SPONTANEOUS CURRENT-LAYER FRAGMENTATION AND CASCADING RECONNECTION IN SOLAR FLARES. II. RELATION TO OBSERVATIONS

For many years the common picture of solar eruptions and flares has been based on the CSHKP model (Magara et al. 1996 and references therein). This model involves reconnection in a global vertical flare current layer formed behind an ejected flux rope/filament (Lin & Forbes 2000 and references therein). The standard model is in agreement with the observed large-scale dynamics of eruptive events. Several questions, however, remain open. In particular, it is not clear by which mechanism the energy accumulated in relatively large-scale (≈1000 km) structures of the magnetic field associated with the flare current sheet (CS) is transferred toward the small dissipation scales. Indeed, the magnetic diffusion in the almost collisionless solar coronal plasma is an essentially kinetic process at scales of the order d_i = c/ω_pi (Büchner 2006). Another open question is the relation between the well-organized dynamics of solar eruptions observed at large scales and the hard X-ray (HXR) and radio signatures of fragmented energy release (Aschwanden 2002; Karlický et al. 2000). Finally, the CSHKP model has been questioned by many authors (e.g., Vlahos 2007) because of its apparent inability to explain the large fluxes of accelerated particles inferred from HXR observations.

Recently, in order to address the issue of energy transfer across a broad range of MHD scales, Bárta et al. (2011, hereafter referred to as Paper I) investigated magnetic reconnection in an extended current layer by means of a high-resolution MHD simulation. Inspired by the “fractal reconnection” conjecture of Shibata & Tanuma (2001) the authors of Paper I performed a 2.5D numerical simulation covering a large range of MHD scales. They conjectured that, in analogy with the vortex-tube cascade in fluid dynamics, a cascade of magnetic flux tubes from large to small scale can provide the mechanism for energy transfer across the scales. Two mechanisms of fragmentation were identified in Paper I: (1) the tearing cascade, in line with the concept of fractal reconnection by Shibata & Tanuma (2001), developed recently into the theory of chain plasmoid instability by Loureiro et al. (2007) and Uzdensky et al. (2010), supported further by numerical MHD simulations by Bhattacharjee et al. (2009), Samtaney et al. (2009), and Huang & Bhattacharjee (2010); and (2) ambient-field-driven coalescence of flux ropes/plasmoids leading to the formation of transversal CSs subjected further to the same chain of processes of cascading fragmentation. Extrapolation of the results leads to a picture in which the plasmoid cascade continues down to the kinetic scale where actual magnetic dissipation and particle acceleration occurs, most likely via the kinetic coalescence of plasmoids (Drake et al. 2005; Karlický et al. 2010). Nevertheless, other processes can appear in the range between MHD and plasma kinetic scales, e.g., the Hall type of reconnection—see recent simulations by Shepherd & Cassak (2010) and Huang et al. (2010).

As shown in Paper I, cascading reconnection forms multiple non-ideal regions (at the resolution limit—see the discussion in Paper I) hierarchically distributed in the fragmenting current layer. As a result, the open questions of energy transfer, fragmented versus organized flare energy release, and particle acceleration seem to be closely related to one another in the presence of cascading reconnection.

The question arises whether this mechanism found by simulations is relevant for actual solar flares. We address this topic in our present paper, deriving critical signatures specific to the model developed in Paper I that allow comparison with observations. It is currently impossible to directly observe small-scale magnetic structures, whose formation in the fragmented flare current layer is predicted by the cascading-reconnection model. Almost all information on the impulsive phase of flares comes from the radiation emitted by accelerated particles. Therefore, we concentrate on such predictions of the MHD model that are connected with specific distribution and dynamics of acceleration regions in the fragmenting flare current layer.

In particular, by magnetic mapping of non-ideal regions embedded in the CS to the photosphere we derive expected consequences of the distribution and dynamics of acceleration regions for flare ribbons. Using the results of the underlying MHD model presented in Paper I, we qualitatively derive the structure and dynamics of the expected ribbon emission. We compare the modeled ribbon structure with the observations directly and by means of its statistical properties used by Nishizuka et al. (2009).

