Neutral sheet normal direction determination

Transcription

1 Advances in Space Research 36 (2005) Neutral sheet normal direction determination T.L. Zhang a, *, W. Baumjohann a, R. Nakamura a, M. Volwerk a, A. Runov a,z.vörös a, K.-H. Glassmeier b, A. Balogh c a Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, Graz 8042, Austria b IGM, Technische Universität Braunschweig, Germany c Imperial College, London, UK Received 14 October 2002; received in revised form 26 August 2004; accepted 26 August 2004 Abstract One of the important parameters in magnetotail studies is the thickness of the current sheet. The determination of this thickness is subject to many assumptions of the geometry, such as the current sheet normal direction. The minimum variance analysis (MVA) has been widely used in studies of the current sheet in the tail or at the magnetopause with single-satellite magnetometer data. With four Cluster spacecraft, we are able to determine the current sheet normal direction by using the crossing timings. In this study, we perform MVA on selected Cluster neutral sheet crossing cases to determine the normal directions. We compare these MVA normal estimates with the timing-derived normals. We find that both MVA and four spacecraft timing analysis give useful data. Published by Elsevier Ltd on behalf of COSPAR. Keywords: Neutral sheet; Magnetotail studies; Cluster 1. Introduction The study of the neutral sheet is of fundamental importance in understanding the physical processes involved in the solar wind interaction with the magnetosphere, because the dynamics of the EarthÕs magnetosphere are greatly influenced by physical processes that occur near the neutral sheet. The neutral sheet is a relatively narrow region within the plasma sheet, where the X component of the magnetic field, measured along the Sun Earth axis, reverses sign and the magnetic field intensity reaches a minimum. This neutral sheet separates the magnetotail into two adjacent hemispheres which have opposite magnetic field polarities. One of the key problems in studying the magnetotail current sheets configuration is the determination of the * Corresponding author. address: tielong.zhang@oeaw.ac.at (T.L. Zhang). current sheet thickness. The minimum variance analysis (MVA), pioneered for space magnetic field applications by Sonnerup and Cahill (1967), has been widely used in determination of the normal direction of the the current sheet in the tail (e.g., Sergeev et al., 1993) or at the magnetopause. When a single spacecraft transverses through a one-dimensional structure, such as the current layer, the variations in magnetic field along that normal direction are zero (or at least minimized). Thus, MVA allows us to make an estimate of the normal to the simple planar current sheet from a set of magnetic field data measured by a single spacecraft. It is known that the accuracy of MVA might be affected by various factors (Dunlop and Woodward, 1998; Sonnerup and Scheible, 1998). Thus, it is interesting to compare the MVA with another independent method in determining the neutral sheet normal. The four spacecraft Cluster mission enables us to do so. Using measurements at four non-coplanar and well-separated spacecraft, the neutral sheet orientation and speed along /$30. Published by Elsevier Ltd on behalf of COSPAR. doi: /j.asr

2 T.L. Zhang et al. / Advances in Space Research 36 (2005) the neutral sheet normal can be determined uniquely (Russell et al., 1983). Here, we assume the neutral sheet speed is constant over the region including the observations and the neutral sheet is planar over the separation scale of the spacecraft. We first find out the times and locations of the neutral sheet crossing, i.e., B x =0, by all four spacecraft. Then with four times and four position vectors, the normal of the neutral sheet and the speed along the normal can be calculated. We note that possible errors for these assumptions may arise due to curvature of the discontinuity, acceleration of the surface. It is the purpose of this study to analyze a number of crossings of the neutral sheet to determine the normal direction with MVA methods and timing method. Then, we compare these timing-derived normals with MVA estimates, and show that they are can be significantly different. 2. Observations For our investigation, we examine the magnetic field data measured by FGM instrument (Balogh et al., 2001) from July to October 2001 when the Cluster satellites were located within the magnetotail and crossings of the neutral sheet were observed. In this section, we will present three neutral sheet crossing cases. These three crossings were selected for their well-defined neutral sheet crossings for all four Cluster spacecraft. Before we go on with our case study, we will first estimate how sensitive the timing normal are to uncertainties in the times at which each spacecraft crosses each boundary. In this study, we restrict ourselves to 4-s spin average data. With such a data resolution, we easily identify the neutral sheet crossing, i.e., B x = 0, with a time accuracy of 2 s. If we take a neutral sheet moving speed of 15 km/s, this will cause a possible uncertainty of 30 km in spacecraft position. This 30 km uncertainty is negligible comparing with 2000 km Cluster spacecraft separation in the tail during In other words, the timing derived normals are well-defined in this study. Here, we will apply the MVA procedure to the magnetic field data of each spacecraft separately. Furthermore, we perform MVA on nested sets of data intervals (2; 4; 6; 8; 10; 12 min), centered at the middle of the current sheet, i.e., B x = 0. Time stationarity can be checked in this manner. When the normal vectors are strictly time stationary, then the results from all the different nested segments should be the same Case 1: September 07, 2001, 1930 UT Fig. 1 shows the magnetic field data with the neutral sheet crossing at 1930 on September 07, The data are shown in GSM Cartesian coordinates with a standard line scheme of solid line, dotted line, dashed line, and dash-dotted line for spacecraft 1 4, respectively. Fig s Magnetic field data showing two neutral sheet crossings at 1930 and 2100 on September 07, The data are shown in GSM coordinates with a regular line styles of: solid line, dotted line, dashed line, and dash-dotted line for spacecraft 1 4, respectively.

