ULF-Sounding of Magnetosphere and Earth - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 323

We reconstruct the plasma density using the data from Table 11.1 with
the following 2 methods of hydromagnetic diagnostics.
A solution of (11.18) determines the dependencies x 1 ( q ) ,x 2 ( q ), a ( q )and s ( q ).
The parameter q in the field-aligned plasma density distribution can be de-
termined at N 3 minimizing the sum of the squares of the deviations of the

calculated and measured FLR-frequencies.

Ω j ( q ) ω (1)

ω (1 j 2

N

Ψ q ( q )= ( q ) − .

j

j =1

However, this method is not effective for finding q because of the weak depen-
dence of Ψ q on q and resulting in severe mistakes in q . Therefore a different
method should be used to find q, e.g., the ratio of two FLR-harmonic frequen-
cies as described above.

Numerical Example

As an example, we estimate the parameters of the magnetospheric plasma
using the data of a meridional chain of 5 stations in England [16]. Four pairs
of adjacent stations allow us to use the gradient method to find the resonant
frequencies of four field-lines crossing the meridian in the centers of arcs be-
tween station pairs. These frequencies are from the spectra of Pc 3,4 pulsations
measured on April 22, 1976. The geomagnetic latitudes of corresponding field
line footpoints are given in Table 11.1 [7].
1 method. Table 11.1 gives f (1)

r =6 . 9mHz, f (2 r =17 . 3mHz, Φ ∗ =
61 . 25 ◦ , w 0 =sin Φ ∗ =0 . 8767 . The numerical solution of the equation
l 2 ( q, w 0 )=17 . 3 / 6 . 9=2 . 51 at w 0 =0 . 8767 gives q

1 . 15. A root can also be
obtained graphically, as shown in Fig. 11.2. At q =1 . 15 and w 0 =0 . 8767 from
(11.11), we have ω 1 =2 . 15 and for Ω = ω (1 r /ω 1 =2 πf (1)

≈

/ω 1 =2 . 02 ×

10 − 2 r

s − 1 . Substituting these results into (11.13), we obtain n 0 ≈ 227 cm − 3 .

2 method. Assume that q =1 . 15 obtained from the first method is
constant within the latitude range 55 ◦ -60 ◦ . We get the normalized FLR-
frequencies ω (1 j ( q =1 . 15) = (2 . 15 , 2 . 18 , 2 . 20 , 2 . 24) s − 1

corresponding to the

geomagnetic latitudes Φ j =61 . 25 ◦ , 59 . 45 ◦ , 58 . 00 ◦ , 55 . 85 ◦ from the bound-

ary problem (11.10). Then we compute Ω j = ω (1 j /ω (1) 10 − 2 , 2 . 85

=(2 . 02 × ×

j

10 − 2 , 3 . 68 10 − 2 , 5 . 29 10 − 2 ) from (11.14) and from (11.15) get the plasma × ×

Table 11.1. FLR-frequencies extracted from the observations at the meridional
chain of 5 stations in England [16]

Φ j 61.25 59.45 58.00 55.85

L j 4.32 3.87 3.56 3.17

f (1) mHz 6.9 9.9 12.9 18.9 r

f (2)
r

mHz 17.3 - - -