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Article

Numerical Simulation of Perkins Instability in the Midlatitude F-Region Ionosphere: The Influence of Background Ionospheric Multi-Factors

by
Yi Liu
1,
Ting Lan
2,*,
Yufeng Zhou
3,*,
Yunzhou Zhu
4,
Zhiqiang Fan
3,
Yewen Wu
5,
Yuqiang Zhang
6 and
Xiang Wang
1
1
School of Artificial Intelligence, Hubei University, Wuhan 430062, China
2
School of Computer, Huanggang Normal University, Huanggang 438000, China
3
Beijing Institute of Applied Meteorology, Beijing 100029, China
4
Institute of Space Science and Applied Technology, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China
5
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
6
Department of Space Physics, School of Electronic Information, Wuhan University, Wuhan 430072, China
*
Authors to whom correspondence should be addressed.
Atmosphere 2025, 16(2), 221; https://doi.org/10.3390/atmos16020221
Submission received: 10 December 2024 / Revised: 13 February 2025 / Accepted: 13 February 2025 / Published: 16 February 2025
(This article belongs to the Section Planetary Atmospheres)
Browse Figures
Figure 1
<p>The evolution process of relative plasma density perturbation at 280 km for 0 s (<b>a</b>), 1200 s (<b>b</b>), 2400 s (<b>c</b>), and 3600 s (<b>d</b>) from random perturbation conditions with a range of 0–500 m.</p> "> Figure 2
<p>Power spectral density of density perturbation in the wave vector domain in the height of 280 km for t = 3600 s. The area composed of solid lines indicates the region where the Perkins instability occurs.</p> "> Figure 3
<p>Time variation of mean field-line-integrated Pedersen conductivity perturbation.</p> "> Figure 4
<p>Neutral density scale height dependence of Perkins instability. The neutral density scale heights are (<b>a</b>) 60 km, (<b>b</b>) 80 km, (<b>c</b>) 100 km, and (<b>d</b>) 120 km, respectively.</p> "> Figure 5
<p>The variation of the mean amplitude of relative density perturbation with a neutral density scale height at 280 km for 3600 s.</p> "> Figure 6
<p>The evolution process of relative field-line-integrated Pedersen conductivity perturbation at 0 s (<b>a</b>), 1200 s (<b>b</b>), 2400 s (<b>c</b>), and 3600 s (<b>d</b>) under the action of GW activity.</p> "> Figure 7
<p>The evolution process of relative field-line-integrated Pedersen conductivity perturbation at 0 s (<b>a</b>), 120 s (<b>b</b>), 600 s (<b>c</b>), and 1200 s (<b>d</b>)under the action of E region polarized electric field.</p> ">
Versions Notes

Abstract

:
A numerical simulation of Perkins instability in the midlatitude F-region ionosphere is developed in this study. The growth of nighttime plasma density perturbation excited by Perkins instability was successfully reproduced. The simulated results show that the ionospheric perturbation structure elongated from northwest (NW) to southeast (SE) was generated from initial random seeding by applying a very large southeastward neutral wind (200 m/s). The domain wave vector direction agreed with the linear Perkins theory. Our simulated results were consistent with the previous observations and simulations. To investigate the influence of background ionospheric multi-factors on the generation of nighttime medium-scale traveling ionospheric disturbance (MSTID), we simulated the evolution process of ionospheric perturbations under initial background ionospheric conditions. The simulated results indicate the importance of neutral scale height on the development of nighttime MSTID and suggest that a smaller neutral scale height would amplify the amplitude of ionospheric perturbations. The influences of gravity wave (GW) activity and polarized electric field seeding from plasma instability in the E region are also discussed in this study. We conclude that the additional seeding processes play a major role in the accelerated Perkins instability and amplify ionospheric perturbations. The electrodynamic coupling process has a greatly significant effect on the growth rate of Perkins instability compared to GW activity.

