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Next: Potential Fields Up: Magnetohydrostatic Equilibria Previous: Hydrostatic Pressure Balance

The Plasma Beta - $\beta $

If the magnetic field is now included then we can anticipate the importance of the Lorentz force by considering the magnitude of the terms in (3.1).
  1. When ther is no magnetic field then pressure gradient and gravitational force balance gives

    \begin{displaymath}
{p_{0}\over L} = \rho_{0} g = {p_{0}\over \Lambda}.
\end{displaymath}

    Thus, we can neglect gravity and the exponential decrease of the pressure with height if $L \ll \Lambda $.

  2. Compare the Lorentz force with the pressure gradient force so that

    \begin{displaymath}
{p_{0}\over L} = {B_{0}^{2}\over \mu L}.
\end{displaymath}

    Hence, we may neglect the pressure gradient term if

    \begin{displaymath}
{\mu p_{0}\over B_{0}^{2}} \ll 1
\end{displaymath}

    and we may alternatively neglect the Lorentz force if

    \begin{displaymath}
{\mu p_{0}\over B_{0}^{2}} \gg 1.
\end{displaymath}

We define the ratio of the gas pressure to the magnetic pressure as

\begin{displaymath}
\beta \equiv {\hbox{gas pressure}\over \hbox{magnetic pressure}} =
{p_{0}\over B_{0}^{}/2\mu} = {2 \mu p \over B_{0}^{2}},
\end{displaymath}

so that
\begin{displaymath}
\beta = {2\mu p_{0}\over B_{0}^{2}}.
\end{displaymath} (3.7)

Example 3.2.1   If $L \ll \Lambda $, so that we may neglect gravity and $\beta \ll 1$ so that we may also neglect the gas pressure, then (3.1) reduces to the low $\beta $ plasma approximation
\begin{displaymath}
{\bf j} \times {\bf B} = 0
\end{displaymath} (3.8)

and the magnetic field is called ``force-free''.

Example 3.2.2  
  1. Coronal active regions where the magnetic field is closed.

    \begin{displaymath}
B = 100G = 10^{-2}{\rm tesla}, \qquad \mu = 4 \pi \times 10^{-7},
\qquad \tilde{\mu} = 0.5,
\end{displaymath}


    \begin{displaymath}
n = 10^{16}m^{-3}, \qquad m_{p} = 1.67 \times 10^{-27}kg, \qquad T =
2 \times 10^{6}K.
\end{displaymath}

    This gives $\rho = 1.67 \times 10^{-11}$kg m$^{-3}$ and $p = 0.55$ pascals. Therefore, the plasma beta is

    \begin{displaymath}
\beta = {2\mu p \over B^{2}} = 0.01.
\end{displaymath}

    Thus, $\beta $ is small in the corona.

  2. Coronal holes have a weaker magnetic field strength and a lower temperature. Typically we take

    \begin{displaymath}
B = 10G = 10^{-3}{\rm tesla}, \qquad \rho = 1.67 \times
10^{-13}{\rm kg m}^{-3}, \qquad T = 10^{6}K.
\end{displaymath}

    Thus, in a coronal hole the plasma beta has a typical value of

    \begin{displaymath}
\beta = 7 \times 10^{-3},
\end{displaymath}

    and is even smaller than the active region value.

To a good approximation the magnetic field in the solar corona is force-free since the plasma beta is much smaller than unity. This is not the case in the convection zone where the plasm beta is usually much larger than unity. The next few sections will consider different force free equilibria.
next up previous
Next: Potential Fields Up: Magnetohydrostatic Equilibria Previous: Hydrostatic Pressure Balance
Prof. Alan Hood
2000-01-11