# Microscopic Current Sheets and Fast Tearing Modes in Plasma Turbulence

Homam Betar, Daniele Del Sarto

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Homam Betar, Daniele Del Sarto. Microscopic Current Sheets and Fast Tearing Modes in Plasma Turbulence. The Astrophysical Journal, 2025, 990 (1), pp.28. 10.3847/1538-4357/adea47. hal-05219245v2

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# Microscopic Current Sheets and Fast Tearing Modes in Plasma Turbulence

Homam Betar<fnref n="1" /> ID and Daniele Del Sarto<fnref n="2" /> ID

Received 2024 August 24; revised 2025 June 26; accepted 2025 June 27; published 2025 August 22

## Abstract

Since the seminal work by W. H. Matthaeus & S. L. Lamkin, a large amount of evidence has been collected over the years that magnetic reconnection can disrupt current sheets formed in turbulence. The details about how this happens, however, are not clear, yet. The observation of plasmoids suggests that tearing-type modes are involved, but their nature of spontaneous linear instabilities developing on a static (or at most steady) magnetic equilibrium poses strong constraints on their growth rate versus the timescale of the current sheet evolution. None of the tearing-based scenarios, which to date are most credited in literature, seems to fulfill both this constraint and other consistency requirements on the equilibrium profile. In revising them and the main hypotheses, which any tearing-based theory for 2D turbulent reconnection must satisfy, we propose a possible explanation—supported by numerical calculations—for why tearing modes may be relevant. This explanation is grounded on the microscopic thickness that current sheets attain in turbulence, which makes the growth rates of tearing modes large enough for the instability to possibly develop. At the same time, this implies that theoretical growth rates obtained from a boundary layer analysis cannot be applied in this case. We discuss a few implications of these elements in solar wind turbulence and in comparison with alternative models for tearing-based turbulent reconnection that are available in literature.

*Unified Astronomy Thesaurus concepts: Space plasmas (1544); Solar wind (1534); Plasma physics (2089); Plasma astrophysics (1261)*

## 1. Introduction

Magnetic reconnection is probably the chief and most frequent mechanism by which magnetic energy is converted into kinetic energy, in space and astrophysical plasmas. These are typically met in a turbulent state. While there is enough evidence, both observational (A. Retinò et al. 2007; K. T. Osman et al. 2014; A. Greco et al. 2016; T. C. Li et al. 2021; R. Wang et al. 2023) and numerical (see, e.g., W. H. Matthaeus & S. L. Lamkin 1986; H. Politano et al. 1989; S. Servidio et al. 2009, 2010; A. F. Rappazzo & E. N. Parker 2013; M. Wan et al. 2013; Y.-M. Huang & A. Bhattacharjee 2016; S. S. Cerri & F. Califano 2017; L. Franci et al. 2017; C. C. Haggerty et al. 2017; L. Comisso et al. 2018; C. Dong et al. 2018; E. Papini et al. 2019a; S. Adhikari et al. 2020; L. Franci et al. 2022a, 2022b; T. Passot et al. 2024), that magnetic reconnection can occur in turbulence, there is not yet a consensus about the theoretical models justifying this.

Here, we focus on the relevance of tearing-type instabilities as a reconnection mechanism in turbulence with a dominant 2D spatial dependence. This interest is motivated by the large number of turbulent reconnection models, which, at some step of their structure, rely on assumptions based on the tearing mode analysis. This includes models—which we will come upon in the following—that, for example, ground on the “fractal reconnection” (K. Shibata & S. Tanuma 2001), on the “plasmoid instability” (N. Loureiro et al. 2005; A. Bhattacharjee et al. 2009; L. Comisso et al. 2016), on the “ideal tearing” (F. Pucci & M. Velli 2014; D. Del Sarto et al. 2016; F. Pucci et al. 2017; E. Papini et al. 2019b), as well as on recent models proposed for the “electron-only” reconnection regime (see, e.g., A. Mallet 2020).

In the first part of this work (Section 2), we recall why the applicability of tearing modes to plasma turbulence should not be given as “granted.” To this end, we discuss a number of hypotheses required for the applicability of tearing modes, as they are acquired in a quite diverse range of literature spanning applications to both astrophysical and magnetically confined fusion plasmas. The notation, which we will use in relation to the normalization of quantities, and which will play a key role for the renormalization procedures we will perform, is introduced in Section 2.1.1.

In Section 3, we focus on the requirement that the growth rate of a tearing mode should be fast enough to let one consider the current sheet as “static.” In this respect, we discuss a possible reason for why the disruption of current sheets developing in turbulent plasmas may be explained in terms of the tearing mode instability, as long as the dynamics remains essentially 2D with respect to the coordinate dependence. All three components of vector quantities are in principle accounted for; the negligibility of some component—if any—depends on the specific orderings entailed by each model. As a quantitative example, we consider a parameter range relevant to the solar wind turbulence close to the terrestrial magnetosphere. We study it with the aid of linear solvers implementing two complementary models of tearing reconnection. We show that both models provide comparable quantitative results. However, while they are both compatible with such a parameter range, only one allows for arbitrary values of the ratio between the electron and magnetic pressure (i.e., of the electron beta parameter). The procedure we follow is of a heuristic type: we first assume the current sheet to be static so as to allow for the onset of tearing modes. Then, we combine it with other hypotheses—supported by observational and numerical evidence—which allow us to make quantitative estimates on the parameters at play. We thus use these estimates to numerically compute the tearing growth rates in the parameter range of interest so as to verify a posteriori that the static condition is respected even for a current sheet evolving over an

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Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹Formerly at Laboratoire De Mécanique, Modélisation et Procédés Propres, UMR7340 CNRS—Université Aix Marseille, Centrale Méditerranée, F-13451, France</footnote>

<footnote n="2">²Institut Jean Lamour, UMR 7198 CNRS—Université de Lorraine F-54000 Nancy, France; daniele.del-sarto@univ-lorraine.fr</footnote>
Alfvénic timescale. In this way, we just prove the formal consistency of the argument, not the correctness of the physical assumptions behind it. This is per se an interesting result since it allows us to recognize some consistency problem in other models of turbulent reconnection, which in the scenario we propose are “fixed” thanks to an ingredient apparently quite neglected in literature: the fact that current sheets generated by turbulence have a microscopic thickness *a* comparable to some of the scale lengths associated with the nonideal effects. These are generally smaller than the current sheet length *L* to which the Alfvénic timescale of the current sheet evolution is typically related. The explanation we suggest accounts for the generation of “plasmons,” of which wide evidence, both experimental and numerical, nowadays exists. At the same time, it is compatible with some further constraints we recognize, and which previous models for the generation of plasmons in turbulence apparently do not account for.

These latter issues are addressed in the final part of the manuscript (Section 4), where we discuss further implications. Notably, the example of solar wind turbulence considered in Section 3 lets us shed light on the role of the Hall term in the achievement of a “fast enough” reconnection rate; this role is formally equivalent to that of other nonideal effects like the ion-sound Larmor radius $\rho_s$ or the electron skin depth $d_e$, provided the scale of the shear length *a* is small enough (Section 4.1). When *a* is microscopic, however, analytical estimates from boundary layer theory are inapplicable (Section 4.2). Then, (Section 4.3) we make a comparison with existing models for turbulent reconnection based on the “plasmoid instability” and on the “ideal tearing,” whose unlikeliness in the low-collision solar wind turbulence we argue. Finally, we discuss the importance of some hypotheses we have done, for which an accurate quantitative analysis is still lacking (Section 4.4).

In the Conclusions (Section 5), we consider possible extensions of the proposed argument to regimes beyond the case considered in Section 3, and we suggest investigations to be done in the future. Details on the reduced models integrated to obtain the numerical results are in Appendices A and B.

## 2. Tearing Modes in Space and Astrophysical Turbulent Plasmas

In this section, we recall the main features of both turbulent reconnection and tearing modes, as they are acquired in a quite diversified range of literature encompassing both astrophysical and laboratory plasmas. This is the starting point on which we develop the argument exposed in Section 3.

Space and astrophysical plasmas are mostly met in a state of developed turbulence (see, e.g., W. H. Matthaeus 2021). They are also low collision, so that for large time intervals of the dynamics the fluid advection plays a dominant role over the resistive diffusion of magnetic lines. In turbulent magnetized plasmas, this effect makes magnetic lines “pile up” close to the vorticity sheets separating vortices, so as to form large aspect ratio current sheets (see, e.g., H. Politano et al. 1995; G. G. Howes 2016). Of course, 3D structures more complex than current sheets can be generated by magnetic dynamo processes in a turbulent plasma and can be equally dragged by the flow. Well-known examples are provided by coronal loops in the solar corona: their typical arc-like shape is believed to be due to convection and buoyancy effects that make magnetic

flux tubes emerge from the inner Sun, where they form (E. N. Parker 1970). Both the arc-like coronal loops and the curtain-like magnetic structures that should replace coronal loops in the recently proposed “coronal veil” scenario (A. Malanushenko et al. 2022) are equally advected by the turbulent plasma flow. Current sheets, coronal loops, or coronal veils can all be disrupted by magnetic reconnection, which transfers magnetic energy to the plasma particles, in the form of both acceleration (ordered kinetic energy) and heating (disordered kinetic energy)—see, e.g., J. F. Drake et al. (2003) or A. Lazarian et al. (2012) and F. Guo et al. (2024) for a more recent review. While there is a large amount of evidence of reconnection processes in turbulent plasmas, there is not any consensus, yet, on the actual mechanism by which they occur.

For example, some models have been proposed (A. Lazarian & E. T. Vishniac 1999; P. Browning & A. Lazarian 2013; A. Lazarian et al. 2015), in which a steady Sweet-Parker type reconnection (E. N. Parker 1957; P. A. Sweet 1958) dissipates magnetic field lines advected by microscopic, weakly stochastic fluctuations. Because of these fluctuations, the steady reconnection occurs in an infinitesimal fraction of the length *L* of the associated current sheet but involves several magnetic lines simultaneously. Other models, believed to be especially relevant to coronal and/or to some magnetospheric reconnection events, rely on instabilities allowed only in a fully 3D geometry (e.g., like the “torus instability,” O. Olmedo & J. Zhang 2010) or like the “fan reconnection” (E. R. Priest & V. S. Titov 1996; D. I. Pontin & E. R. Priest 2022).

We depart from all these models, since we focus on spontaneous reconnecting instabilities that are compatible with the observation of the so-called “plasmons,” namely, “plasma blobs” whose planar section has the same topology of a magnetic island. Because of this, we restrict our analysis to turbulence displaying a dominant 2D spatial dependence and to “tearing-type” instabilities, first introduced by H. P. Furth et al. (1963).

### 2.1. Magnetic Reconnection, Generalized Ohm's Law, and “Nonideal Parameters”

We first recall some key features of the physical effects that can induce magnetic reconnection. This is allowed when “nonideal effects” violate the Lagrangian invariance expressed by ideal Ohm’s law. The latter implies that at ideal-MHD scales magnetic field lines are dragged by the bulk plasma motion (whose dynamics is in turn influenced by the magnetic field through Lorentz forces). For future use, it is convenient to recall the “generalized Ohm’s law,”

$$ 
\begin{aligned}
& E + U \times B \\
& = \frac{1 - \tilde{Z}\varepsilon_m}{1 + \varepsilon_m} \left(\frac{d_i}{L_0}\right) \frac{J \times B}{n} + S_{L_0}^{-1}J \\
& \quad - \left(\frac{\rho_s^2}{d_i L}\right) \left(\frac{B_0^{\text{tot}}}{B_0}\right)^2 \left[\frac{\nabla \cdot \Pi^e}{n} - \tilde{Z}\varepsilon_m \frac{\nabla \cdot \Pi^i}{n}\right] \\
& \quad + \frac{1}{1 + \varepsilon_m} \left(\frac{d_e}{L_0}\right)^2 \frac{1}{n} \left[\frac{\partial J}{\partial t} + \nabla \cdot \left(UJ + JU - \frac{1 - \tilde{Z}\varepsilon_m}{1 + \varepsilon_m} \left(\frac{d_i}{L_0}\right) \frac{JJ}{n}\right)\right]
\end{aligned}
\quad (1) $$

as it is formally derived from a two fluid modeling (see, e.g., N. A. Krall & A. W. Trivelpiece 1973, Section 3.4–3.6).
Equation (1) is expressed with respect to the average bulk velocity $\mathbf{U} \equiv (\sum_{\alpha} m_{\alpha} n_{\alpha} \mathbf{u}_{\alpha}) / (\sum_{\alpha} m_{\alpha} n_{\alpha})$ and density $n \equiv (\sum_{\alpha} m_{\alpha} n_{\alpha}) / (\sum_{\alpha} m_{\alpha})$, which make appear the parameter $\epsilon_m \equiv m_e/m_i$ after quasi-neutrality $n_e = \tilde{Z}n_i$ is used (here, $\tilde{Z}$ is the ion charge number after ionization, coincident with the charge atomic number $Z$ in the case of complete ionization); the pressure tensors are defined with respect to their species fluid velocity, i.e., $\Pi^{\alpha} \equiv m_{\alpha} n_{\alpha} \langle (\mathbf{v} - \mathbf{u}_{\alpha})(\mathbf{v} - \mathbf{u}_{\alpha}) \rangle_{\alpha}$. Lengths are normalized to an arbitrary length scale $L_0$, and velocities are normalized to an Alfvén velocity $c_{A,0} = B_0 / \sqrt{4\pi m_i n_0}$ evaluated with respect to a reference value of the “equilibrium” magnetic field $B_0$ and of the density $n_0$. Microscopic parameters associated with nonideal effects have been introduced: the ion ($d_i$) and electron ($d_e$) inertial lengths, related by $d_e^2 = (m_e/m_i)d_i^2/\tilde{Z}^2$; the ion-sound Larmor radius, related to the electron beta and ion inertial length by $\rho_s^2 = \beta_e d_i^2$, where $\beta_\alpha \equiv \sum_\alpha 8\pi P_\alpha / (B_0^{\text{tot}})^2$, $B_0^{\text{tot}}$ is the reference value for the total magnetic field associated with $B_0$ (in the case, $B_0$ refers just to a component of the magnetic field, as sometimes it happens), and $P_\alpha$ is a reference value for 1/3 of the trace of the pressure tensor $\Pi^\alpha$; the Lundquist number $S_{L_0} \equiv 4\pi c_{A,0} L_0 / (\eta c^2)$, related to the resistivity $\eta$ (or to magnetic diffusivity $\eta_m \equiv c^2\eta/(4\pi)$). For simplicity, the electron-electron and ion-ion viscosity have been neglected, but with some algebra, they could be straightforwardly included, once a collision rate between alike particles is introduced. We see from Equation (1) that the limit $\epsilon_m \to 0$ is consistent with retaining finite-electron inertia effects related to the $\sim d_e^2$ terms, and that the ion pressure contribution is formally independent of the ion mass, since $\epsilon_m \Pi^i = m_e n_i \langle (\mathbf{v} - \mathbf{u}_i)(\mathbf{v} - \mathbf{u}_i) \rangle_i$. Despite the nonrelativistic writing, Equation (1) also gives hints of the fact that, in an electron-positron plasma ($\epsilon_m = \tilde{Z} = 1$), the Hall term does not contribute to Ohm's law (N. Bessho & A. Bhattacharjee 2007) and that the electron and ion pressure contributions may even cancel out.

The nonideal terms are those at the right-hand side of Equation (1) and are all weighted by some dimensionless parameter (e.g., $(d_e/L_0)^2$, $d_i/L_0$, $S_{L_0}^{-1}$, etc.). For shortness, we will call them “nonideal parameters.” We emphasize here the dependence of their numerical value on the chosen normalization scales. When $L_0$ and $c_{A,0}$ correspond to the macroscopic scales of the ideal MHD, all these parameters are by definition small.

As long as an equation of the kind of $\partial\mathbf{B}/\partial t = \nabla \times (\mathbf{V} \times \mathbf{B})$ can be obtained by taking the curl of Equation (1), where the velocity field $\mathbf{V}$ enters in a continuity equation of the kind $\partial\mathcal{N}/\partial t + \nabla \cdot (\mathcal{N}\nabla\mathbf{V}) = 0$, the Lagrangian invariance of $\mathbf{B}/\mathcal{N}$ is granted (C. Truesdell 1950), and the other topological conservations like Alfvén's or the linking theorem follow (see D. Del Sarto et al. 2006; and references therein); if $\mathbf{V}$ is incompressible, then $\mathbf{B}$ alone is a Lagrangian invariant, and the magnetic lines are advected by the flow (W. M. Elsasser 1950). It follows that some nonideal terms do not allow for magnetic reconnection. For example, a polytropic pressure tensor does not contribute to the curl of Equation (1). Also, the Hall term, alone, keeps on granting the Lagrangian invariance of $\mathbf{B}/n_e$ or of $\mathbf{B}/n_i$ (equivalent thanks to quasi-neutrality) with respect to $\mathbf{V} = \mathbf{U} - [(1 - \tilde{Z}\epsilon_m)d_i/(1 + \epsilon_m)L_0]\mathbf{J}/n = (\mathbf{u}_e + \tilde{Z}\epsilon_m\mathbf{u}_i)/(1 + \tilde{Z}\epsilon_m)$ and to $\mathcal{N} \propto n_i = n_e/\tilde{Z}$; in the limit $\epsilon_m \to 0$, this yields the well-known “frozen-in” condition with respect to the electron

flow, i.e., $\mathbf{V} \simeq \mathbf{u}_e + O(\epsilon_m)$. Nevertheless, all nonideal parameters can influence the reconnection rate, and it is not easy to quantify how much: in the case of tearing modes, the reconnection rate coincides with the growth rate $\gamma$, but its functional dependence on the nonideal parameters is nontrivially affected by the spatial gradients of the eigenfunctions, too (see, e.g., H. Betar et al. 2022). However, a power-law dependence of $\gamma$ on the nonideal parameters can be always locally determined in parameter intervals that are “narrow enough.” It is numerically verified that, when the nonideal parameters are small enough and depending on the reconnection regimes, these power laws span several orders of magnitude of the parameter intervals (see H. Betar et al. 2020, 2024; H. Betar & S. Del 2023).

