What Determines the Parameters of a Propagating Streamer: A Comparison of Outputs of the Streamer Parameter Model and of Hydrodynamic Simulations

: Electric streamer discharges (streamers) in air are a very important stage of lightning, 1 taking place before formation of the leader discharge, and with which an electric discharge 2 starts from conducting objects which enhance the background elecric ﬁeld, such as airplanes. 3 Despite years of research, it is still not well understood what mechanism determines the values 4 of streamer parameters, such as its radius and propagation velocity. The Streamer Parameter 5 Model (SPM) is aimed to understand this mechanism, as well as to provide a way to efﬁciently 6 calculate streamer parameters. Previously, we demonstrated that SPM results compared well 7 with a limited set of experimental data. In this Brief Report, we compare SPM predictions to the 8 published hydrodynamic simulation (HDS) results. 9


Introduction
Electric streamer discharges, or simply streamers, are ionized columns in gas or 14 liquid which advance by ionizing the material in front of them with the enhanced field 15 at the streamer tip [1,2]. They are an important stage in the formation of sparks, and 16 thus, especially those propagating in air, play a huge role both in technology and natural 17 phenomena such as lightning and Red Sprites. Quantifying streamer properties at high 18 altitudes is important for understanding how lightning interacts with airplanes [e.g., 3,4]. 19 Beside being affected by diverse background conditions, streamer properties may not 20 simply scale in proportion to air density: in particular, the positive streamer threshold 21 field may have nonlinear dependence on air density [5]. 22 Raether [6], Meek [7], and Loeb and Meek [8] were the first to propose that electrons, 23 when undergoing impact ionization avalanche in high electric field in air, create sufficient 24 space charge to form a streamer. In the process of the avalanche-to-streamer transition, 25 electron diffusion plays a crucial role as it determines the transverse size of the avalanche. 26 The same authors also proposed the mechanism of streamer propagation in air, which 27 is based on photoionization. The mechanism works in the following way: (1) UV 28 photons are generated in the streamer head by de-excitation of N 2 ; (2) photons propagate The physics determining the parameters of a propagating streamer discharge in 36 air, such as its radius and speed, had been a long-standing problem [9]. Even though 37 the lateral spreading of an avalanche is due to electron diffusion in the avalanche-to-38 streamer transition, it may be shown that the diffusion is not the main mechanism due 39 to which the streamer acquires its finite radius [10][11][12]. In the present work, we calculate 40 corrections to streamer parameters due to electron diffusion and demonstrate that they 41 are insignificant.

42
The usual approach to theoretical studies of streamers is the numerical solution of ally intensive as they need to have many spatial grid cells in order to resolve well the thin 46 ionization front. This is complicated by the need to resolve other spatial scales, which are 47 very different: the streamer head, which may have a radius of two orders of magnitude 48 larger than the ionization front thickness, and the streamer length, which may be at least 49 an order of magnitude larger than the radius. Despite considerable development effort,

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HDS still remain challenging as the computational stability and accuracy is achieved 51 only at small grid cell sizes, and therefore, a large number of cells. There exist even 52 more complicated numerical models that attempt to include kinetic effects, such as 53 particle-in-cell (PIC) and hybrid codes. A brief review of the HDS modeling efforts is 54 given in [13].

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With the Streamer Parameter Model (SPM), we attempt to uncover the mechanism 56 responsible for the emergence of streamer parameters, and at the same time develop an 57 efficient algorithm for their computation. In Section 3, we demonstrate that SPM results 58 compare reasonably well to those of HDS. This Brief Report applies SPM to positive 59 streamers, however, SPM also makes predictions about negative streamers: in particular, 60 the negative streamer threshold field is calculated to be E −t ≈ 1 MV/m, which is also 61 observed in experiments [2, p. 362], and, according to the theory of Lehtinen [11,12], is 62 not due to electron attachment process.

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The details of the Streamer Parameter Model (SPM) are given in [11,12], and also in 65 an unpublished manuscript [14]. Here, we give a quick overview of the key points of 66 the model.

