Reconnection and electron temperature anisotropy in sub-proton scale plasma turbulence
aa r X i v : . [ phy s i c s . s p ace - ph ] J a n Accepted for publication in ApJ, Jan 3, 2014
Reconnection and electron temperature anisotropy in sub-proton scale plasmaturbulence
C. T. Haynes , D. Burgess , and E. Camporeale ABSTRACT
Turbulent behavior at sub-proton scales in magnetized plasmas is important fora full understanding of the energetics of astrophysical flows such as the solar wind.We study the formation of electron temperature anisotropy due to reconnection in theturbulent decay of sub-proton scale fluctuations using two dimensional, particle-in-cell(PIC) plasma simulations with realistic electron-proton mass ratio and a guide field outof the simulation plane. A fluctuation power spectrum with approximately power lawform is created down to scales of order the electron gyroradius. In the dynamic mag-netic field topology, which gradually relaxes in complexity, we identify the signaturesof collisionless reconnection at sites of X-point field geometry. The reconnection sitesare generally associated with regions of strong parallel electron temperature anisotropy.The evolving topology of magnetic field lines connected to a reconnection site allowsspatial mixing of electrons accelerated at multiple, spatially separated reconnection re-gions. This leads to the formation of multi-peaked velocity distribution functions with astrong parallel temperature anisotropy. In a three-dimensional system, supporting theappropriate wave vectors, the multi-peaked distribution functions would be expected tobe unstable to kinetic instabilities, contributing to dissipation. The proposed mecha-nism of anisotropy formation is also relevant to space and astrophysical systems wherethe evolution of the plasma is constrained by linear temperature anisotropy instabilitythresholds. The presence of reconnection sites leads to electron energy gain, nonlocal ve-locity space mixing and the formation of strong temperature anisotropy; this is evidenceof an important role for reconnection in the dissipation of turbulent fluctuations.
Subject headings: turbulence – magnetic reconnection – solar wind [email protected] School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK T-5 Applied Mathematics and Plasma Physics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
1. Introduction
Turbulence is almost certainly ubiquitous in astrophysical plasma flows, and it is crucial fora full understanding of energetic particle propagation and the transport and dissipation of energy.Recent advances in the understanding of astrophysical plasma turbulence have followed from theinterpretation of in situ measurements of space plasmas such as the solar wind and magnetosheath.For the solar wind at MHD scales, a nonlinear active turbulent cascade operates which is dominatedby Alfv´enic fluctuations (e.g., review by Horbury et al. 2005). The power law scaling of fluctuationsin the inertial range (with periods of hours to tens of seconds) is approximately Kolmogorov with apower law ∼ f − / , although the turbulence is also characterized by a number of other propertiessuch as anisotropy and intermittency. For a recent review of solar wind turbulence properties inthe inertial range see Bruno & Carbone (2013).At higher frequencies, in a collisionless plasma such as the solar wind, one might expect that theviscous dissipation of hydrodynamic turbulence is replaced by processes operating at particle kineticscales, such as cyclotron or Landau damping. Indeed, in the solar wind, at frequencies correspondingto the characteristic proton scales, the magnetic fluctuation spectrum shows a break, above whichit steepens with a spectral index varying between -2 and -4 (Leamon et al. 1998; Bale et al. 2005;Smith et al. 2006), indicating that ion kinetic processes are at work. At 1 au the frequenciescorresponding to the Doppler shifted proton gyroradius and proton inertial length are usually closeto each other, making it difficult to infer the appropriate scaling. However, the variation of theion-scale break frequency with radial distance from the Sun indicates that it is related to theproton inertial length, rather than the proton gyroradius (Bourouaine et al. 2012). A similar breakof the spectrum at ion scales is also characteristic for ionospheric conditions. (Kelley et al. 1982;Hysell et al. 1994)Dissipation processes at the proton kinetic scale are clearly important, but recent observationalwork has addressed the question whether solar wind turbulent fluctuations extend down to electronscales (the electron thermal gyroradius ρ e ∼ ρ e steepen with a slope of ∼ − T k /T ⊥ against β k is constrained withinboundaries related to the marginal growth of linear instabilities. Typically the relevant constraintsderive from temperature anisotropy instabilities such as the cyclotron, fire hose and mirror insta-bilities (Hellinger et al. 2006), and the VDF evolves in a competitive balance between the effects ofsolar wind expansion, growth of linear instabilities and Coulomb collisions (Hellinger & Tr´avn´ıˇcek2008; Matteini et al. 2012). Although most work has concentrated on proton parameter constraints,similar effects are also seen for electrons (ˇStver´ak et al. 2008). The role of such linear instabilityparameter constraints in solar wind turbulence is still unclear. In this paper we argue that anadditional driver for the electron temperature anisotropy in an expanding plasma flow might bemagnetic reconnection occurring as an element within turbulence.