The larger the number of nonideal parameters retained is, the more complex the analysis becomes. Because of this, analytical estimates are so far available only in limited parameter ranges where two or at most three nonideal effects at a time are considered. In general, a numerical approach is preferable when several parameters are accounted for. Numerical calculations become mandatory for tearing-type instabilities when some of the nonideal parameters become “large” with respect to the reference scale $L_0$: in this limit, the analytical integration procedure based on the boundary layer analysis is not applicable anymore. This is the regime we are interested in (see Section 3 below).

### 2.1.1. Notation and Normalization

Normalization plays a crucial role in the remainder of this work, so it is worthwhile to introduce the alternative notations we will use beside the $c_{A,0}$ defined above in terms of an arbitrary $B_0$ and $n_0$. In a 2D coordinate dependence with respect to $x$ and $y$, we will consider a magnetic field like $\mathbf{B}(x, y, t) = \mathbf{B}_{\perp}(x, y, t) + \mathbf{B}_{\parallel}(x, y, t) \mathbf{e}_z = \mathbf{B}_{\perp}(x, y, t) + (\mathbf{B}_g + \delta\mathbf{B}_z)(x, y, t) \mathbf{e}_z$. The reconnecting current sheet is orthogonal to a uniform guide field $\mathbf{B}_g \mathbf{e}_z$, has characteristic density $n_{cs}$ and the associated magnetic field has characteristic amplitude $B_{cs} \sim |\mathbf{B}_{\perp}|$. We introduce $\epsilon_B \equiv |\mathbf{B}_{\perp}|/B_g$.

If $B_0 = B_{cs}$, then $B_0^{\text{tot}} = \sqrt{B_{cs}^2 + B_g^2}$, so that $B_0^{\text{tot}} \simeq B_g + O(\epsilon_B)$ for $\epsilon_B \ll 1$.

When $B_0 = B_{cs}$ and $n_0 = n_{cs}$ so that $c_{A,0} = c_{A,cs}$, the notation $S_{L_0}$, and $\tau_{A,L_0} = L_0/c_{A,cs}$ will be used and specialized for different values of $L_0$: e.g., $S_L$ and $\tau_{A,L}$ or $S_a$ and $\tau_{A,a}$ for $L_0 = L$ or $L_0 = a$, respectively.

When, instead, $B_0 = B_g$ and $n_0 = n_{cs}$, we will write $c_{A,0} = c_{A,cs}^*$ together with $\tau_{A,L_0}^* = L_0/c_{A,cs}^*$, $S_{L_0}^*$, etc.

Furthermore, this is consistent with introducing the ion cyclotron frequencies as $\Omega_{i,cs}^* = \Omega_{i,cs} B_g / B_{cs} = eB_g / (m_i c)$ and $\Omega_i = \Omega_{i,cs}^*(B_0^{\text{tot}}) / B_{cs}$.

In the following, we will need to compute $\gamma\tau_{A,L}$ from numerical values obtained for $\gamma\tau_{A,a} = f(a, B_{cs})$ or $\gamma\tau_{A,a}^* = g(a, B_g)$. In this case, just the time needs to be rescaled. For example, using $\tau_{A,a}^* = \tau_{A,L}(a/L)(B_{cs}/B_g)$, we obtain $\gamma\tau_{A,L} = g(a, B_g)(L/a)(B_g/B_{cs})$ from the numerical value $g(a, B_g)$. Also, note that if $g(a, B_g)$ can be approximated as a homogenous function of degree $-\lambda_a$ with respect to $a$ and of degree $-\lambda_b$ with respect to $B_g$, then we can generally write $g(a, B_g) = g(L, B_{cs})(L/a)^{\lambda_a}(B_{cs}/B_g)^{\lambda_b}$. This case is met in the boundary layer analysis, for wide intervals of the parameter space, or if we perform a local analysis in the neighborhood of the value of a set of nonideal parameters. We also recall that
such a rescaling argument is used in models like the “ideal tearing mode” (F. Pucci & M. Velli 2014) where a threshold condition on $L/a$ with respect to the parameters appearing in the dispersion relation $f(a, B_{cs})$ is sought by imposing a condition on $\gamma\tau_{A,L}$.

## 2.2. Plasmoids in 2D Turbulence

The interest for turbulence in a 2D coordinate dependence is not only to simplify calculations. On the one hand, this restriction is consistent with the incompressible solar wind turbulence at the MHD scales (see G. P. Zank & W. H. Matthaeus 1992a; and references therein). On the other hand, magnetic reconnection numerically observed in turbulence is typically compared to the tearing mode analysis developed in a 2D planar geometry. Finally, several features of 2D turbulent reconnection are “surprisingly” recovered also in a 3D modeling, especially as far as the occurrence of “plasmoids” is concerned (see, e.g., D. Biskamp 1986; and references therein; or H. Ji et al. 2022). This happens despite striking topological differences between 2D and 3D reconnection existing, although further linear instabilities like the “drift-kink” instability (see G. Lapenta & J. U. Brackbill 1997; W. Daughton 1998; P. H. Yoon et al. 1998; W. Daughton 1999a) and the “oblique tearing” mode (see K. B. Quest & F. V. Coroniti 1981; F. Cao & J. R. Kan 1991; W. Daughton 1999b; G. Lapenta & J. U. Brackbill 2000; D. Borgogno et al. 2005; H. Karimabadi et al. 2005; C. Shi et al. 2020) can be destabilized in a 3D coordinated dependence and can play an important role in mode coupling (D. Borgogno et al. 2005) and in the 3D turbulence (G. Lapenta & J. U. Brackbill 2000; W. Daughton et al. 2011).

When the coordinate dependence of the turbulent dynamics is 2D, there is numerical evidence (see citations in Section 1) that magnetic reconnection occurs via the formation of plasmoids. These structures display the same topology of magnetic islands, are transported by the flow along the reconnecting current sheet, and can mutually interact and merge (D. Biskamp & H. Welter 1964; R. Hautz & M. Scholer 1987). In an earlier terminology, they have been also named “magnetic solitons” (D. Biskamp 1982). In a 3D geometry, planar plasmoid structures are also met, although they map into flux ropes. Of course, they can mutually interact in ways not admitted in a simpler 2D configuration.

Plasmoids are detected not only in simulations of turbulent space plasmas (see references in Section 1) but also by direct observations, especially in planetary magnetotails (M. Hoshino et al. 1994; Q.-G. Zong et al. 2004; T. W. Hill et al. 2008; J. Lin et al. 2008; C. M. Jackman et al. 2011; M. F. Vogt et al. 2014) or in solar mass ejections (S. Tsuneta 1997; C. Foullon et al. 2011; X. Yan et al. 2022) and in dedicated laboratory experiments (E. E. Peterson et al. 2021). These facts make reconnection models accounting for the formation of plasmoids particularly interesting.

Nowadays, and especially after the work by N. Loureiro et al. (2007), who underscored with analytical calculations the connection between plasmoids and tearing modes developing on steadily reconnecting current sheets, plasmoids are typically interpreted as magnetic islands associated with tearing-type instabilities. This association had been already made in the numerical studies by R. Hautz & M. Scholer (1987) and D. Biskamp (1996). In a slab geometry at small wavelengths, these instabilities encompass the so-called “internal kink”

modes met in tokamaks (B. Coppi et al. 1976). Therefore, only the name tearing mode is generally used in a planar geometry (see M. Velli & A. W. Hood 1989): in this case, the constant-psi (tearing) mode of H. P. Furth et al. (1963) and the internal kink mode of B. Coppi et al. (1976) just correspond to the same instability in two different wavelength intervals—see S. V. Bulanov (2016) and H. Betar et al. (2022) for a discussion.

Tearing-type instabilities are appealing because they develop after some threshold condition is achieved, in this case related to the equilibrium current sheet aspect ratio and to the functional dependence on the sheared coordinate (the latter determines the $\Delta'$-instability parameter introduced by H. P. Furth et al. 1963). This is interesting since most reconnection-related events—e.g., solar flares or magnetic substorms—abruptly occur after some building-up phase, much longer than the event itself. We must however mention that some numerical studies show the formation of plasmoids in 3D turbulence without identifying specific signatures of tearing modes (G. Kowal et al. 2017).

In the following, we pursue the rationale of associating plasmoids to tearing-type instabilities, and we thus inquire, from a theoretical point of view, about the possible relevance of the latter to turbulence.

## 2.3. Conditions for the Applicability of Tearing Modes

Even in 2D turbulence, in which plasmoids are observed and the planar geometry is in principle compatible with the onset of standard tearing modes, the relevance of the latter is not granted. This is surmised to the validity of a number of hypotheses, on which the tearing mode theory is grounded. Not always are all these hypotheses explicitly stated—regardless of their likeliness— which is why it is worthwhile to recall them. One is the incompressibility of the flow, which can be ascribed to the parameter range (Section 2.3.3), but most of them (Sections 2.3.1–2.3.2) are related to conditions that the current sheet generated by turbulence should fulfill.

In slab tearing mode theory, a stream function $\psi(x, y, t)$ is introduced such that $\mathbf{B} = B_{cs} \nabla \times \mathbf{e}_z + (B_g + \delta B_z) \mathbf{e}_z$. The typical magnetic equilibrium is of the form $B_{y,0}(x) = B_{cs} \psi_0(x)$, aligned with the in-plane projection of the current sheet, and it is perturbed by $B_{x,1} = B_{cs} \partial_y \psi_1(x, y, t)$. By comparison, in 2D Hall-MHD turbulence, $\mathbf{B} = \delta \mathbf{B}_{\perp} + B_g \mathbf{e}_z$ with $\delta \mathbf{B}_{\perp}$ having rms amplitude $\delta B_{rms}$.

### 2.3.1. Requirements of a Periodic and Static Current Sheet

A problem about the applicability of tearing modes in turbulence is the fact that, formally speaking, tearing modes develop as exponentially growing perturbations sinusoidally modulated with a wavevector $k$ on a (i) *periodic* and (ii) *static* current sheet, for which the threshold condition $\Delta'(ka) > 0$ is fulfilled, where $\Delta'(ka) = \lim_{\epsilon \to 0} [B'_{x,1}(x + \epsilon) - B'_{x,1}(x - \epsilon)]/B_{x,1}(0) = \lim_{\epsilon \to 0} [\psi'_1(x + \epsilon) - \psi'_1(x - \epsilon)]/\psi'_1(0)$ is the instability parameter of H. P. Furth et al. (1963). This condition makes the unstable wavenumbers comprised in an interval $(0, k_m)$ where the order of magnitude of the marginally stable $k_m$ is such that $k_m a \sim O(1)$.

The issue is that current sheets generated in turbulence are neither static nor periodic.

Let us first consider the periodicity issue (i). Since $k \equiv 2\pi m/L$, the normalized “spacing” between nearby modes is $a\Delta k \sim O(a/L)$. The periodicity requirement is generally overcome by assuming that the large aspect ratio $L/a \gg 1$

<figure><img src="image_4.png" /></figure>
allows for the formal limit $L/a \to \infty$. A continuum spectrum of unstable modes can be thus considered in an “infinitely long” current sheet. Then, a maximally growing mode exists (H. P. Furth et al. 1963; M. A. Cross & G. Van Hoven 1971): the fastest mode whose wavevector and growth rate we name $k_M$ and $\gamma_M$. The error about “border effects” due to the finite size of the current sheet may be estimated to be $\sim O(a/L)$ although, we note, a systematic study of this is still lacking. Assuming the continuum spectrum hypothesis for a finite $L/a$, M. Velli & A. W. Hood (1989) noted that, in a Harris pinch profile (E. G. Harris 1962), a value of $L/a \simeq 20$ suffices to destabilize a mode number $m$ corresponding to a wavenumber sufficiently close to $k_M$. This estimate holds also for different equilibria (see H. Betar et al. 2020). These arguments have been used—explicitly or implicitly—in all models available in literature, in which the tearing mode theory is applied to finite length current sheets. This is regardless of whether tearing modes develop on steadily reconnecting sheets (R. Hautz & M. Scholer 1987; K. Shibata & S. Tanuma 2001; T. Tajima & K. Shibata 2002; N. Loureiro et al. 2007; A. Bhattacharjee et al. 2009; P. Cassak & J. Drake 2009; L. Comisso & A. Bhattacharjee 2016; K. A. P. Singh et al. 2019), or on current sheets generated in turbulence (H. Strauss 1986; A. Lazarian et al. 2015; H. Betar et al. 2020) or on current sheets generated by primary tearing-type instabilities (S. Tanuma et al. 2001; N. Loureiro et al. 2005; W. Daughton et al. 2006; J. F. Drake et al. 2006; S. Landi et al. 2015; A. Tenerani et al. 2015; L. Del Zanna et al. 2016; D. Del Sarto & M. Ottaviani 2017; E. Papini et al. 2018, 2019b; A. Tenerani & M. Velli 2020; C. Granier et al. 2022).

Different is the way condition (ii) can be handled. To start with, current sheets in turbulence are generally steady due to the presence of “equilibrium” flows. Then, they generally evolve, dragged by the bulk plasma over a characteristic time $\tau_{cs}$. So, even if equilibrium flows parallel to the current sheet could be neglected (see Section 2.3.2 below), in order to consider them as static with respect to the tearing mode, the growth rate, $\gamma$, must be larger than $\tau_{cs}$. Condition (ii) consequently translates into the formal requirement

$$ \gamma\tau_{cs} \gg 1, \qquad (2) $$

whose verification is nontrivial. The challenge lies in estimating $\tau_{cs}$, a topic to which we will return later in Section 3.1.3.

### 2.3.2. Role of Flows and of the Current Sheet Geometry

Further simplifications are typically made, while applying the classical tearing mode theory to 2D turbulence:

(iii) the current sheets are usually assumed to be straight and symmetric;

(iv) no contribution of a magnetic component parallel to the shear coordinate is considered; and

(v) the current sheet is assumed to be strictly static and not just steady, in the sense that flows parallel to the current sheets are disregarded, at equilibrium.

In principle, condition (iii) may be relaxed. Curvature effects are associated with an asymmetry of the current sheet. Generalized incompressible Sweet-Parker models accounting for an asymmetric magnetic field have been developed (P. A. Cassak & M. A. Shay 2007) and found to agree with turbulence simulations, in regimes in which plasmons were

not observed (S. Servidio et al. 2011). In the case of tearing modes, asymmetry effects are known to provide subdominant corrections to the growth rates (G. Bertin 1982; F. Militello et al. 2004), as long as conditions (i) and (ii) are strictly valid. It is then reasonable to neglect curvature effects, as a first approximation.

Condition (iv) does not seem to us to represent a major issue for the present discussion. A magnetic component along the shear coordinate has been shown by B. V. Somov & A. I. Verneta (1988, 1989) to contribute, to stabilize the tearing mode. While this effect is important in the magnetosphere, it is an unlikely occurrence in turbulent current sheets forming in between vortices in 2D turbulence.

Condition (v) is instead the most critical one, in our opinion. A flow $u_\parallel$ parallel to the current sheet on a scale length $\ell_\parallel$ and sheared in the same direction was shown by S. V. Bulanov et al. (1979) and S. I. Syrovatskii (1981) to stabilize the tearing mode, unless the condition $\gamma \gtrsim u_\parallel/\ell_\parallel$ is satisfied. For $u_\parallel \sim c_{A,L}$, as it is often the case, and $\ell_\parallel \sim L$, the instability criterion by Bulanov and Sirovatskii for the fastest growing mode reads

$$ \gamma_M \tau_{A,L} \gtrsim 1. \qquad (3) $$

Numerical investigations of the stability criterion $\gamma \gtrsim u_\parallel/\ell_\parallel$ for Sweet-Parker current sheets led C. Shi et al. (2018) to distinguish two combined effects: the slowing down of the growth of the magnetic island, due to its inhomogeneous stretching by the sheared flow $u_\parallel$, and the evacuation of the island itself by the outflow—the latter was already pointed out, e.g., by D. Biskamp (1996), N. Loureiro et al. (2007). Corrections accounting for parallel flows in the stabilization of Sweet-Parker current sheets against tearing modes have been also considered in some regimes (A. Tenerani et al. 2015; K. A. P. Singh et al. 2019). Notably, D. Biskamp (1996) inferred from numerical simulations that the instability criterion can be quantitatively refined as $\gamma > 4u_\parallel/\ell_\parallel$: he considered the case of a Sweet-Parker current sheet $L/a \sim S_L^{1/2} \sim S_a$ and specialized the condition $\gamma \gtrsim u_\parallel/\ell_\parallel$ by assuming that the plasmons are generated by a fastest tearing mode $\gamma_M \tau_{A,a} \sim S_a^{1/2}$ (H. P. Furth et al. 1963); he then compared this scaling to the instability threshold $S_L \gtrsim 10^4$ that he already found numerically (D. Biskamp 1986)—the same hypothesis that later would be used to characterize the “plasmoid instability regime” identified by N. Loureiro et al. (2007) via a boundary layer analysis. At least for tearing modes developing on the Sweet-Parker current sheet, the estimate of D. Biskamp (1996) may be considered robust enough, since the threshold condition $S_L \gtrsim 10^4$ was verified by several subsequent works. Just small variations around this value—i.e., multiplying factors ranging from 1/5 to 2/3, depending on the way the threshold condition is numerically evaluated—were found in the dedicated studies of L. Ni et al. (2010) and C. Shi et al. (2018).

