Geoscience Reference
In-Depth Information
For the power plasma distribution (6.75) within the thin ionosphere ap-
proximation we get
i ν 2 c
2 ω 1
w 2 6 −p =0 ,
ω
ν 2 c y 2
y + i
(6.119)
ν 1+3 w 0 1 / 2 η + .
y
| w = ±w 0 =
(6.120)
w 0
with the initial values determined by (6.118) or (6.120) to w = w 0 . Then we
equate the obtained boundary values of y + = y ( w 0 ) to the boundary value of
the admittance at w = w 0 . In doing so we construct the admittance equation
written for the boundary conditions (6.120) in the form
Take the integral of (6.117) or (6.119) numerically beginning with w =
F ( ω, ν )= y + + ν (1 + 3 w 0 ) 1 / 2 η + .
(6.121)
Zeroes of the function F ( ω, ν ) determine the Alfven resonance frequencies.
Consider now the ionosphere of a finite thickness. The coecients of
(6.117) have different characteristic scales in the ionosphere and magne-
tosphere. Therefore, split the interval of integrating into two subintervals.
The first corresponds to the ionosphere and the second to the magnetosphere.
Introduce new variables for the ionospheric interval:
2 X
LR E u,
τ = ( r
R I )
l P
y =
,
with
X = 4 πΣ P
c
Σ P
,
P =
.
σ P
Here
is the height averaged Pedersen conductivity, l P is the thickness of
the ionospheric conductive layer. Then from (6.117) we have
σ P
u 2
d u
d τ
σ P
1
µτ / 4
1 /L
= ik 0 l P X
µτ ) 1 / 2
1 /L ) 1 / 2 ,
(6.122)
σ P
(1
(1
µτ
where the upper (lower) sign corresponds to the northern (southern)
ionosphere; µ = l P / ( LR I ). The boundary conditions (6.118) are given by
u
| τ =0 =0 .
(6.123)
Integrating (6.122) with the boundary conditions (6.123) for both hemi-
spheres, we find u at τ = 1. Calculating the admittance y and returning to
(6.117) we construct the function F ( ω, L ) and obtain the dispersion equation
(6.121).
Roots of (6.121) can be found using, for instance, Newton's method. Let
ω (0 j ( L ) be an zero approximation of the j -th FLR frequency ω j ( L ). Then the
next approximation is
 
Search WWH ::




Custom Search