Magnetic field structures and topologies play an important role in dynamical plasma phenomena, such as solar or stellar flares (e.g. Kusano et al., 2012; Bamba et al., 2013) and magnetospheric substorms (e.g. Consolini and Chang, 2001), and in the transfer of mass, momentum, and energy in space and astrophysical plasmas. A key physical process underlying the rearrangement of field line configurations and topology changes is magnetic reconnection, which is known to occur in the solar corona (e.g. Masuda et al., 1994), solar wind (e.g. Gosling et al., 2005), and magnetotail (e.g. Nagai et al., 2013), and at the magnetopause (e.g. Sonnerup et al., 1981). In order to understand in what manner and how efficiently this process converts energy and transfers mass and momentum across a current layer, it is indispensable to reveal the nature of one-dimensional (1-D) discontinuities (rotational or tangential discontinuities and shocks), the formation, location, and interplay of the X- and O-points in 2-D, and those of magnetic nulls and separators in 3-D (e.g. Cai et al., 2001; Xiao et al., 2006; Wendel and Adrian, 2013), which may be embedded in the current layer in question.

In studies of solar magnetic activities, a number of attempts have been made to reconstruct 3-D force-free magnetic field structures in the corona from nearly instantaneous, remote-sensing (ground-based or space telescope) measurements of the photospheric field (e.g. Wheatland and Leka, 2011; Wiegelmann and Sakurai, 2012; Inoue et al., 2014). In particular, nonlinear force-free field models allow for an estimation of the free magnetic energy available, by comparison with reconstructed potential or linear force-free fields, and thus potentially make it possible to forecast where solar flares could be initiated. However, such reconstructions of fully 3-D fields have not been conducted by use of in situ data taken by spacecraft during a short period, while a 3-D magnetohydrostatic equilibrium field in the magnetosphere has been recovered, for example, by modelling the field using Euler potentials and using an average equatorial profile of the plasma pressure based on long-term, in situ measurements as input and empirical fields as boundary conditions (e.g. Zaharia, 2008).

Recently, Sonnerup and Hasegawa (2011) developed a novel data analysis method for the reconstruction of steady, 3-D, magnetohydrostatic structures using plasma and magnetic field data recorded by two closely spaced spacecraft, which is hereafter referred to as the SH11 method. They developed and benchmarked a primitive version of the numerical code for the SH11 method, using an analytical solution of the 3-D magnetohydrostatic equations (∇⋅B=0 and ∇p=j×B). This type of reconstruction in 3-D space, as well as those in 3-D space-time (2-D space and time) developed recently by Sonnerup and Hasegawa (2010), and Hasegawa et al. (2010a, 2014), is a natural extension of a variety of 2-D reconstruction techniques developed to date (see Sonnerup et al., 2006, 2008 and Hasegawa, 2012 for an overview or reviews) that in principle require data from single spacecraft as input. In this paper, we present a first application of the SH11 method to actual observations in space, along with modest improvements of the reconstruction code.

The present paper is organised as follows. In Sect. 2, the basic equations used in the reconstruction and methodology are briefly summarised. In Sect. 3, an overview is given of THEMIS spacecraft observations at the subsolar magnetopause of a flux transfer event (FTE), which is generated through some time-dependent form of magnetopause reconnection. For overviews and models of FTEs, the readers are referred to Scholer (1995), Raeder (2006), and Paschmann et al. (2013). We apply the SH11 3-D reconstruction method as well as the classical Grad–Shafranov (GS) reconstruction technique for 2-D magnetohydrostatic structures (Sonnerup and Guo, 1996; Hau and Sonnerup, 1999; Hasegawa et al., 2004) to the FTE seen by THEMIS and compare the results. In Sect. 4, particle measurements during and around the FTE are analysed in detail to discuss the generation mechanism of the FTE. A brief summary and discussion is presented in Sect. 5.

We numerically solve the magnetohydrostatic equations using as input the magnetic field and pressure data taken along the paths of two closely separated spacecraft. The assumptions underlying the technique are that the structure to be reconstructed is time-independent and magnetohydrostatic, and moves at a constant velocity relative to the spacecraft. The magnetohydrostatic equations solved in the reconstruction are: ∇⋅B=0, and the force balance relation, ∇p=j×B, which can be written, by use of j=∇×B∇×Bμ0μ0, in the form ∇P=B⋅∇Bμ0, where P=p+B2B22μ02μ0 is the total (magnetic plus plasma) pressure. Equation () assumes that the inertia terms in the MHD equation of motion can be neglected, and expresses the balance between the force from the total pressure gradient and magnetic tension. Equations () and () constitute four scalar equations for the four unknown physical quantities, U=Bx;By;Bz;P. To understand the reasons why we use the total pressure, rather than plasma pressure, as one of the variables, see Sect. 5 in SH11.

