$ is neglected,
\beq
\alpha = {1\over 3} \tau\lra{\bfv\cdot\nt\bfv} \label{n3}\ee
and
\beq
\beta={1\over 3} \tau \lb{v^2}\rb \; .
\label{n4}
\ee
Here $\tau$ is a typical correlation time of the flow $\bfv$, and
${1/2}\lb{\bfv\cnt\bfv}\rb$ is its kinetic
helicity, a measure of the net handedness of cyclonic motions 
[\cite{p55,p79}].
$\beta$ represents turbulent diffusion of $\OB$.  
In this paper, the brackets and overbar represent spatial averages.








%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
%\cite{fbc} may be realistic for a fixed $B_1$, but not  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 in \cite{b2001}
%, 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.



\section{Separate isotropy of $\bf b$ and $\bf v$ need
not imply $\lb\bfv\ts\bfb\rb=0$}

Suppose that $\bfb$ and $\bfv$ are separately
isotropic. This means that 
\beq
\int \bfv f(v^2) d\bfv =0
\label{iso1}
\ee
where $f(v^2)$ is the distribution function for velocity,
which depends only on its magnitude.
Similarly
\beq
\int \bfv g(v^2) d\bfv =0
\label{iso2}
\ee
where $g(v^2)$ is the distribution function for the magnetic field.
However, for joint distributions, 
\beq
f(v^2) = \int {\tilde f}({\bfv},{\bfv b}) d\bfb
\ee
and
\beq
g(b^2) = \int {\tilde f}({\bfv},{\bfv b}) d\bfv
\ee
and it need not be the case that 
$\int v_i b_j {\tilde f}({\bfv},{\bfv b}) d\bfv =0$
even if (\ref{iso1}) and (\ref{iso2}) are true.

The implication is that at least for weak $\bbB$, one
can be justified by taking the kinetic helicity, current
helicity and diffusion tensors as being isotropic, which 
do not invovle cross correlations between magnetic field and velocity,
whilst still allowing $\lb\bfv\ts\bfb\rb\ne 0$.

\section{Comparison to previous work}


Usually $\emfb$ is 
expressed as 
\beq
\emf_i=-\alpha_{ij} \bB_j+\beta_{ijk}\partial_i\bB_j,
\ee
where $\alpha_{ij}$ and $\beta_{ijk}$ are 
the generalized helicity and diffusion dynamo coefficient tensors.
These tensors are composed of correlations of turbulent
quantities, and when they are assumed to be isotropic,
they lead to the $\alpha$ pseudo-scalar 
helicity and $\beta$ diffusion coefficient of mean field dynamo theory.

-discussion of choices of beta

-differences with respect to previous work

-comparing to simulations, limitations.



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%------------------------------------------------------------------------------
\vspace{-.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{emff5.ps} \hfill }

\noindent {Figure 3:
Plot of the $\zeta=2/R_{m,2}$
triplet solution for $h_1(t)$ 
with $k_1=1$ and $k_2=5$.  
The two curves are plotted for $R_{m,2} = 200$,
but the top curve has $\chi\propto 1+h_2$ and the lower curve has 
$\chi \propto (1+h_1 k_1/k_2)^{-1}$. 
The dotted curve is again the asymptotic
empirical fit formala [\cite{b2001}].}




we solve (\ref{8}) and (\ref{6}).
The resulting second order equation is
\beq
h_1\partial_\tau^2 h_1=(2/3)h_1^{5/2}(k_1/k_2)(1+h_2)-(k_1/k_2)^2
\chi h_1^{5/2} -\zeta h_1\partial_\tau h_1+(\partial_\tau h_1)^2
\label{approx}
\ee
Let us  assume $\chi=1/3$. 
Setting $h_1 = h_{10}Exp[nt]$, and ignoring terms $\propto (k_1/k_2)^2$, 
the positive solution gives
\beq
n\sim [\zeta+(4/3)(k_1/k_2)(1+h_2)
%-(2/3)(k_1/k_2)^2
]^{1/2}
-\zeta.
\label{gr}
\ee



