Each of the spoke to the detriment the 33 and highways have caused and continue to cause to Buffalo, from ripping neighborhoods in half to contributing to severe urban sprawl. They also shared their inspiring vision of the future. Brian Dold shared the rich history of the Olmsted Park system and its importance of creating a tight-knit and proud community system.

Stephanie Barber Geter addressed the importance of reconnecting neighborhoods and the opportunity to replace blighted properties and vacant spaces with thriving local business. We want to thank everyone who attended, shared their voice, and continue to support our movement to make Buffalo and the Scajaquada Corridor stronger. If you want to continue to stay involved, please take our survey. Their problems consist in their high time-variability, the impossibility of preparing a controlled initial state, and last not least, the relatively sporadic availability of data.

Their main deficiency is that they are subject to passive point-like observations only taken in small regions of space and at stochastically distributed times when the spacecraft incidentally passes close to a reconnection site. The point-like character of the observations is imposed by the miniscule scale of the spacecraft in comparison to all spatial scales in dilute space plasmas. Though at first glance this seems of advantage, the high time variability of space plasmas together with the notorious low instrumental resolution and the enormous amount of data flow generally inhibit any wanted high space-time mapping.

Nevertheless, over the last four decades of the unmanned space-flight age large amounts of observational data have been collected and accumulated which, already at the present time, allow drawing a preliminary though by far not ultimate picture of collisionless reconnection. Reconnection, understood as the dynamics of magnetic fields in electrically conducting media is a problem of electrodynamics. Its current density shall be determined from the dynamics of the differently charged components of the plasma as function of the self-consistent electric field under the boundary condition that the magnetic fields sufficiently far outside the current layer are directed antiparallel.

At a later occasion we will return to this important point in relation to reconnection as, so far, it has never ever been properly addressed. All fluid physics is contained in Ohm's law J E. The field is forever frozen to the fluid, and there is no reconnection. Moreover, the induction equation tells that an initially non-magnetic plasma in the current sheet can by no means become magnetized unless magnetization currents flow at it boundaries. Reconnection thus requires that the current sheet itself is magnetized. Applying its stationary version dimensionally to a plane current sheet under the condition of continuity of flow just reproduces the reconnection rate of the Sweet-Parker model due to resistive diffusion of the magnetic field, a slow process.

Thus, even though one may speak about the diffusion of the field into the current sheet, the process of reconnection itself lies outside of MHD. The simplest generalized collisionless Ohm's law is provided by two-fluid theory. We added a ponderomotive force term—indexed pmf. All terms on the right enter Faraday's law and produce a substantially more involved induction equation than Equation 9. Application of the curl operation makes all irrotational contributions vanish, leaving just the rotational terms contributing to the inductive electric and magnetic fields.

A reconnection theory can be based on each of the terms. It is instructive to consider each of the terms separately. In collisionless reconnection, the first term is non-zero only when the plasma develops an anomalous resistance. This requires a separate investigation which we delay to the end of this section. Near the reconnecting current sheet electron and ion motions decouple almost completely on the inertial scale of the ions.

Ions decouple from the magnetic field. They constitute the Hall current [ 34 ]. This Hall resistance is an inductive blind resistance and plays no role in the generation of heat. The Hall current evolves mainly in the absence of binary Coulomb collisions and, assuming that anomalous resistance is negligible, has been taken as evidence for collisionless reconnection in Earth's magnetotail [ 35 — 37 ]. It has been reclaimed also for the magnetopause [ 38 ] of providing evidence for the existence of an electron diffusion layer there.

Though an electron diffusion layer is reasonable and is supported by other observations, the claim of a Hall magnetic field and current system is to be taken with care because the Hall magnetic field in these particular observations is stronger than the main field almost everywhere which is inconsistent with the fact that Hall currents are secondary effects and, therefore, their field must necessarily be weak at any location.

Hall currents near a reconnection site generate a Hall magnetic field in the direction of the current flow perpendicular to the symmetry plane of the external magnetic field but parallel to the current and the convection electric field E. This Hall field has quadrupolar structure and, though necessarily being weaker than the external magnetic field, introduces a guide field into reconnection [ 40 ], opening the possibility of accelerating electrons along the guide field in the direction anti-parallel to the convection electric field.

This is shown in Figure 3. Figure 3. Geometry of the Hall effect the X-line in collisionless reconnection. Hall currents J H are centered on the separatrices forming four half-loops. They generate the quadrupolar Hall magnetic field B H. Open arrows show the inflow at low speed V in and the outflow at high speed V out.

The convection electric field points out of the plane. Ions are accelerated in this field out of the plane thereby contributing to the sheet current. This contribution effectively causes a broadening of the current layer. In addition, Hall currents serve as sources of various plasma instabilities like the electrostatic modified-two stream instability or the electromagnetic Weibel instability [ 41 ].

