The nonlinear dynamics of electrons in the vicinity of magnetic field neutral lines during magnetic reconnection, deep inside the “diffusion” region where the electron motion is nonadiabatic, has been numerically analyzed. Test particle orbits are examined in that vicinity, for a prescribed planar two-dimensional magnetic field configuration and with a prescribed uniform electric field in the neutral line direction. On electron orbits, a strong particle acceleration occurs due to the reconnection electric field. Local instability of orbits in the neighborhood of the neutral line is pointed out. It combines with finiteness of orbits due to particle trapping by the magnetic field, and this should lead to the effect of mixing in the phase space, and the appearance of dynamical chaos. The latter may presumably be viewed as a mechanism producing finite “conductivity” in collisionless plasma near the neutral line. That conductivity is necessary to provide violation of the magnetic field frozen-in condition, i.e., for magnetic reconnection to occur in that region.

The problem of magnetic reconnection in the vicinity of the neutral line for
collisionless plasma has an extremely extensive literature; see, for example,
the special issue of Space Science Reviews 2011 and the work (Hesse et al.,
2011) and references therein. According to the prevailing ideas, the
reconnection process proper occurs in the central part of a current sheet
(CS) separating the regions with oppositely directed magnetic field, and is
provided by processes on a very small electronic scale in the direction

By numerical simulation in models of different types it was found that the
behavior of the system in the larger Hall region is quite similar for
different types and intensity of dissipation in the smaller inner region.
Moreover, the reconnection rate itself is insensitive to the nature and
intensity of the dissipation (Hesse et al., 2001; Liu et al., 2017). A fairly
simple qualitative explanation can be given to that. It turns out that the
outflow of the electron fluid from the inner (dissipation) region has the
character of a strong standing whistler wave. The phase velocity of the
whistler

Note, however, that such a situation does not mean that the real nature of electron dynamics and corresponding features of electron orbits, which lie at its basis, may be thought insignificant.

Outside the diffusion region, the process of “magnetic annihilation” – the transformation of electromagnetic energy into the energy of plasma flows, meaning magnetic reconnection in the wider sense, is determined by ion motions, including those that are substantially nonadiabatic in thin layers adjacent to the diffusion region. As was pointed out in Domrin and Kropotkin (2007a, b, c), Kropotkin and Domrin (2009), and Kropotkin (2013), in that region the process is dominated by an “anisotropic” CS, the structure of which is determined by the specific ion orbits. This leads to ion anisotropy of a certain type, dependent on the distance from the central plane. However, inside the diffusion region the process of magnetic reconnection requires an “intermediary”, an additional link. This is the abovementioned Hall MHD structure, based on specific differences between the electron and ion orbits. And finally, deep inside this layer, in the nearest neighborhood of the neutral line, unmagnetized electron orbits determine the dynamics and structure of the reconnecting CS.

It is precisely the specific nature of such orbits, which in previous studies has remained mostly unidentified in a proper perspective, that this paper is devoted to. Basically, the numerical approach applied in this paper differs only weakly from that adopted in a number of earlier studies (Martin, 1986; Burkhart et al., 1991, and references therein). Moreover, the chaotic particle dynamics were then identified numerically by means of Lyapunov characteristic exponent analysis. However, the insight which appeared since that time, concerning the role of the electron zone in the “diffusion region”, on the one hand, and penetration of the nonlinear dynamics and dynamical chaos notions into this research area, on the other hand, has laid the way to a number of new results.

In this paper, the nonlinear dynamics of electrons moving in a magnetic field near its neutral line and in an electric field corresponding to the inflow of plasma into the reconnection zone has been studied numerically in a wide range of determining parameters (Sect. 2). Of course, these dynamics do not obey the adiabatic theory; its important property is a strong particle acceleration near the field neutral line. Another important feature of particle dynamics near the neutral line is pointed out in Sect. 3. Analysis of an equation governing such dynamics, involving the numerical results of Sect. 2, indicates an exponential divergence of orbits starting closely nearby. This feature of local instability, along with finiteness of the orbits in the phase space, in the general theory of nonlinear dynamical systems, should result in the appearance of stochastization, i.e., of dynamical chaos in the system. This is discussed in Sect. 4 in terms of the phenomenon of mixing in the phase space and formation of the collisionless energy dissipation mechanism. In this way we obtain a corroboration on the side of microscopic dynamics that the diffusion region should have a dissipation property in the macroscopic sense, i.e., provide the transformation of electromagnetic energy (a nonzero Poynting vector in the inflow region) into energy of accelerated particles.

We study test particle orbits in a prescribed field configuration. We use a
two-dimensional model of magnetic field

Equations (1) must be supplemented by initial conditions, i.e., by setting the
values

We have studied a set of possible orbits over a wide range of parameters,
including the values

In the simplest situation, it can be assumed that a flow of plasma with cold
electrons arrives at the CS. Then, if an electron starts somewhere away from
the CS, it must be postulated that its velocity

For solution of the Equation set (1) with corresponding initial conditions and for graphical presentation of results, the Wolfram Mathematica package has been used. Accuracy of the codes applied in Mathematica calculations was of course adequately verified by the Wolfram team long ago; it is quite sufficient for the presented graphics. In our work, in some cases that accuracy has been tested by means of varying the parameters used.

First of all, let us turn to the case

Now consider the results of numerical calculations relating to how the electron orbits in general look and how their character changes when the initial conditions change. From a large number of calculations performed, here we illustrate several characteristic series.

