Whistler Anisotropy Instability Research Articles

Whistler turbulence vs. whistler anisotropy instability: Particle-in-cell simulation and statistical analysis

Particle-in-Cell simulations and statistical analysis are carried out to study the dynamic evolution of a collisionless, magnetized plasma with co-existing whistler turbulence and electron temperature anisotropy as the initial condition, and the competing consequences of whistler turbulence cascade and whistler anisotropy instability growth. The results show that the operation of the whistler instability within whistler turbulence has almost no effects on the fluctuating magnetic field energy and intermittency generated by turbulence. However, it leads to a small reduction of the magnetic field wavevector anisotropy and a major reduction of the intermittency of electron temperature anisotropy. Hence, while the overall effect from whistler instability is minor as compared to that of whistler turbulence due to its much smaller field energy, the whistler instability may act as a regulation mechanism for kinetic-range turbulence through wave-particle interactions.

Electron-Driven Instabilities in the Solar Wind

The electrons are an essential particle species in the solar wind. They often exhibit non-equilibrium features in their velocity distribution function. These include temperature anisotropies, tails (kurtosis), and reflectional asymmetries (skewness), which contribute a significant heat flux to the solar wind. If these non-equilibrium features are sufficiently strong, they drive kinetic micro-instabilities. We develop a semi-graphical framework based on the equations of quasi-linear theory to describe electron-driven instabilities in the solar wind. We apply our framework to resonant instabilities driven by temperature anisotropies. These include the electron whistler anisotropy instability and the propagating electron firehose instability. We then describe resonant instabilities driven by reflectional asymmetries in the electron distribution function. These include the electron/ion-acoustic, kinetic Alfvén heat-flux, Langmuir, electron-beam, electron/ion-cyclotron, electron/electron-acoustic, whistler heat-flux, oblique fast-magnetosonic/whistler, lower-hybrid fan, and electron-deficit whistler instability. We briefly comment on non-resonant instabilities driven by electron temperature anisotropies such as the mirror-mode and the non-propagating firehose instability. We conclude our review with a list of open research topics in the field of electron-driven instabilities in the solar wind.

Combined Whistler Heat Flux and Anisotropy Instabilities in Solar Wind

AbstractWhistler heat flux and electron temperature anisotropy instabilities are spontaneously generated in the expanding solar wind. The present study investigates the interplay between the two unstable modes and how each contributes to the dynamical evolution of macroscopic quantities. Whistler heat flux instability is excited in the forward direction along the heat flow, whereas anisotropy instability propagates in both backward and forward directions. Velocity moment‐based quasi‐linear theory is employed in order to investigate the saturation behavior of these instabilities, and it is shown that under certain conditions, the forward propagating whistler heat flux instability dominates, while for other conditions, especially when the temperature anisotropies are sufficiently high, it is shown that the backward mode may also contribute to the dynamical process, and even dominate. These saturation behaviors cannot be predicted by linear growth rate alone and that they imply that in the global‐kinetic models of the solar wind, bidirectional nature of these instabilities must be carefully taken into account.

The Effects of Thermal Electrons on Whistler Mode Waves Excited by Anisotropic Hot Electrons: Linear Theory and 2‐D PIC Simulations

AbstractThe wave normal angle of excited whistler waves was previously considered to be controlled by the parallel plasma beta (β∥h) of anisotropic hot electrons, while the effects of thermal electrons were usually neglected. By combining both the linear theoretical and 2‐D particle‐in‐cell (PIC) simulation models, we have investigated the effects of thermal electrons on the whistler anisotropy instability. In the high‐beta (β∥h ≥ 0.025) regime, the wave normal angle of the dominant whistler mode with the largest growth rate is always 0°, which is not affected by thermal electrons, while, its wave frequency and linear growth rate decrease with the density and temperature of thermal electrons. These results are also confirmed by PIC simulations. In the low‐beta (β∥h ≤ 0.025) regime, with the increase of the density and temperature of thermal electrons, the wave normal angle of the dominant whistler mode turns to zero from a large value. This change could be due to the stronger damping caused by thermal electrons for oblique whistler mode, since oblique wave usually has a smaller cyclotron resonant velocity than parallel wave. PIC simulations also show a consistent result, but reproduce a broad magnetic spectrum, even in the case including sufficient thermal electrons. Furthermore, thermal electrons with large parallel velocities are resonantly accelerated in the perpendicular direction, while parts of hot electrons are trapped and accelerated in the parallel direction. Our study suggests that the wave normal angle of whistler mode in the Earth's magnetosphere could be determined by both anisotropic and thermal electrons.

