"Ideal" tearing and the transition to fast reconnection in the weakly collisional MHD and EMHD regimes
aa r X i v : . [ phy s i c s . p l a s m - ph ] O c t “Ideal” tearing and the transition to fast reconnection in the weakly collisional MHDand EMHD regimes Daniele Del Sarto
Institut Jean Lamour, UMR 7198 CNRS - Universit´e de Lorraine, BP 239 F-54506 Vandoeuvre-le-Nancy, France ∗ Fulvia Pucci
Dipartimento di Fisica, Universit`a di Roma-Tor Vergata, Roma, Italy
Anna Tenerani and Marco Velli
Earth, Planetary and Space Sciences, UCLA, Los Angeles, USA (Dated: October 7, 2018)This paper discusses the transition to fast growth of the tearing instability in thin current sheets inthe collisionless limit where electron inertia drives the reconnection process. It has been previouslysuggested that in resistive MHD there is a natural maximum aspect ratio (ratio of sheet length andbreadth to thickness) which may be reached for current sheets with a macroscopic length L , thelimit being provided by the fact that the tearing mode growth time becomes of the same order as theAlfv´en time calculated on the macroscopic scale ( Pucci and Velli (2014) [54]). For current sheetswith a smaller aspect ratio than critical the normalized growth rate tends to zero with increasingLundquist number S , while for current sheets with an aspect ratio greater than critical the growthrate diverges with S . Here we carry out a similar analysis but with electron inertia as the termviolating magnetic flux conservation: previously found scalings of critical current sheet aspect ratioswith the Lundquist number are generalized to include the dependence on the ratio d e /L where d e is the electron skin depth, and it is shown that there are limiting scalings which, as in the resistivecase, result in reconnecting modes growing on ideal time scales. Finite Larmor Radius effects arethen included and the rescaling argument at the basis of “ideal” reconnection is proposed to explainsecondary fast reconnection regimes naturally appearing in numerical simulations of current sheetevolution. ∗ [email protected] I. INTRODUCTION
Magnetic reconnection is thought to be the mechanism underlying many explosive phenomena observed in bothspace and laboratory plasmas, ranging from magnetospheric substorms, to solar flares and coronal mass ejections, tothe sawtooth crashes observed in tokamaks. The classic picture of reconnection involves current sheets, most oftenassumed to be planar-like and concentrated more narrowly in the third dimension. Often, a guide magnetic field lieswithin the current sheet itself, so that the actual three-dimensional field does not vanish in the sheet. Different modelsfor reconnection occurring in such quasi-2D configurations have been developed, two prominent, different examplesbeing the Sweet-Parker (SP) stationary reconnection scenario and the spontaneous reconnecting modes naturallydeveloping due to the tearing instability of the current sheet itself.
Biskamp (1986) [10] first pointed out the importantrole played by the current sheet aspect-ratio in determining whether a stationary reconnection configuration couldbe reached. He found, via numerical simulations, that the SP current sheet could become unstable to reconnectingmodes once a critical value of the Lundquist number (estimated on the current sheet length or breadth, L ) of about S ≃ was exceeded. A detailed examination of the stability of the SP configuration led to the definition of theplasmoid-chain instability [41], reminiscent of the plasmoid-induced reconnection concept and fractal reconnectionmodels introduced by Shibata et al. (2001) [62]. Recently,
Pucci and Velli (2014) [54] have pointed out that thedivergence of the growth rate of the plasmoid chain instability in the limit of large Lundquist number within resistiveMHD implies that current sheets should never elongate sufficiently to achieve the SP aspect ratio. They have shownthat a critical aspect ratio separates slowly unstable current sheets (with growth rate scaling as a negative, fractionalexponent of the Lundquist number) from violently unstable ones (growth rates scaling with a postive power of S ).They dubbed the instability of the critically unstable current sheet “ideal tearing” (hereafter IT), because the growthrate, normalized to the Alfv´en time along the sheet L , becomes of order unity, and independent of the Lundquistnumber itself.The large predicted growth rates and the presence of critical values for dimensionless numbers such as current-sheet aspect ratio make the described instabilities good candidates to understand and model the mechanisms behindobserved fast reconnection phenomena [68]. Indeed, to date there is no agreed theoretical explanation for the fast timescales over which reconnection events develop in nature, nor for their triggering, while evidence from both experimentsand numerical simulations points to the importance of small scale formation and kinetic effects, [9, 20, 43] which aretheoretically expected to lead to Alfv´enic (or “ideal”) reconnection in 3D configurations as well [13]. Moreover,numerical simulations of tearing mode instabilities have identified a secondary, nonlinear, increase of the reconnectionrate, that has been sometimes interpreted in terms of a nascent plasmoid-unstable SP regime [1, 40] or genericallya secondary “explosive reconnection” regime [9]. A nonlinear increase of the reconnection rate on ideal, Alfv´enictime-scales was also numerically measured by Yu. et al. (2014) [72] in simulations of low mode-number reconnectioninstabilities. Given the recent developments of the theory of large-aspect ratio current sheet instabilities, it is importantto understand whether such augmented fast reconnection rates may indeed be interpreted as fast secondary instabilitiesof the nonlinearly generated current sheets stemming from the primary reconnection event. Specifically, given thatkinetic and two-fluid effects easily become dominant compared to classical, collisional resistivity at small spatial scales,it seems timely to see whether and how such effects modify the transition to an IT regime.The present paper focuses on the extension of the IT scaling arguments to weakly collisional regimes where recon-nection is mediated by electron inertia effects, and on whether such generalized IT regimes might explain the nonlinearoccurrence of fast exponentially growing reconnection rates. We will consider both the incompressible reduced MHD(RMHD − see e.g. [74]) and electron MHD (EMHD [34]) frequency ranges, where the perturbations are dominatedby Alfv´en and whistler modes respectively. The formal similarity between RMHD and EMHD reconnection in slabgeometry, previously discussed in [21, 23], allows a unified treatment for the onset of IT in an electron-inertia drivenframework.Electron inertia has long been considered the most promising alternative to standard resistive reconnection thanks toits greater weight with respect to resistivity in the generalized Ohm’s law of quasi-collisionless plasmas [18, 19, 51, 70].Astrophysical and thermonuclear fusion plasmas are examples of such systems, since their particle mean free pathtipically exceeds the characteristic hydrodynamic lengths by order(s) of magnitude. In general, interspecies collisionsmay be neglected with respect to inertial terms when the characteristic ion-electron collision frequency is negligiblewith respect to the inverse time scale of the phenomena considered [32, 46, 52]. The inertial slab RMHD regimewe focus on here has indeed been widely used to model basic features of magnetic reconnection in tokamak devices,for which the strong guide field approximation, of which we consider the 2D-geometry limit, was first devised, aswell as for astrophysical applications. In EMHD the neglect of collisional resistivity is even more justified, which iswhy EMHD reconnection is mostly studied in purely inertia-driven regimes (see [32] for a discussion of the transitionfrom resistive to inertial EMHD). Because of the large characteristic frequencies involved, EMHD provides a naturalframework for collisionless reconnection. The relation between the convection electron flow and the magnetic field,typical of the EMHD regime, plays a prominent role in explaining the quadrupolar structure of the out-of-planemagnetic field [7], which is often recognized as a distinctive signature for the in situ detection of magnetosphericreconnection [44]. Rogers et al. (2001) [58] also adopted the incompressible, inertia-less, collisionless EMHD modelto explain the opening-up of the reconnection layer in 2D simulations with no guide field. We finally note that thepresent paper does not cover the framework of the so-called Hall- or whistler- mediated reconnection (Appendix A 1),especially relevant to the magnetopause environment [11, 66], and which is known to provide prominent examples offast reconnection rates weakly dependent from both resistivity [42] and electron inertia [12]. This will be consideredin future works.The paper is structured as follows. In Sec.II we summarize the re-scaling arguments leading to the concept of“ideal tearing”. In Sec.III we introduce the model equations for reconnection in the RMHD and EMHD regimes andthe relevant dispersion relations (Sec.III A). In Sec.IV we extend the IT paradigm first to the inertial RMHD andEMHD reconnection regimes (Sec.IV A) and then to include finite Larmor radius (FLR) effects (Sec.IV B). We thendiscuss these results (Sec.V) by comparing the role of inertia to that of resistivity in different natural and laboratoryplasmas (Sec.V A), and by considering an application of the IT model to collisionless steady reconnecting currentsheets (Sec.V B). Then, in Sec.V C we discuss how the re-scaling argument might explain explosive reconnectionregimes nonlinearly observed in simulations of magnetic reconnection. Sec.VI provides a summary and conclusion andin the Appendix A we recall the derivation of the model equations both from a two-fluid model and compared withthe generalized Ohm’s law (Sec.A 1).
