Technique for diagnosing the flapping motion of ... - Wiley Online Library

PUBLICATIONS Journal of Geophysical Research: Space Physics RESEARCH ARTICLE 10.1002/2014JA020973 Key Points: • Technique is developed based on minimum variance analysis • It is able to diagnose flapping type and propagation of kink-like flapping • It might be invalid when the sheet surface is nearly horizontal or vertical

Correspondence to: Z. J. Rong, [email protected]

Citation: Rong, Z. J., S. Barabash, G. Stenberg, Y. Futaana, T. L. Zhang, W. X. Wan, Y. Wei, and X.-D. Wang (2015), Technique for diagnosing the flapping motion of magnetotail current sheets based on single-point magnetic field analysis, J. Geophys. Res. Space Physics, 120, 3462–3474, doi:10.1002/2014JA020973. Received 29 DEC 2014 Accepted 7 APR 2015 Accepted article online 9 APR 2015 Published online 7 MAY 2015

Technique for diagnosing the flapping motion of magnetotail current sheets based on single-point magnetic field analysis Z. J. Rong1,2,3, S. Barabash3, G. Stenberg3, Y. Futaana3, T. L. Zhang4, W. X. Wan1, Y. Wei1, and X.-D. Wang3 1 Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China, 2Beijing National Observatory of Space Environment, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China, 3Swedish Institute of Space Physics, Kiruna, Sweden, 4Space Research Institute, Austrian Academy of Sciences, Graz, Austria

Abstract The magnetotail current sheet is active and often flaps back and forth. Knowledge about the flapping motion of current sheet is essential to explore the related magnetotail dynamic processes, e.g., plasma instabilities. Due to the inability of single-point measurements to separate the spatial-temporal variation of magnetic field, the moving velocity of flapping current sheets cannot be revealed generally until the multipoint measurements are available, e.g., the Cluster mission. Therefore, currently, the flapping behaviors are hard to be resolved only relying on single-point magnetic field analysis. In this study, with minimum variance analysis, we develop a technique based on single-point magnetic field measurement to qualitatively diagnose the flapping properties including the flapping type and the traveling direction of kink-like flapping. The comparison with Cluster multipoint analysis via several case studies demonstrates that this technique is applicable; it should, however, be used with caution especially when the local sheet surface is either quasi-horizontal, or quasi-vertical. This technique will be useful for the planetary magnetotail exploration where no multipoint observations are available. 1. Introduction Numerous earlier studies noticed that the Earth’s magnetotail current sheet (CS) often flaps back and forth, which leads to the multiple local CS crossings being recorded by spacecraft (S/C) [e.g., Speiser and Ness, 1967; Toichi and Miyazaki, 1976; Lui et al., 1978; Sergeev et al., 1998]. The flapping motion of CS, as an exhibition of released energy, is attracting researchers’ attention, particularly after the launch of Cluster mission [Escoubet et al., 2001]. With Cluster multipoint analysis of flapping CS, many studies unambiguously revealed that these local flapping motions, being triggered by impulsive sources in the midnight region, are able to propagate as kink-like waves toward both flanks with velocities of a few tens of kilometers per second [e.g., Zhang et al., 2002, 2005a; Sergeev et al., 2003, 2004; Runov et al., 2005; Petrukovich et al., 2006; Shen et al., 2008; Rong et al., 2010]. It is unclear which mechanisms drive the kinklike waves, although several explanations have been proposed, such as penetration of Alfvénic fluctuations of interplanetary magnetic field [Toichi and Miyazaki, 1976], induction of hemispheric asymmetric plasma flow [Malova et al., 2007], response to enhanced solar wind disturbance [Shen et al., 2008; Forsyth et al., 2009], MHD ballooning-type waves [Golovchanskaya and Maltsev, 2005], drift kink mode of current sheet instability [e.g., Karimabadi et al., 2003a, 2003b; Sitnov et al., 2006; Zelenyi et al., 2009], and MHD waves related to a double-gradient current sheet model [Erkaev et al., 2007]. Readers are referred to the review of Sharma et al. [2008] for more details. The interaction of solar wind with other planets can also result in a magnetotail and a tail current sheet [Blanc et al., 2005, and references therein; Jackman et al., 2014]. For the planets with intrinsic magnetic fields, e.g., Mercury, Jupiter, and Saturn, a terrestrial-like magnetotail is formed by the interaction of the solar wind with a planetary dipole field. For the unmagnetized planets, e.g., Venus and Mars, the direct interaction of solar wind with the planetary ionosphere generates an “induced” magnetotail [Bertucci et al., 2011].

©2015. American Geophysical Union. All Rights Reserved.

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The knowledge about the flapping motion of Earth’s magnetotail current sheet has expanded significantly in the past decade, but very little is still known about the other planets. No one knows whether magnetotail

