Oblique Whistler Mode Waves Research Articles

Narrowband oblique whistler-mode waves: comparing properties observed by Parker Solar Probe at <0.3 AU and STEREO at 1 AU

Aims.Large amplitude narrowband obliquely propagating whistler-mode waves at frequencies of ~0.2fce(electron cyclotron frequency) are commonly observed at 1 AU, and they are most consistent with the whistler heat flux fan instability. We want to determine whether similar whistler-mode waves occur inside 0.3 AU and how their properties compare to those at 1 AU.Methods.We utilized the waveform capture data from the Parker Solar Probe Fields instrument from Encounters 1 through 4 to develop a data base of narrowband whistler waves. The Solar Wind Electrons Alphas and Protons Investigation (SWEAP) instrument, in conjunction with the quasi-thermal noise measurement from Fields, provides the electron heat flux, beta, and other electron parameters.Results.Parker Solar Probe observations inside ~0.3 AU show that the waves are often more intermittent than at 1 AU, and they are interspersed with electrostatic whistler-Bernstein waves at higher-frequencies. This is likely due to the more variable solar wind observed closer to the Sun. The whistlers usually occur within regions when the magnetic field is more variable and often with small increases in the solar wind speed. The near-Sun whistler-mode waves are also narrowband and large amplitude, and they are associated with beta greater than 1. The association with heat flux and beta is generally consistent with the whistler fan instability. Strong scattering of strahl energy electrons is seen in association with the waves, providing evidence that the waves regulate the electron heat flux.

Very Oblique Whistler Mode Propagation in the Radiation Belts: Effects of Hot Plasma and Landau Damping

AbstractSatellite observations of a significant population of very oblique chorus waves in the outer radiation belt have fueled considerable interest in the effects of these waves on energetic electron scattering and acceleration. However, corresponding diffusion rates are extremely sensitive to the refractive index N, controlled by hot plasma effects including Landau damping and wave dispersion modifications by suprathermal (15–100 eV) electrons. A combined investigation of wave and electron distribution characteristics obtained from the Van Allen Probes shows that peculiarities of the measured electron distribution significantly reduce Landau damping, allowing wave propagation with high N ∼ 100–200. Further comparing measured refractive indexes with theoretical estimates incorporating hot plasma corrections to the wave dispersion, we provide the first experimental demonstration that suprathermal electrons indeed control the upper limit of the refractive index of highly oblique whistler mode waves. Such results further support the importance of incorporating very oblique waves into radiation belt models.

Quasi‐linear diffusion coefficients for highly oblique whistler mode waves

AbstractQuasi‐linear diffusion coefficients are considered for highly oblique whistler mode waves, which exhibit a singular “resonance cone” in cold plasma theory. The refractive index becomes both very large and rapidly varying as a function of wave parameters, making the diffusion coefficients difficult to calculate and to characterize. Since such waves have been repeatedly observed both outside and inside the plasmasphere, this problem has received renewed attention. Here the diffusion equations are analytically treated in the limit of large refractive index μ. It is shown that a common approximation to the refractive index allows the associated “normalization integral” to be evaluated in closed form and that this can be exploited in the numerical evaluation of the exact expression. The overall diffusion coefficient formulas for large μ are then reduced to a very simple form, and the remaining integral and sum over resonances are approximated analytically. These formulas are typically written for a modeled distribution of wave magnetic field intensity, but this may not be appropriate for highly oblique whistlers, which become quasi‐electrostatic. Thus, the analysis is also presented in terms of wave electric field intensity. The final results depend strongly on the maximum μ (or μ∥) used to model the wave distribution, so realistic determination of these limiting values becomes paramount.

Nonlinear dynamics of electrons interacting with oblique whistler mode chorus in the magnetosphere

AbstractWe perform test particle simulations for relativistic electrons interacting with a whistler mode chorus packet propagating at oblique angles. By confirming that the energy transport of oblique lower band chorus is nearly along the ambient magnetic field, we apply the gyroaveraging method in calculating equations of motion of electrons. We trace evolution of a delta function of relativistic electrons in a phase space of kinetic energy and equatorial pitch angle and obtain numerical Green's functions of the chorus wave‐particle interactions. Examining the Green's functions in a wide range of kinetic energies, we find that Landau resonance can accelerate MeV electrons efficiently and that higher nth resonances such as n =− 1 and n = 2 also contribute to acceleration of electrons at high equatorial pitch angles (∼70°) and high energies (∼2 MeV). We investigate the rate of energy gain of the cyclotron resonance acceleration and the Landau resonance acceleration and find that the perpendicular component of wave electric field dominates both accelerations for MeV electrons. Furthermore, the proximity between the parallel components of Vp and Vg of oblique whistler mode waves and the nonlinear trapping condition make the interaction time of Landau resonance much longer than that of n = 1 cyclotron resonance, resulting in efficient acceleration of MeV electrons.

