On the Relationship Between Nanoflare Energy and Delay in the Closed Solar Corona

TL;DR

The paper investigates whether nanoflare energy $E$ correlates with the inter-event delay $\\tau_D$ in a self-consistently driven 3D MHD model of a solar active region. It analyzes nanoflares identified on individual field lines by three methods, using nonparametric tests (weighted $t_w$ and Spearman) and a power-law scaling test $E \\propto \\tau_D^{\\alpha}$ with bootstrap resampling to estimate $\\alpha$. Across methods and subsets (including high-energy nanoflares), the results show little to no correlation between $E$ and $\\tau_D$, with $\\alpha$ clustered near zero and delays broadly distributed within energy bins. The findings suggest that nanoflare onset is not determined solely by a local stress threshold and may be driven by complex, multi-strand interactions and triggering across the coronal topology, aligning with an avalanche-like heating scenario and informing coronal heating models.

Abstract

Determining the relationship between nanoflare energies and their delays is the key for understanding the physical mechanism of the events and the plasma response. Nanoflares analyzed in this study were generated self-consistently via prescribed photospheric motions in a 3D multi-strand simulation of a subset of active region magnetic flux. Energies and durations were quantified using three distinct methods. In this study, we investigated the correlation between nanoflare energies (E) and delays ($τ_D$) using two non-parametric, rank-based statistical tests. Across all methods, results consistently show little to no correlation. This is further supported by the distribution of the exponent $α$ in the assumed relation $E \propto τ_D^α$, which peaks near zero, and by broad delay distributions within fixed energy bins. These findings are irrespective of whether delays are correlated with the energy of the preceding or subsequent event. They also hold for a subset of high-energy nanoflares. The absence of correlation suggests that nanoflare onset is not solely determined by a critical value of magnetic stress and may involve triggering by other events, perhaps related to a locally complex topology.

Figures (12)

• Figure 1: A schematic illustrating two theoretical models of the correlation between nanoflare delay and energy: (a) delay proportional to the energy of the previous event, and (b) delay proportional to the energy of the next event. In panel (a), the dashed black line represents a critical stress threshold (a 'ceiling'), beyond which magnetic reconnection is triggered, releasing a portion of the stored free energy. In panel (b), the dashed black line denotes a minimum energy state (a 'floor'), suggesting that each reconnection event drives the system toward complete energy release and a common post-event state.
• Figure 2: 2D histograms of nanoflare energies and delays for events identified using Method B. The left (right) panels correspond to energies from the previous (next) event, respectively. The lower panels provide close-ups of the upper panels.
• Figure 3: Statistical correlation results obtained via bootstrapping of the original nanoflare sample identified using Method B. The first four histograms correspond to the $(\tau_D, E_{previous})$ data set, and the later four to $(\tau_D, E_{next})$. Blue and red histograms represent results from the $t_w$ and Spearman rank correlation tests, respectively. Left panels show the histograms of the probability of uncorrelation (p-values), while right panels display the most probable values of $\alpha$. A substantial fraction of the sampled subsets suggest the presence of weak or no statistical correlation. The extremely low values of $\alpha$ further support the absence of a strong statistical relationship between $\tau_D$ and either $E_{\text{previous}}$ or $E_{\text{next}}$.
• Figure 4: The figure illustrates the spread in nanoflare delays in the original dataset obtained from Method B. The bottom panel shows scatter plots of nanoflare energies versus delays. The middle and top panels display the mean and standard deviation of delays within each energy bin, with the bins defined by the red dashed lines in the bottom panel. The fact that the standard deviations (sigma delays) are comparable to the mean delays in each bin suggests that any correlation between $\tau_D$ and either $E_{previous}$ or $E_{next}$ is likely to be weak or statistically insignificant.
• Figure 5: Cartoon illustrating how the evolving local magnetic geometry disrupts any correlation between nanoflare energies and their delays.