A wave coupling formalism for magnetohydrodynamic (MHD) waves in a non-uniform
background flow is used to study parametric instabilities of the large
amplitude,
circularly polarized, simple plane Alfvén wave in one Cartesian
space dimension.
The large amplitude Alfvén wave (the pump wave) is regarded
as the background flow,
and the seven small amplitude MHD waves (the backward and forward fast
and slow
magnetoacoustic waves; the backward and forward Alfvén waves
and the entropy wave)
interact with the pump wave via wave coupling coefficients which depend
on the gradients
and time dependence of the background flow. The dispersion equation
for the waves
D(k,\omega)=0 obtained from the wave coupling equations reduces to
that obtained
by previous authors. The general solution of the initial value problem
for the waves is
obtained by Fourier and Laplace transforms. The dispersion function
D(k,\omega) is a fifth order
polynomial in both the wave number k and the frequency \omega. The
regions of
instability and the neutral stability curves are determined.
The instabilities depend parametrically on the pump wave amplitude
and on the plasma beta.
The wave interaction equations are also studied from the perspective
of a single master
wave equation, with multiple wave modes, and with a source term due
to the entropy wave.
The wave hierarchies for short and long wavelength waves of the master
wave equation
are used to discuss wave stability. By expanding the dispersion equation
for the different
long wavelength eigenmodes about k=0, yields either the linearized
Korteweg deVries
equation or the Schröidinger equation as the generic wave equation
at long wavelengths.
The equations exhibit an absolute instability for hump or pulse-like
initial data. Some preliminary results on envelope equations are discussed.
Last modified: Thu Jan 18 13:55:17 MST 2001