Otherwise Hall currents do not directly contribute to the excitation of reconnection. This is contrary to what has sometimes been claimed by the defenders of Hall reconnection. These conclusions have been confirmed by subsequent similar simulations [ 44 ] using substantially smaller particle numbers and less resolution but arriving at the same result cf. An important side product of the Hall currents is provided by the necessity of current closure. Such closure proceeds preferably along the separatrix magnetic field with the field-aligned current carried by electrons e.

The field-aligned currents also contribute to the guide field; this is, however, a second-order effect. The main function of the field-aligned current is to couple the reconnection site to its external environment, possibly giving rise to important secondary effects far away from the reconnection region. In the magnetosphere such effects are observed as aurorae during substorms, particle precipitation, chains of electron holes, excitation of near-Earth plasma instabilities and electromagnetic radiation.

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In the solar atmosphere they serve as sources for solar radio emissions like, for instance, Type III bursts. Since no steep gradients neither in density nor temperature are expected near the reconnection site, the pressure term in a weakly inhomogeneous plasma contributes only when the pressure tensor is highly anisotropic or contains non-diagonal elements. If at all, contribution will come only from the electron pressure term P e which, in the center of the current layer, may develop inhomogeneity and possibly even electron-viscosity, generating non-diagonal elements.

The latter, in particular, have been attributed to be important drivers of reconnection cf. It is, however, not completely clear whether non-diagonal pressure terms based on physical effects will indeed be generated at a reconnection site. All wave-particle interaction-based non-linear plasma effects that might generate viscosities here are probably too weak though Divin et al. The opinions are ambiguous on this matter. Numerical particle-in-cell simulations [ 52 ] do indicate that it is indeed the pressure tensor effect which takes responsibility for collisionless reconnection. Karimabadi et al.

The case is a bit obscured by the imprecise definition of the main axes of the pressure tensor near a reconnection site. The natural reference system is thus not that of the magnetic field but that of the current sheet instead, centered at current maximum. Accounting for the three different non-diagonal elements in this geometric representation of the tensors, pure anisotropic pressure effects will naturally become important.

The decision about the role of pressure needs clarification of which frame of reference has to be considered as basic in reconnection and how pressure anisotropy either is generated or maintained in the electron diffusion region. In fact, simulations seem to confirm that both is the case thus, apparently, supporting pressure-driven reconnection in collisionless plasma configurations to be a general, possibly even the canonical mechanism in collisionless non-forced reconnection.

This measure is the unrecoverably dissipated Joule energy, the scalar quantity. The last term is not necessarily zero because in the reconnection region electric fields arise partly due to charge separation. These quantities are useful in mapping the dissipation region both in measurements as in simulations thus identifying the physical reconnection site.

Observations [ 54 ] support this measure in symmetric reconnection settings like in the magnetotail. In non-symmetric reconnection, however, the measure seems not as useful, as has been discussed by Pritchett [ 55 ]. This time is usually long compared with the period of plasma oscillations. Therefore, moderately fast current oscillations could well contribute to inertial resistivity providing sufficient inertial diffusivity for reconnection.

The reconnection rate is then found of order. It is instructive to ask for the role of the reconnection electric field E rec. This is most simply seen in two dimensional reconnection. We explicitly wrote out the only two remaining non-diagonal electron pressure terms neglecting ion pressure. Otherwise, the non-diagonal elements may also have been produced by finite-gyro-radius effects causing non-gyrotropy as argued in [ 49 , 53 , 56 ].

E rec is determined by the divergence of the electron pressure, i. Moreover, an inertial term contributes as well but is important only in the electron diffusion region where inertia comes into play, with the time derivative being of lesser importance. Pritchett [ 57 ], in an important paper, investigated the separate contributions of these terms in two-dimensional non-forced simulations, finding that electron pseudo-viscosity and inertia both contribute to E rec , compensating for the convection term.

Their contributions are located in the electron diffusion region and have a pronounced spatial dependence. They are responsible for the breaking of the frozen-in state of electrons here. This gives support to the view that, in anisotropic plasma, collisionless reconnection is indeed made possible by electron pseudo-viscosity see also [ 49 ] , aided by non-linear electron inertia the non-linear term in the inertial contribution to Equation Apparently except for the ponderomotive term all other terms in the induction equation provide minor corrections.

One may even go further in concluding that any resistive reconnection is probably unrealistic. Resistive dissipation in a plasma will always be secondary to the simple electron pressure effects which generate electron pseudo-viscosity on the electron inertial scale. Reconnection proceeds, presumably, always on the micro-scale of the electron skin depth. Ohm's law Equation 10 contains the flow velocity, density, current and magnetic field. In a mean-field theory of reconnection, the non-linear terms in Ohm's law then contribute additional correlation terms which, when violating the frozen-in condition, can also become sources of turbulent reconnection.

