Magnetic reconnection can vary considerably in spatial
extent. At the Earth's magnetopause, the extent generally corresponds to the
extent in local time. The extent has been probed by multiple spacecraft
crossing the magnetopause, but the estimates have large uncertainties because
of the assumption of spatially continuous reconnection activity between
spacecraft and the lack of information beyond areas of spacecraft coverage.
The limitations can be overcome by using radars examining ionospheric flows
moving anti-sunward across the open–closed field line boundary. We therefore
infer the extents of reconnection using coordinated observations of multiple
spacecraft and radars for three conjunction events. We find that when
reconnection jets occur at only one spacecraft, only the ionosphere conjugate
to this spacecraft shows a channel of fast anti-sunward flow. When
reconnection jets occur at two spacecraft and the spacecraft are separated by
<1 Re, the ionosphere conjugate to both spacecraft shows a
channel of fast anti-sunward flow. The consistency allows us to determine the
reconnection jet extent by measuring the ionospheric flows. The
full-width-at-half-maximum flow extent is 200, 432, and 1320 km,
corresponding to a reconnection jet extent of 2, 4, and 11 Re. Considering
that reconnection jets emanate from reconnections with a high reconnection
rate, the result indicates that both spatially patchy (a few Re) and
spatially continuous and extended reconnections (>10 Re) are
possible forms of active reconnection at the magnetopause. Interestingly, the
extended reconnection develops from a localized patch via spreading across
local time. Potential effects of IMF Bx and By on the reconnection
extent are discussed.
Introduction
A long-standing question in magnetic reconnection is the following: what is
the spatial extent of reconnection in the direction normal to the
reconnection plane? At the Earth's magnetopause, for a purely southward IMF,
this corresponds to the extent in the local time or azimuthal direction. The
extent of reconnection has significant relevance to solar wind–magnetosphere
coupling, as it controls the amount of energy being passed through the
boundary from the solar wind into the magnetosphere and ionosphere.
Magnetopause reconnection tends to occur at sites of strictly antiparallel
magnetic fields as antiparallel reconnection (e.g., Crooker, 1979; Luhmann
et al., 1984) or along a line passing through the subsolar region as
component reconnection (e.g., Sonnerup, 1974; Gonzalez and Mozer, 1974).
Evidence shows either or both can occur at the magnetopause, and the overall
reconnection extent can span from a few to 40 Re (Paschmann et al.,
1986; Gosling et al., 1990; Phan and Paschmann, 1996; Coleman et al.,
2001; Phan et al., 2000b, 2003; Chisham et al., 2002, 2004a, 2008;
Petrinec and Fuselier, 2003; Fuselier et al., 2002, 2003, 2005,
2010; Pinnock et al., 2003; Bobra et al., 2004; Trattner et al., 2004,
2007, 2008, 2017; Trenchi et al., 2008). However, reconnection does not
occur uniformly across this configuration but has spatial variations
(Pinnock et al., 2003; Chisham et al., 2008), and it is reconnection with
high reconnection rates that effectively contributes to the momentum and
energy flow within the magnetosphere. Reconnection with high reconnection
rates is expected to cause rapid magnetic flux generation and fast
reconnection jets. This paper therefore investigates the spatial extent of
reconnection through the extent of reconnection jets.
Numerical models show that reconnection tends to occur at magnetic
separators, i.e., at the junction between regions of different magnetic field
topologies, and global magnetohydrodynamics (MHDs)
models have identified a spatially continuous separator along the
magnetopause (Dorelli et al., 2007; Laitinen et al., 2006, 2007; Haynes
and Parnell, 2010; Komar et al., 2013; Glocer et al., 2016). However,
little is known about where and over what range along the separators
reconnection proceeds at a high rate. Reconnection in numerical simulations
can be activated by introducing perturbations of the magnetic field or can
grow spontaneously with instability or resistivity inherent in the system
(e.g., Hesse et al., 2001; Scholer et al., 2003). When reconnection
develops as patches (due to instabilities or localized perturbations), the
patches can spread in the direction out of the reconnection plane (Huba and
Rudakov, 2002; Shay et al., 2003; Lapenta et al., 2006; Nakamura et
al., 2012; Shepherd and Cassak, 2012; Jain et al., 2013). The patches
either remain patchy after spreading if the current layer is thick or form an
extended X line if the current layer is already thin (Shay et al., 2003).
Studies have attempted to constrain the extent of reconnection based on
fortuitous satellite conjunctions whereby the satellites detect reconnection
jets at the magnetopause at different local times nearly simultaneously
(Phan et al., 2000a, 2006; Walsh et al., 2014a, b, 2017). The satellites
were separated by a few Re in Phan et al. (2000a) and Walsh et
al. (2014a, b, 2017) and >10 Re in Phan et al. (2006), and
this is interpreted as the reconnection being active over a few Re and even
10 Re, respectively. At the magnetopause, reconnection of a few Re is often
referred to as spatially patchy (e.g., Fear et al., 2008, 2010), and
reconnection of >10 Re is spatially extended (Dunlop et
al., 2011; Hasegawa et al., 2016). The term patchy has also been used to
describe the temporal characteristics of reconnection (e.g., Newell and
Meng, 1991). But this paper primarily focuses on the spatial properties. The
extent has been alternatively determined by studying the structures of newly
reconnected flux tubes, i.e., flux transfer events (FTEs) (Russell and
Elphic, 1979; Haerendel et al., 1978). Conceptual models regard FTEs
either as azimuthally narrow flux tubes that intersect the magnetopause
through nearly circular holes, as formed by spatially patchy reconnection
(Russell and Elphic, 1979), or as azimuthally elongated bulge structures or
flux ropes that extend along the magnetopause, as formed by spatially
extended reconnection (Scholer, 1988; Southwood et al., 1988; Lee and
Fu, 1985). FTEs have been observed to be on the order of a few Re wide in
local time (Fear et al., 2008, 2010; Wang et al., 2005, 2007). FTEs have
also been observed across ∼20 Re from the subsolar region to the
flanks (Dunlop et al., 2011). But it is unclear whether these FTEs are
branches of one extended bulge or flux rope or multiple narrow tubes formed
simultaneously. When the satellites are widely spaced, it is in general
questionable whether a reconnection jet and/or FTE is spatially continuous between the satellites or
whether satellites detect the same moving reconnection jet and/or FTE. Satellites
with a small separation may possibly measure the same reconnection jet and/or FTE,
but only provide a lower limit estimate of the extent. A reconnection
jet and/or FTE may also propagate or spread between satellite detections, but
satellite measurements cannot differentiate spatial and temporal effects.
