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The relation between reconnected ﬂux, the parallel electric field, and the reconnection rate in a three-dimensional kinetic simulation of magnetic reconnection

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DOI: 10.1063/1.4833675

Deirdre E. Wendel

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The relation between reconnected flux, the parallel electric field, and the reconnection rate in a three-dimensional kinetic simulation of magnetic reconnection D. E. Wendel, D. K. Olson, M. Hesse, N. Aunai, M. Kuznetsova, H. Karimabadi, W. Daughton, and M. L. Adrian

Citation: Physics of Plasmas (1994-present) 20, 122105 (2013); doi: 10.1063/1.4833675 View online: http://dx.doi.org/10.1063/1.4833675 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/20/12?ver=pdfcov Published by the AIP Publishing

The relation between reconnected flux, the parallel electric field, and the reconnection rate in a three-dimensional kinetic simulation of magnetic reconnection

D. E. Wendel,1 D. K. Olson,1 M. Hesse,1 N. Aunai,2 M. Kuznetsova,1 H. Karimabadi,3,4 W. Daughton,5 and M. L. Adrian1 1NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA 2Institute for Research in Astrophysics and Planetology, University Paul Sabatier, Toulouse, France 3SciberQuest, Inc., Del Mar, California 92014, USA 4Department of Computer and Electrical Engineering, University of California, San Diego, La Jolla, California 92093, USA 5Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 18 September 2013; accepted 4 November 2013; published online 11 December 2013)

We investigate the distribution of parallel electric fields and their relationship to the location and rate of magnetic reconnection in a large particle-in-cell simulation of 3D turbulent magnetic reconnection with open boundary conditions. The simulation’s guide field geometry inhibits the formation of simple topological features such as null points. Therefore, we derive the location of potential changes in magnetic connectivity by finding the field lines that experience a large relative change between their endpoints, i.e., the quasi-separatrix layer. We find a good correspondence between the locus of changes in magnetic connectivity or the quasi-separatrix layer and the map of large gradients in the integrated parallel electric field (or quasi-potential). Furthermore, we investigate the distribution of the parallel electric field along the reconnecting field lines. We find the reconnection rate is controlled by only the low-amplitude, zeroth and first-order trends in the parallel electric field while the contribution from ﬂuctuations of the parallel electric field, such as electron holes, is negligible. The results impact the determination of reconnection sites and reconnection rates in models and in situ spacecraft observations of 3D turbulent reconnection. It is difficult through direct observation to isolate the loci of the reconnection parallel electric field amidst the large amplitude ﬂuctuations. However, we demonstrate that a positive slope of the running sum of the parallel electric field along the field line as a function of field line length indicates where reconnection is occurring along the field line. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4833675]

I. INTRODUCTION

PHYSICS OF PLASMAS 20, 122105 (2013)

20, 122105-1

Magnetic reconnection is a universal phenomenon in magnetized plasmas that converts magnetic field energy into kinetic particle energy. The theory of magnetic reconnection and its redistribution of magnetic ﬂux changes dramatically when it incorporates 3D spatial dependence of fields and par- ticles. While the separatrices that govern the location and rate of reconnection are lines at null points in 2D, at 3D null points they are surfaces called fan planes,1-3 for example. In 2D, reconnection is readily defined as the discontinuous map- ping of field lines and ﬂow of plasma across the separatrices, and the reconnection rate is defined as the out-of-plane elec- tric field at the x-point.4-6 In 3D reconnection with null points, reconnection may entail ﬂux across the fan plane. The case of 3D reconnection in an ambient background magnetic field, however, inhibits the formation of null points and their associated separatrices and separators and therefore requires a more general definition for the occurrence and rate of recon- nection. In this case, the reconnection rate is defined as the maximum of the integral of the parallel electric field along the field lines that thread the diffusion region, or the quasi- potential, while a necessary and sufficient condition for its occurrence is a spatial gradient of the quasi-potential.7-9 From a topological point of view, ﬂux is transported across

