A&A 455, 349-359 (2006)
DOI: 10.1051/0004-6361:20064806
S. Dasso1,2 - C. H. Mandrini1 - P. Démoulin3 - M. L. Luoni1
1 - Instituto de Astronomía y Física del Espacio, IAFE,
CC. 67 Suc. 28, 1428 Buenos Aires, Argentina
2 -
Departamento de Física, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Argentina
3 -
Observatoire de Paris, LESIA, UMR 8109 (CNRS), 92195 Meudon Cedex,
France
Received 3 January 2006 / Accepted 18 April 2006
Abstract
Context. Magnetic clouds are transient magnetic structures expulsed from the Sun that travel toward the external heliosphere carrying a significant amount of magnetic flux and helicity.
Aims. To improve our understanding of magnetic clouds in relation to their solar source regions, we need a reliable method to compute magnetic flux and helicity in both regions. Here we evaluate the sensitivity of the results using different models, methods and magnetic cloud boundaries applied to the same magnetic cloud data.
Methods. The magnetic cloud was observed by the spacecraft Wind on October 18-20, 1995. We analyze this cloud considering four different theoretical configurations (two force free and two non-force free) that have been previously proposed to model cloud fields. These four models are applied using two methods to determine the orientation of the cloud axis: minimum variance and simultaneous fitting. Finally, we present a new method to obtain the axial and azimuthal magnetic fluxes and helicity directly from the observed magnetic field when rotated to the cloud frame.
Results. The results from the fitted models have biases that we analyze. The new method determines the centre and the rear boundary of the flux rope when the front boundary is known. It also gives two independent measurements in the front and back parts for the fluxes and helicity; they are free of model and boundary biases. We deduce that the leading flux of the magnetic cloud had reconnected with the overtaken solar wind magnetic field and estimate the fluxes and helicity present in the full cloud before this reconnection.
Key words: Sun: coronal mass ejections (CMEs) - Sun: magnetic fields - interplanetary medium
Interplanetary Coronal Mass Ejections (ICMEs) are transient solar ejecta that perturb the solar wind. They are expelled from the Sun and travel toward the external heliosphere, reaching the Earth's environment when their trajectories are appropriate. Depending on their magnetic field orientation, they can trigger intense geomagnetic storms.
An important subset of ICMEs is known as interplanetary magnetic clouds (MCs). These huge objects are characterized by a strongly enhanced magnetic field intensity (with respect to ambient values), a smooth and large coherent rotation of the magnetic field vector, and a low proton temperature (e.g., Burlaga et al. 1981; Burlaga 1995; Klein & Burlaga 1982; Burlaga 1991). The global magnetic structure of an MC is a twisted magnetic flux tube; therefore, an MC carries a large amount of magnetic helicity.
In the Sun, magnetic helicity is generated by the dynamo (helical turbulence and differential rotation) and it does not change its sign with the solar cycle. In the northern hemisphere, magnetic features have preferentially negative (left-handed) helicity, while the southern hemispheric features show a preference for the opposite sign (positive, right-handed helicity), for a recent review see Pevtsov & Balasubramaniam (2003). The magnetic helicity injected in each hemisphere cannot be dissipated on short time scales, such as typical flare durations, nor on the time-scale of the solar cycle (Berger 1984). Apart from the cancellation of helicity by reconnection between the two hemispheres and that carried by the solar wind, the coronal helicity only can be released from the Sun by CMEs. Thus, the accumulation of magnetic helicity in the corona is plausibly the origin of the initiation of CMEs (Zhang et al. 2006; Low 1996; Nindos & Andrews 2004).
The conservation of magnetic helicity is also a key property that allows us to link solar phenomena with their interplanetary counterpart (see, e.g., Mandrini et al. 2005; Luoni et al. 2005). Presently, few works have determined the magnetic helicity in interplanetary flux tubes, such as MCs (e.g., Dasso et al. 2005b,a; Gulisano et al. 2005; Nindos et al. 2003; Nakwacki et al. 2005; Mandrini et al. 2005; Dasso et al. 2003; Luoni et al. 2005).
In this paper we analyze the magnetic cloud observed by the spacecraft Wind on October 18-20, 1995. This cloud has been studied by several authors (e.g., Janoo et al. 1998; Lepping et al. 1997; Collier et al. 2001; Hidalgo et al. 2002; Larson et al. 1997). The plasma and magnetic data obtained by the Wind spacecraft indicate that the cloud started to cross the spacecraft at 18:58 UT, on October 18, 1995. While the beginning of the cloud is well defined, there is some ambiguity about the position of the end time, taken by several authors as between October 19, 22:54 UT and October 20, 01:38 UT. Lepping et al. (1997) and Hidalgo et al. (2002) considered an end time at 00:00 UT, on October 20, while Janoo et al. (1998) and Collier et al. (2001) took it as 22:54 UT, on October 19, 1995. However, Larson et al. (1997) chose the end time of this cloud at 01:38 UT on October 20.
The orientation and the size of this cloud have been determined by Lepping et al. (1997) and Hidalgo et al. (2002). The physical parameters (free parameters of a magnetic model) have been fitted by Lepping et al. (1997) under the assumption of a linear force-free field model, and by Hidalgo et al. (2002) using a non-force free model. However, a detailed comparison of the results, in particular the magnetic helicity content, using different models has not yet been done.
