The Development and Application of a Three-Dimensional Corona Discharge Numerical Model Considering the Thunderstorm Electric Field Polarity Reversal Process

Abstract

The study of the ground tip corona discharge is an important part of the lightning strike mechanism and lightning warning research. Because the characteristics of the corona charge distribution are difficult to observe directly, simulation research is indispensable. However, most of the previous models have been unipolar models, which cannot reflect the characteristics of the tip corona discharge under electric field reversal during real thunderstorms. Therefore, the development of three-dimensional positive and negative corona discharge models is of great significance. In this study, a three-dimensional corona discharge numerical model considering the polarity reversal process of the electric field was developed with or without a wind field and simulated the tip corona discharge characteristics under this reversal. The reliability of the model was verified by comparing the observed results. Compared with the unipolar corona discharge model, this model could effectively evaluate the impact of the first half-cycle corona discharge on the second half-cycle opposite-polarity corona discharge and invert the spatial separation distribution characteristics of different polar corona charges released in both cycles under the influence of wind and the spatial electric field distribution characteristics generated by the corresponding corona charges. Comparing unipolar corona discharges under the same wave pattern and amplitude of the background electric field, it was assumed that the unipolar corona discharge occurred in the half cycle after the polarity reversal of an electric field, and there was also an opposite-polarity corona discharge process before it. Due to the influence of the first half cycle, the background electric field required for a corona discharge was smaller, and the corona current was generated earlier, but the end time was equivalent. At the same time, due to the neutralization effect of positive and negative corona charges, the peak value of the total corona charge in the second half cycle was significantly smaller than that of the unipolar model. At different building heights, the peak difference in the corona current and the peak difference in the corona charge between the two models increased linearly with an increase in height. It could be seen that this model had better simulation results and wider application value.

1. Introduction

A corona discharge refers to the phenomenon where the electric field intensity on the tip surface exceeds the air breakdown threshold [1,2]. When the ground environment electric field exceeds the threshold, ground grounding objects (such as grounding towers, trees, buildings, and windmills) will produce corona discharge phenomena [3,4]. When the height of the ground grounding object increases, the probability of charge accumulation increases [5,6,7]. In a thunderstorm environment, the higher the tip of the ground grounding object is, the stronger the corona discharge phenomenon is. The corona charge layer generated by the tip corona discharge will affect the distribution of the space electric field. During thunderstorms, the enhancement of the environmental electric field will cause a corona discharge at the tip of the ground object [8,9,10,11,12]. In an actual situation, when a thunderstorm cloud passes through, the ground electric field has a positive and negative polarity alternating charge. Therefore, a three-dimensional positive and negative corona discharge model has an important practical role in studying the corona discharge of ground grounding objects.

Corona discharges are divided into two types: positive and negative corona discharges. The research on the characteristics of corona discharges is divided into three types: laboratory gap discharge experiments, ground tip observation experiments, and corona discharge numerical simulations.

Luo [13] studied tip corona discharge characteristics through a combination of laboratory tests and numerical calculations and obtained the variation trend of positive and negative corona discharge characteristics along with the electric field strength. Tian [14] obtained the motion law and electric field distribution characteristics of charged particles in an AC corona discharge by establishing a microscopic physical model of an AC corona discharge. Yosho et al. [15] measured the corona discharge characteristics of a needle electrode tip on the ground and obtained the variation characteristics of the tip corona current and the relationship between the corona current and the ground electric field intensity.

