- Email: [email protected]
Observation and theory of coronal loop structure
OBSERVATION AND THEORY OF C O R O N A L L O O P S T R U C T U R E J. A. Klimchuk
Space Science Division, Naval Research Laboratory, Washington, DC 20375, USA
ABSTRACT Following up on an initial study of 10 soft X-ray loops observed by Yohkoh (Klimchuk et al. 1992), we have carefully examined 43 additional Yohkoh loops and 24 EUV loops observed by TRA CE, and we confirm our original finding that most coronal loops have a nearly uniform thickness. This implies that: 1. the magnetic field in these loops expands with height much less than standard coronal models would predict; and 2. the shape of the loop cross section is approximately circular. We have investigated whether these surprising results can be explained by locally enhanced twist in the field, so that observed loops correspond to twisted coronal flux tubes. Our approach is to construct numerical models of fully three-dimensional force-free magnetic fields. To resolve the internal structure of an individual loop embedded within a much larger dipole configuration, we use a nonuniform numerical grid of size 609•215 the largest ever applied to a solar problem, to our knowledge. Our models indicate that twist does indeed promote circular cross sections in the corona, even when the footpoint cross section is irregular. However, twist does not seem to be a likely explanation for the observed minimal expansion with height.
INTRODUCTION Early in the Yohkoh mission, we published a study finding that coronal loops observed by the Soft X-ray Telescope (SXT) have a nearly uniform thickness (Klimchuk et al. 1992). This was a surprising result, since loops are believed to coincide with magnetic field lines, and the field must on average diverge with height above the solar surface. This suggests that a majority of loops should be wider at their tops than at their footpoints. Some expansion is in fact observed, but much less than predicted by extrapolation models of idealized or observed photospheric magnetic flux distributions (see Klimchuk 2000). Longcope (1996, 1998) has suggested that coronal loops lie in the immediate vicinity of magnetic separators, but the expansion properties of such field lines have not been fully investigated (point magnetic charge models seem to indicate minimal expansion, but these models may not accurately represent the real solar field). Our initial study was rather limited, since it involved only 10 loops observed in half resolution mode (4.9 arcsec pixels). We have recently completed a more extensive follow-up study involving 43 loops observed at both half and full (2.45 arcsec) resolution (Klimchuk 2000). In addition, we have completed a companion study of 24 loops observed by the Transition Region and Coronal Explorer (TRA CE), which has much higher spatial resolution (0.5 arcsec pixels) (Watko & Klimchuk 2000). The results of these new studies are reviewed here. In our original paper, we suggested that the uniform thickness might be explained by localized twist in the magnetic field. It is well known that the central portions of straight axisymmetric flux tubes become constricted as the tubes are twisted (e.g. Parker 1977, Zweibel & Boozer 1985, Lothian & Hood 1989). -55 -
J.A. Klimchuk
Because the constriction is greater for weak magnetic fields than for strong fields, and because no constriction is possible at the photospheric footpoints, where line tying applies, one might imagine that an expanding loop within a potential magnetic field configuration would become more and more uniform as the loop is twisted. We have recently computed 3D force-free magnetic field models to investigate this idea (Klimchuk et al. 2000), and those results are also reviewed here. OBSERVED LOOP EXPANSION FACTORS Details of our data analysis procedure can be found in the full papers, but two important points are worth emphasizing. First, we carefully subtract the background emission from the loops before making the width measurements. Second, we attempt to account for spatial resolution effects by correcting the measurements for the point spread function of the telescope and the finite size of the CCD pixels. To characterize the footpoint-to-apex expansion of the loops, we define a parameter called the "expansion factor": r ~ _- r m i d
,
(1)
r foot
where rmid and rfoot are the widths measured at the loop midpoint and footpoint, respectively. Each loop has two expansion factors, one for each leg. Figure 1 shows Fr plotted against loop length for the loops of the new SXT study. Stars and diamonds represent measurements made from full- and halfresolution observations, respectively. The median Fr is 1.30, meaning that the loops are typically 30~ wider at their midpoints than at their footpoints. This is much less than predicted by standard magnetic field models (Klimchuk 2000). Standard models also predict that Fr should increase with loop length, which is not seen in the data. Finally, the full- and half-resolution observations give similar results, suggesting that spatial resolution effects are not important.