2. Methodology
We use the particle-in-cell (PIC) code Parsek2D (Markidis et al. 2009) based on the implicitmoment method for time advance of the electromagnetic fields, and a predictor-corrector methodfor the particle mover. The implicit method allows larger time steps and cell sizes compared withexplicit PIC methods, which are usually constrained (for numerical stability) by the condition ω pe ∆ t <
2, where ∆ t is the time step, and ω pe is the electron plasma frequency. Also Parsek2Dallows a relaxation of the Courant-Friedrichs-Lewy (CFL) condition c ∆ t/ ∆ x <
1, where c is thespeed of light and ∆ x is the cell size. The time step ∆ t = 0 . − e , where Ω e is the electrongyrofrequency (so that the electron cyclotron motion is fully resolved), and the cell size ∆ x = 5 –∆ y ∼ λ D , where λ D is the Debye length. The code is two dimensional in the x - y plane butretains all three vector components for velocities and fields. The electron-proton plasma is initiallyloaded with a uniform, isotropic Maxwellian distribution. The simulation box is 200 ×
200 cells,with periodic boundary conditions and 6400 simulation particles per cell for each species. This largenumber of particles reduces the statistical particle noise so that the dynamic range in Fourier spaceis large enough to resolve the formation of a turbulent cascade. The simulation box is sized to resolvewave vectors ranging from kρ e = 0 . kρ e = 10, where k is wave vector and ρ e the thermal electrongyroradius. The box length is about 1 ion inertial length, and the electron gyro-motion is resolvedwith ∼ ω pi / Ω i ∼ m i /m e = 1836. The ions and electrons are initialized to the same temperature β e = β i = 0 .
5. Thesimulation was run until t = 200Ω − e . Unless quoted otherwise, simulation results are shown inGaussian CGS units with the following normalizations: Velocities are normalized to the speed oflight, time is normalized to 10 ω − pe , and charge per unit mass is normalized to the proton chargeper unit mass. Temperatures are the variance of velocities in each simulation cell. Initial particledensity is assumed to be equivalent to 10 particles per cm .Similar to the method of Camporeale & Burgess (2011), we initialize the simulation with abackground magnetic field B and add random long wavelength magnetic field fluctuations, but herethe background field is in the out-of-plane z direction. The intention is to provide an initial inputof energy at low values of k with properties mimicking a turbulent field, and then follow the decayof this initial perturbation and the development of power at larger wave numbers. The magneticfield is initialized with random fluctuations in all three components for wave vectors k x = 2 πm/L x and k y = 2 πn/L y for m = − , . . . , n = − , . . . ,
3. We do not impose any spectral slopeon the initial fluctuations. The initial electric field is zero, but the abrupt perturbation of themagnetic field acts to initialize the self-consistent evolution of the turbulent decay after a shortperiod at the start of the simulation. This method emphasizes the random nature of turbulence,and in particular has the advantage that no particular linear modes are assumed dominant. Othermethods of initializing the decay of turbulent fluctuations are possible, such as initial equilibria(Karimabadi et al. 2013), Alfv´enic-like fluctuations (Camporeale & Burgess 2011), or superpositionof linear modes (Chang et al. 2011). The requirement to resolve the development of a turbulentcascade means that the initial perturbation has to be relatively large, and a value δB/B = 1 isused here. The choice of a configuration with the background field perpendicular to the simulationplane does not support k k wave vectors (at least on average), but does favor magnetic field linetopologies with islands and X-points with a guide field.We identify magnetic field X-points as potential reconnection sites using the technique de-scribed by Servidio et al. (2009). The vector potential A z is computed from the magnetic field;contours of constant A z represent magnetic field lines in the x - y simulation plane. For each cell wecalculate the Hessian matrix for A z and its eigenvalues. If the eigenvalues are of opposite sign, then 6 –the location is a saddle point, and if the gradient of A z is also zero then this is a potential locationfor reconnection. Since the simulation is discrete in space a threshold is used to indicate a possiblezero gradient. This method may find multiple locations around a single reconnection point, andin this case the cell with the lowest gradient of A z is taken to be the actual center. Additionallywe use in-plane field lines to confirm the calculated positions of reconnection sites, and in order tocompare the field line geometry and motion with the electron bulk flow velocities.
