\section{An analytical estimate for $\eta_{yx}$}
\label{analyticetayx}
With assumptions
\begin{itemize}\itemsep-2pt
\item ideal MHD: $\eta=\nu=0$,
\item drop of pressure term,
\item constant density,
\item first order in mean magnetic field,
\item neglect of all bilinear/quadratic terms in fluctuations (SOCA),
\item vanishing initial conditions of the fluctuations which are due to the influence of shear and $\meanBB$, and
\item vanishing mean flow other than shear flow  $\UUS$,
\end{itemize}
we obtain (arguments $x,z$ temporarily dropped)
%\begin{align}
%       \eta_{yx} = &-\left\langle \bzz_x \left[\int_0^t \left(\bzz_y(y_0+S x \tau,\tau) + S \int_0^\tau \bzz_x(y_0+S x \tau,\tau') d\tau' \right) d\tau\right]_{y_0=y-Sxt} \right.\\
%        &\left. -\,Sx \left( \bzz_x  \left[\int_0^t  \int_0^\tau \partial_y \bzz_y(y_0+S x \tau,\tau') d\tau' d\tau \right]_{y_0=y-Sxt} \right.\right.\\ 
%      &\left.\left. \qquad\quad + \phantom{S}\left(\int_0^t \partial_y \bzz_x(y,\tau) d\tau\right) \left[\int_0^t \bzz_y(y_0+S x \tau,\tau) d \tau\right]_{y_0=y-Sxt} \right.  \right. \\
%      &\left. \left. \qquad\quad+ \, S \left(\int_0^t \partial_y \bzz_x(y,\tau) d\tau\right) \left[ \int_0^t  \int_0^\tau \bzz_x(y_0+S x \tau,\tau') d\tau' d\tau \right]_{y_0=y-Sxt} \right)\right. \\
%       &\left. + \,S^2x^2 \left(\int_0^t \partial_y \bzz_x(y,\tau) d\tau\right) \left[\int_0^t \int_0^\tau \partial_y \bzz_y(y_0+S x \tau,\tau') d\tau'd\tau \right]_{y_0=y-Sxt}  \right\rangle_{xy} .
%       \label{eq:estim}
%\end{align}
%NS: the last line (C6) in the expression above is at least quadratic
%NS: in S, and thus seems incompatible with 4th bullet points above
%MR: No, there I added  "in the equations governing the fluctuating fields"
%
%NS: In fact, even C5 (last but one) appears quadratic in S.
%NS: Shouldn't these be suppressed from this eta_yx expression?
%MR: The problem is, that as S appears in the arguments by y_0 + Sx\tau, one cannot perform
%       an analysis which is from the beginning 1st order in S. So it is more honest to keep
%       all orders as they come and suppress only finally all higher ones.
%       Perhaps bullet 4 should be completely dropped and the result of Sec. 1.2 of derivation.tex
%       be used instead (but even bulkier expressions).
%
%NS: Fine, just wanted to highlight this with regard to the bullet point
%NS: as S^2 * F(S) can readily be dropped (for 1st order in S) as
%NS: such terms, C5 and C6, are manifestly of at least order 2
%NS: But I agree that full treatment is necessary to judge the correct
%NS: orders also in terms like C3, C4
%
%MR: now arbitrary order in S
%AB: wow!
%MR: simplified notation
 \begin{align}
     \eta_{yx} &= -\left\langle \bzz_x(y,t)\, I_1 -Sx \left( \bzz_x(y,t)  \,I_2 + \int_0^t \partial_y \bzz_x(\xi,\tau) d\tau \, I_1 \right) + \,S^2x^2  \int_0^t \partial_y \bzz_x(\xi,\tau) d\tau \, I_2 \right\rangle_{\!\!\!xy} \hspace{-0mm},\\ 
       I_1 &=\int_0^t \left(\bzz_y(\xi,\tau) + S \int_0^\tau \bzz_x(\xi',\tau') d\tau' \right) d\tau\\
       I_2 &= \int_0^t \int_0^\tau \left( \partial_y \bzz_y(\xi',\tau') + S  \int_0^{\tau'}\partial_y \bzz_x (\xi{''},\tau{''}) d\tau{''} \right) d\tau' d\tau .
\end{align}
with $\xi=y+Sx(\tau-t)$, $\xi'=y_0+Sx(\tau'-t)$ etc. The
  magnetic field is in units of $(\rho_0\mu_0)^{1/2}$ and $\left[\uu^{(00)},\bb^{(00)}\right]$ is the background turbulence (i.e., for $\meanBB=\boldsymbol{0}$) without influence of shear ($S=0$), hence
\begin{align}
  \partial_t \bb^{(00)} &= \nab\times\left(\uu^{(00)} \times \bb^{(00)} \right)' +\ffM \label{eq:backturb}\\
  \partial_t \uu^{(00)} &= - \left(\uu^{(00)}\cdot\nab \uu^{(00)} - \jj^{(00)}\times\bb^{(00)}\right)' + \ffK . \nonumber
\end{align}
Remarkably, there is no contribution from $\uu^{(00)}$ to $\eta_{yx}$ under these approximations.
