Current sheets formed in magnetic reconnection events are found to be unstable to highwavenumber perturbations. The instability is very fast: its maximum growth rate scales as S 1/4 LCS/vA, where LCS is the length of the sheet, vA the Alfvén speed and S the Lundquist number. As a result, a chain of plasmoids (secondary islands) is formed, whose number scales as S 3/8 .PACS numbers: 52.35. Vd, 52.35.Py, 94.30.cp, 96.60.Iv Magnetic reconnection is a plasma phenomenon in which oppositely directed magnetic field lines are driven together, break and rejoin in a topologically different configuration. It is an essential element in our understanding of the solar flares and the geotail [1,2,3], where it is directly observed [4,5], as well as of other astrophysical plasmas. On Earth, it plays a crucial role in the dynamics of magnetically confined plasmas in fusion devices [6].There are two standard reconnection models: the Sweet-Parker (SP) model [7,8], and the Petscheck model [9]. The latter is very appealing as it predicts fast reconnection rates similar to the observed ones. However, numerical simulations have consistently failed to reproduce it unless a spatially inhomogeneous anomalous resistivity is used [10] or Hall physics is invoked [3,11]. In contrast, the SP reconnection, characterized by long current sheets (∼ system size) and slow reconnection rates ∝ η 1/2 , where η is the plasma resistivity, is routinely observed both in experiments [12] and in simulations.The break up of current sheets and formation of plasmoids (secondary islands) appears to be a generic feature of reconnecting systems. Plasmoids have been observed both in solar flares [13] and in the geotail [14]. In numerical simulations, plasmoid formation has been reported in many different set ups, from fluid [15,16] to fully kinetic [17,18]. Plasmoids have been popular in theories of magnetic reconnection and related phenomena: e.g., they have been invoked as a plausible mechanism for accelerating reconnection, either by decreasing the effective length of the SP current sheet and/or by triggering anomalous-resistivity mechanisms associated with small-scale plasma effects [19]; a multiple-plasmoid scenario has been suggested to explain the production of energetic electrons during reconnection events [20]; it has been conjectured that periodic ejection of plasmoids in star-disk systems could account for the knot-like structures observed in stellar jets [21].A theoretical understanding of the mechanism whereby the plasmoids are formed has, however, been lacking. It has been believed that plasmoid formation is due to a standard tearing instability [22] of the current sheet, which implies a resistively slow instability growth rate [23]. In this Letter, we show that, while the resistivity is essential for the instability, the growth rate is, in fact, much faster even than the ideal rate.Theoretical estimates for the speed of magnetic reconnection in natural systems are usually based on the idea that a current sheet is formed (Fig. 1), whose length L C...
A conceptual model of resistive magnetic reconnection via a stochastic plasmoid chain is proposed. The global reconnection rate is shown to be independent of the Lundquist number. The distribution of fluxes in the plasmoids is shown to be an inverse square law. It is argued that there is a finite probability of emergence of abnormally large plasmoids, which can disrupt the chain (and may be responsible for observable large abrupt events in solar flares and sawtooth crashes). A criterion for the transition from magnetohydrodynamic to collisionless regime is provided.PACS numbers: 52.35. Vd, 94.30.cp, 96.60.Iv, 52.35.Py Introduction. Magnetic reconnection is the process of topological rearrangement of magnetic field, resulting in a conversion of magnetic energy into various forms of plasma energy [1]. It is believed to cause solar flares and has been studied in tokamaks [2], dedicated laboratory experiments [3] and measured in situ in the Earth's magnetosphere [4]. The basic conceptual underpinnings of the modern understanding of resistive reconnection can be summarised in three points: (i) generic X-point configurations are unstable and collapse into current layers [5,6]; (ii) the structure of resistive current layers is well described by the Sweet-Parker (SP) model [7]: if B 0 is the upstream magnetic field, V A = B 0 / √ 4πρ is the Alfvén speed (ρ the plasma density), L the length of the layer, η the magnetic diffusivity, and S ≡ V A L/η the Lundquist number, then the layer thickness is δ ∼ L/ √ S, the outflow velocity is V A , and the reconnection rate is cE ∼ V A B 0 / √ S -"slow" because it depends on S, which is very large in most natural systems; (iii) when S exceeds a critical value S c ∼ 10 4 , the SP layers are linearly unstable [8] and break up into secondary islands, or plasmoids [9]. This fact has emerged as a defining feature of numerical simulations of reconnection as they have broken through the S c barrier [6,[9][10][11][12][13][14][15][16]. It seems that high-S reconnection generically occurs via a chain of plasmoids, born, growing, coalescing, and being ejected in a stochastic fashion [17,18]. Importantly, recent numerical evidence [11,[13][14][15][16] suggests that plasmoid reconnection is "fast", i.e., independent of S.
A numerical study of magnetic reconnection in the large-Lundquist-number (S), plasmoiddominated regime is carried out for S up to 107 . The theoretical model of Uzdensky et al. [Phys. Rev. Lett. 105, 235002 (2010)] is confirmed and partially amended. The normalized reconnection rate isẼ eff ∼ 0.02 independently of S for S ≫ 10 4 . The plasmoid flux (Ψ) and half-width (wx) distribution functions scale as f (Ψ) ∼ Ψ −2 and f (wx) ∼ w −2x . The joint distribution of Ψ and wx shows that plasmoids populate a triangular region wx Ψ/B0, where B0 is the reconnecting field. It is argued that this feature is due to plasmoid coalescence. Macroscopic "monster" plasmoids with wx ∼ 10% of the system size are shown to emerge in just a few Alfvén times, independently of S, suggesting that large disruptive events are an inevitable feature of large-S reconnection.
A detailed numerical study of magnetic reconnection in resistive MHD for very large, previously inaccessible, Lundquist numbers (10(4)
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