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The fine magnetic field structure of two successive plasmoids previously reported is investigated by magnetic rotation analysis using four Cluster satellite data. Between these two plasmoids, opposite trends of curvature radius (Rc) variations of the magnetic field lines from the boundary to the inner part are found. The different variations of Rc...

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Context 1
... 2 ] Since the concept of magnetotail plasmoid was put forward by Hones in his substorm study [ Hones , 1977], plasmoids in the different regions of the plasma sheet have been widely investigated [e.g., Slavin et al ., 1989, 1995, 2003a, 2003b; Moldwin and Hughes , 1991, 1992; Zong et al ., 2004, 2007; Kiehas et al ., 2012]. Hones [1977] pointed out that the magnetic fi eld lines in plasmoids have a two- dimensional closed structure (magnetic loop, ML) [ Richardson et al ., 1987, Richardson and Cowley , 1985], while Hughes and Sibeck [1987] argued that plasmoids appear as magnetic fl ux rope (MFR) and have helical magnetic fi elds. During the past few years, plasmoids at near-tail distances with X > À 30 R E resulting from multiple X line reconnection (MXR) [ Lee et al ., 1985] have drawn more attention [e.g., Slavin et al ., 2003a, 2003b; Zong et al ., 2004, 2007; Eastwood et al ., 2005; Henderson et al ., 2006; Zhang et al ., 2007; Walsh et al ., 2007; Borg et al ., 2012]. In most of in situ observations, a common feature of plasmoids (either ML or MFR) in the tail plasma sheet is the bipolar B z component signature in the GSM coordinates (the GSM frame will be used throughout this paper unless otherwise stated). The plasmoid is de fi nitely identi fi ed as a MFR when the bipolar B z is accompanied by prominent | B y | enhancement, because commonly, the core fi eld along the MFR axis has its main component along the Y direction [ Slavin et al ., 2003a; Zong et al ., 2004]. The situation becomes ambiguous, however, when the | B y | enhancement is not so prominent; that is, bipolar B z without | B y | enhancement can be produced either by ML crossing where there is no core fi eld along the axis direction, or by MFR edge crossing because the core fi elds only concentrate at the MFR center (see Figure 1 in Zong et al ., 2004). In view of this, we refer to the bipolar B z signature as plasmoid in general but cannot de fi nitely state whether the non-| B y |-enhancement plasmoid is due to a ML or to a MFR. For the large-scale MFR at the distant tail, Hughes and Sibeck [1987] pointed out that only one X line in the near- Earth plasma sheet is needed and that B y component in the plasma sheet is necessary. However, according to the MXR theory [ Lee et al ., 1985], in the near-Earth plasma sheet, ML is the product of antiparallel MXR, while MFR is the result of component MXR. Discrimination between ML and MFR and studying the fi ne magnetic fi eld structure of plasmoids are important to study the properties of MXR occurrence in space plasma. To separate ML from MFR and to study the plasmoids structure, methods such as MFR model data fi tting [ Lepping et al ., 1990; Moldwin and Hughes , 1991; Kivelson and Khurana , 1995; Zhang et al ., 2008], Grad- Shafranov reconstruction [ Hu and Sonnerup , 2002], curlometer electric current measurement [ Dunlop et al ., 2002], the single-point method [ Rong et al ., 2013], and the multipoint analysis methods [ Shen et al ., 2003, 2007] can provide useful analysis tools. For example, given some restriction on the plasmoid con fi guration, MFR models usually will result into a series of parameters fi tting the plasmoid observations with least errors. If the fi tting results do not include obvious axial fi eld, maybe we can classify this plasmoid as a ML. Among these methods, the multipoint magnetic rotation analysis (MRA) developed by Shen et al . [2007] and based on four-spacecraft tetrahedron measurements allows to directly investigate the 3-D geometric structure of magnetic fi eld lines independently of any default restriction. [ 3 ] With the MRA method [ Shen et al ., 2007], we analyze two successive plasmoids previously investigated by Eastwood et al . [2005] and Henderson et al . [2006] but from the different aspects of the curvature radius R c variations and the current density distributions in them. The emphasis lies on the different magnetic con fi guration determined in these two plasmoids, which can help to differentiate the plasmoid structures between the ML type and the MFR type. [ 4 ] At the beginning of 2 October 2003, Cluster observed two successive B z bipolar signatures in the tail plasma sheet at 00:47:00 UT at GSM ( À 16.8771, 7.763, À 3.1343) R E and at 00:51:30 UT at ( À 16.8786, 7.7671, À 3.2046) R E . Figure 1 shows an overview of the Cluster (C1) ion data [ Rème et al ., 2001] and the magnetic fi eld data [ Balogh et al ., 2001] for the interesting time slot containing the B z bipolar signatures. The plasma is found to have the following properties: ion density of ~0.3/cm 3 , ion temperature of ~2 keV, positive B x component, and, except for the regions with bipolar B z signatures, magnetic fi eld intensity ranges between 5 and 15 nT. During the intervals with the two bipolar B z signatures, the plasma beta is less than 1 (the typical beta value in the near-Earth plasma sheet is usually much greater than 1), which is a typical indicator of the existence of plasmoids due to the strong fi elds inside these plasmoids [ Slavin et al ., 2003a; 2012; Henderson et al ., 2006; Zhang et al ., 2007]. These observations indicate that Cluster was located in the northern part of the plasma sheet and that possible plasmoids exist. After the time of 00:51:46 UT (blue vertical line in Figure 1), the earthward velocity of the ion fl ow displays an obvious increase after the passage of the bipolar B z signature. Determined from the ion velocity and the duration of the bipolar B z signature, these two bipolar B z signals have a space transverse scale of about 1 R E . The separation between the Cluster spacecraft is about 300 km. [ 5 ] The fi rst south-then-north B z signature is accompanied by a tailward-then-earthward fl ow. These observations are usually explained as the satellite passing through a tailward- moving X line [ Ueno et al ., 1999]. However, Eastwood et al . [2005] identi fi ed that this south-then-north B z signature instead was moving earthward. They interpreted the observations as an active earthward-moving plasmoid resulting from MXR [ Lee et al ., 1985], which is