bi-Maxwellian Distribution Function Research Articles

Quasilinear evolution of Alfvén-ion-cyclotron and mirror instabilities driven by ion temperature anisotropy

In this paper the simultaneous nonlinear evolution of the Alfvén-ion-cyclotron and mirror instabilities driven by an anisotropic ion distribution function are studied. The ions are modeled by a bi-Maxwellian distribution function. For the sake of generality, two ion components are considered; the initially isotropic component and a population possessing a large temperature anisotropy with perpendicular temperature greater than parallel temperature. Here, perpendicular and parallel are defined with respect to the ambient magnetic field. The analysis is based on quasilinear kinetic theory. It is shown that initially, the mirror mode grows at a slightly faster rate when compared with the ion-cyclotron mode, but the subsequent evolution shows that the ion-cyclotron mode saturates at a much larger intensity. Simultaneously, large perpendicular temperature associated with the anisotropic ions is substantially reduced as the free energy is taken away by the unstable waves, while the parallel temperature increases so as to reduce the anisotropy. The initially isotropic ions, on the other hand, are also heated in the direction perpendicular to the ambient magnetic field vector.

Effect of parallel electric fields on whistler mode waves in an anisotropic Jovian magnetospheric plasma

The effect of parallel electrostatic field on the amplification of whistler mode waves in an anisotropic bi-Maxwellian weakly ionized plasma for Jovian magnetospheric conditions has been carried out. The growth rate for different Jovian magnetospheric plasma parameters forL = 5.6Rj has been computed with the help of general dispersion relation for the whistler mode electromagnetic wave of a drifted bi-Maxwellian distribution function. It is observed that the growth or damping of whistler mode waves in Jovian magnetosphere is possible when the wave vector is parallel or antiparallel to the static magnetic field and the effect of this field is more pronounced at low frequency wave spectrum.

A time-dependent two electron group model for a discharge excited He-Sr recombination laser

A time-dependent two electron group model 2EGM, based on a bi-Maxwellian electron energy distribution function (EEDF), has been developed to simulate the discharge kinetics in a high-current, high-repetition frequency, pulsed He-Sr recombination laser. A comparison is made between the results predicted by 2EGM and 1EGM (Maxwellian) simulations and experimental data corresponding to typical operating conditions of the laser. Results from the 2EGM indicate that during the current pulse, the high-energy tail region of the EEDF is severely depleted due to both inelastic collisions between electrons and ground state helium atoms and incomplete thermalization via Coulomb collisions. This leads to a highly non-Maxwellian EEDF, a feature which cannot be accommodated in the 1EGM simulation. In addition, the He*(23S), Sr, and Sr+ population densities predicted by the 2EGM are shown to be in close agreement with the experimental data, whereas the 1EGM predicts a partitioning of energy between helium and strontium states which is inconsistent with the observed population densities.

Weakly relativistic dielectric tensor in the presence of temperature anisotropy

Analytical expressions for the weakly relativistic dielectric tensor near the electron-cyclotron frequency and harmonies are obtained to any order in finite-Larmor-radius effects for a bi-Maxwellian distribution function. The dielectric tensor is written in ternis of generalized Shkarofsky dispersion functions, whose properties are well known. Relevant limiting cases are considered and, in particular, the anti-Hermitian part of the (fully relativistic) dielectric tensor is evaluated for two cases of strong temperature anisotropy.

Solutions to bi-Maxwellian transport equations for the polar wind

In this study, polar wind solutions are obtained for a broad range of O(+) density, H(+) drift velocity, electron temperature and H(+) temperature boundary conditions. The bi-Maxwellian-based 16-moment set of transport equations is used, since this set is expected to be superior to Maxwellian-based equations in describing large temperature anisotropies and heat flows. The present solutions corroborate earlier results when similar boundary conditions are used. Also, for previously unexplored combinations of boundary conditions, the present solutions are often qualitatively different from any obtained before.

Three-wave coupling coefficient in a drifting bi-Maxwellian plasma

A general electrostatic coupling coefficient which satisfies the Manley-Rowe relations is used to derive an explicit expression for the resonant three-wave coupling coefficient between electrostatic normal modes of a uniformly magnetized, infinite, homogeneous plasma with species described by drifting bi-Maxwellian distribution functions. The limit of this expression is taken when the phase velocities of the three waves are much larger than a species thermal speed, and also when the phase velocities are much smaller than the thermal speed. These are fluid limits and are applicable to the three-wave interaction between some low-frequency electrostatic waves, such as ion acoustic and ion cyclotron modes, in a plasma where Te ≫ Ti.

