Moving boundary approximation for curved streamer ionization fronts: Solvability analysis

TL;DR

This paper derives a curvature correction to the moving boundary approximation for negative streamer ionization fronts, revealing the need to replace ideal conductivity with a constant conductivity model, resulting in a Laplace equation for the potential.

Contribution

It introduces an analytical curvature correction to the streamer front approximation, involving a solvability analysis with unconventional features, and proposes a modified conductivity model.

Findings

Curvature affects the electric field and conductivity relation.

Ideal conductivity approximation is insufficient; a constant conductivity model is better.

The analysis leads to a Muskat-type problem for the streamer interior.

Abstract

The minimal density model for negative streamer ionization fronts is investigated. An earlier moving boundary approximation for this model consisted of a "kinetic undercooling" type boundary condition in a Laplacian growth problem of Hele-Shaw type. Here we derive a curvature correction to the moving boundary approximation that resembles surface tension. The calculation is based on solvability analysis with unconventional features, namely, there are three relevant zero modes of the adjoint operator, one of them diverging; furthermore, the inner/outer matching ahead of the front has to be performed on a line rather than on an extended region; and the whole calculation can be performed analytically. The analysis reveals a relation between the fields ahead and behind a slowly evolving curved front, the curvature and the generated conductivity. This relation forces us to give up the ideal conductivity approximation, and we suggest to replace it by a constant conductivity approximation. This implies that the electric potential in the streamer interior is no longer constant but solves a Laplace equation; this leads to a Muskat-type problem.