A&A 451, 319-330 (2006)
DOI: 10.1051/0004-6361:20054171
S. Régnier1,
-
R. C. Canfield
2
1 - ESA Research and Scientific Support Department, SCI-SH, Keplerlaan 1,
2201 AZ Noordwijk, The Netherlands
2 -
Montana State University, Physics Dept., 264 EPS Building, Bozeman, MT
59717, USA
Received 7 September 2005 / Accepted 30 January 2006
Abstract
To better understand eruptive events in the solar corona, we combine
sequences of multi-wavelength observations and modelling of the
coronal magnetic field of NOAA AR 8210, a highly flare-productive
active region. From the photosphere to the corona, the observations
give us information about the motion of magnetic elements
(photospheric magnetograms), the location of flares (e.g.,
H,
EUV or soft X-ray brightenings), and the type of events
(H
blueshift events). Assuming that the evolution of the
coronal magnetic field above an active region can be described by
successive equilibria, we follow in time the magnetic changes of the
3D nonlinear force-free (nlff) fields reconstructed from a
time series of photospheric vector magnetograms. We apply this
method to AR 8210 observed on May 1, 1998 between 17:00 UT and
21:40 UT. We identify two types of horizontal photospheric motions
that can drive an eruption: a clockwise rotation of the sunspot, and
a fast motion of an emerging polarity. The reconstructed nlff
coronal fields give us a scenario of the confined flares observed in
AR 8210: the slow sunspot rotation enables the occurence of flare by
a reconnection process close to a separatrix surface whereas the
fast motion is associated with small-scale reconnections but no
detectable flaring activity. We also study the injection rates of
magnetic energy, Poynting flux and relative magnetic helicity
through the photosphere and into the corona. The injection of
magnetic energy by transverse photospheric motions is found to be
correlated with the storage of energy in the corona and then the
release by flaring activity. The magnetic helicity derived from
the magnetic field and the vector potential of the nlff
configuration is computed in the coronal volume. The magnetic helicity
evolution shows that AR 8210 is dominated by the mutual helicity
between the closed and potential fields and not by the self helicity
of the closed field which characterizes the twist of confined flux
bundles. We conclude that for AR 8210 the complex topology is a more
important factor than the twist in the eruption process.
Key words: Sun: magnetic fields - Sun: flares - Sun: corona - Sun: evolution
The structure of the Sun's corona is dominated by its magnetic field. To understand eruptive events (flares, coronal mass ejections (CMEs) or filament eruptions), we need to know the evolution of the 3D magnetic configuration (geometry and topology) of the corona. In this study, we combine observations of the solar atmosphere at various heights with models of the coronal magnetic field to determine the sources of flaring activity and the time changes of an active region before and after a flare. We focus our study on a five-hour period which is particularly interesting because (i) it precedes a major flare/CME event and (ii) it is well observed with vector magnetograms.
Most flare models (see review by Priest & Forbes 2002;
Lin et al. 2003) involve magnetic reconnection processes to
explain the rapid conversion of magnetic energy into kinetic energy and thermal
energy (hard X-ray sources, soft X-ray flux, brightening in hot EUV lines or in
H). In the classical CSHKP model (Hirayama 1974; Kopp & Pneuman 1976; Carmichael 1964; Sturrock 1968),
the reconnection process occurs at the location of an X point in 2D, or at the
location of a null point or a separator field line in 3D. In 3D topology (see
Priest & Forbes 2000), the reconnection processes involved in
flares do not occur only in the vicinity of a null point but can also be
associated with other topological elements (e.g., fan surfaces, spine field
lines). The study of the topology of coronal magnetic fields should help us to
answer important questions for the energetics of flares, including (i) how is
magnetic energy stored before the eruption? (ii) Is the stored magnetic energy
enough to power a flare or a CME? Question (i) can be tackled by following the
time evolution of the magnetic energy injected through the photosphere, and the
free magnetic energy available in the corona. To answer question (ii), we need
to understand the temporal and spatial relationship between observed
brightenings and magnetic field changes in the corona.
Table 1:
Photospheric, chromospheric and coronal observations of AR 8210 on May 1, 1998.
is the pixel size,
is the time between
two consecutive observations.