However, current sheets forming close to vorticity sheets typically have a parallel flow $u_\parallel$ sheared also across the current sheet, similarly to the magnetic field. Depending on the symmetry or asymmetry of the velocity component $u_\parallel$ with respect to the shear coordinate—the same of the equilibrium magnetic field, the vorticity sheet can become unstable to “varicose modes” and Kelvin-Helmholtz-type modes. This makes tearing modes potentially compete and/or merge with
fluid instabilities of Kelvin–Helmholtz type (M. Dobrowolny
1972; I. Hofman 1975; M. Dobrowolny et al. 1983; G. Einaudi
& F. Rubini 1984, 1986; X. Chen & P. Morrison 1990;
J. Drake et al. 1994; D. Biskamp & T. Sato 1997; Q. Chen
et al. 1997; D. Biskamp et al. 1998, 2000; J. H. Li & Z. W. Ma
2010; X. Zhang et al. 2011; J. H. Li & Z. W. Ma 2012;
Y.-H. Liu et al. 2014; D. Borgogno et al. 2022; J. De Jonghe &
R. Keppens 2024). In particular, the tearing mode analysis in
the presence of flows has been solved by I. Hofman (1975)
for the equivalent of the small-Δ' limit in absence of
flows. He obtained for a Harris-pinch equilibrium the scaling
$\gamma\tau_{A,a} \sim S_a^{-1/2}[\gamma_{KH}\tau_{A,a}(1 - (\gamma_{KH}\tau_{A,a})^2)]$, valid for $0 \le \gamma_{KH}\tau_{A,a} < 1$, where $\gamma_{KH} \simeq u_\parallel(0)/a$ is the growth rate that a
Kelvin–Helmholtz instability would have if destabilized on the
same velocity profile $u_\parallel(x)$ in absence of a current sheet; this
scaling with $S_a$ was confirmed by complementary analytical
calculations performed by M. Dobrowolny et al. (1983) and by
the numerical study of G. Einaudi & F. Rubini (1986). An
extension of this analysis to a wider spectrum of unstable
wavenumbers (i.e., to the equivalent of the large-Δ' and of the
fastest mode limits) is however lacking. The competition
between the tearing and Kelvin–Helmholtz instability has been
investigated also for Sweet–Parker-type current sheets
(N. Loureiro et al. 2013).

Neglecting sheared flows at equilibrium is probably the strongest approximation we are going to make also in this work. Nevertheless, including these effects is likely to provide corrections, which are maybe dominant but generally not detrimental for—and possibly in favor to—the arguments we develop below.

2.3.3. Role of Fluid Compressibility in Tearing Modes and Relation to the Hall Term

The tearing mode theory was initially developed by
neglecting the Hall term in Ohm's law. This term was later
included, and its effects were studied in several works, but
extending the analytical theory to account for it is not trivial.
The change brought to the linear problem by this term seems
radically different from other nonideal parameters. The Hall
term is related to fluid compressibility, and, as a consequence,
to the number of eigenfunctions of the tearing mode analysis,
especially in the $\beta_e \simeq \beta_i \simeq 1$ regime of interest here. Some
words on this point are coming up, so as to outline these issues
and the strategy with which we are going to model the Hall
physics, next.

H. P. Furth et al. (1963) proved that in the strong guide field
limit the tearing mode can be described in an incompressible
model, corresponding to what later would become known as
“reduced MHD” (H. R. Strauss 1976, 1977; G. P. Zank &
W. H. Matthaeus 1992b). Incompressibility is valid when the
sound speed is larger than the phase velocity $v_\varphi$ of the normal
modes. In MHD, when $v_\varphi \sim c_{A,0}$, this translates into
$\beta \equiv \sum_\alpha \beta_\alpha \gg 1$. However, in the reduced-MHD ordering, in
which the condition $\epsilon_B \equiv |B_\perp|/B_g \ll 1$ holds, incompressibil-
ity was shown to be valid also as long as $\beta \ll 1$ (H. R. Strauss
1976, 1977): intuitively, the flow perpendicular to an almost
uniform guide field is incompressible at the leading order, as
long as the drift ordering holds, and the plasma is polytropic
(so that both the $E \times B$ and diamagnetic drift have null
divergence). Then, G. P. Zank & W. H. Matthaeus (1992b)
showed that, neglecting Hall's term in Ohm's law, a set of

equations formally equivalent to those of incompressible
reduced MHD are appropriate also when $\beta \simeq 1$.

The issue is that, when the Hall term is included, the value
of the plasma beta alone does not justify anymore the fluid
incompressibility, which generally depends on the values of
both $\beta$ and of the normalized ion skin depth $d_i/a$ (see, e.g.,
V. V. Mirnov et al. 2003, 2004; R. Fitzpatrick & F. Porcelli
2004; N. Bian & G. Vekstein 2007; N. H. Bian & D. Tsiklauri
2009). In decaying turbulence, compressibility becomes
important at the small scales where Hall effects dominate
(S. Servidio et al. 2007). In reconnection, the parameter range
in which incompressibility becomes more questionable is
exactly when both $\beta_e \simeq 1$ and $d_i/a \simeq 1$, which is the regime of
interest for solar wind turbulence that we will focus on
(Section 3).

Notably, the $d_i/a \simeq 1$ value asymptotically speaking lays in
between two physically distinct but formally similar tearing
regimes that are both incompressible. One is the slab, reduced-
MHD tearing model met at $d_i/a \ll 1$, in which the Hall term is
negligible when either $\beta_e \lesssim 1$ or $\beta_e \gg 1$; here, the magnetic
field is advected by the incompressible bulk fluid velocity $U$,
which can be assumed to be of the order of the Alfvén velocity
we named $c_{A,a}$, at which energy propagates. The other one is
the electron-MHD (EMHD) regime obtained from the Hall-
MHD description when $d_i/a \gg 1$ (S. Kingsep et al. 1990;
A. Fruchtman & H. R. Strauss 1993; A. V. Gordeev et al.
1994), in which tearing modes are described by a slab model
(S. V. Bulanov et al. 1992; A. Fruchtman & H. R. Strauss
1993; K. Avinash et al. 1998; N. Attico et al. 2000), and the
magnetic field in the ideal limit is advected by an
incompressible electron fluid $u_e$; the in plane components of
this velocity are associated with the fluctuations of the guide
field, which is instead neglected when the Hall term is
disregarded. In EMHD, energy is propagated by whistler
waves rather than by Alfvén waves, and the model is valid for
any value of $\beta_e$, as long as the closure is polytropic.

Density fluctuations are important for the transition from the
velocity field $U$ to $u_e$ in dragging the magnetic lines in the two
regimes. No analytic theory based on the boundary layer
analysis is available when density fluctuations are important,
and $d_i/a$ is large. Nevertheless, in some regimes at $\beta_e \ll 1$, the
Hall term can be shown to contribute in the generalized Ohm's
law of reduced MHD via a term proportional to $(\rho_s/a)^2$,
making the tearing equations independent of $d_i/a$ (see
Appendix A.3).

This is why, in Section 3, we will address the eigenvalue
problem by comparing the results obtained with a slab
reduced-MHD incompressible model, extended so as to
include the $\rho_s$ scale, to those obtained with a slab Hall-MHD
model accounting for the $d_i$ scale, density fluctuations, and
arbitrary values of the plasma $\beta$ (R. Fitzpatrick &
F. Porcelli 2004).

We conclude recalling that several numerical studies,
starting from the linear numerical analysis of T. Terasawa
(1983), pointed out an increase of the reconnection rate when
the current sheet becomes comparable or smaller than the ion
inertial length $d_i$. Among them, we cite M. E. Mandt et al.
(1994), D. Biskamp et al. (1995, 1997), L. Yin et al. (2002),
H. Karimabadi et al. (2004), I. F. Shaikhislamov (2004),
H. Karimabadi et al. (2005), F. Pucci et al. (2017), and
E. Papini et al. (2019b, 2020), and the studies related to the so-
called “Geospace Environmental Modeling challenge”
(J. Birn et al. 2001), which largely focused on this topic (see, e.g., J. Birn & M. Hesse 2001; Z. W. Ma & A. Bhattacharjee 2001; A. Otto 2001; M. A. Shay et al. 2001). This transition to a fast reconnection has been often related to a transition from the Alfvénic to the whistler dynamics, i.e., to a transition from MHD to EMHD reconnection (A. Fruchtman & H. R. Strauss 1993; M. E. Mandt et al. 1994; D. Biskamp et al. 1995; D. Biskamp & T. Sato 1997; N. Bian & G. Vekstein 2007; M. Hosseinpour et al. 2009); this led to the identification of the important role played by dispersive waves in the transition to fast reconnection (B. N. Rogers et al. 2001). Then, it was shown that an inclusion of further kinetic effects related to the anisotropic pressure tensor can combine with the Hall physics to contribute to the increase of the reconnection rate (see, e.g., L. Yin et al. 2002; P. Cassak et al. 2015).

### 3. An Application of Tearing Mode Analysis to Solar Wind Turbulence

We now aim to assess if a tearing instability can be applied, with respect to the condition $γτ_{cs} ≫ 1$ of point (ii) above, to a current sheet generated by Alfvén turbulence in the solar wind assuming “typical” conditions $β_i ≃ β_e ≃ 1$, for example around 1 au (see, e.g., F. Sahraoui et al. 2013; D. Verscharen et al. 2019; L. B. Wilson III et al. 2018).

We pursue the strategy to heuristically assume $γτ_{cs} ≫ 1$ so as to quantify $γτ_{A,a}$ or $γτ_{A,a}^*$ according to some model (i.e., regime) of tearing instability. To this end, it is essential to evaluate the current sheet dimensions, i.e., a and L, relative to the microscopic nonideal scales involved, as well as its characteristic evolution time, $τ_{cs}$. Section 3.1 below presents the specific hypotheses we make to this purpose.

Of course, if the condition $γτ_{cs} ≫ 1$ were not verified, the estimate of $γτ_{cs}$ we obtain this way should be regarded as *inconsistent*, and thus, it should be rejected, since it would contradict the hypothesis (ii) we started from. On the other hand, verifying the condition $γτ_{cs} ≫ 1$ does not necessarily imply that the estimate obtained for $γτ_{cs}$ is correct, but just that it is consistent and thus “possible.”

#### 3.1. Estimate of the Size and of the Evolution Time of the Current Sheet

We now estimate a, L, and $τ_{cs}$ in Alfvénic turbulence for $β_e ≃ β_i ≃ 1$, and we discuss the related hypotheses.

##### 3.1.1. Current Sheet Thickness, a

We start with the estimate of a. Measuring the thickness of a current sheet in turbulence is difficult, and it is also difficult in simulations (S. Servidio et al. 2009; H. Betar et al. 2022). A current sheet is expected to exist as a steady structure, since it should form due to the balance of magnetic stress forces (E. Parker 1972). Under this hypothesis, a rough dimensional estimate in terms of the plasma beta can be obtained (D. Del Sarto & F. Pegoraro 2018) by imposing steady conditions on the first three fluid moments of Vlasov equation combined in an extended MHD model with polytropic electrons and anisotropic ions. We estimate $|u_i| ∼ c_H$ (characteristic “fluid” velocity) and $|∇| ∼ (ℓ)^{-1}$, so that $|∇·Π^α|/(nm_i) ∼ β_0 c_{A,tot}^2/ℓ$, where $c_{A,tot}^2 = c_{A,0}^2(B_0^{tot}/B_0)^2$. Balancing the convective and stress forces in momentum equation, $nm_i·∇u_i ∼ ∇·(P^eI + Π^i + BB)$, we get $c_H^2 ∼ c_{A,tot}^2(β_e + β_i)$ when $β_e + β_i ≫ 1$ and $c_H^2 ∼ c_{A,tot}^2$ when $β_e + β_i ≪ 1$. Balancing the

convective, compressive, and fluid stress terms with the magnetic rotation term in the ion pressure tensor equation, we get $c_H/ℓ ∼ Ω_i$ (see aforementioned reference). Combining these estimates, we find $ℓ ∼ d_i$ for $β_e + β_i ≪ 1$ and $ℓ ∼ \sqrt{ρ_s^2 + ρ_i^2}$ for $β_e + β_i ≫ 1$, whence assuming $a ∼ ℓ$ we can infer

$$a ∼ d_i ∼ ρ_s ∼ ρ_i \quad \text{for} \quad β_e ≃ β_i ≃ 1. \quad (4)$$

It follows from this that in a typical hydrogen plasma $d_e ≃ d_i/42.3 ≃ 0.023 a$.

Equation (4) agrees with further independent estimates based on numerical measures (Y. L. Sasunov et al. 2015; S. S. Cerri & F. Califano 2017; L. Franci et al. 2017; E. Papini et al. 2019a, 2020), on fits of the turbulent spectra in simulations (L. Franci et al. 2016, 2022b), and on other indirect measurements from spacecraft data (T. I. Pulkkinen et al. 1994; T. D. Phan et al. 2010; C. Chen et al. 2014; J. J. Podesta 2017; A. B. Lotekar et al. 2022; I. Y. Vasko et al. 2021, 2022). They are also compatible with specific studies on the current sheet dynamics in astrophysical turbulence (G. G. Howes 2016). It is because of the consistency with all these results that henceforth we will assume the validity of estimates in Equation (4).

The derivation above, which makes use of the full pressure tensor equation, is appropriate only if a pressure anisotropy is allowed: should the closure be isotropic (e.g., polytropic), estimates in Equation (4) would not be obtained. It is nowadays well acknowledged that around X-points both ion and electron pressure anisotropies develop (see H.-J. Cai & L. C. Lee 1997; J. U. Brackbill 2011; J. Egedal et al. 2013). These can be generated by shear flows with a strain (D. Del Sarto et al. 2016; D. Del Sarto & F. Pegoraro 2018), so that anisotropic pressures are generally expected not only at X-points, which are hyperbolic points of the flow, but along the whole vorticity sheet. Below, we neglect for simplicity the role of pressure anisotropy on the tearing growth rate. This may be a further strong approximation. However, as long as pressure anisotropy increases the reconnection rate of tearing-type instabilities, as it is generally believed to do (see, e.g., S. I. Vainshtein & V. A. Mazur 1982; J. Chen & P. Palmadesso 1984; M. Hesse et al. 2001; M. M. Kuznetsova et al. 2001; L. Yin et al. 2002; P. Cassak et al. 2015; C. Granier et al. 2021), this does not go against the argument we develop.

#### 3.1.2. Current Sheet Length, L

The current sheet length L is itself a difficult quantity to measure, but for the rescaling argument that we will use, rough estimates are sufficient.

In general, L is a significant fraction of the vortex circumference associated with the current sheet: simulations suggest it to be comparable to a quarter or to half of a circumference. The estimate $L ∼ k^{-1}$ can be thus done for a vortex whose diameter is comparable to $1/k$ (numerical factors like 2 or $π$ are not necessary, here), which makes L scale dependent. We may thus try to estimate L in terms of both the nonideal parameters and of some length scale. Attempting to estimate the aspect ratio $L/a$ is thus the most natural choice.

In the resistive limit, for example, the well-known Sweet-Parker aspect ratio, $L ∼ S_a a$, or $L ∼ S_a^{1/2} a$, can be obtained by imposing steady conditions on both the momentum equation and resistive Ohm’s law. However, the “high instability” of these steady configurations to tearing modes may pose further consistency problems on which we will come upon later
(Section 4.3.1). All of this leads us to seek alternative criteria for estimating $L$.

The most practical and reliable alternative, sufficient to our purposes, is to refer to several “experimental results,” whether inferred from in situ measurements or from numerical simulations. These suggest that the aspect ratio $L/a$ of reconnecting current sheets spans some (or several) decades. For example, simulations of a 2D turbulence cascade initialized at MHD scales with values of $\beta_e = \beta_i = 1$ suggest values of $L/a \sim 30–40$ (see, e.g., the contour-plots in S. S. Cerri & F. Califano 2017; L. Franci et al. 2017; L. Franci et al. 2022b). Similarly, estimates of $L \gtrsim 20d_i$ can be derived from Figure 1 of E. Papini et al. (2019a), in which the $\rho_i$ scale is not included in the model.

We then note that both the aspect ratio and the current sheet thickness depend on the reconnection regime. For example, 2D reduced-MHD simulations of viscous-resistive turbulent reconnection in reduced MHD led to estimates of $L/a$ spanning some orders of magnitude: a majority of values in the range $L/a \sim 100–1000$ were observed, with a small fraction of cases with $L/a \sim 10–100$ (see S. Servidio et al. 2011; M. Wan et al. 2013). Comparable results, with aspect ratios $L/a \gtrsim 50$ or much larger, can be inferred by the simulations performed in analogous regimes with a much higher resolution by C. Dong et al. (2018). Additionally, values of $L/a \sim 100–1000$ are suggested also by the 2D reduced-MHD simulations of D. Borgogno et al. (2022) in a collisionless, cold regime, where $a \sim d_e$ was estimated. Aspect ratios $L/a > 20$ have been measured also in “electron-only” reconnection regimes: from Figure 1 of F. Califano et al. (2020), where for $d_i = 12d_e$ it is measured $a = 4d_e$, we can estimate $L \gtrsim 10d_i = 120d_e$ and therefore $L/a \gtrsim 30$. Similar results can be derived from a visual estimate of the “electron-only” reconnecting current sheets reported in Figure 6(c) of L. Franci et al. (2022b), where $d_i = 10d_e$, instead.

We conclude by recalling the lowest threshold value of $L/a \sim 20$ suggested by M. Velli & A. W. Hood (1989) for the destabilization of a fastest tearing mode. Since this value is smaller than the actual estimates derived from all simulation results, we will conservatively adopt it in the following, so as to provide a lower bound estimate for $\gamma\tau_{A,L}$ after rescaling $\gamma\tau_A$, $a$ by positive powers of $L/a$.

It is equally important to underscore that in all numerical results quoted above, in which the turbulence cascade simulations were initialized by injecting energy at the ideal-MHD scale $1/k_{\text{inj}}$, it was found that $L \sim k_{\text{inj}}^{-1}$.

### 3.1.3. Current Sheet Evolution Time, $\tau_{cs}$

This is probably the most difficult estimate, also from a theoretical point of view.

In an estimate widely adopted in literature, compatible with the steadiness condition of a current sheet formed due to a force balance, $\tau_{cs}$ is generally assumed to be of the order of the eddy turnover time $\tau_{nl}$ of vortices.

