1. Saffman-Taylor streamer discharges
Fabian Brau, CWI Amsterdam
Alejandro Luque, CWI Amsterdam
Ute Ebert, CWI Amsterdam Eindhoven University of Technology
Streamers, sprites, leaders, lightning:
from micro- to macroscales
Leiden, 11 October 2007

2. Talk overview Minimal PDE model for streamers
Characteristics of single streamers
Interacting streamers : periodic array of streamers in 2D
Numerical solutions + characteristics
FMB explicit solution fits very well the numerical fronts
Conclusions

3. Minimal PDE model for streamers
In dimensionless unit, the minimal PDE model reads
s : Electron density
r : Ion density
Solved in the background of a homogeneous field
Negative streamer
Non attaching gas like nitrogen
Normal condition
→ Electron impact ionization :
Townsend’s Approximation
→ Poisson equation
Electric potential
Electric field
Cf talks of Alajandro Luque and Chao Li

4. Characteristics of single streamers (Valid also for interacting streamers)
Evolution of some initial condition in the E-field produced by two electrodes:
Solutions after the avalanche phase looks like (3D + cylindrical symmetry)
z (mm)
r (mm)
Net charge ( )
z (mm)
r (mm)
Electric Field (kV/cm)
Thin charge layer
E-field enhancement
E-field screening
[Montijn, Ebert, Hundsdorfer, J. Comp. Phys. 2006, Phys. Rev. E 2006, J. Phys. D 2006
Luque, Ebert, Montijn, Hundsdorfer, Appl. Phys. Lett. 2007]

5. Net charge and E-field evolution for single streamers
On branching as a Laplacian instability:
[Arrayas, Ebert, Hundsdorfer, Phys. Rev. Lett. 2002, Ebert et al., Plasma Sour. Sci. Techn. 2006]

6. Interacting streamers as periodic array : previous work
G V Naidis, J. Phys. D 29, 779 (1996)
Streamers with fixed radius
Essentially 1D
Charge density along a line
Study how the interaction affects the charge density, the electric field and the velocity

7. Interacting streamers : periodic array of streamers in 2D
Anode
Direction of
propagation L
Cathode
Neumann boundary conditions for: Potential and Densities
= Symmetry axis

8. Characteristics of Interacting streamers
If : Streamers do not branch
256

9. Net charge and E-field evolution for interacting streamers

10. Characteristics of Interacting streamers
Uniformly translating streamers
Results are robust against changes in the initial condition

11. FMB mathematical setup
Single streamers x
[E. D. Lozansky and O. B. Firsov, J. Phys. D 6, 976 (1973)]
Interacting streamers: periodic array of streamers in 2D
Ideally conducting streamer
Free moving boundary for Hele-Shaw flow
Saffman-Taylor solution y L

12. Hele-Shaw Flow
Radial Symmetry
Hole
Colored Water →
Glycerol

13. Hele-Shaw Flow
Radial Symmetry
Hole
Glycerol
Colored Water →

14. → Hele-Shaw Flow Radial Symmetry Hole Colored Water Glycerol
Channel configuration

15. Saffman-Taylor solution
Family of possible solutions

16. Selection: Boundary condition
The boundary condition doesn’t allow any selection mechanism
[B. Meulenbroek, U. Ebert and L. Schäfer, PRL 95 (2005)
F. Brau, A. Luque, B. Meulenbroek, U. Ebert and L. Schäfer, (2007)]

17. Saffman-Taylor solution
For small surface tension, only the finger with is selected

18. Saffman-Taylor solution
Experiment
Theory

19. Comparison: PDE vs FMB

20. Comparison: PDE vs FMB No free parameter Saffman-Taylor finger
The tip of the Saffman – Taylor finger coincide with the maximum of the net charge of the PDE solution
No free parameter
Saffman-Taylor finger

21. Comparison: PDE vs FMB Maximum of the net charge along y axis
for each value of x

22. Comparison: PDE vs FMB
Saffman-Taylor finger

23. Various evolutions of the streamer fronts + comparison with Saffman-Taylor Front
From the left:
Saffman-Taylor finger

24. Conclusions
Interacting streamers viewed as a periodic array of streamers in 2D present remarkable features:
If the streamer spacing is small enough for a given background electric field, streamers do not branch.
When streamers do not branch they reach a steady state which is an attractor of the dynamics.
This steady state is well approximated by a solution from hydrodynamics: the Saffman-Taylor finger.

25. Predictions If you are in the right part of the phase diagram:
In contrast to single streamers, branchings should be mostly suppressed
After some transient evolution, the velocity should reach a constant value
This value is in 2D. In 3D, we expect that the following linear relation should hold
In physical units:

26. Branching

27. Electric field along the streamer axis