II. THE IDEAL TEARING MODEL
Consider a current sheet of length L and thickness a . As MHD is scale-free, in the classical tearing mode theory it iscustomary to take the width a as normalization length, since typically L/a > a is the only characteristic lengthdefined by the (usually 1D) equilibrium profile. However, when dealing with thin sheets with a arbitrarily small,the distinction between L and a becomes important, as the tearing mode growth rate is only small when measuredwith respect to the “ideal” Alfv´en timescale based on a , but can become large when measured with respect to amacroscopic scale L >> a (the basic idea behind the plasmoid instability and IT, detailed below). From now on,we will label quantities normalized to the scale L with the apex “ ∗ ”, using standard notation for non-dimensionalquantities defined in terms of the (possibly microscopic) shear-scale a .In this notation, the classical linear reconnecting mode on Harris-type current sheets has a maximal growth ratescaling as γ M τ A ∼ S − / where the Lundquist number S = aV A /η m and τ A = a/V A , with V A the Alfv´en speed based onthe characteristic magnetic field strength far from the sheet. In the SP case, predicated on the renormalized Lundquistnumber S ∗ = LV A /η m , one finds immediately that γ M τ ∗ A = γ M L/V A ∼ S ∗ / , i.e. a growth rate which diverges withthe macroscopic Lundquist number S ∗ . Pucci and Velli (2014) [54], aiming to resolve this paradox, incompatible withthe ideal MHD limit, studied large-aspect ratio current sheets with
L/a scaling as a positive fractional power of theLundquist number S ∗ = LV A /η m ≫
1. They showed that when a threshold
L/a ∼ ( S ∗ ) α (1 / > α >
0) is reached,the resistive tearing mode growth rate γ M τ ∗ A becomes of order unity and independent of S ∗ . This regime was named“ideal tearing”, in contrast to the CT theory in which the growth rates scale as a negative power of the a -normalizedLundquist number S . The large aspect ratio limit allowed [54] to evaluate the characteristic CT reconnection ratethrough the fastest growing mode, from which the value α = 1 / S ∗ (different equilibrium profiles may induce small deviations from this value[55]). The renormalization in fact gives γ M τ ∗ A ∼ ( S ∗ ) − / ( L/a ) / (1)and the clock whose rate defines the reconnection speed enters this renormalized theory through τ ∗ A which dependsitself on L , i.e., the clock set on the ideal scale L results slower by a factor a/L (or, as we shall see, ( a/L ) in theEMHD regime) than the clock with which the reconnection rate is measured in the CT theory: it is thus alwayspossible to find a critical exponent α > γ M τ ∗ A ≃ L/a ) ∼ ( S ∗ ) α is imposed. Inother words, the tearing-mode theory, under the assumption of a current sheet whose aspect ratio scales as a powerof the (small) non-ideal parameter ε ∗ which allows reconnection, say a/L ∼ ( ε ∗ ) α , can explain the transition to fastreconnection if the value of α is such that the growth rate of the instability is independent from ε ∗ itself. Noticehowever that the IT criterion may be applied in principle to any reconnection unstable aspect ratio L/a , if L is largeenough with respect to a . It is e.g. the case of tearing unstable current sheets, nonlinearly developed by primaryreconnection events, which we will consider later. We now consider how this happens once electron inertia first, andFLR-type effects second, are taken into account. III. MODEL EQUATIONS
We restrict our analysis to a 2D system in the ( x, y ) plane, and assume for simplicity an electron-proton plasma.Consider the incompressible equations in slab-geometry. We adopt the standard “poisson-bracket” representation[ f, g ] ≡ ∂ x f ∂ y g − ∂ y f ∂ x g = e z · ( ∇ f × ∇ g ). The velocity stream functions ϕ and b are such that U ⊥ = − ∇ ϕ × e z in RMHD and u e ⊥ = − ∇ b × e z in EMHD (see below), where “ ⊥ ” stands for components in the ( x, y ) plane, and u e and U are the electron and bulk plasma velocities, respectively. Analogously, the magnetic stream function ψ isdefined through B = ∇ ψ ( x, y ) × e z + ( B + b ( x, y )) e z , with B uniform in space. We assume an equilibrium in-planemagnetic field B ⊥ = B y ( x/a ) e y with B y ( x/a ) = ∂ x ψ ( x/a ). Equilibrium quantities are labeled with “0”, and weintroduce the fields F ≡ ψ − d e ∇ ψ and W ≡ b − d e ∇ b . Here d e = c/ω pe is the electron-skin-depth.Using a as the reference length and characteristic quantities B ⊥ and n for magnetic field and densities, the modelequations may then be written in non-dimensional form either as: ∂∂t F + [ ϕ, F ] = ρ s [ ∇ ϕ, ψ ] + S − ∇ ψ (2) ∂∂t ∇ ϕ + [ ϕ, ∇ ϕ ] = [ ψ, ∇ ψ ] + R − ∇ ϕ , (3)valid in the RMHD frequency range, or ∂∂t F + [ b, F ] = S − Emhd ∇ ψ (4) ∂∂t W + [ b, W ] = [ ψ, ∇ ψ ] + S − Emhd ∇ b , (5)valid in the EMHD frequency range.In the above, time is normalized to τ A ≡ ( a/d i )Ω − i in RMHD, where Ω i is the ion cyclotron frequency and d i ≡ √ m i c/ ( √ m e ω pe ) is the ion-skin depth ( ω pe being the usual plasma frequency and with obvious notation forthe masses); in EMHD time is normalized to the inverse of the whistler frequency, τ W ≡ ( a/d e ) Ω − e = ( a/d i ) Ω − i .The other parameters on which the tearing reconnection rate depends are the ion-sound Larmor radius, also non-dimensionalized with a i.e. ρ s ≡ c is /a Ω i , where c is is the ion sound speed, i.e. the thermal speed based on electrontemperature and ion mass; R ≡ ( ν ii τ A ) − (Reynold’s number) with ν ii the ion-ion viscosity; S ≡ τ D /τ A (Alfv´enicLundquist number) and S Emhd ≡ τ D /τ W (EMHD Lundquist number) with τ D = 4 πa / ( ηc ) the resistive diffusion time( η is the scalar resistivity). The physical meaning of the terms of Eqs.(2-5) and their relation to both the two-fluidmodel equations and the generalized Ohm’s law are discussed in Appendix A.Note that, calling L MHD and L EMHD the normalization lengths in RMHD and EMHD, the inequality τ ∗ W τ ∗ A = (cid:18) L EMHD d i (cid:19) (cid:18) L EMHD L MHD (cid:19) ≪ , (6)must hold since the characteristic quantities in EMHD must be much smaller than d i and those of RMHD much largerthan d i . A. Linear dispersion relations
We now focus on the collisionless regimes, S − = S − EMHD = 0; we will not consider viscous effects, whose role inMHD has been clarified recently by
