Determining field-aligned currents with the Swarm constellation mission

Abstract

Field-aligned 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 magnetosphere-ionosphere 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 side-by-side are interpreted. The presented algorithm was implemented as described here in the Swarm Level-2 processing facility to provide the automatically estimated radial and field-aligned currents. It was tested with synthetic data in the Swarm Level-1b 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.

Earth Planets Space, 65, 12851294, 2013

Determining eld-aligned currents with the Swarm constellation mission

Patricia Ritter, Hermann Lhr, and Jan Rauberg

GFZ German Centre for Geosciences, Potsdam, Germany

(Received March 1, 2013; Revised August 14, 2013; Accepted September 9, 2013; Online published November 22, 2013)

Field-aligned 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 eld analysis. At the same time FACs can be regarded as a key element for studies of magnetosphere-ionosphere 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 eld data is a straightforward and fast process, applying Ampres integral law to a set of four magnetic eld values. In this method the horizontal magnetic eld components at a quad of measurement points sampled by the two satellites moving side-by-side are interpreted. The presented algorithm was implemented as described here in the Swarm Level-2 processing facility to provide the automatically estimated radial and eld-aligned currents. It was tested with synthetic data in the Swarm Level-1b 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: Field-aligned currents, ionosphere, constellation mission.

1. Introduction

Field aligned currents play an important role in space plasmas. They are able to transfer energy almost loss-less over large distances. In the magnetosphere, they connect distant source regions with the high-latitude ionosphere. There, they drive the entire auroral current system (Untiedt and Baumjohann, 1993). At lower latitudes, FACs ow whenever potential differences between the ionospheres of the two hemispheres build up. Rather well-known 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 near-Earth space. One way to acquire this information is by performing closely-spaced multi-point magnetic eld measurements. ESAs 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 Level-2 Processor is to provide advanced data products that take advantage of the dedicated constellation of three Swarm satellites (Olsen et al., 2013). Field-aligned currents computed along track from the magnetic eld measured at single satellites have always suffered from non-uniqueness. Since the satellite moves through three-dimensional regions of high current density, the recorded eld changes can be interpreted in terms of current density only if certain assumptions on the current geometry and its stationarity are

Copyright c

[circlecopyrt] The Society of Geomagnetism and Earth, Planetary and Space Sci

ences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB.

doi:10.5047/eps.2013.09.006

made (Lhr et al., 1996). With measurements being available only along the orbit direction, i.e. along-track, the current distribution has to be generally assumed constant over the time span of passage and organized in sheets of known orientation (Lhr et al., 1996; Stauning et al., 2001).

More realistic FAC densities can be computed directly and uniquely from magnetic eld measurements if synchronous, multi-point measurements spanning a two-dimensional area in space are available. The two lower Swarm satellites ying side-by-side provide this type of datasets. The benet of constellation processing for the determination of eld-aligned currents has been demonstrated in an ESA-sponsored scientic study during Swarm Mission Phase A (Vennerstrm et al., 2005; Ritter and Lhr, 2006). With the planned constellation of three satellites in near-polar 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 y side-by-side, 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 cross-track spatial derivative directly in the computation and produce more complete results. This allows for the rst time to determine the radial current density and from that eld-aligned currents in the ionosphere unambiguously by directly employing Ampres law as curl-B relation or the surface integral solution (Ritter and Lhr, 2006).

A 3D curl B technique (Dunlop et al., 2002) has been

1285

1286 P. RITTER et al.: FIELD-ALIGNED CURRENTS FROM SWARM CONSTELLATION

Fig. 1. (a) Sketch of a quad of 4 measurements points needed for the calculation of FAC density at the centre. Points Q1, . . ., Q4 are positions on the orbit tracks of satellites SwA and SwB connected by route elements d . (b) Measurement points on SwB orbit are selected at slightly shifted times with respect to SwA, so that a symmetric quad is achieved.

developed for estimating FACs from four-point Cluster data by taking advantage of the four-spacecraft constellation. This technique has been applied at mid-altitudes to estimate FAC densities and compared with success to the results obtained by the single-spacecraft method (e.g. Marchaudon et al., 2009).

The method for FAC determination described here, Ampres integral solution, was adapted and developed further for application to Swarm Level-1b data. The algorithm subsequently presented was implemented in the Level-2 processor to generate one of the Swarm Category-2 (CAT-2) products that are produced automatically by ESAs 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 (Vennerstrm et al., 2005, 2006; Moretto et al., 2006).

