Local structure of directional discontinuities in ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, A04105, doi:10.1029/2010JA016152, 2011

Local structure of directional discontinuities in the solar wind W.‐L. Teh,1,2 B. U. Ö. Sonnerup,3 G. Paschmann,4 and S. E. Haaland5,6 Received 28 September 2010; revised 3 January 2011; accepted 25 January 2011; published 12 April 2011.

[1] We examine local structures of three directional discontinuities (DDs) observed by Cluster in the solar wind, using reconstruction based on the ideal 2‐D MHD equations in a steady state. In this novel application of the technique, our goals are the following: (1) to explain why the minimum variance analysis of the magnetic field (MVAB) often fails to meaningfully predict the vector normal to a DD and (2) to use the reconstructed field maps as an aid in interpreting the differences in the magnetic field profiles recorded by the four Cluster spacecraft. From the maps, we learn that the failure of MVAB as a predictor of the normal direction is due to internal structure such as magnetic islands (flux ropes) within the DDs and also that we can partly understand the differences in the fields observed by the four spacecraft. We find fairly good agreement between the normal directions determined from the four‐point timing approach and from MVAB, provided the constraint hBni = 0 is imposed on MVAB. Because of the island structures, the DDs cannot be readily identified as either tangential or rotational discontinuities, although the approximately Alfvénic flows on both sides favor the latter interpretation. Citation: Teh, W.‐L., B. U. Ö. Sonnerup, G. Paschmann, and S. E. Haaland (2011), Local structure of directional discontinuities in the solar wind, J. Geophys. Res., 116, A04105, doi:10.1029/2010JA016152.

1. Introduction [2] Directional discontinuities (DDs) in the solar wind have the property that the direction of the interplanetary magnetic field changes across them. In previous studies, these DDs have been identified as either tangential discontinuities (TDs) or rotational discontinuities (RDs) [e.g., Smith, 1973; Belcher and Solodyna, 1975; Burlaga et al., 1977]. In the standard MHD definition [e.g., Colburn and Sonett, 1966], TDs are boundaries across which no magnetic connection is present. In other words, the average ^, along the vector, magnetic field component Bn = hBi • n ^, normal to the discontinuity is zero. These discontinuities n can, but do not have to, separate regions of different densities, temperatures, and field strengths. In ideal MHD, plasma transport across a TD does not occur. On the other hand, for isotropic pressure RDs have the same density, temperature and field strength on its two sides and the two

1

Laboratory for Atmospheric and Space Physics, Boulder, Colorado, USA. Now at Space Research Institute, Austrian Academy of Sciences, Graz, Austria. 3 Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire, USA. 4 Max Planck Institute for Extraterrestrial Physics, Garching, Germany. 5 Max Planck Institute for Solar System Research, Katlenburg‐Lindau, Germany. 6 Department of Physics and Technology, University of Bergen, Bergen, Norway. 2

Copyright 2011 by the American Geophysical Union. 0148‐0227/11/2010JA016152

fields are connected via a nonzero value of Bn. The plasma flows across an RD at the Alfvén speed based on Bn. [3] To understand the nature of the DDs has been a long‐ standing issue in solar wind research. Under the assumption of 1‐D structure, we can in principle distinguish between TDs and RDs by determining the normal magnetic field component Bn across the boundary, which should be zero for TDs and nonzero for RDs. But in practice, this often turns out to be difficult because Bn is usually small even for RDs. For single‐ spacecraft data, the normal direction of the DD has in the past been estimated by minimum variance analysis of the magnetic field (MVAB) [e.g., Sonnerup and Scheible, 1998]. For four‐ point measurements of the magnetic field, such as provided by the Cluster mission, the DD orientation can also be estimated by the constant velocity approach (CVA), which allows determination of velocity and orientation of a planar discontinuity from the measured time differences between the discontinuity crossings over the four spacecraft and the separation vectors between them, assuming that the velocity and orientation of the discontinuity remain constant during the encounter [Russell et al., 1983; Harvey, 1998; Haaland et al., 2004]. [4] By use of MVAB normals, a substantial fraction of the solar wind DDs was classified as RDs [e.g., Lepping and Behannon, 1980; Neugebauer et al., 1984]. However, Horbury et al. [2001] found small Bn values for most of the DDs, when the normal direction was calculated from timing, using the spacecraft Geotail, Wind, and IMP‐8, together with the assumption that the DDs move with the plasma bulk speed. They suggested that the discrepancy between MVAB and CVA normals could be caused by surface waves. By analyzing Cluster observations in the solar wind, Knetter et al.