1070-664X/2013/20(12)/122105/8/$30.00

VC 2013 AIP Publishing LLC

layers of neighboring field lines whose endpoint locations move rapidly and differ dramatically but continuously, rather than across well-defined separatrix surfaces whose endpoint locations differ discontinuously.10,11 The increased complex- ity of the topology and the particle behavior in general 3D reconnection presents a challenge to identifying the location and rate of magnetic reconnection in observations and simula- tions, which may be further complicated by the development of turbulence.12-16

Here we present results that identify the location and rate of reconnection in a 3D kinetic particle-in-cell simulation and show that the predictions of both the quasi-potential and topo- logical pictures mentioned above coincide. The VPIC code,17,18 executed on the Cray supercomputer Kraken, simu- lates solutions to the set of Vlasov-Maxwell equations over a domain of 70di (cid:2) 35di (cid:2) 70di, where di ¼ c/xpi is the ion skin depth, with a mass ratio mi/me ¼ 100 and a grid resolution of approximately 0.34 de, where de is the electron skin depth. In practice, boundary conditions are open in x and z, and peri- odic along y, the direction of the initial current layer. The simulation thus represents a large open system over a large range of scales. A weak (4%) perturbation (consistent with the boundary conditions) is imposed on an initial Harris equi- librium with a half-thickness of an ion skin depth. A primary

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:132.239.1.231 On: Wed, 11 Dec 2013 15:43:29Though they are not unique, as any a and b that satisfy the expressions for B and A in Eq. (1) are allowed, each a and b uniquely define a field line if the domain through which B passes is simply connected.22 They are also the ﬂux coordi- nates such that

II. THEORY OF GENERAL MAGNETIC RECONNECTION

Hesse and Schindler7 and Schindler et al.9 show that separatrices are not defined in the most general case of 3D reconnection with a non-vanishing magnetic field. Therefore the classic 2D identification of reconnection with the inter- section of separatrix surfaces is no longer applicable in gen- eral. Working in terms of Euler potentials, they propose a more general definition of reconnection and the reconnection rate that encompasses those instances. Rather than relying on topological definitions, they derive the necessary and suffi- cient condition for reconnection from a measure of the change in magnetic ﬂux.

In the vicinity of a region of interest, the magnetic field and the magnetic vector potential A may be expressed in terms of Euler potentials a(x,y,z) and b(x,y,z) as

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Wendel et al.

and turbulence.19 Turbulence

tearing instability leads to the generation of oblique ﬂux ropes. In addition, a secondary tearing instability of elon- gated electron current layers lead to the formation of sec- these scales ondary ﬂux ropes. Flux ropes generated at quickly grow well above ion scales and overlap, leading to the interaction of highly anisotropic and inhomogeneous structures across multiple scales, a stochastic magnetic field, to be self-generated within the reconnection layer.13,16 For this study, we analyze the volume over one time step for the same run discussed in depth by Daughton et al.13 Reconnection is well underway, at tXci ¼ 98, where Xci is the ion cyclotron frequency. Other parameters of the system are Ti ¼ Te, By ¼ Bx0, and ncell ¼ 120, where Ti and Te are the ion and electron temperature, By is the guide field, Bx0 is the initial reconnecting field, and ncell is the number of particles per cell per species.

continues

We first pursue a topological approach by determining the quasi-separatrix layer where it is predicted reconnection may occur if there is also a parallel electric field.10,11 Even if separatrices and separators do occur in a guide field sim- ulation, they will be very difficult to find and are not rele- vant to the analysis we propose here. By integrating the parallel electric field along field lines, we then produce a 2D map of the quasi-potential as a function of the Euler coordinates at the starting y-plane of the integration. The theory of general magnetic reconnection predicts that reconnection occurs on those field lines that have a gradient with respect to the Euler coordinates of the integral of the parallel electric field while the maximum value of the inte- gral of the parallel electric field gives the reconnection rate.7-9 We then compare the predictions of both analyses. The combination of these two analyses indicates which field lines in the simulation are reconnecting. To determine where it is occurring along the length of a given field line, however, we calculate the partial, running sum of the quasi-potential, i.e., the parallel electric field summed up to a given position along the field line as a function of posi- tion along the field line. This approach serves to smooth over the large amplitude ﬂuctuations in the parallel electric field and thus reveals where along the field line the parallel electric field contributes to the reconnection rate. We also examine the relative contributions of DC and low-order components and of higher-order ﬂuctuations of the parallel electric field to the reconnection rate. Our determination of the reconnection rate from the parallel electric field differs from the analysis in Liu et al.,20 which finds the average parallel electric field over a selected region. Here we make a spatial map of the quasi-potential, for comparison against the quasi-separatrix layer, and find the reconnection rate from an estimate of the maximum value of the quasi- potential. We begin our discussion with a brief description of the theory of general magnetic reconnection and of reconnection on quasi-separatrix layers. We then proceed to discuss our implementation of these ideas to the volume of VPIC simulation data and analyze and compare the results. Finally, we investigate the characteristics of the parallel electric field and the distribution of the reconnec- tion rate along field lines.

