FLR in Plasma Configurations - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 146

r=R i

Ω

L 2

L 1

S a

Fig. 6.1. Asketchofthe Ω volume between two magnetic shells

The region Ω between two magnetic shells (see Fig. 6.1), transforms in the
coordinates into the region T (see Fig. 6.2a). As the field-lines are not

orthogonal to the ionospheres, the meridional cross-section T is a curvilinear
trapezium.
In the non-orthogonal coordinate system, it is possible to achieve the co-
incidence of boundary S a with the coordinate surface by replacing variables

{ y i }

y 1 ,y 2 ,y 3 by x 1 ,x 2 ,x 3 as

x 1 = y 1 ,x 2 = y 2 ,x 3 = K 1 ( y 1 ,y 2 )+ K 2 ( y 1 ,y 2 ) y 3 ,

(6.30)

where

Γ N ( y 1 ,y 2 )+ Γ S ( y 1 ,y 2 )
Γ N ( y 1 ,y 2 )

K 1 ( y 1 ,y 2 )= d
2 Γ S ( y 1 ,y 2 ) , −

d

K 2 ( y 1 ,y 2 )=

Γ S ( y 1 ,y 2 ) ,

Γ N ( y 1 ,y 2 ) −

and Γ N,S is defined in (6.28). By substituting y k ( x, y, z ) from (6.27) into
(6.30), we obtain the transformation equations from the Cartesian coordinate

x 3

y 3

d/2

(a) (b)

T

Π

y 1

L 1 −1 L 2 −1

x 1

-d/2

Fig. 6.2. The Ω volume of Fig. 6.1 is transformed into (a) the curvilinear trapezium

T in the orthogonal system y i and (b) the rectangle Π in the curvilinear system

x i