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.
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.
- What are streamers?
- The minimal streamer model
- Streamer simulations: the need for grid refinements
- Streamer propagation and branching
- Using the code in the future
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.
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:
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.
- 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
- 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.
- This drift leads to the build-up of space
charge densities that modify the externally applied field.
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
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
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
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.
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
- NWO Computational Science kick-off meeting,
November 29, 2002, Amsterdam, The Netherlands.
- NWO Computational Science meeting,
November 28, 2003, Eindhoven, The Netherlands.
- 16th NNV/CPS symposium for Plasma Physics and
Radiation Technology, March 15-17, 2004, Lunteren,
- Workshop on The Multiscale Nature of
Spark Precursors and High Altitude Lightning, May 9-13, 2005, Leiden,
- Annual scientific meeting FOM-Decemberdagen,
December 13-14, 2005, Veldhoven, The Netherlands.
The research codes developed for this project are available on request.
Please contact Willem Hundsdorfer,
Evolution of Negative Streamers in Nitrogen: a Numerical Investigation
on Adaptive Grids.
Thesis, Technical University Eindhoven, 2005.
A. Luque, U. Ebert, C. Montijn, W. Hundsdorfer,
Photoionisation in negative streamers: fast computations and two
Appl. Phys. Lett. 90, 081501 (2007).
C. Montijn, W. Hundsdorfer, U. Ebert,
An adaptive grid refinement strategy for the simulation of
J. Comput. Phys. 219 (2006), 801--835.
C. Montijn, U. Ebert, W. Hundsdorfer,
Numerical convergence of the branching time of negative streamers,
Phys. Rev. E 73, 065401 (2006)
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).
- C. Montijn and U. Ebert
Diffusion correction on the avalanche-to-streamer transition
J. Phys. D: Appl. Phys. 39, 2979-2992 (2006).
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)
- W. Hundsdorfer and C. Montijn,
A Note on Flux Limiting for Diffusion equations
IMA J. Numer. Anal. 24, 635-642 (2004)
- 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" .
Vroege Vonken onder de Virtuele Microscoop,
Nederlands Tijdschrift voor Natuurkunde, 2006.
[Text PDF (in Dutch)]
Third NTvN-prize 2006.