Magnetic flux ropes, characterized by a reversal of the magnetic field in the cross section, are 3-D helical magnetic structures. The magnetic flux rope is usually regarded as the physical model of the flux transfer event (FTE) on Earth's magnetopause, with spatial sizes extending from several ion inertial lengths (di) to a few Earth radii (RE) e.g.,. Recent in situ observations revealed that flux ropes could be flanked by two converging plasma jets, indicating these flux ropes are still active, and possibly generated by multiple, or even sequential, X-line reconnection . Such active flux ropes are relatively less often observed in a statistical study ; thus, X lines flanking flux ropes might be short-lived once these flux ropes convect away from the generation region.

A typical flux rope, with an enhancement of the magnetic field in the core region, can be modeled either as a force-free structure or a non-force-free structure, where magnetic pressure is balanced by plasma thermal pressures e.g.,. There are also some flux ropes with a decrease in magnetic field strength in the center, which are referred to as crater flux ropes e.g.,. proposed that these crater flux ropes are initial stages of typical flux ropes, and they would evolve into typical flux ropes with a reduction of central plasma pressure resulting from the transport of plasma along their axes. Therefore, they are modeled as a non-force-free structure .

The Magnetospheric Multiscale (MMS) mission with its unprecedented high-resolution plasma measurements provides a good opportunity to study the structure of ion-scale flux ropes with a duration of a few seconds in the data e.g.,, which present observations of current filaments, nonideal ion behaviors, wave activities and even flux rope coalescence. An ion-scale crater-like flux rope has also been resolved by MMS, which is interpreted as a result of the depression of transverse magnetic fields in a flux rope simulation .

Small-scale flux ropes, identified primarily by magnetic bipolar structures, have been observed at the trailing edges of Kelvin–Helmholtz (KH) waves e.g.,, where local conditions could be favorable for magnetic reconnection . However, due to the limitation of temporal resolutions, such observations had been unable to provide direct and conclusive evidence. The first direct evidence of ongoing magnetic reconnection at the trailing edges of KH waves, where the local magnetic shear is enlarged and the current sheet is compressed to ion scale, is recorded by MMS , and the MMS observations confirm previous studies from both field and particle measurements. But whether such reconnection can result in the generation of flux ropes has not been further examined by MMS. In this study, we report an ion-scale flux rope at the trailing edge of KH waves probably due to multiple X-line reconnection. The reported flux rope, with a size of ∼ 8.5 di, presents a depression of the magnetic field and an enhancement of the plasma thermal pressure in the core region, which is taken as a feature of a crater-like flux rope. We also analyze the lower hybrid drift waves (LHDWs) and their possible roles in plasma transport at the magnetospheric side of this flux rope.

The MMS observations were made at the dusk flank of the Earth's magnetopause on 27 September 2016. The four MMS spacecraft were located approximately at (0.3, 11.5, -3.4) RE in geocentric solar magnetospheric (GSM) coordinates, and the spacecraft were in a tetrahedron formation with a separation of ∼ 10 km. At this scale, data from individual satellites appear almost identical, and we primarily present the data from MMS1. We use plasma data from the fast plasma investigation (FPI) , magnetic field data from the fluxgate magnetometer (FGM) and the search coil magnetometer (SCM) , and electric field data from the electric field double probes (EDP) .

Figure 1 shows an overview of the fast survey between 19:50 and 20:09 UT, when the solar wind is relatively steady and presents an extended northward interplanetary magnetic field (IMF) from OMNI data: nsw=8.5±0.5 cm-3, Vsw = [-600.0±5.8,-5.2±11.6,-42.5±9.3] km s-1, Pdyn=6.2±0.4 nPa and BIMF=[-4.6±1.4,2.6±3.7,7.9±1.6] nT in GSM. During this time interval, MMS was initially located in the magnetospheric boundary layer, characterized by slow plasma velocities (Fig. 1e and g), and a mixture of magnetospheric ions (> 10 keV) and magnetosheath ions (∼ 1 keV) from ion energy–time spectrograms (Fig. 1i). Then it was followed by significant variations of field and plasma parameters during the boundary layer crossings and, occasionally, MMS recorded relatively cold and dense plasma with a large tailward speed, indicating it had entered into the sheath region. After 20:05 MMS returned into the magnetosphere again.

