Streamers are the first stage of sparks and lightning; they grow due to a strongly enhanced electric field at their tips; this field is created by a thin curved space charge layer. These multiple scales are already challenging when the electrons are approximated by densities. However, electron density fluctuations in the leading edge of the front and non-thermal stretched tails of the electron energy distribution (as a cause of X-ray emissions) require a particle model to follow the electron motion. As super-particle methods create wrong statistics and numerical artifacts, modeling the individual electron dynamics in streamers is limited to early stages where the total electron number still is limited. The method of choice is a hybrid computation in space where individual electrons are followed in the region of high electric field and low density while the bulk of the electrons is approximated by densities (or fluids). We here develop the hybrid coupling for planar fronts. First, to obtain a consistent flux at the interface between particle and fluid model in the hybrid computation, the widely used classical fluid model is replaced by an extended fluid model. Then the coupling algorithm and the numerical implementation of the spatially hybrid model are presented in detail, in particular, the position of the model interface and the construction of the buffer region. The method carries generic features of pulled fronts that can be applied to similar problems like large deviations in the leading edge of population fronts etc.
Deep Dive into Spatially hybrid computations for streamer discharges with generic features of pulled fronts: I. Planar fronts.
Streamers are the first stage of sparks and lightning; they grow due to a strongly enhanced electric field at their tips; this field is created by a thin curved space charge layer. These multiple scales are already challenging when the electrons are approximated by densities. However, electron density fluctuations in the leading edge of the front and non-thermal stretched tails of the electron energy distribution (as a cause of X-ray emissions) require a particle model to follow the electron motion. As super-particle methods create wrong statistics and numerical artifacts, modeling the individual electron dynamics in streamers is limited to early stages where the total electron number still is limited. The method of choice is a hybrid computation in space where individual electrons are followed in the region of high electric field and low density while the bulk of the electrons is approximated by densities (or fluids). We here develop the hybrid coupling for planar fronts. First, to
Terrestrial Gamma-Ray Flashes were first observed accidentally in 1994 [1]; later they were found to be correlated with thunderstorm activity. In 2001, energetic radiation was also observed from normal cloud-to-ground lightning [2] and later from rocket triggered lightning [3]. Meanwhile X-ray bursts were also found in the laboratory near long sparks [4,5,6,7]. Such flashes and bursts are likely to be produced by Bremsstrahlung of very energetic electrons (so called run-away electrons).
One possible source of highly energetic electrons are the streamer discharges that pave the way of sparks and lightning. Streamers are ionized plasma channels that grow into a non-ionized medium due to the self-enhancement of the electric field at their tips. In this high field region, the electron energy distribution is very non-thermal and can have a long tail at high energies; it could act so much as a self-organized local electron accelerator that it is a possible source of X-rays [8,9,10].
The single electron dynamics in streamers can be studied by a particle model, which models the streamer dynamics at the lowest molecular level. It follows the free flight of single electrons in the local field between abundant neutral molecules and includes the elastic, exciting and ionizing collisions of electrons with the molecules in a stochastic Monte Carlo procedure. However, the increasing number of electrons in a growing streamer channel leads to an increasing demand of computational power and storage and excludes the simulation of all electrons in a full long streamer on present day’s workstations.
The generation of highly energetic run-away electrons from streamers therefore has only been modeled in particle models with simplifying assumptions or tricks. In [8] a Monte Carlo simulation treated planar fronts, and 3D aspects were modeled by a simplified electric field profile in the forward direction; in [10] a 3D axis-symmetrical streamer was simulated with super-particles of high weight where many real electrons are replaced by one super-particle. Both simulations indicate that electrons can be accelerated to very high energies in the high field zone at the tip of streamers. However, the 1D simulation [8] does not include the intricate 3D multiscale spatial structure of the streamer head properly [11,12,13], while the super-particle approach [10] has been shown [14] to lead rapidly to numerical artifacts already shortly after a streamer emerges from an ionization avalanche; it will not represent the physically important tail of the electron energy distribution correctly.
Realizing that the number of electrons in the high field zone at the streamer tip is limited, and that only these electrons have a chance to gain high energies, a natural idea is to include only those electrons into a particle model while all others are treated in density or fluid approximation, as the fluid approximation is computationally much more efficient and has been widely used in streamer modeling, see e.g., [11,12,15,16,17,18,19,20,21,22,23,24,25,26,27]. Such a spatially hybrid approach should allow us to follow single electrons in the relevant region while avoiding the introduction of super-particles. We here present essential steps of its implementation.
The concept of a spatial coupling of density and fluid model is shown in Fig. 1. The natural structure of the streamer consists of an ionized channel and an ionization front. In the channel, the electrons are dense enough to be approximated as continuous densities, and the field is too low to accelerate them to high energy. In the ionization front the electron density decreases Figure 1: (Color online) The spatially hybrid scheme is shown in a one dimensional section of the streamer. The solid line shows the electron density n e as a function of the spatial coordinate z. The particle model follows the leading part of the ionization front and the fluid model is applied to the rest. The questions treated in the present paper are: Which fluid model approximates the particle model in the best manner? Where should the model interface be placed and how should it be structured? The figure is reproduced from [28].
rapidly while the field increases, and the electrons gain high energies far from equilibrium. The hybrid model should apply the particle model in the most dynamic and exotic region with relatively few electrons and the fluid model in the remaining region that contains the vast majority of the electrons. The hybrid model is not only suitable for studying possible run-away electrons from streamer heads, but also for studying the role of electron density fluctuations on streamer branching, and for studying the gas chemistry inside the streamer and the electron energy distribution at the streamer tip in general.
To implement such a hybrid computation, we have set first steps in [9,28]. In [9] we studied the Monte Carlo discharge model in detail and compared it with its fluid approximation; we
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