J.A. le Roux, G.P. Zank Bartol Research Institute, University of Delaware, Newark, Delaware, USA H. Fichtner Institute for Theoretical Physics, Ruhr-University Bochum, Bochum, Germany
The current theoretical framework of non-linear diffusive shock acceleration traditionally used to model the formation of anomalous cosmic rays (ACRs) at the termination shock and its affect on the shock structure and position is ill suited for this purpose. The theory, based on the assumption of isotropic or near-isotropic particle distributions, does not deal with the highly anisotropic nature of pickup ion (PUI) reflection and initial acceleration at the shock before diffusive shock acceleration dominates. To address this problem, we adopt a time-dependent HD approach in which a one fluid description of the solar wind with PUI protons, under the dynamic influence of multiply reflected ion (MRI) accelerated PUI protons as the second and ACR protons as the third fluid, is followed. In the process we make use of the semi-analytical theory of MRI acceleration as presented by Zank et al.  to specify appropriate loss terms that describe self-consistently the effect of MRI acceleration on the termination shock structure and position. To close the set of HD equations, the ACR transport equation is solved with a self-consistently determined source of PUI protons based on a combination of PUI transport theory and MRI theory. The ACR radial diffusion coefficient is simulated theoretically on the basis of ACR diffusion tensor theory and a recent MHD transport model for solar wind turbulence. The results show the following: (i) Both MRI accelerated PUI protons and ACR protons might have a significant affect on the position and macrostructure of the termination shock. (ii) The modeled ACR spectrum at the shock has complex features reflecting the non-linearity of the diffusive acceleration process, for example. (iii) This unified approach yields reasonable upstream ACR modulated spectra in the light of Voyager observations.