Ionospheric Alfvén Resonator (IAR) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 158

The latter condition is usually valid at the daytime. The solution of Eq. ( 5.39 ) can

be written in the form n D n if 0 Ǜ P <1and n D n

2 ,ifǛ P >1,

1

while the wave damping factor is

2 ln LJ LJ LJ LJ

LJ LJ LJ LJ

1 1 C Ǜ P

1 Ǜ P

D : (5.41)

The set of values n define dimensionless eigenfrequencies of the normal modes
of the IAR when the Hall conductivities is neglected. As we shall see, the IAR
excitation seems to be the most effective at the nighttime, when the height-integrated
Pedersen conductivity is small compared to wave conductivity † w , so that Ǜ P <1.
On account of Eq. ( 5.17 )forx 0 the dimensional values of the nighttime IAR
eigenfrequencies are

! n
2 D

V AI n
2L ; n D 1;2;:::;

f n D (5.42)

which coincide with the rough estimate made in the beginning of this section. A
simple interpretation of Eq. ( 5.42 ) is that the integer number of half-wavelength
keeps within the IAR length; that is L D n n=2 where the wavelength n D
V AI =f n .
At the daytime condition when the Pedersen conductivity is large enough to
satisfy the inequality Ǜ P >1, the eigenfrequencies can be rewritten as

n ; n D 1;2;:::

V AI
2L

1
2

f n D (5.43)

It should be noted that the former relation can be considered in terms of the fact that
the integer number of half-wavelength and one fourth of the wavelength keep within
the IAR length, that is L D . n =2/.n C 1=2/.
In the remainder of this subsection we focus attention on the attenuation of the
normal modes. The first term on the right-hand side of Eq. ( 5.41 ) corresponds to the
IAR damping factor due to the wave energy leakage through the resonator upper
wall, whereas the second term describes the losses due to the ionosphere Joule
dissipation. It should be noted that in the case Ǜ P ! 1 the damping factor in
Eq. ( 5.41 ) tends to infinity. This only means that here the dispersion due to the Hall
conductivity that has been neglected in Eq. ( 5.37 ) must play a major role.
Another family of roots in Eq. ( 5.36 ) corresponds to the FMS/compressional
mode which is described by

is C x 0 Ǜ P D 0; (5.44)

where s is given by Eq. ( 5.35 ). By analogy with Eq. ( 5.38 ) we can rewrite
Eq. ( 5.44 )as