Turbulent heating due to magnetic reconnection

Turbulent heating due to magnetic reconnection M. A. Shay, C. C. Haggerty, W. H. Matthaeus, T. N. Parashar, M. Wan, and P. Wu Citation: Physics of Plasmas 25, 012304 (2018); doi: 10.1063/1.4993423 View online: https://doi.org/10.1063/1.4993423 View Table of Contents: http://aip.scitation.org/toc/php/25/1 Published by the American Institute of Physics Articles you may be interested in Test-electron analysis of the magnetic reconnection topology Physics of Plasmas 24, 122303 (2017); 10.1063/1.5004613 On drift wave instabilities excited by strong plasma gradients in toroidal plasmas Physics of Plasmas 25, 014502 (2018); 10.1063/1.5000281 Magnetic plasma expulsion Physics of Plasmas 25, 012508 (2018); 10.1063/1.5006887 Observation of trapped-electron-mode microturbulence in reversed field pinch plasmas Physics of Plasmas 25, 010701 (2018); 10.1063/1.5010198 Electron holes in phase space: What they are and why they matter Physics of Plasmas 24, 055601 (2017); 10.1063/1.4976854 Editorial: Preface to the 25th Volume of Physics of Plasmas Physics of Plasmas 25, 010401 (2018); 10.1063/1.5021964

PHYSICS OF PLASMAS 25, 012304 (2018) Turbulent heating due to magnetic reconnection M. A. Shay, 1,a) C. C. Haggerty, 1 W. H. Matthaeus, 1 T. N. Parashar, 1 M. Wan, 2 and P. Wu 3 1 Department of Physics and Astronomy, Bartol Research Institute, University of Delaware, Newark, Delaware 19716, USA 2 Department of Mechanics and Aerospace Engineering, South University of Science and Technology of China, Shenzhen, Guangdong 518055, People s Republic of China 3 School of Mathematics and Physics, Queens University, Belfast BT7 1NN, United Kingdom (Received 28 June 2017; accepted 7 December 2017; published online 2 January 2018) Dissipation of plasma turbulent energy is a phenomenon having significant implications for the heating of the solar corona and solar wind. While processes involving linear wave damping, stochastic heating, and reconnection have been postulated as contributors to heating mechanisms, the relative role that they play is not currently understood. In this manuscript, we establish a theoretical framework for applying reconnection heating predictions to turbulent systems. Kinetic particle-in-cell (PIC) simulations are used to study heating due to reconnection, and these results are then adapted to a turbulent medium. First, the factors controlling the heating of plasmas in reconnection exhausts are examined using laminar reconnection simulations; predictions for heating are determined which require only the plasma conditions just upstream of the reconnection diffusion region as input. The laminar predictions are then applied to PIC simulations of turbulence. Key assumptions are: (1) the plasma conditions just upstream of the diffusion region are consistent with Kolmogorov scaling of turbulent fluctuations at the ion inertial scale and (2) the statistics of the numbers of reconnecting x-lines do not vary significantly between the various turbulent simulations. We find that the reconnection theory predicts quite well the scaling of the ratio of ion to electron heating, in which the statistics of the turbulent reconnection sites are expected to roughly cancel. Separate ion and electron heating rates scale differently from the theory, however. This suggests that the statistics of the turbulent reconnection (e.g., number of x-lines, percentage of x-lines reconnecting) is playing an important role in determining the ion and electron heating. Published by AIP Publishing. https://doi.org/10.1063/1.4993423 I. INTRODUCTION The dissipation of plasma turbulent energy is a phenomenon with wide ranging implications for heliospheric and astrophysical plasmas. In many of these systems, the plasma collision time is often longer than the dynamical timescales of the system, leading to unsolved questions concerning the mechanisms ultimately dissipating the turbulent energy. The situation is complicated by the fact that turbulence is a multiscale phenomenon and therefore dependent on the range of available length scales. To capture the physics of high (effective) Reynolds number, turbulence 1 requires at least several decades of scale. 2 Recently, however, direct numerical simulations including some portion of the inertial range, while also including kinetic physics at the ion and/or electron scales, have become feasible. Such simulations allow the question of dissipation in weakly collisional turbulence to be probed. 3 5 One set of these fully kinetic simulation studies has investigated kinetic damping employing an adaptation of von-karman similarity decay to the case of plasma. 6,7 It was found that while the total heating in the plasma was consistent with a von-karman decay, the relative heating between ions and electrons was strongly dependent on the amplitude of the turbulence (see also Chandran et al. 8 ). For stronger a) shay@udel.edu amplitude of the turbulence, one obtains a larger ratio of ion to electron heating rates. This was explained by stating that the larger amplitude would lead to larger perturbations at the ion inertial scale, which would perturb ion orbits and lead to greater input of energy into random ion motions. Subsequently, there would be less energy left for electron heating, which presumably occurs at scales comparable to the electron inertial length. Similarly, studies of proton and electron heating during whistler and kinetic Alfven wave turbulence, some using kinetic PIC simulations, found that proton heating increased relative to electron heating as the turbulence amplitude was increased. 8 10 Even if macroscopic cascade processes control the total rate of dissipation, important questions remain concerning microscopic dissipation. In particular, what are the heating mechanisms that lead to the above behavior? Many such mechanisms have been postulated, with three prominent examples being linear Landau damping of waves, 3 stochastic heating, 8 and dissipation associated with coherent structures and intermittency. 4 In this manuscript, we focus on the dissipation occurring in or near coherent structures. In plasmas, coherent structures often take the form of current sheets or similar structures that can reconnect, create vortices and exhausts (jets), leading to significant plasma heating. There has been significant work trying to characterize both the generation of coherent structures during turbulence 11,12 and the resulting 1070-664X/2018/25(1)/012304/9/$30.00 25, 012304-1 Published by AIP Publishing.

