Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 93

point x r increases with the growth of k y (see 4.55). Since the field outside the
resonance region decreases exponentially, with decrement proportional to k y ,
the FLR-amplitude decreases at large k y . Inasmuch as the effect vanishes at
k y =0and k y = , it is most vividly expressed at intermediate values of k y .

Phase mixing for Alfven waves noninteracting with FMS-waves was studied
above in (4.2). Suppose here that ξ x ,ξ y and their derivatives are given in plane
z =0and ξ x ,ξ y ∝

∞

iωt + ik y y ) . Waves are sought that depart in the
positive direction of axis z . Note that an analogy exists between the stationary
problem of perturbation propagation along axis z and the temporal problem
(4.43) - (4.45). The latter is considered closely in ( [4], [16], [17])
It is satisfactory here to take advantage of the results obtained in the
study of the time evolution of the initial perturbation. We shall describe the
asymptotic behavior of waves at large z . Omitting the details (see [16], [17]),
we present the conclusions ensuing from this analogy

−

exp(

1. At z the component ξ x can be presented as the sum of two parts.

The first are, usually, a slowly exponentially damping collective modes. The
second are the modes diminishing by the power law

→∞

ξ x = 0(1 /z ) .

Each of them propagates along its respective field line with phase velocity
c A ( x ). The collective mode arises as a surface wave if there is a surface
near which the wave is concentrated and along which it propagates. It can
appear also as a cavity mode which fills the whole volume.
2. Asymptotically the component ξ y decays weakly

ξ y ( z →∞ )= ξ y = 0(1) .

It can be presented as a sum of the collective mode and the modes propa-
gating along each field line with Alfven velocity c A ( x ).
3. Alfven waves are guided by the magnetic field and can propagate over large
distances.
4. A scale shortening is the common property of MHD-waves propagating in
inhomogeneous plasma.

These results obtained for a 1D-system remain valid for more complex
plasma configurations as well, when the many-dimensional inhomogeneity of
the Alfven velocity, the curvature and convergence of the magnetic field-lines
are essential.

Fields in the Vicinity of a Resonance Shell

Consider the behavior of the wave in the vicinity of a resonance field line
proceeding from the system of equations (4.43) - (4.45). The solution will be
sought by the Frobenius method (e.g.,[6], [20]). From (4.45) we readily see
that