The x axis of the Cartesian reconstruction coordinates is chosen to be parallel to, and halfway between, the paths of the two spacecraft, Sc-A and Sc-B, in the frame co-moving with the structure (Fig. A1). The z axis is defined in such a way that the two spacecraft are contained in the x-z plane. The two spacecraft are separated by lz in the z direction, and are assumed to move in the +x direction at a constant speed, VSc, relative to the structure. This velocity is usually approximated by the negative deHoffmann–Teller (HT) velocity, determined by the method as described by Khrabrov and Sonnerup (1998), i.e. VSc=-VHT. Because of the assumed time independence of the structure, temporal variations seen in time series of the data can now be converted into spatial information along the two spacecraft paths, i.e. spatial variations in the x direction.

We use an equilateral triangular integration grid, as shown in Fig. A1, so that central differences can be used (SH11). Under the assumption of time independency, the z derivatives of U, ∂U∂U∂z∂z, on the x axis, namely, at the midpoint, (y, z)=(0,0), between the two spacecraft paths, can be evaluated by use of the data taken at the same x locations, i.e. ∂U∂U∂z∂zi=UB,i-UA,iUB,i-UA,ilzlz. Here UA,i and UB,i are the values at (x, y, z)=(xi, 0, -lz-lz22) on the path of Sc-A and at (x, y, z)=(xi, 0, +lz+lz22) on the path of Sc-B, respectively. The subscript i (= 0, 1, 2, …, Nx=LxLxΔxΔx), represents the ith grid point in the x direction, where Lx is the x length of the reconstruction domain and Δx is the distance between the neighbouring points after interpolation in the x direction of the original data. The x derivatives, ∂U∂U∂x∂x, on the x axis are calculated by use of a set of the interpolated values at points along the x axis, which are averages, UM,i=UA,i+UB,iUA,i+UB,i22, of the two spacecraft data at the same x locations, i.e. ∂U∂U∂x∂xi=UM,i+1-UM,i-1UM,i+1-UM,i-12Δx2Δx if the lowest-order central difference is used. The four unknown y derivatives ∂U∂U∂y∂y are then given by the following equations derived from Eqs. () and () (SH11), ∂U∂y=∂Bx∂Bx∂y∂y∂By∂By∂y∂y∂Bz∂Bz∂y∂yμ0∂P∂P∂y∂y= μ0μ0ByBy∂P∂P∂x∂x-BxBxByBy∂Bx∂Bx∂x∂x-BzBzByBy∂Bx∂Bx∂z∂z-∂Bx∂Bx∂x∂x-∂Bz∂Bz∂z∂zμ0μ0ByBy∂P∂P∂z∂z-BxBxByBy∂Bz∂Bz∂x∂x-BzBzByBy∂Bz∂Bz∂z∂zBx∂By∂By∂x∂x+Bz∂By∂By∂z∂z+By-∂Bx∂Bx∂x∂x-∂Bz∂Bz∂z∂z.

These derivatives can be used to integrate U in the +y direction to obtain the values at (y, z)=(Δy, 0) (point 1 in Fig. A1). Here Δy is the integration step, which should be adjusted to an optimal value that allows for sufficiently accurate reconstruction over a larger domain and, in Fig. A1, is chosen to satisfy lz<2233Δy. In the present study, Δx is set equal to ΔyΔy33. In general, Δx should be comparable to Δy for obtaining the best solution.

The next step is to compute the values at point 2 by use of the extrapolated (or interpolated when lz>2233Δy) and integrated values at points 0b and 1, respectively. This can be done by first rotating the coordinate system by +60∘ about the x axis and then using Eq. () to integrate U in the rotated y axis direction, i.e. along +s in Fig. A1. This kind of rotation-then-integration process is repeated until the integration reaches the two z boundaries, zmax and zmin (point 9 in Fig. A1). The integration in the plus and minus y direction is then performed by use of Eq. () on the basis on the recovered values at y=Δy and y=0, respectively, and is continued until the two y boundaries are reached. The end results are the reconstructed 3-D distributions of the four quantities U, i.e. the 3-D configuration of the magnetic field lines and the pressure distribution, in a rectangular parallelepiped domain.