Tenerani et al. (2015) [64]. In addition, to further simplify the analysis, we startby setting ρ s = 0 in Eqs.(2)-(5). Because of the fact that both the (squared) electron skin depth and the Lundquistnumber weigh non ideal terms in Ohm’s law which allow magnetic lines to reconnect (Appendix A), and of othersimilarities which will be later discussed, let us introduce for future use the notations ε d ≡ d e and ε S ≡ S − . Then,after re-scaling, we will write ε ∗ d = ε d (cid:16) aL (cid:17) , ε ∗ S = ε S (cid:16) aL (cid:17) . (7)After linearizing Eqs.(2)-(3) around an equilibrium ψ ( x/a ) with perturbations of the form ∼ e iky + γt , analyticapproximations to the dispersion relations in both RMHD and EMHD may be obtained by applying the boundarylayer technique, as first shown by Furth et al. (1963) [27].Here we summarize the results valid in the two asymptotic regimes called large (LD) and small (SD) ∆ ′ , whichrespectively correspond to the internal kink and constant- ψ orderings [3]. In RMHD such regimes are respectivelydefined by the conditions ∆ ′ δ > ′ δ < δ is the characteristic reconnection layer width.The inertial RMHD tearing dispersion relations become (see e.g. [51]):RMHD (cid:26) γ LD τ A = kd e γ SD τ A = ( C ∆ ′ ) kd e , (8)where C ≡ Γ(1 / / (2 π Γ(3 / ≃ . γ LD /k ∼ constant , we consider thedispersion relations EMHD ( γ LD τ W = C kd e γ SD τ W = ( C ∆ ′ ) d e , (9)where C ≡ (2Γ (3 / − / ≃ . γ LD growth rate above, which is the one evaluated by Attico et al. (2000) [4] starting from an equilibrium given by ψ ( x ) = x/a for − a < x < a and ψ ( x ) = 1 for | x | ≥ a , has been assumedas the prototype for the more general “LD” EMHD dispersion relation for a generic sheared, even, ψ ( x ) profile. Thereason is that this is the only available formula obtained for this wavelength regime, and, with the same equilibrium,the general γ SD dispersion relation first computed in [14] and quoted in Eq.(9), was exactly recovered.For illustrative purposes in Fig.1 we show the scaling of the growth rate of a given unstable mode ˜ k as a functionof ε d in the RMHD regime. Notice that the whole range of regimes from SD to LD is spanned while varying the valueof d e at given k . Indeed, since δ = δ ( k, ε d ), an interval in the ε d parameter space such that ∆ ′ (˜ k ) δ (˜ k, ε d ) is smaller(SD), equal ( γ M , see Sec.IV A), or greater (LD) than unity, always exists. −6 −5 −4 −3 −2 −1 −6 −5 −4 −3 −2 −1 γ τ A ε d1/2 ε d3/2 k=0.02 FIG. 1. Scaling of γ (˜ k ) τ A in the RMHD regime as a function of d e for a fixed ˜ k . At the increase (decrease) of d e the small(large) ∆ ′ regime is progressively entered. Here ˜ k = k M for d e ≃ × − (lengths in units of a ). As a comment, note that almost ideal growth rates (saturating at ( γ EMHDLD ) ∗ ≃ . τ ∗ W ) − ) were observed innumerical integrations of the EMHD linear system at 0 . . d e < L/a = 2 π and k ∗ = k = 1. Such largevalues of d e are not unreasonable in the collisionless EMHD regime, because of the constraint d e ≪ a ≪ d i (now indimensional units), which must be fulfilled by the equilibrium shear length. With such large values of the reconnectionparameter, we are outside the realm of the asymptotic/boundary layer analysis, but for EMHD this is to be expected,since characteristic EMHD scale lengths must satify ℓ fulfill d e ≪ ℓ , or, given that d i /d e ≃ Z for an ion charge Z , d e ≪ ℓ ≪ d e Z . Similarly large growth rates are found in strongly resistive RMHD regimes S − & .
01, though theseare normally of little interest. Discrepancies with analytical estimations from Eqs.(7)-(8), suggest that at ε d ∼ . ε S ∼ .
01 the boundary layer approach to the linear tearing breaks down.
IV. RESULTSA. Transition to the inertial ideal regime
When
L/a ≫
1, say,
L/a &
20 [67], we can search for the fastest unstable mode k M with corresponding growthrate γ M . As noticed by Battacharjee et al. (2009) [6], the latter can be estimated by imposing the condition γ LD ( k M ) = γ SD ( k M ) ≡ γ M . Approximating ∆ ′ ( k M ) ≃ Kk − p M where K is a constant, from Eqs.(8) and Eqs.(9) we canestimate (see also [55]): RMHD ( k M ≃ ( KC ) p d p e γ M τ A ≃ ( KC ) d pp e , (10)EMHD k M ≃ (cid:18) K C C (cid:19) p d p ) e γ M τ W ≃ ( KC C p ) p d
23 3+2 p p e . (11)Let us now apply the re-scaling argument to evaluate, from Eqs.(10-11) and from the definitions of τ A and τ W , thescaling of the most unstable mode when lengths are normalized to L . Neglecting the numerical coefficients in theparentheses of Eqs.(10-11) we find in RMHD, k ∗ M ≃ ( ε ∗ d ) p (cid:18) La (cid:19) p γ M τ ∗ A ≃ ( ε ∗ d ) p p (cid:18) La (cid:19) pp , (12)and in EMHD k ∗ M ≃ ( ε ∗ d ) p (cid:18) La (cid:19) p γ M τ ∗ W ≃ ( ε ∗ d ) p p (cid:18) La (cid:19) p p . (13)In the RMHD regime it is easy to verify from the analytical estimates δ LD ∼ d e and δ SD ∼ ∆ ′ d e (see e.g. [46]) thatthe fastest growing mode satisfies the condition ∆ ′ ( k M ) δ ( k M ) ∼
1. The characteristic width of the reconnection layerfor the most unstable RMHD mode therefore becomes δ M ≃ d e , (14)which, after rescaling, reads δ ∗ M ≃ ( ε ∗ d ) .The condition for “ideal” tearing is set by searching for the value of α such that when a/L ∼ ( ε ∗ d ) α with α > γ ∗ M becomes independent of ε ∗ d = ε d a /L . Imposing this, we find the exponent α both in RMHD and EMHD,respectively, α RMHD d = 1 + p p , α EMHD d = 3 + 2 p
12 + 16 p . (15)In particular, for a Harris-pinch equilibrium, which has p = 1, we find α RMHD d = 13 , α EMHD d = 528 ≃ . . (16)A set of curves γ ( k ) for different values of d e along the RMHD threshold condition a/L = ( ε ∗ d ) / is plotted inFig.2a, while the corresponding graph for the EMHD regime is in Fig.2b. The independence of γ ∗ M from d e and itsvalue of order unity, namely ≃ . τ ∗ A ) − in RMHD and ≃ . τ ∗ W ) − in EMHD, is evidenced in both regimes.Referring to the example of the Harris-pinch profile and assuming for the EMHD the numerical threshold condition a/L = ( ε ∗ d ) , we then deduce the scalings of the threshold current sheet widths a with respect to d e , which will bediscussed in Sec.VI: (cid:18) ad e (cid:19) RMHD = (cid:18) Ld e (cid:19) , (cid:18) ad e (cid:19) EMHD = (cid:18) Ld e (cid:19) . (17) γ τ A ε d =10 − ε d =10 − ε d =10 − ε d =10 − ε d =10 − ε d =10 − γ τ W ε d =10 − ε d =10 − ε d =10 − ε d =10 − ε d =10 − FIG. 2. RMHD (left frame) and EMHD (right frame) dispersion relations γ ∗ = γ ∗ ( k ∗ , ε ∗ d ), computed for different values of d e and represented as functions of k ∗ a ∗ . For each curve an aspect ratio was chosen, satisfying the threshold condition for aHarris-pinch equilibrium, a/L = ( ε ∗ d ) / in RMHD and a/L = ( ε ∗ d ) / in EMHD. The maximum growth rate on each curve isindependent from d e and of order unity with respect to the characteristic time: γ ∗ M τ ∗ A ≃ .
39 in RMHD and γ ∗ M τ ∗ W ≃ .
37 inEMHD.