The injection of magnetic energy into the corona through the photosphere is considered to be associated with the horizontal displacement of magnetic features on the photosphere: emergence (Nindos & Zhang 2002; Schmieder et al. 1997; Kusano et al. 2002; Ishii et al. 1998) and cancellation (Fletcher et al. 2001; Livi et al. 1989) of magnetic flux, rotation of sunspots (Lin & Chen 1989; Nightingale et al. 2002; Kucera 1982), and moving magnetic features (Moon et al. 2002; Zhang & Wang 2001). The velocity fields can be detrmined using the white light images (Nightingale et al. 2002) or by estimating the small displacements from a Local Correlation Tracking (LCT) technique (November & Simon 1988). Recently several more powerful techniques have been developed to retrieve the full photospheric velocity field from vector magnetograms (Longcope 2004; Welsch et al. 2004; Georgoulis & LaBonte 2006).
In our study, the coronal magnetic field is assumed to be in a force-free equilibrium state at the time of observation. Therefore if the photospheric distribution of vertical electric current density is known in addition to the vertical magnetic field (Sakurai 1982), the nonlinear force-free field (nlff) can be extrapolated in the corona (e.g., Mikic & McClymont 1994; Amari et al. 1997; Wheatland et al. 2000; Yan & Sakurai 2000; Wiegelmann 2004). Inside nlff magnetic configurations, a more realistic distribution of twist and shear can be considered in comparison to other assumptions commonly used to extrapolate the coronal magnetic field (potential, linear force-free fields). These nlff extrapolation methods were applied to solar active regions using one snapshot of the magnetic field (e.g., Yan & Wang 1995; Régnier et al. 2002; Bleybel et al. 2002; Régnier & Amari 2004). Here we propose to study the time evolution of an active region considering that it can be described by succesive nonlinear force-free equilibria. This assumption is justified by considering that the evolution of the active region is sufficiently slow which means that the photospheric velocities of the footpoints are small compared to characteristic speeds in the corona, such as the Alfvén velocity (Antiochos 1987).
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Figure 1: X-ray flux measured by GOES-8 in the wavelength range 0.05-0.4 nm. Gray areas are the flaring periods. The rise (resp. decay) phase of flares are the dark (resp. light) gray areas as defined in Sect. 3.1. |
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Using the above methods to estimate the velocity fields on the photosphere and
the 3D coronal magnetic field, we can estimate the rate of magnetic energy
injected into the corona by photospheric motions and where this energy is
deposited or released in the corona. We can also derive the magnetic helicity
and its evolution to understand the effects of reconnection on the connectivity
of field lines. In this work, we have selected the active region 8210 (AR 8210) observed on May 1, 1998 between 17:00 UT and 21:40 UT for which we have a
good set of data covering the photosphere, the chromosphere and the corona as
well as a high-cadence vector magnetic field observations of good quality. AR
8210 is a well studied active region for its flaring activity on May 1st and
May 2nd (Sterling & Moore 2001b; Wang et al. 2002; Pohjolainen et al. 2001; Sterling & Moore 2001a; Xia et al. 2001; Sterling et al. 2001; Warmuth et al. 2000; Thompson et al. 2000).
We focus our attention on the time period shown in
Fig. 1 by the evolution of the X-ray flux. We
first give an overview of the AR 8210 data (see Sect. 2) we use
to analyse the precursors or signatures of flaring activity
(Sect. 3): X-ray flux, H blueshift events (BSEs),
photospheric velocity fields. In Sect. 4, we describe how to
determine and analyse the 3D magnetic field of AR 8210. We then give a scenario
of the magnetic field evolution during the flaring period
(Sect. 5). The magnetic energy and helicity budgets are derived
in Sects. 6 and 7. In Sect. 8, we
discuss the implications of those processes for flaring activity and solar
eruptions.
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Figure 2:
Images of AR8210: magnetic field ( top left), H![]() |
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In Table 1, we summarize the observations on May 1, 1998 we are using in this study. Between 17:00 UT and 23:00 UT, we have photospheric line-of-sight and vector magnetograms, chromospheric images and spectra, and coronal images. Those data guide the analysis presented in later sections.
SOHO/MDI (Michelson Doppler Imager, Scherrer et al. 1995)
measures the line-of-sight magnetic field strength deduced from the Zeeman
splitting of the Ni I 676.8 nm line. During the period of observation, we have 1 min cadence full-disc magnetograms which allow us to study the dynamics of
photospheric magnetic features. The measurement uncertainty is 20 G. In
Fig. 2 top left, we have the distribution of the longitudinal
magnetic field at 20:10 UT in a field-of-view of
.
