turbulent-reconnection-theory-tests-2020

Turbulent Reconnection 3D Turbulent Reconnection: Theory, Tests & Astrophysical Implications Alex Lazarian, 1,a) Gregory L. Eyink, 2, 3 Amir Jafari, 3 Grzegorz Kowal, 4 Hui Li, 5 Siyao Xu, 1 and Ethan T. Vishniac 3 1) Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, WI 53706, USA 2) Department of Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore, MD 21218, USA 3) Department of Physics & Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA 4) Escola de Artes, Ci ˆ encias e Humanidades, Universidade de S ̃ ao Paulo, Av. Arlindo B ́ ettio, 1000 - Vila Guaraciaba, S ̃ ao Paulo - SP, CEP: 03828-000, Brazil 5) Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: 6 January 2020) Magnetic reconnection, topological change in magnetic fields, is a fundamental process in magnetized plasmas. It is associated with energy release in regions of magnetic field annihilation, but this is only one facet of this process. Astrophysical fluid flows normally have very large Reynolds numbers and are expected to be turbulent, in agreement with observations. In strong turbulence magnetic field lines constantly reconnect everywhere and on all scales, thus making magnetic reconnection an intrinsic part of the turbulent cas- cade. We note in particular that this is inconsistent with the usual practice of regarding magnetic field lines as persistent dynamical elements. A number of theoretical, numerical, and observational studies starting with the Lazarian & Vishniac 1999 paper proposed that 3D turbulence makes magnetic reconnection fast and that magnetic reconnection and tur- bulence are intrinsically connected. In particular, we discuss the dramatic violation of the textbook concept of magnetic flux-freezing in the presence of turbulence. We demonstrate that in the presence of turbulence the plasma effects are subdominant to turbulence as far as the magnetic reconnection is concerned. The latter fact justifies an MHD-like treat- ment of magnetic reconnection on all scales much larger than the relevant plasma scales. We discuss numerical and observational evidence supporting the turbulent reconnection model. In particular, we demonstrate that the tearing reconnection is suppressed in 3D and, unlike the 2D settings, 3D reconnection induces turbulence that makes magnetic re- connection independent of resistivity. We show that turbulent reconnection dramatically affects key astrophysical processes, e.g. star formation, turbulent dynamo, acceleration of cosmic rays. We provide criticism of the concept of “reconnection-mediated turbulence” and explain why turbulent reconnection is very different from enhanced turbulent resistiv- ity and hyper-resistivity, and why the latter have fatal conceptual flaws. PACS numbers: Valid PACS appear here Keywords: Suggested keywords I. PROBLEM OF MAGNETIC RECONNECTION IN ASTROPHYSICS Magnetic fields are ubiquitous in astrophysical systems and critically affect the dynamics and properties of magnetized plasmas over an extended range of scales. These scales are typically much larger than any relevant plasma scales, e.g., the ion inertial length. The magnetic field of the Earth’s magnetosphere, the case-study of manyin situmeasurements, is an exception in the astrophysical context in terms of the involved scales. In what follows, we will discuss magnetic fields at much larger scales. a) Electronic mail: [email protected] ar Xiv:2001.00868v1 [astro-ph. HE] 3 Jan 2020 Turbulent Reconnection2 One textbook concept related to magnetic fields, widely used in astrophysical studies, is that magnetic fields in highly conducting plasmas are nearly perfectly frozen into the fluid and retain their topology for all time (Alfv ́ en, 1942; Parker, 1979). This concept of flux-freezing is at the heart of many theories, e.g., the theory of star formation in magnetized interstellar medium. At the same time, there is ample evidence that magnetic fields in astrophysical systems do change their topology, solar flares being a classical example of such a process (see e.g. Parker, 1970; Lovelace, 1976; Priest and Forbes, 2002). The studies on the process enabling such a topology change, i.e., magnetic reconnection, were historically motivated by observations of the solar corona (Inneset al., 1997; Yokoyama and Shibata, 1995; Masudaet al., 1994), which also created a number of misconceptions. First of all, magnetic reconnection is frequently associated only with the processes of conversion of magnetic energy into heat. In addition, this understanding influ- enced attempts to find very peculiar conditions for flux conservation violation. For instance, in a fundamental study by Priest and Forbes (2002), a number of examples of magnetic configurations that produce fast reconnection are discussed. The attempts to accelerate the reconnection rate for special plasma conditions, e.g., in plasmas with very small collision rates (see e.g. Shayet al., 1998; Drake, 2001; Drakeet al., 2006; Daughton, Scudder, and Karimabadi, 2006; Uzdensky and Kulsrud, 2006; Bhattacharjee, Ma, and Wang, 2003; Zweibel and Yamada, 2009; Yamada, Kulsrud, and Ji, 2010) follow the same logic of seeking special circumstances where the magnetic flux-freezing is violated. We feel that the above treatment of magnetic reconnection is not generic or widely applicable. Solar flares (Sturrock, 1966) are just one vivid example of reconnection activity. In reality, mag- netic reconnection is a ubiquitous process taking place in various astrophysical environments, both collisionaless and collisional (see Shibata and Magara, 2011). For instance, magnetic reconnection is a part of large-scale dynamo acting in stellar interiors (Parker, 1993; Ossendrijver, 2003). Mag- netic reconnection is also required to make the theory of magnetohydrodynamic (MHD) turbulence (Goldreich and Sridhar, 1995) self consistent (Jafari and Vishniac, 2019), etc. The problem of magnetic reconnection is not limited to explaining its typically fast rates. In fact, it is often overlooked that the observations of solar activity indicate that the reconnection rates can significantly vary. This presents serious problems for theories that are based on relating the rate of magnetic reconnection to the peculiar properties of plasmas. These properties do not change, for instance, as the flux gets accumulated prior to a solar flare and gets annihilated during the flare. This may suggest that magnetic reconnection should have a sort of trigger. It is also worth noting that for decades a paradoxical situation existed with the treatment of astro- physical magnetic reconnection. On the one hand, the core of the reconnection community claimed numerous restrictions either on magnetic field configurations or the properties of reconnecting plas- mas. On the other hand, the practitioners of numerical astrophysics used their MHD codes to sim- ulate various astrophysical settings without worrying about the aforementioned restrictions. At the same time, the question of the actual effect of magnetic reconnection on the validity of numerical studies was somehow avoided. The absence of an effective communication between the communities is surprising. For instance, for many years it was considered that in order for reconnection to be fast, it must take place in collisionless environments. However, if collisionless reconnection were the only way to make re- connection rapid, then numerical MHD simulations of many astrophysical processes including those of the interstellar medium (ISM), which is collisional, are in error. Fortunately for numerical prac- titioners, the observations of collisional solar photosphere indicate that the reconnection is fast in these environments as well (see§VII A). In this review, we provide evidence that, in fact, turbulence determines the rate of reconnection in realistic 3D astrophysical systems. The concept of 3D turbulent reconnection was introduced by Lazarian and Vishniac (1999, henceforth LV99). This model was followed by several subsequent theoretical studies, for instance, Eyink, Lazarian, and Vishniac (2011, henceforth ELV11), Eyink (2015), and Jafariet al.(2018). The LV99 theory has been supported by numerical simulations in non-relativistic (see Kowalet al., 2009, 2012; Eyinket al., 2013; Beresnyak, 2017; Oishi et al., 2015; Kowalet al., 2017), as well as relativistic settings (see Takamoto and Lazarian, 2016; Takamoto, 2018). In addition, different pieces of observational evidence, discussed in the review, support the predictions of the 3D turbulent reconnection theory of LV99. Turbulent Reconnection3 To understand the essence of LV99 model, one can recall that the problem of magnetic reconnec- tion in astrophysical systems can be viewed as the problem related to the scale disparity. Reconnec- tion occurs on very large scales, while the dissipation processes take place at the smallest plasma scales, which are set by, e.g., resistivity. LV99 theory shows how turbulence, as a multi-scale phe- nomenon, can indeed resolve the problem of scale disparity as well as other problems that plague the traditional reconnection models. While the idea of turbulent reconnection is nearly coeval with the ideas of 2D collisionless Hall- MHD reconnection, the latter has been substituted more recently by the ideas of tearing/plasmoid re- connection (Shibata & Tanuma 2001). In fact, 2D numerical work has already shown that magnetic reconnection can be accelerated significantly by the tearing instability (Loureiro, Schekochihin, and Cowley, 2007; Lapenta, 2008; Daughtonet al., 2009a,b; Bhattacharjeeet al., 2009; Cassak, Shay, and Drake, 2009). In a sense, this model has some similarities with the turbulent reconnection. Indeed, it departs from the Sweet-Parker Y-point reconnection but allows instabilities to evolve within the current sheet. We will discuss how in realistic 3D configurations this type of instability transfers to turbulent reconnection. One should keep in mind that we deal with 3D turbulent reconnection on large scales at which an MHD-like approximation is applicable. In particular, we consider scales much larger than the ion Larmor radius or skin depth, where an ideal Ohm’s law is generally agreed to be valid. At the same time, usual tests of reconnection are based on thein situmeasurements of magnetospheric reconnection. This type of reconnection occurs on scales comparable to the ion inertial length and therefore is atypical for the large-scale reconnection that occurs in most astrophysical systems. For the reconnection explored at larger scales, e.g., in the solar wind (Gosling, 2012; Lalescuet al., 2015), the measured properties of reconnection are consistent with the general expectations based on the LV99 theory (see§VII B). Occasionally, the process of magnetic reconnection is understood in a very narrow sense, e.g. magnetic reconnection necessarily in low-βplasmas with the magnetic energy much larger than the thermal energy. Here we consider turbulent reconnection as a generic process, which is applicable to plasmas of arbitraryβand for arbitrary arrangements of reconnecting magnetic fluxes. Our review is devoted to the astrophysical reconnection, although the magnetic reconnection is a fundamental physical process and therefore our conclusions can be applied also to the labora- tory reconnection. We discuss the theory and observations of astrophysical turbulence in§II. The LV99 model of 3D turbulent reconnection and its extensions are described in§III and§IV. There we explain how turbulent reconnection and flux-freezing violation in turbulent fluids are interrelated. The numerical tests of turbulent reconnection and the testing of flux freezing violation in turbulent fluids are provided in§V. Both testing with MHD codes and Hall-MHD codes are discussed. The initial tests of selected ideas of turbulent reconnection with 3D kinetic simulations are presented in §VI. The observational evidence supporting turbulent reconneciton is briefly discussed in§VII. The magnetic reconnection in high-Prandtl number medium, reconnection in relativistic fluids, and re- connection in the presence of whistler turbulence are discussed in§VIII. Astrophysical implications of reconnection for turbulent dynamo, star formation, as well as for particle acceleration and gamma ray bursts are covered in§IX. In§X, we discuss the differences between our model of turbulent re- connection and other ideas on the effects of turbulence on reconneciton and resistivity. We explain the reasons for our disagreement with (i) the currently-popular “reconnection-mediated turbulence” idea and (ii) the concepts of turbulent resistivity and turbulent ambipolar diffusion. There we also outline the problems of the mean field approach to studying reconnection, compare turbulent and tearing reconnection, and explain why reconnection rates are not expected to have a “universal” value of0.1Alfven velocity. We present our summary in§XI. Turbulent Reconnection4 FIG. 1. Big power law in the sky from Armstrong, Rickett, and Spangler (1995) extended to scale of parsecs using WHAM data. From Chepurnov and Lazarian (2010), c ©AAS. Reproduced with permission. II. TURBULENCE AS A NATURAL STATE OF ASTROPHYSICAL FLUIDS AND ITS MHD DESCRIPTION A. Observational Evidence Observations of the diffuse warm ISM reveal a Kolmogorov spectrum of electron density fluc- tuations (see Armstrong, Rickett, and Spangler, 1995; Chepurnov and Lazarian, 2010). Similar spectral slopes of supersonic velocity fluctuations are measured using Doppler shifted spectral lines (see Lazarian, 2009, for a review), as well as through thein situmeasurements of the solar wind fluctuations (Leamonet al., 1998). The evidence of turbulence being ubiquitous comes from non- thermal broadening of spectral lines as well as measures obtained by other techniques (see Burkhart et al., 2010). This fact is not surprising as magnetized astrophysical plasmas generally have very large Reynolds numbers. Indeed, the length scales involved are large, while the motions of charged particles in the direction perpendicular to magnetic fields are of the order of the Larmor radius. Plasma flows at these high Reynolds numbers are prey to numerous linear and finite-amplitude instabilities. This induces turbulent motions readily. The precise origin of the plasma turbulence differs from instance to instance. It is sometimes driven by an external energy source, such as supernova explosions in the ISM (Norman and Ferrara, 1996; Ferri ` ere, 2001), merger events and active galactic nuclei outflows in the intercluster medium (ICM) (Subramanian, Shukurov, and Haugen, 2006; Enßlin and Vogt, 2006; Chandran, 2005), and baroclinic forcing behind shock waves in interstellar clouds. In other cases, turbulence is spon- taneous, with available energy released by a rich array of instabilities, such as magneto-rotational instability (MRI) in accretion disks (Balbus and Hawley, 1998; Jafari and Vishniac, 2018b), kink instability of twisted flux tubes in the solar corona (Galsgaard and Nordlund, 1997; Gerrard and Hood, 2003), etc. In all these mentioned cases, turbulence is not driven by reconnection. Nev- ertheless, we would like to stress that the driving of turbulence through the energy release in the reconnection zone is important for the transfer from laminar to turbulent reconnection. All in all, whatever its origin is, the signatures of plasma turbulence are seen throughout astrophysical media. Indeed, observations show that turbulence is ubiquitous in all astrophysical plasmas. The spec- trum of electron density fluctuations in Milky Way is presented in Figure 1, as one dramatic piece of Turbulent Reconnection5 evidence for cosmic turbulence. Similar spectra are reported in Leamonet al.(1998) and Baleet al. (2005) for solar wind, Padoanet al.(2006) for molecular clouds, and Vogt and Enßlin (2005) for the ICM. New techniques for studying turbulence, e.g. the Velocity Channel Analysis (VCA) and Velocity Coordinate Spectrum (VCS) techniques (Lazarian and Pogosyan, 2000, 2004, 2006) have provided important insight into the velocity spectra of turbulence in molecular clouds (see Padoan et al., 2006, 2010), Galactic and extragalactic atomic hydrogen (Stanimirovi ́ c and Lazarian, 2001; Chepurnovet al., 2010, 2015, , see also the review by Lazarian (2009), where a compilation of velocity and density spectra obtained with contemporary HI and CO data is presented). B. Basic Phenomenology of MHD turbulence It is generally accepted that an MHD-like description of turbulence is applicable to the motions of magnetized plasmas at sufficiently large scales (e.g. see discussion in Schekochihinet al., 2009). Standard MHD or a multi-species variant is valid at length-scales much larger than the collisional mean-free-path of the plasma. However, even in a nearly collisionless but magnetized plasma, “kinetic MHD” or “kinetic reduced-MHD” equations are satisfied at scales much larger than the ion gyroradius (Kulsrud, 1983; Kunzet al., 2015). Since collisions in those plasmas are too infrequent to bring the particle distributions to Maxwellian, the pressure tensor is anisotropic and given not by a thermodynamic equation of state but instead by 1D kinetic equations along particle guiding line-centers. This subtlety is irrelevant for the incompressible shear-Alfv ́ en wave modes, which nearly uncouple from the kinetic/compressible modes and satisfy the well-known “reduced MHD” (RMHD) equations 1 . Shear-Alfv ́ en waves are often energetically dominant, as in the solar wind where they contain about 90% of the total energy of plasma and fields (see Dobrowolny, Mangeney, and Veltri, 1980). For this reason, we shall generally assume that MHD is an accurate model at sufficiently large scales or at least a good working approximation within which to explore the effects of turbulence on large-scale reconnection. We shall invoke below in our discussion of strong MHD turbulence the Goldreich and Sridhar (1995, henceforth GS95) theory of Alfv ́ enic turbulence. We believe that the empirical evidence largely supports GS95 over its competitors and gives a good description of MHD turbulence (apart from the phenomenon of small-scale intermittency). We wish to avoid here any controversy regard- ing the “correct” phenomenology, however. It may be shown following LV99, Appendix D, that the predictions on turbulent reconnection are qualitatively independent of the particular phenomenology adopted and, in particular, do not depend sensitively upon the predicted spectral slopes. Therefore, our use of GS95 theory may be regarded by its persistent skeptics as a mere convenience, since it is the simplest model that incorporates the small-scale anisotropy due to a local magnetic field. 1. Derivation of the MHD turbulence relations MHD turbulence was traditionally viewed as the process dominated by the Alfv ́ en wave interac- tions. Below we follow this path to show how to obtain GS95 relations and go beyond them (see also Cho, Lazarian, and Vishniac, 2003a). For Alfv ́ enic perturbations the relative perturbations of velocities and magnetic fields are related in the following way: δB l B = δB l B L B L B = u l u L M A = u l V A ,(1) where B l is the perturbation of the magnetic field Bat the scalel,B L is the perturbation of the magnetic field at the injection scale L, whileu l is the velocity fluctuation at the scalelin the turbulent flow with energy injected with velocityu L . 1 For relativistic turbulence (see Takamoto and Lazarian, 2017) the transfer of energy to fast modes is more important. Turbulent Reconnection6 To understand Alfv ́ en wave damping it is advantageous to present MHD turbulence as a su- perposition of colliding Alfv ́ en wave packets. Consider such packets with parallel scalesl ‖ and perpendicular scalesl ⊥ . The system of reference with respect to the local magnetic field is what is relevant when dealing with MHD turbulence. In the rest of the discussion we operate withl ‖ and l ⊥ that are defined in the frame of wavepackets moving along the local magnetic field. The wave packet collision induces a change ∆E∼(du 2 l /dt)∆t,(2) where the first term is the change of the energy of a packet as it interacts with the oppositely moving wave packet. The time of the interaction should be identified with the time of the passage of the given wave packet through the oppositely moving packet of the sizel ‖ . Thus the interaction time is given by∆t∼l ‖ /V A . The characteristic rate of cascade is due to the change of the structure of the oppositely moving wave packet. The latter has the rateu l /l ⊥ . As a result Eq. (2) gives ∆E∼u l · ̇ u l ∆t∼(u 3 l /l ⊥ )(l ‖ /V A ),(3) The fractional energy change per collision is∆E/E, which measures the strength of the nonlinear interaction: f≡ ∆E u 2 l ∼ u l l ‖ V A l ⊥ .(4) In Eq. (4)fmeasures the ratio of the rate of shearing of the wave packetu l /l ⊥ to the rate of the propagation of the wave packet V A /l ‖ . Two cases can be identified. Forf1the shearing rate is much smaller than the propagation rate and the cascade process is a random walk. This means that א=f −2 ,(5) which is the number of steps required for O(1) cascade of energy and therefore the cascade time is t cas א∼∆t.(6) Forא1, the turbulence corresponds to theweak MHD turbulence. Naturally,אcannot be smaller than unity. The limiting case of≈א1corresponds tostrong MHD turbulence. Consider first the case of weak MHD turbulence, which requires V A u L and very high- frequency oscillations. This cascade thus proceeds by resonant 3-wave Alfv ́ enic interactions that satisfyω 1 +ω 2 =ω 3 in addition tok 1 +k 2 =k 3 ,wherekandωdenote wavevector and angular fre- quency, respectively. Because of the dispersion relationω=s V A k ‖ withs=±1for shear-Alfv ́ en waves travelling parallel and anti-parallel to magnetic fields and because only counter-propagating waves interact, one finds thatk 2,‖ = 0andk 1,‖ =k 3,‖ .Thus resonantly interacting wavepackets have unchangingk ‖ butk ⊥ increases by distortions through collisions. The decrease ofl ⊥ whilel ‖ does not change signifies the increase of the energy change per collision. This eventually makesא of the order of unity and the cascade transitions to strong MHD turbulence. In this case one gets u l l −1 ⊥ ≈V A l −1 ‖ ,(7) which signifies the cascade time being equal to the wave period∼∆t. It was hypothesized by Goldreich and Sridhar (1995) that the further decrease ofl ⊥ entails the corresponding decrease of l ‖ to keep Eq. (7) satisfied, which is thecritical balance condition. The ability to changel ‖ by non-resonant strong cascade means that the frequencies of interacting waves increase. However, this increase has a natural limit because a too high frequency would restore the resonance condition ω 1 +ω 2 =ω 3 and this would shut off the parallel cascade. Thus, the strong turbulent cascade has a tendency to operate at marginal resonance, where the critical balance relation (7) is satisfied. The cascade of turbulent energy satisfies the relation (Batchelor, 1953): ≈u 2 l /t cas =const,(8) Turbulent Reconnection7 which for the hydrodynamic cascade provides  hydro ≈u 3 l /l≈u 3 L /L=const,(9) where the relationt cas ≈l/u l is used. For the weak cascadeא1  w ≈ u 4 l V 2 A ∆t(l ⊥ /l ‖ ) 2 ≈ u 4 L ` ‖ V A ` 2 ⊥ ≈ u 4 L V A L (10) where Eqs. (8) and (6) are used (see LV99). The isotropic turbulence injection at scale` ‖ '` ⊥ 'L results in the third relation in Eq. (10). Taking into account thatl ‖ is constant for weak turbulence, it is easy to see that Eq. (10) provides u l ∼u L (l ⊥ /L) 1/2 ,(11) which is different from the hydrodynamic∼l 1/3 scaling. 2 Forsub-Alfv ́ enic turbulenceinjected isotropically at length scale Lwithu L < V A the transition to the strong regime, i.e.≈א1,happens at the scale (LV99) l trans ∼L(u L /V A ) 2 ≡LM 2 A .(12) Therefore, weak turbulence has a limited, i.e.[L,LM 2 A ], inertial range and transits atl trans into strong turbulence. The velocity at the transition follows from≈ א1condition given by Eqs. (5) and (4): u trans ≈V A l trans L ≈V A M 2 A .(13) The scaling for the strong turbulence range atl < l trans can be easily obtained. Indeed, the turbu- lence cascades over one wave period, which according to Eq. (23) is equal tol ⊥ /u l . Substituting the latter in Eq. (8) one gets  s ≈ u 3 trans l trans ≈ u 3 l l ⊥ =const.(14) The latter is analogous to the hydrodynamic Kolmogorov cascade in the direction perpendicular to the local direction of the magnetic field. This strong MHD turbulence cascade starts atl trans and it has the injection velocity given by Eq. (13). This provides the another way to get the scaling relations for velocity in strong Alfv ́ enic turbulence (LV99) u l ≈V A ( l ⊥ L ) 1/3 M 4/3 A .(15) which can be rewritten in terms of the injection velocityu L as u l ≈u L ( l ⊥ L ) 1/3 M 1/3 A .(16) Substituting this in Eq. (7) we get the relation between the parallel and perpendicular scales of the eddies (LV99): l ‖ ≈L ( l ⊥ L ) 2/3 M −4/3 A .(17) 2 Using the relationk E(k)∼u 2 k it is easy to show that the spectrum of weak turbulence is E k,weak ∼k −2 ⊥ (LV99, Galtier et al., 2000). Turbulent Reconnection8 The relations Eq. (17) and (15) reduce to the GS95 scaling for trans-Alfv ́ enic turbulence if M A ≡1. In the opposite case ofsuper-Alfv ́ enic turbulence, i.e. foru L > V A ,the scales close to the injection scale are essentially hydrodynamic as the influence of magnetic forces is minimal. Thus the velocity is Kolmogorov u l =u L (l/L) 1/3 .(18) The cascade changes its nature at the scale l A =LM −3 A ,(19) for which the turbulent velocity is equal to the Alfv ́ en velocity (Lazarian and Pogosyan, 2016). The cascade rate forl < l A is:  super A ≈u 3 l /l≈V 3 A /l A =const.(20) This case likewise reduces to trans-Alfv ́ enic turbulence, but withl A acting as the injection scale. At scales` < l A l ‖ ≈L ( l ⊥ L ) 2/3 M −1 A ,(21) and u l ≈u L ( l ⊥ L ) 1/3 .(22) The wave description of strong MHD turbulence is very productive in the case when the turbulence is imbalanced, i.e. the flow of wave-energy from one direction along field-lines significantly exceeds the flow from the other direction. An example of imbalanced MHD theory that agrees with numer- ical simulation is the theory by Beresnyak and Lazarian (2008) (see also Beresnyak and Lazarian, 2010). 2. Eddy description of MHD turbulence In our discussion above we revealed the analogy between the Alfv ́ enic and Kolmogorov turbu- lence. This analogy can be extended in the case of strong Alfv ́ enic turbulence. In fact, the GS95 relations can be naturally obtained if one assumes that such turbulence presents a collection of ed- dies that are mixing the conducting fluid perpendicular to the direction of the magnetic field present at the location of the eddies. This description can be justified in the presence of fast reconnec- tion that we are describing in the review and this supports our claim that the MHD turbulence and turbulent magnetic reconnection are intrinsically connected. As it was described in LV99, the reconnection of magnetic fields associated with an eddy takes place within the eddy turnover time (see also§III). In this situation, the random driving is prefer- entially exciting the motions that induce minimal bending of magnetic field lines, i.e. the hydrody- namic type motions of eddies that mix magnetized fluid perpendicular to the direction of magnetic field. This is the picture of MHD turbulence that follows from LV99 description. The key concept in the eddy description is that magnetic perturbations are measured in terms of the local direction of the magnetic field, not the mean field. The concept that is missing in GS95. and this causes a lot of confusion with different authors attempting to test the GS95 scale-dependent anisotropy in the frame of mean magnetic field. In fact, the latter anisotropy only exists in the local frame of the eddies, while the anisotropy in the reference frame of mean magnetic field is scale-independent and is determined by the anisotropy of the largest eddies (see Cho, Lazarian, and Vishniac, 2003a). Incidentally, within the description suggested in LV99, it is clear that the gradients of both ve- locities and magnetic field are expected to be perpendicular to the local magnetic field direction. Therefore by tracing the directions of the gradients one can trace the magnetic field direction in Turbulent Reconnection9 turbulent media. This idea is at the foundations of the new technique of magnetic field tracing that employs gradients of intensities within spectroscopic channel maps (Lazarian and Yuen, 2018b), gradients of synchrotron intensity (Lazarianet al., 2018) and gradients of synchrotron polarization (Lazarian and Yuen, 2018a). The magnetic field tracing by the new technique provides results in good agreement with polarization data (see Hu, Yuen, and Lazarian, 2019), which is an indirect way of confirming both the ubiquitous presence of the MHD turbulence in interstellar medium as well as a successful confirmation of the theoretical predictions. Adopting the picture of eddies perpendicular to the local magnetic field of the eddies, one can claim that in terms of perpendicular motions we expect to have the Komogorov picture, i.e.u ` ∼ l 1/3 ⊥ . The critical balance relations in the picture of the eddies also follow naturally. Indeed, as the eddy provides mixing of magnetic field lines in the perpendicular direction over the time scale l ⊥ /u ` it sends an Alfv ́ en wave with the same period. This is exactly l −1 ‖ V A ∼` −1 ⊥ u l ,(23) wherel ‖ andl ⊥ are eddy axes parallel and perpendicular to thelocaldirection of magnetic field, the magnetic field of the eddy. Note that this requires a more sophisticated ways to analyse numerical simulations. For instance, a way of determining of local magnetic field in numerical simulations using structure functions is discussed in Cho and Vishniac (2000). In fact, the notion of local magnetic field is the essential part of the modern understanding of Alfv ́ enic turbulence and it was added to the MHD turbulence theory by the later studies (LV99, Cho and Vishniac, 2000; Maron and Goldreich, 2001). This notion is natural within the picture of LV99 turbulent reconnection. Indeed, if one considers the aforementioned eddy picture, the use of local magnetic field means that the eddies are only affected by the magnetic field around them and do not know anything about the mean magnetic field obtained by averaging over the entire turbulent volume. It is assumed in GS95 that the energy is injected at the scale Lwith the injection velocity equal to the Alfv ́ en velocity. If the injection velocityu L is smaller than V A ,then the eddies at small scale get more elongated. For M A <1the eddy description is valid only for the strong turbulence, i.e. starting with the scales less thanl trans =LM 2 A . Takingl trans as the injection scale of the turbulence and the velocity at this scale V A M 2 A (see Eq. (13)) as the injection velocity one can repeat the arguments about the correspondence of the time scales of parallel and perpendicular motions to obtain the relations given by Eq.(15) and (17). 3. Compressible MHD turbulence The compressible MHD turbulence can be approximated by the superposition of 3 cascades of basic MHD modes, namely, Alfv ́ enic, slow and fast. The original idea of the decomposition of low amplitude MHD perturbations into the modes can be traced to (Dobrowolny, Mangeney, and Veltri, 1980), while the practical realization for the actual trans Alfvenic turbulence was first performed in (Cho and Lazarian, 2002). We, however, will concentrate on Alfv ́ enic part of the MHD turbulent cascade, as this component is most important for the theory of reconnection that we consider in the review. In non-relativistic MHD turbulence, the backreaction of slow and fast magnetosonic modes (Cho and Lazarian, 2002, 2003; Kowal and Lazarian, 2010) on Alfv ́ enic cascade is insignificant (Cho and Lazarian, 2002; Goldreich and Sridhar, 1995; Lithwick and Goldreich, 2001). Therefore in non-relativistic MHD turbulence the relations above obtained for Alfv ́ enic turbulence are not much affected by the presence of compressible modes. In the rest of this section we discuss how the partial ionization and relativistic effects change the picture of Alfv ́ enic turbulence. C. MHD turbulence in a partially ionized medium In partially ionized astrophysical plasmas in e.g., the early universe, cold interstellar phases, protoplanetary disks, solar atmosphere, the MHD turbulence is subjected to the damping effects due Turbulent Reconnection10 to the presence of neutrals. Since we are interested in the turbulent reconnection associated with Alfv ́ enic turbulence, here we focus on the damping of Alfv ́ enic turbulence. The damping effects in a partially ionized medium mainly arise from the collisional friction be- tween ions and neutrals, i.e., ion-neutral collisional damping (IN) and the viscosity in neutral fluid, i.e., neutral viscous damping (NV) (Lithwick and Goldreich, 2001; Lazarian, Vishniac, and Cho, 2004). Their relative importance depends on the turbulence and plasma parameters (Xu, Lazarian, and Yan, 2015; Xu, Yan, and Lazarian, 2016; Xu and Lazarian, 2017b). In super-Alfv ́ enic tur- bulence with the turbulent energy larger than the magnetic energy, IN dominates over NV if the condition ξ n ν 2 ni ν n Lu −3 L <0.5(24) is satisfied, whereξ n is the neutral fraction,ν ni is the neutral-ion collisional frequency,ν n is the neutral viscosity,Lis the driving scale of turbulence, andu L is the turbulent velocity at L. In sub- Alfv ́ enic turbulence with the turbulent energy smaller than the magnetic energy, under the condition ξ n ν 2 ni ν n Lu −3 L M −1 A <0.5,(25) IN is more important than NV. Here M A =u L /V A is the Alfv ́ en Mach number and V A is the Alfv ́ en speed. An illustration of the above condition in sub-Alfv ́ enic turbulence can be found in Fig. 2, where the typical parameters of the interstellar turbulence are used (Xu and Lazarian, 2017b). Xu, Yan, and Lazarian (2016) found that in the warm neutral medium of the ISM and the solar chromosphere, NV plays a significant role in damping MHD turbulence. ξ i 10 -4 10 -3 10 -2 10 -1 10 0 B [ μ G] 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 NV IN FIG. 2. The parameter space for determining the relative importance between IN and NV in the case of sub-Alfv ́ enic turbulence. From Xu and Lazarian (2017b), c ©IOP Publishing & Deutsche Physikalische Gesellschaft. Reproduced by permission of IOP Publishing. CC BY-NC-SA. When the damping effects are significant, MHD turbulence is suppressed. The damping scale where the MHD turbulence is dissipated can be determined by comparing the turbulent energy cascade rate with the damping rate. Different from the damping scale of linear MHD waves, the damping scale of MHD turbulence has parallel and perpendicular components,k dam,‖ andk dam,⊥ , due to the anisotropy of MHD turbulence. In the frame of local magnetic field, i.e., the magnetic field averaged over the length scale of interest, magnetic fields are mixed by turbulent motions in the perpendicular direction, and the perturbed magnetic fields have wave-like motions in the parallel direction. For strong MHD turbulence, the lifetime of waves is equal to the eddy turnover time. As the turbulence anisotropy increases toward small length scales, when the damping scale of MHD turbulence is much smaller than L, it can be approximated by its perpendicular component in terms of magnitude. Under the damping effects of both IN and NV, a general expression of the damping scale is (Xu Turbulent Reconnection11 and Lazarian, 2017b) k dam,‖ = −(ν n + V 2 Ai ν in ) + √ (ν n + V 2 Ai ν in ) 2 + 8V A ν n l A ξ n 2ν n l A ,(26a) k dam =k dam,‖ √ 1 +l A k dam,‖ (26b) for super-Alfv ́ enic turbulence, and k dam,‖ = −(ν n + V 2 Ai ν in ) + √ (ν n + V 2 Ai ν in ) 2 + 8V A ν n LM −4 A ξ n 2ν n LM −4 A , k dam =k dam,‖ √ 1 +LM −4 A k dam,‖ (27) for sub-Alfv ́ enic turbulence, where V Ai is the Alfv ́ en speed of the ionized gas,ν in is the ion-neutral collisional frequency, andl A =LM −3 A in super-Alfv ́ enic turbulence. When IN is the dominant damping effect, the damping is related to the coupling state between ions and neutrals. The most severe damping arises in the weakly coupled regime in a weakly ionized medium, where neutrals decouple from ions, but ions are still coupled with neutrals due to their frequent collisions with surrounding neutrals. In this situation, the (perpendicular) damping scale for super- and sub-Alfv ́ enic turbulence can be simplified as (Xu and Lazarian, 2017b) k dam,IN,sup,⊥ = ( 2ν ni ξ n ) 3 2 L 1 2 u − 3 2 L (28) and k dam,IN,sub,⊥ = ( 2ν ni ξ n ) 3 2 L 1 2 u − 3 2 L M − 1 2 A .(29) Its relation to the neutral-ion decoupling scale is k dec,ni = ( 2 ξ n ) − 3 2 k dam,IN (30) for both super- and sub-Alfv ́ enic turbulence. The difference between the decoupling and damping scales has been confirmed by two-fluid MHD simulations (Burkhartet al., 2015). The above relation shows that the decoupling of neutrals from ions plays a key role in IN damping of MHD turbulence. After neutrals decouple from ions, the hydrodynamic turbulence carried by neutrals is only sub- ject to viscous damping, and its NV damping scale is smaller than the IN damping scale of MHD turbulence in ions. This differential damping of neutrals and ions can account for the observed linewidth difference of coexistent neutrals and ions in molecular clouds (Li and Houde, 2008; Xu, Lazarian, and Yan, 2015). When NV dominates over IN, the damping scale is approximately (Xu and Lazarian, 2017b) k dam,NV,sup,⊥ = ( ξ n 2 ) − 3 4 ν − 3 4 n L − 1 4 u 3 4 L (31) for super-Alfv ́ enic turbulence, and k dam,NV,sub,⊥ = ( ξ n 2 ) − 3 4 ν − 3 4 n L − 1 4 u 3 4 L M 1 4 A (32) for sub-Alfv ́ enic turbulence. MHD turbulence is damped at the NV damping scale. On the length scales smaller than the NV damping scale and larger than the IN damping scale, magnetic fluc- tuations persist, and the magnetic energy has a spectral formk −1 . This “new regime of MHD turbulence” in the sub-viscous range was first analytically predicted by Lazarian, Vishniac, and Cho (2004) and further confirmed by MHD simulations (Cho, Lazarian, and Vishniac, 2002a, 2003b). Thek −1 spectrum of magnetic energy in the sub-viscous range has also been observed in dynamo simulations in the nonlinear stage of dynamo (Haugen, Brandenburg, and Dobler, 2004; Schekochi- hinet al., 2004). Turbulent Reconnection12 Alfven mode Fast mode Fast Shock dissipation scale energy cascade FIG. 3. A schematic picture of the energy transfer in high-σturbulence. From Takamoto and Lazarian (2017). D. Relativistic MHD turbulence Due to its numerical and theoretical simplicity, MHD turbulence in relativistic force-free regime has been studied first. Relativistic force-free formalism can be used for a system, such as the mag- netosphere of a pulsar or a black hole, in which the magnetic energy density is much larger than that of matter. Scaling relations for relativistic Alfvenic MHD turbulence were first derived by Thompson and Blaes (1998). The predictions of these theory in terms of the spectral slope and anisotropy coincide with those in GS95 theory. These relations were first numerically tested by Cho (2005) who per- formed a numerical simulation of a decaying relativistic force-free MHD turbulence with numerical resolution of512 3 and calculated energy spectrum and anisotropy of eddy structures. Aftert= 3, the energy spectrum with E∼k 5/3 was shown to decrease in amplitude without changing its slope. The anisotropy was also measured and was shown to be consistent with the GS95 expectations, i.e. k ‖ ∼k 2/3 ⊥ . Thus in terms of Alfvenic turbulence the correspondence between the non-relativistic theory and its relativistic counterpart is excellent. Interestingly enough, the relativistic simulations of imbalanced Alfvenic turbulence, i.e. the turbulence with the excess of the energy flux moving in one direction, performed in Cho and Lazarian (2014) provided results in agreement with the theory of non-relativistic imbalanced turbulence in Beresnyak and Lazarian (2008). The situation is somewhat different in the case of relativistic compressible theory. The initial studies of relativistic MHD turbulence (see Zhang, Mac Fadyen, and Wang, 2009; Inoue, Asano, and Ioka, 2011; Beckwith and Stone, 2011; Zrake and Mac Fadyen, 2012, 2013; Garrison and Nguyen, 2015) have not revealed much differences between the non-relativistic and relativistic turbulence. However, more recent studies in Takamoto and Lazarian (2016, 2017) provided the decomposition of the turbulent motions into Alfven, slow and fast modes. The comparison of the obtained results with the analogous decomposition into modes in non-relativistic case in Cho and Lazarian (2002, 2003) and Kowal and Lazarian (2010) revealed significant differences in terms of the coupling of the compressible and incompressible motions in the two cases. These differences are illustrated by Figure 3 where the transfer of energy between Alfven and fast modes is shown in the relativistic case. Note, that the corresponding transfer of energy is suppressed for non-relativistic compressible turbulence. In terms of anisotropy, this energy transfer makes fast modes more similar to Alfven ones. We find that this effect of energy transfer is important for describing turbulent relativistic reconnection. E. Turbulence: Summary of basic facts For our discussion of turbulent reconnection it is important turbulence is ubiquitous in astro- physical environments and that Alfvenic modes usually represent the dominant component of MHD turbulent cascade. These modes are present in collisional and collisionless environments. The tur- bulence in most astrophysical settings has the sources different from the magnetic reconnection. However, it is difficult to understand the properties of turbulent fluids if magnetic reconnection does not happen on the dynamical time of the fluid motions. The partial ionization of plasmas changes the properties of MHD turbulence increasing the scale at which MHD turbulence is damped. The relativistic and non-relativistic MHD turbulence have significant similarities, although they show a difference in the coupling of fast and Alfven modes. Turbulent Reconnection13 More detailed description of the theory and implications of MHD turbulence can be found in a book by Beresnyak and Lazarian (2019). III. ANALYTICAL MODEL OF TURBULENT RECONNECTION A. Derivation of LV99 reconnection rate The 3D model of turbulent reconnection proposed in LV99 provides a natural generalization of the classical Sweet-Parker model (Parker, 1957; Sweet, 1958). Thus we remind the reader first the basic properties of this model. In the Sweet-Parker model,two regionswith uniformlaminarmagnetic fields are separated by thin current sheet. The magnetic fields are frozen in through the two regions and the violation of the flux freezing takes place over a thin slotδ, the latter being determined by Ohmic diffusion. Thus the speed of reconnection is determined roughly by the resistivity divided by the sheet thickness, i.e. V rec1 ≈η/∆.(33) Obviously, the smaller∆the larger the reconnection speed. Small∆, however, prevents plasmas from leaving the reconnection region. The Sweet-Parker model deals with thesteady state reconnectionand therefore the plasma in the diffusion region must be ejected from the edge of the current sheet at the Alfv ́ en speed,V A . As a result, the reconnection speed is limited by the conservation of mass condition V rec2 ≈V A ∆/L x ,(34) where L x is the lateral extend of the current sheet. The conditions imposed by Eq. (33) and (34) are acting in the opposing directions as far as the thickness of of the reconnection region∆is concerned. As a result, the Sweet-Parker reconnection cannot be fast as it faces a contradictory requirements. Therefore the compromise speed of recon- nection is the Sweet-Parker reconnection speed that is reduced from the Alfv ́ en speed by the square root of the Lundquist number,S≡L x V A /η, i.e. V rec,SP =V A S −1/2 .