1. Introduction

A typical wave-like structure associated with irregularity observed in the nighttime midlatitude F region ionosphere has been under investigation since the 1970s [1]. The general morphological features have been studied by all-sky imagers [2,3,4,5] and Global Navigation Satellite System (GNSS) total electron content (TEC) maps [6,7,8,9,10,11,12,13,14,15,16,17]. They reported that banded structures or medium-scale traveling ionospheric disturbances (MSTIDs) are elongated from the northwest (NW) to southeast (SE) and migrate toward the southwest in the Northern Hemisphere with a typical horizontal wavelength of about hundreds of kilometers, a horizontal phase velocity of about hundred meters per second, and for a period of 20–70 min.
Although the plasma instability proposed by Perkins [18] may be the most possible mechanism for nighttime MSTID, the slow growth of instability is incapable of producing a significant MSTID phenomenon. Additional seeding processes are required to accelerate the growth of instability [6,19]. Early and recent observational evidence has revealed that the generation of nighttime MSTID is closely related to local gravity waves and E region irregularity [11,14,15,16,19,20,21,22,23,24,25,26]. Xu et al. [27] and Chen et al. [14] pointed out that gravity waves (GWs) from strong convective weather at a lower atmosphere can propagate upward to the bottom of the F region as a seed disturbance, accelerating the excitation process of Perkins instability for generating MSTID. Huang et al. [11] and Liu et al. [15,25] proposed that a polarized electric field induced by the Hall polarization process in sporadic E (ES) irregularity can map to the F region and accelerate the growth of the F region irregularity. Theoretical explanations have been presented in previous studies [28,29,30,31]. A positive feedback mechanism exists in the E–F coupling system, including ES instability and Perkins instability, and provides a larger growth rate than either alone.
Numerical simulations have been conducted on Perkins instability [32,33,34,35,36,37,38,39,40]. The evolution process of NW–SE structures from random disturbances was successfully shown by two-dimensional and three-dimensional models [38,39,40]. The effect of gravity waves from low atmosphere and polarized electric field from coupling instability has been preliminarily investigated by Huang et al. [33] and Yokoyama et al. [39], but more quantitative analysis is required. In this study, we developed the numerical simulation model of Perkins instability in the midlatitude F-region ionosphere, and the quantitative comparisons about the evolution process of NW–SE structures under the action of gravity wave and polarized electric field from E region instability are discussed.

2. Model Description

In this study, a numerical model of the midlatitude F region was developed. O+ is considered the dominant ion species in the ionospheric F region. The continuity and momentum equations are written as follows [38]:
∂ n ∂ t + ∇ · n v i = 0
q i E + v i × B + m i g − ∇ n k B T n + m i υ in U − v i = 0
where n is the plasma density, v i is the plasma velocity, q i is the ion charge, E is the electric field, B is the geomagnetic field ( B = 4.6 × 10 − 5   T ), m i is the ion mass, g is the gravitational acceleration (g = 9.8 m/s2), k B is the Boltzmann constant ( k B = 1.38 × 10−23 JK−1), T is the plasma temperature, υ in is the collision frequency with neutrals, and U indicates the neutral wind field. v i is calculated by Equation (2).
Following Miller [34] and Yokoyama et al. [38], and considering the ambipolar diffusion in the F region, E is written as:
E = E 0 − ∇ φ − k B T q m i υ in − m e υ en m i υ in + m e υ en ∇ n n − m i q m e υ en m i υ in + m e υ en g ∥
where E 0 is the background electric field that could be ignored at midlatitudes. m e indicates the electron mass, υ en is the collision frequency between electrons and neutrals, and g ∥ is the B-parallel component of g. φ is the electrostatic potential of a polarized electric field and is calculated by the basic equations for Perkins instability [18]:
∂ Σ ∂ t + ∇ ⊥ Σ · g ⊥ × z → ′ Ω − ∇ φ × z → ′ c B = e c g N sin 2 D Ω B H + Σ cos D c B H ∂ φ ∂ y ′ + Σ U z ′ sin D H
∇ ⊥ · Σ ∇ ⊥ φ − U ⊥ × B c − ∂ N ∂ y ′ e g cos D Ω + 2 T e ∇ ⊥ 2 Σ + ∂ Σ ∂ x ′ 2 T e H + M g e cos D = 0
where c is the speed of light, e is the electron charge (e = 1.6 × 10−19 C), D is the geomagnetic dip angle (D = 45°), H is the atmospheric scale height, and M indicate the ion mas. Σ and N are the field-line-integrated Pedersen conductivity and electron density in the ionospheric F region, respectively. Ω = eB/Mc represents the ion gyrofrequency. ⊥ is the B-perpendicular component of variations.
In this study, Equations (1) and (2) are solved on a three-dimensional coordinate system using the total variation diminishing (TVD) method [41]. The unit vectors in the x → , y → , and z → directions are geomagnetic east, geomagnetic north, and upward, respectively. Using the TVD method and successive over-relaxation (SOR) method, φ is calculated numerically by combining Equations (4) and (5) on the three-dimensional coordinate system where the axes of the x → ′ , y → ′ , z → ′ are geomagnetic east, perpendicular to geomagnetic field B, and parallel to geomagnetic field B, respectively. The relationship between x → , y → , z → and x → ′ , y → ′ , z → ′ is given as:
x → ′ = x → y → ′ = y → sin D + z → cos D z → ′ = y → cos D − z → sin D
The model covers altitudes from 150 km to 470 km and horizons from −160 km to 160 km with a grid spacing of 2 km. The background atmospheric and ionospheric parameters were given by MSISE-00 and IRI-16 models at (34.85° N, 136.10° E) at 22:00 LT on 27 June 2012 when the Kp index was less than 3. Table 1 gives the geophysical constants used in the model. The Neumann boundary condition is applied to the top and bottom boundaries of O+, and the periodic boundary condition is applied in the x → , y →   x → ′ , and y → ′ directions. The time step is 0.5 s. Considering the equipotential effect along the geomagnetic field, we set ∂ φ / ∂ z ′ = 0 .

