Electron density fluctuations accelerate the branching of positive streamer discharges in air

A. Luque and U. Ebert

Phys. Rev. E 84, 046411 – Published 25 October 2011

Abstract

Branching is an essential element of streamer discharge dynamics. We review the current state of theoretical understanding and recall that branching requires a finite perturbation. We argue that, in current laboratory experiments in ambient or artificial air, these perturbations can only be inherited from the initial state, or they can be due to intrinsic electron-density fluctuations owing to the discreteness of electrons. We incorporate these electron-density fluctuations into fully three-dimensional simulations of a positive streamer in air at standard temperature and pressure. We derive a quantitative estimate for the ratio of branching length to streamer diameter that agrees within a factor of 2 with experimental measurements. As branching without this noise would occur considerably later, if at all, we conclude that the intrinsic stochastic particle noise triggers branching of positive streamers in air at atmospheric pressure.

Received 8 February 2011

DOI:https://doi.org/10.1103/PhysRevE.84.046411

Authors & Affiliations

A. Luque¹ and U. Ebert^2,3

¹Instituto de Astrofísica de Andalucía (IAA), CSIC, P.O. Box 3004, 18080 Granada, Spain
²Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
³Department of Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

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Vol. 84, Iss. 4 — October 2011

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Images

Figure 1
Scheme of an elementary time step in the lattice model. Each electron is represented here by a dot; note, however, that we do not keep track of every individual particle, but only of the number of particles in each cell. During the time step from $t$ to $t + Δ t$ we move $u_{i j}$ electrons from cell $C_{i}$ to $C_{j}$ . Meanwhile, $r_{i}$ electrons arise in $C_{i}$ from impact ionization and photoionization.Reuse & Permissions
Figure 2
Snapshots of the cross section of the electron density in the $y = 0$ plane in a 2-mm gap between planar electrodes with a potential difference of 16 kV; i.e., the background field of 80 kV/cm is well above the breakdown value. We plot the densities of the simulation cells centered at $y = 0$ with coordinates $x, z$ : the volume of those cells is proportional to their separation from the axis, and therefore in this cross section the relative fluctuations are larger close to the $x = 0$ line. Photoionization creates multiple electron-ion pairs throughout the volume, and the electrons start avalanches. Therefore the electric breakdown extends over the complete volume and no actual streamer is initiated. A movie of this simulation is available in the supplementary material [56].Reuse & Permissions
Figure 3
Highest deviation of the absolute value of the electric field from its azimuthal average at three snapshots. The volume labeled A corresponds to the evolution of the streamer up to $t = 13.5 ns$ , with noise switched off. When we switch on noise, the streamer body (B) develops low-amplitude, small-scale fluctuations. The streamer surface close to the tip (C) has fluctuations with a higher amplitude and an autocorrelation length of about one tenth of a millimeter. The amplitude grows, but in this simulation it remains small compared with the field values of some hundreds of kilovolts per centimeter in that area.Reuse & Permissions
Figure 4
Evolution of $A$ as defined in the text. We show the evolution of three runs: in all runs the system was deterministic and cylindrically symmetrical up to a time of 13.5 ns. After this point we add stochasticity and allow deviations from cylindrical symmetry. In the curves marked as deterministic, we remove noise again at the time marked with the dots while keeping the asymmetric perturbations that have evolved up to that time. Within this limited time frame, $A$ seems go through a transient, fast growth phase until it settles into an exponential growth. The dashed line fits this second phase. For the completely stochastic run the best fit is $A = a e^{t / τ}$ with $a = 5 \cdot 10^{- 20}$ and $τ = 0.48 ns$ . Extrapolating, this predicts branching at $t \sim - τ \log a \approx 21 ns$ .Reuse & Permissions

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