{\it Dynamo Equations-}
We couple (\ref{4time})
to the equations for small and large-scale magnetic helicity 
evolution for a dynamo in which the kinetic energy is
externally forced and $\bbV=0$, 
Neglecting boundary terms, 
the total magnetic helicity, $H^M=\lb\bfA\cdot\bfB\rb$, 
satisfies [\cite{moffatt}] 
$
\partial_t\lb\bfA\cdot\bfB\rb=-2\lb{\bfE}\cdot\bfB\rb,
$
where $\bf E= -\partial_t {\bf A} -\nabla \phi$,
and  $\phi$ is the scalar potential.
%the induction equation  
%$\partial_t {\bf B}= -\curl {\bf E}$, and Ohm's law  
%${\bf E} = - \bfv \ts {\bf B} +\eta \bfJ$, 
%imply
The mean and fluctuating components of magnetic helicity 
satisfy [\cite{b2001,fb02,bf00}]
\beq
\partial_t(\bbA\cdot \bbB)=2{\emfb}\cdot\bbB-2\lambda {\bbB\cdot \curl\bbB},
\label{h1}
\ee
and
\beq
\partial_t\lb\bfa\cdot \bfb\rb=-2{\emfb}\cdot\bbB -2\lambda 
\lb{\bfb\cdot\curl\bfb}\rb.
\label{h2}
\ee
%Following [\cite{fb}] we invoke
%our two scale assumption for these equations
%where $k_2$ is that of the small-scale, taken to be at the 
%forcing wavenumber and $k_1$ is that of the large scale.
%We define $H_1^M\equiv \lb\bbA\cdot\bbB\rb$
%and $H_2^M\equiv\lb\bfa\cdot\bfb\rb$, so that
%$\lb\bfj\cdot\bfb\rb = k_2^2H_2^M$ and 
%$\lb\bbJ\cdot\bbB\rb = k_1^2H_1^M$.
When $\bbB$ is force free, the two-scale
approximation allows us to write 
(\ref{h1}) and (\ref{h2}) as 
\beq
\partial_t H_1^M=2\emf_{||}k_1^{1/2}|H_1^{1/2}|
-2\lambda k_1^2H_1^M
\label{h3}
\ee
and 
\beq
\partial_t H_2^M=-2\emf_{||}k_1^{1/2}|H_1^{1/2}|-2\lambda k_2^2H_2^M.
\label{h4}
\ee
%Then  
%\beq
%-\partial_t\emf_{||}=k_1^{1/2}{H_1^M}^{1/2}(k_2^2H_2^M+H_2^V)-k_1^{3/2}
%({H_1^M}^{1/2})\sigma.
%\label{5}
%\ee
We need to solve (\ref{h3}), (\ref{h4}), and (\ref{4time})
after  converting them into dimensionless form.
We define the dimensionless quantities
$h_1\equiv H_1^M (k_2/v_2^2)$ and $h_2\equiv H_2^M (k_2/v_2^2)$,
$\ren\equiv v_2/\lambda k_2$, $Pr\equiv \nu/\lambda$, 
$\tau\equiv tv_2k_2$, $Q=-\emf_{||}/v_2^2$, 
$\chi={\tilde \beta}/v_2^2$, $\zeta = {\tilde \zeta}/v_2 k_2=(1+Pr)/\ren$, and 
use $H_2^V=-k_2v_2^2$.
For (\ref{h3}), (\ref{h4}) and (\ref{4time}) respectively, 
this gives
\beq
\partial_\tau h_1=2Qh_1^{1/2}(k_1/k_2)^{1/2}-2h_1(k_1/k_2)^2/\ren,
\label{7}
\ee
\beq
\partial_\tau h_2=-2Qh_1^{1/2}(k_1/k_2)^{1/2}-2h_2/\ren,
\label{8}
\ee
and
\beq
\partial_\tau Q= \left(k_1/k_2 \right)^{1/2}h_1^{1/2}(1+h_2)/3
-(k_1/k_2)^{3/2}h_1^{1/2}\chi-\zeta Q.
\label{6}
\ee


%\section

{\it Discussion of Solutions-}
Since $H_2^V < 0$, $H_1^M>0$ and $H_2^M<0$ for
a growing solution.  The solutions of (\ref{7}), (\ref{8}), and (\ref{6}) for
two different $\ren$ are shown
in Figs. 1 \& 2 over different time ranges 
for both $Pr=1$ and $Pr\simeq \ren$.  
Using ${\chi}\propto {\tilde \xi}\propto (1+h_2)$  
captures $\chi$ quenching for fully helical $\bfv$ and $\bfb$,
but Fig.2 shows that the dynamo quenching is only weakly sensitive to $\chi$.
In Figs.1\&2 we also plotted the empirical fit formula
to numerical simulations [\cite{b2001}]
(equation (54) of Ref \cite{b2001}) using our dimensionless parameters. 

