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\def\MHD{{\sc mhd}}
\def\SMHD{{\sc smhd}}
\def\TFM{{\sc tfm}}
\def\NLTFM{{\sc nltfm}}
\def\QKTFM{{\sc qktfm}}
\def\red{\textcolor{red}}
\def\blue{\textcolor{blue}}
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\begin{document}
\section{Derivation of Eq. (C4)}
In general, assuming that all large scale flows except $\UUS$ are absent/suppressed 
\EQL
  \UU &= \UUS+\uuzz+\uuSz+\uuB , \quad \uuz = \uuzz+\uuSz \\
    \BB &= \meanBB+\bbzz+\bbSz+\bbB , \quad \bbz =  \bbzz+\bbSz \\
    \JJ &= \meanJJ+\jjzz+\jjSz+\jjB , \quad \jjz =  \jjzz+\jjSz \\
    \meanEEEE &= \meanEEEE^{(0)} + \meanEEEE^{(B)} \\
\intertext{with Eq. (C3) of the paper, and further}
   \partial_t \bbSz = &\nab\times\left[\left(\UUS+\uuSz\right)\times\bbz + \uuzz\times\bbSz\right]' \\
   \partial_t \uuSz = &- \UUS\cdot\nab\uuz - \uuz\cdot\nab\UUS - \left(\uuSz\cdot\nab\uuz + \uuzz\cdot\nab\uuSz\right)'\\
                               &  + \left(\jjz\times\bbSz + \jjSz\times\bbzz\right)' 
\end{align}
\subsection{Perturbative in $S$}
\EQL
   \partial_t \bbSz = &\nab\times\left[\left(\UUS+\uuSz\right)\times\bbzz + \uuzz\times\bbSz\right]' \\
   \partial_t \uuSz = &- \UUS\cdot\nab\uuzz - \uuzz\cdot\nab\UUS - \left(\uuSz\cdot\nab\uuzz + \uuzz\cdot\nab\uuSz\right)'\\
                               &  + \left(\jjzz\times\bbSz + \jjSz\times\bbzz\right)'
 \intertext{SOCA:}
   \partial_t \bbSz = &- \UUS\cdot\nab\bbzz + \bbzz\cdot\nab\UUS \label{eq:socaSz}  \\ 
   \partial_t \uuSz = &- \UUS\cdot\nab\uuzz - \uuzz\cdot\nab\UUS \\ 
 \intertext{integration with initial condition $=0$:}
   \bbSz = &- \UUS\cdot\nab \int_0^t \bbzz(t') dt' + \left(\int_0^t \bbzz(t') dt' \right)\cdot\nab\UUS  \\
   \uuSz = &- \UUS\cdot\nab \int_0^t \uuzz(t') dt' - \left(\int_0^t \uuzz(t') dt' \right)\cdot\nab\UUS  \\
\intertext{further}
   \partial_t \bbB = &\nab\times\left[\UU\times\left(\meanBB+\bbB\right) +  \uuB\times\bbz\right]' \\
   \partial_t \uuB = &  - \left[ \UU\cdot\nab\uuB + \uuB\cdot\nab\left(\UUS+\uuz\right)\right]' \\ 
                             &  + \left[\jj\times\left(\meanBB+\bbB\right) + \meanJJ\times\bb + \jjB\times\bbz\right]'  \\
 \intertext{first order in $\meanBB$:}
    \partial_t \bbB = &\;\nab\times\left[\left(\UUS+\uuz\right)\times\left(\meanBB+\bbB\right) +  \uuB\times\bbz\right]' \\
    \partial_t \uuB = & - \left[ \left(\UUS+\uuz\right)\cdot\nab\uuB + \uuB\cdot\nab\left(\UUS+\uuz\right)\right]' \\ 
                              &  + \left[\jjz\times\left(\meanBB+\bbB\right) + \left(\meanJJ+\jjB\right)\times\bbz\right]'  \\
 \intertext{SOCA:}
    \partial_t \bbB = &  - \UUS\cdot\nab\bbB + \bbB\cdot\nab\UUS + \meanBB\cdot\nab\uuz - \uuz\cdot\nab\meanBB - \meanBB \nab\cdot\uuz \\
    \partial_t \uuB = &  - \UUS\cdot\nab\uuB - \uuB\cdot\nab\UUS + \jjz\times\meanBB +  \meanJJ \times\bbz
 \intertext{or}
