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

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## Abstract

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## 1. Introduction

## 2. Streamer Parameter Model (SPM)

- SPM1: Electrostatic relationship between ${E}_{e}$, ${E}_{m}$, ${E}_{s}$: the field at the streamer tip is enhanced because of available voltage due to difference in ${E}_{e}$ and ${E}_{s}$: $\Delta U=({E}_{e}-{E}_{s})L$.
- SPM2: Current continuity at the streamer tip: the conductivity current inside the streamer becomes displacement current outside.
- SPM3: Ionization and relaxation balance: the ionization time scale in the region of the highest electric field at the streamer tip is approximately equal to the Maxwellian charge relaxation time inside the streamer.
- SPM4: Photo-ionization and impact ionization balance, which provides the relation between V and a.

## 3. Results

- No photoionization; presence of relatively high background free electron number density of ${n}_{e}={10}^{13}$ m${}^{-3}$.
- No photoionization; presence of relatively low ${n}_{e}={10}^{9}$ m${}^{-3}$.
- With photoionization and ${n}_{e}={10}^{9}$ m${}^{-3}$. Photoionization is treated with three different approximations to the original description by Zheleznyak et al. [20]. These approximations are described in detail in (Bagheri et al. [12] Appendix A). In this article, we label them as “Luque”, “Bourdon2”, and ”Bourdon3”, similarly to [12].

#### 3.1. Case 1

- Velocity V in Figure 1a grows with streamer length (and with time).
- Maximum electric field ${E}_{m}$ decreases with streamer length L, at least for the middle values of L (the discrepancies at low and high L are discussed below).
- Number of electrons N grows with L; the rate of growth also increases with L.
- Radius a grows with length L.

- The HDS did not start at zero L, namely, it started with a small ionized region in the vicinity of the anode. This caused the field ${E}_{m}$ to gradually rise until the streamer was formed (see the low L values in Figure 1b). SPM, on the other hand, provided results only with the streamer already formed.
- At large L, the discrepancy is due to the proximity of the opposite electrode (cathode). The negative image charge of the streamer head, induced in the conducting cathode, enhanced the field in HDS (Figure 1b). The cathode, as we already mentioned, was not taken into account in the SPM.

#### 3.2. Case 2

#### 3.3. Case 3

## 4. Discussion

#### 4.1. Possible Errors Due to Approximations in SPM

- The radius, a, enters the system of SPM1–SPM4 in relations that describe processes at the tip of the streamer. Therefore, the value of a is more relevant to the tip curvature radius, than to the possibly different radius of the channel.
- We assumed that ${n}_{s}=\mathit{const}$ along the axis of the channel. At low external fields ${E}_{e}$, especially those close to the positive streamer threshold ${E}_{+t}\approx 0.45$ MV/m [2] (p. 362), the number of electrons in the channel declines due to attachment. However, for the field ${E}_{e}=1.5$ MV/m, used in this work, attachment in the channel can be neglected, especially when using the attachment coefficient expression from Bagheri et al. [12] which gives lower values than we would get if the 3-body attachment process were included.
- Assumption of ${E}_{s}=\mathit{const}$ along the channel follows from ${n}_{s}=\mathit{const}$ taken together with the assumption of constant current, $J=e{n}_{s}v\left({E}_{s}\right)=\mathit{const}$.

#### 4.2. Shortest Spatial Scale in a Streamer and in HDS

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SPM | Streamer Parameter Model |

HDS | Hydrodynamic simulation(s) |

UV | Ultraviolet |

## Appendix A

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**Figure 1.**Case 1: Initial background electron number density ${n}_{e}={10}^{13}$ m${}^{-3}$, and no photoionization: (

**a**) length L as a function of time t; (

**b**) maximum electric field ${E}_{m}$ as a function of length; (

**c**) total number of produced electrons N as a function of length; (

**d**) streamer radius a as a function of length (HDS results for one team only). Dashed lines denote SPM results with diffusion. HDS results in panels (

**a**–

**d**) were adapted from Figures 5b, 6a,b and 3 in Bagheri et al. [12].

**Figure 2.**Case 2: Initial background electron number density ${n}_{e}={10}^{9}$ m${}^{-3}$, and no photoionization: (

**a**) length L as a function of time t; (

**b**) maximum electric field ${E}_{m}$ as a function of length; (

**c**) total number of produced electrons N as a function of length; (

**d**) streamer radius a as a function of length. Dashed lines denote SPM results with diffusion. HDS results in panels (

**a**–

**d**) were adapted from Figures 8, 9a,b and 10 in Bagheri et al. [12].

**Figure 3.**Case 3: Photoionization is present, initial background electron number density is ${n}_{e}={10}^{9}$ m${}^{-3}$: (

**a**) length L as a function of time t; (

**b**) maximum electric field ${E}_{m}$ as a function of length; (

**c**) total number of produced electrons N as a function of length; (

**d**) streamer radius a as a function of length. Dashed lines denote SPM results with diffusion. HDS results in panels (

**a**–

**d**) are adapted from Figures 13, 14a,b and 15 in Bagheri et al. [12].

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**MDPI and ACS Style**

Lehtinen, N.G.; Marskar, R.
What Determines the Parameters of a Propagating Streamer: A Comparison of Outputs of the Streamer Parameter Model and of Hydrodynamic Simulations. *Atmosphere* **2021**, *12*, 1664.
https://doi.org/10.3390/atmos12121664

**AMA Style**

Lehtinen NG, Marskar R.
What Determines the Parameters of a Propagating Streamer: A Comparison of Outputs of the Streamer Parameter Model and of Hydrodynamic Simulations. *Atmosphere*. 2021; 12(12):1664.
https://doi.org/10.3390/atmos12121664

**Chicago/Turabian Style**

Lehtinen, Nikolai G., and Robert Marskar.
2021. "What Determines the Parameters of a Propagating Streamer: A Comparison of Outputs of the Streamer Parameter Model and of Hydrodynamic Simulations" *Atmosphere* 12, no. 12: 1664.
https://doi.org/10.3390/atmos12121664