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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/261013293 The relation between reconnected flux, the parallel electric field, and the reconnection rate in a three-dimensional kinetic simulation of magnetic reconnection Article in Physics of Plasma · November 2013 DOI: 10.1063/1.4833675 CITATIONS 29 READS 251 8 authors, including: Deirdre E. Wendel National Aeronautics and Space Administration 20 PUBLICATIONS 210 CITATIONS SEE PROFILE Nicolas Aunai Laboratoire de Physique des Plasmas, French National Centre for Scientific Resea... 57 PUBLICATIONS 1,813 CITATIONS SEE PROFILE Homa Karimabadi 318 PUBLICATIONS 8,878 CITATIONS SEE PROFILE All content following this page was uploaded by Homa Karimabadi on 05 August 2014. The user has requested enhancement of the downloaded file. 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 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 15:43:29 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. Adrian 1 1 NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA 2 Institute for Research in Astrophysics and Planetology, University Paul Sabatier, Toulouse, France 3 Sciber Quest, Inc., Del Mar, California 92014, USA 4 Department of Computer and Electrical Engineering, University of California, San Diego, La Jolla, California 92093, USA 5 Los 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 andin situspacecraft 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. V C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4833675] I. INTRODUCTION 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 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 70d i 35d i 70d i , whered i ¼c/x pi is the ion skin depth, with a mass ratio m i /m e ¼100 and a grid resolution of approximately 0.34d e , whered e is the electron skin depth. In practice, boundary conditions are open inxandz, and peri- odic alongy, 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 1070-664X/2013/20(12)/122105/8/$30.00 V C 2013 AIP Publishing LLC20, 122105-1 PHYSICS OF PLASMAS20, 122105 (2013) 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 15:43:29 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- ondary flux ropes. Flux ropes generated at these scales 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, and turbulence. 19 Turbulence continues 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 Daughtonet al. 13 Reconnection is well underway, att X ci ¼98, where X ci is the ion cyclotron frequency. Other parameters of the system are T i ¼T e ,B y ¼B x0 ,andn cell ¼120, where T i and T e are the ion and electron temperature,B y is the guide field,B x0 is the initial reconnecting field, andn cell is the number of particles per cell per species. 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 startingy-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 Liuet 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. II. THEORY OF GENERAL MAGNETIC RECONNECTION Hesse and Schindler 7 and Schindleret 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-