Basically AR 8210 is a sunspot complex of negative polarity (polarity N1) surrounded by
positive polarities (P1-4). AR 8210 also includes parasitic polarities such as N2, which is a new emerged and moving negative polarity.
IVM at MSO (Imaging Vector Magnetograph/Mees Solar Observatory,
Mickey et al. 1996) is a vector magnetograph measuring the
full Stokes profiles of the Fe I 630.25 nm line. The four Stokes parameters,
I = (I, Q, U, V), are measured inside a field-of-view of
pixels with
a pixel size of 1.1
square. The vector magnetograms are built with a
series of 30 polarisation images obtained over 3 min (Mickey et al. 1996). To increase
the signal-to-noise ratio and to suppress the effects of photospheric
oscillations, we average the Stokes profiles over 15 min. In the reduction
process, we take into account the cross-talk between the I and V profiles as
well as scattered light using daily off-limb measurements. A detailed
reduction scheme is given by LaBonte et al. (1999). To infer the magnetic field, the
inversion code follows the radiative transfer of line profiles as in
Landolfi & Degl'innocenti (1982) based on Unno (1956) equations and including magneto-optical
effects. We then obtain the magnetic field:
along the line-of-sight,
and
the strength and the azimuthal angle of the transverse
components (in the plane perpendicular to the line-of-sight). We perform the
transformation into the disc-center heliographic system of coordinates and
resolve the 180
ambiguity existing on the transverse field following
Canfield et al. (1993). The resulting magnetic field in a Cartesian frame is (
Bx,
By, Bz).
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Figure 3: Unsigned magnetic flux for the IVM time series and the associated errors (unit of 1022 G cm2). Gray areas are the flaring periods as defined in Fig. 1. |
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We have performed an analysis of the noise level for the vertical and the
transverse components of the magnetic field on each of the 15 averaged
magnetograms. We proceed as follows: for the vertical magnetic field we plot
the distribution which can be fitted with a Gaussian profile, for the
transverse field we fit the distribution with a distribution. In both
cases, the estimated error is defined as the 3
value associated with
the width (
)
of the fitted distribution (see Leka & Skumanich
1999; Leka 1999). In
Fig. 3, we plot the time evolution of the photospheric unsigned
magnetic flux as well as the associated errors (from the 3
errors on
the Bz component) to show the quality of the data. The estimated formal
errors on Bz range between 25 and 50 G. We observe that the variation of the
magnetic flux does not exceed 10% and the errors are of 2% of the total flux.
The estimated errors on the transverse components range between 40 and 90 G. By
averaging the vector magnetograms over 15 min, we reduce significantly the
noise. For a single magnetogram, the formal errors are
150 G or greater
(see e.g. Leka & Skumanich 1999). The net magnetic flux which
characterizes the imbalance of positive and negative flux is less than 15% for
the IVM data, with an excess of negative flux. For the computation of the
nonlinear force-free equilibria we do not take into account pixels below the
estimated errors on the vertical and transverse components. Therefore the area
that we consider for the computation is different from one time to another. In
Fig. 4, we plot Bz in the IVM field-of-view (background
image) for AR 8210 as well as the black contour representating the area of
pixels used for the computation for 3 examples: typical thresholds (left),
large threshold value in the transverse components (center) and large threshold
value in the vertical component (right). As shown in Fig. 4,
the area of valid pixels is enclosed in the black contour and the variation of
area from one time to another is not significant.
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Figure 4: Areas inside the black contour for which the vertical component and the transverse components are above the thresholds [Bz, Bt]: at 17:13 UT ( left) with a threshold of [25 G, 46 G], at 18:01 UT ( center) with a threshold of [30 G, 90 G], and at 21:29 UT ( right) with a threshold of [55 G, 75 G]. |
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We use a time series of Big Bear Solar Observatory H images to observe
the reponse of the chromosphere to flaring activities of AR 8210 and its
surroundings. Each image observed every 1 min has a field-of-view of
with a pixel size of 1
.
In
Fig. 2 top right observed on may 1, 1998 at 20:10 UT, we
observed strong absorption features such as the sunspot and filaments in the
neighborhood of the active region, and bright regions (plages) associated with
weak magnetic field areas of the active region.In this H
image, we
label the flare sites: A (East part of the sunspot), B (South-East positive
polarity) and F (large area including A and B). The flare sites are identified
by strong intensity enhancement in the BBSO images and/or by emission profiles
as observed in the MCCD data.
