to determine the diffusive current S. The diffusion approximation assumes that the scattering is strong enough to drive the distribution function to a near isotropic state, and that the effective scattering time is much shorter than the time scale for the evolution of Fo. Using Eq. 2.27.6 and Eq. 2.27.7 it follows that the diffusive current has the form:
where eB = ez is the unit vector along Bo , and
The expressions in Eq. 2.27.9 for k// , kl and ka have the same form as in Forman et al. (1974).
2.27.4. Evaluation of the Green function
Introducing the Laplace-Fourier transform:
~ M M d 3r f (k, p, s)= J dt J—^ exp(- st - ik • r )h (r, p, t ) (2.27.10)
the BGK Boltzmann Eq. 2.27.1 reduces to the ordinary differential equation:
f - (s + ik • v + v±h = [- f (k,p,0) + (v// - v±)h - v//Fo], (2.27.11)
where v// = 1/t//, Vl = 1/T and f (k,p,0) is the Fourier transform of the initial data f (r, p,0). For Dirac-delta function for initial data, with f (r,p,0) = AS(r - ro - Vo - </>o ) (2.27.12)
it will be
Cosmic Ray Propagation in Space Plasmas 267
f (k, p,0) = -Ar exp(- ik • rG - Ho )(t -to) • (2.27.13)
Using Eq. 2.27.13 as the source term in Eq. 2.27.11, and integrating Eq. 2.27.11 yields the solution:
f = (OI(t, 0))-1 [(//Fo - (v// - v± ) )(2n, 0)S - L(t, 0))] + Q , (2.27.14) where
Q = (o,0o)exp3-ik • roK"-^)/1 -H(t-t)/ (2.27.15)
Was this article helpful?