Inferring of flux rope orientation with the minimum ... - CiteSeerX

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, A11218, doi:10.1029/2004JA010594, 2004

Inferring of flux rope orientation with the minimum variance analysis technique C. J. Xiao,1,2 Z. Y. Pu,1 Z. W. Ma,3 S. Y. Fu,1 Z. Y. Huang,1 and Q. G. Zong4 Received 19 May 2004; revised 1 August 2004; accepted 17 September 2004; published 27 November 2004.

[1] The accuracy in flux rope orientation inference from both the traditional magnetic-

field-based minimum variance analysis (BMVA) technique and the current-based MVA (CMVA) approach is examined. Four different flux rope models are used in the MVA test. It is found that the directions of eigenvectors of MVA are critically dependent on the spacecraft path relative to the flux rope axis and structure of the flux rope encountered. For force-free flux ropes, the M direction of BMVA best fits the axial orientation, while for non-force-free flux ropes the BMVA may fail as the orientation inference tool. Magnetic field data from a single satellite path through non-force-free flux ropes are often insufficient to determine the rope orientation. Uncertainty may appear, as neither the N nor the M eigenvector is close to the axial direction. On the other hand, the CMVA based on multiple spacecraft measurements may help to eliminate such an uncertainty and shows great effectiveness for study of structures and geometries of the observed flux INDEX TERMS: 2724 Magnetospheric Physics: Magnetopause, cusp, and boundary layers; 2728 ropes. Magnetospheric Physics: Magnetosheath; 2784 Magnetospheric Physics: Solar wind/magnetosphere interactions; 2799 Magnetospheric Physics: General or miscellaneous; KEYWORDS: flux rope, flux rope orientation, minimum variance analysis, MVA Citation: Xiao, C. J., Z. Y. Pu, Z. W. Ma, S. Y. Fu, Z. Y. Huang, and Q. G. Zong (2004), Inferring of flux rope orientation with the minimum variance analysis technique, J. Geophys. Res., 109, A11218, doi:10.1029/2004JA010594.

1. Introduction [2] Much of the plasma and magnetic field structure in space is created in the form of flux ropes [Priest, 1990] via magnetic reconnection. When time-dependent magnetic reconnection takes place at the magnetopause, the interplanetary magnetic field (IMF) becomes connected to the geomagnetic field to form twisted flux tubes which are commonly referred to as the flux transfer events (FTEs) [Russell and Elphic, 1979]. Flux ropes at the magnetopause provide channels for the solar wind plasma to access to the magnetosphere and for the magnetospheric particles to escape to the interplanetary space. Recently, Cluster observations of multiple flux rope events at the high-latitude magnetopause show that the energetic ions originated from the magnetosphere move out to the magnetosheath on the whole along the axis of the flux ropes, while the solar wind plasma flows into the magnetosphere along the opened field lines associated with the tubes [Pu et al., 2003; Pu et al., 2004; Huang et al., 2004]. Determination of the axial

1

Department of Geophysics, Peking University, Beijing, China. Now at National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China. 3 Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, China. 4 Center for Space Physics, Boston University, Boston, Massachusetts, USA. 2

Copyright 2004 by the American Geophysical Union. 0148-0227/04/2004JA010594$09.00

orientation of flux ropes is a key issue in study of its geometry, structure, and the possible formation process. [3] Up to now the most widely used approach for inference of the flux rope axis is the minimum variance analysis (MVA) of magnetic field data measured by spacecraft, which presents the directions of maximum, intermediate, and minimum variance of the time series of the observed magnetic field B (denoted by L, M, and N directions, respectively) [Sonnerup and Cahill, 1968; Sonnerup and Scheible, 1998]. When the MVA is applied to determinate the orientation of flux ropes, it is usually referred to as the principal axis analysis (PAA) approach [Sibeck et al., 1984; Elphic et al., 1980; Elphic and Russell, 1983; Zong et al., 1997; Zong, 1999]. [4] It is well known that by assuming that the flux rope is cylindrically symmetric and that the spacecraft passes closely to the axis of the flux rope, the orientation of the flux rope could be determined by casting the magnetic field data into a principal axis coordinate frame. Three types of flux models have been used in attempt to determine the orientation of flux ropes with the MVA technique: (1) Force-free model by Goldstein [1983] and Lepping et al. [1990], which is widely used in studying CME-associated magnetic clouds [Lepping et al., 1990; Bothmer and Schwenn, 1998]. This model assumes J = aB for a magnetic force-free field, where a could be a constant or variable. One can always find that the L and N directions lie on a plane approximately perpendicular to the axis of the flux ropes. The orientation of the flux rope is then pointed to the M direction. (2) Non-force-free models by Elphic and Russell [1983] and Russell [1990] used to study the twisted magnetic structure of the flux ropes in the Venus

