# On the Problem of Critical Electric Field of Atmospheric Air

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Ambient Conditions

- Air temperature (T) and pressure (p) altitude distributions—Table 1 from the peer-reviewed web resource [8];
- Water vapor number density ([${\mathrm{H}}_{2}\mathrm{O}$])—Figure 2 from the peer-reviewed web resource [8];
- Atomic oxygen number density ([$\mathrm{O}$])—Figure VIII-10 from [9];
- Ozone number density ([${\mathrm{O}}_{3}$])—Figure 1.4 from [10];
- Nitric oxide (II) number density ([$\mathrm{NO}$])—Figure 1 from [11];
- Nitric oxide (IV) number density ([${\mathrm{NO}}_{2}$])—Figure 5 from [12];
- Nitric oxide number density ([${\mathrm{NO}}_{3}$])—Figure 1(b) from [13];
- Nitric oxide (I) number density ([${\mathrm{N}}_{2}\mathrm{O}$])—Figure 8 from [14].

#### 2.2. Evolution Matrix

- 1.
- How does the critical electric field of atmospheric air depend on altitude AMSL?
- 2.
- How does the presence of SGCs influence the critical electric field altitude profile?
- 3.
- How does the effective ionization frequency depend on electric field and altitude AMSL?
- 4.
- How does the composition of charged particles (electrons and ions) vary with both electric field and altitude AMSL?
- 5.
- What is the ratio of detachment frequency to ionization frequency at different electric fields and altitudes AMSL?

## 3. Results

## 4. Discussion

## 5. Conclusions

- 1.
- The critical electric field of atmospheric air, at which the multiplication of charged particles begins, is significantly smaller than the conventional value, mostly due to electrons’ detachment from negative ions. The gap between conventional and nonconventional thresholds increases with increasing altitude AMSL from 15% at the ground level to 50% at the height of 40 km.
- 2.
- The presence of SGCs does not significantly influence the critical electric field.
- 3.
- Close to the critical threshold, the effective ionization frequency is a sharp function of the reduced electric field. The rate of its growth decreases with increasing altitude AMSL which partially compensates for the critical electric field reduction.
- 4.
- Above the critical electric field, ionized air contains some amount of free electrons. Their relative share in “community” of negatively charged particles, which can be expressed via the ratio of effective detachment frequency to the attachment frequency, generally increases with increasing reduced electric field.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AMSL | Above mean sea level |

SGC | Small gas component |

## Appendix A. Model Reactions

Reaction | Frequency, s${}^{-1}$ | Rate Constant k, m${}^{-3}$s${}^{-1}$ or m${}^{-6}$s${}^{-1}$ | Study |
---|---|---|---|

Impact ionization | |||

(1a) $e+{\mathrm{N}}_{2}\Rightarrow 2e+{\mathrm{N}}_{2}^{+}$ | ${\nu}_{i1}=\left[{\mathrm{N}}_{2}\right]k$ | $\begin{array}{c}{10}^{-14.09-402.9/\stackrel{\u02d8}{E}},\stackrel{\u02d8}{E}=80--300\phantom{\rule{4.pt}{0ex}}\mathrm{Td}\\ {10}^{-13.37-618.1/\stackrel{\u02d8}{E}},\stackrel{\u02d8}{E}=300--1000\phantom{\rule{4.pt}{0ex}}\mathrm{Td}\end{array}$ | [19] |

(1b) $e+{\mathrm{O}}_{2}\Rightarrow 2e+{\mathrm{O}}_{2}^{+}$ | ${\nu}_{i2}=\left[{\mathrm{O}}_{2}\right]k$ | $\begin{array}{c}{10}^{-14.31-285.7/\stackrel{\u02d8}{E}},\stackrel{\u02d8}{E}=60--260\phantom{\rule{4.pt}{0ex}}\mathrm{Td}\\ (1+4\xb7{10}^{-10}{\stackrel{\u02d8}{E}}^{3})\times \\ \times {10}^{-13.54-485.7/\stackrel{\u02d8}{E}},\stackrel{\u02d8}{E}=260--1000\phantom{\rule{4.pt}{0ex}}\mathrm{Td}\end{array}$ | [19] |

