Propagation and structure of planar streamer fronts

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Abstract
Streamers are a mode of dielectric breakdown of a gas in a strong electric field: A sharp nonlinear ionization wave propagates into a nonionized gas, leaving a nonequilibrium plasma behind. The ionization avalanche in the tip of the wave is due to free electrons being accelerated in the strong field and ionizing the gas by impact. This chain reaction deeper in the wave is suppressed by the generated free charges screening the field. Simulations of streamers show two widely separated spatial scales: the width of the charged layer where the electron density gradients and the ionization rate are very large [O(μm)], and the width of the electrically screened, finger-shaped, and ionized region [O(mm)]. We thus recently have suggested analyzing first the properties of the charge-ionization layer on the inner scale on which it is almost planar, and then understanding the streamer shape on the outer scale as the motion of an effective interface, as is done in other examples of nonequilibrium pattern formation. The first step thus is the analysis of the inner dynamics of planar streamer fronts. For these, we resolve the long-standing question about what determines the front speed, by applying the modern insights of pattern formation to the streamer equations used in the recent simulations. These include field-driven impact ionization, electron drift and diffusion, and the Poisson equation for the electric field. First, in appropriately chosen dimensionless units only one parameter remains to characterize the gas, the dimensionless electron diffusion constant D; for typical gases under normal conditions D≈0.1–0.3. Then we determine essentially all relevant properties of planar streamer fronts. Technically, we identify the propagation of streamer fronts as an example of front propagation into unstable states. In terms of the marginal stability scenario we then find that the front approached asymptotically starting from any sufficiently localized initial condition (the ``selected front'') is the steepest uniformly translating front solution, which is physical and stable. Negatively charged fronts are selected by linear marginal stability, which allows us to derive their velocity analytically. Positively charged fronts can only propagate due to electron diffusion against the electric field; as a result their behavior is singular in the limit of D→0. For D≲1, these fronts are selected by nonlinear marginal stability and we have to apply numerical methods for predicting the selected front velocity. For larger D, linear marginal stability applies and the velocity can be determined analytically. Numerical integrations of the temporal evolution of planar fronts out of localized initial conditions confirm all our analytical and numerical predictions for the selection. Finally, our general predictions for the selected front velocity and for the degree of ionization of the plasma are in semiquantitative agreement with recent numerical solutions of three-dimensional streamer propagation. This gives credence to our suggestion that the front analysis on the inner (μm) scale yields the moving boundary conditions for a moving ``streamer interface,'' whose pattern formation is governed by the evolution of the fields on the outer (mm) scale.
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