Electron Phase Space Research Articles

Reanalysis of relativistic radiation belt electron fluxes using CRRES satellite data, a radial diffusion model, and a Kalman filter

In this study we perform a reanalysis of the sparse MEA CRRES relativistic electron data using a relatively simple one‐dimensional radial diffusion model and a Kalman filtering approach. By combining observations with the model in an optimal way we produce a high time and space resolution reanalysis of the radiation belt electron fluxes over a 50‐d period starting on 18 August 1990. The results of the reanalysis clearly show pronounced peaks in the electron phase space density (PSD), which can not be explained by the variations in the outer boundary, and can only be produced by a local acceleration processes. The location of the innovation vector shows that local acceleration is most efficient at L* = 5.5 for electrons at K = 0.11 G0.5REand μ = 700 MeV/G. Sensitivity numerical experiments for various values of μ and K indicate that peaks in PSD become stronger with increasing K and μ. To verify that our results are not affected by the limitations of the satellite orbit and coverage, we performed an “identical twin” experiments with synthetic data specified only at the locations for which CRRES observations are available. Our results indicate that the model with data assimilation can accurately reproduce the underlying structure of the PSD even when data is sparse. The identical twin experiments also indicate that PSD at a particular L‐shell is determined by the local processes and cannot be accurately estimated unless local measurements are available.

Multisatellite determination of the relativistic electron phase space density at geosynchronous orbit: An integrated investigation during geomagnetic storm times

An integrated investigation method, which can study the relativistic electron phase space density distribution and check the reliability of employed magnetic field models simultaneously, is developed and applied to the geosynchronous orbit region for 53 geomagnetic storms during a ∼190‐d period. First, to test how the magnetospheric magnetic field affects the study of phase space density, two approaches are taken on handling the magnetic field model: One is to use an existing empirical model through the whole storm period; the other is to select one from a list of existing magnetic field models for each time bin during the period by fitting to multipoint in situ measurements. The magnetic field models in both approaches are again tested by Liouville's theorem, which requires the conserved phase space density for fixed phase space coordinates given no local losses and sources. Then on the basis of the selected magnetic field model, the phase space density is calculated by transforming the flux data from three Los Alamos National Laboratory geosynchronous satellites. By following the procedure developed here and using the cross‐satellite calibration achieved in previous work, we deduce the storm time electron phase space density distribution for the region near geosynchronous orbit, covering a range of L shells with L* centered ∼6. This work establishes the radial phase space density gradient at constant adiabatic invariants as a function of universal time during storm times, and three types of geomagnetic storms are defined according to the degree of energy‐dependent enhancements of energetic electrons during recovery phases. Initial results from this study suggest a source outside geosynchronous orbit for low‐energy electrons and a major source inside for high‐energy electrons.

Comparing electrostatic instabilities driven by mildly and highly relativistic proton beams

An electrostatic instability driven by counter-propagating tenuous proton beams that traverse a bulk plasma consisting of electrons and protons is considered. The system is spatially homogeneous and is evolved in time with a one-dimensional particle-in-cell simulation, which allows for a good statistical plasma representation. Mildly and highly relativistic beam speeds are modeled. The proton beams with a speed of 0.9c result in waves that saturate by the trapping of electrons. The collapse of the phase space holes in the electron distribution scatters these to a flat-top momentum distribution. The final electric fields are weak and the proton beams are weakly modulated. No secondary instabilities are likely to form that could thermalize the proton beams. The proton beams moving with 0.99c initially heat the bulk plasma through a three-wave interaction. Coalescing phase space holes in the bulk proton distribution arising from the saturation of ion acoustic waves transport wave energy to low wavenumbers. Highly relativistic phase space holes form in the electron distribution, which are not spatially homogeneous. The spatial envelope of these electron phase space holes interacts with the fluctuations driven by the phase space holes in the bulk protons, triggering a modulational instability. A Langmuir wave condensate forms that gives rise to strong and long electrostatic wave packets, as well as to a substantial modulation of the proton beams. The final state of the system with the highly relativistic proton beams is thus more unstable to further secondary instabilities that may transfer a larger beam energy fraction to the electrons and thermalize the proton beams more rapidly.