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Length (10 5 km) Fig. i. Expansion factor versus loop length. Stars and diamonds are for full- and half-resolution SXT observations, respectively (from Klimchuk 2000).
The loops of our T R A C E study were observed mostly in the 171 A and 195 .~ bands, and, unlike the SXT loops, several were clearly associated with a flare. The 15 non-flare cases have a median Fr of 0.99 (actually narrower at the midpoints!), and the 9 post-flare cases have a median Fr of 1.13. These are listed in Table 1 together with the median values from both the old and new SXT studies. Estimated uncertainties are given in parentheses. It is rather striking how consistent the results are from the different studies. It is also interesting that the expansion factors measured from the higher resolution T R A CE observations are, if anything, smaller than the SXT values. This is further indication that spatial resolution is not a concern. T W I S T E D FLUX TUBE MODELS To test the idea that the observed thickness uniformity can be explained by localized twist in the magnetic field, we have used the magnetofrictional method to model a twisted coronal flux tube embedded within a much larger dipole potential field configuration. Other modelers have examined active-region-scale twist in the field, which is appropriate to active l'egion evolution (e.g. Sakurai 1979, Van Hoven et al. 1995, Amari -
66
-
Observation and Theory of Coronal Loop Structure Table i. Median Expansion Factors Study SXT (old) SXT (new) T R A C E (non-flare) TRA CZ (post-flare)
1.13 1.30 0.99 1.13
Fr (0.10) (0.12) (0.04) (0.34)
# Loops 10 43 15 9
et al. 1996), but our models are the first to have a truly localized twist. In order to numerically resolve the internal structure of the loop and at the same time include a large volume of surrounding field, it was necessary to adopt a very large nonuniform grid of size 609x 513 x 593.
Fig. 2. Flux tube of 2~r twist as viewed from the side at a 15~ angle to horizontal. Every twentieth grid line is shown (from Klimchuk et al. 2000).
Fig. 3. Flux tube of 27r twist as viewed from directly above. Every twentieth grid line is shown (from Klimchuk et al. 2000).
Figure 2 shows a side view of the inner part of the flux tube after an end-to-end twist of 27r has been applied. Compared to the untwisted potential state, the expansion factor of the tube has decreased from 2.8 to 2.3. This is still much larger than the values given in Table 1. Figure 3 shows the same flux tube, only this time viewed from directly above. From this perspective, the twist has an opposite affect and causes the expansion factor to increase rather than decrease, from 2.0 to 2.4! Based on these results, it seems doubtful that the thickness uniformity of observed loops can be explained by magnetic twist. However, there remains an issue of line-of-sight overlap which prevents us from ruling out the twist explanation with absolute certainty. This can be seen in Figure 3, where the footpoints of the loop overlap with the lower portions of the loop leg. There is another very interesting aspect to the simulations. Each flux tube in the original potential field configuration has a cross section that varies in shape as well as area along the tube. Our twisted flux tube has a circular cross section in the photosphere, by definition, since we rotate a circular patch at the footpoints. Before the twist is applied, the circle maps to an oval at the loop apex (elongated in the vertical direction), but after a twist of 27r, the oval has changed into a near circle. Thus, the primary affect of twist is to circularize the cross section while maintaining a nearly constant area. The physical reason for this can be understood in terms of the magnetic tension associated with the azimuthal field component that is introduced by the twist. Magnetic field lines trace out closed paths when viewed projected onto a cross section, and the tension in these lines will act to make the paths circular. The stronger the twist, the greater the force that organizes the flux into an axisymmetric bundle. This result is quite significant because it offers a natural explanation for why coronal loops are observed -67-
