Whistler Mode Instability Research Articles

Linear and nonlinear interactions of an electron beam with oblique whistler and electrostatic waves in the magnetosphere

Both linear and nonlinear interactions between oblique whistler, electrostatic, quasi‐upper hybrid mode waves and an electron beam are studied by linear analyses and electromagnetic particle simulations. In addition to a background cold plasma, we assumed a hot electron beam drifting along a static magnetic field (Bo). Growth rates of the oblique whistler, oblique electrostatic, and quasi‐upper hybrid instabilities were first calculated. We found that there are four kinds of unstable mode waves for parallel and oblique propagations. They are the electromagnetic whistler mode wave (WW1), the electrostatic whistler mode wave (WW2), the electrostatic mode wave (ESW), and the quasi‐upper hybrid mode wave (UHW). When the angle θ between the wave vector k and Bo is small enough (≤10°), the electrostatic instability is dominant compared with the whistler mode instability. When θ is around 30°, the growth rates of whistler (both electrostatic and electromagnetic) mode waves and electrostatic mode waves are of the same order. When θ increases to 60°, the WW2 mode will be the most unstable mode wave. For a very large θ, (∼ 80°), the WW2 instability still has positive growth rates, and the UHW instability begins to have positive growth rates. Electromagnetic particle simulations were performed for parallel and three oblique cases, θ = 0°, 30°, 60° and 80°. When θ = 0°, whistler mode waves can hardly grow from the thermal fluctuation level because the electron beam which is supposed to provide free energy to the whistler mode waves is quickly diffused in the velocity space by much faster growing ESW. The ESW can lead to a secondary electrostatic instability. With θ = 30°, both electrostatic and whistler mode waves grow simultaneously. Also, the electrons diffused by the whistler mode instability to higher υ∥ velocity regions can lead to a secondary electrostatic instability. When θ is 60°, diffusion of the electron beam is controlled mainly by the WW2 instability. For θ = 80°, both WW2 and UHW grow despite their small growth rates. The simulations agree with linear analyses on the ESW growth rate for θ = 0°, but from our simulation data, growth rates of oblique whistler mode, electrostatic mode and quasi‐upper hybrid mode waves are usually smaller than those predicted by linear analyses. The electrostatic and whistler mode instabilities affect each other through their interactions with the electron beam. While the most intense ESW is generated for parallel propagation, the most intense whistler mode wave is observed at an oblique direction. Modulations between different electrostatic waves are found for θ = 0° after the electric field reaches its saturation level. Modulations between whistler waves (θ = 60°,80°) are also found while their magnetic fields increase nonlinearly and reach their saturation levels. A possible mechanism is proposed to explain the satellite observations of whistler mode chorus and accompanied electrostatic waves, whose amplitudes are sometimes modulated at the chorus frequency.

Effect of AC electric field on the whistler mode instability in the magnetosphere

The dispersion relation for the whistler mode instability in an infinite bi‐Maxwellian magnetoplasma in the presence of a parallel ac electric field has been derived using the method of characteristic solutions. The perturbed and unperturbed particle trajectories have been used to determine the perturbed distribution function. The growth rate has been determined. The earlier known results are recovered under suitable approximations, when the ac field amplitude and frequency are set to zero. It is found that the ac field parameters modify the growth rate and the resonance condition of the charged particles. The growth rate for various frequencies and magnitudes of the ac field, suitable to magnetospheric conditions, has been evaluated. It has been found that the growth rate depends significantly on the variation of the frequency rather than on the variation of the magnitude.

An assessment of lightning or in situ instabilities as a source for whistler mode waves in the night ionosphere of Venus