The plasma sheet and boundary layers under northward IMF: A multi-point and multi-instrument perspective

During conditions of northward interplanetary magnetic field (IMF), the near-tail plasma sheet is known to become denser and cooler, and is described as the cold-dense plasma sheet (CDPS). While its source is likely the solar wind, the prominent penetration mechanisms are less clear. The two main candidates are solar wind direct capture via double high-latitude reconnection on the dayside and Kelvin–Helmholtz/diffusive processes at the flank magnetopause. This paper presents a case study on the formation of the CDPS utilizing a wide variety of space- and ground-based observations, but primarily from the Double Star and Polar spacecraft on December 5th, 2004. The pertinent observations can be summarized as follows: TC-1 observes quasi-periodic (∼2 min period) cold-dense boundary layer (compared to a hot-tenuous plasma sheet) signatures interspersed with magnetosheath plasma at the dusk flank magnetopause near the dawn-dusk terminator. Analysis of this region suggests the boundary to be Kelvin–Helmholtz unstable and that plasma transport is ongoing across the boundary. At the same time, IMAGE spacecraft and ground based SuperDARN measurements provide evidence of high-latitude reconnection in both hemispheres. The Polar spacecraft, located in the southern hemisphere afternoon sector, sunward of TC-1, observes a persistent boundary layer with no obvious signature of boundary waves. The plasma is of a similar appearance to that observed by TC-1 inside the boundary layer further down the dusk flank, and by TC-2 in the near-Earth magnetotail. We present comparisons of electron phase space distributions between the spacecraft. Although the dayside boundary layer at Polar is most likely formed via double high-altitude reconnection, and is somewhat comparable to the flank boundary layer at Double Star, some differences argue in favour of additional transport that augment solar wind plasma entry into the tail regions.

The energization of relativistic electrons in the outer Van Allen radiation belt

The origin and dynamics of the Van Allen radiation belts is one of the longest-standing questions of the space age, and one that is increasingly important for space applications as satellite systems become more sophisticated, smaller and more susceptible to radiation effects. The precise mechanism by which the Earth’s magnetosphere is able to accelerate electrons from thermal to ultrarelativistic energies (E≫0.5 MeV) has been particularly difficult to definitively resolve. The traditional explanation is that large-scale, fluctuating electric and magnetic fields energize particles through radial diffusion1. More recent theories2,3 and observations4,5 have suggested that gyro-resonant wave–particle interactions may be comparable to or more important than radial diffusion. Using data collected simultaneously by multiple satellites passing through the magnetosphere at different distances from the Earth, we demonstrate that the latter of these is the dominant mechanism responsible for relativistic electron acceleration. Specifically, we identify frequent and persistent peaks in equatorial electron phase space density near or inside geosynchronous orbit that provide unambiguous evidence for local wave–particle acceleration. These observations represent an important step towards a more complete physical understanding of radiation belt dynamics and to the development of space-weather models.

On the fluctuation spectrum of plasma

The spectrum of electron phase space density fluctuations of a plasma is calculated by a novel method that parallels conventional calculations of the partition function in statistical physics. Expressions for the electric field fluctuations and the closely related form factor agree with existing results. The method clears up ambiguities about equipartition and provides a new expression for the spectrum of electrostatic phase space density fluctuations about stable non-Maxwellian equilibria.

Modulation of the growth of auroral kilometric radiation by electromagnetic ion cyclotron waves

The role of electromagnetic ion cyclotron (EMIC) waves in the stimulation of auroral kilometric radiation is addressed in this study. Observations of the modulation of electron flux and energy due to EMIC/electrostatic ion cyclotron waves within the auroral kilometric radiation (AKR) source region have been reported in the past. These modulations produce in turn oscillations of the electron phase space density and therefore lead to fluctuations of AKR growth rate that may translate along the magnetic field line at the group velocity of the stimulating waves. For a model electron “horseshoe” distribution function, we use the theory of the cyclotron maser instability to investigate how this process might influence the AKR wave spectrum and fine structure.

Storm time evolution of the outer radiation belt: Transport and losses

During geomagnetic storms the magnetic field of the inner magnetosphere exhibits large‐scale variations over timescales from minutes to days. Being mainly controlled by the magnetic field the motion of relativistic electrons of the outer radiation belt can be highly susceptible to its variations. This paper investigates evolution of the outer belt during the 7 September 2002 storm. Evolution of electron phase space density is calculated with the use of a test‐particle simulation in storm time magnetic and electric fields. The results show that storm time intensification of the ring current produces a large impact on the belt. In contrast to the conventional Dst effect the dominant effects are nonadiabatic and lead to profound and irreversible transformations of the belt. The diamagnetic influence of the partial ring current leads to expansion of electron drift orbits such that their paths intersect the magnetopause leading to rapid electron losses. About 2.5 hr after the storm onset most of the electrons outside L = 5 are lost. The partial ring current pressure also leads to an electron trap in the dayside magnetosphere where electrons stay on closed dayside drift orbits for as long as 11 hours. These sequestered electrons are reinjected into the outer belt due to partial recovery of the ring current. The third adiabatic invariant of these electrons exhibits rapid jumps and changes sign. These jumps produce localized peaks in the L*‐profile of electron phase space density which have previously been considered as an observable indication of local electron acceleration.