Excitation of high-β plasma instabilities at the geostationary orbit: Theory and observations

Abstract The general theory of low frequency plasma instabilities is developed for a plasma consisting of a cold component and a hot high β component. In the limit k⊥rH⪡1, two groups of instabilities can generate ULF oscillations at the geostationary orbit: for the first the magnetic field line curvature can be neglected ( βk 2 ⊥ r 2 H ⪢ a R ) in the dispersion relation, for the second it must be retained ( βk 2 ⊥ r 2 H ≲ a R ). The case of straight field line geometry yi two independent modes: the compressional Alfven wave and the drift mirror mode. The instabilities that can excite them were treated by Pokhotelov et al. (1985) for a bi-maxwellian distribution function. When the existence of a loss cone is also taken into account, a significantly lower threshold is obtained for both the drift anisotropy instability (corresponding to the compressional Alfven wave) and the drift mirror instability. The case of curved field lines yields a strong coupling between the compressional Alfven wave and the drift mirror mode. The theoretical predictions are compared in detail with the Pc5 events of 27 October 1978 and 1 November 1978 observed simultaneously by the geostationary satellite Geos 2 and by the STARE radar system. In both cases the observed anisotropy was too low for the drift mirror mode as well as for the drift anisotropy instability. The calculation of the growth rate for the coupled mode, in which the field line curvature is taken into account, shows that the oscillations generated by this mechanism are growing and propagate in the East—West direction in agreement with the STARE measurements. Thus the coupling between the compressional Alfven waves and the drift mirror mode can explain the observed low frequency particle intensity oscillations.

On the generation of quasi-electrostatic half-electron-gyrofrequency VLF emissions in the magnetosphere

A theoretical study is made of the generation mechanism of quasi-electrostatic VLF emissions observed in the distant magnetosphere, with frequency greater than half the electron gyrofrequency and with wave normal around the cold plasma oblique resonance cone. The two-component plasma is treated, composed of cold electrons and hot electrons with bi-Maxwellian and loss-cone distribution functions. The effects of various plasma parameters on the instability characteristics are examined in order to estimate their relative importance in determining the properties of unstable waves. It is found that both types of hot plasma distribution function can account for quasi-electrostatic half-gyrofrequency emissions. The frequency where the maximum growth rate occurs is mainly determined by the temperature anisotropy of the hot plasma, and the wave normal angle where maximum growth is expected is determined by the temperature of the hot plasma and the ratio of the cold to hot plasma densities. These theoretical considerations form the basis of a suggested plasma model which is able to explain our experimental direction-finding results.

Electromagnetic ion beam instabilities - Hot beams at interplanetary shocks

This paper considers Vlasov instabilities driven by a very hot ion beam streaming parallel to a magnetic field. The linear theory of electromagnetic instabilities driven by such a beam in a homogeneous, collisionless, nonrelativistic plasma is used. Numerical solutions of the full linear dispersion equation for bi-Maxwellian distribution functions are presented. If the thermal speed of the beam is greater than its drift speed, the two dominant modes are the right-hand and left-hand resonant ion beam instabilities. The parametric dependencies of the threshold drift speeds and growth rates for both modes are presented. In particular, the thresholds are shown to be sensitive functions of the beam temperature anisotropies. Arguments are presented that these two modes are driven by the suprathermal ion component at interplanetary shocks, and that the growth of these modes is sufficient to account for the amplitudes of MHD-like waves observed at such shocks.

Longitudinal electron waves for a plasma with a bi-Maxwellian velocity distribution

An original method for calculating and representing the excitation coefficients of longitudinal electron waves is applied to the case of a hot, collisionless, homogeneous and isotropie plasma with a bi-Maxwellian velocity distribution function. Using a Van Kampen treatment we introduce a ‘wave density’ distribution which is represented graphically. When hot electrons are added to a plasma of cool electrons the Landau mode is distorted and its damping increases with the hot/cold temperature ratio θ. The Landau mode separates into two modes for θ ≥ θ0 where θ0 increases as the hot/cold density ratio α decreases. We show that no wave seems to be able to propagate at frequencies below the plasma frequency. Using an abrupt cut-off in the hot electron distribution function, we recover the results for a Maxwellian plus water-bag distribution.

On the influence of a plasma hot component on whistler propagation beyond the plasmapause

Theoretical spectrograms are computed for whistlers propagating beyond the plasmapause. The electron distribution function was modelled as consisting of a hot plus a cold component and an appropriate dispersion equation is used. A collisionless (CL) model is used for the cold electron concentration and for the hot electron component the derived model assumes a bi-maxwellian distribution function with a loss cone at the equator. The results indicate limits on the use of the cold plasma approximation (c.p.a.) in the study of magnetospheric whistler propagation beyond the plasmapause and show that whistler analysis with the c.p.a. may under or overestimate the L value of the path deduced from ground spectrograms, depending on the anisotropy of the hot component.