012304-2 Shay et al. Phys. Plasmas 25, 012304 (2018) reconnection. 13,14 On the other hand, there has also been significant recent progress in understanding how a single reconnection event leads to heating in the reconnection exhaust (e.g., Refs. 15 20). Both satellite observations and scaling studies have found that the bulk plasma heating in the exhaust can be predicted accurately using only the inflowing plasma conditions. 16,18 Given that current sheets and reconnection occur in many locations in a turbulent medium, intriguing questions arise concerning the relationship between two types of theories and/or observations: the heating due to reconnection 15 and the heating due to von-karman decay in turbulence. 6 A major motivation for this manuscript is to explore the relationship between these two apparently disparate heating theories. In this manuscript, therefore, we propose a framework for applying laminar reconnection heating theories to turbulent plasmas. We determine the factors controlling the heating during single reconnection events and generalize these results to the case of turbulence. These predictions are then tested using kinetic PIC simulations of turbulence. Key assumptions in applying the laminar heating predictions to turbulence are: (1) the plasma conditions just upstream of the diffusion region are consistent with the Kolmogorov scaling of turbulent fluctuations from the correlation scale to the ion inertial scale and (2) the statistics of the numbers of reconnecting x-lines do not vary significantly between the various turbulent simulations. We find that the reconnection theory is consistent with the scaling of the ratio of ion to electron heating. The individual ion and electron heating rates scale differently, however. A possible explanation for these results is that the statistics of the turbulent reconnection is playing an important role in determining the separate ion and electron heating rates, but for the ratio, the statistics cancel. By statistics of the turbulent reconnection, we mean the behavior of such factors as the number of x-lines as well as the percentage of those x-lines that are actively reconnecting. Finally, the scaling of the turbulent proton to electron heating has intriguing similarities with previous turbulent heating studies based on the assumption of a von Karman total heating rate. 7 The fact that these two very different paradigms show complementary results raises interesting questions as to whether there may be a universal mechanism, or at least a commonality of basic physics, that describes heating in turbulence and in reconnection. II. LAMINAR RECONNECTION SIMULATIONS We first study the scaling of heating in reconnection exhausts using kinetic-pic simulations of laminar reconnection. This method has been successfully applied to studies of electron and ion heating, 21 and has been favorably compared with satellite observations of reconnection events. 16 We use the parallel PIC code P3D 22 to perform simulations in 2.5 dimensions of collisionless symmetric reconnection. In the simulations, magnetic field strengths and particle number densities are normalized to arbitrary characteristic values B 0 and n 0, respectively. Lengths are normalized to the ion inertial length d i0 ¼ c=x pi0 at the reference density n 0. Time is normalized to the ion cyclotron time X 1 ci0 ¼ðeB 0 =m i cþ 1 : Note that in different simulations, the actual inflow magnetic field varies, so the physical cyclotron time varies between simulations. p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Speeds are normalized to the Alfven speed c A0 ¼ B 2 0 =ð4p m i n 0 Þ. Electric fields and temperatures are normalized to E 0 ¼ c A0 B 0 =c and T 0 ¼ m i c 2 A0, respectively. The coordinate system is a generic simulation coordinates, meaning that the reconnection outflows are along ^x and the inflows are along ^y. Simulations are performed in a periodic domain with the size and the grid scale varied based on simulation and inflow parameters. The initial equilibrium consists of two Harris current sheets superimposed on an ambient population with a uniform density of n up. The initial reconnection magnetic field is given by B x ¼ B r ðtanh½ðy L y =4Þ=w 0 Š tanh½ðy 3L y =4Þ=w 0 Š 1Þ, where w 0 and L y are the half-width of the initial current sheets and the box size in the ^y direction. We simulate a range of inflowing reconnection magnetic fields B r, upstream densities n up, and upstream ion and electron temperatures T iup and T eup. These parameters are given in Table I. Additional information regarding these simulations can be found in two previous publications. 