3. Results
Figure 1 shows magnetic field line configurations for the simulation at times t = 0 and t =200Ω − e . Potential X-point reconnection sites are shown by black crosses where the threshold forthe gradient is satisfied. Sometimes a single X-point will be shown as a cluster of cells satisfying themagnetic geometrical criterion. The end state has a magnetic field which is topologically simplerwith 8 X-point sites compared to 10 initially. When the simulation is run for longer times, beyond t = 200Ω − e , the number of X-points reduces further, with a consequent reduction in the number ofmagnetic islands. During the simulation the field line evolution is highly dynamic, with X-pointsmoving around the simulation region and interacting with magnetic islands and other X-points.Animations show that magnetic islands gain or lose flux via reconnecting field line motion throughX-points, i.e., field lines with island-like connectivity become linked to other X-points or encirclemore than one island. Thus an island may shrink until it disappears or is absorbed by anotherisland, and at this point the separating X-point also disappears. Occasionally new X-points areseen to form, but only temporarily as the field fluctuates. We also see that the sense of reconnectionmay reverse at an X-point, with the movement of field lines changing direction as the surroundingislands shrink or grow. Associated with a change in the sense of magnetic field reconnection therewill also be a change in the sense of the reconnection electric field. In the final state there are someX-points which are in complex geometries and seem on the verge of disappearing if the simulationwere run longer. By examining the time evolution of the local electron gyroradius we find that oncean X-point region develops a scale less than about an electron gyrodiameter it is likely that theX-point will disappear, although sometimes disappearance occurs in more topologically complexregions with several X-points close to each other.Figure 2 shows power spectra of | δB | / | B | as functions of k x and k y at t = 0 (green) and t = 200Ω − e (blue). The k x and k y directions are both perpendicular to the average guide field.The noise floor, as determined from a simulation with no applied perturbation, is shown in red.Starting from the initial energy input at small k values, the power at larger k evolves rapidlyuntil t = 100Ω − e , after which the spectra are relatively time steady over the period simulated.There are no major differences between the spectra in the two directions. Simulations with anin-plane guide field show a similar rapid formation of approximately power law spectra, but with apower anisotropy in the parallel and perpendicular wave vector directions (Camporeale & Burgess2011). The simulation domain size is approximately one ion inertial length λ i , and between this 7 –scale and kρ e = 0 . k − / in agreement withother simulations and observational data (Smith et al. 2006; Alexandrova et al. 2008). Beyond thisdriving scale, the spectra gradually steepens, roughly consistent with observations, until it reachesa power law of approximately k − . , but there appears to be no obvious break point in wavenumber.Note that for kρ e ≥ t = 100Ω − e , and show thatan ensemble of stochastic fluctuations is present within the simulation, and has properties that aresimilar to turbulence. This indicates the formation of a turbulent cascade down to the noise levelof the simulation and scales of order the electron gyroradius.We have also run the same simulation with different sets of random initial perturbations inthe magnetic field, and find qualitatively the same results as presented in the following sections,just with different topological configurations. Additionally, we have found that running the samesimulation but with m i /m e = 400 does not significantly change the form of the power spectra, thetemperature, or temperature anisotropy signatures that we present below. Reconnection is usually understood to produce plasma heating, and so one might expect tofind increased electron temperature around magnetic field X-points. However, over the course ofthe simulation, the changes in electron temperature seen in the vicinity of the reconnection sitesare not significantly different from those at other locations. This indicates that energy dissipationoccurring through the reconnection process does not dominate over dissipation elsewhere. Thelack of a unique, strong correlation between T e changes and the location of reconnection sitesmay be related to recent observations of magnetopause reconnection outflows that show a widevariability in bulk electron heating, explained by a dependence on the Alfv´en speed of the inflow(Phan et al. 2013). Consequently, in order to illustrate the effects of reconnection around magneticfield X-points, we concentrate on the electron drift velocity and temperature anisotropy.Figure 3 shows the electron temperature anisotropy T e k /T e ⊥ over the full simulation domain at t = 97Ω − e , with magnetic field lines shown in black. Although T e does not significantly increase atthe reconnection sites, three reconnection sites have been labeled where there is a strong signature ofparallel temperature anisotropy. Our analysis