These quasi-periodic fluctuations near the magnetic equator can be excited by internal or external mechanisms, one of which is the KH instability. Figure h shows the total pressure perturbations (sum of the thermal and magnetic pressures) and the strong total pressure minimum, created inside the waves, is evident for KH instability . Another indicator for the roll-up of the KH vortex is the low-density and faster-than-sheath signature . Figure 2 shows a scatter plot of VX versus ion density from all four MMS spacecraft between 19:52 and 20:05 UT, and the x direction here is redefined to be tangential to the average magnetopause, following . The data points inside the dotted box are magnetospheric ions, characterized by low densities and slow speeds, and the magnetosheath ions are located in the dashed box, identified by the highest ion number densities. There are some data points in the lower-left region presenting a low-density and faster-than-sheath feature, which is consistent with rolled-up KH vortex. Meanwhile, the magnetic field and plasma parameters at both sides, averaged from data points in the boxes of Fig. 2, are shown in Table 1. The boundary conditions are KH unstable based on the linear theory , and the dominant period of these KH waves is ∼ 44 s (ΔTKH) from ion density and magnetic field perturbations. At the trailing edges of these surface waves, the magnetic shear could suddenly increase, providing favorable conditions for magnetic reconnection, which has been reported in another KH event on 8 September 2015 . In this event, signatures of reconnection at waves' trailing edges are also revealed, and furthermore several bipolar structures are detected by MMS. We will focus on one of them, marked by the vertical dotted line in Figure 1, and investigate its properties in details.

Figure  shows the marked zoom-in plot of the KH wave trailing edge crossing from the magnetospheric side to the magnetosheath during the interval 19:58:01–19:58:07 UT, indicated by the gradual disappearance of the magnetospheric ions (Fig. f). The y component of the magnetic field presented a bipolar signature with a significant depression of the magnetic field in the z direction, suggesting a flux rope (Fig. a), and its duration is about 2.7 s (ΔTFR). The axis direction of this flux rope is consistent with previous full kinetic simulations, which showed flux ropes were formed along the periphery of the vortex . The field magnitude dip at the center of this flux rope suggests it is a crater flux rope, which is accompanied by a plasma density peak (Fig. c) at the BY reversal region around 19:58:03.2 s.

At the magnetospheric side of the flux rope (prior to 19:58:02.5 s), a dawnward ion flow was observed (Fig. d). As suggested by , this flow could be attributed to the passage of the flux rope, which pushes the surrounding plasma to the side, generating flows opposite to its motion on the flank. Inside the flux rope, another duskward ion flow faster than the sheath plasma flow was detected, and then we test the Walén relation to check if it is a reconnection jet . Taking the magnetosheath plasma parameters as reference (the shaded red region), we compare two vectors ΔVi=Vi-Vi,MSH and ΔVA=VA-VA,MSH in the shaded yellow region (19:58:03.7–19:58:05.4 UT) and show the scatter plot in Fig. i: Though the ion velocity changes across the magnetosheath side of the flux rope, the observed velocity change (ΔVi) is about 52 % of the predicted flow change (ΔVA). Therefore, additional evidence from the ion distributions should be used to classify if it is a reconnection jet . Figure  shows the ion velocity distribution functions at 19:58:04.042 and 19:58:06.892 UT in the plane defined by the local B and E×B vectors. The flow in the sheath region is almost along the E×B direction (Fig. b) but, crossing the current sheet, the ions have a field-aligned component, suggesting they are transmitted ion flows (Fig. a). The bulk velocity is close (∼ 77 %) to the flow velocity predicted by the Walén relation (marked by the black dot). This indicates that this jet was a consequence of magnetic reconnection and the X line is located dawnward of the flux rope. The possible reason why the slope of the Walén test is relatively poor could be attributed to the structure of the flux rope, which modified the flow speeds when ions propagated away from the X line.

