CWI > Modeling Analysis and Simulations (MAS) > Nonlinear Dynamics and Complex Systems (MAS 3)Numerical Methods for Leading Edge Dominated Dynamics:Propagation and branching of negative streamer channels 
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This project is supported by NWO in the Computational Science Program.

Description of the project
What are streamers?When initially nonionized or weakly ionized matter is exposed to high electric fields, nonequilibrium ionization processes, socalled discharges, occur. They create a nonequilibrium plasma. The discharges may appear in various forms depending on the spatiotemporal characteristics of the electric field and on the pressure of the medium. For d.c. or pulsed voltages, one distinguishes the dark, glow or arc discharges that are stationary, and transient nonstationary phenomena such as streamers and leaders. These transient phenomena often are the initial state of a discharge that later on becomes stationary.Electrical discharges have become of high interest because of their numerous applications. In industry, they are used for a large number of applications. Because of the reactive radicals they emit, they are used for the treatment of contaminated media like exhaust gasses, ozone generators, treatment of polluted water and as sources of excimer radiation for material processing. They can also be observed in nature: next to conventional lightning, socalled sprites and blue jets in the higher regions of the athmosphere, now draw considerable scientific attention. UpThe minimal streamer modelIn this project we focus on streamers, that are growing plasma filaments and whose dynamics are controlled by highly localized and nonlinear space charge regions. The model we use to investigate the propagation of such discharge channels is the socalled minimal streamer model. This model describes the evolution, of an initial ionization seed placed at the cathode under the influence of an applied electric voltage. It is a fluid model for the dimensionless electron density &sigma and the positive ion density &rho (in the case of an attaching gas negative ions should also be included, but we consider a nonattaching gas like N_{2}), coupled to the Poisson equation for the electric field E and the electric potential Φ:In the evolution of streamers there are three basic mechanisms at work:
UpStreamer simulations: the need for grid refinementsUp to now all the simulations performed on this model have been carried out on uniform grids, fixed in time. One of the main goals of this project is to develop some grid refinement strategy to overcome the limitations of a uniform grid approach.For example, it appeared that, when the background electric was high enough, the streamer tends to grow into an unstable state, leading to spontaneous branching (see Array´s et al., Rocco et al.). In order to investigate the nature of these instabilities it is necessary to use finer grids, which is impossible in a uniform grid approach with the nowadays available computational memory. Moreover, the problem has a clear multiscale structure: different length scales are given by the thickness of space charge layer around the head, the radius of the channel, the channel length and the fact that the whole channel fills only a small part of the total volume. Finally, it would be interesting to investigate the behaviour of streamers on larger domains than done up to now. This however is also impossible due to the limitations of computational memory. Unfortunately, standard grid refinement procedures in regions with steep gradients fail because of the pulled character of the streamer front. This pulled front character means that the longterm dynamics of the streamers are set in the unstable region ahead of the streamer front – the socalled leading edge – where the particle densities decay exponentially and where the gradients are not necessarily steep. Therefore this leading edge should be accounted for in the refinement algorithm. The simulation can also be improved by using separate grids for the different physical processes. For example, charge densities are negligible far away from the streamer. The electric field, however, must be computed on a larger domain, extending from the cathode to the anode, and ideally infinite in the direction parallel to the electrodes. Evaluating the equations describing the electric field is essential there, whereas the equations describing the charge densities do not yield any relevant information. Using the same grid for both parts of the model would therefore involve unnecessary computations. Before these adjustments could be implemented, several difficulties had to be overcome. Most notably, the introduction of multiple grids is complicated by the mutual influence of the different parts of the model. But the improvements lead to a significant gain of time and memory. Simulations that took a month to run five years ago can now be completed within hours. UpStreamer propagation and branchingBelow we can see some results of the simulations in a cylindrical coordinate system (r,z), symmetric around the zaxis. The planar electrodes are placed perpendicular to the zaxis. Between these electrodes we apply a background electric field of 0.4 (corresponding to 80kV/cm) and place an initial ionization seed at the cathode (at z=0). The electron density, the ion denisty, the total charge densities with equipotential lines, and the electric field are plotted at different times.This instability is mathematically similar to the one of a rather thick coral: if one part of the coral lags behind, the food flow is shielded from it by the parts of the coral that are further ahead. Therefore the eminating coral parts get more food, grow more rapidly and get further ahead. The food density around the coral here plays the same mathematical role as the electric potential around the ionzation channel. UpUsing the code in the futureThe simulation code in its current form can now be used to investigate the dependence of the streamer evolution on the parameters. For example, the effect of the background electric field can be investigated. Is there a thershold field below which the streamer would eventually die out instead of growing continuously? Does the streamer always branch, or does this only occur when the background electric field is high enough? And what is the effect of the distance between the electrodes?Currently the simulations of gas discharges are effectively twodimensional. One of the goals of future research is to extend them to three dimensions, and to include the effect of indirect ionization mechanisms through photons created in the active impact ionization zone. This is required to understand sparking in different types of gases. Simulation continues to answer questions and to inspire theory to new phenomenological extrapolations to larger length and time scales which in turn offers new challenges of computations. Together they push the boundary of our knowledge on the complex phenomenon of sparks and lightning.

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Numerical codesThe research codes developed for this project are available on request. Please contact Willem Hundsdorfer, Willem.Hundsdorfer@cwi.nl 
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