The paper is organized as follows. First, in Section 2.1, we briefly summarize the results of the underlying high-resolution MHD simulation of cascading reconnection. Then, in Section 2.2, we present the extension of the underlying MHD model utilized to produce model-specific results/consequences in a form that is comparable with the observations. In particular, we study the distribution and dynamics of dissipative/acceleration regions, represented by the embedded X-points, and their magnetic mapping to the chromosphere/photosphere of the Sun. For the obtained dynamics of acceleration regions and their magnetic footpoints, we qualitatively model the expected emission of flare ribbons (Section 3). We compare the emission structures of the modeled and observed ribbons. Furthermore, we extract selected statistical properties of emission kernels embedded in the modeled ribbons and compare them with the results found by an analysis of the actual ribbon observations (Nishizuka et al. 2009). Finally, in Section 4, we discuss our results with the intention to evaluate the relevance of cascading reconnection/fragmentation processes presented in Paper I to actual solar flares.

In this section, we describe the procedures for simulating observable signatures specific to cascading reconnection. For clarity, we first briefly summarize the main features of the current-layer fragmentation model (Paper I).

2.1. Basic MHD Model

In Paper I, the authors studied cascading reconnection and energy transfer from accumulation (large) to dissipation (small) scales. They used a high-resolution MHD model with a non-ideal, resistive term depending on the current-carrier drift velocity v_D as

$$\begin{equation} \eta (\mbox{\boldmath $r$},t)=\left\lbrace \begin{array}{@{}lll}0 & : & |v_{\rm D}|\le v_{{\rm cr}}\\ C\frac{\left(v_{\rm D}(\mbox{\boldmath $r$},t)-v_{{\rm cr}}\right)}{v_0}& : & |v_{\rm D}|> v_{{\rm cr}} \end{array} \right. \end{equation} \tag{ 1 }$$

with the threshold v_cr set to the higher, more realistic value corresponding to increased resolution. The authors applied the model to the large-scale vertical CS in solar eruptive flares.

The simulations in Paper I were performed in dimensionless variables: spatial coordinates x, y, and z were expressed in units of the CS half-width L_A at the photospheric level (z = 0) and the time was normalized to the Alfvén transit time τ_A = L_A/V_A,0, where $V_{\rm A,0}=B_0/\sqrt{\mu _0 \rho _0}$ is the asymptotic value (x → ∞, z = 0) of the Alfvén speed at t = 0. In order to relate the model to real solar conditions, appropriate scaling was adopted in Paper I. Based on the gravity-introduced scale–height relation L_A = 600 km was found. Due to this relation both the dimensionless and SI units can be used further in our present paper.

In Paper I, the authors described how tearing and driven coalescence instabilities lead to the fragmentation of an originally unstructured flare current layer. As a consequence, a cascade of magnetic flux ropes/plasmoids is formed from large to consecutively smaller scales. The structuring of the current layer leads to the formation of multiple thin embedded CSs. They can host non-ideal regions where particles can be accelerated. Enhanced numerical resolution revealed the structure of dissipation regions: they form many thin channels of non-zero magnetic diffusivity. The spatial distribution and dynamics of these diffusivity channels has been further studied via tracking of the X-points associated with the non-ideal regions. The analysis shows hierarchically structured grouping of the dissipation channels and intermittency in their lifetimes and times for which they are magnetically connected to the photosphere.

The structure, distribution, and dynamics of the non-ideal regions should be reflected in the specific observable features that we are going to derive in the following sub-section.

2.2. Derivation of Model-specific Observable Features

Unfortunately, it is impossible to directly measure the magnetic field or the current density in the coronal CSs (and not at all with the resolution reached in the simulation done in Paper I). Hence, we need some indirect, more subtle comparison between the observed and modeled quantities. Nishizuka et al. (2009) recently presented a possible procedure that may lead to indirect indication of CS fragmentation. These authors studied the structure of emission in flare ribbons, namely, the distribution and dynamics of the embedded bright kernels. They conjectured a possible relation of the resulting power-law distributions found in their statistical analysis of ribbon–kernel properties with the concept of fractal CS (Shibata & Tanuma 2001).

We follow this idea from the other end. We start with the structure, distribution, and dynamics of diffusion/acceleration regions specific to the cascading-reconnection model as they were found in Paper I. Then, we relate these features to the structure and dynamics of emission in flare ribbons. We proceed as follows: as shown in Paper I (Figure 6), the reconnection X-points are associated with thin channels of magnetic diffusivity. Hence, we take the X-points as geometric representatives of dissipation regions. In these regions, electrons can be accelerated, e.g., by the DC electric fields. However, different acceleration mechanisms have also been proposed, e.g., Drake et al. (2005) suggest Fermi-type acceleration in contracting plasmoids. We take this type of acceleration into account with the following consideration: extrapolating the results of numerical simulations of the reconnection cascade to smaller scales, one can imagine that each dissipative region surrounding the studied X-point in fact contains many unresolved small-scale magnetic islands. Electron acceleration in these non-ideal regions is then performed (in line with Drake et al. 2005) via coalescence and shrinkage of these micro-plasmoids. Electrons accelerated by Fermi-type mechanism inside the plasmoids are trapped and can be released to the open field (which connects the CS to the chromosphere) again in the vicinity of the X-points, where they become demagnetized. For further related discussion on electron acceleration in magnetic islands see, e.g., Oka et al. (2010).

