Characteristics of a stable are based on FAST and MIRACLE observations

Abstract

A stable evening sector arc is studied using observations from
the FAST satellite at 1250 km altitude and the MIRACLE ground-based network,
which contains all-sky cameras, coherent radars (STARE), and magnetometers. Both
FAST and STARE observe a northward electric field region of about 200 km width
and a field magnitude of about 50 mV/m southward of the arc, which is a typical
signature for an evening-sector arc. The field-aligned current determined from
FAST electron and magnetometer data are in rather good agreement within the
arcs. Outside the arcs, the electron data misses the current carriers of the
downward FAC probably because it is mainly carried by electrons of smaller
energy than the instrument threshold. Studying the westward propagation speed of
small undulations associated with the arc using the all-sky cameras gives a
velocity of about 2 km/s. This speed is higher than the background ionospheric
plasma speed (about 1 km/s), but it agrees rather well with the idea originally
proposed by Davis that the undulations reflect an E × B motion in
the acceleration region. The ground magnetograms indicate that the main current
flows slightly south of the arc. Computing the ionospheric conductivity from
FAST electron data and using the ground magnetograms to estimate the current
yields an ionospheric electric field pattern, in rather good agreement with FAST
results.Key words: Ionosphere (auroral ionosphere;
ionosphere-magnetosphere interactions) - Magnetospheric physics (auroral
phenomena)

Full text

Characteristics of a stable arc based on FAST and MIRACLE
observations
P. Janhunen1, A. Olsson2, O. Amm1, K. Kauristie1
1Finnish Meteorological Institute, Geophysical Research, Helsinki, Finland
2Swedish Institute of Space Physics, Uppsala Division, Uppsala, Sweden
Received: 16 June 1999 / Revised: 11 October 1999 / Accepted: 14 October 1999
Abstract. A stable evening sector arc is studied using
observations from the FAST satellite at 1250 km
altitude and the MIRACLE ground-based network,
which contains all-sky cameras, coherent radars
(STARE), and magnetometers. Both FAST and STARE
observe a northward electric ®eld region of about
200 km width and a ®eld magnitude of about 50 mV/m
southward of the arc, which is a typical signature for
an evening-sector arc. The ®eld-aligned current deter-
mined from FAST electron and magnetometer data are
in rather good agreement within the arcs. Outside the
arcs, the electron data misses the current carriers of the
downward FAC probably because it is mainly carried by
electrons of smaller energy than the instrument thresh-
old. Studying the westward propagation speed of small
undulations associated with the arc using the all-sky
cameras gives a velocity of about 2 km/s. This speed is
higher than the background ionospheric plasma speed
(about 1 km/s), but it agrees rather well with the idea
originally proposed by Davis that the undulations re¯ect
an EBmotion in the acceleration region. The ground
magnetograms indicate that the main current ¯ows
slightly south of the arc. Computing the ionospheric
conductivity from FAST electron data and using the
ground magnetograms to estimate the current yields an
ionospheric electric ®eld pattern, in rather good agree-
ment with FAST results.
Key words: Ionosphere (auroral ionosphere;
ionosphere-magnetosphere interactions) ±
Magnetospheric physics (auroral phenomena)
1 Introduction
Static or nearly static auroral arcs are a common feature
in the auroral ionosphere, but there are still many open
questions concerning the physical mechanisms responsi-
ble for their creation and maintenance. To make progress
in the understanding of more dynamical cases, it is
necessary to improve our knowledge of single isolated
stable arcs, because many of the dynamical auroral
phenomena such as curls, folds and auroral breakups
may be at least partly described by perturbations of stable
auroral arcs, or as waves propagating along the arcs.
The typical behavior of the horizontal electric ®eld at
and near to arcs in both evening and morning sectors is
rather well known (Aikio, 1995; Marklund, 1984), but
the distribution of the electric ®eld at higher altitudes,
especially above 10000 km, is much less certain. There
are indications that at least in some cases, the potential
structure above inverted-V regions (Lin and Homan,
1979) could be O-shaped rather than U-shaped (Jan-
hunen et al., 1999).
In this work we will study a FAST passage over a
stable arc which is simultaneously detected by four
ground-based all-sky cameras and the STARE bistatic
coherent radar. The instrumentation allows us to
compare the optical and satellite-inferred widths of the
arcs and to compare the horizontal electric ®elds as
measured by the satellite and by the radars. Likewise, we
estimate the westward velocity of horizontal arc undu-
lations from the all-sky images and compare this with
the electric ®eld measurements and discuss it also in
terms of a model in which these undulations propagate
with an EBvelocity persisting higher up in the
acceleration region (Haerendel et al., 1996).
2 Instrumentation
The FAST satellite (Carlson et al., 1998a) was launched
in August 21, 1996 into an 83elliptic orbit with an
apogee of 4175 km and a perigee of 350 km. DC electric
and magnetic ®eld data as well as the electron data from
FAST are used in this study.
We will use data from the MIRACLE (Magne-
tometer, Ionospheric Radar and All-sky Camera Large
Experiment) ground-based instrument network (Syrja
È-
suo et al., 1998) (Fig. 1). The all-sky camera (ASC) data
from the three stations at Kilpisja
Èrvi (KIL), Kevo
Ann. Geophysicae 18, 152±160 (2000) ÓEGS ± Springer-Verlag 2000