Complementary information to understand the topology of this cloud can
be obtained by studying the electron distribution function. Impulsive
electron beams with energies between 0.1-100 keV are good
tracers of the interplanetary magnetic field; they are supposed to
come from the corona, indicating a magnetic connection to the Sun. The
cloud of October 18-20 presents different topologies: regions
magnetically connected to the Sun at both ends, only at one end, and
disconnected (Collier et al. 2001). Larson et al. (1997)
have estimated the semi-length of the magnetic field lines from
in situ observations (at 1 AU) of impulsive electron events
(
1-100 keV) and solar type III radio bursts. From an analysis
of the arrival time and the velocity of these electrons, the
semi-length of the field lines near the centre of the cloud (i.e.,
lines practically parallel to the cloud axis) was estimated as
1.2 AU, a result consistent with a leg still connected to the Sun.
The solar source of this MC was identified as located between two interacting solar active regions (NOAA 7912 and NOAA 7910) by Smith et al. (1997). In a later study, based on the analysis of the coronal soft X-ray evolution, van Driel-Gesztelyi et al. (2000) proposed that the MC was launched from the active region NOAA 7912. This MC was related to a C1.6 long duration event between 5:00 UT and 20:00 UT on October 14, 1995.
The local magnetic configuration of MCs can be modeled using a cylindrical geometry and four different approximations: a linear force-free field (L, Lundquist 1950), a uniformly twisted and non-linear force-free model (G, Gold & Hoyle 1960), a non force-free model with constant current density (H, Hidalgo et al. 2000), and a non force-free model assuming a constant axial component and a linear dependence on the radius for the poloidal component of the current density (C, Cid et al. 2002). These four models provide significantly different helical magnetic configurations with very different distributions of the magnetic twist.
The present work is focused on the determination of the magnetic helicity and flux contained in the MC of October 18-20, 1995, to compare the obtained values when various models and/or methods and/or cloud-sizes are used. We fit the physical parameters for each of the four models (L, G, H, and C). The orientation of the tube is computed with two different methods: a minimum variance (MV) analysis and a simultaneous fitting (SF) of all the parameters. We also present a new method to compute the content of magnetic helicity and flux directly from the magnetic observations, assuming only a cylindrical symmetry for the observed cloud section. We find that the computation of these quantities is very useful to improve the localization of the cloud boundaries. Finally, we explore the sensitivity of the results to variations of the radius of the cloud.
In Sect. 2, we briefly summarize the basic equations. Four models of twisted flux tubes are presented in Sect. 3, where we derive the theoretical expressions for the magnetic helicity and flux. The results of fitting the models to the data are given in Sect. 4. In Sect. 5, we use the new method to directly measure the magnetic helicity and flux of the cloud. We interpret the flux observations in terms of a magnetic shape for the MC, and we analyze the sensitivity of the results to variations of the MC boundaries. In Sect. 6, we give our conclusions.
The relative magnetic helicity within a volume V is obtained by
subtracting the helicity of a reference field
having the same distribution of the normal component Bn on the surface S
surrounding V (Berger & Field 1984):
Several authors have modeled the local magnetic field of an interplanetary flux tube as a straight cylindrical structure having a magnetic field
The component
of the vector potential can be
written as a function of the magnetic flux (Fz) across a surface
perpendicular to the cloud axis as:
In the magnetohydrodynamic (MHD) framework, the magnetic configuration of an MC can be obtained from the balance between the magnetic Lorentz force and the plasma pressure. Several magneto-static models have been used to describe the magnetic configuration of MCs.
When the plasma pressure is negligible compared to the magnetic
pressure, the configuration is called "force-free'' because the
magnetic self-force is null (and so,
is parallel to the
electric current density,
), and the magnetic pressure is
balanced by the tension of the curved magnetic field lines. The
"linear-force-free'' field is a sub-set of solutions from the previous
set, which satisfies
,
with
constant in space. In this section we present four different cylindrical models, two force-free (one linear and one non-linear) and two non force-free.
For the two non force-free models we set the condition
Bz(r=R)=0 in
order to keep the same number of degrees of freedom as with the
force-free models (two degrees of freedom, B0 and ,
for a fixed radius of the tube). For a more general case, including
,
see Dasso et al. (2005b).
The axially symmetric magnetic field corresponding to a linear
force-free configuration was obtained by Lundquist (1950). It has
been shown that this solution is consistent with in situ measurements of interplanetary magnetic flux tubes at 1 AU (Burlaga et al. 1981; Burlaga 1988; Lepping et al. 1990). Thus, the field of MCs is
often relatively well modeled by
The magnetic field line twist per unit length (Eq. (3))
is
.
The constant
determines the twist at the flux tube axis,
.
The axial magnetic flux (Eq. (4)) is given by:
.
It
has been shown that Lundquist's solution extended to the first zero of J0 is enough to describe the magnetic structure of some MCs
(e.g., Lepping et al. 1990). In these cases
.