@b a W ¼ u þ @t c (cid:2) @a @t

E ¼ (cid:3)

U ¼

;

(2)

(1)

Phys. Plasmas 20, 122105 (2013)

(cid:3) rW;

@b @t

Ejjdl;

(4)

(3)

a;b

(cid:3)

where u is the static potential and c is the speed of light. Here we have expressed quantities in cgs units for consis- tency with later calculations. Inserting these terms into the nonideal Ohm’s law, the authors derive an expression for the time rate of change for a and b and for the dependence of W on the field line length s.7,8

To derive the conditions for reconnection, Hesse and Schindler7 find a general expression for the change in mag- netic ﬂux across a non-ideal region. If a localized non-ideal region produces a parallel potential drop along a field line, then the quasi-potential differs on either side of the non-ideal region by

N ¼ W2 (cid:3) W1 ¼ (cid:3)

where Ejj is the parallel electric field. A difference in the time rate of change of the ﬂux coordinates a and b on either side of the non-ideal region, i.e., da/dtj1 6¼ da/dtj2 and db/dtj1 6¼ db/dtj2, reveals that connections between field line elements threading the non-ideal region have changed in

B ¼ ra (cid:2) rb; A ¼ arb:

dadb;

rb þ

where U is the magnetic ﬂux.22 One may then define a quasi- potential W and an electric field E in terms of static and time-dependent contributions as

db dt

(cid:4) (cid:4) (cid:4) (cid:4)1 (cid:4) (cid:4) (cid:4) (cid:4)1

time. Hesse and Schindler7 and Hesse et al.8 show that the difference in evolution of the ﬂux coordinates on either side equates to

(cid:3)

da dt

db dt

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Wendel et al.

(cid:4) (cid:4) (cid:4) (cid:4)2 (cid:4) (cid:4) (cid:4) (cid:4)2

;

(6)

(5)

¼ (cid:3)

@N @b

@N @a

(cid:5) (cid:6)2 @X @x

Q ¼

(cid:4) (cid:4) (cid:4) (cid:4)

They therefore deduce that a necessary and sufficient condi- tion for reconnection is

ra;bNða; bÞ 6¼ 0:

They show, furthermore, that the Hamiltonian form of the expressions in Eq. (5) produces changes of N in a-b space, and therefore new field line connections, in closed loops. It is not N per se, but rather the gradients of N across field lines that form the necessary and sufficient prerequisite for recon- nection. Hesse et al.8 also show that the rate at which ﬂux is reconnected across an entire ﬂux surface is given by max(Ð Ejjdl) ¼ (cid:3)max(N).

