The dispersion equation of a plane ejector
Dispersion equations for boundary stability noise analysis of multilayer and multiple jets
Nazarova, L. I.↗
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Electromagnetic Scattering by Discrete Random Media. I: The Dispersion Equation and the Configuration-Averaged Exciting Field
We consider the scattering of a plane electromagnetic wave obliquely incident on a plane-parallel layer of discrete random medium with non-scattering boundaries. We solve the Lax integral equation for the conditional configuration-averaged exciting field coefficients by assuming a special-form solution, that is, by representing the conditional configuration-averaged exciting field coefficients as a linear combination of the coefficients corresponding to an up-going and a down-going wave. This solution representation is supposed to be valid within the whole domain occupied by the particles, even in the close proximity of the boundaries. By balancing the waves with different propagation directions and wavenumbers we derive two homogeneous systems of equations corresponding to the generalized Lorenz–Lorentz law and two inhomogeneous systems of equations corresponding to the generalized Ewald–Oseen extinction theorem. It is shown that (i) the two homogeneous systems of equations of the generalized Lorenz–Lorentz law reduce to a single homogeneous system of equations corresponding to a semi-infinite discrete random medium at normal incidence; (ii) the dispersion equation is direction and polarization independent; and (iii) the two inhomogeneous systems of equations of the generalized Ewald–Oseen extinction theorem can be reduced to two scalar equations by means of the addition theorem for vector spherical harmonics. It is also shown that the same dispersion equation can be obtained without assuming a special-form solution representation in the proximity of the boundaries.
Adrian Doicu↗
A quartic dispersion equation for internal gravity waves in the thermosphere
A new quartic dispersion equation in the square of the complex vertical wave number is derived by employing the 'shallow atmosphere' approximation and an ion drag approximation. These approximations allow the coefficients of the quartic equation to be given in terms of the corresponding cubic equation (which neglects the Coriolis force and the zonal ion drag component), but modified to take into account these neglected effects. Coupling between the extraordinary viscosity wave mode and the other three wave modes is highlighted and numerical solutions are compared for this quartic equation, an exact eighth order equation and the cubic equation. For the first time the validity of using the 'shallow atmosphere' approximation to describe internal gravity wave motions is demonstrated.
Hickey, M. P.↗
The dispersion equations for multilayer jets and for several jets
Boundary stability, noise formation, and dispersion equation of plane supersonic jet ejector
Sedelnikov, T. KH.↗
New Instabilities in Line Driven Winds
The physical mechanisms which potentially lead to instabilities in line driven winds, the drift instability and the line shape instability, are discussed. A general three dimensional treatment of the stability problem of line driven winds which leads to the general dispersion equation is proposed. From this dispersion equation automatically a third physical mechanism driving instability in stellar winds is deduced; the thermal drift instability which is related to changes in absorption of radiation caused by temperature perturbations. This mechanism results in growing inwardly propagating sound waves.
Martens, P. C. H.↗
The coupling of Alfven and compressional waves
The article studies the hydromagnetic wave propagation characteristics in a mixture of cold and hot plasma in the presence of an inhomogeneous magnetic field. Electron and ion distribution functions with a temperature anisotropy and a density gradient are used to obtain the dispersion equation by solving the Vlasov equation and Maxwell equations. From the solutions of the dispersion equation we find that the Alfven waves can couple to unstable drift mirror waves under certain conditions. The polarization of the coupled waves is studied for varying parameters of temperature anisotropy and the cold to hot density ratio. From detailed comparison of the theoretical results with the low-frequency wave properties observed in the magnetosphere we propose that the storm-associated magnetic field oscillations with periods of 100-600 s might be caused by the coupling of Alfven waves and the unstable drift mirror waves.
Lin, C. S.↗
Theory of 2 omega(sub pe) radiation induced by the bow shock
A new radiation emission mechanism is proposed to explain electomagnetic radiation observed at twice the electron plasma frequency, 2 omega(sub pe), in the upstream region of the Earth's bow shock. This radiation had its origin at the electron foreshock boundary where energetic electron beams and intense narrow-band Langmiur waves are observed. The proposed emission mechanism results from the interaction of the electron beam and Langmuir waves that are backscattered off thermal ions. This interaction is described by a nonlinear dispersion equation which incorporates an effect owing to electron trajectory modulation by the backscattered Langmuir waves. Subsequent analysis of the dispersion equation reveals two important consequences. First, a long-wavelength electrostatic quasi-mode with frequency at 2 omega(sub pe) is excited, and second, the quasi-mode and the electomagnetic mode are nonlinearly coupled. The implication is that, when the excited 2 omega(sub pe) quasi-mode propagates in an inhomgeneous medium with slightly decreasing density, the quasi-mode can be converted directly into an electromagnetic mode. Hense the electomagnetic radiation at twice the plasma frequency is generated. Numerical solutions of the dispersion equation with the choice of parameters that describe physical characteristics of the electron foreshock are presented, which illustrates the viability of the new mechanism.