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The streamer under consideration grows with velocity V from a planar electrode, in 68 constant uniform electric field E e (see Figure 2 in either [11] or [12]). It has a shape of a 69 cylinder (channel), which is attached to the electrode on one end and has a hemispherical V and a.

87
The algebraic equations corresponding to these relations and the references to the works 88 in which they were originally discussed are given in Lehtinen [11,12].

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The system of equations SPM1-SPM4 is sufficient to uniquely determine the set of 90 streamer parameters only if the radius, a, is fixed. This led Lehtinen [11,12]  and the closest analogy to the growth rate that we can find is the streamer velocity, V.

108
Thus, we propose that the parameters of a physical streamer are described by the system 109 of equations SPM1-SPM4, with a 'preferred' or 'most unstable' radius a at which V is smaller. The smaller field yields smaller ionization rate ν t (E m ), thus V again declines. 129 We do not exclude the possibility that there exists a gas in which ν t does not decline fast 130 enough with growing a, and therefore V(a) does not have a maximum. SPM predicts 131 that in such a gas formation of a streamer discharge would be impossible.

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In our earlier work [11,12], we compared SPM predictions to limited experimental 134 results by Allen and Mikropoulos [20]. Only velocity at streamer length L = 12 cm was 135 compared, in a wide range of background fields E e , with discrepancy not exceeding 136 ∼ 30%. In this work, we compare SPM to hydrodynamic simulations (HDS), which 137 were performed by several research groups and presented by Bagheri et al. [13]. Such a 138 comparison is grounded in the hypothesis that in HDS, as in nature, the preferred, i.e., 139 the most unstable, mode of the streamer propagation is also being selected. 140 We consider the same three test cases as Bagheri et al. [13] for positive streamers in 141 dry air at 1 bar and 300 K:  Brief Report, we label them as 'Luque', 'Bourdon2', and 'Bourdon3', similarly to 149 [13].

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Streamer discharges in HDS [13] were simulated between planar electrodes in a square the electron diffusion as a small correction, which is also derived in the Appendix.

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For consistency, we used the same functional dependence on electric field E for 161 ionization rate ν i , attachment rate ν a , electron mobility µ and electron diffusion coefficient 162 D as Bagheri et al. [13]. (In this Brief Report, we often use quantities which are derived 163 from these, namely net ionization rate ν t = ν i − ν a and electron drift velocity v = µE.) 164 To model photoionization in Case 3, we used the same approximations as Bagheri et al.

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[13] instead of the original Zheleznyak et al. [21] expression, which was used in [11,12]. for Case 1 in the form of a plot in [13], but they were available for Cases 2 and 3; we 179 presented SPM calculations of a in Figure 1d anyway.

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For sufficiently fine grids good agreement was reached between several HDS codes 181 [13]. We observe that SPM also produces reasonable agreement with HDS, reproducing 182 the same qualitative features:   In the second case, the background electron number density is n e = 10 9 m −3 .

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The results are presented in Figure 2, with the same notations as in Figure 1. However, SPM in general produced higher fields E m than HDS, as seen in Figure 2b.  were more significant than the type of approximation [13], and only 'Bourdon3' approxi-221 mation HDS results are shown in the Figure. The differences in L(t) and E m (L) due to 222 photoionization approximation choice were presented in Figure 16 of [13]. However, 223 they were too small to draw parallels with analogous differences in SPM.    [11,12], so we just briefly mention them here:

Discussion
Radius a enters system SPM1-SPM4 in relations that describe processes at the tip 232 of the streamer. Therefore, value of a is more relevant to the tip curvature radius, 233 than to the radius of the channel, which may be different.

3.
Assumption of E s = const along the channel follows from n s = const taken together 242 with the assumption of constant current, J = en s v(E s ) = const.

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The last assumption deserves more discussion as it may not be valid in some situations.

