oriented down the specific angular momentum gradient. It means that convection tries to drive the system toward a stat of uniform specific angular momentum and consequently it corresponds to an inward angular momentum transportation (Narayan et al. 2000). We have determined the convective turbulence parameter $\alpha_{c}$ be the mixing length approximation. It can be imagined that a convective differentially-rotating fluid include of many independent fluid blobs. following Grossman et al. (1993) the convectively turbulence viscosity is defined as $$\nu_{turb}=\sigma L_{M}$$ where $\sigma$ is the velocity dispersion of the blobs and $L_{M}$ is the characteristic mixing length corresponding to effective mean free path of the blobs. So we can write the $\nu_{con}$ as follows (Lu et al. 2004): $$\nu_{con}=\frac{L_{M}^{2} }{4} (-N^{2}_{eff})^{\frac{1}{2}}$$ Here $N_{eff}$ is the effective frequency of the convective blobs and it will be:

$$N^{2}_{eff}=N^{2}+k^{2}$$

where $N$ and $k$ are the Brunt-Vaisala frequency and epicyclic frequency respectively, which are defined as $$N^{2}=-\frac{1}{\rho}\frac{ dp_{g}}{dr}\frac{d}{dr}\ln(\frac{p^{\frac{1}{\gamma}}_{g}}{\rho})$$ and $$k^{2}=2\Omega ^{2}\frac{d \ln(r^{2}\Omega)}{d lnr}$$ Also the characteristic mixing length $L_{M}$ could be written in terms of the pressure scale height $(H_{p}=-\frac{d r}{d \ln p})$ and the dimensionless mixing length parameter $l_{m}$ as bellow: $$L_{M}=2^{-\frac{1}{4}}l_{M}H_{p}$$

We have adopt $l_{M}=\sqrt{2}$ as it was estimated by Narayan et al. 