Equatorial electron loss by double resonance with oblique and parallel intense chorus waves

AbstractPuzzling satellite observations of butterfly pitch angle distributions and rapid dropouts of 30–150 keV electrons are widespread in the Earth's radiation belts. Several mechanisms have been proposed to explain these observations, such as enhanced outward radial diffusion combined with magnetopause shadowing or scattering by intense magnetosonic waves, but their effectiveness is mainly limited to storm times. Moreover, the scattering of 30–150 keV electrons via cyclotron resonance with intense parallel chorus waves should be limited to particles with equatorial pitch angle smaller than 70°–75°, leaving unaffected a large portion of the population. In this paper, we investigate the possible effects of oblique whistler mode waves, noting, in particular, that Landau resonance with very oblique waves can occur up to ∼89°. We demonstrate that such very oblique chorus waves with realistic amplitudes can very efficiently nonlinearly transport nearly equatorially mirroring electrons toward smaller pitch angles where nonlinear scattering (phase bunching) via cyclotron resonance with quasi‐parallel waves can take over and quickly send them to much lower pitch angles <40°. The proposed double resonance mechanism could therefore explain the formation of butterfly pitch angle distributions as well as contribute to some fast dropouts of 30–150 keV electrons occurring during moderate geomagnetic disturbances at L = 4–6. Since 30–150 keV electrons represent a seed population for a further acceleration to relativistic energies by intense parallel chorus waves during storms or substorms, the proposed mechanism may have important consequences on the dynamics of 100 keV to MeV electron fluxes in the radiation belts.

Oblique Whistler-Mode Waves in the Earth’s Inner Magnetosphere: Energy Distribution, Origins, and Role in Radiation Belt Dynamics

In this paper we review recent spacecraft observations of oblique whistler-mode waves in the Earth’s inner magnetosphere as well as the various consequences of the presence of such waves for electron scattering and acceleration. In particular, we survey the statistics of occurrences and intensity of oblique chorus waves in the region of the outer radiation belt, comprised between the plasmapause and geostationary orbit, and discuss how their actual distribution may be explained by a combination of linear and non-linear generation, propagation, and damping processes. We further examine how such oblique wave populations can be included into both quasi-linear diffusion models and fully nonlinear models of wave-particle interaction. On this basis, we demonstrate that varying amounts of oblique waves can significantly change the rates of particle scattering, acceleration, and precipitation into the atmosphere during quiet times as well as in the course of a storm. Finally, we discuss possible generation mechanisms for such oblique waves in the radiation belts. We demonstrate that oblique whistler-mode chorus waves can be considered as an important ingredient of the radiation belt system and can play a key role in many aspects of wave-particle resonant interactions.

Nonlinear local parallel acceleration of electrons through Landau trapping by oblique whistler mode waves in the outer radiation belt

AbstractSimultaneous observations of electron velocity distributions and chorus waves by the Van Allen Probe B are analyzed to identify long‐lasting (more than 6 h) signatures of electron Landau resonant interactions with oblique chorus waves in the outer radiation belt. Such Landau resonant interactions result in the trapping of ∼1–10 keV electrons and their acceleration up to 100–300 keV. This kind of process becomes important for oblique whistler mode waves having a significant electric field component along the background magnetic field. In the inhomogeneous geomagnetic field, such resonant interactions then lead to the formation of a plateau in the parallel (with respect to the geomagnetic field) velocity distribution due to trapping of electrons into the wave effective potential. We demonstrate that the electron energy corresponding to the observed plateau remains in very good agreement with the energy required for Landau resonant interaction with the simultaneously measured oblique chorus waves over 6 h and a wide range of L shells (from 4 to 6) in the outer belt. The efficient parallel acceleration modifies electron pitch angle distributions at energies ∼50–200 keV, allowing us to distinguish the energized population. The observed energy range and the density of accelerated electrons are in reasonable agreement with test particle numerical simulations.

