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\end{equation}  Inserting eqn.~\ref{eqn:psi} this can be simplified to  \begin{equation}\label{eqn:ode}  \frac{\rm{d}\omega}{\rm{d}z} = \tan{\atan{\frac{\omega}{z}}-\asin{\frac{r}{R_0}}} \tan{\arctan{\frac{\omega}{z}}-\arcsin{\frac{r}{R_0}}}  \end{equation}  with  \begin{equation}