Low-Frequency Wave Propagation

(522) | |||

(523) | |||

(524) |

Here, use has been made of . Thus, the eigenmode equation (490) reduces to

The solubility condition for Eq. (525) yields the dispersion relation

(526) |

(527) |

(528) |

It is fairly easy to show, from the definitions of the plasma and cyclotron
frequencies [see Eqs. (465)-(468], that

(529) |

(530) |

and

Here, we have made use of the fact that in conventional plasmas.

The dispersion relation (531) corresponds to the *slow* or
*shear* Alfvén wave, whereas the dispersion relation (532)
corresponds to the *fast* or *compressional* Alfvén wave.
The fast/slow terminology simply refers to the ordering of the
phase velocities of the two waves. The shear/compressional
terminology refers to the velocity fields associated with the waves. In
fact, it is clear from Eq. (525) that for both waves, whereas
for the shear wave, and for the compressional wave.
Both waves are, in fact, MHD modes which satisfy the linearized MHD Ohm's law
[see Eq. (387)]

(533) |

(534) |

(535) |

Figure 8 shows the characteristic distortion of the magnetic field associated with a shear-Alfvén wave propagating parallel to the equilibrium field. Clearly, this wave bends magnetic field-lines without compressing them. Figure 9 shows the characteristic distortion of the magnetic field associated with a compressional-Alfvén wave propagating perpendicular to the equilibrium field. Clearly, this wave compresses magnetic field-lines without bending them.

It should be noted that the thermal velocity is not necessarily negligible compared to the Alfvén velocity in conventional plasmas. Thus, we can expect the dispersion relations (531) and (532) to undergo considerable modification in a ``warm'' plasma (see Sect. 5.4).