For low-
plasmas, with
(also referred to as cold
plasmas) the stresses in the plasma are predominantly magnetic.
We seek MHD wave solutions in a cold magnetized plasma.
In treating small-amplitude waves, the MHD equations are linearized,
keeping only terms linear in the amplitude of the wave
(
,
,
and
).
We seek plane wave solutions; i.e., solutions that vary in space and
time as
(assuming that the plane wave propagates
in the ** x**-direction, with
).
Additional assumptions are that the background magnetic field
and plasma density
are uniform, that there are no
background
currents or electric fields, and that there is
no bulk fluid motion.
Our starting equations are:

(3.51) |

After linearizing, and replacing the time and spatial derivatives by and , these equations become

Without loss of generality we assume that lies in the

where the Alfvèn velocity

(3.56) |

A solution for exists only if the determinant of this matrix vanishes. This yields two independent non-trivial solutions for as a function of

The first solution corresponds to

(3.58) |

so that the flow of energy associated with Alfvèn waves is directed along the background magnetic field direction.

The dispersion relation
corresponds to
the *magnetoacoustic mode*.
Substituting the dispersion relation into () yields
the requirement that ** U_{1y}=0** (for
), so that the
fluid motion is in the plane containing
and
.
Because

(3.59) |

so that wave energy may flow at an arbitrary angle to , as opposed to Alfvèn waves (with ).

In a warm plasma,
when
is no longer small relative to unity, the plasma
pressure terms can no longer be ignored.
The pressure gradient term is reinserted in ()
and the adiabatic
equation of state () closes the set of equations.
In this case a linear analysis yields a dispersion relation
with three solutions:

(3.60) |

with the sound speed . These three solutions correspond to the Alfvèn mode, and the fast (

(3.61) |

In the limit of small background magnetic field strengths, fast mode waves become gas sound waves, with the dispersion relation . In the cold plasma limit, fast mode waves become magnetoacoustic waves. In the small-field limit, slow mode waves become magnetoacoustic-like, with the dispersion relation . These only have magnetoacoustic properties for small angles . In the cold plasma limit, slow mode waves (along the field lines) become gas-sound-like, with .