However, Vandas & Geranios (2001) showed that there are some MCs that
seem to be better described using Lundquist's solution beyond the
first zero of J0. The azimuthal magnetic flux (Eq. (5)) is
given by:
.
We obtain the relative helicity per unit length from
Eq. (6), taking
:
A non-linear force-free field with a uniform twist has been used to
model interplanetary flux tubes (e.g., Farrugia et al. 1999). For this
configuration, is given by Gold & Hoyle (1960):
From Eq. (6), and
A non force-free model has been proposed by Hidalgo et al. (2000,2002)
to describe MCs. This model assumes a constant current density such as
,
where
and jz are constants. The magnetic field is:
The twist distribution is
.
The
axial flux is
,
and the azimuthal flux is
.
The relative helicity per unit length is:
A cylindrical magnetic configuration with a current density such as:
,
with a
and jz constants, has been proposed for magnetic clouds by
Cid et al. (2002). This structure has a magnetic field distribution given by:
The twist distribution is
.
The axial flux is
,
and
the azimuthal flux is
.
The relative helicity per unit length is:
In situ measurements of the magnetic field vector components are
obtained along the trajectory of the spacecraft. Since the speed of
an MC is practically in the Sun-Earth direction and it is much larger
than the spacecraft speed (which can be supposed to be at rest during
the cloud transit time), we assume a rectilinear spacecraft trajectory
in the cloud frame.
We analyze the three components of the magnetic field
measured in GSE (Geocentric Solar Ecliptic) coordinates. In this
system of coordinates,
corresponds to the Earth-Sun
direction,
points to the North (perpendicular to the
ecliptic plane) and
is such that the system is right handed.
It is useful to define a local system of coordinates linked to the
cloud in which
is along the cloud axis (with
at this axis). To associate the cloud axis direction with the GSE system,
we define the latitude angle (
)
between the ecliptic plane and
the cloud axis (oriented as
), as
well as the longitude angle (
)
between the projection of the
axis on the ecliptic plane and the Earth-Sun direction (
)
measured counterclockwise. Accordingly
is
such that when
= 90
(
= -90
)
the cloud
axis is parallel to
and it points to the ecliptic
North (South). When
= 0
the cloud axis is on the
ecliptic plane,
= 0
being the case of the axial field
pointing toward the Sun, and
= 90
(
= 270
)
when it points to the terrestrial dusk (dawn).
The rectilinear trajectory of the spacecraft defines a direction ;
then, we define
in the direction
and
completes the right-handed orthonormal base (
). We also define the
impact parameter, p, as the minimum approach distance from the
spacecraft to the cloud axis. Then, we construct a rotation matrix
from the GSE system to the cloud system, and obtain the components
of the observed magnetic field in the cloud
coordinates:
,
,
.
The local system of coordinates is especially useful when p is small
compared to the MC radius. In particular, for p=0 and an MC
described by a cylindrical magnetic configuration
,
we have
and
when the spacecraft
leaves the cloud. In this case, the magnetic field data obtained by
the spacecraft will show:
,
a large and coherent
variation of
(with a change of sign), and an intermediate and coherent variation of
,
from low values at one cloud edge, taking the largest value at its axis and
returning to low values at the other edge.
It is possible to estimate the orientation of an MC
applying the minimum variance (MV) method to the magnetic observations
when p is small compared to the cloud radius. This method finds the
directions (
)
in which the mean quadratic deviation of the
field,
,
is minimum and maximum, as well as the orthogonal direction to both. It is
possible to show that this is equivalent to finding the eigen-vectors of
the covariance matrix Mi,j=
(Sonnerup & Cahill 1967). This symmetric and real matrix has three real
eigen-values with orthogonal eigen-vectors.
The MV method determines the MC axis direction,
,
as the eigen-vector associated with the intermediate
eigen-value. The eigen-vector associated with the lowest eigen-value is
expected to be close to the direction that results from the projection
of the spacecraft trajectory on the plane perpendicular to
.
In particular, in the cloud analyzed in this
paper (see Sect. 4.4) this spacecraft trajectory
projection and this eigen-vector differ by less than
,
compared to the mean value
when different boundaries are chosen for the cloud
(Sect. 4.6). The eigen-vector associated with the highest eigen-value closes the system such that it is right handed. A more complete discussion of
the MV method applied to interplanetary flux tubes can be found in the
appendix of Bothmer & Schwenn (1998).
From the observed bulk velocity of the interplanetary plasma,
we transform the observed time series,
,
to a spatial
series,
,
along
.
Assuming a local invariance
by translation along the MC axis and for a given orientation of this axis,
the data are transformed to a series along
.
Further, assuming a cylindrical symmetry, and for a given impact parameter p,
we can write the field as a function of the spacecraft distance to the MC axis,
r, then we obtain the series
.
We define a residual function (
), comparing
with
,
for each of the models described in Sect. 3, such that:
We use two methods which differ in the number of free parameters in
the minimization. The first method takes the orientation
angles,
and
,
from the MV method. The impact
parameter p is assumed to be zero and we fit the two free parameters
of the models (B0 and
,
Sect. 3).
The results of this method are labeled simply with MV.
In order to test the validity of the MV method, and to determine p,
the second method simultaneously fits (SF) the orientation angles (,
), the impact parameter (p), and the two physical
parameters (B0 and
). The equations for the SF method are given in Hidalgo et al. (2002).
