6.9 Plasma heating (cooling) by wave absorption (emission)

In the previous sections we have extensively discussed the various non-thermal features of particle VDFs in the solar wind and the associated linear instabilities and non-linear effects. The coronal origin of the proton beam, thermal core anisotropy or ion differential streaming still remains unclear. The ion core temperature anisotropy (as illustrated in Figure 3View Image and Figure 16View Image) is commonly believed to be generated by cyclotron resonance through Alfvén-cyclotron wave absorption, a process that presumably takes place in the corona and solar wind. The ion beams may originally come from coronal sources, such as reconnection jets (Feldman et al., 1996) and explosive events, or they may be produced in situ, either by plasma wave absorption and scattering or cumulative collisions (Livi and Marsch, 1987Jump To The Next Citation Point). Also, an electric current, e.g., due to an ion-electron cross-field drift, may produce cyclotron waves (Markovskii and Hollweg, 2002) that in turn could generate the core temperature anisotropy.

Markovskii (2001) argued that ion-cyclotron waves might be generated in coronal holes by a global resonant magnetohydrodynamic wave mode. Particle loss-cone distributions might originate in coronal magnetic mirrors, as represented by expanding coronal funnels (Vocks and Mann, 2003Jump To The Next Citation Point). Parametric decay of large-amplitude Alfvén waves (Gomberoff et al., 2002Araneda et al., 2002Gomberoff et al., 2003) may also lead to cyclotron daughter waves. Inhomogeneity of the field will cause frequency sweeping (Tu and Marsch, 1997Jump To The Next Citation Point) of a primordial spectrum via the radial decline of Ωi,e(r) with solar distance r. Do all these processes operate in the corona? We do not know yet, but it seems likely. Hollweg and Isenberg (2002) provide a comprehensive review of the cyclotron heating mechanism.

Whatever the wave-particle interaction is, according to quasilinear theory the heating due to plasma wave absorption can readily be calculated by taking appropriate moments of the fundamental kinetic Equation (50View Equation), which describes the evolution of the VDF in the wave field. The corresponding rates are equivalent to the work done by the rest-frame electric field on the current density. In their book, Melrose and McPhedran (1991Jump To The Next Citation Point) give a lucid general account of particle heating by electromagnetic fluctuations. Energy and momentum will, as the result of wave-particle interactions, be exchanged between fields and particles. Quasilinear relaxation of fj(v ) will be the consequence, at the expense of the available free energy. The resulting heating rates (Qj ∥,Qj⊥) and acceleration or momentum transfer rate (R j) for any species were calculated by Marsch and Tu (2001aJump To The Next Citation Point), and can be written as follows:

( ) ( ) Rj ∫ + ∞ 3 ∑ +∑ ∞ k ∥ || Qj∥ || = ρ -d-k-(Ωj-)2 ˆℬ ℛ (k, s)|| 2k∥Vj(k, s)|| . (63 ) ( Qj ⊥ ) j −∞ (2π)3 k ∥ M j ( sΩj ) M s=− ∞
Equation (63View Equation) expresses the wave heating rates and acceleration in terms of an integral over the normalised magnetic PSD and sums over the mode number, M, and resonance-order number, s, which denotes the order of the involved Bessel function. The normalised spectral density ℬˆM (k ) was before defined in Equation (52View Equation). The resonant speed was already defined by the expression (54View Equation). The resonance function or wave absorption coefficient, ℛj (k, s), is a functional of the particle distribution function, fj(v ∥,v ⊥) and essentially involves the negative pitch-angle derivative, which is evaluated in the respective wave frame at the Landau resonance (s = 0) or at the cyclotron resonance, for any integer Bessel function index (s = ±1, ±2, ..). This dimensionless coefficient is given by the following expression:
∫ ( ) 2 ∞ 1- + − 2 ∂fj- ℛj(k, s) = sign(k∥)2π dv ⊥ | v⊥ 2(Js−1eM + Js+1eM ) + Vj(k,s)JseMz | − ∂ α .(64 ) 0 v∥=Vj(k,s)
It corresponds to the velocity average of the relaxation rate given in Equation (53View Equation), which is weighted by the pitch-angle gradient at the resonance. As in the case of parallel propagation, for oblique wave propagation the coefficient ℛj can be entirely expressed in terms of reduced VDFs, see, e.g., Marsch (2002), if the dependence of (64View Equation) on v⊥ is smoothed out by replacing this variable in the Bessel functions by the thermal speed Vj⊥. Wave absorption vanishes when the pitch-angle gradient is zero and a plateau is formed, i.e. for ∂fj∕∂ α = 0. Explicit expressions for ℛj, e.g. for a bi-Maxwellian, were given in the paper of Marsch and Tu (2001a).

Of course, the full rates in Equation (63View Equation) can only be evaluated once the VDF, fj(v∥,v⊥), for all particle species and the wave power spectral density (PSD), ℬˆM (k ∥,k ⊥), of all wave modes involved are known. This complexity is an unavoidable feature of kinetic theory as compared with fluid theory, in which only velocity averages and mean wave amplitudes are considered. Note that ℛj plays the role of a “wave opacity”, using a term from radiation transfer theory. The wave PSD in the kinetic domain are not well known for the solar wind (not to speak of the corona), and in particular the electric field near the ion gyrofrequency is notoriously difficult to measure from spacecraft. It was only more recently (Kellogg, 2000Kellogg et al., 2001), that with wave instruments on the modern Cassini and Cluster spacecraft such measurements became possible. As plasma waves and fluctuations, by inelastic pitch-angle scattering according to the diffusion operator (50View Equation), randomise the particle VDFs, the knowledge of the wave PSD is of paramount importance to understand the kinetic evolution of the VDFs, or the possible wave-particle equilibrium.

Given reasonable wave fluctuation levels in the corona, see for example the numbers quoted in Marsch (1992) or in Shukla et al. (1999Jump To The Next Citation Point), these micro-turbulent rates might provide sufficient ion and electron heating, and perhaps acceleration as well. The spectra of the plasma waves, as well as of the VDFs, are of course crucial, yet unknown in the corona. The lack of empirical knowledge forces one to make either assumptions, or to calculate ab initio a wave PSD for each wave mode and particle VDF for each species by help of the kinetic Equation (9View Equation) and the wave transfer equation, which was for example derived in Melrose and McPhedran (1991). Both equations may then be applied to a specific magnetic field structure in the corona, such as to coronal loops or a coronal hole. Limited reduced cases of Equation (63View Equation) were discussed in the literature. For example, heating by high-frequency dispersive Alfvén waves was considered by Shukla et al. (1999), and kinetic aspects of coronal heating were discussed recently by Bingham and Shukla (2004), who specifically investigated lower-hybrid drift modes.

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