# A Mechanism for Large-Amplitude Parallel Electrostatic Waves Observed at the Magnetopause

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## Abstract

**:**

## 1. Introduction

**${B}_{0}$**) and is marked with the subscript, “‖”, X is perpendicular to

**${B}_{0}$**and in the spacecraft spin plane (P1), and Y completes the right-handed system (P2). The top two panels show electron and ion omnidirectional energy spectra; the next two panels show electron and ion number density, electron and ion temperature, respectively. Panels 5 and 6 show ion bulk velocity and magnetic field vector, respectively. Panels 7, 8 and 9 show electric field power spectral density (EPSD) in (mV/m)${}^{2}$/Hz, the electric field in FACs, and total magnetic field fluctuations, respectively. The white line on the spectrum (panel 7) is the ion plasma frequency, and the dotted vertical lines indicate the interval with turbulent region where the magnetopause and magnetosheath plasmas are intermixed.

## 2. Theoretical Model

#### 2.1. Ion- and Electron-Acoustic Soliton Solutions

#### Numerical Results

#### 2.2. Predictions of the Model

#### 2.3. Comparison of Theoretical Predictions with Observations of Large Amplitude Electrostatic Waves at the Magnetopause

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Overview example of ESWs observed during a single period of K–H instability oscillations on 8 September 2015 by MMS spacecraft. From top to bottom, electron and ion omnidirectional energy spectra, electron and ion number density, electron and ion temperature, ion bulk velocity in geocentric solar ecliptic (GSE) coordinates, magnetic field vector in GSE, electric field power spectral density (EPSD) in (mV/m)${}^{2}$/Hz, the electric field in field-aligned coordinates(FAC), and total magnetic field fluctuations. The white line on the spectrum is the ion plasma frequency, the dashed vertical black lines indicate crossings of the vortex-induced current sheets, and the dotted vertical lines indicate the interval with turbulent region where the magnetopause and magnetosheath plasmas are intermixed. (Figure reprinted with permission from [52]).

**Figure 2.**Panel (

**a**) shows variation of Sagdeev pseudo-potential $S(\varphi ,M)$ for the slow ion-acoustic solitons versus the electrostatic potential $\varphi $, for the cold electron density of ${n}_{ec}^{0}$ = 0.04 and for ${n}_{ic}^{0}$ = 0.9, ${n}_{ih}^{0}$ = 0.1, ${\sigma}_{ic}$ = 0.07, ${\sigma}_{ih}$ = 1, ${\sigma}_{ec}$ = 0.0017, ${\sigma}_{e1}$ = ${\sigma}_{e2}$ = 0.13, ${U}_{ih}$ = 1.25, ${U}_{e1}=-{U}_{e2}$ = 5.52, and ${n}_{e1}^{0}$ = ${n}_{e1}^{0}$ = 0.48, and for Mach number M = 0.57, 0.58, 0.60 and 0.61 for the curves 1, 2, 3 and 4, respectively. Curve 4 shows that slow ion-acoustic solitons do not exist for $M\ge $ 0.61. Panel (

**b**) shows soliton profiles of electrostatic potential $\varphi $ versus $\xi $ for the Mach numbers corresponding to curves 1, 2 and 3 of Panel (

**a**). Panel (

**c**) shows profiles of the electric field E versus $\xi $ for the Mach numbers corresponding to curves 1, 2 and 3 of Panel (

**a**). Panel (

**d**) shows FFT power spectra of the unnormalized slow ion-acoustic solitons electric field E (in mV m${}^{-1}$). The x-axis represents the frequency, f, in Hz. The y-axis represents the electric field power spectral density ((mV m${}^{-1}$)${}^{2}$/Hz), normalized with the peak power spectral density, expressed in units of decibel, dB. The power spectral density peaks at f = 616 Hz, 627 Hz, and 865 Hz for M = 0.57 (curve 1), 0.58 (curve 2) and 0.60 (curve 3), respectively.

**Figure 3.**Panel (

**a**) shows variation of Sagdeev pseudo-potential $S(\varphi ,M)$ for the slow electron-acoustic solitons versus the electrostatic potential $\varphi $, for the same plasma parameters as in Figure 2 and for Mach number M = 6.5, 6.7, 7.0 and 7.1 for the curves 1, 2, 3 and 4, respectively. Curve 4 shows that slow electron-acoustic solitons do not exist for $M\ge $ 7.1. Panel (

**b**) shows soliton profiles of electrostatic potential $\varphi $ versus $\xi $ for the Mach numbers corresponding to curves 1, 2 and 3 of Panel (

**a**). Panel (

**c**) shows profiles of the electric field E versus $\xi $ for the Mach numbers corresponding to curves 1, 2 and 3 of Panel (

**a**). Panel (

**d**) shows FFT power spectra of the unnormalized slow electron-acoustic solitons electric field E (in mV m${}^{-1}$) for the Mach numbers corresponding to panel c of Figure 3, and in the same format as that of Figure 2d.

**Figure 4.**Panel (

**a**) shows variation of Sagdeev pseudo-potential $S(\varphi ,M)$ for the fast electron-acoustic solitons versus the electrostatic potential $\varphi $, for the same plasma parameters as in Figure 2 and for Mach number M = 26.45, 26.50, 26.55 and 26.59 for the curves 1, 2, 3 and 4, respectively. Curve 4 shows that fast electron-acoustic solitons do not exist for $M\ge $ 26.59. Panels (

**b**,

**c**) show profiles of electrostatic potential $\varphi $ and electric field E versus $\xi $, respectively, for the Mach numbers corresponding to curves 1, 2 and 3 of Panel (

**a**). Panel (

**d**) shows FFT power spectra of the unnormalized fast electron-acoustic solitons electric field E (in mV m${}^{-1}$) in the same format as that of Figure 2d.

