Fast charged-particle acceleration in incompressible flows

The standard paradigm for the transport and acceleration of cosmic rays and other fast charged particles is based on the well-established Parker equation. The Parker equation can be demonstrated to be a good approximation if the particle distribution function is nearly isotropic in pitch angle, even in the presence of discontinuities in the flow such as shocks and current sheets. However, there are many situations, such as MHD reconnection where the fluid flow velocity is nearly incompressible. In these cases the Parker equation predicts little or no particle acceleration since the energy change is proportional to the divergence of the flow velocity. This energy-change rate is independent of spatial or temporal scales, for isotropic angular distributions. One approach to accelerating particles in nearly incompressible flows is to extend the Parker equation by invoking small-scale propgating waves - giving rise to the venerable 2nd-order Fermi acceleration. A second possibility is to invoke large pitch-angle anisotropies. A third possibility is considered in this paper. I examine the effects of fluid flow acceleration and shear, both of which also accelerate charged particles. The resulting transport equation is Parker's equation augmented by terms proportional to the fluid acceleration and shear, which can be non-zero in incompressible flow. These new terms are of second order in the fluid flow speed U, and hence are generally small. Nonetheless, they will be important in considerations of charged-particle acceleration in nearly incompressible flows such as reconnection. A synthesized divergence-free fluid velocity U(x,y chosen to be similar to that found in reconnection is used to illustrate the acceleration. The acceleration rate is estimated for the inner heliosphere and shown to be greater than the adiabatic cooling rate in the expanding solar wind.