Magnetospheric MHD Resonances and ULF Pulsations - Ultra and Extremely Low Frequency Electromagnetic Fields - page 217

where k n D ! n =V A D ! n C i! 00 =V A is not a function of x and the real and
imaginary parts of ! are given by Eqs. ( 6.39 )-( 6.41 ).
As long as x is fixed, Eqs. ( 6.39 )-( 6.41 ) describe the spectrum of normal Alfvén
oscillations. It should be emphasized that this spectrum depends on x and thus is
continuous. The real part of the frequency ! n .x/ represent the eigenfrequency of
the n-th harmonic. Notice that all the eigenfrequencies are equidistant, whereas the
damping factor ! 00 .x/ is independent of n: From these equations it is clear that
the magnetic shell which has a constant value of x will vibrate as a whole. In this
picture all the segments of the field lines at constant x will covibrate so that it is
sufficient to study the normal oscillations of one of these field lines. The net Alfvén
perturbations can be thus considered as a superposition of the independent normal
oscillations of the field lines/magnetic shells.
It is clear that Eq. ( 6.39 ) describes the frequencies of the half-wave mode in such
a way that an integer number of half-waves lies on the field line while Eq. ( 6.40 )
corresponds to the quarter-wave mode. In this case the field line length equals to
1=4, 3=4, 5=4::: of the wavelength.
This kind of field line oscillations has a close analogy with normal oscillations of
a taut elastic string. The sense of the magnetic force is similar to that of elastic forces
due to tension in a stretched string since the restoring force in Alfvén oscillations
arises due to tension in the magnetic field line. In addition, the shear Alfvén waves
play a role of the transverse elastic waves propagating along the string.
In the framework of the “MHD-box” model, we note that the energy loss is
mostly due to the Joule dissipation at the ends of field lines, that is in the ionosphere.
The interpretation we make is that the Joule dissipation results from the Pedersen
conductivity, which is subject to diurnal variations. It is usually the case that at the
nighttime ionosphere † P is much smaller than † w so that the parameter Ǜ P is small.
On the contrary, at the dayside ionosphere the plasma conductivity is so high that
Ǜ P is greater than unity.
Consider first the extreme case of a small Pedersen conductivity in the conjugate
ionospheres when Ǜ P tends to zero at both ends of the field line, that means that
R C R >0. From here it follows that the set of eigenfrequencies, ! n .x/,is
described by Eq. ( 6.39 ), while the damping factor, ! 00 .x/,inEq.( 6.41 ) tends to
zero. As it is seen from Eq. ( 6.42 ), in this case @ z E x D 0 at the ends of field line.
This means that the transverse electric field E x has a node in the equatorial plane
and it has the antinodes at the ends of the field line. The same is true for the plasma
velocity V y , which is related to E x through Eq. ( 6.30 ). On the contrary, it follows
from Eq. ( 6.42 ) that the transverse magnetic field ıB y has the nodes at the ends
of the field line. To illustrate this, the symmetrical profiles of the first and second
harmonics of ıB y are shown in Fig. 6.2 with solid .n D 1/ and dotted .n D 2/ lines.
Before leaving this case it is useful to return to the analogy between the field
lines and elastic strings. Considering the taut elastic string dead at its two ends and
replacing V A by the velocity of the elastic wave, we note that Eq. ( 6.39 ) can describe
the eigenfrequencies of the elastic string with the length l 1 .