(35) The Lundquist number in astrophysical conditions is enormous, it can be10 16 or larger. In this situation, the Sweet-Parker reconnection rate is absolutely negligible. Due to the disparity between L x and∆the Sweet-Parker model cannot explain most of the ob- served astrophysical reconnection events, e.g. Solar flares. A way to deal with this problem was suggested by Petschek (1964), who speculated that in special conditions the reconnection configura- tions may have magnetic field lines converging to the reconnection zone at a sharp angle making L x and∆comparable. The corresponding configuration of magnetic field is known as X-point recon- nection as opposed to the Sweet-Parker extended current sheet Y-type reconnection. The research exploring X-point reconnection was the dominant trend in the attempts of obtaining fast reconnec- tion for decades. However, the major problem of such configurations is to preserve the X-point configuration over astrophysically large scales. The direction taken by LV99 was to modify the Y-type reconnection in order to make∆macro- scopic and even astrophysically large. Naturally, this is impossible with plasma effects. However, LV99 proposed that turbulence can induce magnetic field wandering that can do the job. The corre- sponding model of magnetic reconnection is illustrated by Figure 4. Similar to the Sweet-Parker model, the turbulent model developed in LV99 deals with a generic configuration, that arises naturally as magnetic flux tubes crossing each other make their way one through another. The outflow thickness∆is determined by the large-scale magnetic field wander- ing. This wandering depends on the level of turbulence. As a result, the rate of reconnection in LV99 model is determined by the level of MHD turbulence. The LV99 reconnection rates is fast, i.e. its rate do not depend on the fluid/plasma resistivity. However, the rates of turbulent reconnec- tion can change significantly depending on the level of turbulence. This is an important prediction Turbulent Reconnection14 x L Sweet-Parker model Turbulent model ∆ ∆ FIG. 4.Upper plot: Sweet-Parker model of reconnection. The thickness of the outflow∆is limited by Ohmic diffusivity that takes place on microscopic scales. The disparity between this scale and the astrophysically large scale L x ∆makes the reconnection slow. Lower plot: LV99 model of turbulent reconnection model. The width of the outflow∆is determined by macroscopic field line wandering and it can be comparable with the scale L x for trans-Alfvenic turbulence. Lazarian and Vishniac (2000). of LV99 theory which make it different from the theories that, for instance, appeal to plasma effects for describing fast reconnection. To quantify the reconnection rate in LV99 model one should use the scaling relations for Alfv ́ enic turbulence from§II. In fact, it is reconnection at low Alfven Mach numbers that corresponds to the classical representation of magnetic reconnection flux tubes. This, in fact required LV99 to generalize the GS95 MHD theory for arbitrary Alven Mach numbers. As the first step consider the problem of magnetic field wandering, as this magnetic wandering determines the thickness of outflow∆. We may note that magnetic wandering determined in LV99 is not only important for magnetic reconnection. In fact, it is a fundamental property of turbulent magnetic field and it is important for different problems, e.g. problems of heat transfer (Narayan & Medvedev 2001, Lazarian 2006) and cosmic ray propatation (Yan & Lazarian 2008, Lazarian & Yan 2014). Using the MHD turbulence relations one can describe the diffusion of magnetic field lines. In- deed, a bundle of field lines confined within a region of widthyat some particular point spreads out perpendicular to the mean magnetic field direction as magnetic field lines are traced in either direction. The rate of field line spread is given by (see more in§III B 3) d〈y 2 〉 ds ∼ 〈y 2 〉 λ ‖ ,(36) whereλ −1 ‖ ≈` −1 ‖ ,` ‖ is the parallel scale. The corresponding transversal scale,` ⊥ , is∼ 〈y 2 〉 1/2 , andxis the distance measured along the magnetic field direction. As a result, using equation (21) one gets d〈y 2 〉 ds ∼L i ( 〈y 2 〉 L 2 i ) 2/3 ( u L V A ) 4/3 (37) where the substitution〈y 2 〉 1/2 for` ⊥ is used. This expression for the magnetic field dispersion is applicable only whenyis small enough for us to use the strong turbulence scaling relations, i.e. when〈y 2 〉< L 2 i (u L /V A ) 4 . For scales less that L i it follows from Eq.(37) that the mean squared magnetic field wandering is equal to: 〈y 2 〉∼ s 3 L i ( u L V A ) 4 ,s < L i (38) which provides a superdiffusive dependence of〈y 2 〉 ∼s 3 , which was one of the major findings in LV99 as far as the properties of magnetic field in turbulence are concerned. Due to the above Turbulent Reconnection15 scaling the probability of wandering magnetic field line to return back to the reconnection layer is very small. The dependence of〈y 2 〉 ∼M 4 A is significant as it shows that if the turbulent kinetic energy is a small portion of magnetic energy of the fluid, the magnetic field lines are nearly straight. This provides a smooth transition to the slow reconnection in the absence of turbulence and, as we discuss later, to the flux freezing being a fair approximation to the low M A fluids. For describing magnetic field wandering over scales larger than the turbulence injection scale, i.e. sL i magnetic field wandering obeys the usual random walk scaling. The corresponding random walk step iss/L i with the mean squared displacement per step equal to L 2 i (u L /V A ) 4 . As a result, 〈y 2 〉 1/2 ≈(L i s) 1/2 (u L /V A ) 2 x > L i ,(39) which provides the other limit for the magnetic field wandering. While the diffusive regime of magnetic field lines given by Eq. (39) was well known in cosmic ray literature (see Schlickeiser, 2002), the superdiffusive one was introduced in LV99 3 . Using Eqs. (38) and (39) one can describe the thickness of the outflow for the reconnection of the anti-parallel magnetic fields in turbulent media, i.e. in the setting generalizing Sweet-Parker reconnection without guide field for turbulent media. Measuringsalong the horizontalx-axis along the magnetic field reversal and combining Eqs. (38) and (39) it is possible to derived the thickness of the outflow∆in the aforementioned two regimes. Then, using the mass conservation Eq. (52) it is straightforward to obtain V rec ≈V A min [ ( L x L i ) 1/2 , ( L i L x ) 1/2 ] M 2 A ,(40) where V A M 2 A is proportional to the turbulent eddy speed. As we discussed earlier, the obtained reconnection rate vary depending on Alfven Mach number M A and for M A ∼1can represent a large fraction of the Alfv ́ en speed in the case when L i and L x are not too comparable. Similar to the Sweet-Parker picture, the contribution of the guide field does not change the dy- namics of magnetic reconnection. Figure 4 presents a XYplane of cross section of the reconnection region perpendicular to the guide field B z shared by two fluxes. The guide field should be ejected together with plasma from the reconnection region. Indeed, both random velocity and magnetic field can be decomposed into components in the direction of the guide field and perpendicular to it. The field wondering in the direction perpendicular to the guide field is affected by the velocity perturbations in the same direction. The guide field does not oppose to the corresponding bending of magnetic field. The outflow velocity is, however, is determined only by the Alfven velocity cor- responding to the perpendicular component of magnetic field. In fact, for the sake of clarity, it could be right to change M A for M A,XY , where the latter denotes the Alfven Mach number correspond- ing to the Alfven velocity in the plane perpendicular to the guide field. This notation of the Alfven Mach number is not common and we will not use it therefore. It was communicated to us by Andrey Beresnyak that the independence of the reconnection rate on the guide field can be also justified from the point of Reduced MHD (RMHD) approach (see Beresnyak 2012). Formally, the RMHD is applicable for describing incompressible fluids when the perturbations are much smaller than V A . However, in practical cases RMHD is well applicable for many collisionless systems, e.g. Solar wind (see Schekochihin et al. 2009), and its practical validity is good even for trans Alfvenic turbulence (Beresnyak & Lazarian 2009). Within the RMHD the dynamics of turbulent motions is not changed if the ratio B 0 /l ‖ stays the same. If no constraines are imposed on the parallel scale of perturbationsl ‖ they increases with the increase of the guide field B 0 as was demonstrated in numerical simulations in Beresnyak (2017). The dynamics in terms of perpendicular perturbationsl ⊥ and field wondering that these perturbations entailed increasing the reconnection outflow stayed the same. 3 This resulted in several important developements not only for reconnection but for the understanding of cosmic ray propa- gation and acceleration (see Lazarian and Yan, 2014). Turbulent Reconnection16 For distances along magnetic field less than the injection scale L i using the definition of cascading rate for strong MHD turbulence given by Eq. (14), we can rewrite the expression for magnetic field wandering (see Eq. (38)) as 〈y 2 〉∼ s ( s V A ) 3 (41) from which it is clear that the increase of V A can be naturally compensated by the increase of the parallel scale along the magnetic field. For the scales larger than the injection scale, the Eq. (39) can be rewriten as 〈y 2 〉∼ L 2 i s V 3 A  w ,(42) where the expression for the cascading of the weak MHD cascade given by Eq.(10) is used. It is also obvious that the simultaneous increase of the parallel scale of both injection and turbulent motions and the Alfven velocity does not change the magnetic field wandering. For the formal applicability of RMHD one should assume that the guide field B z is much stronger than the reconnecing magnetic field B x . The relevant setting corresponds to reconnection of two flux tubes intersecting at a small angleα. The corresponding setting was discussed in LV99 for the reconnection induced by individual eddies within MHD turbulence and it was shown there that the reconnection happens within one eddy turnover time. Here we consider a more generic situation of reconnecting flux tubes with an arbitrary turbulence driving. In the above setting,B x ≈B z sinθ/2and the perturbations at the injection scale L i result in the perturbations in the XYplane with the scale L i,XY ∼L i θ/2. Similarly the pathxmeasured along the magnetic flux tubes projects into the pathx XY ≈xθ/2in the XYplane. If one considers the dynamics of magnetic field lines in the XY-plane the speed of propagation of Alfvenic perturbations alongx-axis is V A θ/2and therefore in this approximation it is obvious that the growth of the outflow region∆as seen in XY-plane does not depend on the angleθbetween reconnecting fluxes, i.e. magnetic reconnection proceeds independently of the value of guide field, but depends only on the B x field and the properties of turbulence in XY-plane. In numerical experiments described in§V the properties of driving as well as value of the re- connecing component of magnetic field were fixed in the XY-plane. Therefore the magnetic field wandering for such settings can be described by Eq. (41) and (42) with V A arising only from B x and L i being the injection scale of turbulence in the XY-plane. Correspondingly, Eq. (40) should use M A ≈δB x /B x , whereδB x is a turbulent perturbation of B x at the injection scale L i . The LV99 theory of turbulent reconnection has provided a number of predictions very different from the contemporary theories of fast magnetic reconnection. First of all, the turbulent reconnec- tion was identified as a generic process that takes place everywhere in the magnetized conducting fluid. A violation of flux freezing implicitly follows from that. Second, LV99 does not prescribe a given the reconnection rate, but provides an expression that shows that the speed of reconnection vary depending on the level of turbulence. This can explain the variability of the energy release in reconnection events observed in e.g. Solar flares. Indeed, many of the observation processes require that the rate of reconnection changes. In LV99 theory this is controlled by the level of turbulence that can vary. For instance, the low rate of turbulent reconnection can explain how magnetic flux can be accumulated prior to a Solar flare. Third, the independence on the plasma parameters makes the turbulent reconnection a generic reconnection process in the astrophysical settings. While the earlier theories of turbulent reconnection were focused X-point reconnection, LV99 showed that Y-type reconnection can be fast. The tearing reconnection that was developed later (see §X C) presented another version of the fast Y-type reconnection. Similar to LV99 it departed from the regular structures prescribed by Sweet-Parker and Petscheck reconnection models. Being Y-type fast reconnection the LV99 necessarily involved a significant volume of magnetized fluid in the reconnection process. This volume reconnection changes the nature of the energetic particle acceleration as it is discussed in§IX C Turbulent Reconnection17 B. Flux freezing violation: Richardson dispersion 1. Alfv ́ en flux freezing Since the seminal work of Alfv ́ en (1942), the “flux freezing principle” has become a powerful tool to estimate solutions of MHD equations in a diverse range of problems (Parker, 1979; Kulsrud, 2005). Besides its utility in plasma physics, in astrophysics as well it has found many applications, for example, in explaining the low angular momentum of stars and the spiral structure of magnetic field lines in the solar wind. Despite its success in addressing diverse problems, however, there have been also fatal failures (see Burlagaet al., 1982; Khabarova and Obridko, 2012; Richardson et al., 2013; Eyink, 2015). For instance, assuming flux freezing, the magnetic topology in a high- conductivity magnetized fluid should not change rapidly, whereas observations of solar flares and Coronal Mass Ejections (CMEs) indicate otherwise. Also, in star formation, the magnetic pressure of in-falling magnetized material should be strong enough to prevent gravitational collapse alto- gether, if magnetic flux-freezing held with good accuracy. Finally, if magnetic field were frozen into the fluid in small-scale astrophysical dynamos, the tangled structure of the generated magnetic field would quench its further growth. In order to account for these difficulties with magnetic flux-freezing, one might find it tempting to appeal to additional effects beside collisional resistivity. For instance, one might consider additional terms, such as Hall effect and electron pressure anisotropy, in the generalized Ohm’s law (electron momentum equation) (Burchet al., 2016; Cassak, 2016). This does not answer the question, unfortunately, how magnetic reconnection events can be fast in fluids with high rates of collisions. Indeed, both observations and theory imply fast magnetic reconnection in the solar photosphere, and other environments such as collisional parts of accretion disks, which cannot be explained by appealing to collisionless mechanisms (Chaeet al., 2010; Singhet al., 2011). The problem persists even for weakly collisional systems where an ideal MHD-type description should be valid at length-scales much larger than the ion gyroradius, above which flux-freezing is often assumed to hold. However, observations indicate that magnetic structures with length scales much larger than ion gyroradius reconnect very rapidly, e.g. at the heliospheric current sheet in the solar wind (Gosling, 2012). It is unclear how collisionless mechanisms at scales below the ion cyclotron radius can lead to fast magnetic reconnection on much larger scales. This is the essence of the “scale problem” in magnetic reconnection (Jiet al., 2019). While the necessity of fast magnetic reconnection was widely accepted, most of the studies as- sumed that the fast reconnection happens for very special configurations of magnetic fields. This way the community could reconcile at least some of the observational evidence of fast reconnec- tion and the concept of magnetic flux freezing. LV99, as discussed previously, identified magnetic reconnection as an intrinsic property of MHD turbulent cascade implying the violation of the flux freezing principle (see Vishniac and Lazarian, 1999, for an explicit statement about the flux freezing in turbulent fluids). Below we provide a description of the Richardson dispersion, the phenomenon that manifests a gross violation of the flux freezing in magnetized turbulent fluids. This numerically confirmed phenomenon (see§V B) provides another way of re-deriving LV99 rate of turbulent reconnection. A more rigorous approach to the flux freezing violation is discussed in§IV. 2. Time dependent Richardson dispersion Another way to understand the violation of flux freezing in turbulent fluids is related to the con- cept of Richardson dispersion. This process initially introduced in hydrodynamic turbulence carries over to MHD turbulence. Richardson (1926) empirically analyzed the dispersion of the volcanic ashes during the eruption and formulated the law that can be derived using hydrodynamic turbulence theory. Note, however, that the corresponding theory was formulated much later by Kolmogorov (1941). Using this theory one can predict that the separation between two particles obeys the equa- tiond/dt[l(t)]∼v(l)∼αl 1/3 , whereαis proportional to a cubical root of the energy cascading rate. The solution of this equationl 2 ∼t 3 describes the Richardson dispersion, i.e. the mean square Turbulent Reconnection18 FIG. 5. Illustration of diffusion for different regimes. Magnetic field Bis perpendicular to the eddies. Up- per plot corresponds to the diffusion on scales larger than the maximal size of turbulent eddies; Middle plot corresponds to the diffusion at scales less than the damping scale of turbulence; Lower plot corresponds to a Richardson dispersion case, i.e. when the dispersion is measured for fluid elements with separations within the inertial range of turbulent motions. From Lazarian (2014). separation between particles increasing in proportion to the cube of time. The accelerated growth of the separation is due to the fact that the larger the separation, the larger the eddies that induce the separation. In Kolmogorov turbulence the larger eddies have the larger velocity dispersion. An interesting feature of this solution above is that the provides the type of fast separation even if the initial separation of particles is zero. Formally this means the violation of Laplacian determinism. Mathematically the above paradox is resolved by accounting to the fact that turbulent field is not dif- ferentiable 4 and therefore it is not unexpected that the initial value problem does not have a unique solution. In any realistic physical settings the turbulence is damped at a non-zero scalel min , which makes the paradox less vivid. At the scales less than thel min the separation of points is not stochas- tic and it follows the Lyapunov growth (see Lazarian, 2006). However, at the scales largerl min but smaller than the injection scale of turbulence L max , i.e. over the inertial range of turbulence, the explosive growth of separation is still present. In terms of motions perpendicular to the local direction of magnetic field, the scaling of veloci- ties in GS95 picture corresponds to the Kolmogorov one. Therefore we do expect that the plasma particles to separate in accordance to the Richardson law over the inertial range. The situation is illustrated by Figure 5. The magnetic field is shown to be perpendicular to the plane of the picture. A cloud of test particles undergoes an ordinary diffusion foraL max , undergoing Lyapunov separation of field lines fora < l min and exhibiting Richardson dispersion over the inertial range [L max ,l min ]. Obviously, the Richardson-type dynamics is not possible without magnetic field lines disentangling themselves via fast reconnection. This corresponds to the notion in LV99 that tur- bulent reconnection is required for turbulent magnetized fluid to preserve fluid-type behavior. We note that if reconnection were slow the intersecting magnetic field flux tubes would form unresolved knots. This inevitably would arrest the fluid motions, making the fluid with dynamically important magnetic field Jello or felt-like substance. The spectrum of magnetic fields would be very mod- ified in this case with a significant increase of energy at small scales. Only acoustic type waves can propagate in this substance and definitely no present-day numerical simulations of MHD type can describe it. This picture of slow reconnection in turbulent media contradicts both to numerical simulations and observations of interstellar turbulence (see§II A and§II B). The turbulent reconnec- tion, on the contrary, predicts the free evolution of eddies that produce Kolmogorov-type cascade for eddies perpendicular to local direction of magnetic field as discussed in§II B. The Richardson dispersion is a numerically proven phenomenon (see§V B) and it was used in ELV11 to re-derive the LV99 rates of reconnection. Consider first the Sweet-Parker reconnection. The Ohmic diffusion induces the mean-square 4 The Kolmogorov velocity field is Holder continuous, i.e.|v(r 1 )−v(r 2 )|.C|r 1 −r 2 | 1/3. Turbulent Reconnection19 displacement across the reconnection layer in a timetgiven by 〈y 2 (t)〉∼λt.(43) whereλ=c 2 /4πσis the magnetic diffusivity. As in our previous treatment the magnetized plasmas is ejected out of the sides of the reconnection layer of length L x at a velocity of order V A . Thus, the time that the lines can spend in the resistive layer is the Alfv ́ en crossing timet A =L x /V A . Therefore, the reconnected field lines are separated by a distance ∆ = √ 〈y 2 (t A )〉∼ √ λt A =L x / √ S,(44) where Sis Lundquist number. Combining Eqs. (34) and (44) one the Sweet-Parker result,v rec = V A / √ S. The difference with the turbulent case is that instead of Ohmic diffusion the Richardson dispersion dominates (ELV11). In this case the mean squared separation of particles follows the Richardson law, i.e.〈|x 1 (t)−x 2 (t)| 2 〉 ≈t 3 , wheretis time,is the energy cascading rate. For sub Alfv ́ enic turbulence≈u 4 L /(V A L i )(see LV99) and therefore one can write ∆≈ √ t 3 A ≈L(L/L i ) 1/2 M 2 A (45) where it is assumed that L < L i . Combining Eqs. (34) and (45) it is easy to obtain v rec,LV99 ≈V A (L/L i ) 1/2 M 2 A .(46) in the limit of L < L i . Similar considerations allowed ELV11 to re-derive the LV99 expression for L > L i , which differs from Eq. (46) by the change of the power1/2to−1/2. 3. Field line wandering Magnetic fields produce a new effect compared to the Richardson dispersion in hydrodynamics. In magnetized fluids one can trace magnetic line separation. This effect has been discussed for decades in the cosmic ray literature as the cause of the perpendicular diffusion of cosmic rays in the galactic magnetic field (Jokipii, 1973). It worth mentioning that the correct quantitative description of the effect obtained only in LV99. Earlier papers assumed that the diffusive separation of magnetic field lines, which is the effect taking place only if magnetic field lines are separated by more than the turbulence injection scale L. It was shown in LV99 that at scales less than Lthe magnetic field lines separate superdiffusively. This changes a lot in terms of both the propagation and the acceleration of cosmic rays (see Lazarian and Yan, 2014). Dealing with magnetic reconnection in§III we have discussed the separation of magnetic field lines. Figure 6 illustrates the spread of magnetic field lines in the perpendicular direction in the situation when magnetic field lines are traced by particles moving along them. This figure confirms the prediction for the magnetic field line separation in Eq. (38), i.e. that the squared separation between the magnetic field lines increases in proportion the cube of the distance measured along the magnetic field lines (see 6). The other important prediction in Eq. (38) is the change of the square of the separation between the magnetic field lines increases with the forth power of Alfven Mach number, i.e.∼M 4 A . The successful testing of this prediction is demonstrated in Fig. 7 that is taken from Xu and Yan (2013). The asterisks correspond to the calculations obtained using the parallel particle velocity. This way of calculating the separation provides a better description of magnetic field wandering. The fitted dependence is M −3.8 A , which is very different from the M 2 A dependence traditionally accepted in the cosmic ray literature. We note that the magnetic field field tracing with particles is not the only way to numerically explore the magnetic field separation. In fact, the analytical predictions of magnetic field line su- perdiffusion were also explored numerically by direct tracing of magnetic field lines first with low resolution in Lazarian, Vishniac, and Cho (2004) and with higher resolution in Beresnyak (2013). Turbulent Reconnection20 10 −2 10 −1 10 0 10 −2 10 −1 10 0 |δ ̃x|/L 〈 ( δ ̃ z ) 2 〉 1 / 2 /L l A l ⊥,m in L |δ ̃x| 1.5 FIG. 6. Tracing magnetic fields with charged particles. The mean squared separation of particle trajectories 〈(δz) 2 〉 1/2 as the particles move along along the magnetic field is measured in thex-direction. From Xu and Yan (2013), c ©AAS. Reproduced with permission. 0.30.40.50.60.7 10 −1 10 0 10 1 M A 3 3 L 〈 ( δ ̃ z ) 2 〉 | δ ̃ x | 3 FIG. 7. The dependence of the magnetic field line separation on the Alfven Mach number M A as measured by particles streaming freely along the magnetic field lines. The fit for the asterisks corresponds to the∼M −3.8 A dependence, very close to the predicted one. From Xu and Yan (2013), c ©AAS. Reproduced with permission. Taken together, these numerical testing provides a solid confirmation of the predictions given by Eq. (38). It is significant that the turbulent magnetic field wandering induced by strong MHD turbulence provides the separation of magnetic field linesδ 2 lines ∼s 3 , where the distancesalong the instantaneous realization of a magnetic field line. Thus for sufficiently largesall parts of the volume of magnetized fluid get connected. In other words, the entire volume becomes accessible to particles moving along magnetic field lines. This is important for a wide variety of astrophysical processes, e.g. the cosmic ray propagation and acceleration (Yan and Lazarian, 2008; Lazarian and Yan, 2014), heat propagation (Narayan and Medvedev, 2001; Lazarian, 2006). We would like to stress again that magnetic wandering cannot be understood using the notion of magnetic field lines preserving their identify in turbulent flows. Magnetic field lines wandering in the turbulent magnetized flow constantly reconnect inducing the exchange of plasmas and magnetic field. In view of this, it is interesting to recall the considerations on the non-trivial nature of astro- physical magnetic field lines that can be found in a prophetic book by Parker (1979). There it was mentioned that a magnetic field line may be traced through a significant part of the galaxy. We add that this is only possible through constant reforming of the magnetic field lines via reconnection. We note that there is a formal analogy between the Richardson diffusion that predicts that the mean squared separation between particlesδ 2 particles growth with time ast 3 and the mean squared separation between the magnetic field linesδ 2 lines that grows with the distancesass 3 . Therefore, Turbulent Reconnection21 rather than use a new term to describe the latter dependence theoretically predicted in LV99, it was decided in ELV11 to term this new type of dispersion, asthe Richardson diffusion in space, which keeping the term “Richardson dispersion” for the original time-dependent process empirically discovered in Richardson (1926). C. Applicability of LV99 approach The derivation of LV99 reconnection rate assumes that all magnetic energy is converted to kinetic energy of the outflow. This may not be true when M A is large. In this case, the energy dissipation in the reconnection layer becomes non-negligible and the decrease of the outflow velocity is expected (see ELV11). The corresponding effect of turbulent dissipation can be estimated from steady-state energy bal- ance in the reconnection layer: 1 2 v 3 out ∆ = 1 2 V 2 A v ren L x −εL x ∆,(47) where kinetic energy of the outflow is balanced against magnetic energy transported into the layer minus the energy dissipated by turbulence. The turbulent dissipation is given byε=u 4 L /V A L i for sub-Alfv ́ enic turbulence Kraichnan (1965). Dividing (47) by∆ =L x v rec /v out , one obtains v 3 out =V 2 A v out −2 u 4 L V A L x L i .(48) The corresponding solutions are easy to get by introducing the ratiosf=v out /V A andr= 2M 4 A (L x /L i )which are the measures of the outflow speed and the energy dissipated by turbu- lence in units of the available magnetic energy. This gives r=f−f 3 .(49) Whenr= 0, the only solution of (49) withf >0isf= 1,which is the the LV99 estimate v out =V A for M A 1. For larger values ofr, f'1−(r/2), in agreement with the formula f= (1−r) 1/2 that follows from Eq. (65) in ELV11. This implies a decrease inv out as compared to V A . Note that formula (49) can be used whenris not too large. The largest value ofrfor which a positive, realfexists isr max = 2/ √ 27≈0.385. Forr max one getsf min = 1/ √ 3≈0.577. This implies that the LV99 approach is limited to sufficiently small M A . For L x 'L i , one may consider values of M A up to0.662. Therefore, the original LV99 approach is accurate for M A .1. While the accuracy of LV99 expressions can be decreases due to the additional dissipation of the turbulent energy, the predicted phenomenon of fast reconnection arising from the turbulence persists for any M A . In fact, one can see that the effect of the reduced outflow velocityincreasesthe recon- nection rate. The reason is that field-lines now spend a time L x /v out exiting from the reconnection layer, greater than assumed in LV99 by a factor of1/f. This implies a thicker reconnection layer∆ due to the longer time-interval of Richardson dispersion in the layer, greater than the LV99 estimate by a factor of(1/f) 3/2 . The net reconnection speedv rec =v out ∆/L x is thus larger by a factor of(1/f) 1/2 . The increased width∆more than offsets the reduced outflow velocityv out . At even larger M A magnetic fields at large scale are weak that they are easily bent and twisted by the fluid turbulent motions. However, at the scales smaller thanl A we expect the original LV99 treatment to be accurate again. D. LV99 model: summary of facts The model of turbulent reconnection suggested in LV99 departed from the earlier attempts to explain astrophysical magnetic reconnection as something that required special conditions. On the contrary, it suggested that reconnection takes place everywhere in realistically turbulent fluids. Turbulent Reconnection22 Moreover, the transition to turbulent reconnection is inevitable as the 3D outflow from the recon- nection region gets turbulent for sufficiently large Lundquist numbers. The model identified turbulent magnetic field wandering as the process that opens up the recon- nection region outflow making reconnection fast, i.e. independent of fluid resistivity or microscopic plasma effects. The derived dependences for the field wandering were successfully tested with the numerical simulations. The model provided analytical expressions relating the reconnection rate with the level of turbu- lence. It also provided the relations between the thickness of the reconnection layer and the level of turbulence. This explains that rate of magnetic reconnection may vary significantly and also provides ways for quantiatative numerical and observational testing of the model. IV.VIOLATION OF FLUX-FREEZING, RENORMALIZATION AND THE SCALE PROBLEM In§III B we discussed the issue of flux freezing on the basis of the simplified treatment of Richardson dispersion, treating the latter more like an empirical fact and using the corresponding scaling relations to re-derive the LV99 reconnection rates. However, the phenomenon of Richardson dispersion presents a much more fundamental property of magnetic fields in turbulent fluids. Below we discuss the violation of flux freezing on the basis of a more rigorous theoretical treatment. A. Spontaneous stochasticity The conventional flux-freezing principle has not only problems with explaining observations. It also encounters serious theoretical difficulties. This can be exemplified by naively extending the concept to turbulent MHD flows with both small resistivity and small viscosity. To understand the subtlety of such singular limits, consider the problem of incompressible fluid turbulence. As is well-known, the kinetic energy dissipation rate, in a fluid with viscosityνand velocity fieldu, does not vanish in the limit of vanishing viscosity (e.g., see Sreenivasan (1984, 1998)). On the contrary, lim ν→0 ν|∇u| 2 >0which implies a “dissipative anomaly”. This in turn requires that velocity gradients must blow up, with∇u→ ∞in the limitν→0. In fact, as Onsager pointed out in a prescient work (see§IV B below), the velocity field in a turbulent fluid must develop H ̈ older-type singularities in the limit as viscosity tends to zero (Onsager, 1949; Eyink and Sreenivasan, 2006; Eyink, 2018). These singularities render all hydrodynamic equations containing velocity gradients naively undefined. Extending Onsager’s analysis to MHD plasmas, turbulent solutions are found to suffer several dissipative anomalies which require diverging gradients of all MHD fields, e.g., velocity field, magnetic field, density etc. (see e.g., Mininni and Pouquet (2009); Caflisch, Klapper, and Steele (1997); Eyink and Drivas (2018). For example, magnetic-field gradients blow up in the limit as magnetic diffusivityηtends to zero, with∇B→ ∞asη→0. Because velocity fields in particular become non-smooth and singular asν→0, the magnetic field lines cannot freeze into the fluid in a standard deterministic sense. Instead a non-deterministic frozen-in property known as “stochastic flux freezing” (§IV C) has been established by Eyink (2009, 2011), which is intrinsically random in nature due to the roughness of the advecting velocity field (Eyink, 2010) and is associated with thespontaneous stochasticityof Lagrangian particle trajectories (Bernard, Gawedzki, and Kupiainen, 1998; Eyink, Lazarian, and Vishniac, 2011; Eyinket al., 2013). The latter concept is crucial and must be briefly explained. In essence, “spontaneous stochas- ticity” means that stochastic Lagrangian particle trajectories remain non-unique and indeterminis- tic (random) for turbulent flows even in the limit of vanishing viscosity and molecular diffusivity (Bernard, Gawedzki, and Kupiainen, 1998). This phenomenon is directly related to Richardson super-ballistic behavior (Richardson, 1926) (see§III B). In a turbulent flow, therefore, infinitely- many magnetic field lines are advected to each point in space even in the ideal limit and must be averaged to obtain the resultant magnetic field (Eyink, 2011; Eyinket al., 2013). Due to Richard- son dispersion, in fact, even a tiny breakdown of magnetic flux freezing at small scales will be explosively amplified into larger scales in the turbulent inertial range (Eyink, 2015, see also§IV D below). The rates of such rapid amplification are non-vanishing in the ideal limit (Eyink, 2011; Turbulent Reconnection23 Eyink, Lazarian, and Vishniac, 2011). This breakdown of conventional magnetic flux freezing has already been verified in numerical simulations of resistive MHD turbulence at high conductivity (e.g. Eyinket al., 2013). To appreciate the importance of (asymptotic) singularities in MHD, we may note that without differentiability of fields even fundamental concepts such as the notion of field line need to be reconsidered. Taking magnetic field B(x)as an example, itsfield linethrough pointxis conven- tionally defined as the integral curveξ(s;x)whose tangent vector at any point is parallel to the vector field at that point, so that, parameterized with arclengths, dξ(s)/ds=B(ξ(s))/|B(ξ(s)|,ξ(0) =x.(50) If the magnetic field satisfies the following inequality for some constant C >0; |B(x)−B(x ′ )|≤C|x−x| h withh= 1(Lipschitiz condition), then there exists a unique field-lineξ(s;x)passing through point x; however, if this condition holds merely with0< h <1(H ̈ older continuity), then there are gener- ally infinitely many such integral curves (see Jafari and Vishniac (2019); Jafari and Vishniac (2019); Jafari, Vishniac, and Vaikundaraman (2019b)). If the Lipschitz condition is not satisfied and hence the uniqueness theorem cannot be applied for a given magnetic field, what would a magnetic field line mean after all? When resistivity is tiny but non-zero, then Bis differentiable and indeed equa- tion (50) has unique solutions. However, these solutions exhibit such extreme sensitivity to initial conditions that the distance|ξ(s;x ′ )−ξ(s;x)|between field lines through neighboring points be- comesindependent ofx ′ −xat distancess|x ′ −x|along the lines (LV99). 5 Such explosive sepa- ration of magnetic field lines is not observed if the field lines exhibit standard “deterministic chaos”, where memory of the initial separation is preserved. Exactly analogous difficulties plague the con- cepts of “Lagrangian fluid particle” or “fluid trajectory”, which are ill-conditioned and asymptoti- cally non-unique if the fluid velocityv(x,t)becomes spatially non-Lipschitz asν→0. The singularities associated with dissipative anomalies lead to subtle effects whose consistent de- scription requires great care, otherwise contradictory conclusions are obtained. Similar difficulties were encountered already many decades ago in quantum field theory and condensed matter physics, which could be alleviated by mathematical methods collectively known as regularization and renor- malization. As we review in§IV B below, such methods can be applied also to plasma turbulence by exploiting a spatial coarse-graining operation, which removes divergences and leads to meaningful equations. Physically speaking, “coarse-graining” simply means averaging quantities in a volume in space. Indeed what is measured in the laboratory as magnetic or velocity fields at a point in space and time is actually always an average over a small volume around that point during a time interval. Paradoxes are avoided by developing a theory based upon such quantities that can be measured in principle rather than idealized pointwise fields that are strictly unobservable. Such renormaliza- tion introduces an “observation scale” or “resolution scale” which is entirely arbitrary. However, obviously, no objective physical fact can depend upon this arbitrary scale. In the language of quan- tum field theory and statistical physics, this principle is called “renormalization-group invariance” (Gross, 1976). B.Renormalization of MHD Equations and Singularities We illustrate Onsager’s renormalization-type arguments in one of the simplest models with realis- tic astrophysical applications, the standard MHD equations for a compressible, magnetized, single- component fluid with mass densityρ,velocityv,magnetic field B,and internal energy density u: ∂ t ρ+∇·(ρv) = 0, 5 We provide more detail on this effect in§III B. Turbulent Reconnection24 ∂ t (ρv) +∇· [ ρvv− 1 4π BB+ ( p+ 1 8π B 2 ) I −2μS−ζΘI ] =0, ∂ t B=∇×(v×B−c J/σ),∇·B= 0, ∂ t ( 1 2 ρv 2 + 1 8π B 2 +u) +∇· [ (u+p+ 1 2 ρv 2 )v+ 1 4π B×(v×B) −κ∇T−2μS·v−ζΘv− c 4πσ B×J ] = 0. Here Iis the identity tensor andp=p(ρ,u)is the thermodynamic equation of state for the pressure and S ij = 1 2 ( ∂v i ∂x j + ∂v j ∂x i − 2 3 (∇·v)δ ij ) ,Θ =∇·v, J=c(∇×B)/4π, with shear viscosityμ,bulk viscosityζ,thermal conductivityκ,and electrical conductivityσ(or magnetic diffusivityη=c 2 /4πσ). See Landauet al.(2013), Chapter VIII,§65-66. The local balance of mechanical energy for the above model is given by the equation: ∂ t ( 1 2 ρv 2 + 1 8π B 2 ) +∇· [ (p+ 1 2 ρv 2 )v+ 1 4π B×(v×B) −2μS·v−ζΘv− c 4πσ B×J ] =p(∇·v)−2μ|S| 2 −ζΘ 2 −η(∇×B) 2 /4π. Just as for the case of incompressible, neutral fluids studied by Onsager, turbulent solutions are characterized bydissipative anomalies(see, e.g. Mininni and Pouquet, 2009) in which 2μ|S| 2 , ζΘ 2 , η(∇×B) 2 90,asμ,ζ,κ,η→0. Similar anomalous dissipation must then appear also in the balances of internal energy and entropy. Such anomalies obviously require diverging space-gradients|∇v|,|∇B|→∞forμ,ζ,κ,η→0, or a “violet catastrophe” in the words of Onsager (1945). To obtain a well-defined dynamical description in this singular ideal limit, one must regularize these UV divergences. For example, one may employ a “coarse-graining” or “block-spin” regular- ization, defined as B ` (x,t) = ∫ d 3 r G ` (r)B(x+r,t) and likewise forv, ρ, u,etc. with G ` (r) =` −3 G(r/`)for some positive, smooth, rapidly decaying kernel G(ρ)normalized as ∫ d 3 ρG(ρ) = 1.Coarse-grained gradients such as∇B ` (x,t)necessar- ily remain finite asμ,ζ,κ,η→0,their UV divergences removed. It should be emphasized again that such a coarse-graining is purely passive and corresponds to observing the flow “with spectacles off” so that eddies of size< `cannot be resolved. The basic principle of renormalization-group invariance is that this regularization scale`is completely arbitrary and no objective physics can depend upon it. Because the coarse-graining operation commutes with space-time derivatives, the effective equations of motion for eddies of size> `are exactly the same as the original MHD equations with just an additional overbar () ` . However, even when there are dissipative anomalies in the microscopic or “bare” balance equa- tions for conserved quantities, the dissipative transport terms such as∇×(η∇×B) ` in the coarse- grained MHD equations can be shown to vanish forμ,ζ,κ,η1at fixed`or for`1at fixed Turbulent Reconnection25 μ,ζ,κ,η. These dissipative terms are thusirrelevantin the renormalization-group sense. It follows, whenμ,ζ,κ,η1,that there exists an “inertial-range” of length-scales`at which the ideal MHD equations hold in the “coarse-grained sense”, i.e. that: ∂ t ρ ` +∇·(ρv) ` = 0,(51) ∂ t (ρv) ` +∇· [ ρvv− 1 4π BB+ ( p+ 1 8π B 2 ) I ] ` =0,(52) ∂ t B ` =∇×(v×B) ` ,∇·B ` = 0(53) ∂ t ( 1 2 ρv 2 + 1 8π B 2 +u ) ` +∇· [ (u+p+ 1 2 ρv 2 )v+ 1 4π B×(v×B) ] ` = 0.(54) Taking the limitμ,ζ,κ,η→0these coarse-grained equations hold for all` >0,which is equivalent to the standard mathematical notion of a “weak”/“distributional” solution, or what Onsager called a “more general description” than standard differential equations. The validity of the ideal MHD equations in this generalized sense in not sufficient, however, to guarantee that naive balances of mechanical energy, internal energy and entropy will hold. To show this clearly, the above ideal equation can be cosmetically rewritten by introducing thedensity- weighted Favre-average ̃ f ` = (ρf) ` /ρ ` andpth-order cumulants τ ` (f 1 ,f 2 ,...,f p )and ̃τ ` (f 1 ,f 2 ,...,f p )for the two types of coarse-graining averages. With these definitions, the effective equations of motion for scales> `become: ∂ t ρ ` +∇·(ρ ` ̃ v ` ) = 0,(55) ∂ t (ρ ` ̃ v ` ) +∇·[ρ ` ̃ v ` ̃ v ` + p ` I+T ` ] = 1 c J ` ×B ` ,(56) ∇· B ` = 0, ∂ t B ` =∇×(v ` ×B ` +ε ` ),(57) and ∂ t ( 1 2 ρ ` | ̃ v ` | 2 + 1 8π |B ` | 2 +u ` +δu ` ) + ∇· [( 1 2 ρ ` | ̃ v ` | 2 +u ` +δu ` +p ` ) ̃ v ` + 1 4π B ` ×(v ` ×B ` ) +T ` · ̃ v ` − c 4π ε ` ×B ` +J h ` ] = 0.(58) Here we have introduced theturbulent stress tensor T ` =ρ ` ̃τ ` (v,v)− 1 4π τ ` (B,B) + 1 8π τ ` (B i ,B i )I, turbulent electromotive force (emf) ε ` = τ ` (v × ,B) :=(v×B) ` −v ` ×B ` ,(59) Turbulent Reconnection26 turbulent internal energy δ u ` = 1 2 ρ ` ̃τ ` (v i ,v i ) + 1 8π τ ` (B i ,B i ) andturbulent enthalpy transport J h ` = τ ` (h,v)−h ` τ ` (ρ,v)/ρ+ 1 2 ̄ρ ` ̃τ ` (v i ,v i ,v) + 1 4π (τ ` (B,B)−τ ` (B i ,B i )I)·τ ` (ρ,v)/ρ + 1 4π (τ ` (v,B)−τ ` (v i ,B i )I)·B ` + 1 4π τ ` (B × ,(v × ,B)) withh=u+pthe enthalpy per volume. Although equations (55)-(58) are exactly equivalent to the coarse-grained ideal MHD equations (51)-(54), they appear “non-ideal”. Integrating out the small-scale eddies has led to “renormalized” equations with new terms T ` ,ε ` , δu ` ,J h ` that would not appear in naive ideal MHD equations for the resolved fieldsρ ` , ̃ v ` ,B ` ,u ` at length-scales> `.The precise condition for validity of the ideal equations (51)-(54) at length-scale`is that the standard dissipative transport terms are negligible compared with these new terms. For example, the ideal magnetic induction equation (57) is valid precisely when |ε ` || (η∇×B) ` |(60) One of Onsager’s key insights was that contributions from coarse-graining such asε ` can be ex- pressed entirely in terms ofspace-increments(see Eyink and Aluie, 2006; Eyink and Drivas, 2018), such as δB(r;x,t) :=B(x+r,t)−B(x,t). It follows that |ε ` |∼δv(`)δB(`),|(η∇×B) ` |∼ηδB(`)/`.(61) In that case, the condition (60) becomes `δv(`)/η1, which precisely defines the inertial-range of scales`for which the ideal induction equation is valid. This condition can be restated as`` η where` η is the resistive dissipation length specified by ` η δv(` η )'η.Similar conditions guarantee validity of all of the other ideal equations as well (see Eyink and Drivas, 2018). From the coarse-grained ideal MHD equations rewritten in this manner, one obtains a balance equation forresolved mechanical-energy: ∂ t ( 1 2 ρ ` | ̃ v ` | 2 + 1 8π |B ` | 2 ) +∇· [ p ` v ` + 1 2 ρ ` | ̃ v ` | 2 ̃ v ` + 1 4π B ` ×(v ` ×B ` ) +T ` · ̃ v ` − c 4π ε ` × B ` ] = p ` ∇·v ` −Q flux ` .(62) where the renormalization has introduced new apparently non-ideal terms. In particular, the right side of this balance contains theturbulent energy flux Q flux ` =−T ` :∇ ̃ v ` −ε ` · J ` −τ ` (ρ,v)· 1 ρ ` ( 1 c J ` ×B ` −∇p ` ) Turbulent Reconnection27 which represents nonlinear cascade of mechanical energy to unresolved scales< `. The energy lost from resolved mechanical modes reappears as total unresolved energyu ∗ ` =u ` +δu ` ,whereu ` is the thermal kinetic energy of microscopic particles andδu ` represents mechanical energy in unresolved turbulent eddies at scales< `.The balance equation for this total unresolved energy orintrinsic internal energycan be obtained by combining Eq.(62) with coarse-grained energy conservation to give ∂ t u ∗ ` +∇·(u ∗ ` ̃ v+J u ` ) =−p∇·v+Q flux ` where J u ` =J h ` +pτ(ρ,v)/ρand where Q flux ` is the same term that appears in Eq.(62) but with the opposite sign. If mechanical energy is dissipated in the limitμ,ζ,κ,η→0,then such dissipation will be ap- parent even to a “myopic” observer who sees only eddies> `and the dissipation rate must become independent of`for`→0.From estimates of the form given by Eq.