This situation can be improved by studying ionospheric signatures of
reconnection and FTEs, since their spatial sizes in the ionosphere can be
obtained from wide-field ground instruments or low-Earth-orbit spacecraft.
The ionospheric signatures include poleward-moving auroral forms (PMAFs),
channels of flows moving anti-sunward across the open–closed field line
boundary (e.g., Southwood, 1985), and cusp precipitation (Lockwood and
Smith, 1989, 1994; Smith et al., 1992). Radar studies have shown that the
flows can differ considerably in size, varying from tens of kilometers
(Oksavik et al., 2004, 2005) to hundreds of kilometers (Goertz et al.,
1985; Pinnock et al., 1993, 1995; Provan and Yeoman, 1999; Thorolfsson
et al., 2000; McWilliams et al., 2000b, 2001) or thousands of kilometers
(Provan et al., 1998; Nishitani et al., 1999; Provan and Yeoman, 1999).
A similarly broad distribution has been found for PMAFs (e.g., Sandholt et
al., 1986, 1990; Lockwood et al., 1989, 1990; Milan et al., 2000, 2016)
and the cusp (Crooker et al., 1991; Newell and Meng, 1994; Newell et
al., 2007). This range of spatial sizes in the ionosphere approximately
corresponds to a range from <1 to >10 Re at the
magnetopause. However, care needs to be taken when interpreting the above
ionospheric features, since they could also form due to other drivings such
as solar wind dynamic pressure pulses (Lui and Sibeck, 1991; Sandholt et
al., 1994). Unambiguous proof of their connection to magnetopause
reconnection requires simultaneous space–ground coordination (Elphic et
al., 1990; Denig et al., 1993; Neudegg et al., 1999, 2000; Lockwood et
al., 2001; Wild et al., 2001, 2005, 2007; McWilliams et al., 2004;
Zhang et al., 2008).
Therefore, a reliable interpretation of reconnection extent has been
difficult due to observation limitations. We will address this by comparing
the extents probed by multiple spacecraft and radars using space–ground
coordination. On the one hand, this enables us to investigate whether
reconnection continuously spans between satellites and how wide
reconnection extends beyond satellites. On the other hand, this helps to
determine whether reconnection is the driver of ionospheric disturbances and
whether the in situ extent is consistent with the ionospheric disturbance
extent.
Methodology
We study the characteristic extent of reconnection jets as the local time
extent of reconnection. We use conjugate measurements between the Time
History of Events and Macroscale Interactions during Substorms (THEMIS)
(Angelopoulos, 2008) and Super Dual Auroral Network (SuperDARN) (Greenwald et al.,
1995). We focus on
intervals when the IMF in OMNI data remains steadily southward. We require
that two of the THEMIS satellites fully cross the magnetopause nearly
simultaneously and that the satellite data provide clear evidence for
reconnection occurring or not. The full crossings are identified by a
reversal of the Bz magnetic field and a change in the ion energy
spectra. The requirements of nearly simultaneous crossings and steady IMF
conditions help to reduce the spatial–temporal ambiguity by satellite
measurements, for which the presence and absence of reconnection jets at different local times likely reflects
spatial structures of reconnection. Reconnection can still possibly vary
between the two satellite crossings, and we use the radar measurements to
examine whether the reconnection of interest has continued to exist and
maintained its spatial size.
Identification of reconnection jets in the magnetosphere is based on
fluid (MHD) evidence of magnetopause reconnection. Reconnection accelerates
plasma bulk flow to Alfvénic speed, producing reconnection jets at the
magnetopause, and the acceleration should be consistent with the prediction
of tangential stress balance across a rotational discontinuity, i.e., the Walen
relation (Hudson, 1970; Paschmann et al., 1979). The Walen relation is expressed as
ΔVpredicted=±1-α11/2μ0ρ1-1/21B21-α2/1-α1-B1,
where ΔV is the change in the plasma bulk velocity vector across the
discontinuity. B and ρ are the magnetic field vector and plasma mass
density. μ0 is the vacuum permeability. α=(p||-p⊥)μ0/B2 is the anisotropy factor wherein p||
and p⊥ are the plasma pressures parallel and perpendicular to the
magnetic field. The magnetic field and plasma moments are obtained from
fluxgate magnetometer (FGM) (Auster et al., 2008) and electrostatic
analyzer (ESA) instruments (McFadden et al., 2008). The plasma mass density
is determined using the ion number density, assuming a mixture of 95 %
protons and 5 % helium. The subscripts 1 and 2 refer to the reference
interval in the magnetosheath and to a point within the magnetopause,
respectively. The magnetosheath reference interval is a 10 s time period
just outside the magnetopause. The point within the magnetopause is taken at
the maximum ion velocity change across the magnetopause. We ensure that the
plasma density at this point is >20 % of the magnetosheath
density to avoid the slow-mode expansion fan (Phan and Paschmann, 1996). We
compare the observed ion velocity change with the prediction from the Walen
relation. The level of agreement is measured by ΔV*=ΔVobs⋅ΔVpredicted/|ΔVpredicted|2, following Paschmann et al. (1986). Here ΔVobs is
the observed ion velocity change. By convention only the velocity
changes with ΔV*>0.5 are classified as reconnection jets
(e.g., Phan et al., 1996; Phan and Paschmann, 2013).
To further ensure that reconnection occurs, we examine the kinetic signature
of reconnection, which is D-shaped ion distributions at the magnetopause. As
magnetosheath ions encounter newly opened magnetic field lines at the
magnetopause, they either transmit through the magnetopause entering the
magnetosphere or reflect at the boundary. The transmitted ions have a cutoff
parallel velocity (i.e., de Hoffmann–Teller velocity) below which no ions
could enter the magnetosphere. The D-shaped ion distributions are deformed
into a crescent shape as ions travel away from the reconnection site (Broll
et al., 2017). We require the satellites to operate in the fast survey or
burst mode in which ion distributions are available at 3 s resolution.
We determine reconnection as active if the plasma velocity change across
the magnetopause is consistent with the Walen relation with ΔV*>=0.5 and if the ions at the magnetopause show a
D-shaped distribution. Reconnection is deemed absent if neither of the two
signatures is detected. We require that at least one of the two satellites
observes reconnection signatures. Reconnection is regarded as ambiguous if
only one of the two signatures is detected, and such reconnection is
excluded from our analysis.