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Figure 1:
Wind observations for the magnetic cloud observed on 18-20 October,
1995 (one minute cadence). From the upper to the lower panel: radial (
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Figure 1 shows the magnetic field components, the bulk velocity and the proton density for an extended range of time around the magnetic cloud. In each panel, the dashed line shows the beginning of the magnetic cloud (18:58 UT, October 18, 1995), while the two dash-dotted lines mark the range of possible final boundaries (from 22:54 UT, October 19 to 01:38 UT, October 20).
We select an end boundary at 00:00 UT on October 20 to apply the MV method; with this boundary we obtain
and
(with a radius R=0.13 AU). The flux tube axis is
almost on the ecliptic plane and perpendicular to the Sun-Earth
direction. Choosing different final boundaries inside the range
marked by the two dash-dotted lines in Fig. 1,
we obtain very similar orientations for the cloud axis (the angles
are modified by less than
).
The MV analysis gives a ratio between the intermediate and the
minimum eigen-values 7, so that the MC axis is well defined. This
is confirmed by the SF method which gives a similar orientation (with,
at most,
difference to the MV result).
The fact that
in Fig. 1
fluctuates around zero is consistent with a trajectory of Wind almost
intersecting the cloud axis (
). In the second and third panels
we have the typical behavior of the azimuthal (
,
global largest variance) and axial (
,
global
intermediate variance) magnetic field components of a cylindrical flux
rope, as described in Sect. 4.1.
The bulk velocity profile (fourth panel) shows that this cloud
is not expanding, it travels with an almost constant velocity of
400 km s-1. A shock front is present
8 h ahead of the
cloud (10:42 UT, October 18). This shock was produced because the
speed of the leading boundary of the cloud relative to the solar wind
is greater than the magneto-acoustic speed (Lepping et al. 1997).
Accordingly, the proton density (bottom panel) is high between the
shock and the outer front of the cloud implying the consequent
accumulation of material at its front. From a 4 minute time cadence
analysis, Lepping et al. (1997) showed that there is a tangential
discontinuity with a magnetic hole at the leading boundary of the
magnetic cloud.
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Figure 2:
Axial component,
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Figure 3:
Field component in the direction orthogonal to both the
spacecraft trajectory and the MC axis.
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Table 1:
Results from fitting of the geometrical and physical
parameters for the magnetic cloud of October 18-20, 1995, choosing the
end time as 20-October-1995 00:00 UT (i.e., at t=29 in
Figs. 2, 3, 5).
The first four rows correspond to the orientation given by the
minimum variance method (MV, Sect. 4.2), while the next
four to a simultaneous fit (SF, Sect. 4.3). The next two rows are the mean and
the maximum deviation from the mean of the parameters
found with the eight above models and methods. The bottom three rows
give the results of the direct method for the in-bound and out-bound parts of
the flux rope, and the full back part of the MC (see
Sect. 5.4). L, G, H, C refer to
the Lundquist, Gold-Hoyle, Hidalgo et al. and Cid et al. models,
respectively, and D to the direct method. The geometrical parameters
are: the angle ()
between the axis of the tube and the
ecliptic plane (i.e., its latitude), the angle (
)
between the
projection of the cloud axis on the ecliptic plane and
the x-axis of GSE (i.e., its longitude), the flux tube radius (R), and the impact
parameter (p) measured in units of R. The two physical parameters
of the models are: the twist angle per unit length (
)
and the
intensity of the field (B0), both computed at the cloud axis. The
quality of the fitting is given by the square root of
(Eq. (16)). Finally, we give the axial magnetic flux (Fz) and quantities defined per unit length along the cloud axis:
the azimuthal magnetic flux and magnetic helicity (
and H/L).
Figures 2 and 3 show the magnetic
observations together with the fitted curves corresponding to the four models and the MV method. The curves from the SF (not shown) are similar. Close to the centre of the cloud models L, G, and H overestimate
,
while model C rather overestimates
near the cloud boundaries.
In the case of model H, a triangular profile is present for the
magnetic field computed along any rectilinear trajectory that crosses
the flux tube axis (p=0, see dotted curve
in Fig. 2). In Fig. 3, the two non-force free models (H and C) are represented by the same straight
line, which gives only the global behavior of
.
The
two force-free models (L and G) underestimate
near the cloud boundaries.
According to the
values (Table 1), the quality
of the fit, in decreasing order, is: C, H, L and G models for both
MV and SF methods. This confirms the visual inspection of
Figs. 2 and 3. However, we have
to be cautious when making this comparison because in models L and G,
Bz is not forced to be zero at the cloud boundary, while this is
the case in models H and C to keep the same degrees of freedom between
models. Finally, as expected, a better quality fit is obtained with a SF in all four cases.
The SF method gives a way to estimate the impact parameter p, which is
of the order of
of the MC radius, while the orientation remains
almost the same. This result, as well as the large angle of rotation
of the magnetic vector seen on the variance plane (larger than
180
,
not shown), justifies the use of p=0 in the MV method.
The orientation of the magnetic cloud is well determined,
being the maximum deviation in
of
and in
of
.
