Magnetic reconnection processes in the near-Earth magnetotail can be highly three-dimensional (3-D) in geometry and dynamics, even though the magnetotail configuration itself is nearly two-dimensional due to the symmetry in the dusk–dawn direction. Such reconnection processes can be induced by the 3-D dynamics of nonlinear ballooning instability. In this work, we explore the global 3-D geometry of the reconnection process induced by ballooning instability in the near-Earth magnetotail by examining the distribution of quasi-separatrix layers associated with plasmoid formation in the entire 3-D domain of magnetotail configuration, using an algorithm previously developed in the context of solar physics. The 3-D distribution of quasi-separatrix layers (QSLs) as well as their evolution directly follow the plasmoid formation during the nonlinear development of ballooning instability in both time and space. Such a close correlation demonstrates a strong coupling between the ballooning and the corresponding reconnection processes. It further confirms the intrinsic 3-D nature of the ballooning-induced plasmoid formation and reconnection processes, in both geometry and dynamics. In addition, the reconstruction of the 3-D QSL geometry may provide an alternative means of identifying the location and timing of 3-D reconnection sites in the magnetotail from both numerical simulations and satellite observations.

There has been a long-standing controversy over whether the magnetic
reconnection or the ballooning instability in the magnetotail actually
triggers the onset of substorms, since both mechanisms found support in
observation and simulation (e.g.,

The overall evolution of the magnetotail-like configuration has been studied
for many years (

In comparison to the low-

Although our previous work has demonstrated in MHD simulations the formation
of plasmoids induced by ballooning instability in the generalized Harris
sheet

Whereas the overall evolution of the magnetotail-like configuration has been
studied in the space community for many years, the irreducible dimensionality
of the reconnection process associated with the evolution of ballooning
instability has never been addressed before in the literature, including the
papers by, e.g.,

Similarly to the magnetic island, the plasmoid presented in this work is
identified in the

The QSL has long been a powerful concept and method for the analysis and
understanding of magnetic structures in the solar
atmosphere

The rest of the paper is organized as follows. We first briefly review our previous simulation results for the plasmoid formation process induced by ballooning instability in Sect.

Our recent MHD simulations are developed to demonstrate the dynamic process
of plasmoid formation induced by nonlinear ballooning instability of the
near-Earth magnetotail. In these simulations, the magnetic configuration of
the near-Earth magnetotail is modeled using the generalized Harris sheet,
which can be defined in a Cartesian coordinate system as

For a sufficiently small magnitude of the

Total pressure contours in the

Magnetic field lines crossing lines

To address these questions in this work, we for the first time apply the
concept of a quasi-separatrix layer (QSL) to the analysis of the geometry of
magnetic reconnection induced by ballooning instability in a generalized
Harris sheet that represents the magnetotail. QSL has been adopted for the
analysis of the reconnection structures involved in the solar corona for a
long time (e.g.,

Mathematically, the squashing degree

A newly developed implementation for efficiently computing the squashing
degree of magnetic field lines in any 3-D domain has been successfully
applied to investigating the evolution of magnetic flux ropes in a coronal
magnetic field extrapolated from a photospheric magnetic
field

In this section, we compute the squashing degrees and analyze the 2-D and 3-D QSL distributions of the magnetic field configuration as well as its evolution in the near-Earth magnetotail, in an attempt to understand the global geometry of the magnetic field and the 3-D nature of the magnetic reconnection process in association with the plasmoid formation process induced by ballooning instability.

We first review the development of QSLs in the equatorial plane of the
magnetotail (i.e., the

Contours of the logarithm of squashing degree in the

In the initial and early stages of ballooning instability evolution, QSLs are
absent in the

Even within the 2-D equatorial plane (

Surface plots for the logarithm of squashing degree in the

We further examine the 3-D distribution of QSLs in the entire simulation
domain of the magnetotail. Not only are QSLs located in isolated regions in
the 2-D plane, but they are also localized in isolated and confined regions
in the 3-D domain (Fig.