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By taking the channel current to be constant along the channel, we assumed that the 245 surface charges on the walls of the channel do not change as the streamer grows, and 246 the new charges are formed only at its head. This assumption seems to be valid for 247 propagating streamers, but breaks down, e.g., for steady-state streamer propagation at 248 E +t [22], in which the charges on the walls of the channel change with time, namely, 249 drop to zero towards the tail of a finite-length streamer as it moves through the air. In 250 the future versions of SPM, we plan to include E s and n s not as single numbers, but as 251 1D variables that vary along the channel, in order to correctly describe such situations.

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This will allow to study, e.g., the nature of positive streamer threshold field and positive 253 streamer inception. Understanding and predicting electric field thresholds for streamer 254 inception in diverse conditions inside clouds is going to contribute to understanding 255 airplane-lightning interactions. (also called net Townsend coefficient), taken at the maximum field E m . We may estimate 262 d as [11,12] 263  threshold of E −t ≈ 1 MV/m, which was calculated by Lehtinen [11,12].

273
The ionization front thickness is the shortest spatial scale that has to be resolved 274 in discretized solution methods, such as HDS. The usual criterion for the choice of grid 275 step ∆x used in HDS is [13,[23][24][25][26]: At the streamer front, this is equivalent to d 0 /∆x = 1/C, or i.e., the ionization front thickness is resolved by N g spatial grid points. For a usual 278 situation V v(E m ), criterion (1) works well even for C ∼ 1 because even then N g 1.

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However, for short or narrow streamers, which propagate slowly, velocity V is rather 280 low and may be even lower than the electron drift speed, v(E m ). Below positive streamer 281 threshold E +t , propagating streamers slow down and eventually stop [27]. When such 282 streamers stop propagating, in addition to declining velocity V, we also have shrinking 283 radius a → 0 as well as increasing electric field E m → ∞, which exacerbates the situation, 284 because d 0 decreases as well. However, in these situations non-local effects [28] also 285 need to be incorporated into the HDS model.  Figure 11 of Bagheri et al. [13]  We solve the following continuity equation for electron number density n, in the 329 presence of impact ionization, diffusion and photoionization: We use the same notations as Lehtinen [11,12]: V is the streamer velocity; ξ = x − Vt 331 is the co-moving coordinate along the streamer axis, with ξ = 0 corresponding to the 332 streamer front; ∂ ξ denotes the derivative in respect to ξ; n(ξ) is the electron number 333 density on the axis; v is the electron drift velocity; ν t is the net ionization rate; s ph (ξ) 334 is the source of free electrons due to photoionization. The upper (lower) sign is for 335 a positive (negative) streamer. In addition to terms included by Lehtinen [11,12], we introduced the diffusion term with coefficient D. Values of ν t , v, D are functions of 337 electric field E, which, in turn, is a function of ξ. 338 If the diffusion term is neglected, the solution of (A1) is The integration constant, C = n e [V ± v(E e )], is obtained from the boundary condition 340 n(∞) = n e , where n e is the initial background electron number density. By equating 341 n(0) = n s , we get the following condition: Without photoionization (the first term on the left-hand side) this expression is the same 343 as equation (4)  should be used instead of SPM4 in the system SPM1-SPM4, when background electron 346 number density n e = 0.

347
Function K a ph (ξ) is given by and is dependent only on ξ and a ph , which is the effective streamer head radius when 349 it acts as the source of photons (Lehtinen [11,12] assumed a ph = a/2) and K(r) is the 350 kernel of the integral transform which turns S i = ν i n ≈ ν t n into s ph [21]. In HDS, K(r) is

352
Let us now tackle the correction due to diffusion and demonstrate that it is small.

353
From now on, we neglect the photoionization term in (A1), since the diffusion is impor-354 tant only in the region where n is already high and the impact ionization term dominates 355 as the source of free electrons. Substitute 356 ∂ ξ n = − ν t ± ∂ ξ v V ± v n into the diffusion term in (A1) and transfer it to the left-hand side: This looks like (A1) without diffusion, with substitution which we can also make in formula (A2) to get condition SPM4 in the next order of