From Fig. 2.23.2 can be seen that for isotropic pitch-angle scattering the general pattern is similar to that of the telegrapher's equation (compare with Fig. 2.23.1), but there are noticeable differences at the same time.
2.23.5. Dispersion relations for the cases with dominant helicity
Bieber et al. (1987) called attention to the possible role of a dominant helicity, which introduces an asymmetry into Dm. Fig. 2.23.3 shows the dispersion relations for a dominant helicity characterized by g = 0.5.
Fig. 2.23.3. Dispersion relations for a focusing geometry (L = 1), with c = + 0.5 (+) and c = - 0.5 (-) helicities. According to Kota (1999).
Kota (1999) notes that the imaginary part of the derivative dvJdk becomes finite at k = 0 in accord with the predictions of Eq. 2.23.2 for a non-zero value of A (Pauls et al., 1993).
2.23.6. Dispersion relations for focusing scattering
Adiabatic focusing becomes important when field lines diverge on a scale comparable with or smaller than the scattering mean free path. According to Kóta (1999) focusing appears in Eq. 2.23.8 through the function G(u). In a focusing geometry, Eq. 2.23.3 suggests that k can be obtained as k = -iLdvo¡dk at k = 0. Since the zeroes eigenfunction Fo is always constant at k = 0, Eq. 2.23.10 immediately leads to k=-vL^/ue~G^l(e~g^j . (2.23.11)
which is identical to the expression inferred by Bieber and Burger (1990) using a Born approximation. Bieber et al. (1987) pointed out that the combined effect of focusing and dominant helicity leads to charge dependence in k This effect is clearly demonstrated in the dispersion relations shown in Fig. 2.23.3 for a focusing length L = 1, and helicities o = 0.5 and o = - 0.5. Both the curvature of Re(vc) and the slopes of Im(vc) at k = 0 indicate different effective diffusion coefficients for the two different signs of helicity.
2.23.7. Dispersion relations for hemispherical scattering
A case of particular importance is the hemispherical scattering when particles are strongly scattered both in the ^ < 0 and ^ > 0 hemispheres but scattering through ^ = 0 is restricted. Such a case is described, for instance, by q ~ 1 or, in another formulation, by the introduction of two distinct levels f+ and f- for the two hemispheres. The equations for f± have been developed and discussed in detail by Isenberg (1997) and Schwadron (1998). Fig. 2.23.4 shows the dispersion relations for q = 0.9 for a rectilinear case, without focusing (left panels), and for a focusing scenario (L = 1).
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