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ionosphere. If there is an intense core field inside the flux tube, as the spacecraft trajectory penetrates into the tube, the axis can either be in the M direction, or in the L direction. (3) ‘‘Grazing’’ FTE model by Farrugia-Elphic-Southwood [Farrugia, 1987; Elphic and Southwood, 1987], commonly applied in studying FTEs at the magnetopause [e.g., Papamastorakis et al., 1989; Bosqued et al., 2001]. This model requires the spacecraft to remain outside the FTE tubes and assumes the plasma flow to be incompressible throughout. In this case the components of the flow and field parallel to the axis of the passing flux tube contain no perturbations so that the axis would be aligned to the N direction of the PAA. Sonnerup et al. [1992], Walthour and Sonnerup [1990], and Walthour et al. [1994] found that the angle between the true axis and the N direction could be as large as 14 when compressibility effects of the plasma are included. This brief review indicates that with the aforementioned three B-based MVA methods anyone of L, M, and N directions in the principal axis coordinate frame could be aligned with the axis of the flux rope. Since one does not know in practice how the spacecraft trajectory is relative to the encountering flux ropes, it is not so easy to infer which one among the L, M, and N directions is the closest-approach to the tube orientation and how large angle between the inferred orientation and the flux rope axis would be. [5] It is commonly recognized that magnetic reconnection generates field-aligned currents during the flux rope formation [Saunders et al., 1984; Ma and Lee, 1999, and references therein]. If there is an ambient magnetic field, component reconnection then yields intense axis-aligned current. The Cluster multipoint measurements of the magnetic field make it possible to calculate current J inside flux tubes [Dunlop et al., 1990; Khurana et al., 1996; Chanteur, 1998; Pu et al., 2003, 2004]. Pu et al. [2004] proposed a new technique, the current minimum variance analysis (CMVA), for estimation of the orientation of a twisted flux tube. They suggested that after conducting the curlometer calculation on four-spacecraft magnetic field data, one may perform further the MVA for the obtained time series of J; if there is an intense axis-aligned current inside a flux rope, the maximum variance direction of the currents coincides with the direction of the maximum current component and lies parallel to the axis of the flux rope [Pu et al., 2003, 2004; Xiao et al., 2004]. [6] The goals of this paper are (1) to examine the accuracy of orientation inference from the traditional magnetic-fieldbased MVA (BMVA) and the current-based CMVA and (2) to understand the different situations in which different flux models are best employed, respectively, in PAA of magnetic flux ropes. Our results further confirm the effectiveness of the CMVA for inferring the flux rope orientation when reliable current data are available. In section 2 we perform numerical tests for the BMVA and CMVA with the force-free model, non-force-free model, and ‘‘grazing’’ model of flux ropes mentioned above. In section 3 we apply both the BMVA and CMVA to a flux rope model produced by multiple X-line reconnection in three-dimensional Hall MHD simulation [Ma and Lee, 1999]. As the previous studies pointed out, the relationship between the flux rope orientation and the PAA direction is extremely dependent on a number of factors, especially the path of the satellite through the structure [Moldwin and Hughes, 1991]. On the

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other hand, our results show that for all satellite paths we tested, the orientation inferred from the CMVA is always more closer to the real axis of the structure than that from the BMVA, except for the case of the force-free model in which the two results are identical since J//B and for the situation where no valuable currents are observed. In section 4 we apply both CMVA and BMVA to the multiple flux rope events observed by Cluster on 26 January 2001 [Pu et al., 2003, 2004; Huang et al., 2004; Xiao et al., 2004]. A brief discussion and main conclusions will be given in section 5.

2. Numerical MVA Tests With Three Existing Flux Tube Models 2.1. Lundquist-Lepping (L-L) Model [7] The force-free model of flux ropes assumes a cylindrically symmetric geometry and constant alpha which can be expressed as [Lundquist, 1950] BZ ¼ B0 J0 ðarÞ;

ð1Þ

Bf ¼ B0 J1 ðarÞ;

ð2Þ

Br ¼ 0;