In Figs. 1\&2 we  also compare the triplet
solution of $h_1$ with the doublet solution [\cite{fb02}]
that results from solving (\ref{7}) and (\ref{8}) but 
with $imposing$ $\emfb=\alpha\bbB-\beta\curl\bbB$ such that  
$Q=Q_{d}\equiv (k_1/k_2)^{1/2}h_1^{1/2}(1+h_2)\tau_c/3-
(k_1/k_2)^{3/2}h_1^{1/2}\chi\tau_c$,
%
%
%$\emf_{||}=
%(-k_1^{1/2}|H_1^M|^{1/2}(k_2^2H_2^M-H_2^V)+k_1^{3/2}(H_1^M/(|H_1^M|^{1/2})\sigma)\tau_c$
%, 
where the correlation time $\tau_c$ is a
free parameter taken to be $\sim 1$.
The triplet solution does not require $\tau_c$.
Fig. 1 shows that the triplet solution  matches the doublet solution
at early times for $\zeta=1$ when oscillations are suppressed, 
but Fig. 2 shows that 
the  $\zeta=2/\ren$ and $\zeta=1$ cases are indistinguishable at late times.
A $\zeta\sim 1$ could arise if the large-scale field were
changing direction rapidly, if $Pr\sim \ren$, 
or if ${\bf T} \ne 0$.
The rise to the first peak of $h_1$ in Fig. 1a
is independent of $\ren$ and there the two $\ren$ triplet solutions
overlap.  This is the  kinematic regime.  
The end of the doublet kinematic regime
occurs at $h_1\simeq 1$, as
seen in Fig. 1.

%High
%resolution simulations
% than Ref. \cite{b2001} 
%are required to test for the subsequent oscillations.



The maximum kinematic growth rate for $h_1$ changes with $\zeta$ 
because it occurs where 
% can be 
%seen from semi-analytic considerations of 
%(\ref{7}), (\ref{8}) and (\ref{6}). 
%The maximum growth rate occurs 
$Q$ is a minimum. 
If we ignore resistive terms such that $h_1 = -h_2$,
and assume that $\zeta<<1$ and $\chi=1/3$, 
then Eq. (\ref{6}) implies that 
%and (\ref{8})
the maximum growth rate occurs when $h_1\simeq 1 - k_1/k_2$.  From Fig 1a, 
the minimum of $Q$ during the first oscillation is 
$\sim -1/3$ (found to be independent of $k_1/k_2$).  Setting 
$\partial_\tau h_1 \sim n h_1$, the maximum kinematic growth rate  from
(\ref{7}) is then $n \sim  (2/3)(k_1/k_2)^{1/2}(1 - k_1/k_2)^{-1/2}
\sim 0.33$, for $k_2=5k_1$.
However, when $\zeta=1$, the minimum of $Q$ from (\ref{6}) 
occurs where $Q=Q_d$. In this case, $n\sim (2/3)(k_1/k_2)(1-k_1/k_2)\sim 0.11$
for $k_2=5k_1$. 
This demonstrates that the kinematic growth rate for
$\zeta<<1$ is $\sim 3$ times that for $\zeta=1$, and 
why the triplet kinematic growth rate with $\zeta=1$ matches that of the 
doublet. Both results are seen in Fig 1.

Inspection of (\ref{7}), (\ref{8}), and (\ref{6}) 
% (\ref{gr})  
reveals why there are oscillations for a positive seed $h_1$
and $\zeta<<1$. As long as $-1< h_2 <0$, $Q$ grows more negative and 
$h_1$ and  $h_2$  grow with mutually opposite signs.
As $h_2$ passes through $-1$ from above,   
$\partial_\tau Q$  changes sign immediately but 
$h_1$ continues to grow positive, albeit more slowly, 
until $Q$ changes sign.  Then, $\partial_\tau h_1$ 
changes sign and $h_1$ decreases. But $\partial_\tau h_2$  
changes sign when $\partial_\tau h_1$  does, 
so when $h_2$ eventually passes back through $-1$
from below,  $\partial_\tau Q$ reverses sign again, and 
eventually $Q$ becomes negative and $h_1$ again grows.  
Finite $\ren$ terms  weakly damp the oscillations.  
This description is consistent with Fig 1a.
If instead, $\zeta \sim 1$, 
%$\zeta^2 \gsim (4/3) (k_1/k_2)$, 
once $\partial_\tau Q$ is depleted 
by the growth of $h_2$, the $\zeta$ term of
(\ref{6}) takes over and $Q$ decays without oscillating.  
Then $h_1$ grows without oscillations (Fig 1b). 
%For $\zeta\sim 1$ the growth rate 
%in (\ref{gr})
%is significantly reduced from the $\zeta=0$ case.
%and the early-time
%growth rates for the doublet and triplet solutions nearly match, as seen
%in Fig. 1b.



