The Alfven-Falthammar formula for the parallel E-field and its analogue in downward auroral-current regions

The authors give an alternative derivation of the Alfven-Falthammar formula for a positive parallel E-field in upward auroral-current regions and its analogue for a negative parallel E-field in downward auroral-current regions. These formulas give new insight into the physics of the Birkeland current system. They find that for downward auroral-current regions, the velocity-space anisotropy in the ion distribution function is such that half the ion energy perpendicular to the magnetic field is greater than the ion energy parallel to the magnetic field. This is just opposite to the case for upward auroral-current regions. These results are compared to recent particle-in cell simulations and FREJA satellite data.