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Both proton ring and shell distributions, providing a positive gradient along the perpendicular direction, can trigger ion Bernstein instabilities and excite magnetosonic (MS) waves in the Earth’s inner magnetosphere. A thorough comparison of the ion Bernstein instability due to the proton ring and shell distributions has not been performed. In this study, we perform and compare the MS wave instabilities under different wave normal angles and parameters of ring and shell distributions. We find that (1) compared to shell-driven MS waves, ring-driven MS waves have narrower frequency and wavenumber ranges. (2) The peak growth rates of shell-driven MS waves are always near the first peaks of the square of first-kind Bessel function Jn2. While, The peak growth rates of ring-driven MS waves are near the first peaks of Jn2, and shift to the cold plasma dispersion relation as ring velocity or wave normal angle increases. (3) MS wave growth rates increase as first peaks of Jn2 approach the cold plasma dispersion in frequency-wavenumber domain. This result can be used to explain dependences on ring (shell) velocity and wave normal angle. (4) The MS wave frequency range and growth rates decrease with increasing ring or shell temperature.

In the present paper, we investigate the effects of latitude-dependent wave power spectrum on the interactions of chorus with electrons. Great errors in evaluating the electron diffusion coefficients and the resultant electron temporal evolutions are introduced by the widely adopted latitudinally constant model, compared with the latitudinally varying model. The latitudinally constant model tends to overestimate (underestimate) the diffusion coefficients for electrons below (above) 200 keV. The overestimation and underestimation are mainly confined in small to intermediate pitch angles, increase with decreasing pitch angles, and can reach up to several orders of magnitude. The large differences in diffusion coefficients significantly alter the net changes of electron phase space densities and the resultant shapes of electron pitch angle distributions. Our simulations demonstrate that the wave power spectrum distribution along the magnetic field line plays an important role in controlling the dynamics of radiation belt electrons.

The frequency chirping of chorus waves is commonly observed in the Earth's inner magnetosphere, but its generation remains an open question. Recently, Liu et al. (2021), https://doi.org/10.1029/2021JA029258 reported two unusual rising‐tone (upward chirping) chorus elements. Although the central frequency of constituent subpackets rises, the frequency of a single subpacket is surprisingly downward chirping. With a gcPIC‐δf $\delta f$ simulation in the dipole field, we successfully reproduce this kind of substructure, which contains alternating signs of chirping. Interestingly, both hole and hill structures are formed around the theoretical resonant velocities in the electron phase space, no matter whether the chirping is upward or downward. However, during each chirping interval, only one structure (either a hole or a hill) is associated with wave excitation: the upward chirping is related to the hole, while the hill contributes to the downward chirping. Our study provides a fresh perspective on the theory of frequency chirping in chorus waves. Plain Language Summary: The frequency chirping is a typical feature of chorus waves in the Earth's inner magnetosphere, which generally contain either rising‐tone (upward chirping) elements or falling‐tone (downward chirping) elements. Previous theory has suggested that the chirping is due to the nonlinear wave‐particle interaction, where the hole or hill structure is formed in the electron phase space. Recently, Liu et al. (2021), https://doi.org/10.1029/2021JA029258 have observed the upward chirping elements with their subpackets of downward chirping. What electron structure is associated with these elements becomes a puzzle. With a one‐dimensional (1D) general curvilinear particle‐in‐cell (gcPIC) δf simulation in the dipole magnetic field, we successfully reproduce this kind of chorus element, whose frequency contains alternating upward and downward chirping. Interestingly, both the hole and hill structures are formed during a chirping interval, but only one of the two structures is responsible for wave excitation and frequency chirping. The structure of hole‐hill combination provides an important clue into the theory of the frequency chirping in chorus waves. Key Points: With a gcPIC‐δf $\delta f$ simulation in the dipole field, we reproduce the upward chirping chorus element, whose subpackets are downward chirpingBoth hole and hill structures can be formed in the ζ−v‖ $\zeta -{v}_{\Vert }$ phase space, no matter whether the frequency is upward or downward chirpingThe time evolution of the hole and hill structures in the phase space leads to the alternating frequency chirping [ABSTRACT FROM AUTHOR]

