Orientation and motion of two-dimensional structures ...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, A06208, doi:10.1029/2004JA010853, 2005

Orientation and motion of two-dimensional structures in a space plasma ¨ . Sonnerup and H. Hasegawa1 B. U. O Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire, USA Received 20 October 2004; revised 8 February 2005; accepted 15 March 2005; published 11 June 2005.

[1] We present a new least squares technique for the determination of axis orientation,

motion, and intrinsic axial electric field for two-dimensional coherent structures in a space plasma. This single-spacecraft method is based on Faraday’s law, which, under the model assumptions, requires the intrinsic axial electric field to be constant in space and time. The technique is applied to two flux transfer events seen by the four Cluster spacecraft, although data from only one of them (Cluster 1) is used in the analysis. For these events, excellent agreement is found with the orientations obtained from optimal Grad-Shafranov reconstruction (Hasegawa et al., 2005), a method that uses data from all four Cluster spacecraft. However, the new method can also fail to produce such agreement. It must therefore be used with caution. ¨ ., and H. Hasegawa (2005), Orientation and motion of two-dimensional structures in a space plasma, Citation: Sonnerup, B. U. O J. Geophys. Res., 110, A06208, doi:10.1029/2004JA010853.

1. Introduction [2] In the study of moving structures in a magnetized collision-free space plasma, the proper frame of the structure is often taken to be the deHoffmann-Teller (HT) frame, i.e., the moving frame in which the remnant electric field is as small as possible, ideally zero. In ideal MHD, the plasma velocities, evaluated in the HT frame, are parallel or nearly parallel to the magnetic field. This statement also implies that in the HT frame, the magnetic structure is time independent or nearly time independent. The HT frame was introduced by deHoffmann and Teller [1950] in a theoretical study of shocks. The concept was first applied to observations in the vicinity of the magnetopause by Aggson et al. [1983] by use of a trial-and-error procedure to find the frame velocity. A least squares analytical procedure was subsequently developed by Sonnerup et al. [1987, 1990] and has been extensively used in many studies of the magnetopause. These studies have revealed that very good HT frames often exist. This result is somewhat surprising because there seems to be no obvious reason to expect such a frame to be possible. As an elementary illustration of this point, consider a long, straight, timeindependent, and dissipation-free magnetic flux rope of circular cross section, with transverse field Bj(r). The rope has some arbitrary (axially symmetric) distribution, Bz(r), of magnetic field along its axis, ^z, and @/@z = 0. It is in magnetohydrostatic equilibrium so that rp = j  B. In a frame moving with the flux rope, the plasma is static, except for (axially symmetric) motion, vz(r), along the flux-rope axis. Unless this axial motion happens to be independent of radius, it is impossible to find a frame of reference in which 1

Now at Tokyo Institute of Technology, Tokyo, Japan.

Copyright 2005 by the American Geophysical Union. 0148-0227/05/2004JA010853$09.00

the radial electric field vanishes. On the other hand, in the absence of resistive dissipation, the axial component of the electric field in the comoving frame must be identically zero. This can be seen as follows: at the center of the flux rope is a purely axial magnetic field line along which no electric field can be sustained without dissipation. Furthermore, the circumferential component of Faraday’s law, r  E = @B/@t = 0, gives @Ez/@r = 0 so that Ez = 0 must hold in the entire flux rope. The axial component of r  E = 0 always gives Ej = 0 in our flux rope so that only the nonremovable radial component of E remains. A velocity that describes the motion of such a two-dimensional, timeindependent structure in the direction transverse to the invariant axis nevertheless exists. It is the purpose of the present paper to develop a method, based on Faraday’s law, for finding not only this velocity but also the orientation of the invariant axis and any intrinsic axial electric field that remains in the moving frame. Application of the method is by no means limited to flux ropes or to dissipation-free systems. In particular, it could be useful in the study of X-type magnetic geometries where Ez 6¼ 0 and dissipation is present near the X.

