Cyclotron resonances and magnetized waves

Take a homogeneous equilibrium magnetic field along z and define the signed cyclotron frequency as charge times field magnitude divided by mass. Ions and electrons then have opposite signs. Choose a gyroangle harmonic convention in which the resonance denominator is frequency minus parallel Doppler shift minus harmonic number times that signed frequency. Integer harmonic labels depend on the gyroangle convention, so comparisons across references must preserve the whole convention rather than copying a sign in isolation.

For nonzero parallel wavenumber, each harmonic selects a parallel velocity. The n = 0 case is the parallel Landau resonance, while nonzero harmonics involve gyration. If parallel wavenumber vanishes, division by it is invalid: the resonance instead requires frequency equal to the harmonic cyclotron frequency. A positive physical frequency can resonate with a negative harmonic number for electrons under the signed-frequency convention. Replacing the signed frequency by its absolute value without relabeling harmonics can select the wrong population.

As the particle gyrates, the perpendicular wave phase oscillates around its guiding center. Expanding that periodic phase gives Bessel-function weights for the cyclotron harmonics. The dimensionless argument is perpendicular wavenumber times the particle's gyroradius. It uses that particle's perpendicular speed, not automatically one chosen thermal speed. When the argument is small, only the lowest permitted harmonics contribute strongly; when it is order one, finite-orbit averaging can substantially alter the response.

Polarization determines which rotating electric-field component can exchange energy efficiently with the gyration. For parallel propagation at small perpendicular orbit scale, the two circular polarizations couple differently to the species and harmonic senses. A resonance equation with an allowed integer does not override a vanishing polarization or Bessel coupling factor. In an inhomogeneous plasma, field variation and orbit drifts broaden or shift these ideal resonances, requiring orbit-based or local approximations with their own ordering.

In an unmagnetized longitudinal wave, the response depends on the parallel velocity slope. A magnetized harmonic generally samples both perpendicular and parallel derivatives of a gyrotropic distribution. Temperature anisotropy, loss cones, and beams can therefore supply free energy through different gradients. A monotonically decreasing parallel marginal alone does not prove that every magnetized mode is stable. The kinetic susceptibility and the mode's polarization and wave energy determine the net exchange.

A cold magnetized dielectric tensor can predict branches and resonance frequencies while omitting thermal cyclotron absorption and finite-orbit effects. Use it when the neglected velocity spread and orbit scale are appropriately small. Conversely, a gyrokinetic low-frequency theory deliberately averages over fast gyration and normally excludes cyclotron-frequency resonances. These are different reductions of kinetic theory, chosen for different frequency ranges. Checking the orderings is essential before comparing a resonance measured near the cyclotron frequency with a low-frequency turbulence model.