A moving boundary model motivated by electric breakdown: II. Initial value problem

TL;DR

This paper investigates a Laplacian growth model with kinetic undercooling to understand the evolution and stability of interfaces in electric breakdown phenomena like sparks and lightning, analyzing both linear and nonlinear dynamics.

Contribution

It introduces a detailed analysis of the initial value problem for a moving boundary model with kinetic undercooling, revealing stability properties and potential for interface branching.

Findings

Perturbations are advected and decay at the back of the circle.

A circle is the asymptotic attractor for small perturbations.

Larger perturbations can cause interface branching and loss of smoothness.

Abstract

An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a Laplacian growth model regularized by a 'kinetic undercooling' boundary condition. Using this model we study both the linearized and the full nonlinear evolution of small perturbations of a uniformly translating circle. Within the linear approximation analytical and numerical results show that perturbations are advected to the back of the circle, where they decay. An initially analytic interface stays analytic for all finite times, but singularities from outside the physical region approach the interface for $t\to\infty$, which results in some anomalous relaxation at the back of the circle. For the nonlinear evolution numerical results indicate that the circle is the asymptotic attractor for small perturbations, but larger perturbations may lead to branching. We also present results for more general initial shapes, which demonstrate that regularization by kinetic undercooling cannot guarantee smooth interfaces globally in time.