the fi rst in situ evidence of the occurrence of MXR in the plasma sheet. The tailward- then-earthward fl ows come from two X lines separately located at the earthside and the tailside of this plasmoid. Two X lines are reconnecting the plasma sheet fi elds and are producing an active plasmoids between them. Moreover, except the region near the B z in fl ection point, the total pressure (fourth panel from the top in Figure 1) is balanced, which implies that this B z bipolar is not a transient magnetic disturbance such as waves [ Lee et al ., 1988]. In our calculation of the total pressure, the dynamic pressure (~0.04 nPa) is taken into account, because the reversed fl ow will induce the antidirected pressure on this plasmoid and contribute to the pressure balance. One point that should be emphasized in this case is that the | B y | enhancement located at the B z in fl ection point is not so prominent. This impedes us from determining whether this plasmoid is a ML or a MFR. To answer this question, a detailed investigation of the observed magnetic fi eld is required. [ 6 ] Compared with the fi rst signal, the second B z bipolar signature has the clear characteristics of a MFR [ Moldwin and Hughes , 1991; Slavin et al ., 2003a; Shen et al ., 2007; Walsh et al ., 2007; Zong et al ., 2004; Borg et al ., 2012; Kiehas et al ., 2012]: B z south-to-north turning is associated with sharp increases in | B x | and | B y |, both of which correspond to components of the strong core fi eld in MFR. The enhancement in | B | overrides the enhancement in | B |, implying that the axis of the MFR largely deviates from the traditional dawn-dusk direction. The existence of faster fl ow following this MFR and the lack of faster fl ow at the earthside indicate that this MFR is a “ fossil ” one (released from the tailward MXR region). According to the analysis of Henderson et al . [2006], this MFR has the following main features: the principal axis direction at ( À X , +Y , À Z ), nearly fi eld-aligned current inside the MFR, and the radial expansion due to the imbalance total pressure. [ 7 ] According to IMAGE (International Monitor for Auroral Geomagnetic Effects) magnetometer array data and the AE index ( AE > 900 nT, not shown here), an intense substorm occurred between 20:00 UT and 24:00 UT on 1 October 2003. These plasmoids observations followed the recovery phase of this substorm. [ 8 ] To clearly investigate the magnetic eld geometric structure and to fi nd the possible differences between these two observed plasmoids, we will compute the time series of the R c of the magnetic fi eld lines and the electric current density by MRA, point-by-point along the path of Cluster. [ 9 ] The main idea of the MRA method is to investigate the 3-D magnetic topology by calculating the rotation rate ( l Á ( ∇ b ) = ∂ l b ) of the magnetic unit vector b ( b = B /| B |) along an arbitrary direction ( l ) [ Shen et al ., 2007; Rong et al ., 2011]. The characteristic directions of ∇ b can indicate the characteristic directions of the magnetic structure. For example, for the typical current sheet crossing, the direction with the largest magnetic rotation corresponds to the ...
Context 2
... | B y | enhancement is not so prominent; that is, bipolar B z without | B y | enhancement can be produced either by ML crossing where there is no core fi eld along the axis direction, or by MFR edge crossing because the core fi elds only concentrate at the MFR center (see Figure 1 in Zong et al ., 2004). In view of this, we refer to the bipolar B z signature as plasmoid in general but cannot de fi nitely state whether the non-| B y |-enhancement plasmoid is due to a ML or to a MFR. For the large-scale MFR at the distant tail, Hughes and Sibeck [1987] pointed out that only one X line in the near- Earth plasma sheet is needed and that B y component in the plasma sheet is necessary. However, according to the MXR theory [ Lee et al ., 1985], in the near-Earth plasma sheet, ML is the product of antiparallel MXR, while MFR is the result of component MXR. Discrimination between ML and MFR and studying the fi ne magnetic fi eld structure of plasmoids are important to study the properties of MXR occurrence in space plasma. To separate ML from MFR and to study the plasmoids structure, methods such as MFR model data fi tting [ Lepping et al ., 1990; Moldwin and Hughes , 1991; Kivelson and Khurana , 1995; Zhang et al ., 2008], Grad- Shafranov reconstruction [ Hu and Sonnerup , 2002], curlometer electric current measurement [ Dunlop et al ., 2002], the single-point method [ Rong et al ., 2013], and the multipoint analysis methods [ Shen et al ., 2003, 2007] can provide useful analysis tools. For example, given some restriction on the plasmoid con fi guration, MFR models usually will result into a series of parameters fi tting the plasmoid observations with least errors. If the fi tting results do not include obvious axial fi eld, maybe we can classify this plasmoid as a ML. Among these methods, the multipoint magnetic rotation analysis (MRA) developed by Shen et al . [2007] and based on four-spacecraft tetrahedron measurements allows to directly investigate the 3-D geometric structure of magnetic fi eld lines independently of any default restriction. [ 3 ] With the MRA method [ Shen et al ., 2007], we analyze two successive plasmoids previously investigated by Eastwood et al . [2005] and Henderson et al . [2006] but from the different aspects of the curvature radius R c variations and the current density distributions in them. The emphasis lies on the different magnetic con fi guration determined in these two plasmoids, which can help to differentiate the plasmoid structures between the ML type and the MFR type. [ 4 ] At the beginning of 2 October 2003, Cluster observed two successive B z bipolar signatures in the tail plasma sheet at 00:47:00 UT at GSM ( À 16.8771, 7.763, À 3.1343) R E and at 00:51:30 UT at ( À 16.8786, 7.7671, À 3.2046) R E . Figure 1 shows an overview of the Cluster (C1) ion data [ Rème et al ., 2001] and the magnetic fi eld data [ Balogh et al ., 2001] for the interesting time slot containing the B z bipolar signatures. The plasma is found to have the following properties: ion density of ~0.3/cm 3 , ion temperature of ~2 keV, positive B x component, and, except for the regions with bipolar B z signatures, magnetic fi eld intensity ranges between 5 and 15 nT. During the intervals with the two bipolar B z signatures, the plasma beta is less than 1 (the typical beta value in the near-Earth plasma sheet is usually much greater