We mainly use the three SuperDARN radars located at Rankin Inlet (RKN;
geomagnetic 72.6∘ MLAT, -26.4∘ MLON), Inuvik (INV;
71.5∘ MLAT, -85.1∘ MLON), and Clyde River (CLY;
78.8∘ MLAT, 18.1∘ MLON) to measure the ionospheric convection
near the dayside cusp. The three radars have overlapping fields of view
(FOVs), enabling a reliable determination of the 2-D convection velocity. The
FOVs cover the ionosphere >75∘ MLAT, covering the
typical location of the cusp under weak and modest solar wind driving
conditions (i.e., Newell et al., 1989) and the high occurrence region of
reconnection-related ionospheric flows (Provan and Yeoman, 1999) with high
spatial resolution. Data from Saskatoon (SAS; 60∘ MLAT,
-43.8∘ MLON) and Prince George (PGR; 59.6∘ MLAT,
-64.3∘ MLON) radars are also used when data are available. The
measurements of these two radars at far-range gates can overlap the cusp. The
radar data have a time resolution of 1–2 min. We focus on observations ±3 h MLT from magnetic noon (approximately 16:00–22:00 UT). The satellite
footprints should be mapped close to the radar FOVs under the Tsyganenko
(T89) model (Tsyganenko, 1989). Footprints mapped using different
Tsyganenko (e.g., T96 or T01; Tsyganenko, 1995, 2002a, b) models have
similar longitudinal locations (difference <100 km), implying
the longitudinal uncertainty of mapping to be small. The latitudinal
uncertainty can be inferred by referring to the open–closed field line
boundary as estimated using the 150 m s-1
spectral width boundary (e.g., Baker et al., 1995, 1997; Chisham and Freeman, 2003). T89 has given
the smallest latitudinal uncertainty for the studied events. We surveyed
the years 2014–2016 during the months when the satellite apogee was on the
dayside and present three events in the paper.
The ionospheric signature of reconnection jets includes fast anti-sunward
flows moving across the open–closed field line boundary. We obtain the flow
velocity vectors by merging line-of-sight (LOS) measurements at the radar
common FOVs (Ruohoniemi and Baker, 1998), and these merged vectors reflect the true ionospheric
convection velocity. However, the radar common FOVs are hundreds of kilometers wide
only, which can be too small to cover the full azimuthal extent of the
reconnection-related flows (which are up to thousands of kilometers wide). We
therefore also reconstruct the velocity field using the spherical elementary
current system (SECS) method (Amm et al., 2010). Similar to the works by Ruohoniemi et al. (1989) and
Bristow et al. (2016), the SECS method reconstructs a divergence-free flow pattern using
all LOS velocity data. We refer to these velocities as SECS velocities. The
accuracy of SECS velocities can be validated by comparing to the LOS
measurements and the merged vectors. SECS velocities work best in regions
with dense echo coverage, and those around sparse echoes are not reliable and
thus are excluded from our analysis.
The third way of obtaining a velocity field is spherical harmonic fit (SHF).
This method uses the LOS measurements and a statistical convection model to
fit the distribution of electrostatic potential, which is expressed as a sum
of spherical harmonic functions (Ruohoniemi and Baker, 1998). The statistical model employed
here is from Cousins and Shepherd (2010). While this method may suppress small-scale or mesoscale velocity
details, such as sharp flow gradients or flow vortices, we compare SHF
velocities with the LOS measurements and merged vectors to determine how
well the SHF velocities depict the velocity details.
As seen in our observations presented below, the longitudinal profile of the
fast anti-sunward ionospheric flows has a nearly bell-shaped curve. We measure
the extent based on the full width at half maximum (FWHM) of the profile at
1∘ poleward of the open–closed field line boundary. The choice of
FWHM is analogous to Shay et al. (2003), in which the reconnection extent is measured as
regions of electron speed above half of the peak electron flow speed during
reconnection. The choice is also supported by magnetopause observations,
in which we find that ionospheric flows with a speed above half of the peak
flow speed map to jets consistent with the Walen relation, while those with a
speed below map to jets much slower than the Walen relation (Sect. 3.1).
However, it should be noted that the magnitude of the widths is always
dependent on the threshold used, and the half maximum is very likely not
the only sensible threshold. Using FWHM excludes ionospheric flows with a
speed below half of the peak flow speed. Those flows, if related to
reconnection, are associated with the comparatively slow generation of open magnetic
flux and a low contribution to geomagnetic activity.
Among the three presented events, the time separations of magnetopause
crossings by two satellites are 1, 2, and 30 min. While the time separation
for the third case is somewhat long, we distinguish spatial and temporal
effects using the radar data. Although the three events occurred under
similar IMF Bz conditions, the reconnection-related flows in the
ionosphere had an azimuthal extent varying from a few hundred kilometers
(Sect. 3.1–3.2) to more than 1000 km wide (Sect. 3.3). This corresponds to
reconnection a few to >10 Re wide, indicating that both
spatially patchy (a few Re) and spatially continuous and extended
reconnections (>10 Re)
are possible forms of active reconnection at the magnetopause.
Interestingly, the extended reconnection was found to arise from a spatially
localized patch that spreads azimuthally. Potential effects of IMF Bx
and By on the reconnection extent are discussed in Sect. 4.
Note that reconnection can happen over various spatial and temporal scales
and our space–ground approach can resolve reconnections that are larger than
0.5 Re and persist longer than a few minutes. This is limited by the radar
spatial and temporal resolution and the magnetosphere–ionosphere coupling
time, which is usually 1–2 min (e.g., Carlson et al., 2004). This
constraint is not expected to impair the result because reconnection above
this scale has been found to occur commonly in statistics (see the
Introduction for spatial characteristics and the following for temporal
characteristics: Lockwood and Wild, 1993; Kuo et al., 1995; Fasel,
1995; McWilliams et al., 2000a).