%Roughly speaking, without resistive terms, 
%$h_2= -h_1$, and the oscillation from (\ref{gr}) proceeds with
%a frequency that depends on $h_1$, namely  
%$\omega \sim h_1(2/3)^{1/2}(k_1/k_2)^{1/2}$.
%The pulses thus have narrow peaks
%and wide bases. 

%\section{ Conclusion}

 {\it Conclusion-}
We have developed an MFD theory in which we  
solve for $\emfb$ dynamically rather 
than imposing its specific form a priori.
We coupled the equation for $\partial_\tau \emfb$  
to  total magnetic helicity evolution, 
and to the large-scale field growth equation.
For force free solutions, the latter is 
the large-scale magnetic helicity equation.
Using a two-scale closure theory that highlights the importance
of pressure terms, we found that the residual helicity emerges 
as the driver of helical MFDs, the residual energy density is 
important for turbulent diffusion, and that 
the turbulent correlation time is not a free parameter.
We compared the results to the imposed $\emfb$  doublet approach
[\cite{fb02}]. Agreement is very good for the $R_m$ dependent 
growth phase and matches simulations [\cite{b2001}].
However, oscillations 
are possible at early times only in the triplet approach.
This can be tested with future numerical experiments.
For $\zeta=1$, the agreement between the two approaches becomes
nearly exact for all times. 
For all $\zeta \ge 0$, $R_m$ independent growth occurs to at least
$\bB^2 \sim (k_1/k_2) v_2^2$, and the $R_m$ dependent
growth phase eventually 
saturates with field strength $\bB^2 \sim (k_2/k_1)v_2^2 $.

%We now invoke the two-scale approximation, where the small-scale
%quantites of interest vary on scale $k_2^{-1}$ and 
%large-scale quantities vary on scale $k_1^{-1}$.
%We can replace the Laplacian operating on fluctuating quantites by $-k_2^2$.
%and that operating on the mean field by $-k_1^2$. 
%Dividing both sides of (\ref{pres}) by $-k_2^2$ gives the
% solution for $p$. 

%If divergences of  bracketed quantities vanish, 
%the 2nd term of (\ref{c3}), when placed in the first term
%of (\ref{timed}), cancels with the 5th term of (\ref{c3}) when the
%latter is placed in the third term of (\ref{timed}).
%If we further assume that 
%correlations of factors of $\bfb$ with $\bfb$ and its gradients
%or $\bfv$ with $\bfv$ and its gradients
%are separately isotropic (but mixed correlations between $\bfv$ and $\bfb$ 
%are not necessarily isotropic), 
%then the 5th and 10th terms of (\ref{c2}) do not contribute 
%when placed into 
%(\ref{timed}).  Then we have


%\beq
%\begin{array}{r}
%\lb\partial_t\bfv\ts\bfb\rb +\lb\bfv\ts\partial_t\bfb\rb=
%{1\over 3}(2\lb\bfb\cdot\curl\bfb\rb)
%-\lb\bfv\cdot\curl\bfv\rb)\bbB 
%\\- 
%{1\over 3}\lb
%\bfv^2 + 2\bfb^2\rb\curl\bbB+
%\lb\bfv\ts\bfb\cdot\nabla\bfv\rb-\lb\nabla p \ts \bfb\rb 
%+(\nu + \lambda)\lb\bfv\ts\nabla^2\bfb\rb.
%\end{array}
%\label{timed3}
%\ee
%The factors 2 will be modified after dealing with the 
%pressure: taking the divergence of (\ref{c2}), ignoring the bracketed
%terms and multiplying by the inverse Laplacican gives

%\beq
%\nabla^2p = \nabla\cdot(\bfb\cdot\nabla \bfb)+
%\nabla\cdot(\bfb\cdot\nabla \bbB)
%+\nabla\cdot(\bbB\cdot\nabla \bfb)
%-\nabla\cdot(\bfv\cdot\nabla \bfv).
%\label{pres}
%\ee