Here they provide a collisionless fluid means of driving reconnection see, e. Similar effects are provided by the ponderomotive term. Expressions for the ponderomotive force for different plasma models are found in the literature. For two-fluid plasmas Lee and Parks [ 59 , 60 ] and Biglari and Diamond [ 61 ] derived some simplified forms. In contrast to the above fluid turbulence, ponderomotive forces account for the slow variability of plasma turbulence.

They account for the role of kinetic effects on fluid scales. Reconnection theory has only recently begun implicitly including fluid and plasma turbulence in simulations [ 62 — 65 ]. Of the above noted possible causes of reconnection of particularly interest are all those which may contribute by effects in plasma which are not covered by a simple two-fluid theory. These are effects due to heating and its reaction on reconnection as well as those which generate anomalous resistance, anomalous viscosity or other anomalous effects like current bifurcation first observed by Ronov et al.

Since non-linearity comes in at this place cf. Uzdensky [ 70 ] pointed out that Coulomb processes may come into play via heating the plasma. This yields an upper limit on for reconnection being collisionless. Thus, should it happen that the length L of reconnection along the antiparallel fields drops below L cl , reconnection went collisionless.

Clearly, this depends on some process which is hidden in the electron mean free path and therefore depends on the mechanism which generates collisions. One concludes that electron heating and dilution of plasma may increase the critical scale until reconnection becomes collisionless. This indicates that, for the parameters in the magnetosphere, Sweet-Parker reconnection is indeed close to the claimed marginality. However, the scale where marginality would set on is much larger than any magnetospheric scale and, in particular, much larger than any observed longitudinal scale of the reconnection region.

This lets one doubt in the meaning of L SP , cl also in the solar corona. Such wave fluctuation levels require extraordinarily strong plasma wave excitation reaching far into the highly non-linear regime. Wave intensities such high have not been observed, not even under extremely disturbed magnetospheric conditions. This condition is easier to satisfy, which indicates that Petschek-like reconnection based on some localized anomalous resistivity is not quite as unreasonable as Sweet-Parker reconnection under nearly collisionless conditions.

Lacking a resistivity under collisionless conditions in plasma, the importance of anomalous effects as a possible reason for anomalous resistance has early attracted attention. We already made use of Sagdeev's formula in Equation Before returning to it we note that, formally, an anomalous conductivity can be defined for each of the terms in Equation 10 by simply defining some equivalent conductivities. This does not make sense for the Hall term, however. One immediately sees that the right-hand side is the projection of the pressure tensor divergence onto the direction of the current.

Similar games can be played at the other terms in Ohm's law. Though this is a formal representation only, it shows that the different terms in Ohm's law may contribute to an equivalent resistance each in its way. They just refer to fluid conditions. This assertion applies also to the ponderomotive terms in Equation 10 for they contain just the slowly variable effects of waves in the momentum exchange between waves and the plasma fluid.

The microscopic interactions which cause the ponderomotive forces are not included here. With this philosophy in mind, Ohm's law becomes an equation for the current density J. Conversely, providing the current density and electric field can be unambiguously measured, W ed can directly be determined.

In this case the non-linear terms contribute to Ohm's law in similar ways as known from dynamo theory. Ohm's law provides the link between dynamo theory and reconnection, with reconnection becoming driven by fluid and current turbulence [ 58 ] and interacting with dynamo effects providing merging in dynamo theory and enabling collisionless reconnection. Though it is clear that turbulence affects the dynamo, it is not clear whether it really causes reconnection. The scales of fluid turbulence are longer than any microscopic plasma scales.

On those scales electrons should remain frozen to the magnetic field. Vice versa, onset of reconnection and the resulting tearing-mode turbulence will necessarily affect any ongoing dynamo by either speeding it up or braking it. In collisionless plasma one might suggest that reconnection-caused turbulence and anomalous resistivity will rather support the dynamo.

On the other hand, in a recent mean field MHD model with the pmf in a turbulent Ohm's law [ 72 ] simulations seem formally to indicated that this kind of turbulence may drive reconnection, even explosively. Since this model is MHD it implies large scales; therefore its physical reality remains unclear.

Microscopic non-linear interaction can become a means of generating reduced electrical conductivities via the interaction of particles with self-consistently excited plasma waves. A spectrum of such waves may retard the electrons from their ballistic free flight motion due to scattering in the self-generated wave fields. The retardation is the cause of a fake friction exerted by the waves on the particles. It results in real though anomalous not based on binary interaction between particles collision frequencies and contributes to an anomalous resistivity.

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Such a resistivity is necessarily anisotropic as it depends on the natural anisotropy of the plasma waves. The idea goes back to Sagdeev cf. Sagdeev [ 73 ] favored ion-acoustic waves as the agents of anomalous collisions in non-magnetic, and electron-cyclotron waves in magnetized plasmas.