2. Theory [3] Two basic assumptions are needed for our development: (1) The structure is two-dimensional in the sense that derivatives in the axial direction, identified by the unit vector k, are much smaller than those in the plane transverse to the axis; (2) When viewed in the proper frame, the structure is time independent or nearly time independent. In such a moving frame, Faraday’s law then requires that the electric field component along the axis be constant, or nearly constant, throughout the structure (in the complete absence of dissipation it must be zero). Note that the proper frame is not unique: its speed along the axis is arbitrary. Here we will use a

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proper frame that has no motion along the axis so that V0  k = 0, where V0 is the frame velocity relative to the spacecraft frame. If the electric field vectors measured in the latter frame are denoted by E (m) , with m = 1, 2, . . . M, then the corresponding field vectors in the proper frame are E0ðmÞ ¼ EðmÞ þ V0  BðmÞ :

ð1Þ

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where dE = (E  hEi) and dB = (B  hBi) denote deviations of an individual measured E or B from its average. [6] The next step is to partially differentiate D2 with respect to the ith component of U0 (i = 1, 2, 3) and put the result equal to zero. Using the summation convention (summation over repeated indices), the resulting three equations can be written as hdBi dBj iU0j þ hdBi dEj ikj ¼ 0:

ð6Þ

(m)

Here B denotes the corresponding measured magnetic field vectors. The measured electric field components along the axis are then

Similarly, partial differentiation of D2 with respect to ki leads to


 E0ðmÞ  k ¼ EðmÞ  k þ V0  BðmÞ  k:

hdEi dBj iU0j þ hdEi dEj ikj  ldij kj ¼ 0;

ð2Þ

[4] If the two model assumptions are ideally satisfied, then E0(m)  k = E0 must be strictly constant for all the data taken as the structure moves past the observing spacecraft. In reality there would almost always be some deviations from the ideal model. In such a situation, Khrabrov and Sonnerup [1998a] proposed that the axis vector, k, the frame velocity, V0, and the average axial field, E0, all be determined by minimization of the variance in the axial field, i.e., by minimization of   m¼M i2 1 X h 0ðmÞ D0 ¼ E  k  E0 M m¼1   m¼M
 i2 1 X h ðmÞ ¼ E  k þ V0  BðmÞ  k  E0 ; M m¼1

where dij is the Kronecker d. In (6) and (7), we recognize the angle brackets as being covariance matrices, e.g., MBB = hdBidBji, MBE = hdBidEji, etc. Assuming MBB, the inverse of MBB, exists we can then solve (6) for U0, the result being U0 ¼ MBB  MBE  k:

ð8Þ

When this expression for U0 is substituted into (7), the result is the following eigenvalue/eigenvector problem ½M0  lI  k ¼ 0;

ð3Þ

ð7Þ

ð9Þ

where I is the identity matrix and M0 is a symmetric matrix, defined by M0 ¼ MEB  MBB  MBE þ MEE :

ð10Þ

subject to the constraints jkj2 = 1 and V0  k = 0. Khrabrov and Sonnerup argued that the term (V0  B(m))  k causes the optimization with respect to V0 and k to produce nonlinear equations, which would have to be solved by some numerical search procedure. Therefore they did not pursue the analysis beyond the formulation of (3). However, the difficulty they identified can be avoided by the simple step of putting the vector (k  V0) = U0 so that (3) becomes the quadratic form

[7] The next step is to examine the quantity k  U0. Given the definition of U0 as k  V0, this dot product should ideally be zero. However, as we show below, in any real application of our method it may not be precisely zero. We first multiply k into (8) from the left to find

  m¼M i2 1 X h ðmÞ D0 ¼ E  k þ BðmÞ  U0  E0 M m¼1

By use of (2) in the form

¼ h½E  k þ B  U0  E0 2 i

E0ðmÞ  k ¼ EðmÞ  k þ U0?  Bm ;

ð4Þ

in E0 and the components of k and U0. In the rightmost member of (4) and in the developments to follow, the averaging operation (M1 )S has been denoted by the angle brackets, h . . . i, and the superscripts m, identifying individual measured values, have been suppressed. The variance D0 can then be minimized with respect to k, U0, and E0, subject to the constraint jkj2 = 1, the result being a set of linear equations. [5] The minimization of D0, subject to the constraint, can be performed by minimization, without constraint, of D1 = {D0  lk  k}, where l is a Lagrange multiplier. In the first step of this process, we put the partial derivative of D1 with respect to E0 equal to zero and solve for E0. The result is E0 = [hEi  k + hBi  U0]. When this optimal value of E0 is substituted into D1, the result is D2 ¼ h½dE  k þ dB  U0 2 i  lk  k;

 k  U0 ¼  k  MBB  MBE  k :

