Energetic Electron Distribution Function Research Articles

Electrostatic lower hybrid turbulence due to magnetic islands and particle heating in magnetopause reconnection regions

This investigation presents a model to understand the magnetopause turbulence in the lower hybrid frequency range at reconnection regions as observed by the magnetospheric multiscale mission. We have modeled 3D electrostatic lower hybrid wave (LHW) in cold magnetized plasma in the presence of magnetic islands. In this model, we have developed the modified nonlinear Schrödinger equation, contemplating that nonlinear effects are due to ponderomotive force. The dynamical equation of the system thus obtained is solved using numerical simulation techniques. The simulation results illustrate the formation and evolution of localized structures and the current sheets. Due to the nonlinear effects and magnetic reconnection, the current sheets become turbulent with time. A semi-analytical model is also given to illustrate the scale size of current sheets. The turbulent power spectra have also been studied. The spectral index of the power spectrum is found to be −2.7 (approximately) in the perpendicular direction and −2.2 (approximately) in the parallel direction, indicating anisotropy. Using the Folker–Plank diffusion equation with the new velocity space diffusion coefficient, we find the distribution function of energetic electrons due to these turbulent structures. We propose that LHW may be responsible for plasma heating in magnetopause.

On the Influence of Propagation Properties of Whistler-Mode Waves in the Earth’s Magnetosphere on Their Cyclotron Amplification

We study the properties of the cyclotron amplification of whistler-mode waves during their propagation in the Earth’s magnetosphere in the presence of large-scale density inhomogeneities such as the plasmapause or density ducts. Wave propagation is considered within the framework of the geometrical optics with the use of cold plasma density profiles measured onboard the Van Allen Probes satellites. Wave amplitude variation due to the cyclotron interactions with energetic electrons having an anisotropic distribution function is studied. The cyclotron growth rate is calculated along the wave trajectory taking into account the wave vector variation for a given analytical distribution function of energetic electrons. We show that in the case of guided propagation in a density duct or near the plasmapause the frequency dependences of the one-hop wave gain and the local growth rate in the equatorial region approximately coincide with each other for low initial wave normal angles (|Θ0| ≲ 30°) and relatively low energies (W0 ≲ 15 keV). The amplification band expands towards the higher frequencies for higher initial propagation angles and electron energies. In the case of nonducted propagation, the efficiency of cyclotron interactions is notably lower, and the frequency dependences of the one-hop wave gain and the equatorial growth rate differ: the waves having the initial wave angle directed towards the Earth are stronger amplified, and this asymmetry increases with increasing electron energy.

Frequency Dependence of Very Low Frequency Chorus Poynting Flux in the Source Region: THEMIS Observations and a Model

AbstractUsing Poynting vector measurements of whistler mode chorus emissions detected by the THEMIS spacecraft within the source region, that is, close to the magnetic field minimum, we found both in individual events and statistically that chorus elements propagating equatorward had systematically higher frequencies and smaller amplitudes compared with simultaneously observed elements propagating away from the equator. We demonstrate similar features in the results of numerical simulations based on backward wave oscillator equations. It can be qualitatively explained by the nonlinear evolution of the energetic electron distribution function during wave generation. The motion of electrons from the equator is accompanied by a decrease in their velocity component along the magnetic field line due to both the adiabatic mirror force and nonlinear wave‐particle interactions. Thus, the frequency of the chorus elements generated by such electrons and propagating equatorward is higher compared with the elements propagating away from the equator.

Whistler Growth Rates in the Magnetosphere According to Measurements of Energetic Electron Fluxes on the Van Allen Probe A Satellite

Measurements of the differential fluxes of energetic electrons in the equatorial region of the magnetosphere in the energy range from units of keV to hundreds of keV carried out on Van Allen Probes make it possible to obtain the distribution function of electrons resonantly interacting with whistlers in the VLF range and to calculate the corresponding local growth rates (or increments) of the waves. The article presents the results of the reconstruction of the distribution function of energetic electrons, as well as the direct calculation of the wave increments without any model assumptions about the form of the distribution function. The data from the Van Allen Probe A satellite on electron fluxes in the energy range of 33–143 keV are used, and the dependence of the increment on frequency along the satellite path is calculated. The obtained increment values are compared with the spatial and temporal variations in VLF emissions detected by the satellite. An important feature of the calculated increments is their significant variation during the time between successive measurements of the fluxes, i.e., at times on the order of 11 s.