3. Simulation and Discussion

3.1. Random Perturbation

In the model, a uniform 37° south of the east neutral wind with a amplitude of 200 m/s was applied as an initial background wind profile. The equivalent electric field U × B is 8.33 mV/m pointing to 53° northeast. An example of the evolution process of relative plasma density perturbation for 280 km at 0 s, 1200 s, 2400 s, and 3600 s from random perturbation conditions with a range of 0–500 m is shown in Figure 1. It is clearly seen that the random perturbation distributions gradually evolved into a northwest–southeast (NW–SE) structure over time. As the structure evolves, the relative plasma density perturbation and the maximum polarized electric field reached 2.41% and 0.21 mV/m at 3600 s, respectively. Figure 2 shows the power spectral density of the density perturbation at t = 3600 s. The peak spectral density occurred at k (2π/37.88 km−1, 2π/92.00 km−1), and the angle between k and the magnetic east was 22.38°, which agrees with the theoretical analysis of the maximum growth rate of Perkins instability. The simulated results are consistent with previous observations and simulations [10,35,36,38,42].
To further investigate the evolution process of Perkins instability, Figure 3 shows the time variation of the mean field-line-integrated Pedersen conductivity. It is shown that the amplitude of perturbation decreased in the early stages since the inappropriate perturbation component that is not satisfied by the Perkins instability theory would not develop. After about 800 s, the system entered a linear stage very quickly and the amplitude of perturbation reached about 0.25 mho. Then, the system began to saturate about 8000 s. The results agree with the previous studies shown in a study by Zhou et al. [35,36] but more quantitative analyses are worthy of further study based on our simulation model.