In a turbulent velocity field $V$, statistically isotropic in the shear plane (so that $V(r)$ can replace $V(r)$ in the spatial averages below), the turnover time of a vortex of a size $l_k \sim 1/k$ is $\tau_{nl}(l_k) \equiv l_k / \langle |V(r) - V(r + l_k)|^2 \rangle^{1/2}$, where $\langle f(r) \rangle$ denotes the spatial average over $r$ of any scalar function $f(r)$ (see, e.g., H. A. Rose & P. L. Sulem 1978). Denoting with $\hat{f}_k$ the $k$ component of the Fourier transform, i.e., $\hat{f}_k = \mathcal{F}[f(r)]_k$, one can approximate $\tau_{nl}(l_k) \sim 1/(k\hat{V}_k)$ (U. Frisch et al. 1978).

In this approximation, $\tau_{nl}$ is related to the advection term $V \cdot \nabla V$ in Navier-Stokes equation. By extension, an analogous eddy turnover time can be associated with the corresponding advection term in other turbulent regimes.

In the case of magnetic field lines dragged by vortices of a size $l_k \sim 1/k$, one estimates $\tau_{cs}(l_k) \sim \tau_{nl}(l_k) \sim 1/(k\hat{u}_{e,k})$, physically related to the advection term $u_e \cdot \nabla B$ in the equation for $\partial B/\partial t$, which is obtained from the curl of Ohm's law in Equation (1) when the Hall term is retained (L. Franci et al. 2017; E. Papini et al. 2019a).

In the presence of a guide field strong enough, mainly oriented perpendicularly to the plane of the turbulence dynamics, the electron and ion fluid velocities are dominated by the same $E \times B$-drift component, which makes the customary estimate $\tau_{nl} \sim 1/(k\hat{U}_k) \simeq 1/(ku_i) + O(\epsilon_m)$ valid, too. Note that at ideal-MHD scales the condition $\tau_{nl} \sim 1/(k\hat{U}_k)$ may follow from $\partial U/\partial t + U \cdot \nabla U \simeq 0$ obtained under force balance conditions. This in principle allows for the identification $\tau_{cs}(l_k) \sim \tau_{nl}(l_k) \sim 1/(k\hat{U}_k)$ also in steady reconnection models á la Sweet-Parker. Among Sweet-Parker type reconnection models, we can include those by P. A. Sweet (1958), E. N. Parker (1957), H. E. Petschek (1964), B. U. Sonnerup (1970), T. Yeh & W. I. Axford (1970), E. R. Priest & B. U. Ö. Sonnerup (1975), and P. A. Cassak & M. A. Shay (2007) as well as collisionless inertia-driven versions where $L/a \sim (L/d_e)$ (J. A. Wesson 1990; S. V. Bulanov et al. 1992; K. Avinash et al. 1998).

For Alfvénic turbulence initialized at the MHD scales $k_{\text{inj}}^{-1} \sim L$ with harmonic perturbations $\delta B_{\perp}$ of rms amplitude $\delta B_{\text{rms}}$ over a uniform guide field $B_g e_z$, the ion velocity is of the order of the Fourier component $k = k_{\text{inj}}$ of the Alfvén velocity, $\hat{c}_{A,k_{\text{inj}}} \equiv \mathcal{F}[B/\sqrt{4\pi m_i n_i}]_{k_{\text{inj}}}$. At the scale $L \sim k_{\text{inj}}^{-1}$ at which $|\delta \hat{B}_{\perp,k}|$ is peaked with amplitude $\sim \delta B_{\text{rms}}$ (see the initialization of decaying turbulence simulations in L. Franci et al. 2015), only the perturbed magnetic field $\delta B_{\perp}$ contributes to the Fourier transform of $B = B_g e_z + \delta B_{\perp}$, whereas density fluctuations $\delta \hat{n}_{i,k_{\text{inj}}}$ are typically negligible with respect to the uniform $\hat{n}_{i,0}$ component (this is due to the incompressibility of the leading drift MHD flow in a polytropic plasma). We thus approximate $\hat{c}_{A,k_{\text{inj}}} \simeq \delta B_{\text{rms}}/\sqrt{4\pi m_i |\hat{n}_{i,0}|}$ (note that the identification $\hat{c}_{A,k_{\text{inj}}} \propto |\delta \hat{B}_{\perp,k_{\text{inj}}}| \sim \delta B_{\text{rms}}$ would not be appropriate, instead, if the fluctuating magnetic energy were not concentrated at the injection scale). Therefore,

$$ \tau_{cs}(L) \sim \tau_{nl}(L) = \frac{1}{k_{\text{inj}} \hat{c}_{A,k_{\text{inj}}}} \approx \frac{\sqrt{4\pi m_i |\hat{n}_{i,0}|}}{k_{\text{inj}} |\delta B_{\text{rms}}|} \quad (5) $$

### 3.1.4. Current Sheet Evolution Time, $\tau_{cs}$, versus Normalization Time of Tearing Modes

The nonlinear time $\tau_{nl}(L)$ of Equation (5) is defined in a statistical sense, using the Fourier components. We now make connection with the normalization quantities of Section (2.1.1). In a classical tearing analysis, both $B_0 = B_{cs}$ and $n_0 = n_{cs}$ are evaluated locally, “far enough” from the neutral line, in a region where ideal MHD holds. For a single current sheet, this is usually compatible with values of the density and of the magnetic field, which are spatially uniform. For example, thanks to the incompressible MHD approximation, if a Harris pinch equilibrium $B_{y,0} = B_{cs} \tanh(x/a)$ is chosen, a reference value $c_{A,cs} = B_{cs}/\sqrt{4\pi m_i n_{cs}}$ uniform in space can be defined.
The same formal definition does not apply to current sheets generated by turbulence: the equilibrium shear length itself, i.e., the current sheet thickness *a*, is microscopic, and the current sheets are unlikely to be separated enough so as to grant the validity of ideal-MHD hypothesis in an intermediate region. Nevertheless, we can consider for each current sheet a local characteristic amplitude $B_{cs}$ like if an in-plane magnetic field $B_{y,0} \simeq B_{cs} \tanh(x/a)$ provided a local approximation of the current sheet profile. Statistically speaking, all current sheets of characteristic length $L \sim k_{inj}^{-1}$ have a comparable amplitude $B_{cs} \sim |\delta\hat{B}_{\perp,k_{inj}}| \simeq \delta B_{rms}$ (L. Franci et al. 2017; E. Papini et al. 2019a; L. Franci et al. 2022a). Moreover, owing to Equation (5), we can choose $n_{cs} \simeq |\hat{n}_{i,0}|$. So, we can normalize speeds to the (statistically representative) in-plane Alfvén velocity associated with turbulent fluctuations at the scale $L$, i.e., we define $c_{A,cs} = |\hat{c}_{A,k_{inj}}|$. We can this way identify the characteristic time that an Alfvén wave takes to travel, respectively, across a current sheet of the aspect ratio $L/a$, i.e., $\tau_{A,a} \equiv a/c_{A,cs}$, and along it, i.e., $\tau_{A,L} \equiv L/c_{A,cs}$. This normalization choice departs from the aforementioned studies of decaying MHD turbulence, where speeds are typically normalized to $c_{A,cs}^*$, i.e., to a “large scale” Alfvén velocity uniform in space and computed with respect to $|\hat{n}_{i,0}|$ (i.e., $n_0 = n_{cs}$) and to the total magnetic field ($B_0 = B_0^{\text{tot}}$) or to the guide field ($B_0 = B_g$). Note that $\tau_{A,a} = a\tau_{A,L}/L$ implies $\tau_{A,L} \gg \tau_{A,a}$ for $L \gg a$, and while $\tau_{A,a}$ is the typical normalization time of the tearing mode theory (in a periodic current sheet, *a* is the relevant length at equilibrium, since $L \sim 1/k$ depends on the perturbation). Moreover, $\tau_{A,L} \simeq \tau_{nl}(L)$ thanks to the identifications above. In conclusion, we estimate $\tau_{cs}(L) \sim \tau_{nl}(L) \sim \tau_{A,L}$.

According to this scenario, a current sheet in Alfvénic turbulence can be considered to be static if $\gamma_M \tau_{A,L} \gg 1$. The threshold condition $\gamma_M \tau_{nl}(L) \sim 1$ for the disruption of a current sheet by tearing modes in turbulence was already pointed out and discussed by V. Carbone et al. (1990). It was also considered in the framework of the “ideal tearing” by F. Pucci & M. Velli (2014) and M. Velli et al. (2015), although for such a model different arguments are used with respect to the present ones, and values $\gamma \tau_{A,L} \lesssim 1$ are a posteriori obtained. It is then of practical interest to quantify “how much” bigger than unity $\gamma_M \tau_{nl}(L) \sim \gamma_M \tau_{A,L}$ must be for the threshold condition $\gamma \tau_{cs} \gg 1$ to be satisfied. Of course, in an asymptotic sense, the conditions $\gamma_M \tau_{A,L} > 1$ and $\gamma_M \tau_{A,L} \gg 1$ shall be read as equivalent. Nevertheless, for practical applications of a stabilization criterion of this kind, a ratio between the compared quantities of the order of 1 or 2 orders of magnitude is usually required. Quantifying this ratio may be discriminant for the interpretation of numerical results with respect to different theoretical models. This is why the “$\gg$” symbol should be preferred in between $\gamma \tau_{cs}$ and 1. However, for the product $\gamma_M \tau_{A,L}$, we will write $\gamma_M \tau_{A,L} \gtrsim 1$, by keeping unity as a lower bound. Indeed, despite of the formal condition in Equation (2), too large values of $\gamma \tau_{A,L}$ with respect to unity are unlikely, since the current sheet would be disrupted by the spontaneous instability as soon as the threshold condition is fulfilled—the same argument because of which the occurrence of Sweet Parker current sheets at large values of $S_L$ are deemed unlikely due to the instability of such current sheets versus (too fast) tearing modes (R. K. Jaggi 1964; F. Pucci & M. Velli 2014; see also Section 4.3.1, next). In this regard, we also note the practical quantitative equivalence of the steadiness condition $\gamma_M \tau_{A,L} \gtrsim 1$ with Equation (3) for the Bulanov

stability criterion against flows, upon which we will return later (Section 4).

Finally, we note that a current sheet generated by turbulence could in principle remain in an almost static condition $\tau_{cs} \gg \tau_{nl}(L)$ for a long time, enough that tearing modes disrupt it. While numerical results suggest that often $\tau_{cs} > \tau_{nl}(L)$, once the current sheet has formed (and note that this would facilitate the fulfillment of $\gamma \tau_{cs} \gg 1$ even for “small” values of $\gamma \tau_{A,L} > 1$), the two values are typically close. Moreover, as we will discuss in Section 4.3.3, the possibility that $\tau_{cs} \gg \tau_{nl}(L)$ turns out to be not compatible with models for which the condition $\gamma \tau_{A,L} < 1$ is implied, instead, and which we will make a comparison to.

### 3.2. Growth Rate for $\beta_e \simeq \beta_i \simeq 1$ and Related Scaling Laws

We can now evaluate the growth rate of the fastest tearing mode with the aid of two numerical eigensolvers adapted from the original version discussed in H. Betar et al. (2020). We consider two complementary models, using different normalizations. One is the two-field model of D. Grasso et al. (2000), which neglects the Hall term—and thus the scale length $d_i$—but which accounts for $d_e$, $\rho_s$, and $\rho_i$. We have also included the resistivity (i.e., $S_a$). This model, formally valid for a sufficiently large guide field, corresponds to a non-ideal-MHD extension of the resisting tearing model by H. P. Furth et al. (1963) in reduced MHD (see Appendix A). Speeds are normalized to $c_{A,cs}$, and times are normalized to $\tau_{A,a}$. The other one is the four-field model developed by R. Fitzpatrick & F. Porcelli (2004), formally valid for arbitrary values of $d_i/a$ and of $\beta_e$—and thus for an arbitrary guide field amplitude. This model, in which speeds are normalized to $c_{A,cs}^*$ and times to $\tau_{A,a}^*$ (see Section 2.1.1), accounts for the scale lengths $d_e$, $d_i$, $\rho_s$ and for $S_a^*$, but lacks ion finite Larmor radius (FLR) effects, i.e., of the scale $\rho_i$. In principle, it allows for a self-consistent description of the Hall term contribution (see Section 4.1 and Appendices A.2–A.3).

We need to quantify $\gamma_M \tau_{A,a}$ and $\gamma_M \tau_{A,a}^*$. No analytical estimate is available, since the boundary layer theory is valid only for all $d_e$, $d_i$, $\rho_s$, $\rho_i \ll a$ and $S_a^{-1}$, $(S_a^*)^{-1} \ll 1$. We thus rely on numerical calculations. For $d_e = \rho_s = \rho_i = a$, $d_e = 0.023a$, and $S_a \gtrsim 10^5$, we, respectively, measure $\gamma_M \tau_{A,a} \simeq 0.2$ in the two-field model, and $\gamma_M \tau_{A,a} \simeq 0.13$ in the four-field model; see Figures 1–2, below. These results are in line with the “$\sim 0.1$ reconnection rate” for fast reconnection, which has been discussed in previous models compatible with reconnecting instabilities (P. A. Cassak et al. 2006; P. Cassak et al. 2015; L. Comisso & A. Bhattacharjee 2016).

We emphasize that these reconnection rates could not be a priori determined in the framework of the boundary layer analysis. To make a comparison with the known theoretical power-law scalings obtainable when *a* is much larger than the microscopic lengths at play, we can estimate a local power-law dependence in the parameter space from a graphical fit of a set of numerically computed values of $\gamma_M \tau_{A,a}$ and $\gamma_M \tau_{A,a}^*$. To do so, we follow the criteria below.

For the two-field model, we look for a scaling like $\gamma_M \tau_{A,a} \sim (d_e/a)^c [(\rho_s/a)^2 + (\rho_i/a)^2]^{c_2} S_a^{-c_3}$ with $c_1, c_2, c_3 \ge 0$. We infer this by an ansatz based on the known asymptotic scalings valid both in the small and large $\Delta'$ limits (F. Pegoraro & T. J. Schep 1986; F. Porcelli 1991) and for the fastest mode (L. Comisso et al. 2013; D. Del Sarto et al. 2016): a symmetric
<figure><img src="image_5.png" /><figcaption>**Figure 1.** Normalized growth rates $\gamma\tau_{A,a}$ as a function of ka in the two-field model (Appendix A) for $d_e = 0.023a$, $\rho_s = \rho_i = a$, and $S_a^{-1} = 0$ (blue color) or $S^{-1} = 10^{-6}$ (red color).</figcaption></figure>

dependence on $\rho_s$ and $\rho_i$ as a power of $\sqrt{\rho_s^2 + \rho_i^2}$ is always observed, at least as long as $d_e/a \ll a$. Departures from it have been numerically observed, but only for $d_e/a \gtrsim 0.1$ (D. Del Sarto et al. 2011), a case not relevant here.

For the four-field model, instead, no systematic numerical study is available. For $\beta_e \simeq 1$, of interest here, the theoretical scalings $\gamma\tau_{A,a}^* \sim (d_i/a)(d_e/\Delta')^2(ka)$ and $\gamma\tau_{A,a} \sim \sqrt{(d_e/a)(d_i/a)}(ka)$ are available in the small-$\Delta'$ and large-$\Delta'$ limits, respectively (V. V. Mirnov et al. 2003, 2004; R. Fitzpatrick & F. Porcelli 2004), but only for $d_e, d_i \ll 1$. We also note that $d_e$ and $d_i$, on the one hand, and $\rho_s$ and $d_i$, on the other hand, are not independent parameters since $d_e = \sqrt{m_e/m_i}(d_i/\tilde{Z})$ and $\rho_s = \sqrt{\beta_e}d_i$ (see Section 2.1). This is why we look for a scaling like $\gamma_M \tau_{A,a}^* \sim (m_e/m_i)^{C_1}\beta_e^{C_2}(d_i/a)^{C_3}(S_a^*)^{-C_4}$ for $C_1, C_2, C_3, C_4 \gtrsim 0$, rather than for scalings with respect to $(d_e/a), (d_i/a), (\rho_s/a)$. Also, note that $\beta_e$ is not a “nonideal” parameter, but rather a dimensionless parameter related to the equilibrium conditions.

We first note (see Figure 1) that, in agreement with the results already discussed for $\rho_s = \rho_i = 0$ in the two-field model in H. Betar et al. (2020), resistivity is completely negligible with respect to $d_e/a = 0.023$ already for $S_a \gtrsim 10^5$, also when $\rho_s = \rho_i = a$. Then, based on the numerical results shown in Figures 2(a)–(c) for the two-field model and in Figures 2(d)–(f) for the four-field model, we make the local estimates in the parameter space

$$ \gamma_M \tau_{A,a} \sim \left(\frac{d_e}{a}\right)^{\frac{3}{5}} \left(\left(\frac{\rho_s}{a}\right)^2 + \left(\frac{\rho_i}{a}\right)^2\right)^{\frac{2}{10}} S_a^0, \quad (2-field) \quad (6) $$

$$ \gamma_M \tau_{A,a}^* \sim \left( \frac{m_e}{m_i} \right)^{\frac{3}{10}} \beta_e^{\frac{1}{5}} \left( \frac{d_i}{a} \right)^{\frac{3}{2}} (S_a^*)^0, \quad (4-field). \quad (7) $$

The numerical calculations for the four-field case have been performed for $B_{cs} = B_g$, so that, numerically speaking, $\tau_{A,a} = \tau_{A,a}^*$ and $S_a = S_a^*$, in this case. Note that the scalings above are meant just as a fit of a local power-law approximation of the growth rate in a neighborhood of the values $d_e = \rho_s = \rho_i = a$, $d_e = 0.023a$, and $S_a \gtrsim 10^5$. We also note that we have conservatively taken a as the normalization scale of tearing modes, rather than the half-thickness $a/2$ of

the current sheet, as it is normally assumed in tearing mode theory (H. P. Furth et al. 1963; H. Betar et al. 2022). Given the number of approximations made, this is not a major issue, but we highlight that because of this choice the value of $\gamma\tau_{A,L}$ we obtain is likely underestimated, rather than the opposite.