%Upon taking the gradient and placing this back into the 
%$-\lb \nabla p \ts \bfb\rb$ term
%of (\ref{timed3}), using isotropy in the manner discussed above, 
%collecting all the terms surviving the
%tensor algebra, 
%and plugging back into (\ref{timedp})
%we then have 
%\beq
%p_{th}= k_2^{-2} (\nabla\cdot(\bfb\cdot\nabla\bfb)-\nabla\cdot
%(\bfv\cdot\nabla\bfv))
%\label{peff}
%\ee 
%\beq
%\begin{array}{r}
%\lb\partial_t\bfv\ts\bfb\rb +\lb\bfv\ts\partial_t\bfb\rb=
%{1\over 3}(\lb\bfb\cdot\curl\bfb\rb(1-2k_1^2/k_2^2)
%-\lb\bfv\cdot\curl\bfv\rb)\bbB 
%\\- 
%{1\over 3}\lb
%\bfv^2 - \bfb^2(1+k_1^2/k_2^2)\rb\curl\bbB
%-
%{\tilde\zeta}
%\lb\bfv\ts\bfb\rb,
%\end{array}
%\label{timed3}
%\ee
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\def\bbA{\bar{\bf A}}
\def\bA{\bar {A}}

\def\bff{{\bf f}}
\def\bfx{{\bf x}}
\def\BS{{\bf S}}
\def\bV{\bar V}
\def\bfa{{\bf a}}
\def\bfh{{\bf h}}
\def\bfS{{\bf S}}

\def\a{{\alpha}}
\def\be{{\beta}}
\def\ga{\gamma}
\def\de{\delta}
\def\ep{\epsilon}
\def\ve{\varepsilon}
\def\ze{\zeta}
\def\et{\eta}
\def\th{\theta}
\def\vt{\vartheta}
\def\io{\iota}
\def\ka{\kappa}
\def\la{\lambda}
\def\vpi{\varpi}
\def\rh{\rho}
\def\vr{\varrho}
\def\si{\sigma}
\def\vs{\varsigma}
\def\ta{\tau}
\def\up{\upsilon}
\def\ph{\phi}
\def\vp{\varphi}
\def\ch{\chi}
\def\ps{\psi}
\def\om{\omega}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\Th{\Theta}
\def\La{\Lambda}
\def\Si{\Sigma}
\def\Up{\Upsilon}
\def\Ph{\Phi}
\def\Ps{\Psi}
\def\Om{\Omega}
\def\lb{\langle}
\def\rb{\rangle}
%





\def\b0{b^{(0)}}
\def\v0{v^{(0)}}
\def\w0{\omega^{(0)}}
\def\bb0{\bfb^{(0)}}
\def\bv0{\bfv^{(0)}}
\def\bw0{\bfw^{(0)}}
\def\bj0{\bfj^{(0)}}
\def\bbj{\bar {\bf j}}
\def\bbm{\bar {\bf m}}




\subsection{Folded field line structure} \label{structure}

The magnetic field, in the course of amplification and entanglement, develops
structure not identifiable in the power spectrum. Define

\begin{equation} \label{eq:sa}
k_\perp^2 = <(\bb \times \nabla \times \bb)^2> / <b^2>^2 \hspace{7mm}
k_\parallel^2 = <(\bb \cdot \nabla \bb)^2> / <b^2>^2

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%------------------------------------------------------------------------------
\vspace{-.1cm} \hbox to \hsize{ \hfill \epsfxsize8cm
\epsffile{emff5.ps} \hfill }

\noindent {Figure 3:
Plot of the $\zeta=2/R_{m,2}$
triplet solution for $h_1(t)$ 
with $k_1=1$ and $k_2=5$.  
The two curves are plotted for $R_{m,2} = 200$,
but the top curve has $\chi\propto 1+h_2$ and the lower curve has 
$\chi \propto (1+h_1 k_1/k_2)^{-1}$. 
The dotted curve is again the asymptotic
empirical fit formala [\cite{b2001}].}




we solve (\ref{8}) and (\ref{6}).
The resulting second order equation is
\beq
h_1\partial_\tau^2 h_1=(2/3)h_1^{5/2}(k_1/k_2)(1+h_2)-(k_1/k_2)^2
\chi h_1^{5/2} -\zeta h_1\partial_\tau h_1+(\partial_\tau h_1)^2
\label{approx}
\ee
Let us  assume $\chi=1/3$. 
Setting $h_1 = h_{10}Exp[nt]$, and ignoring terms $\propto (k_1/k_2)^2$, 
the positive solution gives
\beq
n\sim [\zeta+(4/3)(k_1/k_2)(1+h_2)
%-(2/3)(k_1/k_2)^2
]^{1/2}
-\zeta.
\label{gr}
\ee