3.2. Neutral Density Scale Height Dependency

Neutral density scale height is considered the main controlling factor in the evolution of nighttime MSTID in the midlatitude ionosphere. The anticorrelation between the linear growth rate of Perkins instability and the neutral density scale height is shown in the theoretical work by Perkins [18]. Recent studies have observed a negative relationship between MSTID occurrence and solar activity [16,43,44,45,46]. They suggested that this negative relationship could be attributed to the changes in ion-neutral collision frequency and neutral density scale height with solar activity based on the Perkins instability theory. During a solar maximum, the ion-neutral collision frequency and neutral density scale height are larger than those during a solar minimum. Based on the simulated results from the MSISE00 model in the nighttime ionosphere at midlatitudes, H and υ in are about 51.32 km and 0.20 s−1 during the solar minimum periods, while these are about 77.49 km and 2.19 s−1 during the solar maximum periods.
To quantitatively evaluate the neutral density scale height dependency of Perkins instability in this study, we simulated the plasma density perturbation under the different neutral density scale height conditions. The range of the scale height was assumed from 40 km to 160 km. Note that the range exceeds the neutral scale height variations between the solar maximum and solar minimum periods. The initial density profile adopts the formula given in ref. [47]:
N z = N p e a k exp 1 − Z − exp − Z
where N p e a k = 106 cm−3, Z = (z − z0)/H, z0 = 400 km, and H is the neutral density scale height. Figure 4 shows the distributions of the relative plasma density perturbation at 280 km for 3600 s for four H; H is set at 60 km, 80 km, 100 km, and 120 km, which are labeled (a), (b), (c), and (d), respectively. The variations of relative perturbation with H follow the Perkins instability theory. A perturbation with a smaller H has a larger amplitude. In Figure 4a, the relative perturbation reached nearly 12%, but in Figure 4d, it was less than 2.5%. To further investigate the possible linkage between relative perturbation and H, Figure 5 presents the correlation of the mean amplitude of relative density perturbation with H at 280 km for 3600 s. The amplitude of perturbation decreased exponentially with the increase of H when the value of H was within the range of 40–60 km, and then weakened weakly and finally remained basically unchanged since 120 km. Thus, according to the simulated results in Figure 4 and Figure 5, the development of the Perkins instability is closely related to H. In other words, background conditions in the F region are important for amplifying density perturbations by the Perkins instability. In brief, our simulated results could further explain the negative correlation between nighttime MSTID occurrence and solar activity.

3.3. The Influence of Gravity Wave Activity

Previous reports have indicated that Perkins instability alone cannot explain the observed results since the growth rate is too small [15,25,38,39,48]. In Figure 1, the maximum amplitude of related density perturbation reached about 2.50% at 3600 s, which is smaller than the observed results of the nighttime MSTID event wherein the median amplitude was 7.00% in June [10]. Additionally, the evolution process took a long time from the initial seeding. Additional seeding factors for accelerating the development process of Perkins instability need to be considered.
Several studies in different locations have observed a close linkage between local GW activity and MSTID [9,10,14,16,49]. They explained that the GW activity from a lower atmosphere during strong mountain wave events or deep convective activity can propagate upward in the upper atmosphere and excite ionospheric perturbation by a neutral particle–ion collision process. To further examine the influence of GW activity, we simulated the relative amplitude of the field-line-integrated Pedersen conductivity variation with time under the GW activity seeding. The initial field-line-integrated Pedersen conductivity perturbation distribution is given as follows [33]:
S P x , y = S P p e a k sin k x x + k y y
where S P p e a k = 0.0023 mho when the background field-line-integrated Pedersen conductivity was 0.45 mho. kx and ky are the wave vector components of GW in the x → direction and y → direction, respectively. Considering that GW in the F region has an equatorward-propagating direction, we let k x and k y be −2π/113 km−1 and −2π/113 km−1, respectively [10]. Figure 6 shows the evolution process of the relative field-line-integrated Pedersen conductivity perturbation at 0 s, 1200 s, 2400 s, and 3600 s under the action of GW activity. It could be seen that initial perturbation generated by GW activity seeding gradually increased and the maximum amplitude reached 8.00% at 3600 s. Compared to the simulated results in Figure 1d, the relative amplitude quadrupled at 3600 s in Figure 6d. The observed results of MSTID in the study by Ding et al. [10] also presented the maximum amplitudes of ionospheric perturbation reached 25%, which agrees with the simulated results in Figure 6. Thus, our model demonstrates that GW activity seeding could effectively accelerate Perkins instability and make ionospheric perturbation grow rapidly.