### 3.3. Rescaling $a \to L$ and Growth Rate $\gamma_{M\tau_{A,L}}$

We must now rescale $\gamma_M \tau_{A,a} \simeq 0.2$ (two-field) and $\gamma_M \tau_{A,a}^* \simeq 0.13$ (four-field) to obtain the corresponding values $\gamma_M \tau_{A,L}$. It suffices to rescale the Alfvén times at left-hand side of these expression (see Section 2.1.1). We obtain

$$ \gamma_M \tau_{A,L} \sim 0.2 \times \left(\frac{L}{a}\right), \quad (2-field) \quad (8) $$

$$ \gamma_M \tau_{A,L} \sim 0.13 \times \left(\frac{L}{a}\right) \left(\frac{B_g}{B_{cs}}\right), \quad (4-field). \quad (9) $$

Assuming conservatively an aspect ratio $L/a \simeq 20$, we obtain $\gamma_M \tau_{A,L} \gtrsim 4$ in the two-field model and $\gamma_M \tau_{A,L}^* = \gamma_M \tau_{A,L} \gtrsim 2.6$ in the four-field model for $B_{cs} = B_g$. They provide a likely lower bound to the growth rate of the fastest tearing mode obtained in the framework of this modeling, which comply with the condition $\gamma_M \tau_{A,L} \gtrsim 1$. Note that larger values of $L/a$ would yield larger growth rates; larger values of $L/a$ are suggested by several numerical studies indicating values $L/a > 20$ (see Section 3.1.2), or could be obtained just by normalizing lengths to $L_0 = a/2$ rather than to $L_0 = a$ (see Section 3.2). Also, note that for $B_{cs} < B_g$, as it typically happens in turbulence where $\delta B_{rms} < B_g$ and $B_{cs} \sim \delta B_{rms}$ can be assumed for an injection scale $k_{inj} \sim 2\pi/L$, the numerical value of $\gamma\tau_{A,L}$ is larger than the corresponding $\gamma\tau_{A,L}^*$. At the same time, given the number of assumptions made in Section 3.1—whose accurate verification is however challenging—we do not consider the estimates we obtained to be quantitatively precise. Instead, they should be just viewed as sufficient to demonstrate the plausibility of the condition $\gamma_M \tau_{A,L} \gtrsim 1$, which in turn can justify the applicability of the tearing analysis with respect to a static current sheet (i.e., $\gamma_M \tau_{cs} \gg 1$). Then, if the coupling with Kelvin–Helmholtz type instabilities along the current sheet were considered, several numerical results suggest that higher growth rates would be obtained for the resulting coupled mode, capable of producing plasmoid-like structures (see discussion in Sections 4.4, 5, next).

This conclusion is supported by the numerical simulations of decaying MHD turbulence of L. Franci et al. (2017) and by the more detailed studies of E. Papini et al. (2019a) and L. Franci et al. (2022a): for $\beta_e = \beta_i = 1$ and $\delta B_{rms} \simeq B_g/4$ (see also S. S. Cerri et al. 2017), from which we estimate $B_{cs} \simeq B_g/4$, they report estimated growth rates spanning a wide range of values and satisfying $\gamma_M \tau_{A,L} \gtrsim 1$. For example, from Figure 1(d) of L. Franci et al. (2017) and from the parameters given in S. S. Cerri et al. (2017), we identify $B_g/B_{cs} \sim 4$, $\gamma = \tau_{rec}^{-1}$, and $\tau_{A,L} = \tau_{inj}^{nl}$ so that we can infer that the first reconnection events satisfy $\gamma_M \tau_{A,L}^* \sim 10$, whence $\gamma_M \tau_{A,L} \sim 40$. Even if this value differs by a factor $\sim 15$ from the estimate in Equation (9), we read it as in strong support of the argument we developed, given the number of approximation we made by privileging an underestimate of the results, and given the
<figure><img src="image_6.png" /><figcaption>**Figure 2.** Local scalings of $\gamma_M \tau_A$ computed as described in H. Betar et al. (2020) for $S^{-1} = 0$. Upper frames: two-field model integrated for $d_i = 0$, in which only one among the parameters $d_e$, $\rho_s$, and $\rho_i$ is made to vary around $d_e = 0.023a$ (frame (a)), $\rho_s = a$ (frame (b)), and $\rho_i = a$ (frame (c)) by keeping the other values fixed as $d_e/a = 0.023$ and $\rho_s/a = \rho_i/a = 1$. Lower frames: four-field model integrated for $\rho_i = 0$, in which only one among the parameters $m_e/m_i = \sqrt{d_e/d_i}$, $\beta_e = \rho_s^2/d_i^2$, and $d_i/a$ is made to vary around values corresponding to $d_e = 0.023a$ (frame (d)), $\rho_s = a$ (frame (e)), and $d_i = a$ (frame (f)) by keeping the other values fixed to give the lengths above.</figcaption></figure>

different analytical models concerned: the one of S. S. Cerri et al. (2017); L. Franci et al. (2017) is hybrid Particle-In-Cell, so it accounts for possible ion anisotropization at the ion-scales (see comments at the end of Section 3.1.1), it has $d_e = 0$, but it assumes a finite resistivity $S_L^*$, and it accounts for all the scales $d_i \approx \rho_s \approx \rho_i \sim a$, while we can estimate from the other figures therein reported a typical value $L/a \gtrsim 30$; we recall that the results for the fluid four-field model of Equation (9) assume, instead, $d_e = 0.023a$ and $\rho_i = S_L^{-1} = 0$, all factors that concur, alone, to a correspondingly smaller value of $\gamma_M \tau_{A,a}$, which we rescaled by $L/a = 20$. A somewhat more accurate comparison can be made to the results of E. Papini et al. (2019a) with dedicated numerical calculations. These are discussed in Appendix B.2 and further support the argument above.

## 4. Summary and Discussion

We have proposed a possible explanation for how tearing modes can disrupt current sheets generated in 2D turbulence. These current sheets evolve over a characteristic timescale $\tau_{cs}$, which is typically of the order of the Alfvén time $\tau_{A,L}$ associated with the current sheet length $L$ and in-plane magnetic field. We focused on the fulfillment of the condition $\gamma_M \tau_{cs} \gg 1$, which in our opinion is necessary to make the standard tearing analysis applicable to this problem. The key element of the argument we propose is grounded on the microscopic thickness $a$ of current sheets generated in turbulence, because of which the growth rate of tearing-type modes is large enough—of the order 0.1–1, in units of the

Alfvén time $\tau_{A,a} = (a/L)\tau_{A,L}$—so as to grant the fulfillment of the condition $\gamma_M \tau_{A,L} \gtrsim 1$ after rescaling of lengths to $L$. We supported this argument with numerical calculations performed with two complementary models, both of which allow for the simultaneous inclusion of several nonideal effects. A parameter case compatible with low-collision solar wind turbulence at $\beta_e = \beta_i = 1$ has been considered, in which we estimated $a \approx d_i \approx \rho_s \approx \rho_i$.

The interpretation we propose for the observation of plasmons in turbulence is complementary to other theoretical models available in literature. At the same time, it shares with them some simplifying assumptions, which we have discussed (points (i), (iii-v) in Sections 2.3.1, 2.3.2), and which would deserve some dedicated studies on their own.

It is now worthwhile to discuss some implications of the analysis we have carried out, notably (1) some insights on the role played by kinetic effects in the transition to a fast reconnection (Section 4.1); (2) the nonapplicability of analytical results based on the tearing mode theory (Section 4.2); and (3) the consistency requirement imposed on turbulent reconnection models based on tearing modes by the condition $\gamma \tau_{cs} \gg 1$, which we translated into $\gamma \tau_{A,a} \gtrsim 1$ (Section 4.3). Points (2) and (3), in particular, will be discussed in relation to the main tearing-based interpretations of turbulent reconnection available in literature. A separate discussion will be dedicated to the purely resistive turbulent reconnection scenario (Section 4.3.1–4.3.2) and to the inapplicability of the “ideal tearing” estimates in the
collisionless regime of turbulence (Section 4.3.3–4.3.4). Then, we comment on the potential impact that the further approximations we made may have on the estimates in Equations (8)–(9) we obtained (Section 4.4).

## 4.1. Role of Microscopic Effects in the Transition to a “Fast” Reconnection

An immediate implication of the results discussed in Section 3 concerns the interpretation of the reconnection rate increase typically associated with the inclusion of the Hall term in Ohm's law, i.e., the Hall-induced transition to a fast reconnection (see Appendix A of D. Del Sarto & M. Ottaviani 2017 for further notions of “fast reconnection” used in the literature over the years). This increase as $d_i/a$ grows was typically interpreted in terms of the progressive decrease of the reference time, which is associated with the faster oscillation of the waves propagating energy in each regime (A. Fruchtman & H. R. Strauss 1993; M. E. Mandt et al. 1994; D. Biskamp et al. 1995; B. N. Rogers et al. 2001): Alfvén waves in MHD ($d_i/a \ll 1$), kinetic Alfvén waves in Hall-MHD ($d_i/a \sim 1$ and $\beta_e \sim 1$), and then whistler waves in EMHD ($d_i/a < 1$). We can now single out a complementary element contributing in a more formal way to the reconnection rate increase as $d_i/a$ progressively grows from negligible values (MHD regime) to values close to unity or larger (Hall-MHD regime): this is due to the natural behavior of the tearing growth rate, which tends to values $\gamma\tau_{A,a} \sim O(10^{-1}) - O(1)$ when some of the nonideal parameters approach unity ($d_i/a$, in this case). This is evident in Figure 2(f). Note that the normalization time remains $\tau_{A,a} = a/c_{A,cs}$ while $d_i/a$ changes, meaning that the reference time is formally the same regardless of the value of $d_i/a$: as long as $a$ is the normalization length, identical results hold both if $a$ is fixed and $d_i$ increases, or if $d_i$ is fixed and $a$ decreases (which, physically speaking, is the typical case). This is distinct from the further interpretation we can provide if we notice that the reference time $\tau_{A,a} = a/c_{A,cs}$ reduces to $\tau_{A,d_i} = d_i/c_{A,cs} = \Omega_{i,cs}^{-1}$ as $a$ shrinks down from $a \sim L$ to $a \sim d_i$. Interestingly, similar arguments apply to any other nonideal term $\epsilon$, as it approaches the nonasymptotic regime. For example, if we increase $\rho_s/a$, values $\gamma\tau_{A,a} \sim O(10^{-1}) - O(1)$ are attained for $\tau_{A,a}$ “fixed” (Figure 2). A complementary interpretation of these results is that the reference Alfvén time has decreased due to the spatial dispersion related to the transition from shear Alfvén to kinetic Alfvén waves (B. N. Rogers et al. 2001). In general, however, the growth rate of tearing instabilities increases and becomes “fast” (mean as comparable to the timescale with which waves propagate energy across the current sheet), whenever the equilibrium current sheet thickness attains microscopic “nonideal” scales —and, this is regardless of the kinetic scale involved.

## 4.2. Microscopic Thickness a and Quantitative Link between Reconnection Rate and Energy Cascade in Turbulence

Another implication, dependent solely on the current sheet thickness, is that, at least in the solar wind conditions with $\beta_e \sim \beta_i \sim 1$, analytical estimates based on the boundary layer analysis of tearing mode theory are not applicable even when tearing modes are relevant to the process. A departure from the analytical predictions of the (oblique) tearing instability was already highlighted by W. Daughton et al. (2011) in

simulations of 3D turbulent reconnection, in which the estimate $a \simeq d_i$ was also noted. In reality, this occurs in all turbulent regimes when the current sheet thickness becomes “microscopic,” i.e., comparable to one of the “nonideal length scales.” More formally, this happens whenever one of the nonideal parameters of Ohm’s law in Equation (1) approaches or exceeds unity normalized to the scale $L_0 = a$. Numerical results suggest for us to quantify this threshold inequality as $(d_e/a)$, or $(d_i/a)$, $(\rho_s/a)$, $(\rho_i/a)$, ... $\ge 0.5$ and as $S_a^{-1} \ge 10^{-2}$ for the Lundquist number—see Fig.2 of H. Betar et al. (2020) and the related discussion in D. Del Sarto et al. (2016; H. Betar et al. 2022).

This remark must be confronted with the fact that all attempts made so far to quantify the impact of tearing modes in the turbulent energy cascade use some results of the boundary layer analysis. These attempts include the early models by H. R. Strauss (1988), who used results from the drift tearing analysis in the viscous-resistive regime (H. Strauss 1986); those by V. Carbone et al. (1990) and V. Carbone (1995), who used the resistive tearing scalings in the presence of flows (I. Hofman 1975); all recent models based on the “plasmoid instability” in both resistive (S. Boldyrev & N. F. Loureiro 2017; N. F. Loureiro & S. Boldyrev 2017a; A. Mallet et al. 2017; L. Comisso et al. 2018) and “kinetic” regimes (i.e., those in which $\rho_s$ and/or $d_e$ are involved —see N. F. Loureiro & S. Boldyrev 2017b; A. Mallet et al. 2017; D. Hu et al. 2018; S. Boldyrev & N. F. Loureiro 2019; D. Borgogno et al. 2022; and also A. A. Schekochihin 2022 for a recent review); and also the quantitative modeling based on the “ideal tearing” mode scalings (A. Tenerani & M. Velli 2020).

None of these theoretical models can be applied when one among $d_i/a$ or $\rho_s/a$ or $\rho_i/a$ is comparable to or larger than unity. This is the case of the solar wind turbulence regimes, whose numerical or experimental estimates were mentioned in Sections 3.1.1–3.1.2. The same holds for cases where $d_e/a$ is comparable to unity, like the electron-only reconnection (T. D. Phan et al. 2018) and the cold, collisionless, inertia-driven turbulence studied by D. Borgogno et al. (2022).

There are however certain aspects indicating that the purely resistive (and viscous-resistive) regimes should be considered separately. This distinction arises from the observation that, in these regimes, the estimates of the thickness and aspect ratio of current sheets generated by turbulence as a function of the nonideal parameters remain less definitive. Such uncertainties affect the interpretation of the plasmoid formation (we discuss this in Section 4.3.3, below).

## 4.3. Condition $\gamma\tau_{A,L} \gtrsim 1$ in Other Tearing-based Turbulent Reconnection Models

The last, trivial consequence of our analysis is that, as long as the identification $\tau_{cs} \gtrsim \tau_{A,L}$ is correct and the condition $\gamma\tau_{cs} \gg 1$ is “necessary” to destabilize tearing modes on a static current sheet, the models that are not compliant with $\gamma\tau_{A,L} \gtrsim 1$ are not appropriate. We note, moreover, that the primary criterion to assess the validity of a reconnection model is typically a quantitative comparison of theoretical and numerical/experimental scalings as a function of the nonideal parameters. Unfortunately, the exponents of the associated power laws are often so close, numerically, to be indistinguishable through observations or numerical fits (see, e.g., D. Vech et al. 2018; H. Betar & S. Del 2023). This combines with the technical difficulty to accurately measure key
quantities of interest such as the current sheet aspect ratio (see Sections 3.1.1–3.1.2) or the local reconnection rate (see, e.g., H. Betar et al. 2022 for a discussion of complementary operational definitions). In this context, the conditions $\gamma\tau_{cs} \gg 1$ and $\gamma\tau_{A,L} \gtrsim 1$ provide a possible discriminatory criterion.

Let us then confront some known models with these conditions, by focusing on the less stringent one, $\gamma\tau_{A,L} \gtrsim 1$. The two main models we can compare are the “plasmoid instability” and the “ideal tearing.” Both were first developed in the purely resistive regime and are supported by several independent numerical studies. They appear as complementary, and many questions about their applicability are still open. Of course, the purely resistive regime is not of interest for the low-collision solar wind case of interest here. It is however of fundamental interest, both theoretical and practical, since it concerns a large number of numerical studies (see references cited in Sections 3.1.1–3.1.3), including the original work by W. H. Matthaeus & S. L. Lamkin (1986), with respect to which it is interesting to position the scenario proposed in Section 3. We then recall and analyze the hypotheses and implications of these two models, before proposing, in the light of the condition $\gamma\tau_{A,L} \gtrsim 1$, an interpretation to the numerical results related to them.

### 4.3.1. Resistive Plasmoid Instability: Hypotheses and Open Problems

We start from the “plasmoid instability,” i.e., from tearing modes developing on current sheets steadily reconnecting à la Sweet-Parker: the growth rate of the fastest tearing mode on a current sheet with $L/a \sim S_L^{1/2}$ is $\gamma_M \tau_{A,L} \sim S_L^{1/4}$ (T. Tajima & K. Shibata 2002), that is, diverging with a vanishing resistivity. The scaling $\gamma_{pl} \tau_{A,L} \sim S_L^{1/4}$, later identified as the “plasmoid regime” (N. Loureiro et al. 2007) or “plasmoid instability” (A. Bhattacharjee et al. 2009), once combined with the identification $\tau_{cs} \sim \tau_{nl}(L) \sim \tau_{A,L}$ makes this model compliant with the consistency requirement $\gamma_{pl}\tau_{cs} \sim \gamma\tau_{A,L} \gg 1$, as already underscored by L. Comisso et al. (2016). The same holds for the former “fractal reconnection” model by K. Shibata & S. Tanuma (2001), which also relies on tearing modes on the Sweet-Parker current sheet. However, the “too good” fulfillment of the condition $\gamma_{pl}\tau_{cs} \gg 1$, i.e., the typically too large value of $\gamma_{pl}\tau_{A,L}$, poses another well-known consistency problem to this model: as first noted by R. K. Jaggi (1964) and P. Sweet (1969) and later shown by the first numerical simulations of D. Biskamp (1986), the growth rate of tearing modes in Sweet-Parker current sheets is so large to make the natural occurrence of these current sheets unlikely. The fast disruption of Sweet-Parker current sheets by the “plasmoid instability” has been later verified by several studies, while the attention to the unlikeliness to find Sweet-Parker current sheets in experimentally relevant cases has been more recently (re)drawn by F. Pucci & M. Velli (2014) and by subsequent works (N. Loureiro & D. Uzdensky 2015; D. A. Uzdensky & N. F. Loureiro 2016). Note that the identification of Sweet-Parker reconnection in some numerical simulations of turbulence (S. Servidio et al. 2009, 2010, 2011; C. Dong et al. 2018; E. Papini et al. 2019a) and in dedicated laboratory experiments (H. Ji et al. 2001) always occurred for values $S_L^{-1} \gtrsim 10^{-4}$. This is compatible with the stability criterion of S. V. Bulanov et al. (1979), discussed at point (v) of Section 2.3.2, upon which we will return soon. However, the criticism about the unlikeliness of Sweet-Parker current

sheets with a too large aspect ratio, i.e., for $S_L^{-1} \lesssim 10^{-4}$, leaves open the question about the nature of the instabilities generating plasmoids therein—notably about the applicability of the plasmoid instability scaling.