Burst rate and polarization studies have shown that at least half of the impulsive 100‐Hz signals detected at low altitudes in the night ionosphere of Venus by the Pioneer Venus Orbiter electric field detector are whistler mode waves. At the Earth, whistler mode waves are generated both by lightning and by in situ instabilities, and we investigate the likelihood of either being a source for the VLF bursts at Venus. Because the magnetic field is weak, the nightside ionosphere at Venus can be a high βe plasma, where βe is the ratio of thermal to magnetic energy density for electrons. For βe ≈ 1, electron thermal speeds are comparable to whistler mode phase speeds and Landau resonant instabilities will not apply to the Venus ionosphere. However, gyroresonant instabilities may operate at Venus. Linear stability analysis shows that the damping by ambient electrons is weak for βe ≪ 1 and whistler mode waves could be generated by either lightning or instabilities driven by superthermal electrons if βe is low. On the other hand, the waves would be associated with high βe if they are generated by temperature anisotropy in the ambient electron population. The dependence of 100‐Hz wave intensity on the plasma parameters shows that the waves are most intense for low βe, and hence the waves cannot be generated through temperature anisotropy of the ambient electrons. The 100‐Hz waves are also most intense when the whistler mode damping is weakest, but not when the growth rate due to superthermals is greatest. Moreover, the 100‐Hz wave intensity does not depend on electron density, as would be expected if superthermal electrons were responsible for the waves, since the growth rates are largest when the beam to ambient density ratio is greatest. Consequently, whistler mode instabilities do not appear to explain the 100‐Hz waves at Venus, and atmospheric lightning is the most likely source for these waves.

Improved quasilinear models of parallel whistler-mode instability

Two quasilinear models of parallel whistler-mode instabilities are presented. The first one is based on the assumption of a homogeneous plasma, the second one is relevant to closed magnetospheric field tubes. In contrast to the earlier quasilinear models of these instabilities we take into account in a selfconsistent manner the effects of a finite ratio of wave frequency ω to the electron gyrofrequency Ω. It is pointed out that in many cases relevant to magnetospheric applications, these effects can be negligibly small, so that the applicability of the earlier simplified models obtained for ω Ω → 0 can be justified. The second of the models considered is more reliable for the interpretation of plasmaspheric hiss emissions than for midlatitude hiss emissions generated outside of the plasmapause.

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.

Competing processes of whistler and electrostatic instabilities in the magnetosphere

Competing processes of whistler mode and electrostatic mode instabilities induced by an electron beam are studied by a linear growth rate analysis and by an electromagnetic particle simulation. In addition to a background cold plasma we assumed an electron beam drifting along a static magnetic field. We studied excitation of whistler and electrostatic mode waves in the direction of the static magnetic field. We first calculated linear growth rates for the whistler mode and electrostatic mode instabilities, assuming various possible parameters in the equatorial magnetosphere. We found that the growth rate for the electrostatic instability is always larger than that of the whistler mode instability. A short simulation run with a monoenergetic electron beam demonstrates that a monoenergetic beam can hardly give energy to whistler mode waves as a result of competition with faster growing electrostatic waves, because the beam electrons are trapped and diffused by the electrostatic waves, and hence the growth rates for whistler mode waves become very small. A long simulation run starting with a warm electron beam demonstrates that whistler mode waves are excited in spite of the small growth rates and the coexisting quasi‐linear electrostatic diffusion process. The diffusion of the warm electron beam is explained by three different processes with different time scales. An electromagnetic diffusion follows the electrostatic diffusion, causing pitch angle scattering of the beam electrons to lower pitch angles. At the third stage the diffused electrons at high parallel velocities are further diffused down to a lower parallel velocity range because of the enhanced electrostatic fluctuations. The present simulation demonstrates that many grid points as well as an enormous number of time steps are required to simulate two coexisting instabilities with different time scales. Fewer grid numbers and fewer time steps lead to misunderstanding of the physics involved.

Quasilinear models of oblique whistler-mode instabilities

Two self-consistent quasilinear models of oblique whistler-mode instabilities are presented. The first one is based on the supposition of cyclotron excitation of the waves and predicts constancy of maximal spectral density and an increase of frequencies and bandwidth with increasing wave normal angle θ. The second is based on the supposition of Čerenkov excitation of the waves and predicts an increase of spectral density and a decrease of frequencies and bandwidth with increasing θ.

Excitation of whistler mode instability by high energy electrons in the magnetosphere

Propagation of whistler mode waves through magnetospheric plasma characterised by a double Maxwellian velocity distribution function has been considered. The presence of a high energy electron tail in the composite velocity distribution function provides the needed free energy which excites instability mechanism for propagating whistler mode waves. The presence of the parallel or the antiparallel electric field governs the growth/decay rate variation with normalised parameter k ̃ = kV e ω H . These results are found to be different from those reported, which have used other forms of electron velocity distribution functions. The dependence of growth/decay rate on magnetospheric parameters plays an important role in the interpretation of wave features recorded by in-situ measurements. The presence of a fraction of high energy tail electrons penetrating through magnetospheric plasma exhibits significant change in the growth/decay rate of whistler mode waves.