DIAGNOSE FLAPPING CURRENT SHEET

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flapping at another planet would show similar behavior as that found in the Earth’s magnetotail, although the flapping motion at Venusian magnetotail [Dubinin et al., 2012] has been noticed previously. To conduct comparative studies of the flapping motion could help to reveal the physical processes controlling the dynamics of magnetotail current sheet. The four-point magnetic field measurements made by the Cluster spacecraft can be used to calculate the normal orientation and the velocity of a discontinuity so that the flapping properties of CS can be resolved. This is the reason why studies about the Earth’s magnetotail flapping motion bloom after the launch of the Cluster mission. In contrast to the multipoint measurements of the Cluster mission, most current planetary satellite missions rely on single-point measurements. Therefore, to study planetary magnetotail CS, the major challenge is how to use the single-point magnetic field measurement to resolve the properties of the flapping motion. With single-point magnetic field measurements, the normal direction of a local CS can be estimated by minimum variance analysis (MVA) [Sonnerup and Scheible, 1998]. When kink-like flapping waves are passing, the S/C would sense a deviation of the CS normal from the nominal direction. By checking the variation of the CS normal direction with MVA on several cases, Volwerk et al. [2013] recently argued that the Kronian and Jovian magnetotail current sheets may flap as kink-like waves. However, the flapping type (the multiple crossings can either be generated by a propagating kink-like oscillation of the current sheet or by another mechanism that makes the spacecraft cross the current sheet repeatedly) and the traveling direction of the kink-like waves are not clarified in their study. Being motivated by the study of Volwerk et al. [2013], we try to develop a technique based on single-point magnetic field measurement to diagnose the general properties of the flapping motion including the flapping types and the traveling direction of kink-like waves, although the traveling velocity of the waves is still unresolved. In section 2, we will review the different flapping types and present the methodology of this technique. In section 3, to test the technique, we apply it to several cases and compare the results with multipoint analysis.

2. Methodology 2.1. Flapping Types Without losing generality, we use the Earth’s magnetotail CS to introduce the methodology. For given local orthogonal coordinates, we assume that the undisturbed CS (the undisturbed means flapping motion is absent) nominally lies in x-y plane and +x direction is along the lobe field pointing earthward. As shown in Figure 1, the CS flapping basically consists of two types, i.e., the steady flapping (Figure 1a) and the kinklike flapping (Figures 1b and 1c). For the steady flapping, the S/C just crosses the same sheet repeatedly, the CS normal and the direction of cross-tail current density are basically the same at each crossing, and the flapping does not propagate as waves or it flaps as stationary waves [Rong et al., 2010]. One should note that the diurnal variation of dipole tilt angle can drive the whole CS to flap up and down [Tsyganenko and Fairfield, 2004], but the period with diurnal scale cannot make it the rapid flapping processes we focused on. In contrast, the kink-like flapping is able to propagate as waves. It may propagate along the y axis, in which case the CS normals for neighboring crossings vary significantly in y-z plane (Figure 1b), or may propagate along the x axis, so that the normals vary in x-z plane (Figure 1c). All three flapping types can yield the similar time series of the Bx component. The kink-like flapping propagating along x axis was suggested in earlier studies on Earth’s magnetotail [Speiser, 1973; Fairfield et al., 1981; Yamauchi and Lui, 1997], and it was discussed for a standing wave by Volwerk et al. [2003] in one case study observed by Cluster. However, more Cluster observations verified that the kink-like flapping basically propagates along y axis instead [e.g., Zhang et al., 2002; Sergeev et al., 2003, 2004]. Observations also demonstrated that the flapping sequence might not be induced solely by the kink-like waves; that is, sometimes it comprises the contribution from both steady flapping and kinklike flapping along y axis [e.g., Rong et al., 2010, Figure 14]. In the following subsections, we will try to present a single-point technique to diagnose the flapping type and determine the traveling direction of kink-like waves as well.

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a

10.1002/2014JA020973

n

Steady flapping

j

z

y

b

Kink-like flapping

x

j

j n

n

Bx

n

t

z

c

Kink-like flapping n

n

n

x

y Figure 1. Three different types of flapping motion of a current sheet. In each panel, the dashed line represents the current sheet center, the green line is the relative trajectory of spacecraft to crossing the current sheet, “n” is the normal orientation of the local CS. (a) The steady flapping and (b) the kink-like flapping, respectively, in the y-z plane, and (c) the kink-like waves that propagate along the y axis are shown. In both panels, the By component is assumed to be zero for simplicity, and the cross-tail current density is indicated with a red arrow. Figure 1c shows the type of kink-like flapping, which propagates along x axis. All the types of flapping motion can yield a similar time series of the Bx component.

2.2. Technique To diagnose the flapping motion of a CS, knowledge about the CS orientation is essential. For the single-point magnetic field measurement, the normal direction of the CS can be estimated by minimum variance analysis (MVA) [Sonnerup and Scheible, 1998]. Considering the boundary condition B1n = B2n at the magnetic discontinuity (because of ∇ B = 0), where B1n and B2n are the normal components of magnetic fields at ^ can be determined by minimization of both sides of discontinuity, the normal direction n σ2 ¼ where hBi ¼ N1

N X

N 1X ^ j2 jðBi hBiÞn N i¼1

(1)

Bi and i = 1, 2, 3…N. N is the number of data points. From equation (1), the normal direction

i¼1

^ g (i = 1, 2, 3…N) has ^ is identified as the direction in space along which the field-component set fBi n n minimum variance. After mathematical arrangements, the solution of equation (1) is reduced to finding the eigenvectors of the magnetic variance matrix Mμν ¼ Bμ Bν Bμ hBν i

(2)

where the subscripts μ and ν = 1, 2, and 3 denote the x, y, and z components in a given Cartesian coordinate system. The matrix Mμν has three eigenvalues: λ1, λ2, and λ3 (λ1 ≥ λ2 ≥ λ3 ≥ 0), and the corresponding eigenvectors are x1, x2, and x3. The three eigenvectors are orthogonal and represent the directions of maximum, intermediate, and minimum variance of the field. According to the MVA theory [Sonnerup and Scheible, 1998], the angular error or uncertainty of the eigenvectors can be estimated as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u λ3 λi þ λj λ3 Δφ ¼ Δφ ¼ t 2 ij ji ðN 1Þ λi λj