Probability of relativistic electron trapping by parallel and oblique whistler-mode waves in Earth's radiation belts

We investigate electron trapping by high-amplitude whistler-mode waves propagating at small as well as large angles relative to geomagnetic field lines. The inhomogeneity of the background magnetic field can result in an effective acceleration of trapped particles. Here, we derive useful analytical expressions for the probability of electron trapping by both parallel and oblique waves, paving the way for a full analytical description of trapping effects on the particle distribution. Numerical integrations of particle trajectories allow to demonstrate the accuracy of the derived analytical estimates. For realistic wave amplitudes, the levels of probabilities of trapping are generally comparable for oblique and parallel waves, but they turn out to be most efficient over complementary energy ranges. Trapping acceleration of <100 keV electrons is mainly provided by oblique waves, while parallel waves are responsible for the trapping acceleration of >100 keV electrons.

Stability of relativistic electron trapping by strong whistler or electromagnetic ion cyclotron waves

In the present paper, we investigate the trapping of relativistic electrons by intense whistler-mode waves or electromagnetic ion cyclotron waves in the Earth's radiation belts. We consider the non-resonant impact of additional, lower amplitude magnetic field fluctuations on the stability of electron trapping. We show that such additional non-resonant fluctuations can break the adiabatic invariant corresponding to trapped electron oscillations in the effective wave potential. This destruction results in a diffusive escape of electrons from the trapped regime of motion and thus can lead to a significant reduction of the efficiency of electron acceleration. We demonstrate that when energetic electrons are trapped by intense parallel or very oblique whistler-mode waves, non-resonant magnetic field fluctuations in the whistler-mode frequency range with moderate amplitudes around 3−15 pT (much less intense than the primary waves) can totally disrupt the trapped motion. However, the trapping of relativistic electrons by electromagnetic ion cyclotron waves is noticeably more stable. We also discuss how the proposed approach can be used to estimate the effects of wave amplitude modulations on the motion of trapped particles.

Comparison of formulas for resonant interactions between energetic electrons and oblique whistler-mode waves

Test particle simulation is a useful method for studying both linear and nonlinear wave-particle interactions in the magnetosphere. The gyro-averaged equations of particle motion for first-order and other cyclotron harmonic resonances with oblique whistler-mode waves were first derived by Bell [J. Geophys. Res. 89, 905 (1984)] and the most recent relativistic form was given by Ginet and Albert [Phys. Fluids B 3, 2994 (1991)], and Bortnik [Ph.D. thesis (Stanford University, 2004), p. 40]. However, recently we found there was a (−1)l−1 term difference between their formulas of perpendicular motion for the lth-order resonance. This article presents the detailed derivation process of the generalized resonance formulas, and suggests a check of the signs for self-consistency, which is independent of the choice of conventions, that is, the energy variation equation resulting from the momentum equations should not contain any wave magnetic components, simply because the magnetic field does not contribute to changes of particle energy. In addition, we show that the wave centripetal force, which was considered small and was neglect in previous studies of nonlinear interactions, has a profound time derivative and can significantly enhance electron phase trapping especially in high frequency waves. This force can also bounce the low pitch angle particles out of the loss cone. We justify both the sign problem and the missing wave centripetal force by demonstrating wave-particle interaction examples, and comparing the gyro-averaged particle motion to the full particle motion under the Lorentz force.

Group‐standing of whistler mode waves near the Moon

AbstractNarrowband whistler mode waves with frequencies near 1 Hz have been observed near the Moon. We reveal that the narrowband spectra, the frequency concentration near 1 Hz, and the relations between the wave vector, magnetic field vector, and sunward directions can be explained by a condition in which the group velocity vector is almost canceled by the solar wind velocity vector in the spacecraft frame. Hereafter, we refer to this condition as the group‐standing condition. The spectral density is modified and has a peak at the frequency satisfying the group‐standing condition because of the difference of the frequency width between the solar wind plasma frame and the spacecraft frame. In addition, if the waves were decelerated to be group‐standing, the conservation of the energy flux results in the intensification of the wave amplitude at that frequency. We also derive the analytical expression of the amount of the modifications, which depend on the group velocity. These effects can explain the narrowband spectra near 1 Hz and support the relations between the wave vector, magnetic field vector, and sunward directions. The estimated frequency which satisfies the group‐standing condition is in good agreement with the observed frequency within error bars of the estimation. Considering the group‐standing condition, we suggest that the narrowband waves observed in the spacecraft frame are originated from oblique whistler mode waves in the frequencies near the lower hybrid frequency, which are possibly generated by reflected ions from the lunar magnetic anomalies.