Evolution of the fastest-growing relativistic mixed mode instability driven by a tenuous plasma beam in one and two dimensions

Particle-in-cell simulations confirm here that a mixed plasma mode is the fastest growing when a highly relativistic tenuous electron-proton beam interacts with an unmagnetized plasma. The mixed modes grow faster than the filamentation and two-stream modes in simulations with beam Lorentz factors Γ of 4, 16, and 256, and are responsible for thermalizing the electrons. The mixed modes are followed to their saturation for the case of Γ=4 and electron phase space holes are shown to form in the bulk plasma, while the electron beam becomes filamentary. The initial saturation is electrostatic in nature in the considered one- and two-dimensional geometries. Simulations performed with two different particle-in-cell simulation codes evidence that a finite grid instability couples energy into high-frequency electromagnetic waves, imposing simulation constraints.

Phase space density distributions of energetic electrons in the outer radiation belt during two Geospace Environment Modeling Inner Magnetosphere/Storms selected storms

The phase space density distributions of energetic electrons during two storm periods, including the two storms on 21–23 October 2001 and 4–9 September 2002 selected by Geospace Environment Modeling Inner Magnetosphere/Storms campaign as the radiation belt assessment challenge in 2004 workshop, are presented in this paper. Electron data from the Synchronous Orbit Particle Analysis instrument aboard three Los Alamos National Laboratory geosynchronous satellites as well as the Comprehensive Energetic Particle and Pitch Angle Distribution instrument aboard Polar are used. The Tsyganenko 2001 storm model is chosen for the storm time magnetic field presentation, compared to the best‐fitting magnetic model achieved in a previous study. By tracing the temporally evolving radial distributions, we conclude that while the dropout of electron phase space density during storm main phases appears to be energy‐independent, the enhancement in recovery phases shows an energy‐dependent pattern. The average outwardly decreasing radial gradients of phase space density obtained during the recovery phases of the two storm periods strongly suggest the in situ acceleration is most likely the main source of new energetic electrons, along with a possible contribution from an external source.

Kinetic processes in the laser corona heated by a nanosecond iodine laser

The behaviour of plasma electrons in a laser corona generated by focusing the first harmonic (λ = 1.315 µm) beam of the nanosecond iodine laser Asterix for a power density in the focal spot of 1016 Wcm−2 is studied by solving a 1D Vlasov equation with a small collision term coupled to the Maxwell equations to describe the wave propagation and transformation in a long plasma corona. The temporal evolution of electron phase space is studied in detail with the aim to identify the mechanisms relevant for the wave properties. The dominant wave modes occurring in our model are both the backward and forward propagating Raman waves, each accompanied by a daughter electrostatic wave, which may strongly interact with the plasma electrons. Within the frame of our model we identify several mechanisms of wave transformation accompanying the propagation and the corresponding kinetic phenomena in the phase space. These are visualized by the behaviour of electron distribution function, evolution of electrostatic spectrum and of Raman reflection coefficient.

Remote sensing of a magnetotail reconnection X‐line using polar rain electrons

We report on electron phase space distributions (PSDs) observed near the plasma sheet (PS) boundary layer (PSBL) by the Cluster electron spectrometers when the northern lobe was occupied by significant fluxes of polar rain (PR) electrons. These observations reveal the spatial structure of the electron transition layer (TL) between the polar rain electrons and the PSBL electron population accelerated during reconnection. This TL comprises overlapping spatial dispersion signatures in both energy and pitch angle, which are caused by convection of flux tubes across the magnetic separatrix during the electron time‐of‐flight (TOF) from the X‐line combined with acceleration in the reconnection region. Analysis of this structure allows us to estimate the location of the X‐line. By assuming the PSBL population arises through acceleration of the PR electrons, comparison of their PSD indicates that the electrons gain energy proportional to their initial energy.