ObservationsSpatially patchy reconnection active at one satellite onlyIn situ satellite measurements
On 2 February 2013, THA and THE made simultaneous measurements of the
dayside magnetopause with a 1.9 Re separation in the Y direction around
21:25 UT. The IMF condition is displayed in Fig. 1a and the IMF was
directed southward. The satellite location in GSM coordinates is
displayed in Fig. 1b, and the measurements are presented in Fig. 2. The
magnetic field and the ion velocity components are displayed in the LMN
boundary-normal coordinate system, wherein L is along the outflow
direction, M is along the X line, and N is the current sheet normal. The
coordinate system is obtained from the minimum variance analysis of the
magnetic field at each magnetopause crossing (Sonnerup and Cahill Jr., 1967). Figure 2g–p
show that both satellites passed from the magnetosheath into the magnetosphere,
as seen as the sharp changes in the magnetic field, the ion spectra, and the
density (shaded in pink).
(a) OMNI IMF condition on 2 February 2013. (b) THE and THA
locations projected to the GSM X–Y plane. The inner curve marks the
magnetopause and the outer curve marks the bow shock.
As THE crossed the magnetopause boundary layer (21:22:57–21:23:48 UT), it
detected a rapid, northward-directed plasma jet within the region where the
magnetic field rotated (Fig. 2g and j). The magnitude of this jet
relative to the sheath background flow reached 262 km s-1 at its peak, which
was 72 % of the predicted speed of a reconnection jet by the Walen
relation (366 km s-1, not shown). The angle between the observed and predicted
jets was 39∘. THE also detected kinetic signatures of
reconnection. The ion distributions in Fig. 2k show a distorted D-shaped
distribution similar to the finding of Broll et al. (2017). The distortion is due to
particles traveling in the field-aligned direction from the reconnection
site to a stronger magnetic field region, and Broll et al. (2017) estimated the traveling
distance to be a few Re for the observed level of distortion.
(a) SuperDARN LOS speeds (color tiles) and merged velocity
vectors (color arrows) in the altitude-adjusted corrected geomagnetic
(AACGM) coordinates. The FOVs of the RKN, INV, and CLY radars are outlined
with black dashed lines. The colors of the tiles indicate the LOS speeds away
from the radar. The colors and the lengths of the arrows indicate the merged
velocity magnitudes, and the arrow directions indicate the velocity
directions. Red and anti-sunward-directed flows are the ionospheric signature
of magnetopause reconnection. The dashed magenta lines mark the flow western
and eastern boundaries. The open–closed field line boundary is delineated by
the dashed black curve with the OCB marker. The satellite footprints under
the T89 are shown as the THE and THA markers. (b) Similar to
(a) but showing SECS velocity vectors (color arrows).
(c) Similar to (a) but showing SHF velocity vectors (color
arrows). (d) SuperDARN spectral width measurements (color tiles).
The red contour marks localized enhanced soft electron precipitation.
(e) Time evolution of the northward component of SECS velocities
along 79∘ MLAT. (f) Profile of convection velocities along
79∘ MLAT at 21:29 UT as a function of the distance from magnetic
noon. The profile in black is based on the LOS measurements, and the profile
in red is the northward component of the SECS velocities. The FWHM is
determined based on each profile. (g–j) THE measured magnetic field
(0.25 s resolution), ion energy flux (3 s), ion density (3 s), and ion
velocity (3 s). The ion measurements were taken from ground ESA moments. The
magnetic field and the ion velocity components are displayed in the LMN
boundary-normal coordinate system. The magnetopause crossing is shaded in
pink. (k) THE ion distribution function on the bulk
velocity–magnetic field plane. The small black line indicates the direction
and the bulk velocity of the distributions. (l–p) THA measurements
in the same format as in (g)–(k). (q–z) THA and
THE measurements during a subsequent magnetopause crossing shown in the same
format as in (g)–(p).
THA crossed the magnetopause 1 to 2 min later than THD
(21:24:48–21:25:13 UT). While it still identified a plasma jet at the
magnetopause (Fig. 2l and o), the jet speed was significantly smaller
than what was predicted for a reconnection jet (80 km s-1 versus 380 km s-1 in
the L direction). The observed jet was directed 71∘ away from the
prediction. The ion distributions deviated from clear D-shaped distributions
(Fig. 2p). This suggests that the reconnection jet at THE likely did not
extend to THA.
Ground radar measurements
The velocity field of the dayside cusp ionosphere during the satellite
measurements is shown in Fig. 2a–c. Figure 2a shows the radar LOS
measurements at 21:25 UT, as denoted by the color tiles, and the merged
vectors, as denoted by the arrows. The colors of the arrows indicate the
merged velocity magnitudes, and the colors of the tiles indicate the LOS
speeds that direct anti-sunward (those projected to the sunward direction
appear as black). Fast (red) and anti-sunward flows are the feature of
interest here. One such flow can be identified in the prenoon sector,
which had a speed of ∼800 m s-1 and was directed poleward and
westward. As the merged vector arrows indicate, the velocity vectors have a
major component close to the INV beam directions, and thus the INV LOS
velocities reflect the flow distribution. The flow crossed the open–closed
field line boundary, which was located at 78∘ MLAT based on the
spectral width (Figs. 2d and S1 in the Supplement). This flow thus meets the criteria
for
being an ionospheric signature of magnetopause reconnection jets. Another
channel of fast flow was present in the post-noon sector. This post-noon
flow was directed more azimuthally and was increasingly separated from the
prenoon flow as it moved away from noon (see the region of slow
velocities at >79∘ MLAT around noon). The difference
in flow trajectories implies that these flows were driven by different
magnetic tension forces. They also evolved differently over time as seen in
Fig. 2e, which is discussed below. The flows thus likely originated from
two reconnection regions that were associated with different magnetic field
topologies and different temporal variabilities. Since the satellites were
located in the prenoon sector we focus on the prenoon flow below.
The flow had a limited azimuthal extent. The extent is determined at half of
the maximum flow speed, which was ∼400 m s-1. Figure 2f,
discussed below, shows a more quantitative estimate of the extent. In Fig. 2a, we mark the eastern and
western boundaries with dashed magenta
lines, across which the LOS velocities dropped from red to blue–green.
Figure 2b shows the SECS velocities, denoted by the arrows. The SECS
velocities reasonably reproduced the spatial structure of the flows seen in
Fig. 2a. The flow boundaries are marked by dashed magenta lines,
across which the flow speed dropped from red to blue.