III. THEORY OF MAGNETIC FLIPPING ON QUASI-SEPARATRIX LAYERS

When there are null points, there are separatrices, and reconnection corresponds to a discontinuous mapping of field lines from one footpoint to another. However, as al- ready discussed, Schindler et al.8 show that in a nonvanish- ing magnetic field, the field line mappings are continuous and separatrices do not exist. The concept of magnetic ﬂip- ping describes how reconnection might occur in such a case. In this concept, a volume of field lines passing through a non-ideal region rapidly slip through the plasma, changing their connections along the way. This process first requires a region of large but continuous gradients in the footpoint mapping. A quasi-separatrix layer (QSL) is such a layer of field lines whose footpoints neighbor at one end but map to widely separated locations at the other end. A QSL is a favorable location for magnetic reconnection. If the QSL also supports a parallel electric field, then reconnection occurs for sufficient footpoint motions. In this case, the field line velocity exceeds the local plasma velocity, producing a parallel electric field.10 The field lines undergo slippage within the layer, as represented in Fig. 1, adapted from Priest and D(cid:2)emoulin.10 In the QSL depicted here between two planes, footpoint B2 maps to A2, and footpoint B1 maps to A1. If footpoint B1 is perturbed towards B2, its footpoint must shift a large distance, from A1 to A2. If the ratio of the speed of the footpoint motion of B1 to the Alfven speed exceeds a certain threshold, then the resulting speed of the footpoint motion of A1 can greatly exceed the Alfven speed, producing a parallel electric field.9 The field lines then reconnect while rapidly ﬂipping through the diffusion region (here denoted by shading).

The squashing factor Q identifies the footpoints of field lines belonging to the QSL by measuring the gradients of the

Phys. Plasmas 20, 122105 (2013)

(cid:5) (cid:6)2 @Z @x (cid:4) (cid:4) (cid:4) (cid:4)

(cid:5) (cid:6)2 @Z @z

(7)

Þ Þ

Q is normalized by the ratio of the guide field By between the two end planes to render the measure insensitive to the choice of end planes.11 The magnitude of Q(x,z) plotted in the y0 plane is thus large for those field lines whose end- points (X,Z) in the y1 plane have undergone a large displace- ment as a result of a small displacement in either x or z (and vice versa). Thus, as in Fig. 1, if there is a compression of field lines along x and an expansion along z, for high Q field lines there is a corresponding expansion in Z and compres- sion in X—hence the name “squashing factor.”10

The QSL consists of those field lines with a high Q, a conservative working definition being Q greater than 10 times the mean value over the volume.10 Observers, how- ever, have found a maximum Q (non-normalized) of about 4 in solar ﬂares.21 The exchange along the QSL in which coor- dinates compress or expand produces a twisted ﬂux tube structure as depicted in Fig. 2. The near end of the tube where field lines converge toward the centerline (shown by arrows) during the ﬂipping process is known as the stable manifold, while the far section of the tube where field lines diverge away from the centerline is the unstable manifold.

FIG. 1. A schematic portrait of magnetic field lines undergoing reconnection through ﬂipping along a QSL. A small perturbation of point B1 towards B2 results in a large displacement of the other footpoint at A1 towards A2. If there is a non-ideal electric field, depicted by shading, then the field lines slip rapidly, exceeding the Alfve`n speed, and changing their connections through the layer. The change is, however, topologically continuous, rather than discontinuous. This figure is adapted from Priest and D(cid:2)emoulin.10

mapping of footpoints along the layer. Those field lines that are adjacent at one set of footpoints but diverge widely at the other set of footpoints will have a large value of Q.11 Where one set of footpoints are defined by the positions (x,z) at y0 and the other by (X(x,z),Z(x,z)) at y1, Q provides a measure of the gradient of the mapping between the two sets as

(cid:5) (cid:6)2 @X @z

By y0ð By y1ð

Wendel et al.

FIG. 3. The log of the squashing factor Q in the y ¼ 100 grid point plane of the VPIC simulation for field lines traced to y ¼ 1024 grid points. Q is large for field lines that belong to the QSL.

To derive the QSL, we choose those field lines at y ¼ 100 whose Q values exceed 104, which is just over 10 times the mean value over the domain of seed points at y ¼ 100. Fig. 4 depicts the resulting QSL from a view along the y-axis, where the blue surface represents the QSL, and the pink layer at y ¼ 100 represents footpoints with Q (cid:5) 104. The hyperbolic structure of the QSL is evident, as is the exchange of directions of elongation and compression. The QSL defines a likely location for reconnection. To find where reconnection occurs, we must evaluate the integral of the parallel electric field.

