Fundamentals of Plasma Physics 2025, 24 - Alfven Waves

Waves in a uniform, constant B-field, with wave frequencies smaller than the ion cyclotron frequency, can have either electrostatic character, some might have inductive character. In the former case, the magnetic field must also be constant via Faraday's law, in the latter, the inductive electric fields cause a time-dependency in the B-field. Assuming the wave frequencies are much smaller than the ion cyclotron frequency, the wave falls into the category of Alfven waves, which are normal modes of MHD, include magnetic perturbation and have characteristic velocities of the order of the Alfven velocity. Alfven modes come in two categories. The Magnetosonic mode resembles sound waves, with finite B-field component in z-direction, along with compression and rarefaction of B-field lines. The Shear/Alfven mode involves twisting/shearing/plucking motions with no B-field component in z-direction, itself either classifying as an inertial or kinetic Alfven wave.

The pressure is temporarily assumed to be zero so that all MHD forces are magnetic. The dynamics derive from polarization drift, resulting in a polarization current, and eventually in a perpendicular E-field component with dependence on the Alfven velocity. Standard Maxwell collection follows, using the appropriate conditions for the modes. For finite pressure, the two modes become coupled and an acoustic mode appears. It comes from the MHD e.o.m.

The MHD model ignores parallel electron dynamics and the shear mode dispersion has no dependence on the wave-vector. This dependence is missing in to the 2-fluid model of the shear mode. The parallel behavior of MHD sound waves is problematic, as the the z-component of the E-field is set to 0. Outside of the 2-fluid description, the MHD treatment of the shear mode leads to paradoxical results, but it's descipriton of the magnetosonic mode is adequate.

From a 2-fluid perspective, the shear mode is one of two distinct modes, the occurrence of which is mutually exclusive. The ratios of ion and electron thermal velocities and Alfven velocity are defined as

Below cyclotron frequency, the modes can be decoupled in 2-fluid description too, though it includes the E_z mode. With charge neutrality,

The shear mode has the inertial Alfven wave for β_e << data-preserve-html-node="true" m_e/m_i and kinetic Alfven wave for β_e >> m_e/m_i. It involves incompressible perpendicular motion, so the wave vector is orthogonal to the wave velocity.