Electrons under the dominant action of shock-electric fields

We consider a fast magnetosonic multifluid shock as a representation of the solar-wind termination shock. We assume the action of the transition happens in a three-step process: In the first step, the upstream supersonic solar-wind plasma is subject to a strong electric field that flashes up on a small distance scale $\Delta z\simeq U_1/ \Omega _{\mathrm e}$ (first part of the transition layer), where $\Omega_{\mathrm e}$ is the electron gyro-frequency and $U_1$ is the upstream speed. This electric field both decelerates the supersonic ion flow and accelerates the electrons up to high velocities. In this part of the transition region, the electric forces connected with the deceleration of the ion flow strongly dominate over the Lorentz forces. We, therefore, call this part the demagnetization region. In the second phase, Lorentz forces due to convected magnetic fields compete with the electric field, and the highly anisotropic and energetic electron distribution function is converted into a shell distribution with energetic shell electrons storing about 3/4 of the upstream ion kinetic energy. In the third phase, the plasma particles thermalize due to the relaxation of free energy by plasma instabilities. The first part of the transition region opens up a new thermodynamic degree of freedom never before taken into account for the electrons, since the electrons are usually considered to be enslaved to follow the behavior of the protons in all velocity moments like density, bulk velocity, and temperature. We show that electrons may be the downstream plasma fluid that dominates the downstream plasma pressure.

New results of structured VLF emissions observed simultaneously at two closely located stations near <i>L</i> ~ 5.5

Abstract. Simultaneous records of VLF (very low frequencies) emissions have been carried out at two ground-based stations located at similar geomagnetic latitudes near L ~ 5.5 and spaced in the longitude by ~ 400 km, Kannuslehto (KAN) in Finland and Lovozero (LOZ) in Russia, using quite similar VLF receivers with two calibrated orthogonal air-core loop antennas. We found that the general spectral properties of the VLF chorus emissions at these two stations were similar and typically have right-hand polarization. Contrary to VLF chorus, the short-period VLF emissions (periodic emissions, PE) in which separated spectral elements are repeated with the periodicity of 3–4 s were mostly left-hand polarized. Usually, these waves propagated in the north–south direction. We suppose that PEs are generated inside of the plasmasphere by the cyclotron instability under a quasi-linear relaxation of the energetic electron distribution function. However, sometimes PE occurred only at an individual station. We speculated that this could be due to the influence of the local inhomogeneities to the VLF waves during the propagation through the ionospheric trough to the ground. Unusual series of short-duration (10–100 s) bursts of VLF emissions, lasting several hours, were also found in the morning under very quiet geomagnetic conditions (Kp ~ 0–1). Generally, these emissions were observed simultaneously at KAN and LOZ showing both right-hand and left-hand polarization, and different arrival directions provided the rather extended ionospheric exit area.

Quasiperiodic VLF emissions with short‐period modulation and their relationship to whistlers: A case study

AbstractWe study properties of quasiperiodic (QP) VLF emissions recorded on 24 December 2011 during the VLF campaign in northern Finland. The main attention is paid to interrelationships between different characteristic periods in the QP spectra. In particular, we analyze regular variations in the QP repetition intervals (1–10 min) during the event from 15:30 to 22 UT, their changes during substorms, and short periodic (several second) modulation observed within separate QP elements. We explained the variations of periods of QP emissions in terms of the model of auto‐oscillation regime of the cyclotron instability in the magnetosphere. During the considered event lasting about 7 h we observed a regular increase in the time intervals between the QP elements. We relate this increase with weakening of the magnetospheric source of energetic electrons. Significant variations in the QP period occurred during substorms. These variations can be due to a substorm‐related increase in the energetic‐electron flux and/or due to the precipitation of these electrons into the ionosphere which changes the reflection coefficient of VLF waves. We analyze the fine structure of QP element spectra and reveal the periods related to the time scales of guided propagation of whistler mode waves along the magnetic field line, which suggests that ducted propagation regime took place for the QP emissions. The periods were about 6–9 s for frequencies 1.2–4 kHz, respectively, which was similar to the period of almost simultaneously observed two‐hop whistlers. In the low‐frequency part of QP spectra periodic emissions with shorter periods of about 3 s were observed. Analysis of fine structure of QP elements shows that their formation is affected by both linear effects (i.e., group‐velocity dispersion) and quasi‐linear effects related to the modification of the energetic electron distribution function. It allows us to observe the transition from purely linear to quasi‐linear regime of wave‐particle interactions in the spectra of these short periodic emissions. The nonlinear regime can be understood in terms of passive mode locking in magnetospheric cyclotron maser.

