Empirical models of the Earth's ionosphere field-aligned currents: a review

. In this paper, we present a review of empirical models of the Earth’s ionosphere field -aligned currents (FAC). In total, seven FAC models were considered. Iijima and Potemra’s model was one of the earl iest ones. It described a general structure of FAC distribution and could be used as a FAC common pattern. Papitashvili & co-auth., Lukianova & co-auth., and Weimer improved descriptions of FAC distributions using the larger amount of data from various satellite missions. Their models were published as a set of printed maps of the FAC distributions for various geophysical situations. Recent models (MFACE, AMPS, and EW-FAC) are provided as MATLAB, Python, and IDL software, respectively. These models are the high-resolution (finer spatial grid) description of FAC distributions with improved dependence on the Interplanetary Magnetic Field and Solar wind impacts.


Introduction
The Triad satellite mission provided important magnetic field measurements.It allowed researchers [1][2][3] to reconstruct the spatial distribution of Field-Aligned Currents (FAC).A significant number of the Earth's low-orbit satellite missions were performed during the last fifty years.The collected satellites' magnetic field data yielded the construction of various empirical FAC models.The growing amount of magnetic field observations provided the ability to improve the empirical FAC models.The first models described FAC general structures only; the latest ones presented high-resolution FAC patterns (or even 3D ionospheric currents) with dependence on the solar wind parameters, the Interplanetary Magnetic Field (IMF) strength and orientation, season, and the geomagnetic activity level.
In this paper, we summarize general empirical FACs models, their principal features, and pay the special attention to an ability to use the FAC models by an independent researcher.
Using these, we completed a set of searches on the topic of the Earth's ionosphere fieldaligned currents, i.e., search terms included 'ionosphere', 'Earth', 'field aligned currents', and 'current system'.Then, we excluded duplicates from the search results and kept papers that fulfilled the following criteria.(a) An article should be any kind of 'article', 'research article', 'review', 'report', 'technical report', 'monograph', 'chapter in collective monograph', or 'conference paper'.(b) The model should be an empirical one.There is no way to construct a self-consistent FAC model without the development of a self-consistent magnetospheric model.Earth's magnetosphere modelling is a huge field of research that goes far beyond the scope of the review.(c) The article should discuss the Earth's ionosphere.Recent advances in the modelling of ionospheres and electric current systems of other planets of the Solar system are out of the scope of this review.
We manually analysed these selected papers.The special attention was payed to (a) an ability to use or re-implement a model by an independent researcher; (b) data-sources used to construct a model; (c) the timeline, i.e., the temporal coverage of observations; (d) the spread of a model in the scientific community.

Iijima and Potemra model
Iijima and Potemra [1][2][3] developed one of the earliest empirical models of the Earth's ionosphere FAC.The model is derived from the Triad satellite magnetometer data for July 1973 -October 1974 and describes the FAC system as a composition of the Region 1, Region 2, and the Region 0 currents.The Region 1 and Region 2 currents form the inner and the outer belts, respectively.Those belts are shifted to the night side along the noonmidnight meridian.The maximal values of the Region 1&2 currents are located at the morning and evening sides of those belts.The Region 0 currents are located in the vicinity of the cusp area.The model [1][2][3] is a set of graphical maps.These maps schematically represent the spatial distribution of large-scale in-flowing and out-flowing FACs at 800 km altitude above the Earth's surface.
To the end-user, the model [1] is provided as the following items.(a) Maps of large-scale FAC spatial distribution organized according to the invariant latitude -magnetic local time (MLT).(b) Maps of Region 1 & 2 electric currents density organized according to MLT.(c) Region 1 & 2 FAC density as a function of the kp-index for 13:00-18:00 MLT and 04:00-09:00 MLT sectors (both in-flowing and out-flowing components are presented).(d) Region 1 out-flowing electric currents density as a function of the AE-index for 13:00-18:00 MLT.(e) Region 1 & 2 averaged electric currents density and the total current for the both Afternoon-to-Midnight and Midnight-to-Forenoon sectors.Iijima and Potemra [2] derived FAC averaged distributions and flow directions for |AL| < 100 nT at the dayside cusp region basing on the same data set as in [1].Iijima and Potemra [3] investigated FAC characteristics during substorm conditions.They derived averaged in-flowing and out-flowing FAC distributions and flow directions for both weakly disturbed (|AL| < 100 nT) and disturbed (|AL| ≥ 100 nT) conditions.The paper [3] provided the Region 1 & 2 electric currents density, intensity, and the total current as tables for the both Afternoon-to-Midnight and Midnight-to-Forenoon sectors.

