Diffusive-Compression Acceleration and Turbulent Diffusion of Cosmic Rays in Quasi-Periodic and Turbulent Flows

Multiple scale perturbation methods are used to study the transport and acceleration of energetic charged particles in quasi-periodic, fluid velocity structures in one, two, or three space dimensions, with spatial period l _u, where l_u is much less than the diffusion scale length l_d = κ₀/u₀ and κ₀ and u₀ are characteristic values of the energetic particle diffusion coefficients and fluid speed, respectively. The particle diffusion tensor K is also allowed to vary periodically on the scale l_u. In the case in which the perturbation parameter ε = l_u/l_d = u ₀l_u is small (0 < ≪ 1), the long space and time behavior of the energetic particle distribution function 〈 f 〉 at lowest order is shown to satisfy a modified Fokker-Planck equation. This equation arises from compatibility conditions imposed on the perturbation equations in order to obtain a consistent perturbation expansion that is free of secular terms. The analysis shows that the particles are accelerated stochastically on the large scale as a result of the divergence ∇ δu of the background fluid velocity perturbation δu. The net acceleration of the particles due to the velocity variations can be described in part by a second-order Fermi-like momentum space diffusion term in the long-scale transport equation obtained by averaging over the short-scale variations. The momentum space diffusion coefficient DT describing the effect depends on the two-point correlation of the fluid velocity divergence ∇ δu at different points in the flow. There is also a further energization term in the long-scale transport equation, corresponding to the work done by the scattering center fluid against the differential cosmic-ray pressure gradient that is modified as a result of the short-scale variations. The convective particle streaming is also modified as a result of the short-scale variations. The analysis shows that the effective spatial diffusion tensor for low-energy particles can be significantly modified as a result of turbulent diffusion, whereas higher energy particles with much larger diffusion tensor elements are not significantly affected by turbulent diffusion. Averaging over a random ensemble of short-scale, quasi-periodic velocity structures generalizes the turbulent transport coefficients obtained by previous authors.