This paper describes the details of the algorithms for the multi-satellite and single-satellite FAC determination as implemented in the Level-2 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 scientic validation of the derived eld-aligned current densities in Section 6 and an overview of the coordinate frames used in the algorithm in Section 7.

2. Dual-Satellite Method

The Swarm satellite constellation, in particular the lower pair, provides the opportunity to determine radial and eld-aligned currents uniquely. The technique used for estimating the radial current density employs the horizontal B eld components observed by the lower pair of satellites at 4 measurement points forming a symmetric quad. For the Swarm Phase A study, the traditional form of Ampres law, the curl-B relation, was employed to calculate the vertical current density, jz:

jz =

1 0

Where 0 is the magnetic permeability, Bx and By are the horizontal components of the magnetic eld after removal of the mean magnetic eld from the measurements (hereafter called residual eld).

The mathematical approach used here is the contour integral in the horizontal plane as also described briey in Ritter and Lhr (2006):

j =

1 0 A

B d . (2)

The integral is performed along the closed path of the connecting lines of the quad points. d is a route element along the integration path, B is the residual magnetic eld vector (Bx, By, Bz) in the coordinate frame of the ight direction (VHQ, Velocity-Oriented Horizontal Quad, see Section 7), A is the encircled area, and 0 is the magnetic permeability.

The position, where the obtained current density estimate is assigned to, is the centre of the quad. The coding of this approach involves interpolation procedures and simple rotation and subtraction routines. Figure 1 sketches a quad of 4 measurement points, Q1, . . . , Q4, on the orbit tracks of satellites SwA and SwB, and the integration path encircling the spanned integration area. The current density obtained is a mean value of the current densities within the quad. Also the temporal variations of the currents within the quad points are averaged over about 20 s.

In the paper at hand the integral method used to determine Swarm Level-2 FAC is explained in detail, contrary to Ritter and Lhr (2006) that focused on the description of the curl-B method. The integral method was selected above the curl-B 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 doesnt necessarily need to be rectangular, while the two components of spatial derivatives in the curl-B approach have to be orthogonal. The integral method also avoids a division by vanishing cross-track horizontal distances near the orbit crossover points. In the curl-B technique with its horizontal gradients

Bydx Bx dy

. (1)

P. RITTER et al.: FIELD-ALIGNED CURRENTS FROM SWARM CONSTELLATION 1287

this might lead to unrealistically high current density values in this region.

For the determination of FACs the Swarm magnetic eld vector data provided by the Level-1b processor, BL1b, are

used. They are provided together with positions and timestamps in the NEC coordinate frame (North-East-Center, see Section 7). The following paragraphs will explain the processing steps further.2.1 Data pre-processing

First, the time series are checked for data gaps, interpolated if gaps are minimal, and/or agged accordingly. The FACs are computed, as mentioned before, from residual magnetic eld vector data. That means, the measured magnetic eld is corrected for the core, crustal and magneto-spheric elds at satellite altitude. For calculating the magnetic eld at the orbit positions, initially the following models are utilized: IGRF11 (Finlay et al., 2010) for the core eld, MF7 (Maus et al., 2007) for the lithospheric eld, and part of POMME-6 (Lhr and Maus, 2010) for the external eld. The mean eld, i.e. the sum of BCOR + BLIT BEXT,

is subtracted from the measured magnetic eld data BL1b:

BRES = BL1b BCOR BLIT BEXT (3)

where BCOR is the core magnetic eld, BLIT describes the lithospheric magnetization, and BEXT denotes the magnetic eld from magnetospheric currents. Subsequently the residual B-eld data are low-pass ltered to ensure that only currents with spatial scale lengths >150 km are represented.This scale length corresponds to the east-west separation of the two satellites at the equator. It is recommended to use a cut-off period of 1020 sec for this lter to avoid anti-aliasing. This lter also suppresses Alfvnic wave parts (Ishii et al., 1992) in the data, so that only the stationary part of the FAC is captured by the procedure.2.2 Denition 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 along-track 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 along-

track 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 510 s. This time separation t 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 dened 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.: tFAC = (t1 + t2 + t3 + t4)/4.

Fig. 2. Angles and between route elements along-track and cross-track and the geographic pole.

