Evolution of negative streamers in nitrogen : a numerical investigation on adaptive grids

Plasmas are ionized media, occupying 99% of the universe. Common examples of plasmas are the sun, which is a high-temperature plasma, and neon lights, which are low-temperature plasmas. A high-temperature plasma is at thermal equilibrium, and is driven by a high pressure and temperature of the medium. A low-temperature plasma, on the other hand, is far from equilibrium, and the ionization is generated by electric or electromagnetic fields. Streamers are transient, filamentary, low-temperature plasma channels which, under influence of the self-enhanced electric field at their tip, propagate rapidly into a non- or weakly ionized medium. They are widely used in industry, e.g. for the treatment of exhaust gasses, cleaning of polluted water, and in aerospatial engineering. Streamers are also found in nature, where they play a role in creating the path of lightning. Recent observations showed the existence of sprites, which are very large discharge structures in the higher parts of the atmosphere, composed of a multitude of streamers. One distinguishes streamers according to their polarity: in positive or cathode-directed streamers, positive space charges propagate in the direction of the electric field. In negative or anode directed streamers, on the other hand, it is negative net charge that propagates in the direction of the electron drift, i.e. opposite to the electric field. Experiments show that positive streamers emerge more easily from a point or a wire electrode than negative ones, which require a much higher voltage to emerge. Consequently, industrial applications mainly focus on the use of positive streamers. On the other hand, when streamers emerge in free space from ionization avalanches, they can have both a positive and a negative end. Lightning as well as sprite discharges are examples of such kind of double-ended discharges. Up to now, most experimental and theoretical efforts have been devoted to positive streamers in air because of their applications. However, the cross-sections for photoionization, which is required for the propagation of positive streamers, are not well-known. To define a clear physical signature, it is therefore desirable to study a situation rather independent of photoionization: negative streamers in pure gases. High-voltage experiments to obtain such streamers are currently being set up at the Eindhoven University of Technology in collaboration with the research theme "Nonlinear Dynamics and Complex Systems" at the national research institute for Mathematics and Computer Science (CWI) in Amsterdam, where numerical and analytical research is carried out. This thesis was written at CWI and is concerned with a numerical method for the simulation of negative streamers, and also with an analytical criterion for the emergence of such streamers. The simulation of streamers represents a great computational challenge. First, multiple spatial scales are involved: the non-ionized region into which the streamer propagates is orders of magnitude larger than the ionized channel, which in turn is much larger than the small active region at the streamer tip, which again has an inner layered structure. Secondly, the spatial density gradients in the tip of the streamer grow during the propagation, requiring an increasing accuracy of the numerical method. Finally, another specific difficulty comes from the unstable nature of streamers: any ionized perturbation in the non-ionized, high-field region just ahead of the streamer tip will grow. The dynamics of the streamers are set in this unstable region, the leading edge, where the densities are very low and the density gradients therefore small. The ionization front is pulled into the non-ionized region by the leading edge, which is a main reason for the failure of standard refinement strategies to describe accurately the streamer dynamics. We have developed a numerical algorithm that copes in an efficient way with the inherent computational difficulties. It computes the evolution of the streamer in a fluid approximation. The model consists of continuity equations for the charged particles, which, in pure nitrogen, are electrons and positive ions. These continuity equations tell us that the temporal change of the charged particles is set by their drift, diffusion, and ionization sources and sinks. The drift velocity of the particles as well as the ionization rate depend on the local electric field, which has to be determined through the so-called Poisson equation for the electric potential, whose source term is given by the space charge. This model is nonlinear because the particle motion and generation depend on the field while the field depends on the particle densities. For negative streamers in nitrogen, it is admissible to neglect ionization sources like photoionization, and the only source of charged particles is then ionization by impact of sufficiently energetic electrons with neutral particles. These mechanisms - namely the drift and impact ionization in the local electric field, the diffusion and the space charge effects - in a continuum approximation constitute the so-called minimal streamer model, which is analyzed in this thesis. The algorithm is implemented for a three-dimensional system with cylindrical symmetry, which reduces the computations effectively to two spatial dimensions. The algorithm is based on a decoupling of the numerical grids for the continuity equations on the one hand, and that for the Poisson equation on the other hand. The grids are refined, according to error monitors, at each time step, thereby adapting themselves to the solution. The leading edge is explicitly included in the refinement criterion. Successful test are carried out both on planar and curved streamer fronts. This algorithm enables us to explore a new parameter regime. We can now apply large background electric fields, in which spatial gradients become very large, and still resolve the streamer in an accurate manner. It is now also possible to compute the streamer evolution in low fields and large gas gaps. The results of the simulations exhibit some very interesting features in both cases. Following the evolution of streamers emerging from a single electron in a plane-parallel electrode geometry shows that three physical stages are passed. The emergence of a streamer can occur through an electron avalanche, characterized by the absence of space charge effects, and is therefore linear. Once the amount of space charges is sufficiently large to change significantly the background electric field, the phenomenon becomes non-linear, and a streamer emerges. If the distance to the anode is long enough, the streamer eventually becomes unstable and branches. During the avalanche phase, the electrons drift, diffuse and multiply in the uniform background electric field. If the avalanche