Determining fieldaligned currents with the Swarm constellation mission
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Abstract
Fieldaligned currents (FAC) are the prime mechanism for coupling energy from the solar wind into the upper atmosphere at high latitudes. Knowing their intensity and distribution is of pivotal importance for the selection of quiet time data at high latitudes to be used in main field analysis. At the same time FACs can be regarded as a key element for studies of magnetosphereionosphere interactions. The Swarm satellite constellation, in particular the lower pair, provides the opportunity to determine radial currents uniquely. The computation of FACs from the vector magnetic field data is a straightforward and fast process, applying Ampère’s integral law to a set of four magnetic field values. In this method the horizontal magnetic field components at a quad of measurement points sampled by the two satellites moving sidebyside are interpreted. The presented algorithm was implemented as described here in the Swarm Level2 processing facility to provide the automatically estimated radial and fieldaligned currents. It was tested with synthetic data in the Swarm Level1b format. The resulting currents agree excellently with the input currents of the synthetic model. The data products are computed along the entire orbits. In addition, the L2 processor calculates also FACs with a 1 Hz time resolution individually from the three single Swarm satellites.
Key words
Fieldaligned currents ionosphere constellation mission1. Introduction
Field aligned currents play an important role in space plasmas. They are able to transfer energy almost lossless over large distances. In the magnetosphere, they connect distant source regions with the highlatitude ionosphere. There, they drive the entire auroral current system (Untiedt and Baumjohann, 1993). At lower latitudes, FACs flow whenever potential differences between the ionospheres of the two hemispheres build up. Rather wellknown are the interhemispheric FACs connecting the foci of the two Sq (solar quiet) current systems (e.g. Fukushima, 1979; Park et al., 2011). To improve knowledge about these processes it would be desirable to have reliable measurements of the FACs in nearEarth space. One way to acquire this information is by performing closelyspaced multipoint magnetic field measurements. ESA’s Swarm constellation mission provides this opportunity.
A constellation of three satellites can do more than three single satellites. In this sense, the purpose of the Swarm Level2 Processor is to provide advanced data products that take advantage of the dedicated constellation of three Swarm satellites (Olsen et al., 2013). Fieldaligned currents computed along track from the magnetic field measured at single satellites have always suffered from nonuniqueness. Since the satellite moves through threedimensional regions of high current density, the recorded field changes can be interpreted in terms of current density only if certain assumptions on the current geometry and its stationarity are made (Lühr et al., 1996). With measurements being available only along the orbit direction, i.e. alongtrack, the current distribution has to be generally assumed constant over the time span of passage and organized in sheets of known orientation (Lühr et al., 1996; Stauning et al., 2001).
More realistic FAC densities can be computed directly and uniquely from magnetic field measurements if synchronous, multipoint measurements spanning a twodimensional area in space are available. The two lower Swarm satellites flying sidebyside provide this type of datasets. The benefit of constellation processing for the determination of fieldaligned currents has been demonstrated in an ESAsponsored scientific study during Swarm Mission Phase A (Vennerstrøm et al., 2005; Ritter and Lühr, 2006). With the planned constellation of three satellites in nearpolar orbits (inclination ~87°) at two different heights, one at 530 km and a pair at initially 460 km (Olsen et al., 2013), the mission is particularly well suited to study the complex current systems of the polar ionosphere. The lower pair shall fly sidebyside, separated by only 1.4° in longitude which is equivalent to ~150 kilometres in east/west direction at the equator. The orbits of these two satellites cross near the poles. The simultaneous measurements of the two spacecraft, longitudinally spaced, provide the possibility to include the crosstrack spatial derivative directly in the computation and produce more complete results. This allows for the first time to determine the radial current density and from that fieldaligned currents in the ionosphere unambiguously by directly employing Ampère’s law as curlB relation or the surface integral solution (Ritter and Lühr, 2006).