ð3Þ

where Bz, Bf, and Br are the magnetic field components along the axis of the flux rope, in the azimuthal direction and in the radial direction, respectively; J0 and J1 represent the first two kinds of Bessel functions; and r denotes the radial distance from the axis. The orientation of flux rope (i.e., the z-axis) can be presented in the cylindrical coordinate by q = 0 and f = 0, where q is the polar angle and f is the azimuthal angle, as shown in Figure 1a. [8] In performing the MVA test, we make calculations for a large number of trajectories; each trajectory is characterized by its impact parameter X, the closest distance of the trajectory to the axis of the flux rope normalized by a. The obtained principal axis can be expressed by a pair of (q, f). Figure 1b plots the values of qM and fM as functions of X for the calculated M direction of the BMVA. It is seen that x, the angle between the calculated axis (i.e., the M direction) and the real axis of flux rope (here x  qM), is smaller than 10 when X < 1 and x  20 when X = 5  6. Therefore the BMVA is a valuable means of PAA when the spacecraft passes through the force-free flux ropes in a large range of X. So does the CMVA, since J/m0 = aB in this case. 2.2. Elphic and Russell Model [9] The second flux model we are going to test is the nonforce-free model by Elphic and Russell [1983] and Russell [1990] with an intense core field inside: Bf ¼ BðrÞ sinðaðrÞÞ;

ð4Þ

Bz ¼ BðrÞ cosðaðrÞÞ;

ð5Þ


BðrÞ ¼ B0 exp r2 =b2 ;

ð6Þ



aðrÞ ¼ p=2  1  exp r2 =a2 ;

ð7Þ

where a and b are two characteristic constants.

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Figure 1. Lundquis-Lepping force-free model. (a) The flux rope in a cylindrically coordinate, where X is the impact parameter, the closest-approach distance between the trajectory of a spacecraft and the axis of the flux rope; (b) the values of qM and fM as functions of X for the calculated M direction of the magnetic-field-based minimum variance analysis (BMVA). [10] Figures 2a and 2b show the orientations of three principal axes versus the impact parameter (normalized by a) obtained with the BMVA and CMVA, respectively. It is seen that for X < 2 and 2 < X < 5, the L and M direction obtained from the BMVA better represent the orientation of the rope, respectively, with x  20 – 30 (here x  q). On the other hand, in the CMVA case when x < 2 the L

direction best fits the rope axis with x < 20. Besides, x ! 0 as X ! 0. 2.3. Farrugia-Elphic-Southwood Model [11] The first quantitative model of field and flow perturbations outside of a flux rope was developed by Farrugia et al. [1987] and right after used for BMVA by Elphic and

Figure 2. The principal axis analysis (PAA) results of non-force-free flux rope model. (a) The orientations of three principal axes versus the impact parameter obtained with the BMVA; (b) the orientations of three principal axes versus the impact parameter obtained with the current-based MVA (CMVA). The vertical lines show the boundary of rope. The rectangle boxes indicate the best part fitted to the real axis. 3 of 9

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Figure 3. A flux rope obtained by three-dimensional Hall-MHD simulation. The lines with arrows are the magnetic field lines. The contour is the current Jy along the axial of the rope.

Southwood [1987]. They found that when the plasma surrounding the flux rope is assumed to be incompressible, the component of the flow and field parallel to the axis of the passing flux tube contained no perturbations so that the axis would be aligned to the N direction. Sonnerup et al. [1992] and Walthour et al. [1993, 1994] showed that the angle between the true axis and the N direction of the PAA could be as large as 14 if the compressibility effect is considered. In the case when the spacecraft does not penetrate the flux rope but remains sufficiently outside it, the F-E-S model provides a better prediction of the rope orientation in comparison with the above two models. Because there is no pronounced current existing far away from the flux rope, the CMVA does not apply for these ‘‘grazing’’ cases.

3. Numerical Tests With the Flux Rope Produced by Multiple X-Line Reconnection in ThreeDimensional Hall-MHD Simulation [12] The aforementioned analytical models of flux ropes seem somewhat artificial and too simple. In reality the magnetic field interacts with plasma in a variety of ways. The flux rope structures formed in various conditions may differ from each other in varying degrees. In this section we adopt the flux ropes produced by multiple X-line reconnection (MXR) [Lee and Fu, 1986] in three-dimensional HallMHD simulation [Ma and Lee, 2001; Shi et al., 2004]. For

