Evolution Of Phase Space Density Research Articles

Phase-space density control based on configuration invariant of spacecraft swarm

This paper investigates a phase-space density control based on configuration invariant for large-scale spacecraft swarms. By introducing Jordan-reduced dynamics and the configuration invariant to characterize relative orbits, the macroscopic swarm dynamics is presented with respect to the temporal evolution of phase space density. The density migration of the spacecraft swarm in phase space is modeled as a specific convection–diffusion process, where the diffusion term promotes swarm spreading and the convection term guides the swarm towards a designated region for uniform distribution. Through local density estimation and defining the convection and diffusion terms, the feedback control’s specific form is proposed and its exponential stability is proved. Particularly, the formulation of the convection term outside the target region and the combined kernel function for local density estimation help in avoiding sporadic isolated spacecraft. Numerical results of spacecraft swarm deployment validate the effectiveness and superiority of phase-space density control.

On the Dynamics of Ultrarelativistic Electrons (>2 MeV) Near L* = 3.5 During 8 June 2015

AbstractUnderstanding local loss processes in Earth’s radiation belts is critical to understanding their overall structure. Electromagnetic ion cyclotron waves can cause rapid loss of multi‐MeV electrons in the radiation belts. These loss effects have been observed at a range of L* values, recently as low as L* = 3.5. Here, we present a case study of an event where a local minimum develops in multi‐MeV electron phase space density (PSD) near L* = 3.5 and evaluate the possibility of electromagnetic ion cyclotron (EMIC) waves in contributing to the observed loss feature. Signatures of EMIC waves are shown including rapid local loss and pitch angle bite outs. Analysis of the wave power spectral density during the event shows EMIC wave occurrence at higher L* values. Using representative wave parameters, we calculate minimum resonant energies, diffusion coefficients, and simulate the evolution of electron PSD during this event. From these results, we find that O+ band EMIC waves could be contributing to the local loss feature during this event. O+ band EMIC waves are uncommon, but do occur in these L* ranges, and therefore may be a significant driver of radiation belt dynamics under certain preconditioning of the radiation belts.

The Phase Space Density Evolution of Radiation Belt Electrons under the Action of Solar Wind Dynamic Pressure

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.

Early-Time Non-Equilibrium Pitch Angle Diffusion of Electrons by Whistler-Mode Hiss in a Plasmaspheric Plume Associated with BARREL Precipitation

In August 2015, the Balloon Array for Radiation belt Relativistic Electron Losses (BARREL) observed precipitation of energetic (<200 keV) electrons magnetically conjugate to a region of dense cold plasma as measured by the twin Van Allen Probes spacecraft. The two spacecraft passed through the high density region during multiple orbits, showing that the structure was spatial and relatively stable over many hours. The region, identified as a plasmaspheric plume, was filled with intense hiss-like plasma waves. We use a quasi-linear diffusion model to investigate plume whistler-mode hiss waves as the cause of precipitation observed by BARREL. The model input parameters are based on the observed wave, plasma and energetic particle properties obtained from Van Allen Probes. Diffusion coefficients are found to be largest in the same energy range as the precipitation observed by BARREL, indicating that the plume hiss waves were responsible for the precipitation. The event-driven pitch angle diffusion simulation is also used to investigate the evolution of the electron phase space density (PSD) for different energies and assumed initial pitch angle distributions. The results show a complex temporal evolution of the phase space density, with periods of both growth and loss. The earliest dynamics, within the ∼5 first minutes, can be controlled by a growth of the PSD near the loss cone (by a factor up to ∼2, depending on the conditions, pitch angle, and energy), favored by the absence of a gradient at the loss cone and by the gradients of the initial pitch angle distribution. Global loss by 1-3 orders of magnitude (depending on the energy) occurs within the first ∼100 min of wave-particle interaction. The prevalence of plasmaspheric plumes and detached plasma regions suggests whistler-mode hiss waves could be an important driver of electron loss even at high L-value (L ∼6), outside of the main plasmasphere.

