# Perpendicular Electrical Conductivity in the Topside Ionosphere Derived from Swarm Measurements
## 1 Introduction  
A thorough knowledge of the physical properties of the topside ionosphere is crucially important for understanding the dynamic processes observed therein. Such knowledge would not only make it possible to penetrate the secrets that govern, for example, the energy balance of the ionosphere but would also allow for the realisation of increasingly accurate models that would be functional, in the future, for the prediction of observed phenomena. These phenomena are of practical and intrinsically scientific interest, as they may be upstream in a chain of Space Weather effects in the near-Earth environment, such as influencing positioning and navigation systems [1; 2; 3], generating induced geoelectric fields and associated currents [4; 5], increasing the atmospheric drag, and, thus, affecting low-Earth-orbit satellites [6; 7]. Their knowledge may help to elaborate mitigation strategies aimed at preserving critical infrastructures [8]. One of the most debated and not yet fully unravelled aspects concerns the amplification and dissipation of ionospheric currents. The presence of the geomagnetic fieldintroduces an anisotropy with respect to which current systems can flow mainly parallel and/or perpendicular to the field lines. The amplification of field-aligned currents (FACs) and horizontal Pedersen and Hall currents has long been known to occur at high latitudes [9; 10; 11; 12; 13]. In the same way, significant currents have also been predicted and observed at low and mid latitudes, such as, for example, the equatorial eastward electrojet (EE) [14; 15], the westward counter-electrojet [16; 17; 18], other related flows driven by zonal winds [19; 20], and solar quiet (Sq) current vortexes [18] flowing in both the hemispheres and being connected by inter-hemispheric field-aligned currents (IHFACs) [21; 22]. In light of the studies conducted over the past sixty years, a physical quantity crucial for the characterisation of ionospheric currents is electrical conductivity, \(\sigma\). This quantity describes the ability of a medium, in this case the ionosphere, to conduct a current when an electric field is applied. The link between the electric field, \(\mathbf{E}\); current density, \(\mathbf{J}\); and \(\sigma\) is established by Ohm's law, which, in its generalised form (GOL), when using a single-fluid magnetohydrodynamic description of ionospheric plasma, accounts for electric and magnetic (\(\mathbf{B}\)) fields, pressure gradients, viscous stressed, and electron-ion Coulomb collisions. In the topside ionosphere, GOL can be written in the simple form [23]  
\[\mathbf{J}=\sigma(\mathbf{E}+\mathbf{u}\times\mathbf{B}), \tag{1}\]  
where \(\mathbf{u}\) is the velocity of the neutral wind. In a resistive medium, \(\sigma\) (or equivalently its inverse, resistivity \(\eta=1/\sigma\)) is also required to determine power density dissipated locally by a current, namely \(\mathbf{J}\cdot\mathbf{E}\), which corresponds to the rate of energy in a plasma volume converted into mechanical energy of the particles in the medium traversed due to collisions. It is therefore evident how the study of \(\sigma\) allows one to both characterise the main features of ionospheric currents and quantify their dissipation and the energy exchanges that regulate certain dynamic processes in the ionosphere. Several works in the literature have attempted to characterise \(\sigma\) in the Earth's ionosphere through both models and in situ and ground-based observations. The first theoretical works date back to about a century ago [24]. It was early recognised that \(\sigma\) depends on the direction of the electric field \(\mathbf{E}\) with respect to the geomagnetic field \(\mathbf{B}\)[25]. Parallel to \(\mathbf{B}\), the so-called parallel electrical conductivity, \(\sigma_{||}\), does not depend on \(\mathbf{B}\) itself.