This, together with an always low impact parameter,
implies that the geometry of the crossing is similar in all cases.
Then, for fixed MC boundaries, the determined radius varies very
little (Table 1).
We compare our results with those of previous works that used the same
start and end times. Lepping et al. (1997) found
,
,
p/R=0.08,
AU-1, B0=25.6 nT, using the linear force-free field model (L) but fitting a magnetic field normalized to unity at each position.
These authors fitted hourly averaged magnetic
observations and forced Bz to be null at the cloud boundaries.
Hidalgo et al. (2002), using model H, found
,
,
p/R=0.15,
AU-1, B0=24.3 nT, fitting the toroidal and poloidal components of the current (equivalent to our fit of B0 and
). These authors
analyzed hourly averaged GSE components of the magnetic field. These small differences between our results and those of previous works are due to the different time cadence and minimization
methods used.
From Table 1, we see that the largest
difference between models is found for the parameter ,
as its relative maximum deviation is
(compared to
for B0). However, the magnetic field line twist distribution is strongly model dependent. In the Gold-Hoyle model the twist is constant, in the Lundquist model it is mostly distributed in the
periphery of the tube, in the Hidalgo model it depends inversely on the
distance to the boundary (
(1-r/R)-1), while in the Cid model it
depends on
(1-r2/R2)-1. This implies that the obtained values
for
,
a local quantity, are not directly comparable between
different models. A pertinent comparison can only be done using a global quantity, such as the magnetic helicity.
For the estimated fluxes and helicities we have a bias: the largest
axial fluxes, Fz, are obtained with the two force-free models, while
the largest azimuthal fluxes and helicities are given by the non-force
free models. So the estimated global
quantities,
,
have a maximum
deviation from the mean of 18, 25 and 28%, respectively.
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Figure 4:
Axial magnetic field (B0), axial twist per unit length (![]() ![]() |
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Taking the mean value as reference (for the 4 models),
to change the back boundary to the range used in
Fig. 4 introduces a maximum relative change of 35% in
and of
20% in B0. The above changes of the fitted parameters have implications for the
estimation of the fluxes and helicity, but, since they are
extensive quantities, they are more affected by the flux tube radius (which depends
directly on the selection of the end time). Both fluxes and helicity
show a nearly linear dependence on R (Fig. 4).
We find a maximum relative change (relative to the mean value)
of
50% in Fz,
40% in
,
and
75% in H/L. In particular, H depends strongly on the radius (see Eqs. (9), (11), (13),
and (15)) and, as found, we expect the largest
variations for its value.
The time interval explored for the back boundary is about 3 times larger than the time interval given by the fixed values chosen
as the back boundary by the different previous authors
(Sect. 1.3).
Taking the previous authors' interval, the changes in the
magnitudes are about a factor of 2 lower than using our extended
time interval (the changes can be seen in Fig. 4 between the
dashed line and the right end of the plot).
More precisely, for the previous authors' interval
we find a relative change of 25% in Fz,
20% in
,
and
35% in H/L.
These variations are comparable to the relative changes
introduced when changing fitting methods and models
(relative change of
18% in Fz,
25% in
,
and
28% in H/L from Table 1).
Below we use
and the local invariance of
along the MC axis to define the centre and rear boundary of the twisted flux tube present in the MC.
Previous investigations of MCs (see Sect. 1),
as well as the present one, concluded that they are formed by a twisted
magnetic flux tube (or flux rope). Its magnetic field has two components: a longitudinal one (parallel to the MC axis) and an azimuthal one. Let us cut the flux rope once by a generic plane
,
and let us compute the magnetic flux crossing
.
The longitudinal component cuts
in only one sense
and, thus, its flux contributes a given sign, but the azimuthal component has both positive and negative flux contributions of equal absolute magnitudes across this plane.
In order to have only the
azimuthal flux and to include the data, below we define
as that plane formed by the spacecraft trajectory and
.
Then, we have:
The magnetic field is expected to change significantly along the MC axis on a scale of a few AUs, which is much larger than its radius (0.1 AU). The global radius of curvature of the MC
is also expected to be a few AUs. Then, the hypothesis of a local symmetry by translation along the MC axis is well justified (in particular when data are fitted with models, see
Sect. 3). With this hypothesis, Eq. (17)
reduces to:
If one MC boundary is known, the above flux balance property can be used
to find the MC centre and the other boundary as follows. We define
the accumulative flux per unit length
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Figure 5:
Accumulative flux of
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The accumulative flux Fy/Lis shown in Fig. 5
(thick solid line), together with the observed
component (thin solid line).
Vertical dashed lines mark the time at which Fy/L reaches the
minimum (October 19, 07:26 UT) and the time when the flux cancels
(October 19, 17:37 UT). They correspond to the centre and the rear
boundary of the MC flux rope, respectively.
We note that around the time of the defined centre (2 h) a well defined
sub-structure is present in
.
This sub-structure is
globally anti-symmetric, which is a confirmation of the above
determination of the flux rope centre from global flux balance. It
also indicates the presence of a specific twist profile in the centre
of the main flux rope. Even more striking, the flux balance determines
the rear boundary of the flux rope where there is a discontinuity of
.
