Unfortunately, we know nothing about plasma wave spectra in the corona. Therefore, in kinetic models assumptions have to be made about the spectrum of the waves injected at the coronal base. A power-law is often assumed, the intensity of which is then constrained by extrapolation of the in situ measurements (Tu and Marsch, 1995) to the corona. Furthermore, the important questions of cascading - oblique as well as parallel - remains an open problem (Cranmer and van Ballegooijen, 2003). Large-scale MHD structures may preferentially excite perpendicular short-scale fluctuations Leamon et al. (2000), the dissipation of which may involve strong Landau damping coupled to kinetic processes acting on oblique wavevectors.

The relevant typical wavenumber, , for collisionless dissipation was estimated by Gary (1999), who defined it to be the minimum value at which kinetic damping becomes significant, and determined from linear Vlasov theory for the Alfvén-cyclotron and magnetosonic wave branches. Essentially, the dissipation scale is set by the ion inertial length, whereby a scaling law was found to apply as follows:

Here and are fitting parameters, with being of order unity, respectively, ranges between 0.3 (Alfvén wave) and 0.8 (magnetosonic mode). The cyclotron damping of Alfvén-cyclotron fluctuations increases monotonically with increasing , whereas proton cyclotron damping of magnetosonic fluctuations is essentially zero at low and becomes significant only at . Concerning the spectral index of magnetic fluctuations in the solar wind, Li et al. (2001) argued that collisionless dissipation, because of its exponential dependence of the damping rate on , cannot be the main mechanism for spectral steepening; rather, damped power spectra should decrease more rapidly than any power law as the wavenumber increases. They obtained an analytic expression for the damping rate of the form (with fit parameter, ): The three fit parameters depend upon and vary with the propagation angle of the waves and different values of the plasma beta.For the subsequent theoretical sections of this review, we provide some frequently used definitions. The density of species is , its mass , and its plasma frequency is denoted as . The particle’s gyrofrequency, carrying the sign of the charge, reads , for a background magnetic field of magnitude . The mean thermal speed is , with the temperature . The plasma beta of species is defined as . The mass density is , and fractional mass density, , with the total mass density being . We will also make use of the relation , where the Alfvén speed is based on the total mass density and as usually defined by .

Whatever the wave dissipation process heating the particles may be, it certainly must be more effective for heavy ions than for protons (and electrons), because, as we previously discussed, the minor heavy ions are much hotter than the protons in coronal holes and the fast solar wind (Marsch, 1991a,b; von Steiger et al., 1995). Before we can address wave-particle interactions in more detail, the basics of kinetic theory first need to be discussed. We return to the topic of plasma waves at a later stage of this review in Section 5.

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