ð5Þ

ð11Þ

ð12Þ

where U0? is the component of the actually calculated U0 vector perpendicular to k, the product MBE  k in (11) can be rewritten as
  BE M  k i ¼ hdBi dEj kj i ¼ hdBi d Ej0 kj  U0?j Bj i ¼ hdBi dEj0 kj i  hdBi dBj iU0?j ¼ hdBi dEj0 kj i  MijBB U0?j :

ð13Þ

Substitution of this result into (11) leads to 0

k  U0 ¼ k  MBB  MBE  k; 0

ð14Þ

where MBE = hdBdE0i. In arriving at (14), we have used MBB  MBB = I and k  U0? = 0, both of which are true by definition.

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[8] In the ideal case, Ej0kj is exactly constant so that dEj0kj, the fluctuation of the axial field component about its average, is identically zero. In that case, we see from (14) that k  U0 vanishes identically. This is the reason for not implementing the condition k  U0 = 0 by means of a second Lagrange multiplier. In any good application of the method, dEj0kj should be small. Additionally, there is no obvious reason to assume its correlation with dBi to be perfect. Thus we conclude that in good applications, k  U0 would be very small. We also conclude that it would be inappropriate to let the orientation of k be influenced, via a strict implementation of k  U0 = 0, by the presence of small fluctuations in the axial electric field. [9] The eigenvalue/eigenvector problem defined by equation (9) can now be solved. The predicted axis orientation is k = x3, where x3 is the unit eigenvector corresponding to the smallest eigenvalue, l3. For the ideal case, this eigenvalue is exactly zero; in the presence of small fluctuations, dE30 = dE0  k, in the axial electric field, l can be shown to be of order hjdE30 j2i. This result is obtained by multiplying (9) scalarly from the left by k and then making use of (12) in a manner similar to that used in obtaining (13). [10] Once k is known, U0 is calculated from (8) and the frame velocity is obtained as V0 ¼ k  U0 ;

ð15Þ

i.e., V0 is automatically perpendicular to k as desired. Also, the intrinsic axial electric field becomes E0 ¼ hE0  ki ¼ hEi  k þ hBi  ðk  V0 Þ:

ð16Þ

[11] We emphasize that the derivation presented here requires the magnetic variance matrix, MBB, to be invertible. In other words, none of its three eigenvalues can be zero. A second circumstance in which the method fails is when a perfect deHoffmann-Teller (HT) frame exists [e.g., Khrabrov and Sonnerup, 1998a]. In that case, the electric field data provide no information about axis direction. This fact becomes evident by performing the calculations in the HT frame, in which, for the perfect case, we have E0 = 0. It then follows from (10) that every term in the matrix M0 is identically zero. If a good, but not perfect, HT fame is obtained, there will be small leftover electric fields, E0, in the HT frame. If those fields have a well-defined minimum variance direction, as is sometimes the case, then that direction should be close to the optimal axis orientation obtained from the method described here. However, the agreement is usually not exact because the HT calculation is based on the premise that no systematic electric field is present in the HT frame, whereas the new method only requires the axial electric field component to be as constant as the data permit. The velocity, V0, of the optimal proper frame from our new method therefore does not necessarily coincide exactly with the component of VHT perpendicular to the optimal axis.

3. Summary of Method [12] The new method contains the following steps: [13] 1. Calculate the covariance matrices MEB = hdEdBi, MBE = hdBdEi, and MEE = hdEdEi, as well as MBB = hdBdBi

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and its inverse, MBB. Here the d symbol indicates the deviation of an individual measured quantity from its average during the event, e.g., dB = B(m)  hB(m)i. [14] 2. Construct the symmetric matrix M0 = MEB  MBB  MBE + MEE and find its eigenvalues and eigenvectors. Select the eigenvector x3 corresponding to the smallest eigenvalue, l3, as the predictor of the true axial direction, k. Statistical uncertainty estimates are provided by formulas (8.23) in the work of Sonnerup and Scheible [1998]. [15] 3. Calculate the velocity U0 = MBB  MBE  k and check that U0  k is very small compared with jU0j, as should be the case for a good event. [16] 4. Calculate the proper velocity of the structure examined as V0 = k  U0. Also find the intrinsic axial electric field, E0, from (16).

