Perpendicular Electrical Conductivity in the Topside Ionosphere Derived from Swarm Measurements

Abstract

The study of the physical properties of the topside ionosphere is fundamental to investigating the energy balance of the ionosphere and developing accurate models to predict relevant phenomena, which are often at the root of Space Weather effects in the near-Earth environment. One of the most important physical parameters characterising the ionospheric medium is electrical conductivity, which is crucial for the onset and amplification of ionospheric currents and for calculating the power density dissipated by such currents. We characterise, for the first time, electrical conductivity in the direction perpendicular to the geomagnetic field, namely Pedersen and Hall conductivities, in the topside ionosphere at an altitude of about 450 km. For this purpose, we use eight years of in situ simultaneous measurements of electron density, electron temperature and geomagnetic field strength acquired by the Swarm A satellite. We present global statistical maps of perpendicular electrical conductivity and study their variations depending on magnetic latitude and local time, seasons, and solar activity. Our findings indicate that the most prominent features of perpendicular electrical conductivity are located at low latitudes and are probably driven by the complex dynamics of the Equatorial Ionisation Anomaly. At higher latitudes, perpendicular conductivity is a few orders of magnitude lower than that at low latitudes. Nevertheless, conductivity features are modulated by solar activity and seasonal variations at all latitudes.

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 field introduces 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 (EEJ) [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]

$$\mathbf{J}=\sigma (\mathbf{E}+\mathbf{u}\times \mathbf{B}),$$

(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 $\mathbf{J}·\mathbf{E}$, 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, ${\sigma }_{\left|\right|}$, does not depend on $\mathbf{B}$ itself. 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 [31,33,34,35,36,37,38]. 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, Olsen [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, Amm et al. [47] used Swarm measurements to evaluate ${\sigma }_{\left|\right|}$, ${\sigma }_P$, 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. Giannattasio et al. [55,56,57] carried out statistical studies aimed at characterising ${\sigma }_{\left|\right|}$ 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 ${\sigma }_{\left|\right|}$ and its hemispheric asymmetries with magnetic latitude, Magnetic Local Time, local season, solar, and geomagnetic activity. Their results suggested that particle precipitation is responsible for the enhancement of ${\sigma }_{\left|\right|}$ in the cusp region and in correspondence with the nocturnal main trough. Finally, Giannattasio et al. [59] quantified the power density dissipated by FACs, namely $\mathbf{J}·\mathbf{E}$, taking advantage of local calculations of $\mathbf{J}$ and ${\sigma }_{\left|\right|}$ with Swarm data.

To our knowledge, no one has ever characterised the electrical conductivities perpendicular to the magnetic field, ${\sigma }_P$ and ${\sigma }_H$, in the topside ionosphere via in situ measurements. This is mainly due to their lower magnitude compared to the parallel conductivity in the topside ionosphere. Nevertheless, although small, ${\sigma }_P$ and ${\sigma }_H$ are not null. Therefore, the application of intense electric fields, which were actually observed at those altitudes [60,61] in the direction perpendicular to the geomagnetic field, may lead to the onset or amplification of perpendicular ionospheric currents. The intensity and direction of these currents are influenced by the local structure of conductivity. This is even clearer if one rewrites Equation (1) for the case of a three-dimensional anisotropic medium characterised by the conductivity tensor, $\underline{\sigma }$, as follows [62]:

$$\mathbf{J}=\underline{\sigma }{\mathbf{E}}^{\prime }=\left(\begin{array}{ccc}{\sigma }_P& -{\sigma }_H& 0\\ {\sigma }_H& {\sigma }_P& 0\\ 0& 0& {\sigma }_{\left|\right|}\end{array}\right){\mathbf{E}}^{\prime }.$$

(2)

In Equation (2), ${\mathbf{E}}^{\prime }$ is the electric field in the plasma frame of reference with components $E_i^{\prime }=E_i+{(\mathbf{u}\times \mathbf{B})}_i$, and we assume that $\mathbf{B}$ is directed along the versor $\widehat{\mathbf{z}}$, the component of $\mathbf{E}$ perpendicular to $\mathbf{B}$ is directed along $\widehat{\mathbf{x}}$, and the versor $\widehat{\mathbf{y}}$ is perpendicular to both $\mathbf{B}$ and $\mathbf{E}$ in a left-handed Cartesian coordinate system. Therefore, the knowledge of the conductivity tensor is crucial for reconstructing ionospheric currents, whatever their intensity, and possibly quantifying their dissipation even at Swarm altitude, i.e., between 450 and 500 km.

In this paper, we extend our previous works [55,56] and, for the first time, compute ${\sigma }_P$ and ${\sigma }_H$ in the topside ionosphere from in situ satellite measurements, which is preparatory to an empirical model for $\underline{\sigma }$. This paper is organised as follows: In Section 2 we describe the data set used and the methods employed for the calculation of ${\sigma }_P$ and ${\sigma }_H$. Section 3 is devoted to the description of results, which are discussed in Section 4 considering the previous literature. Finally, in Section 5, we summarise the key points of the work and come to conclusions by indicating future perspectives.

2. Data and Methods

2.1. Swarm Observations and Preliminary Analysis

We used eight years of Level 1b data acquired at 1 Hz sampling by both the Langmuir probes (LPs) of the Electric Field Instrument (EFI, [63]) and the Absolute Scalar Magnetometer (ASM) instruments on board the European Space Agency’s (ESA’s) Swarm A satellite [54]. All data span the time interval from 1 April 2014 to 31 March 2022, just before the occurrence of the manoeuvres in April 2022 that substantially changed the satellite’s altitude. During the selected period, the satellite flew in a low and nearly circular orbit and a polar orbit with an inclination of ∼87.4° and an average altitude of ∼450 km. Due to its orbit, in the selected period, the Swarm A satellite took about 133 days to cover all local times. Thus, the selected eight-year time window ensures adequate coverage of the globe with a robust number of observations at each local time within a 24 h day, even when parcelling out the data by making selections, for instance, on season or solar activity. Level 1b data were downloaded from the LATEST_BASELINES folder of the ESA dissemination server. Specifically, 1 Hz Langmuir probe data are those identified by the label “EFIxLPI” and are obtained from data measured at 2 Hz in harmonic mode [64], (which are identified by the label “EFIx_LP”) by interpolation at full UTC seconds. LP data have been the subject of several calibration and validation studies aimed at understanding the effectiveness of comparing and inter-calibrating data from different satellites [65,66,67,68,69,70,71,72,73]. These studies used for the comparison of incoherent scatter radars, GPS radio occultation, and ionosonde data and pointed out, for example, that LP data generally underestimate values of electron density, while they tend to overestimate observations during low solar activity, particularly at nighttime. Differently, LP electron temperature values show an overall overestimation, particularly at low latitudes. However, we decided to use the original Level 1b data to be consistent with previous work by the same authors aimed at studying electrical conductivity in the topside ionosphere using Swarm’s LP data. Level 1b provides UTC time, the position of the satellite in geographical latitude and longitude, and in situ electron density and temperature observations. The latter were filtered out based on the quality flags provided by the mission team. Swarm LP observations recorded in High-Gain mode are those considered as ‘nominal’ and are recommended for scientific purposes. According to the Swarm’s product definition document (Swarm L1b Product Definition, 2018; available online: https://earth.esa.int/eogateway/documents/20142/37627/swarm-level-1b-product-definition-specification.pdf, accessed on 23 August 2024),we selected nominal LP observations with the following choice of the flags—Flag_LP=1 and Flags_Te or Flags_Ne parameters— equal to 10, 19, or 20. The remaining part was discarded as ‘not nominal’ according to the Swarm’s product definition document. Data missed or discarded after the quality flags check were replaced by not-a-number (NaN) values to guarantee the continuity of the time series. This procedure allowed us to consider only high-gain data from the Langmuir probes. Successively, magnetic data were prepared. We used those identified by the label ‘MAGx_LR’ available in the LATEST_BASELINES folder as conducted for LP data. Then, for UTC time, geographical latitude and longitude of the selected LP data, we picked the corresponding value of the total intensity of the geomagnetic field, namely B, as provided by Swarm A ASM.