Some characteristics of a corona discharge can be obtained through laboratory and observation experiments, but there are limitations such as the inability to study multiple influencing factors at the same time. In contrast, numerical simulation provides a powerful tool to overcome these limitations. In addition, a numerical simulation can also be simulated at different time and space scales, which provides important theoretical support for the in-depth study of corona discharges. Aleksandrov et al. [16,17,18] established a one-dimensional tip corona ion diffusion model and studied the characteristics of a positive corona discharge at the tip of the grounding object through analysis and numerical methods. Bazelyan et al. [19] established a two-dimensional corona electric field model to simulate the influence of the geometric characteristics of different grounding objects on the tip corona discharge. Based on the one-dimensional model established by Aleksandrov et al. [16,17,18], Becerra [20] established a two-dimensional axisymmetric corona model. This study realized the simulation of the tip corona discharge process with a symmetrical structure and considered the corona initiation phenomenon at the top of the tip. Guo et al. [21] established a two-dimensional asymmetric corona discharge model to better simulate the tip corona discharge process under actual thunderstorm clouds. On the basis of the previous two-dimensional model, Guo et al. [22] established a three-dimensional variable grid tip corona discharge model and simulated the corona discharge characteristics for the tip shapes of different buildings. The smaller the radius of the curvature of the tip, the easier it was to produce a corona discharge and the greater the corona current was on the tip. Qie et al. [23] simulated the evolution of corona ions generated by a natural tip using an established one-dimensional numerical model and simulated the evolution of the positive and negative corona current density on the ground. Beccera et al. [24] established a two-dimensional space charge numerical model and obtained the influence of positive and negative space charges on the electric field.

At present, most of the existing numerical simulation studies have mainly been based on the single-polarity corona discharge model, especially considering a positive-polarity corona. Although Qie et al. [23] and Beccera et al. [24] introduced the evolution process of a bipolar charge in their models, their research objects did not involve the corona discharge at the tip of ground buildings, and the models used were only one-dimensional or two-dimensional, which made it difficult to fully reflect the complex discharge process in a real thunderstorm environment. In an actual observation, Soula et al. [25] and Guo et al. [26] pointed out that in the thunderstorm cloud environment, the polarity reversal of the electric field often occurs. In this context, the corona charge formed at an early stage may have a significant impact on the subsequent heteropolar corona discharge. Therefore, in order to further explore the dynamic evolution and interaction mechanism of the corona discharge at the tip of the building during the polarity reversal process of the electric field in the thunderstorm environment, it is particularly important to develop a three-dimensional positive and negative corona discharge model. The model can simultaneously simulate the development process of positive and negative corona discharges, which can provide a theoretical basis and simulation support for a comprehensive understanding of the spatial and temporal distribution of the corona discharge in thunderstorms. The relevant simulation results will be introduced in detail in the subsequent chapters of this article.

The wind field will affect the motion of the corona charge. When a wind field exists, the corona charge will produce a diffusion motion, which will affect the corona discharge characteristics. Bazelyan et al. [27] evaluated the influence of the wind speed on the corona current based on Aleksandrov et al.’s [17] one-dimensional model and Zou et al.’s [28] two-dimensional model. By proposing a three-dimensional variable grid positive corona discharge model, Guo et al. [29] revealed the multiple effects of wind speed changes on corona discharge characteristics.

Therefore, based on the three-dimensional variable grid positive corona discharge model established by Guo et al. [30], a three-dimensional positive and negative corona discharge model was developed, which included two cases with or without a wind field. The corona discharge characteristics of the whole thunderstorm process were comprehensively simulated, and the results will be introduced in this paper.

2. Establishment and Verification of Model

2.1. The Establishment of the Model

In order to reflect the actual electric field variation law when the thunderstorm cloud is in transit as much as possible, this paper referred to the atmospheric electric field data presented in the articles of Yosho et al. [15] and Guo et al. [26] and extracted three typical waveforms and their electric field amplitude variation characteristics. The background electric field E_b included a positive and negative symmetric sawtooth wave (waveform A), a positive and negative symmetric triangular wave (waveform B), and a positive and negative asymmetric triangular wave (waveform C). The positive and negative symmetric sawtooth waves were referenced from the article by Yosho et al. [15], and the positive and negative symmetric triangular waves and the positive and negative asymmetric triangular waves were referenced from the article by Guo et al. [26]. The electric field amplitudes of the three waveforms were also determined according to the above literature: the positive and negative peaks of waveforms A and B were both ±10 kV/m, representing the typical strength of the electric field under common thunderstorm conditions; the positive peak value for waveform C was +10 kV/m, and the negative peak value was −20 kV/m, which reflected the asymmetric electric field situation in an actual thunderstorm. These settings not only considered physical authenticity but also facilitated the comparative analysis of the correlation between the change in the electric field polarity and discharge characteristics. The total simulation time was 4 s, and the time resolution was 0.0004 s. Setting the forward simulation time to 4 s enabled us to control the length of the simulation calculation and improve the simulation efficiency while ensuring a good time resolution. The calculation formula for the background electric field E_b is shown in Equations (1)–(3), and the change trend is shown in Figure 1.