4. Applications [17] In this section, we apply the new method to two flux transfer events (FTEs) that were seen by the Cluster spacecraft on 8 March 2003, in the interval 0700 – 0710 UT, when the spacecraft location had approximate GSE coordinates (7.1, 2.5, 7.4) RE and the spacecraft separation was about 5000 km. An overview of relevant measured quantities is given in Figure 1. Both FTEs have been examined by use of optimal Grad-Shafranov (GS) reconstruction [Hasegawa et al., 2005], which, in addition to maps (transects) of the crosssectional field and plasma structure, produces an optimum axis orientation for each case. Detailed results for a number of FTEs, including transects for the first event (FTE1: 0702:37 – 0703:46 UT), will be presented elsewhere (H. Hasegawa et al., The structure of flux transfer events, manuscript in preparation, 2005) such results for the second event (FTE2: 0707:22 – 0708:27 UT) were given and discussed by Sonnerup et al. [2004]. In both cases, Cluster 1, 2, and 4 (C1, C2, and C4) traversed the FTE near its center, which is desirable for our analysis, while Cluster 3 (C3) was in the magnetosphere, sensing the field and flow draping caused by the FTEs. Since complete plasma measurements are not made by C2 and C4, our present calculations will be based on data from C1. The GS reconstruction showed each FTE to consist of a single, fairly round flux rope embedded in the magnetopause and moving along it at speeds of about 250 km/s. [18] The data actually used in the calculation are velocities from CIS/HIA [Re`me et al., 2001] and magnetic fields from the fluxgate magnetometer (FGM) [Balogh et al., 2001] on board C1. The time intervals are the same as those used for the C1 data entering into the GS reconstruction. The measured (ion) convection electric fields, E(m) = v(m)  B(m), will be used throughout as a proxy for the true field. Note that the method is designed for use with the total electric field. In employing only the convective part, we assume all other terms in the generalized Ohm’s law to give negligible, or constant, axial contributions at all points along the spacecraft trajectory through the structure. We believe this assumption to be valid in the applications presented here. In cases where some of the neglected terms have significant, nonconstant, axial components, the use of only the convection electric field can lead to errors. Accurate triaxial measurements of the full electric field are then needed. Such measurements are not made by Cluster: the E

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Figure 1. Plot of Cluster data during flux transfer events (FTE) on 8 March 2003 (color code: black = C1; red = C2; green = C3; blue = C4). From top to bottom, quantities plotted are ion number density; ion temperature; magnetic field magnitude and GSE components; ion velocity components (GSE). Plasma data are from the CIS/HIA instruments on board C1 and C3; magnetic data from the four fluxgate magnetometer (FGM) experiments. Analysis intervals for the magnetopause traversal and for FTE 1 and FTE 2 are indicated by vertical lines. fields measured by the EFW instruments are in the spacecraft spin plane, with the third component inferred from the assumption E  B = 0. The resulting field incorporates any Hall electric field but not the axial resistive field hj  k (h is the effective resistivity and j the current density). [19] The top part of Table 1 contains a summary of relevant earlier results for the two FTEs, namely, optimal axis, k, from GS, deHoffmann-Teller velocity, VHT, and component of VHT perpendicular to k from C1 data. Vectors are given in terms of their GSE components. [20] The second part of Table 1 shows eigenvalue and eigenvector results from minimum variance analysis of the convection electric field measured by C1 and seen, not in the spacecraft frame but in the HT frame. Since the