simplicity, the strength of the magnetic field, the plasma temperature, and number density on both sides of the current sheet are taken to be equal. In the initial state the magnetic field lines lie in the y-z plane with the center of current sheet located in the plane x = 0. Q is the angle between the magnetic fields on two sides of the current sheet; e.g., in the case of the magnetopause, it represents the angle between the interplanetary magnetic field (BIMF) and the field inside the magnetosphere (BMH). Figure 3 shows the structure and geometry of the flux rope created by MXR with Q = 150 in which the axis of the rope is along the y direction, red lines represent the magnetic field lines, and the color contour represents the strength of the axis-aligned current Jy. The impact parameter X is normalized by the radius of the flux rope. In this simulation model Jy represents the maximum component of the current inside the flux rope. Note that in the three-dimensional Hall-MHD simulation, the By component can be generated (even if initially it sets to be zero) and varied as reconnection develops [Birn et al., 2001; Ma and Lee, 2001; Shi et al., 2004]. [13] Figures 4a and 4b show the PAA results of BMVA and CMVA for the case of Q = 180 with the satellite pass crossing the flux rope along the x (path 1) and z (path 2) directions, respectively. From top to bottom the N direction of BMVA (NB), M direction of BMVA (MB), and L direction of CMVA (LJ) are displayed, successively, as the functions of X (normalized by the current thickness). As mentioned above, the axis of the flux rope is along the

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Figure 4. PAA results for the flux rope models produced by multiple X-line reconnection in Hall-MHD simulations with Q = 180: (a) path 1 and (b) path 2. The top two panels show the N direction and M direction of BMVA, while the bottom one shows the L direction of CMVA. y direction, which corresponds to q = 90 and f = 90. In Figure 4a we see that the N direction of BMVA almost coincides to the real axis. This can be explained by the fact that the By component produced by Hall-MHD simulation is much weaker than the other two components. It is shown in Figure 4b that for the satellite pass parallel to the z axis the N direction of BMVA is almost identical to the real axis, except near the boundary of the rope where departure of q is x  100. As to the CMVA, just the L direction approaches the axial orientation. The CMVA is applicable to a broad range of X with the accuracy as high as x < 5. Outside the tube (X > 1) the CMVA does not work very well since the axis-aligned current vanishes. For an arbitrary satellite path other than paths 1 and 2, PAA results imply that the N direction of BMVA and the L direction of CMVA also best represent the rope orientation, respectively, with the departure angle x smaller than that along path 2. [14] Figures 5a and 5b show the case of Q = 150 for paths 1 and 2, respectively. It is found in Figure 5a that for path 1 the L direction of CMVA coincides well with the rope axis; meanwhile, the M directions of the BMVA can approximate to the axis with the maximum x  20 near the rope boundary. On the other hand, it is shown in Figure 5b that for path 2, the N direction of the BMVA is closed to the axis of rope near and outside the rope boundary; while inside the rope, none of L, M, and N direction of BMVA is close to the real axis of flux rope, The BMVA is nonapplicable in this situation. Nevertheless, one can apply the CMVA for rope orientation when the spacecraft path passing through the flux rope. [15] The BMVA results of Figures 4 and 5 show that for the flux ropes produced by multiple X-line reconnection in the magnetotail and near the magnetopause, PAA of magnetic field data from a single satellite pass through the rope is often insufficient to determine the rope orientation. In the worst case none of the principal axes of the BMVA is close to the rope axis. Uncertainty may often appear, as none of the N and M eigenvector is close to the axial direction when

eigenvalues l2 l3. We will come to this point later in section 4. On the contrary, the current PAA in this section illustrates that the maximum variance eigenvector lies exactly in the direction of Jy, the maximum component of the current. This can be understood as the following: Currents are mostly concentrated in the internal region of the flux rope, along the entire trajectory passing through the interior of the rope, spacecraft certainly sees a maximum variance in Jy .

4. Multiple Flux Ropes Observed by Cluster II on 26 January 2001 [16] On 26 January 2001 the four Cluster spacecraft were traveling outbound in the northern high latitudes from the magnetosphere to the magnetosheath. Between 1110 and 1123 UT, Cluster was staying in the magnetosheath boundary layer (MSBL) outside of but adjacent to the magnetopause. Four spacecraft observed eight events of bursty enhancements of energetic ion flux. Each enhancement of energetic ion flux is closely related to the appearance of a flux rope. Later, around 1130– 1133 in the magnetosheath, the constellation encountered two overlapping pronounced FTE flux tubes. [Pu et al., 2004; Huang et al., 2004; Xiao et al., 2004]. [17] Figure 6 presents an overall view of the calculated current density J, relative error for the calculated currents (DJ/J) which is approximately indicated by the ratio jdiv Bj/jcurl Bj, magnetic field B, and energetic proton flux during the time period of interest. Dunlop et al. [1990] and Chanteur et al. [1998] showed that the ratio jdiv Bj/jcurl Bj has the same behavior as DJ/J when the components of the tetrahedron elongation and planarity parameters (E, P) are both smaller than 0.9, which gives justification of the use of jdiv Bj/jcurl Bj as an indication of the quality of the physical coverage of the current density calculation [Robert et al., 1998]. During 1110– 1140 UT on 26 January 2001 the shortest border of the tetrahedron formed

