Focused acceleration of cosmic-ray particles in non-uniform magnetic fields

The Fokker–Planck equation for cosmic-ray particles in a spatially varying guide magnetic field in a turbulent plasma is analyzed. An expression is derived for the mean rate of change of particle momentum, caused by the effect of adiabatic focusing in a non-uniform guide field. Results of an earlier diffusion-limit analysis are confirmed, and the physical picture is clarified by working directly with the Fokker–Planck equation. A distributed first-order Fermi acceleration mechanism is identified, which can be termed focused acceleration. If the forward and backward-propagating waves have equal polarizations, focused acceleration operates when the net cross helicity of an Alfvenic slab turbulence is either negative in a diverging guide field or positive in a converging guide field. It is suggested that focused acceleration can contribute to the formation of the anomalous cosmic-ray spectrum at the heliospheric termination shock