Electron attachment to neutrals | |||

(2a) $e+{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}^{-}+\mathrm{O}$ | ${\nu}_{a1}=\left[{\mathrm{O}}_{2}\right]k$ | $\begin{array}{cc}\hfill {10}^{-15.42-127/\stackrel{\u02d8}{E}},& \stackrel{\u02d8}{E}=30--90\phantom{\rule{4.pt}{0ex}}\mathrm{Td}\hfill \\ \hfill {10}^{-16.21-57/\stackrel{\u02d8}{E}},& \stackrel{\u02d8}{E}=90--300\phantom{\rule{4.pt}{0ex}}\mathrm{Td}\hfill \end{array}$ | [19] |

(2b) $e+2{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}_{2}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{a2}={\left[{\mathrm{O}}_{2}\right]}^{2}k$ | $\begin{array}{c}\hfill 1.4\xb7{10}^{-41}(300/{T}_{e})\xb7exp(-600/T)\times \\ \hfill \times exp(700({T}_{e}-T)/({T}_{e}\xb7T))\end{array}$ | [17] |

(2c) $e+{\mathrm{O}}_{2}+{\mathrm{N}}_{2}\Rightarrow {\mathrm{O}}_{2}^{-}+{\mathrm{N}}_{2}$ | ${\nu}_{a3}=\left[{\mathrm{O}}_{2}\right]\left[{\mathrm{N}}_{2}\right]k$ | $\begin{array}{c}\hfill 1.07\xb7{10}^{-43}{(300/{T}_{e})}^{2}\xb7exp(-70/T)\times \\ \hfill \times exp(1500({T}_{e}-T)/({T}_{e}\xb7T))\end{array}$ | [17] |

(2d) $e+{\mathrm{O}}_{2}+\mathrm{O}\Rightarrow {\mathrm{O}}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{a4}=\left[{\mathrm{O}}_{2}\right]\left[\mathrm{O}\right]k$ | ${10}^{-43}$ | [17] |

(2e) $e+{\mathrm{O}}_{2}+\mathrm{O}\Rightarrow {\mathrm{O}}_{2}^{-}+\mathrm{O}$ | ${\nu}_{a5}=\left[{\mathrm{O}}_{2}\right]\left[\mathrm{O}\right]k$ | ${10}^{-43}$ | [17] |

(2f) $e+{\mathrm{O}}_{3}\Rightarrow {\mathrm{O}}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{a6}=\left[{\mathrm{O}}_{3}\right]k$ | ${10}^{-17}$ | [17] |

(2g) $e+{\mathrm{O}}_{3}\Rightarrow {\mathrm{O}}_{2}^{-}+\mathrm{O}$ | ${\nu}_{a7}=\left[{\mathrm{O}}_{3}\right]k$ | ${10}^{-15}$ | [17] |

(2h) $e+{\mathrm{O}}_{2}+{\mathrm{O}}_{3}\Rightarrow {\mathrm{O}}_{3}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{a8}=\left[{\mathrm{O}}_{2}\right]\left[{\mathrm{O}}_{3}\right]k$ | the same as in (2b) | [17] |

(2i) $e+\mathrm{M}+\mathrm{NO}\Rightarrow {\mathrm{NO}}^{-}+\mathrm{M}$ | ${\nu}_{a9}=\left[\mathrm{M}\right]\left[\mathrm{NO}\right]k$ | ${10}^{-42}$ | [17] |

(2g) $e+{\mathrm{NO}}_{2}\Rightarrow {\mathrm{O}}^{-}+\mathrm{NO}$ | ${\nu}_{a10}=\left[{\mathrm{NO}}_{2}\right]k$ | ${10}^{-17}$ | [17] |

(2k) $e+{\mathrm{NO}}_{2}(+\mathrm{M})\Rightarrow {\mathrm{NO}}_{2}^{-}(+\mathrm{M})$ | ${\nu}_{a11}=\left[{\mathrm{NO}}_{2}\right]k$ | $3\xb7{10}^{-17}$ | [17] |