$$E_{\mathrm{b}}=-5t+10(kV/m),0s\le t\le 4s$$

(1)

$$E_b=\left\{\begin{array}{l}10t(kV/m),0s\le t\le 1s\\ -10t+20(kV/m),1s<t\le 3s\\ 10t-40(kV/m),3s<t\le 4s\end{array}\right.$$

(2)

$$E_b=\left\{\begin{array}{l}10t(kV/m),0s\le t\le 1s\\ -10t+20(kV/m),1s<t\le 2s\\ -20t+40(kV/m),2s<t\le 3s\\ 20t-80(kV/m),3s<t\le 4s\end{array}\right.$$

(3)

The background electric field waveform will significantly affect the characteristics of positive and negative corona discharges. Therefore, it was necessary to briefly analyze the physical mechanism of positive and negative corona discharges before modeling. Under the two polarities, during the discharge process, electrons, ions, and neutral gas molecules interact with each other, involving multiple physical processes such as collision, recombination, and adsorption.

A positive corona discharge usually begins with the ionization of gas on the tip surface. The initial electrons accelerate to the tip under the action of a strong electric field and collide with neutral molecules to produce more electrons, forming an electron avalanche. Since the velocity of electrons is much larger than that of positive ions, electrons are mainly concentrated in the front region of the avalanche, while positive ions are retained in situ, resulting in charge accumulation and spatial electric field distortion. In the process of an electron avalanche, positive and negative ions recombine violently to release energy, and the excited particles decay to emit photons. These photons can further ionize neutral molecules, generate photoelectrons and trigger secondary electron avalanches, and promote the continuous development of the discharge.

In contrast, a negative corona is mainly caused by electrons escaping from the tip and away from the electrode along the direction of the electric field, causing collision ionization during motion. Since negative ions can be formed through the electron attachment process and their migration speed is slow, the negative corona discharge area is usually more dispersed, and the space charge distribution is wider. In addition, due to the high density of electrons in a negative corona, the local electric field is easily weakened, resulting in a relatively weak discharge intensity but a longer duration.

On this basis, this paper establishes a positive and negative corona discharge model, and the Peek formula [31], Kaptzov’s assumption [32,33,34], and the ion convection–diffusion equation are the three main equations of the three-dimensional positive and negative corona discharge model established in this paper.

First of all, the threshold for the corona onset electric field E_cor is calculated using the Peek formula, with the corresponding calculation formula shown in Equation (4):

$$E_{cor}=27.2(1+{\displaystyle \frac{0.54}{\sqrt{r_{rod}}}})$$

(4)

r_rod is the radius of the curvature of the lightning rod, the value of it is 0.0005 m, and the value of E_cor can be calculated as 3.75 × 10⁶ V/m.

Kaptzov’s hypothesis points out that the electric field on the tip surface remains unchanged at the threshold E_cor after the tip surface coronas. The specific performance is as shown in Equation (5):

$$E_{tip}(t_0)\ge E_{cor}$$

(5)

Here, E_tip(t₀) is the electric field value of the tip surface after a time of t₀.

With the generation of the corona on the tip surface, the generated corona charge will undergo convection–diffusion motion. This mainly involves a conversion between large and small positive and negative ions and neutral aerosol particles, including four processes: diffusion motion caused by different ion (particle) concentrations, composite action between opposite polarity ions, the migration motion of ions (particles) under the electric field force or wind field, and the adsorption of small positive and negative ions by aerosol particles. The convection–diffusion equation is shown in Equation (6):

