deletions | additions       diff --git a/Non-adiabatic convection.tex b/Non-adiabatic convection.tex  index 96f1b5a..2ad2ad3 100644  --- a/Non-adiabatic convection.tex  +++ b/Non-adiabatic convection.tex 
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\begin{equation}  \frac{\nabla_r - \nabla}{\nabla_r - \adgrad} \equiv \zeta = \frac{a_0 \Gamma^2}{1+ \Gamma(1+a_0\Gamma)}.  \end{equation}  $\zeta$ is a function of the convective efficicency only, with $\zeta \rightarrow 0$ as $\Gamma \rightarrow 0$ and $\zeta \rightarrow 1$ as $\Gamma \rightarrow \infty$. It is possible to write a single cubic equation with $\zeta$, the new measure of convective efficicency, as the unknown variable. Once its value its determined the value of the actual gradient is given by $\nabla = (1-\zeta)\, \nabla_r \,+\zeta \,\adgrad$ \,\adgrad$. Note that $\nabla \rightarrow \nabla_r$ for $ \zeta \rightarrow 0$ and $\nabla \rightarrow \ad_nabla$ for $ \zeta \rightarrow 1$ as expected since $\zeta$ is a measure of the convective efficiency. We can write $\Gamma = B \zeta^{1/3}$, where $B\equiv [(\mathscr{A}^2/a_0)(\nabla_r - \adgrad)]^{1/3}$. With these definitions we finally obtain the cubic equation usually solved in MLT non-adiabatic convection:  \begin{equation}  \zeta^{1/3} + B\,\zeta^{2/3}+ a_0\,B^2\,\zeta - a_0\,B^2 =0  \end{equation}