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Figure 1. Plots of magnetic field measurements near directional discontinuities observed by the four Cluster spacecraft in the solar wind for (left) case 1, (middle) case 2, and (right) case 3. Vector components are in GSE. Color codes for the spacecraft are as follows: C1, black; C2, red; C3, green; and C4, blue. The time interval for the minimum variance analysis is enclosed by vertical gray dashed lines. [2004] found that the DDs appeared to be mostly RDs, when the calculation of Bn was based on normal directions from MVAB, but that none of them remained unambiguous RDs when Bn was calculated from CVA (for further discussion, see the work by Neugebauer [2006] and Tsurutani et al. [2007]). These results demonstrate that there can be fundamental problems with the use of MVAB for determining the DD orientations. In MVAB as well as in other analyses, the assumption of a steady, one‐dimensional (1‐D) geometry of the DDs is needed. However, Cluster has revealed that there are differences in the magnetic field profiles even on the short spatial and temporal scales of the spacecraft configuration. Thus, the findings from Cluster suggest the existence of local structures within the DDs. Such structures may result in discrepancies between the normal directions determined from the timing approach and from MVAB. [5] In this paper, we employ ideal MHD reconstruction [Sonnerup and Teh, 2008] to recover 2‐D cross‐sectional maps of three DDs observed by Cluster in the solar wind, under conditions such that no magnetic connection to the bow shock was present. By studying the reconstructed field maps, we explain why the MVAB method fails to determine the DD orientations and we account for some, but not all, of the differences in the magnetic field profiles recorded by the four spacecraft. We also examine the normal directions estimated from CVA and from MVAB, the latter modified by the constraint hBni = 0 (referred to as MVABC) [e.g., Sonnerup and Scheible, 1998], in order to compare these directions with those from the unconstrained MVAB.

2. Case Studies [6] We now present results from our study of the three events, all of which come from the database compiled by

Knetter [2005]. In the analysis, we use magnetic field data from the FGM instrument [Balogh et al., 2001] and plasma data from the CIS/HIA instrument [Rème et al., 2001] at the (4 s) spin resolution imposed by CIS/HIA. For the ion pressure calculations, we only use T?, the plasma temperature perpendicular to the magnetic field, because, in the solar wind, the parallel temperature Tk from CIS/HIA shows large fluctuations (see CIS caveats at http://cluster.cesr. fr:8000/index.php?page = caveats&langue = en). 2.1. Calculations of Bn From MVAB and CVA [7] Figure 1 shows the GSE components of the magnetic field for the three DDs. The average spacecraft separation was ∼3750 km. The observations indicate that there are differences in the details of the magnetic field profiles observed by the four spacecraft. In particular, the field magnitude plots for Cluster 1 (C1) and Cluster 4 (C4) are distinctly different from those for C2 and C3. In Figure 1, the interval enclosed by the two vertical gray dashed lines is used for the minimum variance analysis. For each event and each Cluster spacecraft, Table 1 gives the following: (1) the ^CVA; (2) the MVAB normal CVA normal direction n ^MVAB with the average normal magnetic field, direction n hBni, as well as its statistical uncertainty; (3) the ratio between the intermediate and minimum eigenvalues from ^CVA; and (5) the MVAB, l2/l3; (4) the hBni value based on n ^MVABC. Data from all four MVABC normal direction n Cluster spacecraft are used to produce a single CVA normal direction. As indicated in Table 1, the normal vectors and values of hBni from MVAB are, generally speaking, very different from the corresponding results from CVA. The Bmax values, which are the maximum magnitudes of the magnetic field in the regions immediately upstream or downstream of the DD, are 7.9 nT, 8.1 nT, and 7.4 nT for