We obtain H
spectroscopic data from the Mees Solar Observatory CCD (MCCD) imaging spectrograph (Penn et al. 1991). The MCCD scans were made with its
standard H
flare setup: spectrograph camera spatial scan field of view
,
pixels, 1.87 nm spectral range
(sampled with 50 pixels), completing one scan and one H
monitor image
each 15 s. The most important derived parameter for this study is Doppler
velocity, determined from the shift of the H
line center (see e.g.
Canfield & Reardon 1998; Des Jardins & Canfield 2003). To
maximize our Doppler-velocity sensitivity, we apply to the spectra Fourier
filtering, interpolation, and padding with zeros at the Nyquist frequency. We
then fit the core of the H
line profile with a parabola. From the
shifts of the minima of the fitted profiles of each pixel, we construct Doppler
velocity maps; from the intensity of the minima of each profile, we construct
line-center spectroheliograms. In Fig. 5, we have an example of
a line center H
image (left) and a velocity map (right) obtained by the
MCCD spectroheliograph on May 1, 1998 at 20:15:35 UT. The velocity maps are
used to determine the location of blueshift events (BSEs, see Sect. 3.2).
As for the BBSO image (Fig. 2 top right), we label the sites
of flares as A, B and F.
In the corona, we have access to several sets of data. The primary source of data on May 1, 1998 is the high-cadence full-disc images recorded by SOHO/EIT (Delaboudinière et al. 1995) for the Fe XII 19.5 nm line (see Table 1). We also have a time series of Yohkoh/SXT (Tsuneta et al. 1991) images recorded between 17:16 and 22:16 UT with an average time sampling of 8 min. The soft X-ray images are composite images which combine long exposure time satured images and short exposure time images. Those images are useful to have a tomography of the active region in the corona.
In Fig. 2 bottom, we highlight the coronal topology of AR 8210
which is derived from the EUV (left) and soft X-ray (right) observations: the
line segments represent approximate locations of "separatrix surfaces'' dividing the
active region in several connectivity domains.
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Figure 5:
Left: MCCD H![]() |
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The X-ray flux measured by GOES-8 (see Fig. 1) shows that between 17:00 UT and 21:40 UT on May 1, there are three periods of flaring activity in AR 8210:
Using both MCCD spectrograms and BBSO images, we first identify in the time
series the locations of flare activity as indicated on the H image
(see Fig. 2 top right and Fig. 5 left).
We observe that the H
intensity peaks are located at sites A and B for
the event (1), at site B for the C 2.1 flare of event (2), in the large area F
including sites A and B for the C 2.8 flare of event (2) and at site B for the
event (3). Note that the sites A and B correspond to the areas of
intensity enhancement and not simply to the peak of intensity.
Table 2:
List of blueshift events (BSE) observed in AR 8210 from
MCCD H
observations. The flare locations (A, B and F) are described in Fig. 5.
The velocity maps (see Fig. 5 right) obtained from MCCD data
are used to locate BSEs in AR 8210. BSEs are often observed before or after
flares in the H line. Canfield & Reardon (1998) have observed BSEs with a high
temporal cadence and with high spatial resolution. The authors have concluded
that BSEs are related to the occurence of eruptive phenomena and are certainly
the chromospheric signatures of reconnection processes in the corona. Recently,
Des Jardins & Canfield (2003) have performed a statistical study of BSEs. The authors have shown
that the rate of BSEs increases between 1 and 2 h before an eruptive flare (>C 6 class flares) and that there is a significant drop of the number of
BSEs after flares. In Table 2, we report on 8 periods of BSEs for
which we specify the start time, the time of the maximum velocity and the end
time as well as the location and the relationship with the flaring activity. It
appears that BSEs are at the same location as the flares (see labels in
Fig. 2). The BSEs 1 and 2 occur before the flare period (1) on
site A. The BSEs 3, 4, and 7 are observed during the decay phase of the three C 2.1 flares on site A. The BSE 5 is related to the rise phase of the B-class
flare. The BSE 6 occurs during a time period for which no significant change of
X-ray flux is observed. The BSE 8 covers the period before and after the C 2.8
flare on site F. We can already conclude that BSEs are preferentially observed
after a flare at the flare site and that the duration of these events is
between 5 min and 10 min.
