# Perpendicular Electrical Conductivity in the Topside Ionosphere Derived from Swarm Measurements
## 1 Introduction
If \(\mathbf{E}\) is applied perpendicular to \(\mathbf{B}\), the so-called Pedersen conductivity, \(\sigma_{P}\), drives currents parallel to \(\mathbf{E}\), while the so-called Hall conductivity, \(\sigma_{H}\), drives currents perpendicular to \(\mathbf{E}\)[26; 27]. Over the magnetic equator, a vertical Hall polarisation field in the E-region increases the so-called Cowling conductivity, \(\sigma_{C}=\sigma_{P}[1+(\sigma_{H}/\sigma_{P})^{2}]\)[28]. In the decades that followed, increasingly sophisticated conductivity models have made it possible to reproduce the main structures of FACs and horizontal currents using integrated conductance values for the latter, \(\Sigma_{P}=\int\sigma_{P}dh\), \(\Sigma_{H}=\int\sigma_{H}dh\), where \(dh\) runs between the bottom and top altitudes of the ionosphere [29; 30; 31; 32]. It must be said that the models of \(\sigma\), and thus of the currents, suffer from two main limitations: (1) theoretical models strictly depend on the physical hypotheses that underlie them; (2) empirical models or theoretical models driven by observations while taking into account the dependence on latitude, Magnetic Local Time, and geomagnetic activity were not able to capture the small-, short-, spatial-, and temporal-scale variations that characterise many of the phenomena in the topside ionosphere. The characterisation of \(\sigma\) and/or equivalent current systems also made use of observations using different types of instruments both from the ground and in flight. Electrical conductivity was in fact studied by means of measurements from magnetometers, ionosondes, and incoherent scattering radars [33; 31; 34; 35; 36; 37; 38]. All these measurements suffer from low spatial resolution and relatively long time cadence, preventing the detailed, statistical study of conductivity and current features and catching fast dynamics. Rocket measurements have also been used to conduct studies on the topic [39; 40; 41], but could not provide, for example, direct measurements of electron density that are necessary to compute \(\sigma\) as they do not remain within a certain region for a sufficient time during their flight to detect features with a good resolution. Critical issues related to low spatial resolution also emerge in the case of early studies conducted through direct and indirect measurements (such as the determination of electron density via the energy flux spectrum of precipitating electrons) with balloons and satellites [35; 40; 42; 43; 44; 45; 46; 47]. Over time, studies have been carried out using in situ satellite measurements in the F-region aimed at inferring ionospheric currents and related parameters at lower altitudes. For example, [PERSON] [48] developed an algorithm for determining high-latitude ionospheric currents at an altitude of 115 km from magnetic measurements made by using MAGSAT satellite [49]. Similarly, other authors [47; 50; 51; 52] determined the strength and location of ionospheric currents at a 110 km altitude by using either CHAMP [53] or Swarm [54] satellites. Among these, [PERSON] et al. [47] used Swarm measurements to evaluate \(\sigma_{\parallel}\), \(\sigma_{P}\), and \(\sigma_{H}\) in the E layer. In any case, almost all efforts have focused on the study of \(\sigma\) in the E layer of the ionosphere. Only a few works have characterised \(\sigma\) in the F-region through in situ satellite measurements, which provide important information and observational constraints that could improve the ionospheric models currently used. [PERSON] et al. [55; 56; 57] carried out statistical studies aimed at characterising \(\sigma_{\parallel}\) in the F-region by using in situ Langmuir probe data from the Swarm and CSES-01 [58] missions. They provided climatological high-resolution maps and studied the variation in \(\sigma_{\parallel}\) and its hemispheric asymmetries with magnetic latitude, Magnetic Local Time, local season, solar, and geomagnetic activity.