than 1), which is a typical indicator of the existence of plasmoids due to the strong fi elds inside these plasmoids [ Slavin et al ., 2003a; 2012; Henderson et al ., 2006; Zhang et al ., 2007]. These observations indicate that Cluster was located in the northern part of the plasma sheet and that possible plasmoids exist. After the time of 00:51:46 UT (blue vertical line in Figure 1), the earthward velocity of the ion fl ow displays an obvious increase after the passage of the bipolar B z signature. Determined from the ion velocity and the duration of the bipolar B z signature, these two bipolar B z signals have a space transverse scale of about 1 R E . The separation between the Cluster spacecraft is about 300 km. [ 5 ] The fi rst south-then-north B z signature is accompanied by a tailward-then-earthward fl ow. These observations are usually explained as the satellite passing through a tailward- moving X line [ Ueno et al ., 1999]. However, Eastwood et al . [2005] identi fi ed that this south-then-north B z signature instead was moving earthward. They interpreted the observations as an active earthward-moving plasmoid resulting from MXR [ Lee et al ., 1985], which is the fi rst in situ evidence of the occurrence of MXR in the plasma sheet. The tailward- then-earthward fl ows come from two X lines separately located at the earthside and the tailside of this plasmoid. Two X lines are reconnecting the plasma sheet fi elds and are producing an active plasmoids between them. Moreover, except the region near the B z in fl ection point, the total pressure (fourth panel from the top in Figure 1) is balanced, which implies that this B z bipolar is not a transient magnetic disturbance such as waves [ Lee et al ., 1988]. In our calculation of the total pressure, the dynamic pressure (~0.04 nPa) is taken into account, because the reversed fl ow will induce the antidirected pressure on this plasmoid and contribute to the pressure balance. One point that should be emphasized in this case is that the | B y | enhancement located at the B z in fl ection point is not so prominent. This impedes us from determining whether this plasmoid is a ML or a MFR. To answer this question, a detailed investigation of the observed magnetic fi eld is required. [ 6 ] Compared with the fi rst signal, the second B z bipolar signature has the clear characteristics of a MFR [ Moldwin and Hughes , 1991; Slavin et al ., 2003a; Shen et al ., 2007; Walsh et al ., 2007; Zong et al ., 2004; Borg et al ., 2012; Kiehas et al ., 2012]: B z south-to-north turning is associated with sharp increases in | B x | and | B y |, both of which correspond to components of the strong core fi eld in MFR. The enhancement in | B | overrides the enhancement in | B |, implying that the axis of the MFR largely deviates from the traditional dawn-dusk direction. The existence of faster fl ow following this MFR and the lack of faster fl ow at the earthside indicate that this MFR is a “ fossil ” one (released from the tailward MXR region). According to the analysis of Henderson et al . [2006], this MFR has the following main features: the principal axis direction at ( À X , +Y , À Z ), nearly fi eld-aligned current inside the MFR, and the radial expansion due to the imbalance total pressure. [ 7 ] According to IMAGE (International Monitor for Auroral Geomagnetic Effects) magnetometer array data and the AE index ( AE > 900 nT, not shown here), an intense substorm occurred between 20:00 UT and 24:00 UT on 1 October 2003. These plasmoids observations followed the recovery phase of this substorm. [ 8 ] To clearly investigate the magnetic eld geometric structure and to fi nd the possible differences between these two observed plasmoids, we will compute the time series of the R c of the magnetic fi eld lines and the electric current density by MRA, point-by-point along the path of Cluster. [ 9 ] The main idea of the MRA method is to investigate the 3-D magnetic topology by calculating the rotation rate ( l Á ( ∇ b ) = ∂ l b ) of the magnetic unit vector b ( b = B /| B |) along an arbitrary direction ( l ) [ Shen et al ., 2007; Rong et al ., 2011]. The characteristic directions of ∇ b can indicate the characteristic directions of the magnetic structure. For example, for the typical current sheet crossing, the direction with the largest magnetic rotation corresponds to the normal of the current sheet, while for the MFR crossing, the direction with the least magnetic rotation rate corresponds to the principle axis of the MFR [ Shen et al ., 2007]. Further, if the rotation direction ( l ) is con fi ned to follow the magnetic fi eld direction ( b = B /| B |) , the resulted rotation rate, i.e., b Á ( ∇ b ), is the curvature of the magnetic fi eld line ( ρ c ), and further- more, the curvature radius ( R c = ρ c À 1 ) can be evaluated. Combining the measured b and the calculated R c , how magnetic fi eld lines geometrically con fi gure can be revealed. Both ρ c and R c are signi fi cant indicators re fl ecting how magnetic fi eld lines con fi gure. A key step in MRA is the calculation of the tensor gradient of the magnetic unit vector ( ∇ b , i.e., ∂ j b i , where i and j denote the three components). From four-point measurements of the Cluster mission, ∂ j b i can be evaluated by calculating the fi rst-order coef fi cient of Taylor expansion of measured magnetic vectors (see Appendix C in Shen et al . [2007]) with a relative error ordered L/D , where L is the size of the Cluster tetrahedron and D is the typical spatial transverse scale of the magnetic structure. Empirically, when L / D ≤ 0.1, the calculated result is reliable. The detailed description and application of MRA are given in the study of Shen et al . [2007]. [ 10 ] To check the ability of MRA to recover the rotational characteristics of magnetic fi eld in small-scale magnetic structures such as near-tail plasmoids, we fi rst use MRA to calculate the R c for two modeled magnetic structures with a characteristic scale of 1 R E and at two different distances from the structure center ( Y1 = 0.41 R E , Y2 = 0.81 R E ) (Figure 2a; the cross section of both models): the fi rst modeled structure is the circled magnetic fi elds produced by a straight current I , because they have a similar circled magnetic con fi guration compared to the ideal 2-D ML (Figure 2a); the other one is a kind of MFR model (Elphic-Russell (ER) model) [ Elphic and Russell , 1983]. The separation of the test tetrahedron is set to 600 km to guarantee that in the benchmark L/D (600 km/1 R E ) has the same value as in above in ...
