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The basic equations to be solved are the magnetohydrodynamical equations
and the Poisson equation for the gravitational potential.
In cylindrical coordinates (
,
,
)
with
, the equations are expressed as follows:
![\begin{displaymath}
\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial z}(\rho v_z)+\frac{1}{r}\frac{\partial}{\partial r}(r\rho v_r)=0,
\end{displaymath}](img1464.png) |
(B.11) |
![\begin{displaymath}
\frac{\partial \rho r v_\phi}{\partial t}+
\frac{\partial}...
...}(rB_\phi)B_r
+\frac{\partial B_\phi}{\partial z}B_z \right]
\end{displaymath}](img1473.png) |
(B.14) |
![\begin{displaymath}
\frac{\partial B_z}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}[r(v_zB_r-v_rB_z)],
\end{displaymath}](img1474.png) |
(B.15) |
![\begin{displaymath}
\frac{\partial B_r}{\partial t}=-\frac{\partial }{\partial z}(v_zB_r-v_rB_z),
\end{displaymath}](img1475.png) |
(B.16) |
![\begin{displaymath}
\frac{\partial B_\phi}{\partial t}=
\frac{\partial }{\part...
..._\phi)
-\frac{\partial }{\partial r}(v_r B_\phi -v_\phi B_r),
\end{displaymath}](img1476.png) |
(B.17) |
![\begin{displaymath}
\frac{\partial^2 \psi}{\partial z^2}
+ \frac{1}{r}\frac{\p...
...left(r\frac{\partial \psi}{\partial r}\right) =
4 \pi G \rho,
\end{displaymath}](img1477.png) |
(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.
Next: Hydrostatic Equilibrium
Up: Magnetohydrodynamics
Previous: Basic Equations of Ideal
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Kohji Tomisaka
2007-07-08