(2l) $\begin{array}{c}e+{\mathrm{O}}_{2}+{\mathrm{H}}_{2}\mathrm{O}\Rightarrow \\ \Rightarrow {\mathrm{O}}_{2}^{-}+{\mathrm{H}}_{2}\mathrm{O}\end{array}$ | ${\nu}_{a12}=\left[{\mathrm{O}}_{2}\right]\left[{\mathrm{H}}_{2}\mathrm{O}\right]k$ | $1.4\xb7{10}^{-41}$ | [20] |

Electron detachment from negative ions | |||

(3a) ${\mathrm{O}}^{-}+{\mathrm{N}}_{2}\Rightarrow {\mathrm{N}}_{2}\mathrm{O}+e$ | ${\nu}_{d1}=\left[{\mathrm{N}}_{2}\right]k$ | $\begin{array}{c}\hfill 1.16\xb7{10}^{-18}\xb7exp(-{(48.9/(11+\stackrel{\u02d8}{E}))}^{2})\end{array}$ | [2] |

(3b) ${\mathrm{O}}^{-}+{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}_{3}+e$ | ${\nu}_{d2}=\left[{\mathrm{O}}_{2}\right]k$ | $5\xb7{10}^{-21}$ | [17] |

(3c) ${\mathrm{O}}_{2}^{-}+\mathrm{M}\Rightarrow {\mathrm{O}}_{2}+\mathrm{M}+e$ | ${\nu}_{d3}=\left[\mathrm{M}\right]k$ | $\begin{array}{c}\hfill 1.24\xb7{10}^{-17}\xb7exp(-{(179/(8.8+\stackrel{\u02d8}{E}))}^{2})\end{array}$ | [2] |

(3d) ${\mathrm{O}}^{-}+\mathrm{O}\Rightarrow {\mathrm{O}}_{2}+e$ | ${\nu}_{d4}=\left[\mathrm{O}\right]k$ | $5\xb7{10}^{-16}$ | [17] |

(3e) ${\mathrm{O}}^{-}+\mathrm{NO}\Rightarrow {\mathrm{NO}}_{2}+e$ | ${\nu}_{d5}=\left[\mathrm{NO}\right]k$ | $2.6\xb7{10}^{-16}$ | [17] |

(3f) ${\mathrm{O}}_{2}^{-}+\mathrm{O}\Rightarrow {\mathrm{O}}_{3}+e$ | ${\nu}_{d6}=\left[\mathrm{O}\right]k$ | $1.5\xb7{10}^{-16}$ | [17] |

(3g) ${\mathrm{O}}_{3}^{-}+\mathrm{O}\Rightarrow 2{\mathrm{O}}_{2}+e$ | ${\nu}_{d7}=\left[\mathrm{O}\right]k$ | $3\xb7{10}^{-16}$ | [17] |

(3h) ${\mathrm{NO}}_{2}^{-}+\mathrm{O}\Rightarrow {\mathrm{NO}}_{3}+e$ | ${\nu}_{d8}=\left[\mathrm{O}\right]k$ | ${10}^{-18}$ | [17] |

Ion-ion conversion without nitrogen oxides | |||

(4a) $\begin{array}{c}{\mathrm{O}}^{-}+{\mathrm{O}}_{2}\Rightarrow {({\mathrm{O}}_{3}^{-})}^{*}\Rightarrow \\ \Rightarrow {\mathrm{O}}_{2}^{-}+\mathrm{O}\end{array}$ | ${\nu}_{c1}=\left[{\mathrm{O}}_{2}\right]k$ | $\begin{array}{c}6.96\xb7{10}^{-17}\xb7exp(-{(198/(5.6+\stackrel{\u02d8}{E}))}^{2})\end{array}$ | [2] |

(4b) $\begin{array}{c}{\mathrm{O}}^{-}+{\mathrm{O}}_{2}+\mathrm{M}\Rightarrow \\ \Rightarrow {({\mathrm{O}}_{3}^{-})}^{*}+\mathrm{M}\Rightarrow {\mathrm{O}}_{3}^{-}+\mathrm{M}\end{array}$ | ${\nu}_{c2}=\left[{\mathrm{O}}_{2}\right]\left[\mathrm{M}\right]k$ | $\begin{array}{c}1.1\xb7{10}^{-42}\xb7exp(-{(\stackrel{\u02d8}{E}/65)}^{2})\end{array}$ | [2] |