(KEV) and Muonio (MUO) have a time resolution of
20 s. A 557 nm ®lter was used, thus the images show
only green aurora. The 630 nm ®lter images recorded at
60 s interval are not used in this study. We will also use
the electric ®eld observations of the recently renewed
STARE (Scandinavian Twin Auroral Radar Experi-
ment) radar (Greenwald et al., 1978, Nielsen et al., 1998)
and refer to the IMAGE magnetometer network (Lu
Èhr
et al., 1998) to establish that the level of geomagnetic
disturbances is rather low in this event.
3 Observations
On November 3 1998, 17:35±17:37 UT, FAST moved
from north to south above northern Scandinavia
detecting two rather narrow and moderately intense
(about 10 mW mÿ2) inverted-V type precipitation re-
gions. The FAST electron spectrograms are shown in
Fig. 2. We have performed Maxwellian ®ts to these
spectra (not shown). The peak electron energy within
the arcs is around 5 keV and the characteristic energy
(temperature) about 1.5 keV. As can be seen in Fig. 2,
FAST passes the ®rst one at about 17:35:37±17:35:45
UT and the second one at 17:36:11±17:36:18 UT. We
call these arcs arc 1 and arc 2, respectively. In between
arcs 1 and 2 there are also weaker subvisual arcs. We
want to emphasize that although arc 1 looks very similar
to arc 2 in the FAST data, it does not seem to be an
elongated arc according to the ASC recordings at KIL.
It is rather a single ``blob'' of precipitation, or if it is an
arc, it is generally less intense than arc 2 but has a local
enhancement just at the point where FAST passes
through. Since the ``blob'' is not a temporally stable
feature, in our later analysis we will mostly concentrate
on arc 2 which is a typical stable arc.
In Fig. 3 we show the ®eld-aligned current (FAC) as
determined by two dierent methods from FAST data,
using the magnetic ®eld variation and electron data. The
ion contribution to the FAC is found from the FAST
ion data (not shown) to be negligible in this case. The
inferred upward (positive) FAC density of a few lAm
ÿ2
agrees well between the two methods. The satellite
magnetometer shows that outside the arcs, there is a
downward (negative) FAC, not visible in the electron
data (the FAC estimated from the electron detector is
positive throughout the event), probably because FAST
is at 1250 km altitude and thus the downward current is
carried by upward moving cold electrons which are
below the instrument threshold (25 eV). In regions
where the magnetometer shows a downward FAC the
electron precipitation is small, except around 69.5
GLAT where there is a secondary peak in the upward
FAC carried by precipitating electrons. The secondary
peak is correlated with an upward excursion of the
magnetometer-produced FAC. This is a signature of
counterstreaming electrons.
The passing time as determined from FAST (Fig. 2)
for arc 2 is about 7 s. The velocity of the spacecraft is
7.6 km/s, giving the arc width at FAST altitude of about
53 km, which at ionospheric altitude corresponds to
about 40 km (using the dipole approximation).
The MIRACLE magnetometer and all-sky camera
stations were shown together with the footprint of the
FAST orbit in Fig. 1. We use data from the KIL, MUO
and KEV all-sky cameras. The Abisko (ABK) all-sky
camera also detected the arcs, but the viewing conditions
were not so good. The KIL station is exactly below the
FAST trajectory, and MUO is close to it. Of these,
the KIL station has the best viewing angle of the arc.
The KEV station has excellent viewing conditions and
the arc is close to zenith, but it views the arc about
300 km east of the FAST footpoint.
Both the MUO and KIL recordings show that arc 2 is
centered at 69.90:1geographic latitude. The IM-
AGE magnetometer data shows that the level of
magnetic activity is low (<100 nT). From a close
inspection of the all-sky images at dierent times it is
found that arc 2 hardly moves in the north-south
direction during the event.
In Fig. 4 we show the KIL all-sky camera image at
17:36:20 histogram-equalized, false-colored and map-
ped to 110 km altitude. The histogram equalization
makes it easier to detect the location of the arc while
making the information about exact intensities less
representative.
From the all-sky cameras we can conclude that the
arc width is less than 30 km, which is slightly less than
Fig. 1. Locations of MIRACLE all-sky camera and magnetometer
stations together with the ionospheric projection of the FAST orbit
P. Janhunen et al.: Characteristics of a stable arc based on FAST and MIRACLE observations 153