Particle-in-cell simulations of plasma slabs colliding at a mildly relativistic speed

Plasmas collide at relativistic speeds in many astrophysical and high-energy density laboratory environments. The boundaries that develop between such plasmas and expand at much larger speeds than the ion sound speed cs are not well understood. Here, we address two identical electron–proton plasma slabs that collide with a relativistic speed and a Mach number v/cs of over 400. The collision speed, the plasma temperature and magnetic field are such that the growth rate of the two-stream instability exceeds that of all other instabilities. We model a planar turbulent boundary (TB) with one-dimensional (1D) and 2D particle-in-cell (PIC) simulations. We show that the boundary dissipates its energy via electron phase space holes (EPSHs) that accelerate electrons at the boundary to relativistic speeds and increase significantly the speed of some protons. Our results are put into the context of a dynamic accretion disc and the jet of a microquasar. It is shown that the accelerated electrons could contribute to the disc wind and to relativistic leptonic jets, and possibly to the hard radiation component of the accretion disc.

LF/MF Whistler mode dispersive signals observed with rocket‐borne instruments in the auroral downward current region

The SIERRA sounding rocket launched 14 January 2002, northward from Poker Flat, Alaska, to an altitude of 735 km in active aurora. On the downleg, in the altitude range 500–700 km, the onboard high‐frequency wave experiment detected structured whistler mode signals in the frequency range 100–1000 kHz, lasting nearly 200 s during which the rocket moved approximately 200 km downrange. The most prominent feature consisted of narrowband signals with frequency descending from 500 kHz to 250 kHz in about 0.5 s, with bandwidths ∼10 kHz, with frequency spacings ∼25 kHz, and with amplitudes in the range 10–400 μV/m. The structured whistler mode signals coincide with a region of Alfvénically accelerated electrons of the type often associated with the downward current region, poleward of the inverted‐V electron structures usually associated with the upward current region. However, on short timescales there was no clear correlation between the whistler mode signals and the locally detected electron flux or density. Ray‐tracing calculations indicate that concentrated sources of waves originating on the whistler mode resonance cone and moving upward through a model ionosphere, with speeds of 1000–5000 km/s, to rocket altitudes, exhibit the magnitude of dispersion of the observed structured whistler mode features. We evaluate the strengths and weaknesses of the hypothesis that these concentrated sources could be associated with phase space electron holes shedding electrostatic whistlers as they move upward along the field line at altitudes much higher than the rocket in the downward current region.

Anomalous resistivity of current-driven isothermal plasmas due to phase space structuring

The anomalous electric resistivity of collisionless plasmas is an important issue in the physics of hot plasmas, e.g., in the context of auroral particle acceleration and of reconnection in the solar corona. The linear stability theory of isothermal current driven space plasmas predicts an ion-acoustic instability if the relative drift velocity of the current carrying particles exceeds a certain threshold, which, generally, depends on the plasma parameters. The spectrum of waves, excited by a marginal instability, is very narrow. Hence, the wave power at saturation and the corresponding electric resistivity due to wave-particle interaction cannot be obtained by means of a quasilinear, weak turbulence approach and the nonlinear single mode theory provides too small saturation amplitudes. To solve the nonlinear problem a newly developed unsplit conservative Eulerian Vlasov code is applied to simulate a strongly magnetized current driven plasma, which can be considered in 1D1V (one spatial, one velocity space direction). Instead of periodic boundary conditions, usually used as they are simpler to treat, open boundaries are implemented which allow to maintain a constant current flow. Simulated is a typical almost isothermal (Te=2Ti) hot (κTi=1keV) space plasma for the real mass ratio mi∕me=1836. The initial spontaneous instability is followed by a three-stage nonlinear evolution: First electron trapping leads to the formation of electron phase space holes. Due to a steepening of the leading edges of the potential wells the electron phase space holes gradually become asymmetric, they grow in size and deepen. The phase space holes accelerate until they move much faster than the initial ion-acoustic waves. The interaction of the current carriers with the asymmetric potential wells and causes a nonvanishing net momentum transfer between the particles and the self-generated electric field. After a few ion plasma periods ion trapping starts until, finally, an electrostatic double layer arises. It is found that the nonlinear saturated state of the system is dominated by the particle interaction with coherent phase space structures. The corresponding anomalous resistivity is slightly modulated with an oscillation period τ≈ωpi−1). For a macroscopic description its major part can be parameterized by means of an effective collision rate νeff of the order of 10−2ωpe≈0.5ωpi, where ωpe is the electron and ωpi the ion plasma frequency.

