This paper presents formulation of computationally efficient models of photoionization produced by non-thermal gas discharges in air based on three-group Eddington and improved Eddington (SP 3 ) approximations to the radiative transfer equation, and on effective representation of the classic integral model for photoionization in air developed by Zheleznyak et al (1982) by a set of three Helmholtz differential equations. The reported formulations represent extensions of ideas advanced recently by Ségur et al (2006) and Luque et al (2007), and allow fast and accurate solution of photoionization problems at different air pressures for the range 0.1 < p O 2 R < 150 Torr cm, where p O 2 is the partial pressure of molecular oxygen in air in units of Torr (p O 2 = 150 Torr at atmospheric pressure) and R in cm is an effective geometrical size of the physical system of interest. The presented formulations can be extended to other gases and gas mixtures subject to availability of related emission, absorption and photoionization coefficients. The validity of the developed models is demonstrated by performing direct comparisons of the results from these models and results obtained from the classic integral model. Specific validation comparisons are presented for a set of artificial sources of photoionizing radiation with different Gaussian dimensions, and for a realistic problem involving development of a double-headed streamer at ground pressure. The reported results demonstrate the importance of accurate definition of the boundary conditions for the photoionization production rate for the solution of second order partial differential equations involved in the Eddington, SP 3 and the Helmholtz formulations. The specific algorithms derived from the classic photoionization model of Zheleznyak et al (1982), allowing accurate calculations of boundary conditions for differential equations involved in all three new models described in this paper, are presented. It is noted that the accurate formulation of boundary conditions represents an important task needed for a successful extension of the proposed formulations to two-and three-dimensional physical systems with obstacles of complex geometry (i.e. electrodes, dust particles, aerosols, etc), which are opaque for the photoionizing UV photons.

[1] In the present paper, we demonstrate that the exponential expansion of streamers propagating in fields higher than the critical fields for stable propagation of streamers of a given polarity leads to the exponential growth of electric potential differences in streamer heads. These electric potential differences are directly related to the energy that thermal runaway electrons can gain once created. Using full energy range relativistic Monte Carlo simulations, we show that the exponential growth of potential differences in streamers gives rise to the production of runaway electrons with energies as high as ∼100 keV, with most of electrons residing in energy range around several tens of keVs. We apply these concepts in the case of lightning stepped leaders during the stage of negative corona flash. The computation of electric field produced by stepped leaders demonstrates for the first time that those energetic electrons are capable of further acceleration up to the MeV energies. Moreover, the flux of runaway electrons produced by streamers suggests that stepped leaders produce a considerable number of energetic electrons, which is in agreement with the number of energetic photons observed from satellites in terrestrial gamma ray flashes (TGFs). The results suggest that previously proposed process of relativistic runaway electron avalanche is difficult to sustain in the low-electric fields observed in thunderclouds and is generally not needed for explanation of TGFs. The present work also gives insights on relations between physical properties of energetic electrons produced in streamers and the internal electrical properties of streamer discharges, which can further help development and interpretation of X-ray diagnostics of these discharges.Citation: Celestin, S., and V. P. Pasko (2011), Energy and fluxes of thermal runaway electrons produced by exponential growth of streamers during the stepping of lightning leaders and in transient luminous events,

[1] Terrestrial gamma-ray flashes (TGFs) are energetic photon bursts observed from satellites and associated with lightning activity. Comparison between calculations based on the model of relativistic runaway electron avalanches (RREA) in large-scale weak electric field in thunderstorms and satellite measurements usually shows that the photon spectrum is consistent with source altitudes around 15 km. However, recent observations have located intra-cloud lightning (IC) discharges responsible for TGFs much deeper in the atmosphere (at altitudes $10 km). In the present work, we show that the TGF spectrum as produced by acceleration of electrons in the strong electric field of stepping IC leaders is consistent with the lower altitudes recently discovered. This study reconciles observations and measurements by setting new altitudes for the TGF sources based on mechanism of direct acceleration of electrons in the lightning leader field. Moreover, the photon source beaming geometry is consistently determined from the geometry of electric field lines produced by the lightning leader. Citation: Xu, W., S. Celestin, and V. P. Pasko (2012), Source altitudes of terrestrial gamma-ray flashes produced by lightning leaders, Geophys.

[1] In the sprite halo events produced by cloud-to-ground lightning discharges, the spatial offsets, long delays, and polarity asymmetry related to the inception of sprite streamers are yet to be explained consistently with observations. In the present work, we use a two-dimensional model and a high-resolution one-dimensional plasma fluid model accounting for electron impact ionization, dissociative attachment, and photoionization processes to simulate halo events. In order to monitor the inception of sprite streamers, that cannot be modeled with present computer resources in the framework of fluid models, we use an improved avalanche-to-streamer transition criterion and investigate the response of the lower ionosphere to the charge moment changes induced by lightning discharges as a system of avalanches. On the basis of simulation results, we suggest a new mechanism for the inception of sprite streamers, explaining specifically how they can be triggered spatially outside of and temporally separated from the main sprite halo. The inception of sprite streamers is demonstrated to depend strongly on the charge moment change and the ambient electron density profile, which together determine the size of the streamer initiation region. We show that sprite streamers are mostly triggered by positive cloud-to-ground lightning discharges mainly because of mechanisms related to direction of propagation of electron avalanches. Moreover, the triggering of long-delayed sprites is demonstrated to be a unique property of halos produced by positive cloud-to-ground lightning discharges due to the formation of a long-lasting high-field region. Lightning continuing current can enlarge this high-field region and pull it down to lower altitudes where electron density is lower, extending the streamer initiation region.Citation: Qin, J., S. Celestin, and V. P. Pasko (2011), On the inception of streamers from sprite halo events produced by lightning discharges with positive and negative polarity,