To study the global evolution of the magnetic field on the photosphere, we use
the long-term movie made with MDI 96 min cadence magnetograms from E30
to W30
as well as the MDI 1 min cadence movie around the time of interest. In
addition to that we derive the photospheric velocity fields following
Longcope (2004) using IVM vector magnetograms. We find two characteristic
photospheric motions relevant for the study of the flaring activity: the
rotation of the sunspot, and the emergence and the fast transverse motion of a
parasitic polarity.
Recently, Longcope (2004) has developed a new technique to estimate the
photospheric velocity flow. The minimum energy fit (MEF) method is based
on the vertical component of the ideal induction equation:
From the long-term evolution of AR 8210, we note that the main negative
polarity (N1) is slowly rotating clockwise about its center. The effects of this rotation are
most important in the South-East part of AR 8210 where a positive polarity (P1)
is moving counter-clockwise around N1. These photospheric motions tend to increase the
shear between the sunspot and the positive polarity. The transverse motions due to the
sunspot rotation are not detected because the uncertainty of their measurement
(at 15 min time intervals) is large
(the rotation
of the sunspot is just few degrees per hour, see e.g. Brown et al.
2003).
An other interesting photospheric motion is the emergence of a parasitic
polarity (N2) associated with high transverse velocity field toward the
South-West. We measure a transverse velocity of 0.7 km s-1. The
increase of magnetic flux is estimated to be
33% in 4 h. The
emergence of flux is a precursor of eruption in active region. Therefore we
need to study the time evolution of this polarity to understand the
disturbances created by its emergence inside a pre-existing magnetic
configuration. Note that to compute the magnetic flux associated with the
polarity N2, we have extracted a square of
pixels of the
cross-correlated time series of vector magnetograms only including negative
values of Bz from N2.
In AR 8210, we have observed numerous flares, blueshift events associated with reconnection, and photospheric motions. In the following, we combine those three signatures of eruptions and the time evolution of the nonlinear force-free magnetic configurations to give a scenario of the flare process.
From the three components of the photospheric magnetic field (
), we can derive the
distribution of the electric current density,
and the distribution of the
force-free function,
as follows:
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(2) |
In order to describe the coronal field as a nonlinear force-free equilibrium,
there are several requirements on the properties of the current density
distribution,
.
The electric current should be balanced:
the total electric current should be zero. The positive currents from one
polarity should be equal to the negative currents in the opposite polarity.
These properties can be written as follows:
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(6) |
As an example, we study the distributions for the IVM vector magnetogram at 17:13 UT. The thresholds on the magnetic field components are 25 G for Bz and 46 G for Bt. The ratio of the area of strong field region to the area of weak field region is about 1.2. The electric current imbalance (from Eq. (4)) is 8% and from Eq. (5) the electric current imbalance is 30%.
The imbalance of electric current is plausibly due to the fact that the current in the strong-field regions is detected because the observed fields there exceed the threshold required for Jz calculations while that in weak-field regions is not detected. Note that the imbalance is negative, as one would expect, since Jz is mostly negative in the sunspot N1 where the field strength is high and then Jz well estimated.
To determine the structure of the coronal field we use the nonlinear force-free
approximation based on a vector potential Grad-Rubin (1958) method
by using the XTRAPOL code (Amari et al. 1999,1997). The nlff field in the
corona is then governed by the following equations:
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(7) |
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(10) |
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(12) |
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(13) |
Practically, the boundary conditions on the photosphere are: the observed
vertical component of the magnetic field,
in the disc-center
heliographic system of coordinates allowing the computation in cartesian
coordinates, and the
distribution given by
Eq. (3) in a chosen polarity (we have chosen the negative
polarity which represents the sunspot N1 of the active region). In order to ensure that the entire active
region is included in the field-of-view, we have created composite vector magnetograms by
combining IVM magnetograms (strong-field regions) and MDI magnetograms (surrounding weak-field
regions). We then compute
the nonlinear force-free field for the time series of composite magnetograms
using a cross-correlation technique between
each magnetogram and a non-uniform grid which reduces the computational time.
Those properties insure that we reconstruct the same volume of the corona.
Therefore we can study the time evolution of relevant quantities as the
magnetic energy or the relative magnetic helicity.
