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ω =0 . 1s 1 ,k y =0 . 5and k =0 . 4 . Chosen parameters roughly correspond
to magnetospheric conditions for the McIllwain parameter L
2
5.
reaches the turning (reflection) point at x = x t
determined from condition U ( x t ) = 0 and is reflected in it. In magnetospheric
problems x t
The wave coming from
−∞
x t , which can be found from the approximate equation
k A (
x t )= k y + k 2
.
(4.55)
In the example in Fig. 4.3 x t =10 . 76 and
x t =10 . 53. Below x t and
x t will
not be distinguished.
The wave to the left of the reflection point is a superposition of the propa-
gating incident and reflected waves. In the region to the right of the reflection
point there is the exponential wave damping to the resonance point x = x r
determined by the equation
k A ( x r )= k 2
.
(4.56)
||
At the field line x = x r =14 . 41 the condition of resonance coupling
between FMS and Alfven waves is fulfilled (FLR). Resonance field-lines form
the resonance magnetic surface being a plane x = x r for 1 D -geometry with
homogeneous magnetic field B 0 . The local Alfven velocity c A ( x r ) coincides
with phase velocity projection on the magnetic field
ω
k
V ph =
,
A ( x r )= V ph .
When this point is approached from the left (from the region of small c A val-
ues), magnetic field b x and b y increases and under zero dissipation approaches
infinity at point x r . Actually, the magnetic wave component remains finite
due to dissipation and dispersion.
For the purpose of clarity, we begin with the qualitative exposition of the
results. At oblique wave incidence ( k y
= 0) on the layer in the vicinity of
the reflection point x t the field behaves in the same way as with k y =0
(see Fig. 4.3). There is a standing wave to the left of the reflection point,
while to the right the wave dampens exponentially. Then the amplitude is
increasing (Fig. 4.3) with the wave propagating into the layer and approaching
the resonance point.
The growth is limited by the dissipation or dispersion. However, the longi-
tudinal magnetic field b proves to be limited in the case of straight field lines,
even under ideal conditions. On the other hand, we show in Chapter 6 that
taking into account the curvature of magnetic field lines may result in un-
limited growth of the b component. The magnetic field distributions in Fig.
4.3 are calculated with the account of dissipation caused by the transversal
conductivity. The Alfven wavenumber is, in this case, complex, ω/c A + .
The singularity disappears at k y = 0. But under small dissipation the
resonant field increases with k y becoming progressively less expressed. This
is because of the distance between the reflection point x t and the resonance
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