the FAST estimate of 40 km given above. However, the
half-value width of the energy ¯ux peak of arc 2 (Fig. 2,
panel 7) is about 5 s, which corresponds to a spatial
distance of 30 km at the ionosphere. Thus the optical
and FAST widths are not in disagreement.
We now consider the equatorward side of arc 2. In
this region the electric ®eld is, according to STARE,
about 50 mV/m (corresponding to about 1 km/s elec-
tron ¯ow velocity) and pointing northward. Generally,
STARE receives an echo only when the electric ®eld is
above the Farley-Buneman threshold. The threshold
value depends on the E-region ion temperature but is
usually about 17 mV/m. Near the boundaries of the
region where STARE receives echoes we indeed see that
the electric ®elds are of the order of 17 mV/m. STARE
¯ow velocity vectors are drawn in Fig. 5 only when both
radars' backscatter intensity was at least 2 dB above the
background noise.
FAST electric ®eld data (Fig. 7) also show a north-
ward electric ®eld with a maximum of 50 mV/m and an
average of 40 mV/m on the southern side of arc 2, in
accordance with STARE. Mapped to ionosphere,
40 mV/m corresponds to 50 mV/m so the agreement
with STARE is good. Between the arcs the FAST
electric ®eld is weak, below the STARE threshold. On
the northern side of arc 1 FAST sees a narrow region of
high southward electric ®eld, which is outside the
STARE ®eld of view, however.
Fig. 2. FAST data for 19981103,
17:35±17:37 UT. Panels from top
to bottom are numbered from 1
to 8.1, Eastward magnetic ®eld
de¯ection from the satellite mag-
netometer; 2, downgoing electron
energy ¯ux; 3, pitch angle distri-
bution of electrons, 4, upgoing
electron energy ¯ux, 5, burst
mode data similar to panel 2; 6,
total electron ¯ux; 7, total elec-
tron energy ¯ux; 8, electric ®eld
component along spacecraft or-
bit. Arcs identi®ed from FAST
electrons are marked with red
lines. In panels 6and 7, no red
curves are visible because the
electron ¯ux is downgoing
throughout the event
154 P. Janhunen et al.: Characteristics of a stable arc based on FAST and MIRACLE observations

