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MHD Waves
Let us investigate the small amplitude waves which propagate through
a spatially uniform MHD plasma. We start by combining Eqs. (681)(684) and (687)(688) to form a
closed set of equations:
Next, we linearize these equations (assuming, for the sake of
simplicity, that the equilibrium
flow velocity and equilibrium plasma current are both zero) to give
Here, the subscript 0 denotes an equilibrium quantity. Perturbed quantities
are written without subscripts. Of course, , , and are
constants in a spatially uniform plasma.
Let us search for wavelike solutions of Eqs. (706)(709) in which perturbed
quantities vary like
.
It follows that
Assuming that , the above equations yield
Substitution of these expressions into the linearized equation of
motion, Eq. (711), gives
We can assume, without loss of generality, that the equilibrium magnetic
field
is directed along the axis, and that the wavevector lies
in the  plane. Let be the angle subtended between and
. Equation (717) reduces to the eigenvalue
equation

(718) 
Here,

(719) 
is the Alfvén speed, and

(720) 
is the sound speed. The solubility condition for Eq. (718) is that
the determinant of the square matrix is zero. This yields the dispersion
relation

(721) 
There are three independent roots of the above dispersion relation,
corresponding to the three different types of wave that can propagate through an
MHD plasma. The first, and most obvious, root is

(722) 
which has the associated eigenvector . This
root is characterized by both
and
. It immediately follows from Eqs. (714) and (715) that there is
zero perturbation of the plasma density or pressure
associated with this root. In fact, this root can easily be
identified as the shearAlfvén wave, which was
introduced in Sect. 4.8. Note
that the properties of the shearAlfvén wave in a warm (i.e., nonzero
pressure) plasma are unchanged from those we found earlier in a cold plasma.
Note, finally, that since the shearAlfvén wave only involves plasma
motion perpendicular to the magnetic field, we can expect the
dispersion relation (722) to hold good in a collisionless, as well as a
collisional, plasma.
The remaining two roots of the dispersion relation (721) are written

(723) 
and

(724) 
respectively.
Here,

(725) 
Note that . The first root is generally termed the
fast magnetosonic wave, or fast wave, for short, whereas
the second root is usually called the slow magnetosonic
wave, or slow wave. The eigenvectors for these waves are
. It follows that
and
. Hence, these waves are associated with nonzero perturbations
in the plasma density and pressure, and also involve plasma motion parallel, as
well as perpendicular, to the magnetic field. The latter observation suggests
that the dispersion relations
(723) and (724) are likely to undergo significant modification in
collisionless plasmas.
In order to better understand the nature of the fast and slow waves, let us
consider the coldplasma limit, which is obtained by letting the sound
speed tend to zero. In this limit, the slow wave ceases to exist (in fact,
its phase velocity tends to zero) whereas the dispersion relation for
the fast wave reduces to

(726) 
This can be identified as the dispersion relation for the
compressionalAlfvén wave, which was introduced in Sect. 4.8.
Thus, we can identify the fast wave as the compressionalAlfvén wave
modified by a nonzero plasma pressure.
In the limit , which is appropriate to low plasmas (see
Sect. 3.13), the dispersion relation for the slow wave reduces to

(727) 
This is actually the dispersion relation of a sound wave propagating
along magnetic fieldlines. Thus, in low plasmas the slow
wave is a sound wave modified by the presence of the magnetic field.
The distinction between the fast and slow waves can be further understood
by comparing the signs of the wave induced fluctuations in the plasma and magnetic
pressures: and
, respectively.
It follows from Eq. (716) that

(728) 
Now, the  component of Eq. (711) yields

(729) 
Combining Eqs. (715), (719), (720), (728), and (729), we obtain

(730) 
Hence, and
have the same sign
if
, and the opposite sign if
. Here, is the phase velocity. It is
straightforward to show that
, and
.
Thus, we conclude that in the fast magnetosonic wave the pressure and
magnetic energy fluctuations reinforce one another, whereas the
fluctuations oppose one another in the slow magnetosonic wave.
Figure 16:
Phase velocities of the three MHD waves in the  plane.

Figure 16 shows the phase velocities of the three MHD waves plotted in the
 plane for a low plasma in which . It can be
seen that the slow wave always has a smaller phase velocity than the
shearAlfvén wave, which, in turn, always has a smaller phase
velocity than the fast wave.
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Up: Magnetohydrodynamic Fluids
Previous: Flux Freezing
Richard Fitzpatrick
20110331