In our study we focus on electrons, as we are interested in optical and UV/EUV chromospheric response ascribed to electron beams. These are then transported along the magnetic field lines either upward to the solar corona or downward until they reach dense layers of the solar atmosphere, where their energy is thermalized. Here, we concentrate on the downward transported electrons. In order to study the positions and dynamics of the points where these electrons reach the chromosphere, i.e., the expected bright kernels, we map X-points to the bottom boundary of the simulation box using magnetic field. The magnetic field lines that go through a given X-point represent magnetic separatrices in our two-dimensional model. Therefore, we are seeking intersections of magnetic separatrices with the bottom boundary. In the following, we will refer to these footpoints at the bottom boundary as kernels. Thus, knowing the positions of all X-points (Figure 7 in Paper I) and knowing the magnetic field, we can calculate the positions x_k(t) of all kernels k for each recorded time step t of the simulation. Note that due to the two-dimensional geometry of the model each footpoint/kernel position is uniquely given by its x-coordinate (see Paper I for details).

The calculations of the chromospheric emission in a certain spectral line (e.g., H_α or C iv) in response to the bombardment of electrons accelerated in a distant reconnection region is a difficult task (Kašparová et al. 2009; Varady et al. 2010). In order to obtain at least a qualitative output that could later be compared with observations, we use for ribbon-intensity distribution the expression

$$\begin{eqnarray} \nonumber I_k(x,y,t) & = & I_0 \exp \left(-\frac{\left(x-x_k(t)\right)^2}{\Delta ^2}\right)\\ I(x,y,t) & = & \sum _{k} I_k(x,y,t)\, \end{eqnarray} \tag{ 2 }$$

instead of description of all the complicated processes of electron transport, energy deposition, chromospheric response, and radiative transfer. The intensity I is summed over all kernels k; y is the second (however invariant in our model) coordinate in the photosphere. The kernel size is set to Δ = 0.1 L_A. Since the model is not capable of estimating either the electron-beam flux or the energy thermalized in the kernel, we set the intensity scale to I₀ = 1. We will return to this point later in the discussion.

Let us now apply this procedure to the results of the simulations presented in Paper I.

Figure 1 depicts the above-described procedure of the mapping of X-points to the photosphere along magnetic separatrices and flare-ribbon emission calculation. Obtained results are shown for t = 391.5τ_A. Panels (a)–(c) show how three dissipative regions/X-points embedded in the fragmented current layer are magnetically connected to the bottom boundary and mapped by separatrices to the positions x ≈ ±2800 km (one kernel) and x ≈ ±3750 km (two mutually close kernels).

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A view of the flare ribbons obtained by applying the simple model described by Equation (2) is presented in Figure 1(d). As the figure indicates, in some cases one can observe multiple pairs of flare ribbons. This is close to the idea of chromospheric re-brightening studied by Miklenic et al. (2010). The inner separatrix associated with the internal pair of ribbons is connected to the X-point that appeared temporarily in the transversal CS formed between the merging plasmoid and loop arcade (see also Figure 4 in Paper I). Figures 1(e) and (f) show the emission intensity profile along the x-axis, calculated according to the model in Equation (2). The enlarged detail shown in Figure 1(f) reveals the internal double-peak structure of the outer ribbons. Note that a similar structure is seen in the observed H_α ribbons (Figure 2).