(4c) ${\mathrm{O}}^{-}+{\mathrm{O}}_{3}\Rightarrow {\mathrm{O}}_{3}^{-}+\mathrm{O}$ | ${\nu}_{c3}=\left[{\mathrm{O}}_{3}\right]k$ | $5.3\xb7{10}^{-16}$ | [17] |

(4d) $\begin{array}{c}{\mathrm{O}}_{2}^{-}+{\mathrm{O}}_{2}+\mathrm{M}\Rightarrow {\mathrm{O}}_{4}^{-}+\mathrm{M}\end{array}$ | ${\nu}_{c4}=\left[{\mathrm{O}}_{2}\right]\left[\mathrm{M}\right]k$ | $3.5\xb7{10}^{-43}\times (300/T)$ | [17] |

(4e) ${\mathrm{O}}_{2}^{-}+\mathrm{O}\Rightarrow {\mathrm{O}}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{c5}=\left[\mathrm{O}\right]k$ | $3.3\xb7{10}^{-16}$ | [17] |

(4f) ${\mathrm{O}}_{2}^{-}+{\mathrm{O}}_{3}\Rightarrow {\mathrm{O}}_{3}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{c6}=\left[{\mathrm{O}}_{3}\right]k$ | $4\xb7{10}^{-16}$ | [17] |

(4g) ${\mathrm{O}}_{3}^{-}+\mathrm{O}\Rightarrow {\mathrm{O}}_{2}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{c7}=\left[\mathrm{O}\right]k$ | $3.2\xb7{10}^{-16}$ | [17] |

(4h) ${\mathrm{O}}_{4}^{-}+\mathrm{M}\Rightarrow {\mathrm{O}}_{2}^{-}+{\mathrm{O}}_{2}+\mathrm{M}$ | ${\nu}_{c8}=\left[\mathrm{M}\right]k$ | ${10}^{-16}\xb7exp(-1044/T)$ | [17] |

(4i) ${\mathrm{O}}_{4}^{-}+\mathrm{O}\Rightarrow {\mathrm{O}}^{-}+2{\mathrm{O}}_{2}$ | ${\nu}_{c9}=\left[\mathrm{O}\right]k$ | $3\xb7{10}^{-16}$ | [17] |

(4j) ${\mathrm{O}}_{4}^{-}+\mathrm{O}\Rightarrow {\mathrm{O}}_{3}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{c10}=\left[\mathrm{O}\right]k$ | $4\xb7{10}^{-16}$ | [17] |

(4k) ${\mathrm{N}}_{2}^{+}+2{\mathrm{N}}_{2}\Rightarrow {\mathrm{N}}_{4}^{+}+{\mathrm{N}}_{2}$ | ${\nu}_{c11}={\left[{\mathrm{N}}_{2}\right]}^{2}k$ | $5\xb7{10}^{-41}$ | [17] |

(4l) ${\mathrm{N}}_{2}^{+}+{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}_{2}^{+}+{\mathrm{N}}_{2}$ | ${\nu}_{c12}=\left[{\mathrm{O}}_{2}\right]k$ | $6\xb7{10}^{-17}{(300/T)}^{0.5}$ | [17] |

(4m) ${\mathrm{N}}_{2}^{+}+{\mathrm{O}}_{3}\Rightarrow {\mathrm{O}}_{2}^{+}+\mathrm{O}+{\mathrm{N}}_{2}$ | ${\nu}_{c13}=\left[{\mathrm{O}}_{3}\right]k$ | ${10}^{-16}$ | [17] |

(4n) ${\mathrm{N}}_{2}^{+}+{\mathrm{H}}_{2}\mathrm{O}\Rightarrow {\mathrm{H}}_{2}{\mathrm{O}}^{+}+{\mathrm{N}}_{2}$ | ${\nu}_{c14}=\left[{\mathrm{H}}_{2}\mathrm{O}\right]k$ | $1.8\xb7{10}^{-15}$ | [21] |

(4o) $\begin{array}{c}{\mathrm{O}}_{2}^{+}+2{\mathrm{N}}_{2}\Rightarrow {\mathrm{O}}_{2}^{+}\xb7{\mathrm{N}}_{2}+{\mathrm{N}}_{2}\end{array}$ | ${\nu}_{c15}={\left[{\mathrm{N}}_{2}\right]}^{2}k$ | $9\xb7{10}^{-43}\xb7{(300/T)}^{2}$ | [17] |