2.3 Transformation of B into directions of route elements

The route elements d i (i = 1 . . . 4, see Fig. 1(a)) of

each quad are determined by the vector differences of the position vectors to each quad point in the local time/latitude (LTL, see Section 7) frame. The LT-related longitude, , is estimated using

= + (t/86400) 360 (4) where is the geocentric longitude.

The route elements are estimated from the Cartesian position vectors:

d 1 =

(x1 x2)2 + (y1 y2)2 + (z1 z2)2 d 2 =

(x2 x3)2 + (y2 y3)2 + (z2 z3)2 d 3 =

(x3 x4)2 + (y3 y4)2 + (z3 z4)2 d 4 =

(x4 x1)2 + (y4 y1)2 + (z4 z1)2. (5)

The route elements, d , mark the path from one quad point to the next one; for example, d 1 marks the path from point Q1 to point Q2. Figure 3(top) shows the evolution of the route elements size on one full orbit. Along-track-route elements along track, d 1,3, are constant, whereas those between the satellites, d 2,4, are smallest close to the geographic poles, where the satellites orbits cross.

In order to determine the magnetic eld component in ight direction, the angles between route elements parallel () and route elements transverse () to ight direction and the respective geographic pole are dened as sketched in Fig. 2 for the satellites passage of the northern hemisphere. The equations for deriving the angles and take

1288 P. RITTER et al.: FIELD-ALIGNED CURRENTS FROM SWARM CONSTELLATION

into account the spherical shape of the ionosphere. They are based on spherical geometry and make use of great circle distances. The basic equation is the sine formula for triangles on a sphere:

sin (arc( )) sin ( ) =

sin (arc())sin () . (6)

Where arc( ) and arc() represent the sides of the triangle opposite to the respective angles and .

1 =

asin

sin |2 1| sin 2

sin[acos(cos 1 cos 2+sin 1 sin 2 cos(2 1))]

(7)

2 = asin

sin 1sin 2 sin 1

4 =

asin

sin |3 4| sin 3

sin[acos(cos 3 cos 4+sin 3 sin 4 cos(3 4))]

3 = asin

sin 4sin 3 sin 4

2 =

asin

sin |3 2| sin 3

sin[acos(cos 2 cos 3+sin 2 sin 3 cos(3 2))]

3 = asin

sin 2sin 3 sin 2

1 =

asin

sin |4 1| sin 4

sin[acos(cos 4 cos 1+sin 4 sin 1 cos(4 1))]

sin 1sin 4 sin 1 .

In all equations above, i refers to the co-latitude and i to the longitude of a given quad point in the LTL frame. In the southern hemisphere, the angles are estimated relative to the South pole. However, the equations above can be used for the processing of either hemisphere, as the sign switches of the angles at the equator. When the satellites cross close to the geographic poles, the angles and need to be adapted to account for sign switches of the components. Further details are given in the Detailed Processing Model Document (Swarm Level 2 Processing System Consortium, 2012).

Figure 3(bottom) shows the evolution of angles and after this adaption on one full orbit. Both angles vary smoothly and steadily along the orbit. Only jumps by 180 at the orbit crossings. On the descending arc, takes values within the range [90 . . . 0 . . . 90], on the ascending arc, the angles

range between [90 . . . 180 . . . 90]. The angle

ranges from [0 . . . 90 . . . 0] on the descending arc, whereas it takes values of [180 . . . 90 . . . 180] on the ascending arc.

The angles and are then used to transform the residual magnetic elds from NEC into the ight direction frame VHQ (Velocity-oriented Horizontal Quad):

BQi (i) = BQix cos i BQiy sin i (9)

Fig. 3. Top: Route elements d 1,3 along-track are constant, whereas the cross-track ones, d 2,4 are smallest close to the geographic poles (the crossings are located at indices 1400 (near North Pole), 4150 (near South Pole), 6900 (near North Pole)); bottom: variation of angles and along the same orbit (descending arc (14004150) ascending arc (41516900)).

BQi (i) = BQix cos i + BQiy sin i (10) with i = 1, 2 for spacecraft SwB and with i = 3, 4 for

(8) spacecraft SwA.B (i) is aligned with the axis along ight direction at any one of the quad points Qi. B (i) is aligned with the transverse axis of the quad, i.e. the connection line between two orbit-synchronous measurement points of SwA and SwB (see Eqs. (7), (8)). Qi refer to the quad points dened above. The values of B are fed into the integral equation for the determination of the radial current density.2.4 Calculation of radial currents by integration

From the rotated B vector the four product terms Pi for the integral can be estimated from the sums of the horizontal elds. We use averages of the magnetic eld values at the end points of the route elements:

P1 =

1

2(BQ1 (1) + BQ2 (2)) d 1 (11)

P2 =

4 = asin

1

2(BQ2 (2) + BQ3 (3)) d 2

P3 =

1

2(BQ3 (3) + BQ4 (4)) d 3

P4 =

12(BQ4 (4) + BQ1 (1)) d 4.

