 Express Letter
 Open Access
 Published:
How variable are Birkeland currents?
Earth, Planets and Space volume 75, Article number: 116 (2023)
Abstract
I address the problem of estimating the timerateofchange of highlatitude Birkeland currents by using a stringofpearls formation of satellites. Space series are calculated by linear interpolation of measurements made at the revisit times of the satellites. A lower bound on the total time derivative can be estimated as a function of distance along the orbit. Space series of the vertical component of electric current density, used as a proxy for fieldaligned (Birkeland) current density at high latitude, are estimated from the alongtrack spatial derivative of Swarm magnetic field measurements residual to the CHAOS7 internal field model. The results reveal nonnegligible total time derivatives over periods shorter than 2 mins. Auroral Birkeland current densities derived from singlesatellite traversals of magnetic field gradients can change dramatically in the time it takes a single satellite to cross a largescale current system. In one example, during an overflight by the Swarm satellites of the THEMIS Fort Yukon allsky imager on 1 December 2013, the vertical current density poleward of a visually quiescent auroral arc changes from \(\sim 0.3\ \mu \,\hbox {A}/\hbox {m}^{2}\) upward to \(\sim 1.0\ \mu \,\hbox {A}/\hbox {m}^{2}\) downward in 13.7 s (corresponding to an alongtrack separation of Swarm A and B of 104 km). The variability of Auroral Birkeland currents, between 25 November 2013 and 31 December 2013, as estimated by the median of \(dj_z/dt\), reaches \(15\ \hbox {nA}/\hbox {m}^{2}/\textrm{s}\) in the northern dayside auroral zone and exceeds \(30\ \hbox {nA}/\hbox {m}^{2}/\textrm{s}\) in the prenoon sector of the southern hemisphere.
Graphic Abstract
For a moment, nothing happened.
Then, after a second or so, nothing continued to happen.
 Douglas Adams, The Hitchhiker’s Guide to the Galaxy.
Spacetime ambiguity
Measurements made from an orbiting platform contain artifacts of Dopplershifted structures that appear as timevarying signals. This represents an ambiguity with respect to underlying temporal variations, whether the time series comes from a single satellite or is derived from a cluster of them (see Dunlop et al. 1988; Balogh et al. 1997).
The quotation above is apropos for an assumption often made to circumvent the spacetime ambiguity: that the structure does not change in the time it takes to traverse it. A classic example is the identification of the magnetic fieldaligned currents in the polar regions (Birkeland 1908), global patterns of which were characterized half a century ago with magnetic field measurements from the Triad satellite (see Zmuda and Armstrong 1974, and related papers). Contemporary observations reveal variability in these Birkeland currents at large scales on timescales greater than about 10 min (Coxon et al. 2018). Even further variability in the Birkeland currents is evident in their visible manifestation as the Northern and Southern Lights, which can exhibit rapid and dynamic changes in intensity, colour, location and apparent shape. Estimation of current densities using the European Space Agency’s Swarm spacecraft (FriisChristensen et al. 2008) reveal large differences depending on the technique used (Trenchi et al. 2020), some of which variation may be due to violation of the steadystate assumption.
How variable are Birkeland currents on timescales of about a minute or less? A set of two or more closelyspaced satellites moving along a common orbit in a stringofpearls formation can provide a quantitative answer. Such was approximately the case for the Swarm trio of lowEarthorbiting satellites in the weeks following their launch on 23 November 2013 by a single rocket into a nearcircular, nearpolar orbit.
In what follows, I describe a technique for estimating space series and use it to quantify the time variability of the vertical component of electric current density for Swarm crossings of the auroral zones in late November and December 2013. Case study and statistical analysis show that relative changes in Birkeland currents can exceed 100% in less time than it takes a single satellite to cross an auroral zone.
