1. M W Dunlop, Y Bogdanova, S Chao, H Lühr, N Olsen, Y-Y. Yang.
Activities for constellation data: Cluster coordination
M W Dunlop, Y Bogdanova, S Chao, H Lühr, N Olsen, Y-Y. Yang.

2. Swarm Concept: Calculation of FAC (strictly: vertical currents) Direct modelling of equatorial induced field
Lessons Learned
From Cluster !!
One face:
Ritter et al.
Comparisons between fewer than 4-spacecraft:
stationarity assumption
single face: lack of knowledge of other j-components
quality control difficult
Enormous benefit from additional s/c:
Cluster: desire 4 s/c, bunched
Swarm: exploit initial period of close 3 s/c
‘C’ above ‘A-B’ (common LT)
Spot comparisons with 4-spacecraft at MEO:
Current components through each face: spacecraft configuration and scale are important.
Local extent of FACs can be investigated in some detail using combination of boundary and curlometer analysis.
Best estimates of particular components of curl B (Jf and J||) depend on sampling (orientation to RC).

3. Curlometer: ^B = 0 .B = m0J
Analysis tools: full Curlometer Curlometer is self- stabilising (closing of difference terms) natural estimates of div(B)
Curlometer: ^B = 0 .B = m0J
Uses Ampère’s law to estimate the average current
density through the tetrahedron:
μ0<J>(Δri^ΔRj) = ΔBiΔRj- ΔBjΔRi
e.g. μ0<J>123(Δr12^ΔR13) = ΔB12ΔR13 – ΔB13ΔR12
we also have:
<div(B)>|RiRj^Rk| = |cyclicBiRj^Rk|
i.e. <div(B)>1234(R12R13^R14) = B12R13^R14 + B13R14^R12 + B14R12^R13
This estimates J normal to the face 1ij of the tetrahedron.
curl(B).ds = B.dl
div(B).dv = B.ds

4. Cluster experience: current density
Cluster has often crossed the RC during its 13 years of operations.
Each RC crossing will also encounter adjacent FACs.
Suitable for spot checks of RC strength.
Escoubet et al., 2001
Curlometer: point by point calculation
(divB , s/c configuration and temporal stability).
Direct curl B calculations can identify the FAC structure and Ring Current :
Filamentary small scale signatures: some are FAC, but not all, and temporal behaviour is often present.
Ring current is generally well defined, but requires particular constellations for high accuracy.
Could also test Swarm FAC ‘curlB’ method using pairs of Cluster spacecraft and comparison to the full curlometer (4 s/c).

5. Ring current analysis: FAC
In situ RC may change with growth of eastward current.
Need care with s/c location and sensitivity of current estimates
Global coordinates and matched position data important
Northern connectivity: dusk-side: expect +ve JB from R2 dawn-side: expect -ve JB from R2
Westward current: -ve Jphi
Westward current: -ve Jphi

6. Ring current analysis: FAC
Often alignment is with Jphi inside RC and close to FAC outside
Non-linear gradients in ‘dipole’ field affect linear estimators
Curlometer stable against these effects: subtract ‘zero current’ model field

7. Advanced analysis tools: Magnetic field gradients from rotation and curvature properties of the magnetic field. Generalize use for 2-5 spacecraft: reliable estimates of some components of J.
Three main applications of gradient analysis (no timing assumptions):
Generalised method to calculate spatial gradients from 3-5 spacecraft: suitable for distorted spacecraft constellations and when 3 spacecraft are available (partial result, e.g. one J component).
Full gradient estimates for at least 3 spacecraft in the case where FA currents are expected.
Use of special constraints to obtain full magnetic gradient from at least 2 spacecraft
Comparison to standard ‘curlometer’ methods which employ time shift analysis along orbit track (e.g. Ritter et al. 2006, Grimald et al., 2012):
Swarm ‘vertical’ curlometer: allows a check of FAC component.
Comparative Cluster analysis from 2-3 spacecraft (in-situ RC and FAC coordination)
Key assumptions for special gradient analysis:
Based on Barycentric coordinate representation of the dyadic of B, B, applied through diagonalisation of the volumetric tensor for the spacecraft constellation.
3 s/c FAC:
J = J||b; then may project the known component perpendicular to the constellation plane.
2 s/c full gradient:
Solenoidal condition: .B = 0
Stationarity: dB/dt = (V.)B; B/  t =0
Force free: ^B = m0J = aB
For special orbit constellations, or if V is measured can obtain ideal solutions: tested with ideal dipole field and circular orbits.
Compare to ‘curlometer’ along Swarm obits.