(61) it follows that Q flux ` can be expressed entirely in terms ofspace-incrementsas discussed above, therefore Q flux ` can be non- vanishing as`→0only if the solution fields develop a critical degree of singularity. For example, suppose that for all space-time points(x,t) |δB(r)|≤(const.)B rms ( |r| L ) h B ,(63) and similarly forδv(r)with some exponenth v ,etc. Since ε ` · J=O ( δv(`)(δB(`)) 2 ` ) =O(` h v +2h B −1 ),(64) this term, which represents cascade of magnetic energy, can remain non-vanishing as`→0only if h v + 2h B ≤1. Similarly, the first term in Q flux ` which represents cascade of kinetic energy can remaining non- vanishing only if3h v ≤1and the third term, which generalizes to MHD the “baropycnal work” of Aluie (2013), requires at least one of the inequalitiesh ρ +h v +h B ≤1orh ρ +h v +h p ≤1in order to be non-vanishing. These conditions taken altogether imply thath v <1asμ,ζ,κ,η→0 and thus the velocity fieldcannotremain Lipschitz in the limit. It should be clear from the previous discussion that the balances of other quantities besides me- chanical energy can suffer from dissipative anomalies. In particular, the induction equation (57) contains the turbulent electromotive forceε ` which not only drives cascade of magnetic energy but also vitiates freezing of the resolved magnetic field B ` to the resolved velocityv ` .If instead the magnetic field is referred to the Favre-averaged velocity ̃ v ` ,then violation of flux-freezing is due to the combinationε ` −(1/ρ ` )τ ` (ρ,v)×B ` .Because generallyε ` ·B ` 6= 0,it is not possible to express the turbulent emf asε ` = ∆v ` ×B ` for any choice of vector field∆v ` ,which would allow one to regard lines of B ` as moving with a velocityv ∗ ` = v ` + ∆v ` .Thus,it is not true, as is often stated, that “because the ideal Ohm’s law holds in the turbulent inertial-range, therefore magnetic field-lines are frozen-in at those scales.”The error in this commonplace statement is that no precise operational meaning is ever given to the claim of “flux-freezing in the inertial-range of scales”, so that an experimentalist could in principle test it. As known from exalted areas of physics such as relativity and quantum mechanics, the failure to formulate such claims in operationally meaningful terms can lead to paradoxes and inconsistencies. If one adopts the natural interpretation that the magnetic field B ` of the eddies> `be frozen-in to the velocity fieldsv ` or ̃ v ` at those same scales, then the statements are certainly false. We discuss this later in§X D in relation to the concept of “reconnection-mediated turbulence” which implies that magnetic reconnection takes place only at some small scales where tearing gets important. It is obvious from the discussion above that magnetic reconnection takes place over the entire inertial range rather than some chosen small scales where dissipative processes take place. The reconnection at all scales of the turbulent cascade is necessary to explain the Richardson dis- persion (see§III B) and the numerical simulations demonstrating the gross violation of the classical flux freezing (see§V B. In fact, this idea is implicitly at the very core of the LV99 theory. Turbulent Reconnection28 A clear-minded skeptic might argue that such violations of flux-freezing in the turbulent inertial- range are mere artefacts of the coarse-graining and not “real”. In particular, coarse-graining could be applied even to a laminar ideal MHD flow and would lead to apparent violations of flux-freezing at length-scales`L ∇ ,the “gradient-length” over which the MHD solution fields sensibly vary. If such violations are real, then, invoking renormalization-group invariance, they must be observed for allresolution scales. An experimentalist who measured the velocity and magnetic fields in a laminar ideal MHD flow would find that, as the length-scales resolved by measurements were refined more, standard flux-freezing would hold better. However, in a turbulent MHD flow, deviations from flux- freezing predictions would persist as the experimentalist improved measurement resolution down to smaller length-scales`within the inertial-range, even though ideal MHD holds there in the “coarse- grained sense”. Eventually, the experimentalist would resolve down to the resistive scale` σ and would find the same violations of flux-freezing predictions, but now due to Ohmic electric fields. We note, that the assumption that the turbulence is not damped up to the Ohmic diffusion scale is not a necessary one. In§VIII A we show that fast reconnection is possible in the situation when viscosity is larger than resistivity. Considering the infinite-Reynolds limitμ,ζ,κ,η→0, the ideal MHD equations would be valid atallscales`and the violations of flux-freezing would be due entirely to the turbulent emfε ` .Eyink and Aluie (2006) have shown that magnetic flux conservation can be violated at an instant of time for an arbitrarily small length scale`in the absence of any microscopic plasma non-ideality only if at least one of the following necessary conditions is satisfied: (i) non-rectifiability of advected loops; (ii) unbounded velocity or magnetic fields; or (iii) existence of singular current sheets and vortex sheets that intersect in sets of large enough dimension. Both conditions (i) and (iii) can be expected to hold in MHD turbulence. As we now review, this conclusion that flux-freezing can be violated in turbulent solutions of ideal MHD is corroborated by two entirely independent arguments. The first of these appeals to the phenomenon of spontaneous stochasticity of Lagrangian fluid particle trajectories for non-Lipschitz velocity fields. The second argument invokes the spontaneous stochasticity of the magnetic field- lines themselves and the concept of “slippage” of field-lines relative to the plasma fluid. C. Conventional and Stochastic Flux-Freezing The flux-freezing properties of smooth, laminar solutions of the ideal MHD equations are well- known. One can write the induction equation for magnetic field Bas D t B= (B·∇)v−B(∇·v) +η4B where D t ≡∂ t +u.∇. Assuming that the limiting solution forη→0remains smooth, this equation may be combined with mass balance D t ρ+ρ(∇·v) = 0to yield D t ( B ρ ) = ( B ρ ) ·∇v, which succinctly expresses flux-freezing. The latter equation is solved explicitly by the Lundquist formula, B(x,t) = B 0 (a)·∇ a X(a,t) det (∇ a X(a,t)) ∣ ∣ ∣ ∣ X(a,t)=x , where X(a,t)is the position at timetof the Lagrangian fluid particle that was at pointaat timet 0 , so that d dt X(a,t) =v(X(a,t),t),X(a,t 0 ) =a. The mass-density ratio is represented by the determinantdet (∇ a X(a,t))| X(a,t)=x =ρ(a,t 0 )/ρ(x,t). Turbulent Reconnection29 It is less widely known that analogousstochastic flux-freezingproperties hold even if magnetic diffusivity is non-vanishing, orη >0.In that case, one must consider stochastic Lagrangian trajec- tories that satisfy d ̃ X(a,t) =v( ̃ X(a,t),t)dt+ √ 2η d ̃ W(t) where ̃ W(t)is a 3D Brownian motion that represents the effects of magnetic diffusivity. The exact solution of the magnetic induction equation is then represented by thestochastic Lundquist formula B(x,t) = B 0 (a)·∇ a ̃ X(a,t) det (∇ a ̃ X(a,t)) ∣ ∣ ∣ ∣ ∣ ̃ X(a,t)=x ,(65) where the overbar(·)represents an average over the Brownian motion ̃ W(t)or, equivalently, a path- integral over the stochastic Lagrangian trajectories ̃ X(t)themselves (see Eyink, 2009, 2011). If all solution fields remain smooth in the limitη→0,then the ensemble of stochastic Lagrangian trajec- tories ̃ X(a,t)collapses to the unique deterministic Lagrangian trajectory X(a,t)and conventional flux-freezing is recovered in the limit. This is not the case for turbulent solutions of ideal MHD obtained in the joint limitμ,ζ,κ,η→0. As we have observed in§IV B, the velocity fieldv(x,t)cannot remain uniformly Lipschitz if there is anomalous energy dissipation in that limit. In that case, there is no longer a unique Lagrangian trajectory for each fluid element in the infinite-Reynolds limit. Therefore, one may wonder with such an infinite number of limiting particle trajectories, which trajectory will be selected in the joint limitμ,ζ,κ,η→0? The surprising answer is that Lagrangian trajectories, under such conditions, can remain random—the phenomenon ofspontaneous stochasticity(Bernard, Gawedzki, and Ku- piainen, 1998; Chaveset al., 2003). This effect resembles “spontaneous symmetry-breaking” in condensed matter physics and quantum field theory, where below a critical temperature (such as the Curie temperature for a magnet) the mean order parameter remains non-vanishing even as the external symmetry-breaking field is taken to zero. Spontaneous stochasticity differs greatly from the usual randomness in turbulence theory associated to an ensemble of velocity fields, because the Lagrangian fluid trajectories remain non-unique and stochastic for afixedvelocity fieldv(x,t) andfixedparticle positionaat the initial timet 0 .The Laplacian determinism expected for clas- sical dynamics breaks down because of the explosive separation of particles undergoing turbulent Richardson dispersion. This remarkable prediction is supported by direct empirical evidence for turbulent MHD flows (for details see Eyink, 2011; Eyinket al., 2013) (see also§V B). In consequence of this surprising remnant or persistent randomness of Lagrangian trajectories, the stochastic flux-freezing relation given by Eq. (65) for non-ideal MHD solutions doesnotre- duce to conventional flux-freezing in the limitμ,ζ,κ,η→0.Instead, flux-freezing in such regimes remains valid in only a statistical sense associated to spontaneous stochasticity of the fluid trajec- tories (Eyink, 2011, see also Jafari and Vishniac (2018a)). Magnetic flux conservation in MHD turbulence at large kinetic and magnetic Reynolds numbers, thus holds neither in the conventional sense nor is entirely broken. In ELV11 this stochastic flux-freezing relation was employed, together with GS95 predictions for turbulent Richardson dispersion, to rederive the predictions of LV99 for magnetic reconnection rates in the presence of MHD plasma turbulence. The detailed predictions depend upon the phenomenological theory of MHD turbulence which is adopted, but the qualitative features are independent of such assumptions and, in particular, the prediction that reconnection rates are independent of the precise values of microscopic resistivity or, similarly, of any effective “anomalous resistivity” arising from any other microscopic plasma processes. D.Field-Line Stochasticity and Slippage The original arguments of LV99 rested instead upon the random wandering of magnetic field-lines ξ(s;x)with increasing arclengthsas they move away from their initial positionx. Anticipating the concept of spontaneous stochasticity, LV99 realized that field-lines would disperse explosively, so that separations between neighboring field-linesξ(s;x),ξ(s;x ′ )would become independent of Turbulent Reconnection30 the initial separation|x ′ −x|with increasings. This phenomenon allows field-lines from widely separated space regions to wander into the same thin current sheet with thickness governed byη, even in the limit asη, ν→0.In this way, LV99 argued that violations of flux-freezing by tiny amounts of resistivity can be enhanced to macroscopic scales. Boozer (2018) has emphasized that similar wandering due to deterministic chaos of field-lines can lead to reconnection rates nearly independent of resistivity, but ubiquitous plasma turbulence can produce strictly fast reconnection with no dependence on resistivity whatsoever. We may note, however, a formal inconsistency in the argument of LV99 in the limit of vanishing resistivity. On the one hand, they used GS95 phenomenology of MHD turbulence to estimate the mean-square dispersion of field-lines as proportional toε(s/v A ) 3 ,whereεis the energy dissipation per mass andv A is the Alfv ́ en speed. On the other hand, GS95 theory predicts that the Ohmic electric field R= (η/c)∇×Bhas a magnitude R∼(η/c)B rms /(η 3 /ε) 1/4 ∝η 1/4 →0asη→0. Thus, Ohm’s law E+ (v/c)×B=Ris predicted by GS95 phenomenology to becomepointwise idealin the limitη, ν→0,with not only the width of current sheets but also the magnitude of the non-ideal Ohmic electric field vanishing in that limit. How can magnetic reconnection persist in the zero-resistivity limit when non-ideal electric fields vanish uniformly? One obvious answer is that GS95 theory ignores small-scale intermittency which can produce current sheets with thickness ∝ηand thus Ohmic fields that persist asη→0. However, a deeper solution to this puzzle lies in the recognition that magnetic reconnection is possible even at points where non-ideal electric fields tend to zero! A complete analysis of this problem was presented by Eyink (2015) who studied the rate at which magnetic-field lines “slip” through a non-ideal plasma, or, more accurately, the rate at which magnetic connections between plasma elements change due to non-ideality. As argued already by Axford (1984), such change of magnetic connections is the basal effect that underlies all other reconnection phenomena (diffusion of plasma through separatrices, topology change, conversion of magnetic energy to kinetic energy, etc.) This effect was quantified in Eyink (2015) by theslip velocity∆w ⊥ (s;x), which is the relative velocity between the magnetic field lineξ(s;x)and the fluid velocityvat a distancesin arclength away from the plasma elementxwhich that particular field-line threads. A simple equation was obtained in Eyink (2015) for the growth of slip-velocity ∆w ⊥ (s;x)in arclengthsalong the field-line: d ds ∆w ⊥ − [ (∇ ξ ˆ B) > −( ˆ B ˆ B)·(∇ ξ ˆ B) ] ·∆w ⊥ =− c(∇×R) ⊥ |B| .(66) where ˆ B(x,t) =B(x,t)/|B(x,t)|denotes the magnetic field direction and subscript “⊥” the vector component perpendicular to B.Because this linear ODE can have solutions∆w ⊥ (s;x)6=0 only if the righthand side is non-zero, the above expression indicates that flux-freezing holds if and only if(∇×R) ⊥ =0. In fact, the relation ˆ B×(∇×R) =0has long been known as the general condition for flux freezing (Newcomb, 1958). The quantityΣ=−c(∇×R) ⊥ /|B|represents the rate of development of slip-velocity per unit length of field-line and was dubbed in Eyink (2015) theslip-velocity source. This expression has a topological interpretation as well, arising from the relation∂ t ˆ B= (∂ t B) ⊥ /|B|(Jafari and Vishniac, 2019; Jafari and Vishniac, 2019). The key point in the resolution of the “vanishing Ohmic electric-field puzzle” is that, even if non- ideal electric fields R→0, the corresponding slip-velocity sourceΣcan remain non-vanishing or even diverge to infinity! For example, within GS95 phenomenology of MHD turbulence,Σ∼ η/(η 3 /ε) 1/2 ∝η −1/2 → ∞asη→0.Thus, the original argument of LV99 is self-consistent if GS95 phenomenology is employed throughout. More importantly, the above arguments show that the most intense current sheets due to small-scale intermittency arenotrequired to produce flux-freezing breakdown and that field-lines even far from those strong sheets will slip through the background turbulence at very sizable rates. The slip velocities will certainly be greatest at the most intense current sheets but, because of their rarity, more slippage may occur over time due to the weaker turbulence background. Eyink (2015) argued that such turbulent line-slippage accounts for observed deviations from the Parker spiral model in the solar wind. This line-slippage is the same phenomenon which was termed “reconnection diffusion” introduced in Lazarian (2005) and Turbulent Reconnection31 numerically tested in Santos-Limaet al.(2010). This process was invoked to explain the removal of magnetic flux in star-formation (see more in§IX B). One may roughly translate the quantitative arguments of Eyink (2015), discussed above, into a more qualitative statement by saying that if the magnetic field is not perfectly frozen into the turbulent fluid, i.e., in the presence of stochastic flux freezing, the field should slip through the fluid. The differential equation governing this field-fluid slippage is given by Eq.(66) with the source term (∇×B) ⊥ /B. Interestingly, it turns out that this term also acts as a source in the evolution equation of the unit vector ˆ B, which in turn implies that(∇×B) ⊥ /Bhas a topological interpretation. In fact, it is simple calculus to see that the derivative of the unit vector ˆ B=B/|B|is given by ∂ t ˆ B= ( ∂ t B B ) ⊥ , where(.) ⊥ denotes the perpendicular component with respect to B. The evolution equation of ˆ B, therefore, is easily obtained using the induction equation: ∂ t ˆ B− ∇×(u×B) ⊥ B =− (∇×B) ⊥ B .(67) Jafari and Vishniac (2019) used the relationship between the source term(∇×B) ⊥ /Band the field’s topology, implied by Eq.(67) above, in order to define a measure for the spatial complexity, or self-entanglement, of the magnetic field Bas S(t) = 1 2 (1− ˆ B l . ˆ B L ) rms ,(68) where ˆ B l =B l /B l and ˆ B L =B L /B L with B l and B L as the coarse-grained, or renormalized, components at arbitrary scalesland L. If we take Lof order of the system size andlLin the inertial range,B l (x,t)will be the average field of a fluid parcel of size∼lwhile B L (x,t)can be thought of as the average field of a fluid parcel of scale Ll. The former plays the role of a local field whereas the latter acts as a global field, therefore,1− ˆ B l . ˆ B L is in fact a local measure of the deviation of B l from B L . The RMS value of this quantity measures the global spatial complexity of the field (for a mathematical treatment of magnetic topology and complexity see Jafari and Vishniac (2019)). Magnetic reconnection, field-fluid slippage and magnetic topology change can be studied statistically using the function S(t)and its time derivative∂ t S(t)(Jafari and Vishniac, 2019; Jafari, Vishniac, and Vaikundaraman, 2019b). It also turns out that S(t)is related to Richardson diffusion of magnetic field in turbulence; see Jafari, Vishniac, and Vaikundaraman (2019a). E. Irrelevance of Small Scale Physics in Large Scale Turbulent Reconnection Although the previous arguments were presented in the context of resistive MHD, they are not restricted to plasmas with high collision rates but apply with equal force to inertial-range turbulence in collisionless plasmas, such as in the solar wind. The main difference is that in thegeneralized Ohm’s lawof a collisionless plasma, E+ 1 c v×B=R, the non-ideal electric field Rcontains many additional terms from the Hall effect, electron pressure anisotropy, electron inertia, etc. While these microscopic non-idealities play a crucial role in recon- nection of magnetic reversals near electron- and ion-scales, they are totally negligible and irrelevant to reconnection of structures at scales`ρ i ,the ion gyroradius. This has been demonstrated in detail, starting with either the generalized Ohm’s law (Eyink, 2015) or with the full Vlasov- Maxwell-Landau kinetic equations (Eyink and Drivas, 2018) In both cases, it was shown by a Favre coarse-graining analysis that the generalized Ohm’s law at scales> `takes the form ̃ E ` + 1 c ̃ (v×B) ` = ̃ R ` , Turbulent Reconnection32 and that all of the coarse-grained non-ideal terms on the righthand side are damped out by powers (δ i /`)and(δ i /`) 2 for`δ i ,the ion skin depth. The non-ideal electric field ̃ R ` is thus irrelevant in the renormalization-group sense and anideal Ohm’s lawholds in the coarse-grained sense ̃ E ` + 1 c ̃ (v×B) ` =0 at inertial-range length-scales`ρ i . Exactly as in our discussion of the MHD model, the validity of an ideal Ohm’s law in this gen- eralized sense does not imply that magnetic flux is frozen-in at inertial-range scales. Indeed, re- connection of magnetic structures many orders of magnitude larger than ion scales is observed in the turbulent solar wind (Gosling, 2012; Eyink, 2015) and such reconnection will be observed, obviously, whether ion-scale eddies are resolved or not! The principle of renormalization-group invariance implies once again that the rate of reconnection of such large-scale structures must be independent of the resolution scale`.Within the coarse-grained description, the ideal Ohm’s law at scales`ρ i can be rewritten as E ` + 1 c ̃ v ` ×B ` =−ε v ` −ε n ` where two apparently “non-ideal” terms now appear, one due to velocity fluctuations ε v ` = 1 c ̃τ ` (v × ,B) := 1 c [ ̃ (v×B) ` − ̃ v ` × ̃ B ` ] and the other to density fluctuations ε n ` = 1 n ` [ τ ` (n,E) + ( ̃ v ` /c)×τ ` (n,B)]. Herenandvare the number density and bulk velocity of the ions. Of course, these two terms are actually effects of the ideal Ohm’s law and together they account for all of the reconnection and breakdown of flux-freezing at length-scales`ρ i .Simple analytical estimates show indeed withε ` :=ε v ` +ε n ` that|ε ` |  | ̃ R ` |and also|(∇×ε ` ) ⊥ |  |(∇× ̃ R ` ) ⊥ |so that reconnection and line-slippage at scales`ρ i is due to ideal turbulence effects and not to microscopic plasma non-ideality. Just to be completely clear, this conclusion doesnotmean that large-scale turbulent reconnection in a weakly collisional plasma such as the solar wind will be described by ideal MHD equations. Although an ideal Ohm’s law holds, there are not enough collisions to make the ion distribution function close to a local Maxwellian and to make the ion pressure tensor an isotropic function of local density and temperature. Thus, to provide a detailed quantitative description of large-scale reconnection in the solar wind, one must resort either to the full Vlasov-Maxwell kinetic equations or to gyrofluid models that take advantage of the small ion gyroradius to develop closures for the ion pressure tensor. Ideal MHD should, however, be a suitable model to investigate some of the basic effects of turbulence at inertial-range scales. Since the energetically dominant wave component of solar wind turbulence consists of incompressible shear-Alfv ́ en modes, quantitative measures of field-line stochasticity and turbulent Richardson dispersion should carry over with little change to this collisionless plasma environment. F. Insight into physics of turbulent cascade The explanation of the violation of textbook concept of flux freezing provided above is related to the properties of turbulence first discovered by Onsager (1949). In his prophetic paper he wrote that ideal equations indeed hold in a turbulent inertial range but “the ordinary formulation of the laws of motion in terms of differential equations becomes inadequate and must be replaced by a more general description”. In other words, theideal equations(Euler, MHD, etc.) holds in the inertial range in ageneralized sense, but it does not hold in the standard, naive sense of partial differential Turbulent Reconnection33 equations. In particular, it does not hold in a sense that implies conclusions such as conservation of total kinetic energy for Euler or magnetic flux-freezing for ideal MHD. This profound insight by Lars Onsager is not properly appreciated by the community. It is fre- quently assumed that the modes at scale> Lshould satisfy the standard ideal equations, in the usual sense of partial differential equations, for about one eddy turnover time at that scale,t L L/u L , until small scalesl ν are created where viscosity/resistivity (or microscpic plasma effects in a collisionless plasma) become non-negligible. This is not right way of thinking. In a turbulent flow all scales in the range L > l > l ν are are already present at the initial instant. Low-pass filtering the fields to observe the scales> Ldoes not mean that that scales< Lhave been physically removed. Turbulent motions of the cascade are physically present and cannot be disregarded. For instance, in the solar wind at every instant of time there is a broad-band spectrum with excitation at all scales. If one wants to know what is happening at a given scalel > l ν low-pass filtering the fields at this scale is a purely passive operation of observing only the modes at length scales> land ignoring those at scales< l. But the modes at scales less thanlare still present and cannot be ignored. Therefore the dynamics at the sclaelis not directly affected by the viscosity/resistivity/etc. in the equations for scales larger thanl, but those equations are not the standard ideal equations. In the presentation above dealing with magnetic reconnection we wrote out these ideal equations in physical space as partial differential equations, but had to add the subgrid terms that arise from the modes at scales less thanl. Those subgrid terms are exactly what one gets by coarse-graining the ideal PDEs, i.e. one may say that the ideal equations hold in the coarse-grained sense. This is equivalent (as pointed out by Lars Onsager) to saying that Fourier coefficients of the fields satisfy ordinary differential equations that follow from the ideal PDEs. However, none of these equivalent notions of generalized solution have the usual implications of the ideal PDE’s, such as conservation of total kinetic energy or conservation of magnetic-flux. The necessity of usinga more general descriptionis obvious from a trivial hydrodynamic exam- ple. If the large scales> Lin hydrodynamic turbulenceobeyed Euler equationsin the usual, naive sense, then the kinetic energy at scales> Lwould be conserved in time. But it isnt true. This is the easiest way to see that obeying Euler equations does not mean in this context the usual, naive thing. It is exactly this experimental observation that large eddies decay away, at a rate independent of viscosity, that motivated Onsagers mathematical analysis. In other words, turbulent cascade connects the scales and it is because of the motions associated with this cascade it is not possible to consider motions at large scales as ideal. The real world is decribed at appropriate macroscopic scales by non-ideal fluid or kinetic PDEs with viscosity, resistivity, collisions, etc. If you resolve all motions of a turbulent flow down to those smallest scales of validity, you will see solutions of the non-ideal PDEs. All dissipation and reconnection arises from the nonideal terns within this fine-grained description. However, for many problems, including the problem of astrophysical magnetic reconnection, it is natural to consider only large scalesll ν . At these scales the direct effect of non-ideal, e.g. resistivity, Hall, etc. terms are negligible. However, “subgrid terms” should be accounted for. With these terms the equations are valid in thegeneralized senseand we explicitly accounted for these terms in our quantiatative discussion above. G. Rigorous description of flux freezing violation: summary Turbulent reconnection predicts reconnection taking place at all scales in the turbulent media and this entails flux freezing violation. This section provides a rigorous description of the magnetic field behavior in turbulent fluids and demonstrates clearly that for considering the reconnection at scale lone should account for the turbulent motions at the scale less thanl. This vividly demonstrates and quantifies the violation of the conventional flux freezing. In approach proves that the non-ideal plasma effects can be safely disregarded if magnetic reconnection is studied at scales much larger than the corresponding plasma scales. This section provides strong support for conclusion in LV99, resolves a number of problems re- lated to the earlier treatment of magnetic field in turbulent fluids and establishes solid mathematical foundations of the turbulent reconnection theory. It also clarifies the dynamics of magnetic field Turbulent Reconnection34 in turbulent media and suggests practical ways for evaluating the violation of the magnetic flux freezing. V. TESTING OF TURBULENT RECONNECTION AND FLUX FREEZING VIOLATION A. Testing LV99 predictions with MHD codes Numerical testings of reconnection in the presence of turbulence has a long history. Pioneer- ing two-dimensional studies in a periodic box were done by Matthaeus and Lamkin (1985, 1986). Although the turbulence was only injected initially and not driven, it persevered whole simulation time. The results indicated strong effect of turbulence presence on the reconnection process. For instance, the rates of production of reconnected magnetic islands were insensitive to resistivity. Un- fortunately, the numerical setup precluded calculation of the long-term reconnection speed and no studies were done on how the properties of turbulence itself affect reconnection. More recently, Wat- son, Oughton, and Craig (2007) studied the effects of small-scale turbulence on two-dimensional reconnection, however, no significant effects of turbulence on reconnection were observed. Some possible explanations for such effects was provided by the authors. Of the most important is that the perturbations were injected in the flow-driven merging of the super-Alfv ́ enic regime, resulting in a quick ejection of the fluctuations from the sheet and two-dimensional configuration limiting the degree of freedom of magnetic field lines. Dynamics of magnetic fields is three-dimensional by nature. Any two-dimensional configuration will significantly restrict the magnetic field freedom, making impossible, for example, the conven- tional magnetic field line wandering. Moreover, two-dimensional and three-dimensional magne- tized turbulence are very different in nature (see the discussion in Eyink et al. 2011). Nevertheless, due to the computational challenges and in the absence of any quantitative model to be tested, sim- ulations aimed at studying reconnection have been performed in two-dimensions for a long time. For instance, the two-dimensional study in Matthaeus and Lamkin (1986) stresses the importance of turbulence for modifying the character of magnetic reconnection and specifies heating and trans- port as the effect of particular significance, as well as formation of Petschek-type X-point in two- dimensional turbulence. Kim and Diamond (2001) showed that the transport of magnetic flux is not enhanced to the reconnection zone. At the same time, field wandering described in LV99 is the essential feature of three-dimensional reconnection. The dynamics of field lines is very different in two dimensions (see Eyink et al. 2011). Therefore 2D processes cannot provide us with the guide to what is going on in the real astrophysical situations. Loureiroet al.(2009) studied two-dimensional turbulent reconnection performing high-resolution simulations in a periodic box for Lundquist numbers up to5×10 4 concluding the existence of a critical threshold of the injection rate above which the reconnection rate is enhanced. Kulpa- Dybełet al.(2010) performed numerical simulations of the LV99 model in two dimensions showing that the relations between the reconnection speed and turbulence properties are different than those predicted by the LV99 model, and the reconnection is not fast in the presence of turbulence, since it still depends on the resistivity. In any case, it is obvious that the difference in physics of magnetic turbulence in 2D and 3D makes two dimentional studies a subject of pure academic interest. The actual testing of turbulent reconnection requires one to use the numerical set up of the realistic 3D set up. At the time of publication of the theoretical LV99 model of turbulent reconnection, it was not possible to verify its predictions immediately using numerical simulations. The model is three- dimensional intrinsically, requiring significant computational resources and requires good resolu- tion to resolve all scales from large scales where turbulence is injected to dissipation scales at which magnetic resistivity becomes important. Even though the relations between the reconnection rate and turbulence properties could be tested using relatively low resolutions, any attempt to test the most important prediction of the rate being independent on the magnetic dissipation would be ex- posed to immediate critics. Knowing that numerical simulations would not reach astrophysical Lundquist numbers for decades, it was challenging to undertake such a task. The first extensive testing of LV99 by means of numerical simulations was performed in Kowal Turbulent Reconnection35 FIG. 8. Dependence of the reconnection speed V rec on the injection power P inj (left) and the injection scale k inj (right). Different colors correspond to different turbulence driving methods (black - Fourier driving in velocity, blue and red - real space driving in velocity and magnetic field, respectively). The dotted line corre- sponds to the Sweet-Parker reconnection rate for models withη u = 10 −3 . A unique red symbol shows the reconnection rates for model with driving in velocity performed with higher resolution (512×1024×512) and resistivity coefficient reduced toη u = 5·10 −4 . Reprinted from Kowalet al.(2012) under CC BY license. FIG. 9. Dependence of the reconnection speed V rec on the resistivityηfor models with weak (diamonds) and strong (circles) guide field. From Kowalet al.(2009), c ©AAS. Reproduced with permission. et al.(2009), a decade after the publication of LV99. The tests were done using MHD compressible code in three-dimensional domain with open boundary conditions in the sub Alfv ́ enic turbulence regime. Note, that for simplicity the periodicity was applied along the guide field direction, i.e. Z-direction, but this does not affect our results, as there is no outflow/inflow along Z-direction. Additional tests in terms of different turbulence driving and exploring the parameter space have been performed in Kowalet al.(2012). These pioneering provided new numerical approaches to exploring 3D magnetic reconnection. In particular, the ways of measuring 3D reconnection rate were introduced there. The authors performed numerical models changing three basic parameters, the power P inj and scalek inj of driving, and the resistivity coefficientη. Figure 8 shows the measured dependencies of the reconnection speed V rec on the turbulence parameters P inj andk inj . Even though relatively low resolution was used, the authors confirmed theoretical LV99 model predictions of reconnection rate dependence on the turbulence parameters. Most importantly, however, the numerical studies of Kowalet al.(2009) have proven that the reconnection is indeed fast in the presence of turbulence, since it does not depend on the resistivityη, as shown in Figure 9. Moreover, the same dependence was observed for models with weak and strong guide fields B z . In Kowalet al.(2009), the authors also tested the role of the anomalous resistivity, and found that there is no essential dependence of the reconnection rate on anomalous effects, as well. Apart from the confirmation of these important Turbulent Reconnection36 FIG. 10. Stochastic trajectories that arrive at a fixed point in the archived MHD flow, color-coded red, green, and blue from earlier to later times. From Eyinket al.(2013). relations, Kowalet al.(2012) reported a weak dependence of the reconnection rate on viscosity, V rec ∼ν −1/4 , later studied in more details in Jafariet al.(2018) in the context of large magnetic Prandtl numbers. As expected in the LV99 model, the current sheet is broad with individual currents distributed widely within a three dimensional volume and the turbulence within the reconnection region is sim- ilar to the turbulence within a statistically homogeneous volume. The structure of the reconnection region was analyzed in more details by Vishniacet al.(2012) based on the numerical work by Kowal et al.(2009). The results supported LV99 assumptions about reconnection region being broad, the magnetic shear is more or less coincident with the outflow zone, and the turbulence within it is broadly similar to turbulence in a homogeneous system. B. Demonstration of flux freezing violation As we discussed earlier the LV99 idea of turbulent reconnection is closely related to the gross vi- olation of flux freezing in turbulent environments. As we discussed in§III B Richardson dispersion is a vivid manifestation of the flux freezing violation in turbulent fluids. Richardson dispersion in space was derived in LV99 and it was used for predicting the rate of turbulent reconnection. Figure 6 confirms the corresponding LV99 predictions. Indeed, it is clearly seen that over the inertial range the separation between following magnetic field lines is growing ass 3/2 , wheresis the distance calculated along the magnetic field line. A direct testing of the temporal Richardson dispersion of magnetic field-lines was performed recently in Eyinket al.(2013). For this experiment, stochastic fluid trajectories were tracked back- ward in time from a fixed point in order to determine which field lines at earlier times would arrive to that point and be resistively “glued together”. For the flux-freezing situation the one to one mapping is expected for different times, but the Richardson dispersion prescribes quite a different behavior. To test what is right, many time frames of an MHD simulation were stored so that equa- tions for the trajectories could be integrated backward. The results of this study are presented in Figure 10. It shows the trajectories of the arriving magnetic field- lines, which are clearly widely dispersed backward in time, more resembling a spreading plume of smoke than a single “frozen-in” line. The quantitative results of the time-dependent separation correspond well to the expectations for the temporal Richardson dispersion. This numerical experiment also demonstrates that magnetic reconnection happens at every scale of the cascade as we discussed in§IV F. Turbulent Reconnection37 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 05101520 0.01 0.02 234 5 6 7 0.01 0.02 0.03 1 FIG. 11. The evolution of the current layer width∆(bottom) and the inferred reconnection speed as a func- tion S(top left) and the ratio of B z0 /B y0 (top right). From Beresnyak (2017), c ©AAS. Reproduced with permission. C. Self-driven turbulent reconnection Although studies on reconnection with imposed driven turbulence have been done for a few decades already, the question of self-driven reconnection was approached by aims of numerical sim- ulations within last several years only. The question was first raised in LV99 and further discussed in Lazarian and Vishniac (2009). It was realized, that due to the existence of magnetohydrodynamic instabilities, turbulence could be generated directly by the inhomogeneous current sheet, leaving the imposed external turbulence obsolete. In the self-driven reconnection those inhomogeneouties of current sheet should grow, resulting in variations of the current sheet thickness and the size and strength of the outflow from the local reconnection regions, which interacting develop turbulence. While the reconnection feeds these interactions, so and the turbulence should growth. The devel- oped turbulence, in return, influences the reconnection process, resulting in its enhanced rate. The difficulty of studying self-driven reconnection comes from several factors. First of all, in order for small amplitude fluctuations to not simply propagate out of the box or dissipate, one has to use very small Ohmic resistivitesη10 −3 , corresponding to large Lundquist numbers,S10 3 . However, the thickness of current density scales as∼η 1/2 , e.g. in the Sweet-Parker model, indi- cating necessity of using very large resolutions in numerical models. For instance, for Lundquist number S=V A L/η∼10 6 , assuming V A ∼1and L∼1, the current sheet thickness would be around around10 −3 L, where Lis the longitudinal size if the current sheet. Resolving such a small thickness numerically requires at least resolutions of10 4 cells per unit length L. Moreover, the goal is to reach a steady-state of self-driven reconnection, which cannot be achieved in a periodic box, and one has to use numerical domains with open boundary conditions, which proper implementa- tion is far from trivial. The steady-state also cannot be achieved using very short runs, therefore, one has to simulate a model for at least a few Alfv ́ en times. Finally, one would like to study real- istic three-dimensional configurations, which for the estimated resolutions would require enormous computational resources. There are only a few most powerful supercomputers which could deal with such a huge task nowadays. The numerical simulations with turbulence generated through magnetic reconnection were pre- sented in preprint by Beresnyak (2013) with the final extended version of the paper published in Beresnyak (2017). The studies were done in a three-dimensional periodic box using spectral incompressible MHD simulations up to resolution1536 3 with the highest Lunquist number of S= 6.4×10 4 . The incompressible simulations used correspond the limiting case of high beta plasma, i.e. plasma with sound velocitythe Alfven one. While in 2D the location of the current sheet was remained in the same position in 3D the location of the original current sheet position was Turbulent Reconnection38 largely forgotten as turbulence was developing. The study reported magnetic reconnection that is very different from the magnetic reconnection studied earlier in 2D settings. First of all, the reconnection was observed to proceed at a steady rate, which is in contrast to the to 2D simulations (see Loureiroet al., 2012), exhibiting significant time dependence. The flux ropes were observed in 3D, but, unlike magnetic islands in 2D, the flux ropes were turbulent inside. The number of these structures did not depend on the Lunquist number, contrary to the tearing reconnection described e.g. in Uzdensky, Loureiro, and Schekochihin (2010). Clear turbulent structures the kinetic energy of the order of10 −6 of the magnetic one within several Alfv ́ en times (see Fig. 2 in Beresnyak, 2017). The reported reconnection rate was measured to be∼1.5%of the Alfv ́ en speed (see Fig. 11). The reconnection rate was derived as V rec = d∆/dt, where∆is the measured of the turbulent layer thickness and this result was justified using LV99 theory. 6 . Analysing his data Beresnyak (2017) concluded that the expression for the LV99 reconnection rate V rec =C LV δv 2 L /V A,x should have the constant C LV very close to unity, with C LV = 0.94being claimed. Beresnyak (2017) also studied the properties of generated turbulence, reporting turbulence con- sistent with the Goldreich-Sridhar (Goldreich and Sridhar, 1995), i.e. kinetic energy power slope ∼ −1.5÷−1.7, and scale-dependent anisotropyλ ‖ /λ ⊥ ∼λ −1/3 ⊥ , although, the inertial range of compatible anisotropy was relatively short. Later works include numerical simulations of stochastic reconnection done by Oishiet al.(2015) and Huang and Bhattacharjee (2016), both using compressible MHD approximation, in slightly dif- ferent reconnection configurations, however. Oishiet al.(2015) used a numerical setup starting from the Sweet–Parker setup with periodic box along the current sheet. This setup is somewhat unfortunate, since the Sweet–Parker configuration requires open boundary conditions to allow in- flow and outflow of the magnetic flux in order to maintain steady-state. Due to periodic boundary conditions, the growth of the fluctuations at large scales may be attributed to the presence of the magnetic field setup which is not initially in the equilibrium. Nevertheless, they performed several long time simulations with Lundquist numbers up to S∼3.2×10 6 with the conclusion that for sufficiently large Lundquist numbers S&10 5 , the thickness of current sheetδ SP becomes weakly dependent on S(see their Figure 1). However, it should be noted that simulations with S >10 5 (η <10 −5 ) where done with the maximum effective resolution1024 3 , much lower than the reso- lutions needed to resolve the thickness of current sheet estimated above, therefore, the conclusion might be attributed to the numerical effects. They also estimated the reconnection rate using the decay rate of the magnetic energy and concluding that 3D MHD stochastic reconnection can be fast without recourse to kinetic effects or external turbulence. They did not, however, provide any studied of the properties of reconnection-generated turbulence. Huang and Bhattacharjee (2016) analyzed the statistics of velocity fluctuations generated by re- connection also in the presence of the initial velocity noise, and used, as a starting point, a global Sweet–Parker configuration, but with non-periodic boundaries along the current sheet and a guide field comparable in strength to the reconnecting component, which additionally prevented the in- teractions of the waves. Weak velocity perturbations were injected into such a domain in order to understand their effect on the reconnection rate. The reported turbulent power spectra of kinetic energy with slopes−2.5÷−2.3and observed nearly scale independent anisotropy of velocity fluc- tuations, both in contrast with the predictions of the Goldreich-Sridhar theory. As pointed out later by Kowalet al.(2017), the power spectra and structure functions calculated in Huang and Bhat- tacharjee (2016) from two-dimensional XZ-planes and later averaged cause strong bias, resulting in steeper slopes and much reduced anisotropy, when compared to the fully three-dimensional spec- tral and structure function analysis. While the study in Huang and Bhattacharjee (2016) support the LV99 expectation to the transfer of reconnection to the turbulent regime, some statements of these study contradict to the findings in all other numerical studies by performed by different groups around the world that also studied 3D reconnection. In particular, the statement of the similarity of 2D and 3D magnetic reconnection in Huang and Bhattacharjee (2016) is at odds with the findings by other authors who report that in 3D 6 A somewhat different derivation of the growth of∆is obtained in Lazarian et al. (2015). Turbulent Reconnection39 −0.2−0.10.00.10.2 Z −0.4 −0.2 0.0 0.2 0.4 X ZX-plane −0.2−0.10.00.10.2 Z −0.4 −0.2 0.0 0.2 0.4 Y ZY-plane −0.4−0.20.00.20.4 X −0.4 −0.2 0.0 0.2 0.4 Y Time = 20.0 XY-plane 0 20 40 60 80 100 120 140 160 180 200 j ~ ! j FIG. 12. Slices along three main planes,ZX(left),ZY(center), and XY(right) of the amplitude of vorticity ω=∇×~vat timet= 20.0V A for a model with sound speeda= 1.0and grid sizeh= 1/1024. The turbulent region develops near the initial current sheet and expands in the Ydirection. From Kowalet al. (2017), c ©AAS. Reproduced with permission. 