18,21 All simulations are also initialized with a uniform outof-plane guide field B g. Reconnection is initiated with a small initial magnetic perturbation that produces a single magnetic island on each current layer. Each simulation is evolved until the reconnection becomes quasi-steady. Then, during this period, the simulation data are time averaged over a duration of 100 particle time steps, which is typically on the order of 50 electron plasma wave periods x 1 pe. A typical reconnection simulation is shown in Fig. 1 with fb r ; n up ; T iup ; T eup ¼ 1; 0:2; 1:25; 0:25g: The simulation size is 204:8 d i 102:4 d i. J z is shown in Fig. 1(a), with the black lines being the magnetic field lines. The vertical dashed line shows the location of the vertical cuts along y in Figs. 1(b) 1(e). The exhaust region is determined by inspection of the magnetic fields and ion velocities [Figs. 1(b) and 1(c)], and is denoted by the vertical dashed lines in panels b-e. In panels d-e are shown the ion and electron TABLE I. Initial inflow parameters for the laminar reconnection simulations. Run m i =m e B r B g n up T eup T iup 324 25.0 1.0 1.0 0.2 1.25 0.25 604 25.0 1.0 1.0 0.2 1.25 0.25 606 25.0 2.236 1.0 0.2 1.25 0.25 624 25.0 1.0 1.0 0.2 1.25 0.25 626 25.0 2.236 2.236 0.2 1.25 0.25 629 25.0 0.75 1.0 0.2 1.25 0.25 631 25.0 1.0 0.25 0.2 1.25 0.25 632 25.0 1.0 0.5 0.2 1.25 0.25 633 25.0 1.0 0.75 0.2 1.25 0.25 636 25.0 1.0 2.0 0.2 1.25 0.25 637 25.0 1.0 4.0 0.2 1.25 0.25 681 25.0 0.447 0.45 0.2 0.25 0.05 682 25.0 2.236 2.24 0.2 6.25 1.25 702 100.0 1.0 1.0 0.2 1.25 0.25 706 100.0 2.236 2.236 0.2 1.25 0.25

012304-3 Shay et al. Phys. Plasmas 25, 012304 (2018) temperature by averaging over a rectangle 5 d i upstream of the outermost edge of the exhaust; the rectangle size is 40 d i along x and 10 d i along y. The temperature change from the inflow to the exhaust DT ¼ T ex T up can then be determined. As discussed at length in Shay et al., 18 because of the quasi-steady nature of the reconnection, DT is directly related to the rate of energy conversion from magnetic to thermal in the quasi-steady reconnection exhaust. This relation emerges because the heating rate of plasma in the exhaust is balanced by the influx of cold upstream plasma. This balance leads to the observed property 18 that the total plasma temperature does not show a strong variation with distance from the x-line. The exhaust heating for various simulations can then be directly compared with upstream parameters to study both the scaling and the underlying physics of the heating due to magnetic reconnection. The turbulence simulations used in this study all have a guide field that is typically larger than the fluctuating magnetic field db. For that reason, we limit our heating database to those cases with a significant guide field such that B g > 0:2. We also limit our cases to T e =T i < 1:25 and b i < 2. Scaling of the ion and electron heating relative to the best fitting upstream parameters is shown in Fig. 2. Single pass acceleration of the ions due to contracting B 2 r B 2 ; magnetic islands 15 gives the ion heating of DT i / m i c 2 Ar where B 2 ¼ B 2 r þ B2 g is the total magnetic field and B r is the upstream reconnection magnetic field. This prediction for the ion heating matches well with the observed heating in Fig. 2(a). There is currently no theory for the reconnection electron heating in the guide field case, although progress has been made for antiparallel reconnection where it was found that the electrons perform multiple Fermi reflections and are affected by a parallel potential (Ref. 21, and references therein). Empirically, the best fit for the electron data is DT e / m i c 2 Ar B r B ; as shown in Fig. 2(b). This leads to the predicted heating for the ions and electrons to DT i DT e / B r B, as shown in Fig. 2(c). Note that the scaling of heating determined in this section is solely based on laminar reconnection studies. FIG. 1. Determining the heating inside the reconnection exhaust during laminar magnetic reconnection. (a) Out-of-plane current J z with magnetic field lines. The vertical dashed line shows the location of cuts. Cuts through exhaust: (b) magnetic fields, (c) ion velocities, (d) ion temperatures, and (e) electron temperatures. The dashed lines in (b) (e) show the bounds of averaging region where the exhaust temperature is determined. temperatures. To determine the average quantities in the exhaust, we average quantities in a trapezoid spanning the exhaust between 20 and 40 d i downstream of the x-line; the vertical dashed lines in Figs. 1(b) 1(e) show the bounds of this trapezoid at the location of the