# 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
Thus, at a Swarm altitude and especially at middle and high latitudes, we can safely approximate as negligible the effects of collisions of charged particles with neutrals [55; 56; 57; 77]. The classical calculation of the frequency of Coulomb collisions between plasma particles is based on the idea that a particle with thermal velocity interacts meaningfully only with particles contained in its Debye sphere, simultaneously undergoing many weak collisions. For a totally ionised binary and neutral gas, the electron-ion collision frequency, \(\
u_{e+}\), is given by [78; 79]  
\[\
u_{e+}=n_{e}T_{e}^{-3/2}\bigg{[}34+4.18\log\bigg{(}\frac{T_{e}^{3}}{n_{e}} \bigg{)}\bigg{]}, \tag{8}\]  
where \(n_{e}\) and \(T_{e}\) are electron density and temperature, respectively. Equation (8) exploits the inequality \(m_{+}\gg m_{e}\) and the fact that electrons, which have much smaller inertia, are the bullets, while ions are the targets. The calculation of the electron-electron collision frequency is greatly complicated by the fact that the target cannot be considered relatively immobile. However, in the frame of reference of the centre of mass of the two particles, it is still possible to obtain a result that, with good approximation, gives \(\
u_{ee}\sim\
u_{e+}\)[62]. This finds justification in the fact that the Coulomb force re-sent by the bullet is the same, apart from the sign, with respect to the case where the target is a singly ionised ion. Note that this is strictly valid as, in our case, ions are singly ionised atoms, so each of them has a charge \(+e\), while in general, the ratio \(\
u_{ee}/\
u_{e+}\) is like the inverse square of the particle charge numbers. Thus, in our case, it follows that \(\
u_{e}\simeq 2\
u_{e+}\). In addition, as collisions with neutrals can be neglected, the conservation of collisional momentum transfer between electrons and ions allows writing [62]  
\[n_{e}m_{e}\
u_{e+}=n_{+}m_{+}\
u_{+}, \tag{9}\]  
which, imposing the plasma neutrality condition, brings us to  
\[\
u_{+e}=\frac{m_{e}}{m_{+}}\
u_{e+}. \tag{10}\]  
On the other hand, it can be demonstrated that the ion-ion collision frequency is [62]:  
\[\
u_{++}\sim\bigg{(}\frac{m_{e}}{m_{+}}\bigg{)}^{1/2}\
u_{e+}, \tag{11}\]  
which leads to  
\[\frac{\
u_{+e}}{\
u_{++}}\sim\bigg{(}\frac{m_{e}}{m_{+}}\bigg{)}^{1/2}\ll 1. \tag{12}\]  
Thus, it follows that \(\
u_{+}\simeq\
u_{++}\), and Equations (6) and (7) can be re-written, respectively, as  
\[\sigma_{||}=n_{e}e^{2}\Bigg{(}\frac{1}{m_{e}v_{e}}+\sum_{+}\frac{1}{m_{+}v_{+} }\Bigg{)}. \tag{4}\]0  
\[\sigma_{||}=n_{e}e^{2}\Bigg{(}\frac{1}{m_{e}v_{e}}+\sum_{+}\frac{1}{m_{+}v_{+} }\Bigg{)}. \tag{4}\]1  
By making use of Equation (11), Equations (13) and (14) can be written as follows:  
\[\sigma_{||}=n_{e}e^{2}\Bigg{(}\frac{1}{m_{e}v_{e}}+\sum_{+}\frac{1}{m_{+}v_{+} }\Bigg{)}. \tag{4}\]2  
\[\sigma_{||}=n_{e}e^{2}\Bigg{(}\frac{1}{m_{e}v_{e}}+\sum_{+}\frac{1}{m_{+}v_{+} }\Bigg{)}. \tag{4}\]3  
We recall, from previous works, e.g., [55], that  
\[\sigma_{||}=n_{e}e^{2}\Bigg{(}\frac{1}{m_{e}v_{e}}+\sum_{+}\frac{1}{m_{+}v_{+} }\Bigg{)}. \tag{4}\]4  
It is interesting to notice that according to Equations (15)-(17), \(\sigma_{P}\) and \(\sigma_{H}\) depend on both plasma parameters \(n_{e}\) and \(T_{e}\) (through the collision frequency \(\
u_{e+}\)) and the magnetic field strength. On the other hand, \(\sigma_{||}\) has a strong dependence on \(T_{e}\). All physical quantities that enter the definition of the three conductivities are measured simultaneously by the instruments on board the Swarm mission. This allows us to reliably obtain conductivity at Swarm altitude by applying the equations above. To be consistent with previously published works, we adopt cgs units, according to which conductivity is in [s\({}^{-1}\)] units. We drew maps of electrical conductivity with 1\({}^{\circ}\times\) 1\({}^{\circ}\) binning in QD magnetic latitude-MLT coordinates. Notice that a 1\({}^{\circ}\) magnetic longitude corresponds to 4 min in MLT.