Also current-driven electron-cyclotron modes are stable in the extremely weak magnetic field in the current sheet. Some anomalous collision frequencies for various current driven plasma instabilities have been calculated by Liewer and Krall [ 77 ] and Davidson and Krall [ 78 ]. These papers favor the lower-hybrid drift instability for reconnection [ 79 , 80 ]. It grows in plasma density gradients of a magnetized plasma. It drives the lower-hybrid drift instability as well as a relative of it, the modified-two-stream instability. Such density gradient exist at the boundary of the current sheet and also at the boundary of the electron diffusion region while the magnetic field is weak and electrons are weakly magnetized.

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The lower hybrid drift instability scatters the drifting particles reducing the relative perpendicular diamagnetic velocity between electrons and ions, stabilizing via heating the cooler plasma component. By Sagdeev's argument, it indeed yields high anomalous collision frequencies of the order of the lower-hybrid frequency. Observations of wave spectra [ 81 , 82 ] do not support its importance though measurements in magnetopause crossings [ 83 , 84 ] seem to provide evidence for its excitation even though lacking sufficiently high resolution for locating the waves to either the ion or electron diffusion regions.

This argument may, however, be revised if a strong magnetic guide field is overlaid over the antiparallel fields and current layer. The physics then becomes quite different. Moreover, Fujimoto and Shinohara [ 85 ] report modification of the electric convection field due to non-linear evolution of the lower hybrid instability in the ion diffusion region.

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The additional inductive electric field component accelerates electrons to sustain a thin electron current layer which readily goes unstable with respect to tearing modes. It thus remains undecided whether or not the lower hybrid instability is directly involved into driving collisionless reconnection.

Anomalous collision frequencies require the excitation of some kind of plasma waves. The problem consists in the identification of the most effective instability and its saturation level. Most attempts rely on the lowest order interaction, i. However, wave—wave interactions and strong non-resonant wave-particle interactions may reduce the quasilinear level and deplete the expected anomalous collisions.

In view of the complexity complicated geometry, all kinds of fluid aspects like flows, vortices, boundary layer effects etc. Pioneering work on the stability of a current layer concentrated on the tearing instability [ 86 , 87 ]. Relevant analytical work terminated already in the seventies with papers by Schindler [ 88 ] on the theory of substorms and fundamental work by Galeev and Zelenyi [ 89 , 90 ], followed by Lembege and Pellat [ 91 ].

Birn et al. In contrast to this quasi-fluid theory, Galeev and Zelenyi [ 89 , 90 ] solved the full kinetic stability problem of a two-dimensional current layer with overlaid normal magnetic field including the difference between electron and ion dynamics. It therefore remains to be weakly dipolar in spite of the extended current sheet.

The key idea in this theory is to include the adiabatic oscillatory motion of non-magnetized ions that are trapped in the current sheet and oscillate between the two oppositely directed magnetic fields thereby contributing to Landau damping while electrons remain magnetic due to the presence of the weak normal magnetic field component. To first perturbation order one obtains the condition.

Any kinetic theory, whether analytical or numerical simulations, requires the assumption of an initial equilibrium state. This, in a completely magnetized plasma containing an extended two-dimensional sheet current of width d has been obtained by Harris [ 96 ] under the assumption that the plasma is in thermal equilibrium obeying a Maxwell-Boltzmann distribution function. As initial condition, this distribution function lies also at the base of most particle-in-cell and Vlasov particle simulations of collisionless reconnection.

Whether it is experimentally justified, is an undecided question. Observations suggest e. Such distributions still satisfy the Vlasov equation but are typical for collisionless plasmas. They represent quasi-stationary states far away from thermal equilibrium e. They naturally account for an initial background of particles of higher energy for a recent review of the dynamics of particles in current sheets cf.

One should note that these are two-dimensional equilibria assuming a flat infinitely extended current sheet of thickness d. Such sheets must be distorted in some way in order to undergo reconnection.

## Frontiers | Collisionless magnetic reconnection in space plasmas | Physics

In addition to the stable equilibrium current sheet distortions require a mechanism and an initial magnetic fluctuation level to start from. The distortion then selects some wave number range from this fluctuation level which becomes unstable and grows. Such magnetic fluctuation levels have only recently been calculated, for instance for the Weibel mode [ 41 ], assuming particular plasma response functions in an unmagnetized plasma [ — ] which applies approximately to the very center of the current layer.

Felten and Schlickeiser [ ] recently reported a strongly damped isotropic mode which provides the highest level of magnetic thermal noise in an unmagnetized plasma. All these modes generate a fluctuating magnetic background from which other modes like the tearing may chose to grow. The spontaneous instability which is made responsible for reconnection is, however, the magnetized collisionless tearing instability for collision dominated plasmas first proposed by Furth et al. Preliminary thermal magnetic fluctuation levels for this mode have been provided by Kleva [ ] in tokamak geometry, but no full thermal fluctuation theory is available yet.