We used the non-orthogonal Quasi-Dipole (QD) system of magnetic coordinates [74,75,76], as is common practice when studying processes that are determined or crucially influenced by the geomagnetic field. The position of the Sun was accounted for by using Magnetic Local Time (MLT) instead of UTC time. For this purpose, we used a procedure based on the work of Emmert et al. [75], which essentially works in two steps: (1) it firstly transforms geographical coordinates and satellite altitude to geodetic coordinates; then, (2) it performs the transformation from geodetic coordinates to QD-MLT coordinates [75]. This preliminary analysis allowed us to grid data in bins of QD magnetic latitude versus MLT.

2.2. 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, ${\sigma }_{\left|\right|}$, is defined as [77,78]

$${\sigma }_{\left|\right|}={\displaystyle \frac{n_e{q}_e^2}{m_e{\nu }_e}}+\sum _i{\displaystyle \frac{n_i{q}_i^2}{m_i{\nu }_i}},$$

(3)

where q is the charge, n is the particle density, m is the particle mass, $\nu$ 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

$${\sigma }_{\left|\right|}=n_e{e}^2\left({\displaystyle \frac{1}{m_e{\nu }_e}}+\sum _{+}{\displaystyle \frac{1}{m_{+}{\nu }_{+}}}\right).$$

(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

$${\sigma }_{\left|\right|}=n_e{e}^2\left({\displaystyle \frac{1}{m_e{\nu }_e}}+{\displaystyle \frac{1}{m_{+}{\nu }_{+}}}\right).$$

(5)

In the direction perpendicular to the geomagnetic field and parallel to the electric field, the Pedersen electrical conductivity, ${\sigma }_P$, is defined as [24,62]

$${\sigma }_P=n_e{e}^2\left[{\displaystyle \frac{{\nu }_e}{m_e({\Omega }_e^2+{\nu }_e^2)}}+{\displaystyle \frac{{\nu }_{+}}{m_{+}({\Omega }_{+}^2+{\nu }_{+}^2)}}\right],$$

(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]

$${\sigma }_H=n_e{e}^2\left[{\displaystyle \frac{{\Omega }_e}{m_e({\Omega }_e^2+{\nu }_e^2)}}-{\displaystyle \frac{{\Omega }_{+}}{m_{+}({\Omega }_{+}^2+{\nu }_{+}^2)}}\right].$$

(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 ${\nu }_e={\nu }_{e+}+{\nu }_{ee}+{\nu }_{en}$ and ${\nu }_{+}={\nu }_{++}+{\nu }_{+e}+{\nu }_{+n}$. Under the assumption that collisions of charged particles with neutrals at Swarm height can be neglected, we set ${\nu }_{en}\simeq {\nu }_{+n}\simeq 0$. This hypothesis is supported by previous studies, both theoretical and observational. For example, Aggarwal 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 ∼170 km, and this altitude depends on solar activity and different sunlight conditions. Vickrey et al. [82] found that at ∼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 ∼280 km at moderate and high latitudes and at little higher altitudes at low latitudes. 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, ${\nu }_{e+}$, is given by [78,79]

$${\nu }_{e+}=n_e{T}_e^{-3/2}\left[34+4.18\log \left({\displaystyle \frac{T_e^3}{n_e}}\right)\right],$$

(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 ${\nu }_{ee}\sim {\nu }_{e+}$ [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 ${\nu }_{ee}/{\nu }_{e+}$ is like the inverse square of the particle charge numbers. Thus, in our case, it follows that ${\nu }_e\simeq 2{\nu }_{e+}$. In addition, as collisions with neutrals can be neglected, the conservation of collisional momentum transfer between electrons and ions allows writing [62]

$$n_e{m}_e{\nu }_{e+}=n_{+}{m}_{+}{\nu }_{+e},$$

(9)

which, imposing the plasma neutrality condition, brings us to

$${\nu }_{+e}={\displaystyle \frac{m_e}{m_{+}}}{\nu }_{e+}.$$

(10)

On the other hand, it can be demonstrated that the ion–ion collision frequency is [62]:

$${\nu }_{++}\sim {\left({\displaystyle \frac{m_e}{m_{+}}}\right)}^{1/2}{\nu }_{e+},$$

(11)

which leads to

$${\displaystyle \frac{{\nu }_{+e}}{{\nu }_{++}}}\sim {\left({\displaystyle \frac{m_e}{m_{+}}}\right)}^{1/2}\ll 1.$$

(12)

Thus, it follows that ${\nu }_{+}\simeq {\nu }_{++}$, and Equations (6) and (7) can be re-written, respectively, as

$${\sigma }_P=n_e{e}^2\left[{\displaystyle \frac{2{\nu }_{e+}}{m_e({\Omega }_e^2+4{\nu }_{e+}^2)}}+{\displaystyle \frac{{\nu }_{++}}{m_{+}({\Omega }_{+}^2+{\nu }_{++}^2)}}\right],$$

(13)

$${\sigma }_H=n_e{e}^2\left[{\displaystyle \frac{{\Omega }_e}{m_e({\Omega }_e^2+4{\nu }_{e+}^2)}}-{\displaystyle \frac{{\Omega }_{+}}{m_{+}({\Omega }_{+}^2+{\nu }_{++}^2)}}\right].$$

(14)

By making use of Equation (11), Equations (13) and (14) can be written as follows:

$${\sigma }_P=n_e{e}^2\left[{\displaystyle \frac{2{\nu }_{e+}}{m_e({\Omega }_e^2+4{\nu }_{e+}^2)}}+{\displaystyle \frac{m_e^{1/2}{\nu }_{e+}}{m_{+}^{3/2}({\Omega }_{+}^2+{\nu }_{e+}^2\frac{m_e}{m_{+}})}}\right],$$

