Swarm satellite observations are used to characterize the
extreme behavior of large- and small-scale field-aligned currents (FACs)
during the severe magnetic storm of September 2017. Evolutions of the
current intensities and the equatorward displacement of FACs are analyzed
while the satellites cross the pre-midnight, pre-noon, dusk and dawn sectors
in both hemispheres. The equatorward boundaries of FACs mainly follow the
dynamics of the ring current as monitored in terms of the SYM-H index. The
minimum latitude of the FAC boundaries is limited to 50∘ magnetic latitude (MLat). The
FAC densities are very variable and may increase dramatically, especially in
the nightside ionosphere during the storm-time substorms. At the peak of
substorms, the average FAC densities reach >3µA m-2.
The dawn–dusk asymmetry is manifested in the enhanced dusk-side R2 FACs in
both hemispheres. In the 1 Hz data filamentary high-density structures are
always observed. In the pre-noon sector, the bipolar structures (7.5 km
width FACs of opposite polarities adjacent to each other) dominate, while at
the other local times the upward and downward FACs tend to be latitudinally
separated. The most intense small-scale FACs, up to ∼80µA m-2, are observed just in the post-midnight sector. Simultaneous
magnetic and plasma perturbations indicate that this structure is likely a
current system of a mesoscale auroral arc.
Introduction
Field-aligned currents (FACs) provide electrodynamic coupling of the solar
wind–magnetosphere–ionosphere system. FACs flow along the high-conducting
geomagnetic field lines between different magnetospheric domains and the
high-latitude ionosphere. The current system is driven by the internal
magnetospheric circulation of the plasma and magnetic field within the global
reconnection cycle (Dungey, 1961; Cowley and Lockwood, 1992) and by
additional viscous-like interaction at the flanks of the magnetosphere (Axford,
1964). Configuration of FACs is primarily controlled by the interplanetary
magnetic field (IMF) orientation (Bythrow et al., 1984; Potemra et al.,
1984). Other parameters of the solar wind (velocity, density, IMF strength)
and the ionospheric conductivity also play a role (e.g., Christiansen et al.,
2002; Ridley, 2007; Korth et al., 2010).
Schematic distribution of large-scale FACs was established by Iijima
and Potemra (1976) based on Triad satellite data. Subsequent space
missions allowed construction of comprehensive empirical models of FACs
parameterized by the IMF direction and strength, by season, and by
hemisphere (Weimer, 2001; Papitashvili et al., 2002; Green et al., 2009).
The ionospheric projection of the 3-D FAC system consists of a pair of sheets
elongated approximately along the magnetic latitude, namely, Region 1 (R1)
and Region 2 (R2), with opposite current flow directions in the morning and
evening local time sectors and additional current sheets (R0) located on the
day side poleward of R1/R2. R1 FAC flows into the ionosphere (downward
current) and from the ionosphere (upward current) on the dawn and dusk sides,
respectively. R1 currents, if they reside on closed field lines of the Earth's
magnetic field, are believed to originate in either the boundary layer or in
the plasma sheet (Ganushkina et al., 2015). R2 FAC is considered to be a
diversion of the partial ring current to the ionosphere driven by pressure
gradients in the inner magnetosphere (Cowley, 2000). R0 current is connected
to the dayside magnetopause, and its polarity strongly depends on the IMF By
component. In the Northern Hemisphere, the R0 current flows predominantly
out of the ionosphere for positive IMF By and into the ionosphere for
negative IMF By (Papitashvili et al., 2002; Lukianova et al., 2012).
Additional current associated with the sunward ionospheric flow may
appear inside the polar cap if IMF Bz is northward (Iijima et al., 1984;
Vennerstrøm et al., 2002).
While average large-scale (>150 km) current densities typically
are in units of µA m-2 or less, instantaneous small-scale FACs may
reach several hundred µA m-2 (Neubert and Christiansen, 2003). The
smaller-scale structures are often associated with auroral arcs which are
accompanied by ionospheric conductivity and electric field perturbations
(Aikio et al., 2002; Juusola et al., 2016). In particular, it was shown that
in the evening (morning) sector, there is downward FAC equatorward
(poleward) of the arc and upward FAC above the arc. These two FAC regions
are connected by a poleward (equatorward) horizontal current. Recent studies
also confirmed that the cusp plasma injections are accompanied by pairs of
FACs, upward at lower latitude and downward at higher latitude (Marchaudon
et al., 2006).
Significant differences in the characteristics of FACs at different scales,
especially near noon and midnight, have been found (Gjerloev et al., 2011;
Lühr et al., 2015; McGranaghan et al., 2017). Under stationary conditions
the FAC system evolves in accordance with the reconnection rate, which is
controlled primarily by the solar wind. If a substorm occurs, additional
FACs form a current wedge connecting the cross-tail current and the
nightside westward ionospheric electrojet (Akasofu, 1964; Lui, 1996). The
magnitude of existing large-scale FACs also increases (Iijima and Potemra,
1978; Coxon et al., 2014). The dayside R1 currents are found to be stronger
than their nightside counterpart during the substorm growth phase; at the
same time R1 moves equatorward. After expansion-phase onset, the
nightside R1 currents dominate and their location moves to higher latitudes
(Clausen et al., 2013). Recent studies have also suggested that the substorm
current wedge could also include an R2 current system (Ritter and Lühr,
2008).
Magnetic storms are characterized by a dramatic enhancement of energy
deposition to the Earth's atmosphere. During a magnetic storm, FACs become
highly dynamic because of the enhanced solar-wind–magnetosphere interaction,
release of energy stored previously in the magnetotail, particle
precipitation and ring current build-up. Storm-time FACs are stronger and
more variable compared to non-storm FACs predicted by the climatological
models. Since the intensity and time evolution of FACs vary from storm to
storm, it is of interest to analyze their unique characteristics. However,
relatively few papers focus on observed storm-time FACs. For example,
utilizing the magnetic field measurements by the CHAMP satellite, Wang et al. (2006)
investigated the Northern Hemisphere and Southern Hemisphere dayside and nightside FAC
characteristics during the extreme October and November 2003 magnetic
storms. It was shown that as Dst decreases, the FAC region expands
equatorward, with the shift of FACs on the dayside controlled by the
southward IMF. For both case studies, in the Southern (late spring)
Hemisphere the minimum latitude of the FACs is limited to 50∘
magnetic latitude (MLat) for large negative values of Bz. (The minima are the
same, although in October the IMF Bz drops to -28 nT, while in November
it reaches -50 nT.) In the Northern (late autumn) Hemisphere the equatorward
boundaries of the FAC region are located at 55–60∘ MLat. Using the
global maps from the Iridium constellation, Anderson et al. (2005) studied the FAC
intensities during severe magnetic storms which occurred during solar
cycle 23, with particular attention given to the evolution of FACs in the course
of the storm of August 2000. The results revealed the dawn–dusk asymmetry
of the R1/R2 current sheets, with an increase primarily found on the
duskside. It was also shown that under disturbed conditions the total
current intensity was constrained to be below 20 MA (Anderson and Korth, 2007).