The velocity field reconstructed using the SHF velocities is shown in Fig. 2c
(obtained through the Radar Software Toolkit;
http://superdarn.thayer.dartmouth.edu/software.html, last access:
March 2018). This is an expanded view of the global convection maps in
Fig. S2 focusing on the dayside cusp. Comparing Figs. 2c and S2 reveals that
the employed radars listed in Sect. 2 have contributed to the majority of the
backscatter on the dayside. This is because this event (the same for the
following two events) occurred under non-storm time, wherein the open–closed
field line was confined within the FOVs of the radars used. During storm time
the boundary expands to a lower latitude at which backscatter from a wider
network of radars may be available. The SHF velocities also captured the
occurrence of two flows in the prenoon and post-noon sectors, respectively,
although the orientation of the flows was slightly different from Fig. 2a or
b. The difference is likely due to the contribution from the statistical
potential distribution under the southward IMF. The flow western and eastern
boundaries are again marked by dashed magenta lines.
Figure 2d shows spectral width measurements. Large spectral widths can be
produced by soft (∼100 eV) electron precipitation
(Ponomarenko et al., 2007), and evidence has shown that the longitudinal
extent of large spectral widths correlates with the extent of PMAFs (Moen et
al., 2000) and of poleward flows across the open–closed field line boundary
(Pinnock and Rodger, 2000). Large spectral widths thus have the potential
to reveal the reconnection extent. For the specific event under examination,
the region of large spectral widths, appearing in red, spanned from
10.5 to 14.5 h MLT if we count the sporadic scatter in the post-noon sector.
This does not contradict the flow width identified above because the wide
width reflects the summed width of the prenoon and post-noon flows. In fact, a
more careful examination shows that there might be two dark red regions
(circled in red; the red dashed line is due to the discontinuous backscatter
outside the INV FOV) embedded within the ∼200 m s-1 spectral
widths. These two regions had slightly higher spectral widths than the
surrounding regions (by ∼20–50 m s-1) and possibly corresponded to the
two flows.
Figures 2a–c all show a channel of fast anti-sunward flow in the
prenoon sector of the high-latitude ionosphere, and the flow had a limited
azimuthal extent. If the flow corresponded to a magnetopause reconnection
jet, the reconnection jet would be expected to span a limited local time
range. This is consistent with the THEMIS satellite observation in Sect. 3.1.1,
in which THE at Y=-2.9 Re detected a clear reconnection jet, while
THA at Y=-4.8 Re did not. In fact, if we project the satellite location
to the ionosphere through field line tracing under the T89 model, THE was
positioned at the flow longitude, while THA was to the west of the flow
embedded in weak convection (Fig. 2a).
While this paper primarily focuses on the spatial extent of reconnection,
the temporal evolution can be obtained from the time series plot in Fig. 2e.
Figure 2e presents the northward component of the SECS velocity along
79∘ MLAT (just 1∘ poleward of the open–closed field line
boundary) as a function of magnetic local time (MLT) and time. Here we only
show the northward component of the SECS velocity as this component
represents reconnecting flows across an azimuthally aligned open–closed
field line boundary. Similar to the snapshots, the flow of interest here
appears as a region of red. The time and the location where THA and
THE crossed the magnetopause are marked by crosses. The prenoon flow
emerged from a weak background from 21:22 UT and persisted for
∼>30 min, while the post-noon flow only lasted
for ∼10 min. In the minutes following the onset, the prenoon flow
spread in width, whereby the western boundary of red moved from 10.7
to 10.5 h MLT, and the eastern boundary moved from 11.2 to 11.5 h MLT. After
21:34 UT the spreading ceased and the entire flow moved westward (the western
boundary moved beyond the FOV). Hence the reconnection-related ionospheric
flow, once formed, has spread in width and displaced westward. The spreading
behavior is similar to events studied by Zou et al. (2018) and is
interpreted to relate to spreading of the reconnection extent seen in
simulation studies (see Introduction). The spreading has also been noticed
in the other two events (see Sect. 3.3), indicating that this could be a
common development feature of reconnection-related flows.
A consequence of the flow temporal evolution is that THA, which was
previously outside the reconnection-related flow, became immersed in the
flow from 21:30 UT, while THE, which was previously inside the flow, was left
outside from 21:42 UT (Fig. 2e). This implies that at the magnetopause the
reconnection has spread azimuthally, sweeping across THA, and has slid in the
-y direction away from THE. This is in perfect agreement with satellite
measurements shown in Fig. 2q–z. Figure 2q–z present subsequent
magnetopause crossings made by THA and THE following the crossings in
Fig. 2g–p. THA detected an Alfvénic reconnection jet and a clear D-shaped
ion distribution, and THE detected a jet much slower than the Alfvénic speed
and an ion distribution without a clear D shape. This corroborates the
connection between the in situ reconnection jet with the fast anti-sunward
ionospheric flow and reveals the dynamic evolution of reconnection in the
local time direction. On the other hand, this also sheds light on the nature
of the slow convection outside the fast flow, which corresponds to
sub-Alfvénic jets at the magnetopause.
We quantitatively determine the flow extent in Fig. 2f. Figure 2f shows
the profile of the northward component of the SECS velocity at 21:29 UT as a
function of the distance from magnetic noon. At 21:29 UT
the flow extent has slowed down from spreading and stabilized. The profile
should theoretically be taken just poleward of the open–closed field line
boundary. In practice we smooth the velocity in latitude with a 1∘
window and take measurements 1∘ poleward of the open–closed field
line boundary. The profile has a nearly bell-shaped curve, and the FWHM was
200 km at an altitude of 250 km. Also shown is the INV LOS velocity profile,
which is obtained in a similar manner as the SECS one. The LOS velocity
profile also gives a narrow FWHM, which was 280 km.
While it is commonly assumed that the extent of reconnection jets reflects
the extent of reconnection, we test the assumption by calculating the
distribution of the reconnection electric field in Fig. 3. The reconnection
electric field can be estimated by measuring the flow across the open–closed
field line boundary in the reference frame of the boundary (Pinnock et al., 2003;
Freeman et al., 2007; Chisham et al., 2008), and we follow this procedure to derive its distribution
across local time. A close-up presentation of the open–closed field line
boundary is shown in Fig. 3a–c around the space–ground conjunction time
and longitude. The open–closed field line boundary, drawn as a dashed
black line, is identified following Chisham and Freeman (2003, 2004) and Chisham et al. (2004b, 2005a, b,
c). The boundary was almost along a constant magnetic latitude. The
motion of the boundary is obtained by inspecting the time series of the
spectral width measurements along each radar beam and examples are given for
INV beams 4, 7, and 10 in Fig. 3d–f. Subtracting the speed of the
boundary from that of the flow (in the rest frame) across the boundary gives
the flow speed in the reference frame of the boundary. Assuming that the
flow is E×B drift, the electric field can be derived, and this
is the ionosphere-mapped reconnection electric field. The flow speed across
the boundary is taken from the 1∘ averaged speed at the boundary
latitude (similar to Chisham et al., 2008). Note that a precise determination of the
boundary motion could be subject to radar spatial and temporal resolution,
and the error can be as large as 300 m s-1 or 15 mV m-1.