Phys. Plasmas 20, 122105 (2013)

FIG. 4. The hyperbolic structure of the QSL between y ¼ 100 and y ¼ 1024 grid points based on Q (cid:5) 104 in the VPIC simulation. The blue surface is the QSL, which twists from one orien- tation to another along y, and the pink layer corresponds to the endpoints of the QSL at the start plane at y ¼ 100.

V. RECONNECTED MAGNETIC FLUX AND INTEGRATED PARALLEL ELECTRIC FIELDS

A map of N for a fixed y identifies which field lines lie on a gradient in the quasi-potential and therefore are reconnecting. Fig. 5(a) is the resulting map of the quasi-potential N in x-z space in the starting plane at y ¼ 100 grid steps after integrat- ing the parallel electric field from y ¼ 100 to y ¼ 1024 grid steps. The quasi-potential is here expressed in units of vAB0di/c, where c is the speed of light and vA ¼ B0/(4pn0mi)1/2 is the Alfve`n speed. Figure 5(b) of the current density in the the same plane shows a corresponding structure. (Again, limited range cuts computational costs while producing suffi- cient information for our purposes. To calculate the reconnec- tion rate, we integrate a subset of field lines all the way through the system, as explained below.) The contours of N collocate with the cross-section of the QSL in the same plane. However, because the theory of general magnetic reconnection is formulated in a-b space, we derive an a-b coordinate system

FIG. 2. A front view of the hyperbolic structure of a QSL. The layer is extended along the out-of-page direction and twists from one orientation to another. Here the darker “S” curve lies in the near plane while the lighter curve lies in the far plane. The dashed arrows indicate the direction of motion of field lines that undergo ﬂipping. The section of the layer where field lines move toward the center is the stable manifold while the section where they move away from the center is the unstable manifold.

IV. QUASI-SEPARATRIX LAYER IN VPIC

We first derive the squashing factor Q for the inner part of the volume by integrating field lines across 1=2 of the y- domain (i.e., between y ¼ 100 and y ¼ 1024 grid steps), the direction of the guide field and the initial current layer. Each grid step is (cid:4)0.034 di based on the Harris reference density, so this corresponds approximately to a span from y ¼ 3.4 di to y ¼ 35 di. We limit the integration domain to 1=2 of the y length because the calculations are very computationally ex- pensive and 1=2 of the domain is sufficient for our purposes. While this limitation may omit part of the QSL structure, we find that the integration length is sufficient to achieve a squashing factor ten times the global average, the benchmark we use to define the QSL. The field lines are calculated using both the stream3 function in the Matlab toolbox and, for com- parison, by an in-house Python field tracing routine, and both methods show agreement. By numerically differentiating the endpoints of the resulting field lines at y ¼ 100 and y ¼ 1024, we derive a map of Q in the plane at y ¼ 100 for the region 500 < x < 1500 and 360 < z < 600. To maximize the compu- tational efficiency, we exclude the outer edges of the x-z plane where the current density is much lower. In our calcula- tion, the denominator in Q is (cid:4)1 since the guide field is fairly uniform over the volume. Fig. 3 shows the resulting log scale map of Q, where warmer colors signify a larger Q. The white regions correspond to seed points at y ¼ 100 whose field lines exit the simulation box before reaching y ¼ 1024.

Wendel et al.

bðu; vÞ ¼

b ¼ (cid:3)

where

z ð

(10)

Þr;

(9)

(8)

(cid:8)

FIG. 5. (a) A map of the quasi-potential N in the y ¼ 100 plane, from integra- tion of the parallel electric field between y ¼ 100 and y ¼ 1024 grid points. The quasi-potential is in units of vAB0di/c. The two asterisks in the inset denote the footpoints of the two field lines discussed in Figure 9. (b) The magnitude of the current density in the corresponding plane shows a corresponding structure.

in which to create the map. If U(x,y,z) and V(x,y,z) are func- tions that satisfy

B ¼ FðU; VÞrUðx; y; zÞ (cid:2) rVðx; y; zÞ;

and u(x,z) and v(x,z) are the values that U and V take on the plane y ¼ 100, then in the plane y ¼ 100, a and b are func- tions of u, v, i.e., a(u,v) and b(u,v). Because a and b are not unique, we can choose a ¼ u and find the b(u,v) that satisfies Eq. (1). Stern22 shows that

v ð

f ðu; v0Þdv0;

