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,
. 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,
; current density,
; and
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 (
) fields, pressure gradients, viscous stressed, and electron–ion Coulomb collisions. In the topside ionosphere, GOL can be written in the simple form [
23]
where
is the velocity of the neutral wind. In a resistive medium,
(or equivalently its inverse, resistivity
) is also required to determine power density dissipated locally by a current, namely
, 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
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
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
depends on the direction of the electric field
with respect to the geomagnetic field
[
25]. Parallel to
, the so-called parallel electrical conductivity,
, does not depend on
itself. If
is applied perpendicular to
, the so-called Pedersen conductivity,
, drives currents parallel to
, while the so-called Hall conductivity,
, drives currents perpendicular to
[
26,
27]. Over the magnetic equator, a vertical Hall polarisation field in the E-region increases the so-called Cowling conductivity,
[
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,
,
, where
runs between the bottom and top altitudes of the ionosphere [
29,
30,
31,
32]. It must be said that the models of
, 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
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
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
,
, and
in the E layer. In any case, almost all efforts have focused on the study of
in the E layer of the ionosphere. Only a few works have characterised
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
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
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
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
, taking advantage of local calculations of
and
with Swarm data.
To our knowledge, no one has ever characterised the electrical conductivities perpendicular to the magnetic field,
and
, 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,
and
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,
, as follows [
62]:
In Equation (
2),
is the electric field in the plasma frame of reference with components
, and we assume that
is directed along the versor
, the component of
perpendicular to
is directed along
, and the versor
is perpendicular to both
and
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
and
in the topside ionosphere from in situ satellite measurements, which is preparatory to an empirical model for
. This paper is organised as follows: In
Section 2 we describe the data set used and the methods employed for the calculation of
and
.
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.
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,
and
, 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
and
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
and
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
. 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
s
−1 and
s
−1 (the
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 (
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
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).