### 4.3.2. Ideal Tearing: Hypotheses and Open Problems

A possible, alternative candidate to plasmoid instability in turbulence is the “ideal tearing” introduced by F. Pucci & M. Velli (2014): seeking an upper limit to the aspect ratio of a current sheet before it can be disrupted by a restive tearing, F. Pucci & M. Velli (2014) argued this to be fixed by the condition $\gamma_M \tau_{A,L} \sim S_L^0$, for which the growth rate becomes formally independent of resistivity (whence the adjective “ideal”), regardless of the small—yet nonnull—value of $S_L^{-1}$. Using the results of the tearing mode theory for $S_a^{-1} \ll 1$, they found that in a Harris-pinch current sheet (E. G. Harris 1962) this occurs once a critical value of the aspect ratio as large as $L/a \sim S_L^{1/3}$ is achieved. They also proved numerically a rapid convergence, for finite values of $S_L$ of experimental interest, of the growth rate obtained when $L/a \sim S_L^{1/3}$ to the asymptotic value $\gamma_{id_{[S_L^{-1}]}} \tau_{A,L} \approx 0.623$ formally obtained in the same conditions as $S_L \to \infty$. This threshold criterion on $L/a$ was then extended by A. Tenerani et al. (2015) to the viscous-resistive tearing, and it was proven in D. Del Sarto et al. (2016) to be in principle applicable to any other reconnection regime—also kinetic— and to any magnetic equilibrium profile (see also F. Pucci et al. 2018). The argument indeed easily extends to different nonideal parameter(s) $\epsilon_L \ll 1$ replacing $S_L^{-1}$. Specific threshold conditions $L/a \sim \epsilon_L^{-\alpha}$ for some $0 < \alpha < 1$ can be thus obtained, such that $\gamma_M \tau_{A,L} \approx \gamma_{id_{[\epsilon_L]}} \tau_{A,L} \sim \epsilon_L^0$ is recovered in the formal limit $\epsilon_L \to 0^+$; even an extension to the Hall-resistive regime—not trivial due to the reasons discussed in Section 2.3.3—was developed for $d_i/a \ll 1$ by F. Pucci et al. (2017) and by E. Papini et al. (2019b). In all cases, resistive or not, it is however found that $\gamma_{id_{[\epsilon_L]}} \tau_{A,L} \lesssim 1$. So, the fact that several nonlinear studies (S. Landi et al. 2015; A. Tenerani et al. 2015; L. Del Zanna et al. 2016; E. Papini et al. 2019b; A. Tenerani & M. Velli 2020) have supported, at least in the non-Hall regime, the relevance of the ideal tearing scalings in the case of current sheets evolving over a timescale $\tau_{cs} \sim \tau_{A,L}$ appears at odds with the condition (2), $\gamma\tau_{cs} \gg 1$.

### 4.3.3. Resistive Plasmoid Instability versus Ideal Tearing: A Possible Interpretation

The three “problems” outlined above, namely, the applicability of the plasmoid instability and of the ideal tearing, and the “necessary” character we have ascribed to the requirement $\gamma\tau_{A,L} \gtrsim 1$, can be reconciled if, in this resistive regime, one looks at such a condition as to the Bulanov-Syrovatskii criterion of tearing stabilization by flows (point (v) of Section 2.3.2). This allows for a consistent interpretation of both the numerical results supporting the plasmoid instability and the ideal tearing model, respectively. In this respect, the ideal tearing threshold condition on $L/a$, for which $\gamma_{id_{[\epsilon_L]}} \tau_{A,L} \lesssim 1$, can be interpreted as roughly indicative of the upper limit of a current sheet aspect ratio, which is still stable versus tearing modes, as the current sheet cannot be considered as “static,” yet. Instead, plasmoid instabilities can be destabilized on a Sweet-Parker type current sheet as soon as the aspect ratio $L/a \sim S_L^{1/2}$ becomes large enough to satisfy
the Bulanov–Syrovatskii stability criterion in Equation (3), i.e., for $S_L \gtrsim 10^4$. Let us discuss this in detail.

From Equation (2) and the related discussion, it follows that $\gamma_M \tau_{A,L} \sim 1$ represents a threshold condition for the stability of a current sheet evolving over a timescale $\tau_{cs} \sim \tau_{A,L}$: according to this rationale, the current sheet is still stable at least as long as $\gamma_M \tau_{A,L} \le 1$. An almost identical condition is established when looking at the stability condition with respect to flows sheared along the current sheet expressed by Equation (3). Instead, for tearing modes developing with the growth rate $\gamma_M$ as secondary to a primary tearing mode —like the ones that have been studied in some of the works we are going to consider below—we should likely assume $\tau_{cs} \sim \gamma_I^{-1}$, where $\gamma_I$ is the growth rate of the primary tearing close to the end of its linear stage (D. Del Sarto & M. Ottaviani 2017). This is valid if we admit that secondary tearing modes may develop during the late linear stage of the primary tearing, before it has reached saturation and its growth rate has considerably diminished so that $\tau_{cs}^{-1} \ll \gamma_I$: the onset of the secondary mode before the end of the linear stage of the primary tearing is theoretically possible (D. Del Sarto & M. Ottaviani 2017), and it is supported by numerical simulations (see Figure 4 of E. Papini et al. 2019b). In this case, the “necessary” condition to be confronted with reads $\gamma_M \gg \gamma_I$, with $\gamma_M \le \gamma_I$ providing a stability criterion. Then, if the primary mode is an ideal tearing, the condition $\gamma_M \tau_{A,L} \gg 1$ (or $\gamma_M \tau_{A,L} \gtrsim 1$) is recovered for the secondary mode, since $\gamma_I \sim \tau_{A,L}^{-1}$.

With all this in mind, let us first consider the numerical results in support of the “ideal tearing” scenario. S. Landi et al. (2015), L. Del Zanna et al. (2016), and E. Papini et al. (2019b) identified, for different values of $S_L$, tearing modes developing as secondary instabilities once the current sheet generated by a (resistive) primary ideal tearing achieves an aspect ratio satisfying $L/a \sim S_L^{1/3}$. A. Tenerani et al. (2015) studied numerically the (in)stability of a current sheet of the fixed length $L$, which was forced to shrink at a rate $1/\tau_{A,L}$ (i.e., they imposed $\tau_{cs} \equiv \tau_{A,L}$), and showed that the onset of the tearing mode (and of the secondary tearing modes stemming from it in a cascade process) was compatible with the threshold condition $L/a \sim S_L^{1/3}$. None of these results seems compatible with the case $\tau_{cs} \gg \tau_{A,L}$ of a tearing-unstable current sheet, which is almost static with respect to the timescale $\tau_{A,L}$ (note that the study case of A. Tenerani et al. 2015 surely excludes it), which we then disregard. Therefore, a likely interpretation of all these results, which is compatible with both the incertitude on the numerical estimates reported therein and the “condition of necessity” we attribute to inequality in Equation (3), is the one first proposed in F. Pucci & M. Velli (2014) and further discussed in E. Papini et al. (2018, 2019b; H. Betar et al. 2020): the critical aspect ratio $L/a \sim S_L^{1/3}$ for the onset of the ideal tearing (which can be corrected as $L/a \sim S_L^{1/3}(V_{out}/c_{A,L})^{2/3}$ if one takes account of an outflow nonnecessarily Alfvénic; D. A. Uzdensky & N. F. Loureiro 2016; E. Papini et al. 2018) corresponds to the “last observable” stable current sheet, before a super-Alfvénic tearing mode disrupts it. In particular, for a current sheet whose aspect ratio *increases* over the timescale $\tau_{cs} \sim \tau_{A,L}$, the condition $\gamma_M \tau_{A,L} \gg 1$ is rapidly satisfied in a few *e*-folding times after the critical aspect ratio is achieved. It is however interesting to put it together with other results. We thus consider the results of A. Tenerani et al. (2015) which show that (secondary) Sweet–Parker-type current sheets can be

generated, in place of tearing-unstable current sheets, by primary ideal tearing modes when flows become strong enough, and the results of E. Papini et al. (2019b). The latter show that the current sheet generated by a primary ideal tearing is of Sweet–Parker type, as long as the Lundquist number evaluated with respect to its length remains below the critical threshold $S_L \sim 10^4$, whereas for larger values it is disrupted by a secondary tearing, so fast that its growth rate appears to be independent of $S_L$ (as it is expected for the “ideal tearing” mode). All this appears to shift the logical priority from the $\gamma_M \tau_{A,L} \sim S_L^0$ condition for the onset of a “(quasi-) ideal reconnection regime” toward an (in)stability condition with respect to other physical effects—namely, the presence of flows.

In this light, all numerical results showing the formation of plasmons in resistive turbulence simulations appear compatible with a plasmoid instability, meant in the sense of N. Loureiro et al. (2007) and A. Bhattacharjee et al. (2009), which is destabilized with a $\gamma_M$ “just above” the threshold condition $\gamma_M \simeq u_\parallel/L$ of S. V. Bulanov et al. (1979), with respect to the outflow. Because of Equation (3), this is practically equivalent to the stability condition $\gamma_M \tau_{A,L} \sim 1$ for a current sheet evolving at Alfvénic turbulence (inequality in Equation (2)); and it is also numerically equivalent to the ideal tearing condition $\gamma_M \tau_{A,L} \sim S_L^0$ of F. Pucci & M. Velli (2014) because of the value $\gamma_{id[s]} \tau_{A,L} \simeq 0.6$. In this scenario, the criterion giving the higher threshold on $\gamma_M \tau_{A,L}$ among the static current sheet condition and the effectiveness of tearing modes with respect to flows fixes the growth rate of the reconnecting mode. However, quantifying it precisely enough, so as to compare it unequivocally with the scaling $\gamma_{pl} \tau_{L,A} \sim S_L^{1/4}$, seems extremely difficult. At the same time, for practical purposes, the threshold condition on $L/a$ given by the “ideal tearing” criterion provides a reliable estimate for the largest aspect ratio of a marginally stable current sheet, since such criterion on the scaling of $L/a$ with $S_L$ is obtained in a parameter range in which the boundary layer theory is still applicable. All this is compatible also with the results of E. Papini et al. (2019a) and L. Franci et al. (2022a), who —in the limit of validity of the assumptions $\tau_{cs} \sim \tau_{A,L}$— measured faster than ideal growth rates in turbulence simulations.

### 4.3.4. Condition $\gamma_M \tau_{A,L} \gtrsim 1$ and “Ideal Tearing” Hypothesis in Low-collision Turbulence

In resistive turbulence, values of $S_a$ of experimental interest are in a parameter range compatible with the boundary layer theory. We have seen (Section 4.3.3) that here the ideal tearing condition $L/a \sim S_L^{1/3}$ can be of practical utility even if the requirement $\gamma_M \tau_{A,L} \sim S_L^0$ were subsidiary to the apparently more fundamental conditions in Equations (3) and (2). In the collisionless regime, things are somewhat different. Let us discuss this.

The relevance to turbulence of the estimates of D. Del Sarto et al. (2016) for the collisionless ideal tearing is ruled out by the failure of the boundary layer theory (Section 4.2), which invalidates the theoretical results with which the threshold criterion on $L/a$ is generalized to nonresistive regimes. In principle, however, one might still deduce the condition $L/a \sim \epsilon_L^{-\alpha}$ for some nonideal collisionless parameter $\epsilon_a$, after numerically evaluating—as we did in Section 3.2—a power-law scaling of $\gamma_M \tau_{A,a}$. This approach would be less predictive
scale a

and of questionable practical utility, but it would preserve the
criterion on which the ideal tearing model is constructed. In the
case of the two-field model, for example, this would
correspond to impose γ<sub>M</sub>τ<sub>A,L</sub> = 1 in Equation (6), whence,
proceeding as it is outlined in D. Del Sarto et al. (2016) and
recalled here in Section 2.1.1, one would obtain
(L/a) ~ (d<sub>e</sub>/L)<sup>-3/10</sup>[(ρ<sub>s</sub>/L)<sup>2</sup> + (ρ<sub>i</sub>/L)<sup>2</sup>]<sup>-1/10</sup>. Using the para-
meters of Equation (4), we thus find L/a ≃ 2.7, which is far
too smaller than the actual aspect ratio of reconnecting current
sheets inferred from observations and simulations. This is not
surprising, since the growth rate γ<sub>M</sub>τ<sub>A,a</sub> numerically obtained
in Section 3.2 is already of an order ~O(0.1) – O(1) and thus
not much far from the corresponding γ<sub>id</sub> value, which is likely
to be of the same order of magnitude (see D. Del Sarto et al.
2016). Also, note that, as the numerical value of ϵ<sub>a</sub> increases,
γτ<sub>A,a</sub> approaches saturation (see end of Section 4.1), and its
dependence on the nonideal effects becomes weaker: a scaling
γτ<sub>A,a</sub> ~ ϵ<sub>a</sub><sup>α</sup> with α ≃ 0 may be thus measured already with
respect to the scales (see, e.g., Figure 3 of H. Betar et al.
2022). Should this happen, the independence of γτ<sub>A,L</sub> on ϵ<sub>L</sub>
would persist after rescaling to L, although with a different
numerical value of γ<sub>M</sub>τ<sub>A,L</sub> from that of the corresponding
"ideal tearing": one would still measure γ<sub>M</sub> τ<sub>A,L</sub> ~ ϵ<sub>L</sub><sup>0</sup> but with
γ<sub>M</sub> τ<sub>A,L</sub> ≠ γ<sub>id[1]</sub> τ<sub>A,L</sub> and, generally, with γ<sub>M</sub> τ<sub>A,L</sub> ≳ γ<sub>id[1]</sub> τ<sub>A,L</sub>.

All this suggests that the condition for the “ideal tearing”
mode does not represent a physical constraint for collisionless
turbulent reconnection. Instead, it seems again that the
dominant one among conditions in Equations (3) and (2)
determines the aspect ratio L/a of the reconnecting current
sheet. In the collisionless regime, however, the ideal tearing
threshold deduced from boundary layer theory does not
provide a useful estimate of such an aspect ratio anymore,
differently from the purely resistive case. Nevertheless, the
reconnection rate of the tearing mode developing on such a
current sheet is so large—much larger than 1/τ<sub>nl</sub>(L)—so that,
with respect to the turbulent energy cascade, it is effectively
independent of ϵ<sub>L</sub><sup>0</sup>, as an “ideal tearing” would be.

4.4. Role of Model Approximations in the Estimate of γτ<sub>cs</sub>

Quantitatively speaking, the most critical of the hypotheses
we have done is to have neglected equilibrium flows parallel to
the current sheet and sheared across it (point (v) of
Section 2.3.2). Several results known from previous studies
suggest however that including these effects may go in favor
of the argument we propose: the coupling of tearing modes and
Kelvin–Helmholtz type instabilities, which would result from
such a configuration, generally widens the range of unstable
wavenumbers for a fixed aspect ratio and seems to increase the
numerical value of the growth rate γ<sub>M</sub>τ<sub>A,a</sub> (I. Hofman 1975;
M. Dobrowolny et al. 1983; D. Borgogno et al. 2022; J. De
Jonghe & R. Keppens 2024). These effects need to be
quantified in a future work, so we take here condition (v) just
as an approximation allowing us to provide a proof of concept
in a configuration, which we tried to keep the simplest—yet
the most reasonable—as possible.

Another questionable point can be in the use of extended
fluid models, in place of a full kinetic description, when non-
ideal-MHD effects do not represent “microscopic corrections”
anymore, like for ρ<sub>s</sub>/a ~ ρ<sub>i</sub>/a ≃ 1—see the related discussion
in W. Daughton et al. (2011). Using fluid models, although
“extended,” is a strong approximation, here, but this does not
seems to be a major issue for the argument we have developed,

which is grounded on a linear analysis. We first note that fluid
models sometimes fail to account for the correct cutoff
conditions about the instability wavenumber, when this
depends on kinetic effects. The range of unstable wavevectors
of tearing modes is determined by the spatial profile of the
equilibrium magnetic field, i.e., by the value of Δ'(ka), as long
as an “ideal-MHD” region can be identified. Although this is
not possible when the current sheet width is microscopic, the
fastest growing mode dominating the reconnection rate when
L/a ≫ 1 lays in the ka ≪ 1 range, always admitted by fluid
models. Moreover, the inclusion of kinetic effects combined
with the presence of sheared flows is expected to induce the
generation of pressure anisotropy, which, in turn, should
further increase the reconnection rate (see Section 3.1.1).
Besides, we note that a full Vlasov linear instability analysis
was performed in some early works. Examples are the studies
by B. Coppi et al. (1966), P. L. Pritchett et al. (1991), and
M. Brittnacher et al. (1995), with an application to magnetotail
reconnection, or those by G. Laval et al. (1966), J. F. Drake &
Y. C. Lee (1977), and S. M. Mahajan et al. (1978, 1979), with
applications to magnetically confined plasmas. Of particular
interest for the present discussion is however the more recent
linear analysis performed by W. Daughton (1999b) and
rediscussed in W. Daughton & H. Karimabadi (2005), of a
Harris type current sheet with isotropic (Maxwellian) ion and
electron thermal equilibrium. The results of these works
appear in general agreement with the discussion we have
developed: since τ<sub>A,a</sub> = Ω<sup>-1</sup><sub>i,cs</sub> for a = d<sub>i</sub>, the parameter case
considered in Figure 3 of W. Daughton (1999b) matches
closely the one we have considered in Section 3, except for the
fact it considers m<sub>i</sub>/m<sub>e</sub> = 1 (whence d<sub>e</sub>/a = 1) and for the fact
it also accounts for ρ<sub>i</sub> = a. From this figure (where Ω<sub>i</sub> reads
Ω<sub>i,cs</sub> in our notation), one deduces γ<sub>M</sub>τ<sub>A,a</sub> ≃ 0.19. This is larger
but remarkably close to the γ<sub>M</sub>τ<sub>A,a</sub> ≃ 0.13 value we found with
the four-field model for Ω<sup>2</sup><sub>i</sub> = Ω<sub>i,cs</sub>, m<sub>i</sub>//m<sub>e</sub> = 1836 and
ρ<sub>i</sub> = 0, but the difference, yet within an order of magnitude, is
still in general agreement with our argument.

