FLR in Plasma Configurations - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 143

Here Q s ≡ Q s ( x 0 ,z ), and

R ( z )= k y
2

Q s Q s
( ω s )

Q s + k y

Q s Q s
( ω s )

Q s | x = x 0

x = x 0

k y
n =3

Q n Q s | x = x 0
ω 2

−

ω n ( x 0 ) Q n ( x 0 ,z ) . −

The quantity A is an arbitrary constant which determines the amplitude of
resonance Alfven oscillations.
Singular terms in expressions (6.24) and (6.25) for displacements ξ x and ξ y
are determined (with an accuracy of up to a normalization factor). However,
any regular solution can be added to the regular terms in (6.23) and (6.25)
for b s and ξ y .

Dissipation

Singular solution (6.20) includes terms with ln[ k y ( x x 0 )], which has a branch

point at x = x 0 . When the integration path goes below (above) x 0 , then
iπ (

−

iπ ) is added to the logarithm. Proper rules for integration over the sin-
gular point necessary for making the solution single-valued, can be derived,
for example, by taking account of the Joule dissipation.
The obtained results can be generalized to the case of finite ionospheric
Pedersen conductivity. For this purpose, the method used should be modified.
The necessity for the modification is caused by the fact that when we use
(6.1)-(6.3), we come to the generalized nonselfconjugated eigenvalue prob-
lem. In this case, the eigenfrequency ω occurs to second power in Dungey's
problem. On the other hand, in the case of an infinitely thin ionosphere, the
frequency ω occurs in boundary conditions. Eigenfrequencies become complex
and corresponding eigenfunctions do not satisfy the orthogonality condition
(6.6).
However, an examination of the singularity at a resonance point by the
Frobenius method for partial differential equations still remains valid inclusive
of the Joule dissipation as well. The main result of such investigation is the
following. Taking into account the Joule losses, logarithmic and power singu-
larities are conserved, but real x 0 should be replaced by some complex value
w 0 . When losses are weak, w 0 can be found from the approximate relationship
(5.38):

−

γ s ( x 0 )
ω s ( x 0 ) .

w 0 = x 0 + iδ s , δ s ≈ (6.26)

Here δ s is the half-width of a resonance region; ω s ( x ) is the frequency of the
s -th FLR-harmonic, determined from Dungey's problem without losses; γ s is
the damping rate of the s -th harmonic.
A dissipation results in the damping of oscillations, so
γ s > 0 .