Phase space density analysis of the outer radiation belt energetic electron dynamics

We present an analysis of the electron phase space density in the Earth's outer radiation belt during three magnetically disturbed periods to determine the likely roles of inward radial diffusion and local acceleration in the energization of electrons to relativistic energies. During the recovery phase of the 9 October 1990 storm and the period of prolonged substorms between 11 and 16 September 1990, the relativistic electron phase space density increases substantially and peaks in the phase space density occur in the region 4.0 < L* < 5.5 for values of the first adiabatic invariant, M ≥ 550 MeV/G, corresponding to energies, E > ∼0.8 MeV. The peaks in the phase space density are associated with prolonged substorm activity, enhanced chorus amplitudes, and predominantly low values of the ratio between the electron plasma frequency, fpe, and the electron gyrofrequency, fce (fpe/fce < ∼4). The data provide further evidence for a local wave acceleration process in addition to radial diffusion operating in the heart of the outer radiation belt. During the recovery phase of the 9 October 1990 storm the peaks are more pronounced at large M (550 MeV/G) and large Kaufmann K (0.11 RE) than large M (700 MeV/G) and small K (0.025 RE), which suggests that radial diffusion is more effective below about 0.7 MeV for 5.0 < L* < 5.5 during this period. At low M (M ≤ 250 MeV/G), corresponding to energies, E < ∼0.8 MeV, there is no evidence for a peak in phase space density and the data are more consistent with inward radial diffusion and losses to the atmosphere by pitch angle scattering. During the 26 August 1990 storm there is a net loss in the relativistic electron phase space density for 3.3 < L* < 6.0. At low M (M ≤ 250 MeV/G) the phase space density decreases by almost a constant factor and the gradient remains positive for all L*, but at high M (M ≥ 550 MeV/G) the decrease in phase space density is greater at larger L* and the gradient changes from positive to negative. The data show that it is possible to have inward radial diffusion at low energies and outward radial diffusion at higher energies, which would fill the outer radiation belt.

Nonlinear stationary states of the Vlasov equation in the Fourier-transformed velocity-space

We give an exact expression for the nonlinear stationary solutions of the Vlasov–Poisson equation in the Fourier-transformed velocity-space in both one and three space dimensions. We show that these solutions are entire functions of the amplitude of the electrostatic potential and that they converge to the van Kampen linear continuum eigenfunctions, if this amplitude tends to zero. We establish a correspondence between the nonlinear solutions and the BGK waves over the whole complex energy-plane. The solution corresponding to a phase space electron hole is investigated in detail.

Quasineutral particle simulation technique for whistlers

We present a new hybrid fluid/PIC simulation scheme for whistlers, which eliminates both the speed-of-light time scale and the electron plasma oscillation time scale, and concentrates simulation resources on the resonant parts of electron phase space that control whistler evolution. The code runs with time steps on the order of the electron gyrofrequency, with extremely accurate energy conservation and numerical stability. Examples are shown of application to whistler instability growth and saturation, and ducting of whistlers in density channels.

Canonical perturbation theory for complex electron dynamics in gyrotron resonators

Complex electron dynamics in gyrotron resonators are analyzed in the context of the Hamiltonian formalism. Application of the canonical perturbation theory provides analytical approximate invariants of the electron motion. The latter are used for describing the resonant structure of the electron phase space and the electron rest energies at the output of the cavity. Hysteresis effects are also described through analytic expressions and approximate electron distribution functions are provided. The general case of resonant interaction at an arbitrary harmonic of the electron cyclotron frequency is considered and the effect of a varying frequency mismatch is studied. Also, the case of electron interaction with multiple rf modes is investigated.

Modeling the radiation belt electrons with radial diffusion driven by the solar wind

On the basis of the correlation between the solar wind and radiation belt electron fluxes, we develop a model to simulate the MeV electron phase space density variations from L = 3 to L = 8 by extending the Li et al. (2001) radial diffusion model for geosynchronous electrons. We add L dependence to the Li et al. model and allow for comparison with measurements at more than one L shell while retaining a similar form of their diffusion coefficient. The extended model achieves a prediction efficiency (PE) of 0.61 at L = 4 and 0.52 at L = 6 when the phase space density is converted to differential flux and compared with orbit‐averaged Polar 2 MeV measurements at L = 4 and daily averaged LANL 0.7–1.8 MeV geosynchronous measurements for the year 1998. These results indicate that radial diffusion plays a strong role in the enhancement of radiation belt electrons yet leaves a significant portion of the variance unaccounted for. We have also tuned parameters to model the electron fluxes during four individual geomagnetic storms during 1998 and found that the parameter values must differ from those of the long term and from each other to achieve the best PE. This suggests that the different solar wind drivers have varying degrees of influence on the MeV electron variations during different magnetic storms. This model can be used to forecast the MeV electron variations inside geosynchronous orbit with a reasonably good PE on the basis of real‐time solar wind measurements only.