Since we can use only the lowest-order central difference and two of the four equations () have terms divided by By (and also because of the ill-posed nature of initial value problems), numerical errors develop with increasing number of integration steps, especially in regions where By reverses sign. The reconstruction is thus possible over only a limited range in the y and z directions, while the x length Lx of the reconstruction domain is determined by the path length of the two spacecraft, i.e. VSc times the length of the chosen data interval. Because of the numerical errors, the reconstructed total pressure and/or plasma pressure may take on negative values at some grid point(s), after a number of integration steps. Thus, in each integration step, if the computed total pressure value becomes negative, it is reset to zero. If the plasma pressure p=P-B2B22μ02μ0 computed at the end of all integration steps becomes negative, it is also set to zero. As consequences of these errors and corrections, the reconstructed pressure may not be as strictly preserved along the reconstructed field lines as expected from the magnetohydrostatic force balance equation, and the field may also have nonzero divergence. For details of some newly developed methods to reduce these numerical errors, the readers are referred to Appendix A.

In this section, we present an overview of THEMIS observations on 27 June 2007, ∼ 04:50 UT, when an FTE was encountered at the subsolar magnetopause under a southward interplanetary magnetic field (IMF) condition. The FTE is analysed by applying both the 3-D and GS (2-D) reconstruction techniques to the event. Note that both methods assume the magnetohydrostatic force balance. The primary differences are the spatial dimension of the structure reconstructed, and that the 3-D reconstruction requires data from two spacecraft, whereas the Grad–Shafranov reconstruction (GSR) in principle utilises data from a single spacecraft. Hereafter we use lower-case italic letters (x, y, z) for representing the components and positions in GSM, capital italic letters (X, Y, Z) for those in the 3-D reconstruction coordinate system, and capital roman letters (X, Y, Z) for those in the 2-D GSR coordinate system.

In this section, we investigate magnetic field-aligned and anti-field-aligned fluxes of electrons, as well as of ions observed in the FTE and surrounding regions. Energy-versus-time spectrograms of the ion fluxes can be used to identify signatures of ion acceleration or heating, possibly associated with magnetopause reconnection, while field-aligned streaming electrons can be used as tracers of the topology of the field lines. Figure 8 shows the spectrograms of the differential energy fluxes (eV cm-2 s-1 ster-1  eV-1) in the spacecraft frame based on the THD and THE observations, along with the magnetic field data from the four probes presented in Fig. 1. The THC spectrograms look similar to the ones for THD (Fig. 8e–h) and thus are not shown, while THB (on the magnetosheath side) was too far from the FTE core to be of help in revealing the topological properties. Note that THD traversed the central but somewhat magnetosheath-side part of the FTE, whereas THE was on the magnetospheric side of THD and THC (Fig. 2).

For the 14 min interval shown in Fig. 8, THD observed either the magnetosheath or the magnetopause boundary layer in which FTEs were embedded, while THE observed either the typical outer magnetospheric region or the boundary layers including the magnetosheath boundary layer (MSBL) (e.g. Fuselier, 1995). The magnetosheath, encountered by THD, e.g. after 04:53 UT, was dominated by ions with energies less than a few keV and electrons with energies less than a few hundreds eV (Fig. 8e–h). As discussed in Sect. 3, two FTEs were encountered at around 04:46 and 04:50 UT, when both the ion and electron spectrograms were characterised by the coexistence of the hot (> a few keV) magnetospheric population and heated or accelerated magnetosheath population. During a part of the second FTE interval to which the reconstruction methods were applied (Sect. 3), the field-aligned and anti-field-aligned electron fluxes were approximately balanced at all energies (Fig. 9), suggesting that the field lines are closed, i.e. anchored to the Earth at both ends (e.g. Øieroset et al., 2008; Pu et al., 2013). However, at ∼ 04:50:30 UT near the centre of the FTE flux rope, the anti-field-aligned flux was significantly higher than the field-aligned flux at energies of more than a few keV. Such electron pitch-angle anisotropy within FTEs generated through multiple X-line reconnection has been reported by Øieroset et al. (2011) and Pu et al. (2013). It indicated that the field lines near the FTE centre were open, connected to the northern ionosphere at one end and extending to interplanetary space at the other end, in this particular case.

