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Alfvén Waves

Now consider what happens when there is a magnetic field present. To keep things simple we assume that there is a uniform equilibrium magnetic field ${\bf B}_{0} = B_{0} \hat{\bf z}$, $p_{0} = 0,
{\bf g} = 0$ but $\rho_{0} \neq 0$. This allows us to see the effect of the magnetic field without having to worry about the sound waves. It seems plausible that since the magnetic field has a direction this will be reflected in the wave motions that ensue. They will be anisotropic with the $z$-axis (${\bf B}_{0}$ direction) being preferred in some sense.

Next we assume that there are no pressure or density variations so that (4.6) implies that $\nabla \cdot {\bf v}_{1} = 0$ and the motion is incompressible. Again taking Fourier components, the incompressible assumption reduces to

{\bf k}\cdot {\bf v}_{1} = 0.
\end{displaymath} (4.29)

Now the linearised equation of motion, (4.11) reduces to
-\rho_{0} \omega^{2} {\bf v}_{1} = - \left [ {\bf k}\times ...
...}\times {\bf B}_{0}\right )\right ] \times
{\bf B}_{0}/\mu .
\end{displaymath} (4.30)

From (4.30) it is clear that
{\bf v}_{1}\cdot {\bf B}_{0} = {\bf v}_{1} \cdot \hat{\bf z} = 0.
\end{displaymath} (4.31)

so that the motion is transverse to the direction of the equilibrium magnetic field. Using vector identities the right hand side of (4.30) can be built up. Thus,

{\bf k}\times ({\bf v}_{1}\times {\bf B}_{0}) = ({\bf k}\cd...
... v}_{1}) {\bf B}_{0} = ({\bf k}\cdot {\bf B}_{0}) {\bf v}_{1},

since $({\bf k}\cdot {\bf v}_{1}) = 0$. Next,

{\bf k}\times {\bf k} \times \left ({\bf v}_{1}\times {\bf ...
...ight ) = ({\bf k}\cdot {\bf B}_{0}) {\bf k}\times {\bf v}_{1}.


\left [ {\bf k}\times {\bf k}
\times \left ({\bf v}_{1}\tim... B}_{0}, \\
& = & ({\bf k}\cdot {\bf B}_{0})^{2}{\bf v}_{1},

since ${\bf v}_{1}\cdot {\bf B}_{0} = 0$. Hence, (4.30) reduces to
\rho_{0}\omega^{2}{\bf v}_{1} = ({\bf k}\cdot {\bf B}_{0})^...
..._{0} \omega^{2} = {({\bf k}\cdot
{\bf B}_{0})^{2} \over \mu}
\end{displaymath} (4.32)

We define the Alfvén speed, $c_{A}$, as
c_{A}^{2} = {B_{0}^{2}\over \mu \rho_{0}},
\end{displaymath} (4.33)

so that (4.32) can be written as
\omega = \pm c_{A}({\bf k}\cdot \hat{\bf k}) = \pm m c_{A}.
\end{displaymath} (4.34)

(4.34) describes Alfvén waves that are anisotropic (due to the ${\bf k}\cdot {\bf B}_{0}$ term. Note that

Figure 4.1: The sketch shows the angle, $\theta $, between the direction of propagation and the direction of the equilibrium magnetic field.

From Figure 4.1 we have ${\bf k}\cdot \hat{\bf z} = K \cos
\theta $ so that the phase speed is

c_{ph} = {\omega \over K} = c_{A}\cos \theta .
\end{displaymath} (4.35)

This may be represented by a polar diagram as shown in Figure 4.2.

Figure 4.2: The radius for the angle $\theta $ gives the magnitude of the phase speed.

(4.35) is the equation of a circle with centre at $\theta = 0$ and radius = ${1\over 2}c_{A}$. Compare this with the sound waves with $c_{ph} = c_{s}$, which are isotropic and thus independent of $\theta $.

The group velocity is $\partial \omega /\partial {\bf k}$ so that

{\bf c}_{g} = \left ({\partial \omega \over \partial k}, {\...
...artial \omega \over \partial m}\right ) =
c_{A}\hat{\bf z}.
\end{displaymath} (4.36)

Therefore, the group velocity for the Alfvén wave is always along the equilibrium magnetic field and of magnitude $c_{A}$.

Example 4.3.1   Consider a coronal loop with ${\bf B}_{0} = 10$Gauss ($10^{-3}$tesla), $L = 50$Mm ( $5 \times 10^{7}$m) and $n = 5 \times
10^{14}$m$^{-3}$ ( $\rho_{0} = 8 \times 10^{-13}$kg m$^{-3}$). For these values the Alfvén speed is approximately $c_{A} = 10^{6}$m s$^{-1}$ and the wave number is $k = 2 \pi /L = 1.3 \times
10^{-7}$m$^{-1}$. Therefore, the frequency of oscillations in a coronal loop is

\omega = k c_{A} = 1.3 \times 10^{-7} \times 10^{6} = 0.13

Hence, the period, $\tau$, of oscillation is

\tau = {2 \pi \over \omega} = 50\hbox{seconds}.

Thus, oscillations in a coronal loop with these properties should have a period of approximately 50 seconds.

next up previous
Next: Magnetoacoustic Modes Up: Magnetohydrodynamic Waves Previous: Alternative approach to sound
Prof. Alan Hood