    \partial_t \bbB = & \phantom{--}\!\!\bbB\cdot\nab \UUS - \UUS\cdot\nab \bbB \;+ \text{terms independent of $\bbB$} \\                 
    \partial_t \uuB = & - \uuB\cdot\nab\UUS - \UUS\cdot\nab\uuB + \text{terms independent of $\uuB$}.\\
 \intertext{Let $\UUS = S x \ee_y \longrightarrow$}
    \partial_t \bbB = & \phantom{--}\!\!\bB_x S \ee_y - S x \partial_y \bbB \;+ \ldots \\
    \partial_t \uuB = & -\uB_x S \ee_y - S x \partial_y \uuB + \ldots
\intertext{or}
   \partial_t \bB_{x,z}  + & S x \partial_y \bB_{x,z} = \rB_{x,z} \\
   \partial_t \bB_y  + &S x \partial_y \bB_y =  \rB_y + S \bB_x  \\
   \partial_t \uB_{x,z}  + & S x \partial_y \uB_{x,z} = \sB_{x,z} \\
   \partial_t \uB_y  + & S x \partial_y \uB_y =   \sB_y - S \uB_x 
\end{align}
{solution of equations of type $\partial_t f + S x \partial_y f = r $ by method of characteristics (initial condition $=0$):}
\EQ
      f= \left[\int_0^t r(y_0 + Sx\tau,\tau;x,z) d\tau \right]_{y_0=y-Sxt}  
\EN
(Leave out arguments $x,z$ now.)
{let $\meanBB = -\meanJ_x z \ee_y, \; \meanJ_x = \mbox{const.},  \longrightarrow$}
%NS: question, added
%\blue{NS: 
%s $J_x$ being assumed as uniform;
%that would appear strange, I think.
%MR: Why? J_x can only depend on z, but as the transport coefficients cannot depend on z
%        it is sufficient to consider constant J_x.
%NS: OK, but then mean B is linear in z. Does that not appear
%NS: incompatible with periodicity?
%MR: in the simulation, yes, but for the derivation we need not to specify the BCs in z 
%NS: ok, all good for me then

\EQL
     \partial_t \bB_{x,z}  &+ S x \partial_y \bB_{x,z} = \meanB_y \partial_y \uz_{x,z} \\
     \partial_t \bB_y  &+ S x \partial_y \bB_y = \meanB_y \left(\partial_y \uz_y - \nab\cdot\uuz \right)+ \uz_z \meanJ_x  + S \bB_x \\
     \bB_{x,z}  &=  \left[\meanB_y \int_0^t \partial_y \uz_{x,z} (y_0 + Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}  \\
     \bB_y  &=  \left[\meanB_y \int_0^t  \left(\partial_y \uz_y - \nab\cdot\uuz \right) (y_0 + Sx\tau,\tau) d\tau\right.  \\
     \qquad&+ S \meanB_y \int_0^t \int_0^\tau \partial_y \uz_x (y_0 + Sx\tau',\tau') \, d\tau'  d\tau \\
     \qquad&\left.+ \:\meanJ_x \int_0^t \uz_z (y_0 + Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}
\end{align}
{hence no contribution to $\meanEEE_y \sim \meanJ_x$ !}
\EQL
     &\partial_t \uB_{x,z}  + S x \partial_y \uB_{x,z} = \sB_{x,z} = \begin{cases}  -&\!\!\!\!\jz_z \meanB_y \qquad\qquad\quad\text{for $x$}\\ &\!\!\!\!\jz_x \meanB_y + \meanJ_x  \bz_y \quad\:\text{for $z$}\end{cases} \\
      &\partial_t \uB_{y}  + S x \partial_y \uB_y = \sB_y - S \uB_x =  - S \uB_x - \meanJ_x \bz_z  \\
      & \uB_{x,z} =\left[\int_0^t \sB_{x,z}(y_0 + Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}\\
      & \uB_y      =\left[ - S \int_0^t \uB_x(y_0 + Sx\tau,\tau) d\tau -\meanJ_x \int_0^t \bz_z(y_0 + Sx\tau,\tau) d\tau\right]_{y_0=y-Sxt}.
