Background

Two recently very popular topics in magnetospheric research are fast transient plasma flows, also called bursty bulk flows (BBFs) (e.g., Angelopoulos et al. 1992; Nakamura et al. 2001), and dipolarization fronts (DFs) (e.g., Nakamura et al. 2002; Runov et al. 2009; Schmid et al. 2011). Statistical studies show that BBFs carry significant amount of mass, energy, and magnetic flux earthward (Angelopoulos et al. 1994; Liu et al. 2011).

A statistical examination of BBFs with Geotail observations reveals that BBFs near the neutral sheet are associated with brief B z component enhancements that bear striking resemblance to DFs (Ohtani et al. 2004). The general pattern of B z during BBFs is also reproduced well by global magnetohydrodynamics (MHD) simulations (Wiltberger et al. 2015). Therefore, BBFs and DFs are associated phenomena, but their time scales are different. The time scale of BBFs is ~10 min, while that of DFs is ~1–2 min. In spite of the short durations for DFs, their magnetic flux transport is estimated to be just as significant as BBFs. In particular, Liu et al. (2013, 2014) studied the magnetic flux transport by DFs, referring them as dipolarization flux bundles and proposing them as elementary elements for the substorm current wedge.

Magnetotail observations often reveal the presence of multiple particle populations showing significant deviations from a single fluid (Chen et al. 2000; Parks et al. 2001; Parks 2004). Lui et al. (2005) were the first to present quantitative comparison between single-component (MHD) and multi-component (kinetic) approaches on the transport of mass and energy in BBF events. The result shows significant differences between MHD and kinetic approaches in determining transport properties of BBFs, a result that was further verified by later studies (Lui and Hori 2006; Cao et al. 2013). Therefore, the use of single-fluid approach such as dipolarization flux bundles (Liu et al. 2013, 2014) in evaluating transport properties of BBFs is likely to be inappropriate.

There are attempts to label current disruption events (Takahashi et al. 1987; Lui et al. 1988) as groups of DFs without providing evidence for such an assertion (Zhang et al. 2011; Liu et al. 2013). Careful examination of their characteristics shows that DFs and current disruptions have distinct characteristics and associated physical processes, suggesting that they should not be considered as the same phenomenon (Lui 2014; Yao et al. 2015). In fact, the event analyzed by Zhang et al. (2011) showed the disturbance spreading tailward, which is a feature of current disruptions (Lopez and Lui 1990; Jacquey et al. 1991; Ohtani et al. 1992) and is in contrast to many reported isolated DFs that propagate earthward.

Inherent in the previous studies on magnetic flux transport by BBFs and DFs is the assumption that magnetic flux is carried by plasma flow. This implies the validity of the frozen-in field line condition (FIC) in these structures. In this work, we examine several cases of DFs for the validity of FIC. These cases were based on observations from THEMIS (time history of events and macroscale interactions during substorms) (Angelopoulos 2008). The result shows that the FIC does not hold in most cases. Therefore, whether DFs can be related to magnetic flux transport needs to be investigated carefully. Furthermore, it is estimated that the magnetic flux transport for cases when the FIC holds is insignificant in comparison with what is needed to account for the magnetic flux increase during substorm dipolarization.

Methods

The DF events are first examined for the FIC so that the DF intervals can be considered as line preserving, i.e., particles on the same field line will remain so at later times (Newcomb 1958). This is done in this work by comparing the values of E y component with the values of −(V × B) y , where E y is the y-component of the electric field, V is the plasma bulk flow (practically the proton plasma flow since electrons are much lighter than protons), and B is the magnetic field. The comparison is done only when the electric field magnitude exceeds 0.5 mV/m to eliminate times of low-level electric fluctuations. Two criterion levels are used. The strict criterion is when the deviation between these two values differs by more than 50 % of the (V × B) y component within the DF interval and the lenient criterion (i.e., giving more favorable statistics for DFs with FIC satisfied) is when it is twice the value. This comparison ratio, \(\left| {{{\left[ {E_{y} + \left( {V \times B} \right)_{y} } \right]} \mathord{\left/ {\vphantom {{\left[ {E_{y} + \left( {V \times B} \right)_{y} } \right]} {\left( {V \times B} \right)_{y} }}} \right. \kern-\nulldelimiterspace} {\left( {V \times B} \right)_{y} }}} \right|\), will be denoted as CR. For DFs that satisfy the FIC, the magnetic flux transport for DF is evaluated. The use of electron velocity instead of plasma bulk velocity to claim the validity of FIC and the calculation of magnetic flux transport by DFs will be discussed later in “Summary and discussion ” section.