Heated magnetosheath electrons were seen by THD throughout the FTE in both the field-aligned and anti-field-aligned fluxes (Fig. 8g and h). We note that these bi-directional heated electrons were observed on the magnetosheath side of the FTE centre where Bz<0, and thus were not due to THD crossing into the magnetosphere or its boundary layer. This feature can be taken as a signature of magnetopause reconnection on the field lines traversed by the spacecraft (e.g. Onsager et al., 2001). For this particular event, it indicated that reconnection occurred on both the northern and southern sides of the FTE, i.e. the observed FTE was formed by multiple X-line reconnection (e.g. Hasegawa et al., 2010b). These electron pitch-angle distribution signatures demonstrate that FTEs generated by multiple X-line reconnection can consist of field lines of various topologies, as shown by Pu et al. (2013). On the other hand, at ∼ 04:48:35 and ∼ 04:52:50 UT in the MSBL immediately on the magnetosheath side of the FTE, the field-aligned (southward streaming) electrons showed no heated magnetosheath population while the anti-field-aligned (northward streaming) electrons showed the heated population. This indicated that the MSBL field lines observed there crossed the magnetopause on the southern side only of the FTE (Hasegawa et al., 2010b).

THE was initially in the magnetosphere, dominated by ions and electrons of more than 1 keV, but after ∼ 04:44 UT was in the boundary layers, either earthward (Bz>0) or sunward (Bz<0) of the magnetopause (Fig. 8i–l). Both boundary layers are characterised by the coexistence of magnetospheric and magnetosheath populations. However, the fluxes of magnetospheric electrons and energies of magnetosheath ions were both somewhat lower in the MSBL, seen by THE after ∼ 04:53 UT when the field became southward, than in the boundary layer on the magnetospheric side. The magnetospheric electrons observed near the FTE centre (∼ 04:50:45 UT) had a pitch-angle anisotropy similar to that seen by THD at ∼ 04:50:30 UT; the field lines in the FTE core were open. Interestingly, the field-aligned and anti-field-aligned electron fluxes were roughly balanced throughout the MSBL interval of THE (Fig. 8k, l), suggesting that the MSBL field lines were closed. Possible explanations for this feature are given and discussed in Sect. 5. In summary, the THD and THE particle signatures are all consistent with the view that the FTE resulted from multiple X-line reconnection at the low-latitude magnetopause and consisted of a mixture of closed field lines and open ones with one end connected to the northern ionosphere and the other end extending to interplanetary space.

We have presented the first results of the data analysis technique, developed by Sonnerup and Hasegawa (2011), for reconstructing steady 3-D magnetohydrostatic magnetic field and plasma structures from dual-spacecraft observations. The method was applied to data taken at ∼ 04:50 UT on 27 June 2007 by the THC and THD probes of the THEMIS spacecraft during a flux transfer event (FTE). The event occurred at/around the subsolar magnetopause, when the IMF had a southward component and the two probes were separated by ∼ 390 km. The results can be summarised as follows:

Structure: the 3-D reconstruction results show that a magnetic flux rope with a diameter of ∼ 3000 km was embedded in the FTE. Comparison between the 3-D reconstruction and 2-D GS reconstruction results indicates that the flux rope had a substantial 3-D structure, not well described by 2-D models. The flux rope was elongated roughly in the dawn-dusk direction, with a left-handed chirality, i.e. the core field had a significant dawnward component.

Although the present version of 3-D reconstruction using two-spacecraft data is restricted to applications to steady magnetohydrostatic structures, it is in principle possible to extend the method to those applicable to steady fully MHD or Hall MHD structures. As discussed by Sonnerup and Hasegawa (2011), a more general case would be reconstruction of not only 3-D features but also time dependent effects, which requires the use of data from at least three closely spaced spacecraft. A difficulty associated with these more sophisticated reconstructions is that they require such well calibrated data, for all MHD or Hall MHD parameters (including ion velocity and electric field) and from a larger number of spacecraft, so that precise values of spatial and/or temporal gradients can be estimated. The present method requires magnetic field and pressure data of sufficient quality from only two spacecraft (or the field data alone under low β conditions). Another disadvantage would be that the equations used in the more sophisticated versions could have more singularities, leading to larger numerical errors. These are the issues that should be addressed in future studies.

