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

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 $2E_{\infty }$, 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.

7 pages, 6 figures

References in corpus (5)

• Photoionization in negative streamers: fast computations and two propagation modes
• The multiscale nature of streamers
• Numerical convergence of the branching time of negative streamers
• Construction and test of a moving boundary model for negative streamer discharges
• Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling

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• Review of recent results on streamer discharges and discussion of their relevance for sprites and lightning
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• Interaction of streamers in air and other oxygen-nitrogen mixtures
• Saffman-Taylor fingers with kinetic undercooling
• A moving boundary problem motivated by electric breakdown: I. Spectrum of linear perturbations
• Moving boundary approximation for curved streamer ionization fronts: Solvability analysis
• Moving boundary approximation for curved streamer ionization fronts: Numerical tests
• Macroscopic parameterization of positive streamer heads in air