High‐amplitude VLF transmitter signals and associated sidebands observed near the magnetic equatorial plane on the ISEE 1 satellite

On February 16, 1983, during a period of strong magnetic disturbance (Kp ∼ 5), as the ISEE 1 satellite passed near the magnetic meridian of the Omega navigation transmitter (98°W, 46°N geographic, L ∼ 3.4), the Stanford University VLF wave receiver detected the presence of Omega transmitter signals over the range 2 ≤ L ≤ 3.7 at magnetic latitudes (λm) within 10° of the magnetic equator. Over a 500‐km orbital segment near L ∼ 3.4 and λm ∼ 5°S, the electric field amplitude of the Omega signals reached ∼ 0.6 mV/m, a value &gt; 40dB higher than the signal amplitude both preceding (L ∼ 3.7–3.5) and following (L ≤ 3.2) the high‐amplitude period. Over the range 3.3 ≤ L ≤ 3.7 the transmitter pulses at 13.1 and 13.6 kHz were associated with sideband signals spaced in frequency roughly symmetrically about the carrier and generally reduced in amplitude ∼5 dB with respect to the carrier. The strongest sidebands were generally offset in frequency from the carrier by ∼ ±55 Hz. Simultaneous data from the University of Iowa Plasma Wave experiment indicated that the region of high wave amplitude was located just beyond the outer edge of the plasmapause where low values of cold plasma density (∼40 el/cm³) prevailed. High amplitudes were observed only during the time in which the signal frequency lay within the range of a natural noise band of rising center frequency in which bursts of VLF emissions were occasionally triggered by “knee‐trace” whistlers. On the basis of the group time delay and amplitude distribution of the Omega signals and the presence of knee‐trace whistlers, it is concluded that the high amplitude signals originally propagated along the base of the plasmapause surface into the southern hemisphere and subsequently reflected or scattered back up to the satellite. The high‐wave amplitude was presumably produced through the coherent whistler mode instability as the input waves interacted with gyroresonant energetic electrons near the magnetic equatorial plane, and the presence of sideband signals indicates that this interaction had reached a nonlinear stage. The wave magnetic field in the interaction region is estimated to be approximately 20 mγ. The high amplitudes reached by the signals indicates that particle trapping effects could be responsible for the sideband generation.

Excitation of whistler mode instability due to slow cyclotron interaction in an inhomogenous beam–plasma System

The excitation of whistler wave instability due to slow cyclotron (m = – 1) interaction in an inhomogeneous plasma penetrated by an inhomogeneous beam of electrons is studied. Expressions are obtained for the elements of the plasma and beam dielectric tensors. It is shown that the inhomogeneity in both beam and plasma number densities affects the growth rate of the instability.

Unified theory of whistler mode instability in the presence of a parallel electric field

The propagation characteristics of right-hand circularly polarized whistler mode waves propagating parallel to the external magnetic field in an anisotropic plasma have been reformulated including the effect of a parallel electric field. Analytical expressions for the real frequency and growth rate have been obtained for the full range of the parameters β (the ratio of particle pressure to magnetic pressure of the hot particles), A (temperature anisotropy) and P ( = βA(A + 1)2) without any restriction on the magnitude of the imaginary part of the wave frequency. The effect of cold plasma injection on the marginal instability has also been studied. Possible application of the present theory to the atmospheres of Earth and Jupiter has been discussed.

A theoretical study of plasmaspheric hiss generation

An investigation of the generation of plasmaspheric hiss by the whistler mode instability has been carried out. Using combined ray‐tracing and growth rate calculations, the path‐integrated growth rates of a large number of waves have been calculated using a distribution function of the form F ∝ E−n sinm α. In general, the waves damp out before being magnetospherically reflected, or soon after. This is true of all the waves traced, independent of the distribution functions used for the energetic (resonant) electrons. Thus the growth rates are generally low, since the waves undergo few equatorial transits. A comparison of these results with observed spectra indicates that while some spectral characteristics are in qualitative agreement, there is a major discrepancy in the intensity levels.

Electron whistler mode instability in an inhomogeneous thermal plasma in the presence of an inhomogeneous beam of suprathermal electrons

The excitation of the whistler mode waves propagating obliquely to the constant and uniform magnetic field in a warm and inhomogeneous plasma in the presence of an inhomogeneous beam of suprathermal electrons is studied. The full dispersion relation including electromagnetic effects is derived. In the electrostatic limit the expression for the growth rate is given. It is found that the inhomogeneities in both beam and plasma number densities affect the growth rates of the instabilities.

