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Next: Potential Problems Up: Magnetohydrodynamics MHD Previous: Non-potential current flowing inside

Summary

You must know the MHD equations.
\begin{displaymath}
\rho {D{\bf v}\over Dt} = -\nabla p + {\bf J}\times {\bf B} + \rho
{\bf g},
\end{displaymath} (4.17)


\begin{displaymath}
{\partial {\bf B}\over \partial t} = \nabla \times ({\bf v}\times
{\bf B}),
\end{displaymath} (4.18)


\begin{displaymath}
{\partial \rho \over \partial t} + \nabla \cdot (\rho {\bf v}) = 0,
\end{displaymath} (4.19)


\begin{displaymath}
\nabla \cdot {\bf B} = 0,
\end{displaymath} (4.20)


\begin{displaymath}
{\bf J} = \nabla \times {\bf B}/\mu_{0},
\end{displaymath} (4.21)


\begin{displaymath}
p = \rho R T,
\end{displaymath} (4.22)

and an energy equation.

Simplifications to the MHD equations include looking for steady, static equilibrium solutions that satisfy force balance so that the equation of motion reduces to

\begin{displaymath}
\nabla p = {\nabla \times {\bf B}\over \mu_{0}}\times {\bf B} + \rho
{\bf g}.
\end{displaymath} (4.23)

If gravitational effects are negligible, then we have
\begin{displaymath}
\nabla p = {\nabla \times {\bf B}\over \mu_{0}}\times {\bf B},
\end{displaymath} (4.24)

and finally, if the pressure gradient term is negligible (meaning that the gas pressure $p$ is much smaller than the magnetic pressure $B^{2}/2\mu_{0}$, the magnetic field is called force-free and
\begin{displaymath}
{\nabla \times {\bf B}\over \mu_{0}}\times {\bf B} = 0.
\end{displaymath} (4.25)


next up previous
Next: Potential Problems Up: Magnetohydrodynamics MHD Previous: Non-potential current flowing inside
Prof. Alan Hood
2000-11-06