$$\begin{array}{l}{\displaystyle \frac{\partial n_{+}}{\partial t}}=D·{\nabla }^2{n}_{+}-k_{n_{+}{N}_0}·n_{+}·N_0-k_{n_{+}{N}_{-}}·n_{+}·N_{-}-\alpha ·n_{+}·n_{-}-\nabla ·(n_{+}·{\mu }_{n_{+}}·\overrightarrow{E})\\ {\displaystyle \frac{\partial n_{-}}{\partial t}}=D·{\nabla }^2{n}_{-}-k_{n_{-}{N}_0}·n_{-}·N_0-k_{n_{-}{N}_{+}}·n_{-}·N_{+}-\alpha ·n_{+}·n_{-}-\nabla ·(n_{-}·{\mu }_{n_{-}}·\overrightarrow{E})\\ {\displaystyle \frac{\partial N_{+}}{\partial t}}=D·{\nabla }^2{N}_{+}+k_{n_{+}{N}_0}·n_{+}·N_0-k_{n_{-}{N}_{+}}·n_{-}·N_{+}-\nabla ·(N_{+}·{\mu }_{N_{+}}·\overrightarrow{E})\\ {\displaystyle \frac{\partial N_{-}}{\partial t}}=D·{\nabla }^2{N}_{-}+k_{n_{-}{N}_0}·n_{-}·N_0-k_{n_{+}{N}_{-}}·n_{+}·N_{-}-\nabla ·(N_{-}·{\mu }_{N_{-}}·\overrightarrow{E})\\ {\displaystyle \frac{\partial N_0}{\partial t}}=D·{\nabla }^2{N}_0+k_{n_{-}{N}_{+}}·n_{-}·N_{+}+k_{n_{+}{N}_{-}}·n_{+}·N_{-}-k_{n_{+}{N}_0}·n_{+}·N_0-k_{n_{-}{N}_0}·n_{-}·N_0\end{array}$$

(6)

Here, n₊ and n₋ are the concentrations of small positive and negative ions, N₊ and N₋ are the concentrations of large positive and negative ions, N₀ is the concentrations of neutral aerosol particles, D = 1 m²·s⁻¹ is the turbulent diffusion coefficient of the ions, and μ_n+ and μ_n− are the mobility of small positive ions and small negative ions, respectively, and their values are 1.5 × 10⁻⁴ m²·(V·s)⁻¹ and −2 × 10⁻⁴ m²·(V·s)⁻¹, respectively. μ_N+ and μ_N− are the mobility of small positive ions and small negative ions, and their values are 1.5 × 10⁻⁶ m²·(V·s)⁻¹ and −2 × 10⁻⁶ m²·(V·s)⁻¹, respectively. k_n+N0 = 2.9 × 10⁻¹² m³·s⁻¹ is the adsorption rate of small positive ions by aerosol particles, k_n−N0 = 3.5 × 10⁻¹² m³·s⁻¹ is the adsorption rate of small negative ions by aerosol particles, k_n+N− = 5 × 10⁻¹² m³·s⁻¹ is the recombination coefficient of small positive ions and large negative ions, k_n−N+ = 6 × 10⁻¹² m³·s⁻¹ is the recombination coefficient of small negative ions and large positive ions, α = 1.6 × 10⁻¹² m³·s⁻¹ is the recombination coefficient of small positive ions and small negative ions, t is the time, and E represents the environmental electric field [23,24,35].

When a wind field is added, the ion convection–diffusion equation is shown in Equation (7):

$$\begin{array}{l}{\displaystyle \frac{\partial n_{+}}{\partial t}}=D·{\nabla }^2{n}_{+}-k_{n_{+}{N}_0}·n_{+}·N_0-k_{n_{+}{N}_{-}}·n_{+}·N_{-}-\alpha ·n_{+}·n_{-}-\nabla ·(n_{+}·{\mu }_{n_{+}}·\overrightarrow{E}+n_{+}·v)\\ {\displaystyle \frac{\partial n_{-}}{\partial t}}=D·{\nabla }^2{n}_{-}-k_{n_{-}{N}_0}·n_{-}·N_0-k_{n_{-}{N}_{+}}·n_{-}·N_{+}-\alpha ·n_{+}·n_{-}-\nabla ·(n_{-}·{\mu }_{n_{-}}·\overrightarrow{E}+n_{-}·v)\\ {\displaystyle \frac{\partial N_{+}}{\partial t}}=D·{\nabla }^2{N}_{+}+k_{n_{+}{N}_0}·n_{+}·N_0-k_{n_{-}{N}_{+}}·n_{-}·N_{+}-\nabla ·(N_{+}·{\mu }_{N_{+}}·\overrightarrow{E}+N_{+}·v)\\ {\displaystyle \frac{\partial N_{-}}{\partial t}}=D·{\nabla }^2{N}_{-}+k_{n_{-}{N}_0}·n_{-}·N_0-k_{n_{+}{N}_{-}}·n_{+}·N_{-}-\nabla ·(N_{-}·{\mu }_{N_{-}}·\overrightarrow{E}+N_{-}·v)\\ {\displaystyle \frac{\partial N_0}{\partial t}}=D·{\nabla }^2{N}_0+k_{n_{-}{N}_{+}}·n_{-}·N_{+}+k_{n_{+}{N}_{-}}·n_{+}·N_{-}-k_{n_{+}{N}_0}·n_{+}·N_0-k_{n_{-}{N}_0}·n_{-}·N_0-\nabla ·(N_0·v)\end{array}$$

