In the kinetic shell (or bi-shell) model, the approximation is made that the resonant cyclotron interaction proceeds much faster than any other process affecting the protons, which is consistent with the size of the relaxation rate as defined in Equation (53). In addition, it is assumed that the resonant portions of the VDF are kept in a state of marginal stability, such that resonant protons are organised on nested shells of constant density in velocity space, defined by the condition that the proton energy is conserved in the rest frame moving with the phase speed of the resonant wave, according to the plateau condition, , as derived from Equation (50). This assumption is to some extent supported by the shape of the middle parts of the observed fast proton VDFs, which were shown in Figure 14 and appeared to obey Equation (56). Note however, that most VDFs in their innermost cores, where the strongest resonance due to the highest proton density will take place, hardly reveal a rigid bi-shell shape (see, e.g., the previous Figure 3).
The fixed shells in the model are argued to evolve on the large non-resonant timescale (expansion time), corresponding to the shell-averaged forces of gravity, charge-separation electric field, and magnetic mirror, which all are contained in Equation (9). The crucial assumption then made is that these kinetic shell VDFs are held in a marginally stable state, in which they cannot exchange any more energy with the waves. Isenberg (2004) claims that in this state the evolution of the shells corresponds to the effect of the maximum possible dissipation of the waves.
The earlier work was expanded to include Sunward propagating waves and improved to treat also anti-Sunward protons. These changes in the model yielded more plausible proton speeds and temperatures than obtained previously. Also, the important effects of ion-cyclotron wave dispersion were incorporated into the model. Dispersion created broader resonant shells, consistent with the in situ observations (Marsch and Tu, 2002; Tu and Marsch, 2002).
In this way Isenberg (2004) found that the non-resonant forces acting on dispersive-shell VDFs invariably produced only weak acceleration, and surprisingly perpendicular cooling rather than the observed heating. These model effects were attributed to the weaker inertial force on the Sunward dispersive shells. Since according to the authors the model describes the maximal wave dissipation, it was concluded that heating and acceleration of protons in coronal holes are not caused by the dissipation of parallel propagating ion-cyclotron waves. Relaxing the extreme kinetic shell assumption will not change this negative result, Isenberg (2004) stated and concluded that “...with a less instantaneous interaction, the resonant shells will not be completely filled, but then the dissipation will provide even less proton heating than in our model...”. Apparently, more work needs to be done, and certainly other wave modes, like kinetic Alfvén waves propagating obliquely, should be considered.
One reason for the negative conclusion of Isenberg (2004) is his invalid assumption that Alfvén-cyclotron waves cannot scatter protons through the condition . However, as noted in Subsection 6.2, the simulation of Gary and Saito (2003) showed that such scattering can indeed take place. Of course, their simulations were based on different approximations, and thus it remains to be consistently determined whether protons can be scattered appreciably by Alfvén-cyclotron fluctuations.
Such future research should be conducted in the framework defined by the work of Cranmer (2000, 2001) and Vocks and Marsch (2002). If ion heating is due primarily to Alfvén-cyclotron wave scattering, and frequency sweeping is the mechanism transferring fluctuation energy from longer to shorter wavelengths, do the heavy ions absorb all the fluctuation energy and leave nothing for the protons? Kinetic modelling of ions in the corona seems to show that heavy-ion wave scattering might saturate in a way allowing wave energy to progress to the shorter proton cyclotron wavelengths. This issue will be resumed and further discussed in Subsection 7.2.
Galinsky and Shevchenko (2000) did not make the rigid bi-shell assumption in their model, but started from the full quasilinear Equation (50) for wave-particle interactions in the solar wind with a weakly non-uniform magnetic field. Their method was also based on a scale separation between the length scales of quasilinear diffusion and magnetic-field inhomogeneity, thus allowing them to obtain large-scale kinetic solutions for both the VDF and wave energy spectrum density, without the need to consider the details of the small-scale relaxation process. The numerical solution of their equations illustrated the importance of the diffusion plateaus and indicated the possible existence of a secondary plasma instability for an initially stable proton VDF. Tam and Chang (1999, 2001) also solved the diffusion equations of QLT in a non-uniform model solar wind for electrons and protons and reproduced some observational trends, however, obtained VDFs that hardly resembled the measured ones.
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