Figure h shows the pitch angle spectrum of energetic electrons with energies > 3 keV. Inside the flux rope, almost-isotropic energetic electrons were clearly identified, while the energy flux intensity is weaker than that in the magnetosphere, consisting of the re-closure of the field lines in the flux rope. This is different with large-scale flux rope situations, in which both trapped and untrapped electrons can be observed, suggesting different topologies of magnetic field lines . At the sheath side (from 19:58:05.0 to 19:58:06.2 s), energetic electrons only in the antiparallel directions were detected, suggesting the existence of another X line on the north-dusk side to generate these open field lines, and also this X line was formed earlier than the one at the dawn side. In addition, the bulk electron velocity could reach up to 1200 km s-1 at the magnetospheric side where a local density dip was revealed (Fig. e).

A sketch of the observed flux rope in the equator plane is inferred and summarized in Fig. j. At the trailing edge of KH waves, a flux rope is detected at the distorted magnetopause by MMS, whose relative trajectory is shown by red dotted lines. The projection of the magnetic field lines is shown by solid black lines, while the red arrows present the direction of current flows. Circles with a “dot” and “cross” denote the magnetic field (black) and the current (red) is directing out of or into the plane of paper. The ion jets are marked by a blue arrow. The current flows will be explained later.

To better illustrate the structure of the flux rope in detail, we transform the observations into a local boundary normal (LMN) coordinate by performing a minimum variance analysis on the magnetic field (MVAB) from the interval 19:58:02.0 to 19:58:05.5 UT, which yields L=[-0.22,0.94,-0.25], M=[0.06,0.27,0.96] and N=[0.97,0.19,-0.11] in GSM. The four-spacecraft timing analysis of the burst mode electric field data leads to the direction normal to the boundary N′=[0.97,0.22,0.06] in GSM, with a velocity VN=-225 km s-1. Since the difference of the normal direction between MVAB and the timing method is ∼ 10∘, the size of the flux rope in the normal direction is estimated to be 610 km, or 8.5 di, considering that the nearby magnetosheath density is ∼ 10.2 cm-3. The currents calculated from the curlometer method and direct particle measurements are presented in Fig. d and e. Good agreement is found between two measures, both showing considerable structure and variability in the current density of the flux rope with a maximum value of ∼ 900 nA m-2, while the current from plasma measurements is more structured, as shown by . Therefore, this current is transformed to the directions parallel and perpendicular to the magnetic field (Fig. f). We find these currents are highly filamentary: in the core region of the flux rope (shaded by the yellow color), the current is parallel to the magnetic field, and at the two edges (shaded by the blue color), it is predominately antiparallel to the magnetic field. The azimuthal component (jL) of the central parallel current changes from negative to positive (19:58:02.90–19:58:03:10 UT). By applying a simple thin planar current approximation, the local magnetic field change can be estimated by ΔBM=μ0IL, where IL is the integrated jL inside the current sheet. From either the positive IL or negative IL component, the central current pair could raise the magnetic field in the M direction from about 7.2 to 11.2 nT, which is in good agreement with a local BM peak in Fig. a, with a ∼ 7 nT increase. This provides further evidence that the observed structure is a flux rope. The other jL current pair with an opposite polarity at edges is also revealed, which is co-located with the depression of the magnetic field in the M direction (19:58:02.57–19:58:02.75 UT and 19:58:04.30–19:58:04.65 UT). The integrated filamentary current density (IL) is also compared with the decrease in the magnetic BM component at the magnetospheric edge (17.9 mA m-1 and 24.1 nT) and the sheath edge (16.3 mA m-1 and 26.6 nT), and they are roughly consistent with each other. Thus, the overall result is that the observed magnetic field is depressed inside the flux rope compared to the ambient magnetic field. It is worth noting that most of the current component (jL) at the edges should be part of the field-aligned current, for its maximum value (> 500 nA m-2) is larger than the perpendicular current (∼ 250 nA m-2) shown in Fig. g. This is different from small-scale magnetic holes, which are supported by electron currents perpendicular to the magnetic field .