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(3)

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a

d

+Bx

+Bx

-Bx

-Bx

b

e

+Bx

+Bx

z y

-Bx

-Bx

x

c

nz

nz

t, or -ny

f

t, or -ny

Figure 2. (a–f) Propagation of kink-like flapping in y-z plane. The magnetic field By component in flapping CS is assumed to be zero. The black lines with double arrows at each panel represent the normal orientation yielded from MVA. The green line with an arrow represents the relative path of the spacecraft to the flapping current sheet. The red arrow represents the traveling direction. Figures 2a and 2b or Figures 2d and 2e would yield a similar time series of normal orientations as that shown in Figure 2c or Figure 2f, although the propagation direction is opposite.

The |Δφij| denotes the expected angular uncertainty of eigenvector xi for rotation toward or away from eigenvector xj. The eigenvectors, x1, x2, and x3 are usually written as the letters L, M, and N, respectively. Taking the magnetotail CS as an example, the maximum variance direction, L, is basically along the local ^ of CS, and the lobe field; the minimum variance direction, N, is seen as the normal direction n intermediate variance direction, M, is along the dawn-dusk direction. The quality of normal calculation can be judged by the ratio λ2/λ3. The larger the ratio is, the more accurate the yielded N becomes. Note that the vectors with direction parallel or antiparallel to the calculated eigenvectors are also the valid eigenvectors; thus, both N and N are valid normal directions in terms of MVA. The normal direction yielded from MVA can be wrong sometimes even if λ2 ≫ λ3 [Zhang et al., 2005b]. We will discuss this later; for now, we assume that the normal direction yielded by MVA is absolutely accurate for the technique development. ^ for The flapping types in Earth’s magnetotail can now be investigated. For the steady flapping, the derived n the neighboring CS crossings should show the similar orientation, i.e., quasi parallel or antiparallel to each ^ would other. For the case of kink-like flapping traveling along y axis (x axis), the orientation of n alternately show significant variation in y-z (x-z) plane with minor nx (ny) component. As an example to show the technical procedures, we will investigate the kink-like flapping motion, which is assumed to propagate along y axis. In this case, the x component of normal equals zero, i.e., nx~0. As sketched in Figure 2, in the local coordinates, the spacecraft is crossing a sequence of flapping kink-like waves, which either propagates along y direction (Figures 2a and 2e) or along +y direction (Figures 2b and 2d).The traveling direction is indicated by the red arrows, and the magnetic field By component in flapping CS is ^ and its reverse direction -n ^ are assumed to be zero for simplicity. At each crossing, the normal direction n ^ and -n ^ are valid normal directions from MVA analysis. plotted as the double arrow line, because both n Figures 2a and 2b would yield a similar time series of normal orientations (Figure 2c), although the

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z y a

x

b

-Bx +Bx +Bx -Bx

c

+Bx -Bx

Quasi-horizontal

Quasi-vertical

|nz|>>|ny|

|ny|>>|nz|

+Bx

Overshoot

Figure 3. The invalid cases for this technique. The black line in each panel represents the CS surface, while the line with double arrows in each panel represents the normal orientation yielded from MVA. (a and b) The cases when CS surface is quasi-horizontal (|nz| ≫ |ny|) and quasi-vertical (|ny| ≫ |nz|), respectively, are shown. (c) The case when the front of kink-like waves shows overshoot, where the traveling direction toward y direction is assumed as indicated by the red arrow.

traveling direction at both panels is the opposite. The same is true for Figures 2d and 2e. Thus, it is impossible to determine the traveling direction only relying on a time series of normal orientations, and we have to consider other information. We should remind the readers that based on the time difference of each S/C crossing the discontinuity, the timing analysis of the Cluster tetrahedron [Harvey, 1998] is able to diagnose the normal direction as the moving direction of planar discontinuity relative to tetrahedron, and thus, the CS flapping motion can be diagnosed directly from the time series of normal orientations with timing analysis. It is noticed that the order of the Bx polarity change is different for Figures 2a and 2b. If we designate the polarity variation from Bx to +Bx as ΔBx > 0 while variation from +Bx to Bx as ΔBx < 0, then a parameter k can be defined as k ¼ sign ny nz signðΔBx Þ (4) ^, respectively. It is easy to verify that for either case in Figure 2, if where ny and nz are the y and z components of n the kink-like flapping is propagating toward /+ direction, the yielded value of k at each crossing would be +1/ 1 always; that is, the value of k can be used to represent the traveling direction of kink-like flapping. In contrast, if the S/C just experiences steady flapping in the y-z plane (nx is always minor), k would change sign for neighboring crossings. Therefore, the polarity variation of k can be used to diagnose the flapping type. This technique is not absolutely valid for all cases. As shown in Figure 3, in some cases, the yielded value of k would fail in determining the flapping type: (1) equation (4) is very sensitive to the polarity of the term sign (ny × nz); one has to be very careful to deal with the situation when |nz| ≫ |ny| (Figure 3a) or |ny| ≫ |nz| (Figure 3b). In these cases, the MVA analysis error may allow a polarity change of the minor ny or nz component, and the flapping type is undiagnosed; (2) when the front of kink-like structure is over tilted (we call it overshoot) (Figure 3c), the polarity of the yielded k would be reversed to what is expected. In either case, one has to take the other neighboring crossings into account to determine the global flapping behavior for a sequence of multiple CS crossings. 2.3. Summary of Technique The technique procedures to diagnose the flapping properties can be summarized as follows: 1. Set up local coordinates for the CS, where an undisturbed CS lies in the x-y plane, the +x direction is basically along the lobe field pointing toward Earth, and the nominal CS normal is along the +z direction. This step can be achieved by the MVA analysis; we will detail it in the next section. 2. Use MVA analysis to calculate the normal direction for each CS crossing. 3. Check the series of normal directions to determine the flapping orientation. If the series of normal directions show significant variation in the y-z (x-z) plane with a minor nx (ny) component, then the flapping orientation should be basically in the y-z (x-z) plane.