J.A. Klimchuk to have approximately circular cross sections (Klimchuk 2000). Although circular cross sections are often assumed, there has until now been no obvious reason to expect them. The magnetic field has a very clumpy distribution in the real solar photosphere, so the footpoints of loops are likely to have highly irregular shapes (unlike our model). In the absence of twist, we expect the irregular footpoints to map to similarly irregular cross sections in the corona. Twisted loops, on the other hand, should tend to have circular cross sections irrespective of the shape of their footpoints. In closing, we note that observed loops are actually plasma structures, and we have assumed that they coincide with magnetic flux tubes. In principle, they could be different, though this seems highly unlikely given the efficiency with which thermal energy and plasma flow along but not across the field lines. It has been suggested that the scale height of the plasma might be greater at the axis of a flux tube than at the outer edge. This could produce a plasma loop with a uniform cross section even if the magnetic flux tube is expanding. For this to feasible, however, the scale height at the outer edge would need to be smaller than both the scale height at the axis and the geometric height of the loop. This requires radial temperature stratification, which is not observed, and temperatures in the outer layers that are far too cool for Yohkoh to detect. The puzzle remains! REFERENCES Amari, T., Luciani, J.F., Aly, J.J., & Tagger, M., Very Fast Opening of a Three-dimensional Twisted Magnetic Flux Tube, ApJ, 466, L39 (1996). Klimchuk, J.A., Cross-Sectional Properties of Coronal Loops, Solar Phys., 193, 53 (2000). Klimchuk, J.A., Lemen, J.R., Feldman, U., Tsuneta, S., & Uchida, Y., Thickness Variations Along Coronal Loops Observed by the Soft X-Ray Telescope on Yohkoh, PASJ, 44, L181 (1992). Klimchuk, J.A., Antiochos, S.K., & Norton, D., Twisted Coronal Magnetic Loops, ApJ, 542, 540 (2000). Longcope, D.W., Topology and Current Ribbons: A Model for Current, Reconnection and Flaring in a Complex, Evolving Corona, Solar Phys., 169, 91 (1996). Longcope, D.W., A Model for Current Sheets and Reconnection in X-ray Bright Points, ApJ, 507, 433 (1998). Lothian, R.M., & Hood, A.W., Twisted Magnetic Flux Tubes: Effect of Small Twist, Solar Phys., 122, 227 (1989). Parker, E.N., The Origin of Solar Activity, ARAUA, 15, 45 (1977). Sakurai, T., A New Approach to the Force-Free Field and Its Application to the Magnetic Field of Solar Active Regions, PASJ, 31,209 (1979). Van Hoven, G., Mok, Y., & Mikid, Z., Coronal Loop Formation Resulting From Photospheric Convection, ApJ, 440, L105 (1995). Watko, J.A., & Klimchuk, J.A., Width Variations Along Coronal Loops Observed By TRACE, Solar Phys., 193, 77 (2000). Zweibel, E.G., & Boozer, A.H., Evolution of Twisted Magnetic Fields, ApJ, 295, 642 (1985).
-68 -
Space Science Division, Naval Research Laboratory, Washington, DC 20375, USA
ABSTRACT Following up on an initial study of 10 soft X-ray loops observed by Yohkoh (Klimchuk et al. 1992), we have carefully examined 43 additional Yohkoh loops and 24 EUV loops observed by TRA CE, and we confirm our original finding that most coronal loops have a nearly uniform thickness. This implies that: 1. the magnetic field in these loops expands with height much less than standard coronal models would predict; and 2. the shape of the loop cross section is approximately circular. We have investigated whether these surprising results can be explained by locally enhanced twist in the field, so that observed loops correspond to twisted coronal flux tubes. Our approach is to construct numerical models of fully three-dimensional force-free magnetic fields. To resolve the internal structure of an individual loop embedded within a much larger dipole configuration, we use a nonuniform numerical grid of size 609•215 the largest ever applied to a solar problem, to our knowledge. Our models indicate that twist does indeed promote circular cross sections in the corona, even when the footpoint cross section is irregular. However, twist does not seem to be a likely explanation for the observed minimal expansion with height.