(a–c) Snapshots of spectral width measurements around the
space–ground conjunction time and longitude. The open–closed field line
boundary is shown as a dashed black line. (d–f) Time series of the
spectral width measurements along INV beams 4, 7, and 10 as a function of
latitude, from which the motion of the open–closed field line boundary can
be derived. (g) The electric field along the open–closed field line
boundary in the frame of the boundary (solid) and in the rest frame (dashed)
following Pinnock et al. (2003), Freeman et al. (2007), and Chisham et
al. (2008). The former is the reconnection electric field.
As shown in Fig. 3g, the profile of the reconnection electric field had a
peak in the azimuthal direction with a limited FWHM, and the FWHM is
essentially the same as the flow width just poleward of the boundary
(the difference being less than the radar spatial resolution). This establishes
the relation between our measure of the reconnection jet extent and the
extent of reconnection with high reconnection rates. Regions of high
reconnection rates are localized, although those of low reconnection rates
(>0 mV m-1) can extend over a much broader region. For example,
the western boundary of nonzero reconnection rates was located just at the
edge of INV FOV (considering the 15 mV m-1 uncertainty), and the eastern edge
extended beyond INV FOV, likely into where the post-noon flow originated. A lower estimate of the extent of nonzero reconnection rates is
therefore ∼4 h MLT. It is likely that there were two
components of reconnection at different scales: broad and low-rate
background reconnection and embedded high-rate reconnection.
To infer the reconnection extent at the magnetopause, we project the flow
extent based on the SECS in the ionosphere to the equatorial plane. The
result suggests that the reconnection local time extent was ∼2 Re.
Spatially patchy reconnection active at both satellitesIn situ satellite measurements
On 19 April 2015, under a southward IMF (Fig. 4a), THA and THE crossed
the magnetopause nearly simultaneously (<2 min lag) with a 0.5 Re
separation in Y (Fig. 4b). They passed from the magnetosheath into the
magnetosphere. Both satellites observed jets in the VL component at the
magnetopause (Fig. 5g–p). The jet at THA at ∼18:28:05 UT
had a speed of 84 % and an angle within ∼15∘
of the Walen prediction. The jet at THE at ∼18:26:25 UT had
a speed of 95 % and an angle within ∼29∘ of the
Walen prediction. The ion distributions at THA and THE exhibit clear
D-shaped distributions. Reconnection thus occurred at both local times.
OMNI IMF condition and THEMIS satellite locations on 19 April 2015
in a similar format to Fig. 1.
Ground radar measurements
During the satellite measurements, the radars observed a channel of fast
anti-sunward flow around magnetic noon (Fig. 5a–c). The flow crossed the
open–closed field line boundary at 77∘ MLAT and qualifies as an
ionospheric signature of magnetopause reconnection jets. The flow direction
was nearly parallel to the RKN radar beams, and therefore the RKN LOS
measurements in Fig. 5a approximate to the 2-D flow speed. The flow
eastern boundary can be identified as where the velocity dropped from
red–orange to blue (dashed magenta line). Determining the flow western
boundary requires more measurements of the background convection velocity,
which is beyond the RKN FOV. But we infer that the western boundary did not
extend more than 1.5 h westward beyond the RKN FOV because the PGR and INV
echoes there showed weakly poleward and equatorward LOS speeds around the
open–closed field line boundary. The CLY radar data further indicated that
the anti-sunward flow had started to rotate westward immediately beyond the
RKN FOV. This is because the CLY LOS velocities measured between the RKN and
INV radar FOVs were larger for more east–west-oriented beams (appearing in
yellow) than for more north–south-oriented beams (green). The
rotation likely corresponds to the vortex at the flow western boundary as
sketched in Oksavik et al. (2004).
THEMIS and SuperDARN measurements of reconnection bursts on 19 April 2015
in a similar format to Fig. 2. The velocity time evolution in
(e) and the velocity profile in (f) are taken along
78∘ MLAT.
The more precise location of the western boundary can be retrieved from the
SECS velocities in Fig. 5b and the SHF velocities in Fig. 5c. The SECS
velocities present a flow channel very similar to that in Fig. 5a, while
the flow channel in the SHF velocities was more azimuthally aligned than in
Fig. 5a–b. It can be seen that across the flow western boundary the flow
direction reversed. The equatorward flows are interpreted as the
return flow of the poleward flows, as sketched in Southwood (1987) and Oksavik et al. (2004).
The determined flow extent agrees with the extent of the cusp in Fig. 5d.
The high spectral widths associated with the cusp were located at the
western half of the RKN FOV. They extended westward beyond the RKN FOV into
CLY far-range gates, where they dropped from red to green. This is
consistent with the inferred location and the extent of the anti-sunward fast
flow.
The flow of interest here just emerged from a weak background at the time
when the THEMIS satellites crossed the magnetopause (Fig. 5e). This
implies that the related reconnection just activated at the studied local
time. The flow spread azimuthally until 18:33 UT when it stabilized. We
quantify the stabilized flow extent and the reconnection electric field
extent (Fig. 5f) in a similar way as Figs. 2f and 3g. The FWHM of
the flow is determined to be 432 and 336 km based on the SECS and RKN LOS
data, respectively. While the reconnection electric field had data gaps due
to limited coverage and backscatter availability at the near-range gate, it
implies a western boundary of FWHM consistent with the flow slightly
poleward of it. This is also the western boundary of nonzero reconnection
rates considering the 15 mV m-1 uncertainty. The eastern boundary extended
beyond RKN FOV. The FWHM of the SECS flow profile corresponds to
∼4 Re in the equatorial plane.
The fact that the fast anti-sunward flow had a limited azimuthal extent
around magnetic noon implies that the corresponding magnetopause
reconnection should span a limited local time range around noon.
This is consistent with the THEMIS satellite observation in Sect. 3.2.1,
showing that reconnection was active at Y=0.7 (THA) and 0.2 Re (THE).