This is a confirmation of the rear boundary location since a current sheet, so a discontinuity of
,
is expected at the boundary of two different magnetic structures
(e.g. Aly 1990).
An alternative interpretation of this discontinuity has been proposed by Collier et al. (2001). These authors suggested that it is due to an internal shock traveling approximately along the cloud axis. The origin of the shock is supposed to be magnetic reconnection occurring close to the solar surface. With this interpretation, the flux balance determined above is only a coincidence. Our results rather support that this discontinuity is consistent with the presence of a time-dependent current sheet which separates the flux rope from the open field structure at the back part of the cloud. We further justify this below.
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Figure 6: Schematic 2D view of the magnetic cloud embedded in the solar wind near the Sun ( top) and during the spacecraft passage at 1 AU ( bottom). Because the MC is faster than the surrounding solar wind and their magnetic fields form a large angle (anti-parallel in the scheme), reconnection is forced in front of the MC (see Sect. 5.2). The horizontal dashed line in the bottom figure indicates the spacecraft trajectory. |
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The previous analysis of Fy/L shows that this MC is not simply formed by a flux rope. We summarize below the most plausible physical scenario to create such magnetic structure: reconnection between the original flux rope and the solar wind magnetic field. This scenario is coherent with all our results (Sect. 5.4), but we cannot rule out other possibilities such as reconnection closer to the solar surface (coronal observations are not stringent enough, Luoni et al. 2005).
Some of the MC characteristics, such as the low magnetic variance and the low proton temperature, continue well after the rear boundary of the flux rope (Lepping et al. 1997). Indeed, the accumulative Fy/L shows a strong change in the slope at 01:36 UT on October 20 (36 min after the tick at 30 h on the abscissa of Fig. 5), which was almost the back boundary chosen by some authors (Sect. 1.3). The fact that a portion of the back of the MC maintains the characteristics of a classical cloud, as well as a larger unbalanced flux, is unusual and was not noticed in previous works.
We interpret the lack of flux cancellation in the rear portion of the MC as evidence of a magnetic structure connected with solar wind field lines, which in the past formed the periphery of a larger flux rope. The top panel of Fig. 6 shows a 2D scheme of the initial magnetic configuration. It is formed by a flux rope embedded in a solar wind environment having the conditions that favor reconnection at the front of the MC as the flux rope overtakes the wind field. The bottom panel shows a possible result of magnetic reconnection between the MC and the solar wind field lines. Vandas et al. (1995) and Schmidt & Cargill (2003) have done MHD simulations of this process in comparable but different configurations: in a Parker spiral field with an MC axis orthogonal to the ecliptic, and in a radial unipolar solar wind or in a radial current-sheet with an MC, respectively.
The magnetic field in the MC front is mostly directed north-south
(Fig. 1 of Janoo et al. 1998). If this field is pushed against a field
with a significantly different direction, like the solar wind magnetic
field dominant in the ecliptic plane, it will reconnect with it.
Indeed, our Fig. 1 shows a sudden change of the
bulk velocity at the upstream shock occurring at 10:42 UT on October 18, from 320 km s-1 before the shock to
400 km s-1 after,
indicating a significant relative velocity between the cloud (plus its
sheath) and its environment during the early stage of its travel from
the lower corona to 1 AU. This fact favors the reconnection process
because its efficiency increases with the relative velocity between
the cloud and the solar wind (Schmidt & Cargill 2003).
In low density plasmas, such as those found in the interplanetary medium, numerical simulations show that the Hall effect can significantly increase the reconnection rate
over the classical rate (e.g., Morales et al. 2005). Furthermore,
Farrugia et al. (2001) found that magnetic holes, such as that preceding the
cloud analyzed here, can be a signature of magnetic reconnection.
Thus, assuming a reconnection process in the cloud front, part of the
flux in the front will be removed and will not be observed by a spacecraft crossing the cloud centre. However, the available data do not allow us to localize
the reconnection site during the cloud travel from the Sun to the spacecraft.
What are the major implications for the reconnected field lines and
the remnant flux rope? First, the magnetic stress is able to decrease
because of the propagation of Alfvén waves from the MC into the solar
wind along the new reconnected field lines. This decreases
at the back of the MC as the reconnected field lines tend to
straighten. However, the flux of
cannot decrease
since it would need a further reconnection with almost parallel
magnetic fields (located further inside the MC). In order to compensate
the decrease of
in the magnetic pressure,
has to increase. Furthermore, the new reconnected field
lines are linked to the solar wind at the back of the MC where a faster stream is present (Lepping et al. 1997). All these effects imply a compression of the back of the MC, with an expected effect also on the back of the remnant flux rope. In conclusion, the front reconnection
makes the MC asymmetric, not only in flux but also in field strength.
In this section we show how the direct method can be used
to compute the magnetic fluxes (axial and azimuthal)
and the helicity per unit length directly from the observed magnetic time
series. These computations are done
as a function of the x coordinate along the projection of the
spacecraft trajectory orthogonal to the MC axis. This requires us to
transform the magnetic data in the cloud frame
(Sect. 4.1). Three hypothesis are needed: the
local invariance along the cloud axis, the cylindrical symmetry and a moderately low impact parameter (
,
see last paragraph of this section).