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Table 1. Summary of Results of Directional Discontinuity Normal Directionsa ^MVAB(GSE), hBn1i (nT) n

l2/l3

hBn2i (nT)

^MVABC(GSE) n

−0.5 +0.7 −1.1 +0.1

[0.971, −0.035, −0.235] [0.934, 0.150, −0.324] [0.883, −0.166, −0.439] [0.950, −0.006, −0.311]

b

C1 C2 C3 C4

[0.931, [0.753, [0.934, [0.743,

0.342, 0.647, 0.103, 0.667,

C1 C2 C3 C4

[0.021, −0.899, [0.332, −0.821, [0.152, −0.881, [0.352, 0.857,

C1 C2 C3 C4

[0.483, [0.461, [0.470, [0.455,

−0.128], −0.123], −0.343], −0.053],

−0.844, −0.856, −0.834, −0.864,

−2.3 −3.3 −1.6 −4.3

± ± ± ±

Case 1: 11 Mar 2003 0941:55–0942:30 UT 0.9 6.6 1.1 4.4 2.9 1.9 1.2 3.4

Case 2: 11 Mar 2003 0628:30–0629:30 UTc −0.437], +6.2 ± 0.7 4.6 −0.464], +7.1 ± 0.4 3.9 −0.448], +6.3 ± 0.3 10.9 0.376], −4.2 ± 1.0 4.6

+0.2 −0.1 +0.5 +0.1

[0.798, [0.820, [0.769, [0.805,

0.584, 0.537, 0.613, 0.570,

0.148] 0.197] 0.180] 0.166]

Case 3: 6 Mar 2003 1954:50–1955:41 UTd 0.1 17.4 0.1 18.3 0.1 15.8 0.1 13.7

+0.3 +0.2 +0.6 +0.7

[0.892, [0.907, [0.922, [0.870,

0.259, 0.218, 0.295, 0.260,

−0.372] −0.360] −0.253] −0.420]

0.234], 0.235], 0.289], 0.215],

+6.7 +6.7 +6.9 +6.8

± ± ± ±

a ^MVAB, with statistical errors estimated from equation (8.24) in the work of Sonnerup and Scheible Average normal magnetic field hBn1i based on n ^CVA. [1998]; hBn2i based on n b ^CVA = [0.879, 0.048, −0.474] (GSE). n c ^CVA = [0.811, 0.570, 0.134] (GSE). n d ^CVA = [0.952, 0.290, −0.101] (GSE). n

cases 1, 2, and 3, respectively. For cases 2 and 3, the hBni values from MVAB are comparable to Bmax. On the basis of normal vectors from MVAB, all three DDs would be interpreted as RDs but none remains in the unambiguous RD category, when the normal vectors come from CVA. This is the fundamental result reported by Knetter et al. [2004]. 2.2. Reconstruction Results [8] We have employed ideal MHD reconstruction [Sonnerup and Teh, 2008; Teh and Sonnerup, 2008; Eriksson et al., 2009; Teh et al., 2009] to recover the cross section of the DDs, using magnetic field and plasma data, measured by C1, as inputs. Figures 2a, 3a, and 4a show the resulting cross sections of the three DDs, with the axial field Bz in color. The reconstructions are performed in the de Hoffmann–Teller (HT) frame [e.g., Khrabrov and Sonnerup, 1998], i.e., the moving frame in which the convection electric field has been minimized. The z axis in each map is the invariant axis (∂/∂z ’ 0) and the x‐y plane is the reconstruction plane. As expected for solar wind events, each case has an excellent HT frame. In this frame, the flows are nearly parallel to the magnetic field, and approximately Alfvénic, for all three events, reaching 76%, 83%, and 100%, of the Alfvén speed for cases 1, 2, and 3, respectively. In the maps, the projected magnetic field vectors from C1–C4 are shown as magenta, red, green, and blue arrows, respectively, at points on the spacecraft trajectories. The unit normal vectors from MVAB, MVABC, and CVA are projected as yellow, cyan, and gray arrows, respectively; greater length of an arrow indicates that it is more perpendicular to the invariant (z) axis. According to the field maps, a magnetic island, or more precisely a flux rope, is embedded within each of the three DDs, with width much larger than the Cluster separation (∼3750 km). More islands, and for thinner discontinuities smaller islands, are likely to be present but cannot be recovered by the reconstruction. This limitation is a consequence of the low time resolution of CIS/ HIA and the fact that the spacecraft path (the x axis) in the maps forms a large angle with the DD surface.