Data set

The THEMIS data set used in this study consists of magnetic field measured by the fluxgate magnetometer (FGM) (Auster et al. 2008), plasma velocity based on measurements from the electrostatic analyzer (ESA) (McFadden et al. 2008) covering 5 eV–25 keV and from the solid-state telescope (SST) (Angelopoulos 2008) covering 25 keV to ~1 MeV, and electric field obtained by the electric field instrument (EFI) (Bonnell et al. 2008). Only the y-component of the electric field will be examined and its offset for the events is determined by matching the averaged E y component with the averaged value of −(V × B) y during appropriate quiet time of 5 min prior to DF arrival.

Observations

2009 February 27

The clearest multi-case study of DFs was reported by Runov et al. (2011). One of the DFs has also been reported in Runov et al. (2009), which is on 2009 February 27. This event is shown in Fig. 1. The satellite locations for this event and the others discussed in this work are shown in Table 1. The first column of panels in Fig. 1 shows the 3 s spin-averaged B z component in GSM coordinates for the four spacecraft P1, P2, P3, and P4. The error bars are calculated based on the standard deviation of the mean (SDM) of the spin-averaged values from measurements at a higher resolution of 0.25 s. The second column shows the V x component in GSM coordinates based on the combined measurements from ESA to SST. The third column shows the comparison between the measured 3 s spin-averaged E y component in the dsl coordinates (close to GSE) and the value of −(V × B) y component also in the dsl coordinates using V and B in the dsl coordinates. The error bars on E y are derived from the SDM of the spin-averaged values based on measurements at a higher resolution of 0.125 s. The error bars on −(V × B) y component are based on the SDM of the magnetic field, since no uncertainties are provided for V. The highlighted time intervals are the enhanced B z periods within the DFs.

Fig. 1
figure 1

The B z component, plasma bulk flow, and comparison between E y and −(V × B) y for four THEMIS satellites on 2009 February 27. The parameters E y and −(V × B) y are in dsl (close to GSE) coordinates

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Table 1 Spacecraft locations of dipolarization front events
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The B z profiles in Fig. 1 show the general properties of DFs. In these cases, the ambient B z prior to the DF arrival was quite small, ~2–7 nT. The DFs had sharp B z enhancements to ~20–22 nT for a duration of ~1–2 min. A magnetic dip ahead of a DF, which is a common feature noted previously, can be seen in P1, P2, and P4. Two of the magnetic dips at P1 and P4 reached negative values. The associated flows were earthward and increased significantly further into the DF intervals. There were small tailward flows prior to DF arrival at P2, P3, and P4. Comparison between E y and (V × B) y shows a general agreement. However, there were short intervals that indicate significant discrepancy between the two parameters. For example, there is a noticeable difference during 0751:26–0751:58 UT at P1 with the CR of 0.5–0.8, during 0752:25–0753:18 UT at P2 with the CR of 0.5–0.7, and during 0754:10–0754:20 UT at P4 with the CR of 0.7–1.4. Therefore, under the strict criterion, DFs at P1, P2, and P4 did not satisfy the FIC; and under the lenient criterion, all DFs satisfied the FIC.