Since 2014, comprehensive studies of FAC distributions were carried out
based on high-precision observations onboard the Swarm constellation (e.g., Dunlop
et al., 2015; Juusola et al., 2016; McGranaghan et al., 2017). However, the
Swarm data have not yet been fully utilized for the storm-time FAC analysis. It
is the purpose of this paper to characterize the magnitude and position of
the large- and smaller-scale FACs as their response to the magnetic storm
development. The Swarm observations are used in order to identify various
characteristics of the storm-time FACs for the event of 6–9 September 2017,
which was one of the two most severe magnetic storms of the recent solar
cycle 24 (the previous event was the St. Patrick's Day storm on 17 March 2015). The
September 2017 event is of particular interest because it was a two-step
storm during which two major substorms occurred and the FAC system is
affected by the storm–substorm interplay. In this paper we investigate the
time evolution of the large-scale FAC intensities, the displacement of the
FAC equatorward boundaries and the extreme small-scale (for the 1 Hz data,
the spatial resolution is ∼7.5 km) currents.
Swarm satellitesInstrumentation
The ESA Swarm mission is a constellation consisting of three identical
satellites (hereafter SwA, SwB and SwC, respectively); all are at
low-altitude polar orbits (Friis-Christensen et al., 2008). The Swarm
constellation was launched at the end of 2013 and entered the operational
phase in April 2014. The initial orbit altitudes are 465 km (SwA and SwC) and
∼520 km (SwB) and the inclination is 87.5∘. For By in
September 2017 the orbit altitude decreases to ∼440 and
505 km, respectively. SwA and SwC fly in a tandem separated by
1–1.4∘ in longitude and the differential delay in orbit is
∼3 s. The orbit period is about 93 min (the speed of the
satellites is about 7.5 km s-1) and slightly different between SwA/SwC and the
upper satellite SwB, so that their along-orbit separation in local time
gradually changes. Their orbital planes also gradually drift apart, and the
separation angle increases by ∼20∘ longitude per
year. Slowly drifting in longitude, the orbits cover all the local time
sectors over about 130 d.
The mission has a multi-instrument payload. The main module is the
high-sensitivity vector (fluxgate type) and scalar magnetometers for
determining the magnitude and direction of the total vector and variations
of the geomagnetic field with an accuracy of more than 0.5 nT (Merayo et
al., 2008). Magnetometers make it possible to carry out measurements in a
wide range, including the Earth's main magnetic field and the variations of
the external magnetic field generated by FACs. FACs are detected by their
magnetic perturbations in the orthogonal plane which are obtained after
subtracting the main magnetic field model from the total measured values.
From a single spacecraft the FAC density can be estimated based on one
magnetic component with a technique invoking Ampere's law under assumptions
about the infinite current sheet geometry and the orthogonal crossing of the
current sheet. This method was used for the previous one-satellite missions,
such as Magsat and Ørsted (Christiansen et al., 2002). It is also applied
to each Swarm satellite separately. The dual-satellite estimation method
calculates current density from curl(B) measured quasi-simultaneously at four
locations adapted for SwA and SwC data, where measurements separated
along-track are used to create a “tetrahedron” (Ritter and Lühr, 2006).
The curl(B) method provides more reliable current density estimates, as it
does not require any assumptions about current geometry and orientation. The
FAC outputs of both dual-satellite and single-satellite methods are
considered to be in reasonable agreement (Ritter et al., 2013). However, a
high degree of coherence is typical at auroral latitudes, while in the polar
cap the results based on a dual-spacecraft technique are more reliable (Lühr et
al., 2016). Both algorithms are implemented to generate the Swarm products that
are produced automatically by ESA's processing center as soon as all input
data are available. The products are provided using the dual-satellite
method on the lower pair of satellites SwA and SwC and the single-satellite
solution for each of the Swarm spacecraft individually. The 1 s values (1 Hz
sampling rate) of FAC densities are available via the online Swarm data portal
(ftp://swarm-diss.eo.esa.int, last access: 10 January 2020) as Level 2 data products (Swarm Level 2
Processing System, 2019). In the present study the single-satellite FACs are used
in order to apply the similar method to SwB and SwA/SwC data.
Each satellite is also equipped with the Electric Field Instrument, which
includes the Langmuir probe to provide measurements of ionospheric plasma
parameters: electron density, electron temperature and spacecraft potential
(Knudsen et al., 2003). The plasma data are available at a 2 Hz sampling rate
as the standard product of the Swarm database. Unfortunately, due to technical
problems, measurements of the electric field and ions are rather rare.
Nevertheless, the combination of data provided by a magnetometer and a
plasma analyzer on electrons makes it possible to identify perturbations
associated with FACs. In each Level 2 data file the location of the
satellite is presented in an geographic coordinate system NEC (x – north, y –
east, z – center), where the x and y components lie in the horizontal plane,
pointing northward and eastward, respectively, and z points to the center of
gravity of the Earth. For the purpose of the present study all projections of
the passes are shown in the magnetic local time (MLT) and MLat domain. For this the coordinates are available via the online Swarm
Data Visualisation Tool (VirES).
Orbits on 6–9 September 2017
The polar projection of the satellite orbits (14–15 trajectories per day) as
of 6–9 September 2017 in the Northern Hemisphere and Southern Hemisphere is shown in
Fig. 1. For mid-September 2017 the passes are centered in the pre-midnight,
pre-noon, pre-dusk and pre-dawn sectors. The satellite SwA (orbits of SwC
are very similar) enters the region of MLat > 50∘
between ∼ 09:00 and 12:00 MLT and leaves this region between
∼ 21:00 and 23:00 MLT. The entry (exit) points of the SwB orbit are
between ∼ 15:00 and 17:00 (02:00 and 04:00) MLT. In the Southern
Hemisphere the direction of the tracks in the MLT–MLat framework is
opposite. During a day, the successive projections are systematically
shifted almost parallel to each other; however, at auroral latitudes, they
stay mainly within the same sectors. The MLT ranges covered by the tracks
are presented in Table 1.
