Index of content:
Volume 25, Issue 8, August 1982

Radio frequency current generation by waves in toroidal geometry
View Description Hide DescriptionThe quasilinear generation of current by rf wave fields is investigated in toroidal geometry. A general expression is derived for the amount of current generated for a given wave‐induced flux of particles. This expression should be of great practical value in assessing the various competing effects on current generation in toroidal geometry and is well‐suited for implementation in a toroidal ray tracing code.

Evolution and merger of isolated vortex structures
View Description Hide DescriptionNumerical simulations of the instability, merger, and breaking of two piecewise‐constant finite‐area vortex regions (FAVR’s) are presented. An improved contour dynamical algorithm with node insertion‐and‐removal to maintain the a p r i o r i accuracy is used. Corotating ’’V states’’ (symmetric steady‐state FAVR’s) were found to be unstable when properly perturbed if their centroid‐effective radius ratio, x̄/R, is <1.6, thereby verifying an estimate of Saffman and Szeto. This causes the FAVR’s to approach at an exponential rate, merge, and reform into a stable perturbed elliptical structure with filamentary arms (to conserve angular momentum). For larger x̄/R ratios, regular perimeter oscillations were observed and estimates of an eigenfrequency of the perturbed stable V states were obtained. When regions of different vorticity density merge, the larger‐density region is eventually entrained within the smaller‐density region. These simulations elucidate the self‐consistent close interactions of isolated vortex regions in two‐dimensional high Reynolds number flows.

One‐dimensional turbulent dispersion in convergent and divergent flows
View Description Hide DescriptionA theory is presented that gives an analytical solution to the kinematic interaction of a convergent (or divergent) mean flow and turbulence for spatially constant convergence and homogeneous turbulence of a specified type. The turbulence is characterized by its mean square velocity, ∼(v ^{2}), and by its Lagrangian autocorection function. The problem is formally that of the Langevin equation, and analytical solutions for the time dependence of the ensemble‐average variance of many fluid tracers are presented. These solutions are compared with numerical solutions obtained by calculating the Lagrangian motion of a large number of tracers.

Instabilities induced by variation of Brunt–Väisälä frequency in compressible stratified shear flows
View Description Hide DescriptionThe stability characteristics of a Helmholtz velocity profile in a stably stratified, compressible fluid in the presence of a lower rigid boundary are studied. A jump in the Brunt–Väisälä frequency at a level different from the shear zone is introduced and the variation of the Brunt–Väisälä frequency with respect to the vertical coordinate in the middle layer of the three‐layered model is considered. An analytic solution in each of the layers is obtained, and the dispersion relation is solved numerically for parameters relevant to the model. The effect of shear in the lowermost layer of the three‐layered model for a Boussinesq fluid is discussed. The results are compared with the earlier studies of Lindzen and Rosenthal, and Sachdev and Satya Narayanan. In the present model, new unstable modes with larger growth rates are obtained and the most unstable gravity wave modes are found to agree closely with the observed ones at various heights.

Application of the Wiener–Hermite expansion to turbulence of moderate Reynolds number
View Description Hide DescriptionThe Wiener–Hermite expansion with a renormalized, time‐dependent moving base is used to calculate the evolution of the energy spectra of several homogeneous, isotropic flows. The results compare well with laboratory experiments and with numerical simulation experiments. Wiener– Hermite calculations have been done previously for high Reynolds numbers by Hogge and Meecham [J. Fluid Mech. 85, 325 (1978)]with some success. In this work moderate Reynolds numbers are treated.

Analysis of the decay of temperature fluctuations in isotropic turbulence
View Description Hide DescriptionThe Lagrangian dispersion theory of Durbin is used to analyze experiments by Warhaft and Lumley [J. Fluid Mech. 88, 659 (1978)] and by Sreenivasan e t a l. [J. Fluid Mech. 100, 279 (1980)] on temperature fluctuations in grid‐generated turbulence. Both theory and experiment show that the decay exponent m depends on the ratio of the initial length scales of velocity and temperature; although when this ratio is greater than 2.5 such dependence is negligible. The theory shows that m is not truly constant, but within the range covered by the experiments it is nearly so. The agreement between theory and experiment lends credence to the idea that the decay of fluctuations is controlled largely by turbulent relative dispersion.

Some aspects of strongly heated turbulent boundary layer flow
View Description Hide DescriptionRayleigh scattering and laser Doppler velocimetry were used to study density and velocity statistics in a strongly heated turbulent boundary layer with free stream velocity U _{∞} = 19 m/sec and wall temperature T _{ w } = 1100 °K. Mean and rms fluctuation profiles of density and velocity were found to be self‐preserving, and the velocity profiles were not significantly affected by wall heating. Streamwise velocity profiles, and mean temperature profiles deduced from density data were compared with Townsend’s self‐preserving analysis. It was found that wall heating caused a reduction in Reynolds stress within the thermal layer. This phenomenon was explained with reference to the thermal layer structures observed in high speed schlieren movies.

Resistive wall effect on the stability of planar relativistic Brillouin flow
View Description Hide DescriptionPrevious treatments of relativistic Brillouin flow stability are extended to the case of nonzero anode resistivity, for the case of wave propagation across the equilibrium magnetic field. It is found that transverse magnetic (TM) waves are unstable at frequencies below the equilibrium cyclotron frequency, unlike the case for a perfectly conducting anode. Transverse electric (TE) waves, always oscillatory in the case of a perfectly conducting anode, are damped; an explicit expression for the damping rate, valid for large conductivities, is given.