It is worth noting that a fraction of the low-latitude boundary layer (LLBL) seen by THD and THE was characterised by approximately balanced fluxes of field-aligned and anti-field-aligned electrons (Figs. 8 and 9) and thus appeared to be on closed field lines, even under a southward IMF condition. The formation of such a closed LLBL requires magnetopause reconnection at more than one site, as proposed by Nishida (1989) and Song and Russell (1992), unless diffusion is responsible for its formation. Our observations indeed show that multiple X-line reconnection was involved in the generation of the observed FTE. Since the oppositely directed plasma jets emanating from different reconnection sites may collide, the associated open flux tubes may interact with each other and be entangled in a complex way (Fig. 4 in Nishida, 1989). Louarn et al. (2004) indeed reported a signature of such entangled or interlinked reconnected flux tubes. A possible scenario that can explain our observations is that reconnection of these open flux tubes resulted in the formation of the closed flux tube containing solar wind plasma and of the IMF-type flux tube (Fig. 2 in Nishida, 1989), thus creating the closed portion of the LLBL present in and around the FTE. For another possible way to create the closed field lines in the flux rope, see Fig. 2 in Pu et al. (2013). We point out that in an FTE reported by Øieroset et al. (2011), there was no evidence of reconnection in the central part of the flux rope flanked by two active X-lines where two reconnection jets and thus reconnected flux tubes from the two X-lines were colliding (Øieroset et al., 2014). Thus, the question of whether the Nishida mechanism works in reality remains open.

Another possible consequence of multiple X-line reconnection at the low-latitude magnetopause under southward IMF is less efficiency in the transfer of solar wind energy to the magnetotail than with the transfer resulting from single X-line reconnection (Hasegawa et al., 2010b; Hasegawa, 2012). Because an X-line may exist ahead of a reconnection jet emanating from another X-line, magnetic flux tubes reconnected and eroded on the dayside may be entangled or interlinked (Nishida, 1989) and may not be smoothly transported from dayside to nightside. Thus, if multiple X-line reconnection occurs more frequently under larger geomagnetic dipole tilt, as suggested by Raeder (2006), lowering the energy transfer rate, it may be that the total amount of solar wind energy deposited to the magnetotail is smaller during summer and winter, possibly contributing to seasonal variations of geomagnetic activities. Although it may be difficult to observationally confirm such a possible relationship among the occurrence of multiple X-line reconnection, dipole tilt, and solar wind energy transfer rate, global magnetospheric simulations may help to reveal or refute the connection.

Interesting questions that can be addressed with the 3-D reconstruction method are whether an FTE, as hypothesised originally by Russell and Elphic (1978), exists and to what extent such FTEs contribute to magnetic flux transport toward the tail. In the Russell and Elphic model, FTEs are generated when magnetopause reconnection occurs intermittently and locally on a short segment of single X-line, and are characterised by an elbow-shaped (namely fully 3-D) flux tube connecting the magnetosheath and magnetosphere. Such localised and transient reconnection is indeed shown to be able to produce a bipolar variation in the magnetic field component normal to the nominal current layer (e.g. Semenov et al., 1994; Shirataka et al., 2006). However, unambiguous identification of the Russell–Elphic type FTE would need to demonstrate that the local orientation of the flux tube axis at around the elbow has a nonzero angle with respect to the magnetopause surface. It requires accurate estimation of the magnetopause normal as well as of the orientation of the flux tube or rope. The latter would be possible by careful analysis of the magnetic gradient tensor that can be computed from the reconstructed data at any point in the 3-D reconstruction domain. We also need some reasonable way for determining the boundary of the flux tube or rope and magnetic flux content from the 3-D field data (and other additional information) in order to be able to assess the role of FTEs. These subjects will be pursued in a future study.

The 3-D magnetic field recovered by methods of the type presented here can in principle be used to calculate the spatial gradient of the field (Shi et al., 2005) and current density (j=∇×B∇×Bμ0μ0) at any point in the reconstruction domain, and also to identify magnetic nulls, separators, and quasi-separatrix layers (Priest and Demoulin, 1995; Cai et al., 2001; Xiao et al., 2006; Wendel and Adrian, 2013; Komar et al., 2013), key ingredients in 3-D reconnection that could potentially exist in the domain. Recent particle simulations show that 3-D dynamics can play an important role in the magnetic dissipation in collisionless reconnection (Che et al., 2011; Fujimoto and Sydora, 2012) and the evolution of reconnecting current sheets (Daughton et al., 2011, 2014; Nakamura et al., 2013). Our expectation is that the 3-D reconstruction technique, combined with other multi-spacecraft methods and data from NASA's forthcoming Magnetospheric MultiScale (MMS) mission (Burch and Drake, 2009; Moore et al., 2013), will facilitate our understanding of the 3-D aspects of magnetic reconnection.