(7)

where v is the wind field.

When solving the above ion convection–diffusion equation, the range of the whole simulation domain is set to 500 m × 500 m × 500 m. For the convection–diffusion module, the upper horizontal boundary is set as a convection edge and the remaining boundaries (except for the rod surface boundary) are considered as zero flux boundaries. The surface of the rod where the local electric field is equal to or larger than the corona onset field E_cor is defined as a concentration boundary, generating a corona according to Kaptzov’s assumption. The point where a corona discharge occurs on the tip of a building is a fixed boundary, and the other boundaries of the building are approximated by the forward difference.

In this paper, the upper boundary, ground, and building in the three-dimensional simulation domain were set as the first boundary, and the other four boundaries are set as the second boundary. In this paper, the over-relaxation iteration method was used to calculate the potential value of each lattice point. The corresponding electric field value can be obtained by calculating the potential value. The calculation formula for the electric field is shown in Equation (8):

$$E=-\nabla \phi$$

(8)

where E is the electric field value.

The potential value of each grid point in the three-dimensional space can be obtained by solving the Poisson equation. The formula for obtaining the potential value is shown in Equation (9):

$${\nabla }^2\phi =-{\displaystyle \frac{\rho }{{\epsilon }_0}}$$

(9)

Here, $\nabla$ is the Laplace operator, φ is the electric potential, $\rho$ is the volume density of the corona charge, and ${\epsilon }_0$ is the vacuum capacitance rate (vacuum dielectric constant), which is 8.854 × 10⁻¹² F·m⁻¹.

The volume density of the corona charge can be calculated using the small and large positive and negative ions. According to the above Formulas (8) and (9), the calculation formulas for the electric field and potential can be further obtained. The calculation formula is shown in Equation (10):

$${\nabla }^2\phi =-{\displaystyle \frac{e_0(n_{+}+N_{+}-n_{-}-N_{-})}{{\epsilon }_0}}$$

(10)

Here, $e_0$ is the amount of elemental charge, and ${\epsilon }_0$ is the vacuum capacitance rate.

2.2. Model Validation

In order to verify the accuracy and reliability of the model established in this paper, considering that a change in the background electric field waveform is realized by adjusting the change in the electric field with time, in this section we describe the selection of the positive and negative symmetric sawtooth wave background electric field for comparative verification. In this paper, the observation results of Yosho et al. [15] were selected as the basis for comparative analysis, and the simulation results were compared with them in detail.

Yosho et al. [15] fixed a 1.5 m long platinum wire on a grounding needle electrode with a total height of 5 m and connected it to a measuring device about 10 m away from the needle electrode using a coaxial cable to measure the corona current and the near-ground electric field. The specific measurement device layout is shown in Figure 2. In this paper, the background electric field waveform was set to be a positive and negative symmetrical sawtooth wave. The absolute value of the background electric field peak was set to be 10 kV/m, the three-dimensional space simulation domain was 500 m × 500 m × 500 m, the total simulation time was 4 s, the time resolution was set to be 0.0004 s, and the building height was 5 m.