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Figure 5. PAA results for the flux rope models produced by multiple X-line reconnection in Hall-MHD simulations with Q = 150: (a) path 1 and (b) path 2. The top two panels show the N direction and M direction of BMVA, while the bottom one shows the L direction of CMVA. by the four Cluster spacecraft is 507.5 km and the longest one is 729.7 km. The elongation and planarity parameters are E (0.54,0.77,0.33) and P (0.63,0.63,0.46), respectively; none of E, P is larger than 0.9 [Xiao et al., 2004]. Thus it is reasonable to use the jdiv Bj/jcurl Bj as an indication of the accuracy of DJ/J in the present study. [18] It can be seen in the figure that there are seven events in which intense currents exist inside the flux ropes with the relative error DJ/J being lower than 0.1– 0.2. Outside the ropes the estimated current greatly reduces; meanwhile, DJ/ J often increases up to more than 0.5.

[19] Table 1 presents the axial orientations of all ten flux ropes inferred from the BMVA. The CMVA results for seven events in which intense currents flow inside the tubes are also shown for comparison. Here q and f denote the polar angle and azimuthal angle of the axial direction in the GSE coordinate system; l1, l2, and l3 represent the maximum, intermediate, and minimum eigenvalues of the PAA, respectively. Capitals L, M, and N imply that the axial orientation is to be predicted to coincide with the maximum, intermediate, and minimum variance eigenvector, respectively. The BMVA gives that among ten events listed, the

Figure 6. The current density J, approximate relative error of calculated current DJ/J, magnetic field B, and energetic proton flux in thee GSE coordinate system. 6 of 9

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Table 1. Magnetic-Field-Based Minimum Variance Analysis (BMVA) and Current-Based Minimum Variance Analysis (CMVA) Results for the Observed Events f, deg Event Time, UT 1 2 3 4 5 6 7 8 9 10

1110 1113 1114 1115 1117 1118 1119 1122 1130 1131

BMVA

CMVA

q, deg BMVA

46 (N?) 65 (L) 42 (N?) 41 40 20 54 65 34 35 74 50 50 76 53 78 80 79 43 41 72 57 (M) 67 92 (M) 86 (N?) 78 78 (N?)

l2/l3

72 77 72 84 79

1.9 2.5 39.0 13.4 7.7 36.8 4.7 7.8 19.2 1.1

the same direction. Therefore it is not strange that their orientations were much alike. This is just what the CMVA show us.

l1/l2

CMVA BMVA CMVA 84 83

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14.0 7.0 8.7 17.2 26.4 3.7 13.7

axes of seven ropes are presented by the direction of the N eigenvector and one is presented by the M direction, while the other two (the first and tenth events with question mark) are difficult to determine. In the meantime there are four events in which the value of l2/l3 is less than 6, implying that uncertainty is present while determining the minimum variance direction in these cases. This uncertainty has been found to be rather common in analyses of magnetopause crossings [Sonnerup and Scheible, 1998]. On the other hand, the CMVA indicates that all flux rope axes are parallel to the L eigenvectors of the current PAA. Since the L direction of the CMVA best fits the axial orientation of the flux tube, we list l1/l2 in the last row of Table 1. The fact that most values of l1/l2 are greater than 6 indicates that the PAA performances of CMVA are reasonable. At the first glance of Table 1 one sees that in almost all events (except event 2) the directions of flux rope axes inferred from the BMVA are quite similar to those obtained based on the CMVA. Of special interest is the fact that there are five events for which the rope orientations predicted by the N direction of the BMVA and inferred from the CMVA are nearly identical. One thus has confidence that the BMVA results for these events are reliable. Originally, this was referred to as the evidence of the ‘‘grazing’’ pass [Pu et al., 2004]. However, since Cluster saw intense current flowing inside the tubes, spacecraft trajectories had certainly passed through the flux ropes. The BMVA results of these events may then be similar to the situations shown in Figure 4b. On the other hand, in events 3, 6, and 7 no notable currents were found. The Cluster spacecraft remained outside the flux ropes. For these ‘‘grazing’’ cases the N direction of magnetic field PAA should reasonably represent the axial orientation. In event 10 the uncertainty of BMVA becomes large because l2  l3 . Nevertheless, the orientation estimation with the N direction of the BMVA is fairly similar to that of the CMVA; thus it still adequately reflects the real situation. Finally, in event 9 the M direction of the magnetic field PAA, which fits well the L direction of the CMVA, appropriately presents the axial orientation of the FTE rope. This may be due to the fact that the Cluster spacecraft passed through the central region of the rope. It was shown previously that the directions of the DeHoffmann-Teller velocity obtained for all ten events are quite similar [Pu et al., 2004; Huang et al., 2004]. These ten flux ropes were likely coming from

