# Perpendicular Electrical Conductivity in the Topside Ionosphere Derived from Swarm Measurements
## 4 Discussion  
Electrical conductivity is a fundamental physical property of the ionosphere, as it represents the physical background that allows electric currents to be established and/or amplified when an electric field is applied. To our knowledge, no one has ever characterised the perpendicular electrical conductivities, \(\sigma_{P}\) and \(\sigma_{H}\), in the topside ionosphere via in situ measurements. This has primarily occurred for two reasons: (1) At those altitudes, the perpendicular conductivity is expected to be very small, several orders of magnitude smaller than the parallel conductivity (105); (2) these quantities have never been calculated locally but only in terms of their mean value integrated over a range of altitudes, i.e., the conductance. Nevertheless, local electrical conductivity is of utmost importance because it allows for characterising the local response of the ionospheric medium to the application of electric fields. Additionally, it enables one to compute the rate of dissipation of currents flowing locally. For example, [PERSON] et al. (2019) showed that in the topside ionosphere, the dissipation of FACs is non-zero and exhibits a well-defined structure. This could provide insights into magnetosphere-ionosphere coupling and the energetic budget of the ionosphere. Moreover, the dissipation of FACs was found to be related to the variation of electron temperatures at the same altitude. In addition to FACs, the understanding of \(\sigma_{P}\) and \(\sigma_{H}\) can be also important for studying the pressure-gradient currents, which are ubiquitous in the ionosphere and flow perpendicular to the geomagnetic field. For instance, [PERSON] et al. (2016) used Swarm A data to reconstruct the pattern of pressure-gradient currents, offering a more comprehensive understanding of the complex current systems present in the ionosphere. An important assumption made in computing \(\sigma_{P}\) and \(\sigma_{H}\) is that the dominant ion species at the Swarm altitude is O\({}^{+}\). This makes it possible (1) to calculate the collision  
Figure 13: Maps in QD-MLT coordinates and stereographic projection of Hall electrical conductivity, \(\sigma_{H}\), in the Northern (top row) and Southern (bottom row) Hemispheres during the first (left column) and third (right column) biennia, respectively. frequencies on which conductivity depends by modelling the ionospheric plasma as an electrically neutral and binary gas and (2) to use the mass of oxygen in the above equations by posing \(m_{+}\equiv m_{\text{O}^{+}}\). However, in some specific cases, this assumption might be challenged. For example, in winter nights during periods of very low solar activity, the density of O\({}^{+}\) ions may be comparable, with a density of H\({}^{+}\) ions at the upper transition height lowering down to about 500 km [107; 108; 109; 110; 111]. Nevertheless, we note the following. 1. The average altitude of Swarm A is 450 km, sensibly below the indicated upper transition height of 500 km. Thus, even in the specific case of winter night during very low solar activity, the dominant ion species at Swarm A altitude should still be O\({}^{+}\). 2. The specific case mentioned should only affect the third biennium of the data set used in this work [112; 113]. As explained in the text, the choice of selecting biennia guarantees an even sampling of seasons. Thus, winter nights in the third biennium cover only a fraction of the data set used to pursue our statistical study. Therefore, the specific case may be limiting for local rather than statistical studies. 3. Most important of all, the relevant features of perpendicular conductivity emerging from our study are mainly located on the day side, where O\({}^{+}\) is definitely the dominant species at the Swarm altitude. In light of these considerations, we can reasonably assert that the approximation of O\({}^{+}\) as the dominant ionic species at Swarm altitude is robust for the results obtained in our study.