Iso-surfaces of the logarithm of squashing degree in the 3-D domains
centered at

Iso-surfaces of the logarithm of squashing degree in the broader 3-D
domains, which include two periods of repeating QSL distribution from

Another approach to characterizing the 3-D distribution of QSLs in the
near-Earth magnetotail is to examine the squashing degree contours on various
strategically selected 2-D slices parallel or perpendicular to coordinate
axes. For example, at an earlier time

Contours of the logarithm of squashing degree in the

The above approach also helps in visualizing the development of 3-D
distribution of QSLs over time. At a later time

Contours of the logarithm of squashing degree in the

In summary, the 3-D distribution of quasi-separatrix layers (QSLs), as well
as its evolution directly following the nonlinear development of ballooning
instability in the near-Earth magnetotail, has been thoroughly evaluated and
examined based on previous resistive MHD simulation data on the plasmoid
formation process induced by the ballooning instability. The quasi-separatrix
layers have been identified by locating the regions of high squashing degree
throughout the entire 3-D domain of the model near-Earth magnetotail in
simulation. It is found that the 3-D distribution of QSLs correlates well not
only with the 2-D-mode structures of ballooning instability within the

Whereas the near-Earth magnetotail can become ballooning unstable under
substorm conditions, the nonlinear evolution of ballooning instabilities, by
themselves, may not always lead to the near-explosive growth. The coupling
between ballooning and reconnection could be an alternative, though not the
necessary, route to substorm onset. Previous studies

Although this work was in part motivated by the substorm problem in magnetospheric physics, it should not be seen as one confined only to the space plasma physics community. Rather, with our first application of QSL to the magnetotail configuration represented by the generalized Harris sheet, this work provides new insight into the ubiquitous 3-D reconnections in nature and laboratory by identifying and characterizing 3-D reconnection induced by ballooning instability.

Because the 2-D perception of magnetic reconnection has been the conventional paradigm for interpreting and understanding most phenomena and processes associated with reconnection in both natural and laboratory plasmas since the beginning, our work and results provide a dramatically different and refreshing view on one of the most fundamental processes in all plasmas. It touches the core question as to what exactly defines a reconnection or whether reconnections in two dimensions and three dimensions are qualitatively different. Different answers to such a question can lead to vastly contrasting or contradicting interpretations and conclusions. These issues would continue to be addressed in future work.

The QSL method may potentially be applied to in situ observation data
analysis as well, since it is the knowledge of the magnetic field lines'
connectivity itself only that is required for the calculation of QSLs. The in
situ observation data from both single-point and multi-point spacecraft
measurements, with additional assumptions and modeling, have been used in
various reconstruction methods for the magnetic field-line geometry in the
magnetotail. These include the global MHD simulations of magnetotail
evolution calibrated using the in situ observation data in general
(e.g.,

Data used in this study are available for download at

PZ was responsible for providing the simulation results and the original idea, the planning, coordinating, and executing of the overall QSL analyses, as well as the writing of the entire paper.

ZW performed the QSL calculations and plotted the corresponding results.

JC provided the Fortran and IDL programs for QSL calculations.

XY converted the simulation data into the format for QSL calculations and aided in figure revisions and paper preparation.

RL was responsible for the QSL algorithm implemented in the Fortran program used in the QSL calculation and contributed to paper writing, revisions, and responses to the referees.

The authors declare that they have no conflict of interest.

The computational work used the NSF XSEDE resources provided by TACC under grant no. TG-ATM070010 and the resources of NERSC, which is supported by the DOE under contract no. DE-AC02-05CH11231.

This research has been supported by the National Natural Science Foundation of China (grant no. 41474143) and the U.S. Department of Energy (grant nos. DE-FG02-86ER53218 and DE-SC0018001).

This paper was edited by Christopher Owen and reviewed by Andrei Runov and one anonymous referee.