Quasi-separatrix Layers Induced by Ballooning Instability in Near-Earth Magnetotail

Magnetic reconnection processes in the near-Earth magnetotail can be highly 3-dimensional (3D) 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 3D dynamics of nonlinear ballooning instability. In this work, we explore the global 3D 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 3D domain of magnetotail configuration, using an algorithm previously developed in context of solar physics. The 3D distribution of quasi-separatrix layers (QSLs) as well as their evolution directly follows 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 3D nature of the ballooning-induced plasmoid formation and reconnection processes, in both geometry and dynamics. In addition, the reconstruction of the 3D QSL geometry may provide an alternative means for identifying the location and timing of 3D reconnection sites in magnetotail from both numerical simulations and satellite observations.

This leads to general questions as to where and how reconnection takes place in the 3D configuration, as well as how the global structure of the 3D reconnection process can be characterized and captured in manners different from the more familiar 2D reconnection process. More fundamentally, it has remained unclear whether this 3D reconnection process can be reducible to or interpretable in terms of the conventional 2D reconnection processes.
Whereas the overall evolution of the magnetotail-like configuration has been studied in 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 literature, including the papers by, e.g. Birn and Hones, Jr. [1981]; Hesse and Birn [1991] which were reviewed in the book by Schindler [2007]. There is also a long history of work trying to identify the possible role of out-of-plane instabilities on reconnection (see for example, Pritchett [2013] and Sitnov et al. [2014]). Different from those previous work, in this work we intend to identify the geometry features associated with the intrinsically 3D reconnection process induced by the ballooning instability in near-Earth magnetotail, in light of those questions raised in the previous paragraph.
The quasi-separatrix layer (QSL) is a concept we adopt for the above purposes. In fact, QSL has long been a common and powerful tool for the analysis and understanding of magnetic structures in the solar atmosphere [Titov and Démoulin, 1999;Titov et al., 2002]. Recently the concept of QSL has also been effectively applied to the analysis of laboratory reconnection experiments [Lawrence and Gekelman, 2009]. Previously, we calculated the spatial distribution and the structure of the QSLs, as well as their temporal emergence and evolution, within the equatorial plane [Zhu et al., 2017] simulation results on the formation of plasmoids induced by ballooning instability in the magneotail [Zhu andRaeder, 2013, 2014]. There we found the QSL structures are not invariant along any direction within the 2D equatorial plane; instead they are disconnected and isolated local structures. Those initial findings start to reveal the intrinsic 3D nature of the reconnection induced by ballooning instability in the generalized Harris sheet, which is irreducible to 2D reconnection process in geometry and dynamics within the 2D equatorial plane. In this work, we extend our previous study within the 2D equatorial plane to the entire 3D domain of near-Earth magnetotail. Using a newly developed implementation for efficiently computing the squashing degree of magnetic field lines in any 3D domain [Liu et al., 2016], we obtain the 3D distribution of QSLs as well as their evolution in the near-tail plasma sheet. The intersection of the 3D distribution of QSLs with equatorial plane recovers results from our previous work. More importantly, the calculated 3D distribution of QSLs provides a complete and global view of the geometric structure of the 3D reconnections associated with the plasmoid formation induced by the nonlinear ballooning instability in the near-Earth magnetotail.
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 Sec. 2. Next in Sec. 3 we describe the method we use for efficiently evaluating the squashing degrees of entire magnetic fields. Both 2D and 3D distributions of QSLs revealed from the squashing degree calculation are reported and analyzed in Sec. 4.
Finally, summary and discussion are given in Sec. 5. Our recent MHD simulations are developed to demonstrate the dynamic process of plasmoid formation induced by nonlinear ballooning instability of the near-Earth magnetotail.

Plasmoid formation induced by ballooning instability
In these simulations, the magnetic configuration of near-Earth magnetotail is modeled using the generalized Harris sheet, which can be defined in a Cartesian coordinate system and λ is the characteristic width of the current sheet. The conventional Harris sheet is recovered when F (x) = 1. The configuration can be further specified with a particular B z profile that features a minimum region along the x axis, corresponding to an embedded thin current sheet ( Fig. 1), such as those often found in global MHD simulations and inferred from satellite observations in the near-Earth magnetotail.
For a sufficiently small magnitude of B z minimum, the magnetotail becomes unstable to ballooning instability, whose nonlinear development leads to the formation of tailward receding plasmoids in the magnetotail (