Calculation of the contributions of electrons and ions in the above tearing dispersion relation can be done in either the absence e. The electron contribution is from oscillations of trapped electrons inside the electron inertial length, ions contribute non-magnetically via Landau damping from the entire ion diffusion region.

The expression in the brackets contains the ratio of electron thermal speed to tearing mode phase velocity. Adding ion-Landau damping completes the effective potential in Equation The total potential consists of a well with a hump in its center being formed by ion Landau damping and the additional rigidity of electrons in the normal magnetic field. The tearing, for becoming unstable, has to overcome both these humps, i. Including both contributions from ions and electrons and a non-zero normal magnetic field cf.

Violation of this condition stabilizes the tearing mode. The upper limit is due to the residual magnetization of electrons. The stronger B z 0 the higher is the electron magnetization and consequently the hump in the potential well. When the size of the island reaches the current thickness, i. This is the theoretically maximum possible size of the tearing islands. On the other hand, this is also the size at which strong non-linear effects are expected to come into play. The combination of both limits implies a meta-stability of the tearing mode which has been applied to the stability of Earth's magnetotail during substorms [ 90 , 95 , , , ].

Recent extended three-dimensional simulations [ 63 — 65 , — ] suggest that in three dimensions this inference for tearing stabilization may be invalid. The tearing mode in three dimensions is more easily destabilized due to oblique effects. The lower limit is provided by ion Landau damping. It can be overcome by the presence of a sufficiently strong magnetic field B z normal to the current sheet.

Therefore, the tearing mode will not grow spontaneously when the normal magnetic component vanishes. The current sheet is metastable awaiting the evolution of an increase in B z over the lower bound in the above equation in order to grow. This lower bound depends on the tearing mode wavelength. Preference is given to long-wavelength tearing modes. This linear theory of tearing mode growth does not yet account for a large number of additional effects such as weak collisions which contribute to a collision integral in the Vlasov equation and may both stabilize and destabilize the tearing mode further [ ].

In addition, the dynamics of electrons and ions in the plasmoids plays another essential role when the size of the plasmoids reaches the thickness of the current layer. These particles,oscillating in the plasmoids, are trapped in the plasmoids only for limited times and can be lost when passing the weak magnetic field in the X points, an effect which non-linearly contributes to tearing growth because of heating ions and effectively decreasing the number of ions which contribute to Landau damping. If this happens, it has been shown cf.

Inclusion of collisions requires definition of the structure of the Boltzmann-collision integral in the Vlasov theory. In the simplest way this is done by assuming a BGK structure of the collision term. This leads to a modification of the Galeev-Zelenyi theory [ ]. Collisions, whether weak or strong, help breaking the frozen-in state of the electrons. So one expects that they diminish the stabilizing effect of magnetized electrons in the current sheet which is introduced by both normal magnetic components and also guide fields, which we have not yet discussed here.

Since, however, the theory strongly depends on the assumptions of the structure of the collision term and the form of the anomalous resistance, i. Moreover, anomalous collisions depend on the wave modes and through it on the distribution function which makes the problem non-linear to a higher degree.

The obvious way out is to switch to numerical simulations. This way has been gone since the early seventies. It is thus not surprising that, though the Galeev and Zelenyi [ 89 , 90 ] solution was intriguing, until recently [ ] it has found less attention in the literature than it deserved. It was readily superseded by numerical simulation techniques. Most recent three-dimensional simulations [ 65 , , ] do, however, surprisingly indicate that the idea of explosive reconnection and the evolution of multiply structured X points [ 90 , 95 ] may have been quite realistic.

The three-dimensional investigation does, in addition, reveal that the above theory has to be corrected in two points: changes in topology due to the inclusion of the third dimension, and a different asymptotic limit for the growth rate which in three dimension has a pronounced dependence on the angle with respect to the magnetic field. The tearing mode in this case is oblique. The asymptotic tearing mode growth rate was given by Daughton and Roytershteyn [ ] as.

Companion three-dimensional simulations demonstrate, however, a much more complicated evolution of reconnection. This will be discussed below in relation to guide and normal field reconnection. Another property of interacting plasmas in nature is their magnetization state. Frequently currents flow along a superimposed guide magnetic field which can be both weak and very strong. Such guide field reconnection has different properties. If the fields are weak, as is the case when self-generated quadrupolar Hall magnetic fields overlay over the external field in the ion inertial region, ions are accelerated in the ion inertial region and cause broadening of the current layer.

If the electron inertial region at the center of the current layer remains free of such guide fields, reconnection is going on there in the sublayer of the gross current flow. If, on the other hand, the guide field points in the direction of the current and is externally applied—as is the case at the magnetopause—completely different effects set on. The guide field magnetizes the electrons to some degree thereby setting a threshold on the onset of reconnection. However, the presence of the guide field along the current, together with the external electric field which is also directed along the current, allows for acceleration of the electrons antiparallel to the electric field.