Though the perpendicular current is relatively small in magnitude, the corresponding J×B force is not negligible (Fig. b). The normal component of this force is much larger, and its magnitude varies from -0.015 to 0.015 nPa km-1. To check if it is a force-balanced structure, we plot the pressure profile in Fig. c. Though the magnetic pressure and the plasma thermal pressure (ions and electrons) vary obviously during the flux rope crossings, the total pressure stays almost unchanged. This suggests this flux rope is not magnetically force-free but still in force balance with the magnetic field and plasma pressures . Since the magnetic field is always depressed in such crater flux ropes, a nonzero pressure gradient force is required to balance the J×B force to maintain the force balance, meaning the crater flux rope should intrinsically be a non-force-free structure.

To test if particles are frozen-in in this flux rope, we compare the electric field measured from EDP (Eedp) with the fluid convective electric field (Eci,ce=-Vi,e×B). Since Ece=Eci+J×B/ne and the J×B/ne term is not negligible as shown in Fig. b, it is expected that not all fluid convective electric fields are consistent with the measured electric field. This comparison in LMN among Eedp (in black), Eci (in blue) and Ece (in red) is presented in Fig. d–f. Overall, good agreement of these three lines is found during the selected time interval. The agreement between Eedp and Ece is extremely good when disregarding some small systematic constant offsets at a few mV m-1, indicating that the electrons are largely frozen-in throughout the encounter with the flux rope. In contrast to the electrons, there are intervals where there is significant deviation in the ions. For example, at the magnetospheric edge, the difference of Eci and other electric fields exceeds 20 mV m-1, meaning ions cannot move as fast as electrons, which remain frozen-in as pointed out. These deviations are consistent with the changing of J×B and can be found inside the flux rope and at the edges, while they are more obvious at the magnetospheric side due to smaller densities. This indicates the decoupled motions of ions and electrons are not rare in a crater flux rope.

Various wave activities, for instance ion-scale magnetosonic waves and ion acoustic-like waves, have been revealed during KH wave periods, which play a role in plasma transport and heating . In the presence of the pressure (density) gradient in the non-force-free flux rope reported in this study, lower hybrid drift waves can be excited, as shown in Fig. . Figure c presents the electric field fluctuation in burst mode, whose maximum amplitude can reach ∼ 80 mV m-1, corresponding to a broadband spectrum covering the lower hybrid frequency (Fig. d). Magnetic perturbations parallel to the background magnetic field in the lower hybrid frequency range are also clear and presented in Fig. e. In this study, we mainly focus on LHDWs on the magnetospheric edge from 19:58:02.40 to 19:58:02.75 UT. Figure f presents the fluctuations of the electric field in the wave propagation direction (δEk) and parallel magnetic field (δB||) with a band-pass filtering for frequencies 10 Hz < f < 300 Hz. Then the wave potential is estimated by ΦB=(B0/neeμ0)δB|| , which peaks at ∼ 20 eV as shown in Fig. g. The phase velocity v of the lower hybrid drift waves is found by fitting ΦE=∫Edt⋅v to ΦB. The best fitted ΦE agrees well with ΦB, with a correlation coefficient 0.83, and the phase speeds v is estimated at 498 × [-0.51, 0.40, -0.75] km s-1 in GSM. The good agreement between ΦE and ΦB suggests the constant phase speed is still acceptable, although the background magnetic field changes significantly. The wave length is then about 16.6 km, or equivalently kρe≥0.2, where ρe is the electron thermal gyroradius. This suggests LHDWs may have a larger wave length in a finite β plasma .