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In order to study the dynamics of kernels and associated X-points during the entire recorded interval of evolution (300τ_A–400τ_A) we tracked the moving positions of all X-points and the magnetically associated (mapped) kernels. The kinematics of the kernels (footpoints of separatrices) are depicted in Figure 3. Figure 3(a) presents a global picture of kernel dynamics over the entire recorded interval. Because of the CS symmetry, only the right ribbon positions are displayed. The spatial coordinate x, the distance from the polarity-inversion line (PIL), is limited to the interval where the kernels actually occur. Figure 3(b) shows a detailed view of the area selected in Figure 3(a). In order to reach higher zoom, the global trend (motion) of the group of kernels (i.e., the increasing ribbon separation as the flare proceeds) has been subtracted by applying the transformation $\overline{x}/L_{\rm A}=x/L_{\rm A}-0.0125(t/\tau _{\rm A}-300)$. X-points that are separated only by tiny magnetic islands can map practically to the same kernel, and cannot be distinguished even in the zoomed display. Therefore, for each time we take a set of kernel positions (its size varies with time) and display each kernel in a different color. Figure 3 shows various lifetimes of the kernels and other aspects of their dynamics, such as bifurcation/merging. Note that the lifetime of the kernel is not determined solely by the lifetime of the associated distant X-point but also by the processes in the current layer that can change the magnetic connectivity of the diffusion regions embedded in the current layer to the photosphere (Figure 7 in Paper I).

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We further extracted selected characteristics of the modeled kernel dynamics and compared them with the observations of Nishizuka et al. (2009). For this sake, we process our model results in the same manner as those authors by binning photosphere along the x-axis. For the bin size, we chose Δx = 0.05 L_A. For each time, we integrated the intensity given by Equation (2) over each bin. Thus, we obtained a proxy for the “light curves” of all bins. Figure 4 shows three examples of such light curves for bins centered around positions x = 4.58 L_A (2750 km), x = 4.93 L_A (2950 km), and x = 5.73 L_A (3435 km). Narrow peaks coming from the short-lived X-points are superimposed over the longer-lived bright kernels (c.f. Figure 2(b) in Nishizuka et al. 2009).

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As in Nishizuka et al. (2009), we then determined the peaks at each light curve and recorded their times of occurrence and their maximum intensities. We calculated the statistical distributions of the peak intensities and the time intervals between two consecutive peaks. The results of this analysis are presented in Figure 5. Despite the poor statistics (for our one-dimensional mesh we identified only 68 peaks), we obtained spectral indices s = −1.48 for the peak intensity (panel 5(a)) and s = −1.73 for the time interval between consecutive peaks (Figure 5(b)).

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The “standard” CSHKP picture of solar eruptive flares involves the formation of a global flare CS behind an ejected filament followed by magnetic reconnection in this CS. Nevertheless, deeper study of this scenario invokes further questions. (1) How is the energy accumulated in flares in relatively large-scale structures (global flare current layer) transferred to the dissipative (kinetic) scales? (2) How can we observe regular dynamics in eruption/flare on large scales, just in line with the CSHKP scenario, and the signatures of fragmented energy release at the same moment? and (3) How can the large fluxes of accelerated electrons inferred from HXR observations be reconciled with a single, relatively small diffusion region assumed in the classical CSHKP model?

The simulations presented in Paper I showed that a cascading reconnection resulting from the formation and interaction of plasmoids/flux ropes can address these three fundamental questions at once. This might provide a viable scenario for magnetic energy dissipation in large-scale systems such as solar (eruptive) flares.

In the present paper, we evaluated the relevance of this model for actual solar flares. For this sake, we derived observable model predictions that are specific to the cascading-reconnection scenario and searched for predicted features in the observed data. Since multiple acceleration regions embedded in the global flare current layer are inherently connected via magnetic-field-line mapping with the structured emission in the flare ribbons, a detailed study of the ribbon kernels might provide some evidence for the cascading processes in the current layer above the flaring region. This idea is illustrated in Figure 1, which shows the structuring of the modeled emission in ribbons. Figure 1 further indicates that a larger-scale plasmoid's interaction with the flare-loop arcade can temporarily form a second pair of ribbons. The plasmoid/loop-arcade interaction may play a significant role in flare dynamics. Further indirect evidence of such interaction, based on analysis of a series of X-ray images of the limb flare, has been presented by Kołomański & Karlický (2007) and Milligan et al. (2010).

Similar structuring of H_α emission as presented in Figures 1(d)–(f) is, indeed, visible in Figure 2. Yet another interesting feature in Figure 2 is the dark filament-like structure (indicated by an arrow). Since the original large-scale filament whose eruption initiated the flare was already far away at the time of observation, the absorption feature can be interpreted as a manifestation of one of the secondary plasmoids formed in the global current layer. This is in line with the presented cascading-fragmentation scenario. The secondary plasmoids represent enhanced-density structures. Density increase in connection with consequent faster radiative cooling might lead to detectable H_α absorption. The secondary plasmoid can consequently be manifested as a darker feature at the background of a relatively brighter chromosphere.