This acceleration can become strong enough to excite the Buneman instability which has two effects: it causes anomalous resistance, heating of the bulk electron component, and it produces chains of electron holes along the guide field. Similar effects may already occur for Hall magnetic fields as well. Electron holes have a number of consequences. They structure the plasma in electric field-free regions and localized region of strong electric fields. They split the electron distribution into trapped and passing particles. They cause further heating of the plasma, generate cold electron beams and possibly radiation, transforming a strong current layer into a source of plasma waves and radiation, which might become of interest in astrophysical application.

Strong guide fields, on the other hand, result mainly in these effects. Though reconnection takes also place in this case, it is restricted to the weak current-generated magnetic field and does not dissipate the energy of the strong field. Below, guide field reconnection will be discussed in connection with numerical simulations of reconnection under various conditions.

Since analytical theories of the complete complexity of reconnection are unmanageable, the proper and efficient approach is via numerical simulations. Simulations started in the mid-seventies where there were based mainly on MHD. Particle simulations were for a while inhibited by the necessity to include at least two spatial dimensions and thus large particle numbers.

These remained intractable by the then available computer capacities. Two-dimensional particle-in-cell simulations became available to large extent in the nineties. Now, almost every decade, computational progress is made stepwise by increasing particle numbers, simulation boxes, more sophisticated asymmetric set ups, boundary conditions, flows, and including, recently, the highly desired three dimensions. Contrary to early belief, reconnection is a microscopic plasma process capable of breaking the frozen-in condition prescribed by the electrodynamic induction equation in a moving collisionless plasma.

This breaking depends on the microscopic properties of the plasma in the current layer and its vicinity. Reconnection took place exclusively on the electron scale. In this section we briefly review the recent achievements. Figure 4. Top: Initial state of a two-dimensional thin current sheet J separating two antiparallel fields in a completely magnetized plasma. Open arrows show the plasma inflow from both sides before reconnection sets on.

Bottom: During reconnection the current sheet breaks off, and an X point is formed indicated by the two dashed red separatrices. Connected field lines are shown in yellow forming plasmonds to both sides and producing two plasma jets red arrows. The thin green line across the X point shows the extension of the electron diffusion region. In this region electrons are heated and jetted away from the X point to become two electron jets. These jets disappear at the end of the electron diffusion region where ions take part in the acceleration.

A general problem of any simulation is the initiation of reconnection. In most simulations this is artificially done imposing a disturbance in the center of the current sheet and box. It has turned out that the best approach is to impose an initial weak X point [ , ], an effective approach in both two and three dimensions. Even a weak initial disturbance is, however, already a non-linear disturbance and thus skips all the interesting physics which causes reconnection to start by itself. Such an approach makes sense if one is rather interested in applications, for the reconnection events encountered in nature are without exception well developed have for long left the initial growth phase.

A shortcoming of this approach remains. Since the initial perturbation is necessarily large it skips any possible state where the reconnection process levels non-linearly out at smaller amplitude than the initial disturbance. This approach therefore misses an entire class of fast but weak reconnection events. In those simulations without an initial disturbance e. Unfortunately numerical fluctuations are not localized, and thus those simulations belong rather to the class of a low but finite homogeneous resistivity in the entire simulation box than to really collisionless conditions where anomalous resistances are caused self-consistently by collisionless wave-particle interactions.

## Multi-satellite space mission encounters magnetic reconnection in the Earth’s magnetotail

The two-dimensional simulations by Pritchett [ 57 ] without initial disturbance gave no indication of a saturation at low reconnection amplitudes. This provides confidence in the abbreviated approach of flux perturbation. The most urgent challenges on the numerical simulation of magnetic reconnection by particle-in-cell codes in near-Earth space for the first decades of the Century have been laid down in a widely referenced paper by Birn et al. They concern among others the identification of the reconnection process in the electron inertial layer, the structure of the electron diffusion region, asymmetric reconnection, forced reconnection, and as main challenges three-dimensional reconnection including simulation-technical problems and the transition to macroscopic effects.

In the following we very briefly review the state of the art until about before referring to new evolutions that started in the past 5 years. This is the class of the overwhelming majority of published simulations in two- and, to a lesser extend, also in three-dimensional reconnection. The majority of simulations are of course two-dimensional in space, applying either periodic conditions, which implies the simulation of two antiparallel current sheets, i.

Otherwise open boundary conditions are used where plasma can readily escape and is thus lost. In the former case the simulation is valid only as long as the two sheets do not affect each other. In the latter case plasma flows freely out of the box and is either replaced or not. Limits are reached when the plasmoids generated reach the boundary, the current sheets start interacting which is advantageous for investigating interacting current sheets, however or when the plasma becomes locally too strongly diluted by jetting away from the X point.