The FAST traversal is close to the westward boun-
dary of the STARE ®eld of view. About 200±300 km
east of the FAST trajectory, STARE detects a north-
ward electric ®eld also on the northern side of arc 2,
while the ®eld seen by FAST is much weaker. We do not
know an exact reason for this. One possibility is that the
electric ®eld north of the arc changes in the east-west
direction.
Between arcs 1 and 2 the satellite-measured electric
®eld is anticorrelated with the electron precipitation
quite well. (The region is ®lled with weak subvisual
arcs.) Comparing panel 6 of Fig. 2 with the lower panel
of Fig. 7 we see that whenever there is a peak in the
precipitating energy ¯ux, there is a plateau in the
potential. At arc 1, which is a precipitation blob rather
than a real arc, no plateau is seen.
4 Discussion and conclusions
4.1 Comparisons with previous work
FAST electron data miss the downward ®eld-aligned
current in this case. In other studies where this did not
happen (Carlson et al., 1998b), FAST was always close
to 4000 km altitude, i.e., at regions where the electrons
had gained enough energy from a downward electric
®eld to be detected.
To check that our explanation for the downward
current carriers is a feasible one we can estimate how
large electron density is required to carry the required
downward current density by electrons whose upward
speed does not exceed the instrument threshold, i.e.,
jenv with 1=2mev2<25 eV. For j2lAm
ÿ2(the
largest current, see Fig. 3) we obtain n>4:2cm
ÿ3.
Typical densities at the 1200 km altitude are much
higher, so cold electrons can very well act as current
carriers and still remain undetected.
The FAC structure for arc 2 (upward FAC on the
northern side of a downward one) is in accordance with
a schematic model of evening sector arcs (Fig. 6, taken
from Aikio, 1995).
Optical widths of arcs as compared to FAST have
recently been analyzed using TV-cameras mounted on
an aircraft (Stenbaek-Nielsen et al., 1998). Our all-sky
camera data do not allow a detailed comparison.
In the classi®cation of Marklund (1984), arc 2 is most
likely an evening anticorrelation polarization arc (type
Ia).
4.2 Davis-Haerendel model
When viewing the all-sky images as a movie with
suitable color corrections, it is possible to see that there
are westward-propagating small undulations or weak
bulges. By determining the position of these structures in
the adjacent KIL frames (20 s time resolution) from the
images mapped to the geographic grid we can estimate
that the westward velocity is in the range 1.3±3 km/s, the
most probable value being 2 km/s when FAST passes
over. The uncertainty is mainly due to an apparent
temporal variability of the propagation speed. We
believe that these are the same phenomena that have
been observed earlier using TV cameras (Davis, 1978;
Fig. 3. FACs determined from FAST magnetic ®eld and particle data
plotted versus geographic latitude. Positive FAC is upward. The solid
line represents the FAC computed from the electron data while the
dashed line is the FAC estimated from the satellite magnetometer data
Fig. 4. Histogram-equalized false color image, mapped to 110 km
altitude of the Kilpisja
Èrvi (KIL) all-sky camera (the white dot at the
center of the circle) together with FAST trajectory and position (black
dot) at 17:36:20 UT on November 3, 1998. We see that arc 2 is
centered at about 70geographic latitude and its width, as deduced
from the all-sky camera is 30 km or smaller. The bright area
dominating the southeastern part of the all-sky image is the Moon
P. Janhunen et al.: Characteristics of a stable arc based on FAST and MIRACLE observations 155

Haerendel et al., 1996). They suggested that the velocity
of the undulations would correspond to the EB
velocity, where Eis associated with the ``sides'' of a
U-shaped potential in the acceleration region (of course,
mapped down to the ionosphere to honor the ¯ux tube
convergence). This model uses the assumption that the
parallel potential drop V0estimated from the low-
altitude electron peak energy (V05 kV for arc 2) is the
same as the perpendicular potential drop associated with
the ``side'' of the U-shaped potential (Fig. 8).
To apply the Davis-Haerendel idea in our event, we
note that the EBvelocity at any point, mapped down
to the ionosphere, is vi
B=Bi
pE=BE=
BiB
p. For
each latitude point, we ®nd the maximum of vialong the
®eld line and call it the Davis-Haerendel velocity vDH .We
can do this in both O- and U-shaped potential models
(Janhunen et al., 1999), but the result is practically the
same in both cases. We show the result in Fig. 9. In
regions where there is not enough inverted-V type
precipitation, vDH is not a sensible quantity and those
points have not been plotted in Fig. 9.
We see that inside the equatorward half of arc 2, vDH
attains a maximum of about 4 km/s, but is about 2 km/s
in a region just adjacent to arc 2. It is in the adjacent
region that the speed of the undulations was measured;
thus we conclude that the Davis-Haerendel model
produces an estimate which may be somewhat higher
than the observed value, but taking into account the
observational uncertanties there is no disagreement
between the two.
4.3 FAC determined from ground-based measurements
Instantaneous, two-dimensional distributions of iono-
spheric electrodynamic parameters (i.e., height-integrat-
Fig. 5. MIRACLE plot showing
KIL and KEV all-sky camera
views mapped to 110 km alti-
tude, STARE electron velocity
vectors (black) and IMAGE
ground magnetometer equivalent
current vectors (dark blue)
17:36:20 UT. The color scaling
corresponds to ASC pixel values
(originally in the range 0±255).
The red spots in the southeast are
images of the Moon. Arc 2
extends through both camera
views close to 70geographic
latitude. STARE shows a west-
ward drift of about 1 km/s (cor-
responding to a northward
electric ®eld of about 50 mV/m)
Fig. 6. A schematic model of an evening-sector arc (from Aikio,
1995)
156 P. Janhunen et al.: Characteristics of a stable arc based on FAST and MIRACLE observations

