Saffman-Taylor streamers: mutual finger interaction in an electric breakdown

TL;DR

This paper investigates the interaction of multiple streamers in an electric field using PDE models, revealing that their shapes resemble Saffman-Taylor fingers and providing insights into streamer behavior and finger selection in moving boundary problems.

Contribution

It introduces a PDE-based model for interacting streamers, connects it to Hele-Shaw flow theory, and shows the shape similarity to Saffman-Taylor fingers, advancing understanding of streamer interactions.

Findings

Streamers do not branch when closely spaced and approach uniform translation.

Enhanced electric field at streamer tips is approximately twice the background field.

Streamer shapes match the analytical Saffman-Taylor finger solution.

Abstract

Bunches of streamers form the early stages of sparks and lightning but theory presently concentrates on single streamers or on coarse approximations of whole breakdown trees. Here a periodic array of interacting streamer discharges in a strong homogeneous electric field is studied in PDE approximation in two dimensions. If the period of the streamer array is small enough, the streamers do not branch, but approach uniform translation. When the streamers are close to the branching regime, the enhanced field at the tip of the streamer is close to $2 E_{\infty}$, where $E_{\infty}$ is the homogeneous field applied between the electrodes. We discuss a moving boundary approximation to the set of PDEs. This moving boundary model turns out to be essentially the same as the one for two-fluid Hele-Shaw flows. In two dimensions, this model possesses a known analytical solution. The shape of the 2D interacting streamers in uniform motion obtained from the PDE simulations is actually well fitted by the analytically known ``selected Saffman-Taylor finger''. This finding helps to understand streamer interactions and raises new questions on the general theory of finger selection in moving boundary problems.