Any results depend highly on the settings and thus may differ considerably, in many cases they are incomparable which makes drawing conclusions about the general validity difficult. Natural pair plasmas don't exist in near-Earth space. In reconnection theory they just serve for the most simple particle arrangement and simulation model. This was done first by Jaroschek et al. The simulation was repeated with the only emphasis on the non-Hall aspect in a strongly simplified two-dimensional mini-set-up [ 44 ].

The relativistic treatment becomes necessary in a proper simulation in order to account for retardation, correct local charge neutralization, and to resolve the Debye scale. Reconnection was ignited by imposing a weak initial disturbance of the current sheet. A two-dimensional cut through the pair simulation box is shown on the left in Figure 5 demonstrating that reconnection does indeed evolve.

It, moreover, evolves on a very fast time scale and does not lead to Sweet-Parker reconnection. Several X points form rapidly and, most important, strong quasi-stationary inductive electric fields are generated at the plasmoid edges. Such electric fields have also been found recently in non-pair plasma simulations but were first observed here.

They play a key role in particle acceleration in reconnection. Figure 5. Three-dimensional simulation of non-Hall reconnection in a pair plasma of high resolution in a large domain simulation data taken from Jaroschek et al.

Top: Magnetic flux in color representation only relative scales are given here. Contours indicate the magnetic topology. Several X point are generated separated by plasmoids red blobs , indicating evolution of a tearing mode. The initial reconnection region dominates all secondary X points. These have their own dynamics. Note that the horizontal extension length of the diffusion region is several times its vertical extension indicating jetting.

Bottom : Electric fields red is positive, blue negative. Strong fields evolve around the plasmoids causing electron-positron acceleration. Plasma jetting occurs around each X point thus leads to competition, jet braking and deviation. In this 3-dimensional simulation the X points have finite extension perpendicular to the plane shown.

Right: Orbit of one selected accelerated electron in the electric field near the first right plasmoid. Originally the electron performs a meandering oscillatory gyration in the current sheet between the two opposing magnetic fields until feeling the reconnection electric field, picking up energy and enlarging its gyroradius. At the end of the orbit shown it just enters the strong field on the backside of the plasmoid to become further accelerated. This scale is determined not only by the expansion of the main X point but also by the reaction of the secondary X points and plasmoids.

Each of them ejects quasi-symmetric pair plasma jets into both horizontal directions. The jets interact with the neighboring X points, braking their the expansion and thereby shortening the extension of the jets. This is the reason for the diffusion layer of the main X point to terminate at a shorter scale, while all secondary diffusion regions are much narrower.

The simulations were three-dimensional, however, permitting to infer about the extension of the reconnection site into y -direction. Limitations in particle number and computer time did not allow to extend the box such that no information could be gathered about the further evolution of the reconnection site in y. However, some interesting inferences were drawn on wave excitation and particle acceleration see the respective paragraphs below.

Formation of many X points and plasmoids of different sizes was found. The focus was on the initial evolution of the tearing mode, determination of growth rates, interaction of plasmoids, and formation of the diffusion region. The main conclusion of this study is that pair plasmas in these simulations never reached a final steady state reconnection regime, in agreement with the findings of Jaroschek et al.

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The evolution of the tearing mode is highly dynamic with formation of an elongated unstable current layer which decays into a long series of plasmoids ejected along x and mutually interacting. Average reconnection rates remain fast and are insensitive to the size of the system. The authors conclude that pressure tensor effects cannot prevent the elongation of the current layer in pair plasmas thus inhibiting the final stationary state. Plasmas in Earth's environment consist of electrons and ions.

Particle-in-cell simulation studies of collisionless reconnection in electron-proton plasmas are ubiquitous both in symmetric and non-symmetric current sheet configurations with or without guide fields. Depending on the set-up and the focus of the subsequent analysis they show a large variety of effects. Reviews of the achievements on various aspects onset, Hall fields, current structure, field and current topology, bifurcation, dissipation, heating, acceleration, wave generation etc. Here we just summarize some of the most interesting insights before turning briefly to the most recent investigations.

Hall fields. An important finding in particle-in-cell simulations was the confirmation of the difference between electron and ion dynamics near the current sheet center leading to the generation of Hall currents and fields, as predicted [ 34 ]. Figure 6 shows an example of the fully developed magnetic field structure in a two-dimensional simulation of a thin current sheet with the background magnetic field exhibiting the typical X point-separatrix-plasmoid topology. The quadrupolar Hall field B y is along y out of or into the plane.

In the dark blue regions B y is positive, in the white regions negative. Hall fields maximize just inside but along the separatrices and at the outer plasmoid boundaries where the Hall electrons turn around. Added are confirmation measurements in the Earth's magnetotail [ 37 ]. The top panel shows the bulk velocity-field reversal typical for plasma jetting away from the X point.

The bottom panel contains the magnetic out-of-plane field B y exhibiting the plus-minus sequence expected of a Hall field near the X point.