(4p) ${\mathrm{O}}_{2}^{+}+2{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}_{4}^{+}+{\mathrm{O}}_{2}$ | ${\nu}_{c16}={\left[{\mathrm{O}}_{2}\right]}^{2}k$ | $2.4\xb7{10}^{-42}{(300/T)}^{3.2}$ | [17] |

(4q) ${\mathrm{O}}_{4}^{+}+{\mathrm{N}}_{2}\Rightarrow {\mathrm{O}}_{2}^{+}\xb7{\mathrm{N}}_{2}+{\mathrm{O}}_{2}$ | ${\nu}_{c17}=\left[{\mathrm{N}}_{2}\right]k$ | $4.61\xb7{10}^{-18}\xb7{(T/300)}^{2.5}\xb7exp(-2650/T)$ | [17] |

(4r) ${\mathrm{O}}_{4}^{+}+{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}_{2}^{+}+2{\mathrm{O}}_{2}$ | ${\nu}_{c18}=\left[{\mathrm{O}}_{2}\right]k$ | $3.3\xb7{10}^{-12}\xb7{(300/T)}^{4}\xb7exp(-5030/T)$ | [17] |

(4s) ${\mathrm{O}}_{4}^{+}+\mathrm{O}\Rightarrow {\mathrm{O}}_{2}^{+}+{\mathrm{O}}_{3}$ | ${\nu}_{c19}=\left[\mathrm{O}\right]k$ | $3\xb7{10}^{-16}$ | [17] |

(4t) ${\mathrm{N}}_{4}^{+}+{\mathrm{N}}_{2}\Rightarrow {\mathrm{N}}_{2}^{+}+2{\mathrm{N}}_{2}$ | ${\nu}_{c20}=\left[{\mathrm{N}}_{2}\right]k$ | ${10}^{-20.6+0.0036(T-300)},\phantom{\rule{4.pt}{0ex}}T=300$–$900\phantom{\rule{4.pt}{0ex}}\mathrm{K}$ | [17] |

(4u) ${\mathrm{N}}_{4}^{+}+{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}_{2}^{+}+2{\mathrm{N}}_{2}$ | ${\nu}_{c21}=\left[{\mathrm{O}}_{2}\right]k$ | $2.5\xb7{10}^{-16}$ | [17] |

(4v) ${\mathrm{N}}_{4}^{+}+{\mathrm{H}}_{2}\mathrm{O}\Rightarrow {\mathrm{H}}_{2}{\mathrm{O}}^{+}+2{\mathrm{N}}_{2}$ | ${\nu}_{c22}=\left[{\mathrm{H}}_{2}\mathrm{O}\right]k$ | $2.4\xb7{10}^{-15}$ | [21] |

(4w) $\begin{array}{c}{\mathrm{O}}_{2}^{+}\xb7{\mathrm{N}}_{2}+{\mathrm{N}}_{2}\Rightarrow {\mathrm{O}}_{2}^{+}+2{\mathrm{N}}_{2}\end{array}$ | ${\nu}_{c23}=\left[{\mathrm{N}}_{2}\right]k$ | $1.1\xb7{10}^{-12}\xb7{(300/T)}^{5.3}\xb7exp(-2357/T)$ | [17] |

(4x) ${\mathrm{O}}_{2}^{+}\xb7{\mathrm{N}}_{2}+{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}_{4}^{+}+{\mathrm{O}}_{2}$ | ${\nu}_{c24}=\left[{\mathrm{O}}_{2}\right]k$ | ${10}^{-15}$ | [17] |

(4y) ${\mathrm{H}}_{2}{\mathrm{O}}^{+}+{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}_{2}^{+}+{\mathrm{H}}_{2}\mathrm{O}$ | ${\nu}_{c25}=\left[{\mathrm{O}}_{2}\right]k$ | $4.1\xb7{10}^{-16}$ | [21] |

