1 Introduction

Magnetic reconnection is accepted as the key mechanism for energy release in solar flares and other eruptive events in astrophysical plasmas. However, direct application of magnetic-reconnection theory to the physics of flares (and other large-scale phenomena) faced a crucial issue for a long time: All known micro-physical processes leading to the change of magnetic field topology (i.e. the reconnection) require very thin current sheets ( ≈ 1 m in the solar corona). On the other hand, the typical flare current-layer width estimated from observations [6] is about six orders of magnitude larger. This duality is reflected also in the flare observations which exhibit both coherent large-scale (107 m) dynamics and signatures of the micro-scale, chaotic energy release, at the same moment [1]. In order to overcome this huge scale-gap [9] suggested the concept of fractal reconnection. Nevertheless, this ad hoc suggested scheme had no support in the analytical theory of magnetic reconnection neither has been similar effects convincingly observed in numerical experiments, and thus this interesting approach was for a quite long time mostly ignored. However, quite recently [8] discovered missed solution in the Sweet-Parker’s analysis of the current-sheet stability and they found that each CS with sufficiently high aspect ratio is linearly unstable to the formation of plasmoids (plasmoid instability). Later on, Uzdensky et al. [10] have developed analytical theory of the chain plasmoid instability, related closely to the original Shibata’s and Tanuma’s idea [9].

As the analytical theory has a limited scope we focus in our paper at the research of the anticipated tearing-mode cascade in fully non-linear regime using numerical simulations. At the same moment we are looking for other mechanisms of direct energy cascade (fragmentation/filamentation) in magnetic reconnection.

2 Model

We are interested in processes of filamentation of the originally thick current layer to smaller structures. The range of scales in which these processes should operate is still far from the scales where other terms of generalised Ohm’s law (from which the largest is the Hall-scale) operate. The evolution of magnetized coronal plasma can be thus adequately described by a set of compressible resistive one-fluid MHD equations, including gravity. For details of the model used see [2]. The set of MHD equations is rewritten into its conservative form and numerically integrated. In order to see anticipated filamentation of the current density the AMR technique (see, e.g., [3]) is implemented into our numerical code. The mesh is refined whenever the CS typical width drops below certain number of grid cells. The initial state has been chosen in the form of a vertical generalized Harris-type CS with the magnetic field slightly decreasing with height – see [2] for details. Calculations are performed in a symmetric box with line-tying boundary condition (BC) at the bottom and free BCs otherwise. We assume translational symmetry in the current density direction (y) and presence of the guide field B y in this direction. Anomalous resistivity η is switched-on whenever the current (measured by current-carrier velocity) intensifies above some threshold [2, 5]; η = 0 otherwise. The threshold for the resistivity onset leads to the minimum width of the CSs at the order of couple of cells at the highest resolution. As the ratio between the largest and smallest resolved scales in the simulation is ≈ 104, the Lundquist number is of the order ≈ 108. The model parameters used are the same as in [2].

3 Results

Results of our modelling are shown in Figs. 1 and 2. The figures display magnetic field structure on the background of the current density. Figure 1 shows the situation at t = 300τA (τA being the Alfvén transit time – see [2]). Subsequent zooms from panels (a) to (c) reveals smaller and smaller plasmoids formed by the tearing instability. The plasmoids are separated by consecutively thinner CSs. This picture is in line with the chain plasmoid instability [10]. Figure 2 shows formation of the CS between two merging larger plasmoids at t = 365τA. As the CS become sufficiently long and thin, plasmoid instability takes place again and a new plasmoid is formed in this transversal CS. Situation is better visible in Fig. 2d which represents a projection of Fig. 2c into plane y = 0.

