Laplacian instability of planar streamer ionization fronts - an example of pulled front analysis

TL;DR

This paper develops a new framework for analyzing the transverse stability of planar streamer ionization fronts, combining numerical and analytical methods to understand the Laplacian instability and its dependence on physical parameters.

Contribution

It introduces a novel dynamical systems approach for transverse stability analysis of pulled fronts, including a numerical method and analytical expressions for dispersion relations.

Findings

Dispersion curves derived for various diffusion constants and electric fields.

Numerical solutions confirm the eigenvalue-based stability analysis.

The smallest unstable wavelength is proportional to the diffusion length.

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 transverse stability analysis has to take these features into account. In this paper we introduce a framework for this transverse stability analysis, involving stable and unstable manifolds in a weighted space. Within this 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 analytical expressions for the dispersion relation in the limit of small and large wave numbers and 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.