(15)

$${\sigma }_H=n_e{e}^2\left[{\displaystyle \frac{{\Omega }_e}{m_e({\Omega }_e^2+4{\nu }_{e+}^2)}}-{\displaystyle \frac{{\Omega }_{+}}{m_{+}({\Omega }_{+}^2+{\nu }_{e+}^2\frac{m_e}{m_{+}})}}\right].$$

(16)

We recall, from previous works, e.g., [55], that

$${\sigma }_{\left|\right|}={\displaystyle \frac{e^2{T}_e^{3/2}}{\left[34+4.18log\left(\frac{T_e^3}{n_e}\right)\right]m_e}}.$$

(17)

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 ${\nu }_{e+}$) and the magnetic field strength. On the other hand, ${\sigma }_{\left|\right|}$ 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° × 1° binning in QD magnetic latitude-MLT coordinates. Notice that a 1° magnetic longitude corresponds to 4 min in MLT. All data collected within each bin were filtered out with a standard median filter to remove spikes. The mean of filtered data was set as the value representative of each bin. Statistical uncertainties on electrical conductivity were estimated as the standard error on the mean values within each bin.

3. Results

3.1. Climatological Behaviour of Electrical Conductivity Perpendicular to the Geomagnetic Field

Climatological maps of electrical conductivity were obtained using all the LP and ASM measurements selected as described in Section 2. Figure 1 displays the results for ${\sigma }_{\left|\right|}$ using QD-MLT coordinates. In this map, values range between ∼2.6 × 10¹¹ s⁻¹ and ∼9.4 × 10¹¹ s⁻¹ and are saturated below 3.5 × 10¹¹ s⁻¹ and above 7.5 × 10¹¹ s⁻¹. The estimated statistical uncertainties range between ∼0.1% and ∼13.8%. However, when considering only bins with at least 100 counts, i.e., excluding the bins around poles, the maximum uncertainty drops to ∼0.9%. As observed, relative maxima are present at the magnetic equator and around 06:00 MLT, corresponding to the morning overshoot [87] and, at high latitudes in both hemispheres, around 80° QD magnetic latitude and between 06:00 MLT and 18:00 MLT, corresponding to the polar magnetic cusp [88]. It is noteworthy that the features of ${\sigma }_{\left|\right|}$ align with those of electron temperature, $T_e$, in terms of diurnal, seasonal, and solar activity variations [68,73]. This alignment is due to $T_e$ being the dominant physical quantity in defining ${\sigma }_{\left|\right|}$. These findings are fully consistent with Giannattasio et al. [55], Giannattasio et al. [56], who extensively studied ${\sigma }_{\left|\right|}$ using a shorter Swarm A data set (up to six years of data, compared to the eight years used in this work). For a comprehensive discussion about ${\sigma }_{\left|\right|}$, the reader can refer to those works. Here, we focus on the perpendicular conductivities, ${\sigma }_P$ and ${\sigma }_H$.

The top panel of Figure 2 displays the results obtained for ${\sigma }_P$ in QD-MLT coordinates. In this and all global maps that follow, we represent the values in logarithmic scale, as they span several orders of magnitude. In the case of Figure 2, values of ${\sigma }_P$ range between ∼61 s⁻¹ and ∼7 × 10⁵ s⁻¹ and are saturated above 10⁶ s⁻¹ and below 10² s⁻¹. The estimated statistical uncertainties range between ∼0.6% and ∼25.5%. However, when considering only bins with at least 100 counts, i.e., excluding the poles, the maximum uncertainty drops to ∼2.4%. The bottom panel of Figure 2 displays the results obtained for ${\sigma }_H$ in the same coordinates. In this map, values range between ∼10⁻² s⁻¹ and ∼1.3 × 10⁴ s⁻¹ and are saturated above 10⁴ s⁻¹ and below 10⁻² s⁻¹. The estimated statistical uncertainties range between ∼0.8% and ∼29.3%. However, when considering only bins with at least 100 counts, i.e., excluding the poles, the maximum conditioned uncertainty drops to ∼2.6%. These values are summarised in

Table 1. Climatological values of perpendicular electrical conductivities.

	Minimum Value (s−1)	Maximum Value (s−1)	Minimum Uncertainty (%)	Maximum Uncertainty (%)	Maximum Conditioned Uncertainty (%)
${\sigma }_P$	61	7 × 105	0.6	25.5	2.4
${\sigma }_H$	10−2	1.3 × 104	0.8	29.3	2.6

.

As shown in both panels of Figure 2, relative maxima are observed around the magnetic equator, specifically between ∼7° and ∼13° QD magnetic latitude, and between 11:00 MLT and 15:00 MLT in both hemispheres. These elongated structures appear to trace the crests of the Equatorial Ionisation Anomaly (EIA), which are typical electron density features observed on the day side. The EIA consists of a plasma depletion at the magnetic equator due to the $\mathbf{E}\times \mathbf{B}$ drift. This drift causes plasma to uplift and diffuse poleward along the geomagnetic field lines [89,90,91,92]. Conversely, the relative minima of ${\sigma }_P$ and ${\sigma }_H$ are present in the pre-dawn sector at a ∼±60° QD magnetic latitude and in correspondence with the morning overshoot. Notably, ${\sigma }_P$ and ${\sigma }_H$ exhibit characteristics similar to electron density, $n_e$, in terms of latitudinal and diurnal variations, as can be easily seen by considering Figure 3 in [93]. This indicates that $n_e$, rather than $T_e$ or B, is the dominant physical quantity in driving the ${\sigma }_P$ and ${\sigma }_H$ variation.

Figure 2 clearly illustrates that the features of ${\sigma }_P$ and ${\sigma }_H$ at low latitudes dominate and are orders of magnitude more intense than those at high latitudes. To detect any structures at high latitudes, it is necessary to zoom in on QD magnetic latitudes ≥60°. For this purpose, Figure 3 presents the results obtained for ${\sigma }_P$ in a polar stereographic projection using QD-MLT coordinates. At high QD magnetic latitudes, the relative maxima of ${\sigma }_P$ are ∼10³ s⁻¹, about two orders of magnitude smaller than the same quantity at low QD magnetic latitudes. Moreover, there is also an evident hemispheric asymmetry: in the Northern Hemisphere (left panel in Figure 3), ${\sigma }_P$ is enhanced around noon between 70° and 80° QD magnetic latitude and from 12:00 MLT to 21:00 MLT from 70° and 80° QD magnetic latitude and above. Conversely, in the Southern Hemisphere (right panel in Figure 3), ${\sigma }_P$ is generally smaller, with enhancements localised in the polar cap regions (across almost all MLTs) and within 60–70°, the QD magnetic latitudinal ranges from 06:00 MLT to 16:00 MLT. It should be noted that latitudes above 80° have the lowest counts, resulting in significantly increased error margins. Nevertheless, below 80°, the results are very robust.

