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\subsection{Solutions for non-adiabatic convection}  The fraction of the flux carried by convection can be written using \ref{eq:totalflux} as  \begin{equation}  \frac{F_{c}}{F} = \frac{F -F_{r}}{F} = \frac{\nabla+r \frac{\nabla_r  - \nabla}{\nabla_r}. \end{equation}