Magnetospheric chorus wave simulation with the TRISTAN-MP PIC code

We present the results of particle-in-cell simulations of the whistler anisotropy instability that results in magnetospheric chorus wave excitation. The simulations were carried out using, for the first time for this problem, the 2D TRISTAN-massively parallelized code, widely used before in the modeling of astrophysical shocks. The code has been modified to allow for two populations of electrons: cold electrons (which maintain the wave propagation) and hot electrons (which provide the wave growth). For the hot electrons, the anisotropic form of the relativistic Maxwell–Jüttner distribution is implemented. We adopt the standard approximation of a parabolic magnetic field to simulate the Earth's magnetic field close to the equator. Simulations with different background magnetic field inhomogeneity strengths demonstrate that higher inhomogeneity yields lower frequency chirping rates and, eventually, it suppresses chorus generation. The results are in agreement with other numerical simulations and the theoretical predictions for the frequency chirping rates.

Simulation and Quasi‐Linear Theory of Whistler Anisotropy Instability

AbstractThe whistler anisotropy (or electromagnetic electron cyclotron) instability may be operative in many geomagnetic and heliospherical environments, including the radiation belt, solar wind, and the solar corona. The present investigation carries out a comparative analysis between the two‐dimensional particle‐in‐cell simulation of weakly growing whistler anisotropy instability and the velocity moment‐based two‐dimensional quasi‐linear theory under the assumption of bi‐Maxwellian electron distribution function. It is shown that the simplified quasi‐linear theory provides a qualitative agreement with the more rigorous particle‐in‐cell simulation, but some discrepancies are also found. Possible causes for the differences in either method are discussed, and future improvements on the theory are suggested. Potential applicability of the present finding in the context of the space and astrophysics is discussed.

On the parameter dependence of the whistler anisotropy instability

AbstractThe evolution of the whistler anisotropy instability relevant to whistler‐mode chorus waves in the Earth's inner magnetosphere is studied using kinetic simulations and is compared with satellite observations. The electron distribution is constrained by the whistler anisotropy instability to a marginal stability state and presents an upper bound of electron anisotropy, which agrees with satellite observations. The electron beta β∥e separates whistler waves into two groups: (i) quasi‐parallel whistler waves for and (ii) oblique whistler waves close to the resonance cone for . Landau damping is important in the saturation and relaxation stage of the oblique whistler wave growth. The saturated magnetic field energy of whistler waves roughly scales with the electron beta , shown in both simulations and satellite observations. These results suggest the critical role of electron beta β∥e in determining the whistler wave properties in the inner magnetosphere.

Effects of variations in electron thermal velocity on the whistler anisotropy instability: Particle-in-cell simulations

This paper investigates how the physics of the whistler anisotropy instability (WAI) is affected by variations in the electron thermal velocity vte, referred to here in terms of the ratio v̂te=vte/c, where c is the speed of light. The WAI is driven by the electron condition RT>1, where RT=Te⊥/Te∥ is the temperature anisotropy ratio and ⊥/∥ signify directions perpendicular/parallel to the background magnetic field B0. While a typical value of v̂te in the solar wind is ∼0.005, electromagnetic (EM) particle-in-cell (PIC) simulations often use a value near 0.1 in order to maximize the computational time step. In this study, a two-dimensional (2D) Darwin particle-in-cell (DPIC) code, MDPIC2, is used. The time step in the DPIC model is not affected by the choice of v̂te, making DPIC suited for this study. A series of simulations are carried out under the condition that the electron βe is held fixed, while v̂te is varied over the range 0.1≥v̂te≥0.025. The results show that, with βe held fixed, the linear dispersion properties and the nonlinear saturation amplitude and pitch angle scattering rates associated with the WAI are insensitive to the value of v̂te. A supplementary investigation is conducted which characterizes how the WAI model is affected at various values of v̂te by noise associated with the limited number of particles in a typical PIC simulation. It is found that the evolution of the WAI is more strongly influenced by electrostatic noise as v̂te is decreased. The electrostatic noise level is inversely proportional to the number of particles per computational cell (Nc); this implies that the number of particles required to remove nonphysical effects from the PIC simulation increases as v̂te decreases. It is concluded that PIC simulations of this instability which use an artificially large value of v̂te accurately reproduce the response of a cooler plasma as long as a realistic value of βe is used. Moreover, for a fixed number of particles, the quantitative accuracy of the model improves with increasing v̂te.