Figure 4 displays the results obtained for ${\sigma }_H$ in a polar stereographic projection in QD-MLT coordinates. At high QD magnetic latitudes, the relative maxima of ${\sigma }_H$ are around a few s⁻¹, approximately three orders of magnitude smaller than at low QD magnetic latitudes. The features of ${\sigma }_H$ at high QD latitudes mirror those of ${\sigma }_P$, including the hemispheric asymmetry. In the Northern Hemisphere (left panel in Figure 4), ${\sigma }_H$ is predominantly enhanced around a 80° QD latitude and between 14:00 MLT and 18:00 MLT. In the Southern Hemisphere (right panel in Figure 4), ${\sigma }_H$ is weaker, with faint enhancements localised in the polar cap regions between a 75° and 80° QD magnetic latitude. Also, in this case, the error increases for magnetic latitudes above 80° and decreases significantly below 80°.

3.2. Seasonal Variation of Electrical Conductivity Perpendicular to the Geomagnetic Field

Given that perpendicular conductivities exhibit similar characteristics to electron density, it is natural to investigate the variation in ${\sigma }_P$ and ${\sigma }_H$ with seasons, as electron density demonstrates significant seasonal fluctuations owing to varying sunlit conditions [94]. To accomplish this, we divided the data set based on the seasons, selecting data falling within a three-month long time window centred around: (1) the June solstice to sample summer in the Northern Hemisphere and winter in the Southern Hemisphere; (2) the December solstice to sample winter in the Northern Hemisphere and summer in the Southern Hemisphere; (3) the March equinox to sample spring in the Northern Hemisphere and Autumn in the Southern Hemisphere; (4) the September equinox to sample Autumn in the Northern Hemisphere and spring in the Southern Hemisphere.

In Figure 5, we present global maps of ${\sigma }_P$ in QD-MLT coordinates, from top to bottom: around the December solstice, around the June solstice, around the March equinox, and around September equinox. During the December solstice, the values of ${\sigma }_P$ span from ∼4.6 s⁻¹ to 1.2 × 10⁶ s⁻¹, during the June solstice from ∼0.1 s⁻¹ to 4.3 × 10⁵ s⁻¹, during the March equinox from ∼15.4 s⁻¹ to 1.9 × 10⁶ s⁻¹ and during the September equinox from ∼19.9 s⁻¹ to 1.1 × 10⁶ s⁻¹. Associated statistical uncertainties range between ∼0.7% and ∼59.4% for the December solstice, between ∼0.7% and ∼70.7% for the June solstice, between ∼0.8% and ∼70.7% for the March equinox and between ∼0.4% and ∼70.7% for the September equinox. In all of these cases, when considering bins with at least 100 counts, the maximum conditioned uncertainty significantly decreases down to 4.7% for the December solstice, 5.2% for the June solstice, 4.9% for the March equinox, and 4.9% for the September equinox. These values are resumed in

Table 2. Seasonal values of ${\sigma }_P$.

	Minimum Value (s−1)	Maximum Value (s−1)	Minimum Uncertainty (%)	Maximum Uncertainty (%)	Maximum Conditioned Uncertainty (%)
December solstice	4.6	1.2 × 106	0.7	59.4	4.7
June solstice	0.1	4.3 × 105	0.7	70.7	5.2
March equinox	15.4	1.9 × 106	0.8	70.7	4.9
September equinox	19.9	1.1 × 106	0.4	70.7	4.9

.

Similarly to Figure 5, Figure 6 shows global maps of ${\sigma }_H$ in QD-MLT coordinates, from top to bottom: around the December solstice, around the June solstice, around the March equinox, and around the September equinox. During the December solstice, the values of ${\sigma }_H$ span from ∼0.0 s⁻¹ to 3.1 × ${10}^4$ s⁻¹, during the June solstice from ∼0.0 s⁻¹ to 5.5 × ${10}^3$ s⁻¹, during the March equinox from ∼0.0 s⁻¹ to 6.5 × ${10}^4$ s⁻¹, and during the September equinox from ∼0.0 s⁻¹ to 3.0 × ${10}^4$ s⁻¹. The associated statistical uncertainties range between ∼1.0% and ∼66.7% for the December solstice, between ∼1.0% and ∼70.7% for the June solstice, between ∼1.0% and ∼70.7% for the March equinox, and between ∼1.0% and ∼70.7% for the September equinox. In all of these cases, when considering bins with at least 100 counts, the maximum conditioned uncertainty significantly decreases down to 5.1% for the December solstice, 5.9% for the June solstice, 5.1% for the March equinox, and 5.5% for the September equinox. These values are resumed in

Table 3. Seasonal values of ${\sigma }_H$.

	Minimum Value (s−1)	Maximum Value (s−1)	Minimum Uncertainty (%)	Maximum Uncertainty (%)	Maximum Conditioned Uncertainty (%)
December solstice	0.0	3.1 × 104	1.0	66.7	5.1
June solstice	0.0	5.5 × 103	1.0	70.7	5.9
March equinox	0.0	6.5 × 104	1.0	70.7	5.1
September equinoxes	0.0	3.0 × 104	1.0	70.7	5.5

.

Figure 5 and Figure 6 show features that are strongly dependent on the geomagnetic latitude. At low latitudes, the high ${\sigma }_P$ and ${\sigma }_H$ values, associated with the EIA, are maximised at the March equinox, and the values at the December solstice are sensibly higher than at the June solstice, irrespective of the hemisphere. In other words, perpendicular conductivity shows both the semiannual and the annual anomalies as those characterising the F-layer electron density [95,96,97,98,99,100]. The semiannual anomaly is associated with the observation that NmF2 values at equinoxes are larger than at solstices. NmF2 is the electron density associated with the F2 layer, i.e., the maximum electron density in the ionosphere. Ma et al. [101] conducted a detailed study showing how the amplitudes of the semiannual anomaly depend on solar activity, seasons, and the local time. They highlighted that (1) the amplitudes are larger for years of maximum solar activity than for years of minimum solar activity; (2) the amplitudes are larger in the middle and low latitudes than in the high latitudes; (3) there is a tendency of having larger amplitudes around the March equinox than around the September equinox, especially for daytime during low solar activity years. Figure 5 and Figure 6 are climatological seasonal maps based on magnetic and plasma data going from April 2014 to 31 March 2022 and then including periods of both high and low solar activity. Both figures show larger values of conductivity for the March equinox, and this might also be due to a greater weight of low solar activity values when calculating the mean as the value representative of each bin. The annual anomaly is associated with the observation that NmF2 values at the December solstice are significantly greater than those at June solstice, globally. This further demonstrates how much perpendicular conductivities are tied to the underlying electron density variations. Differently, mid and high latitudes show a marked seasonal variation. There is an increase in perpendicular conductivity during the local summer compared to the local winter. In fact, around the December solstice, features of ${\sigma }_P$ and ${\sigma }_H$ are enhanced in strength and extension in the Southern Hemisphere with respect to the Northern Hemisphere, while around the June solstice, these features are enhanced in the Northern Hemisphere with respect to the Southern Hemisphere. This is particularly evident at high QD magnetic latitudes, where the hemispheric asymmetry in conductivity values exceeds an order of magnitude.