10 0 10 1 10 2 10 3 k 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 P ( k ) Kolmogorov t = 0. 0 t = 0. 5 t = 1. 0 t = 2. 0 t = 5. 0 t = 10. 0 t = 15. 0 t = 20. 0 FIG. 13. Evolution of power spectra of velocity at different times. From Kowalet al.(2017), c ©AAS. Reproduced with permission. the reconnection is evolving differently from that in 2D (see Oishiet al., 2015; Wang, Yokoyama, and Isobe, 2015; Strianiet al., 2016; Beresnyak, 2017; Kowalet al., 2017; Takamoto, 2018; Wang and Yokoyama, 2019; Kowalet al., 2019). This statement in Huang and Bhattacharjee (2016) as well as their finding that the spectral slope and anisotropy of the obtained turbulence are different from expectations of MHD turbulence theory (see§II B) contradict to the LV99 expectations. Below we provide the results of numerical testing of these claims in Huang and Bhattacharjee (2016). Below we discuss the most recent studies of the reconnection-driven turbulence in Kowalet al. (2017). As the initial setup they used a uniform box with a single discontinuity of the X-component of magnetic field placed in the middle of the box along the XZ plane. Although they used periodicity along the X and Z directions, the box was open in the vertical direction, perpendicular to the current sheet, and significantly elongated in Y, with physical dimensions of1.0×4.0×1.0in order to prevent any influence of the open boundary conditions. They let the turbulence grow from the initial noise of amplitude0.01applied to each velocity component for long runs up tot= 20t A . Figure 12 shows three main cuts across the domain of the vorticity amplitude at the final time of one of the investigated models. The choice of different vertical boundaries had some important consequences. Periodicity in all three directions with two separated current sheets imposed, as used in Beresnyak (2017) for in- stance, allows for the possibility of horizontal large scale interactions through the deformations of both current sheets. These interactions, especially at later times, can generate large scale motions Turbulent Reconnection40 10 -3 10 -2 10 -1 10 0 l ⟂ 10 -3 10 -2 10 -1 10 0 l ∥ h= 1=512 l ∥ = l ⟂ l ∥ /l 2=3 ⟂ a= 4:0 t= 0:5 t= 1:0 t= 2:0 t= 5:0 t= 10:0 FIG. 14. Anisotropy scalings for total velocity for models with sound speeda= 4obtained at different moments. For comparison, we show isotropic and strong turbulence scalings (denoted by gray thin lines). From Kowalet al.(2017), c ©AAS. Reproduced with permission. 10 2 10 1 10 0 10 1 10 2 10 3 10 4 Growth Rate ( A ) 10 0 10 1 10 2 10 3 10 4 10 5 Histogram k = 1 k = 10 k = 100 10 2 10 1 10 0 10 1 10 2 10 3 10 4 Growth Rate ( A ) 10 0 10 1 10 2 10 3 10 4 10 5 Histogram k = 1 k = 10 k = 100 FIG. 15. Distribution of the growth rate of the tearing mode (left) and Kelvin-Helmholtz instability (right) for reconnection-driven turbulence simulations by Kowalet al.(2019). Distributions with different colors correspond to three different perturbation wave numbers assumed (see legend). From Kowalet al.(2019). in the perpendicular direction to the current sheet providing additional to the reconnection energy input (see, e.g. Kowal, de Gouveia Dal Pino, and Lazarian, 2011, for a similar setup with many current sheets, in which the current sheet deformations are well manifested). In this case, such large-scale interactions are not allowed, therefore the only energy input comes from the small-scale reconnection events. They, contrary to other studies, investigated the dependence of the properties of developed turbulence on theβ-plasma parameter. They reported Kolmogorov-like power spectra of generated velocity fluctuations (see Fig. 13), which also presented scale-dependent Goldreich– Sridhar anisotropy scaling at later times in largeβmodels (shown in Fig. 14). They claimed that the turbulent statistics are similar to strong MHD turbulence, but are significantly affected by the dynamics of reconnection induced flows, more clearly manifested in the low-βregime, where su- personic reconnection outflows were present. In high-βregime the Goldreich–Sridhar anisotropy scaling manifested at much earlier times, due to the fluctuations generated by reconnection outflows propagating and interacting faster. They also estimated the reconnection rate to be twice as high as those estimated in Beresnyak (2017), which they justified by more diffusive code. Kowalet al.(2019) studied what exactly drives turbulence in their stochastic reconnection mod- els. They considered two instabilities, tearing mode and Kelvin-Helmholtz instability. For each Turbulent Reconnection41 10 6 10 5 10 4 10 3 10 2 10 1 10 0 S= v A 10 4 10 3 10 2 10 1 10 0 10 1 10 2 = B n / B = 10 4 = 10 5 = 10 6 10 0 10 1 10 2 10 3 FIG. 16. Distribution ofξ=B n /Bas a function ofv A δ. Since the specific Lundquist number is S δ = v A δ η , by assuming explicit resistivityηwe can find the stability region limit. Red lines correspond to three values of η= 10 −4 ,10 −5 ,10 −6 . The stable region lays above the red lines. From Kowalet al.(2019). analyzed model they determined the magnetic (in the case of tearing mode) and velocity (in the case of Kelvin-Helmholtz instability) shear location and performed detailed analysis of the field profiles taking into account local geometry, i.e. the local coordinate system oriented with respect to the shear plane. Using such analysis, they could determine a number of quantities important for the development of instabilities. For instance, in order to study the tearing mode, they determined the local thickness of the current sheet, the length of the local sheet, the strength of the transverse and guide components of magnetic field. For Kelvin-Helmholtz instability, they determined the shear amplitudes and the strength of the local magnetic field component which stabilized the instability, or sonic and Alfv ́ enic Mach numbers. These parameters allowed also to estimate the growth rates for both instabilities. The results indicated tearing mode growth is significantly suppressed at all studied wave numbers. See for example Figure 15, where statistical distribution of the growth rates for tearing mode (left panel) and Kelvin-Helmholtz instability (right panel) for three different perturbation wave numbers are shown. Strong suppression of tearing mode can be explained by the fact, that its growth rate actually decreases with the wave numberk, while for Kelvin-Helmholtz instability it increases. It could be also attributed to the presence of the transverse component of the magnetic field, easily generated by turbulence. Figure 16 shows distribution of the transverse component relative strength ξ=B n /Bvsv A δ. It is well seen, significant number of cells within current sheets are stable due to strong transverse component. It is evident that the study in Kowal et al. (2019) support the expectation of the turbulent re- connection theory and disagrees with some of the conclusions in Huang and Bhattacharjee (2016). In particular, the spectrum and anisotropy that are reported are consistent with the expectations of the MHD turbulence theory (see§II B). In addition, Kowal et al (2019) testifies that, in agreement with other earlier studies, the reconnection in 3D is different from that in 2D. In particular, the tear- ing modes that dominate the reconnection dynamics in 2D are replaced by perturbation induced by Kelvin-Helmholtz instability. In other words, the steady state self-driven reconnection is not driven by tearing/plasmoids. The issue why the results in Huang and Bhattacharjee (2016) are different re- quires further studies. One possibility is that the development of a true turbulent state requires more time that the current sheet was evolving in the simulations in Huang and Bhattacharjee (2016). Studies of 3D self-driven turbulent magnetic reconnection with different initial conditions is im- portant. In this respect, it is worth noting other studies of turbulence generating by stochastic recon- nection in different than the Harris current sheet configurations. For example, turbulence developed through reconnection in a three-dimensional cylindrical plasma column was investigated in Striani et al.(2016). An example visualization is shown in Figure 17. The authors report a fragmentation of the initial current sheet in small filaments, which leads to enhanced reconnection rate, independent Turbulent Reconnection42 FIG. 17.Left: Snapshot for cylindrical plasma colum with S= 2.4×10 4 att= 1.0t A showing the three-slice (pseudocolor rendering) of the density. Right: 2D slice on the Y Zplane (atx= 0)of the current density. Reprinted from Strianiet al.(2016) with permission of Oxford University Press. FIG. 18.Left: Temporal evolution of the volume averaged magnetic energy normalized to the initial magnetic energy for case PB-0 (red solid line), case PB-0.5 (dashed line) and case PB-0-NS (dotted line), along with case two-dimensional (black). For all cases the Lundquist number is S= 2.4×10 4 . The black vertical line separates phase I and phase II for case PB-2D.Right: Decay rateγof the magnetic energy as a function of S for case PB-0. For each Svalue,γis computed in phase I (circles) and phase II (triangles). Note that the decay rate follows the Sweet-Parker scaling during phase I, and is nearly independent on Sin phase II. Reprinted from Strianiet al.(2016) with permission of Oxford University Press. of the Lundquist number already at S∼10 3 . Moreover, they observe at the first phase, before the initial perturbation develop turbulence, the classical Sweet-Parker dependence of the reconnection rate on the resistivity. Once the turbulence grows, the reconnection rate increases quickly and be- comes independent of the Lundquist number (see Fig. 18). Apparently this study also shows that 2D and 3D self-driven reconnection are different. D. Testing turbulent reconnection with Hall-MHD code One may ask a question whether the magnetic reconnection that we described above for the MHD case both in the externally driven turbulence and in the case of self-driven turbulence is going to be different from the reconnection in actual plasmas. The LV99 theory gives a negative answer to this question and states that the microphysical plasma affects are not important at the scales much larger than the ion gyroradius or/and the ion skin depthd i =c/ω pi , whereω pi is the ion plasma frequency. This conclusion corresponds to both the common knowledge that the MHD turbulence properties that should not depend on the physics at the dissipation scale and to the accepted practise of representing magnetized flows on the scales much larger thand i using the MHD equations. Turbulent Reconnection43 FIG. 19. Normalized reconnection rate as a function of the reconnection region width. For relatively small width the reconneciton rate is∼0.1V A , but it decreases as the width increases. The value of0.015V A reported in the earlier MHD simulations by Beresnyak (2017) is expected for170d i . Reprinted from Beresnyak (2018) under CC BY license. However, the direct testing of the effects of plasma microphysics for magnetic reconnection on much larger scales is extremely challenging numerically. We address the approaches to this problem in the next section (§VI). In this section we discuss simplified ways to perform this testing. To test the LV99 prediction that plasma effects are not irrelevant for large scale turbulent recon- nection Kowalet al.(2009) performed the simulations of turbulent reconnection where the plasma effects were simulated through anomalous resistivity. No change of the reconnection was observed as the anomalous resistivity was varied. For the case of reconnection with self-generated turbulence Beresnyak (2018) performed Hall- MHD simulations with the resolution up to2304×4608 2 . The simulations had different ratios of the ion-electron depthd i to the grid sized, namely,d i /dwere chosen to be 9.5 and 39. The latter ratio approaches the ratio of the ion to electron skin depths, the former is used to obtain follow the scaling for larger current sheet widths. Figure 19 shows the dependence of the Hall-MHD reconnection ratev r as a function of the width Wof the reconnection layer. Similar to the earlier MHD reconnection simulations in Beresnyak (2017) the reconnection rate in the Hall-MHD simulations was measured as the increase of the reconnection layer thickness 7 v r = dw dt .(69) It can be shown that the differences between the set-ups with differentd i /ddisappear when the width of the reconnection layer gets larger than3.6d i . The current layer in the simulations was getting fully turbulent with magnetic fluctuations ex- hibiting whister turbulence (§VIII C) below thed i scale. The original rate of reconnection in the set up was larger that in the case of MHD self-excited reconnection. However, this rate was decreasing with the increase of the width of the reconnection layer. In other words, these findings support the notion that turbulent reconnection on the scales where one expects the MHD approximation to be valid (see a discussion of this in Eyink, Lazarian, and Vishniac, 2011), does not depend on the un- derlying microphysical plasma processes. This corresponds both to the physical intuition and to the accepted practice of using MHD for modeling the processes at scales much larger than the scales of the proton gyroradius. We note that while the main achievement of the study by Beresnyak (2018) was to demonstrate that the effects of Hall-MHD get negligible for turbulent reconnection at scalesd i , the study is 7 A quantitative study of this measure of reconnection for the simulations with periodic boundary conditions and its relation to the LV99 theory is provided by Lazarianet al.(2015) in section 4(d) there. Turbulent Reconnection44 also of importance for the reconnection at sufficiently small scales. Such type of reconnection can take place e.g. in the Earth magnetosphere and the study demonstrates how the turbulent plasma- scale reconnection gradually transfers into the reconnection that is determined entirely by the MHD turbulence. We, however, expect that the reconnection will depend on the plasmaβand the corre- sponding studies would be useful. E. MHD and Hall-MHD testing of turbulent reconnection: summary 3D magnetic turbulent reconnection is notoriously difficult to study. The initial studies employed the externally driven turbulence and confirmed the analytical dependences for the reconnection rate predicted in LV99. Similarly, the violation of the conventional flux freezing predicted by the turbu- lent reconnection theory was also demonstrated in simulations with externally driven turbulence. How reconnection proceeds in the situation when no turbulence is initially present is the focus of the ongoing research on the spontaneous 3D reconnection. This sort of reconnection is tradi- tionally studied in 2D resulted and such studies in the MHD limit established the tearing/plasmoid reconnection model. However, the 3D simulations clearly show that for sufficiently large Lundquist numbers the reconnection layer in 3D gets turbulent and this makes the reconnection in 2D and 3D are very different as it is established in many independent studies. 3D studies are computationally costly, but, unlike 2D ones, they are astrophysically relevant. Quantitative measurements of the properties of magnetic field in self-driven 3D reconnection are necessary. The first measurements of turbulence spectrum in the 3D reconnection layer, turbu- lence anisotropy, estimates of the growth rate of Kelvin-Helmholtz instability are very encouraging. These results support the predictions of turbulent reconnection theory. More studies of self-driven turbulent reconnection are necessary for exploring the phenomenon of flares induced by magnetic reconnection. The work that shows that 3D self-driven reconnection naturally transfers to turbulent regime testifies that the turbulent reconnection is a generic type of astrophysical large-scale reconnection. At the same time, tearing and various plasma effects can be important only in special circumstances, e.g. in the transient regime before the system gets turbulent. For studies of 3D reconnection it is important to have high resolution. Otherwise, the numerical viscosity can suppress turbulence, completely distorting the real picture of astrophysical reconnection. A recent study of the self-driven turbulent reconnection with the Hall term confirms the prediction of the turbulent reconnection theory that small-scale plasma effects do not change the rate of large scale reconnection in turbulent fluids. This is a confirmation of a practically important conclusion in LV99 that the large scale processes in turbulent fluids can be simulated using MHD codes. VI. TURBULENT RECONNECTION WITH KINETIC SIMULATIONS Kinetic simulations offer another perspective at understanding the role of turbulence on recon- nection. As energy cascades to smaller scales, the interactions between particles and fields could become increasingly important, including many collisionless processes. Fully kinetic simulations can track the motion of ions and electrons rigorously, this self-consistent “closure” is different from the resistivity and viscosity treatment often invoked in fluid studies. In fact, a direct comparison (Makwanaet al., 2015) between the compressible MHD and kinetic simulations of the turbulent cascade process has demonstrated the validity of using particle-in-cell (PIC) codes to study the nonlinear processes in the continuum limit at large scales. As described earlier in this review, it is instructive to quantify the field line diffusion behavior in kinetic simulations of reconnection. In addition to the often pre-existing turbulent background and the collisionless tearing driving scales, 3D large-scale kinetic reconnection process also contains several characteristic scales associated with ions and electrons that naturally lead kinetic instabilities on small scales (e.g. Bowers and Li, 2007; Daughtonet al., 2011; Guoet al., 2015), as well as hydrodynamic instabilities at large scales (see Kowalet al., 2019), can produce fluctuations on various scales, one might expect that the magnetic field diffusion at the microscopic plasma scales to Turbulent Reconnection45 FIG. 20. The global reconnection rate derived from 3D reconnection. The initial turbulence drastically shortens the early stage and causes the reconnection to start quickly. be more complicated than its fluid counterpart. Below we describe some initial results based on one 3D PIC simulation of magnetic reconnection. It is very difficult to get fully turbulent fluid behavior for the outflow from the reconnection region. Therefore in what follows the study is focused on exploring the separation of the field lines, which is related to the Richardson dispersion (see§III B). A. Numerical Method and Initial Set-up 3D kinetic simulations were carried out using the VPIC code (Bowerset al., 2008). VPIC is a first-principles electromagnetic charge-conserving code solving Maxwell’s equations and the Vlasov equation in a fully relativistic manner. It has been carefully designed and optimized on recent peta-scale computers, enables multi-dimensional simulations with more than 1 trillion parti- cles and 5 billion numerical cells. Using the newly implemented Trinity machine at LANL, we have successfully run simulations that contains 5.2 trillion particles with about 17 billion cells. The simulation presented here (Guoet al., 2019) starts from a force-free current sheet with B= B 0 tanh(z/λ)ˆx+B 0 sech(z/λ)ˆy, where B 0 is the strength of the reconnecting magnetic field andλ is the half-thickness of the current sheet. We assume a positron-electron pair plasma withm i /m e = 1. Initiallyλ= 12d e which implies that the growth rate for collisionless tearing is quite small. Here,d e =d i =c/ω pe =c/ √ 4πn i e 2 /m e is the electron/ion inertial length. The initial particle distributions are Maxwellian with uniform densityn 0 and temperature T i =T e =m e c 2 . The magnetization parameter isσ= (Ω ce /ω pe ) 2 = 100. The domain size is L x ×L z ×L y = 1000d e × 500d e ×500d e . The grid sizes are N x ×N z ×N y = 4096×2048×2048. The 150 particles per cell per species were used. For electric and magnetic fields, periodic boundaries along thexandy directions and perfectly conducting boundaries along thez-direction were employed. For particles, we employ periodic boundaries along thexandy-direction and reflecting boundaries along thez- direction. The simulation lastsω pe t≈1050, which is about the time for an Alfv ́ en wave to travel L x . The timescale to traverse the current sheet thickness, however,ω pe τ= 2λ/c= 24, is much Turbulent Reconnection46 FIG. 21. Total current intensity|J|evolution in the{x,z}plane, showing the formation and merging of flux ropes in 3D and development of additional thin sheets. In addition to the initially injected turbulence, additional turbulence is produced as the reconnection proceeds. shorter. Different from the earlier simulations, we inject an array of perturbations with different wave- lengths with the total summed amplitude(d B/B 0 ) 2 ∼0.1to initiate a background turbulence at the beginning of the simulation. (Injection of the initial perturbations follow the procedure described in Makwanaet al.(2015).) These initially injected perturbations have wavelengths longer than the initial thickness of the current sheet. B. Analysis Figure 20 depicts the overall global reconnection rate derived from this simulation, and Figure 21 shows several snapshots of the total current density|J|indicating the evolution of many flux ropes (in 3D) and the development of subsequent thin current sheets. The reconnection rate is evaluated using the method described in (Daughtonet al., 2014a), where the mixing of electrons originating from separate sides of the initial current layer is used as a proxy to rapidly identify the magnetic topology and track the evolution of magnetic flux. It was found that, while the distribution and dynamics of reconnection X-points are strongly modified by the injected fluctuations and self- generated fluctuations due to secondary instabilities, the peak rate is on the order of 0.1 times of the Alfv ́ en speed, consistent with previous 2D and 3D kinetic studies. The reconnection changes by about a factor of 2 from its peak to its final quasi-steady rate. To quantify the magnetic field diffusion during the reconnection, the following procedure was adopted: using an output at a particular time snapshot, the two field lines were traced by starting from a pair positions which are closely spaced (the typical initial separation iss 0 = 0.01d i ). The separation between thems(d i )as a function of field line path lengthl(d i )was calculated. Pairs at different initial distances (above and below) from the initial current sheet layer center are chosen randomly. Typically 1000 initial pairs to enhance the statistics were used. Figure 22 shows some sample magnetic field lines. There are 100 pairs within0.01d e of each other, whose starting loca- tions are randomly chosen within a2×2×2d e box that is2λabove the middle plane of the initial current sheet atx= 500,y= 250,z= 256d e . The time snapshot is att= 1280orω pe t= 126.3, which is in the early stage of reconnection (before the peak, cf. Figure 20). These field lines are in- tegrated for up to500d e in total length and we stop following them when they reach any simulation boundary. The initial turbulence is apparently causing a fraction of magnetic field lines to go cross the current sheet. This could have interesting implications for initiating reconnection. Figure 23 gives the details of magnetic field separation behavior. Overall, we find that magnetic Turbulent Reconnection47 246 248 250 252 254 256 258 260 475 480 485 490 495 500 505 510 z (d i ) x (d i ) t = 1280 FIG. 22. Field line behavior traced in 3D but projected on the{x,z}plane atω pe t= 136.3, before the reconnection rate reaches its peak. The initial locations are aroundx= 500,y= 250,z= 256d e . Some magnetic field lines have wandered through the current sheet, which is atz= 250±3d e (the light shaded area). 0.01 0.1 1 10 0.01 0.1 1 10 100 average separation s (d i ) path length l (d i ) Starting 2 λ above midplane t = 2560 t = 5680 t = 8640 0.01 0.1 1 10 0.01 0.1 1 10 100 average separation s (d i ) path length l (d i ) Starting 4 λ above midplane t = 2560 t = 5680 t = 8640 0.01 0.1 1 10 0.01 0.1 1 10 100 average separation s (d i ) path length l (d i ) Starting 2 λ above midplane t = 5680 0.09 * x 1.24 0.01 0.1 1 10 0.01 0.1 1 10 100 average separation s (d i ) path length l (d i ) Starting 4 λ above midplane t = 5680 0.08 * x 1.27 FIG. 23. Field line diffusion based on the 3D kinetic reconnection simulation. The top row shows the field line separation by starting from2λ(left) and4λ(right) above the initial current sheet central layer. The three curves in the plots are made using three time snapshots att= 2560,5680,8640, which correspond to ω pe t= 252.6,560.3,and852.5, respectively. They represent at-peak, post-peak and quasi-steady stages of reconnection (cf. Figure 20), respectively. The bottom row shows an approximate fit to the separation from two cases, depicting a power-law behavior over a part of the curves. field lines indeed separate from each other at a rate faster than the regular diffusion process. The slopes from the power-law portion cluster around1.24−1.35. In addition, we have chosen two initial heights (2λand4λ) for tracing field lines and their behavior is quite similar. Furthermore, such analyses were done using three different time snapshots capturing the at-peak and post-peak reconnection stages. Within the uncertainties of the fits, both the amplitudes and the exponents stay constant in time. Turbulent Reconnection48 The exponent ranging between1.24−1.35suggests that magnetic field lines exhibit super- diffusive behavior, consistent with the turbulent reconnection theory as discussed earlier. The rela- tively narrow range of this exponent also suggests that the initially injected turbulence might have strongly regulated the diffusion behavior, though the turbulence produced by 3D kinetic reconnec- tion could have impacted the diffusive dynamics. More detailed analyses are needed to differentiate whether the injected and self-generated turbulence could lead to different field line diffusion behav- ior and how they might interact with each other. C. Kinetic simulations of turbulent reconnection: summary By analyzing the magnetic field line behavior based on a large 3D kinetic simulation of recon- nection that is subject to an initial turbulence injection, it was found that: First, the overall global reconnection rate reaches0.1Aflv ́ en speed, similar to previous kinetic simulations without the ini- tial turbulence injection. This is perhaps not surprising, given that the kinetic physics is probably still dominating the dynamics despite the relatively large simulation box. Second, the presence of the initial turbulence, however, drastically shortens the stage through which the peak reconnec- tion rate is achieved. This suggests the turbulence can accelerate the “triggering” process. Third, magnetic field lines exhibit super-diffusive behavior throughout the reconnection process, generally consistent with the super-diffusion theory (LV99, Eyink et al. 2011, Lazarian & Yan 2014). as well as previous 3D MHD reconnection studies (see (see Kowal et al. 2017). More studies are needed to better understand the detailed dynamics as well as to delineate any possible different roles played by the initial background turbulence versus the self-generated turbulence via reconnection. VII. OBSERVATIONS OF TURBULENT RECONNECTION Magnetic reconnection is notoriously difficult to study directly. In most cases one can see the consequences of the reconnection, e.g. heat release or energetic particles accelerated, rather than the actual picture of reconnection. For turbulent large scale reconnection the situation is really dis- advantageous, as most of the in situ magnetic reconnection measurements have been done for the Earth magnetosphere where the thickness of the current sheets is comparable to the ion Larmor radius. In such settings one does not observe MHD-type turbulence and therefore magnetospheric measurements provide information about plasma reconnection dominating small scales rather than large scale reconnection relevant to most of astrophysical settings. Nevertheless, there have been both observations and measurements in the relevant parameter regime and those support the turbu- lent reconnection model. A. Solar Flares Turbulent reconnection provides the most natural explanation for the variations of solar activity. Indeed, the reconnection speed that can be inferred from the observations of the magnetic activity in the solar atmosphere varies dramatically. This is difficult to account on the basis of any plasma reconnection process that prescribes a given reconnection rate. On the contrary, as we discussed earlier, the LV99 predictions suggest that the rate of magnetic reconnection vary with the level of turbulence and this level of turbulence can be affected by the reconnection itself. The latter provides a natural explanation of solar flares (LV99, Lazarian and Vishniac, 2009). One can estimate the reconnection rates for solar reconnection assuming that turbulent velocity dispersion U obs,turb measured during the reconnection events is due to the strong MHD turbulence regime. This is reasonable as we expect strong bending of magnetic field lines in the reconnection region. Therefore: V rec ≈U obs,turb (L inj /L x ) 1/2 ,(70) Turbulent Reconnection49 Similarly, the thickness of the reconnection layer should be defined as ∆≈L x (U obs,turb /V A )(L inj /L x ) 1/2 .(71) For solar atmospheres the expected ratio of∆to the proton gyroradiusρ i is10 6 or even larger. Therefore we expect to deal with the LV99 type of reconnection. The expressions given by Eqs. (70) and (71) should be compared with observations. Using the corresponding data in Ciaravella and Raymond (2008) one can obtain the widths of the reconnection regions in the range from 0.08L x up to 0.16L x and the observed Doppler velocities in the units of V A of the order of 0.1 ((see more in Lazarian et al. 2015). It is easy to see that these values agree well with the predictions given by Eq. (71). In fact, the corresponding comparison was first done in Ciaravella and Raymond (2008) in order to observationally test the LV99 theory. However, the authors assumed that turbulence was isotropically driven, which was unrealistic for the solar flare case. While Ciaravella and Raymond (2008) provide, within the observational uncertanities, a satisfactory quantitative agreement between their observations and the LV99 theory prediction, it is demonstrated in Lazarian et al. (2015) that this agreement can be further improved if the actual anisotropic driving is accounted for. A more recent study exploring the growth of the thickness of the reconnection layer of the mi- croflaring loop is presented in Chitta and Lazarian (2019). This study shows that the turbulent velocity measured using Doppler-broadened lines and the width of the reconnection region are in a good agreement with the predictions of the LV99 theory. A very distinct prediction in the LV99 theory is the spread of magnetic reconnection from an active reconnection region to the adjacent reconnection regions. Indeed, the turbulence induced by the reconnection in one flaring region is expected to destabilize the adjacent regions that are undergoing relatively slow reconnection and induce flares of reconnection in them. It is evident that one should not expect any similar effects in the models of plasma reconnection. In Sychet al. (2009) and Sychet al.(2015) the authors observed the effect similar to one we just described. They explaining quasi-periodic pulsations in observed flaring energy releases at an active region above the sunspot and proposed that the wave packets arising from the sunspots can trigger such pulsations via the LV99 mechanism. Plasmoids associated with tearing reconnection has also been observed (Shibata and Tanuma, 2001) and on the basis of our experience with self-driven turbulent reconnection that we described in§... we associate the plasmoid stage with the onset of turbulent reconnection. B. Reconnection in Solar Wind Reconnection throughout most of the heliosphere is expected to be similar to that in the Sun and therefore we do expect that the LV99 theory can be applicable to it. However, for the measurement interpretation one should treat the small scale and large scale reconnection events separately. For example, there are now extensive observations of strong narrow current sheets in the solar wind (Gosling, 2012). The most intense current sheets observed in the solar wind have widths of a few tens of the proton inertial lengthd i or proton gyroradiusρ i (whichever is larger). The reconnection associated with these small-scale current sheets of exibits exhausts with widths of a few hundreds of ion inertial lengths (Goslinget al., 2007; Gosling and Szabo, 2008). Naturally, such small-scale reconnection events require collisionless physics for their description (Vasquezet al., 2007). Within the LV99 picture these are small scale events representing the microphysics, but not much affecting the large scale reconnection. At the same time there are very large-scale reconnection events in the solar wind, often associated with interplanetary coronal mass ejections and magnetic clouds or occasionally magnetic discon- nection events at the heliospheric current sheet (Phan, Gosling, and Davis, 2009; Gosling, 2012). These events have reconnection outflows with widths up to nearly10 5 of the iond i and exhibit a prolonged, quasi-stationary regime with reconnection lasting for several hours. Such large-scale reconnection is expected within the LV99 theory when very large flux-structures with oppositely- directed components of magnetic field interact with each other in the turbulent environment of the Turbulent Reconnection50 FIG. 24.(Upper panels)Reconnection event in 3D MHD turbulence.(Lower panels). Reconnection event in High-Speed Solar wind. The left panels show magnetic field components and the right panels velocity components. Both velocities and magnetic components are rotated into a local minimum-variance frame of the magnetic field. Red denotes the component of maximum variance. It is the reconnecting component. The component of medium variance in green corresponds to guide-field direction. The minimum-variance direction depicted in blue is perpendicular to the reconnection layer. From Lalescuet al.(2015). solar wind. The “current sheet” producing such large-scale reconnection in the LV99 theory con- tains itself many ion-scale, intense current sheets embedded in a diffuse turbulent background of weaker current. The comparison between the reconnection events in High-Speed Solar Wind and those in MHD turbulence simulations was performed in Lalescuet al.(2015). In the latter simulations the mag- netic field undergo Richardson diffusion as proven in Eyinket al.(2013) and therefore their tur- bulent reconnection corresponds to LV99 theory. Figure 24 shows that an inertial-range magnetic reconnection event in an MHD simulation is similar to those reconnection events observed in Solar Wind. The authors observed stochastic wandering of magnetic field lines in space, breakdown of conventional flux-freezing principle as a result Richardson dispersion of magnetic field lines, and also a broadened reconnection region containing several current sheets. The coarse-grain magnetic geometry in this study shown in Figure 25 appears as a large scale reconnection event in the solar wind, while the detailed picture of magnetic field lines exhibits a much more stochastic turbulent nature of the event. Lalescuet al.(2015) concluded that the breakdown of conventional magnetic flux-freezing as an evidence of reconnection and also verified topology change in both fine-grained and coarse-grained magnetic fields. We believe that further studies of solar wind reconnection is a good way to test LV99 model as well as the related theories of flux freezing violation, slippage of magnetic field lines that we discussed in the review. In view of that we note, that LV99 better applicable to reconnection events Turbulent Reconnection51 FIG. 25.Left: Magnetic field iso-surface at half-maximum value 1.11 in given in yellow. B-lines sampled along the direction of minimal variance are given in green and in the directions of maximal variance are given in red. Right: Same as figure on the left, but in terms of the averaged magnetic field. The blue vectors on the field-lines are velocities measured in the local magnetic field frame. From Lalescuet al.(2015). more distant from the Sun, because of the scales of the events increase. For instance, reconnecting flux structures in the inner heliosheath could have sizes up to∼100 AU, much larger than the ion cyclotron radius∼10 3 km (see Lazarian and Opher, 2009). C. Heliospheric Current Sheet and Parker Spiral The violation of flux freezing required by the model of turbulent reconnection can be also directly observable. For instance, Eyink (2015) analyzed the data relevant to the region associated with the broadened heliospheric current sheet (HCS), noticed its turbulent nature and argued that LV99 magnetic reconnection can explain the properties of the HCS. Naturally, more dedicated studies of the phenomenon are necessary. The Parker’s spiral model (Parker, 1958) of the interplanetary magnetic field remains as one of the applications of the magnetic flux-freezing. The deviations from this model have been observed, however. Using yearly averaged magnetic field strengths calculated from observations of Voyager 1 and 2, Burlagaet al.(2002) found a mean deviation of−2%between the observations and the Parker model. Yearly deviations were around±20%with the maximum error (of about−40%). In addition to magnetic field strengths studied by Burlagaet al.(2002), an earlier work by Burlaga et al.(1982) considered also the magnetic field geometry which revealed notable deviations from the spiral model. This work found a different radial dependence for the radial and azimuthal magnetic field component from that predicted by the Parker model. These observations were based on the data collected by Voyager 1 and 2, while a comparison with earlier Pioneer 10 and 11 investigations during the less active solar activity period (1972through1976) confirmed the spiral model modulo variability on timescales shorter than a solar rotation period. These early observations were recently confirmed by Khabarova and Obridko (2012). The exten- sive averaging used in the latter work eliminated the sizable fluctuations still observed in the daily averages of Burlagaet al.(1982); see their Fig.1. Khabarova and Obridko (2012) interpret their observations as due to a quasi-continuous magnetic reconnection, occurring both at the heliospheric current sheet and at local current sheets inside the IMF sectors. They present extensive evidence that most nulls of radial and azimuthal magnetic field components (i.e., potential reconnection sites) are not associated to the heliospheric current sheet. This interpretation is consistent with the results of Eyink (2015) as magnetic slippage through the fluid as a result of pervasive turbulence in the near-ecliptic solar wind leads to a less tightly wound spiral and a stronger radial field strength than in the Parker model, as observed by Khabarova and Obridko (2012). This is intimately related to the Turbulent Reconnection52 reconnection diffusion proposed by Lazarian (2005) and first studied numerically by Santos-Lima et al.(2010) (see more in§IX B). In addition, Eyink (2015) analyzed the data relevant to the region associated with the broadened heliospheric current sheet (HCS), noticed its turbulent nature and provided arguments on the appli- cability of LV99 magnetic reconnection model to HCS. This seems to be a very promising direction of research to study turbulent reconnection in action using in situ spacecraft measurements. D. Observational testing turbulent reconnection: summary Quantitative studies of magnetic reconnection from observations and in-situ spacecraft measure- ments are notoriously difficult. However, in the last 20 years a number of predictions of the turbu- lent reconnection theory has been successfully tested. For instance, the prediction of the relation between the level of turbulence and the width of the reconnection layer was confirmed with so- lar observations, the deviations of the Parker spiral and heliospheric current sheet were explained with the turbulent reconnection theory. Very importantly, the predicted effect of inducing mag- netic reconnection by a Alfvenic perturbations coming from the neighoring reconnection event was observed. The quantitative studies of the reconnection in Solar Wind showed that the reconnection events identified there have direct analogs in the reconnection events observed in numerical simuations of MHD turbulence. We believe that Solar Wind studies can provide the best testing ground for quantitative comparing the prediction of the theory of turbulent reconnection with the in situ mea- surements of reconnection. At the same time, the magnetospheric reconnection events cannot serve the same purpose as they happen in different regime that is dominated by plasma effects. The ex- isting theory of turbulent reconnection cannot be directly applied for describing such reconnection events. VIII. TURBULENT RECONNECTION IN SPECIAL CONDITIONS A. Stochastic Reconnection For Large Pr m Partially ionized, turbulent regions of the Interstellar Medium (ISM) are often associated with a high magnetic Prandtl number Pr m (see e.g. Xu and Lazarian, 2017b). This regime of large Pr m is also prevalent in the laboratory fusion plasmas, different phases of star formation as well as in galaxies, protogalaxies and clusters which often are filled with hot and rarefied plasmas (Jafari et al., 2018). Lazarian, Vishniac, and Cho (2004) proposed a model for magnetic reconnection in partially ionized gases extending the arguments of stochastic reconnection (LV99) which was concerned with Pr m '1. Another attempt, to study magnetic reconnection in large Pr m regime, was made by Lazarianet al.(2015) who considered the role of viscosity in the diffusion of field lines caused by neutrals in a partially ionized gas. To reach a better agreement with numerical simulations (Kowalet al., 2009, 2012), in a more recent work Jafariet al.(2018) have considered the effects of viscosity on fluid motions along magnetic field lines which is basically a generalization of stochastic reconnection (LV99) for viscous fluids. Stochastic reconnection was proposed in LV99 for highβplasmas with a magnetic Prandtl num- ber of order unity,Pr m '1. It predicts reconnection speeds of order the large scale turbulent eddy velocity. A modified version of stochastic model for Pr m >1in highβplasmas, developed by Jafariet al.(2018), showed that the width of the matter outflow layer and the ejection velocity are both unaffected by viscosity. Nevertheless, if Pr m >1viscosity may suppress small scale reversals near and below the viscous damping scale. In that case, there will be a threshold for the suppression of large scale reconnection by viscosity when the Prandtl numbers is larger than the root of the Reynolds number, that is Pr m > √ Re. For Pr m >1this leads to the spectral index∼ −4/3for length scales between the viscous dissipation scale and eddies larger by roughly Pr 3/2 m . Here, we briefly review this regime of large magnetic Prandtl number studied in Jafariet al.(2018) (see also Jafari and Vishniac, 2018a). Turbulent Reconnection53 The parallel wavelength at the viscous damping scale,λ ‖,d =k −1 ‖,d can be obtained noting that k −1 ⊥,d ∼l ⊥ (ν/l ⊥ u T ) 3/4 . The latter vertical damping scale comes from the Kolmogorov scaling for the viscous damping length scale,l d ∼ −1/4 ν 3/4 with'u 3 T /l ⊥ 'u 2 T V A /l ‖ where we have also usedl ⊥ V A =l ‖ u T . If we define the injection velocity asu L = √ u T V A , then we can write 'u 2 T V A /l ‖ =u 4 L /(l ‖ V A ). Analytical (see e.g., Lazarian, Vishniac, and Cho (2004); Jafariet al.(2018)) and numerical (Cho, Lazarian, and Vishniac (2002b); Cho, Lazarian, and Vishniac (2003a); Schekochihinet al.(2005)) work indicate that a Prandtl number larger than unity will affect MHD turbulence in major ways. In MHD turbulence, there are two dissipation scales; the viscous damping scalel d and the magnetic diffusion scale,l m . In fully ionized collisionless plasmas, both of these dissipation scales are very small. In partially ionized plasmas, they can be very different but always we havel d l m ([Jafari et al.(2018)]; [Lazarian, Vishniac, and Cho (2004)]). The kinetic energy is dissipated at viscous damping scale,l d ∼ν 3/4  −1/4 , l d ∼l ⊥ ( ν l ⊥ u T ) 3/4 ,(72) where we have used the sub-Alfv ́ enic energy transfer rate∼u 4 L /(l ‖ V A )'u 2 T V A /l ‖ (u L = √ u T V A ) with critical balance condition (Goldreich and Sridhar (1995); Lazarian and Vishniac (1999); Jafariet al.(2018))u T l ‖ 'l ⊥ V A . However, at this scale magnetic energy does not dissipate as resistivity is smaller than viscosity by assumption. Instead, the magnetic damping scale is given by l m ∼l ⊥ ( η l ⊥ u T ) 3/4 .