General solution for calculating polarization electric fields in the auroral ionosphere and application examples

We devise an approach to calculate the polarization electric field in the ionosphere, when the ionospheric conductances, the primary (modeled) or the total (measured) electric field, and the Cowling efficiency are given. In contrast to previous studies, our approach is a general solution which is not limited to specific geometrical setups, and all parameters may have any kind of spatial dependence. The solution technique is based on spherical elementary current (vector) systems (SECS). This way, we avoid the need to specify explicit boundary conditions for the searched polarization electric field of its potential which would be required if the problem was solved in a differential equation approach. Instead, we solve an algebraic matrix equation, and the implicit boundary condition that the divergence of the polarization electric field vanishes outside our analysis area is sufficient. In order to illustrate our theory, we then apply it to two simple models of auroral electrodynamic situations, the first being a mesoscale strong conductance enhancement in the early morning sector within a relatively weak southward primary electric field, and a morning sector auroral arc with only a weak conductance enhancement, but a large southward primary electric field at the poleward flank of the arc. While the significance of the polarization electric field for maximum Cowling efficiency is large for the first case, it is rather minor for the second one. Both models show that the polarization electric field effect may not only change the magnitude of the current systems but also their overall geometry. Furthermore, the polarization electric field may extend into regions where the primary electric field is small, thus even dominating the total electric field in these regions. For the first model case, the total Joule heating integrated over the analysis area decreases by a factor of about 4 for maximum Cowling efficiency as compared to the case of vanishing Cowling efficiency. Furthermore, for this case the resulting total electric field structurally shows a strong resemblance to that frequently observed during auroral omega band events.