The contour integral (Eq. (2)) is performed along the closed path of the connecting lines of the quad points. d is a route element along the integration path, B is the measured magnetic eld vector (Bx, By, Bz) in the coordinate frame of the ight direction (VHQ). Practically the integral is performed by adding the product terms Pi:

Bd = (P1 + P2 + P3 + P4). (12)

Division by the magnetic permeability, 0, and the quad area, A, yields the radial current density jr:

jr =

10 A (P1 + P2 + P3 + P4)/1000. (13)

P. RITTER et al.: FIELD-ALIGNED CURRENTS FROM SWARM CONSTELLATION 1289

If the route elements, like the position radii, and the enclosed area A are given in [m] and the magnetic eld readings in [nT], the values need to be divided by 1000 to obtain the current density in units of [A/m2].

The integration area is the area encircled by the route elements d (Fig. 1(a)):

A =

1

2

satellite data, the current is generally assumed to ow in current sheets perpendicular to the magnetic meridian. Ideally, the eld-aligned current, j is computed from the tem

poral gradient of the azimuthal magnetic eld component and the spacecraft velocity along the meridional component in the mean eld-aligned coordinate (MFA, see Section 7) frame:

j =

1 0dt

(d 2 + d 4)

d 1 sin

12(1 1 + 2 2)

+d 3 sin

d BMFAy

V MFAx

12(3 3 + 4 4) . (14)

Here, we allow for a certain distortion of the area framed by the symmetric quad. This radial current density is computed at each of the quads at 1 s intervals along the entire orbits. Close to the geographic poles, no radial currents can be estimated if the horizontal spacecraft distance d 2 or d 4 is too small. Initially, we chose 3 km for the minimum distance of the satellites (Ritter and Lhr, 2006). This is the case for geographic latitudes > 86 and causes a data gap of 15 s near the poles, occurring typically in the quiet polar cap region.2.5 Determination of eld-aligned currents

The Swarm L1b data is given in the Earth-xed coordinate system (NEC): i.e. the three coordinate axes are xed with respect to the rotating Earths surface. The radial current, jr, computed in the previous section from these data is also presented in this system. To obtain the current density owing along the mean magnetic eld, i.e. the eld-aligned current, this radial current component has to be completed by the inclination of the mean magnetic eld. The mean magnetic eld vector BMF at each current position is computed as the sum of the core, crustal and magnetospheric model elds in the NEC frame (see Eq. (3)). The mean eld inclination is estimated from the above models at each current position:

I = tan1

BMFz

BMF2x + BMF2y

. (17)

For the actual implementation in the Level-2 processor, we chose a different approach. To avoid singularities of the velocity in the MFA system near the equators, the measurements are transformed into the spacecraft velocity frame (VSC, see Section 7) and the eld-aligned currents are assumed to ow in sheets perpendicular to the orbit plane. Obviously the real current geometry may deviate from this ideal setting. However, the simplistic approach described here was chosen above more sophisticated methods such as Minimal Variance Analysis to guarantee provide a fast and reliable automatic data processing. Deviations from the assumption given above are accounted for by the estimation of uncertainties detailed in Section 4. The eld gradients and velocities are computed at time spacings of dt = 1 s.

In case of the single-satellite solutions, no ltering of the data is performed. Therefore this technique resolves also small-scale structures.

In the VSC frame the two horizontal velocity components V VSCx and V VSCy are equal:

V VSCx = V NECx cos + V NECy sin (18) V VSCy = V NECx sin + V NECy cos .

Based on this condition, the rotation angle can be derived from the ratio of the difference and sum of the V NEC com

ponents:

= tan1

V NECx V NECy V NECx + V NECy

. (15)

The full eld-aligned current density, j , is obtained by

dividing the radial current reading, jr, by the sine of the inclination angle:

j =

jrsin I . (16)The FAC density comes in units of [A/m2]. To avoid unrealistically large FAC densities around the magnetic equator due to near-horizontal inclinations, no FAC values will be estimated for magnetic eld inclinations, |I|, smaller than

30, i.e. about 15 in latitude off the magnetic equator.

