G.V. Naidis Institute for High Temperatures Russian Academy of Sciences Moscow, Russia

1. Modeling of streamer breakdown of short non-uniform air gaps G.V. Naidis Institute for High Temperatures Russian Academy of Sciences Moscow, Russia Lorentz Center workshop, Leiden, May 2005

2. Streamer-to-spark transition is considered in a • rod-to-plane gap with the length d = 1 cm in air • at pressure 1 bar and temperature 300 K • Two stages of the transition are simulated: • Streamer propagation inside the gap. At U/d > 5 kV/cm streamer bridges the gap, forming a plasma channelwith relatively low electrical conductivity (τpropagation ~ 10 ns) • 2)Evolution of plasma in the channel, governed by kinetic and gas dynamic processes • (τbreakdown ~ 102-104 ns, depending on U)

3. Mechanisms resulting in streamer-to-spark transition: • Thermal mechanism: a lowering of the gas density N inside the channeldue to expansion of the heated plasma (Marode e.a.1979,1985; Bayle e.a.1985). • This factor is ineffective at τbreakdown « τexpansion • = R/Csound~ 6x102 ns (for R = 0.02 cm). • Kinetic mechanism: accumulation of active particles changing the ionization balance (Rodriguez e.a.1991; Eletskiy e.a.1991; Lowke 1992; Aleksandrov, Bazelyan e.a.1998; Naidis 1999). • In the present work both factors are accounted for.

4. Simulation of positive streamer propagation 1)2D model (with axial symmetry): , . • 1.5D model, with constant streamer radius R, • electric field is calculated using the method of disks Calculated streamer parameters: R = 0.02-0.04 cm, Nec = (1.5-3.0)x1014 cm-3

5. Simulation of channel evolution after bridging the gap Telegraph equations for the electric field E and current I : , the capacity and electrical conductivity per unit length are Φ(0) ­ Φ(d) = U - Ucathode, Ucathode = 0.2 kV.

6. Simulation of channel evolution after bridging the gap Gas dynamic equations:

7. Simulation of channel evolution after bridging the gap Kinetic equations for species N, O, NO, N2(A3Σ), N2(a'1Σ), O2(a1Δ), ions O-, O2-, O3-, O2+, O4+, electrons: . (diffusion of species is neglected). The density of energy input versus r is taken as

8. Conditions of applicability of the model 1) Ions stay in the channel (positive charge does not change): τbreakdown«τion drift = d/Vion ~ 4x104 ns 2) Diffusion of species may be neglected: τbreakdown«τdiffusion = R2/6D τbreakdown«τambipolar diffusion = R2/6Da At R = 0.02 cm τdiffusion~ 4x105 ns, τambipolar diffusion ~ 3x104 ns .

9. The electric field distributions along the channel after bridging the gap at U = 19 kV The distribution becomes nearly uniform along the channel at t ~ 102 ns

10. The electric current dependence on time for various applied voltages 0D simulation at E=(U-Ucathode)/dgives the results in agreement with those of 1D model (accounting for the change of plasma parameters along z) 1D (solid) and 0D (dashed) simulations at N = const

11. The electric current dependence on time for various applied voltages 0D simulations: R = 0.02 cm, Ne0 = 2x1014 cm-3

12. The streamer-to-spark transition time R = 0.02 (full) and 0.04 cm (broken), Ne0 = 2x1014 cm-3

13. The densities of neutral species U = 19 kV (simulation at N = const)

14. The densities of charged species U = 19 kV (simulation at N = const)

15. The rates of the processes of generation and loss of electrons Accumulation of oxygen atoms leads to the increase of detachment rate, resulting in the change of sign of the source term for electrons U = 19 kV (simulation at N = const)

16. Gas pressure at the streamer axis for various applied voltages At U = 18 kV τbreakdown ~ τexpansion (~ 103 ns)

17. Gas temperature at the streamer axis for various applied voltages

18. Vibrational temperature of N2 molecules at the streamer axis for various applied voltages

19. Gas density at the streamer axis for various applied voltages

20. Reduced electric field at the streamer axis for various applied voltages

21. Radial distributions of pressure at U = 18 kV

22. Radial distributions of gas temperature at U = 18 kV

23. Radial distributions of gas density at U = 18 kV

24. Radial distributions of gas velocity at U = 18 kV

25. The streamer-to-spark transition time (the effect of initial electron density)

26. Effect of pressure variation pd = 1 bar cm,pR = 0.02 bar cm, Ne0/p2 = 2x1014 cm-3 bar-2

27. Conclusions • 1. Streamer breakdown of atmospheric-pressureair in gaps with lengths d ~ 1cm at constant applied voltage U occurs during one current pulse if U/d > 14 kV/cm. • In this case τbreakdown< 10-4s. • Streamer-to-spark transition at τbreakdown «10-6 s may be described in approximation ofconstant gas density; • at τbreakdown»10-6 s it may be described in approximation • of constantpressure. • Streamer breakdown is observed also at lower U/d. In this case the breakdown is the result of a sequence of streamers, propagating along the same path with frequencies • f < 104Hz, each of these streamers changing slightly the parameters of the medium (temperature, density, etc.).