Geoscience Reference
In-Depth Information
inclination of the geomagnetic field, l P is the length of the field line segment
within the Pedersen layer. Certainly, the boundaries of the layer cannot be
determined exactly, but the decrease of conductivity is very steep and this
inaccuracy is of no practical importance.
By going to the thin ionosphere approximation, integration of (6.79) within
each ionosphere with the conditions (6.80) gives
w
(1 + 3 u 2 ) ε ( u ) e 2 ( u )d u,
e 1 ( w )= ik 0
(6.85)
±w 0
w
e 2 ( w )= e 2 ( w 0 )+ ik 0
ν 2
e 1 ( u )d u.
(6.86)
±w 0
Hereafter, the upper (lower) sign corresponds to the northern (southern)
ionosphere. Solve (6.85)-(6.86) with iterations and put the values of e 1 and
e 2 for ω = 0 as a zero approximation. That is,
e (0 1 ( w )=0 e (0 2 ( w )= e 2 (
±
w 0 ) .
Substitution of the zero approximation into the right-hand side of (6.85)-
(6.86) and ε = i 4 πσ P yields the first approximation
w
e 1 ( w )= 4 π
c
σ P ( u )(1 + 3 u 2 )d u,
e 2 (
±
w 0 )
±
w 0
e 2 ( w )= e 2 (
±
w 0 ) .
In the same manner we obtain the second, third, etc., approximations. With
the condition (6.84) holding, the second-order corrections are small in com-
parison with the first-order ones over the whole ionosphere. Omitting these
terms, we obtain from the second equation the boundary condition at the
upper boundary of the Pedersen layer:
X ±
sin I 0 e 2 ( w ) t w =
ν (1 + 3 w 2 )
e 1 ( w )=
±
w 0 ,
(6.87)
where X ± =4 πΣ P /c , tan I 0 = 2 cot θ 0 , θ 0 is co-latitude of the field line pierce
point on the ionosphere.
Resonance Fields
The resonance fields are written in (6.59) for the covariant components.
Rewrite these equations for the physical components which are found from
the covariant components with the aid of Lame coecients. For the dipole
 
Search WWH ::




Custom Search