Fundamentals of Plasma Physics 2025, 26 - Electromagnetic Mode Analysis

Inductive electric fields (induced by time-dependent currents) don't involve density perturbations. To derive their electrostatic susceptibilities, the electrostatic modes don't need to be derived separately, but instead the relationship between Poisson's equation, Ampere's law and charge-weighted summation of the 2-fluid continuity equation suffices. It results in the constancy of -Σ_σ n_σq_σ + ε_σ∇○E. This provides an initial condition. Small-amplitude perturbations are assumed to have the phase dependence, behaving as a single Fourier mode. A single particle immersed in a constant uniform equilibrium magnetic field and subject to a small-amplitude wave with a periodic electric field, define the velocity as sums of parallel quiver velocity, generalization polarization drift, and generalized E × B drift.

At the no-plasma limit, K becomes the unit tensor, describing the effect of the vacuum displacement current. Descriptions of the curl for both fields follow directly from K.

At θ = 0, the determinant of K×E becomes [(S - n²)² - D²] P = 0 with roots P = 0, n² = R, n² = L with R = S+D, L = S-D. Definitions of the Eigenvectors follow in the usual manner. Resonance happens at n² → ∞, corresponding to λ → 0. Slight dissipation will cause large wave damping. At θ = π/2, the eigenvalue equation becomes ω² = k²c² + Σ_σω²_pσ, which after solving the system, yields the upper hybrid frequency and lower hybrid frequency. For ω << data-preserve-html-node="true" ω_ci, the dispersion relation reduces to n² = S. At the limit that the displacement current can be neglected, this recovers the intertial Alfven wave. Generalized values of θ have resonances at about S sin²θ + P cos²θ ≅ 0.