Short-period VLF emissions as solitary envelope waves in a magnetospheric plasma maser

Space and ground-based experiments have shown evidence of natural short-period VLF emissions in which separate spectral elements are repeated with a periodicity of 2–7 s. Their basic morphological properties are found on the basis of original experimental data. In our opinion, excitation of such emissions is the result of quasi-linear relaxation effects that compensate for natural spectral dispersion. The quasi-linear relaxation of the energetic electron distribution function incrementally changes wave cyclotron instability and hence the VLF emission spectral forms. Some properties of the quasi-linear interaction of whistler waves with magnetospheric radiation belt electrons are studied. It is shown that quasi-linear relaxation can increase the cyclotron instability at the leading edge of an electromagnetic pulse. This effective saturation of absorption facilitates the division of VLF hiss-like emission into separate electromagnetic pulses without spectral modification from one pulse to the next. Some features and manifestations of this effective saturation of absorption are discussed. The results are important for a better understanding of temporal and spatial structures of VLF whistler-mode emissions and energetic electron fluxes.

Universal scaling of the electron distribution function in one-dimensional simulations of relativistic laser-plasma interactions

A formula based on simulations is given for the distribution function of energetic electrons which stream in the forward direction when a relativistically intense laser pulse irradiates a short-scale-length dense plasma at normal incidence. If the energy absorption fraction is known, the entire function can be characterized in terms of the laser intensity I. A number of important characteristics can then be obtained, including the fast-electron energy, current density, and angle of ejection. An objective means for calculating the fast-electron energy is proposed which gives a value of ∼0.6 of the vacuum oscillatory energy. This reduction in energy is shown to be consistent with the energy required to draw a return current. These results could be used to provide the characteristics of the electron source in theory and simulation of intense laser-solid interactions.

Kinetic Alfvén waves turbulence in the Earth’s magnetosphere

The numerical simulations of the model equation governing the nonlinear dynamics of kinetic Alfvén waves in the intermediate-β plasmas are performed. When the nonlinearity arises due to the ponderomotive force driven density perturbations of kinetic Alfvén waves, the model equation turns out to be a modified nonlinear Schrödinger equation. This has been solved numerically by using appropriate boundary conditions. The coherent, damped magnetic filaments with turbulent spectra have been observed. Our results reveal the interesting change in spectral index because of the damping effect. The steeper power spectra follow ∼k−3.4 scaling. Using the Fokker–Planck equation with the new velocity space diffusion coefficient, we find the distribution function of energetic electrons in these turbulent structures. These turbulent structures can be responsible for plasma heating in Earth’s magnetosphere.

Landau damped kinetic Alfvén waves and coronal heating

AbstractSome recent observations of solar corona suggest that the kinetic Alfvén waves (KAWs) turbulence may be responsible for electron acceleration in solar corona and coronal heating. In the present research, we investigate the turbulent spectra of KAW due to filamentation process in the presence of Landau damping and particle energization. We present here the numerical simulation of model equation governing the nonlinear dynamics of the KAW in the presence of Landau damping. When the ponderomotive and Joule heating nonlinearities are incorporated in the KAW dynamics, the power spectra of the turbulent field is evaluated and used for particle heating. Our results reveal the formation of damped coherent magnetic filamentary structures and the turbulent spectra. The effect of Landau damping is to make the turbulent spectra steeper. Two types of scalings k−3.6 and k−4 have been obtained. We have studied the turbulence with different initial conditions. Using the Fokker–Planck equation with the new velocity space diffusion coefficient, we find the distribution function of energetic electrons in these turbulent structures. Landau damped KAWs may be responsible for the acceleration of the energetic electrons in solar corona and coronal heating.