%
%Disregarding such terms, we obtain 
To zeroth and first order in $S$ we obtain
\begin{align}
\eta_{yx} = &- S \left\langle b_x^{(00)}(y,t)  \int_0^t \int_0^\tau b_x^{(00)}(y,\tau') d\tau' d\tau \right\rangle_{\!\!\!xy}  
                  - \left\langle b_x^{(00)} (y,t) \int_0^t b^{(00)}_y(y,\tau) d\tau\right \rangle_{\!\!\!xy} \label{eq:estim2} \\
%MR: corrected
%                  & - S \left\langle x b_x^{(00)}(y,t)  \int_0^t \partial_y b^{(00)}_y(y,\tau) \tau d\tau \right\rangle_{\!\!\!xy},   
                   &-  S \left\langle x b_x^{(00)}(y,t)  \int_0^t \partial_y b^{(00)}_y(y,\tau)(\tau-t) \, d\tau \right\rangle_{\!\!\!xy}   %\label{eq:estim3}\\
%MR: added
                   + S \left\langle x b_x^{(00)}(y,t)  \int_0^t \int_0^\tau \partial_y b^{(00)}_y(y,\tau') \, d\tau' d\tau \right\rangle_{\!\!\!xy}  \label{eq:estim4}\\
                  & + S \left\langle x  \left( \int_0^t \partial_y b^{(00)}_x(y,\tau) d\tau  \right)  \left(  \int_0^t b^{(00)}_y(y,\tau) d\tau \right)  \right\rangle_{\!\!\!xy},   \label{eq:estim5} 
\end{align}
where the second contribution in \eqref{eq:estim2} vanishes in isotropic background turbulence because of $\left\langle b_i^{(00)}(y,t) b_j^{(00)} (y,\tau) \right\rangle_{\!\!xy} \propto \delta_{ij}$.
%MR: added, but can be shortened as we anyway don't provide all details of the derivation
Integration by parts in \eqref{eq:estim4} yields
\begin{align}
%      \int_0^t \partial_y b^{(00)}_y(y,\tau)(\tau-t) \, d\tau = - \int_0^t \int_0^\tau \partial_y b^{(00)}_y(y,\tau') \, d\tau' d\tau 
%AB: full stop
       \int_0^t \partial_y b^{(00)}_y(y,\tau)(\tau-t) \, d\tau = - \int_0^t \int_0^\tau \partial_y b^{(00)}_y(y,\tau') \, d\tau' d\tau.
\end{align}
%MR: Now it would be extremely  nice to have cancellation between this and (C8) but the sign seems not favorable.
%MR: Nishant, could you please check the signs carefully?
%NS: I can try, but getting too little time, see also comments below
Further, we have
\begin{align}
     b_x^{(00)}(y,t)  \int_0^t \int_0^\tau \partial_y b^{(00)}_y(y,\tau') \, d\tau' d\tau  = &\,\partial_y \left( b_x^{(00)}(y,t)  \int_0^t \int_0^\tau b^{(00)}_y(y,\tau') \, d\tau' d\tau \right) \\
%     &- \partial_y b_x^{(00)}(y,t)  \int_0^t \int_0^\tau b^{(00)}_y(y,\tau') \, d\tau' d\tau \label{eq:estim3}
%AB: comma
      &- \partial_y b_x^{(00)}(y,t)  \int_0^t \int_0^\tau b^{(00)}_y(y,\tau') \, d\tau' d\tau, \label{eq:estim3}
\end{align}
%in which the first term on the r.h.s vanishes under averaging over $y$. 
%AB: r.h.s -> rhs
in which the first term on the rhs vanishes under averaging over $y$. 
%MR: added
Hence, for the terms \eqref{eq:estim4} - \eqref{eq:estim5} we obtain
\begin{align}
%    S \left\langle x \left[ \left( \int_0^t \partial_y b^{(00)}_x(y,\tau) d\tau  \right)  \left(  \int_0^t b^{(00)}_y(y,\tau) d\tau \right) - 2 \, \partial_y b_x^{(00)}(y,t)  \int_0^t \int_0^\tau b^{(00)}_y(y,\tau') \, d\tau' d\tau \right]\right\rangle_{\!\!\!xy}
%AB: full stop
    S \left\langle x \left[ \left( \int_0^t \partial_y b^{(00)}_x(y,\tau) d\tau  \right)  \left(  \int_0^t b^{(00)}_y(y,\tau) d\tau \right) - 2 \, \partial_y b_x^{(00)}(y,t)  \int_0^t \int_0^\tau b^{(00)}_y(y,\tau') \, d\tau' d\tau \right]\right\rangle_{\!\!\!xy},
     \label{eq:estim6} 
\end{align}
%
Because of isotropy and mirror-symmetry of the background turbulence, the correlator $\left\langle b_i^{(00)}(y,t) \partial_k b^{(00)}_j(y,\tau) \right\rangle_{\!\!xy}$ vanishes $\forall i,j$. Hence, it is only the factor $x$ in 
\eqref{eq:estim6} which possibly prevents this term from vanishing, in contrast to the first term in \eqref{eq:estim2}, which is based on a  correlator, usually assumed positive definite. 
