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


This project is supported by NWO in the Computational Science Program.

Description of the project

  1. What are streamers?
  2. The minimal streamer model
  3. Streamer simulations: the need for grid refinements
  4. Streamer propagation and branching
  5. Using the code in the future

What are streamers?

When initially non-ionized or weakly ionized matter is exposed to high electric fields, non-equilibrium ionization processes, so-called discharges, occur. They create a nonequilibrium plasma. The discharges may appear in various forms depending on the spatio-temporal 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 non-stationary 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, so-called sprites and blue jets in the higher regions of the athmosphere, now draw considerable scientific attention.


The minimal streamer model

In 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 so-called 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 non-attaching gas like N2), 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:
  1. Free electrons in high electric fields can gain sufficient energy to create additional electron ion pairs through collisions with neutral particles. This is the reaction term in Eq.(1).
  2. Electrons drift antiparallel to the local electric field. This is the drift term in Eq.(1). Next to this convective mechanism there is also a little diffusion of the electrons. The positive ions also drift and diffuse, but since their mobility is two orders of magnitude smaller than that of the electrons, they can be considered as immobile.
  3. This drift leads to the build-up of space charge densities that modify the externally applied field.

Streamer simulations: the need for grid refinements

Up 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 long-term dynamics of the streamers are set in the unstable region ahead of the streamer front – the so-called 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.


Streamer propagation and branching

Below we can see some results of the simulations in a cylindrical coordinate system (r,z), symmetric around the z-axis. The planar electrodes are placed perpendicular to the z-axis. 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.
These results show that the process is deeply nonlinear and far from equilibrium: the electric field drives creation and motion of electrons and ions, while electrons and ions change the field. As a result, ionized channels are formed. Their propagating tips have a specific dynamically emerging inner structure: the interior of the streamer head is electrically screened, it is surrounded by a thin space charge layer that creates a strong field enhancement ahead of the tip. In this strong field zone, creation and motion of charged particles is very rapid. If the space charge layer becomes sufficiently thin in comparison to the channel radius, the structure becomes unstable and the streamer channel splits. This branching state was also seen when running the simulation on very fine grids, which supports the hypothesis that this is not a numerical instability.

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.


Using the code in the future

The 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 two-dimensional. 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.

Online posters and talks

Numerical codes

The research codes developed for this project are available on request. Please contact Willem Hundsdorfer,


PhD Thesis

  • C. Montijn,
    Evolution of Negative Streamers in Nitrogen: a Numerical Investigation on Adaptive Grids.
    Thesis, Technical University Eindhoven, 2005. ISBN 90-386-2371-2


  • A. Luque, U. Ebert, C. Montijn, W. Hundsdorfer,
    Photoionisation in negative streamers: fast computations and two propagation modes,
    Appl. Phys. Lett. 90, 081501 (2007).

  • C. Montijn, W. Hundsdorfer, U. Ebert,
    An adaptive grid refinement strategy for the simulation of negative streamers,
    J. Comput. Phys. 219 (2006), 801--835. [ PDF ]

  • C. Montijn, U. Ebert, W. Hundsdorfer,
    Numerical convergence of the branching time of negative streamers,
    Phys. Rev. E 73, 065401 (2006) [ PDF ]

  • U. Ebert, C. Montijn, T.M.P. Briels, W. Hundsdorfer, B. Meulenbroek, A. Rocco, E.M. van Veldhuizen,
    The multiscale nature of streamers,
    Plasma Sources Science and Technology 15, S118-S129 (2006). [ PDF ]

  • C. Montijn and U. Ebert
    Diffusion correction on the avalanche-to-streamer transition
    J. Phys. D: Appl. Phys. 39, 2979-2992 (2006). [Report PDF]

  • T.M.P. Briels, J. Kos, E.M. van Veldhuizen, C. Montijn, A. Luque, U. Ebert,
    Experiments and calculations on pulsed streamers in air and nitrogen. Refereed Proceedings 5th Int. Symp. on Non Thermal Plasma Technology (ISNTPT5), Ile d'Oleron, France, June 2006 [6 pages, 9 figures].

  • C. Montijn, U. Ebert and W. Hundsdorfer
    Adaptive grid simulations of negative streamers in nitrogen in under- and overvolted gaps
    In: 27th ICPIG, Refereed Procedings, Eindhoven, 2005 [PDF]

  • C. Montijn, B.J. Meulenbroek, U.M. Ebert and W. Hundsdorfer.
    Numerical simulations and conformal analysis of growing and branching negative discharge streamers
    IEEE Trans. Plasma Sci. 33, 260-261 (2005) [Report PDF]

  • W. Hundsdorfer and C. Montijn,
    A Note on Flux Limiting for Diffusion equations
    IMA J. Numer. Anal. 24, 635-642 (2004) [Report PDF]

  • J. Wackers
    A nested-grid finite-difference Poisson solver for concentrated source terms
    J. Comp. Appl. Math. 180 (2005), 1-12.

  • A. Rocco, U. Ebert, W. Hundsdorfer,
    Branching of negative streamers in free fligh
    Phys. Rev. E 66, 035102(R) (2002).

  • U. Ebert, A. Rocco, W. Hundsdorfer, and M. Arrayás,
    A mechanism for streamer branching
    In: 16th ESCAMPIG and 5th ICRP Joint Conference, Grenoble, 2002.

  • M. Arrayás, U. Ebert, W. Hundsdorfer,
    Spontaneous branching of anode-directed streamers between planar electrodes
    Phys. Rev. Lett. 88, 174502 (2002). See also the Physical Review Focus "Sparks branch like coral reefs" and the preview science update in Nature "Lightning forks illuminated" .

Other publications

  • C. Montijn,
    Vroege Vonken onder de Virtuele Microscoop,
    Nederlands Tijdschrift voor Natuurkunde, 2006. [Text PDF (in Dutch)]
    Third NTvN-prize 2006.