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eigenvalue ratio, l2/l3, is reasonably large, the eigenvector x3 (bottom line in the second part) should be fairly close to the true axis direction. Comparison with the optimal axis from GS (top line in first part) does show reasonably good agreement: the angular deviation from the optimal GS axis is 8.4 for FTE 1 and 5.5 for FTE 2. As mentioned already, the results in the middle part suffer from the defect that the components of VHT perpendicular to the axis usually do not give the optimal result for the motion of the FTE structures. Nevertheless, the following fact is remarkable: the small imperfections of the HT fit, i.e., the scatter of the data points around the 45 line in the correlation plots, shown in Figure 2, of the convection electric field components, v(m)  B(m) versus the corresponding components of VHT  B(m), actually contains fairly precise information about the axis orientation for the two FTEs. This fact also indicates the high quality and mutual consistency of the FGM and CIS data from Cluster 1. [21] The third part of Table 1 shows results from our new method. It is seen that the eigenvector x3 (third line in third part) is in excellent agreement with the optimal GS axis (first line in first part): the angle between the two vectors is 3.7 for FTE 1 and 2.7 for FTE 2. The agreement between VHT? (from C1; last line in top part) and the velocity V0 from the new method (line four in bottom part) is also well within errors (angle/magnitude deviations are 2.2/2.5 km/s for FTE 1 and 2.6/4.8 km/s for FTE 2). Because VHT? is not the precise FTE velocity, the velocities V0 in the bottom part should in principle be preferred as predictors of the proper motion of the events. The angle between the axis of FTE 1 and that of FTE 2 is 10.1 for the GS optimal axes and 9.2 for the axes from the new method. [22] Near its bottom, the third part of Table 1 also shows the velocity U0 from equation (8) and its component along the invariant axis, k. This component is seen to be very small (2.2% and 3.2% of jV0j in FTE 1 and FTE 2, respectively), as expected for a good event. The intrinsic, average axial electric field, E0, is seen to be negligibly small (0.0169 mV/m) in FTE 1 but nonnegligible (0.1743 mV/m) in FTE 2. The result for FTE 1 is as expected for a time-invariant 2-D dissipation-free flux rope. As mentioned already, such a rope has a straight, axially oriented field line at its center, along which no electric field can be present. The result for FTE 2 also does not indicate the presence of dissipation. Instead, the nonzero axial field in FTE 2 is likely to be the result of an ongoing slow contraction, along the spacecraft trajectory, of the cross section of the FTE flux rope, with a corresponding slow expansion (one assumes) in the direction perpendicular to the trajectory [Sonnerup et al., 2004]. level of the intrinsic axial [23] The average fluctuation pffiffiffiffiffi field is of the order l3, i.e., 0.06 mV/m for FTE 1 and 0.14 mV/m for FTE 2. Note that l3 is smaller (by 18% for FTE 1 and 25% for FTE 2) for the new method than for the HT based results in the middle part of Table 1. This is a clear indication that the HT method does not give an optimal axis determination. [24] Minimum variance analysis of the magnetic field measured by Cluster 1 during its traversal of the magnetopause, in the interval 0653:11 – 0655:49 UT, gives the magnetopause normal vector n = (0.6444, 0.2446, 0.7245) GSE. The angle between this vector and the optimal GS axis (or the

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Table 1. Comparison of Results From Different Methods for Determination of Invariant Axis, k, and Its Motion, V0 or VHT?a FTE 1 (0702:37 – 0703:46 UT) Symbol

l

k VHT VHT?

k

k V0 U0 U0  k E0

0.2008 0.0670 0.00411

0.1848 0.0235 0.00338

FTE 2 (0707:22 – 0708:27 UT) l

Components GSE

Components GSE

Optimal GS/HT Analysis (0.4732, 0.6430, +0.6021) (226.6, +116.8, +149.6) (168.8, +195.4, +76.0)

(0.3296, 0.7434, +0.5820) (211.4, +109.6+151.2)b (186.3, +166.2, +106.9)

Min Var. of E in HT Frame (+0.2769,+0.6105, +0.7420) 0.6278 (0.7850, +0.5890, 0.1917) 0.1871 (0.5541, 0.5294, +0.6424) 0.0275

(+0.4048, +0.3833, +0.8302) (0.8142, +0.5643, +0.1365) (0.4162,0.7312, +0.5405)