The radial current densities are reported all the way across the equator and thus can be used to monitor the vertical currents driven by the F-region dynamo.

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 Single-Satellite Solution

The single-satellite FAC processing is done in a separate step. For the simple FAC determination from single-

. (19)

The angle is also used for the transformation of the magnetic eld from the NEC into the VSC frame:

BVSCx = BNECx cos + BNECy sin (20) BVSCy = BNECx sin + BNECy cos .

The ionospheric radial current (IRC) density is computed from the horizontal gradients of BVSC (i.e. differences between two successive measurements, e.g. BVSCx = BVSCx2

BVSCx1) and the horizontal velocity components of V VSC (Lhr et al., 1996):

jr =

1

20dt

d BVSCyV VSCx d BVSCx V VSCy

1000. (21)

Since the velocities Vx and Vy have the same values, the magnetic eld variations Bx and By are weighted equally (Lhr et al., 1996). If the velocities are given in [m/s] and the magnetic eld measurements in [nT], the current estimates are obtain in the units [A/m2]. The positions

1290 P. RITTER et al.: FIELD-ALIGNED CURRENTS FROM SWARM CONSTELLATION

assigned to the currents are the centres between the two measurement points.

As for the multi-satellite solution, the radial current component needs to be completed by the inclination of the magnetic eld, as outlined in Eq. (16), to represent the FAC density.

4. Uncertainties of Radial and Field-Aligned Currents

The formal uncertainty calculation for the IRC / FACs is based on the following assumptions:

Biases, bmf, of any magnetic eld reading (SwA,

SwB, or SwC) are 1 nT (Mission Requirements Doc

ument, 2004); they do not change during the 5 s inter

val between two quad point readings;

Resolutions (digitisation noise), rmf, of any measure

ment magnetic eld reading (SwA, SwB, or SwC) are

0.1 nT (Mission Requirements Document, 2004);

Each magnetic eld reading has an uncertainty of

bmf rmf. Positions and hence the route elements have no uncer

tainty (Medium Orbit Determination MOD, 1 m)

These numbers represent performance characteristics of the Swarm Level-1b data products.

The two route elements along-track are equally long.

So are the two route elements across-track. d 1 = d 3

and d 2 = d 4.

For the product terms Pi of the integral (Eq. (11)) the uncertainties along the hemispheres are:

P1

1

2(bmfSW2 2rmfSW2) d 1

Fig. 4. Log10 of uncertainties of IRC ( jr ) and FAC ( j , line interrupted

near the equator) along the satellite path from the southern to the northern hemisphere.

2 rmf2SW1 + 2 rmf2SW2 d 2

+ 2 rmf2SW1 + 2 rmf2SW2

d 1

1000.

(24)

Note that the biases and resolutions of independent measurements are added as squares in the root terms. The second term is constant, since the route element d 1 doesnt vary along the orbit. Hence the variation of the ICR uncertainty depends entirely on the variation of the cross-track route elements d 2. The IRC uncertainty is largest in the high latitude regions, where d 2 gets very small, and minimal around the equator. The obtained formal uncertainties range from 12 to 430 [nA/m2] (see Fig. 4). Uncertainties of the mean magnetic eld models cancel in the integration process, because they are based on scalar potentials.

For the single-satellite solution, an additional uncertainty is due to the unknown orientation of the current sheet. It generally causes an underestimation of the current density. We assume a decit of 15% on average of the current estimate accounting for a tilt angle up to 45. With these assumptions the current calculated in Eq. (21) has an uncertainty of:

jr =

1

20dt

(22)

1

2(bmfSW2 rmfSW2

+ bmfSW1 rmfSW1) d 2

P2

P3

1

2(bmfSW1 2rmfSW1) d 3

1

2(bmfSW1 rmfSW1

+ bmfSW2 rmfSW2) d 4

P4

2 rmf2 + 2 rmf2

|V VSC|

With above assumptions the sum of these terms yields:

i=1:4Pi =

1000

15%( jr). (25)