A sophisticated study of the structure of flare ribbons was presented by Nishizuka et al. (2009). These authors studied the statistical properties of emission kernels found inside the ribbons and found a power-law distribution of the studied kernel characteristics. A power-law distribution can be a signature of self-organized criticality evolution (Aschwanden 2002; Vlahos 2007). It can also indicate a fractal CS decay (Shibata & Tanuma 2001). The latter scenario is in fact favored by the subsequent analysis of Nishizuka et al. (2010), which relates the energy release measured by means of HXR flux and ejection of multiple plasmoids formed in the fragmenting CS.

In our present paper, we have established the connection between the hierarchical distribution of diffusion regions formed by cascading reconnection and the statistical properties of emission kernels from first principles. We studied the dynamics (positions and the mechanisms and times of creation/annihilation) of O- and X-type points in the current layer as well as of the kernel points associated with the diffusion regions (X-points) by means of magnetic-field-line mapping. We then associated each found kernel with the emission according to Equation (2). We chose this simple model because we do not have any information about the accelerated electron fluxes absorbed in the kernel, nor can we easily calculate the chromospheric response in terms of (e.g., H_α) emission. Our choice of the unified peak intensity I₀ = 1 for all kernels can be justified in the following way: the particle acceleration is performed by kinetic processes at the small-scale end of the cascade. The suggested mechanism (see, e.g., Drake et al. 2005) involves, e.g., kinetic coalescence and shrinkage of small-scale magnetic islands/plasmoids. In the dissipative range of scales, the slope of the energy spectrum (see Figure 1 in Paper I) is very steep. Hence, the size of plasmoids at kinetic scale can be expected to be more or less uniform as also indicated by PIC simulations. In such a case, the elementary injections of accelerated particles are, perhaps, roughly of the same magnitude. The intermittent, spiky structure of the observed emission profiles is then the result of the overlapping of many of these elementary injection peaks (Aschwanden 2002). Indeed, we observe this feature in examples of the modeled light curves obtained at fixed points at the bottom boundary (Figure 4). Despite the intensity produced by the elementary particle injection being normalized to I₀ = 1, the overlapping elementary peaks produce spiky intensity profiles ranging from I = 0 to I>3.

In order to directly compare our results with the findings of Nishizuka et al. (2009), we performed the same statistical analysis of the peaks in our artificial light curves. We found the statistical distributions of the emission peak intensities and of the time lag between the consecutive emission peaks (Figure 5). Because of the two-dimensional geometry of the underlying MHD model, we obtained a one-dimensional distribution of emission kernels. Consequently, we found a smaller number of peaks. Despite the resulting poor statistics, we were able to conclude that the slopes of the power-law distributions obtained by our model agree surprisingly well with those found by Nishizuka et al. (2009) for the observed light curves. For the statistical distribution of the peak intensities, we found spectral index s = −1.48 (Nishizuka et al. 2009 obtained s = −1.5) and for the time lag between peaks we obtained s = −1.74 (s = −1.8 in Nishizuka et al. 2009). We are aware that our statistics are, however, rather poor as they operate with only 68 peaks found in total. Thus, it would be premature to take this agreement as conclusive. Nevertheless, the results indicate that the statistical properties of emission kernels resulting from cascading reconnection and of those observed are—at least—not in direct contradiction.

The results of the presented comparison between the modeled features attributed specifically to cascading reconnection and their observed counterparts support—in our view—the relevance of cascading reconnection processes studied in Paper I for reconnection in solar atmosphere. Agreements found between the statistical properties of modeled and observed emission kernels invoke, however, further questions. The underlying 2.5D MHD model ignores structuring of the CS in the y-direction (along the PIL). This result might indicate that wave modes with k_y>0 (in the off-plane direction) may not play too significant a role in current-layer fragmentation in solar flares or, at least, that they do not reach the smallest scales to substantially influence the dynamics (lifetime) of kinetic-scale diffusion regions. This issue can be addressed, however, only within the framework of a full three-dimensional MHD model.

This research was performed with the support of the European Commission through the SOLAIRE Network (MTRN-CT-2006-035484) and the grants P209/10/1680, 205/09/1469, 205/09/1705, P209/10/1706 of the Grant Agency of the Czech Republic, the grant 300030701 of the Grant Agency of the Czech Academy of Science, and the research project AV0Z10030501 of Astronomical Institute of the Czech Academy of Science.

The authors thank the anonymous referee whose comments were very helpful in improving the quality of the paper.