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\begin{document}

{\bf Some comments on the BRRK manuscript -- 3}

\bigskip

1.\\
Let us define the mean field $\ol{F}$ belonging to a field $F$ by
\begin{equation}
\ol{F} (x,y,z) = \frac{1}{4 \Lambda^2} \int_{-\Lambda}^{\Lambda} \int_{-\Lambda}^{\Lambda}
    F (x + \xi,y + \eta, z) \, \dd \xi \dd \eta
\label{eq01}
\end{equation}
with any length $\Lambda$ (which I will connect later with the period length $L$ of the forcing).

A definition of that kind has been used in the mean--field theory of the Karlsruhe dynamo,
with $\Lambda$ specified, e.g., to be equal to the period length of the Roberts flow pattern.
In this context also mean magnetic fields $\bmB$ depending on $x$, $y$ or $z$ have been treated.

The definition (\ref{eq01}) implies the validity of the Reynolds rule according to which averaging commutes
with derivatives with respect to $x$, $y$, $z$ and $t$.

Another Reynolds rule, which we need in what follows, reads $\ol{\ol{F} G} = \ol{F} \, \ol{G}$.
It implies $\ol{\ol{F} G} = 0$ if $\ol{G} = 0$.
This rule is clearly satisfied if $\ol{F}$ is independent of $x$ and $y$,
and it applies as an approximation if $\ol{F}$ depends only weakly on $x$ and $y$.

\bigskip

2.\\
With the definition (\ref{eq01}) we have, e.g., $\ol{x} = x$ and therefore $\ol{\bU^S} = \bU^S$,
that is, $\bU^S$ is no longer a fluctuation.

When averaging (BRRK 15), putting $\bU = \bmU + \bu$ and $\bB = \bmB + \bb$
and accepting $\ol{\ol{F} G} = \ol{F} \, \ol{G}$ we find
\begin{equation}
\frac{\D \bmA}{\D t} = - S \mA_y \hat{\bx} + \bmU \x \bmB + \ol{\bu \x \bb} - \eta \bmJ
\label{eq03}
\end{equation}
(which is different from (BRRK 16), see below).
Using again (BRRK 15) and (\ref{eq03}), introducing $\ba = \bA - \bmA$
and accepting again $\ol{\ol{F} G} = \ol{F} \, \ol{G}$ we just arrive at (BRRK 17).

Note that the term $S x \p \bmA / \p y$ in $\frac{\D \bmA}{\D t}$
on the left--hand side of (\ref{eq03}) in general does not vanish.
Even if $\bmB$ is independent of $x$ and $y$, the vector potential $\bmA$ may well depend on them,
see Comments-BRRK-2, \mbox{item 3}.

\bigskip

3.\\
Let us now restrict attention on the situation in which $\bmB$ and $\bmU$ do not depend on $x$ and $y$.
Then the use of $\ol{\ol{F} G} = \ol{F} \, \ol{G}$ in the derivations done so far is exactly justified.
Let us further assume that the forcing is periodic in $x$ and $y$ with a period length $L$
in both the $x$ and $y$ directions.
If we then specify the definition (\ref{eq01}) of mean quantities
by putting $2 \Lambda = L$, all mean quantities of interest in view of our mean-field coefficients
should be independent of $x$ and $y$
(and could therefore be also considered as averages over all $x$ and $y$).
It is then sufficient to calculate them at $x = y = 0$.

\end{document}