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Averaged B (nT)

20 10

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2

a

0 −10 −20 20

Bx (nT)

10

b

0 −10 20 −20

By (nT)

10

c

0 −10 −20 20

Bz (nT)

10

d

0 −10 −20 18:40:00

18:42:00

18:44:00

18:46:00

18:48:00

18:50:00

18:52:00

18:54:00

18:56:00

18:58:00

19:00:00

The day of 2003−10−13 UT Figure 4. Cluster observations of the flapping motion of Earth’s magnetotail current sheet on 13 October 2003 in GSM coordinates. Panels from top to bottom show (a) the time series of the magnetic field strength and its components averaged over four S/C, the (b) Bx, (c) By, and (d) Bz components measured by each S/C, respectively. The vertical dashed lines label the CS crossings where the average Bx equals zero.

4. Construct the parameter k given by equation (4). If the flapping orientation is in the x-z plane, equation (4) still makes sense as long as “ny” is replaced by “nx.” 5. Diagnose the flapping motion based on the values of k: (1) if the polarity of k keeps almost constant during the multiple crossings, then the flapping motion is the kink like, and the traveling direction can be judged by the polarity of k; (2) if the polarity of k reverses alternately for neighboring CS crossings, the flapping motion is steady.

3. Application To test the validity of this technique, in this section, we apply the technique to several flapping cases detected by Cluster, so that a comparison with Cluster multipoint analysis can be conducted. In these cases, Cluster magnetic field data with 4 s resolution [Balogh et al., 2001] are used in geocentric solar magnetospheric (GSM) coordinates, where +x points toward the Sun and the Earth’s dipole axis is in the x-z plane; thus, +z points nearly northward, and +y completes the right-handed set. The studied cases are selected within |y| < 10 RE, where the +z direction of GSM can be simply seen as the nominal CS normal [Rong et al., 2011]. For the region |y| > 10 RE, the CS warps severely at both flanks; hence, a more accurate nominal normal may refer to the CS model [Tsyganenko and Fairfield, 2004], where the CS configuration is a function of dipole tilt angle and solar wind parameters. In the following subsections, the terms “MVA normal” and “timing normal” represent the CS normal yielded by MVA analysis and timing analysis, respectively. 3.1. Case 1 on 13 October 2003 Figure 4 shows a flapping event of the Earth’s magnetotail current sheet, which occurs on 13 October 2003 when Cluster spacecraft located at [x = 14.7, y = 10.2, z = 2.2] RE. The typical size of Cluster tetrahedron is about ~200 km. Figures 4a–4d show the time series of average magnetic field strength and its components (the average over the measurements of four spacecraft) and the magnetic field vector measured by each spacecraft, respectively. As labeled by the vertical dashed lines, the four S/C transverse the CS successively from south lobe to the north lobe at ~18:48:36 (crossing 1) and back at ~18:52:15

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Table 1. Current Sheet Crossings on 13 October 2003 by C3 a

Time

18:48:27 18:52:27

Δt (s)

b

MVA Normal in GSM

MVA Normal in LC

c

Δα

Δn

e

λ2/λ3

k

64 80

0.24, 0.62, 0.75 0.37, 0.90, 0.22

0.03 ± 0.01, 0.66 ± 0.10, 0.75 ± 0.09 0.03 ± 0.00, 0.98 ± 0.01, 0.22 ± 0.05

13.4° 19.6°

7.7° 3.1°

5.5 19.5

+1 1

a The time when C3 is crossing the CS center (Bx~0). b The length of the used time interval for MVA, centered at the CS c The MVA normal of C3 with error in local coordinates (LCs). d The angular deviation of MVA normal from the timing normal or e

d

center.

its antiparallel direction. The normal angular uncertainty for rotation toward or away from the intermediate direction.