(73) The width of the reconnection layers, created by the viscosity damped turbulence, can be de- termined using the eddy size at the viscous damping scalel d . Consequently, the width of such a reconnection layer would be roughlyk −1 ⊥,d =l d =λ ⊥,d . The corresponding parallel length at the damping scale,λ ‖, d , can be obtained as λ ‖,d 'l 1/2 ‖ (u T V A ) −1/4 ( V A u T ) 3/4 ν 1/2 .(74) Belowl d , the energy cascade rateτ −1 cas is scale independent, and thus the constant magnetic energy transfer rateb 2 l τ −1 cas leads tob l ∼const.; E b (k)∼k −1 .(75) The above expression is the magnetic power spectrum by the assumption that the curvature of field lines is almost constantk ‖ ∼const.. This is consistent with a cascade driven by repeated shearing at the same large scale. Also, the component of the wave-vector parallel to the perturbed field does not change much and therefore any change inkcomes from the third direction (Jafariet al., 2018). To find the kinetic spectrum, we may use the filling factorφ l as the fraction of the volume that contains strong field perturbations with a scalek −1 (Lazarian, Vishniac, and Cho, 2004). Assuming a constant filling factor,φ l ∼const., the kinetic spectrum is given by E v (k)∼k −5 . Otherwise, we find E v (k)∼k −4 (Jafariet al., 2018; Jafari and Vishniac, 2018a). To find the reconnection speed in large magnetic Prandtl number regime, Jafariet al.(2018) used the following equation for the diffusion of field lines (see also Jafari and Vishniac, 2018a): dy dx ' η/V A y + y ay 2 +by 2/3 +c √ ν .(76) wherea= (l ‖ /l 2 ⊥ ),b=l 1/3 (V A /u T ) 2/3 andc=l 1/2 (u T V A ) −1/4 (V A /u T ) 3/4 and finallydx= V A dt. In order to compare to numerical data, we can use eq.(76) with calibrated coefficients Turbulent Reconnection54 FIG. 26. Separation of magnetic field lines of Kowalet al.(2012) compared with the analytical solution of eq.(76) presented by the red line. The magnetic Prandtl number is of order10Jafariet al.(2018), c ©AAS. Reproduced with permission. a=a 1 l ‖ /l 2 ⊥ ,b=b 1 l 1/3 ‖ (V A /u T ) 2/3 andc=c 1 l 1/2 ‖ (u T V A ) −1/4 (V A /u T ) 3/4 . The numerical factorsa 1 = 1.32,b 1 = 0.36andc 1 = 0.41are constants introduced to match the solution to the available numerical data obtained by Kowalet al.(2012) in which the injection power P=u 2 T V A /l ‖ and the Alfv ́ en speed V A are set to unity and the injection wavelengthk inj =l −1 ‖ = 8. For these simulations, numerically, we havea=a 1 /4andb=b 1 andc=c 1 . Fig.(26) shows a com- parison of the solution of eq.(76) with the numerical data obtained by Kowalet al.(2012). As mentioned before, we combine the solutiony=y(t)of eq.(76) with the conservation of mass; V R =V eject (〈y 2 〉) 1/2 /L x where V eject is the ejection speed of plasma out of the reconnection zone. For Pr m <1, one expects V eject ∼V A . However, for Pr m >1, we may expect that the outflow is affected by viscosity if the viscous timescale is shorter than the outflow timescale,〈y 2 〉/ν < L x /V eject . Using〈y 2 〉/l 2 ⊥ ∼L x /l ‖ ,V A l ⊥ 'l ‖ u T and defining the Reynolds number as Re= l ⊥ u T /νwhereu T is the velocity characteristic of largest strongly turbulent eddies, one finds Re < V A V eject .(77) This condition can hardly be satisfied, hence viscosity is not expected to affect the outflow speed (Jafariet al., 2018). But it can still affect the reconnection rate by changing the outflow width. In fact, viscosity can affect reconnection process either by hindering the outflow from the reconnection zone or by narrowing the reconnection channel width (Jafari and Vishniac, 2018a). If we assume that the perpendicular separation is completely inside the turbulence inertial cascade, then viscosity cannot not play any important role. That is ify > l d even forx=L x , viscosity’s effect will be negligible. In the dissipative range, the diffusion equation (76) has the solution 〈y 2 〉' ( η V A ) λ ‖,d exp ( L x λ ‖,d ) .(78) The conditiony.l d then becomes Re 1/2 < ( l ‖ L x )[ 1 + ln(Pr m ) ] .(79) where we have used Re=l ⊥ u T /ν. Consequently, the outflow width,y=〈y 2 〉 1/2 , is unaffected unless the magnetic Prandtl number is exponentially larger than the Reynolds number. For Pr m 1, one generally expects that viscosity will slow down the reconnection speed. Fig. 26 compares the general solution of eq.(76) with the numerical simulations performed by Kowal et al.(2009) and Kowalet al.(2012). Below the damping scale, eq.(76) yields y 2 ∼ η V A λ ‖,d [ exp (x/λ ‖,d )−1 ] ,(80) Turbulent Reconnection55 FIG. 27. The rms and median separation of magnetic field lines plotted versus the distance parallel to the mean field in the viscosity damped turbulence using a384 3 grid by Lazarian, Vishniac, and Cho (2004). The median tracks the mean at a slightly smaller amplitude consistent with the exponential growth. This exponential growth in the dissipative regime can also be seen looking at the analytic result given by eq.(80). From Lazarian, Vishniac, and Cho (2004), c ©AAS. Reproduced with permission. which, forxλ ‖,d , becomes Lyapunov exponential growth whose exponent is inversely propor- tional to the square root of the viscosity. Fig. 27 illustrates this exponential growth of field line separation in a set of numerical simulations preformed by Lazarian, Vishniac, and Cho (2004). Another way that viscosity may affect reconnection is through changing the local small scale reversals. It is a very stringent condition for viscosity to affect the large scale outflow. This may imply that viscosity will almost never be important, thus the question arises that if the small scale effect of viscosity can be significant. Another effect of viscosity is associated with topologically distinct local reconnections at small scales. In stochastic reconnection, the global reconnection speed is the collective effect of numer- ous local reconnection events which occur simultaneously. This sets an upper limit on the global reconnection speed Lazarian and Vishniac (1999): V rec < N eddy v rec,eddy ,(81) wherev rec,eddy is the reconnection speed within an individual eddy and N eddy is the number of independent eddies along a field line. Jafariet al.(2018) obtained the following condition for viscosity to have an appreciable effect on local reversals: V rec < v rec,eddy ( η ν ) 1/2 N eddy 'u T Re 1/4 Pr −1/2 m 1 + ln(Pr m ) L x l ‖ .(82) The implication is that for Pr m = 1, the above condition is not an interesting limit. however, for Renot very large and resistivity appreciably less than viscosity, it can be the controlling limit on reconnection (and not the outflow zone width). In particular the dividing line between viscosity dominated reconnection and unimpeded reconnection is no longer overwhelmingly in favor of unim- peded reconnection. Therefore, the critical magnetic Prandtl number does not scale exponentially with the Reynolds number but close to its square root. The simple model outlined above, however, is based on the assumption of a large degree of intermittency below the Kolmogorov scale. This may not be very realistic. In addition, numerical simulations of the viscous regime have currently low resolutions and their results are affected by a damping scale very close to the eddy scale. The small Reynolds numbers accessible by numerical simulations at present may be the reason why they hardly show any effect associated with viscosity (Jafariet al., 2018). The magnetic structures are sheared below the viscous damping scale with a scale independent energy cascade rate and we find a power spectrum for the magnetic field as E b (k)∼k −1 : eq.(75). It follows that more power concentrates on the small scales than predicted by the Goldreich-Sridhar Turbulent Reconnection56 FIG. 28.Magnetic fluctuation spectrum E(k)k Jafari and Vishniac (2018a): Below viscous damping scale, the hydrody- namic motions are suppressed and we have a flat spectrum for magnetic fluctuations. The width of the current sheet is con- trolled by viscosity in this regime. Above the viscous damping scale, and below the scale set by(damping scale)×Pr 3/2 m , reconnection is inhibited while diffusion is enhanced by magnetic tangles. spectrum (Goldreich and Sridhar, 1995). With a viscosity of order, or slightly larger than the re- sistivity, the the outflow width remains independent of the small scale physics, which is similar to the LV99 model. Even with viscosity appreciably larger than resistivity, the width of the outflow region is unaffected unless the magnetic Prandtl number is exponentially larger than the Reynolds number. The viscosity is even less important when the current sheet instabilities dominate over external turbulence. For significantly large Prandtl numbers, i.e., of order √ Re, the magnetic field perturbations cannot relax and a flat magnetic power spectrum extends below the viscous damping scale; see Fig.28. B. Turbulent reconnection in relativistic fluid The relativistic turbulent simulations in Takamoto, Inoue, and Lazarian (2015) (henceforth TIL15) confirmed a general similarity between the relativistic reconnection and the non-relativistic one. They demonstrated that the turbulent reconnection speed can be as large as0.3c, which thus en- ables highly efficient conversion of magnetic energy into kinetic motions and particle acceleration. The numerical results in TIL15 can be understood on the basis of the expression LV99 expression for turbulent reconnection. This expression can be written as V rec ≈V A min [ ( L x l ) 1/2 , ( l L x ) 1/2 ] ( u L V A ) 2 ,(83) where for sub-Alfvenic turbulent the energy cascading is given by expression (LV99):  inj ≈u 4 L /l V A .(84) However, for the relativistic case one should take into account both the density change in the rel- ativistic plasma and the modification of turbulence properties in the relativistic regime. The former can be obtained from the energy flux conservation requirement. With both analytical considerations Turbulent Reconnection57 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 l s / l in v inj /c A m=5 m=0.04 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1  v c 2 œ /  v i 2 œ v inj /c A m = 0.04 m = 0.1 m = 0.5 m = 1 m = 5 FIG. 29.Left: Variations of plasma density in relativistic reconnection. Right: Generation of compressible modes in relativistic reconnection. From Takamoto, Inoue, and Lazarian (2015), c ©AAS. Reproduced with permission. and numerical simulations provided in TIL15, there is ρ s ρ i ∼1−β ( u L c A ) 2 ,(85) whereβwas found in numerical simulations to be a function of plasma relativistic magnetization parameterσ=B 2 /4πρhc 2 , whereh= 1 + (4(p/ρc 2 )is the specific enthalphy of the ideal rela- tivistic gas, whileρandpare, respectively, density and pressure. The change of the outflow region∆was shown to correspond to the original LV99 calcualtion but with injection inj reduced compared to the value in non-relativistic case. This corresponds to the transfer of larger fraction of energy to the fast modes which are subdominant in inducing magnetic field wandering, as indicated by simulations in Takamoto and Lazarian (2016, 2017) With these modifications, the corresponding expression of the turbulent reconnection can be written as V rec,relativ. ≈V A ( ρ s ρ i ) ( α inj l V 3 A ) 1/2 ×min [ ( L x l ) 1/2 , ( l L x ) 1/2 ] ,(86) whereα <1is the factor accounting for the decrease in the fraction of magnetic energy that induces magnetic field wandering. Figures 29 illustrate the results of the numerical calculations in TIL15. These study shows that with the suggested modifications the LV99 theory reflects the major features of turbulent relativistic reconnection. It is evident from Eq. (86) that the theory of relativistic turbulent reconnection does require fur- ther development in order to decrease the uncertainties related to magnetic turbulence in relativistic fluids. For the time being, it is important for our further discussion that qualitatively relativistic turbulent reconnection is similar to its non-relativistic counterpart. The existing numerical simu- lations in TIL15 provide us with guidance for studying the reconnection in GRBs. In particular, it is clear from the simulations that in highσflows, the turbulent reconnection speed only slowly changes with the injection velocity and does not depend on the guide magnetic field, i.e., the com- mon field component shared by the two reconnected fluxes. The injection scale of the turbulence induced through the kink instability is likely to be comparable to the scale of the magnetic field flux tubes, i.e.l∼L x . In this situation, by extrapolating the results in TIL15, one can claim that the reconnection rate is larger than0.1c A , where the relativistic Alfv ́ en speedc A is very close to the light speed. Turbulent Reconnection58 C. Whistler turbulence and reconnection In a turbulent environment, magnetic field wandering also takes place at scales below the proton Larmor radius. For instance, observations of the Earth’s magnetosphere are frequently on scales less than the ion gyro-radius. At such scales electron MHD or the EMHD approximation can be relevant for describing turbulence and reconnection (see Mandt, Denton, and Drake, 1994). The study of EMHD goes back more than 30 years (Kingsep, Chukbar, and Ian’kov, 1987) and the turbulent spectrum E(k)∼k −7/3 was explored more than 20 years ago (Biskamp, Schwarz, and Drake, 1996; Biskampet al., 1999). A quantitative description of anisotropy in EMHD turbulence anisotropy was studied by Cho and Lazarian (2004, 2009). We can derive the required scalings in analogy the approach that we discussed in§II B for the derivation of MHD turbulence scalings. The EMHD approximation applies below the ion inertial lengthd i =c/ω pi , wherecis the speed of light andω pi is the ion plasma frequency. At these scales the ions constitute a smooth motionless background and electron flows carry all the current. (The ion inertial length is not the same as the ion Larmor radius, but differs by a factor ofβ −1/2 , where βis the plasmaβ. Here we will neglect the difference.) As a result v e =− J ne ∼∇×B,(87) wherenis the electron density andeis the charge, respectively. In this limit the magnetic field is dragged by the motions of the electrons. The magnetic energy cascades to smaller scales at a rate: E= b 2 l t cascade =const,(88) whereb l is the magnetic perturbation at the scalelandv l is the electron velocity. Finallyt cascade ≈ l/v l is the cascading time at the scalel. Here the scalelrefers to the scale perpendicular to the large scale field. We note that Whistler waves have comparable amounts of energy in components parallel to and perpendicular to the mean field and the parallel current is comparable to the perpen- dicular current. The parallel wavevector component,k ‖ , will in general be much smaller than the perpendicular component,k ⊥ , but this does not effect estimates of the coherence time. The energy cascade argument depends only on the perpendicular component. Substitutingt cascade in Eq. (88) one getsb l ∝l 2/3 . The magnetic field wandering can be described as d ds y∼ b l B 0 ,(89) where B 0 is the mean magnetic field,sis the distance alone the field lines andyis the rms perpen- dicular distance between two field lines. Substituting the scaling ofb l and usingyforlone gets the perpendicular deviation of magnetic field linesy∼s 3 . This is a larger exponent than classical Richardson divergence of3/2, which also applies to MHD turbulence. Another way to understand this is to use the critical balance condition, which in this context means equating the cascade rate with the Whistler frequency 1 t cascade ∼ d i l k ‖ V A .(90) This implies that k ‖ ∝l −1/3 ,(91) and that the divergence of field lines follows the rule that two field lines separated by an eddy will, on the average, increase their separation by the width of the eddy over a distance comparable to the length of the same eddy. The speed of reconnection on a scalelis V rec ∼v l ∼d i k ‖ V A ∼ d i l b l B V A ,(92) Turbulent Reconnection59 i.e. it actually increases on smaller scales. In terms of the strength of the energy cascade it is V rec ∼ ( E ρ d 2 i l ) 1/3 .(93) Due to the limited range of whistler turbulence we do not expect this type of reconnection to play big role for large scale astrophysical reconnection. In fact, the existing Hall-MHD simulations of reconnection that show spectrum of whistler turbulence transfer for larger scale reconnection events to the turbulent reconnection in the MHD regime (see§V D). However, we expect this type of reconnection to be present in the events that can be probed by in situ measurements in the Earth magnetosphere. Thus the numerical testing of Eq. (93) is of importance for understanding the magnetospheric physics. D. Special regimes: summary The theory of turbulent reconnection was formulated for the non-relativistic reconnection in MHD regime. The numerical studies of the relativistic regime show that the reconnection in rel- ativistic regime is rather similar to that in non-relativistic one. This arises from the fact that the properties of Alfvenic turbulence and therefore magnetic field wandering are similar in the two regimes. However, the effects of compressibility and the coupling of Alfven and fast modes are more important for the relativistic reconnection. The reconnection at the high Prandtl number demonstrates that the turbulent reconnection is robust process and its rate is only marginally affected by the changes of the microscopic properties of the fluid. Indeed, the study shows that viscosity does not change the reconnection rate at large scales much larger than the viscous dissipation scale. We also note that the particular study is of practical interest, as high Prandtl number turbulence can represent some regimes of turbulence in partially ionized gas. Finally, the very idea of turbulent reconnection theory that magnetic field wandering regulates the outflow region and determines the reconnection rate carries over to scales less than the ion gy- roradius. The corresponding whistler (or Kinetic Alfven wave) turbulence can also induce turbulent reconnection. Such reconnection may be important when reconnection events at the plasma scales are considered. Numerical testing of the predictions that we provide in this section would be of interest e.g. for magnetospheric reconnection research. IX. ASTROPHYSICAL IMPLICATIONS OF TURBULENT RECONNECTIONS A. Nonlinear turbulent dynamo and turbulent reconnection Turbulent reconnection plays an important role for the theory of generating magnetic field by turbulent motions, i.e. the theory of turbulent dynamo. The turbulent dynamo has been recognized as the most promising mechanism to account for the cosmic magnetism (Brandenburg and Subra- manian, 2005). The dynamo process operating on the length scales smaller than the driving scale of turbulence, the so-called “small-scale” turbulent dynamo (SSTD), is especially important for the amplification and maintenance of the magnetic fields in the ISM and intracluster medium. Ear- lier studies on the SSTD have been restricted to the kinematic regime, where the magnetic field is dynamically insignificant and its effect on turbulent motions is negligible (Kazantsev, 1968; Kul- srud and Anderson, 1992). In the kinematic regime, the dynamo growth of magnetic fields can be analytically described by using the Kazantsev theory and considering the microscopic diffusion of magnetic fields, including the resistive diffusion and ambipolar diffusion (Schekochihin, Boldyrev, and Kulsrud, 2002; Kulsrud and Anderson, 1992; Xu and Lazarian, 2016; Xuet al., 2019a). How- ever, the kinematic dynamo gives rise to the magnetic fields with the correlation scale comparable to the dissipation scale of magnetic fluctuations. This cannot explain the observed magnetic fields with the magnetic energy concentrated at a large correlation scale in diverse astrophysical media Turbulent Reconnection60 (Leamonet al., 1998; Vogt and Enßlin, 2005; Becket al., 2016). The nonlinear dynamo, as con- firmed by numerical simulations, accounts for not only the amplification of magnetic field strength but also the increase of its correlation length and thus is astrophysically important. Analytical studies on the nonlinear dynamo is challenging because of the strong back-reaction of magnetic fields on turbulent motions. Numerical simulations reveal that the nonlinear dynamo has a universal linear-in-time growth of magnetic energy and the growth rate is only a small fraction of the turbulent energy transfer rate (Ryuet al., 2008; Choet al., 2009; Beresnyak, Jones, and Lazarian, 2009; Beresnyak, 2012). These numerical measurements can be used for testing the validity of analytical theories of nonlinear dynamo. The efficiency of nonlinear dynamo is determined by the interaction between the dynamo stretch- ing of magnetic fields by turbulent motions and the diffusion of magnetic fields. Earlier theoretical attempts to formulate the nonlinear dynamo focused on the microscopic diffusion of magnetic fields, which is applicable to the kinematic regime. For instance, for the nonlinear dynamo in a conducting fluid, Schekochihinet al.(2002) considered the resistive diffusion as the dominant diffusion mech- anism and concluded that the resulting magnetic energy spectrum peaks at the resistive scale. In the case of a partially ionized medium, Kulsrud and Anderson (1992) adopted the ambipolar diffusion as the dominant diffusion mechanism and found that the magnetic fields amplified by the nonlinear dynamo have the spectral peak at the ambipolar diffusion scale. More recently, Schoberet al.(2015) introduced the “effective magnetic diffusion” (Subramanian, 1999) to replace the microscopic diffu- sion used in the nonlinear dynamo. As a result, the magnetic energy spectrum peaks at the effective resistive scale. However, the theoretical predictions of these models on many characteristics includ- ing the dynamo efficiency, magnetic energy spectrum, and correlation length of magnetic fields are inconsistent with numerical findings. As shown in dynamo simulations, MHD turbulence is developed as a consequence of nonlinear dynamo with a Kolmogorov spectrum of magnetic fluctuations (Brandenburg and Subramanian, 2005; Beresnyak, 2012). We discussed earlier that the violation of flux freezing predicted by the turbulent reconnection theory and supported by numerical simulations (see§III B,§IV A,§V B) in most cases exceeds orders of magnitude the magnetic diffusion that directly follows from non- ideal effects, e.g. resistivity and plasma effects. Therefore the process that was termed in Lazarian (2005) the reconnection diffusion (RD) (see§IX B) dominates over microscopic diffusion processes in MHD turbulence. Naturally, one should include the RD to properly model the nonlinear dynamo and this was done in Xu and Lazarian (2016). The latter paper analytically derived the evolution of magnetic energy Efor the nonlinear dynamo, E ∼ 3 38 t,(94) whereis the scale-independent turbulent energy transfer rate. The linear dependence on time comes from the constant energy transfer rate of both hydrodynamic and MHD turbulence. The small fraction3/38comes from the RD, which causes the inefficiency of nonlinear dynamo, and quantitatively agrees with numerical measurements (Choet al., 2009; Beresnyak, 2012). Note that in the above equation,3/2is used as the slope of the growing magnetic energy spectrum (Kazantsev, 1968). If a much steeper slope4is used (Eyink, 2010; Kraichnan and Nagarajan, 1967), then an even smaller dynamo growth rate is expected. Due to the RD in MHD turbulence, the magnetic energy spectrum peaks at the driving scale of MHD turbulence, where the turbulent energy and magnetic energy are in equipartition. We note, that the effect of magnetic reconnection on turbulent dynamo was also discussed in Kulsrud and Anderson (1992). They adopted Petschek’s model for reconnection (Petschek, 1964), and thus the peak scale of magnetic energy spectrum is much less than that in Xu and Lazarian (2016), which cannot account for the large correlation scale of observed magnetic fields. This peak scale increases with the growth of magnetic energy (Xu and Lazarian, 2016), k p = [ k − 2 3 0 + 3 19  1 3 (t−t 0 ) ] − 3 2 .(95) Herek 0 is the wavenumber where the spectrum peaks at the beginning of nonlinear dynamo at t=t 0 . On smaller length scales, magnetic fields are dynamically important and results in the Turbulent Reconnection61 scale-dependent anisotropy of turbulence. On the other hand, the RD allows efficient diffusion and thus turbulent motions of magnetic fields. Therefore, instead of having a folded structure with magnetic field reversals only at the dissipation scale, the magnetic fields amplified by the nonlinear dynamo are turbulent magnetic fields with field reversals on all length scales within the inertial range of turbulence. Fig. 30 illustrates a comparison between the analytical scalings of magnetic energy spectrum obtained by Xu and Lazarian (2016) and numerical measurements in Brandenburg and Subramanian (2005) in cases with different magnetic Prandtl number P m . We see that the nonlinear dynamo theory based on the RD well explains the behavior of nonlinear dynamo presented in numerical simulations. M(k) E(k) k -5/3 k 3/2 k -5/3 1/L k ν k p (a)P m = 1 M(k) E(k) k -5/3 k 3/2 1/L k p k ν k R k -5/3 k -1 (b)P m >1 k p k 3/2 normalized k (c)P m = 1, figure. 5.1 in Brandenburg and Subramanian (2005) normalized k k p k 3/2 k -5/3 k -1 (d)P m = 50, figure. 5.2 in Brandenburg and Subramanian (2005) FIG. 30.Upper panel: sketches of the magnetic (solid line) and turbulent (dashed line) energy spectra for the nonlinear dynamo. Lower panel: the original figures are taken from Brandenburg and Subramanian (2005) (under license CC BY-NC-SA). The added dash-dotted lines indicate analytical spectral scalings, and the vertical dashed line indicates the peak scale of magnetic energy spectrum. From Xu and Lazarian (2016), c ©AAS. Reproduced with permission. The inefficiency of nonlinear dynamo has important astrophysical implications for the evolution and importance of magnetic fields in both early and present-day universe. In earlier studies on the turbulent dynamo during the primordial star formation, e.g., Schoberet al.(2012), the RD was not taken into account. This leads to an efficient nonlinear dynamo and generation of dynamically significant magnetic fields over a short timescale. Xu and Lazarian (2016) applied the nonlinear dynamo theory including the RD and examined the dynamo evolution of magnetic fields in environ- ments like the first stars. As shown in Fig. 31, among different evolutionary stages, the nonlinear dynamo has the longest timescale due to the effect of RD. Since the resulting dynamo timescale is longer than the system’s free-fall time, it is unlikely that the dynamo-amplified magnetic fields can regulate the gravitational collapse. Dynamo amplification of magnetic fields is also expected in supernova shocks to account for Turbulent Reconnection62 t [Myr] 10 -2 10 -1 10 0 10 1 10 2 10 3 B [G] 10 -20 10 -15 10 -10 10 -5 Dissipation-free Viscous Damping Nonlinear (a)First star t [Myr] 10 -2 10 -1 10 0 10 1 10 2 10 3 l p [pc] 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 Dissipation-free Viscous Damping Nonlinear (b)First star FIG. 31. The time evolution of the magnetic field strength and the correlation length of magnetic field during the formation of the first stars. From Xu and Lazarian (2016), c ©AAS. Reproduced with permission. the strong magnetic fields required for the acceleration of Galactic cosmic rays. Xu and Lazarian (2017b) analytically modeled the nonlinear dynamo in the downstream region of a supernova shock. Fig. 32 displays a comparison between their analytical results and the numerical measurements in MHD simulations of a strong shock wave by Inoue, Yamazaki, and Inutsuka (2009). By adopting the same turbulence parameters used in the simulations, Xu and Lazarian (2017b) found that the nonlinear dynamo theory based on the RD well explains the linear-in-time growth of magnetic energy and the amplification of magnetic field strength by two orders of magnitude seen in the simulations (Fig. 32(a)). Due to the inefficiency of the nonlinear dynamo, the maximum field strength can only be reached at a distance on the order of0.1pc behind the shock front (Fig. 32(b)). All these findings are consistent with the properties of magnetic fields implied by the X-ray hot spots observed in supernova remnants (Uchiyama and Aharonian, 2008). It suggests the nonlinear dynamo involving the RD as the origin of strong magnetic fields in supernova remnants. time [yr] 010002000300040005000 B [ μ G] 1 10 100 1000 (a)Time evolution of B position [pc] 1.21.31.41.51.61.71.81.92 B [ μ G] 1 10 100 1000 (b)Spatial profile of B FIG. 32. Magnetic field amplification in the postshock medium. Compared with the numerical measurements by Inoue, Yamazaki, and Inutsuka (2009), the circles in (a) and the vertical dashed line indicating the posi- tion for the maximum field strength in (b) are analytical results derived by Xu and Lazarian (2017a) using the nonlinear dynamo theory based on the RD. From Xu and Lazarian (2017a), c ©AAS. Reproduced with permission. Turbulent Reconnection63 B. Reconnection diffusion and star formation The Richardson dispersion describes the dynamics of particles and magnetic field lines from the point of view of Lagrangian approach. At the same time, for many problems the description of the effects induced by turbulent eddies in the laboratory system is preferable. This is Eulerian approach to the dynamics of the diffusion of magnetic field and this process was termed “reconnection dif- fusion” (Lazarian, 2005; Lazarian and Yan, 2014; Lazarian, Esquivel, and Crutcher, 2012). The name is designed to stress the importance of the process of magnetic reconnection that enables the diffusion process. Naturally, reconnection diffusion is intrinsically related to the Richardson diffu- sion in time and space that we described in§III B. In many respects, the process has similarities to the normal turbulent diffusion in hydrodynamics, which is not so surprising in view of the fact that the dynamics of eddies perpendicular to magnetic field is similar to the Kolmogorov turbulence. Indeed, such eddies depend on mixing of fluid motions perpendicular to the local direction of the magnetic field (see LV99) and the process takes place only if reconnection events happen through every eddy turnover. For small scale eddies magnetic field lines are nearly parallel. Therefore, when they intersect, the pressure gradient is not V 2 A /l ‖ but rather(l 2 ⊥ /l ‖ )V 2 A . This happens since only the energy of the component of the magnetic field that is not shared is available to drive the outflow. At the same time, the characteristic length contraction of a given field line due to reconnection between adjacent eddies isl 2 ⊥ /l ‖ . Taken together, this gives an effective ejection rate of V A /l ‖ . Since the width of the diffusion layer over the lengthl ‖ isl ⊥ one can write the mass conservation in a form by V rec ≈V A (l ⊥ /l ‖ ). This provides the reconnection rate V A /l ‖ , which coincides with the nonlinear cascade rate on the scalel ‖ . The description above provides an introduction to the mathematical framework of turbulent dif- fusivity in a homogeneous magnetized fluid. The standard theory of star formation is based on the assumption that magnetic field lines preserve their identify and the diffusion of charged particles perpendicular to magnetic field lines is restricted. As a result, the mass loading of magnetic field lines does not change. However, LV99 model suggests that the standard assumptions are not ap- plicable if magnetized fluids are turbulent. As a result, in the presence of MHD turbulence, the diffusion of plasma perpendicular to magnetic field is inevitable. We shall illustrate the reconnec- tion diffusion showing how it allows plasma to move perpendicular to the mean inhomogeneous magnetic field (see Figure 33). This is relevant to star formation. Consider two magnetic flux tubes with entrained plasmas that intersect each other at an angle. Due to reconnection the identity of magnetic field lines changes. Consider a situation that before the reconnection event the plasma pressures P plasma in the tubes are different, but the total pressure P plasma +P magn is the same for two tubes. This is a situation of a stable equilibrium. If plasmas are partially ionized, ambipolar diffusion, i.e. slow diffusion of neutrals in respect to magnetic field can make gradually smoothen the magnetic field pressure gradients. In the presence of turbulence a different much faster process related to turbulent reconnection takes place. Magnetic field lines in the presence of turbulence are not parallel. Such field lines can reconnect and do reconnect all the time in a turbulent flow. The process of reconnection connects magnetic fields with different mass loading and plasma pressures. As a result, plasmas stream along magnetic field lines to equalize the pressure. In the process, the portions of magnetic flux tubes with higher magnetic pressure expand as plasma pressure increases due to the flow of plasma along magnetic field lines. The entropy of the system increases as magnetic and plasma pressures become equal through the volume. In other words, a process of the diffusion driven by turbulent reconnection takes place and this process does not depend on the degree of the ionization of the matter. In the absence of gravity, the outcome of the reconnection diffusion is to make magnetic field and plasmas more homogeneously distributed. Both motions of plasmas along the flux tubes and the exchange of parts of the flux tubes between different eddies contributes to the diffusion. To get a clear picture of what is going on, consider a toy model of two adjacent magnetized ed- dies (see Figure 34). Magnetic flux tubes moving with the eddies reconnect and exchange plasmas and magnetic fields. This induces turbulent diffusion of both magnetic field and plasmas. In the case illustrated by Figure 34, the densities of plasma within the flux tubes can be different and the reconnection and creates a new flux tubes along which plasma redistributes due to the pressure dif- ference. This process induces the diffusion of plasma perpendicular to the mean magnetic field. In Turbulent Reconnection64 FIG. 33. Motion of matter in the process of reconnection diffusion. 3D magnetic flux tubes get into contact and after reconnection plasma streams along magnetic field lines. Left: XY projection before reconnection, upper panel shows that the flux tubes are at angle in X-Z plane. Right: after reconnection. From Lazarian, Esquivel, and Crutcher (2012), c ©AAS. Reproduced with permission. FIG. 34. Reconnection diffusion: exchange of flux with entrained matter. Illustration of the mixing of matter and magnetic fields due to reconnection as two flux tubes of different eddies interact. Only one scale of turbulent motions is shown. In real turbulent cascade such interactions proceed at every scale of turbulent motions. From Lazarian, Esquivel, and Crutcher (2012), c ©AAS. Reproduced with permission. reality, the process above happens at every scale and for turbulence with the extended inertial range, the shredding of the columns of plasmas with different density proceeds at all turbulence scales. The time scale of the eddy turnover in GS95 turbulence isl/v l ∼l 2/3 and therefore, assuming the decreasing the size of an eddy in the process of cascading by a factor of 2, one can present the total time for the cascade fromlto the Kolmogorov dissipation scale as a geometric pregression. Thus the full cascading takes place in about one eddy turnover time. Therefore the speed of plasma motions along the magnetic field is, in most cases, not important for the diffusion. As we discussed in§IV the magnetic flux freezing is not applicable to turbulent fluids. The gross violation of the flux freezing starts at the scales larger than the critical scale of the turbulent damping eddies. As we discussed in§II C in partially ionized gas, the corresponding scale perpendicular damping scale can be significantly larger than the resistive scale. This, however, does not change the dynamics of turbulence at scales larger thanl ⊥,crit . This fact follows both from observations and simulations (see Brandenburg and Lazarian (2013)). Therefore the process of reconnection diffusion is supposed to proceed in partially ionized gas and it will be governed by the dynamics of the large eddies in the MHD regime. Nevertheless, at scales smaller thatl ⊥,crit the evolution will be different. The structures at these scales may survive longer as the mixing is suppressed. This can provide an explanation for the existence of the tiny scale structures in HI observed (Heiles, 1997). We should stress that the reconnection diffusion on the scales less than the turbulence injection scale is very closely related to the Richardson dispersion in time and space that we discussed in §III B. Figure 35 illustrates the Richardson dispersion in space, namely, the process of mixing up magnetic field lines originating from different spatial locations. It also illustrates the loss of the Laplacian determinism for magnetic field lines. In analogy with the example above, the final line spreadl ⊥ does not depend on the initial separation of the field lines. The same loss of magnetic field identity is happening if the diffusion of magnetic field lines is traced in time. This is illustrated Turbulent Reconnection65 FIG. 35. Particle tracing magnetic field lines may start at different initial locations shown as coaxial ellipsoids. However after a period of time the field line spread over a larger volume and the final position of the field lines does not correlate with their initial position. From Lazarian, Esquivel, and Crutcher (2012), c ©AAS. Reproduced with permission. FIG. 36. Microscopic physical picture of reconnection diffusion. Magnetized plasma from two regions is spread by turbulence and mixed up over∆.Left: Description of the process in terms of field wandering in space. Right: Description of the spread in time. From Lazarian, Esquivel, and Crutcher (2012), c ©AAS. Reproduced with permission. in Figure 36. It is obvious that the accelerated separation of magnetic field lines is impossible to visualize without breaking a magnetic field line and reconnecting it with the neighboring field lines. Thus while instantaneously one can visualize a field line, the next moment the identify of this field line is entirely different, which is at odds with the classical concept of flux freezing. In fact, as it is dis- cussed in ELV11, already the resistive term induces stochasticity associated with Ohmic diffusion. Therefore the definition of the magnetic field line on scales affected by resistivity gets not determin- istic. In fact, introduces stochasticity and makes deterministic definition of field lines problematic. The determinism fails completely in the presence of turbulence, however. This was demonstrated explicitly in numerical calculations that we discuss in§V B. The two types of the dispersion are not equivalent, however. The difference is obvious for the problem of the propagation of cosmic rays. There, following magnetic field lines is important for describing cosmic ray perpendicular diffusion that arises from fast moving particles that follow essentially an instantaneous realization of a magnetic field line (see Lazarian and Yan (2014)). The effects of the time dependent Richardson dispersion are negligible in this context. The concept of reconnection diffusion has been applied to different aspects of star formation, from the loss of magnetic support of molecular clouds to the solving the problem of magnetic breaking of accretion disks (see Lazarian, 2014; Gonz ́ alez-Casanova, Lazarian, and Santos-Lima, 2016) Turbulent Reconnection66 and ref. therein. These applications are easy to understand. The entire theory of star formation was developed assuming that the magnetic field are perfectly frozen into ionized component of the partially ionized gas. This was the basic assumption of both the classical studies (see Mestel and Spitzer, 1956; Mestel, 1966) and the elaborate theories that followed (see Shu, Adams, and Lizano, 1987; Mouschovias, 1991; Nakano, Nishi, and Umebayashi, 2002; Shu, Li, and Allen, 2004; Mouschovias, Tassis, and Kunz, 2006). Within this approach, magnetic fields slow down and even prevent star formation if the media is sufficiently magnetized. With the Alfven theorem taken for granted, the change of the flux to mass ratio within forming dense clumps happens due to neutrals which do not feel magnetic field directly, but only through ion-neutral interactions. In the presence of gravity, neutrals get concentrated towards the center of the gravitational potential while magnetic fields resist compression and therefore gradualy leave the forming protostar (e.g. Mestel, 1965a,b). As a result, star formation gets inefficient for magnetically dominated (i.e. subcritical) clouds. This was viewed as a good news, as low efficiency of star formation corresponds to observations (e.g. Zuckerman and Evans, 1974). Therefore the star formation controlled by ambipolar diffusion became a standard paradigm. On closer examination, one can see that ambipolar diffusion does not solve all the problems related to star formation. For instance, for clouds dominated by gravity, i.e. supercritical clouds, ambipolar diffusion is too slow and magnetic fields do not have time to leave the cloud. Therefore for supercritical clouds magnetic field are expected be dragged into the star, forming stars with magnetizations much in excess of the observed ones (see Galliet al., 2006; Johns-Krull, 2007). Serious problems the traditional paradigm faces also with other processes associated with star formation, e.g. the formation of the accretion disks around forming stars. In fact, the paradigm has been challenged by observations (Troland and Heiles, 1986; Shuet al., 2006; Crutcher, Hakobian, and Troland, 2009; Crutcheret al., 2010, see Crutcher, 2012, for a review) and this is suggestive that one cannot really describe the processes of star formation erroneously assuming that magnetic field are frozen in within turbulent fluids. It is important to note that turbulence has already become an essential part of the field of star formation picture (see Vazquez-Semadeni, Passot, and Pouquet, 1995; Ballesteros-Paredes, Hartmann, and V ́ azquez-Semadeni, 1999; Elmegreen, 2000, 2002; Mc Kee and Tan, 2003; Elmegreen and Scalo, 2004; Mac Low and Klessen, 2004; Mc Kee and Ostriker, 2007) but the treatment of the turbulent magnetic fields stayed within the flux freezing paradigm. Without going into details, we can state, that the process of turbulent reconnection induces the reconnection diffusion that is faster than the ambipolar diffusion. This can be illustrated by an in- tentionally simplified arguments. As the most mass in molecular clouds is neutral gas, the diffusion coefficient for the fluid can be estimated asν n ∼v n l n , wherev n andl n are the velocity and mean free path of neutrals, respectively. If turbulence is present, the Reynolds number at the scalel, i.e. Re l ≈v l l/ν n is larger than 1. It means that the turbulent diffusion coefficientκ∼v l lis larger than the molecular diffusion coefficientν(see a more rigorous discussion in Lazarian, 2014). As a result, one can expect that in turbulent cloud, the loss of magnetic flux will be driven through the recon- nection diffusion as suggested in Lazarian (2005). The numerical simulations, e.g. in Santos-Lima et al.(2010), support this idea (see also Le ̃ aoet al., 2013). Another important problem that has been solved using the reconnection diffusion concept was the so-called “magnetic breaking catastrophe” (see Shuet al., 2006). The nature of the problem is easy to understand. If the interstellar magnetic field is frozen the matter forming the accretion disk, the forming Keplerian disk is subject to magnetic torques and expected to lose its angular mo- mentum fast. In Gonz ́ alez-Casanova, Lazarian, and Santos-Lima (2016) it was shown that magnetic reconnection changes the picture of star formation by both inducing the reconnection diffusion of magnetic field and changing the magnetic field configuration around the disk. The configuration of the magnetic field that connects the disk with the ambient interstellar media is frequently called “split monopole” as magnetic field from the ambient interstellar medium converge to the accretion disk, which size is very small compared to the size of the molecular cloud. In this configuration the magnetic field of different directions, i.e. converging from the media to the accretion disk and diverging from the accretion disk to the media come close together. The reconnection changes part of the split monopole configuration to the dipole one. As a result, decoupling the magnetic field from the ambient interstellar or molecular cloud gas takes place. This significantly decreases Turbulent Reconnection67 A V V R R A B V FIG. 37.Left: Shrinking magnetic loop within the turbulent reconneciton region with a particle that gains higherp ‖ as it follows the magnetic field. From Lazarian (2005).Right: Particles with a sufficiently large Larmor radius interacts with converging magnetized flow and increases its perpendicular momentump ⊥ as a result. The current sheet of the reconnection layer is denoted by red color. Due to magnetic reconnection the fluxes are moving towards each other with the velocity V rec . As a result, a charged particle gets deflected by these converging magnetic mirrors and increases itsp ⊥ . The dynamics of a particle is different if viewed in the parallel the guide component of magnetic field and from the direction parallel to the plasma outflow, i.e. parallel to the A-A line in the upper right subpanel. In the former direction the components of magnetic field of the upper and lower magnetic fluxes are parallel. Therefore the particle gyrates in this field. The Larmor radius increases due to thep ⊥ increase and the particle trajectory presents a spiral with increasing radius as shown in the upper left subpanel. The observer viewing the reconnection layer perpendicular to the guide field, i.e. along the A-A direction of the upper right subpanel, reports field reversal at the reconnection layer with magnetic field being towards the observer above the reconnection layer and away from the observer below the reconnection layer. For this magnetic field configuration the particle samples magnetic field at both sides of the magnetic reversal and it changes the direction of its gyration as it crosses current sheet. As a result, the particle moves along the open trajectory perpendicular to the A-A line. The amplitude of its oscillations are increasing asp ⊥ increases as a result of the First Order Fermi acceleration. Reprinted by permission from Springer Nature Customer Service Centre Gmb H: Springer Nature, Space Science Reviews, Lazarianet al.