Projecting THA and THE locations to the ionosphere reveals that both
satellite footprints were located within the flow longitudes. Therefore, the
reconnection at the two satellites was part of the same reconnection around
the subsolar point of the magnetopause. (The THE footprint was equatorward
of THA because the X location of THE was closer to the Earth than THA. The
magnetopause was expanding, and it swept across THE and then THA.) The
reconnection further extended azimuthally beyond the two satellite
locations, reaching a full length of ∼4 Re.
Spatially continuous and extended reconnection active at both
satellitesIn situ satellite measurements
On 29 April 2015, under a prolonged and steady southward IMF (Fig. 6a), THA
and THE crossed the magnetopause successively with a time separation of
∼30 min. The locations of the crossings were separated by
0.1–0.2 Re in the Y direction (Fig. 6b). The satellites passed from the
magnetosphere into the magnetosheath, and the magnetic field data suggest
that the satellites crossed the current layer multiple times before
completely entering the magnetosheath (Fig. 6i–r). We therefore only
display the magnetic field and the plasma velocity in GSM coordinates.
Both satellites detected multiple flow jets, all agreeing with the Walen
prediction with ΔV*>0.5. For example, the jet at
18:49–18:50 UT measured by THA had a speed with 80 % and an angle with
9∘ of the Walen prediction, and the jet at 19:20–19:22 UT by THE
had a speed with 83 % and an angle with 1∘ of the Walen
prediction. The ion distributions at THA and THE exhibit clear D-shaped
distributions.
OMNI IMF condition and THEMIS satellite locations on 29 April 2015
in a similar format to Fig. 1.
Ground radar measurements
In the ionosphere, the radars detected a fast anti-sunward flow as an
ionospheric signature of the magnetopause reconnection jet (Fig. 7a–c).
The flow had a broad azimuthal extent, as delineated by the dashed magenta
lines (Fig. 7a). A similar flow distribution is found in the SECS
velocities (Fig. 7b) and the SHF velocities (Fig. 7c). The flow
propagated into the polar cap as one undivided channel (as opposed to
Sect. 3.1.2), implying that it was one flow structure, at least to the
resolution the radars can resolve. Corresponding to the broad extent of the
flow, the cusp had a broad extent (Fig. 7d). The cusp continuously spanned
the INV and RKN FOVs, and its western and eastern edges coincided with
the western and eastern boundaries of the flow, supporting our delineation
of the flow extent.
THEMIS and SuperDARN measurements of reconnection bursts on
29 April 2015 in a similar format to Fig. 2. The velocity time evolution in
(e) and the velocity profile in (f) are taken along
79∘ MLAT. The two branches of the LOS velocity profile in (f) are based on INV and RKN LOS data. The magnetic field and plasma
velocities measured by spacecraft are displayed in the GSM coordinates.
The wide flow channel in the ionosphere implies that the corresponding
magnetopause reconnection jet should be wide in local time. Based on the
flow distribution, we infer that much of the reconnection should be located
on the prenoon sector, except that the eastern edge can extend across the
magnetic noon meridian to the early post-noon sector. This inference is
again consistent with the inference from the THA and THE measurements that
the reconnection extended at least over the satellite separation (Y=-0.2
(THA) and 0 Re (THE)). Note, however, that the distance between THA and THE
only covered <2 % of the reconnection extent determined from the
ionosphere flow. While the satellite configuration and measurements here
were similar to those in Sect. 3.2, the extent of reconnection was
fundamentally different. This suggests that it is difficult to obtain a
reliable estimate of the reconnection extent without the support of
2-D
measurements, and satellites alone also cannot differentiate spatially
extended reconnection from spatially patchy reconnection.
The flow temporal evolution is shown in Fig. 7e, where the velocities are the
northward component of the SECS data. An overall wide flow channel is seen
during the time interval of interest here with the eastern and western
boundaries located at ∼12.0–12.5 and ∼8.0–8.7 h MLT,
respectively. But between the two satellite observations, the flow
experienced an interesting variation. The velocity at 9.3–12.0 h MLT
dropped by 100–200 m s-1 during 19:02–19:12 UT (red turned orange,
yellow, and then green), while the velocity at 8.6–9.3 h MLT did not change
substantially. The velocity was enhanced again from 19:12 UT. The
enhancement centered at 10.7 h MLT and spread azimuthally towards east and
west. The enhancement spread by 0.7 h MLT over 14 min at its eastern end
(marked by the dashed black line), suggesting a spreading speed of
275 m s-1. The enhancement spread by 1.2 h MLT at its western end,
suggesting a spreading speed of 471 m s-1. It should be noted that all
three components of the IMF stayed steady for an extended time (Fig. 8,
discussed below in Sect. 4), and thus the evolution of the flow and
reconnection was unlikely to be externally driven.
Comparison of the IMF and solar wind driving conditions between the
reconnection events on 2 February 2013, 19 April 2015, and 29 April 2015.
From top to bottom: IMF in GSM coordinates, IMF clock angle, solar wind
speed, and solar wind dynamic pressure. The red vertical lines mark the times
of the satellite–ground conjunction.
This sequence of changes gives an important implication that the spatially
extended reconnection was a result of the spreading of an initially patchy
reconnection. If we map the spreading in the ionosphere to the magnetopause,
the spreading occurred bidirectionally and at a speed of 15 and 26 km s-1 in
the east and west directions based on field line mapping under the T89 model
(the mapping factor was 55). Such an observation is similar to what has
recently been reported by Zou et al. (2018), in which the reconnection also spreads
bidirectionally at a speed of a few tens of km s-1. However, the spreading in
Zou et al. (2018) occurs following a southward turning of the IMF, while the spreading
here occurred without IMF variations. The mechanism of spreading is
explained either as motion of the current carriers of the reconnecting
current sheet or as propagation of the Alfvén waves along the guide field
(Huba and Rudakov, 2002; Shay et al., 2003; Lapenta et al., 2006; Nakamura et al.,
2012; Jain et al., 2013).
It should be noted that reconnection spreading can be a common process of
reconnection that is not limited to extended reconnection. It also occurs
for patchy reconnection as seen in Sect. 3.1 and 3.2. The spreading
speeds were similar across the three events, but the duration of the
spreading process was 2 to 3 times longer in the spatially extended
than the spatially patchy reconnection events. For the extended
reconnection, the spreading process persisted for 14 min, expanding the
extent by 5–6 Re.
