The three basic equations to be solved involve three unknowns: \(\nabla\), \(\nabla'\) and the convective efficiency \(\Gamma\): \[\label{eq:mlteq} \Gamma =\mathscr{A} \, (\nabla - \nabla')^{1/2} \\ \nabla_{r} - \nabla = a_0 \,\mathscr{A} \, (\nabla - \nabla')^{3/2} \\ \Gamma = (\nabla - \nabla')/ (\nabla' - {\nabla_{\mathrm{\!ad}}})\] Here the numerical factor \(a_0\) is of order 1 and differs slightly depending on different implementation of the MLT. These three equation can be rearranged to obtain \[\frac{\nabla_r - \nabla}{\nabla_r - {\nabla_{\mathrm{\!ad}}}} \equiv \zeta = \frac{a_0 \Gamma^2}{1+ \Gamma(1+a_0\Gamma)}.\] \(\zeta\) 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 \(\nabla = (1-\zeta)\, \nabla_r \,+\zeta \,{\nabla_{\mathrm{\!ad}}}\). Note that \(\nabla \rightarrow \nabla_r\) for \( \zeta \rightarrow 0\) and \(\nabla \rightarrow {\nabla_{\mathrm{\!ad}}}\) for \( \zeta \rightarrow 1\) as expected since \(\zeta\) is a measure of the convective efficiency. We can write \[\label{eq:zeta} \Gamma = B \zeta^{1/3},\] where \(B\equiv [(\mathscr{A}^2/a_0)(\nabla_r - {\nabla_{\mathrm{\!ad}}})]^{1/3}\). With these definitions we finally obtain the cubic equation usually solved in MLT non-adiabatic convection: \[\zeta^{1/3} + B\,\zeta^{2/3}+ a_0\,B^2\,\zeta - a_0\,B^2 =0.\]
Once the value of \(\zeta\) is known one can calculate the convective efficiency from \ref{eq:zeta}, the value of the actual gradient from \(\nabla = (1-\zeta)\, \nabla_r \,+\zeta \,{\nabla_{\mathrm{\!ad}}}\). To calculate the ratio of convective to total flux one can use \ref{eq:totalflux} and write \[\frac{F_{c}}{F} = \frac{F -F_{r}}{F} = \frac{\nabla_r - \nabla}{\nabla_r} = \frac{\nabla_r - {\nabla_{\mathrm{\!ad}}}}{\nabla_r}\zeta.\] The convective velocities can be calculated obtaining \(\nabla - \nabla'\) from first eq. in \ref{eq:mlteq} and subsituting the value in \ref{eq:velocity}. Finally the superadiabaticity \(\nabla-{\nabla_{\mathrm{\!ad}}}\) is obtained from \(\nabla = (1-\zeta)\, \nabla_r \,+\zeta \,{\nabla_{\mathrm{\!ad}}}\), which gives: \[\nabla - {\nabla_{\mathrm{\!ad}}}= (1-\zeta)\,(\nabla_r -{\nabla_{\mathrm{\!ad}}}).\]