Discussion of the comments on « the mechanism of the long spark formation »

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Discussion of the comments on “ the mechanism of the long spark formation ”

I. Gallimberti

To cite this version:

I. Gallimberti. Discussion of the comments on “ the mechanism of the long spark formation ”. Journal

de Physique, 1981, 42 (1), pp.155-157. �10.1051/jphys:01981004201015500�. �jpa-00208985�

155

Discussion of the comments on « the mechanism of the long spark formation »

I. Gallimberti

University ^of Padova, Electrical Engineering Dep., ^Via Gradenigo 6A, Padova, Italy (Reçu ^{le 19} septembre 1980, accepté ^{le 22} septembre 1980)

J. Physique ⁴² (1981) 155-157 JANVIER 1981,

Classification

Physics ^Abstracts

52.00

52.80

1 thank E. Barreto and H. Jurenka for their dis- cussion on the _paper « The mechanism of the long spark formation » [1] ; unfortunately ^the papers that describe in detail the mathematical techniques ^for the

simulation of waves of potential gradient along dis charge ^{channels are still} unpublished : they ^would give complete ^{answer to all the} questions ^raised by ^{E. Barr}

reto and H. Jurenka; ^some preprint copies ^may be

made available for researchers interested in the subject.

However, I think that it is worth to discuss here some

of the comments of these authors particularly ^those regarding ^the physical background of the simulation.

For purpose of logical presentation ^{I will discusg}

these comments in the inverted order.

3) The notations not defined in equations (81) ^of

référence ^{[1] are printing errors} (vé ^instead of ve and M instead of me). ^The term e. ⁱⁿ equations (34) represents the internal energy above the thermal level ofparticle$

of s-th kind [2, 3] : ^when applied ^{to electrons} produced by ionization of _gas molecules, ^it represents ^{the ioni-} zation energy [3] ; ^{as it is constant with} respect ^to any derivative equations (81) ^are generated ^from equations (34) ^with only ^a few mathematical manipulations :

the (2 eve/3 k) (ô U/ôx) ^{term comes from the} (Je.E)

term in the third of equations (34) ^{once that the latter}

is divided by (3 ne k/2) ^{in order to reduce to} unity thç

coefficient of (ôTe/ôt).

2) ^The top equation (80) ^{is obtained} by integrating

Maxwell’s equations ^{over a unit} cylindrical ^volume

around the discharge filament under the assumption

of a constant field component Ez along the transmis- sion direction [4, 5] ; Ez ^{should not be} necessarily ^zero

or uniform over the channel cross-section; ^on thç contrary the radial component Er ^{has to be} negligible

inside the filament and dominant outside, ^{so that the} potential within the channel is approximately ^constant

over the whole cross-section, and it is determined by

the surface charge density [5].

These conditions are quite well satisfied over a

large variety ^of discharge conditions [6, 7].

The bottom equation (80) is the Ohm’s law not

only ^for steady ^state conditions, ^but also for any transient condition for which the conductivity ^can

be defined ; ^this happens ^{for wave} frequencies ^much

smaller than the plasma frequency : in fact the electron

mobility (eq. (82)) ^{makes sense} when the electron energy distribution in _any point ^{of the} discharge ^is statistically ⁱⁿ equilibrium with the local field. If the channel inductance is negligible ^with respect ^{to the} resistance, ^{the bottom} equation (80) ^{is derived} directly

from Maxwell’s equations [4].

Equations (80) ^{are therefore} equivalent ^{to Maxwell’s} equations for conditions which are representative ^of

waves in filamentary discharges, ^{and are} adequate ^to

model transient processes with ^a frequency spectrum limited with respect ^{to the} plasma frequency. ^Fur- thermore, equations (80) ^{are identical to Maxwell’s}

équations ^{for an R-C} line ; ^{in this case} the ^{wave pro-} pagation velocity ^is parallel ^{to the} electric field inside the conductor, ^and perpendicular outside : well known fixed values of capacitance ^and resistance for unit length ^{can be defined.} Marode et al. [8, 9] ^have analysed the radial and longitudinal distributions of

charged particles ^{within a} discharge filament : their results for the equipotential lines around a discharge

channel are very similar to those around a resistive conductor; the R-C transmission line equations arç

therefore adequate ^to simulate the current and voltage

transient along ^a discharge filament, ^{when an electro-} magnetic perturbation is somewhere generated.

On the other side, Barreto and Jurenka ^assume in their model [10] ^a monodimensional form of Maxwell’s

equations, ^which is, ⁱⁿ my opinion, ^not adequate ^for

electrical filamentary discharges : in fact the field

equations ^presented in appendix, imply necessarily

V ^x H = 0 and D.E = aE/ax. ^These assumptions

may be justified ^{for a} physical situation uniform and

infinitely ^{extended on a} transversal plane; only ⁱⁿ

this case the radial derivatives are identically ^zero.

However, ^{in a} filamentary discharge, ^{at least on the}

filament surface, the radial derivatives are important

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004201015500

156

(even ^{if H or} E, ^are relatively small). ^It has been verified that these assumptions may lead to errors in the field estimations larger ^{than 100} % [11, 12].

In conclusion the form of Maxwell’s equations proposed by ^Barreto and Jurenka [10] may be used in

a monodimensional, transversally uniform model of wave propagation ; ^{on the} contrary ^{the form} pre- sented in equation (80) ^or (82) ^is adequate ^{to a mono-}

dimensional filamentary ^model.