An interesting property of a magnetic configuration is given by its skeleton. The skeleton (Priest & Forbes 2000) corresponds to all topological elements
inside a 3D magnetic field including null points, spine field lines, separatrix
surfaces and separators. To analyse the evolution of AR 8210, we determine
various
topological elements. First we find the null points on the photosphere by
determining where the magnetic field vanishes and corresponds to a local minima
and for which the transverse components vanish. Around the null point, the
magnetic field has three eigenvalues,
,
that sum
to zero to satisfy Eq. (9). An eigenvector is associated with
each eigenvalue (not necessarily three perpendicular vectors). If one
eigenvalue is positive (resp. negative) and the two others are negative (resp.
positive), the spine is the isolated field line directed away from (resp.
toward) the null and the separatrix surface consists of field lines radiating
toward (resp. away from) the null. The separatrix surfaces give us the
definition of the different connectivity domains that comprise AR 8210.
As shown in Fig. 6, AR 8210 exhibits a complex topology with numerous photospheric null points (triangles) and separatrix surfaces represented by the direction of the fan surfaces (green lines) and the spine (thick white lines). We only plot the topological elements inside a reduced field of view. We obtain 49 null points in the entire field of view: 26 negative nulls and 23 positive nulls. We focus our study on four nulls: PN1-3 and NN1 (PN: positive null, NN: negative null). The null points PN1-3 and their associated separatrix surface will be investigated in the next section. NN1 has a spine field line connected with surrounding negative polarities. The separatrix surface is in the same direction as the South separatrix surface shown on EUV and soft X-ray images (Fig. 2 bottom). The topology does not change dramatically during the evolution of AR 8210 (during the studied time period).
We now analyse the coronal magnetic changes during this time period for the emerging, moving magnetic feature, and the rotating sunspot (see Sect. 3.3). We describe small reconnection processes associated with photospheric motions. By "small'' reconnections we mean reconnection processes which do not modify the configuration of the entire active region, but for which the connectivity of field lines is modified locally.
In terms of 3D magnetic field configurations, the emerging, moving feature is
essential to understand how small reconnection processes can take place in the
corona. A small negative polarity emerges in a pre-existing magnetic
configuration. This pre-existing system can be depicted as two positive
polarities (P3 and P4) and one large negative polarity (N1). The topology of
this tripolar configuration has a separatrix surface dividing the domain into
two different domains of connectivity,
and
(see Fig. 7).
Before the IVM time series, the small parasitic polarity (N2) emerges into the
pre-existing magnetic topology as seen in the long-term MDI evolution. The
field lines are then connected to both connectivity domains as shown in
Fig. 7. During the IVM time series, the parasitic polarity
moves toward the south-west. In the series of coronal field reconstructions, we
see that the
field lines previously connected in the domain
cross
the separatrix surface as the parasitic polarity moves, and then those
field lines reconnect into
.
The positive null point PN3 is
located close to N2: we cannot determine if this null point was already in the
pre-existing configuration or is related to the emergence of the parasitic
polarity N2. The spine associated with PN3 links the two positive polarity and
the footprint of the fan is well described by the separatrix surface defined on
EUV and soft X-ray images (Fig. 2 bottom). The coronal field
reconstructions show that the characteristic
scale height of the field lines involved in this process is less than 15 Mm.
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Figure 6: Basic topological elements for AR 8210 at 17:13 UT. Red (resp. blue) triangles are positive (resp. negative) null points. Spine field lines are thick white lines and separatrix surfaces (or fan surfaces) are defined by two green vectors. Only the projection on the photospheric plane is shown. The characteristic null points are labelled PN1-3 and NN1. |
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Figure 7: Area of interest extracted from AR 8210 (MDI image on the left) showing few characteristic field lines and the separatrix surface between two connectivity domains (straight line). Positive polarities are solid lines and negative polarities are dashed lines. |
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Figure 8: Location in AR 8210 where the effect of the sunspot rotation is stronger. The magnetic configuration is tripolar: positive polarities (P1 and P2) and one negative polarity (N1). The topology is indicated by thte separatrix surface (straight line). Few characteristic field lines are shown for the sake of clarity. |
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As seen in Fig. 2, the magnetic configuration at this location
can be depicted as a tripolar configuration with two positive polarities (P1
and P2) and one negative (N1). Then the topology is similar to the pre-existing
configuration in the above case. The topology also includes a negative null
point NN2 (Fig. 6) with a spine field line marked by the
separatrix surface defined in Fig. 2. The negative polarity is
rotating clockwise and the South positive polarity is moving southward. The
effect is that field lines connected in domain
and anchored in
the negative polarity move toward the separatrix surface and then
reconnect in domain
(see Fig. 8). As for the
moving feature, the same scenario of small reconnection applies. The field
lines involved in the reconnection process have a characteristic height between 15 and 30 Mm (Régnier & Canfield 2004).
