Streamer formation as a pattern formation problem

An important class of problems of spontaneous pattern formation concerns those that have to do with growing interfaces - well known examples are dendritic growth and viscous fingering, but questions similar to those that arise for these problems, also come up for problems in which there is not a microscopically sharp interface, but a smooth though thin reaction or transition zone. Examples are chemical waves, combustion, fronts in nerves, etc. A few years ago, we realized that in low temperature plasmas, a similar type of problem occurs, streamer formation. In collaboration with Christiane Caroli (Paris VI) and Ute Ebert (then a postdoc in Leiden, now at CWI, Amsterdam) we therefore started to investigate to what extent streamer formation can be understood in terms of methods developed in the field of interfacial pattern formation. Further motivation for this idea can be found in my summer school article Three basic issues concerning interface dynamics in nonequilibrium pattern formation

The formation of a streamer occurs when a nonlinear ionization wave propagates into a previously unionized region, creating a nonequilibrium plasma behind it. The formation propagation of such a thin shock-like ionization zone results from a combination of three effects: drift of electrons in the local electric field, generation of free electrons due to impact ionization, and screening of the electric field due to the buildup of the charge in the ionization zone. Triggered by the observation of interface-like profiles in the simulations of Dhali and Williams and of Vitello et al., we have taken the first steps towards developing an interface-like description for streamers, by analyzing planar streamer fronts. Our analysis has identified the propagation of streamer fronts as an example of front propagation into unstable states in virtually all models analyzed. For such problems, it is well known that the velocity cannot be obtained just by analyzing uniformly translating fronts using standard methods. Instead, the velocity and properties of fronts propagating into unstable states is governed by a mechanism of dynamical front selection which has become largely understood in the last decade. Applying this theory has allowed us to derive all essential properties of planar fronts for the model of the recent simulations.

A second important aspect of our results is the following. In the non-ionized region outside the streamer, the electrical potential $\Phi$ obeys the Laplace equation, $\nabla^2 \Phi = 0$, while our analysis shows that the normal velocity of a negatively charged planar streamer front is a weakly nonlinear function of the field $E^+= - \nabla \Phi$ just ahead of it. These two features are indeed the two essential ingredients of the equations for other interfacial pattern forming problems like dendrites - e.g., the enhanced diffusion in front of a dendrite tip is analogous to the field enhancement in front of a streamer. This makes one hopeful that streamers will in principle be amenable to the same type of analysis. Physically, one expects that the interface equations will take the form of a conservation equation for a charge sheet (involving at the minimum transport terms along the sheet, a stretch term due to interface curvature and a term associated with charge transport from the plasma behind), supplemented with an equation for the front speed that includes curvature corrections, and an equation for the degree of ionization created by the front which is not determined by any conservation law (in addition, the plasma relaxation properties in the streamer body may play a role). However implementation of this program is hampered by the fact that the interfacial dynamics of fronts propagating into unstable states exhibits a power law relaxation in time, in contrast to fronts between a stable and a metastable state, where the relaxation is exponentially fast. The standard methods of deriving a moving boundary approximation therefore break down. This problem may not arise, however, in the limit in which the electron idiffusion coefficient is zero.

When Ute got a job at the CWI after her postdoctoral stay in Leiden, she decided to build on our work on streamers by starting a research program on various pattern formation aspects in gas discharges. I therefore do not expect to actively work myself on streamer dynamics in the near future.


References
U. Ebert, W. van Saarloos, and C. Caroli, Streamer Propagation as a Pattern Formation Problem, Phys. Rev. Lett. 77, 4178-4181 (1996).
U. Ebert, W. van Saarloos, and C. Caroli, Propagation and Structure of Planar Streamer Fronts, Phys. Rev. E 55, 1530 (1997).

July 15, 1999



[Pattern formation] [Wim van Saarloos] [Instituut-Lorentz]