Polar maps of the SwA and SwB orbits in the Northern Hemisphere and
Southern Hemisphere on 6–9 September 2017 in the MLT–MLat framework.
Circles are drawn every 10∘ down to 50∘ MLat. Symbols *
and ** indicate, respectively, the entry and exit crossing of the boundary
MLat = 50∘.
MLT range of the tracks in the northern and southern polar
regions.
SatelliteMLT range within which theCenter of the satellites cross the boundary ofMLT range 50∘ (70∘) MLat (hh:mm)*hh:mmhh:mmNorthern Hemisphere SwB02:50–04:30 (01:30–05:10)03:4004:00SwA (SwC)09:20–11:30 (08:40–12:50)10:3010:00SwB15:00–16:50 (14:20–18:10)16:0016:00SwA (SwC)21:00–22:50 (19:40–23:30)22:0022:00Southern Hemisphere SwB03:10–05:00 (01:50–06:20)04:0004:00SwA (SwC)09:10–11:00 (08:30–12:20)10:0010:00SwB14:50–16:40 (14:10–18:00)15:5016:00SwA (SwC)21:20–23:10 (20:00–23:50)22:1022:00
* With an accuracy of 10 min.
Space weather conditions on 6–9 September 2017
At the declining phase of solar cycle 24, starting from 6 September 2017,
strong multiple solar flares occurred. The associated interplanetary coronal
mass ejections collided with Earth's magnetosphere and caused the most
intense magnetic storm of the recent solar cycle. The storm produced strong
geomagnetic disturbances, ionospheric effects, magnificent auroral displays,
elevated hazards to power systems and unstable high-frequency (HF) radio wave propagation
(e.g., Chertok et al., 2018; Clilverd et al., 2018; Curto et al., 2018;
Yasyukevich et al., 2018).
Evolution of the solar wind (SW) parameters and geomagnetic activity is
presented in Fig. 2, showing (from top to bottom) the IMF Bz and
By, the SW proton speed (Vsw) and density (Nsw), the auroral AL and the
equatorial SYM-H geomagnetic indices from the OMNI web service
(https://omniweb.gsfc.nasa.gov/, last access: 1 December 2019). Two SW shock events impact the
magnetosphere. The arrival of the first shock late on 6 September (23:50 UT)
results in a sudden increase in all parameters except the AL index. Since at
that time the IMF Bz turns northward, the initial disturbance is only weakly
geoeffective as a result. At 20:40 UT, 7 September, IMF Bz turns southward, which triggers a substorm growth phase and a ring current build-up. The
second shock arrived at ∼ 23:40 UT on 7 September, with the SW
speed up to 800 km s-1 and strongly negative Bz and By. This shock causes an
abrupt drop in SYM-H down to -150 nT and a spike-like decrease in AL down to
-2200 nT. After 03:00 UT, 8 September, the IMF Bz becomes positive, AL
gradually approaches zero and SYM-H starts to recover until the next
southward turn of Bz. At ∼06 UT on 8 September another
strongly negative Bz period is seen, and the SW speed remains high
(>700 km s-1). This causes the second substorm (AL is -2000 nT)
and ring current intensification (SYM-H is -100 nT). A steady recovery
occurs in the AL index throughout 9 September, while the SYM-H gradually
increases from -75 to -35 nT. The SW parameters are not available for this
day.
From (a) to (f): IMF Bz and By, SW speed and density,
and AL and SYM-H indices on 6–9 September 2017 (5 min values).
Data analysisFAC densities
Statistically the large-scale R1 and R2 FAC densities peak at the
dawn–dusk meridian. At dusk, the orbits of SwB are centered at about 16:00 MLT;
on the night side, the orbits are centered at 04:00 MLT. SwA and SwC cross the
pre-noon sector at about 10:00 MLT, where disturbances associated with
substorms are expected. An example of the FACs measured along the SwB track
is shown in Fig. 3. The 1 s values presented in Fig. 3a provide clear
evidence of strong bursts at the auroral latitudes (55–75∘ MLat).
The auroral FACs exhibit large-amplitude spike-like structures, thus
confirming the existence of filamentary current sheets embedded in the
large-scale current sheets. The intensities of these small-scale FACs vary
in units to tens of µA m-2. Figure 3b depicts the 51-point smoothed
curve (the length of the sliding window is ∼380 km). It can
be seen that the satellite approaching the pole from the dusk observes first
the downward (positive) R2 and then the upward (negative) R1 current; both
are of ∼1µA m-2 density. Above approximately
70∘ MLat FACs become marginal. When the satellite moves
equatorward at the early morning local times, a structure is observed in
which the poleward currents are positive, so they may be associated with the
downward R1 FAC. The most equatorward currents are negative and thus
represent the R2 FAC.
(a) 1 s and (b) smoothed FACs measured by SwB in the
northern polar region between 23:50 UT, 7 September, and 00:13 UT, 8 September. Downward (upward) current is positive (negative).
To demonstrate the global temporal evolution of FACs, in Fig. 4 the current
densities for the four MLT sectors are presented separately for the Northern Hemisphere
(Fig. 4a, c, e, g) and Southern Hemisphere (Fig. 4b, d, f, h). Each red
(blue) point is determined by averaging the 1 s downward (upward) current
densities, when the satellite crosses the region filled with FACs. The upper
(a–d) and lower (e–h) plots represent the data from the day side (10:00 and
16:00 MLT) and night side (04:00 and 22:00 MLT), respectively. For easier visual
association of the evolution of FACs with the storm development, the SYM-H
and AL indices are added in the plots (a, b) representing the day side and
in the plots (e, f) representing the night side, respectively. During 6–9 September, FACs shown in Fig. 4 exhibit three pronounced enhancements,
which are of different intensity depending on the MLT sectors. (Note that
the FAC densities do not show any systematic changes associated with the
orbit oscillation during the day.) All FACs start to increase at the very
beginning of 7 September in association with the SW dynamic pressure front
impinging on the magnetosphere, causing a positive excursion of SYM-H. The
dayside FACs increase abruptly (this is especially well seen in Fig. 4b–c, i.e., at 10:00 MLT, north, and at 16:00 MLT, south), while the nightside FACs
(Fig. 4e–h) respond to the shock with a considerable delay. The nightside
FACs peak in the middle of 7 September, when a moderate substorm
occurs.