(2012), c ©2012. the magnetic torques acting on the disk. The numerical simulations in Santos-Limaet al.(2014); Gonz ́ alez-Casanova, Lazarian, and Santos-Lima (2016) show that the formed disks correspond to observations. Incidentally, turbulent reconnection and therefore reconnection diffusion does violate helicity conservation (LV99). This was demonstrated numerically in Bhat, Blackman, and Subramanian (2014). This once again proves that any attempts to substitute reconnection diffusion with effective “turbulent resistivity” or “eddy resistivity” are grossly incorrect (see more in§X E). C. First Order Fermi acceleration of energetic particles by turbulent reconnection Magnetic reconnection was discussed in the context of particle acceleration before the LV99 model was introduced (e.g., Litvinenko (1996); Shibata and Tanuma (2001); Zenitani and Hoshino (2001)). However, this process was intrinsically inefficient as it was based on the Ohmic-induced electric field of current sheets present within the models of reconnection known at that time. The sweet-Parker reconnection is known to be extremely slow in most of astrophysical circumstnaces and unable to transfer any appreciable part of energy from magnetic field to energetic particles. At the same time, for Petschek reconnection, the fraction of magnetic energy that is available for particle acceleration in the small fraction of the magnetic volume occupied by the Ohmic diffusion region goes to zero as the resistivity goes to zero. Thus for both processes the Ohmic electric field acceleration is negligible. The acceleration process that is induced by the LV99-type of reconnection is based on the electric field arising from the fluid motions, i.e.∼v×B, i.e. disregards the Ohmic-induced electric field. In Turbulent Reconnection68 fact, the testing of particle acceleration within a region of turbulent reconnection in Kowal, de Gouveia Dal Pino, and Lazarian (2012) was performed making the Ohmic-induced electric field equal to zero. The corresponding motions originally considered are illustrated by left panel of Figure 37 that is taken from Lazarian (2005). Magnetic reconnection results in shrinking of magnetic loops and the charged particles entrained over magnetic loops get accelerated. One way to describe the accelera- tion is to consider energetic particles that move along the magnetic loop without collisions. In this setting the second adiabatic invariant ∫ b a p ‖ dsis conserved (see Anosov and Favorskii, 1987). There p ‖ is the component of parallel momentum of a particle and the integration is performed along the magnetic loop with magnetic mirrors at pointsaandb. It is obvious that as the loop shrinks the parallel momentum of the particle is bound to increase. Turbulent magnetic reconnection is associated with the process of decreasing of the overall length of magnetic field lines. Therefore in the reconnection region magnetic loops are systematically shrinking. Thus we expect to have a systematic increase of the parallel momentum of the parti- cles trapped over magnetic field lines within the extended turbulent reconnection region (see the illustration of LV99 reconnection in the lower panel of Fig. 4). There are several important points that should be made at this point. First of all, since the rate of turbulent reconnection is regulated only by the level of turbulence, for the sufficiently high level of turbulence the transfer of energy from the magnetic one is fast. Therefore for a significant amount of energy is available for the acceleration of particles per unit time. Moreover, the turbulent re- connection is a volume filling process and therefore the aforementioned shrinking of magnetic field lines is taking place over a significant layer of the thickness∆(see Fig. 4). In addition,the LV99 reconnection operates in incompressible fluid and therefore the acceleration does not depend on the fluid compressibility. As the particles are trapped in the magnetic field loop, one can consider that the volume available for such particles decreases as the magnetic field line shrink in the process of turbulent reconnection. This type of argument relate this type of acceleration to the traditional acceleration processes, e.g. shock acceleration, that require fluid compressibility. Most important, it is obvious that the described process is the First Order Fermi accceleration process. Indeed, particles in the turbulent reconnection region aresystematically gaining energy. Therefore the turbulent reconnection can be as efficient in terms of particle acceleration as the classical shock acceleration (Krymskii, 1977; Bell, 1978). While Figure 37 illustrates the gist the process of the physical essence of the First Order Fermi acceleration in turbulent reconnection regions, the quantitative description of the process is an still an issue of debates. The non-trivial issue for the description is for the process it is essential that the particle distribution function is anisotropic in terms of the particle momentum. Fast particle scattering that isotropize the momentum distribution decrease the efficiency of acceleration in the turbulent reconnection regions. In fact, the effect of the decreased acceleration efficiency in the presence of collisions is described in Cho and Lazarian (2006) for a similar process that increases only one component of particle angular momentum. In de Gouveia Dal Pino and Lazarian (2005) (henceforth GL05) the description of the acceleration was obtained assuming that the shrinking of magnetic field tubes can be approximated by particle bouncing between the mirrors created by the not-reconnected magnetic flux. These mirrors are approaching each other with the velocity V rec . In this model the increase of the particle energy due to the increase ofp ‖ arising due to particle reflections is somewhat similar to the First Order Fermi acceleration of particles in shocks. Further studies showed that additional processes that also increase the perpendicular component of angular momentump ⊥ are also possible in turbulent reconnection regions. For instance, an illustration of this process from Lazarianet al.(2012) is presented in the right panel of Figure 37. There the particles with sufficiently large Larmor radius interact over its gyroperiod both with the shared component of magnetic field (guide field) of the two reconnecting fluxes and with the reversing components of the two fluxes. The reversing component induces the particle motion along the open trajectory in the direction of the guide field. As the magnetic fluxes converge with V rec a particle gets increase of∼ V rec c p ⊥ over one gyration in the total magnetic field. Due to the magnetic field reversal, the direction of gyration changes which results in the motion of accelerating particles Turbulent Reconnection69 along the guide field. Moving this way a particle can either escape from the reconnection region or be reflected back. Magnetic inhomogeneities can reflect the particle back without changing the cosine of its pitch angle. Looking at Figure 4 one may erroneously conclude that for the increase ofp ⊥ through the process described above the particle radius should exceed∆. In fact, Figure 4 is an idealized schematic aimed only to reflect the effect magnetic field wandering for the widening of the reconnection region. As it was discussed in LV99, the actual structure of turbulent reconnection layer is much more complex. In 3D reconnected magnetic loops move in the opposite directions and cross each other creating turbulent reconnection layers at smaller scales. These new layers, in their turn, present the microcosm of turbulent reconnection events. Therefore, we expect to see the hierarchy of turbulent reconnection layers with the decreasing thickness. These layers can increasep ⊥ of particles with different Larmor radii. It is important that both the increase ofp ‖ andp ⊥ in reconnection regions happens for particle that do not experience scattering. The latter is a relatively slow process that limits the efficiency of the textbook version of the First Fermi acceleration in shocks. If scattering is not required the process of particle acceleration is getting much faster and therefore much more efficient. This makes us confident in the astrophysical importance of the First Order Fermi acceleration in turbulent magnetic reconnection. Naturally, the processes of “collisionless” acceleration in turbulent reconnection layers are com- plex. Therefore we view the calculations of the particle spectra in the pioneering study by de Gouveia Dal Pino and Lazarian (2005) only as the first attempt to deal quantitatively with this in- teresting problem. The differences of the particle acceleration in tearing reconnection and turbulent reconnection were discussed in Beresnyak (2017) with more recent attempts to quantify the accel- eration arising due to 3D turbulent reconenction that can be found in Beresnyak and Li (2016). We believe that more quantitative studies in this direction are necessary. Testing of particle acceleration in a large-scale current sheet with embedded turbulence to make reconnection fast was performed in Kowal, de Gouveia Dal Pino, and Lazarian (2012) and its results are presented in Figure 38. The simulations were performed considering 3D MHD simulations of reconnection with the injection of 10,000 test particles. The exponential growth of particle energy was reported with the acceleration rate is∼E −0.4 (Khiali, de Gouveia Dal Pino, and del Valle, 2015). Both the parallel and perpendicular components of the particle momentum was growing, as it is shown in the subpanel of Figure 38. This suggests a rather complex process of acceleration presumably involving both processes depicted on the panels of Figure 37. The process of turbulent reconnection requires 3D settings, while in 2D the acceleration of mag- netic reconnection happens through the tearing instability. In this case the acceleration shown in the left panel of Figure 37 still can happen, but in this degenerate case instead of 3D loops one has to deal with closed islands. The case of acceleration in islands was easier to demonstrate numerically and it was discussed for the 2D tearing reconnection in the subsequent paper by Drakeet al.(2006). Being published in Nature and appealing to the popular in the community tearing reconnection the paper had more impact that either GL05 or Lazarian (2005) publications. The acceleration arising from the 2D tearing reconnection can also be explained on the basis of the second adiabatic invariant. However, the deficiency of the process of acceleration in 2D compared to 3D is self-evident. The 2D magnetic islands are closed structures and the shrinking of magnetic field lines within these islands is limited. The islands can get from elliptical to circular and then further shrinking gets impossible. Therefore, unless the islands undergo a merger, the acceleration process within an individual islands stops as it gets circular. On the contrary, in 3D turbulent reconnection the loops can shrink all the way with no constraints arising from the artificial dimensional constraints. In fact, later research (see Dahlin, Drake, and Swisdak, 2014, 2015, 2016, 2017) has shown that when the problem of tearing is considered in the space of correct dimensions, i.e. in 3D, the acceleration efficiency is increasing. In 3D islands become the contracting loops as in the original suggestion shown in the left panel of Figure 37. The contracting islands increasep ‖ . The increase ofp ⊥ as shown in the right panel of Figure 37 as the 3D loops undergo intersect. However, as this process takes place for fluids of low viscosity the transition to full turbulence is expected as we discussed in§V C. In other words, we expect the acceleration within 3D reconnection layers to be a generic one. At the same time proving this with Turbulent Reconnection70 FIG. 38. Particle kinetic energy distributions for 10,000 protons injected in the simulated turbulent reconnec- tion region. The colors indicate which velocity component is accelerated (red or blue for parallel or perpendic- ular, respectively). The energy is normalized by the rest proton mass. Reprinted figure with permission from Kowal, de Gouveia Dal Pino, and Lazarian (2012). Copyright (2012) by the American Physical Society. PIC simulations may not be easy. The serious deficiency of present day PIC simulations is that they face problems reproducing high Re number behavior of magnetized fluid in the reconenction regions. Therefore, such simulations, unlike the MHD ones, do not show trully turbulent outflows and, as a result, they are missing the key ingredient of realistic astrophysical reconnection, i.e. turbulence. It is important to understand that in the absence of turbulence the 3D reconnection does not feel additional degrees of freedom and therefore shows the 2D reconnection features. Since it introduction, the new mechanism of acceleration by reconnection has been invoked in many studies (see Jaroschek, Lesch, and Treumann (2004); Zenitani and Hoshino (2008); Zeni- tani, Hesse, and Klimas (2009); Ceruttiet al.(2013, 2014); Sironi and Spitkovsky (2014); Werner et al.(2014); Guo and Giacalone (2015), see also Hoshino and Lyubarsky (2012) for a review). In particular, it was discussed in the context of acceleration of energetic particles in relativistic envi- ronments, like pulsars (e.g. Ceruttiet al.(2013, 2014); Sironi and Spitkovsky (2014); Uzdensky and Spitkovsky (2014)) and relativistic jets of active galactic nuclei (AGNs) (e.g. Giannios (2010)). However, in most of the studies, the authors referred to 2D tearing reconnection, as the accepted process. But as we discussed above, turbulent reconnection is the necessary feature of realistic astrophysical settings and therefore the original acceleration process by 3D reconnection is more relevant to appeal in these studies. The process of fast turbulent reconnection acceleration (TRA) is expected to be widespread. In particular, it has been discussed in Lazarian and Opher (2009) as a cause of the anomalous cosmic rays observed by Voyagers and in Lazarian and Desiati (2010) as a source of the observed cosmic ray anisotropies. Turbulent reconnection was considered for the environments around of black hole sources (GL05, de Gouveia Dal Pinoet al., 2010; Kadowaki, de Gouveia Dal Pino, and Singh, 2015; Singh, de Gouveia Dal Pino, and Kadowaki, 2015; Khiali, de Gouveia Dal Pino, and del Valle, 2015; Khiali and de Gouveia Dal Pino, 2015; del Valle, de Gouveia Dal Pino, and Kowal, 2016; Singh, Garofalo, and de Gouveia Dal Pino, 2018; Kadowaki, de Gouveia Dal Pino, and Medina-Torrej ́ on, 2019). The aforementioned studies are based on non-relativistic turbulent reconnection. However, as we discuss in§VIII the relativistic and non-relativistic reconnection processes are rather similar. D. Stochastic acceleration in MHD turbulence In the previous section we have discussed the acceleration of energetic particles in the large scale reconnection layers. However, as we discussed through the entire review, magnetic reconnection ubiquitous in turbulent fluids. For instance, the dynamics of magnetic field lines depicted in Figure 10 is only possible through the unstoppable sequence of magnetic reconnection events changing the topologies of the magnetic field lines. The reconnection at all scales that happens over one eddy turnover time is picture of magnetic reconnection that was advocated in LV99 and that has been supported by more sophisticated theoretical arguments since then (see§IV A). As a result, we expect the magnetic reconnection to be present through the entire turbulent volume and be an intrinsic, essential part of the turbulent cascade in magnetized fluids. Magnetic reconnection shrinks magnetic field lines and releases the energy stored in magnetic Turbulent Reconnection71 field. For the inertial range turbulence, the energy is being transferred to kinetic energy at the com- parable scale. This kinetic energy bends the neighboring magnetic field flux, increasing magnetic field energy. In the turbulence with steady driving the transfer from magnetic energy to kinetic via the reconnection and its transfer via turbulent dynamo compensate each other. It is evident, that in the regions of magnetic reconnection magnetic field lines shrink and can accelerate particles, while in the regions of dynamo particles lose energy. This provides stochastic acceleration, but with a peculiar one. In particular, the increase of a particle energy as it get accelerated in a reconnection region can be significant. This is especially evident in super Alfvenic turbulence where extended reconnecting regions are produced. In other words, the process of acceleration is a random walk process, but with substantial increments in energy. In Brunetti and Lazarian (2016) this process was applied to the acceleration of cosmic rays in galaxy clusters and was termed 1.5 Fermi acceleration. Later this 1.5 Fermi acceleration was also employed to account for the acceleration of electrons in Gamma-ray bursts (Xu and Zhang, 2017; Xu, Yang, and Zhang, 2018) and pulsar wind nebulae (Xuet al., 2019b), which successfully explains the features of their synchrotron spectra. E. Flares and bursts of reconnection In magnetically dominated media the release of magnetic energy via reconnection must result in the outflow that induces turbulence in astrophysical high Reynolds number plasmas. This, in its turn, inevitably increases the reconnection rate further enhancing turbulence. As a result, we get a recon- nection instability as was discussed in LV99 and elaborated in Lazarian and Vishniac (2009) (see more details in Lazarian (2005)). The applications of the theory range from solar flares to gamma ray bursts (GRBs) (see Lazarianet al., 2003). This idea became the basis of the Internal-Collision- induced MAgnetic Reconnection and Turbulence (ICMART) model by Zhang and Yan (2011), who showed that such a model can overcome several difficulties of the traditional internal shock model. (Rees and Meszaros, 1994; Kobayashi, Piran, and Sari, 1997; Daigne and Mochkovitch, 1998; Ghisellini, Celotti, and Lazzati, 2000; Kumar and Mc Mahon, 2008) and can well interpret the lightcurves and spectra of GRBs (Zhang and Zhang, 2014; Uhm and Zhang, 2014; Xu and Zhang, 2017; Xu, Yang, and Zhang, 2018). ZY11 suggested that the magnetic field reversals required to trigger ICMART events may be achieved through internal collisions among high-σblobs. Denget al.(2015) performed relativistic MHD numerical simulations of collisions of magnetic blobs, and reported significant magnetic dis- sipation with an efficiency above30%. However, the simulations were on the global scale and no detailed turbulent reconnection was observed. Lazarian, Zhang, and Xu (2018) used the advances of relativistic turbulent reconnection to improve the ICMART model. The authors also modified the ICMART model by identifying the kink instability as the most probable mechanism of creating the magnetic configurations prone to reconnection and triggering the turbulent reconnection (see Figure 39). FIG. 39. Illustration of the kink instability in the GRB jet. From Lazarian, Zhang, and Xu (2018). The difference of the model in Lazarian, Zhang, and Xu (2018) from other kink-driven mod- els of GRBs (e.g. Drenkhahn and Spruit (2002); Giannios and Spruit (2006); Giannios (2008); Mc Kinney and Uzdensky (2012)) is that the kink instability also induces turbulence (Galsgaard and Nordlund, 1997; Gerrard and Hood, 2003), which drives magnetic fast reconnection. The new Turbulent Reconnection72 model gets support with the simulations in Singh, Mizuno, and de Gouveia Dal Pino (2016). There the kink instability was numerically studied in jets and the regions of kink were shown to coincide with the sites of magnetic reconnection. The simulations were performed using the ideal MHD, but the authors authors appealed to LV99 reconnection model of reconnection to claim that the reconnection that they observe is not a numerical artifact, but the actual process of magnetic recon- nection. The improved ICMART model was shown to provide a good agreement with observations, including the lightcurves, spectra and polarization. Energy release and particle acceleration arising from turbulent magnetic reconnection associated with the environment of black hole sources, e.g. active galactic nuclei (AGNs) and galactic black hole binaries (GHBs) has been also explored. This research was initiated in GL05, where it was proposed that turbulent reconnection takes place between the magnetic field lines arising from the inner accretion disk and the magnetosphere of the BH. Later, Kadowaki, de Gouveia Dal Pino, and Singh (2015) revisited the aforementioned model and extended the study to explore also the gamma-ray flare emission arising from these sources. This study confirmed the dependence reported in GL05, namely, that if observed fast reconnection is driven by turbulence, there is a correlation between the calculated fast magnetic reconnection power and the BH. In fact, this dependence was show in Kadowaki, de Gouveia Dal Pino, and Singh (2015) to span over the range of mass varying by a factor10 10 . It is interesting that this can explain not only the observed radio, but also the gamma-ray emission from GBHs and low luminous AGNs (LLAGNs). This match was confirmed for an extensive sample of more than 230 sources which include those of the so called fundamental plane of black hole activity (Merloni, Heinz, and di Matteo, 2003) high luminous AGNs whose jet points to the line of sight) and GRBs does not follow the same trend as that of the low luminous AGNs and GBHs, suggesting that the observed radio and gamma-ray emission in these cases is not produced at the cores of the sources. The study by Singh, de Gouveia Dal Pino, and Kadowaki (2015) suggests that the results in Kadowaki, de Gouveia Dal Pino, and Singh (2015) are robust and depend on turbulent reconnection properties rather than on the details of the accretion physics. F. Implications of turbulent reconnection: summary Magnetic reconnection is an part and parcel of the strong Alfvenic turbulent cascade. The recon- nection happens at all scales of the cascade. The implications of this are really numerous and only some of them have been explored so far. Changing the conventional flux freezing paradigm, turbulent reconnection induces the process of reconnection diffusion, i.e. the process of diffusion of magnetic field and matter with the rates that are determined by the level of turbulence. As a result, the loss of magnetic flux in the process of star formation is significantly modified. It becomes independent of the ionization rate of the turbuletn fluid, which is a big change in understanding of star formation. Reconnection diffusion is important for the collapse of molecular clouds as well as for the formation of protoplanetary disks. The calculations that account for the reconnection diffusion can explain observations that are puzzling otherwise. Reconnection diffusion exceeds the magnetic diffusion arising from microscopic plasma effects and resistivity. Therefore it is also an essential process for the magnetic field generation by turbu- lence, i.e. for turbulent dynamo. The predictions of the dynamo theory that accounts for reconnec- tion diffusion correspond well to numerical simulations. During magnetic reconnection magnetic lines shrink. Therefore energetic particles entrained on such magnetic field lines are getting the boost for their parallel component of momentum. This is the First order Fermi acceleration process and it takes place during the reconnection of oppositely directed large scale fluxes. Only 3D turbulent reconnection represents this process correctly. In comparison, the 2D tearing reconnection produces 2D islands, which are degenerate configurations arising only due to the artificially reduced dimension of the process. The 2D closed magnetic islands induce the acceleration that is less efficient compared to the acceleration to the magnetic loops formed in 3D case. If turbulence is just driven in the volume and no large scale magnetic field reversals are induced, the particle acceleration induced by reconnection still take place. However, the overall acceleration Turbulent Reconnection73 is stochastic: particles are accelerated in the regions where magnetic field shrinks due to reconnec- tion to the regions where magnetic field lines stretch due to turbulent dynamo. The two processes are intrinsic to the MHD turbulent cascade. Turbulent magnetic reconnection can be successfully applied to explain different observational phenomena. This is a big mostly unexplored field. The results obtained through applying the theory of turbulent reconnection to AGNs, GRBs etc. are encoraging and call for more research efforts in studying the astrophysical implications of turbulent reconnection. X. ALTERNATIVE IDEAS AND MISCONCEPTIONS IN THE LITERATURE A. Definitions of magnetic reconnection and the understanding of stochastic flux freezing As we mentioned earlier, for decades magnetic reconnection was viewed as something very spe- cial and exceptional, necessarily associated with bursts and flares of magnetic energy release. In fact, a notion that magnetic reconnection requires magnetically-dominated low-beta plasmas can still can be occasionally found in the literature. Thus we mention possible definitions of reconnec- tion. Resistive diffusion is a process at the base of the classical Sweet-Parker reconnection. It definitely changes the magnetic connectivity of plasma elements, but the rate of this reconnection is slow, i.e. it strongly depends on the resistivity. Therefore for most astrophysical settings the Sweet-Parker reconnection rate is negligibly slow. Fast reconnection require some additional effects that allow the magnetic connectivity of plasma elements to proceed at rates independent on resistivity. In 1999 two ideas were competing: the plasma effects, in particular Hall-effect, creating X-point Petschek-type reconnection and 3D tur- bulence. By 2019 the first idea is no more popular and substituted by the tearing idea. At the same time, more evidence, as we have discussed in the review, accommulated in support of LV99 turbulent reconnection model. The reconnection is currently understood as an entire suite of reconnection phenomena, each of which has great physical significance, e.g.: 1. Field-line slippage or reconnection diffusion, which is the basic process underlying all others. This is an important process, e.g., in star formation and in creating deviations from the Parker- spiral model. 2. Topological reconnection, or the change in topology of the field-line passing through a given plasma element. An example is diffusion of plasma across a separatrix surface, as when solar-wind plasma leaks into the magnetosphere. In 3D turbulence this actually happens by the Forbes-Priest ”magnetic flipping” phenomena, not by diffusion across separatrices (Priest and Forbes, 2000). 3. Magnetic energy release, in particular Alfv ́ enic jets driven by field-line tension of topologi- cally reconnected lines. This is just the classic flare-type reconnection. Of course, it requires reconnection in both senses 1) and 2). As we discussed in§III B and§IV the classical flux freezing is violated in the turbulent fluids, where magnetic reconnection happens through the entire turbulent volume at all scales. However, this does not mean that suddenly the fluid getting highly resistivive or/and that the notion of mag- netic field line completely loses its meaning. A common misunderstanding seems to be that turbu- lent violation of flux-feezing is an all or nothing thing. In reality, it was predicted back in LV99 that the reconnection rate depends upon the turbulence intensity, i.e. magnitude of the turbulent velocity fluctuations relative to the local mean Alfven speed. As a result, in low Alfven Mach number turbu- lence, i.e. for M A 1, the flux is “nearly frozen” (see Eyink, Lazarian, and Vishniac, 2011). For instance, by analysing the spacecraft data Eyink (2015) estimated that that the turbulent breakdown of flux-freezing in the solar wind was of the order 5%, exactly what was required to explain the observed deviations from the Parker spiral model. Fluids with low M A provide a gradual transition to the settings where the conventional flux freezing is approximately true. Turbulent Reconnection74 B. Critical importance of 3D simulations Most of the reconnection modeling is currently done using 2D PIC simulations. This is usually justified by the higher resolution that is available for such simulations compared to their 3D counter- parts. We feel that these simulations are missing the physics essential in the large scale astrophysical reconnection processes. The modeling of a physical process using in a reduced dimensions is always a tricky and some- what dangerous exercise. In many instances hydrodynamics and MHD flows radically change their properties if instead of 3D, the dimensions are reduced to 2D. The direction of energy flow in hy- drodynamic turbulence is a notorious example of how much the reduction of the dimensions to 2D can change the physics of the process. Therefore it is essential that modeling in 2D is not taken for granted in any research, unless there are solid physical arguments presented that the reduced dimension does not change the essence of the process. Magnetic reconnection is not an exception from the general rule above which is vivid for the case of turbulent reconnection. Note, that the 2D picture of Sweet-Parker reconnection represents the actual 3D physics well exactly as the fluids are assumed laminar, i.e. the fluid motions are very strongly restricted. As it was shown in LV99, the 3D effect of magnetic field wandering radically changes the nature of magnetic reconnection making it fast. Independent arguments on the importance of 3D geometry for magnetic reconnection are provided by Boozer (2018). The nature of MHD turbulence in 2D and 3D are different (see Eyink et al. 2011). The effect of magnetic field wandering induced by Alfvenic turbulence that is at the core of fast turbulent reconnection in 3D is not present in 2D. Therefore the issue of whether the 2D magnetic reconnec- tion in the presence of turbulence is fast (see Loureiroet al., 2009; Kulpa-Dybełet al., 2010) is of pure academic interest. Even if one can prove that magnetic field wondering in 2D can proceed through magnetic nulls that are present at high Reynolds numbers, the actual 3D physics of turbulent reconnection is, nevertheless, is radically different. In addition, the development of turbulence, its evolution and its effects are radically different in 2D and 3D reconnection layers. Similarly, the development of instabilities and their interplay with turbulence also depend whether the reconnection layer is 3D or 2D. For instance, the tearing insta- bility in 2D is accepted to be the dominant process driving magnetic reconnection. It was demon- strated that in 2D the reconnection proceeds in the plasmoid regime for all numerically explored Lundquist numbers and no MHD turbulence is generated. In this respect one should remember that not every chaotic evolving pattern is turbulence. Turbulence requires a cascade of energy. 2D tear- ing produces an inverse cascade of merging loops, which is different from MHD turbulence. On the contrary, as we discussed in§V A, 3D magnetic reconnection produces turbulence with the spec- trum and anisotropy expected for MHD turbulence (see§II B). Moreover, the turbulence suppresses tearing and changing the nature of reconnection. Therefore, as we discuss below, the 2D modeling of tearing reconnection does not provide a correct picture of the actual astrophysical reconnection. The difficulty of studying 3D reconnection is the significant computational cost of such simula- tion. Therefore, the we are only approaching to the state that PIC simulations can resolve the initial stages of turbulence development in reconnection layers (see§VI). However, these numerical diffi- culties do not justify making conclusions about astrophysical reconnection based on high resolution 2D PIC simulations. The two type of simulations sample different physics. As we demonstrated in §V and§VIII the testing of 3D reconnection is already feasible with MHD codes. The applicability of MHD treatment for magnetic reconnection at scales much larger than the relevant plasma scales is demonstrated e.g. in§IV E. The numerical simulations discussed in§V D (see also Kowal et al. 2009) also support this conclusion. As a result, MHD modeling of 3D reconnection at present is the most realistic approach for exploring astrophysical reconnection. We would not judge to what extend the 2D PIC simulations reproduce the physics of plasma- scale reconnection relevant, e.g. magneotspheric reconnection. The 3D effects may be important if magnetic field wandering induced by whistler or kinetic Alfven turbulence (see§VIII C) plays an important role for such reconnection. Turbulent Reconnection75 C. Comparison of turbulent and tearing reconnection models Turbulent reconnection emerged at the time when the competing explanation for fast reconnection was the regular X-point Petsheck model. It is obvious that the latter reconnection scheme is diffi- cult to realize in any realistic astrophysical setting as the X-point configuration of the reconnection region would have to be preserved over the astronomically large scale Lthat enters the definition of the Lunquist number Sin spite of the action of external forces acting on it in the media. In fact, further research has shown that, even in the absence of external forces, a global X-point configura- tion collapses. Instead, the tearing stochastic scheme, which is closer to spirit to LV99, emerges in 2D configuration studies (see Loureiro, Schekochihin, and Cowley, 2007; Uzdensky, Loureiro, and Schekochihin, 2010). Tearing reconnection was discussed in the literature prior to the more recent surge of interest to the process. The most notable contribution was done by Syrovatskii and his group (see Sy- rovatskii, 1981). It was also quantitatively discussed in LV99 and was shown to be a subdominant process compared to turbulent reconnection. Nevertheless, this is an important process that can play the role of a trigger for a more robust turbulent reconnection process in the settings when the reconnecting magnetic fields are mostly laminar. The mechanism has been widely recognized by the community more recently (Biskamp, 1986; Shibata and Tanuma, 2001; Daughton, Scudder, and Karimabadi, 2006; Daughtonet al., 2009b, 2011, 2014b; Fermo, Drake, and Swisdak, 2012; Loureiro, Schekochihin, and Cowley, 2007; Loureiroet al., 2012; Lapenta, 2008; Bhattacharjee et al., 2009; Cassak, Shay, and Drake, 2009; Huang and Bhattacharjee, 2010, 2016; Shepherd and Cassak, 2010; Uzdensky, Loureiro, and Schekochihin, 2010; Huang, Bhattacharjee, and Sullivan, 2011; B ́ artaet al., 2011; Shen, Lin, and Murphy, 2011; Takamoto, 2013; Wyper and Pontin, 2014). The important feature recently capitalized upon is that in 2D tearing reconnection can pro- ceed at smaller scales at higher rates. Magnetic bubbles so-produced undergo a cascade of merging events producing larger and larger bubbles, the size of which eventually determines the thickness of the outflow region. Unlike the X-point Petsheck model, the tearing reconnection is more diffi- cult to quench by random external forcing as the formed magnetic bubbles, which are also called plasmoids, resist the external pressure that attempts to chock the outflow. Conceptually, tearing reconnection, which invokes a cascade of mergers that increases the size of plasmoids, is closer to turbulent reconnection. In 2D MHD numerical simulations the transfer from Sweet-Parker to tearing reconnection was observed in the two dimensional current sheet starting with a particular Lundquist number larger than S≈10 4 . The resulting reconnection does not depend on the fluid resistivity. The study of tearing momentarily eclipsed the earlier mainstream research of the reconnection community, which attempted to explain fast reconnection appealing to the collisionless plasma effects that were invoked to stabilize the Petschek-type X point configuration for reconnection (Shay and Drake, 1998; Drake, 2001; Drakeet al., 2006). Abandoning the artificially extended X point configura- tions was the natural and right step. Indeed, the stability of X- points in the situation of realistic astrophysical forcing was very dubious, as was pointed out e.g. in LV99. In a sense, tearing intro- duced stochasticity into reconnection and brought the mainstream models of magnetic reconnection towards the LV99 model. The tearing reconnection presents volume-filling Y-type reconnection with broad outflow similar to that in LV99 and in contrast to the Petschek X-point reconnection with a very localized region where reconnection takes place. It also allowed some ideas developed within the framework of LV99 theory to be transferred to the tearing mechanism. For instance, the model of First order Fermi acceleration of cosmic rays in turbulent reconnection (de Gouveia Dal Pino and Lazarian, 2005; Lazarian, 2005) was successfully applied within the tearing mechanism (Drakeet al., 2006). The problems of transferring the ideas obtained in 2D to actual 3D settings of any realistic recon- nection even without turbulence were discussed in a series of recent papers by Boozer (2019a,c,b). Here we focus, however, on the reconnection in astrophysical environments where turbulence is a natural component that is extremely difficult to avoid in any realistic setting. Turbulence makes the 2D and 3D even more different as the nature of MHD turbulence in 3D is very different. For instance, arguments why LV99 model may not be applicable to 2D configurations were pro- vided in Eyink, Lazarian, and Vishniac (2011). Therefore, it is possible that in 2D tearing is the Turbulent Reconnection76 dominant mechanism but in our 3D world the situation appears very different. Numerical tests in a number of recent studies (Kowalet al., 2012; Eyinket al., 2013; Oishiet al., 2015; Beresnyak, 2017; Kowalet al., 2017; Takamoto, 2018) show that both in non-relativistic and relativistic MHD simulations the transfer to turbulent reconnection is inevitable even if the reconnection starts from the laminar state. In particular, in§V A we provide evidence that Kelvin-Hemholtz instability dom- inates over tearing instability in the outflow region. One may argue that the tearing instability can play the role of initiating the reconnection, but the subsequent process proceeds as driven by turbu- lence. From the theoretical point of view, the dominance of turbulent reconnection in astrophysics seems inevitable. In realistic 3D configurations the thicker outflows induced by plasmoid/tearing reconnection inevitably induce turbulence. Indeed, the mass conservation constraint requires that in order to have fast reconnection one has to increase the outflow region thickness∆in proportion to L x , which means proportional to the Lundquist number S. The Reynolds number Reof the out- flow — approximately∆V A /νwhereνis viscosity — grows also as S. As Reincreases the flow gets turbulent. Once the shearing rate introduced by eddies gets larger than the rate of the tearing instability growth, the instability should be suppressed. Potentially, one can increase the domain of tearing reconnection in 3D by increasing the Prandtl number Pt=η/νof turbulence. Indeed, as we discussed earlier, the mass conservation constraint in the case of fast reconnection dictates the increase of the outflow region thickness in proportion to the extent of the zone of the magnetic field contact, which for the Y-type tearing reconnection is proportional to S. This provides Re≈Pt −1 Sand requires very high computational effort to test the high Ptself-driven reconnection. Note, that the increase of the numerical efforts grows in proportion to S 4 , i.e. 3 dimensions + time. This is a steep dependence that frustrates the efforts to increase the Lundquist numbers significantly. However, this is only applicable if there is no external turbulence driving, which is rather unrealistic in astrophysical environments. If turbulence is present, the turbulent reconnection proceeds along the lines discussed in§III. We believe that the reason that the tearing reconnection model is so popular within the recon- nection community is that in 3D the PIC simulations do not yet have enough particles to reproduce turbulent fluid behavior in the reconnection zone. Therefore for most PIC simulations the regime of MHD-like turbulent reconnection is very difficult to achieve. Nevertheless, in§VI we reported the first results supporting turbulent reconnection that were obtained using a PIC code. We expect to see more PIC results relevant to MHD turbulent reconnection in the future. What we say about tearing/plasmoid reconnection should not be interpreted that that this type of reconnection is never present in 3D configurations. Indeed, if the initial magnetic field configuration is not turbulent it takes time given by Eq. (6) for turbulent to develop. Thus we can expect to have observe the transient 3D tearing accompanied with the ejection of plasmoids if the reconnecting fluxes were not turbulent initially (see Huanget al., 2015) However, the reconnection is expected to proceed along the turbulent reconnection path at later times (see Kowalet al., 2017, 2019). D. Objections to the concept of “reconnection-mediated turbulence” As discussed above, tearing/plasmoid instability of current sheets is one possible route to tur- bulence that can enhance magnetic reconnection rates. It has also long been understood that such a tearing instability could occur for microscale current sheets in MHD turbulence. For example, LV99 in their Appendix C considered this possibility and concluded that tearing instability would affect negligibly their estimates for turbulent reconnection rates in the inertial range and at larger scales. Recently, however, an extensive literature has developed suggesting that reconnection in MHD turbulence must necessarily be induced by tearing instability, (see Loureiro and Boldyrev, 2017b; Boldyrev and Loureiro, 2017; Loureiro and Boldyrev, 2017a; Walker, Boldyrev, and Loureiro, 2018; Mallet, Schekochihin, and Chandran, 2017; Mallet, Schekochihin, and Chand ran, 2017; Comissoet al., 2018; Vechet al., 2018). The authors explore the subject of tearing instability and its possible modification of the properties of MHD turbulence. The conclusion of those works is that, at extremely high kinetic and magnetic Reynolds numbers, the microscale current sheets do become tearing unstable and that this instability affects the turbulence at scales smaller than a criti- cal scaleλ c ,somewhat larger than the dissipation scale of turbulence. This new “tearing-mediated Turbulent Reconnection77 regime” of MHD turbulence does not modify the inertial-range properties of MHD turbulence at scalesλ c ,although the authors of the above works emphasize that the range of scales between λ c and the resistive scale (or the ion gyroradius in collisionless plasma turbulence) can be sizable in practice. The detailed conclusions of these papers depend upon extrapolating some properties of laminar tearing instability to turbulent environments, which might be questioned. In fact, We expect to see the suppression of the tearing instability by turbulence as it is described in§V A. However, the essential picture in this literature is fully consistent with the earlier conclusions of LV99, since GS95 scaling would not be altered at scales> λ c .In fact, the conclusions of LV99 do not dependent upon GS95 scaling but hold in a qualitatively similar form for any other spectrum of MHD turbu- lence (see LV99, Appendix D). The possible change of the turbulence spectrum that the authors discuss does not affect turbulent reconnection physics over the wide inertial range of scales> λ c . The possible effects of dynamical alignment (Boldyrev, 2006; Beresnyak and Lazarian, 2006, see more discussion of the effect and its testing in Beresnyak and Lazarian, 2019) do not change the model of turbulent reconnection either. There is a serious misconception, however, that has been promulgated by the aforementioned works, e.g. Loureiro and Boldyrev (2017b); Boldyrev and Loureiro (2017); Loureiro and Boldyrev (2017a); Walker, Boldyrev, and Loureiro (2018); Mallet, Schekochihin, and Chandran (2017); Mallet, Schekochihin, and Chand ran (2017); Comissoet al.