[9] The invariant axes used for the reconstruction are (0.046, −0.948, 0.314), (0.596, −0.729, −0.336), and (0.396, −0.842, 0.365) (GSE), respectively, for cases 1, 2, and 3. Those z axes were chosen to maximize the correlation coefficient between the predictions from the map based on C1 and the measurements from the other three spacecraft, which are not otherwise used in the reconstruction [e.g., Hasegawa et al., 2005]. Figures 2b, 3b, and 4b show comparisons of magnetic fields predicted from the map (red circles) and fields actually measured by the other three spacecraft at points along their trajectories (black circles). Scatterplots (not shown) of the field components at the three other spacecraft (C2, C3, and C4) predicted from the reconstructed map, versus the corresponding actually measured components, provide a quantitative assessment of what is shown in Figures 2b, 3b, and 4b, namely that the reconstructed maps from C1 are validated by the fields measured by the other spacecraft, with correlation coefficients, cc = 0.9854, 0.9954, and 0.9866, for cases 1, 2, and 3. These are high values, but we must remember that the time plots of the field components measured by all four spacecraft have very similar features, albeit with brief time lags: Scatterplots of the measured field components at C2, C3, and C4, versus the corresponding components measured by C1, yield the correlation coefficients of 0.9186, 0.9866, and 0.9809, respectively, for the three events. Since (1 − cc) is a measure of deviation from perfect prediction, we find that, for the three events, this measure is 5.6, 2.9, and 1.4 times smaller for the prediction based on the C1 map than from the hypothesis that all four spacecraft simultaneously see the same structure. [10] To further check the consistency of our field maps, we have also produced maps from the C3 data (C2 and C4 lack CIS/HIA plasma data) and have found the island structure to be a robust feature. The islands also remain robust when deviations of up to 10° of the invariant axis from the optimum orientation are tried.

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Figure 2. For case 1, (a) ideal MHD reconstruction map of the magnetic field around the directional discontinuity with the axial field Bz in color. The reconstruction axes in GSE are ^x = (0.998, 0.034, −0.043), ^ y = (0.030, 0.315, 0.949), and ^z = (0.046, −0.948, 0.314). The reconstruction is done in the de Hoffmann‐Teller (HT) frame with VHT = (−417.5, 41.2, −0.3) (km/s). The magnetic field vectors measured by Cluster 1–4 are projected onto the x‐y plane as the magenta, red, green, and blue arrows, respectively, at points along the spacecraft trajectories. The yellow, cyan, and gray arrows are, similarly projected, unit normal vectors obtained from the minimum variance analysis of the magnetic field (MVAB), MVABC, and constant velocity approach (CVA) methods. The gray bar denotes the MVAB interval, and the Cluster tetrahedron configuration is shown by the blue lines. (b) Comparison of magnetic fields predicted from the map (based on C1) and the measurements from Cluster 2–4. The black (red) circles are the observations (predictions). The interval enclosed by the vertical dashed lines corresponds to that of the gray bar in Figure 2a. [11] We emphasize that our study is not an attempt to use reconstruction from data measured by C1 to predict what should be seen at the other three spacecraft. This prediction is used solely as a consistency check for the map and for the purpose of making an optimal choice of the invariant axis. As stated already, our dominant purpose is to obtain the internal structure of the DD, expecting that it will help us understand why such structure can degrade the ability of MVAB to make a meaningful prediction of the normal vector and field component.