Average FAC densities in the four local time sectors
covered by the Swarm data on 6–9 September 2017. The left columns of the plots
correspond to the Northern Hemisphere (NH) and the right columns correspond
to the Southern Hemisphere (SH). The upper plots (a–d) and the lower plots
(e–h) show the dayside and nightside FACs, respectively. The SYM-H and AL
indices are added in plots (a, b) and (e, f), respectively. The centered
MLTs (10:00, 16:00, 22:00 and 04:00) are shown in the right upper corner of each plot. The
downward and upward FACs (and the corresponding error bars) are shown in red
and blue, respectively.
At the very beginning of 8 September, in association with the first deep
drops of SYM-H and AL, a step-like increase is seen at all MLTs except the
pre-noon sector. The peak of the dayside and nightside FACs reaches 2.5 and 3.5 µA m-2, respectively. For a particular crossing the standard
deviation exceeds 5–6 µA m-2, while the standard error is
about 0.3 µA m-2. The dayside FACs (Fig. 4a–d) stay
enhanced during the whole day of 8 September. The nightside FACs (Fig. 4e–h) more closely follow the evolution of AL, so that the current intensities
decrease in accordance with the first storm-time substorm recovery. The next
increase in the nightside FACs occurs at ∼ 12:00 UT on 8 September, when the second major substorm and the second drop in SYM-H are observed.
On the day side the response of FACs to this substorm is marginal, although
the current densities remain elevated throughout the day. All FACs fall to
pre-storm levels by 9 September.
Comparison of the evolution of FAC intensity with the SW and geomagnetic
parameters during the period of 6–9 September reveals that the storm-time
FACs are, on average, several times larger than the quiet-time ones.
Better correspondence exists between the nightside FACs (compared to the
dayside ones) and the substorm activity as monitored by the AL index.
Dynamics of the equatorward boundary of the FAC region
It is well established that the enhanced SW input and the pile-up of open
magnetic flux during a geomagnetic storm result in the equatorward
expansions of the polar cap and the auroral oval as a whole (e.g., Milan et
al., 2004). Following the magnetospheric dynamics FACs also move
equatorward. Figure 5 shows the evolution of the equatorward boundary (EqB) of
FACs on 6–9 September. For the comparison the SYM-H and AL indices are
added. The EqB parameter is determined as the lowest MLat at which FACs are
terminated. The procedure of the 20-point sliding window (the scale is about
150 km) moving along a track from the Equator to the pole is applied to the
1 s FAC values and the corresponding MLats. EqB is selected as the lowest
MLat of the window if 90 % of FAC values within the window exceed |0.1|µA m-2. Then the results are checked visually in order
to avoid the erroneously calculated latitudes, which may happen, e.g., if a
significant latitudinal gap between R1 and R2 occurs. When calculating EqB,
no separation between the upward and downward FACs is made.
MLat of the FAC equatorward boundaries (EqB) in the
Northern Hemisphere (a) and Southern Hemisphere (b) for the sectors centered at around
04:00, 10:00, 16:00 and 22:00 MLT. EqB for each sector is shown by dots of different
colors; blue dots representing the nightside (∼ 22:0 MLT) EqB
are connected by a line. The SYM-H and AL index (black line) is added to the
upper and lower plots, respectively. The vertical lines mark the beginning of
the main and recovery phases.
Even visual comparison of the SYM-H and EqB evolutions in Fig. 5 reveals
generally coherent behavior of these two parameters. In particular, during a
period preceding the storm main phase (before 8 September, when SYM-H is
mainly positive) EqB is located much lower than during the end of the recovery
phase (after ∼ 12:00 UT on 9 September, when SYM-H is still
negative). Before the SYM-H attains the negative values below -20 nT at
22:00 on 7 September, FACs are observed mainly poleward of 60∘
MLat in both hemispheres. Moderate equatorward shifts of EqB are associated
with the modest substorms that occurred before the storm main phase in the middle
of 6 and 7 September. Prior to the main phase, in both hemispheres the pre-noon
(10:00 MLT) EqB is found considerably poleward compared to the EqB locations at
other MLTs. The effect is well seen during the two time intervals: from
∼ 22:00 UT, 6 September till 06:00 UT, 7 September and at 12:00–24:00 UT,
7 September. Both intervals are dominated by the northward IMF (cf. Fig. 2),
so that a shrinking of the polar cap and a poleward shift of the auroral
oval are expected. With regard to the positions of FACs, the displacement of
its equatorward boundary is the largest only in the pre-noon sector, while
the other local times remain less affected.
Upon arrival of the SW shock at the very end of 7 September, EqB is abruptly
shifted equatorward, then tends to recover until the middle of 8 September,
and then drops again following the second intensification of the storm. At
the very beginning of 8 September EqB is found at its lowest position at
50∘ MLat. A drop in EqB occurs simultaneously with the peak of the
first substorm intensification and the lowest SYM-H (-160 nT). The second
substorm reaches its peak slightly before the second minimum of SYM-H (at
12:50 and 13:55, respectively). During this second activation the EqB is
shifted again as low as 50∘ MLat (although SYM-H is only -100 nT).
As seen in Fig. 5, the evolution of EqB tends to follow the gradual change
in SYM-H rather than abrupt drops in AL related to the substorm activations
(see also Fig. 2 for AL). Unlike the current density, which is enhanced
throughout the storm and exhibits several spike-like increases in accordance
with AL, the temporal variations of EqB are relatively smooth. A relatively
small difference in evolution on the dayside and nightside EqBs is observed. At
the peaks of the storm, EqB is at about 50∘ MLat, while during the
late recovery phase, EqB is shifted poleward as high as 70∘ MLat.
Possible expansions of the FAC region during the substorm growth phase and
then its contraction after onset are difficult to resolve with the Swarm data.
The equatorward displacement of FACs roughly correlates with the storm
intensity as monitored by the SYM-H index, while the storm-time substorms can
modify this relationship. In Fig. 6, separately for the main and recovery
phases, the correlations between SYM-H and the nightside EqB are shown. Data
from both the Northern Hemisphere and Southern Hemisphere are included. The
correlation coefficients (cc) for the main and recovery phases are very similar
(cc = 0.88 and 0.87), while the corresponding regression equations are
considerably different. During the storm main phase, the equatorward
expansion of EqB is governed by the equation MLat =63.1+0.1⋅SYMH. When the recovery phase begins, the poleward shift of EqB is described
by the expression MLat =79.5+0.3⋅SYMH. The faster poleward
recovery of EqB compared with its equatorward expansion is due to the fast
decrease in substorm activity on 9 September.