Parametric Dependence of the Formation of Electron Butterfly Pitch Angle Distribution Driven by Magnetosonic Waves

AbstractUsing the full relativistic test particle (TP) simulation code, we investigate the parametric dependence of electron scattering and phase space density evolution driven by magnetosonic (MS) waves at L = 4.5 both inside and outside the plasmapause. The scattering effects caused by Landau resonance, bounce resonance, and the transit‐time effect are all involved in the study. The net scattering effects are evaluated in the form of diffusion coefficients with different combinations of MS wave parameters, such as frequency bandwidth and wave normal angle, and ambient plasma density. The results demonstrate that (1) Landau resonance and the transit‐time effect dominate the electron scattering inside and outside the plasmapause, respectively, while both are modulated by bounce resonant scattering; (2) bounce resonant scattering becomes more important with narrowband MS waves; (3) electron scattering induced by MS waves is highly sensitive to wave normal angle. The temporal phase space density (PSD) evolution obtained from 2‐D kinetic Fokker‐Planck simulations shows that MS waves with larger wave normal angles are more likely to generate electron butterfly pitch angle distributions (PADs) for hundreds of keV electrons outside the plasmapause. Our study suggests that the electron butterfly distribution has important implications for revealing the combined scattering of MS wave‐particle interactions, and the combination of the multiple scattering mechanisms should be carefully incorporated in future global modeling of radiation belt dynamics.

Local heating of radiation belt electrons to ultra-relativistic energies

Electrically charged particles are trapped by the Earth’s magnetic field, forming the Van Allen radiation belts. Observations show that electrons in this region can have energies in excess of 7 MeV. However, whether electrons at these ultra-relativistic energies are locally accelerated, arise from betatron and Fermi acceleration due to transport across the magnetic field, or if a combination of both mechanisms is required, has remained an unanswered question in radiation belt physics. Here, we present a unique way of analyzing satellite observations which demonstrates that local acceleration is capable of heating electrons up to 7 MeV. By considering the evolution of phase space density peaks in magnetic coordinate space, we observe distinct signatures of local acceleration and the subsequent outward radial diffusion of ultra-relativistic electron populations. The results have important implications for understanding the origin of ultra-relativistic electrons in Earth’s radiation belts, as well as in magnetized plasmas throughout the solar system.

Accounting for Variability in ULF Wave Radial Diffusion Models

AbstractMany modern outer radiation belt models simulate the long‐time behavior of high‐energy electrons by solving a three‐dimensional Fokker‐Planck equation for the drift‐ and bounce‐averaged electron phase space density that includes radial, pitch‐angle, and energy diffusion. Radial diffusion is an important process, often characterized by a deterministic diffusion coefficient. One widely used parameterization is based on the median of statistical ultralow frequency (ULF) wave power for a particular geomagnetic index Kp. We perform idealized numerical ensemble experiments on radial diffusion, introducing temporal and spatial variability to the diffusion coefficient through stochastic parameterization, constrained by statistical properties of its underlying observations. Our results demonstrate the sensitivity of radial diffusion over a long time period to the full distribution of the radial diffusion coefficient, highlighting that information is lost when only using median ULF wave power. When temporal variability is included, ensembles exhibit greater diffusion with more rapidly varying diffusion coefficients, larger variance of the diffusion coefficients and for distributions with heavier tails. When we introduce spatial variability, the variance in the set of all ensemble solutions increases with larger spatial scales of variability. Our results demonstrate that the variability of diffusion affects the temporal evolution of phase space density in the outer radiation belt. We discuss the need to identify important temporal and length scales to constrain variability in diffusion models. We suggest that the application of stochastic parameterization techniques in the diffusion equation may allow the inclusion of natural variability and uncertainty in modeling of wave‐particle interactions in the inner magnetosphere.

Energetic Electron Scattering due to Whistler Mode Chorus Waves Using Realistic Magnetic Field and Density Models in Jupiter's Magnetosphere

AbstractWe evaluate energetic electron scattering in pitch angle and energy using realistic magnetic field and density models due to whistler mode chorus waves in Jupiter's magnetosphere and study their dependences on various wave and background parameters. We calculate the bounce‐averaged diffusion coefficients by considering the latitudinal variation of total electron density and ambient magnetic field intensity, using the VIP4 internal magnetic field and CAN current sheet model. The electron phase space density evolution due to chorus waves is simulated at M shell of 10, using the central wave frequency at 0.1fce and wave amplitude of 30 pT. Under the typical values of the ratio between the plasma frequency and electron cyclotron frequency, chorus waves could cause fast pitch angle scattering loss of energetic electrons from tens to several hundred keV in several hours, and gradual acceleration of relativistic electrons at several MeV in several days. The electron pitch angle scattering at ~500 keV and the acceleration at several MeV are both enhanced using the latitudinally varying density and VIP4 + CAN magnetic field model compared to the electron evolution using the constant density and dipole magnetic field model. Our sensitivity study indicates that the electron scattering at higher energy is caused by waves at lower frequencies or in a lower‐density background plasma, and the scattering is faster for waves at smaller wave normal angles. The electron diffusion is mainly caused by waves at lower latitudes, but the waves at higher latitudes (>30°) contribute to the electron loss at higher energies (>2 MeV).