INTRODUCTION Early in the Yohkoh mission, we published a study finding that coronal loops observed by the Soft X-ray Telescope (SXT) have a nearly uniform thickness (Klimchuk et al. 1992). This was a surprising result, since loops are believed to coincide with magnetic field lines, and the field must on average diverge with height above the solar surface. This suggests that a majority of loops should be wider at their tops than at their footpoints. Some expansion is in fact observed, but much less than predicted by extrapolation models of idealized or observed photospheric magnetic flux distributions (see Klimchuk 2000). Longcope (1996, 1998) has suggested that coronal loops lie in the immediate vicinity of magnetic separators, but the expansion properties of such field lines have not been fully investigated (point magnetic charge models seem to indicate minimal expansion, but these models may not accurately represent the real solar field). Our initial study was rather limited, since it involved only 10 loops observed in half resolution mode (4.9 arcsec pixels). We have recently completed a more extensive follow-up study involving 43 loops observed at both half and full (2.45 arcsec) resolution (Klimchuk 2000). In addition, we have completed a companion study of 24 loops observed by the Transition Region and Coronal Explorer (TRA CE), which has much higher spatial resolution (0.5 arcsec pixels) (Watko & Klimchuk 2000). The results of these new studies are reviewed here. In our original paper, we suggested that the uniform thickness might be explained by localized twist in the magnetic field. It is well known that the central portions of straight axisymmetric flux tubes become constricted as the tubes are twisted (e.g. Parker 1977, Zweibel & Boozer 1985, Lothian & Hood 1989). -55 -
J.A. Klimchuk
Because the constriction is greater for weak magnetic fields than for strong fields, and because no constriction is possible at the photospheric footpoints, where line tying applies, one might imagine that an expanding loop within a potential magnetic field configuration would become more and more uniform as the loop is twisted. We have recently computed 3D force-free magnetic field models to investigate this idea (Klimchuk et al. 2000), and those results are also reviewed here. OBSERVED LOOP EXPANSION FACTORS Details of our data analysis procedure can be found in the full papers, but two important points are worth emphasizing. First, we carefully subtract the background emission from the loops before making the width measurements. Second, we attempt to account for spatial resolution effects by correcting the measurements for the point spread function of the telescope and the finite size of the CCD pixels. To characterize the footpoint-to-apex expansion of the loops, we define a parameter called the "expansion factor": r ~ _- r m i d
,
(1)
r foot
where rmid and rfoot are the widths measured at the loop midpoint and footpoint, respectively. Each loop has two expansion factors, one for each leg. Figure 1 shows Fr plotted against loop length for the loops of the new SXT study. Stars and diamonds represent measurements made from full- and halfresolution observations, respectively. The median Fr is 1.30, meaning that the loops are typically 30~ wider at their midpoints than at their footpoints. This is much less than predicted by standard magnetic field models (Klimchuk 2000). Standard models also predict that Fr should increase with loop length, which is not seen in the data. Finally, the full- and half-resolution observations give similar results, suggesting that spatial resolution effects are not important.
. . . . . . . . .
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Length (10 5 km) Fig. i. Expansion factor versus loop length. Stars and diamonds are for full- and half-resolution SXT observations, respectively (from Klimchuk 2000).
The loops of our T R A C E study were observed mostly in the 171 A and 195 .~ bands, and, unlike the SXT loops, several were clearly associated with a flare. The 15 non-flare cases have a median Fr of 0.99 (actually narrower at the midpoints!), and the 9 post-flare cases have a median Fr of 1.13. These are listed in Table 1 together with the median values from both the old and new SXT studies. Estimated uncertainties are given in parentheses. It is rather striking how consistent the results are from the different studies. It is also interesting that the expansion factors measured from the higher resolution T R A CE observations are, if anything, smaller than the SXT values. This is further indication that spatial resolution is not a concern. T W I S T E D FLUX TUBE MODELS To test the idea that the observed thickness uniformity can be explained by localized twist in the magnetic field, we have used the magnetofrictional method to model a twisted coronal flux tube embedded within a much larger dipole potential field configuration. Other modelers have examined active-region-scale twist in the field, which is appropriate to active l'egion evolution (e.g. Sakurai 1979, Van Hoven et al. 1995, Amari -
66
-
Observation and Theory of Coronal Loop Structure Table i. Median Expansion Factors Study SXT (old) SXT (new) T R A C E (non-flare) TRA CZ (post-flare)
1.13 1.30 0.99 1.13
Fr (0.10) (0.12) (0.04) (0.34)
# Loops 10 43 15 9
et al. 1996), but our models are the first to have a truly localized twist. In order to numerically resolve the internal structure of the loop and at the same time include a large volume of surrounding field, it was necessary to adopt a very large nonuniform grid of size 609x 513 x 593.
Fig. 2. Flux tube of 2~r twist as viewed from the side at a 15~ angle to horizontal. Every twentieth grid line is shown (from Klimchuk et al. 2000).
Fig. 3. Flux tube of 27r twist as viewed from directly above. Every twentieth grid line is shown (from Klimchuk et al. 2000).