Estimation of space series
A space series is a collection of measurements at different positions made at the same time.^{Footnote 1} The spatial resolution of the series is determined by the ratio of the satellite speed to the sample rate of the underlying measurements’ time series, whereas the time resolution at a given point is determined by the revisit times of the satellites.^{Footnote 2}
Video 1 (Additional file 1) illustrates the relationships between the time series and space series for magnetic field perturbations (with respect to a model field) measured by the three Swarm satellites in a pearlsonastring orbital configuration. Grid lines represent alongtrack distance and time. The gray curves represent the satellites’ time series measurements of the perturbations, which were made between 06:24 UTC and 06:36 UTC on 13 December 2013. Swarm B makes its measurement at a given position first, followed by Swarm A about 30 s later and Swarm C about a minute after Swarm A. By interpolating in time between pairs of curves, a space series can be constructed (white curve) at a given time spanning the orbit between the leading satellite and the last trailing satellite. Depending on the time, the estimate at a given position is from an interpolation between measurements made by Swarm B and A or by Swarm A and C. No extrapolation is performed for measurements ahead of the leading satellite or behind the last satellite of the formation such that the space series moves along the orbit as a sliding window. Once a series has been constructed, the alongtrack component of its spatial gradient can be calculated at a given time using symmetric finite differencing. Gradients at the leading and trailing data points of the series are undefined.
A model is needed to describe how measurements vary between satellite revisit times. The choice here is to interpolate measurements linearly in time. This has the benefit of allowing reconstruction of a space series at arbitrary times. For convenience the time sample rate of a space series is set equal to the instrument sample rate. A second benefit to linear interpolation is that the time derivative of the measurements is easily calculated, it being constant between any pair of satellite visits at a given point along the orbit. Although it is unrealistic to expect measurements to actually vary linearly in time, the result has a useful interpretation: the time derivative represents a lower bound on highest rate at which the measurements could have varied between revisit times.
The sample positions are those of the leading satellite along the orbit, at the socalled alongtrack distance, which is calculated by setting the distance to zero at the first time and integrating the alongtrack component of satellite velocity with respect to time. It remains to decide how to represent the positions of the trailing satellites.
Were each satellite following precisely the same orbit, it would suffice to determine the initial distance lags of the trailing satellites and integrate numerically the satellites’ velocities. This method fails to produce sensible results when the orbits are slowly changing, as was the case with Swarm following launch. After many orbits, the satellites have slightly different ephemeres (quasidipole latitude, for example) for a given alongtrack coordinate.
A robust method for determining the positions of the trailing satellites is to assign the alongtrack distance of the trailing satellites according to their distance from the leading satellite. This takes into account the slightly different orbital elements.
With the alongtrack positions of each satellite precomputed for a given time interval, the space series are calculated using linear interpolation of measurements made by two satellites at the nearest alongtrack distance to the requested position at two times. The ephemeres of the series are computed using linear interpolation of the ephemeres of the corresponding satellites in a similar way.
Data analysis
This scheme has been programmed in C for the Swarm stringofpearls formation. Source code is available on my GitHub account under a GPL license. Docker can get the software up and running quickly on several operating systems. Although the focus here is Birkeland currents, the software can produce space series and statistics for electron density, electron temperature, satellite potential, and ion drift as well.
Magnetic residuals are estimated by subtracting the CHAOS7 internal magnetic field model (Finlay et al. 2020) from the Swarm highresolution magnetic field measurements (Jørgensen et al. 2008). Version 7.12 of the CHAOS spherical harmonic coefficients are used. To save time, model fields are calculated along each satellite’s orbit at a cadence of one model vector every four seconds, and these are linearly interpolated to the 50 samplespersecond rate of the magnetic field data.
To reduce the amount of computer memory and processing time needed to compute each space series, the magnetic residuals are then downsampled, by averaging, to two vectors per second. Magnetic residuals are transformed into the satellitetrackaligned frame. This frame’s unit vectors are defined with respect to the North, East, and Centre (NEC) location and velocity of the satellite as \(\hat{x}=\hat{v}_{\textrm{sat}}\), \(\hat{y}\propto \hat{C}\times \hat{x}\), and \(\hat{z}=\hat{x}\times \hat{y}\). Vertical current densities are estimated from Ampere’s law (ignoring the displacement current) using the alongtrack gradient of the horizontal crosstrack magnetic residual (\(\delta B_y\)), under the assumption that the satellite traverses a onedimensional current system at normal incidence. The resolution of the 1 Hz Swarm magnetic field measurements is \(\sim 0.17\ \hbox {nT}\) (TøffnerClausen et al. 2016), which corresponds to an uncertainty in the derived 2 Hz current densities of about \(0.05\ \mu \,\hbox {A}/\hbox {m}^{2}\).
Video 2 (Additional file 2) shows space series and allskyimager (ASI) imagery for an overflight of the Swarm trio of the Fort Yukon THEMIS site on 1 December 2013 around 10:40 UTC. The trailing satellite (Swarm C) follows 289 km behind the leading satellite (Swarm B), and the maximum crosstrack separation is 10 km at the end of the interval.
