Magnetic reconnection is a crucially important process for energy conversion in plasma physics, with the substorm cycle of Earth's magnetosphere and solar flares being prime examples. While 2D models have been widely applied to study reconnection, investigating reconnection in 3D is still, in many aspects, an open problem. Finding sites of magnetic reconnection in a 3D setting is not a trivial task, with several approaches, from topological skeletons to Lorentz transformations, having been proposed to tackle the issue. This work presents a complementary method for quasi-2D structures in 3D settings by noting that the magnetic field structures near reconnection lines exhibit 2D features that can be identified in a suitably chosen local coordinate system. We present applications of this method to a hybrid-Vlasov Vlasiator simulation of Earth's magnetosphere, showing the complex magnetic topologies created by reconnection for simulations dominated by quasi-2D reconnection. We also quantify the dimensionalities of magnetic field structures in the simulation to justify the use of such coordinate systems.

A long-standing issue in analysing data from magnetospheric simulations has been the identification of topological features in the magnetic field, especially in relation to reconnection and flux transfer events. These features include magnetic neutral lines; separators; and, loosely, X and O lines. In a 2D configuration, the problem is tractable via the use of a flux function (see

Reviewing the definitions of the topological structures of reconnecting 3D magnetic fields by

As, for example,

Closely related to the concept of separators are the magnetic X and O lines. As the intersections of separatrix surfaces, the separators delimit different magnetic domains. In the classic picture of reconnection, inspecting the magnetic field lines on a plane normal to the tangent of the reconnection line displays an X topology, with the in-plane components of

For magnetic field structures such as current sheets (CSs), null lines and flux ropes, it is useful to select a suitable coordinate system that can exploit the properties of the structure to simplify the problem at hand. Methods such as minimum variance analysis (MVA,

A schematic view of Earth's magnetosphere under southward-IMF driving, with zoomed-in detail panels of the dayside magnetopause and the magnetotail, sketched on the Vlasiator run described by

Global magnetospheric simulations use global coordinate systems such as the Geocentric Solar Ecliptic (GSE) or Geocentric Solar Magnetospheric (GSM) systems, similarly to spacecraft observations. However, local magnetic structures are not necessarily aligned with the axis of the simulation coordinate system. In particular CSs, flux ropes and reconnection topologies can develop unconstrained by the domain boundaries and can be oriented in virtually any direction. In this situation, methods developed to analyse spacecraft data can inspire new ways to investigate data from global simulations.

To review our region of interest, Fig.

Schematic drawing of the

In this paper we introduce and discuss alternative, physically motivated local methods for finding X and O lines in the magnetic field of global 3D magnetospheric simulations by inspecting the magnetic field on a suitable local coordinate basis. To extend the global proxies used in

Let us consider two methods reviewed by

MGA produces a set of basis vectors where the eigenvector that corresponds to the largest eigenvalue (

Both of these eigensystems have the same eigenvalues, but the eigenvectors differ and are not necessarily aligned with each other. We note here that, for a 1D structure, both of these eigensystems have only one well-defined eigenvector. For example, in a 1D

MDD also gives us a way to define the local dimensionality of a structure

Definitions for

These quantities are defined to lie in the range

Neutral point classification in a well-defined local coordinate system. On the plane where

In this study, the methods discussed above will be applied to a global simulation of Earth's magnetosphere performed with the hybrid-Vlasov code Vlasiator. Vlasiator is a supercomputer-scale 6D (3D+3V – three spatial dimensions and three velocity-space dimensions) ion-hybrid plasma simulation

We use the global Vlasiator simulation described by

Despite the solar wind and IMF being constant, the kinetic physics involved in the simulation still produce a remarkably dynamic environment.

If the eigenvalues are not well-separated, the directions obtained from MGA and MDD may be ambiguous

The electric current density

To summarize the definition of our local LMN coordinate system

Lastly,

Absolute value of the dot products of the primary and secondary vectors

The sign of

Figure

While the above local coordinate system is defined where there is some variation in

In

A Vlasiator tail CS (

The local coordinate system, as given above, allows us to define an LMN system at each spatial cell of the simulation, analogously to the definition at each time in a time series for spacecraft observations. Finding

In

Finding the surfaces with

Above, and in general, we assume the existence of the normal component

To use the LMN coordinate system and the components of magnetic field

Schema of LN sign ambiguity and its effect. The signs of L and N vectors are not fixed by the construction and may vary arbitrarily. On the left is a correct evaluation of a

When there is no consistent orientation available, naïve contouring does not work in an off-the-shelf manner. For example, in the case of the upstream solar wind, where Vlasiator shows very little variation in IMF, the LMN coordinate system is dominated by numerical precision artefacts, leading to inconsistent orientation showing up as noise in these contouring methods.