cut. For this study, we focus on the total temperature of each species, which is 1/3 of the trace of the corresponding temperature tensor. In reconnection simulations with a guide field, there can be significant density variation throughout the exhaust, so averaging only the temperature in the exhaust may overemphasize low density regions. For this reason, the average temperature in the exhaust is determined by first averaging the pressure and then dividing by the average density, or T ex ¼hpi=hni. We calculate the upstream III. THEORETICAL FRAMEWORK FOR APPLYING RECONNECTION HEATING PREDICTIONS TO TURBULENCE We now examine this basic scaling of reconnection heating in the context of a turbulent system. The question is whether it is possible to arrive at turbulent heating predictions without invoking the von-karmen phenomenology. 6,7 A simplicity of the reconnection heating predictions is that they only depend on the inflow conditions just upstream of the reconnection diffusion region. If one can determine the typical value of these upstream conditions in the turbulence, then this theory can be applied on average to the turbulent system as a first pass. To lay out a framework for this application, we examine the complete reconnection of two islands of magnetic flux shown in Fig. 3. Note that the radii of the two magnetic islands are different, hence the notations þ and. We make the assumption that the average magnetic field and density in the two islands are the same. Note that this

012304-4 Shay et al. Phys. Plasmas 25, 012304 (2018) assumption can be generalized to differing magnetic fields and densities in a straightforward manner using theories of asymmetric reconnection. 18,23 After the two islands reconnect completely, the volume of plasma which has interacted with a reconnection site will be p ð Þ 2 þ p ½ð þ Þ 2 ð þ Þ 2 Š¼2p þ. We assume that all plasma interacting with the reconnection site is heated as predicted from the laminar reconnection results DT ¼ M T c 2 Ar ðb r B Þc ; where the coefficient M T and the exponent c are determined from the laminar reconnection studies in Section II; c ¼ 2 for ions, while c ¼ 1 for electrons. The total thermal energy added to the islands from the reconnection is thus e x ¼ 2p DT þ n up, which can be simplified to e x ¼ M c T 2 þ B 2 B r r : (1) B Note that the subscript x one is to show that it is for a single x-line. We then sum over all of the active x-lines during the turbulence simulation (written with subscript i). e ¼ M T 2 X i þ i B2 ir x line i B c ir : (2) B i Given a normalized reconnection rate of R, the time to completely reconnect the two magnetic islands is Dt ¼ =ðrc Aup Þ. Note that for most kinetic plasmas, R 0:1 for strongly reconnecting systems (e.g., Ref. 24). A recent kinetic PIC study of turbulence 25 found that the average R 0:1, but a turbulent plasma would have many x-lines that are not reconnecting or reconnecting very weakly. 13 Determining the average heating rate Q ¼ e=dt gives Q ¼ M T 2 X x line i þ i R i c iar B 2 ir B c ir : (3) B i FIG. 2. Heating during laminar reconnection: (a) ion heating, (b) electron heating, and (c) the ratio of ion to electron heating. FIG. 3. Two reconnecting islands of magnetic flux of differing radii þ and. Equations (2) and (3) can be used to estimate the heating due to reconnection occurring within turbulence, if the average reconnection inflow conditions can be determined. If reconnection makes a major or dominant contribution to the overall heating in turbulence, then this procedure may lead to reasonable predictions of the characteristics of heating in turbulence. The required inflow conditions may be estimated in the following way. First, the width of the diffusion region is typically expected to scale with d i c=x pi, 26 and also reconnection has been shown to accelerate or onset when the width of the diffusion region narrows to d i scales. 27,28 We therefore assume that the reconnection diffusion region is associated with turbulent fluctuations at a scale of d i.we write the turbulent magnetic fluctuation amplitude at this scale as db di : We assume that on average, energy is equipartitioned between the bulk ion p flow and the magnetic field, giving dz di c Adi ¼ db di = ffiffiffiffiffiffiffiffiffiffiffiffi 4pm i n. Note again that the Alfven p speed based on the reconnecting field is c Ar ¼ B r = ffiffiffiffiffiffiffiffiffiffiffiffi p 4pm i n and on the guide field is c Ag ¼ B g = ffiffiffiffiffiffiffiffiffiffiffiffi 4pm i n. Defining s ci ¼ð eb g m i c Þ 1 as the inverse ion cyclotron time based on the guide field and s nl ðd i Þ¼ðdZ di =d i Þ 1 as the turbulent nonlinear time at the scale size d i ; we can link the