Nonlinear interactions between relativistic radiation belt electrons and oblique whistler mode waves

Abstract. Resonant interactions between relativistic charged particles and oblique whistler mode waves are explored in this work, and it is shown that nonlinear phase trapping could happen in a gyrophase averaged sense, consistent with previous studies of interactions between nonrelativistic electrons and oblique whistler mode waves. A dimensionless parameter χ is derived to represent the ratio of wave-induced motion to the adiabatic motion of the particle. We show that phase trapping is likely to occur when the wave-induced motion dominates the adiabatic motion, which is caused mainly by the background fields. A mapping of probable regions of nonlinear interactions is shown based on the parameter χ. We show that the nonlinear interactions might be important near the equatorial plane for even moderate wave amplitude, and the latitudinal range for nonlinear interactions to occur is largest for electrons with local pitch angles around 50 degrees, consistent with previous findings. The results are important for understanding the nonlinear dynamics of relativistic radiation belt electrons and the generation of chorus waves.

Oblique Whistler-Mode Waves in the Inhomogeneous Magnetospheric Plasma: Resonant Interactions with Energetic Charged Particles

A consistent account is given of the theory of resonant interactions between energetic charged particles and a whistler-mode wave propagating obliquely to the non-uniform geomagnetic field in the inhomogeneous magnetospheric plasma. The basic equations for the wave field and charged particle dynamics are presented, with the emphasis being placed on the parameters governing the problem. A Hamiltonian approach is consistently used in the analysis of the particle equations of motion which are discussed in detail and solved analytically in various cases. Two applications of the theory are considered. First, we calculate the growth (or damping) rate for a whistler-mode wave propagating obliquely to geomagnetic field in the magnetosphere. Secondly, we estimate the proton precipitation into the upper atmosphere induced by a VLF transmitter signal.

Competing processes of plasma wave instabilities driven by an anisotropic electron beam: Linear results and two‐dimensional particle simulations

A magnetoplasma made up of a background of isotropic electrons and protons and a beam of anisotropic electrons (T⊥/T‖ > 1), with drift velocity parallel to the ambient magnetic field, can feed different unstable modes. Numerical solution of the dispersion equation in the wavenumber plane (k‖, k⊥) reveals several unstable modes with islets of oblique growth unconnected to the unstable parallel modes. The mechanisms involved in the growth and saturation of the fastest instability do not necessarily exclude later developments of other instabilities. Competing processes among the coexisting unstable waves are expected. Linear analysis or one‐dimensional computer simulations cannot properly address this problem; there would not be interaction of modes propagating at different angles to the static magnetic field. Linear and nonlinear evolutions of the modes are studied via computer simulation with a two‐dimensional particle code (KEMPO2). Results show that several modes are excited with different timescales. Some modes are only temporarily excited, while others persist in the system. The distribution function is significantly changed by the first excited modes, the beam‐mode waves. In later stages of development, forward and backward oblique whistler mode waves are excited. A final stage is characterized by coexistence of several waves with different frequencies and wavelengths. The beam‐mode waves have frequencies close to the plasma frequency, whereas the frequencies of the whistler mode waves fall near a half of the cyclotron frequency. These results may be relevant to the interpretation of wave activities observed in the Earth's magnetosphere by the GEOS 1, GEOS 2, and GEOTAIL satellites or in the Uranian bow shock by Voyager 2.

Acceleration of particles in the upper ionosphere and the magnetosphere due to decay interactions of whistlers, part 1

A numerical analysis of the three-wave decay interaction of oblique whistler mode waves with different types of ion waves is presented. It is shown that at altitudes h < 2000 km the most effective mechanism is the decay interaction of whistlers with quasielectrostatic ion-cyclotron waves (the increment is Γ ~ 25s−1 for an initial whistler pump wave amplitude of ℰ10 = 10 mVm-1) as well as with Alfvén waves and ion-ion hybrid waves (Γ ~ 10s−1 for the same pump amplitude ℰ10). When taking into account the media inhomogeneity and the decay wave stabilization due to pump wave amplitude depletion, the magnitudes of excited ion waves are found to be E3 ~ 50-120μV/(mHz½) for ℰ10 = 10 mVm-1. At such amplitudes parametrically excited ion waves can provide transverse ion acceleration with a rate of 0.5eVs−1 and longitudinal electron acceleration up to 5eV at a distance of the order of 2000 km, the flux reaching 3 × 107 el/(cm2s sr eV).