3. Summary and Discussion [12] We have analyzed three Cluster directional discontinuities in the solar wind in terms of the DD orientation and cross section. Each event has an excellent HT frame in which the flows are found to be nearly Alfvénic. Table 1 summarizes our results for the normal directions from CVA and MVAB, together with the eigenvalue ratio (l2/l3) and normal field component hBni for each event and each spacecraft. The overall results for hBni demonstrate that the

three DDs would be classified as RDs based on MVAB but none of them would remain in that category for CVA. This is the conclusion reached by Knetter et al. [2004]. In addition to hBni, we also compute hVni, the average plasma velocity along the CVA normal in the HT frame. The results for C1 (C3) are −2.4 (1.8) km/s, 1.0 (2.3) km/s, and 2.6 (8.5) km/s for cases 1, 2, and 3, respectively. These nonzero values of the normal velocity component are very small; if real, they would suggest the presence of plasma transport across the discontinuity, which is not allowed for an ideal TD structure. However, what is seen could also be plasma circulating around the island without net plasma transport from one side of the DD to the other. On the basis of the hVni value, we can estimate the hBni value using the Walén relation and the results for C1 (C3) are −0.4 (0.2) nT, 0.2 (0.3) nT, and 0.3 (0.9) nT for cases 1, 2, and 3, respectively. The results are fairly consistent with the hBni values obtained from the CVA method, expect C3 in case 1. This consistency indicates that the hVni is consistent with the Alfvén speed based on the hBni value from the CVA normal.

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Figure 3. For case 2, (a) ideal MHD reconstruction map of the directional discontinuity with the axial field Bz in color. The reconstruction axes in GSE are ^x = (0.802, 0.526, 0.282), ^y = (−0.029, −0.438, 0.899), and ^z = (0.596, −0.729, −0.336). The reconstruction is done in the HT frame with VHT = (−418.3, 47.7, 9.8) (km/s). (b) Comparison of magnetic fields predicted from the map (based on C1) and the measurements from Cluster 2–4. Format is the same as in Figure 2. [13] By use of steady, ideal MHD in 2D, we reconstructed the cross sections of the three DDs from C1 data alone, with the results shown in Figures 2a, 3a, and 4a. We then compared the measurements by the other three spacecraft with the predictions from the reconstructed field map. Good agreement was obtained, as shown in Figures 2b, 3b, and 4b. One likely reason for the discrepancies is time aliasing of the maps. In each event, the reconstructed map shows a magnetic island (flux rope) embedded in the DD with a width ∼2.5 times larger than the average spacecraft separation. These results demonstrate that the failure of MVAB to, even approximately, determine the DD orientation is due to the presence of such magnetic island structures within the DD. [14] Table 2 summarizes the angles, , between the invariant axis and the normal directions obtained from MVAB, MVABC, and CVA, and the angles, , between the normal directions from the three different methods, for each event and each spacecraft. In a two‐dimensional geometry, we expect the normal direction to lie in the reconstruction plane, i.e., to be perpendicular to the invariant axis. The normal directions from MVABC for each spacecraft and from CVA are indeed more or less perpendicular to the invariant (z) axis, as indicated by the angles 2 and 3 in the table: For these two methods there is therefore qualitative consistency between the invariant axis obtained from optimization of the correlation coefficient and the normal

directions obtained from MVABC and CVA. With perhaps one exception, the MVAB normal vectors do not have such consistency: In case 3, the MVAB normal is instead more or less parallel to the z axis. Additionally, we see that the an^MVABC and n ^CVA directions are much gles 3 between the n ^MVAB and n ^CVA smaller than, the angles 2 between the n directions. In particular for case 2, the angle 3 is less than 5° for all spacecraft, while 2, and also 1, are large. Here ^MVAB and n ^MVABC directions. 1 is the angle between the n For case 3, it is near 90° for all four spacecraft, indicating ^MVAB direction is instead close to the invariant axis that the n (see also 1 in Table 2). The explanation for this behavior is that the axial field (the ‘guide field’) has relatively small variations while the field variance in the normal direction (taken from MVABC or CVA) is larger as a consequence of the spacecraft encounter with the magnetic island. This is essentially the situation discussed by Sonnerup and Scheible [1998] and Sonnerup et al. [2008]. ^MVABC and n ^CVA [15] The angle 3 between the vectors n is substantial for cases 1 and 3, ranging from 9° to 15°. This deviation could be (1) due to the complicated configuration of the magnetic island embedded within the DD, or (2) due to the location of the Cluster spacecraft paths through the island. We assume the CVA normal direction is representative of the overall DD orientation. With regard to the first explanation, the field maps demonstrate that the magnetic island configuration embedded within the DD is

