The study, carried out by Liza Hovhannisyan of the A. I. Alikhanyan National Laboratory in Yerevan, Armenia, uses the CORSIKA Monte Carlo code, a well-established tool for simulating cosmic-ray air showers that includes the effects of atmospheric electric fields on particle transport. The simulations model a vertically oriented, uniform electric-field layer 2000 meters thick above the detector level, subdivided into twenty 100-meter slices so that the vertical growth of the electron population can be tracked in detail. Four high-altitude observational sites with distinct atmospheric densities were chosen, all of which have recorded thunderstorm-related particle enhancements: Aragats in Armenia at roughly 3000 meters above sea level, Nor Amberd at about 2000 meters, Lomnický Štít in the High Tatras at approximately 2600 meters, and the LHAASO site in China at around 4400 meters. Across the four configurations, the simulated field regions span altitudes of roughly 4000 to 6400 meters, where air densities range from 0.66 to 0.89 kilograms per cubic meter.

The electric fields applied in each configuration were tuned to the local runaway threshold, which scales with the normalized air density. Because thinner air at high altitude lowers the threshold field, the same absolute field strength means something quite different at LHAASO than it does at Nor Amberd. The simulations therefore used field ranges of 180 to 210 kilovolts per meter at LHAASO, 200 to 230 at Aragats, 210 to 240 at Lomnický Štít, and 230 to 260 at Nor Amberd, varied in steps of 10 kilovolts per meter. Each simulation injected a single seed electron with an initial energy spectrum drawn from the EXPACS model, following a power law between 1 and 300 megaelectronvolts, and between 500 and 1000 events were run for each field configuration to ensure statistical stability. Electrons and gamma rays were tracked until their energies fell below 0.05 megaelectronvolts, with particle numbers recorded at every layer boundary.

Within the supercritical region where the applied field exceeds the local runaway threshold, the number of runaway electrons grows approximately exponentially with distance, and the characteristic e-folding length of that growth is the central parameter of avalanche theory. A widely used empirical relation, introduced by J. R. Dwyer in 2003, links this avalanche length to the electric-field excess above the threshold through a single proportionality coefficient, with Dwyer reporting a value of about 7.2 times ten to the third kilovolts. When Hovhannisyan fitted this standard relation separately to the effective avalanche lengths extracted from the vertical electron profiles at each station, the results told a revealing story. The station-specific coefficients ranged from 8.6 to 11.2 times ten to the third kilovolts, all larger than the original value, and the spread between stations persisted even after fit-derived uncertainties were taken into account. A single global fit across all sixteen simulation points yielded a best-fit coefficient of about 9.58 times ten to the third kilovolts, but with a coefficient of determination of only about 0.90, leaving noticeable inter-station scatter.

The remedy was to let the coefficient itself depend on air density. By introducing a generalized relation with an additional density-dependent term, in which the avalanche length is multiplied by the normalized air density raised to a fitted exponent, the joint fit across all four stations produced a coefficient of about 6.39 times ten to the third kilovolts and an exponent of minus 0.923, with an uncertainty of plus or minus 0.083. That exponent, statistically indistinguishable from minus one, indicates an approximately inverse dependence on atmospheric density beyond what is already captured by the density-scaled runaway threshold. The improvement in descriptive power was striking: the coefficient of determination rose from roughly 0.90 to roughly 0.99, and plots comparing simulated and predicted avalanche lengths showed much better agreement, particularly at the larger avalanche lengths where the standard formulation deviated most from a one-to-one relation.

The study then turned to a second, complementary stage of avalanche evolution: what happens to the particles after they leave the accelerating field. Additional simulations for the Aragats configuration placed the detector level 25, 50, 100, and 200 meters below the lower boundary of the electric field, keeping all other parameters fixed. The results revealed sharply divergent transport behavior between the two particle species. The electron spectrum was progressively attenuated with increasing field-free distance, with a pronounced suppression of the high-energy component, and the electron population decreased rapidly and monotonically. Gamma rays, by contrast, persisted over larger distances, attenuated more gradually through stochastic interactions such as Compton scattering, photoelectric absorption, and pair production. At a separation of 100 meters, the electron component was already strongly depleted while a substantial gamma-ray component remained, making that distance a natural calibration point.