Dynamics of the magnetospheric cyclotron ELF/VLF maser in the backward-wave-oscillator regime. II. The influence of the magnetic-field inhomogeneity

We study the influence of the magnetic-field inhomogeneity on the nonlinear dynamics of the absolute instability of whistler-mode waves in the Earth’s magnetosphere in the presence of a step-like deformation in the distribution function of energetic electrons. Development of this instability, implying the transition of the magnetospheric cyclotron maser to the regime of a backward-wave oscillator (BWO), was proposed earlier as a generation mechanism of magnetospheric chorus emissions. We analyze the results of numerical simulations of the simplified nonlinear equations describing the magnetospheric-BWO dynamics in the case of low efficiency of wave-particle interactions. We found that the case of an inhomogeneous magnetic field where the system length is much greater than the length characterizing the linear stage of the BWO regime has important specific features compared with the case of a homogeneous medium. The main feature of the nonlinear wave dynamics in the magnetospheric BWO in an inhomogeneous magnetic field consists in the fact that for a sufficiently large excess over the generation threshold, a sequence of separate wave packets, i.e., discrete elements, is formed. The frequency within each packet varies in time, and these discrete elements are close in their properties to the chorus elements observed in the magnetosphere. The results of calculations confirm the quantitative estimates of parameters of chorus emissions, which were performed earlier on the basis of the BWO model.

Dynamics of the Magnetospheric Cyclotron ELF/VLF Maser in the Backward-Wave-Oscillator Regime. I. Basic Equations and Results in the Case of a Uniform Magnetic Field

We consider the nonlinear dynamics of absolute instability of whistler-mode waves in the Earth's magnetosphere in the presence of a step-like deformation in the distribution function of energetic electrons. Development of this instability, implying the transition of the magnetospheric cyclotron maser to the regime of a backward-wave oscillator (BWO), was proposed earlier as a generation mechanism of magnetospheric chorus emissions. We derive simplified nonlinear equations describing the dynamics of the magnetospheric BWO in the case of low efficiency of wave-particle interactions. Numerical simulations of these equations confirm qualitative similarity of the laboratory and magnetospheric BWOs and justify quantitative estimates of parameters of chorus emissions.

X-ray spectropolarimetry of high-temperature plasmas

X-ray spectropolarimetry is a powerful new tool for investigating the anisotropy of high-temperature plasmas. It is sensitive to the energetic electron distribution function and magnetic field, which, in general, have not been measured adequately. This diagnostic is based on the theoretical modeling of x-ray line polarization-dependent spectra measured simultaneously by spectrometers with different sensitivities to polarization. The increasing sophistication of spectroscopic monitoring of high-temperature plasmas has led to the necessity to account for both non-Maxwellian electron distribution functions and polarization properties of lines. This diagnostic can be applied to a broad range of plasmas, from low-density astrophysical plasmas to dense z-pinch plasmas. This work shows how x-ray spectropolarimetry complements the usual spectroscopic monitoring of hot plasmas and demonstrates the importance of accounting for x-ray line polarization in plasma diagnostics.

A generation mechanism for chorus emission

Abstract. A chorus generation mechanism is discussed, which is based on interrelation of ELF/VLF noise-like and discrete emissions under the cyclotron wave-particle interactions. A natural ELF/VLF noise radiation is excited by the cyclotron instability mechanism in ducts with enhanced cold plasma density or at the plasmapause. This process is accompanied by a step-like deformation of the energetic electron distribution function in the velocity space, which is situated at the boundary between resonant and nonresonant particles. The step leads to the strong phase correlation of interacting particles and waves and to a new backward wave oscillator (BWO) regime of wave generation, when an absolute cyclotron instability arises at the central cross section of the geomagnetic trap, in the form of a succession of discrete signals with growing frequency inside each element. The dynamical spectrum of a separate element is formed similar to triggered ELF/VLF emission, when the strong wavelet starts from the equatorial plane. The comparison is given of the model developed using some satellite and ground-based data. In particular, the appearance of separate groups of chorus signals with a duration 2-10 s can be connected with the preliminary stage of the step formation. BWO regime gives a succession period smaller than the bounce period of energetic electrons between the magnetic mirrors and can explain the observed intervals between chorus elements.Key words. Magnetospheric physics (Energetic particles · trapped). Space plasma physics (wave-particle interactions; waves and instabilities)