(crossing 2). Using the timing analysis [Harvey, 1998] to calculate the normal direction of the CS yields the normal [0.25, 0.78, 0.58] for crossing 1 and [0.26, 0.81, 0.53] for crossing 2. The normal orientations indicate that the CS is evidently tilted (seen as a major ny component). Both normal orientations are basically antiparallel to each other, which imply that the CS is just moving backward and forward across the spacecraft. The crossing order, i.e., C3-C2-C1-C4 for crossing 1 and C4-C1-C2-C3 for crossing 2, also indicates that the four S/C are just repeatedly crossing the same sheet. Therefore, the pair of crossings 1 and 2 is caused by steady flapping. We now show how to use our single-point technique to diagnose the flapping type. Following the technique procedures, we first use the MVA analysis to compute the normal orientation and set up the local coordinates. Taking the measurements of C3 as an example, we perform MVA analysis on nested sets of C3 data with different time intervals, centered at CS center, i.e., Bx~0. In this way, the time stationarity of MVA results can be checked [Sonnerup and Scheible, 1998]. Finally, we choose the appropriate interval to yield a normal which does not strongly depend on the data interval. In the GSM frame, the minimum and maximum variance directions for crossing 1 are N1 = (0.24, 0.62, 0.75) and L1 = [0.89, 0.44, 0.08], respectively, and for crossing 2 are N2 = [0.37, 0.90, 0.22] and L2 = [0.93, 0.36, 0.08], respectively. The associated ratio λ2/λ3, the adopted time intervals Δt and Δα, and the angular deviation of the MVA normal from the timing normal or its antiparallel direction are listed in Table 1. The normal angular uncertainty, Δn, for rotation toward or away from the intermediate direction can be evaluated by equation (3) [Sonnerup and Scheible, 1998; Volwerk, 2006] and is also tabulated in Table 1. With the evaluation of angular uncertainty, the normals with error are shown in local coordinates in Table 1. The Δα indicates MVA and timing analysis yields a consistent orientation, and the ratio λ2/λ3 or the angular uncertainty for both crossings indicates that the normal is well qualified by MVA. Since the terms ny and “nz” in equation (4) are expressed in local coordinates of undisturbed CS, setting up the reasonable local coordinates is necessary for the further analysis. Because the MVA normals at both crossings indicate that the flapping orientation is basically in the y-z plane, the lobe field direction pointing earthward ^ Y′; ^ Z′ ^ for the undisturbed CS would not be affected by the flapping motion, so that the local coordinates X′; could be defined in the GSM frame like this way: The +x direction is defined as the averaged maximum ^ ¼ ðL1 þ L2 Þ=2≈½0:91; 0:40; 0, because the maximum variance direction is usually variance direction X′ seen as the local lobe field orientation. Although the GSM +z direction, z^, could be seen as the nominal CS ^ may not be perpendicular to X′ ^ absolutely. To construct the local orthogonal normal as we stated above, Z ^ ^ ^ ≈ [0.40, 0.92, 0]. Finally, the +z direction coordinates, the +y direction could be calculated as Y′ ¼ Z X′ ^ ¼ X′ ^ Y′ ^ ≈ [0, 0, 1]. can be found as Z′

In the local coordinates, the Bx polarity variation still keeps the same as that in GSM (not shown here). After the transformation of MVA normal into local coordinates, as being tabulated in Table 1, we find that the normal shows significant variation in the y′-z′ plane. The evaluated k value for crossing 1 is +1 but 1 for crossing 2, which indicates that the flapping motion should be steady flapping, showing the consistence with the timing analysis. Certainly, our technique can be also checked valid if we transform the timing normal into the local coordinates and compute the polarity of k via equation (4). 3.2. Case 2 on 12 September 2001 Figure 5 shows another flapping event, which occurs on 12 September 2001 when the Cluster spacecraft were located at [x = 18.9, y = 3.0, z = 0.7] RE. The typical separation distance of the spacecraft is about RONG ET AL.


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Averaged B (nT)

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2

a

0 −10 −20 20

Bx (nT)

10

b

0 −10 −20 20

By (nT)

10

c

0 −10 −20 20

Bz (nT)

10

d

0 −10 −20 14:15:00

14:16:30

14:18:00

14:19:30

14:21:00

14:22:30

14:24:00

14:25:30

14:27:00

14:28:30

14:30:00

The day of 2001−09−12 UT Figure 5. (a–d) Cluster observations of the flapping motion of Earth’s magnetotail current sheet on 12 September 2001 in GSM coordinates. The format is the same with Figure 4.

~1000 km. The four S/C successively cross the CS from south lobe to the north lobe at ~14:19:00 and then back at ~14:22:45. The timing analysis yields the normal [0.07, 0.07, 1] for the first crossing sequence and [0.08, 0.17, 0.98] for the second crossing sequence, which indicates that the CS surface basically lies in GSM x-y plane during the period of two crossing sequences. Both normal orientations are basically antiparallel to each other, which imply that the flapping motion is steady. We apply our single-point technique for each S/C to study this case as is done for case 1. MVA analysis of the four S/C shows that the maximum variance directions for both crossings are basically toward the same orientation; thus, the average of the maximum variation direction over the eight crossings by four S/C ^ ≈ [0.99, 0.07, 0.07], and accordingly, the +y direction gives the +x direction of the local coordinates as X′

^ ′ ≈ [0.07, 0, 1], respectively. Obviously, for this and +z direction can be calculated as Ŷ ′ ≈ [0.07, 1, 0] and Z case, the local coordinates are very close to GSM. The interval used for performing MVA, the MVA normal in local coordinates, the angular deviation of the MVA normal from the timing normal, the angular uncertainty of the MVA normal, the ratio λ2/λ3, and the yielded k via equation (4) are tabulated in Table 2. The minor angular deviation Δα indicates that the yielded MVA normal for each S/C at each crossing is basically consistent with the timing normal. Moreover, the ratio λ2/λ3 and the uncertainty of MVA normal

Table 2. Current Sheet Crossings on 12 September 2001

a

S/C

Time

Δt (s)

MVA Normal in GSM

MVA Normal in LC

Δα

Δn

λ2/λ3

k

C1

14:18:48 14:23:06 14:18:56 14:22:45 14:19:37 14:22:13 14:18:58 14:22:55

64 s 120 s 64 s 64 s 40 s 80 s 64 s 80 s

0.08, 0.23, 0.97 0.08, 0.27, 0.96 0.12, 0.15, 0.98 0.12, 0.45, 0.89 0.13, 0.00, 0.99 0.10, 0.28, 0.96 0.14, 0.13, 0.98 0.01, 0.11, 0.99