B. Kinetic effects in the transition to the inertial ideal tearing: FLR corrections
We now briefly consider the role played by other kinetic effects important at small spatial scales a ≪ L where thetransition to “ideal” tearing takes place. Since ion-ion viscosity effects have been already discussed by Tenerani et al.(2015) [64] we focus on FLR effects, which enter in our set of equations through the so-called gyrofluid corrections,an example of which is provided by the ρ s term in Eqs.(2-3).At small scales ℓ ≪ L the fluid description formally breaks down, but it has been shown that gyrofluid RMHDmodels capture the essential physics of gyrokinetic reconnection [73]. A good agreement between our collisionlessRMHD equations at ρ s = 0 and a drift-kinetic model for magnetic reconnection was already pointed out [49]. Theion-sound Larmor radius was shown to increase the inertia-driven tearing reconnection rate both linearly [47, 48, 51]and nonlinearly [15–17, 24, 29, 31, 46]. Notice that the RMHD equations have been extended to include also ion FLReffects, ρ i ≡ v ith / Ω ci ( v ith being ion thermal velocity), related to the ion-sound Larmor radius by ρ s = ρ i T e /T i . Theseeffects are usually introduced in RMHD equations, notably in Eq.(3), by making some closure assumption on theion kinetic response obtained from the transport equation. Different models are then available, but also those whosedifferent Hamiltonian properties were compared by Welbroeck et al. (2009) [69], were shown to provide numericalresults in a remarkably good agreement [24, 30]. Also notice that the isothermal assumption behind the definition of ρ s and ρ i has been shown to be in good agreement with the numerical results from gyrokinetic models for electrons,during the whole linear reconnection stage [50]. Interestingly, in a certain parameter range, the theoretically predictedscalings of tearing modes [47, 51] display a symmetric dependence on the two FLR effects, as the latter enter in thedispersion relation as powers of ρ τ = ρ s + ρ i . Even if appreciable discrepancies from these predictions are seen as theratio ρ τ /d e increases [24], at ∆ ′ d e ≫ min [1 , ( d e /ρ τ ) / ] a good agreement is found [16].In the regime ρ τ ≫ d e , Comisso et al. (2013) [17] recently pointed out the existence of a maximum growth rate inthe continuum spectrum limit (i.e. continuous k ) of unstable tearing modes, corresponding, in our notation, to k M .The generalization of the result they obtained for the Harris-pinch case to generic equilibria, is obtained as describedin Sec.IV A, by starting from their Eqs.(26)-(27) instead of our Eqs.(8). We find k M ≃ d p e ρ p τ , γ M τ A ≃ d p p e ρ p p τ . (18)Then, applying the rescaling arguments, we obtain γ FLRM τ ∗ A ∼ O (1) when aL ∼ ( ε ∗ d ) p p ( ρ ∗ τ ) . (19)We then see that, depending on the value of the ratio d e /ρ τ , the inclusion of FLR corrections may imply an evenlarger critical aspect ratio for the transition to “ideal” tearing, with respect to the cold-plasma limit. Indeed, if wenow assume ρ τ ≃ Ad e and we compare the threshold condition of Eq.(19) with that of Eq.(15) for the RMHD, wesee that, with obvious notation, the two are related through ( a/L ) FLR ∼ A / ( a/L ) RMHD . Since usually
A > A ∼
10 in tokamak plasmas and it may be even larger in the magnetosphere this implies a broadening of theideally unstable current sheet with respect to the cold plasma case, when kinetic effects are taken in account. γ τ A d e2 =10 −3 d e2 =5x10 −4 d e2 =10 −4 d e2 =10 −5 d e2 =10 −8 d e2 =10 −3 d e2 =5x10 −4 d e2 =10 −4 d e2 =10 −5 d e2 =10 −8 S −1 =10 −8 FIG. 3. Dispersion relations γ vs. k for different values of d e and for S − = 10 − (upper panel) and S − = 0 (lower panel).Some orders of magnitude of separation between the purely inertial and purely resistive growth rates (tipically about 3 −
4, atleast) are needed in order for ε S to be really negligible. V. DISCUSSIONA. Collisionless ideal tearing in space, solar and laboratory plasmas
In order to discuss the relevance of electron inertia and resistivity in various natural and laboratory environmentswhere low-collision reconnection occurs, different plasma parameters, including ε ∗ S and ε ∗ d , are shown in Table A 1.We recall that the condition for purely collisionless reconnection ( S − = 0) is given by γ d τ A ε d ≫ ε S , with γ d reconnection rate of the sheer inertia-driven regime. Notice that this condition becomes less critical when approachingthe ideal regime ( a/L ≪ γ ∗ d → ε S /ε d = ( a/L ) ε ∗ S /ε ∗ d (Eqs.(7)): if ε ∗ S /ε ∗ d ≪
1, then we can assume the IT model applied to large aspect-ratio current sheets as essentiallyinertia-driven. This means, for example, that the magnetotail is in an essentially inertia-dominated tearing regime.On the other hand, fusion devices, for which a ≃ L , may operate in conditions in which the resistive contribution totearing reconnection is not negligible even if ε ∗ S /ε ∗ d ∼ ε S /ε d ∼ − − − , because of the smallness of γ d τ ∗ A , whichremains of the same order of γ d τ A ≪ ε d , enters in the the dispersion relation of tearing modes with a less favorable scalingwith respect to resistivity, ε S . For practical purposes, at a/L ∼ ε S is sufficiently small ( ε − S . − ) and ε d is at least 3 − ε S . The case in which theinertial γ d may dominate over the resistive γ S , is exemplified in Figs.3, where some examples of the inertial-resistivegrowth rate are represented, for which only an implicit analytical expression for γ is available (see e.g. Eq.(16) of[46]). The dispersion relations displayed are obtained by numerical integration of the linearized Eqs.(2-3). Theseexamples show that no appreciable differences in the inertial-resistive growth rates with ε S = 10 − are observedbetween ε d = 10 − and ε d = 10 − . At higher values of S − , both the inertial and the resistive contributions to theinertial-resistive growth rate become appreciable, and for S − & − the resistive contribution to the growth rate isrelevant even for d e approaching unity. B. Ideal tearing and stability of steady-state reconnecting current sheets in the collisionless regime
Both in MHD [70] and in EMHD [5, 14], the reconnection rate of a steady state current sheet has been evaluatedin the collisionless regime, as a generalization of the classic Sweet-Parker configuration. In both cases the samescaling in ε d of the stationary Sweet-Parker-like reconnection rate γ SP was obtained with respect to the respectivenormalization times, ( τ EMHDSP ) − τ ∗ W ∼ ( ε ∗ d ) / and ( τ RMHDSP ) − τ ∗ A ∼ ( ε ∗ d ) / . This implies that both in collisionless Notice that Wesson’s result [70] was specialized to a geometric configuration corresponding to the m = 1 mode in a tokamak, but hisreasoning is easily adapted to the standard planar sheet configuration. RMHD and EMHD, the aspect ratio scaling of a steady current sheet of length L is ( a/L ) SP ∼ ( ε ∗ d ) / . By comparingthe scaling of this ratio with the threshold conditions for the onset of “ideal” tearing (Eqs.(15)) the same qualititativebehavior, though with different scalings, is evidenced in both RMHD and EMHD. In RMHD the width of the steadyreconnecting layer corresponds to a much thinner current sheet than that which is unstable to ideal tearing: at agiven length L , the collisionless Sweet-Parker sheet width, a SP , is related to the ideal-tearing unstable one, a IT , bythe relation a ∗ SP ≃ ( a ∗ IT ) (1+2 p ) / (1+ p ) . Using the same reasoning we can estimate from Eq.(15) a ∗ SP ≃ ( a ∗ IT ) (6+8 p ) / (3+2 p ) for EMHD. If we now neglect the effect of the flow along the neutral line on the growth rate (cfr. also [54] for whyflows may be neglected), this means that both in RMHD and EMHD a collisionless Sweet-Parker-type current sheetis always unstable on ideal time scales. C. “Secondary” ideal tearing and “explosive reconnection”
