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Next: About this document ... Up: Magnetohydrodynamic Waves Previous: Alfvén Waves

Magnetoacoustic Modes

Next we investigate the effect of a magnetic field and a gas pressure but still neglect gravity. Again consider the magnetic field and gas pressure to be uniform such that ${\bf B}_{0} = B_{0} \hat{\bf z}$ and $p_{0} =
$constant. Fourier analysing the pertubations, the linearised equation of motion becomes
\begin{displaymath}
-\rho_{0}\omega^{2}{\bf v}_{1} - -\gamma p_{0}{\bf k}({\bf ...
...1} \times \hat{\bf z})\right
\}\right ] \times \hat{\bf z}.
\end{displaymath} (4.37)

There are three components to (4.37) giving the three components of the velocity, namely $v_{x}, v_{y}$ and $v_{z}$. However, it is easier to deal with $\hat{\bf z} \cdot$ (4.37) to eliminate the Lorentz force, ${\bf k}\cdot $(4.37) and % latex2html id marker 1270
${\bf k}\cdot \left
(\hbox{(\ref{fourmotion})} \times \hat{\bf z}\right )$. We assume that all perturbations are of the form $\hbox{exp}\left (i \omega t - i ({\bf k}
\cdot {\bf r})\right )$.

Firstly, $\hat{\bf z} \cdot$ (4.37) gives

\begin{displaymath}
\omega^{2} v_{z} = c_{s}^{2} m ({\bf k}\cdot {\bf v}_{1}).
\end{displaymath} (4.38)

Next, ${\bf k}\cdot $(4.37) gives
\begin{displaymath}
\omega^{2}({\bf k}\cdot {\bf v}_{1}) = c_{s}^{2}K^{2}({\bf ...
...v}_{1}\times \hat{\bf z}\right )\right ] \times
\hat{\bf z}
\end{displaymath} (4.39)

Using vector identities as in Section 4.3, (4.39) reduces to
\begin{displaymath}
\omega^{2}({\bf k}\cdot {\bf v}_{1}) = \left (c_{s}^{2} + c...
... ) K^{2}({\bf k}\cdot {\bf v}_{1}) - m K^{2} c_{A}^{2} v_{z}.
\end{displaymath} (4.40)

(4.38) and (4.40) are two linear equations for $v_{z}$, the component of the perturbed velocity along the equilibrium magnetic field, and $({\bf k}\cdot {\bf v}_{1})$, the divergence of the perturbed velocity which gives a measure of the compression of the plasma. Since the system is homogeneous, there is only a solution if the determinant of the coefficients is zero. This gives the dispersion relation
\begin{displaymath}
\omega^{4} - K^{2}\left (c_{s}^{2} + c_{A}^{2}\right ) \omega^{2} +
K^{2}m^{2}c_{s}^{2}c_{A}^{2} = 0.
\end{displaymath} (4.41)

(4.41) is the dispersion relation for magnetoacoustic modes.

Example 4.4.1   If ${\bf B}_{0} = 0$, then $c_{A}^{2} = 0$ and (4.41) gives

\begin{displaymath}
\omega^{4} - K^{2}c_{s}^{2}\omega^{2} = 0.
\end{displaymath}

If $\omega^{2} \ne 0$, then we simply get $ \omega^{2} = K^{2}
c_{s}^{2}$ and the magnetoacoustic modes reduces to sound waves.

Example 4.4.2   If $p_{0} = 0$ so that $c_{s}^{2} = 0$, then (4.41) now reduces to

\begin{displaymath}
\omega^{4} - K^{2}c_{A}^{2}\omega^{2} = 0.
\end{displaymath}

Again if $\omega^{2} \ne 0$ then the dispersion relation becomes
\begin{displaymath}
\omega^{2} = K^{2}c_{A}^{2}.
\end{displaymath} (4.42)

However, this is not the anisotropic Alfvén wave of the previous section.