The ASI instrument look direction has been calibrated using the locations of up to 20 of the brightest stars in the field of view for each of the 1,200 images over the period 10:00 UTC to 11:00 UTC. The auroral emissions are projected to the mean Swarm altitude of 492 km by tracing magnetic lines of force of the CHAOS\(\)7.12 core and static model fields from an assumed emission altitude of 110 km. Fieldline tracing is done using the gsl_odeiv2_step_msadams integrator from the GNU Scientific Library (Galassi et al. 2002). It is necessary to ensure accurate calibration of the ASI orientation and accurate fieldline tracing. For example, a \(0.1^\circ\) error in look direction of an ASI pixel near zenith corresponds to a horizontal error of approximately 1 km at the projected altitude. The satellite configuration is shown atop the ASI imagery in Video 2 (Additional file 2), along with geographic graticules.
Shown on the left in Video 2 (Additional file 2) are space series for \(\delta B_y\) (top panel), \(j_z\) (middle panel) and the THEMIS ASI signal (bottom panel) along the satellite track. Measurement positions are shown on the abscissa in quasidipole magnetic latitude (QDLat) and magnetic local time (MLT), as defined in Emmert et al. (2010). The gray solid curve in each panel is the apparent space series obtained from assuming a steadystate system, shown for the Swarm B (leading satellite) measurements. The thick white curve in each panel shows estimated space series at the displayed time. Displacements of the trailing satellites relative to the leading satellite are shown. The x and y coordinates represent the alongtrack and horizontal crosstrack displacements. The z coordinate represents the difference in altitudes of the satellite pairs. Shown also are maximumabsolute and rootmeansquare differences, which represent lower bounds of the error in the static assumption with respect to the space series estimates at each time step.
The ASI signal for the steadystate assumption is representative of a profile for a satellitealigned keogram (see Gillies et al. 2015). ASI image processing consists first of estimating a background image constructed from the minimum pixel value for each pixel over the interval from 14:00 UTC to 15:00 UTC, a period when very little auroral signal is present. This image is subtracted from each offsetcorrected level 1 ASI image. To improve signaltonoise, the mean of the 5 nearest pixels at each point along the satellite track is plotted.
The example in Video 2 (Additional file 2) corresponds to a visually quiescent auroral arc exhibiting no obvious crosstrack spatial variation. The upward current density near the peak of the auroral emission is consistent for all three satellites. There are notable differences between the static assumption for Swarm B and the dynamic space series poleward of the brightest arc, in particular associated with possible alongtrack motion of the narrow, intense downward current system as the last satellite approaches the arc. This is suggestive of the Alfvénic aurora (e.g., Chaston et al. 2008).
Figure 1 provides a zoom on the space and time series from Video 2 (Additional file 2) at a time (10:41:44.5 UTC) when Swarm A (whose position is denoted by the vertical dotted line in the middle panel) observes a downward current density of \(\sim 1.0\ \mu \,\hbox {A}/\hbox {m}^{2}\). This was observed 13.7 s following Swarm B’s observation of a \(\sim 0.3\ \mu \,\hbox {A}/\hbox {m}^{2}\) upward current density. The two measurements are within a crosstrack distance of 3 km.
Video 3 (Additional file 3) shows space series and ASI imagery for an overflight of the Swarm trio of the Rankin Inlet THEMIS site on 25 December 2013 around 05:20 UTC. The frame format is similar to that shown in Video 2 (Additional file 2). The ASI imagery is shown projected at the mean Swarm altitude of 497 km by tracing CHAOS\(\)7.12 model field lines from an assumed emission altitude of 150 km, an altitude for which the ASI signal peaks appear to line up slightly better with the positive vertical current density peaks than for an emission altitude of 110 km. A perfect alignment between these signals is not expected, largely due to the way narrow sheets of auroral emission appear to span latitude for offzenith look directions.