Yoon, Peter H.↗
Plasma instabilities stimulated by HF transmitters in space
Diffuse incoherent signal returns are often observed on Alouette and ISIS topside ionograms in addition to coherent echoes of electromagnetic and electrostatic waves. These diffuse signals, which at times can be the dominant features on topside ionograms, have been attributed to sounder-induced temperature anisotropies which drive the Harris instability. Previous theoretical investigations were based on the electrostatic approximation to the dispersion equation. The present paper will present calculations indicating that when the electromagnetic terms are retained in the dispersion equation and when the sounder-stimulated perpendicular electron temperature approaches 1 keV, then the whistler mode can have a temporal growth rate larger than the electrostatic electron cyclotron harmonic wave mode central to the diffuse resonance problem. Present sounders lack the power and antenna lengths to generate whistler mode waves in this manner. In addition, such waves would have large group velocities and would quickly leave the vicinity of the sounder. Experiments to investigate the wave growth, propagation, and damping of such stimulated waves are planned for the 1990s using a highly flexible sounder on the Space Shuttle and a receiver on a subsatellite.
Benson, Robert F.↗
Numerical modeling for primary atomization of liquid jets
In the proposed numerical model for primary atomization, surface-wave dispersion equations are solved in conjunction with the jet-embedding technique of solving mean flow equations of a liquid jet. Linear and approximate nonlinear models have been considered. In each case, the dispersion equation is solved over the whole wavelength spectrum to predict drop sizes, frequency, and liquid-mass breakup rates without using any empirical constants. The present model has been applied to several low-speed and high-speed jets. For the high-speed case (the LOX/H2 coaxial injector of the Space Shuttle Main Engine Preburner), predicted drop sizes and liquid breakup rates are in good agreement with the results of the CICM code, which have been calibrated against measured data.
Przekwas, A. J.↗
Electrostatic ion cyclotron velocity shear instability
A local electrostatic dispersion equation is derived for a shear flow perpendicular to an ambient magnetic field, which includes all kinetic effects and involves only one important parameter. The dispersion equation is cast in the form of Gordeyev integrals and is solved numerically. Numerical solutions indicate that an ion cyclotron instability is excited. The instability occurs roughly at multiples of the ion cyclotron frequency (modified by the shear), with the growth rate or the individual harmonics overlapping in the wavenumber. At large values of the shear parameter, the instability is confined to long wavelengths, but at smaller shear, a second distinct branch at shorter wavelengths also appears. The properties of the instability obtained are compared with those obtained in the nonlocal limit by Ganguli et al. (1985, 1988).
Lemons, D. S.↗
Lower-hybrid instabilities driven by an ion velocity ring
The lower-hybrid instabilities in high-beta (ratio of plasma to magnetic pressure) plasmas driven by ring-ion distributions in velocity space are investigated. A dispersion equation including electromagnetic effects is derived. In the low-beta limit, analytic expressions are obtained which illuminate the physical nature of the instabilities. The complete dispersion equation is solved numerically as a function of ring speed and plasma beta for several types of ring distribution. Electromagnetic effects are important for relatively energetic rings even in the low-beta regime, suppressing growth rates and shifting the angle of propagation to more oblique angles. Stabilization by thermal effects is also discussed. Application of these results to the earth's bow shock, AMPTE, comets and solar flares is suggested.
Akimoto, K.↗
Selection of Optical Glasses Using Buchdahl's Chromatic Coordinate
This investigation attempted to extend the method of reducing the size of glass catalogs to a global glass selection technique with the hope of guiding glass catalog offerings. Buchdahl's development of optical aberration coefficients included a transformation of the variable in the dispersion equation from wavelength to a chromatic coordinate omega defined as omega = (lambda - lambda(sub 0))/ 1 + 2.5(lambda - lambda(sub 0)) where lambda is the wavelength at which the wavelength is calculated and lambda(sub 0) is a base wavelength about which the expansion is performed. The advantage of this approach is that the dispersion equation may be written in terms of a simple power series and permits direct calculation of dispersion coefficients. While several promising examples were given, a systematic application of the technique to an entire glass catalog and analysis of the subsequent predictions was not performed. The goal of this work was to apply the technique in a systematic fashion to glasses in the Schoft catalog and assess the quality of the predictions.