Nonlinear Wave Growth Analysis of Whistler‐Mode Chorus Generation Regions Based on Coupled MHD and Advection Simulation of the Inner Magnetosphere

AbstractWe show the regions where nonlinear growth of whistler‐mode chorus waves is preferred to occur in the inner magnetosphere. A global magnetohydrodynamics (MHD) simulation was used to obtain large‐scale electric and magnetic fields under the southward interplanetary magnetic field condition. With the electric and magnetic fields obtained by the MHD simulation, we ran a comprehensive inner magnetosphere‐ionosphere model to solve the evolution of phase space density of electrons. Hot electrons originating from the tail region drift sunward and penetrate deep into the inner region due to a combination of convection and substorm‐associated electric fields. Cold electrons also drift sunward, resulting in a contraction of the plasmasphere. We obtained the following results. (1) The whistler waves can first grow due to the linear mechanism (pitch angle anisotropy) in the premidnight‐prenoon region outside the plasmapause, followed by rapid, nonlinear mechanism accompanied with rising‐tone chorus elements. (2) When the solar wind speed is high, the whistler waves grow more efficiently due to linear and nonlinear mechanisms over a wider area because of deep penetration of hot electrons and the large contraction of the plasmasphere. This is consistent with the observation that the outer belt electrons increase for the fast solar wind. (3) For slow solar wind, the linear growth is mostly suppressed, but the nonlinear growth can still take place when external seed waves are present. This may explain the persistence of dawn chorus and large‐amplitude chorus waves that are often observed in the premidnight‐postdawn region in relatively weak geomagnetic activities.

The Radiation Belt Electron Scattering by Magnetosonic Wave: Dependence on Key Parameters

AbstractMagnetosonic (MS) waves have been found capable of creating radiation belt electron butterfly distributions in the inner magnetosphere. To investigate the physical nature of the interactions between radiation belt electrons and MS waves, and to explore a preferential condition for MS waves to scatter electrons efficiently, we performed a comprehensive parametric study of MS wave‐electron interactions using test particle simulations. The diffusion coefficients simulated by varying the MS wave frequency show that the scattering effect of MS waves is frequency insensitive at low harmonics (f < 20 fcp), which has great implications on modeling the electron scattering caused by MS waves with harmonic structures. The electron scattering caused by MS waves is very sensitive to wave normal angles, and MS waves with off 90° wave normal angles scatter electrons more efficiently. By simulating the diffusion coefficients and the electron phase space density evolution at different L shells under different plasma environment circumstances, we find that MS waves can readily produce electron butterfly distributions in the inner part of the plasmasphere where the ratio of electron plasma‐to‐gyrofrequency (fpe/fce) is large, while they may essentially form a two‐peak distribution outside the plasmapause and in the inner radiation belt where fpe/fce is small.

Understanding the Mechanisms of Radiation Belt Dropouts Observed by Van Allen Probes

AbstractTo achieve a better understanding of the dominant loss mechanisms for the rapid dropouts of radiation belt electrons, three distinct radiation belt dropout events observed by Van Allen Probes are comprehensively investigated. For each event, observations of the pitch angle distribution of electron fluxes and electromagnetic ion cyclotron (EMIC) waves are analyzed to determine the effects of atmospheric precipitation loss due to pitch angle scattering induced by EMIC waves. Last closed drift shells (LCDS) and magnetopause standoff position are obtained to evaluate the effects of magnetopause shadowing loss. Evolution of electron phase space density (PSD) versus L* profiles and the μ and K (first and second adiabatic invariants) dependence of the electron PSD drops are calculated to further analyze the dominant loss mechanisms at different L*. Our findings suggest that these radiation belt dropouts can be classified into distinct classes in terms of dominant loss mechanisms: magnetopause shadowing dominant, EMIC wave scattering dominant, and combination of both mechanisms. Different from previous understanding, our results show that magnetopause shadowing can deplete electrons at L* < 4, while EMIC waves can efficiently scatter electrons at L* > 4. Compared to the magnetopause standoff position, it is more reliable to use LCDS to evaluate the impact of magnetopause shadowing. The evolution of electron PSD versus L* profile and the μ, K dependence of electron PSD drops can provide critical and credible clues regarding the mechanisms responsible for electron losses at different L* over the outer radiation belt.