The aurora shown in Video 3 (Additional file 3) is highly structured in both time and space. In this example, more than three weeks after the one shown in Video 2 (Additional file 2), Swarm C trails Swarm B by \(\sim 1083\) km, and their crosstrack separation exceeds 45 km at the end of the interval. Some of the error between the static assumption and the space series can be attributed to horizontal crosstrack structure associated with the misalignment of the satellite orbits. Estimation of time derivatives are less reliable in this case. The sequence of rising squarewave features equatorward of 70\(^\circ\) QDLat results from a flashing light near the southern horizon. The intense, narrow spike on the equatorward side of the main auroral emissions between 05:23:00 UTC and 05:23:40 UTC is starlight. This spike does not appear for projections of imagery corresponding to an emission altitude of 110 km, incidating that uncertainty in the average emission altitude of each pixel contributes significantly to errors in comparisons of satellite measurements with imagery. In any case, the auroral imagery provides validation of the upward vertical current density shown in the space series. This example demonstrates the importance of precise orbit control in the design of multisatellite missions for estimating space series of auroral and ionospheric parameters.
Statistics of Birkeland current variability
Space series are calculated here for intervals of 24 h from midnight to midnight. Due to satellite commissioning activities, several days in late November and early December 2013 have full operations of the vector field magnetometers for only two satellites. Space series derived from these twosatellite cases and the remaining threesatellite cases between 25 November and 31 December 2013 are included in the statistical results below. Data from 13 December 2013 have been excluded owing to the presence of a large number of artifacts in the results. Data from 1 January 2014 and later are not included, by which time the alongtrack and horizontal crosstrack separations of the satellites are quite large and continue to grow.
Statistical binning in QDLat and MLT is done on a grid of approximately “equalarea” bins. The bin size is defined as one third of the solid angle of a polar cap spanning quasidipole latitude from 89\(^\circ\) to 90\(^\circ\). To reduce the influence of outliers on the results, statistics are estimated using the median as implemented in the GNU Scientific Library (Galassi et al. 2002).
Results are illustrated in Fig. 2. The panels in the top row represent results for the northern hemisphere, and the panels in the bottom row represent results for the southern hemisphere. The lefthand column (a) shows median vertical current densities \(j_z\) in units of \(\mu \,\hbox {A}/\hbox {m}^{2}\). Despite the short time interval of just over 1 month, there are clear indications of the region 1 and region 2 current systems of Iijima and Potemra (1978). This validates the use of the vertical component of electric current density as a proxy for Birkeland (magneticfieldaligned) currents at high latitudes. In the vicinity of the auroral oval, the median magnitudes are of order a few tenths of a microampere per square meter. The magnitudes are greater in the summer (southern) hemisphere than in the winter (northern) hemisphere.
The second column (b) of Fig. 2 shows the perbin median of the magnitude of the time derivative of the vertical current density. Rates of change peak between \(5\ \hbox {nA}/\hbox {m}^{2}/\textrm{s}\) and \(15\ \hbox {nA}/\hbox {m}^{2}/\textrm{s}\) in the northern hemisphere, and exceed \(30\ \hbox {nA}/\hbox {m}^{2}/\textrm{s}\) in the southern hemisphere. The currents are more variable on the dayside in a region morphologically similar to the magnetic cuspcleft.
To illustrate the extent to which the static assumption is violated during auroral zone crossings, the perbin median of the magnitude of the percentage change in current density relative to the static assumption calculated from Swarm B measurements (\(j_z  j_{\textrm{z},\textrm{B}}/j_{\textrm{z},\textrm{B}}\times 100\%\)) is shown in the third column (c) of Fig. 2. To avoid excessively large percentage changes associated with small \(j_{\textrm{z},\textrm{B}}\) in the denomenator, only those measurements for which the Swarm B current density magnitude is at least \(0.05\ \mu \,\hbox {A}/\hbox {m}^{2}\) are retained in the statistics. The orientation of the Swarm orbits is such that in the northern hemisphere the satellites move from dayside to nightside, whereas in the southern hemisphere they move from nightside to dayside. The orbits precess slowly with a tendency for postnoon and postmidnight local times in late November and prenoon and premidnight local times in late December. An increase in errors from postnoon to prenoon in the northern dayside auroral zone likely has a significant contribution associated with the increasing alongtrack satellite separations as time goes on. For reference, the median revisit time of the trailing satellites with respect to measurements made by the leading satellite is shown in the righthand column (d) of Fig. 2. This shows that the space series technique as applied to the Swarm data obtains samples, in effect, corresponding to periods spanning from a few seconds (shortly after launch) to about 2 min (near the end of 2013).
It is a limitation of the technique as implemented for Swarm (but not generally, to the extent that satellite orbits are precisely controlled for this purpose) that horizontal gradients are assumed to be zero. Yet the variation in satellite separation does not appear to affect significantly the time derivative shown in the middle column of Fig. 2, which shows large values both postnoon and prenoon yet smaller values premidnight than postmidnight. This indicates that errors in current density (righthand column of Fig. 2) associated with assuming no horizontal crosstrack spatial gradients are typically smaller than errors associated with the static assumption.