will focus on these three sites in order to understandhow this particular signature arises, and how it is associated with magnetic reconnection.Figure 4 shows an enlarged detail for reconnection sites 2 and 3. Electron average velocitystreamlines are shown in white and magnetic field lines in black. The geometry of the magneticfield lines and electron flows in Fig. 4 resemble typical X-point reconnection with inflow and outflowregions. The regular flow configuration is perhaps surprising at this scale, given that the size of the 8 –reconnection site is ∼ λ D (25 cells), compared to the electron thermal gyroradius of ∼ λ D . Thevelocity and the magnetic field patterns are not symmetric, and for site 2 the flow and field patterncenters are displaced from one another by ∼ λ D . The corresponding displacement is larger forsite 3, possibly due to the larger asymmetry imposed by the surrounding islands. Reconnectionsites 2 and 3 are both at the junction of merging magnetic islands. Reconnection site 1 is locatedwithin a more complex magnetic topology, and the sense of field line motion changes as the localislands around it disappear. Despite this, this site shows a large electron parallel temperatureanisotropy and an electron flow signature similar to the other two sites.Figure 5 again shows reconnection site 2, with field lines plotted in black, electron stream-lines plotted in white, and B z − B plotted as a color map. Subtracting the initial guide fieldfrom B z reveals the shape of the out-of-plane quadrupolar signature, as seen in two-fluid HallMHD simulations (Sonnerup 1979; Terasawa 1983; Karimabadi et al. 2004), hybrid simulations(Karimabadi et al. 1999), and full particle simulations simulations (Lapenta et al. 2011). Thisquadrupolar signature arises from the circular motion of electron currents in the region, as theydecouple from the ion flow, which enhances the out-of-plane magnetic field. Anticlockwise electronmotion creates a negative enhancement in the magnetic field, as can be seen in the top-left andbottom-right of Fig. 5, whereas clockwise electron motion creates a positive enhancement, as at top-right and bottom-left. The asymmetry of the X-point configuration is also seen in the asymmetricquadrupolar Hall signature. It is known that the presence of a guide field can result in an asym-metric reconnection field pattern due to the nonlinear interaction between guide field and Hall fieldcomponents (Karimabadi et al. 1999; Eastwood et al. 2010). However, asymmetry can be causedby other factors, such as density gradients and asymmetric in-flows driving the reconnection, bothof which are present in this simulation. It is due to these signatures that the reconnection showncan be described as Hall reconnection and is mainly due to the interaction of decoupled electronand ion flows. The reconnection arises spontaneously, driven by the plasma dynamics introducedby the initial magnetic perturbation.Figure 6 shows the distinctive shape of the region of increased electron temperature anisotropygenerated around reconnection site 2. Two main areas of strong temperature anisotropy are locatedto the top-left and the bottom-right of the center of reconnection in the outflow regions. Ananimation of the time development of the temperature anisotropy and magnetic field lines showsthat the anisotropy increases with the reconnection rate and appears to grow out from the centerof reconnection. (See animation of Fig. 6 online.)In a 2-D geometry the rate of reconnected flux, i.e., the reconnection rate, is equal to the outof plane electric field E z , since E = −∇ V − ∂ A ∂t , (1)where A is the magnetic vector potential and V is the electrostatic potential. Field lines in 2-Dare contours of constant A z , so the rate of in-plane reconnection depends only on A z . With z as 9 –the ignorable coordinate, it follows that: E z = − ∂A z ∂t , (2)so that E z at the center of a reconnection site corresponds to the reconnection rate.The centers of the identified reconnection events were tracked during their motion in thecourse of the simulation and E z recorded, as also were parameters such as electron temperatureand anisotropy. Parameter values were averaged over a box size of 15 cells square, centered on thereconnection site. Measured reconnection rates were consistent with the animations of magneticfield line motion, further evidence that reconnection was occurring. Figure 7 shows time series of theresulting data for reconnection site 2. Figure 7(a) shows reconnection rate, Fig. 7(b) shows averageelectron temperature and Fig. 7(c) shows average electron anisotropy in terms of the parameter(1 − T e k /T e ⊥ ). In this figure an anisotropy parameter value less than zero corresponds to a paralleltemperature anisotropy.As previously noted, the electron temperature is not distinctly greater around reconnectionsites when compared with other temperature variations in the simulation. Comparing the timeprofiles of Fig. 7, T e varies by ± t = 100Ω − e . There is a smallcorrelation with reconnection rate, but after this time the electrons around the reconnection siteexperience overall cooling and the correlation becomes weak. However, a correlation is evidentbetween anisotropy and reconnection rate, throughout the entire