Magnetospheric MHD Resonances and ULF Pulsations - Ultra and Extremely Low Frequency Electromagnetic Fields - page 218

1

2

0.5

0

1

−0.5

−1

0.5 1

z/l 1

Fig. 6.2 Profiles of the standing Alfvén waves at the magnetospheric shell in the extreme cases

Ǜ P ! 0. The first and second harmonics of the transverse magnetic field ıB y are shown with

solid .n D 1/ and dotted .n D 2/ lines

In the inverse case of Ǜ P !1 , the set of eigenfrequencies is defined by
Eq. ( 6.39 ) as before. Moreover, the damping factor in Eq. ( 6.41 ) is equal to zero
as well. In this case E x D 0 at the ends of field line. The interpretation we
make is that the Joule dissipation of energy in the ionosphere can be neglected
since the electromagnetic field cannot penetrate into the ionosphere due to its
infinite conductivity. The components E x and V y have an antinode in the equatorial
plane and the nodes at the ends of the field line, whereas ıB y has the antinodes at
the ends of the field line. Notice that the analogous eigenfrequencies have the elastic
string dead in its middle.
If Ǜ P is finite and nonzero, Eqs. ( 6.39 )-( 6.41 ) generally describe the spectrum of
damped Alfvén oscillations, which are similar to oscillations of the stretched elastic
string with energy losses at the claimed end points.
If a homogeneous confined space is studied, it is usually the case that the
spectrum of normal field oscillations is discrete. On the basis of the “MHD-box”
model, we have found, however, that the spectrum of the Alfvén oscillations is
continuous. It is not surprising that there is one-dimensional (1D) inhomogeneity
across straight field lines. In some sense, the actual Earth magnetic field is
inhomogeneous across the magnetic shells. This implies that the spectrum of the
FLR of the Earth magnetic field depends on the magnetic shell, which is a function
of the McIllwain parameter L. Under nominal magnetospheric conditions one may
expect an increase of the oscillation period with L or with radial distance, at
least at auroral latitude. Below we show that this conclusion is consistent with the
observations. It can be shown that in a curvilinear magnetic field the major features
of the FLR are the same except for the effect of polarization splitting of the FLR-
spectrum. This effect is due to the difference of the convergency/divergency rate of
the magnetic field lines within the meridional and equatorial plains. The interested
reader is referred to the text by Leonovich and Mazur ( 1993 ) and Leonovich ( 2000 )
for details about the dependence of the FLR-resonance frequencies on polarization.