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

Abstract

Streamer ionization fronts are pulled fronts propagating into a linearly unstable state; the spatial decay of the initial condition of a planar front selects dynamically one specific long time attractor out of a continuous family. A stability analysis for perturbations in the transverse direction has to take these features into account. In this paper we show how to apply the Evans function in a weighted space for this stability analysis. Zeros of the Evans function indicate the intersection of the stable and unstable manifolds; they are used to determine the eigenvalues. Within this Evans function framework, a numerical dynamical systems method for the calculation of the dispersion relation as an eigenvalue problem is defined and dispersion curves for different values of the electron diffusion constant and of the electric field ahead of the front are derived. Numerical solutions of the initial value problem confirm the eigenvalue calculations. The numerical work is complemented with an analysis of the Evans function leading to analytical expressions for the dispersion relation in the limit of small and large wave numbers. The paper concludes with a fit formula for intermediate wave numbers. This empirical fit supports the conjecture that the smallest unstable wave length of the Laplacian instability is proportional to the diffusion length that characterizes the leading edge of the pulled ionization front.

Comment on “Mechanism of Branching in Negative Ionization Fronts”

“…A different, extensive analysis with different results can be found in [5]. Below we show that the expansion and calculation in [1] are inconsistent, that the result contradicts a known analytical asymptote, and that it does not fit the cross-checked numerical results presented in [5]. Furthermore, we find in [5] that the most unstable wavelength does not scale as D 1=3 e as claimed in [1], but as D…”

“…This is attempted in [1] in the limit of large field jE 1 j ahead of the front. A different, extensive analysis with different results can be found in [5]. Below we show that the expansion and calculation in [1] are inconsistent, that the result contradicts a known analytical asymptote, and that it does not fit the cross-checked numerical results presented in [5].…”

“…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 …”

Underlying mechanisms of transient luminous events: a review

“…According to the new mode of thought, the breakdown field may arise at the front of electron density/screeningionization wave originating from the lower ionosphere after +CG and then propagating downward in background plasma (Luque and Ebert, 2009, 2010. The computer simulations have shown that this wave can be compressed in transverse direction (Derks et al, 2008) and then be transformed into downward-propagating streamer head. This interesting result may be indicative of any kind of plasma instability and thus must be supported by an analytical condition that has not been established yet.…”

Grassmannian spectral shooting

“…The numerical construction of pure-point spectra is important in determining the linear stability of coherent structures. Examples of such structures are: ground and higher excited states of molecules in quantum chemistry (Johnson [56], Hutson [51], Gray and Manopoulous [38], Manopoulous and Gray [69], Chou and Wyatt [23,24], Ledoux [64], Ledoux, Van Daele and Vanden Berghe [65], Ixaru [55]); nonlinear travelling fronts in reaction-diffusion such as autocatalysis or combustion (Billingham and Needham [9], Metcalf, Merkin and Scott [72], Doelman, Gardner and Kaper [31], Terman [96], Gubernov, Mercer, Sidhu and Weber [41]); nerve impulses (Alexander, Gardner and Jones [2]); neural waves (Coombes and Owen [25]); solitary waves or steady flows over compliant surfaces (Pego and Weinstein [82], Alexander and Sachs [3], Chang, Demekhin and Kopelevich [21], Kapitula and Sandstede [57], Bridges, Derks and Gottwald [14], Allen [4], Allen and Bridges [5]); laser pulses (Swinton and Elgin [95]); nonlinear waves along elastic rods (Lafortune and Lega [62]); ionization fronts (Derks, Ebert and Meulenbroek [28]) or spiral waves (Sandstede and Scheel [89]). …”