Hydromagnetic Resonators - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 113

Substituting these expressions into (5.35) yields

1
Σ P

Λ n
nπ

1
Σ P

Σ P →∞

δ n ≈− + at (5.36)

and

nπ Σ P +

Σ P at

Λ n

Σ P →

δ n ≈− 0 . (5.37)

Comparing (5.36) and (5.37) with (5.24) and (5.25) we get a relationship of
the half-width δ n with decrement γ n :

d ω An
d x

− 1

x = x n

γ n
ω An Λ n = γ n

δ n ≈− .

(5.38)

We obtained (5.38) using the explicit expressions for γ n and δ n .Moreover,
these formulae can be applied in a rather wider field than could been assumed
when deriving them. Let us present other approach independent on the form
of the explicit solution.

Curved Field-Line

Let us consider field-lines that are not obligatory straight. Let their position
be determined by one coordinate of a curvilinear coordinate system. Gener-
ally, a complex resonant frequency ω n ( x ) corresponds to each field-line. These
frequencies are found from the dispersion equation ∆ ( ω, x )=0(seeChap-
ter 6). The latter, as in the case considered above, can be continued to the
complex plane w = x + iu . Then, assuming low dissipation, we first find the
resonance point x n ( ω ) at a given frequency ω , and then determine the real
function ω A ( x ). For a weak dissipation, we have

w n ( ω )= x n ( ω )+ iδ n ( ω ) ,

where ω is real and

ω n ( x )= ω An ( x ) − iγ n ( x ) .

Functions δ n ( ω )and γ n ( x ) are small. Expanding ∆ by δ n at a fixed x, we
have

∆ ( ω A ,x n )
∂∆ ( ω An ,x n ) /∂x ,

δ n = i

and expanding by γ n

∆ ( ω An ,x n )
∂∆ ( ω An ,x n ) /∂ω .

γ n = − i