{\it Dynamo Equations-}
We couple (\ref{4time})
to the equations for small and large-scale magnetic helicity 
evolution for a dynamo in which the kinetic energy is
externally forced and $\bbV=0$, 
Neglecting boundary terms, 
the total magnetic helicity, $H^M=\lb\bfA\cdot\bfB\rb$, 
satisfies [\cite{moffatt}] 
$
\partial_t\lb\bfA\cdot\bfB\rb=-2\lb{\bfE}\cdot\bfB\rb,
$
where $\bf E= -\partial_t {\bf A} -\nabla \phi$,
and  $\phi$ is the scalar potential.
%the induction equation  
%$\partial_t {\bf B}= -\curl {\bf E}$, and Ohm's law  
%${\bf E} = - \bfv \ts {\bf B} +\eta \bfJ$, 
%imply
The mean and fluctuating components of magnetic helicity 
satisfy [\cite{b2001,fb02,bf00}]
\beq
\partial_t(\bbA\cdot \bbB)=2{\emfb}\cdot\bbB-2\lambda {\bbB\cdot \curl\bbB},
\label{h1}
\ee
and
\beq
\partial_t\lb\bfa\cdot \bfb\rb=-2{\emfb}\cdot\bbB -2\lambda 
\lb{\bfb\cdot\curl\bfb}\rb.
\label{h2}
\ee
%Following [\cite{fb}] we invoke
%our two scale assumption for these equations
%where $k_2$ is that of the small-scale, taken to be at the 
%forcing wavenumber and $k_1$ is that of the large scale.
%We define $H_1^M\equiv \lb\bbA\cdot\bbB\rb$
%and $H_2^M\equiv\lb\bfa\cdot\bfb\rb$, so that
%$\lb\bfj\cdot\bfb\rb = k_2^2H_2^M$ and 
%$\lb\bbJ\cdot\bbB\rb = k_1^2H_1^M$.
When $\bbB$ is force free, the two-scale
approximation allows us to write 
(\ref{h1}) and (\ref{h2}) as 
\beq
\partial_t H_1^M=2\emf_{||}k_1^{1/2}|H_1^{1/2}|
-2\lambda k_1^2H_1^M
\label{h3}
\ee
and 
\beq
\partial_t H_2^M=-2\emf_{||}k_1^{1/2}|H_1^{1/2}|-2\lambda k_2^2H_2^M.
\label{h4}
\ee
%Then  
%\beq
%-\partial_t\emf_{||}=k_1^{1/2}{H_1^M}^{1/2}(k_2^2H_2^M+H_2^V)-k_1^{3/2}
%({H_1^M}^{1/2})\sigma.
%\label{5}
%\ee
We need to solve (\ref{h3}), (\ref{h4}), and (\ref{4time})
after  converting them into dimensionless form.
We define the dimensionless quantities
$h_1\equiv H_1^M (k_2/v_2^2)$ and $h_2\equiv H_2^M (k_2/v_2^2)$,
$\ren\equiv v_2/\lambda k_2$, $Pr\equiv \nu/\lambda$, 
$\tau\equiv tv_2k_2$, $Q=-\emf_{||}/v_2^2$, 
$\chi={\tilde \beta}/v_2^2$, $\zeta = {\tilde \zeta}/v_2 k_2=(1+Pr)/\ren$, and 
use $H_2^V=-k_2v_2^2$.
For (\ref{h3}), (\ref{h4}) and (\ref{4time}) respectively, 
this gives
\beq
\partial_\tau h_1=2Qh_1^{1/2}(k_1/k_2)^{1/2}-2h_1(k_1/k_2)^2/\ren,
\label{7}
\ee
\beq
\partial_\tau h_2=-2Qh_1^{1/2}(k_1/k_2)^{1/2}-2h_2/\ren,
\label{8}
\ee
and
\beq
\partial_\tau Q= \left(k_1/k_2 \right)^{1/2}h_1^{1/2}(1+h_2)/3
-(k_1/k_2)^{3/2}h_1^{1/2}\chi-\zeta Q.
\label{6}
\ee


