and Schlickeiser (1999a) note that this result agrees with Steinacker and Miller (1992), where both coefficients were calculated using quasi-linear theory directly. The combining of Eq. 2.20.11 and Eq. 2.20.12 with Eq. 2.20.7 allows one to calculate the CR bulk speed V(p) determined by Eq. 2.20.8, if the scattering frequencies as a function of wave frequency are specified. In particular, if the spectrum of waves as a function of wave number is steep enough, we may approximate aa as unity for the resonant wave with the lowest wave number and zero for the others.
2.21. Non-resonant pitch-angle scattering and parallel mean-free-path
2.21.1. The problem of the non-resonant pitch-angle scattering
According to Ragot (1999) the scattering of charged particles through the zero pitch-angle cosine, has long remained a challenging problem in the theory of CR particle transport, in a magnetic turbulence composed of plasma waves superimposed on a larger-scale regular magnetic field (see e.g., Bieber et al. 1994; Ragot, 1999 and references therein). The quasilinear theory (Vedenov et al., 1962; Jokipii 1966), which usually describes this problem of particle transport only includes the resonant interactions between the waves and the particles and, when the real spectral shape of the turbulence is taken into account, can fail to produce any significant scattering through ^ = 0 at low particle's rigidities. More sophisticated, nonlinear theories (see references in Ragot, 1999) require enhanced levels of fluctuations to achieve this scattering through ^ = 0, which are not necessarily observed each time the particles are efficiently scattered. The occurrence of this problem of pitch-angle scattering at precisely ^ = 0 is, however, somewhat surprising. Indeed, why should the point where ^ = 0 always be the critical point, when ^ is defined with respect to the main magnetic field Bo, while locally the real field lines are not along Bo ? The answer to this problem according to Ragot (1999) can be formulated in a quite simple form. In order to correctly describe the pitch-angle scattering through ^ = 0, one must take into account the lower-frequency waves even if they are not in resonance with the particles, because they determine the local variations of the field line direction. The frequencies at which resonant interactions can take place between waves and particles depend on the particles' rigidity and the dispersion relation of the waves. It appears that the most dramatic effect - extremely weak scattering and resulting divergence of the parallel mean free path - occurs when the resonant frequencies fall in the dissipation range of the turbulence, leaving the waves of the inertial range out of the wave-particle interaction description, despite the fact that these later lower-frequency waves are responsible for the local variations of the field line direction. This is the case, in particular, for low-rigidity CR in the solar wind. They cannot gyro-resonate with MHD waves in the inertial range of the turbulence, below a few hundreds of MV when their pitch-angle cosine ^ approaches 0, and below « 1 MV for any Gyro-resonance between waves of frequency œ (parallel wavenumber k// ) and particles of gyro-frequency
occurs when the condition knz -m,t ± n|Q|t = 0 (2.21.2)
is satisfied for some integer n ^ 0, j = ±1 denoting forward and backward propagating waves. At small less than the ratio Alfven speed va over particle speed v, no transit-time damping (TTD) interaction (n = 0 resonance) is possible either, with the fast magneto-sonic component of the spectrum. As a consequence the quasi-linear theory, which only takes into account the resonant interactions (gyro-resonant and TTD), predicts a very low pitch-angle scattering and a very long mean-free-path along the direction of the magnetic field lines. The original quasilinear prediction (Jokipii, 1966) gave a short parallel mean-free-path, but this was owing to the absence of cutoff in the turbulence spectrum. Latter measurements in the solar wind (Coroniti et al., 1982; Denskat et al., 1983) showed a strong steepening of the spectrum above the ion gyro-frequency Q po ~ 1 Hz, with a spectral index y going from y ~ -1.7 to y ~ -2.9, which is responsible for the 'divergence' of the mean-free-path below about 100 MV. Besides this problem of cutoff and divergence the original quasi-linear mean-free-path in fact never gave the right dependence of X as a function of rigidity R: X kept decreasing with decreasing R, whereas the observational data show little dependence of X on R for rigidities between 10-1 and 103 MV, which is known as the 'flatness problem'.
If it seems relatively clear that the problem of scattering through ^ = 0 arises from the exclusion of the lower-frequency waves, the precise, quantitative prediction of the parallel mean free path for solar CR or small energy galactic CR, and the solution of the 'flatness problem', require a more detailed description of the process of wave-particle interaction, and of the turbulence. We do not know for sure what the real composition of the turbulence is. However, it is likely that the wave turbulence is made of Alfven and fast magneto-sonic waves, because in a magnetized, low but finite ¡3 plasma (¡3 = 2c2s /v^ , cs being the sound speed) like the solar wind, these two types of waves are the less heavily damped. As for the distributions of k -vectors for these waves, Ragot (1999a) make following assumptions: as in the papers by Schlickeiser and Miller (1998), Ragot and Schlickeiser (1998a,b) and Ragot (1999b), a slab Alfven turbulence (parallel propagating, with k along Bo) and isotropic fast magneto-sonic waves. Ragot (1999a) presents the main lines of the derivation of the non-resonant pitch-angle scattering process with these waves, and shows that it very efficiently scatters the particles through ^ = 0. Detailed calculations for this process can be found in the paper by Ragot (1999b), and fits of the parallel mean free path as a function of the rigidity, deduced from measurements for solar CR, are presented in Ragot, 1999c (see below, Section 2.21.5). The case of oblique Alfven waves is also briefly considered in Ragot (1999c). In Ragot (1999a) the slab Alfven waves, for reasons of symmetry, do not contribute to the non-resonant interaction process. Ragot (1999a) thus ignores them in the evaluation of this effect.
2.21.2. Derivation of the non-resonant scattering process
According to Ragot (1999a) gyro-resonance is very inefficient at scattering low rigidity CR, because most of the energy is in the waves that have much too low frequencies to be in gyro-resonance with these particles. The limit between gyro-resonant and non gyro-resonant waves is given, for a linear dispersion relation of the waves coj = jkva , valid well below kc = Qpojva , by:
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