The rescaling argument at the basis of ideal tearing may thus provide a fairly general paradigm to describe explosivegrowth rate increases observed in the nonlinear stage of simulations of reconnection at
L/a not much larger than unity[1, 9, 72], when an X -point collapses into two Y -points and the current sheet between the two becomes tearing unstable,eventually leading to the so-called plasmoid-chain instability. During this stage, even before an ideal growth rate isachieved, a secondary growth rate may be measured, which is arbitrarily large (possibly up to the inverse macroscopictime scale, in the ideal tearing limit).Let be L Y the length and a Y the width of a secondary current-sheet between two Y -points, generated in thenonlinear stage of the tearing of a current sheet with aspect ratio a/L . Focusing on the dynamics of this secondarycurrent sheet, the CT growth rates would refer lengths to a Y , whereas we now need to label with “ ˜ ... ” the quantitiesnormalized to L Y , since the latter plays the role of macroscopic length for the secondary dynamics (cfr. Sec.II). Evenwhen we consider a primary tearing mode with L/a &
1, the secondary current sheet develops with a much smallerthickness (corresponding to the singular layer thickness of the original tearing instability) so that we can assume themost unstable tearing mode to be destabilized: accounting for FLR effects, the re-normalized, most unstable, tearingmode growth rate on the secondary current sheet (cfr. Sec.IV B) therefore becomes dependent from the ratio L Y /a Y , γ FLRM ˜ τ A ∼ ˜ ε p p d ˜ ρ p p τ (cid:18) L Y a Y (cid:19) pp . (20)Analogously, we can rewrite the correspective for the resistive inviscid and viscous, high-Prandtl number RMHDregimes, respectively discussed in [54] and [64, 65], γ resM ˜ τ A ∼ ˜ ε p pS (cid:18) L Y a Y (cid:19) p p , (21) γ viscM ˜ τ A ∼ ˜ ε p pS ˜ R p p (cid:18) L Y a Y (cid:19) . (22)The occurrence of a secondary, ideal tearing mode developing as a consequence of a primary tearing in a largeaspect ratio current sheet in the resitive RMHD regime was first numerically evidenced by Landi et al. (2015) [38]and discussed in depth in [65].Let us now focus on a primary tearing mode of a small aspect ratio current sheet, assuming for simplicity a ∼ L . Inthis case the primary reconnection rate can not be estimated with that of the most unstable mode γ M but the specificLD or SD regime in which the unstable wave-number falls must be taken in account, instead. Let us now comparesuch primary reconnection rate to the secondary one, as estimated from Eqs.(20-22). We immediately recognize that,even before the ideal tearing threshold is reached, the re-scaling argument predicts an increase in the growth rate, measured with respect to the primary mode macroscopic scale L , by some positive power of ( L Y /a Y ) > L/L Y ) >
1. Comparing Eqs.(20-22) we see that, for equal equilibrium profiles (same p ≥ Comisso et al. (2013) [17], assuming an aspectratio so close to unity that a single primary mode m (i.e. k = 2 πm /L ) is excited in the SD, constant- ψ regime,in the whole range of parameters in which d e and ρ τ are varied (∆ ′ ρ τ d e < L , grows with γ I τ ∗ A ≃ k ∗ ( ε ∗ d ) ρ ∗ τ (∆ ′ ) ∗ ( L/a ), with some (∆ ′ ( k ∗ )) ∗ of order unity.For the secondary mode we may now use Eq.(20). Assuming for simplicity (but with no loss of generality) that the0secondary current sheet resembles a Harris-pinch profile to specify some value of p (here p = 1), the secondary growthrate, expressed again in terms of the scale L , is given by Eq.(20) opportunely rescaled, γ II τ ∗ A ∼ ( ε ∗ d ) ρ ∗ τ ( L/a Y ) . Adominant increase of the reconnection rate is therefore provided by the ratio L/a Y ≫
1. In particular, in this examplewe obtain γ ∗ II /γ ∗ I ∼ ( a/a Y )( L/a Y ) ( k ∗ ∆ ′∗ ) − .Of course, a more detailed analysis would be required to verify whether the re-scaling argument summarized byEqs.(20-22) and the corresponding threshold conditions for the ideal tearing suffice to explain the explosive reconnec-tion regimes observed in the above mentioned numerical studies. However, the qualitative considerations about thescalings provided in Fig.(3) of [8] and in Fig.(2) of [9] seem encouraging. Because of the normalization assumed inthese articles, the increase of the growth rates with decreasing plasma β implies for the linear growth rate a scaling γ I τ ∗ A ∼ d e and for the nonlinear one a scaling γ II τ ∗ A ∼ d e at fixed ρ s , thus suggesting (cfr. Eq.(18) and Eq.(20) for p = 1) that an ideal tearing regime was observed in the nonlinear stage of the simulations discussed by Biancalani etal. (2012) [9]. Future studies will elucidate whether the explosive reconnection predicted by Eq.(20) and that studiedin [9] are effectively the same phenomenon.To conclude this Section, we finally notice that, provided the ratio L Y /a Y is large enough to destabilize a mostunstable mode γ M (which is typical for secondary current sheets developed from the collapse of an X -point in theresistive regime [65]) the measured growth rate would be that of an exponentially growing instability, which in theresistive regime has the same scaling with S as the Sweet-Parker reconnection rate (i.e. ∼ S − / ). VI. SUMMARY
We have extended the analysis of [54] to collisionless regimes, both in RMHD and EMHD, by providing the scalingthreshold values a/L ∼ ( d e /L ) α at which a current sheet with L/a &
20 reconnects on the ideal macroscopic times ofthe model. For the Harris-pinch equilibrium profile the exponents measured after numerical solution of the eigenvalueproblem are α RMHD d = 1 / α EMHD d ≃ /
16, in excellent agreement with the analytical estimations obtainedby starting from the SD and LD dispersion relations. In RMHD, FLR corrections typically reduce the width of thecritical aspect ratio for the transition to “ideal” tearing. In the parameter range ∆ ′ d e ≫ min[1 , ( d e /ρ τ ) / ] and for theHarris-pinch case, such an aspect ratio becomes ( a/L ) FLR ∼ ( ε ∗ d ) / ( ρ ∗ τ ) / , instead of ( a/L ) d ∼ ( ε ∗ d ) / in the ρ s = 0limit. Since this implies a broadening of the critical reconnection current layer by a factor ( d ∗ e ) − / ( ρ ∗ τ ) / ∼ A / ,when ρ τ ≃ Ad e with A >
1, as it is usually the case, FLR effects are expected to correspondingly lower the instabilitythreshold.The collisionless IT model has been applied to discuss the instability of steady collisionless reconnecting currentsheets, which, just as in the resistive case, should not be observable as they become unstable to inertia-driven tearingmodes on ideal time-scales. We notice however that the threshold current sheet to the IT, found to be thinner inRMHD than in EMHD (Eqs.(17)), leaves the open question of how the Alfv´enic and whistler-dominated frequencyregimes relate to the Hall-MHD framework, which in principle encompasses both as two of its limits, opposite one toeach other (see Appendix A 1).We have also pointed out the relevance and importance of inertia-driven vs. resistive reconnection: the condition S − ≫ d e γ provides a stringent constraint on when resistivity may be neglected which is often overlooked, for example,when applying Vlasov models of reconnection to tokamak plasmas.We have finally discussed how the rescaling argument at the basis of the IT model may explain the “explosive”reconnection rate increase observed during the nonlinear stage of primary reconnection events, as secondary elongatedcurrent sheets are generated during the collapse of an X -point [1, 9, 40]. The IT regime may thus be in principleachieved also during secondary reconnection events involving the thin, elongated current layers nonlinearly generatedby classical tearing processes [45] or in kinetic turbulence [61]. Notice that large aspect ratio current-layers aregenerally expected to develop because of the “exponentiation” of neighboring magnetic field lines [13]), and evidenceof such exponential thinning of current sheets was recently provided, in the coronal heating context, by the numerical3D simulations of [56]. This model provides therefore a promising key to interpretate reconnection rates, which bothin laboratory and astrophysics are observed to be orders of magnitude faster than what is predicted by the CT theory.The simplicity of the rescaling argument at the basis of the IT model should not betray its non trivial reach. Thedominant trend of recent research on magnetic reconnection, aiming at predicting almost ideal reconnection rates,focuses indeed on the role played by kinetic processes and secondary instabilities, whereas the model first consideredby [54] has the appealing feature of relying on simple and well known results.1 Appendix A: Discussion of the model equations