Characteristic energy range of electron scattering due to plasmaspheric hiss

AbstractWe investigate the characteristic energy range of electron flux decay due to the interaction with plasmaspheric hiss in the Earth's inner magnetosphere. The Van Allen Probes have measured the energetic electron flux decay profiles in the Earth's outer radiation belt during a quiet period following the geomagnetic storm that occurred on 7 November 2015. The observed energy of significant electron decay increases with decreasing L shell and is well correlated with the energy band corresponding to the first adiabatic invariant μ = 4–200 MeV/G. The electron diffusion coefficients due to hiss scattering are calculated at L = 2–6, and the modeled energy band of effective pitch angle scattering is also well correlated with the constant μ lines and is consistent with the observed energy range of electron decay. Using the previously developed statistical plasmaspheric hiss model during modestly disturbed periods, we perform a 2‐D Fokker‐Planck simulation of the electron phase space density evolution at L = 3.5 and demonstrate that plasmaspheric hiss causes the significant decay of 100 keV–1 MeV electrons with the largest decay rate occurring at around 340 keV, forming anisotropic pitch angle distributions at lower energies and more flattened distributions at higher energies. Our study provides reasonable estimates of the electron populations that can be most significantly affected by plasmaspheric hiss and the consequent electron decay profiles.

Electron scattering by magnetosonic waves in the inner magnetosphere

AbstractWe investigate the importance of electron scattering by magnetosonic waves in the Earth's inner magnetosphere. A statistical survey of the magnetosonic wave amplitude and wave frequency spectrum, as a function of geomagnetic activity, is performed using the Van Allen Probes wave measurements and is found to be generally consistent with the wave distribution obtained from previous spacecraft missions. Outside the plasmapause the statistical frequency distribution of magnetosonic waves follows the variation of the lower hybrid resonance frequency, but this trend is not observed inside the plasmasphere. Drift and bounce averaged electron diffusion rates due to magnetosonic waves are calculated using a recently developed analytical formula. The resulting timescale of electron energization during disturbed conditions (when AE* > 300 nT) is more than 10 days. We perform a 2‐D simulation of the electron phase space density evolution due to magnetosonic wave scattering during disturbed conditions. Outside the plasmapause, the waves accelerate electrons with pitch angles between 50° and 70° and form butterfly pitch angle distributions at energies from ~100 keV to a few MeV over a timescale of several days; whereas inside the plasmapause, the electron acceleration is very weak. Our study suggests that intense magnetosonic waves may cause the butterfly distribution of radiation belt electrons especially outside the plasmapause, but electron acceleration due to magnetosonic waves is generally not as effective as chorus wave acceleration.

Effect of chorus normal angle on dynamic evolution of radiation belt energetic electrons

Gyroresonance between chorus waves and radiation belt electrons is usually evaluated by using the Gaussian wave normal angle (X=tanθ) distribution. In this study, we examine the influence of peak wave normal angle (Xm=tanθm) on chorus-electron interaction in the heart of radiation belts L=4.5. By varying Xm=0,2,4,6, we calculate the bounce-averaged diffusion coefficients, and then simulate the phase space density (PSD) evolution of radiation belt energetic electrons induced by both dayside and nightside chorus waves. Numerical simulations show that diffusion coefficients move to the low pitch angle region and drop rapidly by a factor of ∼10 as Xm increases by 2 each time, consequently leading to the smaller amplitude and narrower pitch angle region in electron PSD evolution. The current results demonstrate that chorus-electron interaction in the outer radiation belt is largely dependent on peak wave normal angles.

Controllable surface electrostatic velocity filter for polar molecules

We propose a scheme of a surface electrostatic velocity filter capable of preparing cold polar molecules on the surface of a substrate by selecting a low-velocity component of an effusive beam from a thermal gas reservoir. Using ND3 as a molecular sample, the dependence of the performance of the filter on the parameters of both the filter setup and the incident molecular beam is investigated by using a theoretical model and Monte Carlo simulations. A detailed study of the guiding process of molecules, including the evolution of phase space density of the packet in the filter, is carried out and shows that the beam selection process is mainly completed in the front part of the filter.

Temporal evolution of relativistic electrons induced by fast magnetosonic waves in the radiation belts

We study the influence of fast magnetosonic (MS) waves on the temporal evolution of the phase space density (PSD) of relativistic electrons in the outer radiation belts. We evaluate the bounce-averaged pitch angle, momentum and cross-diffusion rates, and find that all the diffusion rates are most pronounced within a limited region of pitch angles: ∼50°–70°. Consequently, numerical solutions of a 2D momentum–pitch angle diffusion equation show that significant increases in electron PSDs at energies of 1 MeV or above occur primarily within such a limited region of pitch angles. Moreover, neglecting cross-diffusion rates generally produces relatively large overestimates in the PSD evolution. The current results suggest that MS waves are primarily responsible for acceleration of relativistic (>1 MeV) electrons around relatively high pitch angles.