Correlations between the SYM-H index and the latitudinal
position of the nightside (∼ 22:00 MLT) EqB: black dots and open
triangles correspond to the main and recovery phases, respectively.
Small-scale FACs
It is known that FACs appear on a wide range of scales, from large-scale
sheet-like currents of hundreds of kilometers in width to very small-scale
filamentary currents of hundreds of meters in width. The quasi-instantaneous
amplitudes of the small-scale component are often much larger than the
stationary R1/R2 FACs. The current intensity varies inversely with scale, so
that large-scale currents are typically a few µA m-2, whereas the
smaller-scale currents (down to 10 km) are a few tens of µA m-2 (Neubert and
Christiansen, 2003; Luhr et al., 2015; McGranaghan et al., 2017). To obtain
the time series of the Swarm peak current densities on 6–9 September 2017, the
largest positive and negative 1 s values were selected from each crossing in
a given MLT time sector irrespective of the hemisphere. The obtained peak
values are presented in Fig. 7. First of all, from this figure one can see
that the small-scale peaks may be more than an order of magnitude larger
than the FACs averaged over a track (cf. Fig. 4). On 6 September, only two
outliers of about +20 and -30µA m-2 are
observed. Both are from the pre-midnight sector and are associated with a
moderate substorm that occurred in the middle of this day. During the disturbed
period, starting with the compression of the magnetosphere on 7 September,
the amplitudes of peaks tend to increase. Two intense substorms occurring
during the storm main phase cause an additional strengthening of small-scale
FACs at all MLTs. At ∼ 00:00 UT on 8 September, the upward and
downward currents at early morning local times attain their extremes
of 70–80 µA m-2. The second major substorm that occurred in the middle
of 8 September is also accompanied by the peaks, which are more pronounced
on the dusk side, where the upward FAC reaches about -50µA m-2.
Note that some peaks can be missed due to the temporal and spatial gaps
between the satellite tracks.
The largest downward (positive) and upward (negative) 1 s
current densities for four MLT sectors on 6–9 September. The vertical solid
lines mark the beginning of the storm main phase at 22:00 on 7 September
(the time when SYM-H attains its stable negative values <-20 nT;
the period of SYM-H <-20 lasts till the end of 9 September), the
peaks of the first and second major substorms (the time when AL attains its
minimum).
When for each crossing within a certain MLT sector, the minimum (i.e., peak
upward current) and maximum (i.e., peak downward current) 1 s FACs are
selected, it appears that in some cases these peaks are observed at very
close latitudes, while in other cases the minimum and maximum are spaced in
latitude. In Fig. 8, the correlations between the MLats, at which the most
intense small-scale FACs of opposite polarities are observed, are presented
for each MLT sector. The x axis (y axis) corresponds to the MLat of the
downward (upward) peak selected in each crossing. The magnitudes of minima
and maxima are not accounted for. From Fig. 8 one can see that the correlation
between the latitudinal positions of the upward and downward peaks varies with
MLT. The highest correlation coefficient (cc = 0.94) is found in the
pre-noon sector (Fig. 8b). This is indicative of a large population of the
paired, closely adjacent small-scale currents of opposite polarity (called
hereafter the bipolar structure). At dusk (Fig. 8a) the correlation
coefficient decreases down to 0.78. Almost the same correlation (cc = 0.75)
is observed in the pre-midnight sector (Fig. 8c). In the early morning hours
(Fig. 8d) the correlation is much weaker (cc = 0.53), implying that the
extreme upward and downward currents appear less frequently in pairs but rather
are spatially (or temporary) separated. Different mechanisms of the
small-scale FAC formation on the day side and night side can be the cause of
this spatial distribution and variability.
Correlations between magnetic latitudes at which the upward
and downward peak FACs are observed: (a) dusk, 16:00 MLT; (b) pre-noon, 10:00 MLT;
(c) pre-midnight, 22:00 MLT; (d) post-midnight/early morning, 04:00 MLT.
Small-scale FACs of extreme amplitudes
During the storm under consideration a pair of the most intense upward and
downward small-scale FACs is revealed by SwB at around 00:10 UT on 8 September, when the satellite traverses the auroral latitudes from north to south
over the geographic area of the Barents Sea, about 20∘ magnetic
longitude to the east from the IMAGE magnetometer network
(http://space.fmi.fi/image, last access: 1 December 2019). The network produces the IL index, which is
a simple estimate of the total westward currents crossing the IMAGE chain. The
IL index (Fig. 9) shows that the extreme FACs are observed during the first
period of the storm-time substorm intensifications, several minutes before
the IL drops from -1500 to -3700 nT.
The 10 s IL index at 00:00–00:50 UT, 8 September. The time of
the extreme FAC observation is shown by the grey line.
The 1 s FACs and plasma parameters (the electron density, Ne, temperature,
Te, and the spacecraft electric potential, Usc) measured by SwB at
00:08–00:12 UT on 8 September are shown in Fig. 10. As show in Fig. 10a, at 00:10:18–00:10:19 UT the satellite observes the bipolar
current structure of extreme density consisting of the poleward downward (81 µA m-2) and equatorward upward (-66µA m-2) FACs. The
paired upward and downward FACs are of relatively comparable values; thus, they
are balanced and likely closed locally. In Fig. 10a the original 1 s values
are superimposed on the smoothed curve, which reveals a signature of the
downward R1 and upward R2 FACs. The bipolar structure is located at the edge
of the downward FAC.
The 1 s values of (a) FAC density, (b)Ne, (c)Te, and
(d)Usc measured along the SwB track at 00:08:00–00:12:00 UT, 8 September. In the upper plot the 21-point smoothed FAC density is also
shown. Geographic and geomagnetic coordinates are shown at the top.
