Alfven Wave Equations

The transverse magnetic-tension wave of MHD, and the eigenmodes it forms in a nonuniform confined plasma.

The Basic Wave

The shear Alfven wave is a transverse oscillation in which magnetic tension provides the restoring force, analogous to a wave on a plucked string where the field line is the string. It is incompressible, propagates along the field at the Alfven speed v_A = B / sqrt(mu0 rho), and does not perturb the field magnitude or plasma pressure. Its dispersion relation is simply omega = k_parallel v_A.

The Alfven Continuum

In a nonuniform plasma v_A varies across flux surfaces, so the local Alfven frequency omega_A(r) = k_parallel(r) v_A(r) forms a continuum of frequencies. A wave launched at a fixed frequency resonates with whichever surface matches it, where the amplitude sharpens and energy is absorbed by continuum (phase-mixing) damping. This resonant absorption is the basis of Alfven-wave heating.

Discrete Eigenmodes

Toroidal geometry breaks the continuum by coupling neighboring poloidal harmonics, opening frequency gaps. Discrete global eigenmodes can exist inside these gaps free of continuum damping, most importantly the toroidal Alfven eigenmode in the gap formed between m and m+1 harmonics. Because they are weakly damped, such modes are easily driven unstable by resonant fast ions through inverse Landau damping.

Relevance

In a burning plasma, fusion-born alpha particles and beam ions can destabilize Alfven eigenmodes and expel fast ions before they thermalize, degrading self-heating. For the Hyperion D-T breeder concept, the high Alfven speed set by strong field and low density places these modes in the design-stage stability analysis; conclusions are computational for a machine not yet built.