Finally, we take % latex2html id marker 1296
${\bf k} \cdot \left ( (\ref{fourmotion}) \times
\hat{\bf z}\right )$ to give the third component of the linearised equation of motion as

\begin{eqnarray*}
& &\omega^{2}\left [ {\bf k} \cdot \left ({\bf v}_{1} \times ...
...}\cdot {\bf v}_{1}) {\bf k}\cdot ({\bf k} \times \hat{\bf z}).
\end{eqnarray*}



Now most of these terms are zero and the only remaining terms give

\begin{displaymath}
\left [ \omega^{2} - m^{2}c_{A}^{2}\right ] \left \{ {\bf k}\cdot ({\bf v}_{1} \times {\bf k}) \right \} = 0.
\end{displaymath} (4.43)

Hence, we have
\begin{displaymath}
\omega^{2} = m^{2} c_{A}^{2}.
\end{displaymath} (4.44)

This is simply the Alfvén waves dispersion relation that we obtained in the previous section. Thus, the Alfvén wave is not influenced by the presence of a uniform equilibrium gas pressure.

Returning to (4.41), there are two solutions for $\omega^{2}$ namely

\begin{displaymath}
\omega^{2} = {K^{2}\over 2}\left (c_{s}^{2} + c_{A}^{2}\rig...
... -
4K^{2} c_{s}^{2}c_{A}^{2}\cos^{2}\theta \right \}^{1/2},
\end{displaymath} (4.45)

where $m = K \cos \theta$. The positive sign corresponds to the fast magnetoacoustic wave and the negative sign to the slow magnetoacoustic wave. Note that phase speeds for both the fast and slow magnetoacoustic waves depend on the angle between the direction of propagation and the equilibrium magnetic field, that is $\theta $.

For $\theta = 0$, so that wave propagation is parallel to ${\bf B}_{0}$, $m = K$ and

\begin{displaymath}
c_{ph}^{2} = c_{s}^{2}, c_{A}^{2}.
\end{displaymath}

The slow wave has

\begin{displaymath}
c_{ph}^{2} = \hbox{min}(c_{s}^{2}, c_{A}^{2}),
\end{displaymath}

while the fast wave has

\begin{displaymath}
c_{ph}^{2} = \hbox{max}(c_{s}^{2}, c_{A}^{2}).
\end{displaymath}

For $\theta = \pi /4$ the phase speeds become

\begin{displaymath}
c_{ph}^{2} = {c_{s}^{2} + c_{A}^{2}\over 2} - {1\over 2} \left (
c_{s}^{4} + c_{A}^{4}\right )^{1/2},
\end{displaymath}

for the slow wave and

\begin{displaymath}
c_{ph}^{2} = {c_{s}^{2} + c_{A}^{2}\over 2} + {1\over 2} \left (
c_{s}^{4} + c_{A}^{4}\right )^{1/2},
\end{displaymath}

for the fast wave.

Finally, for $\theta = \pi /2$, so that $m = 0$, we have

\begin{displaymath}
c_{ph}^{2} = 0,
\end{displaymath}

for the slow wave and

\begin{displaymath}
c_{ph}^{2} = {c_{s}^{2} + c_{A}^{2}},
\end{displaymath}

for the fast wave. So the slow wave is like the Alfvén wave in that it cannot propagate across the magnetic field.

Example 4.4.3   In the absence of a magnetic field, so that $c_{A} = 0$, the slow waves disappears (since $\omega^{2} = 0$) and the fast wave becomes the sound wave.

For $c_{s} = 0$, so that the gas pressure is negligible, the slow wave again disappears and the fast mode has the same dispersion relation as the Alfvén wave except that it is isotropic.

Example 4.4.4 (4.41)   and (4.44) both come from (4.37). If (4.41) is satisfied, then (4.44) cannot be zero as well. In this case the velocity components must be restricted so that
\begin{displaymath}
{\bf k}\cdot ({\bf v}_{1} \times \hat{\bf z}) = 0.
\end{displaymath} (4.46)

On the other hand, if (4.44) is satisfied, then (4.41) cannot vanish for the same value of $\omega$. Thus, since both (4.38) and (4.40) must be zero, we must have the following restriction on the velocity components, namely
\begin{displaymath}
({\bf k}\cdot {\bf v}_{1}) = 0 \hbox{ and } v_{z} = 0.
\end{displaymath} (4.47)

Thus, the Alfvén wave is incompressible, and so does not have any perturbations in gas pressure and density, and the velocity is perpendicular to the equilibrium magnetic field.


next up previous
Next: About this document ... Up: Magnetohydrodynamic Waves Previous: Alfvén Waves
Prof. Alan Hood
2000-01-11