012304-5 Shay et al. Phys. Plasmas 25, 012304 (2018) upstream reconnection conditions with the turbulent values in the following way: c Ar B r dz di d i s ci c Ag B g c Ag d i s nl ðd i Þ ¼ a nl; (4) where we have defined the quantity a nl s ci s nl ðd i Þ. A straightforward way to estimate the nonlinear time is to adopt the average cascade timescale evaluated at some scale length : Applying the Kolmogorov phenomenology for the scale dependent amplitude, one finds dz Z 1=3 ; (5) k from which one finds, 29 associating the scale with d i, s nl ðd i Þ¼ d i ¼ d 2=3 i s nl: (6) dz di k Here, k is the energy containing scale, and Z and s nl are the turbulence amplitude and the nonlinear time at the energy containing scale. A final simplifying assumption can be made in applying the reconnection heating predictions to the turbulent system. In all of the turbulence simulations used in this study, the mean guide field is much greater than the fluctuating field, i.e., B g B r ; so that the total magnetic field in the reconnection scaling predictions B B g. With this approximation, and using Eq. (4), the heating predictions derived from reconnection scaling can be applied to turbulent heating in the form e i DT i / m i c 2 B 2 r Ar B 2 / a4 nl ; (7) e e DT e / m i c 2 B r Ar B / a3 nl ; (8) giving e i DT i / B r e e DT e B / a nl: (9) IV. TURBULENCE SIMULATIONS We examine a set of turbulence simulations used in previous turbulence studies of heating. 6,7 These 2.5-dimensional simulations also use the kinetic PIC code P3D and have the same normalizations as the laminar reconnection simulations described in Sec. II. Parameters of the simulations are given in Table II. These runs were initialized with MHD-like velocity and magnetic field fluctuations (solenoidal velocity and uniform density) excited at wavenumbers k L within an annular band, where k L ¼ k=ð2p=lþ; k L ¼ 1 corresponds to a single wavelength across the simulation domain. The exceptions are the Parashar runs listed in Table II under the family OTVdbB and PAPJ15, which were Orszag-Tang simulations. 7 In all cases, the initial energy is concentrated near the largest excited wavenumbers. In both types of simulations, the energy then cascades to fill in the complete wavenumber range available and produces turbulence. TABLE II. Turbulence simulation parameters given in normalized units. These simulations have been used in previous publications. 6,7 Listed are: proton pand electron b (b ¼ 2 nt=b 2 ); turbulence amplitude Z 0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðhv 2 iþhb 2 iþ,wherev and b are turbulent fluctuation values of the velocity and the magnetic field, and hi is an average over the entire simulation domain; a uniform guide field B 0 along z; a square simulation domain of size L; the number of grid points N in each direction; particles per grid ppg; wavenumber band of initial conditions, with k L ¼ k=ð2p=lþ; Alfven ratio r A ¼hv 2 i=hb 2 i; and normalized cross helicity r c ¼ 2 hv bi=ðhv 2 iþhb 2 iþ. Run Family b i b e Z 0 B 0 L N x ¼ N y ppg k L r A r c run1 Wu EA PRL13 0.1 0.1 1.414 5.0 25.6 2048 300 {2,4} 0.2 0.0 run2 Wu EA PRL13 0.1 0.1 1.414 5.0 25.6 2048 300 {2,4} 1.0 0.0 run3 Wu EA PRL13 0.1 0.1 1.414 5.0 25.6 2048 300 {2,4} 5.0 0.0 run4 Wu EA PRL13 0.1 0.1 2.828 5.0 25.6 2048 300 {2,4} 1.0 0.0 run5 Wu EA PRL13 0.1 0.1 1.414 5.0 51.2 4096 300 {2,4} 1.0 0.0 run6 Wu EA PRL13 0.1 0.1 2.828 5.0 25.6 2048 300 {2,4} 0.2 0.0 run7 Wu EA PRL13 0.1 0.1 2.121 5.0 25.6 2048 300 {2,4} 1.0 0.0 run8 Wu EA PRL13 0.1 0.1 4.243 5.0 25.6 2048 300 {2,4} 1.0 0.0 run9 Wu EA PRL13 0.1 0.1 2.828 5.0 25.6 2048 300 {2,4} 5.0 0.0 run10 Wu EA PRL13 0.1 0.1 1.414 5.0 25.6 2048 300 {2,4} 1.0 0.8 run11 Wu EA PRL13 0.1 0.1 1.414 5.0 102.4 8192 300 {2,4} 1.0 0.0 run12 Wu EA PRL13 0.1 0.1 3.536 5.0 25.6 2048 300 {2,4} 1.0 0.0 run805.1 PAPJ15 0.08 0.08 3.131 5.0 1.28 64 200 {1,2} 1.0 0.5 run805.2 PAPJ15 0.08 0.08 2.334 5.0 2.56 128 200 {1,2} 1.0 0.5 run805.3 PAPJ15 0.08 0.08 2.085 5.0 5.12 256 200 {1,2} 1.0 0.5 run805.4 PAPJ15 0.08 0.08 2.017 5.0 10.24 512 200 {1,2} 1.0 0.5 run805.5 PAPJ15 0.08 0.08 2.000 5.0 20.48 1024 200 {1,2} 1.0 0.5 run809.1 OTVdbB 0.08 0.08 2.031 5.0 20.48 1024 200 {1,2} 1.0 0.5 run810.1 OTVdbB 0.08 0.08 3.041 5.0 20.48 1024 200 {1,2} 1.0 0.5 run811.1 OTVdbB 0.08 0.08 4.052 5.0 20.48 1024 200 {1,2} 1.0 0.5 run812.1 OTVdbB 0.08 0.08 5.064 5.0 20.48 1024 200 {1,2} 1.0 0.5 Turb808 k13 0.25 0.25 0.986 5.0 25.6 2048 400 {1,3} 1.0 0.0 Turb812 k13 0.25 0.25 2.464 5.0 25.6 2048 400 {1,3} 1.0 0.0 Turb813 k13 0.25 0.25 3.942 5.0 25.6 2048 400 {1,3} 1.0 0.0 Turb814 k24lb 0.25 0.25 2.456 5.0 