Effect of seed electron injection on chorus-driven acceleration of radiation belt electrons

Using the hybrid finite difference method, we solve the Fokker-Planck equation to study the effect of seed electron injection on acceleration of radiation belt electrons driven by chorus waves. Numerical results show that in the absence of injection chorus waves can accelerate electrons at large pitch angles (alpha(e)>60 degrees), producing enhancements in the phase space density (PSD) of (1-2 MeV) electrons by a factor of 100-1000 within 1-2 days. In the presence of injection, chorus waves yield increase in PSD of electrons by accelerating the injected seed electrons. Meanwhile, the PSD evolution increases as the pitch angle increases but decreases as electron energy increases. Moreover, the PSD evolution can extend to higher energies with a time scale of 1-2 days for 1-2 MeV energies. When the injection increases by a factor of 10 higher than the initial value and remains for about two days, maximum values of PSD for 1 or 2 MeV increase to 6 or 3 times respectively higher than those without injection in two days. The current results suggest that the injected seed electrons play an important role in the evolution of the radiation belt electrons.

Rapid acceleration of radiation belt energetic electrons by Z‐mode waves

We present the first simulation of the effect of Z‐mode waves on the outer radiation belt electron dynamics. We calculate bounce‐averaged diffusion rates in pitch angle and momentum and then use them as inputs to solve a 2‐D momentum‐pitch angle diffusion equation. Numerical results show that the phase space density (PSD) of 1 MeV electrons can enhance substantially and very rapidly (e.g., 30 minutes). In particular, the momentum diffusion rate exceeds the pitch angle and cross diffusion rates at 0.5 MeV and above, a behavior completely different from that for EMIC and chorus waves. Consequently, momentum (instead of pitch angle or cross) diffusion plays a dominant role in the dynamic evolution of energetic electrons. Moreover, the PSD evolution is found to be very dependent upon the assumed initial particle distributions. These results provide further insights on the interplay between acceleration mechanisms of outer radiation belt electrons.

On relaxation and transport in gyrokinetic drift wave turbulence with zonal flow

We present a theory for relaxation and transport in phase space for gyrokinetic drift wave turbulence with zonal flow. The interaction between phase space eddys and zonal flows is considered in two different limits, namely for K>>1 and K ≃ 1 where K is the Kubo number. For K>>1, the growth of an isolated coherent phase space structure is calculated, including the associated zonal flow dynamics. For K ≃ 1, mean field relaxation dynamics is considered in the presence of phase space granulations and zonal flows. In both limits, it is shown that the evolution equations for phase space structures are structurally similar to a corresponding Charney-Drazin theorem for zonal momentum balance in a potential vorticity conserving, quasi-geostrophic system. The transport flux in phase space is calculated in the presence of phase space density granulations and zonal flows. The zonal flow exerts a dynamical friction on ion phase space density evolution, which is a fundamentally new zonal flow effect.

Electron flux changes in the outer radiation belt by radial diffusion during the storm recovery phase in comparison with the fully adiabatic evolution

[1] The radial diffusion process can play an important role in redistributing the radiation belt electron fluxes. In this work, we have performed 1-D radial diffusion simulations to examine the evolution of the phase space density (PSD) of the outer radiation belt electrons and to estimate the corresponding fluxes during the storm recovery phase. The key element that distinguishes our simulations from previous works is the initial condition for PSD, which is characterized by a steep radial gradient across the trapping boundary. In our simulations, this condition is formed as a result of the drift loss effect of particles during the storm main phase, and the simulations of radial diffusion were run for the storm recovery phase. We performed the study for three classes of geomagnetic storms of different intensities, i.e., the moderate (−100 nT < Dstmin ≤ −50 nT), strong (−150 nT < Dstmin ≤ −100 nT), and severe (Dstmin ≤ −150 nT) storms. The effects of radial diffusion in PSD are notable in the following respects. First, the radial diffusion process is very significant for the initial few hours of the storm recovery phase. Second, the effect of the radial diffusion occurs in both inward and outward directions, thus affecting a wide range of L regions. The inward diffusion causes the PSD peak to move inward. The regions outside of the initial trapping boundary are refilled with finite PSD by the outward radial diffusion. Consequently, the combination of these effects results in different levels and patterns in the directional and omni-directional fluxes from those expected from a fully adiabatic evolution throughout the entire storm period. Last, the details of the PSD evolution, and thus its effect on the corresponding flux levels and patterns, differ among storms of different intensities. We report these differences in detail.