Fundamentals of Plasma Physics 2025, 23 - Motion in Small-Amplitude Oscillatory Fields

A particle in a small-amplitude EM field in a plasma with large uniform, steady state B-field and no steady-state E-field will develop an oscillatory motion with some perturbation which can be neglected in the field effects. The small oscillatory fields cause small particle velocities, so the magnetic field's contribution will also be "first-order" small and hence insignificant. The oscillatory electric field decomposes into Fourier modes,

In general, small amplitude waves are analyzed by making the appropriate physical assumptions reducing the general Maxwell-Lorentz equations to the simplest set. After an equilibrium solution is found, find the dependent variables that are assumed to have a specific perturbation for at least one of them. Solving the system of differential equations give the responses for said perturbation. The resulting partial differential equation is rewritten with all dependent quantities expanded to first order.

The simplest plasma waves are described by two-fluid theory in unmagnetized plasma. It can be applied to magnetized plasma if all fluid motions are parallel to the equilibrium B-field. For unmagnetized plasma: m_σn_σdu_σ/dt = q_σn_σE - ∇P_σ. Linearized, this gives the classical expression for the electrical field. Its divergence of the previous equations gives the linearized continuity equation. For non-trivial normal modes, it's necessary to have 1 + χ_e + χ_i = 0 as a dispersion relation, prescribing a functional relation between ω and k. Normal modes have two limiting behaviors depending on the regime.

This determines the behavior of the ions.

Waves involving finite A have coupled electric and magnetic fields, and are generalization of vacuum EM waves, so they're considered EM, transverse and incompressible. No electrostatic potential is involved for neutral plasma, but A induces electric currents. As they are solenoidal, the -∇ϕ term can be eliminated through ∂_t∇×(m_σn_σu_σ1) =-q_σn_σ ∂_tB₁. By the Maxwell equations, the limit of no plasma reduces to the standard vacuum EM wave, and the unmagnetized plasma wave dispersion is ω² = ω²_p + k²c². Waves that satisfy this can be used to measure plasma density.