One way to explore these approximations is to look at the dispersion relations they yield. Kota (1999) supposes to consider the solution for f (z,p, t) as a sum of eigenfunctions F (k ,p)exp(ikz - vt). Eq. 2.23.2 then transforms into av2 + iAkv - v/t = -v2(k2 - ik/L)/3 . (2.23.3)

2.23.2. Dispersion relations for diffusion and telegrapher's equations

The resulting dispersion relations v(k), evaluated in Kota (1999) from Eq.

2.23.3, are shown in Fig. 2.23.1. Here there is covered only the half plane, obviously Re(v) is an even and Im(v) is an odd function of k. To use dimensionless quantities, Kota (1999) takes the particle speed, v, and the scattering time, t, to be unity (v = t = 1). These dispersion relations of the approximations can then be compared to those obtained from the full Eq. 2.23.1. In general, the full equation has an infinite number of eigenfunctions and eigenvalues (see Earl, 1974). In Kota (1999) it was focused on the two lowest eigenvalues which are the most important in determining the evolution of the particle density and anisotropy. Clearly the value of a appears in v1 (k = 0), whilst a non-zero A would appear as a non-zero (imaginary) value of dvJdk at k = 0. The dispersion relations for the 'billiard-ball' scattering were first given by Fedorov and Shakhov (1993). 'Hemispherical' scattering was considered by Kota (1994).

Fig. 2.23.1. Dispersion relations for the diffusion (dotted line) and the telegrapher's equation (solid lines). From Kota (1999).

Fig. 2.23.1. Dispersion relations for the diffusion (dotted line) and the telegrapher's equation (solid lines). From Kota (1999).

For the sake of simplicity Kota (1999) assume that both Dm and the focusing length L are independent of location, which corresponds to an exponentially diverging geometry (according to Earl, 1981). The pitch-angle scattering coefficient Dm is allowed to be arbitrary function of ^, so it was assumed that

In this formulation o accounts for helicity (Bieber et al., 1987) and t represents the effective scattering time so that the resulting spatial diffusion coefficient along the magnetic field lines will be

according to Hasselman and Wibberenz (1970). The Fokker-Planck equation including focusing can then be rewritten as f + vJf = - vtj] f + jL d* = ,Gf ,-0 (f)Dmf ], (2.23.6)

where dG/ dß = A(ß)/ L (Kunstmann, 1979). In terms of the eigenfunctions F (k ,ß) reads as vF + ikvßF - eGß)ß e~G ß)DßßdF ^

The eigenvalues v = Vj(k) (j = 0,1,2,...) are complex in general. Slow spatial variation corresponds to k ~ 0. At k = 0 the lowest eigenvalue is always vo = 0, corresponding to the completely homogeneous and isotropic solution, and all the other eigenvalues are real and the eigenfunctions are identical with the eigenfunctions of the scattering operator (Earl, 1974). Moving from k to k + Sk the eigenfunctions and eigenvalues change to F + SF and v + Sv, yielding vSF + ikvjSF - eG(J-j{e~G(JDJJSJ\ = SvF + iSkvjF . (2.23.8)

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