Volume 24, Issue 4, April 1981
Index of content:

Global stability of transient drop extraction to Marangoni instabilities
View Description Hide DescriptionThe method of energy is used to study the stability of transport in a drop which is characterized by an impulsive change in boundary temperature or concentration. Instabilities driven by interfacial‐tension gradients are considered for diffusive base states in which a spherical drop is either stagnant or has an internal circulation at high Peclet number. Two stability criteria are considered for the stagnant drop; strong stability, for which an exponential decay of the disturbance energy is guaranteed; and marginal stability, for which the disturbance energy is less than or equal to its initial value. The strong stability criterion is obtained for the circulating drop. In this latter case, properties of the solution to the dynamic equations are used to constrain the class of admissible functions in the energy theory. Stability curves generated in the Marangoni number‐time plane clearly show the presence of both an onset time and a decay time for the disturbances. Comparison with limited available experimental data shows that predictions based upon the theory are observed qualitatively.

Effect of velocity distribution on the stability of developing flow in a pipe
View Description Hide DescriptionA spatial stability study of the developing flow in a pipe shows that the critical frequency and wavenumber for the Hornbeck profile are larger than those for the Sparrow profile, but the critical Reynolds number is smaller. Results for the Hornbeck profile are found to be closer to the experimental data.

On Stokes flow of a Newtonian fluid through a pipe with stationary random surface roughness
View Description Hide DescriptionThe problems of Stokes’ flow through corrugated pipes are considered. The corrugations are of small amplitude e and are modeled by stationary random noises. A perturbation solution [up to O(e^{2})] is developed showing that the pressure drop enhancement is an order O(e^{2}) effect and is always positive. Furthermore, for slowly varying corrugations, the pressure drop enhancement depends on the statistics of the corrugations em(z) only through its mean square e^{2}<m ^{2}≳ and can be predicted by the simpler lubrication approach.

Rotational distribution of para‐H_{2} in He shock wave
View Description Hide DescriptionThe rotational distribution and collision number Z _{ r } of para‐H_{2} in the He shock wave are studied using the master equation and the quantum mechanical rate constant obtained by Rabitz and Zarur. For high Mach number, at the initial stage, the rotational distribution indicates a merging pattern of two Boltzmann distributions at the lower and higher rotational levels corresponding to the pre‐shock temperature T _{1} and the reduced temperature T _{2} T _{ c }/(T _{2}+T _{ c }), where T _{2} and T _{ c } are the post‐shock and rate‐constant temperatures, respectively. The rotational distribution at the final stage indicates a non‐Boltzmann distribution with underpopulation at the higher levels, which is also observed for low Mach number. The rotational collision number Z _{ r } over the period of the rotational relaxation is found to show a step pattern with the plateaus corresponding to the rotational collision number Z _{ j j′} of the two‐level systems (j j′) = (02), (24), (46),....

Transport equation for the joint probability density function of velocity and scalars in turbulent flow
View Description Hide DescriptionThe transport equation for the joint probability density function of velocity and scalars is shown to provide a good basis for modeling turbulent reactive flows. As in the equation for the probability density function of the scalars alone, nonlinear reaction schemes can be treated without approximation. The advantage of considering the joint probability density function equation is that convection (by both the mean and fluctuating velocities) appears in closed form. Consequently, the gradient‐diffusion assumption for turbulent transport is avoided. Closure approximations are presented for the terms involving the fluctuating pressure and viscous and diffusive mixing. These models can be expected to be reliable since they are compatible with accurate and proven Reynolds‐stress models. The resulting modeled transport equation for the joint probability density function can be solved by the Monte‐Carlo method for inhomogeneous flows with complex reactions.

Thermal mixing layer downstream of half‐heated turbulence grid
View Description Hide DescriptionAdditional experimental results relative to the temperature in the thermal mixing layer downstream of a partially heated turbulence grid are given. Comparison is made of the distributions of mean temperature and temperature intensity with previous data. New results correspond to the characteristics of the higher moments, to the probability density functions at various points within the mixing layer, and the thickness of the temperature interface on the cold side of the layer. The present inability to predict the peak value of the temperature intensity is discussed.

Similar solutions in modified cumulant expansion
View Description Hide DescriptionA method of matching is applied to the energy spectrum equations in the modified cumulant expansion theory for Burgers and two‐ and three‐dimensional Navier–Stokes turbulence, in order to investigate the asymptotic behavior of the energy spectrum in the limit of large Reynolds number and time. It is shown analytically that the spectrum has a similar structure with respect to both Reynolds number and time, and that there is no single similarity law which is valid throughout the wavenumbers, but instead there are several subranges of the wavenumbers which have different similarity laws.