(2018); Vechet al.(2018). This definitely requires our comment in view of our discussion of flux freezing in§IV. The possible effects of tearing instability discussed above have also been described as “reconnection-mediated turbulence” or a modification of MHD turbulence “due to reconnection” at scales smaller thanλ c . There is an implicit assumption in such statements that fast magnetic reconnection does not occur at inertial-range scales larger than this critical value and that it only happens at scalesλ c due to tearing instability. For example, Loureiro and Boldyrev (2017b) regard it as a major surprise that in MHD turbulence “the anisotropic, current-sheetlike eddies become the sites of magnetic reconnection before[their emphasis] the formal Kolmogorov dissipation scale is reached” and they argue that such fast reconnection sets in only because “Sweet-Parker current sheets above a certain critical aspect ratio ... are violently unstable to the formation of multiple magnetic islands, or plasmoids.” In our opinion, both of these assumptions are false. First, there are many other viable pathways to MHD turbulence and fast reconnection besides tearing instability. More importantly, reconnection occurs atall scalesin MHD turbulence, for magnetic eddies ofall sizes(see more discussion in §IV F). As we have discussed in e.g. in§IV and§IV F, that fast reconnection at all scales is part and parcel of the MHD turbulent cascade and it is absolutely essential for the dynamical consistency of MHD turbulence. There are also many observable consequences, such as deviations from the Parker spiral model (Eyink, 2015) and reconnection jets/exhausts seen at all scales in the solar wind (Gosling, 2012). In fact, it has recently been conceded by one of the authors of Mallet, Schekochihin, and Chandran (2017); Mallet, Schekochihin, and Chand ran (2017) that “magnetic fields reconnect at every scale over a time comparable with the correlation time associated with this scale (55–57) and thus, cannot preserve their identity over more than one parallel correlation scale of the Alfv ́ enic turbulence” (Meyrandet al., 2019, p. 5). This physical effect underlies the new theory proposed in Meyrandet al.(2019) for fluidization of compressible modes in collisionless plasma turbulence. We therefore believe that statements about modifications of MHD turbulence “by reconnection” reflect not just a poor choice of terminology but in fact promote a real misunderstanding of the physics of turbulent magnetic reconnection. We therefore urge the community to adopt the accu- rate terminology of “tearing-mediated MHD turbulence” for such possible small-scale effects, as in the recent papers (Walker, Boldyrev, and Loureiro, 2018; Comissoet al., 2018), and not the physically misleading language of “reconnection-mediated turbulence”. MHD turbulence always involves magnetic reconnection and at all scales (see§III B,§IV F). A separate issue is whether the theoretical treatment of “tearing-mediated turbulence” is accurate in the cited papers (Loureiro and Boldyrev, 2017b; Boldyrev and Loureiro, 2017; Loureiro and Boldyrev, 2017a; Walker, Boldyrev, and Loureiro, 2018; Mallet, Schekochihin, and Chandran, 2017; Mallet, Schekochihin, and Chand ran, 2017; Comissoet al., 2018; Vechet al., 2018). Indeed these various works reach somewhat divergent conclusions among themselves. Walker, Boldyrev, and Loureiro (2018) present numerical evidence in a special 2D setting that tearing instability can Turbulent Reconnection78 compete with nonlinear evolution and Vechet al.(2018) interpret a break observed around0.1−1 Hz in solar wind magnetic energy spectra in terms of such effects. Existing theoretical treatments, however, assume that tearing rates are the same as for laminar current sheets and are unaffected by the surrounding turbulent environment. The validity of this assumption is not so clear. We presented evidence in§V A that tearing instability is subdominant and suppressed even for the case of 3D freely-evolving reconnection. We note that another questionable point of the “tearing-mediated turbulence” that this model is based on the assumption that the dynamical alignment steadily increases with the decrease of the scale of turbulent motions over the entire turbulence inertial range it gets so large that the tearing instability gets efficient. This idea of ever increasing dynamical alignment, however, is challenged in a number of papers (see Beresnyak and Lazarian, 2019, and ref. therein) through the analysis of the high resolution numerical data. In fact, the data available to us indicates that the dynamical alignment is a transient effect at the vicinity of the injection scale and therefore it cannot proceed to the scales at which the tearing instability is expected. In other words, numerical simulations testify that the turbulence scaling that we described in§II B are correct. These scaling corresponds to MHD turbulence without any dynamical alignment over the ineritial range. At the same time, the predictions for the spectral slope and and the change of anisotropy with the scale predicted in the turbulence model with the dynamical alignment (Boldyrev, 2006) are inconsistent with numerical simulations (see Beresnyak, 2014; Beresnyak and Lazarian, 2019). This surely leaves a lot of questions, e.g what the cause of the dynamical alignment that is seen over a limited range of scales is not yet clear. 8 However, the discussion of these issues is far beyond the scope of our review. Naturally, more work will be required to properly test the theories of “tearing-mediated turbu- lence” and their underlying assumptions. We feel that it is important to stress that while the hypo- thetical tearing range is a reconnection range, the turbulence inertial-range at larger scales is also a reconnection range. The latter encompasses a much broader range of scales. For instance, in the solar wind the range of scales that is relevant to reconnection of magnetic structures is much larger than ion scales, including events of the type extensively catalogued in Gosling (2012). In any case, it is clear that tearing instability is not necessary for fast reconnection in MHD tur- bulence. Reconnection rates independent of resistivity are observed at all inertial-range scales in current 3D MHD turbulence simulations (see§V B) and in simulations of large-scale magnetic re- connection in the presence of driven turbulence (see§V A), but the Lundquist numbers of microscale current sheets in those simulations are far below the critical values∼10 4 or higher required for tear- ing instability. Even at the astronomically high Reynolds numbers that would be required to make microscale current sheets tearing unstable, the effects of such tearing will be limited to tiny length scales< λ c and irrelevant for the astrophysically significant reconnection at much larger scales. E. Objections to the concept of turbulent resistivity and the mean-field approach to reconnection LV99 and further papers on turbulent reconnection predict dramatic changes of the dynamics of magnetic fields in turbulent fluids compared to their laminar counterparts. These changes sometimes are erroneously associated with the concept of “turbulent resistivity”. We strongly object to this as- sociation and claim that the description of magnetic phenomena based on the “turbulent resistivity” idea is misleading and hasfatalproblems. It is obvious that, if we know the reconnection rate, e.g. from LV99, then an eddy-resistivity can always be adjusted and tuned by hand to achieve the known required rate. But this is engineering with little scientific justification. While the tuned reconnection rate will be correct by construction, this unphysical model will make other predictions that will be wrong. For instance, the required large eddy-resistivity will smooth out all turbulent magnetic structure that are below an “eddy- resistive scale”` eddy . This is in contrast to the physical reality, where turbulence produces strong 8 One can argue that if this effect is not related to turbulence driving, it could be that other instabilities, e.g. Kelvin-Helmholtz one, stop this alignment before the tearing instability can become efficient. Turbulent Reconnection79 small-scale inhomogeneities, such as current sheets, down to the dissipation micro-scale. Natu- rally, field-lines in the flow smoothed by assumed eddy-resistivity will not show the super-diffusive Richardson-type separation at scales below` eddy . These are just a few examples how the wrong concept of “eddy resistivity” misrepresents the reality of magnetic turbulence. As a result, no cor- rect understanding of particle transport/scattering/acceleration in the turbulent reconnection zone is possible. In addition, it is possible to show that in the case of relativistic reconnection, turbulent resistivities will introduce acausal, faster than light propagation effects. The important point to remember, as stressed in the classical monograph of Tennekes and Lum- ley (1972) on fluid turbulence, is that the concept of “turbulent diffusivity” is both a very crude approximation and, also, “a property of the flow, not of the fluid”. One must coarse-grain the turbu- lent flow at some length-scale`in order to see such effects in the averaged equations for the larger eddies. As we discussed in§IV B, coarse-graining the MHD equations by eliminating modes at scales smaller than some length`will indeed introduce a “turbulent electric field”, i.e. an effective field induced by motions of magnetized eddies at smaller scales. However, as is well known in the fluid turbulence community, the resulting turbulent transport is not “down-gradient” and cannot be accurately represented by an enhanced diffusivity. This is what Tennekes and Lumley (1972) call the “gradient-transport fallacy”. The physical reason arises from the fact that turbulence lacks separation in scales. This makes it impossible to use a simple “eddy-resistivity” description of the process. As a result, energy is not only absorbed by the smaller eddies, but also supplied by them, a phenomenon called “backscatter”. One should also remember that the turbulent electric field both destroys and creates magnetic flux. In a steady state MHD turbulence the processes of annihilation of magnetic field by reconnection and creation of magnetic field by turbulent dynamo are in balance. In the language of cascades, the turbulent electric field creates both the forward cascade of magnetic energy and the inverse cascade of magnetic helicity. Attempts are often made to incorporate these effects into the description of MHD turbulence and reconnection by the “mean-field theory” that was originally developed in an attempt to understand turbulent magnetic dynamos. In mean-field electrodynamics, the effects of turbulence are described using parameters such as anisotropic turbulent magnetic diffusivityβ ijkl and magneticα-tensorα ij so that the “turbulent electric field” becomesα ij B j +β ijkl ∂ k B l . Here the underlying assumption is that there is a wide separation L ∇ ` T between the scale of variation L ∇ of mean-fields and the size` T of the largest turbulent eddies, with Brepresenting the field averaged over turbulent ensembles or spatially coarse-grained over some (arbitrary) length-scale`satisfying L ∇ `` T . This mean-field approach was used in Guo, Diamond, and Wang (2012) to formulate a model of turbulent reconnection alternative to LV99. Another model of turbulent reconnection based on the mean-field approach is presented in Higashimori and Hoshino (2012). However, the obtained expressions for fast reconnection, unlike those in LV99, are grossly contradicted by the numerical tests of turbulent reconnection that we discussed in§V A. This is not surprising, because the mean- field theory’s fundamental assumption of scale-separation is badly violated in MHD turbulence. There is, in addition, a much deeper conceptual problem with attempts to explain fast reconnec- tion by averaging fields over space-time or over ensembles and by invoking an enhanced resistivity. Such averaging is a purely passive operation on the turbulent fields that, obviously, can alter none of the objective physics but merely changes the mathematical description. The results depend entirely upon how one performs the averages. It is not possible to claim an “explanation” of the rapid pace of magnetic reconnection using such an approach, unless it is demonstrated that the reconnection rates obtained in the theory are strictly independent of the length and timescales of the averag- ing. This is the idea of “renormalization group invariance” that we invoked in our discussion in §IV. Such invariance is never demonstrated in applications of mean-field electrodynamics to turbu- lent flows without scale-separation. On the contrary, it is by systematically applying this principle in our coarse-graining RG analysis that we derive the necessary conditions for fast reconnection independent of any microscopic plasma non-idealities (see§IV E). Likewise, the stochastic flux-freezing that we discussed in§IV C, a concept intrinsically con- nected to fast turbulent reconnection, is very different from the dissipation of magnetic field by resistivity. The latter can be modelled in a Lagrangian description by field-line stochastic Brownian motion with dispersion growing∝ηtin time. By contrast, the stochastic flux-freezing observed at very high Reynolds numbers corresponds to field-line dispersion growing by Richardson super- Turbulent Reconnection80 ballistic separation or, assuming GS95 scaling, separation∝εt 3 .Other crucial properties are also quite distinct. For example, it is well-known that magnetic helicity undergoes an inverse cascade but no forward cascade in MHD turbulence and is thus very well-conserved (Berger, 1984; Aluie, 2017). This conservation is the cornerstone of the theory of large-scale astrophysical dynamos, as shown by Vishniac and Cho (2001). Richardson dispersion of magnetic field-lines is entirely compatible with conservation of magnetic helicity, whereas enhanced turbulent resistivity would dissipate magnetic helicity. We can summarize that “eddy resistivity” is a very misleading concept that has no sound scien- tific basis whatsoever for MHD turbulence. As a result, it cannot be applied with any confidence to astrophysical problems. Analogous criticisms apply to the hyper-resistivity concept (Bhattacharjee and Hameiri, 1986; Hameiri and Bhattacharjee, 1987; Strauss, 1986; Diamond and Malkov, 2003), which has also been invoked to explain fast reconnection within the context of mean-field electro- dynamics. Apart from the aforementioned problems of using the mean-field approach while dealing with reconnection, we would like to point out that the derivation of hyper-resistivity is questionable from a different point of view. The required parallel electric field is derived from magnetic helicity conservation. Integrating the resulting expression by parts one derives a term proportional to the magnetic helicity current which looks like an effective resistivity. The derivation uses, however, several implicit assumptions. Critical among them is the assumption that the magnetic helicity of mean fields and of small scale, statistically stationary turbulent fields are separately conserved, up to tiny resistivity effects. This disregards the magnetic helicity fluxes through open boundaries that are essential for stationary reconnection. F. On the “turbulent ambipolar diffusion” idea We note that the concept of reconnection diffusion in partially ionized gas (see§IX B) should not be mixed up with that of “turbulent ambipolar diffusion” (Fatuzzo and Adams, 2002; Zweibel, 2002) that was also developed to explain magnetic flux removal from molecular clouds. The latter concept is based on the idea that turbulence can create gradients of neutrals and those can accelerate the overall pace of ambipolar diffusion. The questions that naturally arise are (1) whether this process can proceed without magnetic reconnection and (2) what is the role of ambipolar diffusion in the process. Heitschet al.(2004) performed numerical simulations with 2D turbulent mixing of a layer with magnetic field perpendicular to the layer and reported fast diffusion that was of the order of tur- bulent diffusivity number V L L, independent of ambipolar diffusion coefficient. However, this sort of mixing can happen without reconnection only in a degenerate case of 2D mixing with exactly parallel magnetic field lines. In any realistic 3D case turbulence will bend magnetic field lines and the mixing process does inevitably involve reconnection. Therefore the 3D turbulent mixing in magnetized fluid must be treated from the point of view of reconnection theory. If reconnection is slow, the process in Heitschet al.(2004) cannot proceed due to the inability of magnetic field lines to freely cross each other (as opposed to the 2D case!). This should arrest the mixing and makes the conclusions obtained in the degenerate 2D case inapplicable to the 3D diffusion. If, however, reconnection is fast as predicted in LV99, then the mixing and turbulent diffusion will take place. However, such a diffusion is not expected to depend on the ambipolar diffusion processes, which is, incidentally, in agreement with results in Heitschet al.(2004). In short, the answers to the questions above are that turbulent diffusion in partially ionized gas is (1) impossible without fast turbulent reconnection and (2) independent of ambipolar diffusion physics. In this situation we believe that it is misleading to talk about “turbulent ambipolar diffu- sion” in any astrophysical 3D setting. In fact, the actual diffusion in turbulent media is controlled by magnetic reconnection and is independent of ambipolar diffusion process! Turbulent Reconnection81 G. On the universal0.1V A reconnection rate For decades the reconnection research has been motivated mostly by Solar observations and there for explaining a number of the observed phenomena the reconnection rate of∼0.1V A was required. Therefore providing models that would explain this “Holy Grail” number became the motivation for an extensive research effort. Therefore when this number was achieved, first with the Hall-MHD 2D reconnection simulations, and later, with 2D PIC tearing mode simulations for some researchers this signified the final successful resolving the long standing problem of magnetic reconnection. The reality, however, is different. The astrophysical reconnection does show a variety of re- connection rates, which are rather difficult to accommodate if one assumes that a universal mi- crophysical plasma process regulates the physics of the reconnection. First of all, the evidence is accumulating that the Solar flares (see§ref sec:observations1) and other explosive reconnecton phe- nomena (see§IX C) require both periods of fast and slow reconnection. The deviations of magnetic field from the Parker spiral (see§sec:observations3) and the reconnection diffusion of magnetic field demonstrated in numerical simulations (see§V B) definitely do not fit either into the paradigm of the universal reconnection at the0.1V A or the requirement of the plasma effects to be essential for fast magnetic reconnection. The correspondence of the canonical value of0.1V A reported in the PIC simulations and realistic Solar physics settings was recently questioned in Beresnyak (2018) (see§V D). The author pointed out that the reconnection sheet in Solar corona is3×10 4 ion skin depths. His simulations showed the significant decrease of the reconnection rate as the width of the reconnection layer was increasing from the value of less than10, the latter range being typical for most of the present-day PIC studies. In other words, the correspondence of the rates obtained in the PIC simulations with the “desired” 0.1V A value is coinsidental. The turbulent reconnection theory does not predict a single preferred value for the magnetic reconnection rate. On the contrary, the turbulent reconnection rate is being controlled by the level of turbulence. This turbulence may be driven by external processes or by reconnection itself or both externally and self-driven. In fact, we expect the role of self-driving to increase as the plasma beta is decreasing and there is more energy available to drive turbulence. The predicted rate of reconnection within one eddy turnover time is sufficient for explaining the observations that we are aware of. At the same time, the estimates in Eyink, Lazarian, and Vishniac (2011) show that for calm low-turbulence regions of Solar corona the rate of turbulent reconnection is formally less that that induced by the Sweet-Parker process. As we discussed in this review, the process of turbulent reconnection is intrinsically related to the violation of the classical flux freezing based on the famous Alfv ́ en (1942) theorem. Instead we have the so-called “stochastic flux freezing” (see§IV C), which is a special way of flux freezing violation in turbulent fluids. The latter is necessary to make the description of magnetized turbulent fluid self-consistent. In other words, one should not consider magnetic reconnection as the process that involves annihilation of nearly anti-parallel magnetic flux tubes and that is accompanied by a tangible energy release. Yes, the latter is how the reconnection problem was initially formulated more than half a century ago. By now this picture that is motivated by Solar flares is just one of the incarnations of the ubiquitous reconnection process that takes place in astrophysical turbulent fluids. As a result, the idea of universal0.1V A rate of magnetic reconnection is ill founded and it contra- dicts to the theoretical expectations, numerical testing as well as to the available observational data. The actual astrophysical reconnection rate can be both larger and smaller than this “preferred” rate. XI. PAST, PRESENT AND FUTURE OF TURBULENT RECONNECTION THEORY It has been 20 years since the introduction of the turbulent reconnection model by LV99. By now the models that the LV99 model was competing at its start, such as the 2D X-point Hall MHD recon- nection model, have been abandoned by the community. The currently popular model of plasmoid or tearing reconnection shares common features with the original LV99 idea. In fact, we believe that the tearing may be important for the onset of turbulent reconnection in the situation where Turbulent Reconnection82 magnetic fields are initially not sufficiently turbulent. At the same time, the numerical simulations with both non-relativistic and relativistic MHD codes testify that the steady-state 3D reconnection is generically turbulent (see§V A,§V C,§VIII B). Since its establishment, the idea of 3D turbulent reconnection has become an essential part of a bigger picture of the magnetic field dynamics in turbulent fluids. Deep connections have been identified between the turbulent reconnection and theories of MHD turbulence (see§IV,§V B), turbulent dynamo (see§IX A), magnetic field stochasticity (see§III B), etc., which have shown that turbulent reconnection is an intrinsic part of magnetic field dynamics in realistic magnetized turbulent fluids. Being the fundamental property of turbulent and magnetized flows, the concept of turbulent reconnection has induced significant changes in understanding of fundamental astrophysical pro- cesses, including cosmic ray propagation and acceleration (see§IX C), as well as the star forma- tion (see§IX B). Naturally, the work in terms of exploring astrophysical consequences of turbulent reconnection are expected to bring more fruits, as many brunches of astrophysical theories were developed based on the assumption of classical flux freezing concept. It has been well established by now that the concept is not applicable to turbulent fluids (see§V B). The turbulent reconnection described in LV99 is ubiquitous in magnetized fluids on scales where the MHD approximation is valid. Recent research shows that it is applicable to both turbulent par- tially ionized and fully ionized plasmas, as well as to turbulent relativistic and non-relativistic fluids (see§VIII). On smaller scales, for instance in magnetospheric research, the magnetic reconnection is different. Some ideas of turbulent reconnection can be carried over to this regime, e.g. in the form of whistler turbulence rather than MHD turbulence being the driver of the flux freezing violation (see§VIII C). At the same time, one should clearly realise that the regime where the thicknesses of reconnection layers are comparable to the relevant plasma scales is very different from the typical settings of large-scale astrophysical reconnection. The scale separation is frequently discussed as the biggest challenge of magnetic reconnection. This problem is indeed present in some models of reconnection, e.g. the classical Sweet-Parker reconnection. As a result, in the latter case the magnetic reconnection is slow. In turbulent flu- ids magnetic reconnection takes place on all scales, and therefore talking “reconnection-mediated turbulence” by assuming that magnetic reconnection in turbulent fluids happens only at some cho- sen scale, e.g. the scale of tearing, is not meaningful (see§X D). Turbulent reconnection provides much required solid theoretical foundations for the MHD modeling of astrophysical processes tak- ing place in realistically turbulent fluids when the scales involved much larger that the ion gyrora- dius. Indeed, magnetic reconnection is an essential feature of astrophysical flows and if properly resolving plasma scales were essential for the large-scale dynamics, the modeling of most astro- physical processes would be unrealistic. Indeed, is not feasible and is not feasible in any foreseable future to model with PIC simulations star formation, evolution of magnetized interstellar medium in galaxies and intracluster medium in clusters of galaxies! The turbulent reconnection justifies MHD modeling of various astrophysical processes provided that at the scales that are studies the fluids remain turbulent. To explain the observed properties of magnetized fluids, in the absence of understanding the na- ture of turbulent reconnection and flux freezing violation that it entails, a number of “engineering” concepts, e.g., “turbulent resistivity”, “hyper-resistivity”, have been introduced in the astrophysical literature. These concepts have poor theoretical foundations and are frequently in direct contra- dictions with numerical simulations and observations. On the contrary, the theory of turbulent reconnection provides a way to properly describe the processes in turbulent astrophysical fluids (see §X E). The above statements about the universality and power of turbulent reconnection theory should not be understood in the way that we already know all the answers. In fact, there are still a number of fundamental issues to be addressed. For instance, the studies of turbulent relativistic reconnec- tion are still at their infancy (see§II D). Many issues are still unclear in terms of turbulence in collisionless magnetized plasmas and the reconnecton that accompanies this turbulence (see§IV E). The combined theoretical and numerical efforts are required to achieve the progress along these directions. While turbulent reconnection in MHD approximation correctly addresses the large scale behavior Turbulent Reconnection83 of astrophysical magnetic fields, it is important to achieve the description of reconnection processes on all scales. At the moment, the turbulent reconnection studies do not have much overlap with the research in magnetized plasmas. We expect that the gap between the two fields to decrease in future. Reconnection at the scales where both collisionless effects and turbulent dynamics are important present the next theoretical, observational and numerical challenge, the challenge that we are starting to address. The field of reconnection is the vast field of research. Apparently, different types of reconnection takes place in different environments. For instance, Sweet-Parker reconnection has its own domain, but at high Lundquist numbers its nature changes. Based on the 2D numerical study one could assume that it transforms to the tearing/plasmoid reconnection. However, the 3D studies testify that the generic reconnection model is the turbulent one. This still leaves some parameter space for plasmoid reconnection. For instance, we believe that the plasmoid stage is important for the initial stages of reconnection when the reconnecting fluxes are laminar at the beginning. Similarly, the extensive magnetospheric reconnection research demonstrates that its physics is very different from the aforementioned types of reconnection. This is not surprising as the characteristic widths of magnetispheric current sheets are comparable with the characteristic plasma scales, i.e. the ion inertial length. Therefore, there is no surprise that collisionless plasma effects are getting important for the magnetospheric reconnection. Some ideas of turbulent reconnection can be transferred to this special small-scale reconnection. For instance, we discussed in the review the effect of opening up of the reconneciton outflow arising from field wandering that induced by whistler of kinetic Alfven modes. At the same time, it is clear that the reconnection on astrophysical scales is the domain where the MHD approximation is valid and therefore the LV99 turbulent reconnection is applicable. Further research should define better the conditions and regimes for different types of reconnection. ACKNOWLEDGMENTS AL acknowledges the support of NSF AST 1816234 and NASA TCAN AAG1967 grants. GK acknowledges the support of the Brazilian agency CNPq (no. 304891/2016-9). The final work on the review was done at Kavli Institute for hospitality during the Astroplasma19 event and AL thanks Anatoly Spitkovsky for organizing productive discussions on MHD turbulence and turbulent reconnection during this event. AL thanks Andrey Beresnyak for fruitful discussions. HL gratefully acknowledges the support of Do E/OFES and LANL/LDRD programs. Figures in Sec. VI based on the PIC simulations were made by F. Guo and P. Kilian. SX acknowledges the support for Pro- gram number HST-HF2-51400.001-A provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. We acknowledge exchanges with Nuno Loureiro and Alex Schekochihin on the nature of magnetic reconnection in a turbulent cascade as well as with Jim Drake on the field wandering in the whislter turbulence. ETV and AJ thank the AAS for support. We thank the anonymous referee for the numerous useful suggestions that improved the quality of our presentation. Alfv ́ en, H., “Existence of Electromagnetic-Hydrodynamic Waves,” Nature150, 405–406 (1942). Aluie, H., “Scale decomposition in compressible turbulence,” Physica D: Nonlinear Phenomena247, 54–65 (2013). Aluie, H., “Coarse-grained incompressible magnetohydrodynamics: analyzing the turbulent cascades,” New Journal of Physics19, 025008 (2017). Anosov, D. V. and Favorskii, A. P.,Encyclopedia of Mathematics. ISBN 978-1-556-08000-5. Springer, 1987, p. 488, edited by M. Hazewinkel, Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet ”Mathematical Encyclopaedia” (Springer, 1987). Armstrong, J. W., Rickett, B. J., and Spangler, S. R., “Electron density power spectrum in the local interstellar medium,” The Astrophysical Journal443, 209–221 (1995). Axford, W., “Magnetic field reconnection,” in Magnetic Reconnection in Space and Laboratory Plasmas, Geophysical Mono- graph Series, Vol. 30, edited by J. E. W. Hones (American Geophysical Union, Washington, D. C., 1984) pp. 1–8. Balbus, S. A. and Hawley, J. F., “Instability, turbulence, and enhanced transport in accretion disks,” Reviews of Modern Physics70, 1–53 (1998). Bale, S. D., Kellogg, P. J., Mozer, F. S., Horbury, T. S., and Reme, H., “Measurement of the Electric Fluctuation Spectrum of Magnetohydrodynamic Turbulence,” Physical Review Letters94, 215002 (2005). Turbulent Reconnection84 Ballesteros-Paredes, J., Hartmann, L., and V ́ azquez-Semadeni, E., “Turbulent Flow-driven Molecular Cloud Formation: A Solution to the Post-T Tauri Problem?” The Astrophysical Journal527, 285–297 (1999), ar Xiv:astro-ph/9907053 [astro- ph]. B ́ arta, M., B ̈ uchner, J., Karlick ́ y, M., and Sk ́ ala, J., “Spontaneous Current-layer Fragmentation and Cascading Reconnection in Solar Flares. I. Model and Analysis,” The Astrophysical Journal737, 24 (2011). Batchelor, G. K.,The Theory of Homogeneous Turbulence, Cambridge: Cambridge University Press, 1953, Cambridge Science Classics (Cambridge University Press, 1953). Beck, M. C., Beck, A. M., Beck, R., Dolag, K., Strong, A. W., and Nielaba, P., “New constraints on modelling the random magnetic field of the MW,” Journal of Cosmology and Astroparticle Physics5, 056 (2016). Beckwith, K. and Stone, J. M., “A Second-order Godunov Method for Multi-dimensional Relativistic Magnetohydrodynam- ics,” The Astrophysical Journal Supplement Series193, 6 (2011). Bell, A. R., “The acceleration of cosmic rays in shock fronts - I.” Monthly Notices of the Royal Astronomical Society182, 147–156 (1978). Beresnyak, A., “Universal Nonlinear Small-Scale Dynamo,” Physical Review Letters108, 035002 (2012). Beresnyak, A., “Asymmetric Diffusion of Magnetic Field Lines,” The Astrophysical Journal Letters767, L39 (2013). Beresnyak, A., “Spectra of Strong Magnetohydrodynamic Turbulence from High-resolution Simulations,” The Astrophysical Journal Letters784, L20 (2014), ar Xiv:1401.4177 [astro-ph. GA]. Beresnyak, A., “Three-dimensional Spontaneous Magnetic Reconnection,” The Astrophysical Journal834, 47 (2017). Beresnyak, A., “The Rate of Three-dimensional Hall-MHD Reconnection,” in Journal of Physics Conference Series, Journal of Physics Conference Series, Vol. 1031 (2018) p. 012001. Beresnyak, A., Jones, T. W., and Lazarian, A., “Turbulence-Induced Magnetic Fields and Structure of Cosmic Ray Modified Shocks,” The Astrophysical Journal707, 1541–1549 (2009). Beresnyak, A. and Lazarian, A., “Polarization Intermittency and Its Influence on MHD Turbulence,” The Astrophysical Journal Letters640, L175–L178 (2006), ar Xiv:astro-ph/0512315 [astro-ph]. Beresnyak, A. and Lazarian, A., “Strong Imbalanced Turbulence,” The Astrophysical Journal682, 1070–1075 (2008). Beresnyak, A. and Lazarian, A., “Scaling Laws and Diffuse Locality of Balanced and Imbalanced Magnetohydrodynamic Turbulence,” The Astrophysical Journal Letters722, L110–L113 (2010). Beresnyak, A. and Lazarian, A.,Turbulence in Magnetohydrodynamics(De Gruyter, 2019). Beresnyak, A. and Li, H., “First-Order Particle Acceleration in Magnetically-driven Flows,” The Astrophysical Journal819, 90 (2016), ar Xiv:1406.6690 [astro-ph. HE]. Berger, M. A., “Rigorous new limits on magnetic helicity dissipation in the solar corona,” Geophysical & Astrophysical Fluid Dynamics30, 79–104 (1984). Bernard, D., Gawedzki, K., and Kupiainen, A., “Slow modes in passive advection,” Journal of Statistical Physics90, 519– 569 (1998). Bhat, P., Blackman, E. G., and Subramanian, K., “Resilience of helical fields to turbulent diffusion - II. Direct numerical sim- ulations,” Monthly Notices of the Royal Astronomical Society438, 2954–2966 (2014), ar Xiv:1310.0695 [astro-ph. GA]. Bhattacharjee, A. and Hameiri, E., “Self-consistent dynamolike activity in turbulent plasmas,” Physical Review Letters57, 206–209 (1986). Bhattacharjee, A., Huang, Y.-M., Yang, H., and Rogers, B., “Fast reconnection in high-Lundquist-number plasmas due to the plasmoid Instability,” Physics of Plasmas16, 112102 (2009). Bhattacharjee, A., Ma, Z. W., and Wang, X., “Recent Developments in Collisionless Reconnection Theory: Applications to Laboratory and Astrophysical Plasmas,” in Turbulence and Magnetic Fields in Astrophysics, Lecture Notes in Physics, Berlin Springer Verlag, Vol. 614, edited by E. Falgarone and T. Passot (2003) pp. 351–375. Biskamp, D., “Magnetic reconnection via current sheets,” Physics of Fluids29, 1520–1531 (1986). Biskamp, D., Schwarz, E., and Drake, J. F., “Two-Dimensional Electron Magnetohydrodynamic Turbulence,” Physical Review Letters76, 1264–1267 (1996). Biskamp, D., Schwarz, E., Zeiler, A., Celani, A., and Drake, J. F., “Electron magnetohydrodynamic turbulence,” Physics of Plasmas6, 751–758 (1999). Boldyrev, S., “Spectrum of Magnetohydrodynamic Turbulence,” Physical Review Letters96, 115002 (2006), ar Xiv:astro- ph/0511290 [astro-ph]. Boldyrev, S. and Loureiro, N. F., “Magnetohydrodynamic Turbulence Mediated by Reconnection,” The Astrophysical Jour- nal844, 125 (2017). Boozer, A. H., “Why fast magnetic reconnection is so prevalent,” Journal of Plasma Physics84(2018). Boozer, A. H., “Fast magnetic reconnection and the ideal evolution of a magnetic field,” Physics of Plasmas26, 042104 (2019a), ar Xiv:1812.05673 [physics.plasm-ph]. Boozer, A. H., “Magnetic reconnection with null and X-points,” ar Xiv e-prints , ar Xiv:1907.08062 (2019b), ar Xiv:1907.08062 [physics.plasm-ph]. Boozer, A. H., “Problematic nature of plasmoid theory of magnetic reconnection,” ar Xiv e-prints , ar Xiv:1904.09836 (2019c), ar Xiv:1904.09836 [physics.plasm-ph]. Bowers, K. and Li, H., “Spectral Energy Transfer and Dissipation of Magnetic Energy from Fluid to Kinetic Scales,” Physical Review Letters98, 035002 (2007). Bowers, K. J., Albright, B. J., Yin, L., Bergen, B., and Kwan, T. J. T., “Ultrahigh performance three-dimensional electro- magnetic relativistic kinetic plasma simulations,” Physics of Plasmas15, 055703 (2008). Brandenburg, A. and Lazarian, A., “Astrophysical Hydromagnetic Turbulence,” Space Science Reviews178, 163–200 (2013). Brandenburg, A. and Subramanian, K., “Astrophysical magnetic fields and nonlinear dynamo theory,” Physics Reports417, 1–209 (2005). Turbulent Reconnection85 Brunetti, G. and Lazarian, A., “Stochastic reacceleration of relativistic electrons by turbulent reconnection: a mecha- nism for cluster-scale radio emission?” Monthly Notices of the Royal Astronomical Society458, 2584–2595 (2016), ar Xiv:1603.00458 [astro-ph. HE]. Burch, J., Torbert, R., Phan, T., Chen, L.-J., Moore, T., Ergun, R., Eastwood, J., Gershman, D., Cassak, P., Argall, M.,et al., “Electron-scale measurements of magnetic reconnection in space,” Science352, aaf2939 (2016). Burkhart, B., Lazarian, A., Balsara, D., Meyer, C., and Cho, J., “Alfv ́ enic Turbulence Beyond the Ambipolar Diffusion Scale,” The Astrophysical Journal805, 118 (2015). Burkhart, B., Stanimirovi ́ c, S., Lazarian, A., and Kowal, G., “Characterizing Magnetohydrodynamic Turbulence in the Small Magellanic Cloud,” The Astrophysical Journal708, 1204–1220 (2010). Burlaga, L. F., Lepping, R. P., Behannon, K. W., Klein, L. W., and Neubauer, F. M., “Large-scale variations of the interplan- etary magnetic field - Voyager 1 and 2 observations between 1-5 AU,” Journal of Geophysical Research87, 4345–4353 (1982). Burlaga, L. F., Ness, N. F., Wang, Y.-M., and Sheeley, N. R., “Heliospheric magnetic field strength and polarity from 1 to 81 AU during the ascending phase of solar cycle 23,” Journal of Geophysical Research (Space Physics)107, 1410 (2002). Caflisch, R. E., Klapper, I., and Steele, G., “Remarks on singularities, dimension and energy dissipation for ideal hydrody- namics and MHD,” Communications in Mathematical Physics184, 443–455 (1997). Cassak, P., “Inside the black box: Magnetic reconnection and the magnetospheric multiscale mission,” Space Weather14, 186–197 (2016). Cassak, P. A., Shay, M. A., and Drake, J. F., “Scaling of Sweet-Parker reconnection with secondary islands,” Physics of Plasmas16, 120702 (2009). Cerutti, B., Werner, G. R., Uzdensky, D. A., and Begelman, M. C., “Simulations of Particle Acceleration beyond the Classical Synchrotron Burnoff Limit in Magnetic Reconnection: An Explanation of the Crab Flares,” The Astrophysical Journal770, 147 (2013). Cerutti, B., Werner, G. R., Uzdensky, D. A., and Begelman, M. C., “Three-dimensional Relativistic Pair Plasma Reconnec- tion with Radiative Feedback in the Crab Nebula,” The Astrophysical Journal782, 104 (2014). Chae, J., Goode, P., Ahn, K., Yurchysyn, V., Abramenko, V., Andic, A., Cao, W., and Park, Y., “New Solar Telescope observations of magnetic reconnection occurring in the chromosphere of the quiet Sun,” The Astrophysical Journal Letters 713, L6 (2010). Chandran, B. D. G., “AGN-driven Convection in Galaxy-Cluster Plasmas,” The Astrophysical Journal632, 809–820 (2005). Chaves, M., Gawedzki, K., Horvai, P., Kupiainen, A., and Vergassola, M., “Lagrangian dispersion in Gaussian self-similar velocity ensembles,” Journal of statistical physics113, 643–692 (2003). Chepurnov, A., Burkhart, B., Lazarian, A., and Stanimirovic, S., “The Turbulence Velocity Power Spectrum of Neutral Hydrogen in the Small Magellanic Cloud,” Ar Xiv e-prints (2015). Chepurnov, A. and Lazarian, A., “Extending the Big Power Law in the Sky with Turbulence Spectra from Wisconsin Hα Mapper Data,” The Astrophysical Journal710, 853–858 (2010). Chepurnov, A., Lazarian, A., Stanimirovi ́ c, S., Heiles, C., and Peek, J. E. G., “Velocity Spectrum for H I at High Latitudes,” The Astrophysical Journal714, 1398–1406 (2010). Chitta, L. P. and Lazarian, A., “Turbulent onset of fast magnetic reconnection observed in the solar atmosphere,” The Astro- physical Journalsubmitted(2019). Cho, J., “Simulations of Relativistic Force-free Magnetohydrodynamic Turbulence,” The Astrophysical Journal621, 324– 327 (2005). Cho, J. and Lazarian, A., “Compressible Sub-Alfv ́ enic MHD Turbulence in Low-βPlasmas,” Physical Review Letters88, 245001 (2002). Cho, J. and Lazarian, A., “Compressible magnetohydrodynamic turbulence: mode coupling, scaling relations, anisotropy, viscosity-damped regime and astrophysical implications,” Monthly Notices of the Royal Astronomical Society345, 325– 339 (2003). Cho, J. and Lazarian, A., “The Anisotropy of Electron Magnetohydrodynamic Turbulence,” The Astrophysical Journal Let- ters615, L41–L44 (2004), ar Xiv:astro-ph/0406595 [astro-ph]. Cho, J. and Lazarian, A., “Particle Acceleration by Magnetohydrodynamic Turbulence,” The Astrophysical Journal638, 811–826 (2006), ar Xiv:astro-ph/0509385 [astro-ph]. Cho, J. and Lazarian, A., “Simulations of Electron Magnetohydrodynamic Turbulence,” The Astrophysical Journal701, 236–252 (2009), ar Xiv:0904.0661 [astro-ph. EP]. Cho, J. and Lazarian, A., “Imbalanced Relativistic Force-free Magnetohydrodynamic Turbulence,” The Astrophysical Jour- nal780, 30 (2014). Cho, J., Lazarian, A., and Vishniac, E. T., “New Regime of Magnetohydrodynamic Turbulence: Cascade below the Viscous Cutoff,” The Astrophysical Journal Letters566, L49–L52 (2002a). Cho, J., Lazarian, A., and Vishniac, E. T., “Simulations of Magnetohydrodynamic Turbulence in a Strongly Magnetized Medium,” The Astrophysical Journal564, 291–301 (2002b). Cho, J., Lazarian, A., and Vishniac, E. T., “MHD Turbulence: Scaling Laws and Astrophysical Implications,” in Turbulence and Magnetic Fields in Astrophysics, Lecture Notes in Physics, Berlin Springer Verlag, Vol. 614, edited by E. Falgarone and T. Passot (2003) pp. 56–98. Cho, J., Lazarian, A., and Vishniac, E. T., “Ordinary and Viscosity-damped Magnetohydrodynamic Turbulence,” The As- trophysical Journal595, 812–823 (2003b). Cho, J. and Vishniac, E. T., “The Anisotropy of Magnetohydrodynamic Alfv ́ enic Turbulence,” The Astrophysical Journal 539, 273–282 (2000). Cho, J., Vishniac, E. T., Beresnyak, A., Lazarian, A., and Ryu, D., “Growth of Magnetic Fields Induced by Turbulent Motions,” The Astrophysical Journal693, 1449–1461 (2009). Turbulent Reconnection86 Ciaravella, A. and Raymond, J. C., “The Current Sheet Associated with the 2003 November 4 Coronal Mass Ejection: Density, Temperature, Thickness, and Line Width,” The Astrophysical Journal686, 1372–1382 (2008). Comisso, L., Huang, Y.