The power gap of chorus waves at ∼0.5fce has been discovered for decades, but its generation mechanism is still under debate. Previous studies have revealed that electron plateau distributions are vitally important for the gap formation. By analyzing over one‐year Van Allen Probes data, we have studied chorus waves with (banded events) and without a power gap (no‐gap events), and their correlations with the electron plateau distribution. Although there is no significant difference in the morphology of velocity distributions in banded and no‐gap events, banded chorus events are typically accompanied by a plateau component with about one order higher number density than no‐gap events. The plateau components can cause severe damping at ∼0.5fce through cyclotron resonance rather than Landau resonance, and the gap frequency is roughly determined by the bulk velocity of plateau components. Our study provides new observational constraints on the generation mechanisms of power gap. Plain Language Summary: The generation mechanism of power gap at ∼0.5fce of chorus waves has been a long‐standing problem for decades. Although its generation mechanism is still under debate, there is a consensus that the electron plateau in the parallel velocity distribution is a key factor to solve this problem. Here, we try to find out what role the electron plateau plays in the gap formation based on a statistical analysis of Van Allen Probes data. First of all, we find banded chorus events indeed have a more pronounced plateau shape (about one order higher number density) than no‐gap events, confirming the importance of electron plateau. We further find that the electron plateau can cause the severe wave damping at ∼0.5fce via cyclotron resonance rather than Landau resonance, and the gap frequency is roughly determined by the bulk velocity of electron plateau based on the cyclotron resonance condition. We also compare the morphology of electron velocity distributions in banded and no‐gap events, but do not find an obvious difference between them. Our study provides new observational constraints on the generation mechanisms of the power gap, which may help scientists finally solve this problem. Key Points: Banded chorus events typically have a more pronounced plateau shape than no‐gap events in the parallel electron distributionThe electron plateau can cause severe damping at ∼0.5fce via cyclotron resonance, and roughly determine the gap frequencyThere is no significant difference between the normalized electron velocity distributions in banded and no‐gap events [ABSTRACT FROM AUTHOR]

Abstract The frequency chirping of chorus waves is commonly observed in the Earth’s inner magnetosphere, but its generation remains an open question. Recently, Liu et al. (2021), https://doi.org/10.1029/2021JA029258 reported two unusual rising‐tone (upward chirping) chorus elements. Although the central frequency of constituent subpackets rises, the frequency of a single subpacket is surprisingly downward chirping. With a gcPIC‐δf simulation in the dipole field, we successfully reproduce this kind of substructure, which contains alternating signs of chirping. Interestingly, both hole and hill structures are formed around the theoretical resonant velocities in the electron phase space, no matter whether the chirping is upward or downward. However, during each chirping interval, only one structure (either a hole or a hill) is associated with wave excitation: the upward chirping is related to the hole, while the hill contributes to the downward chirping. Our study provides a fresh perspective on the theory of frequency chirping in chorus waves.

Abstract The power gap of chorus waves at ∼0.5fce has been discovered for decades, but its generation mechanism is still under debate. Previous studies have revealed that electron plateau distributions are vitally important for the gap formation. By analyzing over one‐year Van Allen Probes data, we have studied chorus waves with (banded events) and without a power gap (no‐gap events), and their correlations with the electron plateau distribution. Although there is no significant difference in the morphology of velocity distributions in banded and no‐gap events, banded chorus events are typically accompanied by a plateau component with about one order higher number density than no‐gap events. The plateau components can cause severe damping at ∼0.5fce through cyclotron resonance rather than Landau resonance, and the gap frequency is roughly determined by the bulk velocity of plateau components. Our study provides new observational constraints on the generation mechanisms of power gap.

