2019ApJ...881..123W

A steady-state, analytical model for the acceleration of energetic charged particles owing to cosmic ray viscosity and fluid shear in relativistic jets is explored. The model extends the work of Webb et al. to alternative forms of the mean scattering time τ (r, p). The flow velocity profile u = u(r) e_z of the jet is independent of distance z along the axis of the jet. u(r) is a monotonic decreasing function of cylindrical radius r about the jet axis. The scattering time τ(r,p)=τ₀(p/p₀)^α^ is a power-law function of the particle momentum p as measured in the fluid frame. The solutions are eigenfunction expansions involving J₀ Bessel functions and power-law functions of p. The solutions are used to discuss particle acceleration in shear flows in jets, and to determine if high-energy cosmic rays (i.e., with kinetic energies T ∼ EeV) can be accelerated to these energies in candidate AGN jet sources. Green's function solutions involving J_m Bessel functions and more general boundary conditions at the outer edge of the jet are described. We use a time-dependent model to assess the effects of cosmic ray inertia in limiting the upper particle momentum p_max(t) due to cosmic ray viscosity and from second-order Fermi acceleration due to Alfven waves. The model describes the competition between energy gains due to momentum space diffusion and energy losses of the particles due to synchrotron losses or inverse Compton losses.