A [1 1 Tt

where the angular brackets in Eq. 2.27.27 denote an average over p. For the case of isotropic scattering ( =T±),Eq. 2.27.27 simplifies to

2.27.5. Long-scale, large-time asymptotes

From Fedorov et al. (1995), Kota (1994) and Webb et al. (2000), the long time asymptotes for Fo can be obtained by investigating the dispersion equation

associated with the singular eigen-solutions of Eq. 2.27.27. In particular, the diffusive behavior of the solution follows from the large space-scale ( k ^ 0 ) and long time ( 5 ^ 0) behavior of Eq. 2.27.29). For example, consider the case of isotropic scattering ( t// = ti ) for which D(k, s) = 1 - (a) = 0 is the singular manifold. Using the expansion of the Bessel functions in Eq. 2.27.25 for |k±rg sin^ << 1, Webb et al. (2001) obtain iv// X

Using Eq. 2.27.30 Webb et al. (2001) find a —„J 11+1 (ki/ k// I ^

for the approximate value of a over pitch-angle at long wavelengths, where b = (Vg)-1, h =

From Eq. 2.27.31 and Eq. 2.27.32 the dispersion equation D(k, s ) = 1 - = 0 for |«T << 1 and krg << 1 has the approximate solution s = -

where k// and k± are the parallel and perpendicular diffusion coefficients in Eq. 2.27.9 for T/ = t±=t . Eq.. 2.27.33 is the dispersion equation for the diffusion equation obtained from Eq-s 2.27.5, 2.27.8 and 2.27.9, but with no drift terms, since the background state is uniform.

On the other hand, if k|/k2 << 1 the term I2 can be dropped in Eq. 2.27.31, and the dispersion Eq. 2.27.29 has the approximate solution:

The latter dispersion equation is equivalent to the equation:

In the space-time domain, Eq. 2.27.35 becomes the telegraph equation of Gombosi et al. (1993). Clearly, to obtain an equivalent telegraph equation including perpendicular diffusion, one needs to retain terms o(k4) in Eq. 2.27.30.

2.27.6. Pitch-angle evolution and perpendicular diffusion

It is instructive to consider the integral Eq. 2.27.20 under the assumption that |k±rg| << 1, so that the approximation described by Eq. 2.27.30 applies for a. Eq. 2.27.30 can be re-written in the form:

Using the usual Fourier space map

dt and using the approximation described by Eq. 2.27.30 for a, Eq. 2.27.36 reduces to the approximate, integro-differential evolution equation dj° + v f _ \ ( + vjz + vi)in2flV2= { _ ) + s_ifQ^ dt 2 (+ vjz + v±)2 + Q2 1 /A o Jo} //

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