Since the total simulation time of the observation was at the minute level and there were certain limitations in the calculation of this model due to the calculation of the cornerstone, only a simulation study at the second time scale could be carried out at the time. Considering that this paper mainly focuses on the influence of the electric field waveform on the corona discharge, we selected observation results from a specific period of time for simulation verification, and the total simulation time was set to 4 s. Figure 3 shows the comparison results under approximate environmental conditions. In Figure 3a, the first image shows the observation results over the whole time period, while the second image shows the electric field waveform of a time period within it. In Figure 3b, the first image shows the background electric field waveform used in this model, the second image shows the change trend of the tip corona current, and the third image shows the change trend of the near-surface electric field with time.

It can be seen from Figure 3 that the tip corona current reached its maximum at the peak of the background electric field. When the background electric field was converted from a positive polarity to a negative polarity, there was a certain time delay in the generation of the negative corona current. This was mainly due to the fact that the electric field threshold for the negative corona discharge was not reached after the electric field direction was reversed, which led to the current generation lagging behind the electric field change. The variation trend of the corona current simulated by the model proposed in this paper was basically consistent with that from the observation results, but there were some differences in the peak value.

In the observation, the corona current was measured using a coaxial cable at a distance of 10 m from the needle electrode, and the model proposed in this paper was simulated based on the current near the tip. In a thunderstorm environment, a coaxial cable will produce a shielding effect, resulting in corona current attenuation, especially in long-distance transmission [36], which may lead to errors between the measured value and the actual value. At the same time, the tip material will also affect the current size [37]. This model assumed that the tip was a good conductor, and platinum wire was used in the observation experiment. The latter had low conductivity, so the corona current was different. In addition, the corona current may have been affected by the background electric field in the previous period.

In the lightning experiment conducted by Yosho et al. [15], in the second half of the observation, when the ground electric field increased from 1.5 kV/m to 14 kV/m, the corona current generated on the tip at a height of 5 m increased from 8 μA to 10 μA, indicating that the corona current can reach several microamperes. Due to the difference between the model setting and the observation, the current could not be completely consistent, but the change trend was basically consistent, so the model had a certain amount of reliability.

In addition, the near-ground electric field changes were numerically simulated at 5 m, 10 m, and 15 m from the ground. The results show that the ground electric field was obviously distorted when the horizontal distance was 5 m. When the horizontal distance was 10 m, the position was consistent with the observation point. Although there was still some distortion, it was close to the observation result. When the horizontal distance increased to 15 m, the difference between the ground electric field and the background electric field was the smallest, indicating that the smaller the horizontal distance, the greater the distortion of the ground electric field. In addition, in the background electric field reduction stage, there was a certain morphological difference between the simulated electric field change curve and the observed data, which may have been related to factors such as the change rate of the electric field; the change in the ground electric field may also have been affected by the electric field in the previous time period. However, based on the overall trend, it can be seen that the variation in the near-surface electric field simulated by the model with time was consistent with the observation results, meaning the model could reasonably simulate the evolution characteristics of the near-surface atmospheric electric field and could simulate more corona discharge characteristics.

In summary, the variation trend of the corona current and ground electric field for the model proposed in this paper was basically consistent with the observation results. Therefore, the model established in this paper had high accuracy and reliability as a whole.

Using the established three-dimensional positive and negative corona discharge model, the corona discharge characteristics of a single building with a height of 10 m at a horizontal wind speed of 10 m/s were simulated, so as to verify the accuracy and applicability of the model. This paper only studied the influence of the horizontal wind speed on the model. The tip positive and negative corona discharge characteristics of a single building with a height of 10 m at a horizontal wind speed of 10 m/s were studied when the wind field was added. The wind field affected the corona charge generated by the tip corona discharge and caused it to migrate, thus changing the spatial distribution of the corona charge and the corona electric field. Figure 4 and Figure 5 show the spatial distribution maps of the corona charge, including the XZ section (side view) and XY section (top view), respectively. Figure 6 shows the spatial distribution map of the corona electric field as the XZ section (side view).

It can be seen from Figure 4 that when a wind field did not exist, the corona charge was symmetrically distributed at each moment. However, it can be seen from Figure 5 that the presence of a wind field caused the corona charge generated at the tip of the building to shift significantly. When the background electric field was 0, the tip no longer produced a corona discharge. At this time, the corona charge shifted and left the tip due to the influence of the wind field. At the time of the negative peak of the background electric field, positive and negative corona charges existed at the same time. At this time, the wind field formed a positive corona charge region on the right side of the tip. It can be seen from Figure 6 that when there was no wind field, the corona electric field was consistent with the corona charge distribution, both of which were symmetrically distributed. Under the action of the wind field, the corona electric field also shifted, which was closely related to the change in the corona charge. At the negative peak of the background electric field, positive and negative electric field regions coexisted on the right side of the tip. Therefore, the existence of a wind field will significantly affect the characteristics of the tip corona discharge, and the model established in this paper can reasonably reflect this effect and has high reliability.