1) ^{The set of} equations (83), (84) ^{is a} quasi-lineat system [13], which admits a plane ^wave solution y

modulated by non-linear self-interactions. When the

perturbation which launches the ^wave is relatively

small, ^the propagation velocity ^{can be} approximately

calculated with a dispersion ^{relation, if the} system ^is linearized. It has been shown [14] that, if the initial

steady-state ^condition Yo is ^{constant in} space and time, solutions of equation (83) ^exist only ^for B(YO) ^{= 0.}

However equation (83) ^{can be} linearized around _any

stationary linear solution Yo (which ^{is not} necessarily constant) : ^{in this case} solutions exist under the condition A(Yo) oYlox ⁺ B(Yo)

^{0. In the actual} calculations, ^the Yo ^{vector had three constant compo-}

nents (electron density neo, drift velocity veo and electron temperature 7§o) ^{and a fourth} component

(the potential Uo) linearly varying; ^it corresponds practically ^{to a constant} applied ^field Eo, ^{which deter-}

mines the steady-state ^{values of} veo and 7§o : ⁱⁿ fact,

the existence condition A( Yo) ^ô Ylôx +’B(YO) ^{0 assu-}

mes the form

which are the classical equations for electron drift

velocity ^and temperature ^under stationary ^swarm

conditions [2]. The existence condition represents therefore the physical relationships ^between the Yo components ^and fixes their values once Eo ^{is fixed.}

As observed by ^{Barreto and} Jurenka, ^if Eo ⁼ 0,

the existence condition becomes B(YO) ⁼ 0, ^and

solutions may exist only for the trivial case of Yo ^{= 0.}

However, ^{there is} experimental evidence of a non-zero

longitudinal ^field along ^the discharge ^{channels, which} implies ^{an almost} stationary resistive conduction

regime in ^the weakly ^ionized plasma.

From the mathematical point ^of view, ^if Eo :0 ⁰

and B(YO) :0 ^0, the reductive perturbation ^{method of}

Taniuti cannot be applied ^{in its} original form ; ^however

the equation system (83) ^can be linearized around Yo

with the same relations proposed by Taniuti : it beco-

mes

where

For these equations ^{the wave} propagation ^constants

can be calculated with the dispersion ^relation

where 7 is the unit matrix and k the wave number at the angular frequency ^{co. Such} relation makes possible

to calculate the phase velocity and the attenuation

coefficient, in the form (85), without the choice of à

particular ^series expansion, ^{or the full} application ^of

Taniuti’s perturbation ^{method. It} is obvious that

in this _{way any} non-linear modulation effect is neglect-

ed : however, it has been verified with the computer solution of the complete equation system ^{that the} propagation ^{constants are in} general calculated with

a satisfactory approximation [15].

Taniuti’s method can be fully applied ^{if the term} B(Y) ^is expressed ^{as a} function of the ionization

coefficient a instead of the ionization frequency [15] :

the term e, is assumed to represent the total energy lost by ^{one electron} per ionizing ^collision [16], ^and B( Y ) ^becomes

For Yo

(no, ^0, To, Uo), (and ^{hence for} Eo ^{= 0 and} To ^{= thermal} energy) B(Y.)

^{0 : the} perturbation technique ^can be therefore applied ^{to a} system ^which incorporates ionization and dissipation.

References

[1] GALLIMBERTI, I., « The mechanism of the long spark ^forma-

tion _{», J.} Physique Colloq. ⁴⁰ (1979) ^C7-193.

[2] GILARDINI, A., Low energy collisions _{in gases} (J. Wiley, ^New York) ^1972.

[3] MITCHNER, N., KRUGER, ^C. H., Partially Ionized Gases (J.

Wiley, ^New York) ^1973.

[4] SUZUKI, T., ^« Theoretical Analysis ^to propagation ^of ionizing

waves », CRIEPI Report ^E176003 (1977).

[5] MARODE, E., ^{Ph. D.} Thesis, University ^{of Paris} (1972).

[6] LES RENARDIERES GROUP, ^{« Positive} discharge ⁱⁿ long ^air gaps

at Les Renardieres : 1975 results and conclusions », Electra

53 (1977) ^31-153.

157

[7] LES RENARDIERES GROUP, ^« Research on long ^air gap dis-

charges », Electra 35 (1974) ^49-156.

[8] MARODE, E., GALLIMBERTI, I, HILHORST, D., GALLIMBERTI, B.,

« The radial distribution of _space charge ^{in a} filamentary discharge », Gaseous Electronic Conf., ^Buffalo NY, ^1978.

[9] GALLIMBERTI, I., MARODE, E., HILHORST, D., GALLIMBERTI,

« The structure of the predischarge channels in the spark

breakdown », to be published.

[10] BARRETO, E., JURENKA, H., REYNOLDS, ^S. I., ^« The formation of small sparks », J. Appl. Phys. ⁴⁸ (1977) ^4510.

[11] DAVIES, ^A. J., EVANS, ^C. J., ^« Field distortion in _gaseous dis- charges », Proc. IEE 114 (1967) ^1574.

[12] GALLIMBERTI, I., GLEIJESES, B., « The field computation ⁱⁿ filamentary gas discharges ». ^{Atti Ist. Veneto Scienze,}

Lettere ed Arti, CXXXVI, ^1978.

[13] JEFFERY, A., TANIUTI, T., Non-linear wave propagation (Acade- mic Press, New York) ^1964.

[14] TANIUTI, T., YAJIMA, N., ^« Perturbation Method for a non-

linear wave modulation », J. Mat. Phys. ¹⁰ (1969) ^1369.

[15] GALLIMBERTI, I., LANZARO, A., ^{« The} propagation ^{of waves of} potential gradient along preionized ^channels », to be

published.

[16] GALLIMBERTI, I., « A computer ^{model for streamer} propaga-

tion _», J. Phys. D., Appl. Phys. ⁵ (1972) ^2179.