For all these reasons, the results we obtained in Section 3.3,
although probably quantitatively inaccurate compared to a
kinetic modeling, are reliable enough to provide an interpreta-
tion of a phenomenon not yet fully understood. We stress
however that the quantitative estimates of Equations (8)–(9)
are strongly “model dependent” (see the results of the two-
fluid versus those of the four-field model). This is why we tried
to be as conservative as possible, so as to make choices aiming
at preferably biasing the final estimate of γ<sub>M</sub>τ<sub>A,L</sub> by rounding it
down, rather than off—to the best of our awareness/
understanding of the problem, at least. We thus expect that
any possible quantitative change should not invalidate the
essence of the argument proposed.

5. Conclusions

One contribution of this work is hopefully for the
advancement of the theoretical interpretation of the mechan-
isms of turbulent reconnection, at least in a 2D coordinate
dependence. In Section 3, we have shown that a tearing mode
can be fast enough to grow on a timescale shorter than the
evolution time of the large aspect ratio current sheet, driven by
a turbulence cascade initialized at the ideal-MHD scale. The
idea of the relevance of tearing modes is not new at all, dating
back at least to the '80s, but the argument we have developed
seems to us capable of fixing a number of inconsistencies that
are implied by former models of turbulent reconnection relying
on tearing modes. The model we propose is however surmised
to the validity of a number of hypotheses, listed as (i), (iii), (v)
in Section 2.3. While they are relevant to all tearing-based
models of 2D turbulent reconnection, their validity in
turbulence has not been quantitatively addressed, yet. It will
be interesting to do so in a number of further dedicated
investigations.

From the discussion of Section 4, we have drawn subsidiary
nontrivial implications. One is the identification of the role
played by nonasymptotic values of the nonideal parameters
(i.e., by values approaching unit once the parameters are
normalized to the current sheet thickness) in the increase of the
tearing growth rate associated with the inclusion of micro-
scopic effects: growth rates of the order of $1/\tau_{A,a}$, inverse of
the time that an Alfvén wave takes to propagate energy across
the current sheet, can be thus achieved regardless of the
microscopic effect at play. This remark concerns also the
interpretation of the growth rate increase associated with the
Hall term and normally attributed to the transition to whistler
dominated dynamics (Section 4.1). Another implication
concerns the interpretation of the role possibly played by the
“ideal tearing” and “plasmoid instability” models in different
turbulent reconnection scenarios (Section 4.3).

A further contribution of this discussion is—unfortunately
—in having highlighted (Section 4.2) that the theoretical
scalings of the boundary layer analysis cannot be used to
develop tearing-based models for the dissipation rate of
magnetic energy in the turbulent cascade —contrary to what
is done, instead, in the majority of theoretical models of this
kind. The boundary layer decomposition fails indeed when
some parameter associated with the nonideal effects becomes
of the order of the unity ($\rho_s/a$ or $d_i/a$ in the case we
considered). This is the case for current sheets generated in
low-collision turbulence, also when a full 3D coordinate
dependence is accounted for (see, e.g., W. Daughton et al.
2011). This conclusion, i.e., the nonapplicability of analytical
calculations of the tearing mode theory in turbulent reconnec-
tion, complements the similar remark made by W. Daughton
et al. (2011), who already pointed out the failure of asymptotic
estimates for oblique tearing modes in 3D turbulence
simulations. Although noted in a 3D coordinate dependence
and in a kinetic regime, this discrepancy involved the
parameter range $a \sim d_i \sim \rho_i \sim \rho_s$ practically equivalent to the
one we focused on, here.

Numerical estimates are thus necessary, but they need to be
computed case by case, depending on the regime (model) and
on the parameter range of pertinence. Note, however, that the
numerical fit of fractional scalings, if it is not guided by some
theoretical argument, is in general subject to an unavoidable
incertitude (see the examples discussed in D. Vech et al. 2018;
H. Betar & S. Del 2023). Its outcome should be thus regarded
as approximate and indicative. These issues are met already in
simulations of a single current sheet. Even more complex is to
extrapolate quantitative data of this kind from turbulent
simulations. Therein, several dissipation channels of the
energy exist (see, e.g., J. M. TenBarge & G. G. Howes
2013; Y. Yang et al. 2017a; W. H. Matthaeus et al. 2020;
R. Bandyopadhyay et al. 2021; P. A. Cassak et al. 2023),
although an enhanced dissipation of magnetic energy, likely
linked to magnetic reconnection, has been shown to be
concentrated in current sheets (H. Politano et al. 1995;

D. Sundkvist et al. 2007; V. Zhdankin et al. 2013; Y. Yang
et al. 2017b; E. Camporeale et al. 2018; E. Papini et al. 2020).

In any case, in the limits of validity of the argument we
proposed, any tearing mode satisfying $\gamma_M \tau_{cs} \gg 1$ corresponds
to a reconnection rate so fast that it would be seen as
“instantaneous” by the turbulence dynamics at the scale $L$.
This means that although there is a dependence of the growth
rate $\gamma_M$ on the parameter “$\epsilon_L$” ruling the reconnection
instability, this dependence does not become manifested at
the nonlinear timescale of turbulence, $\tau_{cs} \sim L/(\langle u_i \rangle_L)$.

This point of view agrees with the discussion in E. Papini
et al. (2018, 2019b), but important differences must be pointed
out with respect to the “ideal tearing” scenario. For practical
applications, this result would be indeed undistinguishable
from the $\gamma_M \tau_{A,L} \sim O(\epsilon_L^0)$ condition of the “ideal tearing”
regime, but, differently from that, it is not postulated nor is it
related to a condition $a/L \sim \epsilon_L^\alpha$ on the current sheet aspect
ratio, which is obtained (only) for $\epsilon_a \ll 1$. In the scenario we
propose, it is instead the current sheet width $a$ to be determined
by the turbulence dynamics and by some plasma parameters
like $\beta_e$ and $\beta_i$. The microscopic value of $a$ makes $\epsilon_a \sim O(1)$ for
some nonideal parameter $\epsilon$; the boundary layer theory
inapplicable; and $\gamma_M \tau_{A,a} \sim O(1)$. Then, it follows $\gamma_M \tau_{A,L} \gtrsim 1$
after rescaling $a \to L$, for some value $L \gg a$ large enough.
Here, the “threshold” value of $L/a$ appears to be fixed by the
fulfillment of the instability criteria $\gamma_M \tau_{cs} \gg 1$ (Equation (2))
and $\gamma_M \tau_{A,L} \gtrsim 1$ (Equation (3)), for some given $\gamma_M \tau_{A,a} \sim O(1)$.
In the turbulent regime dominated by resistivity, where the
relevant nonideal parameter (the inverse Lundquist number
$S_a^{-1}$) is still small enough to allow for the asymptotic estimates
provided by F. Pucci & M. Velli (2014; Section 4.3.3), the
threshold condition $\gamma_{id[s]} \tau_{A,L} \sim S_L^0$ yielding $\gamma_{id[s]} \tau_{A,L} \simeq 0.6$ is
numerically equivalent to the threshold condition $\gamma_M \tau_{A,L} \simeq 1$, which we deduce from the requirement in Equation (3) (see
Section 4.3). Therefore, the condition on $L/a$ in powers of $S_L$
obtained in the resistive ideal tearing scenario approximately
separates tearing-stable and -unstable current sheets also in the
scenario we propose, where the requirements in Equations (2)–(3) are assumed as necessary and equivalent to $\gamma_M \tau_{A,L} \gtrsim 1$, a
condition in this context apparently “more fundamental” than
$\gamma_M \gtrsim \gamma_{id[s]}$. Instead, the ideal tearing threshold criterion fails in
collisionless turbulence (Section 4.3.4), where the estimates
for the collisionless “ideal tearing” of D. Del Sarto et al.
(2016) are not appropriate due to the inapplicability of the
boundary layer theory (see Section 4.2).

The argument developed in Section 3 is quite general. It
could be in principle applied also in 3D turbulence, as long as
spontaneous reconnecting instabilities are involved. The
analysis of Section 3 could be adapted to obtain the equivalent
of the estimates of Equation (9) in a kinetic modeling, and for
oblique tearing modes, too, in regimes like those discussed by
W. Daughton (1999b), G. Lapenta & J. U. Brackbill (2000),
and W. Daughton et al. (2011).

Even more straightforward would be the adaptation of the
model to 2D reconnection regimes different from the one we
considered in Section 3. For example, with the possible
adaptations required by the change of hypothesis (v) (see
Sections 2.3.2 and 4.4), the discussion we developed applies
also to the reconnection regime studied by D. Borgogno et al.
(2022). They investigated the $\rho_s = \rho_i = 0$ limit of the same
two-field nonlinear model we have considered, that is, the
nonlinear model corresponding to Equations (A1)–(A3) of

<figure><img src="image_7.png" /></figure>
Appendix A: in the aforementioned work, current sheets of a thickness $a \sim d_e$ were measured in decaying turbulence simulations initialized with $d_e = 5 \times 10^{-4} k_{inj}^{-1}$. For these values, too, the growth rates in absence of flows have values $\gamma_M \tau_{A,a} \gtrsim 0.2$ (see Figure 3 of H. Betar et al. 2022). Comparable or even larger values of the growth rate $\gamma_M \tau_{A,a}$ were measured in the presence of flows (D. Borgogno et al. 2022). This would imply an even “easier” fulfillment of the $\gamma_M \tau_{A,L} \gg 1$ condition (Section 4.4).

The argument we discussed should equally apply to turbulent reconnection at the electron scales, where, again, $a \sim d_e$ is typically measured (see, e.g., H. Karimabadi et al. 2013; F. Califano et al. 2020; C. Granier et al. 2022; L. Franci et al. 2022b). “Electron-only-reconnection” turbulence is usually modeled in a kinetic framework. Gyrofluid models accounting for ion-FLR effects succeeded however to recover an electron-only turbulent reconnection in EMHD (C. Granier et al. 2024). An extended EMHD modeling could be in principle used (H. Betar & S. Del 2023). The difference with respect to the example of Section 3 would be both in the model equations and in the fact that the whistler time evaluated with respect to a (and to the relevant magnetic field amplitude) should replace the Alfvén time $\tau_{A,a}$.

Finally, in all these regimes, it would be interesting to consider the implications of the interpretation we have proposed for the energy transfer in turbulence. For example, one could look at the implications for turbulence’ spatial anisotropy, by following the lines traced by authors who already addressed this problem in the framework of both tearing mode based theories (see references cited in Section 4.2) and without them (see, e.g., G. Kowal et al. 2017; S. Landi et al. 2019). In particular, it may be worthwhile to consider the role played by the possible “scale invariance” of turbulence with respect to magnetic reconnection, suggested by the effective condition $\gamma_M \tau_{A,L} \sim \gamma_M/(k_{inj} \hat{c}_{A,k_{inj}}) \sim O(\epsilon_L^0)$ discussed above.

The relevance of all these results may go beyond solar wind turbulence and may concern turbulence in astrophysical and high temperature laboratory plasmas, too.

## Acknowledgments

We are grateful to an anonymous reviewer for several useful comments. The final version of this work has benefited from stimulating criticisms received from, and of discussions had with many colleagues during the “2024 Arcetri workshop on plasma astrophysics,” which took place while this manuscript was under peer review. We thank in particular Luca Del Zanna, Luca Franci, Simone Landi, Lorenzo Matteini, William Matthaeus, Emanuele Papini, Sergio Servidio, and Marco Velli. Discussions over time on these topics with Maurizio Ottaviani, Francesco Pegoraro, Fulvia Pucci, Alexander Schekochihin, and Anna Tenerani are also gratefully acknowledged. This work has been partially funded by the FR-FCM through the AAP 2023 “Evolution of current sheets in low-collision plasmas” and AAP 2024 “Energy conversion processes mediated by the current and vorticity dynamics in low-collision plasmas.” Part of this work was performed while one of the authors (H.B.) received financial support from the AIM4EP project (ANR-21-CE30-0018), funded by the French National Research Agency (ANR). The authors thank the “Maison de la simulation Lorraine” for partial time allocation on the cluster Explor under the project No. 2019M4XXX0978.

## Appendix A: Scalar Field Models

The restriction to a 2D coordinate dependence allows for a description of zero-divergence vectors in terms of stream functions. The usual formalism of reduced MHD can be thus adopted, in which the in-plane magnetic field $\mathbf{B}_{\perp}$ is uniquely determined by the flux function $\psi(x, y, t)$, and the incompressible component of the bulk plasma velocity is expressed at the dominant $\mathbf{E} \times \mathbf{B}$-ordering by the stream function $\varphi(x, y, t)$. Depending on other hypotheses, like the guide field magnitude and the relative orderings between quantities, further scalar functions can be added to account for additional vector components (e.g., those parallel to the guide field or related to the fluid in-plane compressibility) or for independent quantities (e.g., the particle or gyrocenter densities, the temperature, the pressure or heat tensor components, etc.). The corresponding models can be thus categorized in terms of the number of scalar fields involved. The simplest model for the description of linear tearing modes is the two-field model by H. P. Furth et al. (1963), of which we consider here a collisionless extension with FLR effects. A four-field model is instead used to take account of the Hall effect in Ohm’s law. The equations of both models will be written below by using the usual “Poisson brackets,” customarily defined as $[f, g] = (\nabla f \times \nabla g) \cdot \mathbf{e}_z = (\partial f / \partial x)(\partial g / \partial y) - (\partial f / \partial y)(\partial g / \partial x)$, where $\nabla = (\partial / \partial x, \partial / \partial y, 0)$.

The magnetic equilibrium considered to obtain the results of Figures 2 is the one introduced in F. Porcelli et al. (2002) and in dimensional units reads $\mathbf{B}_{eq} = \nabla \psi_0 \times \mathbf{e}_z + B_g \mathbf{e}_z$ with $\psi_0 = (aB_{cs})/(2 \cosh^2(x/a))$.

### A.1. Two-field Model

The two-field model integrated to obtain the results in Figures 2 (a)–(c) corresponds to the linear version of the gyrofluid model equations first considered in D. Grasso et al. (2000), where $B_{cs}/B_g = \epsilon_B \ll 1$, and fluctuations are ordered so that $\delta B_{\perp} \sim B_{cs}$ and $\delta B_z \ll B_{cs}$. Here, ion-FLR effects (related to $\rho_i$) are included by approximating the ion gyrokinetic operator with Padé approximants (G. W. Hammett et al. 1992), as done in T. J. Schep et al. (1994; B. N. Kuvshinov et al. 1994). The corresponding set of linear equations, linearized around a tearing-unstable magnetic equilibrium $\psi_0(x/a)$, with uniform density and no equilibrium flows (i.e., $\varphi_0(x) = 0$) for perturbations of the kind $\sim f_1(x)e^{iky+\gamma t}$, reads (in dimensionless units referring lengths to a and times to $\tau_{A,cs}$)

$$ \frac{\partial}{\partial t}(\psi_1 - d_e^2 \nabla^2 \psi_1) + [\varphi_1 \psi_0 - d_e^2 \psi_0''] = \rho_s^2 [U_1, \psi_0] + S_a^{-1} \nabla^2 \psi_1, \quad (A1) $$

$$ \frac{\partial U_1}{\partial t} = [\psi_0, \nabla^2 \psi_1] + [\psi_1, \psi_0''] + R_a^{-1} \nabla^4 U_1, \quad (A2) $$

$$ U_1 - \rho_i^2 \nabla^2 U_1 = \nabla^2 \varphi_i. \quad (A3) $$

Equation (A1) corresponds to the z-component of generalized Ohm’s law. In the limit $\rho_i = 0$, Equation (A2) can be read as the z-component of the vorticity equation, but in the presence of ion-FLR effects, it more properly corresponds to the continuity equation for the charge density, in which corrections due to the polarization density associated with the
gyrocenters are retained. Like in D. Grasso et al. (2000), we have here formally included the normalized ion viscosity $\nu$, which gives the normalized Reynold's number $R_a^{-1} = ac_{A,cs}/\nu$. Note that also other models, like the nonlinear three-field gyrofluid model of D. Grasso et al. (2010; L. Comisso et al. 2012), converge to Equations (A1)–(A2) in the linear limit and for a homogeneous equilibrium density. The solver integrating Equations (A1)–(A2) is described in H. Betar et al. (2020). Here, it has been adapted to solve $\phi_1$ once $U_1$ is given, according to Equation (A3).

