S-R4-5
Modeling of inception and structure formation of discharge based onpercolation transition
パーコレーション転移モデルによる放電の突発性、構造形成の解明	 
Akira Sasaki
佐々木明
Quantum Beam Science Directorate, Japan Atomic Energy Agency
6-1-7 Umemidai, Kizugawa-shi, Kyoto, 619-0215, Japan
日本原子力研究開発機構量子ビーム応用研究部門 〒619-0215	  京都府木津川市梅美台8-1-7
As an application of the concept of the plasma phase transition, we show a simulation based on thepercolation model, which reproduces the sudden structure formation of discharge. Taking the dynamics ofthe ionization, the simulation is also found to reproduce the observed pre-discharge activities anddevelopment of the stepped leader.

1.

Introduction
Discharge is useful from light sources,
environmental technologies such those for
disposal of waste gas and water, plasma
processing, to nuclear fusion. However, forinstance, high pressure discharge occurs suddenly,
through a complex path with branching anddetouring. Such discharges show highly
stochastic behavior, which has not yet been fully
understood. We present a simulation analysis ofdischarge based on the percolation transition [1],which reproduces interesting behaviors ofdischarge.
2. Percolation model
We perform calculation of the discharge between
plane electrodes. In the percolation model, thedischarge area is divided into cells using meshes,and each cell is determined to be either insulator orconductor, which corresponds to neutral gas andionized plasma, respectively. The current andpotential are calculated for each mesh point basedon the Kirchhoff’s current law.
In the original model with random ionization,
the threshold concentration of plasma region isdetermined to be 0.59 in the case of
two-dimensional square lattice. When theconcentration exceeds this threshold, the discharge
occurs immediately through a path sometimes withbranching and detouring, which is similar to thoseexperimentally observed [2]. The result suggested
that not only the development of tip of streamers
but random remote ionization in the medium shouldbe the origin of the sudden structure formation ofdischarge [3]. On the other hand, branching ofgrowing streamers has been explained on the basisof Laplacian instability [4], and its structure has

been reproduced using fractal models [5]. Thus amodel, which is applicable to a wide variety ofdischarge including streamers and leaders, isdesired, based on better understating of theirunderlying physics.
In the present study, the model is extended toinclude the dynamics of ionization. We calculate
the formation and extinction of ionized cell basedon a probability, and the spatial distribution of theionized cells for each time step. We assume that theionization rate increases as a square of local electricfield as E2, and the extinction rate decreases as thecurrent j as 1/(1+j/j0) beyond a reference value of j0.
Furthermore, the time dependent circuit equation issolved taking geometrical capacitance andinductance into account.
314659804

535

772808

628

801

1000

Fig.1. Profiles of current showing the development ofa leader. The number in each figure corresponds to
1/100 of calculation steps. Open squares show theionized cells, and the current is shown by the color.

3. Result
Figure 1 shows an example of the results. Thecalculation is carried out using 31x31 meshes andassuming the voltage is applied between upper andlower plane electrodes. 5 meshes long tip is placedat the center of the upper electrode.
It is shown that firstly pre-discharge activity witharrested leaders [6] appears from the tip, then aleader starts to grow, and finally breakdown occurs.
Spatial and temporal evolution of the discharge
for same run is shown in fig.2. Before the growth ofthe leader, flashes are seen in the streak image,which can also be identified as the current pulses.
This current originates from the formation andextinction of the ionization region, which causes
charge and discharge of spatial capacitance.
The growth of the leader is shown to be stepwise,
because clusters of the ionized region are produced,
and the growth of the leader occurs when the leader
and cluster are connected. Furthermore, as thelength of the leader increases, the electric field at
the tip increases. This leads to increasing
probability of ionization and the speed of growth ofthe leader, along its propagation.
pre discharge

At the high concentration, the leader develops
immediately so that breakdown occurs after shorttime. At the low concentration, only less frequentpre-discharge activities are seen without the growthof the leader.
4. Summary
We use the percolation model to simulate
stochastic behavior of discharge, which could not
be taken into account in the previous models [7].
Comparisons with experiments are being planned toimprove the present model and to determine the rateof formation and extinction of the ionized region
from the experimental conditions such as applied
electric field and material constants.

stepped leader
breakdown

current (arb.)

Fig.3. Statistics of the calculated discharge, showing thebreakdown probability (open circle and green line), andbreakdown time from each run (closed circle) and its
averaged value (red line).

Acknowledgments
This work was supported by Grant-in-Aid for
Scientific Research (B) 23340185 from Japan
Society for the Promotion of Science (JSPS).
0200

400
600
steps (x100)

800

1000

Fig.2. Simulated streak image from temporal andspatial evolution of the discharge and corresponding
current at the upper electrode.

As we determine the distribution of ionizedregion using a probability, each run shows different
result, from which statistical behavior of discharge
can be investigated as shown in fig.3.
It is shown that the probability of discharge is
one at high concentration, which decreases rapidly
when the concentration of the ionized region
becomes lower than a limiting value. At the sametime, the dispersion of the breakdown timeincreases significantly.

References[1] D. Stauffer and A. Aharony: Introduction topercolation theory (Taylor & Francis, 1985).
[2] A. Sasaki, et al. Phys. Rev. Lett. 105, 075004(2010).
[3] T. Torii, et al., Geophys. Res. Lett. 31, L05113(2004).
[4] Arrayás, et al. Phys. Rev. Lett. 88, 174502(2002).
[5] L. Niemeyer, et al. Phys. Rev. Lett. 52, 1033
(1984).
[6] M. Seeger, et al. J. Phys. D41, 185204 (2008).
[7] M. Seeger, et al. J. Phys. D42, 185205 (2009).