A 3D curl B technique (Dunlop et al., 2002) has been developed for estimating FACs from fourpoint Cluster data by taking advantage of the fourspacecraft constellation. This technique has been applied at midaltitudes to estimate FAC densities and compared with success to the results obtained by the singlespacecraft method (e.g. Marchaudon et al., 2009).
The method for FAC determination described here, Ampère’s integral solution, was adapted and developed further for application to Swarm Level1b data. The algorithm subsequently presented was implemented in the Level2 processor to generate one of the Swarm Category2 (CAT2) products that are produced automatically by ESA’s processing centre as soon as all input data are available. The implemented processor was tested against the synthetic dataset generated for our Phase A study (Vennerstrøm et al., 2005, 2006; Moretto et al., 2006).
This paper describes the details of the algorithms for the multisatellite and singlesatellite FAC determination as implemented in the Level2 processor (Sections 2 and 3). An estimate of the uncertainties of provided FAC values is detailed in Section 4. Section 5 presents an outline of the output products and their formats. Finally we give a scientific validation of the derived fieldaligned current densities in Section 6 and an overview of the coordinate frames used in the algorithm in Section 7.
2. DualSatellite Method
In the paper at hand the integral method used to determine Swarm Level2 FAC is explained in detail, contrary to Ritter and Lühr (2006) that focused on the description of the curlB method. The integral method was selected above the curlB solution, because the requirements concerning the geometry of the four measurement points needed for the calculation of the current density is less stringent. The integration area spanned by the quad points doesn’t necessarily need to be rectangular, while the two components of spatial derivatives in the curlB approach have to be orthogonal. The integral method also avoids a division by vanishing crosstrack horizontal distances near the orbit crossover points. In the curlB technique with its horizontal gradients this might lead to unrealistically high current density values in this region.
For the determination of FACs the Swarm magnetic field vector data provided by the Level1b processor, B^{L1b}, are used. They are provided together with positions and timestamps in the NEC coordinate frame (NorthEastCenter, see Section 7). The following paragraphs will explain the processing steps further.
2.1 Data preprocessing
2.2 Definition of quad positions and resampling of data
For the estimation of the radial current density a regular quad of measurement points is most suitable. The proper choice of these quads is vital for the quality of the current density. The distance of the quad points in alongtrack direction should be comparable, at least in the most interesting regions, with the longitudinal separation, dy, of the orbits. This separation is ~150 km at the equator, decreasing at high latitudes and it vanishes near the geographic poles (dy ~ 150 km cos (lat)). Hence the time between alongtrack quad points was chosen 5 sec corresponding to ~40 km. To avoid collision at the orbit crossovers, the satellites pass the equator with a time separation of 5–10 s. This time separation Δt is taken into account when selecting the readings of symmetric quad points.
The synchronization of SwB with SwA is done for the northern and the southern hemisphere passages separately. The time shift is chosen that the satellites meet virtually at the crossover of their orbits. In this way, quads are defined at sampling intervals of 1 s along the time series of each hemispheric orbit arc. The current positions in the quad centres and time stamps are estimated from the mean of the coordinates (ITRF, Conventional Terrestrial Reference Frame, see Section 7) of all four quad points, e.g.: t_{fac} = (t_{1} + t_{2} + t_{3} + t_{4})/4.
2.3 Transformation of B into directions of route elements
B_{ℓ}(α_{ i }) is aligned with the axis along flight direction at any one of the quad points Q_{ i }. B_{ℓ}(β_{ i }) is aligned with the transverse axis of the quad, i.e. the connection line between two orbitsynchronous measurement points of SwA and SwB (see Eqs. (7), (8)). Q_{ i } refer to the quad points defined above. The values of B_{ℓ} are fed into the integral equation for the determination of the radial current density.
2.4 Calculation of radial currents by integration
2.5 Determination of fieldaligned currents
More details of the processing approach used for estimating the Swarm FAC products can be found in the Detailed Processing Model Document (Swarm Level 2 Processing System Consortium, 2012).
3. Determination of the SingleSatellite Solution
As for the multisatellite solution, the radial current component needs to be completed by the inclination of the magnetic field, as outlined in Eq. (16), to represent the FAC density.
4. Uncertainties of Radial and FieldAligned Currents

