simulation of streamer propagation using a pic-mcc code. application to sprite discharges

Simulation of streamer propagation using a PIC-MCC code.

Application to Sprite discharges.Olivier Chanrion and Torsten Neubert

Danish National Space Center - Juliane Maries Vej 30, DK-2100 Copenhagen Ø, [email protected]

The multiscale nature of sparks precursors and high altitude lightning. May 9-13, 2005, Leiden

Outline

• Discharge model.

• Numerical model.

• Negative streamer simulation.

• Negative and positive streamer simulation.


The Discharge Model

The model :

- Electrons move and suffer collisions with neutrals.

- Ions ( produced by ionisation ) are assumed immobile.

- Non relativistic kinetic for electrons.

- Electrostatic model.


Governing Equations

Kinetic

Equations for particles

( Vlasov-Boltzmann )

Collision terms :

Fields

Equation for the electric potential

( Poisson )

( with convenient boundary conditions )

Densities given by :


Numerical Methods


Numerical Methods

1 - Push ( trajectories ) : Leap-Frog scheme.2 - Collisions : Monte Carlo, [Nambu, JJAP,94] scheme based on the cross section of each scattering process. - subcycling if the collision frequency is high. - resampling to limit the particle number increase.

3 - Weighting ( density ) : PIC ( particle in cell ) scheme.4 - Field : Solved on a Cartesian mesh with finite element.

- FE array inverted with a direct ( Choleski ) or indirect ( SOR ) method.

Based on a standard PIC-MCC method. [Birdshall, IEEE TPS, 1991]


Code ValidationCalculation of typical swarm parameters for gas discharge physics

, ( Mobility ) defined by Vd / E where Vd is the mean speed of electrons, and E the external field.

, ( Effective ionisation coefficient ) with :

, ( Electronic temperature ), due to collisions in the background electric field E.

Comparison with a Boltzmann solver ( Boeuf / Pitchford )

kTe

, ( Townsend ionisation coefficient ) = / Vd where is the ionisation frequency.

, ( attachment coefficient ) = / Vd where is the attachment frequency.


Electron Avalanche Transition Into a Streamer

Initial conditions

- Neutral density :

- Initial field Em :

- Initiated by a Gaussian electron bead. ( as initiated by a single electron at t=0)

- No background ionisation

- No photo ionisation.

=> typical characteristics of negative streamer propagation :

- electron avalanche / negative streamer head propagate upward.

- self-consistent electric field.


Branching Streamer

-Cylindrical computation-Initial conditions chosen close to air at altitude ~70km, after a +CG lightning.

- Neutral gaz density : - Initial electric field :


Electron Distribution Function.

- plot of the reduced distribution function inside the head of the streamer.


Photoionization Model

The photoionization model is the particle version of the model used in [Liu & Pasko, JGR, 2004]

The emissivity of photons that will ionize oxygen is assumed to be proportional to the ionization rate:

The coefficient is assumed to be a function of E/p [Zheleznyak, High Temp, 1982][Zheleznyak, High Temp, 1982].

=> In our code, when an ionization occurs, we create a photon of frequency chosen randomly in

if a random number

The mean free path for this photon to ionize oxygen is given in by

A ion-electron pair is then created at a distance from the preliminary ionization event chosen randomly accordingly this mean free path.

where p and pq are resp. the gas pressure and the quenching pressure of N2

is the excitation frequency ( which lead to ionizing radiation )

the ionization frequency, the probability to ionize through absorption,

and the ionization rate calculated by our MCC scheme.


Negative and Positive Streamers Propagation

- Neutral gaz density : - Initial electric field :

-Cylindrical computation with photoionization-Test case from [Liu & Pasko, JGR, 2004]

-Initiated by a Gaussian electron bead of peak density 5.1011 m-3 and of characteristic length 3 m.


The Rescaling Technique

=> have to be improved ...

To avoid the exponential growth of the particle number we use a rescaling technique from [Kunhardt & Tzeng, Phys Rev A, 1998].


Optical EmissionsMCC => production rates of different excitation states of N2 or N2+ due to collisions.

Spontaneous emissions of photons come from transition between different excitation states.

=> Differential system solved using a exponential scheme.


Conclusions

• We have :

– 1D/2D/2D cylindric Parallel PIC-MCC model of discharge.– Simulation of negative streamer propagation until branching point.– Simulation of the beginning of the positive streamer propagation.– Calculation of some optical emissions.

• We do not have :

– Relativistic description of electrons. – Magnetic field interactions.

• Future needs :

– Validation of the streamer dynamics.– Validation of the photoionization model.– Improve the resampling of particles.

simulation of streamer propagation using a pic-mcc code. application to sprite discharges

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