We now investigate the relationship between the small reconnections due to the sunspot rotation and the flaring activity in AR 8210 obtained in Sect. 3.
In the above sections, we have identified two sites of reconnection inside AR 8210. But only the magnetic changes associated with the rotation of the sunspot are related to the flaring activity observed in the active region. Several properties can explain the observed differences between the two sites of reconnection: the speed of transverse photospheric motions, the magnetic field strength of the magnetic field lines involved in the process, the scale height of the magnetic field lines and the topology.
For the emerging polarity, the photospheric motion is fast, the field strength
is -450 G and the height is less than 15 Mm. In terms of topology, the
reconnection occurs along a separatrix surface. Assuming that a flaring activity
is related to this reconnection process, the magnetic energy stored in the
region of the parasitic polarity is not enough to trigger a C-class flare (or
above)
The flares are related to the rotation of the sunspot and
southward motion of the opposite polarity. The magnetic field strength on the
photosphere is -1500 G at the location of the flare for
field lines whose height is between 15 and 30 Mm. The site of reconnection is
determined by the existence of H
BSEs, EUV and soft X-ray enhancements
and coronal magnetic field changes from a time series of nlff fields. The
nature of the reconnection process is defined by the topology of the nlff
field: the magnetic changes occur close to a spine field line. All these
elements allow the magnetic configuration to store more magnetic energy than at
the location of the emerging parasitic polarity.
The magnetic energy in a volume V associated with a magnetic configuration
is given by
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Figure 9: Top: time evolution of magnetic energy contained in the potential field (dashed line) and in the nlff field (solid line) magnetic configurations in the coronal volume above the photosphere (unit of 1028 erg s-1). Bottom: time evolution of the rate of change of the free magnetic energy (solid line) and of the energy rate due to transverse motions (dashed line) on the photosphere (unit of 1028 erg s-1). |
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From the reconstructed 3D magnetic fields, we compute the magnetic energy for
the potential field,
,
the magnetic energy for the nonlinear
force-free field,
,
and the rate of change of the free magnetic energy
given by
In Fig. 9 top, we plot the rate of change of the magnetic energy
contained in the corona for the potential field (dashed line) and for the nlff field (solid line). We notice that the rate of change of the potential
and nlff energies always have the same sign. We also observe that
both rates are negative before the flare periods (1) and (2), for which the
impulsive phase is short (<5 min), and that the energy first increases
during the flares and then decreases during the decay phases. By integrating the
rates of change of energy, we conclude that at the end
of the time series more energy is stored in AR 8210 by 10%.
In Fig. 9 bottom, we plot the time evolution of the rates given by Eqs. (18) and (19). The rate of change of coronal energy due to transverse motions represents how the magnetic energy is injected through the photosphere into the corona. For the flare periods (1) and (2), the energy injected through the photosphere into the corona is correlated with the increase of magnetic energy in the coronal volume at the same time as the flares occur. The injection of energy is followed by a release of a comparable amount of energy during the decay phases of the flares.
The flare period (3) is somewhat different. We have an increase of the energy of
the free energy budget and of the energy due to transverse motions some 20 min before the flare. We also have an increase of the nlff magnetic
energy right before the flare and not during the flare as observed in two
previous time periods. It seems that there is delay between the injection of
energy into the corona and the response of the corona. The delay is 20 min.
The magnetic helicity describes the complexity of a magnetic configuration in
terms of topology and of linkage of field lines:
![]() |
(20) |
In addition, we derive other relevant helicities.
Following Berger (1999), we decompose the magnetic field into two fields:
![]() |
(22) |
![]() |
(23) |
In 1984, Berger & Field have derived
another formula for the relative helicity given by
![]() |
Figure 10:
Time evolution of magnetic helicities (unit of 1042 G2 cm4)
as
![]() ![]() ![]() ![]() |
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In Fig. 10, we plot the time evolution of the relative magnetic
helicity given by the Finn & Antonsen formula (solid line), the mutual
helicity between the potential and closed fields (dashed line), the self helicity of
the closed field (dot-dashed line), the vacuum helicity (dot-dot-dot-dashed
line) and the sum of the mutual helicity and the self helicity (stars). As
expected from Eq. (26), the sum of the self and mutual helicities
(stars) follows the evolution of the relative magnetic helicity (solid line).
