Kaijun Liu

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.

Comparison of quasilinear diffusion coefficients for parallel propagating whistler mode waves with test particle simulations

[1] We present a comparison between the classical quasilinear diffusion coefficients and those calculated using a general test particle code. The trajectories of a large number of electrons are followed as they traverse a numerically‐ constructed, broadband, small‐amplitude wave field, using a general relativistic test particle code. The change in each electron’s pitch angle and energy is shown to be stochastic and the resulting diffusion of the entire population is found to be in excellent agreement with quasilinear theory. We also demonstrate that the diffusion coefficients presented by Summers, derived specifically for parallel propagating waves, are a factor of two larger than the test particle results if the power spectral density is one‐sided (w > 0). Our results demonstrate the general validity of using quasilinear theory to describe the effects of broadband small amplitude waves on radiation belt electrons. Citation: Tao, X., J. Bortnik, J. M. Albert, K. Liu, and R. M. Thorne (2011), Comparison of quasilinear diffusion coefficients for parallel propagating whistler mode waves with test particle simulations, Geophys. Res. Lett., 38, L06105,

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>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, ...

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.

Analytic Model of the IBEX Ribbon with Neutral Solar Wind Based Ion Pickup Beyond the Heliopause

Energetic neutral atom (ENA) full sky maps obtained with the Interstellar Boundary Explorer revealed a surprising narrow band of increased intensity, the Ribbon, which has been attributed to the ordering effect of the interstellar magnetic field immediately outside the heliosphere. Among models to explain the enhanced ENA Ribbon intensity, Heerikhuisen et al. base theirs on neutral solar wind origin. It reflects the Ribbon angular and energy distribution correctly, but experiences inherent challenges from the long-term stability of the pickup ion (PUI) ring in velocity space, required for the observed ENA fluxes, and from time variations observed after less than one year due to the long integration length. We provide a simplified analytic model of the neutral solar wind, PUI production beyond the heliopause, and subsequent ENA production. We include convection of the PUIs with the interstellar flow toward the heliopause perpendicular to the interstellar magnetic field, thus far not in any model, extinction of the outward propagating neutral solar wind, and of the PUIs and ENAs on their way inward. Based on hybrid simulations of PUI driven instabilities, with injection rates from this model, scattering and isotropization of the PUIs is noticeably weaker than previously thought, yet too fast for ENA production. Assuming a narrow PUI velocity ring, we find a strong concentration of the ENA origin just outside the heliopause and Ribbon intensities comparable with the observations. Conversely, an isotropic PUI distribution produces ENA fluxes factor of ten too low, thus reemphasizing the need of very slow scattering.

Fast magnetosonic waves driven by shell velocity distributions

Using linear dispersion theory and particle-in-cell simulations, we explore the ion Bernstein instability driven by the shell-type ion velocity distribution which is related to the excitation of fast magnetosonic waves in the terrestrial magnetosphere. We first demonstrate a novel idea to construct the shell velocity distribution out of multiple Maxwellian ring-beam velocity distributions. Applying this technique, we find that the convergence of the linear theory instability can be achieved with only a moderate number of ring-beam components. In order to prove that such an approximation is legitimate and the linear theory instabilities evaluated are indeed valid, we use the exact shell distribution to carry out a number of one dimensional particle-in-cell simulations corresponding to multiple wave propagation angles adjacent to the direction at which the most unstable waves are expected to grow. The agreement between the linear dispersion analysis and the simulation results is generally very good: enhanced waves are organized along the linear theory dispersion curves in the frequency-wave number space, and relative wave amplitudes are ordered as the linear theory growth rates very well. However, the simulations show a few extra branches that are not expected from the linear dispersion analysis. A close examination of these extra branches suggests that they are not simulation artifacts and particularly related to the ring/shell-type distributions with large ring/shell speed (v>∼1.5 vA, where vA is the Alfven speed). In addition, our results show that substantial wave growth can occur at nonintegral harmonics of the proton cyclotron frequency at wave normal angles substantially far away from the perpendicular direction, which may provide an alternative explanation of the off-harmonic peaks of some fast magnetosonic waves observed in space.

Study of EMIC wave excitation using direct ion measurements

With data from Van Allen Probes, we investigate electromagnetic ion cyclotron (EMIC) wave excitation using simultaneously observed ion distributions. Strong He band waves occurred while the spacecraft was moving through an enhanced density region. We extract from helium, oxygen, proton, and electron mass spectrometer measurement the velocity distributions of warm heavy ions as well as anisotropic energetic protons that drive wave growth through the ion cyclotron instability. Fitting the measured ion fluxes to multiple sinm-type distribution functions, we find that the observed ions make up about 15% of the total ions, but about 85% of them are still missing. By making legitimate estimates of the unseen cold (below ∼2 eV) ion composition from cutoff frequencies suggested by the observed wave spectrum, a series of linear instability analyses and hybrid simulations are carried out. The simulated waves generally vary as predicted by linear theory. They are more sensitive to the cold O+ concentration than the cold He+ concentration. Increasing the cold O+ concentration weakens the He band waves but enhances the O band waves. Finally, the exact cold ion composition is suggested to be in a range when the simulated wave spectrum best matches the observed one.