The bipolar current structure is accompanied by plasma perturbations. A
narrow peak in Ne up to 77×103 cm-3 (Fig. 10b) and an
increase in Te up to ∼104 K on average (Fig. 10c), that
is, ∼50 % above their ambient values, are observed almost
simultaneously with a pair of extreme FACs. (It should be noted that the Te
values presented here are based on the current processing of the satellite
data and may be still uncalibrated. However, this hardly affects the
relatively small-scale perturbations.) The elevated Te is observed in a
wider region slightly poleward of the enhanced Ne. The plasma disturbances
are clearly seen in Usc, which is proportional to -k⋅Te (k is the
Boltzmann constant). Note that the level of noise for the Usc channel
is much lower compared to that for the Te channel (0.4 % and 2 % for
Usc and Te, respectively). Figure 10d shows that a reduction of
Usc starts at 00:09:56 UT and then peaks at 00:10:08 (-12 V) and 00:10:20 UT (-8 V); the average decrease is -5 V. The region where the Te and
Usc are perturbed is several times wider than the region occupied by the
pair of extreme FACs.
If the localized increase in Ne indicates conductance enhancement (likely
due to precipitating electrons), the observed plasma and current
perturbations are similar to those associated with auroral arcs (Opgenoorth
et al., 1990; Lyons, 1992; Johnson et al., 1998; Aikio et al., 1993; Juusola,
et al., 2016). In particular, Aikio et al. (2002) studied the current system
of arcs in the evening sector, where the background electric field is
northward. It was shown that for arcs located within the northward
convection, electric field currents flow downward on the equatorward side of
the arcs, then poleward, and then upward from the arcs. The arcs are associated
with an enhanced northward-directed electric field region on the equatorward
side of the arc. An enhancement in the electric field starts already several
tens of kilometers equatorward of the arc edge.
During the storm under consideration the bipolar FAC pattern observed at
00:10 UT is located in the morning sector, where the background electric
field is expected to be southward. This is confirmed by the SuperDARN-based
convection model (http://vt.superdarn.org/tiki-index.php?page=ASCIIData, last access: 12 October 2019),
which predicts in the region of the SwB observations the magnitudes of the
southward and westward components to be about 6.5 and 0.5 mV m-1,
respectively. As mentioned in Sect. 2.1, unfortunately the in situ Swarm
electric field is unavailable. Only the reported characteristics of the
electric field associated with arcs can be used for qualitative analysis. In
particular, for morning-side arcs an enhanced southward electric field on
the poleward side of the arc is expected. In this case the current pattern
consists of a downward FAC on the poleward side of the arc connected to an
upward current above the arc by an equatorward ionospheric closure current.
This is exactly what is seen in Fig. 10a: when SwB flies away the pole, it
first observes a positive spike (downward FAC) and then a negative spike
(upward FAC). Since the width of the region of enhanced Ne is
∼30 km, the arc is relatively narrow. Comparing Fig. 10a and
b one can see that the paired FACs is located on the poleward side of
the region of enhanced Ne. Note that in Fig. 10b a sharp increase in Ne up
to ∼80×103 cm-3 is preceded by a weaker
spike-like drop down to ∼30×103 cm-3. A
decrease in Ne (which is usually much less pronounced than an increase due
to precipitating electrons) is associated with a downward FAC observed at
the opposite boundary of the arc. Elevations of Te may be created by
electric fields which can arise within a narrow region adjacent to the
northern side of the auroral arc as observed by Aikio et al. (2002).
Discussion
Observations of the LEO Swarm multi-satellite mission are used in order to
identify various characteristics of the storm-time FACs for the severe event
of 6–9 September 2017. During the storm main phase two major substorms
occurred, so that the FAC system evolved under conditions of the
storm–substorm interplay. In mid-September 2017 the separation between the
upper and lower Swarm satellites was about 6 h in local time. Within the
sectors centered at 04:00, 10:00, 16:00 and 22:00 MLT the northern and southern polar
regions were covered by about 60 tracks along which the 1 Hz measurements of
FACs were carried out. These observations made it possible to reveal the
evolution of the large-scale FAC intensities, the displacement of the FAC
equatorward boundaries and some features of the extreme small-scale
FACs.
Large-scale characteristics of FACs
The evolution of large-scale characteristics of FACs during the September 2017
storm is in general agreement with regularities observed previously by
CHAMP during the intense 2003 geomagnetic storms (Wang et al., 2006). The common
feature of all storm times is the equatorward motion of FACs generally
correlating with the storm intensity. During the September 2017 storm the
global coverage of the high latitudes by the precise measurement onboard the
Swarm satellites made it possible to reveal that the FACs were enhanced at all
MLTs starting from the time of the first SW shock arrival at the very
beginning of 7 September, although the northward IMF and the prolonged
period of geomagnetic quietness lasted almost a day. After this quietness a
storm abruptly commenced at ∼ 22:00 UT on 7 September. During
the two-step main phase FACs exhibit three pronounced enhancements, and the
evolution of FACs depends on the MLT sectors. On the dayside FACs strengthen
after the sudden commencement and in response to the first drop in SYM-H,
while the response to the second drop in SYM-H is relatively weak. On the
night side the current intensities follow mainly the substorm dynamics as
monitored in terms of the AL index, promptly respond to the onset of
storm-time substorms and strengthen at the peaks of substorms. At the same
time, during the period between the major substorms, when AL is fully
recovered but SYM-H is not, FACs stay considerably enhanced.
The September 2017 storm is characteristic of a considerable equatorward
expansion of the FAC region as low as 50∘ MLat in both
hemispheres. The latitudinal displacement of FACs is more gradual and smooth
than the changes in current intensity. For comparison, during the 2003
storms the minimum latitudes of peak current density are limited to
52–56∘ MLat (Wang et al., 2006). It should be noted that these
authors defined the latitudinal positions of peak current density but not
the most equatorward boundary of the FAC region; thus, the actual FAC region
may expand to lower latitudes. The lowest latitudinal position of the
storm-time FACs was found by Fujii et al. (1992). For the storm of March 1989 the
equatorward boundary of the FAC system reached as low as 48∘ MLat.
Similar to the 2003 storms, in 2017 the latitudinal positions of EqB
generally follow the SYM-H variations. FACs are shifted further equatorward
during the storm-time substorms. Even a relatively minor substorm that occurred
prior to the storm causes a considerable equatorward displacement of FACs. The
lowest latitude of EqB is observed when both the SYM-H and AL indices reach
their minimums.