New Method (+0.2755, +0.6007, +0.7505) (0.8458, +0.5225, 0.1077) (0.4568, 0.6051, +0.6521) (166.4, +197.5, +66.6) (166.4, 74.5, 194.7)

(+0.1239, +0.5865, +0.8004) (0.9208, +0.3686, 0.1275) (0.3698,0.7212, +0.5857) (176.4,+172.6, +101.2) (170.9, 59.7, 196.1)

5.9 0.0169

0.3371 0.1335 0.0205 8.6 0.1743

a Vector components are GSE; velocities, VHT and U0, are in km/s; electric fields, E0, are in mV/m, and eigenvalues, l, are in (mV/m)2. Optimal GS reconstruction uses data from all four Cluster spacecraft. All other results are based on C1 alone. b The result VHT = (233.9, +50.8, +166.1) km/s, quoted by Sonnerup et al. [2004] for FTE 2, was derived from combined C1 and C3 data.

axis from the new method) is 91.5 (or 88.3) for FTE 1 and 88.4 (or 89.4) for FTE 2. These results are consistent with the expectation that the axis should be tangential to the magnetopause surface, although it must be remembered that n itself has an angular uncertainty, the purely statistical part of which is about ±3.5. Note that the interval used for the determination of n was chosen to exclude the partial return of B to its magnetospheric state that follows immediately after the interval and that shows evidence of reconnectionassociated plasma jetting (see Figure 1). [25] The component of VHT?(V0) along n is 5.9(10.7) km/s for FTE 1 and 2.0(+1.9) km/s for FTE 2. A negative velocity component along n means that the magnetopause is

moving earthward, which is consistent with the transition from magnetosphere to magnetosheath conditions seen by C1, C2, and C4 in the interval 0653 – 0656 UT. The optimal GS maps (see Sonnerup et al. [2004] for the FTE 2 map) indicate that the magnetopause motion was small during the two FTEs. Another indication of this is that C3 remained in the magnetosphere during most of the 15-min period following the transition to magnetosheath conditions seen by the other three spacecraft (see Figure 1). [26] Figure 3 shows polar plots of the axis orientations for the two FTEs from optimal GS analysis, from minimum variance analysis of the leftover electric field in the HT frame, and from our new method. For FTE 2 we also show

Figure 2. DeHoffmann-Teller (HT) scatter plot for FTE 1 (left) and FTE 2 (right). The plots show a component-by-component (GSE) comparison of individual measured values of the convection electric field, Ec = v(m)  B(m), and values of the corresponding electric fields, EHT = VHT  B(m). 5 of 7

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Figure 3. Polar plot of axis directions for FTE 1 (left) and FTE 2 (right). The ‘‘bull’s eye’’ represents the vector n  (k  n), where n is the magnetopause normal from minimum variance analysis of the magnetic field measured by C1 in the interval 0653:11 –0655:49 UT and k is the axis direction from optimal GS analysis. Concentric circles about the bull’s eye show angular deviations of 3, 6, 9, and 12. The optimal GS axis is denoted by a dot while the axis from the new method is shown by the multiplication sign and the axis from minimum variance analysis of the electric field in the HT frame by the plus sign. The open square represents the axis derived from remote sensing [Khrabrov and Sonnerup, 1998b] of FTE 2 by C3. Statistical error ellipses are from equation 8.23 in the work of Sonnerup and Scheible [1998].

the axis obtained from ‘‘remote sensing’’ of the FTE by C3 [Khrabrov and Sonnerup, 1998b; Sonnerup et al., 2004]. For FTE 1, the field perturbations at C3 were too small for the successful application of the remote sensing method. The ‘‘bull’s eye’’ point in each plot represents the orientation of the vector n  (k  n) with k from the optimal GS calculation. The upward direction in the plot is along n. It is seen that the axes from the various methods are clustered around the bull’s eye, with the optimal GS axis and the axis from our new method showing the smallest deviations. Ellipses in the figure represent estimates of the purely statistical errors from Sonnerup and Scheible [1998]. Possible systematic errors are not included. Commensurate estimates for the optimal GS axis orientation or for the axis from remote sensing have yet to be developed. The sizes of the ellipses shown are sufficiently large so that the excellent agreement of the axis from the new method with the optimal GS axis and with the bull’s eye could in principle be a fortuitous coincidence. However, the consistency of the results for both FTEs and the fact that the axis is nearly perpendicular to the calculated magnetopause normal vector in both cases speak against such a conclusion. More experience with the new method is needed in order to obtain a reliable assessment of its accuracy.