3.4. The Influence of Polarized Electric Field Excited by Plasma Instability in the E Region

The importance of the E–F electrodynamic coupling process on the generation of nighttime MSTID has been studied based on the observations of previous research [10,11,14,22,25,26,50]. It has been observed that nighttime MSTID frequently occurred during summer at midlatitudes by using ground-based and satellite instruments, which is consistent with the seasonal and local time variations of local E region irregularity occurrence. Similar morphological features confirmed the existence of a tight relationship between nighttime local MSTID and E region irregularity. Saito et al. [50] proposed that a polarized electric field excited by the Hall polarization process in the E region can be mapped to the F region by magnetic field lines to accelerate the evolution process of Perkins instability. First, theoretical analysis was given in Cosgrove and Tsunoda’s study [31]. They suggested that E–F electrodynamic coupling instability could provide a larger linear growth rate than Perkins instability alone, which may succeed in accounting for the nighttime observations at midlatitudes. A three-dimensional model of coupled Perkins instability and ES-layer instability was first shown in a study by Yokoyama et al. [39]. The simulated results showed the importance of a polarized electric field in the generation of nighttime MSTID. In our work, we also simulated the relative amplitude of field-line-integrated Pedersen conductivity variation with time under the polarized electric field seeding. The distribution of polarized potential induced by plasma instability in the E region is given as follows:
φ x , y = φ p e a k cos k x x + k y y
where φ p e a k = 1 V, and k x and k y are the wave vector components of a polarized electric field in the x → direction and y → direction, respectively. Considering that the polarized electric field is in the first quadrant and third quadrant [15,22], we let k x and k y be −2π/90 km−1 and −2π/45 km−1, respectively. Figure 7 presents the evolution process of the relative field-line-integrated Pedersen conductivity perturbation at 0 s, 120 s, 600 s, and 1200 s under the action of the E region’s polarized electric field. The random perturbation distributions in Figure 7a rapidly evolved into a NW–SE structure over time and the maximum amplitude reached 4.00% at 1200 s, which has grown by a factor of ten in Figure 1b. Compared to the simulated results in Figure 6b under the GW activity seeding, we found that a polarized electric field excited by plasma instability in the E region more easily motivates Perkins instability and subsequently produces larger ionospheric perturbations. Corresponding observational evidence was presented in the study by Huang et al. [11]. Their statistical results showed the weak correlation of nighttime MSTID and GW in the mesosphere by using the simultaneous observations from OI 630.00 nm and OI 557.7 nm. In a word, E–F electrodynamic coupling may be the main reason for the frequent occurrence of nighttime MSTID in summer in the midlatitude region.

4. Conclusions

In this paper, we established a numerical model of Perkins instability in the midlatitude ionosphere. The evolution process of plasma density perturbation from the initial random perturbation condition was simulated in this work. The main conclusions can be summarized as follows:
  • The initial random distributed density structure gradually developed into a banded structure aligned in the NW–SE direction, which is consistent with GPS–TEC and all-sky imager observations. Moreover, the relative amplitude of density perturbation grew steadily after about 8000 s.
  • The domain wave vector direction satisfied the linear growth theory of Perkins instability. Inappropriate perturbations would be suppressed based on our simulated results.
  • The neutral scale height dependency of Perkins instability has been quantitatively revealed for the first time. Simulated results in this study showed a negative correlation between the neutral scale height and the growth rate of Perkins instability. Smaller neutral scale height H would amplify the amplitude of ionospheric perturbation, which is in agreement with the statistical results of MSTID occurrence variations with solar activity. The relative perturbation reached nearly 6.00% during the solar minimum periods, but during the solar maximum periods, it was less than 1.50% at 3600 s.
  • A comparative analysis of the effects of GW activity and polarized electric field seeding from plasma instability in the E region on the generation of MSTID is quantitatively presented in this study for the first time. An additional seeding process would greatly accelerate the development of Perkins instability. The maximum amplitude of relative perturbation reached 4.00% at 1200 s under the E region’s polarized electric field seeding, wherein the initial polarized potential perturbation amplitude was 1 V with a horizontal wavelength of about 100 km, while it was less than 1.00% at 1200 s under the GW activity seeding wherein the initial field-line-integrated Pedersen conductivity relative perturbation was 0.51% with a horizontal wavelength of about 160 km. The E–F electrodynamic process is more likely to be the main controlling factor for the generation of nighttime MSTID in the midlatitude ionosphere rather than the modifications of GW activity.

Author Contributions

Methodology and investigation, Y.L., Y.Z. (Yunzhou Zhu), and Y.Z. (Yufeng Zhou); data curation, and writing—original draft preparation, Y.L., Y.Z. (Yuqiang Zhang), and Z.F.; writing—review and editing, X.W., T.L., and Y.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (NSFC grant No. 42204161) and the Hubei Natural Science Foundation (grant No. 2023AFB200).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The simulated data used in this study are available through Zenodo [51].