The electromagnetic ion beam instability upstream of the Earth's bow shock

The linear theory of the electromagnetic ion beam instability for arbitrary angles of propagation has been studied. The parameters considered in the theory are typical of the solar wind upstream of the earth's bow shock when a ‘reflected’ proton beam is present. Maximum growth occurs for propagation parallel to the ambient magnetic field B, but this instability also displays significant growth at wavevectors oblique to B. Oblique, unstable modes seem to be the likely source of the compressive magnetic fluctuations recently observed in conjunction with the ‘diffuse’ ion population. An energetic ion beam does not directly give rise to linear growth of either ion acoustic or whistler mode instabilities.

On the modifications of the whistler mode instability in the magnetosphere in the presence of parallel electric field by cold plasma injection

Dispersion relation for the whistler mode waves in an anisotropic warm magnetoplasma in the presence of a parallel electric field has been derived. The modifications in the whistler mode instability produced by cold plasma injection to an infinite uniform anisotropic plasma (A &lt;1) in the presence of a parallel electric field have been studied. This study has been performed with the plasma parameters similar to those observed in the equatorial magnetosphere at 6.6 RE. The electric field is found to have pronounced effect for small values of nc/nw and extends the range of instability in k space toward higher wavelengths. The cold plasma injection increases the range of instability in k space toward lower wave lengths. The effects of electric field seem to be insignificant for large values of nc/nw.

Effect of a parallel electric field on the whistler mode instability in the magnetosphere

A dispersion relation for the whistler mode wave propagation in an anisotropic warm magnetoplasma in the presence of a weak parallel electrostatic field has been derived from linearized coupled Boltzmann‐Maxwell equations with collision frequency independent of particle velocity. An expression for the growth rate has been derived in the presence of an electric field and for small temperature anisotropy (A&lt;1) for the whistler mode wave. Under suitable approximation we recover the earlier known results. The modifications introduced in the growth rate by the electric field and temperature anisotropy have been discussed using observed plasma parameters at 6.6 RE. It is observed that growth (damping) of whistler mode waves is possible when the wave vector is parallel (antiparallel) to the static electric field. The effect of the electric field is more pronounced at low frequency (higher wave number) wave spectrum.

Beam-plasma instability of a finite temperature electromagnetoplasma

The beam-plasma instability of the collisional magnetoplasma in the presence of a weak longitudinal electric field has been studied. The dispersion relation and the growth rate for the whistler mode instability have been derived and computed. The earlier known results are recovered when the parallel electric field (E0) is reduced to zero. The effect of a parallel electric field is to increase the growth rate. However, the effective electron collision frequency quenches the instability caused by the electrostatic field and the beam. The reversal in the direction of electric field decreases the growth rate. Further, it has been shown that the whistler mode instability can also be excited when the beam particles have no transverse velocity spread. This study may be useful to explain the observations of v.l.f. triggered emissions in the magnetospheric and ionospheric plasma.

Heating and acceleration of ?-particles in the solar wind

The non-linear saturation of whistler mode instability in the solar wind is considered here. The resulting heating and acceleration are calculated. It is shown that this instability heats predominantly the α-particles and the ratio of heating rates of α-particles to ions have been calculated. This instability is shown to be effective in accelerating α-particles than ions. However, this process cannot account for V α/V i ≥ 1.

Critical electron pitch angle anisotropy necessary for chorus generation

Simultaneous wave, resonant-particle, and ambient-plasma data from OGO 5 for chorus emissions on August 15, 1968, were found consistent with the theoretical critical pitch-angle-anisotropy condition for whistler-mode instability by Doppler-shifted electron cyclotron resonance. Local generation, as determined by wave normal measurements, occurred only when the pitch-angle anisotropy of resonant electrons required for instability substantially exceeded the critical anisotropy defined by Kennel and Petschek (1966).

Some comments on the whistler mode instability at Jupiter

The paper considers the possibility that electrons with energies exceeding 21 MeV at Jupiter interact resonantly with obliquely propagating whistler mode waves. The equatorial pitch angle distributions are deduced from the Pioneer 10 and 11 data points of Van Allen et al. (1975). For the L shell subgroup 7, there is observed a 'hat-shaped' pitch angle distribution which is similar to that found within the earth plasmasphere.