Volume 25, Issue 4, April 1982
Index of content:

On viscous dissipation rates of velocity component kinetic energies
View Description Hide DescriptionIt is shown by an example that in the balance equation for the kinetic energy of a velocity component in a Newtonian viscous fluid, the term usually interpreted as viscous dissipation rate can have negative regions, although the second law of thermodynamics requires that total kinetic energy dissipation rate be positive at each point in space‐time. After exploration of an alternate form, it is concluded that the conventional form is correct, and that the thermodynamic requirement is simply not relevant for kinetic energies of individual velocity components.

Allowable discontinuities in a Gerstner wave field
View Description Hide DescriptionVelocity discontinuity surfaces can be inserted at will in a Gerstner edge wave on an inclined boundary in a rotating fluid.

Mixing by a turbulent wake of a uniform temperature gradient in the approach flow
View Description Hide DescriptionThe mean temperature distribution is analyzed in the turbulent wake of a circular cylinder when there is a linear temperature gradient flow upstream of the cylinder. Using the same variable eddy diffusivity derived from published temperature measurements in the wakes of heated cylinders, it is shown that the mean temperature distribution has a self‐similar form which agrees with experimental results of Alexopoulos and Keffer. But, the location and magnitude of the maximum temperature p e r t u r b a t i o n s are in error by 25% and 50%, respectively. The reasons for the limitations of the eddy diffusivity model in wake flows are discussed. The variance of temperature fluctuations ( )^{2̄} in a linear temperature gradient mixed by homogeneous grid turbulence are then analyzed on the basis of Durbin's theory. It is shown that the measurements of ( )^{2̄} are more plausibly fitted by the curve ( )^{2̄}∝x ^{1/2} than by Sullivan’s analysis of the same data.

Injection of a relativistic electron beam into neutral hydrogen gas
View Description Hide DescriptionThe injection of a relativistic electron beam (0.8 MeV, 6 kA, 150 nsec) into hydrogen gas of 190 Pa pressure results in a plasma with density n _{ e }≃10^{20} m^{−3} and temperature k T _{ e }≲k T _{ i }≃3.5 eV. The results of the measurements show good agreement with computations based on a model combining gas ionization and turbulent plasma heating. It is found that a quasistationary state exists in which the energy lost by the beam (about 6% of the total kinetic energy of the beam) is partly used to further ionize and dissociate the gas and for the other part is lost as line radiation.

Experimental test of modulation theory and stochasticity of nonlinear oscillations
View Description Hide DescriptionThe initial nonlinear evolution of two unstable waves in a cold beam‐plasma system can be described by a single wave with a slow modulated amplitude and phase. The modulational calculation is tested quantitatively on the analogous wave‐particle interaction in a traveling wave tube. The model is found to be valid for one autocorrelation time following the onset of nonlinearities. Stochastic behavior follows the breakdown of the modulational approximation, i.e., after one autocorrelation time, in agreement with reasonance overlap theory.

An alternative derivation of the parallel ion viscosity
View Description Hide DescriptionA set of double‐adiabatic fluid equations with additional collisional relaxation between the ion temperatures parallel and perpendicular to a magnetic field are shown to reduce to a set involving a single temperature and a parallel viscosity. This result is applied to a recently published paper [R. V. Bravenec, A. J. Lichtenberg, M. A. Leiberman, and H. L. Berk, Phys. Fluids 24, 1320 (1981)] on viscous flow in a multiple‐mirror configuration.

Nonlinear evolution of Buneman instability. II. Ion dynamics
View Description Hide DescriptionThe nonlinear evolution of the one‐dimensional Buneman instability after saturation due to electron trapping is studied both by analysis and by computer simulation. A nonlinear dispersion relation proposed previously is shown to describe the nonlinear evolution even after the electron trapping sets in. After electron trapping is completed, amplitude oscillation at a frequency close to ω_{ p i } (ion plasma frequency) appears due to the exchange of energy between the ions and the electric field. The maximum energy acquired by the ions reaches more than 10% of the initial electron drift energy.

Plasma oscillations and sound waves in collision‐dominated two‐component plasmas
View Description Hide DescriptionCharge, mass, and electron density fluctuation spectra of strongly correlated, fully ionized two‐component plasmas within the framework of the Mori–Zwanzig memory function formalism are analyzed. All dynamical correlation functions are expressed in terms of the memory functions of the ion and electron velocity autocorrelation functions by a generalized effective field approximation which preserves the exact initial values (i.e., static correlations). The theory reduces correctly to the mean field (or collisionless Vlasov) results in the weak coupling limit, and yields charge density fluctuation spectra in good agreement with available computer simulation data, as well as reasonable estimates of the transport coefficients. The collisional damping and frequency shift of the plasma oscillation mode are sizeable, even in the long wavelength limit. The theory also predicts the propagation of well‐defined sound waves in dense plasmas in thermal equilibrium.