Kinetic theory of turbulence for parallel propagation revisited: Low-to-intermediate frequency regime

A previous paper [P. H. Yoon, “Kinetic theory of turbulence for parallel propagation revisited: Formal results,” Phys. Plasmas 22, 082309 (2015)] revisited the second-order nonlinear kinetic theory for turbulence propagating in directions parallel/anti-parallel to the ambient magnetic field, in which the original work according to Yoon and Fang [Phys. Plasmas 15, 122312 (2008)] was refined, following the paper by Gaelzer et al. [Phys. Plasmas 22, 032310 (2015)]. The main finding involved the dimensional correction pertaining to discrete-particle effects in Yoon and Fang's theory. However, the final result was presented in terms of formal linear and nonlinear susceptibility response functions. In the present paper, the formal equations are explicitly written down for the case of low-to-intermediate frequency regime by making use of approximate forms for the response functions. The resulting equations are sufficiently concrete so that they can readily be solved by numerical means or analyzed by theoretical means. The derived set of equations describe nonlinear interactions of quasi-parallel modes whose frequency range covers the Alfvén wave range to ion-cyclotron mode, but is sufficiently lower than the electron cyclotron mode. The application of the present formalism may range from the nonlinear evolution of whistler anisotropy instability in the high-beta regime, and the nonlinear interaction of electrons with whistler-range turbulence.

Whistler anisotropy instabilities as the source of banded chorus: Van Allen Probes observations and particle-in-cell simulations.

Magnetospheric banded chorus is enhanced whistler waves with frequencies ωr<Ωe, where Ωe is the electron cyclotron frequency, and a characteristic spectral gap at ωr≃Ωe/2. This paper uses spacecraft observations and two-dimensional particle-in-cell simulations in a magnetized, homogeneous, collisionless plasma to test the hypothesis that banded chorus is due to local linear growth of two branches of the whistler anisotropy instability excited by two distinct, anisotropic electron components of significantly different temperatures. The electron densities and temperatures are derived from Helium, Oxygen, Proton, and Electron instrument measurements on the Van Allen Probes A satellite during a banded chorus event on 1 November 2012. The observations are consistent with a three-component electron model consisting of a cold (a few tens of eV) population, a warm (a few hundred eV) anisotropic population, and a hot (a few keV) anisotropic population. The simulations use plasma and field parameters as measured from the satellite during this event except for two numbers: the anisotropies of the warm and the hot electron components are enhanced over the measured values in order to obtain relatively rapid instability growth. The simulations show that the warm component drives the quasi-electrostatic upper band chorus and that the hot component drives the electromagnetic lower band chorus; the gap at ∼Ωe/2 is a natural consequence of the growth of two whistler modes with different properties.

