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MHD Equations

In MHD, the magnetic induction ${\bf B}$ and the plasma velocity ${\bf v}$ are determined by the equation of motion
\begin{displaymath}
\rho {D{\bf v}\over Dt} = \rho {\partial {\bf v}\over \part...
...){\bf v} = -\nabla p +
{\bf J}\times {\bf B} + \rho {\bf g},
\end{displaymath} (4.1)

and the ideal induction equation
\begin{displaymath}
{\partial {\bf B}\over \partial t} = \nabla \times ({\bf v}\times
{\bf B}),
\end{displaymath} (4.2)

In addition, there is the equation of mass continuity
\begin{displaymath}
{\partial \rho \over \partial t} + \nabla \cdot (\rho {\bf v}) = 0,
\end{displaymath} (4.3)

and the remainder of Maxwell's equations
\begin{displaymath}
\nabla \cdot {\bf B} = 0,
\end{displaymath} (4.4)


\begin{displaymath}
{\bf J} = \nabla \times {\bf B}/\mu_{0},
\end{displaymath} (4.5)

plus
\begin{displaymath}
p = \rho R T,
\end{displaymath} (4.6)

and an energy equation. This may, for example, be taken as an isothermal plasma.

Note,

  1. In the equation of motion we have an extra magnetic force, ${\bf J}\times {\bf B}$ (the Lorentz force), that is normal to both the magnetic field and the current. On using (4.5), this force may be written in the form
    \begin{displaymath}
{\nabla \times {\bf B}\over \mu_{0}}\times {\bf B} = {1\ove...
...ght ){\bf B} - \nabla \left
({B^{2}\over 2\mu_{0}}\right ).
\end{displaymath} (4.7)

    The second term on the right hand side represents a magnetic pressure force, $B^{2}/2\mu_{0}$, acting from regions of high magnetic pressure to low magnetic pressure. The first term is interpreted as a magnetic tension force which acts when the field lines are curved. Magnetic tension is similar to the force exerted by a stretched elastic band. Interpreting the Lorentz force in terms of these two basic quantities is important since it is possible to see how the plasma will respond to a given magnetic field configuration.

  2. In the Maxwell equation,

    \begin{displaymath}
{\nabla \times {\bf B}\over \mu_{0}} = {\bf J} + {\partial \over
\partial t}\left (\epsilon_{0}{\bf E}\right ),
\end{displaymath}

    which is called Ampère's Law, the last term is negligible, except for extremely rapid time variations. Dropping this term gives (4.5). In order of magnitude, the Maxwell equation,

    \begin{displaymath}
\nabla \times {\bf E} = -{\partial {\bf B}\over \partial t},
\end{displaymath}

    which is called Faraday's Induction Law, can be approximated by $E/L =
B/t$, where $t$ and $L$ are typical time and length variations of the plasma. Comparing the left hand side with the last term (the displacement current) in Ampère's law, we find that

    \begin{displaymath}
{B\over L} >> \mu_{0}\epsilon_{0}{E\over t}.
\end{displaymath}

    Thus,

    \begin{displaymath}
c^{2} >> {L^{2}\over t^{2}} = V^{2},
\end{displaymath}

    where we have used

    \begin{displaymath}
c^{2} = 1/\mu_{0}\epsilon_{0}
\end{displaymath}

    for the speed of light and the typical plasma velocity is $V = L/t$. Hence, (4.5) is valid provided the plasma flows are small compared with the speed of light.

  3. A plasma moving with a velocity ${\bf v}$ experiences an electric field ${\bf v}\times {\bf B}$ in addition to ${\bf E}$ so that Ohm's Law is
    \begin{displaymath}
{\bf J} = \sigma\left ({\bf E} + {\bf v}\times {\bf B}\right ).
\end{displaymath} (4.8)

    Normally, the left hand side is negligible so that (4.8) reduces to
    \begin{displaymath}
0 ={\bf E} + {\bf v}\times {\bf B}.
\end{displaymath} (4.9)

    Using (4.9) to eliminate ${\bf E}$ the Maxwell equation

    \begin{displaymath}
\nabla \times {\bf E} = -{\partial {\bf B}\over \partial t},
\end{displaymath}

    reduces to (4.2).


Subsections
next up previous
Next: Effect of on Up: Magnetohydrodynamics MHD Previous: Solar Observations
Prof. Alan Hood
2000-11-06