8. Advanced analysis tools: Two spacecraft demonstration: FT
Comparison between the 2-point analysis result (left panels) and 4-point analysis result (left panels) on the 11 Oct 2003 tail flux rope event with the Cluster magnetic data

9. Advanced analysis tools: Two spacecraft demonstration: sensitivity of s/c pair.
C1/C4
C3/C4
C2/C3

10. Swarm-Cluster coordination: separation strategy
Changing perigee height
Orbit roll-over, distorted constellations (multi-scale).

11. Swarm-Cluster coordination:
High latitude FAC conjunctions; low latitude RC signals mapped every orbit.
SM equator crossing: : SC2 behind by 30 mins.

12. Cluster-Swarm Key configurations: 2014
Initial Cluster configuration set for the ring current: to access curl(B).
Orbit tilt samples range of LT giving good local coordination between Cluster and Swarm.
Example, 2014 April 24: shows small Cluster constellation during RC and close LT alignment with Swarm during Cluster FACs.
Inset shows grouping of 3 Swarm spacecraft. (A, B, C = black, red, green)

13. Cluster-Swarm Key configurations: 2014
Initial Cluster configuration set for the ring current: to access curl(B).
Orbit tilt samples range of LT giving good local coordination between Cluster and Swarm.
4 UT
4:10 UT
Example 2, 2014 April 22: shows close LT alignment with Swarm during Cluster FACs (4:00-4:15 UT).
Inset shows close grouping of 3 Swarm spacecraft. (A, B, C = black, red, green)

14. Example, 2014 April 22: Cluster 4 s/c Curlometer: signature at 4:00 UT, followed by Swarm pass (4:05-4:15).

15. Comparison of FA current estimates
Example, 2014 April 22: Swarm, single s/c FACs for A, B & C; Swarm 2 s/c L2 technique;
Application of Curlometer 3 s/c estimate to Swarm configuration.

16. Comparison of FAC estimates: detail
Curlometer at cluster compared to 3 s/c
FA component for two s/c planes: persistent signature
J||
J||
3,4,1
J||
1,2,3
Example, 2014 April 22: Comparison to Curlometer 3 s/c estimate.
1st period well correlated to 2 s/c Swarm method.

17. Comparison of FAC estimates: detail
Curlometer at cluster compared to 3 s/c
FA component for two s/c planes: persistent signature
J||
J||
3,4,1
J||
1,2,3
J||
Example, 2014 April 22: Comparison to Curlometer 3 s/c estimate.
Provides current vector normal to the plane of the s/c configuration.

18. Comparison of FAC estimates: detail
Curlometer at cluster compared to 3 s/c
FA component for two s/c planes: persistent signature
J||
J||
3,4,1
J||
1,2,3
Example, 2014 April 22: Comparison to Curlometer 3 s/c estimate.
Application of gradient analysis to calculate current gives the same result.

19. Cluster-Swarm Key configurations: cusp
Second target for cusp encounter is more rarely encountered through the tilt in the cluster orbits, but achieves close configurations on the same orbits as the RC crossings.
2013 Dec 25: Cluster passes through cusp 13:30-14:30 UT; then through the FAC +(inner)RC 15:30-17 UT.
Good LT alignment with Swarm at Cusp crossing.

20. Cluster-Swarm Key configurations: cusp
Second target for cusp encounter is more rarely encountered through the tilt in the cluster orbits, but achieves close configurations on the same orbits as the RC crossings.
2013 Dec 25: Four spacecraft Cluster magnetic field data showing a clear J|| through cusp.
Also Cluster FAC signature is consistent with R2 connectivity (+/- 20 nAm-2).

21. Swarm passes across the cusp then later orbit crosses FACs at the same time as Cluster.
2013 Dec 25: Cluster passes through cusp 13:30-14:30 UT; then through the FAC +(inner)RC 15:30-17 UT.
Good LT alignment with Swarm at Cusp crossing.

22. Cluster-Swarm Key configurations:
2013 Dec 25: Comparison with Swarm residual data (subtraction of Chaos-4 plus model).
Cluster passes through Cusp just after high latitude passage of Swarm.
Cluster enters FACs as Swarm reaches high latitude.

23. Cluster-Swarm Key configurations:
J||
2013 Dec 25: Cluster passes through Cusp just after high latitude passage of Swarm.
Swarm s/c in string of pearls but 3 s/c method still identifies the high latitude currents.