0.02 ± 0.00, 0.23 ± 0.01, 0.97 ± 0.00 0.01 ± 0.01, 0.28 ± 0.03, 0.96 ± 0.01 0.06 ± 0.00, 0.15 ± 0.02, 0.99 ± 0.00 0.02 ± 0.01, 0.45 ± 0.05 0.89 ± 0.03 0.06 ± 0.00, 0.00 ± 0.03, 1.00 ± 0.00 0.02 ± 0.00, 0.28 ± 0.04, 0.96 ± 0.01 0.08 ± 0.00, 0.12 ± 0.02, 0.99 ± 0.00 0.07 ± 0.01, 0.12 ± 0.05, 0.99 ± 0.01

9.5° 5.8° 5.8° 16.7° 5.1° 6.3° 5.3° 5.0°

0.8° 2.1° 1.1° 3.5° 1.6° 2.4° 0.9° 3.1°

325.0 28.8 198.7 19.8 149.1 33.1 277.4 20.1

+1 +1 +1 +1 +1 +1 +1 +1

C2 C3 C4 a

The format is the same as in Table 1.

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Journal of Geophysical Research: Space Physics 1 Averaged B (nT)

20 10

2

3

10.1002/2014JA020973

4

a

0 −10 −20 20

Bx (nT)

10

b

0 −10 −20 20

By (nT)

10

c

0 −10 −20 20

Bz (nT)

10

d

0 −10 −20 13:50:00

13:56:00

14:02:00

14:08:00

14:14:00

14:20:00

14:26:00

14:32:00

14:38:00

14:44:00

14:50:00

The day of 2004−08−05 UT Figure 6. (a–d) Cluster observations of the flapping motion of Earth’s magnetotail current sheet on 05 August 2004. The format is the same as in Figure 4.

indicate that these MVA normal orientations are well qualified. With the MVA normal, we find that the yielded k via equation (4) for each S/C is +1 for both crossing sequences, which implies that the flapping would be kink-like traveling toward Ŷ ′ direction. Obviously, it contradicts with the steady flapping motion drawn by the timing analysis. The apparent contradiction is not surprising. Because the CS basically lies in the x-y plane, i.e., |nz| ≫ |nx|, |ny|, it is difficult to judge whether the flapping orientation is in the x-y or y-z plane. The nx and ny components could change signs easily due to the MVA analysis error even it is very minor (see Figure 3a). 3.3. Case 3 on 5 August 2004 As shown in Figure 6, during the period of 13:50–14:50 on 5 August 2004, the four Cluster spacecraft record a multiple CS crossing event when Cluster is averagely located at [x = 16.0, y = 9.2, z = 2.7] RE in GSM. The typical scale of Cluster tetrahedron at this time is ~1200 km. This event has been studied previously by Zhang et al. [2005a]. The time for each CS crossing is marked by a vertical dashed line. The successive crossings of the four S/C at each CS crossing allow using the timing analysis to calculate the normal direction of the CS. As can be seen in Table 3, the timing normals show a clear deviation from +z (GSM) implying that the CS is significantly tilted. Meanwhile, the timing normal always contains negative ny with moderate positive nx at each crossing, which demonstrates that the tilted CS is always moving toward y direction of GSM, to be exact, dawnward (because of positive nx), suggesting a sequence of kink-like flapping waves passing by.

Table 3. Current Sheet Crossings on 5 August 2004 by Cluster Crossing 1 2 3 4

a

b

Timing Normal

MVA Normal in GSM

MVA Normal in LC

0.59, 0.62, 0.52 0.56, 0.76, 0.33 0.21, 0.30, 0.93 0.49, 0.80, 0.34

0.64, 0.72, 0.28 0.64, 0.76, 0.13 0.46, 0.74, 0.50 0.47, 0.67, 0.57

0.11 ± 0.00, 0.96 ± 0.01, 0.28 ± 0.03 0.06 ± 0.00, 0.99 ± 0.00, 0.13 ± 0.02 0.03 ± 0.01, 0.87 ± 0.02, 0.50 ± 0.04 0.06 ± 0.01, 0.82 ± 0.04, 0.57 ± 0.06

Δt (s)

Δα

Δn

λ2/λ3

k

80 56 160 100

14.8° 12.3° 38.9° 15.6°

1.7° 1.1° 2.6° 4.3°

62.5 215.0 14.2 9.2

+1 +1 +1 +1

a The b

normals yielded by timing analysis in GSM coordinates. The MVA normals of C3 with error in local coordinates.

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10

2

3

4

5

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6

7

8

a

0 −10

Bx (nT)

10

b

0 −10

By (nT)

10

c

0 −10

Bz (nT)

10

d

0 −10 08:02:00

08:10:00

08:18:00

08:26:00

08:34:00

08:42:00

08:50:00

08:58:00

09:06:00

09:14:00

09:22:00

The day of 2004−08−03 UT Figure 7. (a–d) Cluster observations of the flapping motion of Earth’s magnetotail current sheet on 3 August 2004 in GSM coordinates. The format is the same as in Figure 4.