## A.2. Four-field model

The four-field model integrated to obtain the results in Figures 2 (d)–(f) is the model first introduced by R. Fitzpatrick & F. Porcelli (2004), in which compressive Alfvén waves are assumed to be faster than any other mode, ions are cold ($\rho_i = 0$), and only electrons contribute to the total pressure and then to the plasma beta:

$$c_{\beta}^{2} \equiv \frac{\beta_{e}}{1 + \beta_{e}}, \qquad \rho_{s}^{2} = \beta_{e} d_{i}^{2}. \qquad (A4)$$

The dimensionless model equations have lengths normalized to a and times normalized to $\tau_{A,cs}^*$ (see Section 2.1.1) while $B_{cs}/B_g$ is finite and arbitrary, with $B_0 = B_g$. We recall these equations for convenience, although using a slightly different notation than the one with which they were introduced: in particular, we define $v_{\perp} = -\nabla\varphi \times e_z$ in place of $v_{\perp} = \nabla\varphi \times e_z$ used by R. Fitzpatrick & F. Porcelli (2004), we introduce the contribution of an isotropic resistivity (terms related to $(S_a^*)^{-1}$), below), and we have included in Equations (A7)–(A8) the contribution of the perpendicular and parallel ion viscosities, $\nu_{\perp}$ and $\nu_{\parallel}$, yielding the Reynold's numbers $(R_{\perp,a}^*)^{-1}$ and $(R_{\parallel,a}^*)^{-1}$ originally not accounted for in the model.

$$\begin{aligned} \frac{\partial}{\partial t}(\psi - d_e^2 \nabla^2 \psi) + [\varphi, \psi - d_e^2 \nabla^2 \psi] \\ = d_i[\psi, b] + S_a^{-1}(\nabla^2 \psi - \nabla^2 \psi_0) \end{aligned} \qquad (A5)$$

$$\begin{aligned} \frac{\partial b}{\partial t} + [\varphi, b] = c_{\beta}^{2}[V_z, \psi] + c_{\beta}^{2}d_{i}[\nabla^{2}\psi, \psi] \\ + (S_{a}^{*}) - 1](\nabla^{2}b - \nabla^{2}b_{0}) \end{aligned} \qquad (A6)$$

$$\begin{aligned} \frac{\partial}{\partial t} \nabla^2 \varphi + [\varphi, \nabla^2 \varphi] \\ = [\psi, \nabla^2 \psi] + (R_{\perp,a}^*)^{-1}(\nabla^4 \varphi - \nabla^4 \varphi_0) \end{aligned} \qquad (A7)$$

$$\frac{\partial}{\partial t} V_z + [\varphi, V_z] = [b, \psi] + (R_{\parallel,a}^*)^{-1}(\nabla^2 V_z - \nabla^2 V_{z,0}). \qquad (A8)$$

Equation (A5) is the z-component of Ohm's law, which describes the time evolution of $B_{\perp}$ through the evolution of $\psi$. Equation (A6) corresponds to the evolution equation for $b \equiv \delta B_z/B_g$, which this time is not ordered small versus $B_{\perp}/B_g \sim \delta B_{\perp}/B_g$. Equation (A7) is the z-component of the vorticity equation, which determines the evolution of the incompressible in-plane bulk velocity. Equation (A8) describes the evolution of the bulk velocity along the guide field. Compressibility effects related to the presence of the Hall term are retained in the model through Equations (A6) and

(A8), since the perpendicular fluid compressibility and the particle density have been expressed in terms of the scalar fields b and $V_z$ thanks to the ordering adopted in the model (see R. Fitzpatrick & F. Porcelli 2004 for details).

Linearizing the equations above around $\psi_0(x/a)$ and $b_0 = V_{z,0} = \varphi_0 = 0$, the system in Equations (A5)–(A8) becomes

$$\begin{aligned} \frac{\partial}{\partial t}(\psi_1 - d_e^2 \nabla^2 \psi_1) + [\varphi_1, \psi_0 - d_e^2 \nabla^2 \psi_0] \\ = d_i[\psi_0, b_1] + (S_a^*)^{-1}\nabla^2 \psi_1 \end{aligned} \qquad (A9)$$

$$\begin{aligned} \frac{\partial b_1}{\partial t} = c_{\beta}^2 [V_{z,1}, \psi_0] + c_{\beta}^2 d_i [\nabla^2 \psi_0, \psi_1] \\ + c_{\beta}^2 d_i [\nabla^2 \psi_1, \psi_0] \\ + (S_a)^{-1} \nabla^2 b_1 \end{aligned} \qquad (A10)$$

$$\begin{aligned} \frac{\partial}{\partial t} \nabla^2 \varphi_1 = [\psi_0, \nabla^2 \psi_1] + [\psi_1, \nabla^2 \psi_0] \\ + (R_{\perp,a}^*)^{-1} \nabla^4 \varphi_1 \end{aligned} \qquad (A11)$$

$$\frac{\partial}{\partial t} V_{z,1} = [b_1, \psi_0] + (R_{\parallel,a}^*)^{-1} \nabla^2 V_{z,1}. \qquad (A12)$$

The linear system in Equations (A9)–(A12) was already derived by V. V. Mirnov et al. (2003, 2004).

## A.3. $\rho_s$ and the Hall Term in Ohm's Law

The scale lengths $\rho_s$ and $d_i$ are related via the parameter $\beta_e$ (defined in terms of $B_0^{tot}$), but the physical connection between the electron pressure and the Hall term is not trivial. Different derivations allow for a formally identical inclusion of the contributions related to $\rho_s$ in the reduced-MHD equations. The derivations differ because of the orderings among quantities (notably, among the components of $\delta B$ and $B_g$) and because of the closure assumed for the pressure. In some cases, the Hall term is completely neglected; in other cases, it is not. It is worthwhile to recall these issues.

The same set of two-field linear equations, Equations (A2)–(A2), can be recovered from appropriate limits of both a gyrokinetic approach (F. Pegoraro & T. J. Schep 1986; T. J. Schep et al. 1994; B. N. Rogers et al. 2007; A. Zocco & A. A. Schekochihin 2011) and of other reduced-MHD models, in which more than two scalar fields are retained (see R. Fitzpatrick & F. Porcelli 2004; N. H. Bian & D. Tsiklauri 2009; F. L. Waelbroeck et al. 2009; C. Granier et al. 2024). Moreover, the corresponding nonlinear set with $\rho_i = 0$ can be derived from some “different” two-fluid models. Assuming the ordering $b \sim \varepsilon_B^2$ (see Section 2.1.1), the contribution of the Hall term can be completely neglected, and the $\rho_s^2$-term comes from a gyrofluid closure of the anisotropic electron pressure tensor (T. J. Schep et al. 1994; D. Del Sarto et al. 2006; H. Betar et al. 2022). Adopting instead a polytropic closure, the same $\rho_s^2$-term is given by the gradient of the electron pressure in Ohm's law, which has been projected along the total magnetic field by retaining the dominant $\sim \varepsilon_B^2$ contribution in the parallel electric field component (X. Wang & A. Batthacharjee 1993; R. G. Kleva et al. 1995). In other fluid derivations starting from a four-field model (see Appendix A.2), the same $\rho_s^2$-term is related to a $b \sim \varepsilon_B$ contribution associated with a component of the Hall term in
Ohm's law, which for sufficiently small values of $\beta_e$ is proportional to the parallel component of the fluid vorticity, i.e., to $\nabla^2\varphi$ (V. V. Mirnov et al. 2003; R. Fitzpatrick & F. Porcelli 2004; V. V. Mirnov et al. 2004; N. Bian & G. Vekstein 2007; N. H. Bian & D. Tsiklauri 2009). In these two-fluid models with $\beta_e \ll 1$, all formally identical, the scale $d_i$ does not appear explicitly, being reabsorbed into $\rho_s^2 = \beta_e d_i^2$, and the role of the Hall-physics is not manifested. At the same time, in all these models, the fluid velocity field is incompressible and is associated with the stream function $\varphi$.

However, as pointed out by G. P. Zank & W. H. Matthaeus (1992b), N. H. Bian & D. Tsiklauri (2009), V. V. Mirnov et al. (2003, 2004), and R. Fitzpatrick & F. Porcelli (2004), for $\beta_e \sim O(1)$, the fluid incompressibility formally fails, and the scale $d_i$ must be retained if the Hall term is included. Because of this, in the four-field model of Appendix A.2, which formally accounts for fluid compressibility, the parameters $c_\beta$, related to $\beta_e$, and $d_i$ are separately retained.

## Appendix B
### Benchmark Tests for the Four-field Model

A systematic numerical study of the tearing mode physics described by the system in Equations (A9)–(A12) will be carried out elsewhere. It is however interesting to consider here two parameter ranges, where the model is tested: one is the case in which the Hall term is formally expected to lead to the $\rho_s$ contribution in the reduced MHD in Equations (A1)–(A2) for $\rho_i = 0$ (Appendix B.1); the other one is the Hall-reconnection regime investigated by E. Papini et al. (2019a; Appendix B.2).

#### B.1. Convergence Test for the Four-field Model

We consider the limit $(S_a)^{-1} = (R_{\perp,a}^*)^{-1} = (R_{\parallel,a}^*)^{-1} = 0$ and $\rho_i = 0$ in which Equations (A9)–(A12) converge to Equations (A1)–(A2). This can be taken as a convergence test for the solver adapted to integrate the four-field linear system. Analytically speaking, this convergence occurs when $\beta_e \ll 1$, while $d_i^2 \sim 1/\beta_e \gg 1$ for $\rho_s$ fixed. Once $[\nabla^2\psi, \psi]$ is eliminated using Equation (A7), Equation (A6) for $S_a^{-1} = 0$ reads

$$ \frac{\partial b}{\partial t} + [\varphi, b] = c_{\beta}^{2}[V_z, \psi] - c_{\beta}^{2}d_{i}\left(\frac{\partial}{\partial t}\nabla^{2}\varphi + [\varphi, \nabla^{2}\varphi]\right). \quad (B1) $$

For $\beta_e \to 0$, we have $c_\beta^2 \to 0$ while $c_\beta^2 d_i \gg c_\beta^2$ and $c_\beta^2 d_i^2 = \rho_s^2 \sim \text{const}$. Let us introduce the total time derivative $d/dt \equiv \partial/\partial t + [\varphi, \cdot]$. The leading order in $\beta_e \ll 1$ in Equation (B1) reads

$$ \frac{db}{dt} = -c_{\beta}^{2} d_{i} \frac{d\nabla^{2}\varphi}{dt}, \quad (B2) $$

which can be integrated to give

$$ b = -c_{\beta}^{2} d_{i} \nabla^{2} \varphi + \text{cst.} \quad (B3) $$

Equation (A12) yields $dV_z/dt \simeq O(c_\beta^2 d_i) \to 0$, consistently with the approximation giving Equation (B2). Combining

Equation (B3) with Equation (B1), we obtain

$$ \frac{\partial}{\partial t}(\psi - d_e^2 \nabla^2 \psi) + [\varphi, \psi - d_e^2 \nabla^2 \psi] = c_{\beta}^2 d_i^2 [\nabla^2 \varphi, \psi]. \quad (B4) $$

Equation (B4) and Equations (A4)–(A7) correspond to the $\rho_i = 0$ collisionless limit of the nonlinear two-field model, which leads to Equations (A1)–(A2). This convergence is shown in the linear limit of interest here in Figure (3), in which the growth rates $\gamma\tau_{A,a}$ computed with the two numerical solver are shown as a function of $(d_e/a)$ for tearing modes in the small-$k$ (i.e., $ka \ll 1$; left frame) and large-$k$ (i.e., $ka \to k_m a$; right frame) wavevector ranges. The excellent agreement is shown for a range of values of $d_e/a$ while keeping fixed $\rho_i = 0$, $S^{-1} = 0$, and $\rho_s = 5 \times 10^{-2}$ with $\beta_e = 2.5 \times 10^{-9}$ and $d_i = 1000$.

It is also important to emphasize that the disappearance of the variable $b \sim \delta B_z/B_g$ from the system makes the amplitude $B_g$ “redundant”: since all remaining quantities are ordered like $|B_\perp|/B_g \sim B_{cs}/B_g$, we can change the reference magnetic field from $B_0 = B_g$ to $B_0 = B_{cs}$ by just making both $\varphi$ and $\psi$ ordered $\sim O(1)$ in Equation (B4), like they are in Equations (A1)–(A2), where $B_0 = B_{cs}$. This is the choice we have made in the numerical calculations of Figure 3, in which the timescale $\tau_{A,a}^*$ of the four-field model coincides with the timescale $\tau_{A,a}$ of the two-field model.

#### B.2. A Case of Comparison of the Four-field with the Hall-MHD Model

We now consider a parameter case compatible with the initialization of the nonlinear Hall-MHD simulations performed by E. Papini et al. (2019a). In that work, turbulent reconnection was studied in Hall-resistive MHD, in a 2D spatial coordinate dependence. Magnetic reconnection was shown to develop on current sheets of a characteristic thickness of the order $a \sim d_i$, after a turbulent cascade was initialized by injecting energy at the wavelength $k_{inj} = 0.2/d_i$, corresponding to $\ell_{inj} = 2\pi/k_{inj} \approx 31.4d_i$, with in-plane magnetic perturbations of an average amplitude $B_{rms} \approx 0.24B_g$. The numerical parameters of reference are $\beta_e = \beta_i = 1$. A normalized resistivity $S_d^* = 10^3$ and a normalized isotropic viscosity $(R_d^*)^{-1} = 10^{-3}$ were also considered. The apex “*” refers to the notation introduced in Section 2.1.1, where the magnetic field is normalized with respect to $B_g$, and the reference density can be identified as the $n_0 = n_{cs}$ we use. E. Papini et al. (2019b) also use $L_0 = d_i$. Based on their numerical results, we further identify $B_{cs} = \delta B_{rms} = 0.24B_g$, $a = d_i$, and $L = \ell_{inj} \approx 31.4d_i$.

In Figure 4, we show the scalings of the fastest growing mode $\gamma_M \tau_{A,a}^*$ obtained in a neighborhood of the parameters $d_e = 0$, $\rho_i = 0$ (in Hall-MHD, the electron mass is neglected, and ion-FLR effects are not retained, despite the $\beta_i = 1$ value), $d_i = \rho_s = a$, and $(S_a^*)^{-1} = (R_a^*)^{-1} = 10^{-3}$, where $R_a^{-1} = R_{\perp,a}^{-1} = R_{\parallel,a}^{-1}$. The information on the ratio $\delta B_{rms}/B_g \sim B_{cs}/B_g$ selected by E. Papini et al. (2019b) is retained by fixing a normalized equilibrium $\psi_0(x) = (\delta B_{rms}/B_g) \cosh^{-2}(x/d_i) = 0.24 \cosh^{-2}(x/d_i)$ while $\varphi_0 = b_0 = V_{z,0} = 0$. The numerical growth rate scalings, shown in Figure 4, in a neighborhood of the chosen parameters, are this way naturally normalized to $\tau_{A,a}^*$. The characteristic growth rate $\gamma_M \tau_{A,a}^* = \gamma_M \tau_{A,d_i}^* \approx 0.057$ obtained for these parameter values is very close to the mean value
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 3.** Comparison of the growth rates numerically obtained by integrating the eigenvalue problem in the collisionless regime (i.e., with $d_e \neq 0$) with both the two-field and four-field model equations, in a formal regime with $c_3^2 \ll 1$, $d_i \gg 1$ but $c_3^2 d_i^2 = \rho_s^2 \ll 1$, where the two sets of equations are meant to converge. The parameters besides $d_e$, which have been used, are $S^{-1} = \rho_i = 0$, and $\rho_s = 5 \times 10^{-2}$ in the two-field model (blue circles) and $S^{-1} = \rho_i = 0$, and $\rho_s = \sqrt{\beta_e d_i^2} = \sqrt{(2.5 \times 10^{-9}) \times (10^3)^2} = 5 \times 10^{-2}$ in the four-field model (red crosses). The numerical results are shown for both a small value of ka (i.e., ka = 0.07, left frame), corresponding to the so-called small wavelength or “large-$\Delta'$” limit (left frame), and a “large” value of ka (i.e., ka = 1.75, left frame), corresponding to the so-called large wavelength or “small-$\Delta'$” limit (right frame). The discrepancies between the growth rates evaluated with the two models, corresponding to relative errors of the order of $10^{-2}$, are not appreciable from the graphics.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 4.** Local scalings of $\gamma_M \tau_A$ computed as described in H. Betar et al. (2020) for viscous-resistive Hall-MHD ($d_e = 0$) as function of $\beta$ (a), $d_i$ (b), $S$ (c), and $\nu$ (d). In this parameter range, the dependence on the normalized viscosity $\nu$ seems to be negligible. This appears to be somewhat consistent with analogous results found in the standard two-field reduced-MHD model, where, for comparable values of $R^{-1}$ and $S^{-1}$, resistivity dominates (see N. Bian & G. Vekstein 2005; H. Betar et al. 2020).</figcaption></figure>

$\langle\gamma\rangle \tau_{A,d_i}^* \simeq 0.08$, which E. Papini et al. (2019a) obtained with a statistical average of the reconnection rates measured in nonlinear Hall-MHD simulations of turbulent reconnection. In

particular, the scaling law we numerically obtained,

$$ \gamma_M \tau_{A,a}^* \sim \beta_e^8 \left(\frac{d_i}{a}\right)^{\frac{3}{5}} (S_a^*)^{-\frac{1}{5}} (R_a^*)^0, \quad (B5) $$
can be rewritten in terms of the length *L* and of the magnetic field amplitude $B_{cs}$ by using $\tau_{A,a}^* = \tau_{A,L}(a/L)(B_{cs}/B_g)$. Then, for $L \simeq 30a$ (inferred from the figures) and $B_g/B_{cs} \simeq 4$, we obtain $\gamma_M \tau_{A,L} \simeq 0.057(L/a)(B_g/B_{cs}) \simeq 0.057 \times 120 \simeq 7$.

## ORCID iDs

Homam Betar https://orcid.org/0000-0001-5207-3170

Daniele Del Sarto https://orcid.org/0000-0001-7489-8843

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