Diffusion of energetic particles in focusing fields

We address the modification of the effective spatial diffusive coefficient of energetic charged particles in magnetic field configurations where the divergence of the field lines and the consequent weakening of the field strength lead to adiabatic focusing. Particles propagating along the magnetic field lines and undergoing pitch angle scattering and adiabatic focusing are considered. In the presence of significant focusing the conditions for the applicability of a diffusive description are not strictly valid, thus focusing modifies the effective parallel diffusion coefficient kappa(parallel to). We calculate the appropriate spatial diffusion coefficients from a method based on the use of adjoint Green functions. The correspondence between this method and the method based on the velocity correlation function < v(i) (0) v(j) (t) > developed by Kubo [1957] is discussed. We target the modulation of galactic and anomalous cosmic rays, which can be best approximated by the assumption of a constant, or slowly varying, spatial gradients, for which the method of adjoint Green functions is most suitable. We show that this assumption leads to an effective kappa(parallel to) identical to that derived by Bieber and Burger [1990] from a Born approximation. The derivation also results in a nonsteady analytical solution to the Fokker-Planck equation, which describes a distribution of constant spatial gradient moving at a constant speed. We also discuss the case of hemispherical scattering, when scattering is effective within each of the mu < 0 and mu > 0 hemispheres (where mu is the cosine of pitch angle) but is restricted between the two hemispheres. A refinement to the hemispherical equation of Isenberg [1997] and Schwadron [1998] is suggested.