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\section{Implementation of the $\qp$-term}

On the rhs of the momentum equation we have $\half\nab\qp\BB^2$, and so,
with $\qp=\qp(\beta,z)$, we have
\begin{eqnarray}
\half\nab[\qp(\beta,z)\BB^2]
\!&=&\!\half\qp\nab\BB^2+\half\BB^2\nab\qp\\
\!&=&\!\half\qp\nab\BB^2+\half\BB^2\left[{\dd\qp\over\dd\beta^2}\nab\beta^2+\nab\qp\right]
\\
\!&=&\!\half\qp\nab\BB^2+\half\BB^2\left[{\dd\qp\over\dd\beta^2}\Beq^{-2}(
\nab\BB^2-\BB^2\ln\rho)+\nab\qp\right]
\\
\!&=&\!\half\left[\qp+{\dd\qp\over\dd\beta^2}{\BB^2\over\Beq^2}\right]\nab\BB^2
+\half\BB^2\left[-{\dd\qp\over\dd\beta^2}{\BB^2\over\Beq^2}\nab\ln\rho+\nab\qp\right]
\end{eqnarray}

\section{Implementation of the $\qg$-term}

\begin{eqnarray}
%\half\nab_z[\qg(\beta,z)\BB^2]
%=\half\left[\qp+{\dd\qp\over\dd\beta^2}{\BB^2\over\Beq^2}\right]\nab_z\BB^2
%+\half\BB^2\left[-{\dd\qp\over\dd\beta^2}{\BB^2\over\Beq^2}\nab_z\ln\rho+\nab_z\qp\right]
%\half{\partial\over\partial z}[\qg(\beta,z)\BB^2]
%=\half\left[\qg+{\dd\qg\over\dd\beta^2}{\BB^2\over\Beq^2}\right]{\partial\over\partial z}\BB^2
%+\half\BB^2\left[-{\dd\qg\over\dd\beta^2}{\BB^2\over\Beq^2}{\partial\over\partial z}\ln\rho+{\partial\over\partial z}\qg\right]
%JW: do you mean this?
%AB: yes, and there should be no 1/2 factor
{\partial\over\partial z}[\qg(\beta,z)\BB^2]
=\left[\qg+{\dd\qg\over\dd\beta^2}{\BB^2\over\Beq^2}\right]{\partial\over\partial z}\BB^2
+\BB^2\left[-{\dd\qg\over\dd\beta^2}{\BB^2\over\Beq^2}{\partial\over\partial z}\ln\rho+{\partial\over\partial z}\qg\right]
\end{eqnarray}

%Thus, altogether we have 
In particular, when $\qp/2+\qg=a_{\rm g}\qp$ with
$a_{\rm g}=\const$, we have
\begin{eqnarray}
\pmatrix{
\partial[(\qp/2)\BB^2]/\partial x\cr
\partial[(\qp/2)\BB^2]/\partial y\cr
\partial[(\qp/2+\qg)\BB^2]/\partial z}
=\half\pmatrix{
\partial(\qp\BB^2)/\partial x\cr
\partial(\qp\BB^2)/\partial y\cr
\partial(a_{\rm g}\qp\BB^2)/\partial z}
\end{eqnarray}
The choice $a_{\rm g}=0.2$ means that
$\qg/\qp=-\half(1-a_{\rm g})=-0.4$.

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%Equilibria
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%\bibitem[Biskamp \& M\"uller(1999)]{BM99}
%Biskamp, D., \& M\"uller, W.-C.\yprl{1999}{83}{2195}

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