24. Cluster-Swarm Key configurations:
J||
2013 Dec 25: Cluster enters FACs as Swarm reaches high latitude. .

25. Cluster-Swarm Key configurations:
J||
2013 Dec 25: Cluster enters FACs as Swarm reaches high latitude.
Matched FAC signature at Cluster and Swarm at 16:10 UT.

26. Cluster-Swarm Key configurations: 2014
Later Cluster configurations: set of encounters for at least 6 months
2014 Sept 23: Close LT alignment during RC crossing.

27. Conclusions: Swarm-Cluster coordination:
Test connection through R2 FAC: clarification using the Swarm polar coverage.
Opportunities for both direct conjunctions and statistical comparisons with Cluster.
Special Cluster phase established for start of Swarm operations: science and validation.
The adaption of advanced Cluster tools assists preparation for Swarm data interpretation with 2-4 spacecraft.
Tests of various techniques using both Cluster and Swarm:
Alternative techniques tested for multi-spacecraft analysis and gradient estimates
identify quality indicators
Comparative measurements from Cluster provide spot checks of, e.g., RC or FAC signals

29. Cluster-Swarm Key configurations: 2014
Cluster configuration priority should be given for the ring current, because this is where we can calculate curl(B). In the FACs we can localise them, but accurately measuring curl(B) is difficult, due to the fine structure of the FACs.
Move Cluster 2 along track to optimise the configuration.

30. Cluster-Swarm Key configurations: 2015 GSE
Second priority for R2 FACs is naturally encountered through the tilt in the cluster orbits, which stay above the RC on exit for some hours; maintaining the same radial distance (and changing LT).
Good chance for repeated Swarm conjunctions

31. Cluster-Swarm Key configurations: cusp
Third target for cusp encounter is more rarely encountered through the tilt in the cluster orbits, but achieves close configurations on the same orbits as the RC crossings.

32. Cluster-Swarm Key configurations: cusp
South cusp? Showing sensitivity of cluster configuration

33. Swarm-Cluster conjunctionZ
GSM Coordinates X
Cusp
Swarm-A&B
Auroral Region (~65° ilat, 23:30 MLT)
Brief conjunction with Swarm-A&B and
Cluster 3,4 and 1 suitable for FAC work
at :24 UT
Cluster 3,4 and 1 X
Notes:
An assessment of how well field aligned currents can be
determined using default Cluster separations is planned
Cluster 2’s orbit track is not shown here Y

34. Conjunction with Cluster edge of RC, Swarm under the MFL-footprints of Cluster
Interval: , 20:00: hour
Similar conjunction (Cluster possibly crossing FAC’s)
Interval: , 03:00:00 + 3hours.

35. Using planned LTOF Shift C2 back 30-40 minutes
Ring plane in view direction

36. 2016 configuration
C2 is requires phase shift of > 1 hour to bring it back.
C1-C3-C4 have larger separation

37. Optimised Ring current formation: extended tetrahedron
C3-C1-C4 spacecraft should form triangle with plane as perpendicular to direction of Earth as possible
C2 should be moved along track such that it is along same radial direction as the triangle formed by the other spacecraft: R/r ~ 5 – 10 aspect ratio.
Higher latitude configuration: natural evolution of equatorial C1-3-4 triangle should be ok to capture FACs
2014 orbit cuts through wide local time of field lines
mapping to RC
This is less in later years due to orbit evolution R r
R = ~3000 km
r ~ 600 km C1 C2 C3 C4

38. Targets
Ring current, cusp and high latitude currents - the suggested approach:
The main target is RC: targeting an extended tetrahedra at equatorial crossing. C2 re-phased by about minutes; same radial line.
Other 3 spacecraft phased to form an equilateral triangle: separation ~ 600 km.
The orbit in 2014 also provides reasonable configurations at higher latitude; cutting through MFLs mapping to outer ring current. 
Ring current configurations (extended tetrahedra at perigee) are prime target; discount phasing for high latitudes to FA configurations.
Cusp encounters more infrequent: often natural configurations similar to RC.

39. Swarm-Cluster conjunctionZ
Swarm-Cluster conjunction
GSM Coordinates
Cusp X
Swarm-A&B X
Cluster 3,4 and 1
Ring Current Region (~55° ilat footprint; L ~ 3)
Swarm-A&B sample ring current FAC at ~00:00 MLT
Cluster 3,4 and 1 sample RC FAC but at ~17:00 MLT
These are simultaneous, at :58 UT Y