-M., Lingam, M., Hirvijoki, E., and Bhattacharjee, A., “Magnetohydrodynamic Turbulence in the Plasmoid-mediated Regime,” The Astrophysical Journal854, 103 (2018). Crutcher, R. M., “Magnetic Fields in Molecular Clouds,” Annual Review of Astronomy & Astrophysics50, 29–63 (2012). Crutcher, R. M., Hakobian, N., and Troland, T. H., “Testing Magnetic Star Formation Theory,” The Astrophysical Journal 692, 844–855 (2009), ar Xiv:0807.2862 [astro-ph]. Crutcher, R. M., Wandelt, B., Heiles, C., Falgarone, E., and Troland, T. H., “Magnetic Fields in Interstellar Clouds from Zeeman Observations: Inference of Total Field Strengths by Bayesian Analysis,” The Astrophysical Journal725, 466–479 (2010). Dahlin, J. T., Drake, J. F., and Swisdak, M., “The mechanisms of electron heating and acceleration during magnetic recon- nection,” Physics of Plasmas21, 092304 (2014), ar Xiv:1406.0831 [physics.plasm-ph]. Dahlin, J. T., Drake, J. F., and Swisdak, M., “Electron acceleration in three-dimensional magnetic reconnection with a guide field,” Physics of Plasmas22, 100704 (2015), ar Xiv:1503.02218 [physics.plasm-ph]. Dahlin, J. T., Drake, J. F., and Swisdak, M., “Parallel electric fields are inefficient drivers of energetic electrons in magnetic reconnection,” Physics of Plasmas23, 120704 (2016), ar Xiv:1607.03857 [physics.plasm-ph]. Dahlin, J. T., Drake, J. F., and Swisdak, M., “The role of three-dimensional transport in driving enhanced electron accelera- tion during magnetic reconnection,” Physics of Plasmas24, 092110 (2017), ar Xiv:1706.00481 [physics.plasm-ph]. Daigne, F. and Mochkovitch, R., “Gamma-ray bursts from internal shocks in a relativistic wind: temporal and spectral properties,” Monthly Notices of the Royal Astronomical Society296, 275–286 (1998). Daughton, W., Nakamura, T. K. M., Karimabadi, H., Roytershteyn, V., and Loring, B., “Computing the reconnection rate in turbulent kinetic layers by using electron mixing to identify topology,” Physics of Plasmas21, 052307 (2014a). Daughton, W., Roytershteyn, V., Albright, B. J., Karimabadi, H., Yin, L., and Bowers, K. J., “Influence of Coulomb collisions on the structure of reconnection layers,” Physics of Plasmas16, 072117 (2009a). Daughton, W., Roytershteyn, V., Albright, B. J., Karimabadi, H., Yin, L., and Bowers, K. J., “Transition from collisional to kinetic regimes in large-scale reconnection layers,” Physical Review Letters103, 065004 (2009b). Daughton, W., Roytershteyn, V., Karimabadi, H., Yin, L., Albright, B. J., Bergen, B., and Bowers, K. J., “Role of electron physics in the development of turbulent magnetic reconnection in collisionless plasmas,” Nature Physics7, 539–542 (2011). Daughton, W., Scudder, J., and Karimabadi, H., “Fully kinetic simulations of undriven magnetic reconnection with open boundary conditions,” Physics of Plasmas13, 072101 (2006). Daughton, W. S., Akcay, C., Billey, Z., Finn, J., Zweibel, E., and Gekelman, W. N., “Onset and Evolution of Magnetic Reconnection in Line-Tied Systems,” AGU Fall Meeting Abstracts , SH22A-04 (2014b). de Gouveia Dal Pino, E. M., Kowal, G., Kadowaki, L. H. S., Piovezan, P., and Lazarian, A., “Magnetic Field Effects Near the Launching Region of Astrophysical Jets,” International Journal of Modern Physics D19, 729–739 (2010). de Gouveia Dal Pino, E. M. and Lazarian, A., “Production of the large scale superluminal ejections of the microquasar GRS 1915+105 by violent magnetic reconnection,” Astronomy & Astrophysics441, 845–853 (2005). del Valle, M. V., de Gouveia Dal Pino, E. M., and Kowal, G., “Properties of the first-order Fermi acceleration in fast magnetic reconnection driven by turbulence in collisional magnetohydrodynamical flows,” Monthly Notices of the Royal Astronomical Society463, 4331–4343 (2016), ar Xiv:1609.08598 [astro-ph. HE]. Deng, W., Li, H., Zhang, B., and Li, S., “Relativistic MHD Simulations of Collision-induced Magnetic Dissipation in Poynting-flux-dominated Jets/outflows,” The Astrophysical Journal805, 163 (2015). Diamond, P. H. and Malkov, M., “Dynamics of helicity transport and Taylor relaxation,” Physics of Plasmas10, 2322–2329 (2003). Dobrowolny, M., Mangeney, A., and Veltri, P., “Properties of magnetohydrodynamic turbulence in the solar wind,” Astron- omy & Astrophysics83, 26–32 (1980). Drake, J. F., “Magnetic explosions in space,” Nature410, 525–526 (2001). Drake, J. F., Swisdak, M., Che, H., and Shay, M. A., “Electron acceleration from contracting magnetic islands during reconnection,” Nature443, 553–556 (2006). Drenkhahn, G. and Spruit, H. C., “Efficient acceleration and radiation in Poynting flux powered GRB outflows,” Astronomy & Astrophysics391, 1141–1153 (2002). Elmegreen, B. G., “Star Formation in a Crossing Time,” The Astrophysical Journal530, 277–281 (2000), ar Xiv:astro- ph/9911172 [astro-ph]. Elmegreen, B. G., “Star Formation from Large to Small Scales,” Astrophysics and Space Science281, 83–95 (2002). Elmegreen, B. G. and Scalo, J., “Interstellar Turbulence I: Observations and Processes,” Annual Review of Astronomy & Astrophysics42, 211–273 (2004), ar Xiv:astro-ph/0404451 [astro-ph]. Enßlin, T. A. and Vogt, C., “Magnetic turbulence in cool cores of galaxy clusters,” Astronomy & Astrophysics453, 447–458 (2006). Eyink, G., Vishniac, E., Lalescu, C., Aluie, H., Kanov, K., B ̈ urger, K., Burns, R., Meneveau, C., and Szalay, A., “Flux- freezing breakdown in high-conductivity magnetohydrodynamic turbulence,” Nature497, 466–469 (2013). Eyink, G. L., “Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models,” Journal of Mathematical Physics50, 083102 (2009). Eyink, G. L., “Fluctuation dynamo and turbulent induction at small Prandtl number,” Physical Review E82, 046314 (2010). Eyink, G. L., “Stochastic flux freezing and magnetic dynamo,” Physical Review E83, 056405 (2011). Eyink, G. L., “Turbulent General Magnetic Reconnection,” The Astrophysical Journal807, 137 (2015). Eyink, G. L., “Review of the Onsager” Ideal Turbulence” Theory,” ar Xiv preprint ar Xiv:1803.02223 (2018). Eyink, G. L. and Aluie, H., “The breakdown of Alfv ́ en’s theorem in ideal plasma flows: Necessary conditions and physical Turbulent Reconnection87 conjectures,” Physica D: Nonlinear Phenomena223, 82–92 (2006). Eyink, G. L. and Drivas, T. D., “Cascades and dissipative anomalies in compressible fluid turbulence,” Physical Review X8, 011022 (2018). Eyink, G. L., Lazarian, A., and Vishniac, E. T., “Fast Magnetic Reconnection and Spontaneous Stochasticity,” The Astro- physical Journal743, 51 (2011). Eyink, G. L. and Sreenivasan, K. R., “Onsager and the theory of hydrodynamic turbulence,” Reviews of modern physics78, 87 (2006). Fatuzzo, M. and Adams, F. C., “Enhancement of Ambipolar Diffusion Rates through Field Fluctuations,” The Astrophysical Journal570, 210–221 (2002). Fermo, R. L., Drake, J. F., and Swisdak, M., “Secondary Magnetic Islands Generated by the Kelvin-Helmholtz Instability in a Reconnecting Current Sheet,” Physical Review Letters108, 255005 (2012). Ferri ` ere, K. M., “The interstellar environment of our galaxy,” Reviews of Modern Physics73, 1031–1066 (2001). Galli, D., Lizano, S., Shu, F. H., and Allen, A., “Gravitational Collapse of Magnetized Clouds. I. Ideal Magnetohydrody- namic Accretion Flow,” The Astrophysical Journal647, 374–381 (2006), ar Xiv:astro-ph/0604573 [astro-ph]. Galsgaard, K. and Nordlund, ̊ A., “Heating and activity of the solar corona. 3. Dynamics of a low beta plasma with three- dimensional null points,” Journal of Geophysical Research102, 231–248 (1997). Galtier, S., Nazarenko, S. V., Newell, A. C., and Pouquet, A., “A weak turbulence theory for incompressible magnetohy- drodynamics,” Journal of Plasma Physics63, 447–488 (2000). Garrison, D. and Nguyen, P., “Characterization of Relativistic MHD Turbulence,” Ar Xiv e-prints (2015). Gerrard, C. L. and Hood, A. W., “Kink unstable coronal loops: current sheets, current saturation and magnetic reconnection,” Solar Physics214, 151–169 (2003). Ghisellini, G., Celotti, A., and Lazzati, D., “Constraints on the emission mechanisms of gamma-ray bursts,” Monthly Notices of the Royal Astronomical Society313, L1–L5 (2000). Giannios, D., “Prompt GRB emission from gradual energy dissipation,” Astronomy & Astrophysics480, 305–312 (2008). Giannios, D., “UHECRs from magnetic reconnection in relativistic jets,” Monthly Notices of the Royal Astronomical Society 408, L46–L50 (2010). Giannios, D. and Spruit, H. C., “The role of kink instability in Poynting-flux dominated jets,” Astronomy & Astrophysics 450, 887–898 (2006). Goldreich, P. and Sridhar, S., “Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence,” The Astrophysical Journal438, 763–775 (1995). Gonz ́ alez-Casanova, D. F., Lazarian, A., and Santos-Lima, R., “Magnetic fields in early protostellar disk formation,” The Astrophysical Journal819, 96 (2016). Gosling, J., “Magnetic reconnection in the solar wind,” Space science reviews172, 187–200 (2012). Gosling, J. T., Phan, T. D., Lin, R. P., and Szabo, A., “Prevalence of magnetic reconnection at small field shear angles in the solar wind,” Geophysical Research Letters34, L15110 (2007). Gosling, J. T. and Szabo, A., “Bifurcated current sheets produced by magnetic reconnection in the solar wind,” Journal of Geophysical Research (Space Physics)113, A10103 (2008). Gross, D. J., “Applications of the Renormalization Group to High-Energy Physics,” in Methods in Field Theory, Les Houches 1975, Session XVIII, edited by R. Balian and J. Zinn-Justin (North-Holland Publishing, Amsterdam, 1976) pp. 141–250. Guo, F. and Giacalone, J., “The Acceleration of Electrons at Collisionless Shocks Moving Through a Turbulent Magnetic Field,” The Astrophysical Journal802, 97 (2015). Guo, F., Li, H., Li, X., Kilian, P., and Daughton, W., “Particle Energization in 3D Kinetic Magnetic Reconnection with Turbulence,” (2019), (in preparation). Guo, F., Liu, Y.-H., Daughton, W., and Li, H., “Particle Acceleration and Plasma Dynamics during Magnetic Reconnection in the Magnetically Dominated Regime,” The Astrophysical Journal806, 167 (2015), ar Xiv:1504.02193 [astro-ph. HE]. Guo, Z. B., Diamond, P. H., and Wang, X. G., “Magnetic Reconnection, Helicity Dynamics, and Hyper-diffusion,” The Astrophysical Journal757, 173 (2012). Hameiri, E. and Bhattacharjee, A., “Turbulent magnetic diffusion and magnetic field reversal,” Physics of Fluids30, 1743– 1755 (1987). Haugen, N. E., Brandenburg, A., and Dobler, W., “Simulations of nonhelical hydromagnetic turbulence,” Physical Review E70, 016308 (2004). Heiles, C., “Tiny-Scale Atomic Structure and the Cold Neutral Medium,” The Astrophysical Journal481, 193 (1997). Heitsch, F., Zweibel, E. G., Slyz, A. D., and Devriendt, J. E. G., “Turbulent Ambipolar Diffusion: Numerical Studies in Two Dimensions,” The Astrophysical Journal603, 165–179 (2004). Higashimori, K. and Hoshino, M., “The relation between ion temperature anisotropy and formation of slow shocks in colli- sionless magnetic reconnection,” Journal of Geophysical Research (Space Physics)117, A01220 (2012). Hoshino, M. and Lyubarsky, Y., “Relativistic Reconnection and Particle Acceleration,” Space Science Reviews173, 521–533 (2012). Hu, Y., Yuen, K. H., and Lazarian, A., “Intensity Gradients Technique: Synergy with Velocity Gradients and Polarization Studies,” ar Xiv e-prints , ar Xiv:1908.09488 (2019), ar Xiv:1908.09488 [astro-ph. GA]. Huang, Y.-M. and Bhattacharjee, A., “Scaling laws of resistive magnetohydrodynamic reconnection in the high-Lundquist- number, plasmoid-unstable regime,” Physics of Plasmas17, 062104 (2010). Huang, Y.-M. and Bhattacharjee, A., “Turbulent Magnetohydrodynamic Reconnection Mediated by the Plasmoid Instability,” The Astrophysical Journal818, 20 (2016). Huang, Y.-M., Bhattacharjee, A., Guo, L., and Innes, D., “Plasmoid Instability Mediated Turbulent Reconnection, Simula- tions and Observations,” in Solar Heliospheric and INterplanetary Environment (SHINE 2015)(2015) p. 26. Huang, Y.-M., Bhattacharjee, A., and Sullivan, B. P., “Onset of fast reconnection in Hall magnetohydrodynamics mediated Turbulent Reconnection88 by the plasmoid instability,” Physics of Plasmas18, 072109–072109 (2011). Innes, D. E., Inhester, B., Axford, W. I., and Wilhelm, K., “Bi-directional plasma jets produced by magnetic reconnection on the Sun,” Nature386, 811–813 (1997). Inoue, T., Asano, K., and Ioka, K., “Three-dimensional Simulations of Magnetohydrodynamic Turbulence Behind Rela- tivistic Shock Waves and Their Implications for Gamma-Ray Bursts,” The Astrophysical Journal734, 77 (2011). Inoue, T., Yamazaki, R., and Inutsuka, S.-i., “Turbulence and Magnetic Field Amplification in Supernova Remnants: In- teractions Between a Strong Shock Wave and Multiphase Interstellar Medium,” The Astrophysical Journal695, 825–833 (2009). Jafari, A. and Vishniac, E., “Introduction to Magnetic Reconnection,” ar Xiv e-prints , ar Xiv:1805.01347 (2018a). Jafari, A. and Vishniac, E., “Mathematical theory of physical vector fields,” ar Xiv e-prints (2019), ar Xiv:ar Xiv:1909.04836. Jafari, A. and Vishniac, E., “Topology and stochasticity of turbulent magnetic fields,” Physical Review E100, 013201 (2019). Jafari, A., Vishniac, E., and Vaikundaraman, V., “Magnetic stochasticity and diffusion,” Physical Review E100, 043205 (2019a). Jafari, A., Vishniac, E., and Vaikundaraman, V., “Statistical Analysis of Stochastic Magnetic Fields,” ar Xiv e-prints (2019b), ar Xiv:1909.04624 [astro-ph. HE]. Jafari, A. and Vishniac, E. T., “Magnetic Field Transport in Accretion Disks,” The Astrophysical Journal854, 2 (2018b). Jafari, A., Vishniac, E. T., Kowal, G., and Lazarian, A., “Stochastic Reconnection for Large Magnetic Prandtl Numbers,” The Astrophysical Journal860, 52 (2018). Jaroschek, C. H., Lesch, H., and Treumann, R. A., “Relativistic Kinetic Reconnection as the Possible Source Mechanism for High Variability and Flat Spectra in Extragalactic Radio Sources,” The Astrophysical Journal Letters605, L9–L12 (2004). Ji, H., Angelopoulos, V., Antiochos, S., Arons, J., Bale, S., Begelman, M., Beresnyak, A., Bhattacharjee, A., Blackman, E., Brennan, D., Brown, M., Buechner, J., Burch, J., Carter, T., Cassak, P.,et al., “Major Scientific Challenges and Opportunities in Understanding Magnetic Reconnection and Related Explosive Phenomena in Self-Organizing Plasmas,” White Paper for Plasma 2020 Decadal Survey (2019). Johns-Krull, C. M., “The Magnetic Fields of Classical T Tauri Stars,” The Astrophysical Journal664, 975–985 (2007), ar Xiv:0704.2923 [astro-ph]. Jokipii, J. R., “The Rate of Separation of Magnetic Lines of Force in a Random Magnetic Field,” The Astrophysical Journal 183, 1029–1036 (1973). Kadowaki, L. H. S., de Gouveia Dal Pino, E. M., and Medina-Torrej ́ on, T. E., “A statistical study of fast magnetic reconnec- tion in turbulent accretion disks and jets,” ar Xiv e-prints , ar Xiv:1904.04777 (2019), ar Xiv:1904.04777 [astro-ph. HE]. Kadowaki, L. H. S., de Gouveia Dal Pino, E. M., and Singh, C. B., “The Role of Fast Magnetic Reconnection on the Radio and Gamma-ray Emission from the Nuclear Regions of Microquasars and Low Luminosity AGNs,” The Astrophysical Journal802, 113 (2015). Kazantsev, A. P., “Enhancement of a Magnetic Field by a Conducting Fluid,” Soviet Journal of Experimental and Theoretical Physics26, 1031 (1968). Khabarova, O. and Obridko, V., “Puzzles of the Interplanetary Magnetic Field in the Inner Heliosphere,” The Astrophysical Journal761, 82 (2012). Khiali, B. and de Gouveia Dal Pino, E. M., “Very high energy neutrino emission from the core of low luminosity AGNs triggered by magnetic reconnection acceleration,” Ar Xiv e-prints (2015). Khiali, B., de Gouveia Dal Pino, E. M., and del Valle, M. V., “A magnetic reconnection model for explaining the multi- wavelength emission of the microquasars Cyg X-1 and Cyg X-3,” Monthly Notices of the Royal Astronomical Society 449, 34–48 (2015). Kim, E.-j. and Diamond, P. H., “On Turbulent Reconnection,” The Astrophysical Journal556, 1052–1065 (2001). Kingsep, A. S., Chukbar, K. V., and Ian’kov, V. V., “Electron magnetohydrodynamics,” Voprosy Teorii Plazmy16, 209–250 (1987). Kobayashi, S., Piran, T., and Sari, R., “Can Internal Shocks Produce the Variability in Gamma-Ray Bursts?” The Astro- physical Journal490, 92 (1997). Kolmogorov, A., “The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds’ Numbers,” Akademiia Nauk SSSR Doklady30, 301–305 (1941). Kowal, G., de Gouveia Dal Pino, E. M., and Lazarian, A., “Magnetohydrodynamic Simulations of Reconnection and Particle Acceleration: Three-dimensional Effects,” The Astrophysical Journal735, 102 (2011). Kowal, G., de Gouveia Dal Pino, E. M., and Lazarian, A., “Particle Acceleration in Turbulence and Weakly Stochastic Reconnection,” Physical Review Letters108, 241102 (2012), ar Xiv:1202.5256 [astro-ph. HE]. Kowal, G., Falceta-Gonc ̧alves, D. A., Lazarian, A., and Vishniac, E. T., “Statistics of Reconnection-driven Turbulence,” The Astrophysical Journal838, 91 (2017). Kowal, G., Falceta-Gonc ̧alves, D. A., Lazarian, A., and Vishniac, E. T., “What Drives Turbulence in Stochastic Reconnec- tion?” The Astrophysical Journal , submitted (2019), ar Xiv:1909.09179 [astro-ph. HE]. Kowal, G. and Lazarian, A., “Velocity Field of Compressible Magnetohydrodynamic Turbulence: Wavelet Decomposition and Mode Scalings,” The Astrophysical Journal720, 742–756 (2010). Kowal, G., Lazarian, A., Vishniac, E. T., and Otmianowska-Mazur, K., “Numerical Tests of Fast Reconnection in Weakly Stochastic Magnetic Fields,” The Astrophysical Journal700, 63–85 (2009). Kowal, G., Lazarian, A., Vishniac, E. T., and Otmianowska-Mazur, K., “Reconnection studies under different types of turbulence driving,” Nonlinear Processes in Geophysics19, 297–314 (2012), ar Xiv:1203.2971 [astro-ph. SR]. Kraichnan, R. H., “Inertial-Range Spectrum of Hydromagnetic Turbulence,” Physics of Fluids8, 1385–1387 (1965). Kraichnan, R. H. and Nagarajan, S., “Growth of Turbulent Magnetic Fields,” Physics of Fluids10, 859–870 (1967). Krymskii, G. F., “A regular mechanism for the acceleration of charged particles on the front of a shock wave,” Akademiia Turbulent Reconnection89 Nauk SSSR Doklady234, 1306–1308 (1977). Kulpa-Dybeł, K., Kowal, G., Otmianowska-Mazur, K., Lazarian, A., and Vishniac, E., “Reconnection in weakly stochastic B-fields in 2D,” Astronomy & Astrophysics514, A26 (2010), ar Xiv:0909.1265 [astro-ph. GA]. Kulsrud, R. M., “MHD description of plasma,” in Handbook of plasma physics, Vol. 1 (North Holland Publishing, Amster- dam, 1983) p. 115. Kulsrud, R. M.,Plasma physics for astrophysics / Russell M. Kulsrud. Princeton, N.J. : Princeton University Press, c2005. (Princeton series in astrophysics), Plasma Physics for Astrophysics (Princeton University Press, 2005). Kulsrud, R. M. and Anderson, S. W., “The spectrum of random magnetic fields in the mean field dynamo theory of the Galactic magnetic field,” The Astrophysical Journal396, 606–630 (1992). Kumar, P. and Mc Mahon, E., “A general scheme for modellingγ-ray burst prompt emission,” Monthly Notices of the Royal Astronomical Society384, 33–63 (2008). Kunz, M. W., Schekochihin, A., Chen, C., Abel, I., and Cowley, S., “Inertial-range kinetic turbulence in pressure-anisotropic astrophysical plasmas,” Journal of Plasma Physics81(2015). Lalescu, C. C., Shi, Y.-K., Eyink, G. L., Drivas, T. D., Vishniac, E. T., and Lazarian, A., “Inertial-Range Reconnection in Magnetohydrodynamic Turbulence and in the Solar Wind,” Physical Review Letters115, 025001 (2015). Landau, L. D., Bell, J., Kearsley, M., Pitaevskii, L., Lifshitz, E., and Sykes, J.,Electrodynamics of continuous media, Vol. 8 (elsevier, 2013). Lapenta, G., “Self-Feeding Turbulent Magnetic Reconnection on Macroscopic Scales,” Physical Review Letters100, 235001 (2008). Lazarian, A., “Astrophysical Implications of Turbulent Reconnection: from cosmic rays to star formation,” in Magnetic Fields in the Universe: From Laboratory and Stars to Primordial Structures., American Institute of Physics Conference Series, Vol. 784, edited by E. M. de Gouveia dal Pino, G. Lugones, and A. Lazarian (2005) pp. 42–53, ar Xiv:astro- ph/0505574 [astro-ph]. Lazarian, A., “Enhancement and Suppression of Heat Transfer by MHD Turbulence,” The Astrophysical Journal Letters645, L25–L28 (2006). Lazarian, A., “Obtaining Spectra of Turbulent Velocity from Observations,” Space Science Reviews143, 357–385 (2009). Lazarian, A., “Reconnection Diffusion in Turbulent Fluids and Its Implications for Star Formation,” Space Science Reviews 181, 1–59 (2014). Lazarian, A. and Desiati, P., “Magnetic Reconnection as the Cause of Cosmic Ray Excess from the Heliospheric Tail,” The Astrophysical Journal722, 188–196 (2010). Lazarian, A., Esquivel, A., and Crutcher, R., “Magnetization of Cloud Cores and Envelopes and Other Observational Consequences of Reconnection Diffusion,” The Astrophysical Journal757, 154 (2012). Lazarian, A., Eyink, G. L., Vishniac, E. T., and Kowal, G., “Turbulent Reconnection and Its Implications,” Ar Xiv e-prints (2015). Lazarian, A. and Opher, M., “A Model of Acceleration of Anomalous Cosmic Rays by Reconnection in the Heliosheath,” The Astrophysical Journal703, 8–21 (2009). Lazarian, A., Petrosian, V., Yan, H., and Cho, J., “Physics of Gamma-Ray Bursts: Turbulence, Energy Transfer and Recon- nection,” Ar Xiv Astrophysics e-prints (2003). Lazarian, A. and Pogosyan, D., “Velocity Modification of H I Power Spectrum,” The Astrophysical Journal537, 720–748 (2000). Lazarian, A. and Pogosyan, D., “Velocity Modification of the Power Spectrum from an Absorbing Medium,” The Astrophys- ical Journal616, 943–965 (2004). Lazarian, A. and Pogosyan, D., “Studying Turbulence Using Doppler-broadened Lines: Velocity Coordinate Spectrum,” The Astrophysical Journal652, 1348–1365 (2006). Lazarian, A. and Pogosyan, D., “Spectrum and Anisotropy of Turbulence from Multi-frequency Measurement of Synchrotron Polarization,” The Astrophysical Journal818, 178 (2016). Lazarian, A. and Vishniac, E., “Fast Reconnection of Magnetic Fields in Turbulent Fluids,” in Revista Mexicana de As- tronomia y Astrofisica Conference Series, Vol. 9, edited by S. J. Arthur, N. S. Brickhouse, and J. Franco (2000) pp. 55–62. Lazarian, A. and Vishniac, E. T., “Reconnection in a Weakly Stochastic Field,” The Astrophysical Journal517, 700–718 (1999). Lazarian, A. and Vishniac, E. T., “Model of Reconnection of Weakly Stochastic Magnetic Field and its Implications,” in Revista Mexicana de Astronomia y Astrofisica Conference Series, Revista Mexicana de Astronomia y Astrofisica, vol. 27, Vol. 36 (2009) pp. 81–88. Lazarian, A., Vishniac, E. T., and Cho, J., “Magnetic Field Structure and Stochastic Reconnection in a Partially Ionized Gas,” The Astrophysical Journal603, 180–197 (2004). Lazarian, A., Vlahos, L., Kowal, G., Yan, H., Beresnyak, A., and de Gouveia Dal Pino, E. M., “Turbulence, Magnetic Reconnection in Turbulent Fluids and Energetic Particle Acceleration,” Space Science Reviews , 86 (2012). Lazarian, A. and Yan, H., “Superdiffusion of Cosmic Rays: Implications for Cosmic Ray Acceleration,” The Astrophysical Journal784, 38 (2014). Lazarian, A. and Yuen, K. H., “Gradients of Synchrotron Polarization: Tracing 3D Distribution of Magnetic Fields,” The Astrophysical Journal865, 59 (2018a), ar Xiv:1802.00028 [astro-ph. GA]. Lazarian, A. and Yuen, K. H., “Tracing Magnetic Fields with Spectroscopic Channel Maps,” The Astrophysical Journal853, 96 (2018b), ar Xiv:1703.03119 [astro-ph. GA]. Lazarian, A., Yuen, K. H., Ho, K. W., Chen, J., Lazarian, V., Lu, Z., Yang, B., and Hu, Y., “Distribution of Velocity Gradient Orientations: Mapping Magnetization with the Velocity Gradient Technique,” The Astrophysical Journal865, 46 (2018), ar Xiv:1802.02984 [astro-ph. GA]. Turbulent Reconnection90 Lazarian, A., Zhang, B., and Xu, S., “Gamma-ray Bursts Induced by Turbulent Reconnection,” ar Xiv e-prints (2018). Le ̃ ao, M. R. M., de Gouveia Dal Pino, E. M., Santos-Lima, R., and Lazarian, A., “The Collapse of Turbulent Cores and Reconnection Diffusion,” The Astrophysical Journal777, 46 (2013), ar Xiv:1209.1846 [astro-ph. HE]. Leamon, R. J., Smith, C. W., Ness, N. F., Matthaeus, W. H., and Wong, H. K., “Observational constraints on the dynamics of the interplanetary magnetic field dissipation range,” Journal of Geophysical Research103, 4775 (1998). Li, H.-b. and Houde, M., “Probing the Turbulence Dissipation Range and Magnetic Field Strengths in Molecular Clouds,” The Astrophysical Journal677, 1151–1156 (2008). Lithwick, Y. and Goldreich, P., “Compressible Magnetohydrodynamic Turbulence in Interstellar Plasmas,” The Astrophysi- cal Journal562, 279–296 (2001). Litvinenko, Y. E., “Particle Acceleration in Reconnecting Current Sheets with a Nonzero Magnetic Field,” The Astrophysical Journal462, 997 (1996). Loureiro, N. F. and Boldyrev, S., “Collisionless Reconnection in Magnetohydrodynamic and Kinetic Turbulence,” The As- trophysical Journal850, 182 (2017a). Loureiro, N. F. and Boldyrev, S., “Role of Magnetic Reconnection in Magnetohydrodynamic Turbulence,” Physical Review Letters118, 245101 (2017b). Loureiro, N. F., Samtaney, R., Schekochihin, A. A., and Uzdensky, D. A., “Magnetic reconnection and stochastic plasmoid chains in high-Lundquist-number plasmas,” Physics of Plasmas19, 042303 (2012). Loureiro, N. F., Schekochihin, A. A., and Cowley, S. C., “Instability of current sheets and formation of plasmoid chains,” Physics of Plasmas14, 100703–100703 (2007), ar Xiv:astro-ph/0703631 [astro-ph]. Loureiro, N. F., Uzdensky, D. A., Schekochihin, A. A., Cowley, S. C., and Yousef, T. A., “Turbulent magnetic reconnec- tion in two dimensions,” Monthly Notices of the Royal Astronomical Society399, L146–L150 (2009), ar Xiv:0904.0823 [astro-ph. SR]. Lovelace, R. V. E., “Dynamo model of double radio sources,” Nature262, 649–652 (1976). Mac Low, M.-M. and Klessen, R. S., “Control of star formation by supersonic turbulence,” Reviews of Modern Physics76, 125–194 (2004), ar Xiv:astro-ph/0301093 [astro-ph]. Makwana, K. D., Zhdankin, V., Li, H., Daughton, W., and Cattaneo, F., “Energy dynamics and current sheet structure in fluid and kinetic simulations of decaying magnetohydrodynamic turbulence,” Physics of Plasmas22, 042902 (2015). Mallet, A., Schekochihin, A. A., and Chand ran, B. D. G., “Disruption of Alfv ́ enic turbulence by magnetic reconnection in a collisionless plasma,” Journal of Plasma Physics83, 905830609 (2017), ar Xiv:1707.05907 [physics.plasm-ph]. Mallet, A., Schekochihin, A. A., and Chandran, B. D. G., “Disruption of sheet-like structures in Alfv ́ enic turbulence by magnetic reconnection,” Monthly Notices of the Royal Astronomical Society468, 4862–4871 (2017), ar Xiv:1612.07604 [astro-ph. SR]. Mandt, M. E., Denton, R. E., and Drake, J. F., “Transition to whistler mediated magnetic reconnection,” Geophysics Re- search Letters21, 73–76 (1994). Maron, J. and Goldreich, P., “Simulations of Incompressible Magnetohydrodynamic Turbulence,” The Astrophysical Journal 554, 1175–1196 (2001). Masuda, S., Kosugi, T., Hara, H., Tsuneta, S., and Ogawara, Y., “A loop-top hard X-ray source in a compact solar flare as evidence for magnetic reconnection,” Nature371, 495–497 (1994). Matthaeus, W. H. and Lamkin, S. L., “Rapid magnetic reconnection caused by finite amplitude fluctuations,” Physics of Fluids28, 303–307 (1985). Matthaeus, W. H. and Lamkin, S. L., “Turbulent magnetic reconnection,” Physics of Fluids29, 2513–2534 (1986). Mc Kee, C. F. and Ostriker, E. C., “Theory of Star Formation,” Annual Review of Astronomy & Astrophysics45, 565–687 (2007), ar Xiv:0707.3514 [astro-ph]. Mc Kee, C. F. and Tan, J. C., “The Formation of Massive Stars from Turbulent Cores,” The Astrophysical Journal585, 850–871 (2003), ar Xiv:astro-ph/0206037 [astro-ph]. Mc Kinney, J. C. and Uzdensky, D. A., “A reconnection switch to trigger gamma-ray burst jet dissipation,” Monthly Notices of the Royal Astronomical Society419, 573–607 (2012). Merloni, A., Heinz, S., and di Matteo, T., “A Fundamental Plane of black hole activity,” Monthly Notices of the Royal Astronomical Society345, 1057–1076 (2003). Mestel, L., “Problems of Star Formation - I,” Quarterly Journal of the Royal Astronomical Society6, 161 (1965a). Mestel, L., “Problems of Star Formation - II,” Quarterly Journal of the Royal Astronomical Society6, 265 (1965b). Mestel, L., “The magnetic field of a contracting gas cloud. I,Strict flux-freezing,” Monthly Notices of the Royal Astronomical Society133, 265 (1966). Mestel, L. and Spitzer, L., J., “Star formation in magnetic dust clouds,” Monthly Notices of the Royal Astronomical Society 116, 503 (1956). Meyrand, R., Kanekar, A., Dorland, W., and Schekochihin, A. A., “Fluidization of collisionless plasma turbulence,” Pro- ceedings of the National Academy of Science116, 1185–1194 (2019). Mininni, P. D. and Pouquet, A., “Finite dissipation and intermittency in magnetohydrodynamics,” Physical Review E80, 025401 (2009). Mouschovias, T. C., “Magnetic Braking, Ambipolar Diffusion, Cloud Cores, and Star Formation: Natural Length Scales and Protostellar Masses,” The Astrophysical Journal373, 169 (1991). Mouschovias, T. C., Tassis, K., and Kunz, M. W., “Observational Constraints on the Ages of Molecular Clouds and the Star Formation Timescale: Ambipolar-Diffusion-controlled or Turbulence-induced Star Formation?” The Astrophysical Journal646, 1043–1049 (2006), ar Xiv:astro-ph/0512043 [astro-ph]. Nakano, T., Nishi, R., and Umebayashi, T., “Mechanism of Magnetic Flux Loss in Molecular Clouds,” The Astrophysical Journal573, 199–214 (2002), ar Xiv:astro-ph/0203223 [astro-ph]. Narayan, R. and Medvedev, M. V., “Thermal Conduction in Clusters of Galaxies,” The Astrophysical Journal Letters562, Turbulent Reconnection91 L129–L132 (2001), ar Xiv:astro-ph/0110567 [astro-ph]. Newcomb, W. A., “Motion of magnetic lines of force,” Annals of Physics3, 347–385 (1958). Norman, C. A. and Ferrara, A., “The Turbulent Interstellar Medium: Generalizing to a Scale-dependent Phase Continuum,” The Astrophysical Journal467, 280 (1996). Oishi, J. S., Mac Low, M.-M., Collins, D. C., and Tamura, M., “Self-generated turbulence in magnetic reconnection,” Ar Xiv e-prints (2015). Onsager, L., “The distribution of energy in turbulence [abstract],” in Minutes of the Meeting of the Metropolitan Section held at Columbia University, New York, November 9 and 10, 1945, Physical Review, Vol. 68 (American Physical Society, College Park, MD, 1945) pp. 286–286. Onsager, L., “Statistical hydrodynamics,” Nuovo Cimento Suppl.6, 279–287 (1949). Ossendrijver, M., “The solar dynamo,” Astronomy and Astrophysics Reviews11, 287–367 (2003). Padoan, P., Juvela, M., Kritsuk, A., and Norman, M. L., “The Power Spectrum of Supersonic Turbulence in Perseus,” The Astrophysical Journal Letters653, L125–L128 (2006). Padoan, P., Kritsuk, A. G., Lunttila, T., Juvela, M., Nordlund, A., Norman, M. L., and Ustyugov, S. D., “MHD Turbulence In Star-Forming Clouds,” in American Institute of Physics Conference Series, American Institute of Physics Conference Series, Vol. 1242, edited by G. Bertin, F. de Luca, G. Lodato, R. Pozzoli, and M. Rom ́ e (2010) pp. 219–230. Parker, E. N., “Sweet’s Mechanism for Merging Magnetic Fields in Conducting Fluids,” Journal of Geophysical Research 62, 509–520 (1957). Parker, E. N., “Dynamics of the Interplanetary Gas and Magnetic Fields.” The Astrophysical Journal128, 664 (1958). Parker, E. N., “The Generation of Magnetic Fields in Astrophysical Bodies. I. The Dynamo Equations,” The Astrophysical Journal162, 665 (1970). Parker, E. N.,Cosmical magnetic fields. Their origin and their activity, International series of monographs on physics (Clarendon Press, 1979). Parker, E. N., “A solar dynamo surface wave at the interface between convection and nonuniform rotation,” The Astrophys- ical Journal408, 707–719 (1993). Petschek, H. E., “Magnetic Field Annihilation,” NASA Special Publication50, 425 (1964). Phan, T. D., Gosling, J. T., and Davis, M. S., “Prevalence of extended reconnection X-lines in the solar wind at 1 AU,” Geophysical Research Letters36, L09108 (2009). Priest, E. and Forbes, T., “Magnetic reconnection : MHD theory and applications,” in Magnetic reconnection : MHD theory and applications(2000). Priest, E. R. and Forbes, T. G., “The magnetic nature of solar flares,” Astronomy and Astrophysics Reviews10, 313–377 (2002). Rees, M. J. and Meszaros, P., “Unsteady outflow models for cosmological gamma-ray bursts,” The Astrophysical Journal Letters430, L93–L96 (1994). Richardson, J. D., Burlaga, L. F., Decker, R. B., Drake, J. F., Ness, N. F., and Opher, M., “Magnetic Flux Conservation in the Heliosheath,” The Astrophysical Journal Letters762, L14 (2013). Richardson, L. F., “Atmospheric diffusion shown on a distance-neighbour graph,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character110, 709–737 (1926). Ryu, D., Kang, H., Cho, J., and Das, S., “Turbulence and Magnetic Fields in the Large-Scale Structure of the Universe,” Science320, 909– (2008). Santos-Lima, R., de Gouveia Dal Pino, E. M., Kowal, G., Falceta-Gonc ̧alves, D., Lazarian, A., and Nakwacki, M. S., “Magnetic Field Amplification and Evolution in Turbulent Collisionless Magnetohydrodynamics: An Application to the Intracluster Medium,” The Astrophysical Journal781, 84 (2014), ar Xiv:1305.5654 [astro-ph. CO]. Santos-Lima, R., Lazarian, A., de Gouveia Dal Pino, E. M., and Cho, J., “Diffusion of Magnetic Field and Removal of Mag- netic Flux from Clouds Via Turbulent Reconnection,” The Astrophysical Journal714, 442–461 (2010), ar Xiv:0910.1117 [astro-ph. GA]. Schekochihin, A. A., Boldyrev, S. A., and Kulsrud, R. M., “Spectra and Growth Rates of Fluctuating Magnetic Fields in the Kinematic Dynamo Theory with Large Magnetic Prandtl Numbers,” The Astrophysical Journal567, 828–852 (2002). Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E., and Tatsuno, T., “Astrophys- ical Gyrokinetics: Kinetic and Fluid Turbulent Cascades in Magnetized Weakly Collisional Plasmas,” The Astrophysical Journal Supplement Series182, 310–377 (2009). Schekochihin, A. A., Cowley, S. C., Hammett, G. W., Maron, J. L., and Mc Williams, J. C., “A model of nonlinear evolution and saturation of the turbulent MHD dynamo,” New Journal of Physics4, 84 (2002). Schekochihin, A. A., Cowley, S. C., Kulsrud, R. M., Hammett, G. W., and Sharma, P., “Plasma Instabilities and Magnetic Field Growth in Clusters of Galaxies,” The Astrophysical Journal629, 139–142 (2005). Schekochihin, A. A., Cowley, S. C., Taylor, S. F., Maron, J. L., and Mc Williams, J. C., “Simulations of the Small-Scale Turbulent Dynamo,” The Astrophysical Journal612, 276–307 (2004). Schlickeiser, R.,Cosmic Ray Astrophysics(Springer, Berlin, 2002). Schober, J., Schleicher, D., Federrath, C., Glover, S., Klessen, R. S., and Banerjee, R., “The Small-scale Dynamo and Non-ideal Magnetohydrodynamics in Primordial Star Formation,” The Astrophysical Journal754, 99 (2012). Schober, J., Schleicher, D. R. G., Federrath, C., Bovino, S., and Klessen, R. S., “Saturation of the turbulent dynamo,” Physical Review E92, 023010 (2015). Shay, M. A. and Drake, J. F., “The role of electron dissipation on the rate of collisionless magnetic reconnection,” Geophysics Research Letters25, 3759–3762 (1998). Shay, M. A., Drake, J. F., Denton, R. E., and Biskamp, D., “Structure of the dissipation region during collisionless magnetic reconnection,” Journal of Geophysical Research103, 9165–9176 (1998). Shen, C., Lin, J., and Murphy, N. A., “Numerical Experiments on Fine Structure within Reconnecting Current Sheets in Turbulent Reconnection92 Solar Flares,” The Astrophysical Journal737, 14 (2011). Shepherd, L. S. and Cassak, P. A., “Comparison of Secondary Islands in Collisional Reconnection to Hall Reconnection,” Physical Review Letters105, 015004 (2010). Shibata, K. and Magara, T., “Solar Flares: Magnetohydrodynamic Processes,” Living Reviews in Solar Physics8, 6 (2011). Shibata, K. and Tanuma, S., “Plasmoid-induced-reconnection and fractal reconnection,” Earth, Planets, and Space53, 473– 482 (2001). Shu, F. H., Adams, F. C., and Lizano, S., “Star formation in molecular clouds: observation and theory.” Annual Review of Astronomy & Astrophysics25, 23–81 (1987). Shu, F. H., Galli, D., Lizano, S., and Cai, M., “Gravitational Collapse of Magnetized Clouds. II. The Role of Ohmic Dissipation,” The Astrophysical Journal647, 382–389 (2006), ar Xiv:astro-ph/0604574 [astro-ph]. Shu, F. H., Li, Z.-Y., and Allen, A., “Does Magnetic Levitation or Suspension Define the Masses of Forming Stars?” The Astrophysical Journal601, 930–951 (2004), ar Xiv:astro-ph/0311426 [astro-ph]. Singh, C. B., de Gouveia Dal Pino, E. M., and Kadowaki, L. H. S., “On the Role of Fast Magnetic Reconnection in Accreting Black Hole Sources,” The Astrophysical Journal Letters799, L20 (2015). Singh, C. B., Garofalo, D., and de Gouveia Dal Pino, E. M., “Magnetic reconnection and Blandford-Znajek process around rotating black holes,” Monthly Notices of the Royal Astronomical Society478, 5404–5409 (2018), ar Xiv:1807.04519 [astro-ph. HE]. Singh, C. B., Mizuno, Y., and de Gouveia Dal Pino, E. M., “Spatial Growth of Current-driven Instability in Relativistic Rotating Jets and the Search for Magnetic Reconnection,” The Astrophysical Journal824, 48 (2016). Singh, K., Shibata, K., Nishizuka, N., and Isobe, H., “Chromospheric anemone jets and magnetic reconnection in partially ionized solar atmosphere,” Physics of Plasmas18, 111210 (2011). Sironi, L. and Spitkovsky, A., “Relativistic Reconnection: An Efficient Source of Non-thermal Particles,” The Astrophysical Journal Letters783, L21 (2014). Sreenivasan, K. R., “On the scaling of the turbulence energy dissipation rate,” The Physics of fluids27, 1048–1051 (1984). Sreenivasan, K. R., “An update on the energy dissipation rate in isotropic turbulence,” Physics of Fluids10, 528–529 (1998). Stanimirovi ́ c, S. and Lazarian, A., “Velocity and Density Spectra of the Small Magellanic Cloud,” The Astrophysical Journal Letters551, L53–L56 (2001). Strauss, H. R., “Hyper-resistivity produced by tearing mode turbulence,” Physics of Fluids29, 3668–3671 (1986). Striani, E., Mignone, A., Vaidya, B., Bodo, G., and Ferrari, A., “MHD simulations of three-dimensional resistive reconnec- tion in a cylindrical plasma column,” Monthly Notices of the Royal Astronomical Society462, 2970–2979 (2016). Sturrock, P. A., “Model of the High-Energy Phase of Solar Flares,” Nature211, 695–697 (1966). Subramanian, K., “Unified Treatment of Small- and Large-Scale Dynamos in Helical Turbulence,” Physical Review Letters 83, 2957–2960 (1999). Subramanian, K., Shukurov, A., and Haugen, N. E. L., “Evolving turbulence and magnetic fields in galaxy clusters,” Monthly Notices of the Royal Astronomical Society366, 1437–1454 (2006). Sweet, P. A., “The topology of force-free magnetic fields,” The Observatory78, 30–32 (1958). Sych, R., Karlick ́ y, M., Altyntsev, A., Dud ́ ık, J., and Kashapova, L., “Sunspot waves and flare energy release,” Astronomy & Astrophysics577, A43 (2015). Sych, R., Nakariakov, V. M., Karlicky, M., and Anfinogentov, S., “Relationship between wave processes in sunspots and quasi-periodic pulsations in active region flares,” Astronomy & Astrophysics505, 791–799 (2009). Syrovatskii, S. I., “Pinch sheets and reconnection in astrophysics,” Annual Review of Astronomy & Astrophysics19, 163– 229 (1981). Takamoto, M., “Evolution of Relativistic Plasmoid Chains in a Poynting-dominated Plasma,” The Astrophysical Journal775, 50 (2013). Takamoto, M., “Evolution of three-dimensional relativistic current sheets and development of self-generated turbulence,” Monthly Notices of the Royal Astronomical Society476, 4263–4271 (2018). Takamoto, M., Inoue, T., and Lazarian, A., “Turbulent Reconnection in Relativistic Plasmas and Effects of Compressibility,” The Astrophysical Journal815, 16 (2015). Takamoto, M. and Lazarian, A., “Compressible Relativistic Magnetohydrodynamic Turbulence in Magnetically Dominated Plasmas and Implications for a Strong-coupling Regime,” The Astrophysical Journal Letters831, L11 (2016). Takamoto, M. and Lazarian, A., “Strong coupling of Alfv ́ en and fast modes in compressible relativistic magnetohydrody- namic turbulence in magnetically dominated plasmas,” Monthly Notices of the Royal Astronomical Society472, 4542– 4550 (2017). Tennekes, H. and Lumley, J. L.,A first course in turbulence(MIT press, 1972). Thompson, C. and Blaes, O., “Magnetohydrodynamics in the extreme relativistic limit,” Physical Review D57, 3219–3234 (1998). Troland, T. H. and Heiles, C., “Interstellar Magnetic Field Strengths and Gas Densities: Observational and Theoretical Perspectives,” The Astrophysical Journal301, 339 (1986). Uchiyama, Y. and Aharonian, F. A., “Fast Variability of Nonthermal X-Ray Emission in Cassiopeia A: Probing Electron Acceleration in Reverse-Shocked Ejecta,” The Astrophysical Journal Letters677, L105 (2008). Uhm, Z. L. and Zhang, B., “Fast-cooling synchrotron radiation in a decaying magnetic field andγ-ray burst emission mechanism,” Nature Physics10, 351–356 (2014). Uzdensky, D. A. and Kulsrud, R. M., “Physical origin of the quadrupole out-of-plane magnetic field in Hall- magnetohydrodynamic reconnection,” Physics of Plasmas13, 062305 (2006). Uzdensky, D. A., Loureiro, N. F., and Schekochihin, A. A., “Fast Magnetic Reconnection in the Plasmoid-Dominated Regime,” Physical Review Letters105, 235002 (2010). Uzdensky, D. A. and Spitkovsky, A., “Physical Conditions in the Reconnection Layer in Pulsar Magnetospheres,” The Turbulent Reconnection93 Astrophysical Journal780, 3 (2014). Vasquez, B. J., Abramenko, V. I., Haggerty, D. K., and Smith, C. W., “Numerous small magnetic field discontinuities of Bartels rotation 2286 and the potential role of Alfv ́ enic turbulence,” Journal of Geophysical Research (Space Physics) 112, A11102 (2007). Vazquez-Semadeni, E., Passot, T., and Pouquet, A., “A Turbulent Model for the Interstellar Medium. I. Threshold Star Formation and Self-Gravity,” The Astrophysical Journal441, 702 (1995). Vech, D., Mallet, A., Klein, K. G., and Kasper, J. C., “Magnetic reconnection may control the ion-scale spectral break of solar wind turbulence,” The Astrophysical Journal Letters855, L27 (2018). Vishniac, E. T. and Cho, J., “Magnetic Helicity Conservation and Astrophysical Dynamos,” The Astrophysical Journal550, 752–760 (2001). Vishniac, E. T. and Lazarian, A., “Reconnection in the Interstellar Medium,” The Astrophysical Journal511, 193–203 (1999), ar Xiv:astro-ph/9712067 [astro-ph]. Vishniac, E. T., Pillsworth, S., Eyink, G., Kowal, G., Lazarian, A., and Murray, S., “Reconnection current sheet structure in a turbulent medium,” Nonlinear Processes in Geophysics19, 605–610 (2012). Vogt, C. and Enßlin, T. A., “A Bayesian view on Faraday rotation maps Seeing the magnetic power spectra in galaxy clusters,” Astronomy & Astrophysics434, 67–76 (2005). Walker, J., Boldyrev, S., and Loureiro, N. F., “Influence of tearing instability on magnetohydrodynamic turbulence,” Physical Review E98, 033209 (2018). Wang, S. and Yokoyama, T., “Diffusion regions and 3D energy mode development in spontaneous reconnection,” Physics of Plasmas26, 072109 (2019), ar Xiv:1906.03964 [astro-ph. SR]. Wang, S., Yokoyama, T., and Isobe, H., “Three-dimensional MHD Magnetic Reconnection Simulations with a Finite Guide Field: Proposal of the Shock-evoking Positive-feedback Model,” The Astrophysical Journal811, 31 (2015), ar Xiv:1508.03140 [astro-ph. SR]. Watson, P. G., Oughton, S., and Craig, I. J. D., “The impact of small-scale turbulence on laminar magnetic reconnection,” Physics of Plasmas14, 032301 (2007). Werner, G. R., Uzdensky, D. A., Cerutti, B., Nalewajko, K., and Begelman, M. C., “The extent of power-law energy spectra in collisionless relativistic magnetic reconnection in pair plasmas,” Ar Xiv e-prints (2014). Wyper, P. F. and Pontin, D. I., “Dynamic topology and flux rope evolution during non-linear tearing of 3D null point current sheets,” Physics of Plasmas21, 102102 (2014). Xu, S., Garain, S. K., Balsara, D. S., and Lazarian, A., “Turbulent Dynamo in a Weakly Ionized Medium,” The Astrophysical Journal872, 62 (2019a). Xu, S., Klingler, N., Kargaltsev, O., and Zhang, B., “On the Broadband Synchrotron Spectra of Pulsar Wind Nebulae,” The Astrophysical Journal872, 10 (2019b), ar Xiv:1812.10827 [astro-ph. HE]. Xu, S. and Lazarian, A., “Turbulent Dynamo in a Conducting Fluid and a Partially Ionized Gas,” The Astrophysical Journal 833, 215 (2016). Xu, S. and Lazarian, A., “Magnetic Field Amplification in Supernova Remnants,” The Astrophysical Journal850, 126 (2017a), ar Xiv:1710.07717 [astro-ph. HE]. Xu, S. and Lazarian, A., “Magnetohydrodynamic turbulence and turbulent dynamo in partially ionized plasma,” New Journal of Physics19, 065005 (2017b). Xu, S., Lazarian, A., and Yan, H., “The Line Width Difference of Neutrals and Ions Induced by MHD Turbulence,” The Astrophysical Journal810, 44 (2015). Xu, S. and Yan, H., “Cosmic-Ray Parallel and Perpendicular Transport in Turbulent Magnetic Fields,” The Astrophysical Journal779, 140 (2013). Xu, S., Yan, H., and Lazarian, A., “Damping of Magnetohydrodynamic Turbulence in Partially Ionized Plasma: Implications for Cosmic Ray Propagation,” The Astrophysical Journal826, 166 (2016). Xu, S., Yang, Y.-P., and Zhang, B., “On the Synchrotron Spectrum of GRB Prompt Emission,” The Astrophysical Journal 853, 43 (2018). Xu, S. and Zhang, B., “Adiabatic Non-resonant Acceleration in Magnetic Turbulence and Hard Spectra of Gamma-Ray Bursts,” The Astrophysical Journal Letters846, L28 (2017). Yamada, M., Kulsrud, R., and Ji, H., “Magnetic reconnection,” Reviews of Modern Physics82, 603–664 (2010). Yan, H. and Lazarian, A., “Cosmic-Ray Propagation: Nonlinear Diffusion Parallel and Perpendicular to Mean Magnetic Field,” The Astrophysical Journal673, 942–953 (2008), ar Xiv:0710.2617 [astro-ph]. Yokoyama, T. and Shibata, K., “Magnetic reconnection as the origin of X-ray jets and Hαsurges on the Sun,” Nature375, 42–44 (1995). Zenitani, S., Hesse, M., and Klimas, A., “Two-Fluid Magnetohydrodynamic Simulations of Relativistic Magnetic Recon- nection,” The Astrophysical Journal696, 1385–1401 (2009). Zenitani, S. and Hoshino, M., “The generation of nonthermal particles in the relativistic magnetic reconnection of pair plasmas,” The Astrophysical Journal Letters562, L63 (2001). Zenitani, S. and Hoshino, M., “The Role of the Guide Field in Relativistic Pair Plasma Reconnection,” The Astrophysical Journal677, 530–544 (2008). Zhang, B. and Yan, H., “The Internal-collision-induced Magnetic Reconnection and Turbulence (ICMART) Model of Gamma-ray Bursts,” The Astrophysical Journal726, 90 (2011). Zhang, B. and Zhang, B., “Gamma-Ray Burst Prompt Emission Light Curves and Power Density Spectra in the ICMART Model,” The Astrophysical Journal782, 92 (2014). Zhang, W., Mac Fadyen, A., and Wang, P., “Three-Dimensional Relativistic Magnetohydrodynamic Simulations of the Kelvin-Helmholtz Instability: Magnetic Field Amplification by a Turbulent Dynamo,” The Astrophysical Journal Letters 692, L40–L44 (2009). Turbulent Reconnection94 Zrake, J. and Mac Fadyen, A. I., “Numerical Simulations of Driven Relativistic Magnetohydrodynamic Turbulence,” The Astrophysical Journal744, 32 (2012). Zrake, J. and Mac Fadyen, A. I., “Spectral and Intermittency Properties of Relativistic Turbulence,” The Astrophysical Journal Letters763, L12 (2013). Zuckerman, B. and Evans, N. J., I., “Models of Massive Molecular Clouds,” The Astrophysical Journal Letters192, L149 (1974). Zweibel, E. G., “Ambipolar Drift in a Turbulent Medium,” The Astrophysical Journal567, 962–970 (2002). Zweibel, E. G. and Yamada, M., “Magnetic Reconnection in Astrophysical and Laboratory Plasmas,” Annual Review of Astronomy & Astrophysics47, 291–332 (2009).