3 1942 T.L. Zhang et al. / Advances in Space Research 36 (2005) Table 1 September 07, 2001, 1930 UT: the minimum variation directions determined from all 4 S/C with different time intervals S/C X 3 Y 3 Z 3 k 2 /k 3 Cl a b c d e f C2 a b c d e f C3 a b c d e f C4 a b c d e f The ratio of the intermediate to the minimum eigenvalues, k 2 /k 3 is also given. All Cluster spacecraft measure similar magnetic profiles during the interval. Table 1 shows the minimum variance directions determined from all 4 S/C with nested time intervals: (a) 2 min; (b) 4 min; (c) 6 min; (d) 8 min; (e) 10 min; (f) 12 min, center at neutral sheet crossing. The ratio of the intermediate to the minimum eigenvalues, k 2 /k 3, is also given in this table. The large values of the ratio indicate that the normals to the discontinuities are well-defined. We see that the minimum variance analysis for the magnetic field data yielded very good consistent normal direction for various time intervals and spacecraft. In comparison, we determine the neutral sheet normal direction with the four spacecraft crossing times. We find that the normal vector is [0.137, 0.296, 0.945]. In Fig. 2, the normals of the neutral sheet, determined from both MVA and timing methods, are shown in three projections. Small dots show the minimum variation directions determined from nested time intervals for different Cluster S/C (data from Table 1). The cross is the average for all the data from minimum variance analysis with value of [0.038, 0.911, 0.406]. The stability of the normal orientation determined from MVA is striking. Further considering the good contrast of the eigenvalues, k 2 /k 3, we are supposed to have a very well-defined normal direction with the MVA method. Nevertheless, when we check time sequences of the neutral sheet crossings by considering the Cluster spacecraft tetrahedron geometry (figure not shown here), we find that the minimum variance direction does not represent the neutral sheet normal in this case. The asterisk, shown in Fig. 2, reveals that the normal direction determined from crossing timing deviates much from the minimum variance direction. We note that the intermediate variance vector is in the direction of [0.068, 0.406, 0.896], which is less deviated from the normal direction. It is likely that small field fluctuations within the current sheet, visible in the B z component around 1930 UT in Fig. 1, can shift the minimum variance eigenvector direction, which might explain the disagreement. However, at first sight this neutral sheet crossing seems to be a good candidate for MVA method. In addition, the contrast of the eigenvalues, k 2 /k 3, and time-stationarity test with nested time intervals, are further evident for believable normal direction from MVA. This case illustrates that caution must be taken when one use the minimum variance vector as neutral sheet normal direction. Fig. 2. Neutral sheet normal direction for the crossing of 1930 UT September 07, Small dots show the minimum variation directions determined from nested time intervals for different Cluster S/C (data from Table 1). The cross is the average for all the data from minimum variance analysis. In comparison, the asterisk shows the direction determined from the neutral sheet crossing timings.

4 T.L. Zhang et al. / Advances in Space Research 36 (2005) Table 2 September 07, 2001, 2100 UT : the minimum variation directions determined from all 4 S/C with different time intervals S/C X 3 Y 3 Z 3 k 2 /k 3 Cl a b c d e f C2 a b c d e f C3 a b c d e f C4 a b c d e f The ratio of the intermediate to the minimum eigenvalues, k 2 /k 3, is also given Case 2: September 07, 2001, 2100 UT Fig s Magnetic field data showing the neutral sheet crossings at 2255 UT on September 14, The data are shown in GSM coordinates with a regular line styles of: solid line, dotted line, dashed line, and dash-dotted line for spacecraft 1 4, respectively. The magnetic field data for the Case 2 are shown in Fig. 1. Like in Case 1, we perform the MVA procedure to the nested sets of magnetic field data intervals for each spacecraft. The minimum variance vectors are listed in Table 2. The ratio of the intermediate to the minimum eigenvalues, k 2 /k 3, indicates that the normals are well-defined with an averaged direction of [0.003, 0.134, 0.989]. The time-stationarity test with nested data intervals indicates the stability of the normal direction, although satellites crossed somewhat different parts of the current sheet. Using the four spacecraft neutral sheet crossing timings, we determine the neutral sheet direction of [0.053, 0.142, 0.989]. We show in Fig. 3 the three projections of the minimum variance directions and the normal direction determined by timings. Experience indicates that if the results are essentially the same for several neighboring ÔnestedÕ data segments, they are perhaps believable. We find good agreement between the MVA and Fig. 3. Same as Fig. 2, but for the crossing of 2100 UT September 07, 2001.