Arbitrarily choosing the magnetic field measurements of C3, we perform technique to study this event with the same procedures as being done in Cases 1 and 2. The average of the maximum variation direction over all ^ ≈ [0.80, 0.59, 0.06], and accordingly, the the four crossings gives the +x direction of the local coordinates as X′

^ ′ ≈ [0.05, 0.04, 1], respectively. The interval +y and +z directions can be calculated as Ŷ ′ ≈ [0.59, 0.80, 0] and Z for performing MVA, the MVA normal in local coordinates, Δα, Δn, the ratio λ2/λ3, and the yielded k via equation (4) are tabulated in Table 3. The Δα indicates that the yielded MVA normal orientation is basically consistent with timing normal except for crossing 3. The angular uncertainty Δn and the ratio λ2/λ3 indicate that the MVA is well derived for each crossing. The yielded k always keeps +1 at each crossing (although the deviation of MVA normal from timing normal is up to 40°), which suggests that the crossings 1–4 are induced by kink-like flapping waves traveling toward Ŷ ′ direction. Therefore, our technique shows consistence with the multipoint analysis. We also applied the technique to the other S/C, and the consistence with the multipoint analysis still holds. Certainly, if we use the timing normal to calculate the k value, it is easy to find that the yielded k still consistently keeps +1, which indicates the validity of our technique. 3.4. Case 4 on 3 August 2004 Figure 7 shows a kink-like flapping event, which occurs on 3 August 2004 when the Cluster S/C were located at [x = 16.2, y = 10.0, z = 1.2] RE in GSM. The spatial scale of the Cluster tetrahedron is ~1200 km at this time. This event has been studied previously by Petrukovich et al. [2006] and Shen et al. [2008]. With timing analysis based on multipoint measurements, we can calculate the CS normal. As being tabulated in Table 4, the yielded timing normal indicates that the normal directions vary significantly in the y-z plane, keeping ny negative always. The positive nx with moderate values does not show much variation. The normal orientation at each crossing suggests that the CS is moving toward y direction in the GSM frame or toward the dawnward (because of positive nx). Note that at crossing 7, the yielded normal with minor negative nz, which probably indicates the kink-like front, is almost vertical to the x-y plane with a bit overshoot (see Figure 3). We apply our technique to the magnetic field measurements made by C3. In the GSM frame, the MVA analysis ^ ≈ [0.85, 0.52, 0.09], Ŷ ′ = [0.52, 0.86, 0], and for the eight CS crossings gives the local coordinates as X′ ^ ′ = [0.08, 0.05, 1]. The MVA normal for each crossing, the ratio λ2/λ3, the adopted time interval Δt Z RONG ET AL.


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a

Table 4. Current Sheet Crossings on 3 August 2004 by Cluster Crossing 1 2 3 4 5 6 7 8

Timing Normal

MVA Normal in GSM

MVA Normal in LC

Δt (s)

Δα

Δn

λ2/λ3

k

0.50,0.77,0.40 0.48, 0.70, 0.53 0.60, 0.76,0.24 0.46, 0.69, 0.56 0.79, 0.46, 0.40 0.44, 0.54, 0.72 0.47, 0.88, 0.02 0.47, 0.83, 0.31

0.42,0.83,0.37 0.46, 0.76, 0.45 0.51, 0.86, 0.03 0.42, 0.75, 0.51 0.12, 0.84, 0.53 0.48, 0.70, 0.52 0.45, 0.89, 0.07 0.51, 0.85, 0.16

0.10 ± 0.01, 0.92 ± 0.02, 0.37 ± 0.04 0.04 ± 0.00, 0.89 ± 0.01, 0.45 ± 0.02 0.01 ± 0.01, 1.00 ± 0.00, 0.02 ± 0.03 0.02 ± 0.00, 0.86 ± 0.03, 0.51 ± 0.05 0.49 ± 0.04, 0.65 ± 0.1, 0.58 ± 0.08 0.09 ± 0.00, 0.85 ± 0.02, 0.51 ± 0.03 0.07 ± 0.00, 1.00 ± 0.00, 0.07 ± 0.05 0.01 ± 0.00, 0.99 ± 0.00, 0.16 ± 0.02

180 200 120 200 200 160 240 180

5.9° 5.8° 14.4° 5.2° 85.7° 15.2° 2.7° 8.9°

2.3° 1.0° 1.8° 3.1° 7.6° 1.9° 0.3° 0.9°

16.2 65.9 36.2 8.7 2.8 24.5 503.4 100.9

+1 +1 +1 +1 1 +1 1 +1

a

The format is the same as in Table 3.

centered at the CS center, the angular deviation of the MVA normal from the timing analysis, the angular uncertainty of MVA normal, and the yielded k value are tabulated in Table 4. The MVA normal, except for crossing 5, has a consistent orientation compared with the timing normal, and the yielded k basically keeps +1. Nonetheless, at crossing 5, the MVA normal shows evident deviation from timing normal, although the ratio λ2/λ3 (~2.8) and angular uncertainty (Δn~7.6°) suggest that the MVA quality is acceptable. Zhang et al. [2005b] argued that disagreement with the timing normal could be induced by the fluctuation of Bz near crossing 5. In addition, the ignorable nz at crossings 3 and 7 implies that the local CS is strongly tilted or vertical to the x′-y′ plane, which suggests that the yielded k polarity is less credible than at the other crossings. Thus, our technique is uncertain in diagnosing the flapping type of crossings 3 and 7. However, if we assume that the sequence of crossings 1–8 is induced by a same flapping type, we can still argue that the multiple crossings are induced by kink-like flapping, which is traveling in the y′ direction or dawnward. The technique applied to the other S/C yields similar results. Again, if we use the timing normal to calculate the k value via equation (4), the yielded k still almost keeps +1, which consistently indicates that the flapping motions are kink-like waves traveling dawnward. 3.5. Statistics To show the statistical significance of this technique, significant flapping events detected by Cluster are selected based on the following criteria: 1 0.8 0.6 0.4

n

z

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2

0 ny

0.2

0.4

0.6

0.8

1

Figure 8. The current sheet normals determined from MVA in the local coordinates. The sign of black plus means that the normal yields a consistent diagnosis of flapping types with our technique, while the blue circle means that normal yields an inconsistent diagnosis.