simulation, indicating that thereconnection process is responsible. Not all reconnection sites show such a marked increase inanisotropy, which suggests a more complex physical process is operating rather than a simple, localone which would produce an absolute correlation with reconnection rate.We have examined the ion distribution functions in the vicinity of the reconnection site, andthere are no significant changes in ion temperature or temperature anisotropy. The ion distributionfunctions remain isotropic. This is not unexpected considering the minor role of the ion dynamicsover the timescale of the simulation. We now discuss the velocity distributions seen near reconnection site 2 shown in Fig. 6. Foursub-regions are chosen on different sides of the magnetic separatrices corresponding to electronin-flow (B and C) and electron out-flow (A and D). Figure 8 shows the electron distribution in theregion of inflow box A, where the anisotropy is highest. Figure 8(a) shows the distribution in the v y − v x plane, with a black cross at the electron bulk velocity; Fig. 8(b) shows the distribution inthe v z − v x plane, with the black arrow indicating the direction of the magnetic field; and Fig. 8(c)shows the distribution in the v ⊥ - v k plane. All distributions are normalized to unity with redcontours indicating higher particle density than blue; velocities are normalized to the speed of light c . 10 –From Fig. 8(c) it can be seen that thermal width of the distribution in the parallel directionis approximately 1.5 times that in the perpendicular direction, in agreement with the anisotropyratio of approximately 2 . v k side and another on thepositive v k side of the distribution. The drift velocity in the positive y direction is consistent withthe flow pattern of Fig. 6. The symmetry of the v z - v x distribution around the magnetic fielddirection seen in Fig. 8(b) indicates that the electron distributions are approximately gyrotropic.Figure 9 shows plots of the v ⊥ - v k distribution functions for the other three boxes B, C, andD marked in Fig. 6. These all show multi-peaked structures. For example, Fig. 9(c) box D has theappearance of a core plus beam distribution, as the peak on the positive v k side of the distributionis much larger. The distribution for box C (Fig. 9(b)) even shows a triple peaked distribution.Thus the regions of largest temperature anisotropy in Fig. 3, which occur around reconnectionsites, seem to correspond to the presence of distribution functions with a mix of multiple peaks.This, in itself, suggests that the reconnection process is forming one or both of these peaks, possiblyby accelerating a subset of particles to form a second peak. In order to determine the formation mechanism of these multi-peaked distributions, we trackparticle trajectories of electrons selected from the different peaks of the distributions, to determinewhence these separate populations of electrons originate. Although the complex physics cannot beunderstood merely in terms of single particle motions, this exercise will allow us to show whetherone or both of the distribution function peaks have been produced by electrons being acceleratedor decelerated as they approach and interact with the reconnection site.We show data for two electrons, labeled E1 and E2, which were tracked throughout the simu-lation. Both electrons were located in box A (Fig. 6), and were chosen from a large set of recordedparticles that interacted with reconnection site 2. Electron E1 (Figs. 10 and 11) was chosen fromthe particles in the peak on the positive v k side of the distribution shown in Fig. 8(c). Electron E2(Figs. 12 and 13) was chosen from the particles in the peak on the negative v k side of the distribu-tion. In these figures time has been normalized to Ω e calculated using magnetic field B . Fig. 10(a)shows v z versus time for electron E1. Velocity components v x and v y are shown in Fig. 10(b) in blueand green, respectively. The electric field as experienced by electron E1 is shown in the Fig. 10(c)and (d), with E x and E y plotted in blue and green, and E z in red.Figure 11 shows the trajectory taken by electron E1 before and during its encounter with thereconnection site. Magnetic field line contours for the whole simulation box are shown in black at t = 55Ω − e . Traced in white is the electron position for the interval t = 0 to t = 100Ω − e . The greenand blue crosses on the trajectory mark the start and end locations, respectively. In Fig. 10 theblack crosses marked on the v z and E z time-series, corresponds to the time at which the magnetic 11 –field lines are shown, with a corresponding red cross marked on the electron trajectory at the sametime (Fig. 11). It is important to remember that the magnetic field evolves dynamically over thetime interval of the electron trajectories. Thus, the magnetic field line configuration shown in thesefigures is only illustrative of the magnetic environment at a specific time late in the trajectories.Animations have been used extensively to analyze the electron trajectories relative to the dynamicfield line geometry.Figure 10(a) shows that as electron E1 encounters the reconnection region after t = 55Ω − e ,the parallel z component of its velocity increases. This is