starts from a single electron and the field is homogeneous, the equation for the electrons has an analytical solution, which can be used to derive analytical expressions for the spatial moments of the ions. This allows us to find an analytical approximation for the electric field, and hence determine when the space charge effects have become so strong, that the transition to a streamer takes place. We have thus derived a criterion for the avalanche to streamer transition, which includes the effect of diffusion. The traditional criterion for the transition, Meek's criterion, postulates that, in a specific gas at a specific pressure, the travel time and distance of the electron avalanche before turning into a streamer only depend on the applied background field. The inclusion of diffusion shows that this is not the case and that diffusion can in fact considerably delay the emergence of a streamer. Once the streamer has emerged, the evolution is nonlinear. At this point our grid refinement strategy provides us with a powerful tool to compute the further streamer propagation. The streamer is characterized by the enhanced conductivity of its body, which is therefore partially shielded from the exterior electric field. This shielding requires a space charge layer at the streamer tip, which in turn enhances the electric field ahead of the tip. The streamer extends in this self-enhanced field. We investigate the evolution and branching of streamers in both cases of overvolted and undervolted gaps. These are distinguished by the ability of the background electric field to provide an electron with a sufficient amount of energy to ionize a neutral atom or molecule when colliding with it. In an overvolted gap, the background electric field is sufficiently high for this to happen, and the streamer penetrates a highly unstable state. Its radius continues to grow up to branching, giving it a conical shape. Moreover, the spatial density gradients become very steep, thereby requiring a very high accuracy from the numerical method. In an undervolted gap, the electrons only multiply in the small region ahead of the streamer where the field is sufficiently enhanced, giving the streamer a more filamentary shape. For a sufficient field enhancement, a sufficient amount of charge in the streamer head is required. The accumulation of charge in the head depends both on the initial distribution of ionization and on the boundary conditions on the electrode. We study different cases and eventually, in all cases, the streamer branches provided the gap is sufficiently long. The branching state of the streamer has not been analyzed much up to now, mainly due to a lack of accurate numerical tools which now have become available through the work presented in this thesis. Indeed, the refinement algorithm enables us to reach the branching state with sufficient numerical accuracy within a reasonable computational time, and more importantly, within the limits of the computational memory. First, we here establish that the time of branching converges for identical initial and boundary conditions when using finer and finer numerical grids. Such tests were out of reach up to now. The convergence of branching times allows us now to derive quantitative predictions under given conditions. We find that the branching times converge for sufficiently fine numerical grids both for the underand the overvolted case. An interesting detail is that in the undervolted case, the branched state is always the same while in the overvolted case, different branched states are reached on different grids after a similar evolution time. This suggests that in the second case, several branched states are accessible from the unstable head state. The outcome of such a nonlinear bifurcation process then will depend on minor details (like the numerical grid) as is well known even to the general public as the unpredictability of "chaos theory". Another reason not to analyze the details of the branched state is the assumed cylindrical symmetry in our calculations. Within the present thesis, the streamer splits not into branches but into concentric rings as the space of linear perturbations has been restricted to cylinder symmetrical ones. When a larger space of linear perturbations is admitted, the branching instability can be expected after a similar time of evolution, but to a different state. The physically relevant question that can be answered with the present analysis is: can we characterize a generic unstable state of the streamer head that leads to branching? This indeed seems to be the case: numerical experiments in a fixed external electric field with a variety of initial ionization distributions and boundary conditions on the electrode always seem to evolve to a very similar state of the streamer head immediately before branching. This particular head state would then be an intermediate at tractor of the dynamics that is followed by branching. However, this hypothesis requires further numerical and analytical studies. There is another insight that can be gained from the present numerical studies, namely a verification of a reduced model for well developed streamers that is currently being studied analytically at CWI. Such a model for moving ionization boundaries consists of several building blocks: 1) The ionization front at the streamer tip propagates with a velocity that is a function of the electric field ahead of it. 2) The width of the space charge layer is a decreasing function of the electric field and saturates at high fields. 3) The conductivity in the interior of the streamer is so high that it approaches Lozansky and Firsov's limit of ideal conductivity. For the dependence of front velocity and width on the electric field, analytical predictions have been derived for planar fronts. Their validity for curved fronts can be tested on the numerical results. Furthermore, analytical results show that a planar front is dynamically unstable and will branch due to a Laplacian instability, while the analysis of curved fronts is underway. The limit of a planar front is never reached in the simulations, but a limit of small curvature where the radius of curvature of the streamer head is much larger than the front width does occur. Numerical studies do reveal for which curvature the Laplacian instability sets in and are therefore complementary to the analytical studies. We conclude that the minimal streamer model analyzed in this thesis already exhibits very complex behavior and is better adapted for explorative systematic studies than a model including many more physical features from the start. The predictions of this model should now be tested on experiments on negative streamers in nitrogen while more features like the less well-known photo-ionization should be included to predict the behavior of streamers in air. Also, the step towards fully three-dimensional simulations should be made.