25.6 2048 400 {2,4} 1.0 0.0 Turb815 k24lb 0.25 0.25 3.930 5.0 25.6 2048 400 {2,4} 1.0 0.0 Turb816 k24sb 0.10 0.10 2.456 5.0 25.6 2048 400 {2,4} 1.0 0.0 Turb817 k24sb 0.10 0.10 3.930 5.0 25.6 2048 400 {2,4} 1.0 0.0 The Wu et al. simulations run for 200 X 1 ci, while the Matthaeus et al. simulations are run for 125 X 1 ci, where in this case, X 1 ci is the ion cyclotron based on the mean magnetic field of the turbulent simulations. The Orszag-Tang simulations are run for 18 nonlinear times. All the simulations run long enough that they show saturated proton/electron heating, indicating that most of the available fluctuation energy has been converted to heat. Here, we examine the change in electron and ion thermal energies e i and e e, once this saturated state is achieved. In Fig. 4 are shown e i and e e versus two different powers of the scaling parameter a nl s ci =s nl. Although the simplistic reconnection predictions give e i / a 4 nl and e e / a 3 nl ; these powers of a nl do not organize the data into a straight line as effectively as the scaling e i / a 3 nl and e e / a 2 nl. On the other hand, the scaling of the ratio e i =e e / a nl matches the simplistic theory quite well, as seen in Fig. 5. There are many reasons why the individual electron and ion heating in the turbulence does not exhibit the predicted scaling. One clear set that should be examined are associated with simplifying assumptions used to apply the laminar

012304-6 Shay et al. Phys. Plasmas 25, 012304 (2018) FIG. 4. Scaling of ion and electron heating e i and e e versus powers of a nl for turbulence simulations. (Left) e versus predicted power of a nl from the direct application of reconnection heating theory [Eqs. (7) and (8)]; (right) e versus best fit integer powers of a nl. reconnection results to the turbulence. In this study, we assume that all of the magnetic islands reconnect completely, and that the statistics of the reconnection are the same across the simulations in time. Currently, no systematic study of the variation of reconnection statistics either in time or across simulations has been performed for kinetic PIC simulations. Studying such statistics is a significant undertaking, which will be explored in subsequent studies. One aspect that can be examined in a relatively straightforward fashion is the FIG. 5. e i =e e plotted versus a nl for the turbulence simulations. The best fit line slope and the correlation coefficient are included. total amount of reconnection that occurs, because it is now possible to find x-lines and their rates automatically in kinetic PIC simulations. 25 By comparing the total reconnection that occurs in the simulation, we can get a feel as to whether it is reasonable to assume that all of the islands reconnect. We begin by estimating the amount of reconnected magnetic flux necessary to fully reconnect all of the initial magnetic islands. We examine run11 of the Wu simulations, which is the largest simulation performed for this study. The initial p turbulence amplitude is Z0 2 ¼ 2; so we take db rms ffiffi 2 as an upper bound on the turbulent magnetic fluctuations. The initial system has about N ¼ 30 magnetic islands. As an estimate of the magnetic flux in each island, we assume that the islands are roughly space filling p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and circular, giving a radius of each island of R L 2 =ðn pþ; where L ¼ 102.4 is the length of each side of the square simulation domain. The magnetic flux available to reconnect in each island is roughly w i db rms R. Two magnetic islands reconnecting completely together will reconnect dw ¼ w i flux, so reconnecting all of the islands would be dw all ðn=2þdb rms R 220: The magnetic flux reconnected by an x-line i from t 0 ¼ 0 to t is dw i ðtþ ¼ Ð t 0 je izðt 0 Þjdt 0. The absolute values are necessary because E z can be negative depending on the magnetic topology near the x-line. Summing over x-lines gives dwðtþ ¼ Ð t 0 P i je izðt 0 Þjdt 0, where we have swapped the sum