5. Discussion [27] The results presented in this paper can be summarized as follows:

[28] 1. Following the proposal by Khrabov and Sonnerup [1998a], we have developed a new, least-squares method for determination of the invariant axis, k, motion, V0, and intrinsic axial electric field, E0, of certain approximately coherent and two-dimensional plasma/field structures moving past a single observing spacecraft. The method is based on Faraday’s law, which requires the axial electric field to be constant in any frame of reference moving with such a structure. This is the intrinsic axial electric field associated with the structure. If nonzero, it would strictly speaking be caused by nonideal effects such as resistivity, although in practice it may also be the result of intrinsic time variations (see item 5 below). In 2-D reconnection configurations, resistive dissipation is thought to be confined to a small ‘‘diffusion’’ region surrounding the X point in the magnetic field. Outside of this region the electric field is mainly convective. As long as the spacecraft trajectory is near but does not traverse the diffusion region, our new method can in principle be used with the convective electric field to produce the reconnection field, E0. [29] 2. The new method, the steps of which are summarized in section 3, is restricted to events for which no perfect deHoffmann-Teller (HT) frame of reference exists and for which (in the present version) the magnetic variance matrix is invertible. The unlikely case where this matrix is not invertible requires separate investigation. A perfect HT frame is also essentially never found in practice but one must not conclude from this fact that our new method almost always works. Successful application (see item 4

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below) requires that the electric field remnant in the HT frame have a well-defined minimum variance direction, which, judging from our experience to date, may be the exception rather than the rule. [30] 3. The method has been tested by application to two flux transfer events (FTEs) encountered by Cluster. These applications indicate that for some events this single-spacecraft method can give results that are in excellent agreement with those obtained from the optimal Grad-Shafranov (GS) method [Hasegawa et al., 2005; Sonnerup et al., 2004], which is based on measurements by all four Cluster spacecraft. One can infer from this fact that the model assumptions were well satisfied. The agreement between the multispacecraft method and the new single-spacecraft method also provides persuasive evidence that the axis determination is accurate for these events. However, we have found that the results are usually of low quality for ordinary magnetopause traversals, in which no FTE structures are embedded. It has also failed for some of the FTEs we have examined. The new method must therefore be used with caution. The smallness of the velocity component jk  U0j from (14) and of the smallest eigenvalue, l3, from (9), along with the smallness of the eigenvalue ratio l3/l2, are quantitative indications of the quality of the results. [31] 4. The new method is closely related to a simpler one, in which the electric fields remaining in the deHoffmannTeller (HT) frame are used in the usual minimum variance process to find the invariant axis as the direction in which the corresponding electric field component (in the HT frame) is as nearly constant as the data permit. However, we have argued that the new method in principle will give more accurate results; comparison with the optimal GS axes shows this to be the case, at least for the two FTEs examined here (see Figure 3). [32] 5. In FTE 1, the intrinsic axial electric field is zero, as expected for a dissipation-free structure. The presence of a significant intrinsic axial electric field in FTE 2 also does not indicate the presence of dissipation or other deviations from ideal MHD. Instead, it is likely to be the result of time variations caused by ongoing deformation of the cross section of the structure.

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[33] Acknowledgments. We thank A. Balogh for use of FGM data and H. Re`me for use of CIS data from the Cluster misssion. The research was supported by National Aeronautics and Space Administration under grants NAG5-12005 and NNG05GG26G to Dartmouth College. [34] Arthur Richmond thanks Christopher J. Owen and another reviewer for their assistance in evaluating this paper.

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H. Hasegawa, Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Dokayama, Meguro, Tokyo, 152-8551, Japan. ([email protected]) ¨ . Sonnerup, Thayer School of Engineering, Dartmouth College, B. U. O 8000 Cummings Hall, Hanover, NH 03755, USA. (bengt.u.o.sonnerup@ dartmouth.edu)

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