(cid:7) f ðu; vÞ ¼ ðB?= rUjj (cid:2) rVjj

and the perpendicular and parallel components refer to the plane r defined by u and v (here the plane y ¼ 100). Here we make the simple choice U ¼ x þ y and V ¼ z þ y, which means a ¼ u ¼ x. Then according to Eq. (9)

Byðx; z0Þdz0:

There is excellent agreement between the QSL and the boundaries of the region of enhanced N. Fig. 6 shows the

Phys. Plasmas 20, 122105 (2013)

FIG. 6. (a) A map of N in terms of the Euler variables a ¼ x and b ¼ (cid:3)Ð By(x,z0) dz0. (b) The map of N overlaid with the footpoints of the Q (cid:5) 104 QSL, shown in purple.

map of N along with the footpoints of the QSL (in purple, Fig. 6(b)) as a function of a and b in the y ¼ 100 end plane. Other than an inversion, there is very little change from the mapping in the x-z plane because of the large guide field. According to the theory of general magnetic reconnection, reconnection occurs where there are gradients in N as a func- tion of a and b. Therefore, it occurs on those field lines that lie on the edges of the area of large N, which also show a high correspondence with field lines that lie on the QSL. General magnetic reconnection also predicts that the gra- dients in N with respect to a and b occur in closed loops because of the Hamiltonian form of the evolution equations. This prediction also concurs with the shape of the region of large N and the QSL, albeit in a very stretched and elongated shape. The QSL appears to follow the loop-like structure of elevated N and has the appearance of being bifurcated, which in turn correlates with the appearance of a bifurcated current sheet in Fig. 5(b). Therefore, this mapping allows us to iden- tify reconnecting field lines as those field lines on the QSL that correlate with large gradients in N.

VI. DISTRIBUTION OF THE PARALLEL ELECTRIC FIELD ALONG FIELD LINES AND THE RECONNECTION RATE

The question arises as to how the parallel electric field that contributes to the global reconnection rate is distributed along the field lines that support reconnection. In this simula- tion, Ejj is highly ﬂuctuating, owing to plasma processes leading to turbulence13 and, on grid scales, to statistical noise from low particle counts per cell. Figs. 7(a) and 7(b) com- pare Ejj along a field line that is reconnecting and one that is not. Following Ejj along a field line without integrating, it is

Wendel et al.

FIG. 7. (a) E|| along a reconnecting field line on the QSL. (b) E|| along a non-reconnecting field line. (c) The same reconnecting and non-reconnecting field lines colored by E||.

difficult to make a judgment where reconnection is occur- ring, as Fig. 7 attests. Moreover, it is very difficult to distin- guish in this way any difference between the reconnecting and non-reconnecting signatures, much less where reconnec- tion might be occurring along the field line. The spatial plot in Fig. 7(c) of the same reconnecting and non-reconnecting field lines that are color-coded according to Ejj does appear to reveal a higher average Ejj along the reconnecting field line. Nevertheless, it is not clear to what extent the reconnec- tion is patchy, corresponding to the peaks in the ﬂuctuations, and to what extent the ﬂuctuations contribute to the global reconnection rate. Fig. 8 shows a view along z of the set of QSL field lines colored by Ejj. As Fig. 8 illustrates, some of the ﬂuctuations also organize into Bernstein-Greene-Kruskal (BGK) mode electrostatic solitons, or electron holes.23,24 The question also arises, therefore, as to what contribution the ﬂuctuations make to the reconnection rate.