5. Discussion and Conclusions [20] In the previous sections we have examined the accuracy of flux rope orientation inference from both the traditional magnetic-field-based MVA technique and the current-based MVA approach. Four flux rope models were used: The force-free model by Lundquist [1950] and Lepping et al. [1990], the non-force-free model by Elphic and Russell [1983] and Russell [1990], the ‘‘grazing’’ FTE model by Farrugia [1987] and Elphic and Southwood [1987], and the flux rope models produced by multiple X-line reconnection in Hall-MHD simulations [Ma and Lee, 2001; Shi et al., 2004]. We have also applied both the BMVA and CMVA to infer the axial orientations of ten flux ropes observed by Cluster II on 26 January 2001. The relationship between the axis orientation and the directions of the eigenvectors of PAA are found to be critically dependent on the spacecraft paths relative to the flux ropes and the structures of the flux ropes encountered. It is shown that for the force-free flux ropes, the M direction of BMVA best fits the axial orientation. On the other hand, for non-force-free flux ropes sometimes the BMVA may fail as the PAA tool. This can be explained as the following. [21] In the MVA test performed in the present study, we first construct the databases of the magnetic field and its self-consistent current based on the flux rope models investigated and then compare the variance of the axisaligned component with those of the other two principal directions. If the variance of the axis-aligned component is much less (greater) than or in between the other two, the N direction (L direction) or the M direction represents the rope axis. Figures 7a and 7b plot the three components of the magnetic field seen along a path with X = 1.8 in Figure 5b in the simulation domain coordinate system and the principal axis system, respectively. Since the spacecraft path remains outside the rope, the variance in the axial component By is apparently much smaller than the other two. Therefore the N direction of the BMVA indicates the axial direction of the rope. When the variance of the axis-aligned component is much less than the maximum one and significantly larger than the minimum one, the M direction fits the rope axis, see Figures 7c and 7d. On the other hand, if the variance of the axis-aligned component is comparable with one of the other two (or both) as Figures 7e and 7f show, all three principal axes deflect considerably from the real rope axis, hence the BMVA becomes nonapplicable. As to the CMVA, if the spacecraft path passes through the flux rope composed of primarily axis-aligned current, the variance of the axisaligned component current is much larger than the other two, the L-direction of the current MVA always best represents the rope orientation. [22] It is necessary to point out that the CMVA tests in sections 2 and 3 are based on the current data which are precisely self-consistent with the magnetic fields of the theoretical and numerical flux rope models being tested. In practice, one obtains currents from the multiple-spacecraft magnetic field measurements through the curlometer

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Figure 7. (a and b) The magnetic field along path 2 with X = 1.8 in Figure 5b in the simulation domain coordinate system and the principal axis system, respectively; (c and d) the magnetic field along path 1 with z = 0; (e and f) the magnetic field along path 2 with X = 0.1. or linear interpolation approach [Dunlop et al., 1990; Khurana et al., 1996; Chanteur, 1998; Pu et al., 2004]. This would require that the size of the constellation tetrahedron is considerably less than the dimension of flux ropes and that the spacecraft trajectories passing through the interior of the ropes. Even in these desirable situations, the calculated currents inevitably possess errors of about 10 to 20%. The usage of the CMVA therefore suffers a fairly restriction. Nevertheless, the CMVA can serve as a supplement of the BMVA, which helps to cross-check the estimations and plays the actor in case the BMVA is inapplicable. [23] A few conclusions can be drawn from the present study: [24] 1. When the spacecraft passes through a force-free flux rope, the BMVA provides a valuable technique to infer the flux rope axis. The M direction of the PAA best fits the real orientation.

[25] 2. If the spacecraft passes through a flux rope composed of primarily axis-aligned current, the L direction of the current PAA best fits the axial orientation, and hence the CMVA can serve as a supplement of the BMVA, provided that reliable current data are available. [26] 3. When satellite remains sufficient outside a flux rope, the BMVA presents a unique tool for inferring the flux rope orientation. The N direction of the magnetic field PAA well approaches the axial direction. [27] 4. The PAA of magnetic field data from a single satellite path through a non-force-free flux rope is often insufficient to determine the rope orientation. Uncertainty may appear, as none of the N and M eigenvector is close to the axial direction. The CMVA based on multiple-spacecraft measurements may help to eliminate such an uncertainty in these cases.