Earth's radiation belt and ring current are donut-shaped regions of energetic and relativistic particles, trapped by the geomagnetic field. The strengthened solar wind dynamic pressure (Pdyn) can alter the structure of the geomagnetic field, which can bring about the dynamic variation of radiation belt and ring current. In the study, we firstly utilize group test particle simulations to investigate the phase space density (PSD) under the varying geomagnetic field modeled by the International Geomagnetic Reference Field (IGRF) and T96 magnetic field models from 19 December 2015 to 20 December 2015. Combining the observation of the Van Allen Probe, we find that the PSD of outer radiation belt electrons evolves towards different states under different levels of Pdyn. In the first stage, the Pdyn (~7.94 nPa) results in the obvious rise of electron anisotropy. In the second stage, there is a significant reduction in PSD for energetic electrons at all energy levels and pitch angles under the action of intense Pdyn (~22 nPa), which suggests that the magnetopause shadowing and outward radial diffusion play important roles in the second process. The result of the study can help us further understand the dynamic evolution of the radiation belt and ring current during a period of geomagnetic disturbance. [ABSTRACT FROM AUTHOR]

• Weakened magnetic field is pushing radiation belt particles towards the Earth in SAA. • The average deepening rate of radiation belt particles is 4.3 km/year. • The deepening is increased by 434–590 km in last 120 years. • Deepening in the SAA region is depended on its initial longitude. • Deepening in the SAA region is independent of its equatorial pitch angle. The geomagnetic field has an unusual weak spot over South America and the South Atlantic Ocean, called South Atlantic Anomaly (SAA). We know that the magnetospheric particles trapped in the geomagnetic field move to the deeper altitudes over the SAA, resulting in the lower inner boundary of the radiation belt. However, over the past 400 years, the magnetic field in the SAA region has decreased consistently. In some recent studies, a rapid decrease of the magnetic field in the SAA region and the possibility of splitting of SAA have been reported. This weakened geomagnetic field has a bearing on the lower altitudinal reach (deepening) of the radiation belt particles. In this context, we performed simulations to examine and quantify the lowest altitudes encountered by the radiation belt particles over the SAA region. We found that particles of energies 100 keV-50 MeV, initially entering from an altitudes of 1–2 Earth radii are moving closer to the Earth at the rate of 3.8 - 4.7 km/year, which results in approximately 434 - 590 kms decrease in their altitude over the SAA region during 1900 to 2020. Furthermore, we noticed that the lowest altitude encountered by the particle in the SAA region is dependent on the initial longitude and independent of its initial local pitch angle corresponding to its entry in the Earth's magnetosphere. [ABSTRACT FROM AUTHOR]

Abstract In the Earth’s inner magnetosphere, there exist regions like plasmasphere, ring current, and radiation belts, where the population of charged particles trapped along the magnetic field lines is more. These particles keep performing gyration, bounce and drift motions until they enter the loss cone and get precipitated to the neutral atmosphere. Theoretically, the mirror point latitude of a particle performing bounce motion is decided only by its equatorial pitch angle. This theoretical manifestation is based on the conservation of the first adiabatic invariant, which assumes that the magnetic field varies slowly relative to the gyro-period and gyro-radius. However, the effects of gyro-motion cannot be neglected when gyro-period and gyro-radius are large. In such a scenario, the theoretically estimated mirror point latitudes of electrons are likely to be in agreement with the actual trajectories due to their small gyro-radius. Nevertheless, for protons and other heavier charged particles like oxygen, the gyro-radius is relatively large, and the actual latitude of the mirror point may not be the same as estimated from the theory. In this context, we have carried out test particle simulations and found that the L-shell, energy, and gyro-phase of the particles do affect their mirror points. Our simulations demonstrate that the existing theoretical expression sometimes overestimates or underestimates the magnetic mirror point latitude depending on the value of L-shell, energy and gyro-phase due to underlying guiding centre approximation. For heavier particles like proton and oxygen, the location of the mirror point obtained from the simulation deviates considerably (∼ 10°–16°) from their theoretical values when energy and L-shell of the particle are higher. Furthermore, the simulations show that the particles with lower equatorial pitch angles have their mirror points inside the high or mid-latitude ionosphere.

Earth’s radiation belt and ring current are donut-shaped regions of energetic and relativistic particles, trapped by the geomagnetic field. The strengthened solar wind dynamic pressure (Pdyn) can alter the structure of the geomagnetic field, which can bring about the dynamic variation of radiation belt and ring current. In the study, we firstly utilize group test particle simulations to investigate the phase space density (PSD) under the varying geomagnetic field modeled by the International Geomagnetic Reference Field (IGRF) and T96 magnetic field models from 19 December 2015 to 20 December 2015. Combining the observation of the Van Allen Probe, we find that the PSD of outer radiation belt electrons evolves towards different states under different levels of Pdyn. In the first stage, the Pdyn (~7.94 nPa) results in the obvious rise of electron anisotropy. In the second stage, there is a significant reduction in PSD for energetic electrons at all energy levels and pitch angles under the action of intense Pdyn (~22 nPa), which suggests that the magnetopause shadowing and outward radial diffusion play important roles in the second process. The result of the study can help us further understand the dynamic evolution of the radiation belt and ring current during a period of geomagnetic disturbance.