For insight to these questions, we calculate the running sum of Ejj in units of vAB0di/c up to each position along the field line and plot this as a function of position along the field line for both reconnecting and non-reconnecting field lines. The footpoint of the chosen reconnecting field line corre- sponds to the red asterisk in Fig. 5(a) while that of the non-reconnecting field lines corresponds to the blue asterisk in Fig. 5(a). This shows us where reconnection is occurring and how steady it is along the field line. Fig. 9 shows that the sum for the reconnecting field line almost immediately exceeds that of the non-reconnecting line so that we know that the difference does not owe to the fact that the recon- necting field line is longer. Though there are small amplitude ﬂuctuations in the sum and there are some changes in the

Phys. Plasmas 20, 122105 (2013)

FIG. 8. A view along the -z direction of QSL field lines colored by E||, show- ing ﬂuctuating bands of positive and negative E||, like electron holes.

slope, the value is always increasing on average, suggesting that a low-order, rather than highly ﬂuctuating, contribution to Ejj controls the reconnection rate. To quantify this idea, in Fig. 10 we plot the DC and linear components of Ejj as a function of field line length for both the reconnecting and non-reconnecting field lines. Though the value of Ejj decreases with distance along the field line for both types of lines, it is always positive for the reconnecting field line with a much higher average. In fact, the total integral of the DC and linear trend of the reconnecting field line along the full length of the line is almost exactly equal to N, and the inte- gral over just the higher-order ﬂuctuations is vanishingly small. Therefore, including the electron holes, are not contributing to the global reconnection rate. the electron holes are also observed on We note that non-reconnecting field lines with a negligible parallel elec- tric field. These facts argue against the possibility that a dif- ference in the number of positive and negative layers

the ﬂuctuations,

FIG. 9. The running sum of E|| (in units of vAB0di/c) along a reconnecting (solid) and non- reconnecting (dashed) field line as a function of position along the field line smoothes the data. The difference is apparent near the be- ginning of the field lines, revealing what appears to be a relatively steady contribution to the reconnection rate. Reconnection is occurring anywhere along the field line where the running sum is increasing.

Wendel et al.

which has a peak value at 1500 grid steps of about 5.4 in units of vAB0di/c. Beyond this point the sum starts leveling off, signifying reconnection may have decreased or shut off, and that the field line may have reentered an ideal region. Normalizing by this length (about 1500 grid cells, or 51 di) gives a normalized reconnection rate of about 0.1 vAB0/c. The field lines inside the high N region will remain inside the diffusion region for their entire length. However, we expect field lines on the edges of the region to leave the dif- fusion region at some point. Thus the determination of their potential drop across a nonideal region, from one ideal region to another, follows the prescription of Hesse and Schindler.7 An estimate of the reconnection rate based on averaging the ion inﬂow speed Uiz and the Alfve`n speed vA in a region centered on the inﬂow/outﬂow region gives a reconnection rate of hUiz/vAi (cid:4) 0.1, which provides corrobo- ration of the value derived from N.

VII. SUMMARY AND CONCLUSIONS

We have sought an organizing principle to determine the location and rate of reconnection in a large-scale kinetic simulation of general 3D magnetic reconnection with a guide field and open boundary conditions. The field line evolution appears to be turbulent and the guide field inhibits the forma- tion of null points in the 3D topology, making it challenging to identify where reconnection is happening, though clearly it is. The concept of reconnection through magnetic ﬂipping on a QSL provides a description of how reconnection can occur in this case. We derive the QSL along one-half the length of the system to identify field lines likely to engage in magnetic ﬂipping and find that it consists of a twisted, hyper- bolic layer extending through the system. A parallel electric field along the QSL indicates which field lines undergo slip- page. Therefore, we create a 2D map of the integrated paral- lel electric field for each field line. We find that the QSL correlates well with the boundaries of the region of field lines that have a pronounced quasi-potential. This implies that field lines with significant value of the integrated parallel electric field are, in all likelihood, threading current sheets— the natural locations of enhanced Q factors. The QSL carry- ing a parallel electric field therefore correlates well with the prediction of the theory of general magnetic reconnection that a necessary and sufficient condition for 3D reconnection in the absence of null points is a gradient of the quasi- potential, or integrated parallel electric field, with respect to the Euler variables. The prediction that the reconnection occurs in loop-like structures also agrees with the elongated, but generally loop-like shape of the contours of N. The tur- bulence produces finer structure within the loop. The maxi- mum N gives a normalized reconnection rate across the QSL of about 0.1 vAB0/c.