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[28] Acknowledgments. We thank Y. Lin, T. L. Zhang, X.G. Zhang, Q. Q. Shi, and J. Vogt for useful discussions. The authors are grateful to the Cluster II FGM team, the RAPID team, and the Chinese Cluster Data Center for offering four satellite data. This work was supported by the NSFC major project 40390150 and the national key project G200000784. [29] Lou-Chuang Lee thanks Joachim Vogt for his assistance in evaluating this paper.

References Birn, J., et al. (2001), Geospace Environmental Modeling (GEM) magnetic reconnection challenge, J. Geophys. Res., 106, 3715. Bosqued, J. M., et al. (2001), Cluster observations of the high-latitude magnetopause and cusp: Initial results from the CIS ion instruments, Ann. Geophys., 19, 1545. Bothmer, V., and R. Schwenn (1998), The structure and origin of magnetic clouds in the solar wind, Ann. Geophys., 16, 1 – 24. Chanteur, G. (1998), Spatial interpolation for four spacecraft: Theory analysis methods for multi-spacecraft data, ISSI Sci. Rep. SR-001, edited by G. Paschmann and P. W. Daly, Eur. Space Agency, Paris. Dunlop, M. W., A. Balogh, D. J. Southwood, R. C. Elphic, K.-H. Glassmeier, and F. M. Neubauer (1990), Configurational sensitivity of multipoint magnetic field measurements, in Proceedings of the International Workshop on Space Plasma Physics Investigations by Cluster and Regatta, ESA SP – 306, pp. 23 – 28, Eur. Space Agency, Paris. Elphic, R. C., and C. T. Russell (1983), Magnetic flux ropes in the Venus ionosphere: Observations and models, J. Geophys. Res., 88, 58. Elphic, R. C., and D. J. Southwood (1987), Simultaneous measurements of the magnetopause and flux transfer events at widely separated sites by AMPTE UKS and ISEE 1 and 2, J. Geophys. Res., 92, 13,666 – 13,672. Elphic, R. C., C. T. Russell, J. A. Slavin, and L. H. Brace (1980), Observations of the dayside ionopause and ionosphere of Venus, J. Geophys. Res., 85, 7679. Farrugia, C. J., R. C. Elphic, D. J. Southwood, and S. W. H. Cowley (1987), Field and flow perturbations outside the reconnected field region in flux transfer events: Theory, Planet Space Sci., 35(2), 227 – 240. Goldstein, H. (1983), On the field configuration of a magnetic cloud, Solar Wind Five, edited by M. Neugebauer, NASA Conf. Publ., CP-2280, 731. Huang, Z. Y., et al. (2004), Multiple magnetic reconnection events observed by Cluster (I), Chin. J. Geophys., 2, 1. Khurana, K. K., E. L. Keplo, M. G. Kivelson, and R. C. Elphic (1996), Accurate determination of magnetic field gradients from four-point vector measurements—II: Use of neutral constraints on vector data obtained from four spinning spacecraft, IEEE Trans. Magn., 32(5), 5193 – 5205. Lee, L. C., and Z. F. Fu (1986), Multiple X line reconnection: 1. A criterion for the transition from a single X-line to multiple X-line reconnection, J. Geophys. Res., 91, 6807. Lepping, R. P., J. A. Jones, and L. F. Burgala (1990), Magnetic field structure of interplanetary magnetic clouds at 1 AU, J. Geophys. Res., 95, 11,957. Lundquist, S. (1950), Magnetohydrostatic field, Ark. Fys., 2, 361. Ma, Z. W., and L. C. Lee (1999), A simulation study of generation of fieldaligned currents and Alfven waves by 3-D magnetic reconnection, J. Geophys. Res., 104, 10,117. Ma, Z. W., and L. C. Lee (2001), Hall effect on the generation of fieldaligned currents in three-dimensional magnetic reconnection, J. Geophys. Res., 106, 25,951 – 25,960. Moldwin, M. B., and W. J. Hughes (1991), Plasmoids as magnetic flux ropes, J. Geophys. Res., 96, 14,051. Mulligan, T., and C. T. Russell (2001), Multispacescraft modeling of the flux rope structure of interplanetary coronal mass ejections: Cylindrically symmetric versus nonsymmetrical topologies, J. Geophys. Res., 106, 10,581. Papamastorakis, I. G., G. Paschmann, W. Baumjohann, B. U. O. Sonnerup, and H. Luhr (1989), Orientation, motion, and other properties of flux