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1. The crossing data set based on C3 4 s magnetic field measurement, used in previous study [Rong et al., 2014], is scanned during 2001, 2002, and 2004 with restriction |ygsm| ≤ 10 RE. Because, the typical scale of Cluster tetrahedron is ~800–~2000 km during these magnetotail seasons, which could yield significant crossing difference for the four Cluster spacecraft, and it is suitable for the timing analysis. The timing analysis for the CS normal can be seen as bench mark to evaluate our technique. 2. For the C3 measurement, the time interval of neighboring CS crossings should be larger than 1 min, and each crossing (Bx reverse the sign) is significant (|ΔBx| > 10 nT) and rapid (Δt < 300 s). 3. Plot the time series of magnetic field and select the flapping events when all four spacecraft cross the same current sheet. Totally, we selected 42 flapping events with 110 CS crossings. The flapping types for 3472


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these events had been identified by the timing analysis, which are seen as the bench mark for our technique evaluation. We performed the MVA analysis on each C3 CS crossing to calculate the k value with our technique. If the calculated k value is consistent with the flapping type as being determined from the timing analysis, then the crossing will be labeled as black plus, or else it is blue circle, as shown in Figure 8. We found that, out of the total 110 crossings, there are 81 consistent crossings; in other words, the accuracy of our technique is ~74%. Figure 8 shows the normal orientations of all the crossings in the ny/nz map of the local coordinates. In this figure, the inconsistent crossings, labeled as blue circles, basically have a quasi-horizontal |nz|~1 or quasi-vertical |ny|~1 orientations.

4. Discussion and Conclusion This study provides a new single-point technique to qualitatively diagnose the properties of the flapping motion of a CS, including the flapping type and the traveling direction of kink-like flapping. Several cases were studied and a statistics is conducted in comparison with Cluster multipoint analysis. The results demonstrate that this technique is applicable but should be used with caution. Our technique is highly dependent on the accuracy of the CS normal estimation. For the single-point measurements, we have no better way other than MVA to infer the CS normal, and the MVA normal is usually accurate when λ2/λ3 ≫ 1. Statistically speaking, the MVA normal is usually valid and consistent with the Cluster multipoint timing analysis [e.g., Zhang et al., 2005b; Sergeev et al., 2006]. However, as argued by Zhang et al. [2005b], the minimum variation direction of MVA sometimes shows evident disagreement with the timing normal and cannot represent the CS normal. Two cases referred by Zhang et al. [2005b] could result in the disagreement: (1) MVA shows good contrast of eigenvalues (λ2/λ3 ≫ 1) but fails to yield a consistent normal direction. (2) MVA shows bad quality, λ2/λ3~1, and the angular uncertainty of minimum variation direction becomes significant. The second case is easy to be avoided if one restricts the value of λ2/λ3, while the first case is a bit difficult to be picked out. Zhang et al. [2005b] ascribed the failure of the first case to the fluctuations of the magnetic field, which is noticed here at crossing 7 of Case 4. Nonetheless, at crossing 3 of Case 3, without evident field fluctuation, we also find a significant disagreement with the timing analysis (MVA applied to the other S/C shows similar results). Certainly, the timing analysis just gives an average normal of the CS over the scale of the Cluster tetrahedron. It allows the possibility of the local CS showing evident normal deviation from the timing normal. Anyway, caution must be taken when one uses the minimum variance vector as the CS normal. Our technique is strongly dependent on the polarity of sign(ny × nz) in the local coordinates. The cases studies and the statistics in section 3 demonstrate that our technique favors the moderate-tilted CS, because the polarity of sign(ny × nz) is unreliable when the CS surface is either quasi horizontal (|nz| ≫ |ny| ~ 0) or quasi vertical (|ny| ≫ |nz| ~ 0). In either case, any minor error of MVA normal can probably change the polarity of ny or nz. Therefore, one has to treat it carefully and has to take the neighboring CS crossings into account for the final decision.

Acknowledgments The data for this paper are available at Cluster Science Archive (http://www. cosmos.esa.int/web/csa/access). The authors are thankful to the Cluster MAG team for providing the magnetic field data. This work is supported by the Chinese State Scholarship Fund (201304910018), Chinese Academy of Sciences (KZZD-EW-01-2), National Key Basic Research Program of China (973 Program) grant 2011CB811405, the National Natural Science Foundation of China grant (41104114, 41374180, 41321003, and 41131066). Larry Kepko thanks Martin Volwerk and another reviewer for their assistance in evaluating this paper.

RONG ET AL.

This technique would have wide application in the planetary magnetotail explorations. We believe, with our technique and available magnetic field data from the planetary missions, e.g., Messenger, Venus Express, Maven, Cassini, and Galileo, the knowledge of the planetary magnetotail flapping motion will increase substantially. Application to a planetary data set will be our next study.

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