Cubic turbulence: A model problem
View Description Hide DescriptionMany of the equations encountered in fluid dynamics and plasma physics are cubically nonlinear. The properties of a relatively simple statistical theory of cubically nonlinear equations, analogous to Kraichnan’s direct interaction approximation for quadratically nonlinear equations, are explored. Its predictions are compared with the numerical solution of a set of equations called the random‐coupling model; good agreement is found.

Modified Poisson eigenfunctions for electrostatic Bernstein–Greene–Kruskal equilibria
View Description Hide DescriptionThe stability of an electrostatic Bernstein–Greene–Kruskal equilibrium by Lewis and Symon’s general linear stability analysis for spatially inhomogeneous Vlasov equilibria, which employs eigenfunctions and eigenvalues of the equilibrium Liouville operator and the modified Poisson operator, is considered. Analytic expressions for the Liouville eigenfuctions and eigenvalues have already been given; approximate analytic expressions for the dominant eigenfunction and eigenvalue of the modified Poisson operator are given. In the kinetic limit three methods are given: (i) the perturbation method, (ii) the Rayleigh–Ritz method, and (iii) a method based on a Hill’s equation. In the fluid limit the Rayleigh–Ritz method is used. The dominant eigenfunction and eigenvalue are then substituted in the dispersion relation and the growth rate calculated. The growth rate agrees very well with previous results found by numerical simulation and by modified Poisson eigenfunctions calculated numerically.

Energy transport by weak electrostatic drift fluctuations
View Description Hide DescriptionEnergy transport from weak electrostatic microfluctuations, driven unstable by density and temperature gradients perpendicular to a uniform magnetic field in a Vlasov plasma, is calculated. Using the local approximation and assuming resonance broadening as the saturation mechanism, a consistent procedure is used to evaluate and compare the cross‐field energy transport due to enhanced microfluctuations from the universal, lower hybrid, and ion‐acoustic density drift, and the ion, electron, and lower hybrid temperature drift instabilities. If a temperature gradient is the primary source of free energy, the resultant enhanced fluctuations cause a cross‐field thermal energy loss rate which is much greater than the energy loss rate associated with cross‐field particle transport.

Lower‐hybrid‐drift instability with axis encircling ions
View Description Hide DescriptionThe stability of a cylindrical plasma to electrostatic flute perturbations in the ion‐cyclotron to lower‐hybrid frequency range is considered. The analysis exploits the large growth rates of these modes to reduce the governing radial eigenvalue problem from an integral equation to a second‐order differential equation, but allows treatment of arbitrary ion Larmor radius to plasma radius. The fastest growing modes of the ion guiding‐centers‐on‐axis model are shown to connect onto the lower‐hybrid‐drift modes of local slab theory. Numerical and analytical results for growth rates and most unstable mode numbers are obtained.

Resistive instabilities in reversed‐field confinement configurations without shear
View Description Hide DescriptionA resistive mode with azimuthal mode number m = 1 is shown to exist in reversed‐field confinement configurations without shear. The mode has an almost constant radial profile from the magnetic axis to the radius where the magnetic field vanishes; its growth rate scales with the 1/3 power of the plasma resistivity, and the corresponding perturbed displacement parallel to the field lines is much larger than the perpendicular one in the resistive region. This mode may lead to the splitting of the plasma column into many rings, and may also be a triggering mechanism for the rotational m = 1 instability that is observed in q‐pinch discharges.

Langmuir collapse in a weak magnetic field
View Description Hide DescriptionWith magnetic fields that are not too weak, Langmuir collapse times can be prolonged and the packet geometry significantly distorted.

Two‐dimensional simulation of the formation of the Princeton spheromak
View Description Hide DescriptionVarious schemes proposed for the creation of the spheromak in the Princeton S1 experiment are simulated by our two‐dimensional, time‐dependent, compressible, resistive hydromagnetic code. In these schemes, the toroidal fields and poloidal currents in the plasma are induced by a solenoidal discharge in a core, while a toroidal coil inside the core produces the major part of the initial poloidal fields, as well as the main plasma toroidal current. Poloidal fields are reversed by programming current reversal in the toroidal coil. Poloidal field reconnection, toroidal field compression, and plasma accumulation into a spheromak geometry are achieved in the various schemes with or without the aid of pinching coils. For several schemes, using proper programming of currents, realistic parameter values give effective spheromak formation.

