Propagation of MHD-Beams - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 251

Γ 2 m Γ 2g

Im k Im k

k S

Γ g Im = 0

Im κ = 0

g

Im k g = 0

Γ m

Im k S = 0 Im k S = 0

k 0 e

1/2

k 0 e 1/2

k g = 0
Re

Re

g

k S = 0

Γ 1

Γ 1

k 0 e m 1/2

k 0 e

1/2

m

− k 0 e m 1/2

Γ 1 Re k

Re k

k 0 e m 1/2 −

Re k S = 0

1/2

− k 0 e

g

k 0 e

1/2

Im k S = 0 − Im k S = 0

g

(a)

(b)

k g =

Im 0

Fig. 8.7. The integration path Γ 1 of the integral (8.1) and its deformation to two
branch cuts. Deformation of the primary contour Γ 1 to contours Γ 2 m and Γ 2 g

Equations (8.6), combined with (8.40), gives

1
2 π

1
2 π

G ( r ) = R ( k ) e ikx dk, G ( g ) = T ( k ) e ikx dk. (8.42)

Γ 1 Γ 1

The dependence of b ( g ) ( x ) for arbitrary distribution b ( i ) ( x ) in the incident
beam can be found in the form of Green integrals.
It follows from the consideration of a sequence of extending contours pass-
ing between the poles T ( k )( R ( k )) that at x> 0 the integration path Γ 1 can
be transformed to the contour Γ 2 = Γ 2 m + Γ 2 g (see Fig. 8.7). This procedure
is formally non-applicable to the integral of R ( k ), but it is valid for
R ( k )= R ( k )

R ( − ∞ ) .

R (

∞

) is determined in (7.142) and (8.B.7), while the integral of R (

∞

)
evidently yields either the reflected Alfven wave (7.142) or FMS-wave (8.B.7)
caused by an incident Alfven wave in the form of delta function δ ( x ).
A magnetic field on the ground surface is determined by a matrix G ( g ) ( x )
which may be written in the form
G ( g ) = G m + G g +
n

G n e ik ( n ) x +
n

G nS e ik ( n ) S x , (8.43)

where

G n = i Res ( T ( k )) G nS = i Res ( T ( k ))

| k = k ( n ) , | k = k ( n ) S ,

(8.44)

k 0 √ ε m,g + i∞

1
2 π

∆ T m,g exp ( ikx ) dk. G m,g = (8.45)

k 0 √ ε m,g