At high latitudes, local seasons take into account the fact that, at the December solstice, it is winter in the Northern Hemisphere and summer in the Southern Hemisphere; at the June solstice, it is summer in the Northern Hemisphere and Winter in the Southern Hemisphere; at the March equinox, it is spring in the Northern Hemisphere and Autumn in the Southern Hemisphere; while at the September equinox, it is Autumn in the Northern Hemisphere and spring in the Southern Hemisphere. In light of this, the seasonal maps of ${\sigma }_P$ (Figure 7) and ${\sigma }_H$ (Figure 8) show considerable variation and a significant hemispheric asymmetry. This variability prevents the direct comparison of ${\sigma }_P$ and ${\sigma }_H$ maps using the same level of saturation for different seasons. In the Northern Hemisphere, during winter, ${\sigma }_P$ ranges from 4.6 s⁻¹ to 2.1 × 10³ s⁻¹, with a mean value of 224.1 s⁻¹, while ${\sigma }_H$ ranges from 0.0 s⁻¹ to 1.9 s⁻¹, with a mean value of 0.1 s⁻¹. During summer, ${\sigma }_P$ ranges from 533.1 s⁻¹ to 4.0 × 10³ s⁻¹ with a mean value of 1.2 × 10³ s⁻¹, while ${\sigma }_H$ ranges from 0.2 s⁻¹ to 4.6 s⁻¹ with a mean value of 0.7 s⁻¹. During spring, ${\sigma }_P$ ranges from 15.4 s⁻¹ to 3.5 × 10³ s⁻¹ with a mean value of 737.4 s⁻¹, while ${\sigma }_H$ ranges from 0.0 s⁻¹ to 2.9 s⁻¹ with a mean value of 0.3 s⁻¹. During autumn, ${\sigma }_P$ ranges from 20.8 s⁻¹ to 2.3 × 10³ s⁻¹ with a mean value of 749.2 s⁻¹, while ${\sigma }_H$ ranges from 0.0 s⁻¹ to 1.6 s⁻¹ with a mean value of 0.4 s⁻¹. Thus, in the Northern Hemisphere, during summer, the mean values of ${\sigma }_P$ and ${\sigma }_H$ are at least five times higher than in winter and about twice higher than during spring and autumn. The same substantial variations are observed in the Southern Hemisphere. In the Southern Hemisphere, during winter, ${\sigma }_P$ ranges from 0.1 s⁻¹ to 7.4 × 10³ s⁻¹ with a mean value of 71.6 s⁻¹, while ${\sigma }_H$ ranges from 0.0 s⁻¹ to 15.5 s⁻¹ with a mean value of still around 0.0 s⁻¹. During summer, ${\sigma }_P$ ranges from 464.7 s⁻¹ to 1.2 × 10⁴ s⁻¹ with a mean value of 1.7 × 10³ s⁻¹, while ${\sigma }_H$ ranges from 0.1 s⁻¹ to 22.7 s⁻¹ with a mean value of 1.0 s⁻¹. During spring, ${\sigma }_P$ ranges from 30.9 s⁻¹ to 9.4 × 10³ s⁻¹ with a mean value of 566.1 s⁻¹, while ${\sigma }_H$ ranges from 0.0 s⁻¹ to 22.1 s⁻¹ with a mean value of 0.2 s⁻¹. During autumn, ${\sigma }_P$ ranges from 23.5 s⁻¹ to 2.6 × 10⁴ s⁻¹ with a mean value of 744.6 s⁻¹, while ${\sigma }_H$ ranges from 0.0 s⁻¹ to 75.4 s⁻¹ with a mean value of 0.4 s⁻¹. Thus, in the Southern Hemisphere during summer, the mean values of ${\sigma }_P$ and ${\sigma }_H$ are up to an order of magnitude higher than those measured during winter and about five times higher than that during spring and autumn. Apart from the values themselves, we report the presence of seasonally varying features of ${\sigma }_P$ and ${\sigma }_H$. Concerning a possible connection with the semiannual anomaly, the Northern Hemisphere shows features that are similar to those highlighted at low latitudes by Figure 5 and Figure 6, with an intensification of conductivities for the March equinox, which can be related to the larger amplitudes of the semiannual anomaly characterising this period. Unfortunately, the same cannot be said for the Southern Hemisphere, for which the intensification of conductivities is seen around the September equinox and more centred in the polar cap. This is now an open question, but we have to remark that at high latitudes, the primary source of ionisation is not the solar EUV radiation; the plasma dynamics, specifically the transport processes, at these latitudes play a key role and cannot be neglected. Additionally, it is important to point out that studies of the semiannual anomaly in the topside ionosphere using in situ satellite measurements showed that its hemispheric dependence appears contradictory [102,103].

3.3. Electrical Conductivity Variation Perpendicular to the Geomagnetic Field Due to Solar Activity

We studied also the dependence of perpendicular conductivities, ${\sigma }_P$ and ${\sigma }_H$, on solar activity during the eight years of Swarm observations considered here. To this aim, we evaluated the level of solar activity by taking advantage of the daily solar radio flux at a 10.7 cm wavelength (F10.7) as a proxy [104]. The F10.7 time series was downloaded from the Space Physics Data Facility of the NASA Goddard Space Flight Center. Figure 9 reports F10.7 (blue line) from 1 April 2014 to 31 March 2022 in solar flux units (sfu). We recall that 1 sfu = ${10}^{-19}$ erg·s⁻¹·cm⁻²·Hz⁻¹ in cgs units. F10.7 spans the range between 63.4 sfu, at the minimum of the current solar cycle reached in December 2019, and 255.0 sfu, which was reached in June 2015. During the period considered in this study, the solar activity started near the maximum of the 24th solar cycle and then descended to the minimum that started in the 25th solar cycle. After the minimum, the Swarm measurements used in this work sampled part of the new cycle’s rise, which is expected to peak between 2024 and 2025. We examined the cumulative distribution function (CDF, not shown in this work) of the F10.7 time series. We found that the flux values at the 25th percentile and 75th percentile are, respectively, 70.3 sfu and 100.0 sfu. This means that 25% of F10.7 values are below 70.3 sfu and, likewise, 25% of F10.7 values are above 100.0 sfu. Thus, these values could be used as thresholds to select periods with low and moderate/high solar activity. However, a selection based only on CDF values could lead to an oversampling of one season over another, and thus mix the seasonal variation of ${\sigma }_P$ and ${\sigma }_H$ with that associated with the different levels of solar activity. On the other hand, inspecting biennial changes of ${\sigma }_P$ and ${\sigma }_H$ may be an optimum compromise between the two: (1) two years of Swarm A observations provide a statistical data coverage robust enough to sample Swarm A measurements in bins of QD magnetic latitude versus MLT 1° × 1° wide; (2) two years of data ensure even sampling of the seasons so that they have the same effect on each biennium. On this basis, we have identified two biennia in our data set as representative of low and medium–high solar activity: the third biennium (between 1 April 2018 and 31 March 2020) and the first biennium (between 1 April 2014 and 31 March 2016), respectively. These two-year periods are highlighted with shaded regions in Figure 9. The orange-shaded area between the red dashed lines indicates a period of medium–high solar activity, while the green-shaded area between the black dashed lines denotes a period of low solar activity. This selection aligns with the 25th and 75th percentiles of the CDF for selecting solar activity levels. Specifically, nearly all F10.7 values in the orange region exceed 100.0 sfu, and almost all F10.7 values in the green region fall below 70.3 sfu.