Although the storm of September 2017 is considerably weaker (Dst≈-100 nT) than the storms that occurred in 1989 (Dst≈-600 nT) and 2003
(Dst≈-400 nT), the FAC region expands approximately to the same
latitudes. This effect may be interpreted in terms of saturation, when the
FAC region does not expand lower than ∼50∘ MLat
independently of the storm severity. Linear dependence between latitudinal
boundaries of the FAC sheets upon the dayside merging electric field and the AE
and Dst indices has been reported by Xiong et al. (2014). It was also
pointed out that toward high activity a saturation of equatorward expansion
seems to set in.
In September 2017, prior to the storm main phase, when the IMF Bz is northward,
the pre-noon EqB is located at higher latitudes (∼75∘ MLat) compared to the other MLT sectors (∼65∘ MLat). Surprisingly, in the course of the storm main phase, no
considerable difference between the latitudinal positions of EqB in
different MLT sectors is found. After ∼ 12:00 UT on 9 September,
in the late recovery phase (SYM-H is -50 nT), both the dayside and nightside
EqB recover to their undisturbed position (about 70∘ MLat). The
coherent behavior of EqB is rather unexpected because Wang et al. (2006)
found that the poleward recovery of FACs on the night side is slower than on
the day side. Previous analysis of the latitudinal shift of the polar cap
boundaries based on the IMAGE observations during a magnetic storm has also shown
that, if the IMF Bz turns northward, the dayside boundary recovers much
faster than the nightside boundary (Lukianova and Kozlovsky, 2013). This is
because the dynamics of the nightside boundary depends on the energy accumulated
in the magnetotail during the previous period of the storm main phase.
However, it seems that the storm of September 2017 does not show the same
regularity. The reason may be that during the storm main phase the two major
substorms occurred, so that the energy stored in the tail was released
more quickly. Comparing the evolution of the FAC densities and the equatorial
boundary positions during the storm recovery, one can see that the densities
decay much faster than the boundaries return to their quiet-time positions.
High FAC intensity is associated with the auroral oval. Previous studies
based on particle precipitation and optical observations have shown that the
oval radius increases when the ring current is intensified during magnetic
storms (e.g., Meng, 1982; Yokoyama et al., 1998). Significant variations in
the location of the aurora take place during the substorm cycle. Substorms
occurring on expanded auroral ovals during magnetic storms are most intense,
since they close the most magnetospheric open magnetic flux, and the presence
of the enhanced ring current increases the open flux threshold at which
substorm onset is favored (Milan et al., 2009). It was also shown that
changes in oval radius associated with dayside and substorm driving occur on
timescales of minutes and hours, while changes associated with the ring
current are more protracted, as the ring current dissipates slowly (Milan,
2009).
The Swarm observations, although they are instantaneous, reveal a tendency of
the dawn–dusk asymmetry FACs. The dawn–dusk asymmetry is revealed by
comparing the upward and downward FACs, which are summed for all crossings over
dusk and dawn separately. While the summed FAC intensities are comparable
between the two hemispheres, the positive and negative densities at dusk
and dawn are slightly imbalanced and the net current is nonzero. It seems
that the dusk-side downward (R2) FACs are larger than the dusk-side upward
(R1) and dawn-side R1 and R2 currents. The observed imbalance in FACs is
likely related to an intensification of the partial ring current, which is
connected to the R2 FAC at dusk. Strengthening of the partial ring current may
also lead to asymmetric dusk-side inflation of the geomagnetic field lines.
The dawn–dusk asymmetry in strength and the equatorward displacement of R1
and R2 at the peak of the major storm in August 2000 have been reported by
Anderson and Korth (2007). This study utilized the global distributions
of FACs generated at a 10 min cadence separately for the Northern Hemisphere and
Southern Hemisphere by the AMPERE project which is based on the fleet of
Iridium satellites. Although the Swarm observations are unable to provide the
instantaneous global FAC distribution, the responses of FACs in certain MLT
sectors on the dawn side are different from those on the dusk side. Note
that the results in Table 2 are calculated by using the 1 Hz FAC values, and
their averages do not necessarily represent the large-scale R1/R2 FACs.
Nevertheless, for the storm of September 2017, the dawn–dusk asymmetry is
manifested in the enhanced average density of the downward FACs on the dusk
side. This feature is consistent with the global observations by AMPERE,
from which the asymmetry of large-scale FACs can be identified. At the same
time, almost no difference in the equatorward shift of the dusk-side and dawn-side FACs is observed by Swarm.
Small-scale FACs
Due to their large amplitudes, small-scale FACs play an important role in
the energy input to the upper atmosphere. In several previous studies, the
FACs associated with arcs were estimated as 1–10 µA m-2 (Bythrow
and Potemra, 1987; Elphic et al., 1998; Janhunen et al., 2000; Lühr et al.,
2016). A larger range of current densities, varying between 4 and >40µA m-2, has been observed (Aikio et al., 2002), and even more
intense small-scale FACs, up to hundreds of µA m-2, at the edges of
arcs have been measured by MEO satellites (Marklund et al., 1982; Bythrow et
al., 1984). Such a large range of the FAC estimates is likely related to its
different scales (and different techniques), because for arcs with very
sharp electron density gradients, the FACs associated with ionospheric
currents flow in narrow regions at arc edges. If the real widths are
smaller, the current densities are expected to be larger.
Filamentary structures of high densities are always presented in the Swarm
observations. The narrow high-density currents are averaged out when
integrated over a FAC region, so that multilayer structures of steady
large-scale FACs of the R1/R2 type depicted by Iijima and Potemra (1978) can
be revealed after a proper smoothing. From a statistical study of the
temporal and spatial-scale characteristics of different FAC types derived
with the Swarm satellites, Luhr et al. (2015) have shown that small-scale (up to
some 10 km) FACs are carried predominantly by kinetic Alfvén waves. A
persistent period of small-scale FACs was of order 10 s, while large-scale FACs
can be regarded as stationary for more than 1 min. Neubert and Christiansen (2003) studied the morphology of very small-scale FACs from a survey of
Ørsted satellite 25 Hz data. These FACs are distributed in a broad region around
the pre-noon and cusp regions and in the pre-midnight sector. It was found
that at the considered timescale, instantaneous currents may reach the
largest values up to 1000 µA m-2, while the average current
densities reach a maximum of 10 µA m-2. McGranaghan et al. (2017)
demonstrated a local time dependence in the relationships between large
(>250 km) and small FAC scales (10–150 km width; density is up to 0.5 µA m-2). It was found that linear relationships exist near dawn and dusk
local times, while at noon and midnight local times no similar regularity is
seen. The results are based on all available data from the Swarm satellites and
the AMPERE irrespective of the level of geomagnetic activity.