ed Hall and Pedersen conductances, and horizontal and
®eld-aligned currents) can be obtained from ground-
based measurements of the magnetic ®eld disturbance
and coherent scatter radar observations of the iono-
spheric electric ®eld using the method of characteristics
(Amm, 1998). In addition to the measurements, an
estimate of the Hall- to Pedersen conductance ratio is
needed. However, the eect of this estimate on the ®nal
results has been shown to be small in most cases (Amm,
1995). Moreover, the value of the conductance ratio can
be assessed from the ground magnetic disturbance level
(e.g., Lester et al., 1996). For the present event, the
STARE radar did not provide sucient backscatter to
apply a full 2D analysis. Thus, and since the arc under
study is essentially a 1D structure, we apply a one-
dimensional version of the method of characteristics
which yields the same output quantities on a north-
south pro®le (see Inhester et al., 1992).
The FAC distribution resulting from this analysis,
carried out in the latitude range between 69 and 71,is
shown in Fig. 10, together with the FAC obtained from
the FAST magnetometer. Both results show consider-
able upward FAC in the vicinity of arc 2. However, the
magnitude of the FAC is considerably larger when
inferred from the ground-based data than from the
FAST data, and its peak is shifted slightly towards
the south. One reason for these dierences can be the
smaller spatial resolution of the ground-based data
which is limited to about 50 km for ground magneto-
meters. Another possibility is that the arc may not be
completely uniform in the east-west direction, since the
ground-based pro®le is located about 4of longitude
Fig. 7. FAST electric ®eld along spacecraft orbit (upper panel)and
integrated potential along the orbit (lower panel)
Fig. 8. A schematic ®gure of the Davis-Haerendel suggestion to
explain the high speed of east-west undulations which propagate along
the edges of an arc (reproduced from Haerendel et al., 1996, Fig. 8)
Fig. 9. The perpendicular velocity according to Davis-Haerendel
suggestion, applied to our event. The FAST potential from Fig. 7 (in
kV) is shown as a visual aid (dotted line)
P. Janhunen et al.: Characteristics of a stable arc based on FAST and MIRACLE observations 157

eastward of the FAST satellite path. Another discrep-
ancy between the FAST and the ground-based data is
that the magnetometers show the largest eastward
electrojet (equivalent) currents not at arc 2, but 50±
100 km south of it; this can be seen e.g., from the
location of zero of interpolated and upward-continued
vertical variation magnetic ®eld component (not
shown). Accordingly, to carry this current, a moderate
conductance (5±10 S) required in this region results from
the analysis with the method of characteristics (data not
shown), although the electron precipitation detected by
FAST is weak there. It should be noted that such an
electrojet ¯ow south of an optical arc is a typical
situation in ground-based observations in the evening
sector (Marklund et al., 1982). One possible explanation
for the origin of the conductance south of the arc is that
it is caused by proton precipitation. According to FAST
data (not shown), in this region the ion energy ¯ux is still
increasing when the upper limit of the energy range
(25 keV) is reached; thus the magnetospheric protons
are hot and could provide signi®cant ionization that
results in the observed conductance values. The ioniza-
tion production pro®les by proton and electron precip-
itation are known (Kirkwood and Osepian, 1995), but
since we do not have measurements of high energy ions,
we cannot study this quantitatively here. More low-
orbiting satellite/ground-based conjunction studies are
needed to resolve the dierence between the conduc-
tance distribution as obtained by ground-based mea-
surements, and the one expected from the satellite's
precipitating electron measurements.
An alternative approach is to solve the electric ®eld
from the ionospheric electrodynamic equations, if the
ground magnetograms and the ionospheric conductivi-
ties are known. We can determine the conductivities
from FAST electron data using the formulas given by
(Robinson et al., 1987). The computed conductivities are
shown in the top panel of Fig. 11. The resulting electric
®eld is shown in the bottom panel, together with the
northward electric ®eld measured by FAST. Taking into
account the observational inaccuracies the agreement is
rather good. The discrepancy in the electric ®eld in the
69.0±69.5 geographic latitude could be reduced by
assuming that the conductivities are slightly enhanced
here e.g., due to proton precipitation. The STARE
electric ®eld cannot be directly compared with this since
STARE has only a very few data points which are exactly
under the FAST trajectory. If one averages the STARE
northward electric ®eld in the east-west direction, one
obtains a result which is markedly dierent from the
FAST or the Robinson-model curves: the STARE ®eld is
not small north of and especially inside arc 2. We do not
have a good explanation for this, other than that STARE
Fig. 10. The FAC determined from STARE and ground magneto-
meters by the method of characteristics (solid line) is compared here
with the FAC determined from FAST magnetometer (dashed line).
Positive FAC is upward
Fig. 11. Top, height-integrated ionospheric Hall (solid line)and
Pedersen (dashed ) conductivity as determined from FAST electron
data using formulas of Robinson et al. (1987). Conductivities smaller
than 2 mho have been set to 2 mho. Bottom, northward electric ®eld
inferred from the ground magnetograms and the conductivities in the
top panel (smoother curve), and FAST northward electric ®eld (jagged
curve)
158 P. Janhunen et al.: Characteristics of a stable arc based on FAST and MIRACLE observations

