Propagation of MHD-Beams - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 267

π

4 + nπ < Im ¯ k< 3 π

− + nπ

4

=0 . When u =0 , function w k has a root k = 0 which is not the
root of (8.38). Equation (8.A.3) results from (8.A.2) when changing k for

for n

k

−

Thus, roots of (8.36) do not exist.

Appendix M agnetic mode. F inite σ g

Equation (8.36) at σ g

∞

= has the form

k exp 2 k + iτ K

2

iτ K k d g
k d g −

iτ K
2

− =

κ g .

(8.B.1)

Equation (8.B.1) is obtained at Re k A −

k 2 > 0. We investigate (8.B.1) at
Re κ g > 0, Im κ g > 0 . Assume that roots satisfy the condition

k d g |

| 1 , (8.B.2)

then

1

k d g 1

.

2 i
( k d g ) 2 ≈

i
( k d g ) 2

κ g = k d g

− −

Substituting this expression for κ g into (8.B.1), we get

exp ξ

−

1

exp ξ − Aξ = iτ K , (8.B.3)

ξ

where A = τ K d g / 2 ,ξ =2 k . Equation (8.B.3) is solved by successive ap-
proximations with respect to parameter τ K with fixed A . Then, in the first
approximation,

exp ξ 0 = Aξ 0 . (8.B.4)
We restrict our consideration to the band 0 < Im ¯ k< 4 π .At A>e =2 . 71 this
equation has two real roots, the bigger one can be estimated by the formula

ξ 0 ≈ ln ( A (ln A (ln A... ))) . (8.B.5)

Complex roots appear at A<e . We present (8.B.4) as two real equations for
Re ξ 0 = x 0 and Im ξ 0 = y 0 :

exp x 0 cos y 0 − Ax 0 =0 ,

exp x 0 sin y 0 −

Ax 0 =0 .

(8.B.6)

The values of y 0 and x 0 can be found from
exp ( y 0 cot y 0 ) sin y 0
y 0

= A, x 0 = y 0 cot y 0 .