The centre of the flux rope is defined at the time when Fy/L is
minimum (Sect. 5.1,
Fig. 5), and we set the coordinate origin
there (x=0). Then, we split the time series of
in two subseries for
and
.
The first subseries corresponds to the in-bound path (the path when the
spacecraft is going toward the centre of the cloud, x<0) and the
second to the out-bound path (when the spacecraft has reached the
minimum distance to the cloud axis and is going out the MC, x>0).
Differences in the results obtained with these two branches are due
to the non-cylindrical symmetry of the flux rope, and also to the
systematic bias due to reconnection of part of the MC flux
with the solar wind magnetic field (Sect. 5.2).
The spacecraft trajectory approaches the MC axis up to the minimum
distance p, so we are only able to evaluate the following quantities on the out-bound path:
If the impact parameter is finite, the core of the flux rope is not
present in the data, so the fluxes and helicity will be underestimated.
Since
and
in the
core (
), the relative underestimation of both fluxes given by
Eqs. (20) and (21) is of the order of (p/R)2
compared to the fluxes given by Eqs. (4) and (5).
If Fz is known, the underestimation of the helicity in the core
is of the order of (p/R)4; however, an underestimation of Fz (a cumulative magnitude from r=0) by using Fz,e is present in the integral, and it increases the underestimation
to the order (p/R)2. Then, the underestimation of both global
fluxes and helicity is of the order of
for
the October 18 MC. This is much smaller than the uncertainties present
on these quantities (e.g. the difference between the in and out-bound
values), and we neglect them below.
We note that, for MCs observed with a higher p/R, a correction can be added
by using a fitted model (Sect. 3) as follows.
The simultaneous fit of the model parameters to the data
lets us estimate the impact parameter p, as well as an extrapolation
of the magnetic field in the unobserved core (Sect. 4.3).
This extrapolated field can be used
to estimate Fz(p),
and H(p) in Eqs. (20)-(22), and so to add a correction for the unobserved core.
![]() |
Figure 7:
Accumulative azimuthal flux per unit length (
![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
In this section, we present the results obtained with the direct method applied to the MC of October 18-20, 1995. Each result is first described, and then interpreted within the physical scenario of Sect. 5.2.
The accumulative flux
Fy,e(x)/L, shown in
Fig. 7, is asymmetric in x between the
in-bound (solid line) and the out-bound (dotted line) paths because at the
back of the cloud there is a magnetic compression due to the excess of
flux (as consequence of the flux reconnected in its front, see
Sect. 5.2 and Fig. 6). The vertical
solid line marks the well-determined initial boundary of the cloud at
AU (note the strong change of the slope of
Fy,e /L). An azimuthal flux
1021 Mx/AU is accumulated in the front part of the flux
rope. The vertical dotted line indicates the position
(=0.081 AU) where the azimuthal flux cancels the in-bound flux (Fig. 5).
Thus, the rear boundary of the flux rope is closer to
the centre than its front with an asymmetry factor
of
.
Furthermore, in accordance with Fig. 5,
there is a strong change of the slope at an out-bound distance
AU. Integrating from x=0 up to this extended boundary
we obtain
1021 Mx/AU (a factor 2.3 larger than the closed azimuthal flux). Following Sect. 5.2, this flux is interpreted
as the total azimuthal flux of the flux rope before its reconnection
with the solar-wind magnetic field.
The in- and out-bound
Fz,e(x) fluxes are very close, from the
centre to about the middle of the central part of the flux rope
(Fig. 7). In the outer part the difference has
the opposite sign than for Fy,e/L. This is a consequence of
the compression of the rear of the flux rope: it enhances
,
but it also flattens the flux rope, extending it in the
directions; then, the Bz flux spreads over a larger area, giving a lower measured Bz for the out-bound (than for the in-bound). The computed fluxes, assuming cylindrical symmetry, are:
1021 Mx and
1021 Mx, thus a relative balance of 51% in the flux rope. As with Fy,e/L, a clear change of the
Fz,e(x) slope is present at the back of the MC (
). The extra Bz
flux at the back of the MC (between
and
)
is small:
1021 Mx. This was expected
in the interpretation of Sect. 5.2 since Alfvén waves transport the Bz component away from the MC on field lines reconnected with the solar wind.
The accumulated magnetic helicity per unit length (He/L) is the
most symmetric magnitude (between in- and out-bound paths)
of the three quantities shown
in Fig. 7. However, there is a strong
asymmetry when we consider the amount accumulated at the flux rope
boundaries:
1042 and
1042 Mx2/AU. There is less
helicity at the rear of the flux rope because it is compressed,
so Bz is lower.
When considering the full out-bound part
of the MC, the asymmetry is reversed, and we get
1042 Mx2/AU so about a factor 2 larger than
.
This shows that taking only the flux rope part
of the MC, the magnetic helicity can be significantly underestimated.
In fact, even with the full out bound part, the magnetic helicity
content before reconnection is
still underestimated since part of the helicity is transported away in
the reconnected field lines. Thus, part of the mutual helicity associated
with
,
which encircles the internal Bz flux before
reconnection, is lost at the back part due to the propagation of
torsional Alfvén waves.
