Resonance type | Wave number range | -range | propagation |

Proton cyclotron | all | quasi-parallel | |

Electron Landau | all | oblique | |

Proton Landau | oblique | ||

The work of Gary and Borovsky (2004) showed that parallel Alfvén-cyclotron fluctuations of sufficiently short wavelength lead to strong proton cyclotron resonance. Yet as the wavevector component decreases, the proton cyclotron interaction ceases, and the electron Landau resonance may become effective at oblique propagation. Gary and Nishimura (2004) investigated the dispersion and damping properties of Alfvén-cyclotron waves associated with the transition from the proton-cyclotron to the electron-Landau resonance regime.

If the wavevector component, , of an Alfvén-cyclotron waves becomes smaller, the resonant ion-wave interactions gradually decreases. However, if concurrently becomes substantial, then Landau resonance will play an increasingly important role. The waves with propagation strongly oblique to the field are called “kinetic Alfvén waves” (KAW). They have been studied by many authors, e.g., Hollweg (1999d) (and references therein) who calculates their approximate dispersion relation for an electron-proton plasma:

where the effective sound speed is given by , and corresponds to the ratio of specific heats, i.e., is equal to for a simple hydrogen plasma. Note that the Alfvén wave dispersion is, via the perpendicular wave vector, modified and for KAW also depends on the electron inertial length and the effective gyroradius that is based on the ion and electron temperatures. Equation (44) contains various limiting cases discussed in the literature and explained by Hollweg (1999d). The complete kinetic dispersion properties were calculated with the Vlasov theory for Maxwellian particles. Selected results for are presented in Figure 13 (after Gary and Nishimura, 2004) and shown in dependence on such key wave parameters as propagation angle and electron to proton temperature ratio, , which quantifies the relative importance of electron versus ion Landau damping.Gary and Nishimura (2004) also used particle-in-cell simulations (for earlier simulations see references therein) to examine the electron kinetic response to the waves being subject to electron Landau damping. Their computations show heating of the electrons in the parallel direction and formation of a field-aligned electron beam.

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