A study of radial diffusion of energetic electrons in the Earth's radiation belts: Data satellite “Cosmos-1686”

The distribution function of energetic electrons in the inner and outer radiation belts of the Earth have been calculated from electron spectra and flux data for energetic electrons (with energy 0.03–2 Mev) for a number of satellites (Intercosmos-19, Cosmos-1686, ets). The distribution functions show evidence of changes thought to be due to diffusion processes. The present study determins coefficients of radial diffusion on different L-shells by several methods. Coefficients of radial and pitch-angle diffusion of electrons have been determined by independent methods using the particles and electron diffusion wave measurement data. The present study examines the rols of electric and magnetic fields fluctuations for redistributing relativistic and non relativistic electrons on different L-shells. Coefficients of radial diffusion in the outer radiation belt are calculated using the diffusion wave observed for energetic electrons. The new results are compared with values obtained earlier for coefficients of radial diffusion from data of many experiments on satellites and calculations and modelling for the years 1962–1993.

Analytic model for the runaway distribution function in the presence of spatial diffusion

A steady-state kinetic model for runaway electrons in the presence of radial diffusion in a stochastic magnetic field is adopted and solved for a constant magnetic diffusivity. The model is constructed to recover the correct runaway production rate in the absence of spatial diffusion. The parallel energetic electron distribution function f is found by matching the solutions from three regions in parallel velocity space and is employed to form moments of f. Upper and lower bounds on the spatial diffusion are obtained by using these moments and the model exhibits the strong sensitivity to collisionality needed to explain the difference between similar plasmas with little or no hard x-ray signal and those with significant hard x-ray signals.

The Monte Carlo Calculations for the Distribution Function of Energetic Electrons in a Plane-Parallel Hollow Cathode

The energy distribution of high-energy electrons was calculated from the plane-parallel hollow-cathode glow discharge of helium gas. Calculations showed an energy distribution function similar to that of a cylindrical cathode whose function had already been obtained by the authors. However, few electrons whose energy equalled the cathode fall potential existed in the central region, contrary to the many electrons in that region in the cylindrical cathode.

Calculations for the Distribution Function of Energetic Electrons in a Helium Hollow Cathode Glow Discharge with the Monte Carlo Method

The distribution function of energetic electrons is calculated by tracing their trajectories with the Monte Carlo method. Electrons were made to have inelastic and elastic collisions with helium atoms in the ground state. Calculations were made for a cylindrical hollow cathode whose inner diameter is 1 cm. The electron density increased rapidly with decreasing energy near the lower limit of its energy. The number density of electrons whose energy equals the cathode-fall potential has a peak at the central axis. Their velocities are found to be directed along a collisionless trajectory. The broad angular distribution of the velocity was found in slower electrons. The peak of the number density at the central axis decreased with increasing pressure. The energetic electrons can hardly penetrate into the central region when the gas pressure is 1.3 kPa.

On the problem of quasi-electrostatic whistler mode waves: A possible interpretation of discrete plasmaspheric emissions

In a number of VLF experiments with low-orbiting Intercosmos satellites, high-intensity, discrete VLF emissions at frequencies above the local LHR were observed predominantly between L=2.3 and 4.0. The frequency of their intensity maximum varies continuously with geomagnetic latitude, and approximately parallels the value of a quarter of the equatorial electron gyrofrequency. An attempt is made to interprete these emissions in terms of quasi-electrostatic whistler-mode waves, generated in the near-equatorial region and propagating downward in the quasi-resonance mode. The generating mechanism is supposed to be the kinetic instability connected with the loss-cone and temperature anisotropy of the distribution function of energetic electrons. Some features of the discrete plasmaspheric emissions are discussed on the basis of theoretical considerations.