• PAD of magnetospheric trapped particles is studied using test-particle simulations. • When adiabatic invariants are conserved, particles follow butterfly-type PAD. • All three adiabatic invariant violate when gyro-radius of trapped particles is large. • Particles follow Gaussian type PAD when adiabatic invariants are not conserved. • Theoretically estimated bounce/drift period is invalid as adiabatic invariant violate. This article aims to understand the pitch angle distributions (PADs) of the charged particles in the Earth's inner magnetosphere using test-particle simulations. The emphasis is on characterizing the variation in pitch angle of the charged particles trapped along the Earth's magnetic field lines. These charged particles undergo gyration, bounce, and azimuthal drift motions in the Earth's inner magnetosphere. They are trapped until their pitch angle falls into the loss-cone to get lost into the upper atmosphere. We have developed a three-dimensional test-particle simulation model in which the relativistic equation of motion is solved numerically to track these trapped particle's trajectories. We have examined the pitch angle distributions of the electron, proton, and oxygen for the cases where adiabatic invariants are conserved and non-conserved. For this purpose, we have considered the particle's trajectory in the latitudinal range of [ - 40 ° , 40 ° ] for one complete drift around the Earth. We found that when adiabatic invariants are conserved, the particles possess butterfly-type pitch angle distributions. Whereas, when adiabatic invariants are not conserved, the particle's pitch angle distribution bunches toward the 90°-peaked distribution. The situation of non-conservation of adiabatic invariants demonstrated in the present simulation is arising due to larger gyro-radius (few Earth radii) over which the ambient magnetic field is not constant. We have noticed that in the static dipolar magnetic field, all three, 1st, 2nd, and partly 3rd adiabatic invariants are non-conserved when gyro-radius is larger. The information on the change in the pitch angle distribution pattern from butterfly-type to 90°-peak distribution will be useful to understand the pitch angle distributions observed by the recent spacecraft in the Earth's radiation belts. [ABSTRACT FROM AUTHOR]

• 3D test-particle simulation model is developed to examine trapped particle dynamics. • RK4 can simulate only bounce motion of the protons due to more numerical dissipation. • RK6 can simulate trapped particle motion of energy 5 keV to 250 MeV at L = 2–6. • The simulation results of bounce and drift periods are in agreement with theory. • RK6 can be applied to the time-varying and non-analytical magnetospheric studies. This article aims to understand the motion of the charged particles trapped in the Earth's inner magnetosphere. The emphasis is on identifying the numerical scheme, which is appropriate to characterize the trajectories of the charged particles of different energies that enter the Earth's magnetosphere and get trap along the magnetic field lines. These particles perform three different periodic motions, namely: gyration, bounce, and azimuthal drift. However, often, the gyration of the particle is ignored, and only the guiding center of the particle is traced to reduce the computational time. It is because the simulation of all three motions (gyro, bounce, and drift) together needed a robust numerical scheme, which has less numerical dissipation. We have developed a three-dimensional test particle simulation model in which the relativistic equation of motion is solved numerically using the fourth and sixth-order Runge-Kutta methods. The stability of the simulation model is verified by checking the conservation of total kinetic energy and adiabatic invariants linked with each type of motion. We found that the sixth-order Runge-Kutta method is suitable to trace the complete trajectories of both proton and electron of a wide energy range, 5 keV to 250 MeV for L = 2 – 6. We have estimated the bounce and drift periods from the simulations, and they are found to be in good agreement with the theory. The study implies that a simulation model with sixth-order Runge-Kutta method can be applied to the time-vary, non-analytical form of magnetic configuration in future studies to understand the dynamics of charged particles trapped in Earth's magnetosphere. [ABSTRACT FROM AUTHOR]