Lu and Hamilton model for solar flares over a rewiring complex network

TL;DR

This paper extends the Lu-Hamilton SOC framework for solar flares by embedding the grid of magnetic flux tubes into a dynamically rewiring complex network, where topological neighbors govern energy redistribution. By introducing a controllable driving-rewiring balance, it demonstrates a transition in the dissipated-energy distribution from scale-free $P(E) \sim E^{-\alpha_E}$ to exponential $P(E) \sim e^{-\beta_E E}$ as rewiring dominates, with a crossover near $p(d) \approx 0.55$ and $\alpha_E$ in the $1.07$–$1.39$ range under mixed dynamics. The nonlocal topology induced by rewiring accelerates avalanche production, alters waiting-time implications, and challenges the ubiquity of scale-free flare statistics, offering a framework to explore solar-cycle-dependent flare behavior and larger interconnected active regions. These results highlight the importance of topology dynamics in magnetic energy release and point to future work on waiting-time statistics and system-spanning interactions.

Abstract

We present a modified Lu \& Hamilton-type model where the neighborhood relations are replaced by topological connections, which can be dynamically altered. The model represents each grid node as a flux tube, as in the classic model, but with connections evolving to capture the complex effects of magnetic reconnection. Through this framework, we analyze how the dissipated energy distribution changes, particularly focusing on the power-law exponent $α_E$, which decreases with respect to the original model due to rewiring effects. When the system is dominated by rewiring, it presents an exponential distribution exponent $β_E$, showing a faster decay of dissipated energy than in the original model. This leads to microflare-dominated dynamics at short timescales, causing the system to lose the scale-free behavior observed in both the original model (Lu \& Hamilton 1991) and in configurations where energy release is primarily driven by forcing rather than rewiring. Our results reveal a clear transition from power-law to exponential regimes as the rewiring probability increases, fundamentally altering the energy distribution characteristics of the system. In contrast, when considering topological neighbors instead of local ones, the model's dynamics become intrinsically nonlocal. This leads to scaling exponents comparable to those reported in other nonlocal dynamical systems.

Figures (7)

• Figure 1: 2D Cartesian model representing the original LH91 model. Each grid point is associated with a field $B_{i,j}$, and with four neighbors in vertical and horizontal directions as shown.
• Figure 2: Diagram of the system where (a) is the original lattice before the rewiring processes. In (b) a random node is selected, and its neighbors are identified; the selected node is red, yellow nodes are neighbors which cannot be selected because they are at the boundary, green nodes are neighbors which can be selected. In (c), one of the green nodes has been selected to move its connection to another, randomly selected node; the magenta node is the node which has just lost a connection, and the cyan node is the node who has gained that connection. In (d), the rewiring has been carried out, conserving the number of neighbors, highlighting the modified nodes, and (e) shows the final, rewired state.
• Figure 3: (a) LH total energy at time $t$. (b) Total energy for the whole system at time $t$, where the colors represent the different relations in percentage between driving-rewiring as a mechanism for the avalanche, i.e. light green: 99.9-0.1, purple: 99-1, black: 90-10, red: 80-20, orange: 70-30, yellow: 60-40, blue: 50-50, cyan: 40-60, magenta: 30-70, green: 20-80, brown: 10-90. Using $10^7$ iterations and with $N=64$, $Z_c=1$.
• Figure 4: Dissipated energy from a 80-20 system. The red region in the lower panel shows a particular avalanche with total energy dissipated $E$, peak energy $P$, and duration $T$.
• Figure 5: Spatial distribution of one avalanche: the top row shows the original model without rewiring, and the bottom row corresponds to an 80-20 rewiring configuration. Panels (a) and (e) depict the beginning of the avalanche; (b) and (f) show the system at time $t_3$; (c) and (g) show the state at the peak of the avalanche; and (d) and (h) show the full extent of the avalanche-affected area, with impact frequency visualized on a color scale: white (0 times affected), black (1 time), and a rainbow gradient for larger values.