Axisymmetric Case

The basic equations to be solved are the magnetohydrodynamical equations and the Poisson equation for the gravitational potential. In cylindrical coordinates ($z$, $r$, $\phi$) with $\partial / \partial \phi=0$, the equations are expressed as follows:

\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial z...
...ho v_z)+\frac{1}{r}\frac{\partial}{\partial r}(r\rho v_r)=0,
\end{displaymath} (B.11)

$\displaystyle \frac{\partial \rho v_z}{\partial t}$ $\textstyle +$ $\displaystyle \frac{\partial}{\partial z}( \rho v_z v_z) +
\frac{1}{r}\frac{\partial}{\partial r}(r\rho v_z v_r)=$  
  $\textstyle -$ $\displaystyle c_s^2 \frac{\partial \rho}{\partial z}
-\rho\frac{\partial \psi}...
...c{\partial B_r}{\partial z}
-\frac{\partial B_z}{\partial r}\right)B_r\right],$ (B.12)

$\displaystyle \frac{\partial \rho v_r}{\partial t}$ $\textstyle +$ $\displaystyle \frac{\partial}{\partial z}( \rho v_r v_z) +
\frac{1}{r}\frac{\partial }{\partial r}(r\rho v_r v_r)=$  
  $\textstyle -$ $\displaystyle c_s^2 \frac{\partial \rho}{\partial r}
-\rho\frac{\partial \psi}...
...c{\partial B_r}{\partial z}
-\frac{\partial B_z}{\partial r}\right)B_z\right],$ (B.13)

\frac{\partial \rho r v_\phi}{\partial t}+
+\frac{\partial B_\phi}{\partial z}B_z \right]
\end{displaymath} (B.14)

\frac{\partial B_z}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}[r(v_zB_r-v_rB_z)],
\end{displaymath} (B.15)

\frac{\partial B_r}{\partial t}=-\frac{\partial }{\partial z}(v_zB_r-v_rB_z),
\end{displaymath} (B.16)

\frac{\partial B_\phi}{\partial t}=
\frac{\partial }{\part...
-\frac{\partial }{\partial r}(v_r B_\phi -v_\phi B_r),
\end{displaymath} (B.17)

\frac{\partial^2 \psi}{\partial z^2}
+ \frac{1}{r}\frac{\p...
...ft(r\frac{\partial \psi}{\partial r}\right) =
4 \pi G \rho,
\end{displaymath} (B.18)

where the variables have their ordinary meanings. Equation (B.11) is the continuity equation; equations (B.12), (B.13) and (B.14) are the equations of motion. The induction equations for the poloidal magnetic fields are equations (B.15) and (B.16) and for the toroidal magnetic field is equation (B.17). The last equation (B.18) is the Poisson equation.

Kohji Tomisaka 2009-12-10