An alternative that does not require choices on orientation is to use the local data of

The line where

Further, we may evaluate the derivative

We take the southward-IMF initial configurations of

Figure

Figure

Figure

To validate the observed null lines with both methods in the case of the tail CS, Fig.

Dawn flank of the magnetosphere, with FOTE detections as outlined cells, using the

We may note that the local coordinate systems spanned by our LMN basis describe well the in-plane X and O lines in the simulation. However, this leads to the observation that the X and O line axes might not be aligned with the

To extend our analysis to the global Vlasiator magnetosphere, we use the generic FOTE method that is not limited by construction of consistently oriented neighbourhoods. The FOTE method reveals a sinuous and alternating neutral line structure on the magnetopause flanks, shown in Fig.

Regions with small values of

Overview of local MDD dimensionality, shown on the planes

The applicability of the LMN coordinate system should be inspected in terms of dimensionality of the magnetic structures to ensure that the structures inspected are locally quasi-2D. Figure

Local dimensionality of a Vlasiator simulation obtained via the MDD analysis; ternary histogram over most of the magnetosphere. Each cell outside of the inner boundary with

An interesting finding in the dimensionality parameters in the 6D Vlasiator run can be observed in the lack of dominantly 3D structures, as shown in Fig.

Here, we have presented two local methods to acquire X and O topologies of the magnetic field from plasma simulations of Earth's magnetosphere, enabling fast and efficient identification of possible reconnection sites and flux transfer events. The algorithms work on O(N) complexity, i.e. operating once per grid cell (and possibly using a contouring algorithm of similar complexity) without requiring field line or separator tracing such as that used by

Using the dimensionality measures introduced by

The construction of the local coordinate system involves some choices. The first in-plane direction

For the contouring method, the limitation of requiring consistently oriented local coordinates is a slight issue. It could be mitigated by automated construction of local neighbourhood charts of consistent orientation, as well as automatically finding the extents where the local coordinate charts are sensible. As presented here, the contouring method relies on manual restriction of neighbourhoods for analysis. The FOTE method is given as a generalized alternative, operating cell-wise in the full domain, with the caveat that, in contrast to the contouring method, the FOTE method does not produce topological connectivity for the in-plane null lines; that is, it can be used to mark cells containing null lines but not which neighbouring cells those lines connect to. In contrast, the contouring method outputs line topologies from the isocontours of

In comparison to topological methods, such as the magnetic skeleton method of

The presented FOTE and contouring methods are fundamental to investigate magnetic structures in a physically meaningful coordinate system, supporting reconnection studies in quasi-2D current sheets. The quantification of dimensionalities of magnetic structures may also provide new insights into plasma processes: for example, the magnetospheric tail lobes were found to be characterized by

Vlasiator is an open-source model, made available via Zenodo by

MA, GC, IZ and FTK conceptualized the study, with MA, GC and IZ developing the methodology. MA wrote the original draft, implemented and applied the analysis tools, and performed validation and visualization. SH contributed to conceptualization and validation. UG, MB, YPK and KP developed and performed the relevant supercomputing simulation. JS and MA, along with UG, MB, YPK and KP, provided data curation. MP provided project administration, funding acquisition and supervision. All the authors reviewed the paper.

At least one of the (co-)authors is a member of the editorial board of

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We acknowledge the European Research Council for the grants used to develop Vlasiator and PRACE for the supercomputing resources used for running the global simulations (see below for grant information). The authors wish to thank CSC – IT Center for Science in Finland, the Finnish Computing Competence Infrastructure (FCCI) and the University of Helsinki IT4SCI group for supporting this project with computational and data storage resources. The scientific colour maps by

This research has been supported by the Research Council of Finland (grant nos. 336805, 328893, 335554, 322544, 339327, 339756 and 345701); the European Research Council, FP7 Ideas: European Research Council (grant no. 200141); the European Research Council, H2020 European Research Council (grant no. 682068); and the Partnership for Advanced Computing in Europe AISBL (grant no. 2019204998).Open-access funding was provided by the Helsinki University Library.

This paper was edited by Christopher Mouikis and reviewed by two anonymous referees.