During the September 2017 storm one of the Swarm satellites managed to observe a
pair of the most intense small-scale 7.5 km width FACs of opposite polarity,
the magnitudes of which are approximately +80 and -70µA m-2.
These upward and downward FACs are adjusted to each other and separated in a
fraction of a degree in MLat. The bipolar FAC structure occurs in the region
approximately between R1 and R2, just prior to the abrupt substorm
intensification in the vicinity of the newly developed ionospheric westward electrojet. The
polarity reversal captured by the Swarm data for 2 consecutive seconds implies
a quite localized current closure through the ionosphere mostly via Pedersen
horizontal currents. Although without optical and electric field data one
could not draw a strict conclusion, the small-scale bipolar FAC patterns
accompanied by localized enhancements in Ne and Te are likely associated
with mesoscale discrete aurora. One-to-one correspondence of small-scale
FACs with localized electron precipitation events has been previously
observed (e.g., Fukunishi et al., 1991). The SwB observations are in
agreement with the disturbances expected for the arcs that occurred on the
morning side, where the ambient electric field is southward. The observed
features resemble those reported by Kozlovsky et al. (2007) and Aikio
et al. (2002), but bear in mind that the latter are related to the evening
sector, where the background electric field is northward. Based on
Swarm/THEMIS All Sky Imager observations, Wu et al. (2017) associated multiple auroral arcs
with up–down current pairs. For these arcs unipolar and multipolar FAC
systems with current densities of about a few µA m-2 have been
observed. Arcs in unipolar FAC systems have a typical width of 10–20 km and
a spacing of 25–50 km. Arcs in multipolar systems are wider and more
separated. In the bipolar structure of extreme intensity observed by SwB in
8 September, the current density exceeds the values observed by Wu et al. (2017) at least by a factor of 10, while the spatial extent of FACs is
smaller. This difference implies the existence of sharp electron density
gradients at arc edges. Usually, the arcs consist of auroral rays and bright
spots moving along the arcs, and these spatial irregularities may produce the
extreme small-scale FACs. This study has shown that under disturbed
conditions, FACs forming the arc current system may reach hundreds of µA m-2 on the spatial scale of less than 10 km.
Statistically, the bipolar structures dominate pre-noon. In the
post-midnight MLTs they are observed less frequently. While the
interpretation of the bipolar structure in terms of the mesoscale arc
pattern seems reasonable, the small-scale FACs are often a result of
reconnection processes distributed over the dayside magnetopause and even in
the tail for negative Bz. In contrast to post-midnight, in the pre-noon
sector, where cusp/cleft currents are expected, the bipolar structures are
quite frequent. This may be a signature of the plasma injections which are
accompanied by pairs of FACs generated due to flux transfer event (FTE)
formation (Southwood, 1987) or multiple reconnection at the magnetopause.
Magnetic topologies associated with FTEs were previously observed by the MEO
satellites (Marchaudon et al., 2004, 2006; Pu et al., 2013). These
small-scale FACs are possibly a consequence of turbulence and instabilities
associated with the process of opening previously closed magnetospheric
field lines and merging them with the interplanetary magnetic field
(Watermann et al., 2009). The regularity presented in Fig. 8 shows that
during the September 2017 magnetic storm the bipolar structures dominate
exactly in the region where the signatures of FTEs and the reconnection
lines that formed at the magnetopause are expected. At the same time, a pair of
the most intense FACs is observed on the night side.
Conclusion
Characteristics of FACs inferred from the 1 Hz Swarm observations during the
severe magnetic storm of 6–9 September 2017 are presented. This storm is the
two-step one with an about 22 h preliminary phase, and the intense substorms
occurred in the course of the storm main phase. The satellites cross the
pre-midnight, pre-noon, pre-dusk and pre-midnight sectors. The following
features of the storm-time FACs are found.
The evolution of the current intensities and the latitudinal positions of the
equatorward boundaries of the FAC region are mainly controlled by a
storm–substorm interplay. The FACs become enhanced starting from the SW
shock arrival despite the prolonged period of the northward IMF. The
evolvements of the nightside FACs are combinations of the modulations
related to the geomagnetic storm and substorm. Their densities are more
responsive to the substorm development, while the dayside FACs are
intensified in response to the SW shock and then stay enhanced. At the peak
of the substorm, the FAC densities averaged over a track within a given MLT
sector reach 3±0.25µA m-2, while the undisturbed
level is about 0.2±0.02µA m-2.
The equatorward displacement of FAC sheets correlates with the storm
intensity as monitored by the SYM-H index. The correlation coefficients for
the main and recovery phases are about 0.9, while in the course of the main
phase the rate of equatorward expansion of FACs is slower than their
poleward displacement during the recovery phase. This is likely due to the
relatively fast decrease in substorm activity. The minimum latitude of the
equatorward FAC boundaries is limited to 49–50∘ MLat. Although the
storm of September 2017 is relatively weak (Dst is about -100 nT), the FAC
region expands approximately to the same latitudes as those observed for the
more severe storms.
The filamentary structures of high-density FACs are always presented in the
Swarm observations. A bipolar structure (i.e., the adjacent upward and downward
small-scale FACs), ∼80µA m-2, 7.5 km width, is
observed in the vicinity of the newly developed westward electrojet just
prior to the substorm onset. Simultaneous plasma perturbations indicate that
the FAC pattern is likely associated with the mesoscale auroral arc.
Data availability
The data used for the publication of
this research are freely available from the Swarm Science Team web site
(ftp://swarm-diss.eo.esa.int, ESA, 2018) or the Swarm visualization tool
(https://vires.services/, last access: 1 December 2019).
Competing interests
The author declares that there is no conflict of interest.
Acknowledgements
Swarm data are available through the European Space Agency Online platform
(ftp://swarm-diss.eo.esa.int, last access: 1 December 2019), after registration. We acknowledge the
Swarm Science Team for providing the Level 2 data and the Swarm visualization tool
(https://vires.services/, last access: 1 December 2019). The OMNI data on the solar wind, interplanetary
magnetic field and geomagnetic indices are obtained from NASA/GSFC's Space
Physics Data Facility's CDA web service (http://omniweb.gsfc.nasa.gov/, last access: 12 October 2019).
Review statement
This paper was edited by Dalia Buresova and reviewed by five anonymous referees.
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