%MR: added
On the other hand, truly Galilean-invariant turbulence should not exhibit an explicit $x$ dependence. Given that  the forcing in our simulations  indeed obeys Galilean invariance, deviations from it in $\uu$ and $\bb$ can only emerge
due to the ``memory'' of the turbulence, which is made ``everlasting" by the absence of dissipative damping.
Thus, for the purpose of interpreting our numerical results, we may disregard \eqref{eq:estim6} and assume that only the first term in \eqref{eq:estim2} determines the sign of $\eta_{yx}$. 

Comparing with \cite{SB15a} by setting $\nu=\eta=0$ in their Eqs. (32)-(35) we find the following agreements:
%\begin{itemize}
%AB: added \itemsep-1pt
\begin{itemize}\itemsep-1pt
%\item no contribution from $\uu^{(00)}$ (or $W_u$ in their terms) to $\eta_{yx}$, only from $\bb^{(00)}$ (or $W_b$)
%AB: comma
\item no contribution from $\uu^{(00)}$ (or $W_u$ in their terms) to $\eta_{yx}$, only from $\bb^{(00)}$ (or $W_b$),
\item
        for positive correlation $\left\langle b_x^{(00)}(y,t;x,z)\, b_x^{(00)}(y,\tau';x,z)  \right\rangle_{\!\!xy}$,
        $\eta_{yx}$ has the opposite sign of $S$ and is thus unfavorable for the MSC dynamo.
        (Note that they employ a sign-inverted definition of $S$.)
\end{itemize}
%MR: deleted now.
%However, their result does not contain terms with explicit dependence on
% $x$ like \eqref{eq:estim6}.
%MJK Nobody will be able to understand the implications of this analysis, unless we clearly state it.
%NS: explicit x-dependence does not seem correct, most likely due to
%NS: some error. I recall old discussions with Karl-Heinz on Galilean
%NS: invariance (GI) stuff, where he finally had agreed that a proper analysis
%NS: would not violate the GI. We, in our (long) approach have never
%NS: seen this happening. So, I would not be able to agree with this,
%NS: but it is true that I have not done this calculation (could make
%NS: a serious attempt only after December), so comments above are based
%NS: merely on insight.
%NS: In a rough (skethcy) calculation (following Matthias and upto 1st order
%NS: in S), I only get only the first term on RHS of C6, which I thought
%NS: would be enough for the arguments of this paper. More detailed analysis
%NS: does not seem to be within the scope of this paper.
%NS: Unfortunately I don't see having any time to discuss this thing,
%NS: but have tried to highlight the reason why many might find it
%NS: (x-dependence) suspect. Nevertheless, if you still feel too strongly
%NS: about keeping this (other than 1st term of C6, more dangerously C13),
%NS: then please proceed with resubmission etc, as I can't come up with
%NS: anything mathematically  to convince in a short time.
In summary, our analytical result is in
%MR: added
qualitative
 accordance with the numerical ones for the magneto-kinetic forcing
cases. 
For the kinetic ones, however, the analytics predicts a vanishing contribution to $\eta_{yx}$, while
the numerical experiments do produce a contribution, even with a favourable sign for dynamo action, albeit
too weak to be the main driver of the dynamo.
%MR: added
%MJK This added part is problematic, as now we introduce even more notations from SB15a, which are not properly explained.... we already have some above, which are not too clear either. Also the message is somewhat unclear, even if the notations were explained. Whereto "in that respect" refers? "...ideal limit ... dissipative terms..."
%AB: "in that respect" refers to "analytics predicts a vanishing contribution to $\eta_{yx}$", I thought.
%MJK There must be a clearer way of expressing this. Also, we cannot use notations that are not defined. and obviously eta_{xy} is now eta_{yx}
As such vanishing contribution is in agreement with the ideal limif of \cite{SB15a}, 
%But, being still in accordance with the ideal limit of \cite{SB15a} in that respect, 
we conclude that 
%this contribution 
it
is caused by the
dissipative terms, most likely, by $\eta$, as their result (32) shows. It also reveals that there must be an ``optimal" magnitude of $\eta$ which maximises the contribution 
%MJK Added this
to $\eta_{yx}$ from kinetic fluctuations and shear (their 
%$(\eta_{xy})_u^S$
$(\eta_{yx})_u^S$),
because it 
vanishes again in the limit $\eta\rightarrow\infty$. To be too far from the optimal $\eta$ in numerical setups might explain the absence of a dynamo, enabled by 
%$(\eta_{xy})_u^S$.
$(\eta_{yx})_u^S$.