Oblique whistler-mode waves in the presence of electron beams

The problems of oblique whistler-mode propagation, instability and damping for hot anisotropic plasma in the presence of electron beams are considered using the quasi-longitudinal and quasi-electrostatic approximations. For the quasi-longitudinal approximation, the solution of the real part of the dispersion equation is presented as a result of a perturbation of the whistler-mode refractive index in a cold, dense plasma N 0. This enables us to make a direct comparison between different perturbations of N 0 due to electron finite density, temperature, anisotropy and electron beams. For the quasi-electrostatic approximation, the solution is found in the case when the net electron current due to electron beams is either equal to zero or close to zero. In the case of quasi-longitudinal whistler-mode propagation in the vicinity of the magnetopause, the corrections to N 0 due to electron beams are generally small and can be neglected. For quasi-electrostatic whistler-mode propagation in the same region of the magnetosphere the contribution of electron beams is more significant but does not result in qualitative changes of the basic properties of the waves. For the values of plasma parameters observed near the magnetopause by AMPTE-UKS we can expect only parallel or almost parallel whistler-mode waves to be unstable at frequencies ω below the half electron gyrofrequency Ω. This prediction of the theory is compatible with actual simultaneous observation of the wave spectrum in the magnetopause region. Wave observations at ω >; 0.5Ω could be explained if we would assume larger values for the anisotropy of the electron distribution, or that the waves can penetrate from a region of stronger magnetic field. The first possibility is considered in detail and it shows that these waves can be unstable both within the quasi-longitudinal and quasi-electrostatic approximations. The conditions for instability for quasi-electrostatic waves are more stringent than those for quasilongitudinal ones.

Marginal stability of oblique whistler-mode waves

The problem of marginal stability of oblique whistler-mode waves propagating in a hot, anisotropic plasma is considered both within the quasilongitudinal and quasielectrostatic approximations. It is pointed out that the most favourable region for quasilongitudinal whistler-mode instability lies at frequencies ω above the half electron gyrofrequency Ω unless the wave normal angle is very close to zero. In the opposite limiting case, the waves are expected to be excited at ω < Ω 2 . This can provide a possible explanation of a gap at ω ∼ Ω 2 in the spectrum of chorus emissions observed in the magnetosphere. Quasielectrostatic whistler-mode instabilities are expected for values of anisotropy which seem to be too large for magnetospheric conditions.

Computer experiments on whistler and plasma wave emissions for Spacelab‐2 electron beam

A funnel‐shaped VLF emission observed in the Spacelab‐2 electron beam experiment is studied by computer experiments using an electromagnetic particle code. The computer experiments demonstrated that broad‐band plasma wave emissions extending from a low frequency to the local plasma frequency are excited by a spatially confined thin electron beam. The emissions have parallel phase velocities satisfying the Landau resonance condition with the electron beam. Parts of the emissions below the electron cyclotron frequency can propagate away from the beam as oblique whistler mode waves. The computer experiments show a very clear funnel‐shaped emission, and show a remarkable similarity with that in the Spacelab‐2 electron beam experiment. The funnel‐shape of the whistler emissions is interpreted in terms of the perpendicular group velocity and, as a possible additional mechanism, the wave damping in the background plasma.

The guidance of oblique whistler mode waves along magnetospheric field lines

An approximate analytical expression for wave normal angle θg≠0, at which whistler-mode group velocity is directed along magnetic field line in a hot anisotropic plasma, is derived. This expression is simple enough for practical applications in the magnetospheric conditions and is compatible with the results of numerical analysis for wave frequencies below but not close to the half of electron gyrofrequency.

The whistler mode in a Vlasov plasma

In this study, properties of small-amplitude parallel and oblique whistler mode waves are investigated for a wide range of plasma parameters by numerically solving the full electromagnetic Vlasov dispersion equation. To investigate the cold-plasma and electrostatic approximations for the whistler mode, the results are compared with results obtained using these descriptions. For large wavelengths (kc/ωi≲100), the cold-plasma description is often accurate, while for short wavelengths (kc/ωi≳100) and sufficiently oblique propagation, the electrostatic description is often accurate. The study demonstrates that in a Vlasov plasma the whistler mode near resonance has a group velocity more nearly parallel to the magnetic field than that predicted by cold-plasma theory.