This asymmetry matters for a phenomenological quantity called the free path distance, introduced in earlier work to characterize how far energetic particles can propagate beyond the electric-field region and still reach ground-based detectors. Because the maximum electron energy at the field boundary cannot be measured directly, the formulation estimates it from the maximum gamma-ray energy at the detector through an empirical electron-gamma energy coefficient. For the 100-meter calibration separation, the simulated gamma-ray spectra changed little over the propagation distance, with characteristic maximum energies of about 29 to 31 megaelectronvolts at both levels, validating the use of the detector value as a proxy. Electron spectra, however, shifted markedly toward lower energies: at Aragats, the characteristic maximum dropped from roughly 38 megaelectronvolts at the field exit to around 16 megaelectronvolts after 100 meters of propagation through air.

To quantify the electron-gamma coefficient, the study employed a bootstrap-based statistical procedure with 2000 Poisson resampling realizations of the energy histograms, applying a criterion that defines the characteristic maximum energy as the highest energy with at least three consecutive bins containing more than five counts, which suppresses sensitivity to isolated outliers. The station-specific median coefficients clustered tightly, from about 1.31 at Aragats to 1.46 at Lomnický Štít, and pooling all stations yielded a median of about 1.38, in close agreement with an independent global least-squares optimization that gave 1.375. The adopted value of 1.37 replaces the earlier empirical coefficient of 1.2, which the bootstrap analysis showed systematically underestimates the free path distance at all four stations. The second parameter, the altitude-dependent electron energy-loss rate, was obtained from the NIST ESTAR stopping-power database scaled by local air density, yielding values between roughly 0.17 and 0.23 megaelectronvolts per meter across the sites. Applying the recalibrated formulation reproduced free path distances of 85 to 105 meters, consistent with the 100-meter calibration scale, and an independent test at a 50-meter separation reconstructed a median of 69.9 meters, correctly identifying the short-distance regime.

The practical payoff of this work lies in the interpretation of observations. Mountain observatories such as Aragats, Lomnický Štít, and LHAASO routinely detect particle bursts during thunderstorms, and inferring the structure and strength of the electric fields inside thunderclouds from those detections depends on having reliable models of both avalanche growth and post-field transport. By showing that a single density-independent coefficient leaves systematic residuals across sites spanning a substantial range of atmospheric densities, and by demonstrating that a simple power-law correction removes nearly all of that scatter, the study provides a unified scaling that can be applied across different altitudes and storm conditions. The recalibrated free path distance formulation likewise gives experimentalists a firmer bridge between the gamma-ray spectra they measure and the electron populations that produced them at the edge of the accelerating field.

The author notes that the results are derived for idealized, uniform electric-field configurations and should not be generalized to arbitrary thunderstorm field structures without further validation, a caveat that reflects the persistent difficulty of measuring the spatial and temporal structure of fields inside storm clouds. Even so, the near-perfect fit achieved by the density-dependent formulation, together with the stability of the recalibrated transport coefficients across four very different atmospheric environments, marks a meaningful step toward a common physical framework for the growing family of high-energy atmospheric phenomena. All simulation inputs, analysis scripts, and processed outputs are publicly archived, allowing other groups to reproduce the workflow and extend the calibration to new observational sites and more realistic field geometries.

Subject of Research: Density-dependent scaling and transport of relativistic runaway electron avalanches in thunderstorm electric fields

Article Title: Relativistic runaway electron avalanches: unified density-dependent scaling and transport

Article References: Hovhannisyan, L. (2026). Relativistic runaway electron avalanches: unified density-dependent scaling and transport. Geoscientific Model Development, 19(19), 9289-9300. https://doi.org/10.5194/gmd-19-9289-2026

Image Credits: AI Generated

DOI: 10.5194/gmd-19-9289-2026

Keywords: relativistic runaway electron avalanches, thunderstorm electric fields, cosmic rays, gamma-ray glows, terrestrial gamma-ray flashes, CORSIKA simulations, air density, avalanche length, free path distance, thunderstorm ground enhancements, Monte Carlo modeling, high-altitude observatories