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The evolution process of relative plasma density perturbation at 280 km for 0 s (a), 1200 s (b), 2400 s (c), and 3600 s (d) from random perturbation conditions with a range of 0–500 m.
Figure 1. The evolution process of relative plasma density perturbation at 280 km for 0 s (a), 1200 s (b), 2400 s (c), and 3600 s (d) from random perturbation conditions with a range of 0–500 m.
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Figure 2. Power spectral density of density perturbation in the wave vector domain in the height of 280 km for t = 3600 s. The area composed of solid lines indicates the region where the Perkins instability occurs.
Figure 2. Power spectral density of density perturbation in the wave vector domain in the height of 280 km for t = 3600 s. The area composed of solid lines indicates the region where the Perkins instability occurs.
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Figure 3. Time variation of mean field-line-integrated Pedersen conductivity perturbation.
Figure 3. Time variation of mean field-line-integrated Pedersen conductivity perturbation.
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Figure 4. Neutral density scale height dependence of Perkins instability. The neutral density scale heights are (a) 60 km, (b) 80 km, (c) 100 km, and (d) 120 km, respectively.
Figure 4. Neutral density scale height dependence of Perkins instability. The neutral density scale heights are (a) 60 km, (b) 80 km, (c) 100 km, and (d) 120 km, respectively.
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Figure 5. The variation of the mean amplitude of relative density perturbation with a neutral density scale height at 280 km for 3600 s.
Figure 5. The variation of the mean amplitude of relative density perturbation with a neutral density scale height at 280 km for 3600 s.
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Figure 6. The evolution process of relative field-line-integrated Pedersen conductivity perturbation at 0 s (a), 1200 s (b), 2400 s (c), and 3600 s (d) under the action of GW activity.
Figure 6. The evolution process of relative field-line-integrated Pedersen conductivity perturbation at 0 s (a), 1200 s (b), 2400 s (c), and 3600 s (d) under the action of GW activity.
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Figure 7. The evolution process of relative field-line-integrated Pedersen conductivity perturbation at 0 s (a), 120 s (b), 600 s (c), and 1200 s (d)under the action of E region polarized electric field.
Figure 7. The evolution process of relative field-line-integrated Pedersen conductivity perturbation at 0 s (a), 120 s (b), 600 s (c), and 1200 s (d)under the action of E region polarized electric field.
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Table 1. Geophysical constants used in the model.
Table 1. Geophysical constants used in the model.
ParameterValue
Magnetic field4.6 × 10−5 T
Geomagnetic dip angle45°
Ion gyrofrequency299 s−1
Ion mass(oxygen)2.67 × 10−26 kg
Atmospheric scale height45,000 m
Temperature916 K
Gravitational acceleration9.8 ms−2
Boltzmann constant1.38 × 10−23 JK−1
Electron charge1.6 × 10−19 C
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MDPI and ACS Style

Liu, Y.; Lan, T.; Zhou, Y.; Zhu, Y.; Fan, Z.; Wu, Y.; Zhang, Y.; Wang, X. Numerical Simulation of Perkins Instability in the Midlatitude F-Region Ionosphere: The Influence of Background Ionospheric Multi-Factors. Atmosphere 2025, 16, 221. https://doi.org/10.3390/atmos16020221

AMA Style

Liu Y, Lan T, Zhou Y, Zhu Y, Fan Z, Wu Y, Zhang Y, Wang X. Numerical Simulation of Perkins Instability in the Midlatitude F-Region Ionosphere: The Influence of Background Ionospheric Multi-Factors. Atmosphere. 2025; 16(2):221. https://doi.org/10.3390/atmos16020221

Chicago/Turabian Style

Liu, Yi, Ting Lan, Yufeng Zhou, Yunzhou Zhu, Zhiqiang Fan, Yewen Wu, Yuqiang Zhang, and Xiang Wang. 2025. "Numerical Simulation of Perkins Instability in the Midlatitude F-Region Ionosphere: The Influence of Background Ionospheric Multi-Factors" Atmosphere 16, no. 2: 221. https://doi.org/10.3390/atmos16020221

APA Style

Liu, Y., Lan, T., Zhou, Y., Zhu, Y., Fan, Z., Wu, Y., Zhang, Y., & Wang, X. (2025). Numerical Simulation of Perkins Instability in the Midlatitude F-Region Ionosphere: The Influence of Background Ionospheric Multi-Factors. Atmosphere, 16(2), 221. https://doi.org/10.3390/atmos16020221

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