Hydromagnetic Resonators - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 115

Consider a magnetic shell crossing the magnetic equator at a geocentric
distance LR E ( R E is the Earth's radius). The origin of the rectangular coor-
dinate system is taken at the top of a field-line lying on the selected L -shell.
Let x -axis be directed along the equatorial radius, y is azimuthal, z is di-
rected along the field-line. Consider the wave disturbances with an azimuthal
wavenumber m . Then near the equatorial plane, the wavenumber is

k y ≈ m/LR E .

L − 3 , and let the plasma density ρ 0 ∝ The main geomagnetic field B 0 ∝

L −ν , then the Alfven velocity c A ∝ L − 3+ ν/ 2 . The field-line length is l z ≈

πLR E in a quasi-dipolar field and the wavenumber is k A ∼

π/l z ≈ 1 /LR E .

L − 4+ ν/ 2

Then the frequency of the fundamental harmonic ω A 1 ∝ and its

radial derivative ω A 1 ∝− ν/ 2) L − 5+ ν/ 2 . We denote their ratio by (4 −

ω A 1
ω A 1 ≈

LR E

Λ 1 = −

ν/ 2 . 4 −

Substituting Λ 1 and k y into p (5.39), we have

m 2

p = m 2 Λ 1 = m 2 LR E
4

ν/ 2 ,

A =

ν/ 2 .

− −

4

The half-width δ 1 is

γ 1
ω A 1 Λ 1 ,

δ 1 = −

where γ 1 /ω A 1 is determined in (5.23).
Near the ionosphere (let it be the northern ionosphere for definiteness)
relations (5.39) with the boundary condition (5.10) are given by

b 0 1+ x 2
2

ln x ,

b ≈

b 0 pk A
x

ib 0 pk A Σ P ln x,

Σ P ,

b x ≈ b y ≈−

c
4 π

b y
Σ P ,

c
4 π

b x
Σ P ,

E x ≈ E y ≈− (5.40)

with Σ P = Σ P /Σ A .
The two upper frames of Fig. 5.4 show the distributions of amplitudes
(left) and phases (right) of b x and b y components near the resonance shell cal-
culated according to the approximate relations (5.40). b y is normalized by the
condition b y ( x 1 ) = 1 at the resonance point x = x 1 . The chosen parameters
roughly correspond to the magnetospheric conditions for a middle-latitude
magnetic shell. Two bottom frames of Fig. 5.4 are results of numerical inte-
gration of (5.29)-(5.30) showing the field distribution produced by a mono-
chromatic source located at the magnetospheric boundary. At x =0,the
electric field E y ( x = 0) = 0. We take E y at x = l x such as to satisfy the
condition b y ( x = x 1 ) = 1 at the resonance point.