**Figure 5.**Plot of parallel electric field from MMS1 highest resolution HMFE Electric Field, Level 2 burst mode data sampled at 65 kHz. The y-axis shows the parallel electric field in mV/m and the x-axis represents the time in milliseconds after the start of the event.

**Table 1.**Properties of the slow ion-acoustic, slow electron-acoustic and fast electron-acoustic solitons for the case of cold electron density of ${N}_{ec}$ = 0.04 ${N}_{0}$ for the magnetopause parameters observed by MMS spacecraft by Wilder et al. [52]: ${N}_{0}$ = 20 cm${}^{-3}$, cold ions density ${N}_{ic}$ = 18 cm${}^{-3}$, cold ions temperature ${T}_{ic}$ = 40 eV, hot ions density ${N}_{ic}$ = 2 cm${}^{-3}$, hot ions temperature ${T}_{ih}$ = 600 eV, hot ion beam speed ${V}_{ih}$ = 300 km s${}^{-1}$, ${N}_{e1}$ = ${N}_{e2}$ = 0.48 ${N}_{0}$, cold electron density ${N}_{ec}$ = 0.04 ${N}_{0}$, ${V}_{e1}$ = $-{V}_{e2}$ = 1325 km s${}^{-1}$, ${T}_{e1}$ = ${T}_{e2}$ = 80 eV, and ${T}_{ec}$ = 1 eV. Then, we get the hot ion thermal speed, ${C}_{ih}$ = 240 km s${}^{-1}$, and hot ion Debye length, ${\lambda}_{di}$ = 40 m.

Modes | Polarity | Mach Number | V (km s${}^{-1}$) | W (m) | E (mV m${}^{-1}$) | $\mathbf{\Phi}$ (V) | ${\mathit{f}}_{\mathit{peak}}$ (Hz) |
---|---|---|---|---|---|---|---|

Slow ion-acoustic | Positive | 0.57 | 137 | 83 | 40 | 2.4 | 616 |

0.58 | 139 | 72 | 60 | 3.6 | 627 | ||

0.60 | 144 | 59 | 120 | 5.7 | 865 | ||

Slow electron-acoustic | Negative | 6.5 | 1560 | 158 | 10 | 1.3 | 3659 |

6.7 | 1608 | 134 | 15 | 1.8 | 3772 | ||

7.0 | 1680 | 102 | 30 | 2.5 | 5911 | ||

Fast electron-acoustic | Negative | 26.45 | 6348 | 265 | 12 | 3.1 | 7445 |

26.50 | 6360 | 211 | 21 | 3.8 | 7459 | ||

26.55 | 6372 | 169 | 33 | 4.5 | 14,946 |

**Table 2.**Properties of the slow ion-acoustic, slow electron-acoustic and fast electron-acoustic solitons for the case of cold electron density of ${N}_{ec}$ = 0.01 ${N}_{0}$, ${N}_{e1}$ = ${N}_{e2}$ = 0.495 ${N}_{0}$ and for all other plasma parameters as given in Table 1.

Modes | Polarity | Mach Number | V (km s${}^{-1}$) | W (m) | E (mV m${}^{-1}$) | $\mathbf{\Phi}$ (V) | ${\mathit{f}}_{\mathit{peak}}$ (Hz) |
---|---|---|---|---|---|---|---|

Slow ion-acoustic | Positive | 0.70 | 168 | 123 | 120 | 11 | 504 |

0.75 | 180 | 97 | 320 | 24 | 541 | ||

0.765 | 184 | 87 | 400 | 28 | 551 | ||

Slow electron-acoustic | Negative | 4.20 | 1008 | 177 | 1 | 0.15 | 1513 |

4.30 | 1032 | 131 | 2 | 0.23 | 2324 | ||

4.35 | 1044 | 114 | 3 | 0.27 | 2351 | ||

Fast electron-acoustic | Negative | 26.30 | 6312 | 400 | 5 | 1.5 | 4738 |

26.40 | 6336 | 240 | 15 | 3.0 | 7134 | ||

26.50 | 6360 | 165 | 30 | 4.2 | 9548 |

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**MDPI and ACS Style**

Lakhina, G.S.; Singh, S.; Sreeraj, T.; Devanandhan, S.; Rubia, R.
A Mechanism for Large-Amplitude Parallel Electrostatic Waves Observed at the Magnetopause. *Plasma* **2023**, *6*, 345-361.
https://doi.org/10.3390/plasma6020024

**AMA Style**

Lakhina GS, Singh S, Sreeraj T, Devanandhan S, Rubia R.
A Mechanism for Large-Amplitude Parallel Electrostatic Waves Observed at the Magnetopause. *Plasma*. 2023; 6(2):345-361.
https://doi.org/10.3390/plasma6020024

**Chicago/Turabian Style**

Lakhina, Gurbax Singh, Satyavir Singh, Thekkeyil Sreeraj, Selvaraj Devanandhan, and Rajith Rubia.
2023. "A Mechanism for Large-Amplitude Parallel Electrostatic Waves Observed at the Magnetopause" *Plasma* 6, no. 2: 345-361.
https://doi.org/10.3390/plasma6020024