In Figure 10 we present a global map of ${\sigma }_P$ in QD-MLT coordinates for the first (top panel) and third (bottom panel) biennia of the entire selected time window used in this study. During the first biennium, which represents a period of moderate–high solar activity, ${\sigma }_P$ values range from 10.7 s⁻¹ to 2.8 × 10⁶ s⁻¹. The estimated statistical uncertainties range between ∼0.9% and 70.6%. However, when considering only bins with at least 100 counts, excluding the magnetic poles, the maximum uncertainty decreases to 4.7%. In the third biennium, which corresponds to a period of low solar activity, ${\sigma }_P$ values range between 0.6 s⁻¹ and 3.7 × 10⁵ s⁻¹. The estimated statistical uncertainties range between ∼0.5% and 70.5%. When considering only bins with at least 100 counts, excluding the magnetic poles, the maximum conditioned uncertainty decreases to 5.0%. These values are resumed in

Table 4. Values of ${\sigma }_P$ for the first and third biennia, respectively representing, medium–high and low solar activity.

	Minimum Value (s−1)	Maximum Value (s−1)	Minimum Uncertainty (%)	Maximum Uncertainty (%)	Maximum Conditioned Uncertainty (%)
First Biennium	10.7	2.8 × 106	0.9	70.6	4.7
Third Biennium	0.6	3.7 × 105	0.5	70.5	5.0

.

Compared to the climatological case shown in the top panel of Figure 2, during moderate–high solar conditions, ${\sigma }_P$ increases and its characteristic features around the magnetic equator expand up to ±25° QD magnetic latitude. Moreover, these structures persist until nearly midnight MLT. Conversely, during low-solar-activity conditions, the features typical of ${\sigma }_P$ become visibly weaker and less extended compared to the climatological case. Indeed, the most intense features of ${\sigma }_P$ do not exceed a 12° QD magnetic latitude and are confined within 15:00 MLT and 16:00 MLT. A similar behaviour is observed for ${\sigma }_H$, as illustrated in Figure 11. During the first biennium, under moderate-high solar activity conditions, ${\sigma }_H$ values range from 0.0 s⁻¹ to 1.1 × 10⁵ s⁻¹. The estimated statistical uncertainties range between ∼1.1% and 99.7%. However, when considering only bins with at least 100 counts, i.e., excluding the magnetic poles, the maximum uncertainty falls to 4.9%. In the third biennium, under low-solar-activity conditions, ${\sigma }_P$ values range between 0.0 s⁻¹ and 5.3 × 10³ s⁻¹. The estimated statistical uncertainties cover the widest range between ∼0.8% and 109.4%. Again, when considering only bins with at least 100 counts, i.e., excluding the magnetic poles, the maximum conditioned uncertainty collapses to 5.2%. These values are resumed in

Table 5. Values of ${\sigma }_H$ for the first and third biennia, respectively, representing medium–high and low solar activity.

	Minimum Value (s−1)	Maximum Value (s−1)	Minimum Uncertainty (%)	Maximum Uncertainty (%)	Maximum Conditioned Uncertainty (%)
First Biennium	0.0	1.1 × 105	1.1	99.7	4.9
Third Biennium	0.0	5.3 × 103	0.8	109.4	5.2

.

The structures of ${\sigma }_H$ closely mirror those of ${\sigma }_P$, exhibiting similar behaviour in response to solar activity. During periods of increased solar activity, both ${\sigma }_P$ and ${\sigma }_H$ intensify and expand in QD latitude, extending toward midnight MLT. Conversely, during periods of low solar activity, their features become more subdued and confined.

Similarly to the case of seasonal maps, at high latitudes, biennial maps of ${\sigma }_P$ (Figure 12) and ${\sigma }_H$ (Figure 13) spread over a wide range of values in the transition from the first to the third biennia. This makes it impossible to compare maps of ${\sigma }_P$ and ${\sigma }_H$ setting the same level of saturation for different solar activity conditions. In the Northern Hemisphere, during the first biennium, ${\sigma }_P$ ranges from 0.6 × 10³ s⁻¹ to 7.2 × 10³ s⁻¹ with a mean value of 3.0 × 10³ s⁻¹, while ${\sigma }_H$ ranges from 0.3 s⁻¹ to 11.0 s⁻¹ with a mean value of 2.8 s⁻¹. During the third biennium, ${\sigma }_P$ ranges from 11.0 s⁻¹ to 1.0 × 10³ s⁻¹ with a mean value of 0.4 × 10³ s⁻¹, while ${\sigma }_H$ ranges from 0.0 s⁻¹ to 0.5 s⁻¹ with a mean value of 0.1 s⁻¹. Thus, in the Northern Hemisphere during the first biennium, the mean values of ${\sigma }_P$ and ${\sigma }_H$ are about one order of magnitude higher than that retrieved during the third biennium. In the Southern Hemisphere, during the first biennium, ${\sigma }_P$ ranges from 10.7 s⁻¹ to 16.0 × 10³ s⁻¹ with a mean value of 2.1 × 10³ s⁻¹, while ${\sigma }_H$ ranges from 0.0 s⁻¹ to 38.2 s⁻¹ with a mean value of 1.8 s⁻¹. During the third biennium, ${\sigma }_P$ ranges from 0.6 s⁻¹ to 2.7 × 10³ s⁻¹ with a mean value of 0.3 × 10³ s⁻¹, while ${\sigma }_H$ ranges from 0.0 s⁻¹ to 3.7 s⁻¹ with a mean value of 0.1 s⁻¹. Thus, in both hemispheres, during the first biennium, the mean values of ${\sigma }_P$ and ${\sigma }_H$ are about one order of magnitude higher than those retrieved during the third biennium.

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, Giannattasio et al. [59] 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, Lovati et al. [106] 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 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 $m_{+}\equiv m_{O^{+}}$. 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.

• 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⁺.
• 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.
• 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.