\end{align}
{Now express $\meanEEEE^{(B)}$ in terms of $\uu,\bb$ to first order in $\meanBB$:}
\EQL
      & \meanEEEE^{(B)} = \left\langle \uuz \times \bbB + \uuB\times\bbz\right\rangle_{xy} \\
      & \meanEEE^{(B)}_y = \left\langle \uz_z \bB_x - \uz_x \bB_z +  \uB_z \bz_x - \uB_x \bz_z\right\rangle_{xy} = \ldots -\eta_{yx} \meanJ_x . \\
\intertext{The only contribution $\sim \meanJ_x$ comes from $\uB_z$: }
      & \meanJ_x \left[\int_0^t \bz_y(y_0+Sx\tau,\tau) d\tau\right]_{y_0=y-Sxt} , \label{eq:uzBJx}\\
\intertext{hence}
      & \eta_{yx} = -\left\langle \bz_x(y,t) \left[\int_0^t  \bz_y(y_0+Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}\right\rangle_{xy} \\
\intertext{With}
      & \bbz = \bbzz + \nab\times\left[\UUS \times \int_0^t \bzz(y,\tau) d\tau\right] \\
      & \bz_x = \bzz_x - S x  \int_0^t \partial_y \bzz_x(y,\tau) d\tau \\
      & \bz_y = \bzz_y - S x  \int_0^t \partial_y \bzz_y(y,\tau) d\tau + S \int_0^t \bzz_x (y,\tau) d\tau
\end{align}
\EQL
      \left[\int_0^t  \bz_y(y_0+Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt} &=  \\
      & \hspace{-3cm}\left[\int_0^t  \bzz_y(y_0+Sx\tau,\tau) d\tau  \right.\\
      & \hspace{-3cm}-  S x \int_0^t  \int_0^\tau \partial_y \bzz_y(y_0+S x \tau,\tau') d\tau' d\tau \\
      & \hspace{-3cm}\left.+ S \int_0^t  \int_0^\tau \bzz_x(y_0+S x \tau,\tau') d\tau' d\tau \right]_{y_0=y-Sxt} .    
\end{align}
{Hence} 
\EQL
      \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} .
\end{align}
\subsection{Arbitrary order in $S$}
{instead of \eqref{eq:socaSz}: }
\EQL
   \partial_t \bbSz = &- \UUS\cdot\nab\bbz + \bbz\cdot\nab\UUS  \\ 
   \partial_t \uuSz = &- \UUS\cdot\nab\uuz - \uuz\cdot\nab\UUS \\ 
\end{align}
component-wise:
\EQL
       &\partial_t \bSz_{x,z}  + S x \partial_y \bSz_{x,z} = \rSz_{x,z} = - S x \partial_y \bzz_{x,z}\\
     &\partial_t \bSz_y  + S x \partial_y \bSz_y =   S \bSz_x  + \rSz_y = - S x \partial_y \bzz_y + S \bSz_x  + S \bzz_x \\
    &\partial_t \uSz_{x,z}  + S x \partial_y \uSz_{x,z} = \sSz_{x,z} = -  S x \partial_y \uzz_{x,z}\\
     &\partial_t \uSz_y  + S x \partial_y \uSz_y =   - S \uSz_x  + \sSz_y = -  S x \partial_y \uzz_y - S \uSz_x  - S \uzz_x \\
     &\bSz_{x,z}  =  -Sx\left[\int_0^t \partial_y \bzz_{x,z} (y_0 + Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}  \\
     &\uSz_{x,z}  =  -Sx\left[\int_0^t \partial_y \uzz_{x,z} (y_0 + Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}  \\
     &\bSz_{y}  =  -Sx\left[\int_0^t \partial_y \bzz_y (y_0 + Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}  \\ 
