Hydromagnetic Resonators - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 112

ω .Let ω be real. Suppose that the function k A ( x ) is continued on the complex
plane w = x + iu and search for w n = x n + iδ n from (5.31). It will be shown
in the next section that the field distribution in the vicinity of the resonant
point has a form of Lorentz's curve and the parameter δ n is the half-width of
this resonance curve.
At δ n = 0 the singularity of (5.29)-(5.30) shifts from the real axis into

the complex plane and while integrating of (5.29)-(5.30) over the real axis
x the solution remains finite. Estimate δ n at small losses. Use Taylor-series
expansion

d
d x k A ( x n )+

k A ( x n + iδ n )= k A ( x n )+ iδ n ···

= k A ( x n ) 1+ i δ n

,

Λ n +

··· (5.32)

where

Λ n = k A ( x n ) /k A ( x n )= ω An ( x n ) /ω An ( x n ) −

c A ( x n ) /c A ( x n )= 1

2 ρ ( x n ) /ρ ( x n )

−

= (5.33)

and is a scale of the FLR-frequency at point x n . It is taken into account

here that the external magnetic field B 0 is independent of x . Substitution of
the Taylor expansion into (5.31), yields in the first approximation

| Λ n |

nπ
l z

k A ( x n )= ± . (5.34)

The sign here is determined by the sign of frequency ω. In the next approxi-
mation we have an equation for the determination of δ n :

k A ( x ) δ n

Λ n ≈

κ n .

For definiteness, put a plus sign in (5.34). Suppose the function k A ( x ) − nπ/l z

has a simple zero at the point x n , then

Λ n κ n ( x n )

δ n ≈

k A ( x n ) .

(5.35)

By means of (5.15) and (5.17), with ω = ω A ( x n ) we obtain

1
Σ P

1
l z

1
Σ P

Σ P →∞

κ n ≈− + at ,

l z Σ P

+ Σ P at

1

Σ P →

κ n ≈− 0 .