2009 March 5

The event on 2009 March 5 is shown in Fig. 2 with observations from four satellites P2, P3, P5, and P4. For this day, the B z component prior to the DF arrival was quite different among the satellites, ranging from ~3 to ~10 nT, so was the peak B z detected within the DF intervals, ranging from ~12 to ~30 nT. Magnetic dips ahead of DFs were relatively small and did not reach negative values for all four satellites. Similar to the previous event, only earthward flows were seen within the DF interval at all satellites. However, this event also shows some significant tailward flows immediately after the DF interval for P5 and P4. Most importantly, for all four satellites, this event shows significant departures between E y and −(V × B) y within the DF intervals. Some significant departure also existed even after the DF interval at P2, P3, and P4. The CRs for DFs at P2, P3, P5, and P4 are 0.7–10.6, 1.6–3.0, 0.9–2.6, and 0.7–2.5, respectively. In other words, the FIC did not hold for all these cases even under the lenient criterion.

Fig. 2
figure 2

The B z component, plasma bulk flow, and comparison between E y and −(V × B) y for four THEMIS satellites on 2009 March 5

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2009 March 15

Figure 3 shows the event on 2009 March 15. The three satellites are P5, P4, and P3. For this day, the B z component prior to the DF arrival was quite high before the magnetic dip ahead of the DFs, ranging from ~12 to ~14 nT, so were the peak B z within the DFs from ~17 to 23 nT. All three satellites detected tailward plasma flow during the DF intervals. Comparison between E y and −(V × B) y shows major departures of these two parameters for all three satellites, indicating a breakdown of the FIC. The CRs for DFs at P5, P4, and P3 are 0.8–6.2, 0.6–3.2, and 2.5–3.2, respectively. Therefore, all these cases did not satisfy the FIC under the lenient criterion.

Fig. 3
figure 3

The B z component, plasma bulk flow, and comparison between E y and −(V × B) y for three THEMIS satellites on 2009 March 15

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Other cases

The other DF cases have also been examined and the associated plots are given in the Additional file. Additional file 1: Fig. S1 shows cases for 2009 March 9 event and Additional file 1: Fig. S2 shows cases for 2009 March 19 and March 31 events. For 2009 March 9 event, CRs for P2, P4, and P3 are 1.0–1.3, 0.7–1.1, and 0.7–0.9, respectively. For 2009 March 31, CRs for P3, P2, and P4 are 0.5–2.6, 0.6–0.8, and 1.5–3.5, respectively. For 2009 March 19, CR for P4 is 0.8–0.9. Therefore, under the strict criterion, all seven DFs did not satisfy the FIC whereas under the lenient criterion, the FIC is invalid for only two DFs. Overall, there are 18 cases that can be investigated with reliable measurements. Out of these, FIC was not satisfied for 9 of them under the lenient criterion and for 17 of them under the strict criterion. This small sample indicates that FIC is invalid for ~50 and ~94 % of DFs under the lenient and strict criterion, respectively.

Magnetic flux transport rate and accummulated flux transport

For the cases in which the FIC holds for at least the lenient criterion, one can then proceed to investigate the transport of magnetic flux Φ by DFs. The transport rate \({\rm d}\varPhi /{\rm d}t\) and the accumulated flux transport Φ during the DF interval are given in Fig. 4 for the event on 2009 February 27. The peak magnetic flux transport rate was not steady and fluctuated within the range 35–42 kWb/s/R E while the accumulated magnetic flux transport within the DF intervals was in the range of 0.7–2.8 MWb/R E. Similar plots for the other events are shown in Additional file 1: Figs. S3, S4. The peak magnetic flux transport rate for these other cases varied from ~8 to 42 kWb/s/R E, while the accumulated magnetic flux transport within the DF intervals ranged from 0.1 to 0.8 MWb/R E. For a DF with 3 R E wide in the dawn-dusk direction, the range of accumulated magnetic flux transport is ~0.3–8 MWb.

