Geoscience Reference
In-Depth Information
5.9 Coupling of Alfven and FMS-Waves
Features
Turn now to the case of k y
= 0, when Alfven and FMS-waves interact. This
results in the appearance of a number of peculiarities in the behavior of the
MHD-waves, the principal of which is the energy transfer from the FMS-
mode to the Alfven mode. When the FMS-oscillation frequency matches the
local FLR-frequency, energy transfer between the modes becomes more ef-
fective. The spectrum of the MHD-oscillation contains, in the general case,
discrete and continuous parts. Energy of normal modes of the discrete part
are conserved because of zero dissipation. If ω A ( x )= πc A ( x ) /l x is everywhere
a non-constant function, the continuous spectrum includes ω, satisfying at
some x to the equation ω 2 = ω A ( x ) . The frequency content of the continuous
spectrum is independent of k y , i.e. continuous spectrums for a finite azimuthal
wavenumber k y
=0and k y = 0 coincide with each other. An arbitrary dis-
turbance of b can be presented in the coupling case as the integral over the
continuous spectrum. An asymptotic time dependence is given by [18]
t 1 exp[
b
±
A ( x ) t ] .
The behavior in the uncoupling case is quite different. A frequency of the
normal mode ω n of the discrete spectrum can coincide with a frequency of the
continuous spectrum. Then, two types of oscillations exist for long times: first,
each magnetic shell oscillates at its own frequency ω A ( x ) and goes down as 1 /t.
And the second, global oscillations of the whole cavity exist at the resonant
frequency ω n , damping only due to dissipation. However, it can be shown that
at k y
= 0 there is no normal mode transferred into the uncoupled cavity mode.
A similar effect was found in electrostatic oscillations in cold inhomogeneous
plasma ([21], [22]). A more complicated MHD-model is considered in ([7], [14],
[26], [27]).
The problem of the time evolution of an initial disturbance is solved with
the Laplace transform over time. In order to perform such calculations, it
requires the analysis of the Green function in a manner similar to one applied
for the uncoupling case in Section 5.8. In the coupling case the Green function
is many-valued and its Riemann surface is many-sheeted. If, on the complex
plane ω , there is a cut along the intervals of the continuous spectrum, a one-
valued branch of the Green function is found at each sheet. A physical sheet of
the Riemann surface is determined by the boundary conditions at this sheet.
The poles of the Green function correspond to the resonance frequencies of the
global modes. At k y
= 0 at the physical sheet, there are no poles of the Green
function transferring into the poles of the Green function of the uncoupling
mode (see Section 5.8).
However, at the non-physical sheet adjacent to the physical sheet, there
is a pole of the Green function near the spectral cut at ω = ω m , and under
k y
0 it approaches the cavity mode eigenfrequency. If this pole is near the
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