Pattern formation in electric discharges

Electric discharges with low current densities can be described by a minimal model of reaction, drift and diffusion of two charged species coupled nonlinearly to an externally imposed electric field. The intrinsic ionization-field-coupling of the model leads to pattern formation both in stationary or periodically driven barrier discharges and in transient streamer discharges. In particular, we discuss negative streamer discharge fingers extending into a non-ionized area. They belong to the class of Laplacian growth phenomena. For quantitative predictions, a systematic derivation of the corrections to the velocity of a moving boundary due to curvature or relaxation effects is still missing. This is because negative streamer fronts are so-called pulled fronts, for which standard approximation schemes fail. We present new approaches and results.