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

The story begins with the physics of cosmic rays themselves. Most primary cosmic rays are protons, electrically charged particles whose trajectories are bent by Earth’s magnetic field before they ever reach the atmosphere. When they arrive, they collide with atomic nuclei in the air and generate cascades of secondary particles. Among these secondaries are muons, short-lived charged particles produced at high energies that travel at nearly the speed of light and penetrate deep into the atmosphere. As Sunil Gupta of the Tata Institute of Fundamental Research (TIFR) in Mumbai, one of the authors of the study, explains, the cosmic rays detected at ground level consist predominantly of these highly penetrating muons. The GRAPES-3 facility includes a 560-square-metre muon telescope that records around four billion muons every day, providing an extraordinarily dense stream of data.

Crucially, those muons are not evenly balanced in electric charge. Because the primary cosmic rays are mostly positively charged, positive muons are more abundant in the atmosphere than negative ones. Physicists quantify this imbalance with the muon charge ratio, the number of positive muons divided by the number of negative ones, which normally exceeds one. That small excess is what allows GRAPES-3 to do something it was never originally designed to do. Inside a thundercloud, electric charges become separated, producing intense electric fields and large potential differences. When muons cross these regions, the field acts on positive and negative particles in opposite ways, generally slowing positive muons while accelerating negative ones. If the two charges were present in equal numbers, these effects would cancel almost perfectly, leaving no detectable change in the total muon flux.

Hari Haran Balakrishnan of TIFR, first author of the new study, puts the point starkly: if nature provided equal numbers of positive and negative muons throughout the telescope’s field of view, the thunderstorm phenomenon simply could not be observed with the current setup. Because positive muons dominate, however, each thunderstorm leaves a small but measurable imprint on the total muon count, and from that imprint researchers can infer the electrical potential inside the cloud. Gupta describes muons as an ideal gift for these kinds of studies, likening them to an electric current flowing through the atmosphere. In an earlier analysis, the team reported a striking result: a potential of 1.3 gigavolts measured in a single thunderstorm, a figure that dwarfs anything obtained by conventional means.

The idea that thunderstorms might generate potentials on that scale is not new. As early as the 1920s, the physicist Charles Thomson Rees Wilson, who would later win the 1927 Nobel Prize in Physics, suggested that thunderclouds could reach such enormous voltages. Direct measurements, however, had previously topped out at only around 130 million volts, more than an order of magnitude lower. The difficulty is practical as much as conceptual: sampling the interior of a large, rapidly evolving storm generally requires instruments carried by aircraft or weather balloons into or near active weather, a hazardous and spatially limited approach. Muons invert the problem entirely. They pass through the atmosphere continuously, they can be monitored from the safety of the ground, and they sample the whole volume of the cloud along their paths.

But the earlier GRAPES-3 observations carried an unexpected puzzle alongside their headline result. Between April 2011 and December 2020, the telescope detected 487 thunderstorm events, and their distribution across the sky was strikingly lopsided. Fully 81.5 percent of the events appeared in the eastern part of the instrument’s field of view, while only 13.7 percent appeared in the west, a difference of almost six to one. The obvious suspicion was local weather: perhaps storms in this part of India genuinely favour the east. An independent array of instruments that monitors the atmospheric electric field said otherwise. Actual thunderstorms, measured directly, showed no comparable preference for the eastern sky, which meant the asymmetry had to originate in the detector rather than in the atmosphere.

The answer lies in an effect that atmospheric and cosmic-ray physicists have known about for nearly a century: the east-west effect. Because cosmic-ray protons are charged, Earth’s magnetic field filters their arrival directions. Lower-energy positive particles arriving from the east are screened out more strongly than those arriving from the west, so the mix of cosmic rays, and therefore of secondary muons, depends on direction. This directional threshold, known as the geomagnetic cutoff, varies significantly across GRAPES-3’s field of view. The consequence is that the muon charge ratio, the very quantity that makes thunderstorm sensing possible in the first place, is itself direction-dependent, and the telescope is therefore more sensitive to thunderstorms in some parts of the sky than in others.

To test whether this could account for the observed lopsidedness, the researchers turned to computer simulations that model the production and propagation of muons through the atmosphere. The simulations showed that the balance between positive and negative muons changes markedly with direction: the muon charge ratio is about 1.37 in the east but only 1.14 in the west. That difference, they found, is more than enough to explain the event counts. As Gupta emphasizes, it is not that the voltage is different; thunderstorms to the east are not necessarily more electrically powerful than those to the west. Rather, the larger imbalance between positive and negative muons arriving from that direction makes the telescope more responsive to their electrical effects. The simulations confirmed the mechanism in both directions: when the direction-dependent charge ratio was included, a strong east-west asymmetry emerged, closely resembling the observations, and when the same ratio was imposed in both directions the asymmetry disappeared entirely.

The resolution of the puzzle matters well beyond bookkeeping for one detector. Understanding this directional sensitivity is essential if GRAPES-3 is to serve as a reliable, calibrated probe of thunderstorm electricity, because any inferred potential must account for how efficiently the instrument responds along each line of sight. With that correction in hand, the technique offers something no balloon campaign can match: continuous, ground-based monitoring of the largest electric fields in nature, drawn from a particle flux that arrives whether or not anyone is watching. The broader significance is considerable, since thunderstorm electrification is linked to lightning initiation, to terrestrial gamma-ray flashes, and to the still-uncertain role of storms in the global atmospheric electrical circuit. What began as an instrument for studying particles from the cosmos has ended up illuminating one of the most familiar and, remarkably, still one of the most mysterious phenomena on our own planet. The same muons that carry messages about distant astrophysical sources now carry, quite literally, the voltage of the storm overhead.

Subject of Research: Using cosmic-ray muons to probe electric fields and potentials inside thunderclouds

Article Title: From the cosmos to thunderstorms

Article References: From the cosmos to thunderstorms. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: cosmic rays, muons, thunderstorms, GRAPES-3, electric fields, geomagnetic field, muon charge ratio, atmospheric electricity, lightning, particle physics, TIFR, JCAP