Electromagnetic oscillating two‐stream instability of plasma waves
View Description Hide DescriptionNear the resonance (ω_{o}≊ω_{ p }) a high‐amplitude plasma wave decays into a purely growing electromagnetic perturbation and two Langmuir wave sidebands. The growth rate of the instability γ≃ω_{ p } k ^{2} _{0} v ^{2} _{0}/ 8k ^{2} v ^{2} _{ e } for a Langmuir pump wave of finite wavenumber k _{0}≳k v _{ e }/c and the threshold due to the finite extent, x _{0}, of the pump is v ^{2} _{0}/v ^{2} _{ e }≳4k ^{2} v ^{2} _{ e }(k ^{2} x _{0}+β)/ ω^{2} _{ p }βx _{0}; v _{0} and v _{ e } are the oscillatory and thermal velocities of electrons and β is the x component of propagation vector k of the perturbation. This instability may be a possible source of spontaneous generation of random magnetic field in laser‐produced plasmas.

Existence of a negative potential solitary‐wave structure and formation of a double layer
View Description Hide DescriptionA stationary negative potential with a solitary‐wave structure can exist in a nonequilibrium plasma with T _{ e }≫T _{ i }. The peak value of the potential can exceed the electron temperature, T _{ e }/e. The existence of such a structure is responsible for the formation of a double layer in the presence of current.

Resistive magnetic tearing in a finite length pinch
View Description Hide DescriptionThe effects of axial boundary conditions on the tearing mode of a cylindrical pinch are described. A flexible inertial ’’line‐tying’’ model is used to restrict the available perturbations of a force‐free current‐channel equilibrium, similar to that in a reversed‐field pinch. The resistively unstable mode is found to be azimuthally symmetric (m = 0), and the external (ideal) and internal (resistive) solutions are detailed, the latter being found by a unique numerical procedure. The solutions can be either ’’constant‐ψ’’ or fast growing, depending on the width of the current channel. Threshold behavior and subsequent growth rates are calculated. Finally, an application of this reconnection mechanism to the flare instability of a narrow solar coronal loop is described.

Self‐similar oscillations of a Z pinch
View Description Hide DescriptionA new analytic, self‐similar solution of the equations of ideal magnetohydrodynamics describes cylindrically symmetric plasmas conducting constant current. The solution indicates that an adiabatic Z pinch oscillates radially with a period typically of the order of a few acoustic transit times. A stability analysis, which shows the growth rate of the sausage instability to be a saturating function of wavenumber, suggests that the oscillations are observable.

Plasma heating from upper‐hybrid mode conversion in an inhomogeneous magnetic field
View Description Hide DescriptionLarge‐amplitude short‐wavelength electrostatic fields can be generated at the upper‐hybrid resonance layer due to the mode conversion process when an incident wave of appropriate frequency falls on a plasma. As a consequence, a host of nonlinear and kinetic phenomena can take place. Among them, the generation of energetic electrons due to the breaking of adiabatic electron motion in the wave field, the enhancement of electron second harmonic cyclotron heating at the double resonance (ω_{ u h } = 2ω_{ c e }), and parametric instabilities in which the upper‐hybrid mode decays into another upper‐hybrid mode and a low‐frequency mode. Investigations of these phenomena through the use of computer simulations are presented. The parametric process is observed to produce a substantial number of energetic ions localized at the resonance layer.

Kinetic description of Alfvén wave heating
View Description Hide DescriptionHeating of tokamak plasmas by Alfvén waves is studied by means of a linearized kinetic model which takes into account electron inertia and Landau damping, finite ion gyroradius, the equilibrium current, and magnetic shear. In cylindrical geometry, a fourth‐order set of differential equations in r for the perturbed fields E _{ r } and E _{⊥} is solved numerically for modes driven by a sheet current of single helicity and frequency ω, located between the plasma edge and a conducting wall. Realistic profiles of density, temperature, and safety factor are employed. The energy deposition and density fluctuations as functions of r and the total impedance to be expected in experiments on the p r e t e x t tokamak are computed, and optimum conditions for heating are investigated. Mode conversion to the kinetic Alfvén wave and its damping are observed in the computed solutions. The plasma impedance is sensitive to the profiles and mode numbers chosen, and, with two exceptions, is consistent with previous work based on magnetohydrodynamics. Kinetic effects can produce ’’high Q’’ resonant absorption, both for frequencies below the Alfvén continuum (corresponding to discrete stable kink modes) and for frequencies such that the Alfvén resonance approaches the plasma edge (corresponding to normal modes of the kinetic shear wave in cold plasma).