Two‐dimensional modeling of the formation of spheromak configurations
View Description Hide DescriptionA reduced set of two‐dimensional magnetohydrodynamic equations has been derived describing the axisymmetric time evolution of a stable plasma evolving slowly due to resistive diffusion and changing boundary conditions. The equations are restricted to low b but allow changing topology. They are integrated in time to demonstrate a possible spheromak formation method. External circuit equations are integrated simultaneously with the plasma equations to determine the electromagnetic boundary conditions self‐consistently. The effects of a finite conductivity vacuum chamber are included.

Effect of finite electron drift velocity on the resonant excitation of a nonuniform plasma
View Description Hide DescriptionAn exact analytic solution of the external resonant excitation of a nonuniform plasma in the presence of a zero order finite electron drift is presented. The process of linear mode conversion of the long wavelength external radiation, of frequency w, into a Langmuir wave limits the peak amplitude of the cold plasma resonance at w = w_{ p }, where w_{ p } is the local electron plasma frequency. The finite electron drift alters the effective group velocity of the Langmuir wave, and thus it modifies the peak amplitude of the resonance in a significant manner. For drifts toward the overdense side an amplitude enhancement is obtained, while for drifts toward the underdense side a severe quenching takes place. The effect is governed by a single scaled drift parameter u = (2v _{ d }/3v̄) (3w_{ p } L/v̄)^{1/3}, and significant modifications arise when ‖u‖≳1. Here, v _{ d } is the small drift velocity, v̄ is the electron thermal velocity, and L is the density scale length.

Transport of runaway and thermal electrons due to magnetic microturbulence
View Description Hide DescriptionThe ratio of the runaway electron confinement to thermal electron energy confinement is derived for tokamaks where both processes are determined by free streaming along stochastic magnetic field lines. The runaway electron confinement is enhanced at high runaway electron energies due to phase averaging over the magnetic perturbations when the runaway electron drift surfaces are displaced from the magnetic surfaces. Comparison with experimental data from LT‐3, Ormak, PLT, ST, and TM‐3 indicates that magnetic stochasticity may explain the relative transport rates of runaways and thermal electron energy.

Optical ray retracing of Brillouin backscatter from a nonisothermal plasma
View Description Hide DescriptionThe optical ray retracing of stimulated Brillouin backscatter from a laser fusion plasma has been studied, including the effects of density gradient and flow velocity and allowing the acoustic wave to be weakly damped. The pump is comprised of two plane uniform beams propagating at some angle with respect to each other, producing an interference pattern that behaves as an active grating for the scattered radiation. When the convective loss due to plasma inhomogeneity is small, Bragg diffraction of the scattered radiation results in the reproduction of the same transverse spatial structure as the pump, i.e., the two beams are retraced. However, when the convective loss dominates, the Bragg diffraction is severely affected and the scattered radiation no longer reproduces both beams of the pump with equal intensities or in the correct backward directions.

Potential double layers formed by ion beam reflection in magnetized plasmas
View Description Hide DescriptionExperimental observation of a new potential double layer in a collisionless magnetoplasma are presented. The double layer is formed when an ion beam is injected along converging field lines and reflected at a boundary drawing electron saturation current. The double layer is detached from the boundaries, V‐shaped for magnetized ions, establishes its potential self‐consistently to the beam potential (f_{ d }≲V _{ b }), is stationary over a wide potential range (1∼ef_{ d }/k T _{ e }≲25), and is produced free from ionization effects. Distribution functions measured on the high potential side show drifting Maxwellians subject to the Buneman instability (v _{ d }/v _{ e }≳1) and self‐consistently produced trapped particles.

Electron cyclotron radiation measurements on the PLT tokamak
View Description Hide DescriptionA grating polychromator and Fourier transform spectrometer have been used to study the electron cyclotron radiation emitted by the PLT plasma. Results are presented which illustrate the power of these diagnostics in following the spatial and temporal changes in the plasma electron temperature. Specifically, data showing electron heating during neutral beam injection, fluctuations associated with sawteeth, m = 2 oscillations, and minor disruptions as well as data affected by radiation from runaway electrons are presented. A detailed comparison of temperature profiles obtained by Thomson scattering and electron cyclotron radiation measurements is made.