The Alfvén wave is the dominant low frequency transverse mode of a magnetized plasma. The Alfvén wave propagates along the magnetic field, and displays a continuous spectrum even in a bounded plasma. This is essentially due to the degeneracy of the wave characteristics, i.e. the frequency (ω) is primarily determined by the wave number in the direction parallel to the ambient magnetic field (k||) and is independent of the perpendicular wavenumbers. The characteristics, that are the direction along which the wave energy propagates, are identical to the ambient magnetic field lines. Therefore, the spectral structure of the Alfvén wave has a close relationship with the geometric structure of the magnetic field lines. In an inhomogeneous plasma, the Alfvén resonance (ω−cAk||=0; cA is the phase velocity of the Alfvén wave) constitutes a singularity for the defining wave equation; this results in a singular eigenfunction corresponding to the continuous spectrum. The aim of this review is to present an overview of the perturbation theory for the Alfvén wave. Emphasis is placed on those perturbations of the continuous spectrum which lead to the creation of point spectra. Such qualitative changes in the spectrum are relevant to many plasma phenomena. The first category of perturbations consists of nonideal effects such as the finite conductivity, kinetic effects arising from the finite electron inertia, and finite gyroradius. These effects add singular perturbations to the mode equation, and modify the spectrum dramatically. These modification, viz. the conversion of the continuous to the point spectrum, can lead to interesting physical phenomenon. A case in point is that of an electron beam propagating in a plasma which Cherenkov emits a left-hand circularly polarized Alfvén wave. The helicity of the ambient magnetic field imparts a frequency shift to the eigenmodes changing the critical velocity for Cherenkov emission. It, then, becomes possible for a sub-Alfvénic electron beam to excite a nonsingular Alfvén wave corresponding to a point spectrum. The second category comprises of geometric perturbations associated with higher dimensional inhomogeneity of the ambient field. Forbidden bands occur when a periodic modulation is applied. In a chaotic magnetic field, the weak localization of the wave occurs, resulting in a point spectrum.
The development of magnetic field line wander by plasma turbulence