Consider in more detail the orbits discussed above: we set again

An initial course of the

The orbit for

Having presented in Fig. 2d a plot of time dependence for the particle
kinetic energy

The orbit pattern at

The orbits of electrons that are “cold” at the start retain a similar
character up to the values

The orbit pattern at

With an even greater initial distance from the

The orbit pattern at

When specifying nonzero initial speeds

So, in general, for all variations of the parameters, we observe the same
effect: acceleration of electrons by the electric field when they appear in
the vicinity of the neutral line; the picture is somewhat different for a
single or multiple intersection of the neutral plane. At very small

We return to the case of an orbit starting very close to the

Now for any small deviation

On a small time interval

So if we fix some time moment

In such a situation we may speak of the presence of a

A clear confirmation of instability at

Note that the diverging orbits turn out to be limited (generally speaking, in the phase space) because of cyclotron rotation.

As has been shown earlier (see, e.g., Burkhart, 1990), the motion of an electron in the neutral line vicinity being nonadiabatic, is described by nonlinear equations, and on the orbit there is a “Speiser” meander section on which a fast particle acceleration occurs, in the electric field corresponding to the inflow of plasma into the reconnection area. This should result in a strong conversion of energy of the electromagnetic field into energy of the electron flows.

We point out here that this effect is accompanied by another one which is very important from the point of view of a general theory of nonlinear dynamical systems.

It is well known that even in the case of a relatively simple system of low dimension, there may exist such domains in its phase space, where the orbits are stochastic. We point out that an electron in our model field, in the neutral line vicinity is just such a system.

Consider a pair of orbits in the

Turning back to our case, we have seen in Sect. 3 that in the neutral line
vicinity there is an area in the phase space where local instability acts. It
means that the distance

Note that the existence of local instability, in the terms of a positive Lyapunov characteristic exponent, was first pointed out for this problem in Martin (1986).

As has been noted earlier, the local instability of orbits takes place
along with finiteness of motion in the phase space due to the particle
trapping by the magnetic field. From the general theory of nonlinear
dynamical systems (see, e.g., Tabor, 1989; Ott, 2002; Usikov et al., 1988) it
then follows that the system possesses the corresponding Kolmogorov–Synai
entropy of the order of the maximum positive Lyapunov exponent,

In the case of a somewhat similar situation, a

If in the model we are considering, with a nonuniform

However, in the model considered here, in the presence of a reconnection
electric field, those effects, as we have seen, are supplemented by the local
instability effect near the neutral line. A qualitative explanation may be
given following the notions of, for example, Usikov et al. (1988) concerning the role
in the effect of stochasticity generation which is played by a separatrix
existing in the phase space of a nonlinear system. “Slow” oscillations over

According to modern knowledge (see, for example, Gaspard, 1998)

Thus we obtain an indication of the fundamental dynamical basis of the
collisionless magnetic reconnection process, which

Now we note an analysis somewhat similar to ours was recently carried out in Zenitani and Nagai (2016). A set of electron orbits passing in the vicinity of the magnetic field neutral line has been identified there under conditions when this field itself, as well as the electric field and the plasma characteristics, is determined in numerical simulation of the entire three-dimensional plasma system. Modeling was carried out by means of a particle-in-cell code applied to a time-dependent spontaneous magnetic reconnection process in which an electric field arises in a self-consistent manner as a result of development of an initially small perturbation, rather than being arbitrarily given, as is done here.

Electron orbits are grouped in this work according to some of their basic
characteristics. There are classes of orbits qualitatively corresponding to
those obtained by us, with “Speiser” sections and fast acceleration in the
vicinity of the neutral line. Along with them, orbits of a substantially
different type, named “Speiser orbits without intersection of the central
plane”, were obtained. Those orbits appear due to the fact that in the model
(Zenitani and Nagai, 2016) there are such electric fields that are absent
here: the polarization field

However, the approach used in Zenitani and Nagai (2016) does not
allow the systematic tracing of how the orbit nature and the acceleration
gained by an electron depend on the distance to the neutral line, as we have
done here. This dependence could not be traced down to very small

One more important point, which has a wider relation to particle-in-cell
(PIC) codes, is that the formation of a small-scale (

An additional comment should be made here, in particular in relation to
self-consistent models of magnetic reconnection. In fact it is just assumed
that any electron orbit under study is located deep inside the small
electron diffusion region. There is no attempt to relate the adopted
dimensionless parameters to physical scales characteristic for that domain.
True, this is an ambiguous task since those scales depend in particular on
the (numerical) model adopted for such a comparison. In any case it is reasonable
to postulate that such a small electron diffusion region exists. Note, however,
that if the current carried by unmagnetized electrons themselves were large,
then the magnetic field configuration would be more complicated: it would
have a spatial scale of the same order as the electron oscillations about the
neutral plane of the CS. Then a study based on test particle orbits in a
prescribed field would be inapplicable. So another assumption, implicitly
made, is that the currents forming the magnetic field configuration, are
carried mainly by ions, and not by unmagnetized electrons; correspondingly,
the field configuration spatial scales are sufficiently large. This actually
is the case for the existing self-consistent models of magnetic reconnection
(e.g., Hesse et al., 2011; Bessho et al., 2014; Liu et al., 2017). This may
also eliminate the problem with those electric fields mentioned above which
appear in self-consistent models, the polarization field

Magnetic reconnection in collisionless plasma may be considered as a complex
of two main problems. The first problem is fast energy conversion. As has
been shown in earlier studies and has been pointed out in the Introduction, this
may be understood as a result of “magnetic field annihilation” dominated by
an “anisotropic” thin current sheet, the structure of which is determined
by the

No data sets were used in this article.

The author declares that they have no conflict of interest. The topical editor, Christopher Owen, thanks one anonymous referee for help in evaluating this paper.