Ion-ion conversion involving nitrogen oxides | |||

(5a) ${\mathrm{O}}^{-}+{\mathrm{N}}_{2}\mathrm{O}\Rightarrow {\mathrm{NO}}^{-}+\mathrm{NO}$ | ${\nu}_{n1}=\left[{\mathrm{N}}_{2}\mathrm{O}\right]k$ | $2\xb7{10}^{-16}$ | [17] |

(5b) ${\mathrm{O}}^{-}+\mathrm{NO}+\mathrm{M}\Rightarrow {\mathrm{NO}}_{2}^{-}+\mathrm{M}$ | ${\nu}_{n2}=\left[\mathrm{NO}\right]\left[\mathrm{M}\right]k$ | ${10}^{-41}$ | [17] |

(5c) ${\mathrm{O}}^{-}+{\mathrm{NO}}_{2}\Rightarrow {\mathrm{NO}}_{2}^{-}+\mathrm{O}$ | ${\nu}_{n3}=\left[{\mathrm{NO}}_{2}\right]k$ | $1.2\xb7{10}^{-15}$ | [17] |

(5d) ${\mathrm{O}}_{2}^{-}+{\mathrm{N}}_{2}\mathrm{O}\Rightarrow {\mathrm{O}}_{3}^{-}+{\mathrm{N}}_{2}$ | ${\nu}_{n4}=\left[{\mathrm{N}}_{2}\mathrm{O}\right]k$ | ${10}^{-18}$ | [17] |

(5e) ${\mathrm{O}}_{2}^{-}+{\mathrm{NO}}_{2}\Rightarrow {\mathrm{NO}}_{2}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{n5}=\left[{\mathrm{NO}}_{2}\right]k$ | $8\xb7{10}^{-16}$ | [17] |

(5f) ${\mathrm{O}}_{2}^{-}+{\mathrm{NO}}_{3}\Rightarrow {\mathrm{NO}}_{3}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{n6}=\left[{\mathrm{NO}}_{3}\right]k$ | $5\xb7{10}^{-16}$ | [17] |

(5g) ${\mathrm{O}}_{3}^{-}+\mathrm{NO}\Rightarrow {\mathrm{NO}}_{2}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{n7}=\left[\mathrm{NO}\right]k$ | $2.6\xb7{10}^{-18}$ | [17] |

(5h) ${\mathrm{O}}_{3}^{-}+{\mathrm{NO}}_{2}\Rightarrow {\mathrm{NO}}_{2}^{-}+{\mathrm{O}}_{3}$ | ${\nu}_{n8}=\left[{\mathrm{NO}}_{2}\right]k$ | $7\xb7{10}^{-16}$ | [17] |

(5i) ${\mathrm{O}}_{3}^{-}+\mathrm{NO}\Rightarrow {\mathrm{NO}}_{3}^{-}+\mathrm{O}$ | ${\nu}_{n9}=\left[\mathrm{NO}\right]k$ | ${10}^{-17}$ | [17] |

(5j) ${\mathrm{O}}_{3}^{-}+{\mathrm{NO}}_{2}\Rightarrow {\mathrm{NO}}_{3}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{n10}=\left[{\mathrm{NO}}_{2}\right]k$ | $2\xb7{10}^{-17}$ | [17] |

(5k) ${\mathrm{O}}_{3}^{-}+{\mathrm{NO}}_{3}\Rightarrow {\mathrm{NO}}_{3}^{-}+{\mathrm{O}}_{3}$ | ${\nu}_{n11}=\left[{\mathrm{NO}}_{3}\right]k$ | $5\xb7{10}^{-16}$ | [17] |

(5l) ${\mathrm{O}}_{4}^{-}+\mathrm{NO}\Rightarrow {\mathrm{NO}}_{3}^{-*}+{\mathrm{O}}_{2}$ | ${\nu}_{n12}=\left[\mathrm{NO}\right]k$ | $2.5\xb7{10}^{-16}$ | [17] |

(5m) ${\mathrm{NO}}^{-}+{\mathrm{O}}_{2}\Rightarrow {\mathrm{O}}_{2}^{-}+\mathrm{NO}$ | ${\nu}_{n13}=\left[{\mathrm{O}}_{2}\right]k$ | $5\xb7{10}^{-16}$ | [17] |

