   Next: Magnetoacoustic Modes Up: Magnetohydrodynamic Waves Previous: Alternative approach to sound

# 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 , but . 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 -axis ( direction) being preferred in some sense.

Next we assume that there are no pressure or density variations so that (4.6) implies that and the motion is incompressible. Again taking Fourier components, the incompressible assumption reduces to (4.29)

Now the linearised equation of motion, (4.11) reduces to (4.30)

From (4.30) it is clear that (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, since . Next, Finally, since . Hence, (4.30) reduces to (4.32)

We define the Alfvén speed, , as (4.33)

so that (4.32) can be written as (4.34)

(4.34) describes Alfvén waves that are anisotropic (due to the term. Note that
• is perpendicular to both , the equilibrium magnetic field, and , the direction of propagation. Thus, Alfvén waves are transverse waves.

• There are no disturbances in the pressure and density and so that the motion is incompressible.

From Figure 4.1 we have so that the phase speed is (4.35)

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

(4.35) is the equation of a circle with centre at and radius = . Compare this with the sound waves with , which are isotropic and thus independent of .

The group velocity is so that (4.36)

Therefore, the group velocity for the Alfvén wave is always along the equilibrium magnetic field and of magnitude .

Example 4.3.1   Consider a coronal loop with Gauss ( tesla), Mm ( m) and m ( kg m ). For these values the Alfvén speed is approximately m s and the wave number is m . Therefore, the frequency of oscillations in a coronal loop is Hence, the period, , of oscillation is Thus, oscillations in a coronal loop with these properties should have a period of approximately 50 seconds.   Next: Magnetoacoustic Modes Up: Magnetohydrodynamic Waves Previous: Alternative approach to sound
Prof. Alan Hood
2000-01-11