The climatological results reported in Figure 1 and Figure 2 agree well with what is known from the literature [105,114]. For example, parallel Pedersen and Hall conductivities are consistent with the conductivity height profiles in the dayside shown in Lizunov et al. [115] when fixing the altitude to 450 km, i. e., the Swarm altitude. In the topside ionosphere, the electrical conductivity parallel to the magnetic field is much higher than that perpendicular to the magnetic field, while in the plane perpendicular to the magnetic field, the Pedersen conductivity is higher than the Hall conductivity. Indeed, this behaviour can be explained by the different mechanisms governing the motion of charged particles in the ionosphere. In particular, the parallel conductivity is dominant because charged particles, particularly electrons, can move freely along the magnetic field lines with minimal collisions. As a consequence, this direction offers the least resistance to particle motion, as the magnetic field does not impede movement along its lines. When considering the conductivities perpendicular to the magnetic field, the Pedersen conductivity exceeds the Hall conductivity. The Pedersen conductivity involves particle motion that is influenced by both the electric and magnetic fields, leading to collisions and a moderate level of resistance. However, these collisions are less frequent compared to those in the Hall direction because the Pedersen direction aligns partially with the electric field, allowing for more straightforward movement compared to the purely perpendicular (Hall) direction. The Hall conductivity, which arises due to the Hall effect, is the lowest because charged particles are deflected by the magnetic field, causing them to spiral and experience a higher rate of collisions. These frequent collisions significantly reduce the effective mobility of the particles, resulting in lower conductivity.

The dependence of conductivity on seasons and solar activity can be attributed to changes in ionospheric density and composition, which affect collision frequencies. During periods of high solar activity, increased ionisation enhances the number of charged particles, thus influencing the overall conductivity profiles. Furthermore, seasonal changes impact the neutral density and temperature, further modifying the collision rates and conductivities. These characteristics, which delineate the conductivities both parallel and perpendicular to the magnetic field, are readily identifiable in our climatological distributions, as well as in distributions reflecting seasonal and solar activity variations. For instance, we have seen that the equinoctial pattern shown by perpendicular conductivities seems to be strictly connected with the semiannual anomaly characterising the electron density, especially at middle and low latitudes. Concerning the high latitudes, the discussion becomes a little more complicated. The Northern Hemisphere reflects pretty well the semiannual electron density pattern, while the Southern Hemisphere does not. This is a point that deserves further in-depth analysis. In any case, our results showed that the assumptions underlying our calculations of conductivities at Swarm altitude are reasonable and provide a scenario that aligns well with the theoretical expectations.

As mentioned, this is the first time that statistical maps of Pedersen and Hall conductivities at about 450 km of altitude are shown, with a study of their variation with season and solar activity. This makes it impossible to find studies and models in the literature that calculate these quantities under the same conditions imposed in this work. However, it is possible to show, to some extent, the consistency of our results on the basis of previous works. For example, parallel, Pedersen and Hall conductivities can be retrieved by using the model of the World Data Center (WDC) for Geomagnetism in Kyoto. This model is freely usable online and provides the height profile of conductivities, height-integrated values (i.e., conductances), and maps for specific days and LT or UT hours. The model implements the conductivity equations from Maeda [105] with the collision frequencies in Banks and Kockarts [116] and uses the outcomes of the International Reference Ionosphere (IRI) model by Bilitza et al. [114] and the atmospheric model by Rees and Fuller-Rowell [117]. By running the WDC model to retrieve conductivity height profiles, for example, for the date 21 June 2018 (at the June solstice and at a fairly central point in the time window used in this work) at the location 15° of geographic latitude, 0° of geographic longitude, between 400 and 500 km of altitude and at noon UT (corresponding to a QD magnetic latitude of 4.2° and 11.8 MLT), we find ${\sigma }_{\left|\right|}\sim {10}^{11}$ s⁻¹ and ${\sigma }_P\sim {10}^4$ s⁻¹ (the ${\sigma }_H$ profile is not provided by the WDC model in the selected range of altitudes), which is consistent with our results (see Figure 1 and Figure 5). Proceeding in the same way, we checked that the results obtained in our study (particularly for the dependence of conductivities on seasons) are consistent with those provided by the WDC model. We, however, note that this comparison is made to the best of our capabilities, as the WDC model provides (1) local vertical conductivity profiles (at fixed geographic longitude and latitude and LT); (2) conductance maps (at a fixed geographic longitude and latitude and LT or UT). Neither of these provides maps that are directly comparable with our conductivity maps, so we can only limit ourselves to a fairly approximate comparison based on orders of magnitude.

The dependence of conductivities on solar activity and seasons allows the indirect investigation of the distinct electrical structures and physical processes they are associated with. For instance, at Swarm altitude, where the reconstructed distributions of the conductivities are referenced, we can observe the EIA and the FACs, both of which are influenced by the anisotropic conductivities of the ionosphere. Looking at the equatorial region in the F-region at Swarm altitude, the EIA is still relevant but is more spread out. The vertical plasma drifts caused by the daytime eastward electric field and the Earth’s magnetic field ($\mathbf{E}\times \mathbf{B}$ motion) create peaks in electron density at about ±15° QD magnetic latitude. These plasma drifts are significantly influenced by the Hall and Pedersen conductivities, which play a crucial role in modulating the structure and behaviour of the EIA, in conjunction with zonal and meridional neutral winds. At low latitudes, the Pedersen conductivity facilitates the vertical movement of plasma. At the equator, plasma is vertically lifted by $\mathbf{E}\times \mathbf{B}$ and then diffuses along the geomagnetic field lines toward higher latitudes under the influence of gravity and pressure gradient forces, thus leading to the formation of the EIA’s characteristic crests. During periods of high solar activity, the increased solar flux enhances the electron density, which in turn leads to enhanced Pedersen and Hall conductivities, resulting in more pronounced equatorial ionisation anomalies. Seasonal changes impact both the Hall and Pedersen conductivities through variations in both the charged and neutral components of the ionosphere–thermosphere system. Indeed, the neutral density, collision frequencies, and thermospheric winds also show specific patterns exhibiting seasonal variations. These seasonal variations can alter both the effectiveness of the plasma drifts and the resulting ionisation distribution. For instance, at low latitudes, perpendicular conductivities exhibit annual and semiannual anomalies (Figure 5 and Figure 6) that affect not only the ionospheric plasma but also the neutral atmosphere at those altitudes, as demonstrated by satellite observations and models [118,119,120]. Although perpendicular conductivities have been obtained through observed plasma parameters (in situ electron density and temperature) and the collision frequencies between charged and neutral particles have been neglected in the derivation of corresponding formulas, the coupling with the neutral atmosphere is implicit in the plasma parameters’ variations. Indeed, neutral density acts as a background for the electron density (and drives the vertical distribution of the ionic and molecular species), and the same is true for the neutral temperature which is a lower limit for the electron temperature.