Griffin, DeVon W.↗
Effects of unequal particle number densities on Alfven waves
Analytic plasma theory and numerical solutions of the dispersion equation are used to show that the assumption that the linear properties of the waves are determined by a charge-neutral plasma in the absence of the nonthermal particles, while the nonthermal particles cause growth or additional damping superposed onto the background, is seriously flawed even for stable plasmas. Even when the nonthermal particles do not contribute significantly to the dispersion equation, unequal thermal electron and ion number densities (due to the presence of the nonthermal particles) may cause fundamental low wave number modifications to the Alfven modes, including the creation of a new resonance and severely modified dispersion. These results are found for both cold and warm plasmas. Previous work on Alfven waves should be reevaluated in view of these results.
Cairns, I. H.↗
Collisionless tearing in a field-reversed sheet pinch assuming nonparallel propagation
The problem of collisionless linear tearing is examined assuming a wave vector with a component normal to the equilibrium field. The geometry is defined and the general form of the linear dispersion equation is calculated. The linear theory results when k is parallel to B are reviewed, and Ampere's law is calculated for the external adiabatic region when k times B does not equal zero, using two-fluid theory. A solution is obtained for the approximate form of the perturbed currents and vector potential assuming quasi-parallel k. The resonant current contributions within the singular layer are calculated, obtaining an estimate of the dispersion equation. The form of the adiabatic currents within the singular layer is calculated, showing that an x-z current system persists even in the limit k perpendicular to B goes to zero. Finally, the perturbed vector potential solutions across the singular layer are matched to obtain the shape of the complete eigenfunction.
Quest, K. B.↗
Linear density drift instabilities in very low beta plasmas A different approach
This paper reports a study of the linear Vlasov electromagnetic dispersion equation for density drift instabilities in very low beta plasmas. A uniform magnetic field and a weak uniform density gradient are assumed. This paper differs from most other studies of this topic in three important ways: (1) no low frequency or long wavelength approximations are made, (2) no gauge condition is imposed and (3) a modification of the local approximation is used which is argued to be less arbitrary than the usual local approximation. Numerical solution of the resulting dispersion equation yields stabilization of the universal density drift instability at values of beta somewhat higher than those obtained from the local approximation, and near maximum growth shows no evidence of coupling between the universal and Alfven modes.
Gary, S. P.↗
On the stability of nongyrotropic ion populations - A first (analytic and simulation) assessment
The wave and dispersion equations for perturbations propagating parallel to an ambient magnetic field in magnetoplasmas with nongyrotropic ion populations show, in general, the occurrence of coupling between the parallel (left- and right-hand circularly polarized electromagnetic and longitudinal electrostatic) eigenmodes of the associated gyrotropic medium. These interactions provide a means to driving linearly one mode with free-energy sources of other modes in homogeneous media. Different types of nongyrotropy bring about distinct classes of coupling. The stability of a hydrogen magnetoplasma with anisotropic, nongyrotropic protons that only couple the electromagnetic modes to each other is investigated analytically (via solution of the derived dispersion equation) and numerically (via simulation with a hybrid code). Nongyrotropy enhances growth and enlarges the unstable spectral range relative to the corresponding gyrotropic situation. The relevance of the properties of nongyrotropic populations to space plasma environments is also discussed.
Brinca, A. L.↗
Pedersen density drift instabilities
This paper describes the linear kinetic theory of electrostatic-drift instabilities driven by Pedersen and density-drift velocities. The model uses a uniform magnetic field B; a weak, uniform density gradient in the x direction; and a weak, uniform electric field in the y direction. Weak charged-neutral collisions are represented by the addition of BGK model terms to the Vlasov equation. The resulting local dispersion equation is used to study the properties of the associated instabilities at ka(i) greater than about 1 (where k is the wave number and a(i) is the ion gyroradius). Results show that the E x B gradient drift instability at ka(i) = about 1 may grow in the auroral ionosphere primarily in the vicinity of 200 km and only if the electron density is sufficiently small.
Gary, S. P.↗
Linear analysis of an axially grooved rectangular gyrotron for harmonic operation
In an axially grooved rectangular waveguide the linearized Vlasov equation is solved to find the perturbed distribution function resulting from the electromagnetic forces on the electrons. The resulting beam current and the propagating electromagnetic waves of the cold tube are used in the inhomogeneous Maxwell's equation to derive the general dispersion equation. This equation is then transformed into the electron beam frame and the resulting linear growth rate of amplification is calculated. By maximizing the linear growth rate, the operational parameters of the gyrotron are then optimized.
Ferendeci, A. M.↗