ULF-Sounding of Magnetosphere and Earth - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 321

with y 2 | w = ±w 0 =0.Here ω = ω/Ω is a normalized frequency and w 0 =
(1

1 /L ) 1 / 2 . Integrating the set (11.10) with Cauchy's initial data

−

y 1 ( − w 0 , ω )=1 , 2 ( − w 0 , ω )=0 ,

we get the function ∆ ( ω, q, w 0 )= y 2 ( w 0 , ω ). Equating ∆ to zero, we get

∆ ( ω, q, w 0 )=0 . (11.11)

for the FLR-frequencies. The roots of (11.11) determine the dependence of
FLR-frequencies ω ( k r = Ω ( n 0 ,w 0 ) ω k ( q, w 0 )on q , the parameter that controls
the field-aligned plasma density distribution. Note that the ratio of frequencies
of FLR-higher harmonics to the FLR-fundamental frequency

l k ( q, w 0 )= ω ( k )

ω k ( q, w 0 )
ω 1 ( q, w 0 )

r
ω (1)

=

r

depends only on q and w 0 .

Single-Field Line MHD-Diagnostics

Suppose that the FLR-fundamental f (1)

r and second f (2 r harmonic frequencies
are known for a central point Φ ∗ =( Φ 1 + Φ 2 ) / 2 located between two close
points at geomagnetic latitudes Φ 1 >Φ 2 ,( Φ 1 − 2 ◦ ) along a geomagnetic

meridian where pulsations are measured. It is convenient to use as initial
parameters the ratio of the first two harmonics f (2)

Φ 2 ≈

/f (1) and one of the

r r

resonance frequencies, e.g. f (1)

.For q we have the equation

r

f (2)

r
f (1)

W ( q, w 0 )= l 2 ( q, w 0 ) − =0 . (11.12)

r

It can be solved graphically or numerically. The initial approximation q (1)
is then improved with the iterative procedure

W ( q ( n ) ,w 0 ) ∂W ( q ( n ) ,w 0 )
∂q

− 1

q ( n +1) = q ( n )

− → q ( w 0 ) .

n →∞

Hence, we may obtain Ω = ω (1 r /ω 1 as a function of the measured fundamental
frequency and calculated normalized frequency on which Ω depends. From
(6.103) the frequency Ω = 106 /n 1 / 0 L 4 and for the concentration n 0 we have
n 0 = 106
ΩL 4

2

. (11.13)