To further illuminate how and where Ejj is distributed along the QSL field lines that do support reconnection, we follow the partial, or running, sum of the parallel electric field incrementally along the field lines. It is clear that the field lines that are reconnecting carry a relatively steady DC con- tribution to the reconnection rate that is absent on field lines that are not reconnecting, though both types of field lines

Phys. Plasmas 20, 122105 (2013)

FIG. 10. The parallel electric field filtered to remove all but linear trends in the reconnecting (red) and non-reconnecting (blue) field lines. The recon- necting field line has a non-zero average, and the final reconnection rate owes entirely to the sum over the DC and linear contributions to E|| while the ﬂuctuations contribute nothing.

accounts for the growing running sum in Fig. 9. We also observe that the increase in Ejj along the field line in the direction of the current (Jy > 0) is consistent with JjjEjj > 0, or the conversion of magnetic to particle energy.

Finally, we wish to determine the reconnection rate across the ﬂux surfaces of the QSL for the global system. Hesse et al.8 shows that this is given by the maximum value of N along the ﬂux surface in a-b space. From Fig. 6, we dis- cern that this is just over 4, which yields a normalized recon- nection rate of about 0.1 when divided by the length of the field line. However, in the event that some information has been lost by dint of the fact that we have only integrated halfway through the system, we integrate a small subset of field lines near the center of the simulation volume all the way from y ¼ 1 to y ¼ 2048. Fig. 11 shows the result for one of the field lines with a value of N close to the maximum,

FIG. 11. The running sum of E|| in units of vAB0di/c (solid line) and the cor- responding value of E|| in units of vAB0/c (dashed line) for a reconnecting QSL field line determined from integrating along the entire length of the sys- tem. The peak value of the running sum of E|| is about 5.4. When normalized by the length of the nonideal electric field (about 1500 grid cells, or 51 di), the normalized reconnection rate is about 0.1 vAB0di/c, which is about the maximum value anywhere on the QSL. Therefore, this is approximately the reconnection rate across the ﬂux surface of the QSL.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:132.239.1.231 On: Wed, 11 Dec 2013 15:43:29have ﬂuctuating positive and negative values of the parallel electric field. This conclusion does not necessarily challenge the accepted view that electron-scale, non-ideal processes in a localized region about the reconnecting field line govern reconnection. Rather, the implication is that the observed ﬂuc- tuations do not account for the underlying parallel electric field that governs the global reconnection rate. However, the temperature anisotropy in the generalized Ohm’s law can sup- port a parallel electric field over broader regions not confined to electron scales. When Liu et al.20 investigate the relative importance of temperature anisotropy and electron-scale con- tributions, such as electron inertia and nongyrotropic pressure, they find that the relative importance of the contributions depends greatly on initial conditions, due to the chaotic nature of the fields. Here we demonstrate that the reconnection rate results entirely from the DC and linear trends in Ejj and that higher-order ﬂuctuations do not contribute appreciably on the larger scale even though they may have local effects. Moreover, it is clear reconnection is occurring along a field line wherever the running sum of Ejj is increasing.

The ramification is that the location and rate of recon- nection in simulations and observations can be identified with low-order coherent structures even when fields appear turbulent. This conclusion is consistent with the discussion in Leonardis et al.19 While it is certainly possible that the ﬂuctuations contribute to the physics that sustains the average reconnection electric field, it is rather that average, DC field that governs the reconnection rate. The results also impact the inference of reconnection sites and rates in observations, of importance to the upcoming MMS (Magnetospheric Multiscale) mission. Clearly if the medium is not the global reconnection rate cannot be inferred from a local parallel electric field measurement, but rather must be estimated from the spatial average along a field line or from some other means.

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D. E. Wendel would like to thank Vadim Uritsky and John Dorelli for helpful discussions. This research was supported by a NASA Goddard Space Flight Center Innovation Fund award and by the MMS Science Interdisciplinary Science grant to the Goddard Space Flight Center. D. Olson and N. Aunai were supported by the NASA Postdoctoral Program. HK’s contributions were