An example of wave collapse arising in dispersive
magnetohydrodynamics is analyzed by means of both
direct numerical simulations and envelope formalism.
It is shown that in spite of the presence of various
types of waves and of purely hydrodynamic effects,
the evolution of a longitudinally homogeneous Alfven
wave beam propagating along an ambient magnetic field
is accurately described by a cubic nonlinear Schroedinger
equation with an external potentialproportional to the
initial wave intensity. In this description, an
axisymmetric beam can only collapse on its axis when
its transverse extension significantly exceeds the
typical scale of the modulational instability of the
carrying wave. An axisymmetric configuration is however
unstable with respect to azimuthal perturbations, leading
to off-center collapse, even in situations that are smooth
when axisymmetry is preserved. In the case of a wave
packet with a finite longitudinal extension, a minimal size
is required for blowup, associated with the formation of
strongly anisotropic magnetic structures.
The regime of long Alfven wave is governed by a system
of equations derived by a reductive perturbative expansion.
Its validity extends beyond that of the envelope formalism.
Preliminary results based on adaptive mesh simulations
indicate that a wave train with a small enough amplitude
still blows up in this description. A main advantage of
this approach is that the model can be extended to the case
of collisionless plasmas for which kinetic effects are
usually not negligible.
This is a joint work with D. Laveder,T. Passot, C. Sulem,
D.S. Wang and X.P. Wang.
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