Diffuse Ion Spatial Profiles Along the Magnetic Field Line and Along the Bow Shock Normal: a Comparative Study at the Earth's Quasi-Parallel Bow Shock

The DSA (diffusive shock acceleration) or first-order Fermi acceleration has been studied for more than four decades, yet many details are still unclear. We analyze three individual upstream ion events under various solar wind plasma conditions in order to compare the spatial evolution of partial ion densities in front of the Earth's bow shock along the magnetic field line and along the bow shock normal. For this purpose we use observations of energetic ions upstream of the Earth's quasi-parallel bow shock by Cluster at times of large inter-spacecraft separation distance ( 1-1.5 Re). Using a bow shock model we determine the distance of SC1 and SC3 from the bow shock surface parallel to the magnetic field and in the bow shock normal direction. The CIS-HIA instrument onboard Cluster provides partial energetic ion densities in 4 energy channels between 10 and 32 keV. Using the differences of the partial energetic ion densities observed on SC1 and SC3 and the distances of the spacecraft from the bow shock (i.e., along the magnetic field and along the bow shock normal, respectively,) we determine the spatial gradient of partial energetic ion densities at various distances from the bow shock. The gradient in all energy channels decreases exponentially with distance and the e-folding distance of the gradients depends approximately linearly on energy. We demonstrate for the first time that at each ion energy the e-folding distance along the bow shock normal direction can be considerably smaller compared to the e-folding distance along the magnetic field line.