Gradients are estimated from two readings of the same satellite and are assumed to have a resolution of 1 rmf. The velocity components in Eq. (21) are constant along the orbit:

|V VSC| = V VSCx = V VSCy 5.3 km/s. (26) Hence the uncertainty estimate of the single-satellite radial current can be quantied as

jr = 15 nA/m2 15%( jr). (27)

bmf2SW1 + bmf2SW2

2 rmf2SW1 + 2 rmf2SW2 d 1

+

2 rmf2SW1 + 2 rmf2SW2 d 2 (23)

Using the integration area given in Eq. (14) in the simplied form A = d 1 d 2 we obtain for the uncertainty of the

integral Eq. (13):

jr =

1 0

bmf2SW1 + bmf2SW2

P. RITTER et al.: FIELD-ALIGNED CURRENTS FROM SWARM CONSTELLATION 1291

Table 1. 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 [A/m2]

IRC Error CDF DOUBLE uncertainty of current density [A/m2]

FAC CDF DOUBLE Field-aligned current (FAC) density [A/m2]

FAC Error CDF DOUBLE uncertainty of FAC density [A/m2]

Flags CDF UINT4 ags related to IRC/FAC processing Flags F CDF UINT4 ags passed through from L1b Flags B CDF UINT4 ags passed through from L1b Flags q CDF UINT4 ags passed through from L1b

Table 2. List of problems reported by the processing Flag Flags.

Digit-Nr. Value Meaning of digit Flags > 01 0/N Data gap < 5 sec; data were interpolated linearly;

2 0/N Data is in lter tuning range due to larger data gap before and/or after the gap (gap > 5 sec);

0 for single-satellite processing (no lter employed).3 0/N no EST (external part of DST; Maus and Weidelt, 2004) data were available because no DST was available for magnetospheric eld 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 eld calculation; instead default value was used.5 0/N no Em (merging electric eld; Kan and Lee, 1979) data was available because solar wind data were not available for magnetospheric eld calculation; instead default value was used.6 0/N no interplanetary magnetic eld, IMF, was available for magnetospheric eld calculation; instead default value was used.7 0/N no solar ux parameter, F10.7, was available for magnetospheric eld calculation;instead default value was used.8 0/N No magnetospheric eld coefcients were available; magnetospheric eld is set to 0; resulting FACs are slightly less reliable.9 0/1 IRC = NaN and FAC = NaN because latitude || > 86 near geogr. pole

10 0/1 FAC = NaN because inclination |I| < 30 near magn. Equator

Uncertainties due to the mean magnetic eld subtracted in the preprocessing step may also play a role and are estimated to account for 5% of the current density.

The FAC uncertainty is determined by dividing the ICR uncertainty by the sine of the inclination angle:

j =

jrsin I . (28)

The inclination-induced uncertainty is small in the high latitude regions and increases towards the equator (see Fig. 4).

5. Description of the Level-2 IRC/FAC Product

The radial and eld-aligned current densities are calculated by the Swarm Level-2 processor as an automatically computed product. The product is provided using the dual-satellite method on the lower pair of satellites SwA and SwB (Swarm L2 product name: FAC TMS 2F), and the single-satellite solution for each of the Swarm spacecraft SwA, SwB, and SwC individually (Swarm L2 product name: FACxTMS 2F, x=A,B,C). The IRC/FACs data are given with a time resolution of 1 Hz. For the dual-satellite solution the data are ltered, hence the scale size of the resulting current density is >150 km. The unltered 1 Hz

single-satellite solution has a scale length of >15 km. The dual satellite data product and the three single satellite solutions are provided in separate les. The main part of each of the L2-FAC products consists of timestamp, position, IRC density, FAC density, product ag and Level-1b quality ag. These parameters are listed in Table 1.

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 18 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 single-satellite processing (2 measure

ment points involved).

At each current position, the ag values of the measurement points concerned were added. The values of the digits have the following interpretation:

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

1292 P. RITTER et al.: FIELD-ALIGNED CURRENTS FROM SWARM CONSTELLATION

is good.1 . . . N: One or more measurement points had the problem and a work-around 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.

Digits 910 report whether the NaN value of the current density at this position is intended or not:

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.

Table 2 lists the problems reported by the 10 digits of Flags.

For each current estimate, the Level-1b ag values of the measurement points concerned were added in the same way, as described for the processing ag.

6. Scientic Validation of the Level-2 FAC Product

To validate the radial current algorithm we used the synthetic dataset as employed for the phase A study (Vennerstrm et al., 2005, 2006; Moretto et al., 2006). For generating this test dataset, a global Magneto-Hydro-Dynamics (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 magneto-sphere. The resulting eld-aligned 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 eld perturbations generated by these currents was derived. The magnetic eld components were then computed at spherical grid points. For a verication of the processing algorithms synthetic magnetic eld 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 dual-satellite method and the single-satellite 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.