It is worth noting that the ratio of LHDW potential to the electron temperature (eΦ/kBTe) is ∼ 17 %, suggesting that the electrons could be effectively scattered by the wave electric field. We plot the electron pitch angle distributions of differential energy flux at 19:58:02.227, 19:58.02.647 and 19:58.02.887 UT. In the second panel when LHDWs are active, we find a significant decrease in electron energy flux from a few tens of eV to about 200 eV, especially at 0 and 180∘ pitch angle, which is associated with a local plasma density dip. This result suggests LHDWs could play important roles in plasma transport by this simple wave scattering picture.

The cross-field diffusion coefficient of LHDWs (D⟂) can be estimated from the quasi-linear theory , which is given by D⟂=12ρce2νan1+TiTe≈12ρce2π8121ωLHmime1+ωpe2Ωe2EnTi1+TiTe, where νan,ρce,ωLH,ωpe,Ωe and E=ε0δE2/2 are the anomalous collision frequency, electron gyroradius, lower hybrid frequency, electron plasma frequency, electron gyrofrequency and the wave electric field energy density, respectively. Its value is about 1 ×108 m2 s-1. This implies a diffusion time of about 2.6 s over a diffusion region with its width at one wave length, which is sufficient for the observed local plasma transport. It is worth noting that the transport due to LHDWs may be only important at edges of the observed flux rope, as the estimated diffusion coefficient is about 1 order of magnitude smaller for the diffusive buildup of the magnetopause boundary layer and also smaller than that in the magnetopause reconnection region .

The electron transport efficiency of LHDWs is determined by its saturation level. There are several candidate saturation mechanisms, which have been reviewed by . Considering Te<Ti and Vi<vth,i (Vi, vth,i are the ion bulk and thermal velocity) in a finite β plasma, the electron resonance broadening model is more suitable here , which predicts 2EnTi≈25memiΩe2ωpe2TiTe1/4Vivth,i2.

We estimate the normalized fluctuating electric field energy density (E/nTi) from observations to be ∼ 4 × 10-7, which is comparable to the predicted value (∼ 2 × 10-7). Thus, the LHDWs are damped as a larger number of electrons are permitted to involve the electron-wave resonance. This process also results into significant energy dissipations, which would be another aspect for the anomalous transport properties of LHDWs.

Another possible explanation for the observed electron distribution is the parallel acceleration, meaning the electrons can resonate with LHDWs. This requires the phase speed of LHDWs to be comparable to the local electron thermal speed (vth,e). The resonant condition is written as v||=ωLH/k||=mi/me×ωLH/k⟂ . However, the estimated v|| is far larger than vth,e. This suggests the parallel acceleration is not the dominant process for the observed electron pitch angle distributions compared to wave scattering.

In this study, an ion-scale magnetic flux rope at the trailing edge of KH waves observed by MMS has been investigated, which is probably generated by multiple X-line reconnection. Highly filamentary currents in this flux rope and their induced magnetic field are discussed: in the core region, a local increase in the magnetic field is supported by the parallel currents, and the current filaments at the edges induce a larger opposing field that causes overall |B| depressions interior to the flux rope. As the magnetic field is depressed inside this flux rope, plasma thermal pressures increase towards the center to maintain the force balance, corresponding to a non-force-free structure. Decoupling of electron and ion motions is also observed inside the current sheet. Intense LHDWs are also found at the magnetospheric edge of the flux rope, whose wave potential reaches to ∼ 17 % of the electron temperature. Therefore, these waves could effectively cause diffusive electron transport, with an estimated diffusion coefficient at ∼ 1. ×108 m2 s-1, consisting of the local plasma density dip. This indicates LHDWs can play a role in the plasma transport in a crater flux rope.

Though we have presented the relation between the magnetic variations and current filaments, and the role of LHDWs in transporting electrons inside a crater-type flux rope, the later evolution of the flux rope is still not well understood. In addition, the reported flux rope is located at the trailing edges of the KH vortex, which will be gradually rolled-up with time. Thus, the following the evolution of this flux rope with the KH vortex still requires further investigations.