Particle-in-cell simulations of velocity scattering of an anisotropic electron beam by electrostatic and electromagnetic instabilities

The velocity space scattering of an anisotropic electron beam (T⊥b/T∥b&amp;gt;1) flowing along a background magnetic field B0 through a cold plasma is investigated using both linear theory and 2D particle-in-cell simulations. Here, ⊥ and ∥ represent the directions perpendicular and parallel to B0, respectively. In this scenario, we find that two primary instabilities contribute to the scattering in electron pitch angle: an electrostatic electron beam instability and a predominantly parallel-propagating electromagnetic whistler anisotropy instability. Our results show that at relative beam densities nb/ne≤0.05 and beam temperature anisotropies Tb⊥/Tb∥≤25, the electrostatic beam instability grows much faster than the whistler instabilities for a reasonably fast hot beam. The enhanced fluctuating fields from the beam instability scatter the beam electrons, slowing their average speed and increasing their parallel temperature, thereby increasing their pitch angles. In an inhomogeneous magnetic field, such as the geomagnetic field, this could result in beam electrons scattered out of the loss cone. After saturation of the electrostatic instability, the parallel-propagating whistler anisotropy instability shows appreciable growth, provided that the beam density and late-time anisotropy are sufficiently large. Although the whistler anisotropy instability acts to pitch-angle scatter the electrons, reducing perpendicular energy in favor of parallel energy, these changes are weak compared to the pitch-angle increases resulting from the deceleration of the beam due to the electrostatic instability.

Whistler anisotropy instability: Spectral transfer in a three‐dimensional particle‐in‐cell simulation

A three‐dimensional (3‐D) particle‐in‐cell (PIC) simulation of the whistler anisotropy instability is carried out for a collisionless, homogeneous, magnetized plasma with βe=0.10. This is the first 3‐D PIC simulation of the evolution of enchanced fluctuations from this growing mode driven by an anisotropic electron velocity distribution with T⟂e/T∥e&gt;1 where ⟂ and ∥ represent directions perpendicular and parallel to the background magnetic field Bo, respectively. The early‐time magnetic fluctuation spectrum grows with properties reflecting the predictions of linear theory with narrowband maxima at kc/ωe≃ 1 and k×Bo=0, and a wave vector anisotropy in the sense of k⟂&lt;&lt;k∥. Here ωe represents the electron plasma frequency. At later times the fluctuations undergo both a forward transfer to shorter wavelengths, also with k⟂&lt;&lt;k∥, and an inverse transfer to longer wavelengths with wave vector anisotropy k⟂&gt;&gt;k∥. The inverse transfer is consistent with a prediction of nonlinear three‐wave coupling theory.

Multiscale whistler waves within Earth's perpendicular bow shock

We present observations of intense whistler waves made by Polar in the frequency range from a few Hz to 600 Hz within Earth's nearly perpendicular bow shock. The long duration burst data provided by Polar reveal the detailed properties of whistler waves in context with the macrostructure of the layer of this supercritical shock. We show that the pedestal and ramp have superposed quasiperiodic, large amplitude precursor substructure occurring at a cadence of ∼3 sec, which is near the ion cyclotron period. With increasing penetration into the shock front, the amplitude of this substructure increases and ultimately reaches downstream values. The nonlinear substructure is shown to be concentrated regions of intense whistler wave activity. Power spectra in the whistler range show strong enhancements in two distinct bands: a relatively broadband lower frequency component occurring near the lower hybrid frequency (a few tens of Hertz) and a higher frequency component at a few hundred Hertz. The lower frequency component is composed of right‐hand polarized whistler wave packets propagating quasiparallel to the magnetic coplanarity plane at oblique angles with respect to both the magnetic field and shock normal, with respective anglesθkb varying from 50°–70° and θkn ∼ 50°. These waves generally have relatively large amplitude (δB/B0∼ 0.1–0.4) magnetic fields ranging from a few nT to 15 nT. Given their preferential upstream propagation near the magnetic coplanarity plane, they are likely generated by a kinetic cross‐field streaming instability driven by the relative drift between the reflected ion beam and the electrons. The high‐frequency component appears to be the shock analog of whistler “lion roars” often observed in the magnetosheath. The lion roars occur within the foot and into the shock ramp in regions where sufficiently intense low‐frequency whistlers exist. These are right‐hand circularly polarized wave packets lasting up to ∼10 cycles, with amplitudes reaching 1 nT, and propagating nearly parallel or antiparallel with respect to the background magnetic field (θkb ⪅ 30° or θkb ⪆ 150°). These packets, which appear regularly with a cadence of a few tens of Hertz, occur within local magnetic field (and also electric field) minima of the lower frequency whistler waves. These waves are likely generated by the whistler anisotropy instability driven by preferential perpendicular anisotropies in the electron distribution function induced by the strong magnetic field of the ramp substructure. The growth rate for the instability is estimated at γ ∼ ωci, which is rather low. We suggest that the low‐frequency whistler waves play a key role in the growth and spatial distribution of the lion roar by forming a quasiperiodic series of magnetic field ducts (troughs) that confine their propagation nearly along the surface of the shock. This allows enough time for the lion roar to grow to observed amplitudes before they leave the region of growth. The low‐frequency waves may also be contributing to the growth of the lion roar by inducing enhancements in the electron anisotropy within magnetic troughs.

