A less ambitious approach was taken by (Vocks and Marsch, 2002) who have shown that it may be meaningful to simplify the full problem and reduce the kinetic VDF further by an integration over . This procedure, used before by Dum et al. (1980) to solve dispersion relations, yields two reduced VDFs which are defined as follows:

where a negative points in the Sunward direction. The evolution equations for these reduced VDFs are obtained by taking the corresponding moments of the Boltzmann Equation (9) and using the methods of Vocks (2002). To break the chain of higher-order moments appearing in the original diffusion equation we make the Gaussian approximation (Marsch, 1998), which reads: This relation would be exact for a bi-Maxwellian. Of course, this does not imply that is Gaussian itself. Empirical motivation for the factorization (66) stems from the solar wind in situ observations, yielding that at any parallel speed the protons perpendicular speeds are distributed as a Gaussian (Marsch and Goldstein, 1983), despite the fact that the VDFs can be skewed, and that there may be proton beams drifting along the mean field.Using the reduced VDFs, one can construct a gyrotropic, 2-D model VDF by introducing the effective perpendicular thermal speed, which leads with the Gaussian approximation to a convenient model VDF based solely on the reduced VDFs:

Making use of such reduced VDFs for protons and minor ions in the solar corona and solar wind, Vocks and Marsch (2001) first developed a semi-kinetic hybrid model for solar wind expansion in coronal funnels. We quote the pair of reduced Boltzmann equations, which according to the work of Vocks (2002) and Vocks and Marsch (2001) have the form:

Here the symbol again denotes the cross-sectional area of the magnetic flux tube considered in Equation (31). The last term on the left hand side is related to the expansion of the flux tube and corresponds to the mirror force a charged particle experiences in an inhomogeneous magnetic field. The terms on the right hand side denote the wave-particle interactions and Coulomb collisions, which are not quoted here explicitly. To calculate them for the reduced distribution functions is a tedious task, algebraically as well as numerically, and requires to take partial moments of the wave diffusion operator (50), respectively, of the collision operator (13). This lengthy procedure is described in detail by Vocks (2002).The semi-kinetic model has been applied by Vocks and Marsch (2001) and Vocks and Marsch (2002) to calculate the plasma dynamics and VDFs of heavy ions in the solar corona. The numerical model includes ion-cyclotron wave-particle interactions and Coulomb collisions as calculated by use of the Landau collision integral. The reduced ion VDFs only depend on the height coordinate , ion speed and time , and can numerically be solved with reasonable effort for a coronal funnel with an expanding magnetic field (mirror geometry).

The numerical results obtained for heavy ions in a coronal funnel show good agreement with SOHO observations and yield strong heating of the heavy ions. This is illustrated in the Figure 27. It was found that the heavy ions are heated preferentially with respect to the protons, and that sizable temperature anisotropies and ion beams or heat fluxes form, qualitatively similar than the weak-tail solar wind proton VDFs shown in Figure 3. The reduced model VDFs of the heavy ions develop distinct deviations from a Maxwellian, which tend to increase with height owing to the declining efficiency of Coulomb collisions. The wave damping/growth rate indicates that the VDFs can reach a limit of marginal stability over a wide range of resonance speeds, where wave emission or absorption gets weak. Wave growth or damping completely vanishes for particles being on the quasilinear plateau, where the pitch-angle gradient is zero in the wave frame. Then the resonance function defined in Equation (64) is by definition equal to zero.

Similar results were recently found in direct numerical simulation by Hellinger et al. (2005), who used
the so-called expanding box model for the non-uniform solar wind (Hellinger et al., 2003) and presented
kinetic hybrid simulations of the interaction of left handed outward propagating Alfvén waves with
protons, alpha particles and a tenuous population of oxygen O^{5+} ions. The Alfvén waves were initially
non-resonant with all the ions. Then radial expansion brings the ions to local cyclotron resonance (through
the frequency sweeping mechanism suggested by Tu and Marsch, 1997), first the O^{5+} ions, then alpha
particles, and finally protons. These simulations show that oxygen ions are efficiently heated in the direction
perpendicular to the background field, but are only slightly accelerated. Oxygen scattering
lasts for a finite time span but then saturates, mainly due to the marginal stabilization with
respect to the oxygen-cyclotron instability that is generated by the temperature anisotropy.
During their scattering the oxygen ions can only absorb a limited amount of the available wave
energy.

Liewer et al. (2001) also simulated the Alfvén wave propagation and ion-cyclotron interactions in the expanding solar wind. Recently, Xie et al. (2004) carried out hybrid simulations of heavy ions to analyse the multiple ions resonant heating and acceleration by Alfvén-cyclotron waves in the corona and solar wind. For the solar wind parameters used by Hellinger et al. (2005) in their simulations, the presence of minor heavy ions had a minimal influence on the major species. These simulations do not support the claim made by Cranmer (2000) that minor ions would effectively prevent the cyclotron-wave absorption of alpha particles and protons.

Resonant heating and acceleration of ions in coronal holes driven by cyclotron resonant waves were also
studied by Ofman et al. (2002) in one-dimensional hybrid simulations of an initially homogeneous,
collisionless plasma. They used a model of corona including kinetic protons, a tenuous component of oxygen
ions, and massless fluid electrons. Spectra of ion-cyclotron resonant Alfvén waves were imposed, and the
effects of various power-law spectra scaling like or were analysed. The resulting ion heating
was found to strongly depend on the power contained in the ion resonance frequency range. Usually, the
minor O^{5+} ions were easily heated and became anisotropic, however the protons remained nearly
isotropic and were mostly heated weakly. For the parameters used, the oxygen temperature
ratio, , reached values of up to ten within several thousand proton cyclotron periods.
Whereas shell-like VDFs were transiently present, the long-term shape was more that of a
bi-Maxwellian.

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