# Perpendicular Electrical Conductivity in the Topside Ionosphere Derived from Swarm Measurements
## 2 Data and Methods
### Electrical Conductivity Perpendicular to the Geomagnetic Field in the Topside Ionosphere  
The Earth's magnetic field introduces an anisotropy in the dynamics of ionospheric plasma that is also reflected in an anisotropy of electrical conductivity. Specifically, in the direction parallel to the geomagnetic field, the electrical conductivity, \(\sigma_{||}\), is defined as [77; 78]  
\[\sigma_{||}=\frac{n_{e}q_{e}^{2}}{m_{e}v_{e}}+\sum_{i}\frac{n_{i}q_{i}^{2}}{m_ {i}v_{i}}, \tag{3}\]where \(q\) is the charge, \(n\) is the particle density, \(m\) is the particle mass, \(\
u\) is the collision frequency, and the subscripts \(e\) and \(i\) refer to electrons and ions, respectively. Thus, the first term in the right-hand side (RHS) of Equation (3) describes the contribution of electrons to conductivity, while the second term of RHS sums up the contribution of all ions (both positive and negative) to conductivity. At Swarm altitude (about 450-500 km above Earth), i.e., in the topside ionosphere, one can make some assumptions corroborated by observations [23; 26; 29; 55; 56; 57; 77; 79; 80; 81; 82; 83; 84]. Let us suppose that at those altitudes, positively charged particles are composed of singly ionised atoms, such that \(|q_{i}|=|q_{e}|=e\), and that the contribution of negative ions is negligible, so the sum in Equation (3) runs only over positive particles (subscript \(+\)). We further assume that the condition of quasi-neutrality of ionospheric plasma is realised, whereby \(n_{e}\simeq\sum_{+}n_{+}\). With these prescriptions, Equation (3) becomes  
\[\sigma_{||}=n_{e}e^{2}\Bigg{(}\frac{1}{m_{e}v_{e}}+\sum_{+}\frac{1}{m_{+}v_{+} }\Bigg{)}. \tag{4}\]  
We further assume that the dominant ionic species at Swarm altitude is O\({}^{+}\)[85; 86]. Thus, only one term of the sum in Equation (4) is to be considered, namely  
\[\sigma_{||}=n_{e}e^{2}\bigg{(}\frac{1}{m_{e}v_{e}}+\frac{1}{m_{+}v_{+}}\bigg{)}. \tag{5}\]  
In the direction perpendicular to the geomagnetic field and parallel to the electric field, the Pedersen electrical conductivity, \(\sigma_{P}\), is defined as [62; 24]  
\[\sigma_{P}=n_{e}e^{2}\Bigg{[}\frac{v_{e}}{m_{e}(\Omega_{e}^{2}+v_{e}^{2})}+ \frac{v_{+}}{m_{+}(\Omega_{+}^{2}+v_{+}^{2})}\Bigg{]}, \tag{6}\]  
where \(\Omega_{e}=eB/m_{e}\) and \(\Omega_{+}=eB/m_{+}\) are the particle gyro-frequencies for electrons and ions, respectively, and \(B\) is the geomagnetic field strength. In the direction perpendicular to both geomagnetic and electric fields, the Hall electrical conductivity, \(\sigma_{H}\), is defined as [78]  
\[\sigma_{H}=n_{e}e^{2}\Bigg{[}\frac{\Omega_{e}}{m_{e}(\Omega_{e}^{2}+v_{e}^{2} )}-\frac{\Omega_{+}}{m_{+}(\Omega_{+}^{2}+v_{+}^{2})}\Bigg{]}. \tag{7}\]  
In principle, the collision frequencies in Equations (5)-(7) include collisions of the indicated species with the other species that characterise the ionospheric plasma (in general, electrons, ions and neutrals), so \(v_{e}=v_{e+}+v_{e}+v_{en}\) and \(v_{+}=v_{++}+v_{+e}+v_{+n}\). Under the assumption that collisions of charged particles with neutrals at Swarm height can be neglected, we set \(v_{en}\simeq v_{+n}\simeq 0\). This hypothesis is supported by previous studies, both theoretical and observational. For example, [PERSON] et al. [81] modelled electron collision frequency between 50 and 500 km of altitude by taking advantage of both experimental and theoretical values. They found that electron-ion collisions dominate above \(\sim\)170 km, and this altitude depends on solar activity and different sunlight conditions. [PERSON] et al. [82] found that at \(\sim\)100 km of altitude, the effect of electron-neutral collisions is already negligible in the daytime due to the ionisation produced by solar flux. During the nighttime, electron-ion collisions dominate over electron-neutral collisions above \(\sim\)280 km at moderate and high latitudes and at little higher altitudes at low latitudes.