Whistler anisotropy instability with a cold electron component: Linear theory

The whistler anisotropy instability is driven by an electron temperature anisotropy T⊥/T∥ &gt; 1 where ⊥ and ∥ denote directions perpendicular and parallel, respectively, to the background magnetic field Bo. Here kinetic linear theory in a magnetized, homogeneous, collisionless plasma model is used to study this instability when the electron velocity distribution may be represented as the sum of a hot, anisotropic bi‐Maxwellian and a cold, isotropic component. The critical β∥e, the value at which the maximum growth rate of the instability changes from propagation parallel to Bo to oblique propagation, decreases with increasing nc/ne, where nc is the cold electron density and ne is the total electron density. At parallel propagation the maximum growth rate increases with nc/ne up to nc/ne ≃ 0.8, but then diminishes with further increases of the relative cold electron density. Introduction of a cold electron component can reduce the hot electron anisotropy necessary to excite this instability by up to a factor of 2.

Constraints on electron temperature anisotropies in sheath regions of interplanetary shocks

A statistical study of electron data observed within magnetosheath regions after interplanetary (IP) shocks driven by magnetic clouds has been performed to understand the constraints on temperature anisotropies of suprathermal electron component due to collective effects of electron wave-particle interactions with plasma micro-instabilities. Results show that although shock heating and acceleration, downstream selective reflection and leakage into upstream region, and wave-particle scattering all can contribute to suprathermal electron temperature anisotropy in downstream sheath regions of IP shocks, the consequent whistler anisotropy instabilities triggered by the sufficiently large temperature anisotropies can constrain temperature anisotropy increase to a more nearly isotropic distribution as plasma beta increases.

Empirical model of whistler anisotropy instability

Empirical formulae for the real frequency and growth rate associated with the whistler anisotropy instability are obtained. The electrons are assumed to have an anisotropic distribution function, with Maxwellian parallel distribution. Under such an assumption complex roots of the dispersion relation depend only on two dimensionless parameters, namely, the temperature anisotropy factor A = T⊥e/T∥e − 1, where T⊥e and T∥e are the perpendicular and parallel electron temperatures, respectively, and parallel electron beta, β∥ = (8πnT∥e/B2)1/2, where n and B are the plasma density and magnetic field intensity, respectively. Comparison against exact numerical roots show that analytical formulae describe the whistler instability over a wide range of parallel electron beta and temperature anisotropy factor. The present result may be useful for circumstances in which the use of exact numerical roots becomes impractical, such as in the radiation belt quasi-linear transport coefficient calculation.

Whistler anisotropy instability at low electron β: Particle-in-cell simulations

The whistler anisotropy instability is studied in a magnetized, homogeneous, collisionless plasma model. The electrons (denoted by subscript e) are represented initially with a single bi-Maxwellian velocity distribution with a temperature anisotropy T⊥e/T∥e&amp;gt;1, where ⊥ and ∥ denote directions perpendicular and parallel to the background magnetic field Bo, respectively. Kinetic linear dispersion theory predicts that, if the ratio of the electron plasma frequency ωe to the electron cyclotron frequency Ωe is greater than unity and β∥e≥0.025, the maximum growth rate of this instability is at parallel propagation, where the fluctuating fields are strictly electromagnetic. At smaller values of β∥e, however, the maximum growth rate shifts to propagation oblique to Bo and the fluctuating electric fields become predominantly electrostatic. Linear theory and two-dimensional particle-in-cell simulations are used to examine the consequences of this transition. Three simulations are carried out, with initial β∥e=0.10, 0.03, and 0.01. The fluctuating fields of the β∥e=0.10 run are predominantly electromagnetic, with nonlinear consequences similar to those of simulations already described in the literature. In contrast, the growth of fluctuations at oblique propagation in the low electron β runs leads to a significant δE∥, which heats the electrons leading to the formation of a substantial suprathermal component in the electron parallel velocity distribution.