%\section

{\it Discussion of Solutions-}
Since $H_2^V < 0$, $H_1^M>0$ and $H_2^M<0$ for
a growing solution.  The solutions of (\ref{7}), (\ref{8}), and (\ref{6}) for
two different $\ren$ are shown
in Figs. 1 \& 2 over different time ranges 
for both $Pr=1$ and $Pr\simeq \ren$.  
Using ${\chi}\propto {\tilde \xi}\propto (1+h_2)$  
captures $\chi$ quenching for fully helical $\bfv$ and $\bfb$,
but Fig.2 shows that the dynamo quenching is only weakly sensitive to $\chi$.
In Figs.1\&2 we also plotted the empirical fit formula
to numerical simulations [\cite{b2001}]
(equation (54) of Ref \cite{b2001}) using our dimensionless parameters. 

In Figs. 1\&2 we  also compare the triplet
solution of $h_1$ with the doublet solution [\cite{fb02}]
that results from solving (\ref{7}) and (\ref{8}) but 
with $imposing$ $\emfb=\alpha\bbB-\beta\curl\bbB$ such that  
$Q=Q_{d}\equiv (k_1/k_2)^{1/2}h_1^{1/2}(1+h_2)\tau_c/3-
(k_1/k_2)^{3/2}h_1^{1/2}\chi\tau_c$,
%
%
%$\emf_{||}=
%(-k_1^{1/2}|H_1^M|^{1/2}(k_2^2H_2^M-H_2^V)+k_1^{3/2}(H_1^M/(|H_1^M|^{1/2})\sigma)\tau_c$
%, 
where the correlation time $\tau_c$ is a
free parameter taken to be $\sim 1$.
The triplet solution does not require $\tau_c$.
Fig. 1 shows that the triplet solution  matches the doublet solution
at early times for $\zeta=1$ when oscillations are suppressed, 
but Fig. 2 shows that 
the  $\zeta=2/\ren$ and $\zeta=1$ cases are indistinguishable at late times.
A $\zeta\sim 1$ could arise if the large-scale field were
changing direction rapidly, if $Pr\sim \ren$, 
or if ${\bf T} \ne 0$.
The rise to the first peak of $h_1$ in Fig. 1a
is independent of $\ren$ and there the two $\ren$ triplet solutions
overlap.  This is the  kinematic regime.  
The end of the doublet kinematic regime
occurs at $h_1\simeq 1$, as
seen in Fig. 1.

%High
%resolution simulations
% than Ref. \cite{b2001} 
%are required to test for the subsequent oscillations.



The maximum kinematic growth rate for $h_1$ changes with $\zeta$ 
because it occurs where 
% can be 
%seen from semi-analytic considerations of 
%(\ref{7}), (\ref{8}) and (\ref{6}). 
%The maximum growth rate occurs 
$Q$ is a minimum. 
If we ignore resistive terms such that $h_1 = -h_2$,
and assume that $\zeta<<1$ and $\chi=1/3$, 
then Eq. (\ref{6}) implies that 
%and (\ref{8})
the maximum growth rate occurs when $h_1\simeq 1 - k_1/k_2$.  From Fig 1a, 
the minimum of $Q$ during the first oscillation is 
$\sim -1/3$ (found to be independent of $k_1/k_2$).  Setting 
$\partial_\tau h_1 \sim n h_1$, the maximum kinematic growth rate  from
(\ref{7}) is then $n \sim  (2/3)(k_1/k_2)^{1/2}(1 - k_1/k_2)^{-1/2}
\sim 0.33$, for $k_2=5k_1$.
However, when $\zeta=1$, the minimum of $Q$ from (\ref{6}) 
occurs where $Q=Q_d$. In this case, $n\sim (2/3)(k_1/k_2)(1-k_1/k_2)\sim 0.11$
for $k_2=5k_1$. 
This demonstrates that the kinematic growth rate for
$\zeta<<1$ is $\sim 3$ times that for $\zeta=1$, and 
why the triplet kinematic growth rate with $\zeta=1$ matches that of the 
doublet. Both results are seen in Fig 1.