is not measuring exactly the same region and that the arc
may be too narrow for STARE to resolve. To ®gure out a
possible cause for the discrepancy, it would be worth-
while in the future to try to ®nd more exact arc
conjunctions between FAST and STARE.
5 Conclusions
1. A stable arc (called arc 2 in the text) was studied
using FAST and MIRACLE. In Marklund's classi®ca-
tion, the arc is most likely an evening anticorrelation
polarization arc.
2. FAST gives an estimate of 30 km for the width of
the arc (the half-value width of the precipitating energy
¯ux). This is the same as the upper limit for the width
given by the all-sky camera data.
3. STARE and FAST agree upon the northward
electric ®eld region southward of the arc. This region is a
few hundred km wide and the typical electric ®eld is
50 mV/m in this case. This is a typical feature of an
evening-sector arc (Aikio et al., 1993).
4. The ®eld-aligned current (FAC) determined from
FAST electron data and magnetometer are in rather
good agreement within the arcs. Outside the arcs, the
electron data misses the downward FAC probably
because it is carried by electrons that have smaller
energy than the instrument threshold. This is not strange
since FAST is only at 1250 km altitude for this event.
5. The westward speed of small undulations of the arc
(about 2 km/s) is larger than the background EB
velocity (1 km/s). Assuming that the speed would be
generated within the acceleration region using the
Davis-Haerendel idea would give about 2±4 km/s speed
depending on the exact position, thus this idea seems to
®t in our event, within the observational uncertainties.
6. Solving the ionospheric electrodynamics using
ground-based data with the one-dimensional method of
characteristics gives a FAC pattern which peaks in the
vicinity of arc 2.
7. Ground magnetometer data indicate that the main
eastward electrojet current ¯ows 50±100 km south of arc
2. This requires a higher conductivity than that pro-
duced by electron precipitation. Possibly, the enhanced
conductivity could be due to proton precipitation.
8. By computing the ionospheric conductivities from
FAST electron data and using ground magnetometers to
estimate the current, the resulting ionospheric electric
®eld pattern is in rather good agreement with FAST.
The pattern diers from that inferred from STARE
especially at the arc. This may be due to the fact that the
region where STARE receives enough backscatter in this
event does not exactly coincide with the FAST trajec-
tory or that the arc is too narrow for STARE to resolve.
Acknowledgements. We thank J.P. McFadden and C.W. Carlson
for providing the FAST data and for giving valuable interpretation
hints. We thank all the participating institutes of the IMAGE
magnetometer network. The work of A.O. was partly supported by
the Knut and Alice Wallenberg Foundation and the Swedish
Natural Science Research Council. The work of O.A. was
supported by a grant from the German Science Foundation, and
by a DAAD postdoctoral fellowship HSP III ®nanced by the
German Federal Ministry for Research and Technology.
Topical Editor G. Chanteur thanks G. Marklund and A. D.
Aylward for their help in evaluating this paper.
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