Eqs.(2-5) are derived with different approximations from the electron and ion momentum equations, which wewrite here below, again non-dimensionalized using a and τ A (and the electric field normalized to a fraction V A /c ofthe reference magnetic field): d e (cid:18) ∂ u e ∂t + u e · ∇ u e (cid:19) = − d i (cid:18) E + u e × B − J S (cid:19) − ρ s ∇ · Π e n e (A1) d i (cid:18) ∂ u i ∂t + u i · ∇ u i (cid:19) = d i (cid:18) E + u i × B − J S (cid:19) − ρ s ∇ · Π i n i (A2)Here the kinetic pressure has been normalized to a reference value P for the electron plasma pressure. This explainsthe weight ρ s in front of the ion pressure force in Eq.(A2), even though the ion thermal Larmor radius is ρ i =( T i /T e ) / ρ s . As discussed in [23] for the purely collisionless regime, Eqs.(2-5) may be indeed obtained, underappropriate approximations and closures for the pressure tensors (and after re-normalization to τ W for the EMHDequations), from Eqs.(A1-A2) coupled with Maxwell’s equations using quasi-neutrality, n e = n i . Such an approachis essentially the one via which electron inertia effects were first included in reconnection models in the full MHD[18, 19]) and RMHD frameworks [59]. Within this approach, inclusion of resistive diffusion S − is straightforward,and the perpendicular ion-ion viscosity too can be retained in the form given in Eq.(3) if the hypothesis of a strongguide field is also assumed (for a recent discussion see [64]). Derivation of the EMHD equations follows simply fromEqs.(4)-(5), since ion dynamics is completely neglected [34].It can be verified that both Eq.(2) and Eq.(4) represent the z -component of electron momentum equation (Eq.A1)in the RMHD and EMHD regime respectively, ψ and −∇ ψ expressing the z component of the vector potential A and of the electron current density J .In RMHD, the ρ s contribution on the r.h.s. of Eq.(2) expresses thermal effects related to electron compressibilityalong the magnetic field lines (see e.g. [28, 36]): in the usual, strong guide field limit, b is completely neglected sinceis ordered b ∼ ǫ with ǫ ≡ |∇ ψ | /B z ≪
1, and to leading order ( ∼ ǫ ) both u e and u i are given by the incompressible E × B -drift velocity. As consequence, the stream function ϕ corresponds to the normalized electrostatic potentialwhile the ρ s term appears in the electron momentum equation as a result of the diamagnetic corrections to u e inthe Lorentz force and the z component of the gyrotropic electron pressure tensor [59]. For this reason this term isconsidered to be an FLR-type contribution. However, the cancellation between the diamagnetic drift contribution tothe z -component of u e · ∇ u e and the z -component of the gyrotropic pressure tensor is required in the derivation onlyif we do not order ρ s and d e with respect to ǫ ; in that case Eqs.(2-3) contain terms up to the second order in ǫ . Ifinstead we remember that in the slab, strong guide field, RMHD ordering, β e ∼ ǫ and that ρ s = β e d i /
2, then we mayorder ρ s ∼ d e ∼ ǫ . This is sufficient to re-obtain Eqs.(2-3) even by assuming a scalar electron pressure tensor, if wedisregard any contribution of order ǫ or higher, since from u e, ⊥ ≃ E × B /B + ∇ P e × B / ( eB ) we would obtain( u e × B ) · e z = [ ϕ − ρ s U, ψ ]; our equations will now retain terms up to ǫ .In EMHD, instead, the convection velocity field (i.e. u e ⊥ ) appearing in the second term of Eq.(4) is due to themagnetic field component b , since the current density is carried by electrons only, which drive the dynamics through u e ∝ J ∝ ∇ × B in the incompressible regime that we consider here. As a consequence, b acts as a stream function forthe in-plane electron dynamics, and resistivity, when included, enters also in the equivalent of the vorticity equation.For the same reason, the in-plane components of electron momentum equation, taken in the polytropic, incompressiblelimit, completely close the system of EMHD equations: Eq.(5) is the z -component of the rotational of Eq.(A1), andthe field W is proportional to the z -component of the electron generalized vorticity, defined by the curl of the electronfluid canonical momentum ∇ × ( u e + e A / ( m e c )).The EMHD equation for the electron generalized vorticity is mirrored in RMHD by the equation for the fluidvorticity alone (Eq.(3)), of which ∇ ϕ represents the z -component (see also [58]). This happens because in theAlfv´enic frequency range the plasma moves at the bulk velocity U ≃ u i + O ( m e /m i )): Eq.(3) is therefore the curlof Eq.(A2), under the assumption of incompressibility, which allows expression of the perpendicular fluid velocity interms of the stream function ϕ . If the plasma fluid is assumed to be incompressible but without imposing the strongguide field condition, this function can not be interpeted as the electrostatic potential. With no guide field howevera separate analysis would be required to include ρ s -type contributions. The delicate point about the applicability ofEqs.(2)-(5) lies indeed in the validity of the incompressibility assumption and in its relationship with the ordering ofthe parallel fluctuations of the magnetic field, which weighs the importance of Hall’s term in Ohm’s law, discussedbelow.2
1. Comparison with the generalized Ohm’s law and Hall’s term
Since reconnection models are usually discussed in relation to the non-ideal terms in Ohm’s law rather than inthe framework of the full two-fluid equations for ions and electrons, it is worth to make here reference also to thegeneralized Ohm’s law, written with respect to the average plasma velocity U . The standard text-book form obtainedby combining Eqs.(A1)-(A2) (see e.g. [37], p.91) while neglecting O ( m e /m i ) corrections, reads, after normalizingagain lengths to a and times to τ A , E + U × B = d i J × B n + S − J + d e n (cid:26) ∂ J ∂t + ∇ · (cid:18) U J + JU − d i JJ n (cid:19)(cid:27) − ρ s d i ∇ · Π e n . (A3)Here n = n e = n i is the average plasma density and Π e is the electron pressure tensor of Eq.(A1), measured inthe electron rest frame. The ion pressure tensor contribution is neglected since it is O ( m e /m i ) smaller when thetemperatures of the two species are comparable. Note that it has been recently shown by Kimura et al. (2014) [33]that the (often neglected) term ∇ · ( JJ /n ) is necessary to respect energy conservation of the 1-fluid system in thecollisionless limit ( S − = 0).The generalized Ohm’s law is essentially the rewriting of the electron momentum equation with respect to U and J , that replace u e . We then recognize the essential difference between the dynamics of the bulk plasma and of themagnetic induction, and the role that the Hall-term J × B has in this: while the plasma always moves at the fluidvelocity of ions, the magnetic induction evolves (with the rotational of Eq.(A3)) as dragged by the fluid velocity ofthe electrons, u e = ( U − d i J /n ). In particular, the term u e × B describes the convection of magnetic field lines bythe electron fluid in the collisionless limit neglecting electron inertia. As well known [25], the RMHD and EMHDsets of equations for slab reconnection without electron temperature effects may be therefore seen as two extremelimits with respect to the Hall term ( d i -term), in Ohm’s law: the RMHD regime described by Eqs.