1. PDE simulations with adaptive grid refinement for negative streamers in nitrogen Carolynne Montijn Work done in cooperation with: U. Ebert W. Hundsdorfer J. Wackers

2. Outline The minimal streamer model Numerical implementation: local uniform grid refinements Streamer propagation: –Avalanche phase: a diffusion based correction on the avalanche to streamer transition –Stable streamer regime: comparison with planar front theory –Branching streamers: influence of the electron inflow and the applied field Conclusions

3. The minimal streamer model (N 2, -): fluid approximation Poisson equation for the electric field 3-dimensions with radial symmetry Characteristic length, time and density scales depend on pressure, e.g. in N2: Fluid approximation for the particle densities, in dimensionless units:  = 0  = 0 or  z = 0  =  0 = |E 0 |/L z  z = 0  r =0  r = 0 r = L r E0E0 z = L z r = 0 z = 0  = n e / n 0  = n + / n 0

4. Streamers in nature Low pressure, large systems Streamers in the lab High pressure, small systems 4cm [T.M.P Briels & E.M. v. Veldhuizen, TUE] [H.C. Stenbaek-Nielsen, Alaska Fairbanks]

5. The minimal streamer model: typical solution r (mm) z (mm) electrons (cm -3 ) net charge (cm -3 ) field (kV/cm) ions (cm -3 ) High field, E 0 =0.5

6. The minimal streamer model: numerical challenge Multiscale character of the problem Poisson eq. on whole domain, particles only on part of the domain, sharp ionization front. z r electrons r z net charge Steep gradients, large system size – high accuracy needed, in particular, for instabilities, – limitation of computational memory.

7. The minimal streamer model: influence of the gridsize Electron density at T=250, E 0 =0.5 Electron density, net charge density and electric field strength on axis in the streamer head High resolution needed for the front velocity to be well captured

8. Numerical method decoupling of the domains Whole computational domain

9. Numerical method decoupling of the domains Whole computational domain Grids for continuity equations (σ, ρ and E) Grids for Poisson equation (  ) Decoupling of the computational grids σ, ρ E  x=4  x=2  x=1  x=1/2  x=1/4  x=1/8

10. Numerical method : performance of the code Grids for Poisson equation are refined  considerable gain in memory Number of gridpoints for the continuity equations E 0 =0.5,  x=0.25 E 0 =0.15,  x=2 In high fields, limitations come mainly from the Poisson equation In low fields, the gain in memory is considerable

11. Avalanche: – field approximately constant, – no space charge effects, – Gaussian electron density distribution propagates with constant velocity and has a diffusive widening The different propagation phases (E 0 =0.5)

12. Streamer phase: – appearance of a charged layer with width w and radius r e, – electric field enhanced ahead of streamer, – interior of the streamer shielded from external field The different propagation phases (E 0 =0.5)

13. Branching streamer: – the front becomes flatter, – approaching the limit of a planar front, – which is known to be unstable (Arrayás e.a., 2004) The different propagation phases (E 0 =0.5)

14. The avalanche phase Initial phase of the propagation of a free electron under influence of a sufficiently high electric field. Characterised by the absence of space charge effects Transition to streamer phase when space charges become important Meek’s criterion for avalanche to streamer transition does not include diffusion

15. The avalanche phase: a diffusion based correction on the avalanche to streamer transition No space charge effects:  approximately constant electric field  linearization of the continuity equations: with and The electron density can be computed analytically ! A good analytical approximation for the ion densities has been found  Criterion for the emergence of a streamer taking into account the diffusion [Raizer,1991]

16. Criterion for the transition from avalanche to streamer phase: f(E 0, ) T trans =g(E 0,D,  0 ) Compare to Meek’s criterion: f(E 0, )T trans  20 The avalanche phase: a diffusion based correction on the avalanche to streamer transition

17. The avalanche phase: influence of the boundary condition with electron inflow without electron inflow Using Neumann boundary conditions at the cathode leads to a fast transition to the streamer regime E 0 =0.15, f(E 0 )=1.9·10 -4 E 0 =0.5, f(E 0 )=0.07

18. Stable streamer propagation: front velocity Planar front velocity: + + + + v num vfvf E max E 0 =0.15 E 0 =0.5

19. Stable streamer propagation: width and radius of the space charge layer + + + + w num w1w1 E 0 =0.15E 0 =0.5 Numerical results can be fitted with, in agreement with w rere

20. Stable streamer propagation: maximum space charge field + + + + E max,num E planar E 0 =0.15 E 0 =0.5

21. Branching streamers : Influence of the electron inflow, E 0 =0.5 Electrons Electric field small init. cond. wide init. cond. Without inflow, t=450 With inflow, t=300 With inflow, t=250

22. Branching streamers : Influence of the electron inflow, E 0 =0.15 Electrons Electric field small init. cond. with inflow, t=33000 wide init. cond. w ith inflow, t=31500 Electrons Electric field

23. Branching time as a function of gridsize

24. The grid refinements give a powerful tool for the streamer simulations –Gain in computational time –Gain in computational memory  Large streamers in low fields  High resolution in high fields (needed to capture the front velocity) Diffusion based correction for the avalanche to streamer transition Streamer propagation depends on applied field and also on electron inflow Simulations agree with results for planar fronts and support reduced model for fully developed streamer [Meulenbroek, Rocco, Ebert] (This analytical model also exhibits branching) Future: –more detailed investigation of the propagation mechanism, especially the instabilities –Couple to Monte-Carlo code (Li) Negative streamers in N 2 : conclusions