Weimer-2001 model
Weimer [4] developed the empirical model of FAC using the polar orbiting Dynamic Explorer 2 (DE2) satellite observations for August 1981 -March 1983.The model projects FAC density values to 115 km altitude above the Earht's surface.It describes FAC spatial distributions in corrected geomagnetic latitude (MLAT) -MLT coordinates and considers FAC as a function of IMF parameters, solar wind speed, solar wind plasma density, and geomagnetic dipole tilt angle.
To the end-user, the model is provided as the set of maps.These maps represent FAC distributions for eight IMF Clock angles, three dipole tilt angles, different IMF strength BT, solar wind velocity, and solar wind density values.In addition, the model [4] used the ALindex as an optional input parameter suitable for the description of sub-storm events [4].

Edwards-Weimer Field Aligned Currents (EW-FAC) model
Edwards & co-authors [5] derived the empirical FACs model from Ørsted, CHAMP (CHAllenging Minisatellite Payload), and Swarm satellites' magnetic field measurements.In total, these satellite observations cover fifteen years period.The resulting model projects FAC to 110 km altitude.It provides the spatial 2D FAC distribution in MLAT -MLT coordinates.The model input parameters are (a) Solar Wind Electric Field (SWE), (b) IMF Clock Angle, (c) geomagnetic dipole tilt angle, (d) solar activity characterized by one of the following indexes -F10.7,S10.7, M10.7, or Y10.7, (e) target hemisphere (Northern geographic hemisphere is the default value).Some of these parameters could be recalculated from others.SWE could be calculated from IMF strength BT, and Solar Wind Velocity (SWV).The required BT could be calculated from IMF BY and BZ components provided in Geocentric Solar Magnetospheric (GSM) coordinates.The geomagnetic dipole tilt angle could be derived from the given date.
To the end-user, the model is provided as IDL byte-code published via the GitHub repository at https://github.com/VTMIST/EW-FAC.The repository provides byte-code only (no source codes are provided).To use the model, the interested parties should have IDL or the IDL runtime environment.The model was not tested with free and open-source alternatives like GDL (GNU Data Language).

Papitashvili model
Papitashvili & co-authors [6] derived the empirical FACs model from Ørsted and MagSat satellites' magnetic field measurements for April 1999 -October 2001 and November 1979 -April 1980, respectively.To construct the model, the dataset was organized according to the (a) hemisphere, (b) season, (c) IMF Clock angle, and (d) IMF strength BT considered for three cases -BT ≤ 1 nT (median ~ 0 nT), 1 nT ≤ BT ≤ 4 nT (median ~ 2.7 nT), BT > 4 nT (median ~ 5.7 nT).The paper [6] reports that only a small portion of the dataset was collected for |B| > 12 nT conditions.Due to this, the model is mostly valid for |B| ≤ 12 nT.
The resulting model is a set of FAC maps projected at 115 km altitude.These maps are distributed as the electronic supplement (multipage pdf-file) to the original article [6].The model provides FAC patterns at the Northern and Southern Hemispheres for summer, winter, and equinox conditions taking into account the IMF strength and orientation (BY and BZ components which characterize the IMF Clock Angle).

Lukianova model
Lukianova & Christiansen [7] proposed an approach to model global distributions of ionosphere electric fields (electric potentials) basing on high-resolution FAC maps derived from CHAMP (August 2000 -September 2010) и Ørsted (September 2013 -August 2016) satellites' measurements.The original paper [7] did not provide FAC distributions.Instead, electric potential contour maps were presented for the Northern and Southern Hemisphere for different combinations of IMF directions.Lukianova & co-authors [8] constructed FAC contour maps from MagSat, Ørsted, and CHAMP data.Later, Lukianova & Bogoutdinov [9] derived FAC maps from Swarm observations.This model provided FAC distributions for the Northern and Southern Hemispheres with dependence on the season and IMF orientation.

Model of Field-Aligned Currents through Empirical Orthogonal Functions Analysis (MFACE)
He & co-authors [10][11][12] developed the Model of Field-Aligned Currents through Empirical Orthogonal Functions Analysis (MFACE) using the CHAMP satellite magnetic field observations during 2001 -2010.The model provides FAC patterns at 110 km altitude above the Earth's surface for the Northern and Southern Hemispheres.MFACE considers FAC distributions as a function of the IMF magnitude and orientation (IMF BY and BZ components in GSM coordinates), solar wind velocity, day of the year (seasonal dependence), and magnetic activity (AE index).Additionally, the MFACE defines the position of the Aurora Current Center (ACC).
The software implementation of the model requires the following input: To the end-user, the MFACE model is provided via the SourceForge repository at the URL https://sourceforge.net/projects/mface.It contains Matlab sources and 32-and 64bit precompiled binaries for Windows, Mac OS X, and Linux operating systems.The MATLAB Compiler Runtime (version 8.1, R2013a) is required to run the model's binaries.The model was not tested to work with free and open-source alternatives like GNU Octave.