Context 3
... Hones [1977] pointed out that the magnetic fi eld lines in plasmoids have a two- dimensional closed structure (magnetic loop, ML) [ Richardson et al ., 1987, Richardson and Cowley , 1985], while Hughes and Sibeck [1987] argued that plasmoids appear as magnetic fl ux rope (MFR) and have helical magnetic fi elds. During the past few years, plasmoids at near-tail distances with X > À 30 R E resulting from multiple X line reconnection (MXR) [ Lee et al ., 1985] have drawn more attention [e.g., Slavin et al ., 2003a, 2003b; Zong et al ., 2004, 2007; Eastwood et al ., 2005; Henderson et al ., 2006; Zhang et al ., 2007; Walsh et al ., 2007; Borg et al ., 2012]. In most of in situ observations, a common feature of plasmoids (either ML or MFR) in the tail plasma sheet is the bipolar B z component signature in the GSM coordinates (the GSM frame will be used throughout this paper unless otherwise stated). The plasmoid is de fi nitely identi fi ed as a MFR when the bipolar B z is accompanied by prominent | B y | enhancement, because commonly, the core fi eld along the MFR axis has its main component along the Y direction [ Slavin et al ., 2003a; Zong et al ., 2004]. The situation becomes ambiguous, however, when the | B y | enhancement is not so prominent; that is, bipolar B z without | B y | enhancement can be produced either by ML crossing where there is no core fi eld along the axis direction, or by MFR edge crossing because the core fi elds only concentrate at the MFR center (see Figure 1 in Zong et al ., 2004). In view of this, we refer to the bipolar B z signature as plasmoid in general but cannot de fi nitely state whether the non-| B y |-enhancement plasmoid is due to a ML or to a MFR. For the large-scale MFR at the distant tail, Hughes and Sibeck [1987] pointed out that only one X line in the near- Earth plasma sheet is needed and that B y component in the plasma sheet is necessary. However, according to the MXR theory [ Lee et al ., 1985], in the near-Earth plasma sheet, ML is the product of antiparallel MXR, while MFR is the result of component MXR. Discrimination between ML and MFR and studying the fi ne magnetic fi eld structure of plasmoids are important to study the properties of MXR occurrence in space plasma. To separate ML from MFR and to study the plasmoids structure, methods such as MFR model data fi tting [ Lepping et al ., 1990; Moldwin and Hughes , 1991; Kivelson and Khurana , 1995; Zhang et al ., 2008], Grad- Shafranov reconstruction [ Hu and Sonnerup , 2002], curlometer electric current measurement [ Dunlop et al ., 2002], the single-point method [ Rong et al ., 2013], and the multipoint analysis methods [ Shen et al ., 2003, 2007] can provide useful analysis tools. For example, given some restriction on the plasmoid con fi guration, MFR models usually will result into a series of parameters fi tting the plasmoid observations with least errors. If the fi tting results do not include obvious axial fi eld, maybe we can classify this plasmoid as a ML. Among these methods, the multipoint magnetic rotation analysis (MRA) developed by Shen et al . [2007] and based on four-spacecraft tetrahedron measurements allows to directly investigate the 3-D geometric structure of magnetic fi eld lines independently of any default restriction. [ 3 ] With the MRA method [ Shen et al ., 2007], we analyze two successive plasmoids previously investigated by Eastwood et al . [2005] and Henderson et al . [2006] but from the different aspects of the curvature radius R c variations and the current density distributions in them. The emphasis lies on the different magnetic con fi guration determined in these two plasmoids, which can help to differentiate the plasmoid structures between the ML type and the MFR type. [ 4 ] At the beginning of 2 October 2003, Cluster observed two successive B z bipolar signatures in the tail plasma sheet at 00:47:00 UT at GSM ( À 16.8771, 7.763, À 3.1343) R E and at 00:51:30 UT at ( À 16.8786, 7.7671, À 3.2046) R E . Figure 1 shows an overview of the Cluster (C1) ion data [ Rème et al ., 2001] and the magnetic fi eld data [ Balogh et al ., 2001] for the interesting time slot containing the B z bipolar signatures. The plasma is found to have the following properties: ion density of ~0.3/cm 3 , ion temperature of ~2 keV, positive B x component, and, except for the regions with bipolar B z signatures, magnetic fi eld intensity ranges between 5 and 15 nT. During the intervals with the two bipolar B z signatures, the plasma beta is less than 1 (the typical beta value in the near-Earth plasma sheet is usually much greater than 1), which is a typical indicator of the existence of plasmoids due to the strong fi elds inside these plasmoids [ Slavin et al ., 2003a; 2012; Henderson et al ., 2006; Zhang et al ., 2007]. These observations indicate that Cluster was located in the northern part of the plasma sheet and that possible plasmoids exist. After the time of 00:51:46 UT (blue vertical line in Figure 1), the earthward velocity of the ion fl ow displays an obvious increase after the passage of the bipolar B z signature. Determined from the ion velocity and the duration of the bipolar B z signature, these two bipolar B z signals have a space transverse scale of about 1 R E . The separation between the Cluster spacecraft is about 300 km. [ 5 ] The fi rst south-then-north B z signature is accompanied by a tailward-then-earthward fl ow. These observations are usually explained as the satellite passing through a tailward- moving X line [ Ueno et al ., 1999]. However, Eastwood et al . [2005] identi fi ed that this south-then-north B z signature instead was moving earthward. They interpreted the observations as an active earthward-moving plasmoid resulting from MXR [ Lee et al ., 1985], which is the fi rst in situ evidence of the occurrence of MXR in the plasma sheet. The tailward- then-earthward fl ows come from two X lines separately located at the earthside and the tailside of this plasmoid. Two X lines are reconnecting the plasma sheet fi elds and are producing an active plasmoids between them. Moreover, except the region near the B z in fl ection point, the total pressure (fourth panel from the top in Figure 1) is balanced, which implies that this B z bipolar is not a transient magnetic disturbance such as waves [ Lee et al ., 1988]. In our calculation of the total pressure, the dynamic pressure (~0.04 nPa) is taken into account, because the reversed fl ow will induce the antidirected pressure on this plasmoid and contribute to the pressure balance. One point that should be emphasized in this case is that the | B y | enhancement located at the B z in fl ection point is not so prominent. This impedes us from determining whether this plasmoid is a ML or a MFR. To answer this question, a detailed investigation of the observed magnetic fi eld is required. [ 6 ] Compared with the fi rst signal, the second B z bipolar signature has the clear characteristics of a MFR [ Moldwin and Hughes , 1991; Slavin et al ., 2003a; Shen et al ., 2007; Walsh et al ., 2007; Zong et al ., 2004; Borg et al ., 2012; Kiehas et al ., 2012]: B z south-to-north turning is associated with sharp increases in | B x | and | B y |, both of which correspond to components of the strong core fi eld in MFR. The enhancement in | B | overrides the enhancement in | B |, implying that the axis of the MFR largely deviates from the traditional dawn-dusk direction. The existence of faster fl ow following this MFR and the lack of faster fl ow at the earthside indicate that this MFR is a “ fossil ” one (released from the tailward MXR region). According to the analysis of Henderson et al . [2006], this MFR has the following main features: the principal axis direction at ( À X , +Y , À Z ), nearly fi eld-aligned current inside the MFR, and the radial expansion due to the imbalance total pressure. [ 7 ] According to IMAGE (International Monitor for Auroral Geomagnetic Effects) magnetometer array data and the AE index ( AE > 900 nT, not shown here), an intense substorm occurred between 20:00 UT and 24:00 UT on 1 October 2003. These plasmoids observations followed the recovery phase of this substorm. [ 8 ] To clearly investigate the magnetic eld geometric structure and to fi nd the possible differences between these two observed plasmoids, we will compute the time series of the R c of the magnetic fi eld lines and the electric current density by MRA, point-by-point along the path of Cluster. [ 9 ] The main idea of the MRA method is to investigate the 3-D magnetic topology by calculating the rotation rate ( l Á ( ∇ b ) = ∂ l b ) of the magnetic unit vector b ( b = B /| B |) along an arbitrary direction ( l ) [ Shen et al ., 2007; Rong et al ., 2011]. The characteristic directions of ∇ b can indicate the characteristic directions of the magnetic structure. For example, for the typical current sheet crossing, the direction with the largest magnetic rotation corresponds to the normal of the current sheet, while for the MFR crossing, the direction with the least magnetic rotation rate corresponds to the principle axis of the MFR [ Shen et al ., 2007]. Further, if the rotation direction ( l ) is con fi ned to follow the magnetic fi eld direction ( b = B /| B |) , the resulted rotation rate, i.e., b Á ( ∇ b ), is the curvature of the magnetic fi eld line ( ρ c ), and further- more, the curvature radius ( R c = ρ c À 1 ) can be evaluated. Combining the measured b and the calculated R c , how magnetic fi eld lines geometrically con fi gure can be revealed. Both ρ c and R c are signi fi cant indicators re fl ecting how magnetic fi eld lines con fi gure. A key step in MRA is the calculation of the tensor gradient of the magnetic unit vector ( ∇ b , i.e., ∂ j b i , where i and j denote the three components). From four-point measurements of the Cluster mission, ∂ j b i can be evaluated by calculating the fi rst-order coef fi cient of Taylor expansion of ...
