Hydromagnetic Resonators - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 117

• The amplitude of the longitudinal magnetic component b is approximately

k y a
k A x 1 |

δ 1 |

times smaller than the b y amplitude.

Energy Dissipation

Now we consider energy losses in the FLR-region. The time average energy
flux carried by the wave along axis x is given by the projection of the real
part of the complex Poynting vector

c
8 π [ E

c
8 π E y b ∗

b ∗ ] x =

S x = × .

(5.41)

When the wave propagates through the resonance region, energy losses
can be determined as a difference between the Poynting vectors left and right
of the resonance region

∆S x =Re( S x ( x n − 0) − S x ( x n + 0)) ,

where x n is the coordinate of a resonance point. Substituting E y and b from
(5.39) into (5.41), we find

k y Λ n b 0
k A ( x n )

ω
8 π

∆S x ≈

2 , (5.42)

× Im { ln [ k y ( x − − x n − iδ n )] − ln [ k y ( x + − x n − iδ n )] }| Q n |

where b 0 = b ( x = x n ) .
Equation (5.42) proves to be essentially more accurate than can be ex-
pected from the comparison of the numerical simulation and singular part
of the solution (see (5.40)) shown in Figure 5.4. The regular component of
the solution is responsible for the large error in estimates of fields based just
on the terms with logarithmic singularity. This component is continuous in
the resonance range and its contribution to the discontinuity of the Poynting
vector vanishes with the resonance half-width decreasing.
Analytical continuation of the logarithm from the right-hand semi-axis

x x n < 0 must be carried out in a

complex plane w = x + iu with the singular point bypassed from below at
δ n > 0 and from above at δ n < 0. Therefore at

− x n > 0 onto the left-hand one x −

| δ n || x − x n | , the logarithms

in (5.42) are

ln | k y ( x + − x n ) | , + >x n ,

ln | k y ( x − − x n ) | + iπ · sign( δ n ) , − <x n .