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Figure 4. For case 3, (a) ideal MHD reconstruction map of the directional discontinuity with the axial field Bz in color. The reconstruction axes in GSE are ^x = (0.917, 0.342, −0.205), ^y = (0.048, 0.416, 0.908), and ^z = (0.396, −0.842, 0.365). The reconstruction is done in the HT frame with VHT = (−537.2, 52.2, 4.4) (km/s). (b) Comparison of magnetic fields predicted from the map (based on C1) and the measurements from Cluster 2–4. Format is the same as in Figure 2. more complicated in cases 1 and 3 than in case 2, as shown in Figures 2a, 3a, and 4a. With regard to the second explanation, Figures 2a and 4a show that the Cluster tetrahedron passed through the bottom part (upper part) of the island for case 1 (case 3). By contrast, for case 2 Figure 3a shows that the Cluster tetrahedron became temporarily embedded within the island. Comparison with the results in Table 2 suggests that the normal directions obtained from MVABC and CVA are more consistent with each other when the Cluster tetrahedron moves entirely inside the island and/or becomes centered on it. In Figure 4a, Cluster 3 is the spacecraft closest to the center of the island and has the smallest angle, 3 = 9°, while the other three spacecraft have angles in the range 16°–19°. In addition, we also calculated the normal vectors obtained from the cross product of the upstream and downstream magnetic fields. The cross‐ product vectors form average angles with the corresponding CVA vectors of 11°, 6°, and 23°, for cases 1, 2, and 3, respectively, i.e., the two vector sets are, at best, only approximately consistent. [16] Our second goal is to interpret the differences in the magnetic field profiles observed by the four Cluster spacecraft by use of the field maps. In the following, we discuss the measured magnetic field profiles for each case.

Figure 2a, this feature is the result of the C1–C3 spacecraft passing through a region of low transverse field, below the island. We see that C1 was closer to this region than C2 but in spite of that, the dip is considerably deeper at C2 than at Table 2. Summary of Angles for DD Normal Directions and Invariant Axes 1a

2e

3f

C1 C2 C3 C4

Case 1: 11 Mar 2003 0941:55–0942:30 UT 109° 90° 99° 23° 26° 128° 102° 99° 33° 41° 99° 87° 99° 17° 9° 128° 93° 99° 44° 45°

15° 11° 12° 11°

C1 C2 C3 C4

Case 2: 11 Mar 2003 0628:30–0629:30 UT 35° 90° 89° 125° 124° 18° 88° 89° 105° 105° 28° 93° 89° 120° 116° 123° 90° 89° 33° 35°

1° 4° 4° 2°

Case 3: 6 Mar 2003 1954:50–1955:41 UT 90° 84° 83° 79° 87° 84° 82° 80° 89° 84° 83° 80° 92° 84° 85° 81°

16° 16° 9° 19°

C1 C2 C3 C4

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3 c

1d

^MVAB. Angle between invariant z axis and n ^MVABC. Angle between invariant z axis and n c ^CVA. Angle between invariant z axis and n d ^MVAB and n ^MVABC. Angle between n e ^MVAB and n ^CVA. Angle between n f ^MVABC and n ^CVA. Angle between n a

3.1. Case 1 [17] As shown in Figure 1 (fourth panel of first column), C1–C3 all observed a dip in the magnetic field strength with the strongest dip seen by C3. According to the field map in

