Alfv´en-cyclotron waves and kinetic Alfv´en waves

5.4 Alfvén-cyclotron waves and kinetic Alfvén waves

Waves that interact with the thermal ions are of primary importance for the transport of thermal energy in the solar corona and wind. At wavelengths near the thermal proton gyroradius, $rp = vp∕Ωp$ , or of the order of the proton inertial length, $l = c∕ω = V ∕Ω p p A p$ , there are three prominent normal wave modes: the Alfvén-cyclotron wave, the magnetosonic-whistler wave, and the ion-acoustic wave (Gary, 1993 ). For the solar wind and corona the wave properties and dispersion of course depend on the plasma $β$ , with the parameter range $0.005 < β < 2$ being especially important. Recently, detailed numerical studies were carried out with respect to the parametric dependence on $β$ , and on the ion temperature anisotropies and drifts. The onset of Landau or cyclotron damping in particular depends sensitively on $β$ . Gary and Borovsky (2004

) derived for parallel propagation a typical wavenumber $kd$ at which strong resonant dissipation sets in. It is roughly given by $kdlp = 0.1$ for $β = 1$ . The detailed dependence on the proton $βp$ is for a Maxwellian plasma shown in Figure 12

, which was derived by solving the kinetic dispersion relation (41

). Gary and Borovsky (2004

) obtained three distinct damping regimes. Their results are summarised in Table 2. The dissipation of solar wind magnetic fluctuation spectra was also studied with respect to a comparison of dispersion versus damping by Stawicki et al. (2001 ).

Table 2:

Alfvén-cyclotron wave damping regimes


Resonance type	Wave number range	$β$ -range	propagation

Proton cyclotron	$kd < k ∥$	all $βp$	quasi-parallel
Electron Landau	$k∥ < kd$	all $βe$	oblique
Proton Landau	$k∥ < kd$	$0.1 < βp$	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 $k∥$ 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, $k∥$ , of an Alfvén-cyclotron waves becomes smaller, the resonant ion-wave interactions gradually decreases. However, if $k⊥$ 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 $∘ ---------------------- Cs = (γekBTe + γpkBTp)∕mp$ , and $γe,p$ corresponds to the ratio of specific heats, i.e., is equal to $5 ∕3$ 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 $βp = 0.01$ are presented in Figure 13

(after Gary and Nishimura, 2004

) and shown in dependence on such key wave parameters as propagation angle $θ$ and electron to proton temperature ratio, $Te∕Tp$ , 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.

Figure 13:

The dispersion relation of kinetic Alfvén waves for $βp = 0.01$ versus $k∥lp$ . The solid and dashed lines represent the real frequency, and the dotted chains represent the corresponding damping rates after Gary and Nishimura (2004 ). Left: Dispersion results for parallel and very oblique propagation. Right: Three dispersion curves relating to different temperature ratios.