One may note that in these observations the Hall field undulates around a finite stationary guide field of 6 nT amplitude. Figure 6. Hall fields, observation and simulation. Central panel: Two-dimensional simulation of reconnection including different electron and ion dynamics data taken from Vaivads et al. Shown is only the magnetic field with background field lines drawn in thin white. Dark blue regions indicates positive Hall- B y , white regions negative Hall- B y. The Hall fields concentrate along and inside the separatrices and boundaries of the two plasmoids where the Hall electrons dashed yellow lines turn away from the X point.

Hall fields twist the original magnetic field. They can be considered as self-generated magnetic guide fields in the ion diffusion region. Such guide fields embedded into the convection electric field E y black circles may accelerate electrons out of the plane thereby strengthening the current in the ion diffusion region. Open arrows show convective inflow, yellow arrows outward jetting of plasma.

Top panel: Plasma velocity measured along the spacecraft orbit long red arrow in the central panel in Earth magnetotail during a substorm reconnection event. Flow reversal is seen during passage near the X point indicating the two plasma jets emanating from X. Note the presence of a weak 6 nT magnetic guide field in y direction. The pronounced differences in electron and ion flows in the ion inertial diffusion region has been nicely confirmed in particle-in-cell simulations [ ] as shown in a summary plot in Figure 7.

The left part of this figure shows the usual ion jetting away from the X point, also clearly indicating that only a small fraction of the ion flow is passing the X point. At the boundaries of the ion jet discontinuities develop where the ion flow suddenly turns around.

These are the separatrices. The lower panels show the profiles of the jet and current velocities. Similarly, ion current velocities are decelerated in the X point region, while electron speed become completely deflected from positive to negative velocities, an effect of reconnection, such that the reconnection current in the X point region is carried almost solely by the electrons. Pritchett [ ] also included the third dimension with open boundary conditions in order to investigate the extent of the electron diffusion layer and distortion of its two-dimensional symmetry.

No such distortion was detected except for the evolution of an electric component E z that is required by pressure balance. These conclusions are valid in the absence of guide fields. When guide fields are included see the corresponding section on guide field reconnection below the Hall fields become distorted, asymmetric and compressed.

This has been demonstrated by Daughton and Karimabadi [ ] and Karimabadi et al. A thorough comparison between different theory based simulations hybrid, Hall-MDH and non-Hall hybrid, where the Hall term is removed has been undertaken by Karimabadi et al. Reconnection was found to be independent on the Hall effect even in these ion-kinetic simulations including an anomalous resistivity thereby anticipating strictly Hall-free simulation results [ 42 ]. Including the Hall effect reconnection turns becoming asymmetric, and the X line grows in the direction of electron drift with current carried by electrons.

Figure 7. Top panels: Velocities near the X line. Left: Left half negative x of space for ion velocity. Right: Right half positive x of space for electron velocity. Electron inflow continues across the ion inertial region until close to the center of the current sheet mapping the electron Hall current flow. Bottom: Jetting and currents exhibiting different ion and electron dynamics. Left: Electron and ion jetting velocities.

Ion mass flow in jets barely reaches 0. Right: Current speeds indicate deceleration for the ions at the X point. Electron velocities are inflected such that the current in the center is carried about solely by electrons. Hall fields were observationally inferred first by Fujimoto et al. They represent self-generated guide fields. Only along the separatrices they approach the electron diffusion region near the X point.

For this reason electrons become accelerated in E y in the direction opposite to E y. This acceleration amplifies the current in those domains where the Hall magnetic field is remarkable, an effect that leads to current bifurcation outside the reconnection site in the ion diffusion region. Bifurcation was observed first in [ 66 ] and [ , ] in Earth's magnetotail current sheet.

Though several different mechanisms have been proposed to produce current bifurcation cf. Asymmetric reconnection effects. Most cases of reconnection occur in interaction of plasmas with unlike properties. Such reconnection is non-symmetric. A famous example is reconnection between the interplanetary solar wind magnetic field and the geomagnetic at Earth's magnetopause. In contrast symmetric reconnection like that in the tail of the magnetosphere is a rare case. Theoretically one expects that asymmetric reconnection affects mainly the weaker magnetic field side than the high field side. This was demonstrated in asymmetric simulations under conditions prevalent at the magnetopause [ ].

Figure 8 gives an impression on the non-forced asymmetric case with no guide field imposed. The most interesting effect is probably that the strong-field magnetosphere remains well separated from the distorted region by a slightly deformed but stable magnetopause which itself is adjacent to two legs of the separatrix system. The two newly formed plasmoids and the X point lie entirely on the weak field side.

The second lower Hall dipole is completely suppressed as there is no electron inflow from below. Electrons and ions flow in from the top and become diverted into jets along the magnetopause. This is shown in the lower part of Figure 8. Figure 8.