(2-3) at ρ s = 0corresponds to neglecting Hall’s term entirely, whereas the EMHD framework is recovered when the fluid dynamicsis restricted to electrons only ( U ≃ u i ≃ ℓ ≪ d i and Ω i . ω ≪ Ω e , so that Eq.(A3) becomesthe only relevant equation for our fluid system. It is however interesting to remark that in the strong guide fieldordering, both ions and electrons in-plane velocities are equal at the leading order in ǫ to the E × B -drift. By directcomparison of the z -component of u e × B = ( U − d i J /n ) × B with [ ϕ − ρ s U, ψ ] (cfr. previous Section), is immediateto recognize that the Hall term survives in the ordering with ρ s ∼ ǫ through the diamagnetic-drift contribution to u e, ⊥ , ρ s [ U, ψ ] = d i ( J × B ) · e z /n . This expresses the balance between kinetic and magnetic pressure forces not onlyat equilibrium but also for the perturbations.We conclude by recalling that, when Hall’s term is retained while still considering the bulk plasma response tofield evolution (i.e. ion momentum equation is not neglected, so that J = − ne u e ), an intermediate regime is entered,which is sometimes called “Hall-mediated reconnection” (HMR) or even “whistler mediated reconnection ” [42]. Theseregimes are not of concern in this paper, since they can not be recovered in the framework of two-field models. Thedecoupling of ion and electron motions at the ion inertial scale (i.e. for ℓ . d i ) requires more than two scalar fields to beretained to account for two-fluid effects (also notice that Eqs.(2)-(5) do not contain d i as a characteristic scale length).As discussed by Fruchtman et al. (1993) [26], first, and more recently by
Bian et al. (2007) [7] and
Hosseinpur etal. (2009) [32], Hall term effects are retained by relating the magnitude of b , as generated by Hall’s term in Eq.(A3),to the compressible component of U ⊥ , absent in our incompressible model. By Helmholtz decomposition, this shouldenter through an irrotational contribution, U ⊥ = ∇ ϕ × e z + ∇ χ , related to the scalar field χ ; in turn, the components u ez and U z should also be retained. This immediately highlights the most delicate point concerning the J × B termin Ohm’s law, already pointed out at the end of the previous Section: due to the direct relation between b and χ , the(in)compressibility assumption plays a major role in determining the extent of Hall physics retained in the model.Remarkably, if ∂ z = 0, the in-plane incompressibility ∇ · U ⊥ = 0 is admitted both in the E × B -drift regime of thelow- β limit, where b is neglected with respect to the strong guide field, and in the high- β limit, where the large kinetic(electron) pressure implies the smallness of both ∇ · U = 0 and ∇ · u e = 0. Not to be here confused with the EMHD regime of Eqs.(4)-(5), though there has been some ambiguous notation for different regimesin the past. Also note that, in some works, what we here name (resistive) HMR was even refered to as the “collisionless reconnection”regime (see e.g. [75]), due to the weak dependence from S found in the Hall-dominated reconnection rate (see e.g. [11]). TABLE I. Characteristic plasma parameters of magnetized plasma environments where MHD reconnection may occur. Physicalquantities are expressed in cgs units and temperatures are expressed in eV . For magnetotail reconnection parameters, typicalconditions in the plasma sheet during a substorm growth phase have been considered. For the tokamak devices the value areestimated from design (ITER) or measurements (JET) near to the q = 1 surface, whose circumference on a poloidal sectiongives an estimation of the typical, reconnecting current sheet length, L . Source for the parameters, as labelled in the Table’sthird row, are: [63] (I); [2, 35, 60] (II); [53, 57] (III); [71] (IV).Low Corona Magnetotail Tokamak MRX device(Sun at ∼ R ⊙ ) (Central plasma sheet) ITER JETSources: I II III IV L − −
900 50 10 − n e − . − (2 − × B −
100 10 − . × . × (1 − × T e
86 10 − × × − ε ∗ S ≡ ( S − ) ∗ − − − − − − − − × − − × − ε ∗ d ≡ ( d e /L ) − − − − − − − − − − ACKNOWLEDGMENTS
The authors are grateful to Francesco Pegoraro for discussions and comments. DDS is in debt with MaurizioOttaviani for many interesting discussions, and in particular for having pointed out the possible importance of theideal tearing during the non-linear stage of primary reconnection instabilities, and with Alessandro Biancalani fordiscussions about the explosive reconnection regime and for having kindly provided details about the numericalsimulations performed in [8, 9]. This research was partially supported by the joint training PhD program in Astronomy,Astrophysics and Space Science between the University of Rome “Tor Vergata” and “Sapienza”. [1] Ali, A., J. Li, Y. Kishioto (2014), On the abrupt growth dynamics of nonlinear resistive tearing mode and the viscosityeffects,
Phys. Plasmas , (5), 05312.[2] Angelopoulos, V., A. Runov, X.-Z. Zhou, D.L. Turner, S.A. Kiehas, S.-S. Li, I. Shinohara (2014), Electromagnetic energyconversion at reconnection fronts, Science , (5), 1478.[3] Ara, G., B. Basu, B. Coppi, G. Laval, M.N. Rosenbluth, B.V. Waddell (1978), Magnetic reconnection and m = 1 oscillationsin current carrying plasmas, Annals of Physics , (2), 443–476.[4] Attico, N., F. Califano, F. Pegoraro (2000), Fast collisionless reconnecion in the whistler frequency range, Phys. Plasmas , (6), 2381–2387.[5] Avinash, K., S.V. Bulanov, T. Eisrkepov, P. Kaw, F. Pegoraro, P.V. Sasarov, A. Sen (1998), Forced magnetic field linereconnection in electron magnetohydrodynamics, Phys. Plasmas , (8), 2849.[6] Battacharjee, A., Y.-M. Huang, H. Yang, B.N. Rogers (2009), Fast reconnection in high-Lundquist-number plasmas dueto the plasmoid instability, Phys. Plasmas , (11), 112102.[7] Bian, N.H., G. Vekstein (2007), Is the “out-of-plane” magnetic perturbation always a quadrupole in the Hall-mediatedmagnetic reconnection?, Phys. Plasmas , (12), 120702.[8] Biancalani, A., B. Scott (2012), Nonlinear growth acceleration in gyrofluid simulations of collisionless reconnection, 38 th EPS Conference on Plasma Physics , Vol. 35 , 27 June − EuroPhys. Lett. , (1), 15005.[10] Biskamp, D. (1986), Magnetic reconnection via current sheets, Phys. Fluids , (5), 1520–1531.[11] Birn, J., J.F. Drake, M.A. Shay, B.N. Rogers, R.E. Denton, M. Hesse, M. Kuznetsova, Z.W. Ma, A. Battacharjee, A.Otto, P.L. Pritchett (2012), Geospace Enverinmental Modeling (GEM) Magnetic Reconnection Challenge, J. Geophys.Res. , (A3), 3715–3719.[12] Biskamp, D., E. Schwarz, J.F. Drake (1995), Ion controlled collisionless magnetic reconnection, Phys. Plasmas , (21),3850.[13] Boozer, A.H. (2012), Magnetic reconnection in space, Phys. Plasmas , (9), 092902.[14] Bulanov, S.V., F. Pegoraro, A.S. Sakharov (1992), Magnetic recoonection in electron magnetohydrodynamics, Phys. FluidsB , (8), 2499–2508.[15] Cafaro, E., D. Grasso, F. Pegoraro, F. Porcelli, A. Saluzzi (1998), Invariants and geometric structures in nonlinear Hamil-tonian magnetic reconnection, Phys. Rev. Lett. , (20), 4430.[16] Comisso, L., D. Grasso, E. Tassi, F.L. Waelbroeck (2012), Numerical investigation of a compressible gyrofluid model for collisionless reconnection, Phys. Plasmas , (4), 042103.[17] Comisso, L., D. Grasso, F.L. Waelbroeck, D. Borgogno (2013), Gyro-induced acceleration of magnetic reconnection, Phys.Plasmas , (9), 092118.[18] Coppi, B. (1964), “Inertial” instabilities in plasmas, Phys. Lett. , (3), 226–228.[19] Coppi, B. (1964), Addendum on inertial interchange modes, Phys. Lett. , (3), 213–214.[20] Daughton, W., V. Roytershteyn, H. Karimabadi, L. Yin, B.J. Albright, B. Bergen, K.J. Bowers, (2011), Role of electronphysics in the development of turbulent magnetic reconnection in collisionless plasmas, Nature Phys. , , 539–542.[21] Del Sarto, D., F. Califano, F. Pegoraro (2003), Secondary instabilities and vortex formation in collisionless-fluid magneticreconnection, Phys. Rev. Lett. , (23), 235001.