     &\qquad \quad\;+ S\left[\int_0^t  \bzz_x (y_0 + Sx\tau,\tau) d\tau +\int_0^t  \bSz_x (y_0 + Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}\\
     &\uSz_{y}  =  -Sx\left[\int_0^t \partial_y \uzz_y (y_0 + Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}  \\ 
     &\qquad \quad\;-  S\left[\int_0^t  \uzz_x (y_0 + Sx\tau,\tau) d\tau + \int_0^t  \uSz_x (y_0 + Sx\tau,\tau) d\tau\right]_{y_0=y-Sxt}\\
\intertext{for use in \eqref{eq:uzBJx}}
     &\left[\int_0^t \bz_y(y_0+Sx\tau,\tau) d\tau\right]_{y_0=y-Sxt} \\&= \left[\int_0^t \bzz_y(y_0+Sx\tau,\tau) d\tau +  \int_0^t \bSz_y(y_0+Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt} \\
     & \left[\int_0^t \bSz_y(y_0+Sx\tau,\tau) d\tau \right]_{y_0=y-Sxt}  \\&=  \left[ \int_0^t \left( -Sx  \int_0^\tau \partial_y \bzz_y (y_0 + Sx\tau',\tau') d\tau' \right. \right. \\ & \left.\left. + S\int_0^\tau  \bzz_x (y_0 + Sx\tau',\tau') d\tau' + S \int_0^\tau \bSz_x (y_0 + Sx\tau',\tau') d\tau' \right)d\tau \right]_{y_0=y-Sxt}  \label{eq:interm}\\
     & \bSz_x (y_0 + Sx\tau',\tau')  =   -Sx \int_0^{\tau'}\partial_y \bzz_x (y_0 + Sx\tau{''},\tau{''}) d\tau{''}   \\
\end{align}
{hence from \eqref{eq:interm}}
\EQL
     = \left[  \right.&\left.-Sx  \int_0^t \int_0^\tau \partial_y \bzz_y (y_0 + Sx\tau',\tau') d\tau' d\tau \right. \\
     &   + S  \int_0^t \int_0^\tau  \bzz_x  (y_0 + Sx\tau',\tau') d\tau' d\tau         \\
     &   \left.- S^2x  \int_0^t \int_0^\tau \int_0^{\tau'}\partial_y \bzz_x (y_0 + Sx\tau{''},\tau{''}) d\tau{''} d\tau' d\tau \right]_{y_0=y-Sxt}
\end{align}
so essentially one time integral more.
%MR: now worked out \eta_{yx} explicitly
{Hence} 
\EQL
      \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_0+Sx\tau,\tau) d\tau\right]_{y_0=y-Sxt} \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_0+Sx\tau,\tau) d\tau\right]_{y_0=y-Sxt}  \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.\left. \qquad\quad+ \, S \,\bzz_x \left[\int_0^t \int_0^\tau \int_0^{\tau'}\partial_y \bzz_x (y_0 + Sx\tau{''},\tau{''}) d\tau{''} d\tau' d\tau  \right]_{y_0=y-Sxt}  \right\}\right. \\
      & \left. + \,S^2x^2 \left\{ \left[\int_0^t \partial_y \bzz_x(y_0+Sx\tau,\tau) d\tau\right]_{y_0=y-Sxt}  \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. + \,S\, \left[\int_0^t \partial_y \bzz_x(y_0+Sx\tau,\tau) d\tau\right]_{y_0=y-Sxt}  \left[\int_0^t \int_0^\tau \int_0^{\tau'}\partial_y \bzz_x (y_0 + Sx\tau{''},\tau{''}) d\tau{''} d\tau' d\tau  \right]_{y_0=y-Sxt} \right\} \right\rangle_{xy} .