Excitation of banded whistler waves in the magnetosphere

Banded whistler waves can be generated by the whistler anisotropy instability driven by two bi-Maxwellian electron components with T{sub {perpendicular}}/T{sub {parallel}} > 1 at different T{sub {parallel}} For typical magnetospheric condition of 1 < {omega}{sub e}/{Omega}{sub e} < 5 in regions associated with strong chorus, upper-band waves can be excited by anisotropic electrons below {approx} 1 keV, while lower-band waves are excited by anisotropic electrons above {approx} 10 keV. Lower-band waves are generally field-aligned and substantially electromagnetic, while upper-band waves propagate obliquely and have quasi-electrostatic fluctuating electric fields. The quasi-electrostatic feature of upper-band waves suggests that they may be more easily identified in electric field observations than in magnetic field observations. Upper-band waves are liable to Landau damping and the saturation level of upperband waves is lower than lower-band waves, consistent with observations that lower-band waves are stronger than upper-band waves on average. The oblique propagation, the lower saturation level, and the more severe Landau damping together would make upper-band waves more tightly confined to the geomagnetic equator (|{lambda}{sub m}| < {approx}10{sup o}) than lower-band waves.

All whistlers are not created equally: Scattering of strahl electrons in the solar wind via particle‐in‐cell simulations

Solar wind observations show that suprathermal electrons (70 eV ≲ Energy ≲ 1 keV) of the magnetic‐field‐aligned “strahl” component have broader pitch‐angle distributions than are predicted by adiabatic theories of solar wind expansion. Magnetosonic‐whistler fluctuations propagating toward the Sun at k × Bo = 0 (where Bo is the background magnetic field) have a strong cyclotron resonance with suprathermal electrons propagating in the anti‐Solar direction along Bo. This resonance enables strong pitch‐angle scattering; thus whistlers are a likely source of the observed strahl broadening. Particle‐in‐cell simulations in a magnetized, homogeneous, collisionless plasma of electrons and protons are used to study the response of a strahl‐like electron component to whistler fluctuation spectra. If the whistler anisotropy instability is excited via the initial application of T⊥/T∥ &gt; 1 to the electron core component, the resulting electron scattering leads to strahl pitch‐angle distributions which decrease in width as electron energy increases. In contrast, if a power spectrum of whistler fluctuations proportional to k−3 is initially applied to the simulations, the resulting electron scattering leads to strahl pitch‐angle distributions which increase in width as electron energy increases.

Linear theory of electron temperature anisotropy instabilities: Whistler, mirror, and Weibel

A collisionless, homogeneous plasma in which the electron velocity distribution is a bi‐Maxwellian with T⊥e &gt; T∥e, where the directional subscripts refer to directions relative to the background magnetic field Bo, can support the growth of two distinct instabilities. Linear dispersion theory predicts that the whistler anisotropy instability is excited with maximum growth rate γm at k × Bo = 0 and real frequency ωr greater than the proton cyclotron frequency, whereas the electron mirror instability is excited at propagation oblique to Bo and zero real frequency. In an unmagnetized plasma with a similarly anisotropic electron distribution the electron Weibel instability may be excited with zero real frequency and maximum growth rate in the direction of the minimum temperature. Here linear theory is used to compare dispersion and threshold properties of these three growing modes. For 0.10 ≤ β∥e ≤ 1000, the whistler has a larger γm and a smaller anisotropy threshold than the electron mirror, so that the former mode should dominate in homogeneous plasmas for most physical values of electron β. Threshold conditions describing electron temperature anisotropies and parallel wave numbers at given maximum growth rates are presented for each instability.