012304-7 Shay et al. Phys. Plasmas 25, 012304 (2018) and the integral without loss of generality. Bringing the sum inside the integral makes the numerical calculation of dw much more tractable, because it is no longer necessary to determine exactly when each x-line appears and disappears. The difficulty in applying this formulation to kinetic- PIC simulation results is the presence of numerical fluctuations of E z. X-lines with average E z nearly zero may have significant je z j due to these fluctuations. To address this issue, we use the fact that for a reconnecting X-line, one expects E z and J z to have the same sign. Multiplying the reconnection rate by signðe z J z Þ averages out these fluctuations while at the same time retaining the absolute value of E z ; giving ð t X dwðtþ ¼ je iz ðt 0 Þj signðj iz E iz Þ dt 0 ; (10) 0 i where J iz is the current at x-line i. This dwðtþ is calculated for run11 in Table II and is shown in Fig. 6. The x-lines are determined at each time automatically using the methods formulated in Ref. 25. By the end of the simulation, the system has reconnected almost four times the flux necessary to reconnect all of the initial islands once. This is not surprising given the strong turbulence amplitude in this case, the generation of secondary islands, and the fact that some islands merge multiple times. We emphasize that this kind of analysis is preliminary. A thorough study of the total reconnected flux and its relationship to heating will be performed in a future study. V. DISCUSSION In this manuscript, we have established a framework for applying reconnection heating theories to turbulent systems. A compelling aspect of this framework is the simplicity of the reconnection heating prediction, being only dependent on the plasma parameters just upstream of the diffusion region. However, applying such a formula requires assumptions about the average properties upstream of the diffusion region at actively reconnecting x-lines in the turbulence. In this study, we assumed that such average values are FIG. 6. For run11 in Table II, the total reconnected magnetic flux since the beginning of the simulation using Eq. (10). The horizontal line is an estimate of dw necessary to fully reconnect all initial magnetic islands. consistent with Kolmogorov scaling predictions for turbulence fluctuations at the scale of the ion inertial length. We note that an improved estimate of the local nonlinear time appropriate to reconnection might depart from the Kolmogorov estimate given in Eq. (5). That estimate is an average ignoring intermittency associated with coherent structures. Given that coherent structures such as current sheets are extreme values, one would expect that active reconnection sites will have amplitudes larger than the Kolmogorov estimate. We will not pursue that refinement here. Another major assumption underlying our scaling of e i and e e in Fig. 4 is that the variation of the parameters in the simulations performed did not change the percentage (fractional volume) of the plasma heated by reconnection; note, however, that Fig. 6 makes it clear that most or all of the islands fully reconnect. If, for example, the number of x- lines and thus the percentage of plasma exposed to reconnection heating is changed as the fluctuation amplitude dz is changed at the energy containing scale, then it might be expected that the scaling of e i and e e would not match the reconnection predictions. However, the fact that e i =e e matches well the predictions based on reconnection suggests that reconnection may be an important driver of plasma heating in these simulations. By dividing the ion and electron heating, we have canceled out the dependence on the statistical distributions of the reconnection sites inherent in the sums in Eqs. (2) and (3). We also note that a similar conclusion was found for the ratio of ion to electron heating in the kinetic plasma whistler turbulence studies by Gary et al. 9 It also has not escaped our attention that the heating rate predicted by Eq. (3) shows a structure similar to the Von Karman approach, i.e., under the right circumstances, both are cubic in the amplitude of the plasma fluctuation amplitude. A further comment regarding the meaning of the quantity a nl is warranted. In previous studies, this quantity has been associated with the balance of linear Vlasov and nonlinear cascade effects, 29 and subsequently, with the effects of turbulence strength on proton heating. 7 The present analysis suggests that these influences of the ratio of cyclotron to nonlinear timescale may also extend into the realm of reconnection physics. In particular, we note that the analysis of proton heating in the context of turbulence 7 began with the assumption of a von Karman total heating rate, an assumption that probably does not apply to the large scale reconnection geometry that is typically adopted for examining the dynamics of a single X-line. Significant future work needs to be performed to determine how fluctuation levels at the energy containing scale modify the statistical properties of the number and spatial density of reconnection sites, and the duration in time of the reconnection at each site. This is an ongoing topic of research and requires the development of sophisticated algorithms which can mitigate the effect of noise inherent in kinetic PIC simulations, automatically determine the appropriate plasma conditions upstream of the diffusion region, and determine how the reconnection is modifying the magnetic topology of the system. These topics will undoubtedly