Figure 5 shows the comparison of the resulting IRC and the model input current at the northern hemisphere. The solid red curve shows the radial current computed by the FAC processor on the 2nd pass across the northern hemisphere on 2000-04-05. The blue curve shows the radial current retrieved from the input model at the computed current positions of the same polar pass for reference. The satellites cross the polar region from the left to the right side. Since the model currents cover only a region down to 60

of latitude, signatures equatorward of 60 are meaningless in terms of currents. The root mean square (rms) of the differences between the computed radial current and the ref-

Fig. 5. Comparison of the resulting IRC and the model input current on a polar passage across the northern hemisphere.

Fig. 6. Comparison of the resulting IRC and the model input current on a polar passage across the southern hemisphere.

erence current on a prole [30] across the geographic

North Pole is 28.9 nA/m2. The agreement of the model input and processor result is excellent.

Figure 6 shows that the comparison of results at the southern hemisphere is on a similar high level. The solid red curve shows the radial current computed by the FAC processor on the 3rd pass across the southern hemisphere on 2000-04-05. The root mean square (rms) of the differences between the computed radial current and the reference current on a prole [30] across the geographic South Pole is

26.8 nA/m2.

A proper scientic validation of the multi-point FAC determination is not possible because there are no real magnetic eld 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 in-ight operation the multi-satellite FAC estimates will also be compared with the single-satellite FACs (from SWA and/or SWB separately).

P. RITTER et al.: FIELD-ALIGNED CURRENTS FROM SWARM CONSTELLATION 1293

of the Earth.

MFA (Mean Field-Aligned): (x, y, z) The MFA frame is a local coordinate system dened by the ambient magnetic eld. It is particularly useful for describing electric currents in the topside ionosphere. The origin is the local measurement point of the magnetic eld. The z axis is aligned with the unperturbed magnetic eld 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 single-satellite FACs calculation. The x and y components lie in the horizontal plane, pointing 45 away from the actual ight direction. z points to the centre of the Earth.

VHQ (Velocity-oriented Horizontal Quad): (, ) The frame describes the orientation of the route elements with respect to the direction towards the pole in the LTL frame. The angle describes the angle between an along-track route element and the direction towards the pole, whereas denotes the angle between a route element transverse to the ight direction and the poleward direction.

LTL (Local TimeLatitude) frame: (r, , ) describe the geocentric position with respect to the local time (LT) frame, where r is the radial distance from the Earths centre, the colatitudes and a local time related longitude.

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 L2-FAC prototype was sponsored by the European Space Agency (ESTEC) through contract No. 4000102140/10/NL/JA.

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Fig. 7. Observed signatures of the radial current density (CHAMP): log10(FAC) on the nightside (top) and dayside (bottom). On the night-side, radial current due to plasma bubbles signatures near the magnetic equator are visible.

In order to give an impression of the size and distribution of eld-aligned currents expected along Swarm orbits, we present in Fig. 7 FAC density magnitudes derived from CHAMP magnetic eld measurements. Since the range of FAC intensities is so large, a logarithmic scale (log 10) was chosen. Data are from 02 Jan. 2001. The local time of the orbit is 23 h (top panel) and 11 h (bottom panel). As expected, there are strong FACs observed in the auroral regions, larger amplitudes in the southern (summer) hemisphere than in the northern. At middle and low latitudes also FACs are owing but at much reduced intensity. On the nightside FAC signatures are primarily related to plasma bubbles near the magnetic equator or to mid-latitude ionospheric irregularities (e.g. Medium Scale Travelling. Disturbances (MSTIDs), near 40 gm lat. in Fig. 7). On the dayside at middle latitudes the FAC density is on average somewhat enhanced. Broad peaks are found around 30

of mag. latitude. They coincide reasonably well with the typical positions of the Sq foci where potential differences between the hemispheres are expected to be largest.

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 Earth-xed Cartesian system used for describing the orbit ephemeris. The origin of the frame is the Earths 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 (North-East-Center 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

1294 P. RITTER et al.: FIELD-ALIGNED CURRENTS FROM SWARM CONSTELLATION

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P. Ritter (e-mail: [email protected]), H. Lhr, and J. Rauberg