The magnetic helicity found with the full out-bound data,
1042 Mx2/AU, is within the range
found for the models [3.7,6.1]
1042 Mx2/AU in
Sect. 4.5. These results were obtained
fitting the data from
to
(Table 1). This approach also assumes a cylindrical
symmetry, but sets the centre in the middle between
and
,
which is not the centre of the flux rope
(Sect. 5.1). Moreover, the fits introduce
model-dependent deviations from the observed data.
Then, the estimations from
Eqs. (20)-(22) should be more reliable.
Finally, we compare the MC helicity to the helicity variation
before and after the solar ejection. The relevant quantity is the
total helicity present in the CME/MC before it significantly reconnects
with the solar wind magnetic field. Our best estimate is
.
The estimated length of the cloud is
2 AU (Larson et al. 1997); then, the magnetic helicity content in the MC
is, at least, 10
1042 Mx2. This amount is twice
the value found by Luoni et al. (2005) (
5
1042 Mx2) for the difference between the pre- and post-eruption
helicity in active region NOAA 7912. This was estimated from
a linear force-free field model fitted to best represent the observed coronal loops before and after the solar ejection. This is an indication that the coronal model used underestimates
the amount of magnetic helicity present in the corona.
We have computed the large-scale magnetic fluxes and helicity for the magnetic cloud (MC) observed by Wind on October 18-20, 1995, following two different approaches. The first one fits an analytical model to the observed in situ magnetic field vector. We have tested four models, already proposed in the literature. The second method, introduced in this paper, computes the magnetic fluxes and helicity directly from the observed data, rotated in the MC system of coordinates.
We have determined the MC orientation both from the standard minimum variance analysis (MV) and a simultaneous fitting (SF) of the geometrical and physical parameters. The four different models included in our study are a linear force-free field (L), a uniformly twisted field (G), a field with constant current (H), and a field with a linearly distributed current (C).
The front boundary of the MC is well defined at 18:58 UT on October 18, so we always keep this value. Fixing the back boundary at 00:00 UT on October 20 and
considering the different models using MV and SF, we have found a maximum
deviation from the mean value of 18% in the axial flux (Fz),
of 25% in the azimuthal flux ()
and of 28% in the magnetic
helicity (H). However, because of the ambiguity present in defining the
back boundary, we explore the effects of changing its position.
If we restrict the boundary location to the range previously used by
several authors (Sect. 1.3),
the change of position of the back boundary has an effect comparable
to the change when using a different model and method.
The direct integration method takes as input the observed magnetic data transformed to the MC coordinate system (Sect. 4.1). The centre of the flux rope is determined by the extremum of the flux of the azimuthal field (more precisely the flux of the field component both orthogonal to the MC axis and the spacecraft trajectory). This direct method relaxes the need for a model to compute the fluxes and helicity; then, it avoids the biases introduced by a specific model. It also permits two independent measurements on both sides of the flux rope centre: one in the in-bound and another in the out-bound part.
The direct method allows us to determine the centre and the rear boundary of the flux rope present at the time of the observations. The rear boundary is confirmed by the presence of a strong discontinuity of the magnetic field, which implies a current sheet as expected between two magnetic regions of different connectivities. The fluxes and helicity associated with the flux rope are smaller than those estimated from the fitted models, which include a larger part of the MC.
As an outcome of the application of the direct method we can better understand the structure of the MC. We conclude that this MC was most plausibly formed by a large flux rope after its ejection from the Sun, but its front partially reconnects with the solar wind magnetic field. This reconnection removes magnetic flux from the front part of the MC and creates the corresponding amount of open field lines at the back of the MC. At the time of the observations, only the core of the original flux rope remains. The back part no longer belongs to the flux rope but it is still formed by the field and plasma that have some typical characteristics of MCs (low field variance, low proton temperature). We conclude that the direct method provides a better estimate of the fluxes and magnetic helicity in the original flux rope, before reconnection, by using the extended out-bound results (D-MV back in Table 1).
The extended magnetic configuration (flux rope and open field at the
back) is more representative of the flux tube expelled from the Sun.
Still, magnetic reconnection in the front allows the axial flux and the
helicity to go away from the front of the MC, but also partly from
its back. So our estimation using the extended out-bound measured
values is expected to be only a lower limit. Moreover, we assume a cylindrical symmetry, and with an MC more extended in the transverse direction to the spacecraft trajectory, the axial flux and helicity will be even larger (by a factor roughly proportional to the aspect
ratio). Thus, in the October 18 MC we determine a minimum helicity
content 1043 Mx2. This minimum is still a factor
of two larger than the coronal helicity variation estimated by
Luoni et al. (2005). This indicates that the coronal value is even
more underestimated.
Acknowledgements
This research has made use of NASA's Space Physics Data Facility (SPDF). This work was partially supported by the Argentinean grants: UBACyT X329, PIP 6220 (CONICET), and PICTs 03-12187 and 03-14163 (ANPCyT). S.D. and C.H.M. are members of the Carrera del Investigador Científico, CONICET. C.H.M. and P.D. thank CNRS (France) and CONICET (Argentina) for their cooperative science program N0 18302.