the self similar method to solving above complicated differential equations. This powerful technique is a dimensional analysis and scaling law and is widely used in astrophysical fluid mechanics. Following to AM12, and other similar works (Ghanbari et al 2009 and samadi et al 2014, 2016), self-similarity in the radial direction is assumed: $$\Sigma=c_{0}\Sigma_{out} (\frac{r}{r_{out}})^{s-\frac{1}{2}}$$ $$V_r(r)=-c_1 \sqrt{\frac{G M_{\ast}}{r_{out}}}(\frac{r}{r_{out}})^{-\frac{1}{2}}$$ $$V_\varphi(r)= r \Omega (r) =c_2 \sqrt{\frac{G M_{\ast}}{r_{out}}}(\frac{r}{r_{out}})^{-\frac{1}{2}}$$ $$c_\mathrm{s}^{2}=c_3 \frac{G M_{\ast}}{r_{out}}(\frac{r}{r_{out}})^{-1}$$ $$c_\mathrm{r, \varphi, z}^{2} =\frac{B^{2}_{r, \varphi, z}}{4\pi\rho}= 2 \beta_{r, \varphi, z} c_3 \frac{G M_{\ast}}{r_{out}}(\frac{r}{r_{out}})^{-1}$$ $$H(r)=c_{4}r_{out}(\frac{r}{r_{out}})$$ $$\rho=\frac{1}{2}\frac{c_0 \Sigma_{out}}{c_4}\frac{1}{r_{out}}(\frac{r}{r_{out}})^{s-\frac{3}{2}}$$ $$B_{r,\varphi,z}= 2\sqrt{\frac{\pi\beta_{r, \varphi, z} c_3 c_0 \Sigma_{out} G M_\ast}{c_4}} \frac{1}{r_{out}}(\frac{r}{r_{out}})^{\frac{s}{2}-\frac{5}{4}}$$ where the constant $c_0$, $c_1$, $c_2$, $c_3$ and $c_4$ are dimensionless constants and will be determined later. $\Sigma_{out}$ and $r_{out}$ have been exploited in order to write equations in non-dimensional form. Substituting the above self-similar transformation in the MHD equations of the system, we’ll obtain the following system of coupled ordinary equations, which should be solve to having $c_0$, $c_1$, $c_2$, $c_3$ and $c_4$: $$\dot{m}=c_0 c_1$$

$$-\frac{1}{2}c_1^{2}=c_2^{2}-1-[(s-\frac{3}{2})+\beta_\varphi(s+\frac{1}{2})+(s-\frac{3}{2})\beta_z]c_3$$

$$-\frac{1}{2}c_1 c_2=-\frac{3}{2}(s+\frac{1}{2})(\alpha+g\alpha_c)c_2 c_4 \sqrt{c_3}+(s+\frac{1}{2})c_3\sqrt{B_r B_\varphi}-s l^2 c_1 c_2$$

$$c_4=\frac{1}{2}(s-\frac{3}{2})\sqrt{B_r B_z}+\frac{1}{2}\sqrt{(s-\frac{3}{2})^{2} c^{2}_{3} B_r B_z+4(1+B_r+B_\varphi)c_3}$$

$$(\frac{1}{\gamma-1}+s-\frac{3}{2})[(s-1)\alpha_c c^{\frac{3}{2}}_{3}c_4+c_1 c_3]=-\frac{1}{4}s \zeta c_1$$

$$+f[\frac{9}{4}(\alpha+g\alpha_c)c_4 c^{2}_{2} c^{\frac{1}{2}}_{3}  + \frac{1}{2}\frac{\alpha}{P_m}c_4 c^{\frac{3}{2}}_{3}\beta_\varphi (s-\frac{1}{2})^{2}$$

$$+\frac{1}{2}\frac{\alpha}{P_m}c_4 c^{\frac{3}{2}}_{3}\beta_z (s-\frac{5}{2})^{2}-\frac{\alpha}{P_m}c_4 c^{\frac{3}{2}}_{3}\sqrt{\beta_z \beta_\varphi} (s-\frac{5}{2})(s-\frac{1}{2})]$$

$$\alpha_c=\frac{l^{2}_M}{4\sqrt{2 c_3}c_4 (s-\frac{5}{2})^{2}}\sqrt{(s-\frac{5}{2})c_3[(s-\frac{5}{2})\frac{1}{\gamma}-(s-\frac{3}{2})]-c^{2}_{2}}$$

where $\dot{m}$ is the dimensionless mass 