one example of many particles that weretracked, and all show similar behavior. Electrons experience an acceleration, due to E z , alongthe guide field direction as they approach the reconnection site. The large increase in negative E z in Fig. 10(d) results in a force on electron E1 in the positive z direction; E z at reconnectionsite 2 is mainly negative throughout the simulation (Fig. 7(a)). Figure 10(a) also indicates thatelectron E1 passes very close to the center of reconnection, since the oscillation in v z decreases inamplitude, indicating that the in-plane components of the magnetic field have become almost zero.In summary, reconnection site 2 is responsible for the positive parallel peaks in Figs. 8 and 9.Figures 12 and 13, in the same format as Figs. 10 and 11 respectively, show trajectory infor-mation for electron number E2, which is from the peak on the negative v k side of the distributionof Fig. 8. This particle has experienced acceleration in the negative z direction before it encountersthe reconnection site at approximately t = 85Ω − e . However, it, like electron E1, experiences a pos-itive acceleration after it enters the region around reconnection site 2, consistent with the negative E z . However, despite this acceleration the particle v z remains negative. We have examined 181particle trajectories taken from the negative v k peak of the distribution and they all show a similarhistory; there are a total of 9901 simulation particles in the distribution with v k <
0. From Figs. 11and 13 electrons E1 and E2 have very different trajectory histories, but are eventually colocatedbut with very different parallel velocities.Since reconnection site 2 has mainly negative E z for most of the simulation, it accelerateselectrons in the positive z direction. We have used this fact to confirm this mechanism of tempera-ture anisotropy generation, by examining the positions at previous time-steps of groups of particlesfrom the negative peaks of Figs. 8 and 9. The trajectories of these particles trace a region whoseshape is towards the center of reconnection only from the top left part of the separatrix and thebottom right part of the separatrix, consistent with the shape of the region of enhanced parallelanisotropies in these locations (Fig. 6).In order to determine why a triple peaked distribution is formed, as shown in box C (Fig. 9),particles from the central peak were also tracked. Although not shown here, these particles againshow positive increases in v z near the reconnection site. So the central peak is formed of particlesthat start with a negative v z but as they enter the reconnection site they are only acceleratedenough to finish in the center of the distribution. So double or triple peaked distributions can beformed by electrons with different trajectory histories passing through multiple acceleration regions, 12 –but arriving at the same location within a reconnection site.
4. Conclusion
We have presented the results of 2-D simulations using realistic proton to electron mass ratioof the turbulent decay of large scale fluctuations with an out-of-plane guide field. As in previoussimilar work with the guide field in the simulation plane (Camporeale & Burgess 2011) a fluctuationpower spectrum with approximately power law form quickly evolves, until t = 100Ω − e , after whichthe spectra are relatively time steady over the period simulated. The spectra extends to small scalesof order the electron gyroradius. Animations of the magnetic field evolution show that X-points(i.e., potential reconnection sites) evolve dynamically, responding to the motion of surroundingmagnetic islands in the turbulence. As reconnection occurs the topology of field lines can change asthey move through the X-points, from closed within a single magnetic island to circulating aroundseveral islands. The sense and rate of field line motion can change at any one particular X-pointas the islands surrounding it grow or shrink. During the course of the simulation a number of theinitial X-points disappear, and this is most likely to happen after the scale of the X-point becomesless than the local electron gyrodiameter. When the simulation is run for longer times the numberof X-points reduces further, with a consequent reduction in the number of magnetic islands. Thusthe simulation sees a relaxation of the initial magnetic topology, as well as a redistribution of powerfrom large to short scales.The regions around X-points have signatures which indicate that magnetic reconnection isoccurring, with the motion of field lines and the pattern of electron bulk drifts consistent withreconnection inflows and outflows. Where there is a clear pattern of reconnection associated electrondrifts we also observe a quadrupolar signature in B z , similar to that found in Hall reconnection. Thisis consistent with the scale of the X-point region being smaller than the ion inertial and gyro-scales,so that the electron and ion motion are effectively decoupled. Generally there are asymmetries inthe quadrupolar signature and flow pattern due to the guide field, density gradients and inflows.Because of the size of the simulation (the largest scale is of order the ion inertial length) and theinitial number and shape of the islands, we do