Context 4
... density directions. In BRN orthogonal coordinates linked to the magnetic fi eld line, B lies along the magnetic fi eld direction, R lies along the direction of magnetic curvature, and N completes the right-hand orthogonal set, as displayed in Figure 2b. As we can see in Figure 3a, it is interesting to note that R c decreases from the boundary (the beginning and ending of the fi gure) to the inner part of the fi rst plasmoid. R c has a value of about 2 R E at the boundary and a minimum value of 0.4 R E at the B z in fl ection point (red vertical line). In contrast, as shown in Figure 3b, the R c of the magnetic fi eld shows an opposite variation trend; that is, it increases from the outer toward the center (red vertical line) in the region of the MFR before 00:51:46 UT (blue vertical line). R c reaches a maximum of ~0.9 R E at the center. It should be noted that the point of maximum R c does not coincide with the B z in fl ection point but has an offset toward the trailing part. A satellite can have variable ways crossing MFR due to the different directions of the motion of MFR relative to the satellite [ Borg et al ., 2012]. In this case, the asymmetric B z signal (less and shorter negative B z but larger and longer positive B z ) implies that Cluster has the chance to cross MFR at path Y3 as shown in Figure 2a. Thus, Cluster will fi rst meet the red magnetic in fl ection point and then meet the green innermost point (which has the maximum R c ). After the blue line, R c does not display the same decreasing trend toward the boundary as shown for the leading part of this plasmoid but remains at a higher value of ~0.6 R E . [ 12 ] Another difference in these two plasmoids is the current density distribution, shown in Figures 3a and 3b. In most regions (the regions outside of the two blue vertical lines in Figure 3a) of the fi rst plasmoid except the region near the B z in fl ection point (the regions between the two blue vertical lines in Figure 3a), the total current intensity is less than 10 nA. This value is comparable with the current density in the plasma sheet but is far smaller than that generally encountered in MFR [ Slavin et al ., 2003b]. In these regions, the dominant component of current intensity is J N (| J N |/(| J B | + | J R |) ≥ 1). In the region between the two blue lines, J B dominates the current density. The current density in the second plasmoid (Figure 3b) is more intense and mainly lies along the magnetic fi eld direction (| J B |/(| J R | + | J N |) > 1) before 00:51:46 UT (blue vertical line); however, after the blue line, the current density becomes suddenly weaker and is no longer fi eld- aligned. The MRA calculation of current density in the second plasmoid obtains the similar results as the curlometer calculation showed in Henderson et al . [2006]. [ 13 ] Eastwood et al . [2005] showed the rst observa- tional evidence of MXR in the tail plasma sheet by identi- fying the fi rst bipolar B z as an active plasmoid with possible MFR structure. However, MRA here shows that R c decreases gradually from its edge to its inner part. The time series of R c is more in agreement with the R c characteristics of a two-dimensional ML (as seen in Figure 2c). Recently, Yang et al. (The fi ne structure of fl ux ropes in geomagnetotail: Cluster observations, Journal of Geophysical Research , under review, 2013) statistically found that the inward increasing R c is a certainly general feature of the MFR fi elds as shown in Figure 2d. These clearly demonstrate that the fi rst plasmoid can be represented by a ML structure. As shown in Figure 2b, the magnetic fi eld in an ideal ML forms closed concentric circles and is not helical. In this con fi guration, the radius to the axis center ( ρ ) approximates the R c of the magnetic fi eld. Thus, the inner magnetic fi eld in the ML has a smaller R c than the outer magnetic fi eld. As a result, the fi rst plasmoid is preliminary identi fi ed as a ML type. In the ideal ML case, the R c is zero at the central axis and has maximum value at the edge, and the ratio of the R c to the maximum R c will gradually decrease from 1 at the edge to 0 at the central axis. In this case, the R c has a maximum value of about 2 R E at the edge (the beginning and ending of Figure 3a) and a minimum value of 0.4 R E at the center (red vertical line). The ratio of the minimum R c to the maximum R c is 0.2 which is small. This means that the trajectory of Cluster crosses near the center of this ML. On the contrary, the R c variation before the vertical blue line in the second plasmoid (Figure 3b) is similar to the MFR scenario shown in Figure 2d: R c decreases gradually from the inner part to the edge. This trend indicates that the magnetic fi eld in this plasmoid becomes more curled with increasing distance from the center. However, the R c in the region after the blue line remains at high value and does not obey the trend shown in Figure 2d. By checking the ion velocity (Figure 1), we fi nd that the fl ow after this region has higher earthward speed than the fl ow before the blue line. This faster fl ow would push the magnetic fi eld earthward and introduce a magnetic pileup region at the trailing edge of this MFR. In this region, the curled MFR magnetic fi eld lines will be pushed by the after-neighboring faster fl ow to extend along the Z direction and then to become straighter. Thus, the straighter magnetic fi elds have a higher R c . Through the comparison of the different R c variations in two plasmoids by MRA, we can de fi nitely identify if the plasmoids absent of prominent core fi eld is a ML or MFR con fi guration. [ 14 ] The different magnetic con fi gurations in ML and MFR are induced by the different distributions of current density in them. As shown in Figure 3a, most regions (the regions outside the two vertical blue lines) of the fi rst ML-type plasmoid is current-scarce with two components of J R and J B less than J N except the region near the center, while in Figure 3b, the second MFR-type plasmoid is current-abundant with intense fi eld-aligned current. To simplify the discussion, we place an ideal circular two-dimensional ML in a cylindrical coordinate system (Figure 2b), with the ML axis in the Z direction. It is known that in a cylindrical coordinate system, the current density is expressed as [ 15 ] The three axis directions in local BRN coordinates correspond to φ ^ , À ρ , and Z ^ directions in the cylindrical coordinate system (Figure 2b); thus, we get J 1⁄4 ∂ B ρ À ∂ B z , J R ρ ∂ φ z ∂ z , and J N ρ ∂ ρ ρ ∂ φ Z . For an ideal circled ML, owing to the circular magnetic con fi guration of B = [0, B φ ( ρ ), 0], J R and J B would