5 1944 T.L. Zhang et al. / Advances in Space Research 36 (2005) Table 3 Same as Tables 1 and 2, but for the crossing of September 14, 2001 S/C X 3 Y 3 Z 3 k 2 /k 3 Cl a b c d e f C2 a b c d e f C3 a b c d e f C4 a b c d e f timing method in determining the neutral sheet normal in this case Case 3: September 14, 2001, 2255 UT In Fig. 4, the magnetic field data of a fast neutral sheet crossing is presented. Using four spacecraft neutral sheet crossing timings, we found that the neutral sheet is extremely tilted with a direction of [ 0.086, 0.979, 0.185]. Shown in Table 3, the minimum variance analysis yielded widely differing normals for the various neutral sheet crossing at various spacecraft. Thus, the normal direction is not time-stationary. The ratio of the intermediate to the minimum eigenvalues, k 2 /k 3, indicates less well-defined normal direction. Especially for the Cluster 3, k 2 and k 3 are nearly the same, indicating the uncertainty in the corresponding eigenvectors is large with respect to rotation about the remaining eigenvector, i.e., the maximum variance. This rotation about the maximum variance is clearly illustrated in Fig Concluding remarks The neutral sheet can be considered as a rotational discontinuity. However, to determine the normal direction with the MVA method is not a easy task. Although the maximum variance direction, i.e., the tail-aligned direction due to the cross-tail currents, is well-defined and very steady, the variance in one of the tangential component, the cross-tail direction, can be very small. According to the usual practice, a result is said to be significant if the intermediate variance eigenvalue k 2, is one order higher than the minimum variance eigenvalue k 3. Furthermore, if the results are essentially identical for nested set of data intervals, they are perhaps believable. Using four Cluster spacecraft magnetic field measurement, we determine the neutral sheet normal direction and compare with the normal direction derived from MVA. Indeed, in Case 3, we show that when the eigenvalue ratio, k 2 /k 3, is poor and the normal direction is not stationary, an erroneous direction of the neutral sheet might be resulted from the MVA. In contrast, we show in Case 2 the well-defined neutral sheet normal directions from the MVA, in agreement with the timing method. The Case 1 is somehow particular intriguing, The large values of the eigenvalue ratio, k 2 /k 3, indicate that the normals are well-defined. In addition, time-stationarity test with nested data intervals, are further evident for believable normal direction from MVA. However, the small field fluctuations within the current sheet could give an erroneous direction of the neutral sheet using the minimum variance analysis. Fig. 5. Same as Fig. 3, but for the crossing of September 14, The minimum variance analysis yielded widely differing normals for the various neutral sheet crossing at various spacecraft. Thus, the normal direction is not time-stationary.

6 T.L. Zhang et al. / Advances in Space Research 36 (2005) The choice of method may depend on the particular case. Ideally, different methods should be compared with one another. We found that the results from MVA and timings agree for some neutral sheet crossings and disagree for others. We emphasis that caution make be taken when one use the minimum variance vector as neutral sheet normal direction. Acknowledgments The work by K.H.G. was financially supported by the Bundesministerium für Bildung und Forschung and the Zentrum für Luft und Raumfahrt under Contract 50 OC References Balogh, A., Carr, C.M., Acuna, M.H., Dunlop, M.W., Beek, T.J., Brown, P., Fornacon, K.H., Georgescu, E., Glassmeier, K.öH., Harris, J., Musmann, G., Oddy, T., Schwingenschuh, K. The Cluster magnetic field investigation: overview of in-flight performance and initial results. Ann. Geophys. 19, , Dunlop, M.W., Woodward, T.I. Multi-spacecraft discontinuity analysis: orientation and motion. in: Paschmann, G., Daly, P.W. (Eds.), Analysis Methods for Multi-Spacecraft Data. The International Space Science Institute, Bern, Switzerland, pp , Russell, C.T., Mellott, M.M., Smith, E.J., King, J.H. Multiple spacecraft observations of interplanetary shocks: four spacecraft determination of shock normals. J. Geophys. Res. 88, , Sergeev, V.A., Mitchell, D.G., Russell, C.T., Williams, D.J. Structure of the tail plasma/current sheet at 11R e and its changes in the course of a substorm. J. Geophys. Res. 98, , Sonnerup, B.U.Ö., Cahill, L.J. Magnetopause structure and attitude from Explorer 12 observations. J. Geophys. Res. 72, , Sonnerup, B.U.Ö., Scheible, M. Minimum and maximum variance analysis. in: Paschmann, G., Daly, P.W. (Eds.), Analysis Methods for Multi-spacecraft Data. The International Space Science Institute, Bern, Switzerland, pp , 1998.