Methodology
To address these questions in this work, we for the first time, apply the concept of quasiseparatrix layer (QSL) to the analysis of the geometry of magnetic reconnection induced by ballooning instability in a generalized Harris sheet that represents the magneotail.
QSL has been adopted for the analysis of the reconnection structures involved in the solar corona for a long time (e.g. Titov and Démoulin [1999]; Titov et al. [2002]). It has also been effectively applied to the analysis of laboratory reconnection experiments [Lawrence and Gekelman, 2009]. A QSL is a 3D structure defined by steep gradient in the field line connectivity, which are quantified by mapping field lines across a specified volume.
A surface, S, must first be defined to enclose this volume. Divide S into two subspaces, S 0 and S 1 , where S 0 and S 1 represent the surfaces on which field lines enter and leave the volume respectively. The initial footpoint is defined as r 0 = (u 0 , v 0 ) in S 0 . One then traces the field line from the initial footpoint through the enclosed volume until the field line leaves the volume through S 1 at the point r 1 = (u 1 , v 1 ). The Jacobian transformation matrix and the norm of the mapping from (u 0 , v 0 ) to (u 1 , v 1 ) are defined as A QSL is the region where the gradient of this mapping is large compared to the average mapping, i.e. N >> 1.
Mathematically, the squashing degree Q is defined as Q = N 2 /|∆| where ∆ is the determinant of the Jacobian matrix [Titov et al., 2002;Priest and Demoulin, 1995]. tubes. A high squashing degree corresponds to a large variation in the cross-sectional area of an elemental flux tube from one footpoint to another. Quasi-separatrix layers turn into separatrices in the limit the layer thickness goes to zero, or the corresponding squashing degree goes to infinity. The physical significance of QSL is that current sheets preferentially form on these layers for reconnection.
A newly developed implementation for efficiently computing the squashing degree of magnetic field lines in any 3D domain has been successfully applied to investigating the evolution of magnetic flux ropes in coronal magnetic field extrapolated from photospheric magnetic field [Liu et al., 2016]. The method utilizes the field-line mappings between a cutting plane and the footpoint planes to give optimal results for mapping the squashing factor in the cutting plane. In order to avoid spurious high squashing degree structures for field lines touching the cutting plane, a new plane perpendicular to the particular field line can be introduced and switched to using the same method. We adopt this new method to recover our previous results on 2D QSL distribution based on the calculation of bald patches. We further use the new method to find the 3D distribution of QSLs in the entire domain.

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

2D spatial distribution of QSLs in equatorial plane
We first review the development of QSLs in the equatorial plane of magnetotail (i.e. z = 0 plane) based on the computation of squashing degrees, as shown in Fig. 4, for the same time sequence of nonlinear ballooning development that leads to the formation of tailward receding plasmoids in the magnetotail (Fig. 2). Similar results on QSLs are also obtained in our previous work, where the QSLs are identified based on the computation of bald patches [Zhu et al., 2017]. Here the QSLs are identified as the boundaries of white patches in a plane, on which the squashing degree becomes singularly large.
In the initial and early stage of ballooning instability evolution, QSLs are absent in the 11.0, and 14.3, which can be first seen from the squashing degree contours within the y − z planes (Fig. 10, upper row). This is in contrast with the earlier time at t = 190, when QSLs only appear in two y − z planes along the x axis (Fig. 9, upper row). At the same time, the three QSL regions also show up in the x − z planes, individually or together, depending on where the plane locates in the y direction (Fig. 10,

Summary and discussion
In summary, the 3D distribution of quasi-separatrix layers (QSL), 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 3D domain of the model near-Earth magnetotail in simulation. It is found that the 3D distribution of QSLs correlates well not only with the 2D mode structures of ballooning instability within the x − y plane, but also with the 3D ballooning mode structures as projected onto x − z and y − z planes, both spatially and temporally during the evolution of the magnetotail configuration. Such a close correlation demonstrates a strong coupling between the ballooning and the corresponding reconnection processes. It also further confirms the intrinsic 3D nature of the ballooninginduced plasmoid formation and reconnection processes, in both geometry and dynamics.
In addition, the reconstruction of the 3D QSL geometry may provide an alternative means for identifying the location and timing of 3D reconnection sites in magnetotail from both numerical simulations and satellite observations.
Whereas the near-Earth magnetotail can become ballooning unstable under substorm conditions, the nonlinear evolution of ballooning instabilities, by themselves, may not lead to the near-explosive substorm onset. Previous studies [Pritchett and Coroniti, 1999, 2010, 2013Zhu et al., 2004], have demonstrated the persistent presence of ballooning instabilities in generalized Harris sheet and magnetotail configurations. The models have varied from the global scales in the ideal MHD models, to the meso scales of 2-fluid X -16 ZHU ET AL.: BALLOONING INSTABILITY INDUCED QUASI-SEPARATRIX LAYERS models, and eventually to the microscopic scales of kinetic models of plasmas. Since the intrinsic 3D nature of the reconnection process reported in this work derives from the nature of ballooning instability, the global 3D geometry structure of the ballooning-induced reconnection process is expected to persist in presence of 2-fluid and kinetic effects, particularly on the macroscopic scales where both MHD and kinetic models should agree.
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 3D reconnections in nature and laboratory by identifying and characterizing 3D reconnection induced by ballooning instability.
Because the 2D 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 reconnection in two dimension and three dimension 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.