disappear, and ! the residual current J N would come from J N 1⁄4 1 ρ ∂ ð ρ ∂ B ρ φ Þ Z ^ , which theoretically represents the distribution model of the current in ML. For the outer regions (outside the two vertical blue lines) of the actual ML here, J B and J R do not disappear completely, because ML cannot be a perfect circle in nature, while the lower J R and J B values with respect to the main component J N indicate that the current in these regions coincides with the above-mentioned theoretical result. Due to the weak total current J T , most of these ML outer regions could be seen as current-free. Obviously, this ML con fi guration needs a strong current along the central axis to support it just as is observed for J N near the ML center in Figure 3a. However, it is unexpected that J B overwhelms J N near the center, which indicates that this ML cannot be represented by a simple magnetic structure produced by a strong axis current and that it has a more complicated current-abundant structure in the central region. We can see that the central region of this ML is characterized by the B y increase and abundant fi eld-aligned currents, and there are two B x dips at both sides of the B z in fl ection point (Figure 1), which is very different from the depression of the magnetic intensity in the distant tail ML [ Richardson et al .,1987, Richardson and Cowley , 1985]. Slavin et al . [1995] and Zong et al .[1997] interpreted similar observations as plasmoids with a force- free magnetic fl ux rope (MFR) core, and based on these observations, Zong et al .[2004] gave the possible signals of the plasmoid with a MFR core in their Figure 1b. So we prefer to interpret the above-mentioned “ more complicated current-abundant structure in the central region ” as a possible MFR core. Because the majority of this plasmoid shows the feature of ML and the inside MFR core only occupies a narrow central region, it is fi nally identi fi ed as a ML with a possible MFR core. The difference is that the outer fi elds enveloping the MFR core are helical in the case of Slavin et al . [1995] and are looped in this plasmoid. The little enhancement of | B y | corresponding to B z in fl ection implies that Cluster only sweeps the edge of the MFR core, and then the R c during two blue lines does not display an obvious increase. The intense current aligned to the principal axis at the core of this MFR core will contribute to support the outside loop fi elds. [ 16 ] In the second MFR-type plasmoid, the angles between intense current density and magnetic fi eld directions are in the range of 0 ο À 45 ° at the leading part of the MFR and 135 ο À 180 ° at its trailing part before the blue line (Figure 3b). The direction of the current is nearly aligned to the magnetic fi eld lines, so this MFR can be described as nearly force-free. Interestingly, the direction of the current is found to change from fi eld-aligned in the leading part to anti fi eld-aligned in the trailing part of the MFR, and the parallel/antiparallel currents display an asymmetry: antiparallel currents cross the B z in fl ection point and reach the negative B z region. This phenomenon has been so far rarely observed. Considering that MFR is the “ fossil ” signature of MXR, these ...
Context 5
... 2 ] Since the concept of magnetotail plasmoid was put forward by Hones in his substorm study [ Hones , 1977], plasmoids in the different regions of the plasma sheet have been widely investigated [e.g., Slavin et al ., 1989, 1995, 2003a, 2003b; Moldwin and Hughes , 1991, 1992; Zong et al ., 2004, 2007; Kiehas et al ., 2012]. Hones [1977] pointed out that the magnetic fi eld lines in plasmoids have a two- dimensional closed structure (magnetic loop, ML) [ Richardson et al ., 1987, Richardson and Cowley , 1985], while Hughes and Sibeck [1987] argued that plasmoids appear as magnetic fl ux rope (MFR) and have helical magnetic fi elds. During the past few years, plasmoids at near-tail distances with X > À 30 R E resulting from multiple X line reconnection (MXR) [ Lee et al ., 1985] have drawn more attention [e.g., Slavin et al ., 2003a, 2003b; Zong et al ., 2004, 2007; Eastwood et al ., 2005; Henderson et al ., 2006; Zhang et al ., 2007; Walsh et al ., 2007; Borg et al ., 2012]. In most of in situ observations, a common feature of plasmoids (either ML or MFR) in the tail plasma sheet is the bipolar B z component signature in the GSM coordinates (the GSM frame will be used throughout this paper unless otherwise stated). The plasmoid is de fi nitely identi fi ed as a MFR when the bipolar B z is accompanied by prominent | B y | enhancement, because commonly, the core fi eld along the MFR axis has its main component along the Y direction [ Slavin et al ., 2003a; Zong et al ., 2004]. The situation becomes ambiguous, however, when the | B y | enhancement is not so prominent; that is, bipolar B z without | B y | enhancement can be produced either by ML crossing where there is no core fi eld along the axis direction, or by MFR edge crossing because the core fi elds only concentrate at the MFR center (see Figure 1 in Zong et al ., 2004). In view of this, we refer to the bipolar B z signature as plasmoid in general but cannot de fi nitely state whether the non-| B y |-enhancement plasmoid is due to a ML or to a MFR. For the large-scale MFR at the distant tail, Hughes and Sibeck [1987] pointed out that only one X line in the near- Earth plasma sheet is needed and that B y component in the plasma sheet is necessary. However, according to the MXR theory [ Lee et al ., 1985], in the near-Earth plasma sheet, ML is the product of antiparallel MXR, while MFR is the result of component MXR. Discrimination between ML and MFR and studying the fi ne magnetic fi eld structure of plasmoids are important to study the properties of MXR occurrence in space plasma. To separate ML from MFR and to study the plasmoids structure, methods such as MFR model data fi tting [ Lepping et al ., 1990; Moldwin and Hughes , 1991; Kivelson and Khurana , 1995; Zhang et al ., 2008], Grad- Shafranov reconstruction [ Hu and Sonnerup , 2002], curlometer electric current measurement [ Dunlop et al ., 2002], the single-point method [ Rong et al ., 2013], and the multipoint analysis methods [ Shen et al ., 2003, 2007] can provide useful analysis tools. For example, given some restriction on the plasmoid con fi guration, MFR models usually will result into a series of parameters fi tting the plasmoid observations with least errors. If the fi tting results do not include obvious axial fi eld, maybe we can classify this plasmoid as a ML. Among these methods, the multipoint magnetic rotation analysis (MRA) developed by Shen et