[22] Del Sarto, D. , F. Pegoraro, F. Pegoraro (2005), Current layer cascade in electron-magnetohydrodynamic reconnection andelectron compressibility effects, Phys. Plasmas , (1), 012317.[23] Del Sarto, D., F. Califano, F. Pegoraro (2006), Electron parallel compressibility in the nonlinear dynamics of two-dimensional magnetohydrodynamic reconnection, Mod. Phys. Lett. B , (16), 931-961.[24] Del Sarto, D., C. Marchetto, F. Pegoraro, F. Califano (2011), Finite Larmor radius effects in the nonlinear dynamics ofcollisionless magnetic reconnection, Plasma Phys. Contoll. Fusion , (3), 035008.[25] Fruchtman, A., Y. Maron (1991), Fast magnetic-field penetration into plasma due to the Hall field, Phys. Fluids B , (7),1546.[26] Fruchtman, A., H.R. Strauss (1993), Modification of short scale-length tearing modes by the Hall field, Phys. Fluids B , (5), 1408–1412.[27] Furth, H.P., J. Killeen, M.N. Rosenbluth (1963), Finite-resistivity instabilities of a sheet pinch, Phys. Fluids , (4), 459-484.[28] Grasso, D., F. Pegoraro, F. Porcelli, F. Califano (1999), Hamiltonian magnetic reconnection, Plasma Phys. Control. Fusion , , 1497-1515.[29] Grasso, D., F. Califano, F. Pegoraro, F. Porcelli (2001), Phase mixing and island saturation in Hamiltonian reconnection, Phys. Rev. Lett. , (22), 5051.[30] Grasso, D., E. Tassi, F.L. Waelbroeck (2010), Nonlinear gyrofluid simulations of collisionless reconnection, Phys. Plasmas , (8), 082312.[31] Hirota, M., Y. Hattori, P.J. Morrison (2015), Explosive magnetic reconnection caused by an X-shaped current-vortex layerin a collisionless plasma, Phys. Plasmas , (5), 052214.[32] Hosseinpur, M., N. Bian, G. Vekstein (2009), Two-fluid regimes of the resistive and collisionless tearing instability, Phys.Plasmas , (1), 012104.[33] Kimura, K., P.J. Morrison (2014), On energy conservation in extended magnetohydordynamics, Phys. Plasmas , (8),082101.[34] Kingsep, S., K.V. Chukbar, V.V. Yan’kov (1990), Electron magnetohydrodynamics, Reviews of Plasma Physics , edited byB. Kadomtsev (Consultant Bureau, New York, 1990), , 243.[35] Kivelson, M., C.T. Russell, Introduction to Space Physics, Cambridge University Press, 1995.[36] Kleva, R.G., J.F. Drake, F.L. Waelbroeck (1995), Fast reconnection in high temperature plasmas, Phys. Plasmas , (1),23.[37] Krall, N.A., A.W. Trivelpiece (1973), Princples of Plasma Physics , McGraw-Hill, New York.[38] Landi, S., L. Del Zanna, E. Papini, F. Pucci, M. Velli (2015), Resistive magnetohydordynamic simulations of the idealtearing mode,
Astrophys J. , (1), 131.[39] Lentini, M., V. Pereyra (1974), A variable order finite difference method for nonlinear multipoint boundary value problems, Mathematics of Computation , (128), 981–1003.[40] Loureiro, N.F., S.C. Cowley, W.D. Dorland, M.G. Haines, A.A. Schekochihin (2005), X-point collapse and saturation inthe nonlinear tearing mode reconnection, Phys. Rev. Lett. , (23), 235003.[41] Loureiro, N.F., A.A. Schekochihin, S.C. Cowley (2007), Instability of current sheets and formation of plasmoid chains, Phys. Plasmas , (10), 0703.[42] Mandt, M.E., R.E. Denton, J.F. Drake (1994), Transition to whistler mediated magnetic reconnection, Geophys. Res. Lett. , (1), 73–76.[43] Moser, A.L., P.M. Bellan (2012), Magnetic reconnection from a multiscale instability cascade, Nature , (2), 379–381.[44] Øieroset, M., T.D. Phan, M. Fujimoto, R.P. Lin, R.P. Lepping (2001), In situ detection of collisionless reconnection inEarth’s magnetotail, Nature , (6845), 414–417.[45] Ottaviani, M., F. Porcelli (1993), Nonlinear collisionless magnetic reconnection, Phys. Rev. Lett. , (23), 3802.[46] Ottaviani, M., F. Porcelli (1995), Fast nonlinear magnetic reconnection, Phys. Plasmas , (11), 4104–4117.[47] Pegoraro, F., T.J. Schep (1986), Theory of resistive modes in the balloning representation, Plasma Phys. Contoll. Fusion , (4), 647.[48] Pegoraro, F., F. Porcelli, T.J. Schep (1989), Internal kink modes in the ion-kinetic regime, Phys. Fluids B , (2), 364–374.[49] Pegoraro, F., D. Borgogno, F. Califano, D. Del Sarto, E. Echkina, D. Grasso, T. Liseikina, F. Porcelli (2004), Developmentsin the theory of collisionless reconnection in magnetic configurations with a strong guide field, Nonlin. Proc. Geophys. , ,567–577.[50] Perona, A., L.-G. Eriksson, D. Grasso (2010), Electron response to collisionless magnetic reconnection, Phys. Plasmas , (4), 042104.[51] Porcelli, F., Collisionless m = 1 tearing mode (1991), Phys. Rev. Lett. , (4), 425.[52] Porcelli, F., D. Borgogno, F. Califano, D. Grasso, F. Pegoraro (2004), Magnetic reconnection: collisionless regimes, Phys.Scripta , T107 , 153–158. [53] Porcelli, F., D. Boucher, M.N. Rosenbluth (1996), Model for the sawtooth period amplitude, Plasma Phys. Controll.Fusion. , (12), 2163.[54] Pucci, F., M. Velli (2014), Reconnection of quasi-singular current sheets: the “ideal” tearing mode, Astrophys. Journ.Lett. , (2), L19.[55] Pucci, F. et al. (to be submitted), The effect of different equilibrium profiles on “ideal” tearing scaling laws.[56] Rappazzo, F.A., E.N. Parker (2013), Current sheets formation in tangled coronal magnetic fields, Astrophys. Journ. Lett. , (1), L2.[57] Rebut, P.H., R.J. Bickerton, B.E. Keen (198), The Joint European Torus: installation, first results and prospect, Nucl.Fusion , (9), 1011.[58] Rogers, B.N., R.E. Danton, J.F. Drake, M.A. Shay (2001), Role of dispersive waves in collisionless magnetic reconnection, Phys. Rev. Lett. , (19), 1954004.[59] Schep, T.J., F. Pegoraro, B.N. Kuvshinov (1994), Generalized two fluid theory of nonlinear magnetic structures Phys.Plasmas , (9), 2843–2852.[60] Sergeev, V., D.G. Mitchell, C.T. Russell, D.J. Williams (1993), Current sheet measurements within a flapping plasmasheet, J. Geophys. Res. , (A10), 17345–17365.[61] Servidio, S., F. Valentini, F. Califano, P. Veltri (2012), Local kinetic effects in two-dimensional plasma turbulence, Phys.Rev. Lett. , (4), 045001.[62] Shibata, K., S. Tanuma (2001), Plasmoid-induced-reconnection and fractal reconnection, Earth Planets Space , (2),473–482.[63] Shibata, K., T. Magara (2011), Solar flares: magnetohydrodynamic processes, Living Rev. Solar Phys. , ,6.[64] Tenerani, A., A.F. Rappazzo, M. Velli, F. Pucci (2015), The tearing instability of thin current sheets: the transition tofast reconnecion in presence of viscosity, Astrophys. J. , (2), 145.[65] Tenerani, A., M. Velli, A.F. Rappazzo, F. Pucci (2015), Self-similar current sheet collapse triggered by “ideal” tearing, arXiv preprint , arXiv:1506.08921 .[66] Vaivads, A., Y. Khotyaintsev, M. Andr´e, A. Retin`o, S.C. Buchert, B.N. Rogers, P. D´ecr´eau, G. Paschmann, T.D. Phan(2014), Structure of the magnetic reconnection diffusion region from four-spacecraft observations, Phys. Rev. Lett. , (10),105001.[67] Velli, M. , A.W. Hood (1989), Resistive tearing in line-tied magnetic fields: slab geometry, Solar Phys. , (1), 107-124.[68] Velli, M., F. Pucci, F. Rappazzo, A. Tenerani (2015), Models of coronal heating, turbulence and fast reconnection, Phil.Trans. R. Soc. A , (2042), 20140262.[69] Waelbroeck, F.L., R.D. Hazeltine, P.J. Morrison (2009), A Hamiltonian electromagnetic gyrofluid model, Phys. Plasmas , (3), 032109.[70] Wesson, J.A., (1990), Sawtooth reconnection, Nucl. Fusion , (12), 2545.[71] Yamada, M., J. Yoo, J. Jara-Almonte, H. Ji, R.M. Kulsrud, C.E. Myers (2014) Conversion of magnetic energy in themagnetic reconnection layer of a laboratory plasma, Nature Communications , , 4774.[72] Yu, Q., S. G¨unter, K. Lackner (2014), Formation of plasmoids during sawtooth crashes, Nucl. Fusion , (7), 072005.[73] Zacharias, O., L. Comisso, D. Grasso, R. Kleiber, M. Borchardt, R. Hatzky (2014), Numerical comparison between agyrofluid and gyrokinetic model investigating collisionless magnetic reconnection, Phys. Plasmas , (06), 062106.[74] Zank, G.P., W.H. Matthaeus, (1992), The equations of reduced magnetohydrodynamics, J. Plasma Phys. , (01), 85–100.[75] Zweibel, E., M. Yamada (2009), Magnetic reconnection in astrophysical and laboratory plasmas, Ann. Rev. Astron. As-trophys. ,47