Inspection of (\ref{7}), (\ref{8}), and (\ref{6}) 
% (\ref{gr})  
reveals why there are oscillations for a positive seed $h_1$
and $\zeta<<1$. As long as $-1< h_2 <0$, $Q$ grows more negative and 
$h_1$ and  $h_2$  grow with mutually opposite signs.
As $h_2$ passes through $-1$ from above,   
$\partial_\tau Q$  changes sign immediately but 
$h_1$ continues to grow positive, albeit more slowly, 
until $Q$ changes sign.  Then, $\partial_\tau h_1$ 
changes sign and $h_1$ decreases. But $\partial_\tau h_2$  
changes sign when $\partial_\tau h_1$  does, 
so when $h_2$ eventually passes back through $-1$
from below,  $\partial_\tau Q$ reverses sign again, and 
eventually $Q$ becomes negative and $h_1$ again grows.  
Finite $\ren$ terms  weakly damp the oscillations.  
This description is consistent with Fig 1a.
If instead, $\zeta \sim 1$, 
%$\zeta^2 \gsim (4/3) (k_1/k_2)$, 
once $\partial_\tau Q$ is depleted 
by the growth of $h_2$, the $\zeta$ term of
(\ref{6}) takes over and $Q$ decays without oscillating.  
Then $h_1$ grows without oscillations (Fig 1b). 
%For $\zeta\sim 1$ the growth rate 
%in (\ref{gr})
%is significantly reduced from the $\zeta=0$ case.
%and the early-time
%growth rates for the doublet and triplet solutions nearly match, as seen
%in Fig. 1b.



%Roughly speaking, without resistive terms, 
%$h_2= -h_1$, and the oscillation from (\ref{gr}) proceeds with
%a frequency that depends on $h_1$, namely  
%$\omega \sim h_1(2/3)^{1/2}(k_1/k_2)^{1/2}$.
%The pulses thus have narrow peaks
%and wide bases. 

%\section{ Conclusion}

 {\it Conclusion-}
We have developed an MFD theory in which we  
solve for $\emfb$ dynamically rather 
than imposing its specific form a priori.
We coupled the equation for $\partial_\tau \emfb$  
to  total magnetic helicity evolution, 
and to the large-scale field growth equation.
For force free solutions, the latter is 
the large-scale magnetic helicity equation.
Using a two-scale closure theory that highlights the importance
of pressure terms, we found that the residual helicity emerges 
as the driver of helical MFDs, the residual energy density is 
important for turbulent diffusion, and that 
the turbulent correlation time is not a free parameter.
We compared the results to the imposed $\emfb$  doublet approach
[\cite{fb02}]. Agreement is very good for the $R_m$ dependent 
growth phase and matches simulations [\cite{b2001}].
However, oscillations 
are possible at early times only in the triplet approach.
This can be tested with future numerical experiments.
For $\zeta=1$, the agreement between the two approaches becomes
nearly exact for all times. 
For all $\zeta \ge 0$, $R_m$ independent growth occurs to at least
$\bB^2 \sim (k_1/k_2) v_2^2$, and the $R_m$ dependent
growth phase eventually 
saturates with field strength $\bB^2 \sim (k_2/k_1)v_2^2 $.

%We now invoke the two-scale approximation, where the small-scale
%quantites of interest vary on scale $k_2^{-1}$ and 
%large-scale quantities vary on scale $k_1^{-1}$.
%We can replace the Laplacian operating on fluctuating quantites by $-k_2^2$.
%and that operating on the mean field by $-k_1^2$. 
%Dividing both sides of (\ref{pres}) by $-k_2^2$ gives the
% solution for $p$. 

%If divergences of  bracketed quantities vanish, 
%the 2nd term of (\ref{c3}), when placed in the first term
%of (\ref{timed}), cancels with the 5th term of (\ref{c3}) when the
%latter is placed in the third term of (\ref{timed}).
%If we further assume that 
%correlations of factors of $\bfb$ with $\bfb$ and its gradients
%or $\bfv$ with $\bfv$ and its gradients
%are separately isotropic (but mixed correlations between $\bfv$ and $\bfb$ 
%are not necessarily isotropic), 
%then the 5th and 10th terms of (\ref{c2}) do not contribute 
%when placed into 
%(\ref{timed}).  Then we have


%\beq
%\begin{array}{r}
%\lb\partial_t\bfv\ts\bfb\rb +\lb\bfv\ts\partial_t\bfb\rb=
%{1\over 3}(2\lb\bfb\cdot\curl\bfb\rb)
%-\lb\bfv\cdot\curl\bfv\rb)\bbB 
%\\- 
%{1\over 3}\lb
%\bfv^2 + 2\bfb^2\rb\curl\bbB+
%\lb\bfv\ts\bfb\cdot\nabla\bfv\rb-\lb\nabla p \ts \bfb\rb 
%+(\nu + \lambda)\lb\bfv\ts\nabla^2\bfb\rb.
%