3. Model Application

3.1. Comparison of Positive and Negative Corona Discharge Model and Unipolar Corona Discharge Model

In this section, the established positive and negative corona discharge model is compared with the unipolar corona discharge model. The background electric field waveform was a positive and negative symmetrical triangular wave. The absolute value of the background electric field amplitude was 10 kV/m. The simulation time was 4 s. The time resolution was 0.0004 s. In order to simplify the geometric modeling process, the building was considered to be equivalent to an independent lightning rod with a height of 5 m and a radius of 0.0005 m. The environment surrounding the lightning rod was assumed to be an open and uniform area. The comparison results are shown in Figure 7. In the change trend diagram for the background electric field, the diagram of the positive and negative polarity model overlaps with the diagram of the unipolar model in the negative half cycle.

It can be seen from Figure 7 that compared with the unipolar corona discharge model under the same waveform and amplitude of the background electric field, the negative corona current of the positive and negative corona discharge model was generated earlier, the peak value of the negative corona current was larger, and the duration of the negative corona discharge was longer. The reason for this was that due to the influence of the first half cycle, the background electric field required for a corona discharge was smaller, and the corona current was generated earlier, but the end time was equivalent. At the same time, due to the neutralization effect of positive and negative corona charges, the peak value of the total corona charge in the second half cycle was significantly smaller than that of the unipolar model.

In summary, the positive and negative corona discharge model can demonstrate the influence of background electric field changes before and after a reversal of polarity. Compared with the unipolar corona discharge model, it considers the corona discharge characteristics more comprehensively. Therefore, the development of the positive and negative corona discharge model has important practical significance.

3.2. Comparison of Positive and Negative Corona Discharge Models and Unipolar Corona Discharge Models at Different Building Heights

The characteristics of the tip corona discharge at different building heights are different. The building was simplified to an independent lightning rod with a radius of 0.0005 m, and the heights were set to 5 m, 10 m, 20 m, 30 m, and 60 m, respectively. The environment surrounding the lightning rod was assumed to be an open and uniform area. Figure 8 shows the variation characteristics of the tip corona current corresponding to buildings at different heights under the two models.

It can be seen from Figure 8 that the peak value of the corona current in both models appeared at the peak value of the background electric field. The tip corona current increased with an increase in the building height, and the generation time for the negative corona current also increased with an increase in the building height. However, at different heights, the peak value of the negative corona current in the positive and negative polarity model was larger than that in the unipolar model, the generation time for the negative corona current was earlier than that in the unipolar model, and the duration of the negative corona discharge was longer. Due to the influence of the first half cycle, the background electric field required for a tip corona discharge in the positive and negative polarity model was smaller, the negative corona current was generated earlier, and the end time was equivalent.

Therefore, the absolute value of the difference (ΔI) between the negative corona current obtained using the positive and negative polarity model and the unipolar model was fitted with the building height (H) to obtain a fitting curve, as shown in Figure 9. The fitting curve reached a significant level of 0.05.

It can be seen from Figure 9 that the corona current difference had a linear relationship with the height of the building, and the corona current difference increased with an increase in the height of the building. The relationship between the two was obtained as follows:

ΔI = 0.090·H + 1.025

where the unit for ΔI was μA, the unit for H was m, and the applicable range for the relationship was 5–60 m.

Figure 10 shows the variation characteristics of the tip corona charge corresponding to buildings of different heights under the two models.

It can be seen from Figure 10 that the higher the height of the building, the more significant the increase in the corona charge released. At different building heights, the negative corona charge in the positive and negative polarity model was smaller than the negative corona charge in the unipolar model. The reason for this was that the positive and negative charges were neutralized when the polarity of the electric field was reversed, resulting in the final amount of the corona charge released. The amount was negative, and the absolute value of the negative corona charge was slightly smaller than that of the positive corona charge released.