not see the formation of narrow (small aspect ratio)current layers with embedded X-points. A larger simulation, with initial fluctuation injection atlarger scales, or with initial power anisotropy may produce a different geometry of initial X-points innarrow current sheets as seen in MHD simulations (Servidio et al. 2009) and some PIC simulations(Karimabadi et al. 2013)Animations of the evolution of the electron temperature anisotropy ratio T e k /T e ⊥ indicateenhanced parallel anisotropy at some X-points, and the dynamic appearance of regions of enhancedparallel anisotropy in reconnection outflow regions during periods of strong reconnection. Thereis not a unique one-to-one correspondence between X-points and regions of enhanced anisotropy,but this behavior is frequently observed. We have shown that the enhanced anisotropy is due tomulti-peaked velocity distribution functions. This is the first time (to our knowledge) that such 13 –velocity space structures have been reported at this scale and in turbulence. Further investigationreveals that such distributions are not unique to reconnection outflow regions, but can be foundelsewhere in the simulation.In order to determine how these velocity space features are formed, and whether reconnectionsites are responsible, electrons from the peaks of the distribution were tracked. It was found thatelectrons are accelerated by the reconnection electric field E z , in the direction of the guide field,when they are close to a reconnection site. Acceleration can occur in both positive and negative z directions depending on the sense of reconnection at a particular X-point. Particle trackingallows us to give the following explanation of the mechanism (see Fig. 14): The main peak of thedistribution is generated by the local reconnection site, with the direction being set by the sense ofreconnection, i.e. the sign of E z . The outflow of electrons with the shifted v z distribution will thenpotentially mix with the surrounding population of electrons. Large anisotropies therefore formaround a reconnection site whose outflow area already has a population of electrons, acceleratednear another reconnection site, shifted in v z in the opposite sense. This produces the doublepeaked distributions which are seen. This mechanism explains why not all reconnection sites inthe simulation show this large temperature anisotropy signature, it depends on both the currentdirection of reconnection for the site, and the presence of a population of electrons oppositely shiftedin velocity in its outflow region. This in turn depends on the magnetic topology of field lines allowingelectron trajectories to connect different reconnection sites. In this model the reconnection sitesact as both acceleration regions and mixing zones. It is also possible in this scenario to explain thepresence of triple-peaked distribution functions, which are sometimes seen.We expect the multi-peaked distributions may be unstable preferentially for parallel/obliquepropagating waves, but given the guide field direction and 2-D nature of our simulation it is unlikelythat the unstable waves are supported. In a full 3-D simulation we suggest that these multi-peaked distributions would produce additional waves via beam or anisotropy instabilities. It isnot clear what the full effect of this would be in terms of electron scattering or magnetic field linetopology, given that in a 3-D simulation the reconnection sites themselves would have their ownthree dimensional dynamics. In future work we will consider what type of instabilities and wavesmight be associated with these distributions, and their consequences.The simulation results indicate that turbulence may play an active role in increasing elec-tron parallel temperature anisotropy. This has implications for the study of the evolution of solarwind parameters which has highlighted the importance of kinetic linear instabilities in limitingtemperature anisotropy in response to Coulomb collisions, and the expansion of the solar wind(Camporeale & Burgess 2008; ˇStver´ak et al. 2008; Matteini et al. 2012). Our results indicate thatreconnection can be another driver of electron temperature anisotropy. The simulation has possiblelimitations due to the size of the simulation box and the large amplitude of the initial fluctuations.These have been adopted due to constraints of realistic mass ratio, and the requirement to resolvea turbulent cascade above the noise floor of the simulation. Thus our results are more appro-priate to, for example, the large amplitude turbulence behind the quasi-parallel terrestrial bow 14 –shock (Retin`o et al. 2007) or in current sheets in the solar wind where some evidence of enhanceddissipation exists (Osman et al. 2011).Finally, the power spectrum of fluctuations that we observe develops rapidly after the start ofthe simulation, and has a power law form which is relatively time-steady. It does not seem directlyinfluenced by reconnection, the dynamic behavior of X-points or the evolution of the electrontemperature anisotropy. A full analysis of the fluctuations contributing to the power spectrum andfound during the relaxation process will be addressed in future work. Since the electron behavioris crucially dependent on the topological evolution of the magnetic field via reconnection, it seemspossible that dissipation at the smallest scales in a collisionless plasma might be strongly influencedby how topological complexity is carried to small scales.CTH is supported by an STFC (UK) studentship; DB is partially supported by STFC (UK)grant ST/J001546/1. REFERENCES