\end{align}
\EQL
      \eta_{yx} = &-\left\langle \bzz_x \left[\int_0^t \left(\bzz_y(\xi,\tau) + S \int_0^\tau \bzz_x(\xi',\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(\xi',\tau') d\tau' d\tau + S \int_0^t \int_0^\tau \int_0^{\tau'}\partial_y \bzz_x (\xi{''},\tau{''}) d\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(\xi,\tau) d\tau\right]_{y_0=y-Sxt} \left[\int_0^t \left( \bzz_y(\xi,\tau) + S  \int_0^\tau \bzz_x(\xi',\tau') d\tau' \right) d\tau  \right]_{y_0=y-Sxt} \right\}  \right. \\
      & \left. + \,S^2x^2 \left[\int_0^t \partial_y \bzz_x(\xi,\tau) d\tau\right]_{y_0=y-Sxt}  \left[\int_0^t \int_0^\tau \partial_y \bzz_y(\xi',\tau') \right. \right. \\
      & \left.\left. + \,S  \int_0^t \int_0^\tau \int_0^{\tau'}\partial_y \bzz_x (y_0 + Sx\tau{''},\tau{''}) d\tau{''} d\tau' d\tau  \right]_{y_0=y-Sxt} \right\rangle_{xy} ,
\end{align}
with $\xi=y_0+Sx\tau$, $\xi'=y_0+Sx\tau'$, $\xi''=y_0+Sx\tau''$
\EQL
      \eta_{yx} = &-\left\langle \bzz_x(y,t) \, I_1 -Sx \left\{ \bzz_x(y,t)\, I_2 + \left[\int_0^t \partial_y \bzz_x(\xi,\tau) d\tau\right]_{y_0=y-Sxt} \, I_1 \right\}  \right. \\
      & \left. + \,S^2x^2 \left[\int_0^t \partial_y \bzz_x(\xi,\tau) d\tau\right]_{y_0=y-Sxt}  I_2 \right\rangle_{xy},\\ 
\intertext{with}
      & I_1=\left[\int_0^t \left(\bzz_y(\xi,\tau) + S \int_0^\tau \bzz_x(\xi',\tau') d\tau' \right) d\tau\right]_{y_0=y-Sxt}\\
      & I_2= \left[\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  \right]_{y_0=y-Sxt}.
\end{align}
\end{document}

Assume from Fourier transform
\EQL
    & f=\int\!\!\int \hat{f}(k_y,\omega) \ee^{\ii(k_y y -\omega t)} dk_y d\omega \\
 \intertext{then}
    & \left[\int_0^t f(y_0+Sx\tau,\tau) d\tau\right]_{y_0=y-Sxt} =  \int\!\!\int \hat{f}(k_y,\omega) \left[ \int_0^t  \ee^{\ii\left(k_y y_0+(Sxk_y -\omega)\tau\right)}  d\tau \right]_{y_0=y-Sxt}  dk_y d\omega \\
    &= \int\!\!\int -\ii \frac{\hat{f}(k_y,\omega)}{Sxk_y -\omega} \ee^{\ii k_y y}  \left( \ee^{-\ii \omega t}  - \ee^{-\ii  Sx k_y t} \right) dk_y d\omega\\
    & \int\!\!\int \frac{\ee^{\ii(k_y y -\omega t)} }{Sxk_y -\omega} dk_y d\omega =  \int\!\!\int \frac{\ee^{\ii(k_y y - Sxk_y t + Sxk_y t -\omega t)} }{Sxk_y -\omega} dk_y d\omega \\ 
    &= \ii t \int  \ee^{\ii k_y( y - Sx t)}  \left[\operatorname{Ei}\big(\ii t(Sxk_y -\omega) \big) \right]_{-\infty}^{\infty} dk_y\\   
    & \left[\int_0^t \!\!\int_0^\tau  f(y_0+Sx\tau,\tau') d\tau'  d\tau\right]_{y_0=y-Sxt} =
\end{align}
%MR: this doesn't help though
Here, $\left\langle x b_y^{(00)}(y,t) \partial_y b^{(00)}_x(y,\tau) \right\rangle_{\!\!xy}$ is the $[2,1,2]$
component of the correlator $\left\langle x b_i^{(00)}(y,t) \partial_k b^{(00)}_j(y,\tau) \right\rangle_{\!\!xy}$ and because of isotropy, stationarity and mirror-symmetry of the background turbulence,
the latter has the form $c_1 \delta_{ij} e_k + c_2 \delta_{ik} e_j - \frac{1}{3}(c_1+c_2) \delta_{jk} e_i$ with the unit vector $\ee=[1,0,0]$ in $x$ direction and coefficients $c_i$ depending only on $t-\tau$.
\end{document}

