Reabsorption of self-generated turbulent energy by pick-up protons in the outer heliosphere

It is well known that newly created pick-up ions are first injected into a toroidal distribution in the velocity space which comoves with the solar wind. This distribution is, however, unstable with respect to excitation of resonant MHD waves, and thus by wave-driving some specific amount of the initial kinetic energy of pick-up ions is handed over to the ambient wave fields, while the pick-up ion velocity distribution thereby is rearranged. In this paper we describe the developement of the wavenumber spectrum of pick-up proton-generated turbulence and the pick-up proton energy spectrum resulting from resonant reabsorption of this turbulence, by solving simultaneously the coupled system of two differential equations for the spectral turbulence power and the pick-up proton spectral intensity. We show that the majority of pick-up generated wave energy in the range of wavenumbers above the wavenumber at which the waves are produced is reabsorbed by the pick-up protons themselves.