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17 –Fig. 1.— Initial and final magnetic field line configurations at (a) t = 0 and (b) t = 200Ω − e .Lengths in units of λ D . 18 – −1 −8 −6 −4 −2 k ρ e | δ B | / | B | k −6.5 k −8/3 Noise floorInitial spectrumk x spectrumk y spectrum Fig. 2.— Power spectra of | δB | / | B | as functions of k x and k y , at t = 200Ω − e (blue solid anddotted lines), initial spectrum (green dash-dot line) and noise floor (red dashed line). A k − / gradient is shown to indicate the typical gradient seen in other works where k extends to smallervalues.Fig. 3.— Magnetic field lines (black) and electron temperature anisotropy ( T e k /T e ⊥ ) at time t =97Ω − e for the full simulation domain. The three X-point regions discussed in the text are marked. 19 –Fig. 4.— Enlarged detail showing magnetic field lines (black) and electron streamlines (white) for(a) reconnection site 2, and (b) reconnection site 3. The position of the reconnection sites aremarked in Fig. 3. 20 –Fig. 5.— Out of plane magnetic field component ( B z − B ) showing quadrupolar signature aroundreconnection site 2. Magnetic field lines shown in black, and electron streamlines in white.Fig. 6.— Electron temperature anisotropy ratio T e k /T e ⊥ in the region of reconnection site 2.Magnetic field lines are shown in black and electron streamlines are shown in white. The fourmarked regions are discussed in the text. This figure is available as an animation in the electronicedition of the Astrophysical Journal . The animation shows the time development of T e k /T e ⊥ andmotion of field lines through the reconnection site. 21 – −8 −4 Ω e−1 ) Reconnection rate: E z .10/( ω pe .c.m i )Average Electron Temperature: T e .k B /(m e .c )Average Electron Temperature Anisotropy: (1−T e|| /T e ⊥ ) a)b)c) Fig. 7.— Averaged values around reconnection site 2 for (a) reconnection rate, (b) electron tem-perature, and (c) temperature anisotropy parameter (1 − T e k /T e ⊥ ) as a function of time. v x /c v y / c Box A−0.02 0 0.02−0.04−0.03−0.02−0.0100.010.020.030.040.05 v x /c v z / c Box A−0.02 0 0.02−0.04−0.03−0.02−0.0100.010.020.030.040.05 v || /c v ⊥ / c Box A−0.04 −0.02 0 0.02 0.04−0.04−0.0200.020.040.060.08 (a) (b) (c)
Fig. 8.— Electron VDF for reconnection site 2, box A (cf. Fig. 6) for (a) v x − v y , (b) v x − v z , and(c) v k − v ⊥ planes. 22 – Box Bv || /c v ⊥ / c −0.04 −0.02 0 0.02 0.04−0.0200.020.040.060.08 Box Cv || /c v ⊥ / c −0.04 −0.02 0 0.02 0.04−0.04−0.0200.020.040.060.08 Box Dv || /c v ⊥ / c −0.04 −0.02 0 0.02 0.04−0.0200.020.040.060.08 (a) (b) (c) Fig. 9.— Electron VDF in v k − v ⊥ plane for reconnection site 2, for (a) box B, (b) box C and (c)box D, as marked in Fig. 6. 23 – −0.02−0.0100.01 V z /c−0.03−0.02−0.0100.010.020.03 V x /cV y /c−10−505 x 10 −7 E x .10/( ω pe .c.m i )E y .10/( ω pe .c.m i )0 10 20 30 40 50 60 70 80 90 100−2−101 x 10 −7 Time ( Ω e−1 ) E z .10/( ω pe .c.m i ) c)d)b)a) Fig. 10.— Time series of particle velocity components and electric field components (as experiencedby the particle) for electron E1 which is chosen from the positive v k peak in the distribution functionfor box A (Figs. 6, 8). 24 – λ D ) Y ( λ D ) Fig. 11.— Trajectory (white) of electron E1 as it approaches and interacts with reconnection site2. The start and end locations are shown by green and blue crosses respectively. Magnetic fieldlines (black) are plotted at the time indicated by the black cross in Fig. 10. The position of electronE1 at the time of the plotted field lines is shown with a red cross. 25 – −0.04−0.03−0.02−0.0100.01 V z /c−0.04−0.0200.020.04 V x /cV y /c−1−0.500.51 x 10 −6 E x .10/( ω pe .c.m i )E y .10/( ω pe .c.m i )0 10 20 30 40 50 60 70 80 90 100−2024 x 10 −7 Time ( Ω e−1 ) E z .10/( ω pe .c.m i ) a)b)c)d) Fig. 12.— Time series of particle velocity components and electric field components (as experiencedby the particle) for electron E2 which is chosen from the negative v k peak in the distribution functionfor box A (Figs. 6, 8). 26 – λ D ) Y ( λ D ) Fig. 13.— Trajectory (white) of electron E2 as it approaches and interacts with reconnection site2. The start and end locations are shown by green and blue crosses respectively. Magnetic fieldlines (black) are plotted at the time indicated by the black cross in Fig. 12. The position of electronE2 at the time of the plotted field lines is shown with a red cross. 27 –
Mixing Region
EzEz
Reconnection E z = −V e x B −> PositiveElectrons accelerated in negative zdirection.Reconnection E z = −V e x B −> NegativeElectrons accelerated in positive zdirection. Fig. 14.— Schematic of mechanism for electron temperature anisotropy production due to recon-nection in turbulence. Electrons accelerated at a region with a positive reconnection electric field E z , gaining v k <
0, can propagate along reconnected field lines towards another reconnection sitewith E z negative. Other electrons accelerated more locally gain v k >>