Active and passive methods for the study of non-equilibrium plasmas using electrostatic waves

We study the possibility of measuring the properties of a non-equilibrium plasma by means of a double-dipole radio-frequency probe, consisting of two small dipoles immersed in the plasma, and separated by a distance one or two orders of magnitude greater than the Debye length. This type of probe can be used either in an active or in a passive mode. The theory of both modes is presented, taking the example of an isotropic plasma in which the electrons have a biMaxwellian distribution function; the first Maxwellian represents the major population of thermal electrons, and the second a minor population of suprathermal electrons. In the active mode, one dipole emits artificial signals which the other receives; thus we study the propagation of electrostatic waves between the two dipoles. In the passive mode both dipoles are used to receive random signals induced by the natural electric microfield in the plasma, and we compute the cross-spectrum of these signals. By combining the active and passive techniques, it is possible to measure the electron density and temperature of the thermal electrons, and also to get some information about the distribution function of the suprathermal electrons.

On the kinetic instabilities of uniform magnetized plasmas with generalized loss-cone distribution functions

A general proof is given that in uniform magnetized plasmas described by generalized loss-cone distribution functions (loss-cone index l, thermal velocity α∥, and perpendicular spread α⊥), electromagnetic, electrostatic, or coupled-mode instabilities are insensitive to the separate values of l and (α⊥/α∥); they depend rather, on the effective thermal anisotropy Aeff ≡ (T⊥/T∥)eff-1, where (T⊥/T∥)eff ≡ (l + 1) (α2⊥/α2∥). In the case of parallel propagation this statement is limited only by the linearization assumption; in the oblique propagation case, the additional condition λ⊥/rL ≫ 1 is required (λ⊥ = 1/k⊥, where k⊥ is the wave vector perpendicular to the external magnetic field, and rL is the Larmor radius). Thus, dispersion relations and their solutions obtained by using simple bi-Maxwellian distribution functions can be used directly for the complex case of generalized loss-cone distribution functions by simply replacing the anisotropy factor, A ≡α2⊥/α2∥-1, by Aeff defined above. This result explains earlier conclusions that the growth rate of the whistler instability is independent of the explicit value of the loss-cone index l, for a given thermal anisotropy.

Transport equations for multispecies plasmas based on individual bi-Maxwellian distributions

The authors have derived a closed system of transport equations for an anisotropic plasma of arbitrary degree of ionisation. The system is based on an anisotropic bi-Maxwellian species distribution function, and therefore, should provide a better description of flow conditions characterised by large temperature anisotropies. The method used to derive the transport equations is an extension of Grad's method and corresponds to a 16-moment approximation for the species distribution function. The relevant collision terms were calculated for an arbitrary inverse-power interaction potential and for a resonant charge exchange interaction between an ion and its parent neutral. In the collisionless limit, the system of equations reduces to the collisionless transport equations which include the effects of collisionless 'viscosity' and heat flow.

Cyclotron instabilities of a magnetized electron plasma with anisotropic temperatures

The dispersion relation for an electron plasma in a magnetic field is investigated for a bi-Maxwellian distribution function. A new set of solutions for non-perpendicular propagation is found. The linear growth rates are computed and the instability regions in the (k, cos θ) plane are determined. An approximate analytical treatment of the problem is also given for certain ranges of the parameters.

Representation of dielectric function of magnetized plasma

A representation of the dielectric function of a uniform magnetized plasma with bi-Maxwellian or model loss-cone distribution functions, which is convenient for computational purposes in the case of large temperature anisotropy or propagation at large angles with the magnetic field, is presented.

Numerical Solution of Dispersion Equation for a Bi-Maxwellian Plasma

In the presence of a uniform magnetic field, longitudinal oscillations of a plasma with a bi-Maxwellian distribution function are studied numerically. A dispersion equation which involves a function Z(ζ) = π−1/2 ∫ −ωωdxexp(−x2)(x − ζ), where ζ is complex, is derived. A root of the equation is obtained by using a formula which combines the asymptotic expansion of Z(ζ) with the power series of Z(ζ). The real part of roots gives a mode of cold plasma oscillations and modes of multiple oscillations of the cyclotron frequency. These modes are coupled and hybrid modes exist. Instabilities arise along long strips which are along modes of multiple oscillations. If an anisotropy of the bi-Maxwellian distribution is constant, then instabilities are growing while the temperature of the plasma is increasing.