The mutual helicity is the main contribution to the relative magnetic helicity
of AR 8210. This fact means that the way the field lines crossing between each
other is the essential part of the helicity and NOT the twist or the writhe
inside AR 8210. That is not surprising given the complex topology of AR 8210 as
shown in Figs. 2 and 6. We
note that the helicity of the vacuum field is nearly constant with a positive value
of
G2 cm4. We consider that the vacuum helicity or
helicity of the potential field is mostly related to the complex topology of AR 8210. In a simple connected domain, the self helicity of the magnetic field
should be zero as no twist or writhe are inside a potential field
configuration, but in cases involving separated connectivity domains the
self helicity indicates the complex topology of the field (Berger & Field 1984). We
note that the vacuum helicity is positive following the chirality rules
of active regions (Longcope et al. 1998; Pevtsov et al. 1995), and the relative helicity is mostly
negative.
Variations of magnetic helicity are expected if reconnection processes take place
in a magnetic configuration (see e.g. Biskamp 1997).
From Fig. 10, we note that before a flare negative relative helicity is
injected in the configuration and that after a flare the relative helicity is
decreasing. Again for the period (3), there is a delay of 20 min
between the injection of negative helicity and the release of helicity after the
flare.
In Sect. 5.1, we have identified two areas of interest in terms of magnetic configuration: an emerging, moving magnetic feature, and a rotating sunspot. Each of those phenomena is often seen to be a precursor of flares, CMEs and filament eruptions. In Table 3, we summarize the properties of the flaring or nonflaring activity in AR 8210 on May 1, 1998 associated with C-class flares. By combining a large dataset from the photosphere to the corona with coronal magnetic field models, we have determined the main ingredients of these flares: topology and photospheric motions.
Table 3: Summary of the properties of flare or nonflare activity in AR 8210 for the sunspot rotation and the fast moving polarity. The magnetic energy and the magnetic helicity are global quantities therefore we just mention the characteristic evolution of both quantities before and after a flare associated with BSEs. (SW = South-West)
The moving feature and the rotating sunspot are associated with the same basic magnetic configuration: three sources (two positive polarities and one negative). In this simple magnetic system, a separatrix surface divides the space into two domains of connectivity. We can compare this topology with the topological studies done by Brown & Priest (1998) and Longcope & Klapper (2002). Using a potential field method, Brown & Priest have analysed the topology of three unbalanced sources. They have found several topological states which all contain null points, separatrix surfaces and separators. In Fig. 2 of Longcope & Klapper, the same result is obtain for two negative polarities, N4 and N6, and one positive, P3. The authors have defined this topological structure as a broken fan (equivalent to two different separatrix surfaces part of the same dome). What we called a separatrix surface in Figs. 7 and 8 is the projection on the photospheric plane of the separatrix surface (including the separator field line) dividing the broken fan into two domains of connectivity. For the moving feature, the topological element dividing the domains is the fan surface. For the rotating sunspot, the spine field line has the same photospheric footprint as the separatrix surface. Those types of reconnection can be fast as shown by Parnell & Galsgaard (2004).
In both the moving feature and the sunspot rotation, the origins of reconnections lie in the photospheric motions of field lines footpoints. For the moving feature, the parasitic polarity emerges into the pre-existing three-source magnetic configuration and the fast displacement of this polarity leads to small reconnection processes. For the sunspot rotation, the field lines existing in the three-source configuration are moved toward the separatrix surface by the clockwise rotation and generate reconnections.
In this article, we have focused our study on small eruptive events which did not dramatically modify the magnetic configuration of the active region. In this study the most important ingredient is to use a good time series of vector magnetograms before and after flaring activity. A similar study can be done for M or X-class flares with the development of vector magnetic field measurements on the photosphere or in the chromosphere by Solar B/SOT (Solar Optical Telescope), SDO/HMI (Helioseismic and Magnetic Imager) or ground-based observatories (MSO, NSO/SOLIS, THEMIS, GREGOR, Huairou Observatory).
Acknowledgements
The authors would like to thank M. Berger, G. Fisher, Y. Li, D. Longcope and D. McKenzie for fruitful discussions and comment as well as Mees Solar Observatory observers who have provided us the IVM observations. S.R. research is funded by the European Commission's Human Potential Programme through the European Solar Magnetism Network. S.R. and RCC research has been supported by AFOSR under a DoD MURI grant "Understanding Solar Eruptions and Their Interplanetary Consequences''.