Proton velocity ring-driven instabilities in the inner magnetosphere: Linear theory and particle-in-cell simulations

Linear dispersion theory and electromagnetic particle-in-cell (PIC) simulations are used to investigate linear growth and nonlinear saturation of the proton velocity ring-driven instabilities, namely, ion Bernstein instability and Alfven-cyclotron instability, which lead to fast magnetosonic waves and electromagnetic ion cyclotron waves in the inner magnetosphere, respectively. The proton velocity distribution is assumed to consist of 10% of a ring distribution and 90% of a low-temperature Maxwellian background. Here two cases with ring speeds vr/vA=1 and 2 (vA is the Alfven speed) are examined in detail. For the two cases, linear theory predicts that the maximum growth rate γm of the Bernstein instability is 0.16Ωp and 0.19Ωp, respectively, and γm of the Alfven-cyclotron instability is 0.045Ωp and 0.15Ωp, respectively, where Ωp is the proton cyclotron frequency. Two-dimensional PIC simulations are carried out for the two cases to examine the instability development and the corresponding evolution of the particle distributions. Initially, Bernstein waves develop and saturate with strong electrostatic fluctuations. Subsequently, electromagnetic Alfven-cyclotron waves grow and saturate. Despite their smaller growth rate, the saturation levels of the Alfven-cyclotron waves for both cases are larger than those of the Bernstein waves. Resonant interactions with the Bernstein waves lead to scattering of ring protons predominantly along the perpendicular velocity component (toward both decreasing and, at a lesser extent, increasing speeds) without substantial change of either the parallel temperature or the temperature anisotropy. Consequently, the Alfven-cyclotron instability can still grow. Furthermore, the free energy resulting from the pitch angle scattering by the Alfven-cyclotron waves is larger than the free energy resulting from the perpendicular energy scattering, thereby leading to the larger saturation level of the Alfven-cyclotron waves.

Regime transition of ion Bernstein instability driven by ion shell velocity distributions

Linear kinetic dispersion theory is used to investigate a regime transition of the ion Bernstein instability driven by a proton velocity distribution with positive slopes with respect to the perpendicular velocity, ∂fp(v∥∼0,v⊥)/∂v⊥>0. The unstable waves arising from this instability are ion Bernstein waves with proton cyclotron harmonic dispersion. However, in the inner magnetosphere, particularly inside of the plasmapause where plasmas are dominated by a cold background, the instability leads to ion Bernstein waves which approximately follow the cold plasma dispersion relation for fast magnetosonic waves and are, therefore, fast magnetosonic-like. Subsequently, the relevant waves have been termed fast magnetosonic waves and many studies have assumed the cold plasma dispersion relation to describe them. On the other hand, how the dispersion properties of ion Bernstein waves become fast magnetosonic-like has not yet been well understood. To examine this regime transition of the instability, we perform linear dispersion analyses using a two-component proton model of fp(v) = fM(v) + fs(v), where fM is a Maxwellian velocity distribution and fs is an isotropic shell velocity distribution. The results show that the unstable waves are essentially ion Bernstein waves; however, when the shell proton concentration becomes sufficiently small (less than 10), the unstable waves approach the cold plasma dispersion relation for fast magnetosonic waves and become fast magnetosonic-like. Although a reduced proton-to-electron mass ratio of 100 has been used for convenience, which reduces the number of unstable modes involved by lowering the lower hybrid frequency, this does not change the overall regime transition picture revealed in this study.

Stochastic analysis of pitch angle scattering of charged particles by transverse magnetic waves

This paper describes a theory of the velocity space scattering of charged particles in a static magnetic field composed of a uniform background field and a sum of transverse, circularly polarized, magnetic waves. When that sum has many terms the autocorrelation time required for particle orbits to become effectively randomized is small compared with the time required for the particle velocity distribution to change significantly. In this regime the deterministic equations of motion can be transformed into stochastic differential equations of motion. The resulting stochastic velocity space scattering is described, in part, by a pitch angle diffusion rate that is a function of initial pitch angle and properties of the wave spectrum. Numerical solutions of the deterministic equations of motion agree with the theory at all pitch angles, for wave energy densities up to and above the energy density of the uniform field, and for different wave spectral shapes.