Comment on “Mechanism of Branching in Negative Ionization Fronts”

Abstract

When the fingers of discharge streamers emerge from a planar ionization front due to a Laplacian instability, their initial spacing is determined by the band of unstable transversal Fourier perturbations and generically dominated by the fastest growing modes. The Letter [1] therefore aims to calculate the temporal growth rate sðkÞ of modes with wave number k, when the electric field far ahead of the ionization front is E 1 . In earlier work [2][3][4], sðkÞ was determined in a pure reaction-drift model for the free electrons, i.e., in the limit of vanishing electron diffusion D e ¼ 0. For negative streamers in pure gases like nitrogen or argon, electron diffusion D e > 0 should be included into the discharge model. This is attempted in [1] (8) reproduces the diffusive layer for large jE 1 j, but the nonlinear term is incomplete. Then the dispersion relation is calculated by the expansion (11)-(13) about the planar ionization front. Here the expansion of the electron density n e starts in order 2 (where is the small expansion parameter), while the expansions of ion density n p and field E start in order . The absence of order in the expansion of n e is unexpected, not explained, and in contradiction with the calculation for D e ¼ 0 in [4].Jumping to the result of [1], the dispersion relation in Eq. (21) Furthermore, in [5], dispersion curves sðkÞ for a range of fields E 1 and diffusion constants D e are derived as an eigenvalue problem for s; they are plotted in Fig. 1. In one case, the curve is confirmed by numerical solutions of an initial value problem; the curves are also consistent with the analytical small k asymptote. The results for positive s are conveniently fitted as sðkÞ ¼ c (23)] for the spacing of emergent streamers does not hold; rather our physical arguments and the numerical data in [5] suggest that the fastest growing mode is

Multiple scales in streamer discharges, with an emphasis on moving boundary approximations

“…A (heuristic) analysis for s(k) was given in [71,72]. The asymptotes (8) confirm these results for k 1, and correct them for k 1, see also [73]. On the other hand, the analytical asymptote (8) for k 1 fits the numerical results for the dispersion relation well, while the asymptote for k 1 parametrizes the dispersion relation s(k) deep in its negative range, but not close to the band of wavenumbers of positive growth rate [64].…”

Laplacian Instability of Planar Streamer Ionization Fronts—An Example of Pulled Front Analysis

“…In principle, such an approach combined with proper matched asymptotic expansions could work. However, the calculation in [5] was intrinsically inconsistent [18], disagrees with our asymptotic limit for small k and therefore fits the numerical results even worse than formula (6.1), cf. Fig.…”

Transverse instability of a plane front of fast impact ionization waves

“…The solution of this problem was discussed in numerous works [4][5][6][7][8][9][10][11][12]. The most general analytical results were obtained in [12], when the stability of steady state plane impact ionization waves in reverse biased semiconductor p + -n-n + structures was ana lyzed.…”