Therefore, the value of the difference (ΔQ) between the negative corona charges in the positive and negative polarity model and the unipolar model was fitted with the building height (H) to obtain a fitting curve, as shown in Figure 11. The fitting curve reached a significant level of 0.05.

It can be seen from Figure 10 that the negative corona charge difference had a linear relationship with the height of the building. The negative corona charge difference increased with an increase in the height of the building. The relationship between the two was obtained as follows:

ΔQ = 3.337 × 10⁻⁴·H + 2.332 × 10⁻⁴

where the unit for ΔQ was mC, the unit for H was m, and the applicable range for the relationship was 5–60 m.

In summary, the tip corona discharge characteristics of buildings at different heights were in line with those in an actual situation, and the previous unipolar model could not simulate the tip corona discharge characteristics under electric field reversal, with the ability to do so being an advantage of this model.

4. Discussion

Compared with the previous unipolar corona discharge model, the positive and negative corona discharge model established in this paper had more advantages in terms of the scope of application and physical integrity. The unipolar model is usually based on the simulation of a positive or negative corona discharge, ignoring the influence of heteropolar corona discharges, and makes it difficult to accurately reflect the temporal and spatial distribution characteristics of a corona discharge under complex background electric field conditions. Especially in a thunderstorm environment, where the electric field changes rapidly and the polarity is reversed, the unipolar model often fails to capture the dynamic characteristics of charge polarity conversion and space charge accumulation and dissipation, and its descriptive ability has certain limitations. The positive and negative corona discharge model established in this paper comprehensively considers the migration, recombination, and attachment of positive and negative ions and can describe the evolution process of positive and negative corona discharges more comprehensively. The simulation results show that the first half-cycle corona discharge has a significant effect on the second half-cycle heteropolar corona discharge, which is often ignored in the unipolar model. At the same time, the model established in this paper also provides theoretical support for studying the discharge intensity, space charge accumulation, and its influence on the near-earth electric field during different polarity conversion processes.

Therefore, the model not only has more applicability and physical authenticity under the condition of a multi-polar and multi-waveform background electric field, but also provides theoretical support for studying the influence of a corona discharge on the near-ground electric field. Furthermore, the establishment of this model can provide basic support and a theoretical basis for the subsequent revision of atmospheric electric field measurement instruments and the optimization of lightning warning systems.

5. Conclusions

In this paper, a three-dimensional corona discharge numerical model considering the electric field polarity reversal process was developed for two cases with or without a wind field. The reliability of the model was verified by comparing the model with observation results. Furthermore, the positive and negative polarity model proposed in this paper was compared with the previous unipolar model. The results show that the corona discharge in the positive and negative polarity model is affected by the first half cycle. The required background electric field is smaller, the negative corona current is generated earlier and its peak value is larger, and the negative corona discharge duration is longer. At the same time, due to the neutralization effect of positive and negative corona charges, the peak value of the total corona charge in the second half cycle is significantly lower than that of the unipolar model. For example, when the building height was 5 m, the peak negative corona current in the positive and negative polarity model was −20.672 μA, and the peak negative corona current in the unipolar model was −19.609 μA. The peak negative corona current in the positive and negative polarity model was 5.42% larger than that in the unipolar model. The peak value of the negative corona charge in the positive and negative polarity model was—2.69 × 10⁻³ mC, and the peak value of the negative corona charge in the unipolar model was—4.11 × 10⁻³ mC. The peak negative corona charge in the positive and negative polarity model was 34.55% smaller than that in the unipolar model.

Based on the established three-dimensional positive and negative corona discharge model, the characteristics of a corona discharge at the tip of buildings at different heights under the two models were compared, including the characteristics of the corona current and total corona charge released. The results show that the corona current and corona charge increase with an increase in the building height, and the generation time for a negative corona current increases with an increase in the building height, and the duration of a negative corona discharge increases with an increase in the building height. In addition, the absolute value of the negative corona current difference and the negative total corona charge difference between the two models increased linearly with an increase in the building height, which proves that the model proposed in this paper is not only applicable to buildings of different heights but also has high accuracy and reliability.