%0 Book Section %1 statphys23_0733 %A Brau, F. %A Ebert, U. %A Schafer, L. %A Luque, A. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K boundary conformal cusps mapping moving regularization statphys23 topic-4 %T Spark ionization fronts as Laplacian moving boundaries: boundary approximation and solutions %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=733 %X The formation of ionized fingers (so-called streamers) in the early stages of sparks and lightning are processes involving many scales, from microscopic collisions of electrons with molecules up to macroscopic dielectric breakdown patterns. The self-focussing of the electric field at the tip of an ionized finger resembles the self-focussing of mechanical forces at the tip of a fracture line 1. On intermediate lengths, the ionized finger emerges from a continuum model for electron and ion densities, nonlinearly coupled to the electric field. Numerical solutions of the continuum model show that a thin electric screening front around the finger emerges. Here we concentrate on the dynamics of this front approximating it as a moving boundary coupled to a Laplacian field. Similar models arise in many fields of the natural sciences; examples include viscous fingering in Hele-Shaw flow, growth of corals or bacterial colonies, or solidification fronts in undercooled melts. In simplest approximation such problems are mathematically ill-defined: generically unphysical cusps appear on the boundary within finite time, $t_c$, and a regularizing boundary condition is required. The continuum model for ionization fronts indicates such a boundary condition; it is a mixed Dirichlet-Neumann boundary condition which introduces a regularization length and that appears similarly in so-called kinetic undercooling in solidification fronts 2,3. Using conformal mapping methods, we show that a circle is a uniformly translating solution of this problem and we construct a single PDE governing small perturbations of this solution. When the radius of the circle equals the regularization length, the PDE can be solved analytically, which shows rigorously that the uniformly propagating solution is linearly convectively stable. For planar interfaces, we also prove that this boundary condition prevents the formation of unphysical cusps in the linear perturbation theory. We present numerical results for the nonlinear dynamics of the temporal evolution of planar and curved interfaces between the two phases. These simulations show that cusps do not form up to a time $t$ significantly larger than $t_c$. We also show, for curved interfaces, that, when the perturbations are large enough, the circle is no longer an attractor of the dynamics: the regularization mechanism takes place through branching of the interface which also avoids the formation of cusps. 1) U. Ebert et al., Plasma Sources Sci. Techn. 15, S118-S129 (2006).\\ 2) B. Meulenbroek, U. Ebert and L. Schäfer, Phys. Rev. Lett. 95, 195004 (2005) (2005). \\ 3) F. Brau, U. Ebert, L. Schäfer, B. Meulenbroek, in preparation.

@incollection{statphys23_0733, abstract = {The formation of ionized fingers (so-called streamers) in the early stages of sparks and lightning are processes involving many scales, from microscopic collisions of electrons with molecules up to macroscopic dielectric breakdown patterns. The self-focussing of the electric field at the tip of an ionized finger resembles the self-focussing of mechanical forces at the tip of a fracture line [1]. On intermediate lengths, the ionized finger emerges from a continuum model for electron and ion densities, nonlinearly coupled to the electric field. Numerical solutions of the continuum model show that a thin electric screening front around the finger emerges. Here we concentrate on the dynamics of this front approximating it as a moving boundary coupled to a Laplacian field. Similar models arise in many fields of the natural sciences; examples include viscous fingering in Hele-Shaw flow, growth of corals or bacterial colonies, or solidification fronts in undercooled melts. In simplest approximation such problems are mathematically ill-defined: generically unphysical cusps appear on the boundary within finite time, $t_c$, and a regularizing boundary condition is required. The continuum model for ionization fronts indicates such a boundary condition; it is a mixed Dirichlet-Neumann boundary condition which introduces a regularization length and that appears similarly in so-called kinetic undercooling in solidification fronts [2,3]. Using conformal mapping methods, we show that a circle is a uniformly translating solution of this problem and we construct a single PDE governing small perturbations of this solution. When the radius of the circle equals the regularization length, the PDE can be solved analytically, which shows rigorously that the uniformly propagating solution is linearly convectively stable. For planar interfaces, we also prove that this boundary condition prevents the formation of unphysical cusps in the linear perturbation theory. We present numerical results for the nonlinear dynamics of the temporal evolution of planar and curved interfaces between the two phases. These simulations show that cusps do not form up to a time $t$ significantly larger than $t_c$. We also show, for curved interfaces, that, when the perturbations are large enough, the circle is no longer an attractor of the dynamics: the regularization mechanism takes place through branching of the interface which also avoids the formation of cusps. 1) U. Ebert et al., Plasma Sources Sci. Techn. 15, S118-S129 (2006).\\ 2) B. Meulenbroek, U. Ebert and L. Schäfer, Phys. Rev. Lett. 95, 195004 (2005) (2005). \\ 3) F. Brau, U. Ebert, L. Schäfer, B. Meulenbroek, in preparation.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Brau, F. and Ebert, U. and Schafer, L. and Luque, A.}, biburl = {https://www.bibsonomy.org/bibtex/2023566b350f6e3826f5978faf3e6737a/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {158deafa78f57624f5ae6070730e373e}, intrahash = {023566b350f6e3826f5978faf3e6737a}, keywords = {boundary conformal cusps mapping moving regularization statphys23 topic-4}, month = {9-13 July}, timestamp = {2007-06-20T10:16:28.000+0200}, title = {Spark ionization fronts as Laplacian moving boundaries: boundary approximation and solutions}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=733}, year = 2007 }