al . [2007] and based on four-spacecraft tetrahedron measurements allows to directly investigate the 3-D geometric structure of magnetic fi eld lines independently of any default restriction. [ 3 ] With the MRA method [ Shen et al ., 2007], we analyze two successive plasmoids previously investigated by Eastwood et al . [2005] and Henderson et al . [2006] but from the different aspects of the curvature radius R c variations and the current density distributions in them. The emphasis lies on the different magnetic con fi guration determined in these two plasmoids, which can help to differentiate the plasmoid structures between the ML type and the MFR type. [ 4 ] At the beginning of 2 October 2003, Cluster observed two successive B z bipolar signatures in the tail plasma sheet at 00:47:00 UT at GSM ( À 16.8771, 7.763, À 3.1343) R E and at 00:51:30 UT at ( À 16.8786, 7.7671, À 3.2046) R E . Figure 1 shows an overview of the Cluster (C1) ion data [ Rème et al ., 2001] and the magnetic fi eld data [ Balogh et al ., 2001] for the interesting time slot containing the B z bipolar signatures. The plasma is found to have the following properties: ion density of ~0.3/cm 3 , ion temperature of ~2 keV, positive B x component, and, except for the regions with bipolar B z signatures, magnetic fi eld intensity ranges between 5 and 15 nT. During the intervals with the two bipolar B z signatures, the plasma beta is less than 1 (the typical beta value in the near-Earth plasma sheet is usually much greater than 1), which is a typical indicator of the existence of plasmoids due to the strong fi elds inside these plasmoids [ Slavin et al ., 2003a; 2012; Henderson et al ., 2006; Zhang et al ., 2007]. These observations indicate that Cluster was located in the northern part of the plasma sheet and that possible plasmoids exist. After the time of 00:51:46 UT (blue vertical line in Figure 1), the earthward velocity of the ion fl ow displays an obvious increase after the passage of the bipolar B z signature. Determined from the ion velocity and the duration of the bipolar B z signature, these two bipolar B z signals have a space transverse scale of about 1 R E . The separation between the Cluster spacecraft is about 300 km. [ 5 ] The fi rst south-then-north B z signature is accompanied by a tailward-then-earthward fl ow. These observations are usually explained as the satellite passing through a tailward- moving X line [ Ueno et al ., 1999]. However, Eastwood et al . [2005] identi fi ed that this south-then-north B z signature instead was moving earthward. They interpreted the observations as an active earthward-moving plasmoid resulting from MXR [ Lee et al ., 1985], which is the fi rst in situ evidence of the occurrence of MXR in the plasma sheet. The tailward- then-earthward fl ows come from two X lines separately located at the earthside and the tailside of this plasmoid. Two X lines are reconnecting the plasma sheet fi elds and are producing an active plasmoids between them. Moreover, except the region near the B z in fl ection point, the total pressure (fourth panel from the top in Figure 1) is balanced, which implies that this B z bipolar is not a transient magnetic disturbance such as waves [ Lee et al ., 1988]. In our calculation of the total pressure, the dynamic pressure (~0.04 nPa) is taken into account, because the reversed fl ow will ...
Context 6
... fying the fi rst bipolar B z as an active plasmoid with possible MFR structure. However, MRA here shows that R c decreases gradually from its edge to its inner part. The time series of R c is more in agreement with the R c characteristics of a two-dimensional ML (as seen in Figure 2c). Recently, Yang et al. (The fi ne structure of fl ux ropes in geomagnetotail: Cluster observations, Journal of Geophysical Research , under review, 2013) statistically found that the inward increasing R c is a certainly general feature of the MFR fi elds as shown in Figure 2d. These clearly demonstrate that the fi rst plasmoid can be represented by a ML structure. As shown in Figure 2b, the magnetic fi eld in an ideal ML forms closed concentric circles and is not helical. In this con fi guration, the radius to the axis center ( ρ ) approximates the R c of the magnetic fi eld. Thus, the inner magnetic fi eld in the ML has a smaller R c than the outer magnetic fi eld. As a result, the fi rst plasmoid is preliminary identi fi ed as a ML type. In the ideal ML case, the R c is zero at the central axis and has maximum value at the edge, and the ratio of the R c to the maximum R c will gradually decrease from 1 at the edge to 0 at the central axis. In this case, the R c has a maximum value of about 2 R E at the edge (the beginning and ending of Figure 3a) and a minimum value of 0.4 R E at the center (red vertical line). The ratio of the minimum R c to the maximum R c is 0.2 which is small. This means that the trajectory of Cluster crosses near the center of this ML. On the contrary, the R c variation before the vertical blue line in the second plasmoid (Figure 3b) is similar to the MFR scenario shown in Figure 2d: R c decreases gradually from the inner part to the edge. This trend indicates that the magnetic fi eld in this plasmoid becomes more curled with increasing distance from the center. However, the R c in the region after the blue line remains at high value and does not obey the trend shown in Figure 2d. By checking the ion velocity (Figure 1), we fi nd that the fl ow after this region has higher earthward speed than the fl ow before the blue line. This faster fl ow would push the magnetic fi eld earthward and introduce a magnetic pileup region at the trailing edge of this MFR. In this region, the curled MFR magnetic fi eld lines will be pushed by the after-neighboring faster fl ow to extend along the Z direction and then to become straighter. Thus, the straighter magnetic fi elds have a higher R c . Through the comparison of the different R c variations in two plasmoids by MRA, we can de fi nitely identify if the plasmoids absent of prominent core fi eld is a ML or MFR con fi guration. [ 14 ] The different magnetic con fi gurations in ML and MFR are induced by the different distributions of current density in them. As shown in Figure 3a, most regions (the regions outside the two vertical blue lines) of the fi rst ML-type plasmoid is current-scarce with two components of J R and J B less than J N except the region near the center, while in Figure 3b, the second MFR-type plasmoid is current-abundant with intense fi eld-aligned current. To simplify the discussion, we place an ideal circular two-dimensional ML in a cylindrical coordinate system (Figure 2b), with the ML axis in the Z