Quantum‐kinetic theory of electron heating in plasmas by high‐frequency electromagnetic waves
View Description Hide DescriptionThe heating rate of electrons in laser‐irradiated plasmas is derived from the quantum‐mechanically extended Vlasov equation. The heating of electrons is shown to be due to (i) the electron velocity space diffusion by plasma waves, which are induced by the laser field (classical turbulent heating), (ii) the microscopic quantum mechanical interaction between each electron and the laser field in the electric field of the plasma waves (collisionless inverse bremsstrahlung), and (iii) collisional inverse bremsstrahlung in the static plasma field. A quasilinear equation for the changes of the electron distribution function by these three concomitant mechanisms is derived, and the heating rates of the electrons are calculated. It is shown that the collisionless inverse bremsstrahlung is dominant for hot plasmas in most experimental cases. The so‐called anomalous heating is identified as being due to collisionless inverse bremsstrahlung, and is explicitly expressed in terms of the wavenumber of the plasma wave and the plasma parameters, when the turbulence is due to the ion acoustic instability and is stabilized by ion trapping.

Ponderomotive force effects on slow‐wave coupling
View Description Hide DescriptionLocalized plasma density depressions are observed to form near a multi‐ring slow‐wave structure when the value of the nonlinearity parameter, s = ω^{2} _{ p e }‖E _{ z }‖^{2}/8πω^{2} nκT, is of order unity. Consequent changes in the wave propagation and coupling efficiency are reported. For large enough values of s, the coupling efficiency may be reduced by 50% from the linear value.

Nonlinear aspects of wave propagation near the upper‐hybrid mode conversion layer
View Description Hide DescriptionIt has been found that the effect of self‐modulation by a large‐amplitude wave can significantly modify the mode conversion process at the upper‐hybrid resonance. Under the condition ω^{2} _{ p e }<3Ω^{2} _{ c }, an incoming extraordinary plane wave of sufficiently large amplitude can be mode converted to soliton‐like hot plasma waves. These waves can build up to very high intensities and allow nonlinear processes, such as parametric decay which is otherwise not possible due to high excitation thresholds, to occur close to the upper‐hybrid resonance. The investigation involves determining the modulational stability of the incoming extraordinary wave using the warm fluid model, and the hot plasma wave using the Vlasov model. An equation governing wave propagation in a nonuniform plasma taking into account the effect of self‐modulation is derived. This equation is analyzed to obtain the physical picture described.

Space charge waves in a cylindrical waveguide with arbitrary wall impedance
View Description Hide DescriptionProperties of the space charge waves in a solid relativistic electron beam propagating in a cylindrical waveguide are investigated, including the important influence of arbitrary wall impedance. The stability analysis is carried out within the framework of the linearized Vlasov–Maxwell equations. In order to examine the influence of axial momentum spread on stability behavior, it is assumed that all electrons have a Lorentzian distribution in the axial canonical momentum. One of the most important features of the analysis is that, for short axial wavelength perturbations, the eigenfunction can be described by a Bessel function. Moreover, the condition for zero phase velocity of the space charge wave is also obtained, in connection with collective ion acceleration. Space charge wave properties for a dielectric loaded waveguide are also investigated. For the appropriate choice of the dielectric constant ε and thickness of the dielectric material, it is shown that strong mode‐coupling occurs, exhibiting the growth rate of instability comparable to the beam plasma frequency. The physical mechanism of instability is Cerenkov radiation.

Resistive interchange modes in reversed‐field pinches
View Description Hide DescriptionA new regime is found in which the growth rate of the resistive interchange mode in cylindrical geometry is considerably smaller than previously believed. This stabilization is due to coupling to the external, tearing stable, region. Considering the crossover point from the conventional resistive interchange to the new regime as a critical beta, β_{ c }, it is found that β_{ c }∼7% for classical resistivity with T _{ e } = 10 eV, in agreement with recent numerical studies. However, β_{ c } scales as T ^{−3/5} _{ e }, giving β_{ c }∼1% for T _{ c } = 1 keV. Marginally stable pressure profiles for totally localized modes in the Bessel function model for a reversed‐field pinch are also computed, and show that for tearing stable equilibria the central pressure may be up to 400 times the pressure at the wall.

Whistler instability in the Elmo Bumpy Torus
View Description Hide DescriptionA governing equation for whistler modes in the Elmo Bumpy Torus is presented and solved perturbatively. To lowest order, a discrete spectrum of real frequencies is determined. Finite growth rates are found to first order. Attention is focused on trapped electron resonances. In the frequency range of interest, the destabilizing terms involve summations over a large number of resonances. A threshold condition is obtained, and typical growth rates, corresponding to plausible machine parameters, are presented.