(5n) ${\mathrm{NO}}^{-}+{\mathrm{NO}}_{2}\Rightarrow {\mathrm{NO}}_{2}^{-}+\mathrm{NO}$ | ${\nu}_{n14}=\left[{\mathrm{NO}}_{2}\right]k$ | $7.4\xb7{10}^{-22}$ | [17] |

(5o) ${\mathrm{NO}}^{-}+{\mathrm{N}}_{2}\mathrm{O}\Rightarrow {\mathrm{NO}}_{2}^{-}+{\mathrm{N}}_{2}$ | ${\nu}_{n15}=\left[{\mathrm{N}}_{2}\mathrm{O}\right]k$ | $2.8\xb7{10}^{-20}$ | [17] |

(5p) ${\mathrm{NO}}_{2}^{-}+{\mathrm{O}}_{3}\Rightarrow {\mathrm{NO}}_{3}^{-}+{\mathrm{O}}_{2}$ | ${\nu}_{n16}=\left[{\mathrm{O}}_{3}\right]k$ | $1.8\xb7{10}^{-17}$ | [17] |

(5q) $\begin{array}{c}{\mathrm{NO}}_{2}^{-}+{\mathrm{NO}}_{2}\Rightarrow {\mathrm{NO}}_{3}^{-}+\mathrm{NO}\end{array}$ | ${\nu}_{n17}=\left[{\mathrm{NO}}_{2}\right]k$ | $4\xb7{10}^{-18}$ | [17] |

(5r) $\begin{array}{c}{\mathrm{NO}}_{2}^{-}+{\mathrm{NO}}_{3}\Rightarrow {\mathrm{NO}}_{3}^{-}+{\mathrm{NO}}_{2}\end{array}$ | ${\nu}_{n18}=\left[{\mathrm{NO}}_{3}\right]k$ | $5\xb7{10}^{-16}$ | [17] |

(5s) $\begin{array}{c}{\mathrm{NO}}_{3}^{-}+\mathrm{NO}\Rightarrow {\mathrm{NO}}_{2}^{-}+{\mathrm{NO}}_{2}\end{array}$ | ${\nu}_{n19}=\left[\mathrm{NO}\right]k$ | $3\xb7{10}^{-21}$ | [17] |

(5t) $\begin{array}{c}{\mathrm{NO}}_{3}^{-}+{\mathrm{H}}_{2}\mathrm{O}+{\mathrm{O}}_{2}\Rightarrow \\ \Rightarrow {\mathrm{NO}}_{3}^{-}({\mathrm{H}}_{2}\mathrm{O})+{\mathrm{O}}_{2}\end{array}$ | ${\nu}_{n20}=\left[{\mathrm{H}}_{2}\mathrm{O}\right]\left[{\mathrm{O}}_{2}\right]k$ | $7.5\xb7{10}^{-41}$ | [22] |

(5u) $\begin{array}{c}{\mathrm{NO}}_{3}^{-*}+\mathrm{NO}\Rightarrow {\mathrm{NO}}_{2}^{-}+{\mathrm{NO}}_{2}\end{array}$ | ${\nu}_{n21}=\left[\mathrm{NO}\right]k$ | $1.5\xb7{10}^{-17}$ | [17] |

(5v) $\begin{array}{c}{\mathrm{NO}}_{3}^{-}({\mathrm{H}}_{2}\mathrm{O})+{\mathrm{H}}_{2}\mathrm{O}+{\mathrm{O}}_{2}\Rightarrow \\ \Rightarrow {\mathrm{NO}}_{3}^{-}{({\mathrm{H}}_{2}\mathrm{O})}_{2}+{\mathrm{O}}_{2}\end{array}$ | ${\nu}_{n22}=\left[{\mathrm{H}}_{2}\mathrm{O}\right]\left[{\mathrm{O}}_{2}\right]k$ | $4\xb7{10}^{-41}$ | [22] |

(5w) $\begin{array}{c}{\mathrm{NO}}_{3}^{-}{({\mathrm{H}}_{2}\mathrm{O})}_{2}+{\mathrm{O}}_{2}\Rightarrow \\ \Rightarrow {\mathrm{NO}}_{3}^{-}({\mathrm{H}}_{2}\mathrm{O})+{\mathrm{H}}_{2}\mathrm{O}+{\mathrm{O}}_{2}\end{array}$ | ${\nu}_{n23}=\left[{\mathrm{O}}_{2}\right]k$ | $1.6\xb7{10}^{-19}$ | [22] |