direction. It is known that in a cylindrical coordinate system, the current density is expressed as [ 15 ] The three axis directions in local BRN coordinates correspond to φ ^ , À ρ , and Z ^ directions in the cylindrical coordinate system (Figure 2b); thus, we get J 1⁄4 ∂ B ρ À ∂ B z , J R ρ ∂ φ z ∂ z , and J N ρ ∂ ρ ρ ∂ φ Z . For an ideal circled ML, owing to the circular magnetic con fi guration of B = [0, B φ ( ρ ), 0], J R and J B would disappear, and ! the residual current J N would come from J N 1⁄4 1 ρ ∂ ð ρ ∂ B ρ φ Þ Z ^ , which theoretically represents the distribution model of the current in ML. For the outer regions (outside the two vertical blue lines) of the actual ML here, J B and J R do not disappear completely, because ML cannot be a perfect circle in nature, while the lower J R and J B values with respect to the main component J N indicate that the current in these regions coincides with the above-mentioned theoretical result. Due to the weak total current J T , most of these ML outer regions could be seen as current-free. Obviously, this ML con fi guration needs a strong current along the central axis to support it just as is observed for J N near the ML center in Figure 3a. However, it is unexpected that J B overwhelms J N near the center, which indicates that this ML cannot be represented by a simple magnetic structure produced by a strong axis current and that it has a more complicated current-abundant structure in the central region. We can see that the central region of this ML is characterized by the B y increase and abundant fi eld-aligned currents, and there are two B x dips at both sides of the B z in fl ection point (Figure 1), which is very different from the depression of the magnetic intensity in the distant tail ML [ Richardson et al .,1987, Richardson and Cowley , 1985]. Slavin et al . [1995] and Zong et al .[1997] interpreted similar observations as plasmoids with a force- free magnetic fl ux rope (MFR) core, and based on these observations, Zong et al .[2004] gave the possible signals of the plasmoid with a MFR core in their Figure 1b. So we prefer to interpret the above-mentioned “ more complicated current-abundant structure in the central region ” as a possible MFR core. Because the majority of this plasmoid shows the feature of ML and the inside MFR core only occupies a narrow central region, it is fi nally identi fi ed as a ML with a possible MFR core. The difference is that the outer fi elds enveloping the MFR core are helical in the case of Slavin et al . [1995] and are looped in this plasmoid. The little enhancement of | B y | corresponding to B z in fl ection implies that Cluster only sweeps the edge of the MFR core, and then the R c during two blue lines does not display an obvious increase. The intense current aligned to the principal axis at the core of this MFR core will contribute to support the outside loop fi elds. [ 16 ] In the second MFR-type plasmoid, the angles between intense current density and magnetic fi eld directions are in the range of 0 ο À 45 ° at the leading part of the MFR and 135 ο À 180 ° at its trailing part before the blue line (Figure 3b). The direction of the current is nearly aligned to the magnetic fi eld lines, so this MFR can be described as nearly force-free. Interestingly, the direction of the current is found to change from fi eld-aligned in the leading part to anti fi eld-aligned in the trailing part of the MFR, and the parallel/antiparallel currents display an asymmetry: antiparallel currents cross the B z in fl ection point and reach the negative B z region. This phenomenon has been so far rarely observed. Considering that MFR is the “ fossil ” signature of MXR, these antidirected currents may be related to the Hall currents generated from multiple X lines if this MFR has just been released from the reconnection region [ Deng et al ., 2004]. Because the quadrupole magnetic fi elds in the magnetic reconnection occupy about 30% of the total magnetic fi eld magnitude, the Hall currents strictly perpendicular to the quadrupole magnetic fi elds can be nearly aligned to the total magnetic fi eld [ Pritchett , 2001]. If the reconnection at the tailside of MFR is more intense than that at the earthside, the Hall currents from the tailside reconnection will have the chance to overwhelm the Hall currents from the earthside reconnection and to reach the negative B z region. With regard to the magnetic pileup region behind the blue line, the force-free con fi guration is destroyed by the pushing of the neighboring plasma, which resembles the destruction of the force-free con fi guration at the leading edge of the MFR reported by Slavin et al . [2003b]. [ 17 ] In the near-Earth plasma sheet, ML is the product of antiparallel MXR, while MFR is the result of component MXR. Cluster observed the ML at 00:47 UT and the MFR at 00:51 UT. During these 4 min, Cluster moved 300 km toward the central of the plasma sheet. It is interesting how two different types of plasmoids can be observed in such a short time interval with not so large space separation. Our proposed explanation is that the shear angles between the pair of magnetic fi elds in the north sheet and in the south sheet may become larger with the distance from the innermost region of the plasma sheet. At the time around 00:47 UT, Cluster meets the outer north plasma sheet fi eld which has the large shear angle (~180°) with its counterpart in the south plasma sheet. A large shear angle will favor the formation of ML by the occurrence of the antiparallel MXR. Cluster entered the inner plasma sheet and the fi elds there 4 min later, with the less shear angle favoring MFR formation by component MXR. The fl ux rope core within the ML also supports this explanation: the inner and less-sheared plasma sheet fi elds are fi rst reconnected at two separated X lines to form fl ux rope core; with the outward development of MXR, the outer and large sheared plasma sheet fi elds are reconnected to form ML. However, this explanation needs further investigation of the vertical distribution of the shear angles between the pair of magnetic fi elds in the asymmetric plasma sheet [ Cowley , 1981]. [ 18 ] In summary, this study has presented the analysis of two successive tail plasmoids observed by Cluster, following an intense substorm recovery phase. Based on the MRA, the different R c variations of the fi eld lines from the edge to the inner part of these two plasmoids are exhibited: the decreasing R in the fi rst plasmoid is found to be consistent with the features of ML, while the increasing R c toward axis center in the second plasmoid is ...

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