(2000) and Lu et al. (2004). Convection is present whenever $N^{2}_{eff}< 0$. $\alpha_{c}$ can be written in the form similar to normal viscosity as: $$\alpha_{c}=\frac{\nu_{con}}{c_{s} H}$$

By integrating along $z$ of the hydrostatic balance, we have: $$\Omega^{2}_{k} H^{2}-\frac{c_\mathrm{r}}{\sqrt{\Sigma}}\frac{d}{dr}(\sqrt{\Sigma} c_\mathrm{z})H = c^{2}_\mathrm{s}+\frac{1}{2}(c^{2}_\mathrm{r}+c^{2}_\mathrm{\varphi})$$

In order to complete the problem we need to introduce energy equation. We assume the generated energy due to viscosity and resistivity into the volume is balanced by the advection cooling, outward energy by convection and energy loss of outflow $(Q^{adv}+Q^{rad}+Q^{conv}+Q^{wind}=Q^{diss})$. Thus, $$\Sigma V_{r} T \frac{d S}{dr}+\frac{1}{r}\frac{d}{dr}(r F_{con})= f(\nu+g\nu_{con})\Sigma r^{2} (\frac{d \Omega}{dr})^{2}+\frac{\eta}{4\pi}\mathrm{J}^{2}$$ $$-\frac{1}{2}\zeta \dot{m}_\mathrm{w}(r)V_\mathrm{k}^{2}(r)$$

where $F_{con}$, $S$, $f$ and $T$ are the convective energy flux, the specific entropy, advection parameter and temperature respectively. Also we consider $Q^{diss}-Q^{rad}=fQ^{diss}$. Their corresponding relations are: $$F_{con}=-\nu_{con} \Sigma T \frac{d S}{dr}$$ where $$T\frac{d S}{dr}=\frac{1}{\gamma-1}\frac{d c^{2}_{s}}{dr}-\frac{c^{2}_{s}}{\rho}\frac{d \rho}{dr}$$ here $\gamma$ is the specific energy heats, $J=\nabla \times B $ is the current density and $\eta$ is the magnetic diffusivity due to turbulence. The two first term on the right hand side of the energy equation corresponds to the dissipation energy by viscosity, convection and resistivity $Q_{diss}= f(\nu+g\nu_{con})\Sigma r^{2} (\frac{d \Omega}{dr})^{2}+\frac{\eta}{4\pi}\mathrm{J}^{2}$. We can write the magnetic resistivity turbulence in the form of viscosity and convection turbulence as 