Fig. 4
figure 4

The B z component, the magnetic flux transport rate, and the accumulated magnetic flux transport for four THEMIS satellites on 2009 February 27

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Summary and discussion

We have investigated quantitatively the validity of the FIC for 18 cases of DFs using the measurements from THEMIS and events published by Runov et al. (2011). The FIC is evaluated by comparing the y-component of the in situ electric field with that of the product −V × B in dsl (close to GSE) coordinates. The result shows that out of 18 cases, the FIC is invalid for 9 cases under the lenient criterion of CR greater than 2 and for 17 cases under the strict criterion of CR greater than 0.5. Therefore, most DFs did not satisfy the FIC. The dominance of DFs not satisfying the FIC is probably due to the large inertia of protons, such that their flow cannot change fast enough to match the rapid fluctuations of electric field encountered in DFs.

For cases that satisfy the FIC, the magnetic flux transport rate and the total accumulated magnetic flux within the dipolarization front intervals are evaluated. The peak magnetic flux transport rate is found to be in the range of ~8–42 kWb/s/R E, while the accumulated magnetic flux transport is in the range of 0.1–2.8 MWb/R E.

The magnetic flux needed for dipolarization in the near-earth region during substorms has been estimated by Angelopoulos et al. (1994). The amount needed is ~0.36 GWb. This estimate is derived by judging the amount of magnetic flux transport from dayside magnetic reconnection during the substorm growth phase. Table 5 in that article stated 0.385 GWb for a “typical” substorm. A slightly larger estimate of ~0.5–1 GWb was later given by Angelopoulos et al. (2013). With these estimates, the range of accumulated magnetic flux transport amounts to ~0.1–2.2 % per DF for the 0.385 GWB and ~0.09–0.84 % per DF for the 1 GWB. In other words, it would take at the minimum ~50 DFs to provide the magnetic flux needed for a typical substorm. Therefore, the magnetic flux transport provided by DFs is insignificant in accounting for the magnetic flux needed for substorm dipolarization. This conclusion is consistent with the estimate made in the study by Yao et al. (2015).

Attempts have been made previously to justify magnetic flux transport of DFs by claiming that electrons are still magnetized and the transport can be estimated using the E × B drift of electrons (Angelopoulos et al. 2013; Liu et al. 2013). However, in order to justify line preserving under this condition with electron flow V e, it is necessary to demonstrate that E + V e × B = 0 for all three components of this vector equation, not just one component. This check was not performed by these previous publications. Therefore, it is questionable whether the use of electron motion to compute magnetic flux transport is justified, especially when almost all DFs do not satisfy the strict criterion for the FIC.

In a related matter, Angelopoulos et al. ( 2013) calculated the energy conversion for these DFs using J y E y, where J y denotes the y-component of the current density. The value of J y was estimated using \((\partial{B_z}/\partial{t})/V_{xe}\) where V xe is the x-component of the electron flow. This method implicitly assumes line preserving property as well, which was not demonstrated in the work. A proper way to evaluate the current density to estimate the electromagnetic energy conversion is using a tetrahedron configuration like cluster. This was done by Lui et al. (2007) who evaluated all the terms in the generalized Ohm’s law during current disruption activity unconnected with magnetic reconnection.

Conclusion

DFs are interesting features in the magnetotail and may transport magnetic flux to the inner magnetosphere. However, the FIC that allows magnetic flux being transported by plasma motion does not hold for many DFs. Even though a small sample of DFs is examined here, the result indicates that the FIC does not hold for a majority of DFs. Therefore, it is necessary to isolate only those cases for which the FIC holds to evaluate the magnetic flux transport. For DFs that satisfy the FIC, the peak magnetic flux transport rate is found to be in the range of 8–42 kWb/s/R E, while the accumulated magnetic flux transport during the DF intervals is in the range of 0.1–2.8 MWb/R E. Adopting a width of 3 R E extent for each DF that satisfies the FIC, it would require 50–1000 of them to account for the magnetic flux needed for near-earth dipolarization in a typical substorm. In other words, contribution of magnetic flux transport from DFs is negligible for substorms.