Figure 2 shows a side view of the inner part of the flux tube after an end-to-end twist of 27r has been applied. Compared to the untwisted potential state, the expansion factor of the tube has decreased from 2.8 to 2.3. This is still much larger than the values given in Table 1. Figure 3 shows the same flux tube, only this time viewed from directly above. From this perspective, the twist has an opposite affect and causes the expansion factor to increase rather than decrease, from 2.0 to 2.4! Based on these results, it seems doubtful that the thickness uniformity of observed loops can be explained by magnetic twist. However, there remains an issue of line-of-sight overlap which prevents us from ruling out the twist explanation with absolute certainty. This can be seen in Figure 3, where the footpoints of the loop overlap with the lower portions of the loop leg. There is another very interesting aspect to the simulations. Each flux tube in the original potential field configuration has a cross section that varies in shape as well as area along the tube. Our twisted flux tube has a circular cross section in the photosphere, by definition, since we rotate a circular patch at the footpoints. Before the twist is applied, the circle maps to an oval at the loop apex (elongated in the vertical direction), but after a twist of 27r, the oval has changed into a near circle. Thus, the primary affect of twist is to circularize the cross section while maintaining a nearly constant area. The physical reason for this can be understood in terms of the magnetic tension associated with the azimuthal field component that is introduced by the twist. Magnetic field lines trace out closed paths when viewed projected onto a cross section, and the tension in these lines will act to make the paths circular. The stronger the twist, the greater the force that organizes the flux into an axisymmetric bundle. This result is quite significant because it offers a natural explanation for why coronal loops are observed -67-
J.A. Klimchuk to have approximately circular cross sections (Klimchuk 2000). Although circular cross sections are often assumed, there has until now been no obvious reason to expect them. The magnetic field has a very clumpy distribution in the real solar photosphere, so the footpoints of loops are likely to have highly irregular shapes (unlike our model). In the absence of twist, we expect the irregular footpoints to map to similarly irregular cross sections in the corona. Twisted loops, on the other hand, should tend to have circular cross sections irrespective of the shape of their footpoints. In closing, we note that observed loops are actually plasma structures, and we have assumed that they coincide with magnetic flux tubes. In principle, they could be different, though this seems highly unlikely given the efficiency with which thermal energy and plasma flow along but not across the field lines. It has been suggested that the scale height of the plasma might be greater at the axis of a flux tube than at the outer edge. This could produce a plasma loop with a uniform cross section even if the magnetic flux tube is expanding. For this to feasible, however, the scale height at the outer edge would need to be smaller than both the scale height at the axis and the geometric height of the loop. This requires radial temperature stratification, which is not observed, and temperatures in the outer layers that are far too cool for Yohkoh to detect. The puzzle remains! REFERENCES Amari, T., Luciani, J.F., Aly, J.J., & Tagger, M., Very Fast Opening of a Three-dimensional Twisted Magnetic Flux Tube, ApJ, 466, L39 (1996). Klimchuk, J.A., Cross-Sectional Properties of Coronal Loops, Solar Phys., 193, 53 (2000). Klimchuk, J.A., Lemen, J.R., Feldman, U., Tsuneta, S., & Uchida, Y., Thickness Variations Along Coronal Loops Observed by the Soft X-Ray Telescope on Yohkoh, PASJ, 44, L181 (1992). Klimchuk, J.A., Antiochos, S.K., & Norton, D., Twisted Coronal Magnetic Loops, ApJ, 542, 540 (2000). Longcope, D.W., Topology and Current Ribbons: A Model for Current, Reconnection and Flaring in a Complex, Evolving Corona, Solar Phys., 169, 91 (1996). Longcope, D.W., A Model for Current Sheets and Reconnection in X-ray Bright Points, ApJ, 507, 433 (1998). Lothian, R.M., & Hood, A.W., Twisted Magnetic Flux Tubes: Effect of Small Twist, Solar Phys., 122, 227 (1989). Parker, E.N., The Origin of Solar Activity, ARAUA, 15, 45 (1977). Sakurai, T., A New Approach to the Force-Free Field and Its Application to the Magnetic Field of Solar Active Regions, PASJ, 31,209 (1979). Van Hoven, G., Mok, Y., & Mikid, Z., Coronal Loop Formation Resulting From Photospheric Convection, ApJ, 440, L105 (1995). Watko, J.A., & Klimchuk, J.A., Width Variations Along Coronal Loops Observed By TRACE, Solar Phys., 193, 77 (2000). Zweibel, E.G., & Boozer, A.H., Evolution of Twisted Magnetic Fields, ApJ, 295, 642 (1985).
-68 -