Biases, bmf, of any magnetic field reading (SwA, SwB, or SwC) are ± 1 nT (Mission Requirements Document, 2004); they do not change during the 5 s interval between two quad point readings;

Resolutions (digitisation noise), rmf, of any measurement magnetic field reading (SwA, SwB, or SwC) are ±0.1 nT (Mission Requirements Document, 2004);

Each magnetic field reading has an uncertainty of bmf ± rmf.

Positions and hence the route elements have no uncertainty (Medium Orbit Determination MOD, ~1 m)

The two route elements alongtrack are equally long. So are the two route elements acrosstrack. dℓ_{1} = dℓ_{3} and dℓ_{2} = dℓ_{4}.
5. Description of the Level2 IRC/FAC Product
List of parameters contained in the IRC/FAC data product.
Variable Name  Data Type  Description  

Timestamp  CDF EPOCH  Time stamp in UTC  
Latitude  CDF DOUBLE  geographic latitude [deg.]  
Longitude  CDF DOUBLE  geographic longitude [deg.]  
Radius  CDF DOUBLE  geographic radius [m]  
IRC  CDF DOUBLE  radial current density [μA/m^{2}]  
IRC_Error  CDF DOUBLE  uncertainty of current density [μ A/m^{2}]  
FAC  CDF DOUBLE  Fieldaligned current (FAC) density [μ A/m^{2}]  
FAC_Error  CDF DOUBLE  uncertainty of FAC density [μ A/m^{2}]  
Flags  CDF UINT4  flags related to IRC/FAC processing  
Flags_F  CDF UINT4  flags passed through from L1b  
Flags_B  CDF UINT4  flags passed through from L1b  
Flags_q  CDF UINT4  flags passed through from L1b 
The processing Flag (‘Flags’) has 10 individual and independent digits, each of which gives information about a different problem that may have occurred during the processing chain at each position of current density:
Digits 1–8 report problems that may have occurred at one or more of the measurement points used for the computation of a current density value. Values 0N mark the number of points that were affected by that problem: N = 1… 4 for normal FAC processing (4 measurement points involved),
N = 1… 2 for singlesatellite processing (2 measurement points involved).

0: default, none of the measurement points involved had the problem and processing was executed normally; the computed current density at that position is good.

1…N : One or more measurement points had the problem and a workaround was performed; the computed current density at that position might not be as good as it would be without the problem. It is not important, which one of the points had the problem.

0: default; if a NaN occurs at this record, the reason for the NaN is not known and results from computational problems. Normally this should not occur.

1: the NaN at this current position is intended: the current position is either near the geographic pole or near the magnetic equator.
List of problems reported by the processing Flag ‘Flags’.
DigitNr.  Value  Meaning of digit Flags > 0  