A11218

transfer event structures on September 4, 1984, J. Geophys. Res., 94, 8852. Priest, E. R. (1990), The equilibrium of magnetic flux ropes, in Physics of Magnetic Flux Ropes, Geophys. Monogr. Ser., vol. 58, edited by C. T. Russell, E. R. Priest, and L. C. Lee, p. 1, AGU, Washington, D. C. Pu, Z. Y., et al. (2003), Multiple magnetic reconnection events observed by Cluster II, in Proceedings of the Magnetic Reconnection Meeting, Sci. Rep. 280, edited by L. Rickard and M. Rick, pp. 60 – 64, Swedish Inst. of Space Phys. (IRF), Uppsala, Sweden. Pu, Z. Y., et al. (2004), Multiple flux rope events at the high-latitude magnetopause: Cluster/Rapid observation on January 26, 2001, Surv. Geophys., in press. Robert, P., M. W. Dunlop, A. Roux, and R. C. Elphic (1998), Accuracy of current density determination, in Analysis Methods for Multi-Spacecraft Data, edited by G. Paschmann and P. W. Daly, pp. 395 – 418, Eur. Space Agency, Paris. Russell, C. T. (1990), Magnetic flux ropes in the ionosphere of Venus, in Physics of Magnetic Flux Ropes, Geophys. Monogr. Ser., vol. 58, edited by C. T. Russell, E. R. Priest, and L. C. Lee, p. 413 – 213, AGU, Washington, D. C. Russell, C. T., and R. C. Elphic (1979), Observation of magnetic flux ropes in the Venus ionosphere, Nature, 279, 618. Saunders, M. A., C. T. Russell, and N. Sckopke (1984), Flux transfer events: Scale and interior structure, Geophys. Res. Lett., 11, 131. Shi, Q. Q., Z. Y. Pu, C. J. Xiao, H. Zhang, S. Y. Fu, Q. G. Zong, and Z. X. Liu (2004), Simulation studies of high-latitude magnetospheric boundary dynamics, Surv. Geophys., in press. Sibeck, D. G., G. L. Siscoe, J. A. Slavin, E. J. Smith, S. J. Bame, and F. L. Scarf (1984), Magnetotail flux ropes, Geophys. Res. Lett., 11, 1090. Sonnerup, B. U. O., and L. J. Cahill Jr. (1968), Explorer 12 observations of the magnetopause current layer, J. Geophys. Res., 73, 1757. Sonnerup, B. U. O., and M. Scheible (1998), Minimum variation analysis, in Analysis Methods for Multi-Spacecraft Data, edited by G. Paschmann and P. W. Daly, p. 185, Eur. Space Agency, Paris. Sonnerup, B. U. O., L.-N. Hau, and D. W. Walthour (1992), On steady field aligned double-adiabatic flow, J. Geophys. Res., 97, 12,015. Walthour, D. W., and B. U. O. Sonnerup (1990), Remote sensing of 2D magnetopause structures, in Physics of Magnetic Flux Ropes, Geophys. Monogr. Ser., vol. 58, edited by C. T. Russell, E. R. Priest, and L. C. Lee, p. 247, AGU, Washington, D. C. Walthour, D. W., B. U. O. Sonnerup, G. Paschmann, H. Luhr, D. Klumpar, and T. Potemar (1993), Remote sensing of two dimensional magnetopause structures, J. Geophys. Res., 98, 1489. Walthour, D. W., B. U. O. Sonnerup, R. C. Elphic, and C. T. Russell (1994), Double vision: Remote sensing of a flux transfer event with ISEE 1 and 2, J. Geophys. Res., 99, 8555. Xiao, C. J., et al. (2004), Multiple magnetic reconnection events observed by Cluster (II)—Current calculating, Chin. J. Geophys., 47, 635. Zong, Q.-G. (1999), Energetic oxygen ion in the geospace—An indicator for magnetospheric dynamics, dissertation, Inst. for Geophys. and Meteorol., Tech. Univ. Braunschweig, Braunschweig, Germany. Zong, Q.-G., et al. (1997), Geotail observation of energetic ion species and magnetic field in plasmoid-like structure in the course of an isolated substorm event, J. Geophys. Res., 103, 11,409 – 11,428.



S. Y. Fu, Z. Y. Huang, Z. Y. Pu, and C. J. Xiao, Department of Geophysics, Peking University, Beijing, 100871, China. (zypu@pku. edu.cn) Z. W. Ma, Institute of Plasma Physics, Chinese Academy of Sciences, P. O. Box 1126, Hefei 230031, Anhui, China. Q. G. Zong, Center for Space Physics, Department of Astronomy, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA.

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