Closing remarks
Swarm early mission magnetic field data (Fig. 2 columns c and d) reveal that single satellite estimates of Birkeland currents can be erroneous by tens of percent within 10 s of the measurement and by up to 100% within one to two minutes. Accurate determination of time derivatives potentially has implications for investigations of Alfvén waves, Poynting flux, ionospheric conductance, and auroral structure.
Linear time interpolation is not robust against aliasing, and it does not capture correctly the time variation associated with alongtrack drift of spatial structures. Neither does it account for crosstrack drift of structures with crosstrack gradients. These issues point to the need for large numbers of closelyspaced orbital platforms in two and three dimensions to fully characterize space and time variability and to capture the time history of the system (see Burleigh et al. 2022).
Obtaining massively parallel measurements from fleets of relatively inexpensive cubesats capable of orbit control is a feasible approach. Such missions need to be designed with the capability to maintain precise control of orbital parameters. Accurate measurement of crosstrack and alongtrack ion drift could help to estimate partial time derivatives in a frame of reference moving with the plasma, rather than the total time derivatives along the orbit as estimated here. In terms of measuring Birkeland currents, one imagines multicluster missions consisting of small groups of spacecraft capable of measuring \(\nabla \times \textbf{B}\) at many points along a trajectory simultaneously to resolve spacetime ambiguity during a traversal of the current system. Such capability would enable measuring space series of the vector current density, rather than just the vertical component of it.
The gradually increasing separation of the Swarm satellites following launch was orchestrated to support observations of east–west magnetic field gradients associated with crustal magnetism. After several months of orbital maneouvres, Swarm A and C were closely spaced near an altitude of \(\sim 460\) km separated by \(1.4^\circ\) of longitude, with Swarm C leading Swarm A by 4 s to 10 s throughout the mission. (Swarm B was placed at an altitude of \(\sim 510\) km, its orbital plane precessing slowly with respect to those of the lower pair.) In the vicinity of the geographic poles, Swarm A and C have small crosstrack separations. This suggests an opportunity for using space series to study the morphology of finescale variability of Birkeland currents and other ionospheric parameters with a longterm dataset. The larger offset of the magnetic and geographic poles in the southern hemisphere should provide sufficient coverage of magnetic latitude and local time to cover significant portions of the auroral oval. This would have application in filtering highlatitude magnetic field measurements used in global field models (e.g., Finlay et al. 2020).
Availability of data and materials
Swarm data are available from http://swarmdiss.eo.esa.int. THEMIS ASI level 1 and 2 data are availble from http://themis.ssl.berkeley.edu/data/themis/thg/. ASI calibration was carried out using https://github.com/JohnathanBurchill/AllSkyCameraCal. Swarm magnetic field residuals and CHAOS7 model fieldline tracing were computed using https://github.com/JohnathanBurchill/chaos. Quasidipole magnetic coordinates were calculated using https://github.com/JohnathanBurchill/QuasiDipole. Swarm space series were calculated using https://github.com/JohnathanBurchill/spaceseries.
Notes
Measurements may be deemed simultaneous if they agree to within the timing accuracy, say 10 ms.
A different implementation might interpret an instrument’s sample rate another way.