Fig. 1
figure 1

Current density (color scale) and magnetic field structure at t = 300τA. Increasing zoom reveals subsequently smaller plasmoids separated by correspondingly thinner CSs (tearing cascade)

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Fig. 2
figure 2

Current density and magnetic field structure at t = 365τA. Panels (a)–(c): Increasing zoom shows the micro-plasmoid formed in the CS between two merging larger plasmoids (fragmenting coalescence). Panel (d): 2D projection of panel (c) into plane y = 0. Red and green lines represent locations of B x  = 0 and B z  = 0, respectively, ‘x’ and ‘o’ symbols show locations of X- and O-type null lines. Note the anisotropic scales on x- and z-axis

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4 Discussion and Conclusions

The huge gap between energy input and dissipation scales in large systems like solar-flare current layers represents a big challenge for application of magnetic-reconnection theory to energy release in such large-scale phenomena. The idea of fractal reconnection [9] that recently developed into theory of chain plasmoid instability [10] clearly addresses this issue. Nevertheless, its convincing confirmation by numerical experiments was not available until recently (e.g., [2, 7] and references therein). The reason why earlier numerical simulations met difficulties with finding this predicted process most likely lies in the limited resolution of previous experiments. Thanks to the AMR technique used in our simulation the domain of simulated scales spans over almost five decades. Such a wide scale-range enables investigation of the scaling law for the cascading process. In [2] the authors found for 1-D scaling of magnetic energy density along the CS the power-law \({\mathcal{E}}_{M}(k) \propto {k}^{s}\) with s ≈ − 2. 14. Under some plausible assumptions it can be compared with the scaling found in analysis by Uzdensky et al. [10] for number of plasmoids of size L:

$$N(L) \propto {L}^{-2}$$
(1)

Assuming (for simplicity) that the magnetic field strength is linearly increasing from the null-point in the center of the plasmoid towards its edge, where its reaches a typical value B 0

$$B(x) = {B}_{0} \frac{x} {L}$$
(2)

we get for magnetic energy in a circular-shaped single plasmoid of size L

$${E}_{M}^{SP}(L) ={ \int \nolimits \nolimits }_{\mathrm{plasmoid}}\frac{{B}^{2}(x)} {2{\mu }_{o}} \mathrm{d}V = \frac{1} {2{\mu }_{o}}{ \int \nolimits \nolimits }_{0}^{L}{B}_{ 0}^{2} \frac{{x}^{2}} {{L}^{2}}2\pi x\mathrm{d}x = \frac{\pi } {4{\mu }_{o}}{B}_{0}^{2}{L}^{2}$$
(3)

Since for magnetic energy at scale L holds

$$\mathrm{d}{E}_{M}(L) = {E}_{M}^{SP}(L)N(L)\mathrm{d}L = {\mathcal{E}}_{ M}(k)\mathrm{d}k\ ,$$
(4)

taking into account scaling law in Eq. (1) and applying relations between scale and corresponding wavenumber \(k = \frac{2\pi } {L}\), \(\mathrm{d}k = -\frac{2\pi } {{L}^{2}} \mathrm{d}L\), we finally arrive to power spectrum for magnetic energy density in the form

$${\mathcal{E}}_{M}(k) \propto {k}^{-2}$$
(5)

The scaling law for magnetic energy density is power-law, again, with the spectral index \(s = -2\) quite close to that found in our simulated data (\(s = -2.14\)).

Moreover – and above all – our high-resolved simulations revealed the significance of the opposite process – the plasmoid coalescence – for further fragmentation of the current structures. We have found that the coalescence is accompanied by formation of intense, thin CSs between the merging plasmoids that may later become unstable to further plasmoid formation. Although the coalescence eventually leads to the formation of a single larger structure from the two smaller, this process is inherently connected with further fragmentation in the CS between the merging plasmoids. Consequently, also this process contributes – quite unexpectedly – to the energy transport towards small scales (direct cascade). Since these CSs are transversal to the original large-scale current layer the simulation reveals truly 2D nature of the turbulent cascade towards small scales. Similar process has been recently observed also in kinetic (PIC) simulations [4].