9° 8° 6° 9°

2 b

b

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C1. This behavior is most likely caused by variations along the z axis because C2 was located further away from the reconstruction plane than the other two spacecraft (C3 and C4). The distances along the z axis were z = +3064 km, +1450 km, +652 km, for C2, C3, and C4 (with C1 at z = 0). 3.2. Case 2 [18] The field strength observed by C1 and C4 is approximately constant and is larger than that observed by C2 and C3. As shown in Figure 3a, C1 and C4 were passing through the center of the island while the paths of C2 and C3 were at the edge of the island. This fact can explain the weaker field observed by C2 and C3 within the DD (Figure 1). The lower field seen by these two spacecraft downstream of the DD, may again be attributed to 3‐D effects (z = +2793 km, +3447 km, and +2499 km for C2, C3, and C4) but temporal effects may also play a role. 3.3. Case 3 [19] The enhanced field strength observed by C3 is caused by the fact that the spacecraft is closest to the center of the island, where the axial (z) field is a maximum, as shown in Figure 4a. However, we find that the C2 field strength is not consistent with that at C1 and C4. The reason may again be 3‐D effects, since the C2 distance from the reconstruction plane along the z axis is +2591 km, which is farther away from z = 0 than the other two spacecraft (C3 at +1805 km and C4 at +2146 km). [20] In conclusion, we find that the failure of the MVAB method to determine the DD orientation is caused by the presence of magnetic islands (flux ropes) and other structures within the DD. These structures increase the variance of the field component along the average (CVA) normal direction, often making it comparable to, or even larger than, the variance of the axial (z) field. We have shown examples (case 2; C3 and case 3; C1, C2, C3, and C4) where l2/l3 exceeded 10, a number sometimes used a rule of thumb for a reliable MVAB prediction of the normal direction [e.g., Sonnerup and Scheible, 1998], but where the minimum‐ variance direction formed angles with the corresponding z axis ranging from 1 = 28° to 1 = 6°, instead of the expected value of 90°. In each case, the predicted normal field component Bn was unreasonably large, which, in studies based on single‐spacecraft data, can provide a useful warning that the normal vector predicted by MVAB is in serious error. On the other hand, we have one event (case 1; C3) where l2/l3 has the very low value of 1.9 but where 1 = 99° and Bn is near zero, both of which suggest that MVAB has predicted a reasonable normal direction. These results illustrate that extreme care must be taken before one uses the eigenvalue ratio to accept (or reject) the minimum variance direction from MVAB as a useful predictor of the direction normal to a current sheet: As our reconstructions show, the presence of 2‐D or 3‐D structure within the sheet can easily contaminate the prediction. We have used three events to illustrate this conclusion but to date have reconstructed four additional events, all showing similar behavior, namely internal magnetic structures that corrupt the MVAB prediction to various extents. The contamination will also affect the CVA, MVABC, and cross product analyses, all of which are based on the assumption of planar geometry. However, we expect the errors caused by the magnetic

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islands to be much less significant for these analyses than for MVAB. [21] From the reconstructed field maps, we can also understand some, but not all, of the differences in the magnetic field profiles observed by Cluster. One important remaining question is this: How should the three DDs be classified? We argue that, on the basis of the maps, they cannot readily be called either TDs or RDs, because TDs and RDs are assumed to be locally 1‐D, or nearly 1‐D, structures. However, our results indicate that, in general, solar wind DDs are not 1‐D but have substantial 2‐D and, to some extent, also 3‐D features. The reconstructions do not show any clear indication of any substantial net magnetic flux crossing the discontinuities so that there is no clear support in the maps for the interpretation of the discontinuities as RDs. However, the maps do not extend far in the ±y direction and do not exclude the possibility that a net magnetic flux crosses the discontinuity at other sites. Perhaps arguing for such a scenario is the approximately Alfvénic nature of the plasma flow (as seen in the reconstruction frame of reference). One should also remember that the ideal TD (i.e., a 1‐D structure with Bn ≡ 0 and Vn ≡ 0) is a singular case that is unlikely to be found in real events, where it would be destroyed, e.g., by reconnection (tearing modes with unequal rates) at multiple X points. [22] Acknowledgments. This study is part of a team effort dedicated to DDs and performed under the auspices of the International Space Science Institute (ISSI), Bern, Switzerland. We thank ISSI for its support, and the other ISSI team members, T. Horbury, J. Vogt, and R. Wicks for helpful comments. [23] Philippa Browning thanks Geza Erdos and another reviewer for their assistance in evaluating this paper.

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