Abbreviations
 ASI:

Allsky imager
 ESA:

European Space Agency
 MLT:

Magnetic local time
 NEC:

North, east, centre reference frame
 QDLat:

Quasidipole magnetic latitude
 UTC:

Universal time coordinated
References
Balogh A, Dunlop M, Cowley S, Southwood D, Thomlinson J, Glassmeier K, Musmann G, Lühr H, Buchert S, Acuna M et al (1997) The cluster magnetic field investigation. Space Sci Rev 79(1):65–91
Birkeland K (1908) On the cause of magnetic storms and the origin of terrestrial magnetism, vol 1. H. Aschehoug, Oslo
Burleigh M, Lynch K, Zettergren M, Clayton R (2022) Spatiotemporal limitations of datadriven modeling: an isinglass case study. J Geophys Res Space Phys 127(9):e2021JA030242
Chaston C, Salem C, Bonnell J, Carlson C, Ergun R, Strangeway R, McFadden JP (2008) The turbulent alfvénic aurora. Phys Rev Lett 100(17):175003
Coxon JC, Milan SE, Anderson BJ (2018) A review of Birkeland current research using ampere. Electr Curr Geosp Beyond. https://doi.org/10.1002/9781119324522.ch16
Dunlop M, Southwood D, Glassmeier KH, Neubauer F (1988) Analysis of multipoint magnetometer data. Adv Space Res 8(9–10):273–277
Emmert J, Richmond A, Drob D (2010) A computationally compact representation of magneticapex and quasidipole coordinates with smooth base vectors. J Geophys Res Space Phys. https://doi.org/10.1029/2010JA015326
Finlay CC, Kloss C, Olsen N, Hammer MD, TøffnerClausen L, Grayver A, Kuvshinov A (2020) The chaos7 geomagnetic field model and observed changes in the south atlantic anomaly. Earth Planets Space 72(1):1–31. https://doi.org/10.1186/s40623020012529
FriisChristensen E, Lühr H, Knudsen D, Haagmans R (2008) Swarman earth observation mission investigating geospace. Adv Space Res 41(1):210–216
Galassi M, Davies J, Theiler J, Gough B, Jungman G, Alken P, Booth M, Rossi F, Ulerich R (2002) GNU scientific library. Network Theory Limited Godalming
Gillies DM, Knudsen D, Spanswick E, Donovan E, Burchill J, Patrick M (2015) Swarm observations of fieldaligned currents associated with pulsating auroral patches. J Geophys Res Space Phys 120(11):9484–9499
Iijima T, Potemra TA (1978) Largescale characteristics of fieldaligned currents associated with substorms. J Geophys Res Space Phys 83(A2):599–615
Jørgensen JL, FriisChristensen E, Brauer P, Primdahl F, Jørgensen PS, Allin TH, Denver T et al (2008) Small satellites for earth observation. In: Valenzuela A (ed) The swarm magnetometry package. Springer, Dordrecht, pp 143–151
TøffnerClausen L, Lesur V, Olsen N, Finlay CC (2016) Inflight scalar calibration and characterisation of the swarm magnetometry package. Earth Planets Space 68:1–13. https://doi.org/10.1186/s4062301605016
Trenchi L, Team FM, Kauristie K, Käki S, Vanhamäki H, Juusola L, Blagau A, Vogt J, Marghitu O, Dunlop M, et al (2020) ESA fieldaligned currents–methodology intercomparison exercise. In: MW Dunlop, H Lühr (eds.) Ionospheric MultiSpacecraft Analysis Tools Approaches for Deriving Ionospheric Parameters. Frascati, European Space Agency, pp 167–188
Zmuda AJ, Armstrong JC (1974) The diurnal flow pattern of fieldaligned currents. J Geophys Res 79(31):4611–4619
Acknowledgements
The author thanks David Knudsen for financial support. Megan Gillies from the University of Calgary Auroral Imaging Group provided technical advice on THEMIS ASI calibrations. The SwarmAurora conjunction finder (https://swarmaurora.com) was used to find the events shown in Video 2 (Additional file 2) and Video 3 (Additional file 3). The author is grateful to two anonymous reviewers for their insightful critiques of the manuscript.
Funding
This study was carried out with financial support from the Canadian Space Agency.
Author information
Authors and Affiliations
Contributions
JKB thought of the idea, developed the source code, analyzed the data, made the visualizations, interpreted the results, and wrote the paper.
Corresponding author
Ethics declarations
Competing interests
The author declares to have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Additional file 1: Video S1. Animation of the measurements of magnetic field perturbations (gray curves) from three Swarm satellites in a pearlsonastring orbital configuration, as well as the time development of the space series (white curve) interpolated in time from those measurements.
Additional file 2: Video S2. Space series of δB_{y}, j_{z}, and ASI signal during a Swarm overflight of the Fort Yukon THEMIS allsky imager on 1 December 2013 around 10:40 UTC.
Additional file 3: Video S3. Space series of δB_{y}, j_{z}, and ASI signal during a Swarm overflight of the Rankin Inlet THEMIS allsky imager on 25 December 2013 around 05:20 UTC.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Burchill, J.K. How variable are Birkeland currents?. Earth Planets Space 75, 116 (2023). https://doi.org/10.1186/s40623023018678
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s40623023018678
Keywords
 Ionosphere
 Magnetic field
 Electric current density
 Swarm
 Spacetime series