During high solar activity, increased Extreme Ultra Violet (EUV) radiation enhances ionisation, leading to higher electron densities and stronger equatorial ionisation anomalies. This enhancement is more significant for the Pedersen conductivity, which directly impacts the vertical plasma drifts [121]. The Hall conductivity also increases, influencing the lateral spread of the ionisation peaks. Seasonal changes impact the EIA due to variations in solar zenith angle and thermospheric winds, which modulate the distribution and intensity of ionisation by affecting both conductivities. In summary, the roles of Hall and Pedersen conductivities are fundamental in understanding, for example, the behaviour of the EIA at Swarm altitudes. Their influence on plasma drifts and ionisation distribution highlights the complex interplay between solar activity, seasonal variations, and ionospheric conductivities, shaping the equatorial ionisation anomaly [122,123]. While the features reconstructed using direct measurements from the Swarm constellation align well with theoretical expectations, it is essential to acknowledge the limitations and uncertainties inherent to the study. The spatial and temporal variability of ionospheric parameters, such as local fluctuations in electron density and temperature, can introduce uncertainties in conductivity measurements. The omission of minor ion species, which can significantly affect conductivity values, may lead to the underestimation or overestimation of actual conductivities, especially under varying ionospheric conditions. Additionally, the ionosphere–thermosphere system is influenced by various factors, including neutral winds, pressure gradients, and magnetic field variations, leading to complex interactions and feedback mechanisms that are challenging to model accurately. For example, while our study shows a clear relationship between increased EUV flux and enhanced ionisation, the actual response of the ionosphere can vary significantly depending on other concurrent factors. Moreover, our approximations assume a certain degree of homogeneity and stability in the ionospheric environment, which may not always hold true. Real-world ionospheric conditions are subject to rapid changes and localised disturbances, potentially leading to deviations from average descriptions. Therefore, while our study provides valuable insights into the roles of Hall and Pedersen conductivities in shaping the EIA, it is crucial to interpret these findings within the broader context of ongoing research. At high latitudes, the ionospheric conductivities and associated currents are influenced by different processes. The auroral electrojet currents and FACs are prominent features. The Pedersen conductivity at high latitudes can be significantly enhanced by particle precipitation and auroral activity, leading to strong horizontal currents. The Hall conductivity, which is also affected by particle precipitation, contributes to the perpendicular current systems but generally has a lower magnitude compared to the Pedersen conductivity. The dependence on solar activity and seasonal changes is evident here as well, with increased solar activity leading to enhanced auroral activity and corresponding increases in both Hall and Pedersen conductivities. However, at the Swarm altitude, to which the conductivity maps we have reconstructed refer, the auroral electrojet currents cannot be observed because they flow at significantly lower altitudes. At around 450 km altitude, however, their effect on the magnetic field is measurable. At these altitudes, we can certainly observe the effects of FACs, particle precipitation, and large-scale plasma motions driven by the solar and magnetospheric electric fields typical of this altitude. Our findings reveal that at high latitudes, both Pedersen and Hall conductivities exhibit an enhancement on the dayside, particularly in the region associated with the polar cusp, as can be seen in the spatial–temporal distributions of conductivity in the Northern Hemisphere (see Figure 7 and Figure 8). The increase in conductivities around the polar cusp is due to the continuous flux of charged particles from the solar wind, which can penetrate to lower altitudes in the ionosphere because the magnetic field lines in this region are open. The charged particles ionise the region, enhancing its conductivity. Another interesting region at high latitudes characterised by an enhancement of both Pedersen and Hall conductivity values is primarily concentrated in the duskside sector of the Northern Hemisphere and in the nightside sector of the Southern Hemisphere. This pattern is consistently observed in the distributions of conductivities across different seasons and levels of solar activity. This enhancement could be the result of convective currents associated with global movements of ionospheric plasma driven by solar and magnetospheric electric fields.

5. Summary and Conclusions

This study has, for the first time, characterised the electrical conductivity perpendicular to the geomagnetic field in the topside ionosphere through in situ measurements. The analysis was conducted using data from the Swarm A satellite, calculating the Pedersen and Hall conductivities according to Equations (15) and (16). The motivations behind this study are manifold. Firstly, the study allows us to extend and complete information from previous studies. In the works of Giannattasio et al. [55,56,57] parallel electrical conductivity has been extensively studied by using up to six years of in situ measurements in the topside ionosphere acquired by Swarm A and CSES-01 [58] missions. Here, a characterisation of the perpendicular conductivity is carried out. Parallel and perpendicular conductivities are, indeed, crucial for obtaining the conductivity tensor. The key findings underscore the significant role of solar activity and seasonal variations in modulating both Pedersen and Hall conductivities across different latitudinal regions. In the equatorial region, particularly within the F-region at Swarm altitudes, our results highlight the intricate dynamics influencing the EIA. The vertical plasma drifts induced by the daytime eastward electric field and the Earth’s magnetic field generate peaks in electron density. These drifts are modulated by the Hall and Pedersen conductivities. During periods of high solar activity, increased ionisation enhances Pedersen conductivity, leading to more pronounced EIA peaks. Seasonal changes further impact these conductivities, altering the effectiveness of plasma drifts and the resultant ionisation distribution. Our climatological distributions, aligned with theoretical expectations, illustrate the distinct behaviours of conductivities parallel and perpendicular to the magnetic field. The parallel conductivity, dominant due to minimal collisions along magnetic field lines, contrasts sharply with the perpendicular conductivities, where Pedersen conductivity exceeds Hall conductivity. These findings validate our methodological approach and provide a robust framework for understanding ionospheric conductivity variations. Implications of these results extend beyond academic interest. The detailed conductivity maps and their seasonal and solar activity-dependent variations offer valuable insights for improving ionospheric models, which are crucial for applications in Space Weather forecasting and satellite communication systems. The ability to predict changes in ionospheric conductivity can enhance the accuracy of GPS and other satellite-based navigation systems, mitigate the effects of Space Weather on technological infrastructure, and inform the development of more resilient communication networks.

In conclusion, this study enhances our understanding of ionospheric conductivities by examining the interactions between solar activity, seasonal variations, and ionospheric dynamics. The results align with the theoretical models used and identify specific patterns and anomalies at Swarm altitudes, particularly concerning Pedersen and Hall conductivities. These insights are valuable for improving Space Weather forecasting and satellite communication systems. However, the use of data from a single altitude (around 450 km) limits the generalizability of the findings across different altitudes, which is essential for a comprehensive understanding of ionospheric electrodynamics. Future research should focus on incorporating multi-altitude data to provide a more complete picture of ionospheric current systems and their interaction with geomagnetic field lines. Additionally, future studies should aim to characterise the Cowling conductivity, which is influenced by the Pedersen and Hall conductivities analyzed in this study. Another objective will be to reconstruct the current density distribution in the topside ionosphere at Swarm altitudes, using the tensor relationship outlined in Equation (2). This approach will refine our understanding of ionospheric processes and their implications for practical applications. Finally, it might be interesting to build a perpendicular conductivity model at a specific altitude or within a range of altitudes (those satisfying the assumptions described in Section 2.2). This could be achieved by using the IRI model for estimating electron density and temperature at different QD magnetic latitudes, MLTs, and seasons and taking into account geomagnetic activity, together with a geomagnetic field model (empirical or based on expansion in spherical harmonics, or a mix of the two) for estimating the magnetic field strength and applying Equations (15) and (16).