Average Magnetic field and Polar current System (AMPS) model
Laundal & co-authors [13][14] developed the empirical Average Magnetic field and Polar current System (AMPS) model using CHAMP (August 2000 -September 2010) and Swarm (December 2013 -August 2016) satellites magnetic field observations.The model describes the 3D ionosphere electric current system consisting of both FAC and height-integrated equivalent horizontal currents (also known as 'shield' currents).AMPS describes electric currents as a function of the Solar wind velocity, IMF, geomagnetic dipole tilt angle, and F10.7 index.These currents are mapped to 110 km altitude above the Earth's surface.
The AMPS model inputs are (a) 20-mins averages of 1-min resolution solar wind velocity VSW, (b) IMF BY and BZ components in GSM coordinates, (c) geomagnetic dipole tilt angle, and (d) F10.7 index.
The AMPS outputs are (a) field-aligned currents, (b) divergence-free currents, (c) divergence-free horizontal currents, (d) curl-free horizontal currents, (e) total horizontal currents, and (f) corresponding on-ground magnetic field perturbations.
To the end-user, the model is provided as the Python package and source codes.The model package pyamps is available at PyPI via the URL https://pypi.org/project/pyamps/.It may be installed using the Python package (PIP).Source codes are available at GitHub via the URL https://github.com/klaundal/pyAMPS.

Discussion
We summarized features of the analyzed FAC models in Table 1.The listed models are ordered according to the year of the corresponding initial publication.As one can see, Iijima and Potemra [1][2][3] published the earliest research based on in-situ satellite measurements.Their results are still used as a qualitative model describing the general patterns of Region 1, Region 2, and Region 0 currents.It is not suitable as a quantitative model due to the limited amount of observations.However, there are ionospheric models that numerically fit values of FAC spatial distributions to conform patterns to Iijima and Potemra [1][2][3] and an ionosphere electric potential model (e.g., like [15]).From this point of view, the model [1-3] may be considered as a starting pattern to estimate quantitatively FACs distributions.Weimer-2001 [4] provided a more detailed FAC distribution in comparison with [1][2][3].The use of solar wind plasma parameters as model input allows better specification of the Sun's impact on the ionosphere.However, this model did not gain the popularity of the model [1][2][3] at least as far as we know.Papitashvili & co-authors [6] developed a detailed set of FAC maps representing the reference situations.For that time, it was the most detailed publically available set of FAC maps for the given parametrization.Lukianova & co-authors [8][9] constructed two separate models based on two different datasets.The principles 'under the hood' of models [8][9] are similar.The specific altitude was not mentioned in papers [8][9].As a result, we used the 'NA' symbols in Table 1 instead of the altitude.At the same time, we assume that this altitude should be close to the altitude of the satellite orbit.All the models [1-3, 4, 6, 8-9] were provided as graphical images ordered according to the specific set of parameters.To use these models, the interested parties should digitize those, filter out the noise, and perform interpolations for arbitrary values of given parameters.MFACE [10][11][12], AMPS [13][14], and EW-FAC [5] models were presented as software (see Fig. 1).These models provide FAC values for the same reference altitude and differ mainly by the input.It reflects different approaches to account for the Solar wind and IMF impact on the ionosphere.

Conclusions
This paper presents the review of the empirical models of field-aligned currents.The review is based on the publications indexed in main bibliographic databases (Web of Science, Scopus), scientific search engines, and social networks.Here we focus on the empirical FAC models only because theoretical FAC simulations require self-consistent modelling of the near-Earth's and the solar environments.These topics go far beyond the scope of this paper.

E3S
Web of Conferences 458, 01031 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345801031 (a) magnetic latitude in AACGM coordinates; (b) magnetic local time; (с) day of the year; (d) IMF BX, BY, and BZ components in GSM coordinates; (e) solar wind velocity; (f) AE-index (the AEindex could be recalculated from the solar wind parameters).The model output consists of the following parts.(a) FAC density at 110 km altitude.(b) Geomagnetic latitude at the reference point of the aurora current centre in AACGM coordinates.(c) EOF1 -empirical orthogonal function -component of the solution.The EOF1 component generally represents the Region 1 & 2 pattern with account for the IMF BZ component.(d) The EOF2 component which is generally attributed to the IMF BY-related variations as well as to the cusp region current.(e) The EOF0 -mean FACs density.

Fig. 1 .
Fig. 1.The scheme of FAC models ordered according to the form of distribution