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Fig. 2.19.2 10-MeV proton mean free path for the solar wind model depicted in Fig. 2.19.1. Results for a1 = 0.05 (solid) and a1 = 0 (dashed) at r > 10 rs are shown. According to Vainio et al. (2003a).

The solid curve in Fig. 2.19.2 representing cascading Alfven waves connects a very short mean free path close to the solar surface (r < 2rs) with a larger, spatially almost constant, value at larger distances from the Sun. Thus the model may offer a consistent explanation of both efficient SEP acceleration at coronal shocks (requiring small X) in small SEP events, where efficient generation of the Alfven waves by the energetic protons themselves is not possible, and of the subsequent rapid interplanetary propagation from the acceleration site to the observer. As was shown by Laitinen et al. (2003), the self-consistent modeling of the Alfven wave propagation and the solar wind expansion in case of no cascade term in the wave transport equation (a1 = 0) produces too small SEP mean free paths in the solar wind. From the other hand, Vainio et al. (2003a) have demonstrated that cascading can dramatically increase the values of the mean free path in the solar wind (in accordance with observations).

2.20. Bulk speeds of CR resonant with parallel plasma waves

2.20.1. Formation of the bulk speeds that are dependent on CR charge/mass and momentum

According to Vainio and Schlickeiser (1999) the quasi-linear interaction of CR particles with transverse parallel propagating plasma waves occurs via gyro resonance. To interact efficiently with a circularly polarized wave the particle must gyrate around the mean magnetic field in the same sense and with the same frequency as the electric field of the wave when viewed in the rest frame of the

particle's guiding center (GC frame). Augmented with the dispersion relations of the relevant wave modes, this condition determines the wave numbers and frequencies of the waves resonant with particles of a given type (charge/mass) and velocity. The intensity of the waves at these wave numbers, in turn, determines how fast the given particle is diffusing in momentum space. If the particle's guiding center moves much faster than the waves relative to the plasma, one may neglect the plasma-frame wave frequency in the (Doppler-shifted) GC-frame wave frequency and make the so called magneto-static approximation (e.g., Jokipii, 1966). This approximation, however, does not give correct results for particles with pitch-angles close to 90°. Since the description of particle scattering in this region determines the fundamental CR transport parameter, the spatial diffusion coefficient (Schlickeiser and Miller, 1998), one has to abandon the magneto-static approximation at least when computing this parameter from the assumed/observed spectrum of magnetic fluctuations. Vainio and Schlickeiser (1999) studied the effect of finite phase speeds of the waves on another transport coefficient, the bulk speed of the CR, which is the effective speed of the waves that scatter the CR particles. Dispersive waves, therefore, can give rise to bulk speeds that are dependent on CR charge/mass and momentum. It was also studied how this affects the scattering-center compression ratio in low Mach number parallel shock waves.

2.20.2. Dispersion relation and resonance condition

The dispersion relations of parallel transverse waves in a cold electron-proton plasma can be described according to Vainio and Schlickeiser (1999) with the equation (e.g., Steinacker and Miller, 1992):

QeQ p

where k is the wave number and © is the wave frequency,

mec mpC

are non-relativistic electron and proton gyro-frequencies ( qe and me are the electron charge and mass; qp and mp are the proton charge and mass; B is the background magnetic field magnitude), c is the speed of light, and

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