012304-8 Shay et al. Phys. Plasmas 25, 012304 (2018) be intensely scrutinized by the reconnection/turbulence community in the future. There are additional complications which may impact the role that reconnection plays in heating turbulent plasmas. A major one is associated with the plasma dynamics in the reconnection exhausts. If this plasma is strongly adiabatically compressed, it could have a significant impact on the heating. In addition, if a magnetic island/flux rope is populated by already heated plasma which reaches high values of b, this could ultimately impact the heating by modifying the exhaust velocities. The role of observations is an important question in the context of this work. Coherent structures (current sheets) have been shown to be prevalent in the solar wind and distributed in a way very similar to what is found in strong turbulence simulations. 30 Furthermore, plasma is preferentially hotter in or near these current sheets. 31 However, the percentage of these current sheets that is actually reconnecting is unknown. Although localized or isolated reconnection has been the subject of intense observational scrutiny, 32,33 observational studies of reconnection as an element of turbulence are quite rare. In the turbulent magnetosheath, reconnecting current sheets inferred from E B drifts have been observed with evidence of electromagnetic energy being converted into flows and heating. 34 One major observational test of the framework described in this study will be conclusive evidence that small scale reconnection events are indeed a ubiquitous element of turbulence in either the solar wind or the Earth s magnetosheath. The magnetospheric multiscale mission (MMS) is well suited to studying this question as MMS enters its extended mission. For example, coherent structures have been found to be correlated with electron heating in the turbulent magnetosheath. 35 Focusing on the heating in turbulence, it has been shown that to account for the radial evolution of temperature in the solar wind, a local energy source for heating must be present; 36,37 also, the empirically measured heating rates for protons and electrons are different. 38 An observational test of the scaling of heating rates seen in our study, however, would require empirically estimating the effect of turbulence amplitude on heating rates, which to our knowledge has not yet been performed. There can be no doubt that reconnection plays an important role in plasma turbulence, but at the same time it is clear that there are numerous factors that may cause variations in the details of this complex relationship. The approach adopted here has been to show that ideas about scaling of the heating rates of protons and electrons developed in the study of reconnection exhausts can be usefully adapted to understand heating in plasma turbulence. The results are especially encouraging with regard to the ratio of proton to electron heating rates. To achieve a correspondence between the reconnection and turbulence perspectives requires making only a few reasonable assumptions, e.g., concerning the scaling of turbulence fluctuation amplitude with scale. Therefore, we may conclude that the heating in reconnection exhausts and the heating occurring in turbulent plasmas are closely related. It may be that the specific mechanism of reconnection exhaust heating also occurs in plasma turbulence, and is a major contributor to the total heating in turbulence. An alternative possibility is that the scaling of heating in reconnection exhausts is typical for the scaling of several processes, or perhaps, all processes that contribute to turbulent heating, so that it is the scaling and not the mechanism that is common. In either case, the present results further demonstrate the deep relationship that must exist between reconnection and turbulence. Only through further research will these relationships and their ramifications become better understood. ACKNOWLEDGMENTS This research was supported by NSF Grant No. AGS- 1460130 and NASA Grant Nos. NNX08A083G (MMS IDS), NNX14AC39G (MMS Theory and Modeling team), NNX14AI63G (Heliophysics Grand Challenge Research), NNX15AW58G, and NNX17AI25G. M.W. acknowledges the support from the National Natural Science Foundation of China (NSFC Grant No. 11672123). 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