(5x) $\begin{array}{c}{\mathrm{NO}}_{3}^{-}{({\mathrm{H}}_{2}\mathrm{O})}_{2}+{\mathrm{H}}_{2}\mathrm{O}+{\mathrm{O}}_{2}\Rightarrow \\ \Rightarrow {\mathrm{NO}}_{3}^{-}{({\mathrm{H}}_{2}\mathrm{O})}_{3}+{\mathrm{O}}_{2}\end{array}$ | ${\nu}_{n24}=\left[{\mathrm{H}}_{2}\mathrm{O}\right]\left[{\mathrm{O}}_{2}\right]k$ | $3\xb7{10}^{-41}$ | [22] |

(5y) $\begin{array}{c}{\mathrm{NO}}_{3}^{-}{({\mathrm{H}}_{2}\mathrm{O})}_{3}+{\mathrm{O}}_{2}\Rightarrow \\ \Rightarrow {\mathrm{NO}}_{3}^{-}{({\mathrm{H}}_{2}\mathrm{O})}_{2}+{\mathrm{H}}_{2}\mathrm{O}+{\mathrm{O}}_{2}\end{array}$ | ${\nu}_{n25}=\left[{\mathrm{O}}_{2}\right]k$ | $1.1\xb7{10}^{-18}$ | [22] |

## Appendix B. Evolution Matrix Components

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**Figure 2.**Altitude profiles of conventional (${E}_{b}/N$) and critical (${E}_{c}/N$) reduced electric fields calculated for different atmospheric air compositions. At the bottom left part, the insertion shows an enlarged fragment of the nonconventional reduced breakdown field altitude dependence where its profile splits into several modes corresponding to different types of air composition.

**Figure 3.**Dependence curves of the model-predicted effective ionization frequency ${\nu}_{\mathrm{eff}}$ (

**a**) on the reduced electric field $E/N$ for different altitudes AMSL and (

**b**) on the altitude AMSL h for different values of the electric field E. Vertical dashed lines stand for reduced critical electric fields ${E}_{c}/N$ (in panel (

**a**)) and altitudes (in panel (

**b**)) at which the discharge development begins.

**Figure 4.**Dependence curves of charged particles composition on reduced electric field $E/N$ for different altitudes AMSL. The lower reduced electric field limits correspond to critical breakdown fields ${E}_{c}$ at the considered altitudes. Components with negligible relative fractions are not shown.

**Figure 5.**Dependence curves of charged particles composition on altitude AMSL h for several fixed electric fields E. The lower altitude limits correspond to heights at which $E={E}_{c}$. Components with negligible relative fractions are not shown.

**Figure 6.**Dependence curves of the ratio ${\nu}_{d}^{\mathrm{eff}}/{\nu}_{a}$ on (

**a**) reduced electric field $E/N$ for different altitudes AMSL and (

**b**) on altitude h for several fixed values of E. Vertical dashed lines stand for reduced critical electric fields ${E}_{c}/N$ (in panel (

**a**)) and altitudes (in panel (

**b**)) at which the discharge development begins.

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**MDPI and ACS Style**

Syssoev, A.; Iudin, D.; Iudin, F.; Klimashov, V.; Emelyanov, A.
On the Problem of Critical Electric Field of Atmospheric Air. *Atmosphere* **2021**, *12*, 1046.
https://doi.org/10.3390/atmos12081046

**AMA Style**

Syssoev A, Iudin D, Iudin F, Klimashov V, Emelyanov A.
On the Problem of Critical Electric Field of Atmospheric Air. *Atmosphere*. 2021; 12(8):1046.
https://doi.org/10.3390/atmos12081046

**Chicago/Turabian Style**

Syssoev, Artem, Dmitry Iudin, Fedor Iudin, Vitaly Klimashov, and Alexey Emelyanov.
2021. "On the Problem of Critical Electric Field of Atmospheric Air" *Atmosphere* 12, no. 8: 1046.
https://doi.org/10.3390/atmos12081046