1  0/N  Data gap < 5 sec; data were interpolated linearly;  
2  0/N  Data is in filter tuning range due to larger data gap before and/or after the gap (gap > 5 sec); 0 for singlesatellite processing (no filter employed).  
3  0/N  no EST (external part of DST; Maus and Weidelt, 2004) data were available because no DST was available for magnetospheric field calculation; instead default value was used.  
4  0/N  no IST (internal part of DST; Maus and Weidelt, 2004) data were available because no DST was available for magnetospheric field calculation; instead default value was used.  
5  0/N  no Em (merging electric field; Kan and Lee, 1979) data was available because solar wind data were not available for magnetospheric field calculation; instead default value was used.  
6  0/N  no interplanetary magnetic field, IMF, was available for magnetospheric field calculation; instead default value was used.  
7  0/N  no solar flux parameter, F10.7, was available for magnetospheric field calculation; instead default value was used.  
8  0/N  No magnetospheric field coefficients were available; magnetospheric field is set to 0; resulting FACs are slightly less reliable.  
9  0/1  IRC = NaN and FAC = NaN because latitude ǀθǀ > 86° near geogr. pole  
10  0/1  FAC = NaN because inclination ǀIǀ < 30° nearmagn. Equator 
For each current estimate, the Level1b flag values of the measurement points concerned were added in the same way, as described for the processing flag.
6. Scientific Validation of the Level2 FAC Product
To validate the radial current algorithm we used the synthetic dataset as employed for the phase A study (Vennerstrøm et al., 2005, 2006; Moretto et al., 2006). For generating this test dataset, a global MagnetoHydroDynamics (MHD) model (GGCM, Raeder, 2003) had been run at the Community Coordinated Modeling Centre (CCMC) to simulate the interaction of the solar wind with the magnetosphere. The resulting fieldaligned currents were closed in the ionosphere. By employing an empirical model for the ionospheric conductivity the spatial distribution of the electric potential was deduced, and Hall and Pedersen currents in the ionosphere could be computed. The 3D distribution of magnetic field perturbations generated by these currents was derived. The magnetic field components were then computed at spherical grid points. For a verification of the processing algorithms synthetic magnetic field measurements were derived along predicted Swarm orbits by cubic spline interpolation of the model data on the grid points.
The FAC output of both the dualsatellite method and the singlesatellite method (using SWA, SWB and SWC separately) are compared to the FAC densities of the input model of the test dataset. For this purpose the FACs of this input model are sampled along the estimated FAC positions of the data product for validating the current densities resulting from the FAC algorithm.
A proper scientific validation of the multipoint FAC determination is not possible because there are no real magnetic field data available that could represent Swarm measurements. This task has to be performed as part of the Swarm validation activity during the early mission phase. During the inflight operation the multisatellite FAC estimates will also be compared with the singlesatellite FACs (from SWA and/or SWB separately).
This example gives an impression of the rich variety of different FAC sources that can be investigated with the help of the Swarm Level 2 data product described here. These data will be important for correctly characterizing the electrodynamics in the ionosphere.
7. Coordinate Frames
In the FAC processing, the following coordinate frames are used:
ITRF: (x, y, z) The IERS Conventional Terrestrial Reference Frame (ITRF) is an Earthfixed Cartesian system used for describing the orbit ephemeris. The origin of the frame is the Earth’s centre of mass. The x axis points towards the IERS Reference Meridian (close to Greenwich); the z axis points to the Reference North Pole; the y axis completes the triad.
NEC (NorthEastCenter ITRF): (x, y, z) The x and y components lie in the horizontal plane, pointing northward and eastward, respectively. z points to the centre of gravity of the Earth.
MFA (Mean FieldAligned): (x, y, z) The MFA frame is a local coordinate system defined by the ambient magnetic field. It is particularly useful for describing electric currents in the topside ionosphere. The origin is the local measurement point of the magnetic field. The z axis is aligned with the unperturbed magnetic field which points from the southern to the northern hemisphere; the y axis is perpendicular to the magnetic meridian pointing predominantly eastward; the x axis completes the triad having an outward component.
VSC (spacecraft velocity): (x, y, z) This frame is introduced for singlesatellite FACs calculation. The x and y components lie in the horizontal plane, pointing 45° away from the actual flight direction. z points to the centre of the Earth.
VHQ (Velocityoriented Horizontal Quad): (α, β) The frame describes the orientation of the route elements with respect to the direction towards the pole in the LTL frame. The angle α describes the angle between an alongtrack route element and the direction towards the pole, whereas β denotes the angle between a route element transverse to the flight direction and the poleward direction.
LTL (Local TimeLatitude) frame: (r, θ, λ) describe the geocentric position with respect to the local time (LT) frame, where r is the radial distance from the Earth’s centre, θ the colatitudes and λ a local time related longitude.
Notes
Acknowledgments
The authors acknowledge the valuable help and comments by the reviewers. The CHAMP mission was sponsored by the Space Agency of the German Aerospace Center (DLR) through funds of the Federal Ministry of Economics and Technology, following a decision of the German Federal Parliament (grant code 50EE0944). The development of the Swarm L2FAC prototype was sponsored by the European Space Agency (ESTEC) through contract No. 4000102140/10/NL/JA.
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