Figure 7f quantifies the extent of the flow and reconnection electric field.
The FWHM extent was 1320 km based on the SECS data. Despite the presence of
data gaps, the LOS measurements suggest a western and eastern boundary
consistent with the SECS data. The reconnection electric field had a similar
FWHM to the flow, although regions of nonzero reconnection rates again
extended beyond the available coverage, indicating an overall extent
>4 h MLT. The extent corresponds to a reconnection extent of
∼11 Re.
Discussion
The above events definitely show that the local time extent of magnetopause
reconnection can vary from a few to >10 Re. Here we
investigate whether and how the extent may depend on the upstream driving
conditions. Figure 8 presents the IMF, the solar wind velocity, and the solar
wind pressure taken from the OMNI data for the three events. The red vertical
lines mark the times when the reconnection was measured. The three events
occurred under similar IMF field strengths (5–6 nT), similar IMF Bz
components (-2–3 nT), and similar dynamic pressures (1–2 nPa), implying
that the different reconnection extents were unlikely due to these
parameters. The solar wind speeds had a slight decreasing trend as the
reconnection extent increased. This is different from Milan et al. (2016),
who identified a large solar wind speed as the cause of a large reconnection
extent. However, Milan et al. (2016) studied reconnection under very strong
IMF driving conditions when |B|∼15 nT, while our events occurred
under a more typical moderate driving (|B|∼5–6 nT).
The spatially patchy reconnection events had an IMF Bx of a larger
magnitude than the extended reconnection event (4 vs. 0 nT). The spatially
patchy reconnection events also had an IMF By component of a smaller
magnitude (2 vs. 5 nT, and therefore a clock angle closer to 180∘),
and with more variability on timescales of tens of minutes, than the extended
reconnection event. The IMF Bx and By components are known to
modify the magnetic shear across the magnetopause and to affect the
occurrence location of reconnection. Studies have found that at the dayside
low-latitude magnetopause small By|/|Bz relates to antiparallel and large By|/|Bz to component reconnection (Coleman et al., 2001; Chisham
et al., 2002; Trattner et al., 2007). Large Bx|/|B, i.e., cone angle, also favors the formation of high-speed
magnetosheath jets (Archer and Horbury, 2013; Plaschke et al., 2013) of a
few Re in scale size, resulting in a turbulent magnetosheath environment for
reconnection to occur (Coleman and Freeman, 2005). The steady IMF condition
may allow reconnection to spread across local times unperturbedly, eventually
reaching a wide extent. Thus our preliminary analysis suggests that the
reconnection extent may depend on the IMF orientation and steadiness,
although whether and how they influence the extent needs to be further
explored.
Summary
We carefully investigate the local time extent of magnetopause reconnection
by comparing measurements of reconnection jets by two THEMIS satellites
and three ground radars. When reconnection jets are only observed at one of
the two satellite locations, only the ionosphere conjugate to this
spacecraft shows a channel of fast anti-sunward flow. When reconnection jets
are observed at both spacecraft and the spacecraft are separated by
<1 Re, the ionosphere conjugate to both spacecraft shows a channel
of fast anti-sunward flow. The fact that the satellite locations are mapped
to the same flow channel suggests that the reconnection is continuous
between the two satellites and that it is appropriate to take the satellite
separation as a lower limit estimate of the reconnection extent. Whether
reconnection can still be regarded as continuous when the satellites are
separated by a few or >10 Re is questionable and needs to be
examined using conjunctions with a larger satellite separation than what
has been presented here.
The reconnection extent is measured as the FWHM of the ionospheric flow. In
the three conjunction events, the flows have FWHM of 200, 432, and 1320 km
in the ionosphere, which corresponds to ∼2, 4, and 11 Re at
the magnetopause (under the T89 model) in the local time direction. The flow
extent is confirmed to be related to reconnection with a high reconnection
electric field. The result provides strong observational evidence that
magnetopause reconnection can occur over a wide range of extents, from
spatially patchy (a few Re) to spatially continuous and extended
(>10 Re). Interestingly, the extended reconnection is seen to
initiate from a patchy reconnection, whereby the reconnection grows by
spreading across local time. The speed of spreading is 41 km s-1, summing the
westward and eastward spreading motion, and the spreading process persists
for 14 min, broadening the extent by 5–6 Re.
Based on the three events studied in this paper, the reconnection extent may
be affected by the IMF orientation and steadiness, although the mechanism is
not clearly known. For the observed modest solar wind driving conditions,
spatially extended reconnection is suggested to occur under a smaller IMF
Bx component and a larger and steadier IMF By component than
spatially patchy reconnection. The IMF strength, the Bz component, and
the solar wind velocity and pressure are about the same for the extended and
the patchy reconnection. This finding, however, could be limited by the
number of events under analysis, and further study is needed to achieve an
understanding of how solar wind controls reconnection extent. Reconnection
can vary with time, even under steady IMF driving conditions.
Data availability
Data products from SuperDARN, THEMIS, and OMNI are
available at http://vt.superdarn.org/ (last access: March 2018),
http://themis.ssl.berkeley.edu/index.shtml (last access: March 2018)
and the GSFC–SPDF OMNIWeb website.
The supplement related to this article is available online at: https://doi.org/10.5194/angeo-37-215-2019-supplement.
Author contributions
The study was conceived and designed by YZ, BMW, and YN.
VA, JMR, KAM, and NN acquired the data. YZ, BMW, and YN analyzed and
interpreted the data. YZ drafted the paper. YZ, BMW, YN, and KAM performed
critical revisions.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
This research was supported by the NASA Living
With a Star Jack Eddy Postdoctoral Fellowship Program, administered by
UCAR's Cooperative Programs for the Advancement of Earth System Science
(CPAESS), NASA grant NNX15AI62G, NSF grants PLR-1341359 and AGS-1451911, and
AFOSR FA9550-15-1-0179 and FA9559-16-1-0364. The THEMIS mission is supported
by NASA contract NAS5-02099. SuperDARN is a collection of radars funded by
national scientific funding agencies. SuperDARN Canada is supported by the
Canada Foundation for Innovation, the Canadian Space Agency, and the
Province of Saskatchewan. We thank Tomoaki Hori for useful discussions on the
SECS technique.
Review statement
This paper was edited by Minna Palmrot and reviewed by Karlheinz Trattner and two anonymous referees.
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