The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 38

Magnetic field lines disturbed by the linearly polarized Alfvén wave are shown
in Fig. 1.15 with wavy lines. It follows from Eq. ( 1.57 ) that for waves propagating
along B 0 . k B 0 >0/ the vectors V ? and ı B ? are antiparallel, while for waves
propagating antiparallel to B 0 these vectors are parallel. This relation makes it
possible to detect a direction of the Alfvén wave energy propagation, using only
a single-point measurement (Glassmeier 1995 ).
It should be emphasized that the transverse displacements in the plasma can
propagate along the magnetic field lines because the conducting plasma is frozen
into the magnetic field. The transverse displacements in plasma result in curvature
and stretching of the field lines shown in Fig. 1.15 . Magnetic forces in a conducting
medium act on the field lines in analogy to the quasielastic forces; that is, the
magnetic forces act in such a way to tighten the field lines. In some sense, the Alfvén
oscillations of the field lines are similar to oscillations of stretched strings.
To summarize, we note that the shear Alfvén wave has a field-aligned electric
current parallel to the undisturbed field B 0 , an electric field E parallel to the
perpendicular wave vector k ? , and a magnetic component ı B ? perpendicular to
B 0 and the vector k . This shows that the Alfvén wave is a purely transverse wave
with respect to the magnetic field B 0 . These properties of the Alfvén wave have
an important role in magnetospheric plasma dynamics. The arbitrary perturbations
of the plasma parameters can thus propagate along the magnetic field lines at the
Alfvén velocity [Eq. ( 1.60 )], which in turn depends on the magnetic field induction
and the plasma density. The guiding by field lines, which is probably the major
property of the Alfvén waves, holds in inhomogeneous plasmas and even under the
finite curvature of the field lines.

1.4.3 Fast and Slow Magnetosonic Waves

Contrary to the Alfvén wave, the next wave mode can be excited in the plasma if
both the magnetic perturbation ı B and the mass velocity V are in that plane which
contains the undisturbed field B 0 and wave vector k . A schematic representation
of the field components is shown in Fig. 1.16 . As is seen from this figure, all the
vectors, i.e., ı B , V , B 0 , and k are situated in the x;y plane whereas the electric field
is normal to that plane. As before ı B is perpendicular to k . Taking into account the
field polarization, Eq. ( 1.54 ) is reduced to

!ı B D . V B 0 / z . k O z /: (1.62)

where the subscript z denotes the vector projection on z axis. Notice that the vector
k O z is anti-parallel to y axis as shown in Fig. 1.16 . We may also eliminate ı from
Eqs. ( 1.52 ) and ( 1.53 ) to yield

0 0 ! 2 V D 0 0 c s . k V / k C ! B 0 . k ı B /:

(1.63)