Effect of translational temperature anisotropy on the rotational distribution of N_{2} in the Ar free jet
View Description Hide DescriptionThe effect of the translational temperature anisotropy T _{∥}≳T _{⊥} on the rotational distribution y _{ j }, temperature T _{ r }, and collision number Z _{ r } of N_{2} in the Ar free jet is studied at the lower density level P _{0} d = 15 Torr‐mm using the ellipsoidal velocity distribution and the rotational transition cross section of the power law. The effect of T _{∥}≳T _{⊥} is appreciable. The overpopulation of the higher rotational levels j above the Boltzmann distribution fitted to the distribution of the lower j, i.e., the degree of nonequilibrium for the case T _{∥}≳T _{⊥} = T _{ i } (i≡isentropic) is slightly less pronounced than for the case T _{∥} = T _{⊥} = T _{ i } but more pronounced than for the case T _{∥} = T _{⊥}≳T _{ i }. The ratio y _{ j }/y _{0}, T _{ r }, and the terminal Z _{ r } for T _{∥}≳T _{⊥} = T _{ i } are larger than for T _{∥} = T _{⊥} = T _{ i } but smaller than for T _{∥} = T _{⊥}≳T _{ i }.

A generalized kinetic energy principle
View Description Hide DescriptionUsing three single‐particle adiabatic invariants, we derive an energy principle which generalizes that for the usual guiding center plasma in order to describe the low‐frequency stability of a plasma containing an energetic nonhydromagnetic component (such as the annular electrons in an Elmo Bumpy Torus device).

A boundary‐coupled generalization of the Newcomb stability criterion
View Description Hide DescriptionIn magnetohydrodynamic problems in which there are boundaries in the axial or azimuthal symmetry directions, the energy principle involves multiple nonorthogonal trial perturbations. The subsequent Euler–Lagrange coupling of these allowed excitations necessitates a generalization of the Newcomb necessary‐and‐sufficient stability criterion. This extension, in which the absence of a conjugate point (instead of a simple zero) provides stability, is described in this paper. An important aspect of the proof involves the treatment of certain asymmetries which arise in the presence of the anisotropy caused by the magnetic field. The general method described here has applications to finite‐length axial laboratory pinches and to astrophysical plasmas with rooted magnetic fields.

A sufficient condition for stability of axisymmetric collisionless systems
View Description Hide DescriptionA sufficient condition for the stability of an axisymmetric mirror machine is obtained using the guiding center plasma model. The criterion is particularly suited to configurations with comparable transverse and longitudinal length scales, so‐called short, fat equilibria. Under this ordering, the stabilizing effects of plasma compression is significant. These effects are included through the use of a positive definite lower bound on their contribution to an energy principle.

Free boundary field‐reversed configuration (FRC) equilibria in a conducting cylinder
View Description Hide DescriptionHighly elongated field‐reversed configuration (FRC) equilibria are computed in a straight conducting cylinder for the pressure profile p′(ψ) = c H(ψ), where H(x) is the Heaviside function. The equilibria are found by inverting the Grad–Shafranov equation by means of a Green’s function and by solving the resulting nonlinear integral equation. Long equilibria are obtained only for values of the constant c very near a critical value: the equilibria change from 2:1 elongated to infinitely long as c varies by only 0.3%. This criticial value of c is predicted by the average beta condition.

Ideal magnetohydrodynamic stability limits of a high‐β tokamak with superimposed helical fields
View Description Hide DescriptionThe stability of a high‐β tokamak with a superimposed single helicity stellarator field is investigated in the context of the sharp boundary surface current model. The analysis treats external ideal magnetohydrodynamic modes including both pressure‐driven ballooning effects and current‐driven kink instabilities. The results show that the addition of an l = 2 field is unfavorable, lowering the allowable Ohmic heating current and decreasing the maximum stable β. The addition of an l = 3 field, however, is favorable. Both the allowable Ohmic heating current and maximum stable β are increased as the l = 3 amplitude increases.

Stabilizing effect of a normal magnetic field on the collisional tearing mode
View Description Hide DescriptionA set of coupled eigenmode equations in a current sheet with a normal magnetic field is solved numerically for the collisional tearing mode. It is found that the normal magnetic field has a stabilizing effect. Furthermore, in the case of a very weak normal component, the classical results of Furth, Killeen, and Rosenbluth are recovered.

Parametric instabilities of electron cyclotron waves in plasmas
View Description Hide DescriptionThe parametric instabilities of electron cyclotron waves in (i) the EBT ring and (ii) the large tokamaks, e.g., PLT, have been studied. In EBT, the electron cyclotron pump of finite wavenumber k _{0} decays into two Bernstein modes at the second harmonic cyclotron layer and can account for the heating of the ring in the initial phase. The coupling coefficient for this decay vanishes for a dipole pump, whereas the convective threshold with finite k _{0} is approximately 200 W/cm^{2}. For large tokamaks, the convective threshold for various decay channels turns out to be approximately 200 kW/cm^{2} at 3 mm wavelength.

The direct‐interaction approximation and statistically steady states of three nonlinearly coupled modes
View Description Hide DescriptionThe direct‐interaction approximation is used to find statistically steady states of a system of three modes with complex frequencies coupled by a quadratic nonlinearity. These states are compared to the exact predictions of an ensemble of realizations with Gaussian distributed initial conditions. The direct‐interaction approximation is shown to be reasonably successful in this context.

Statistics of charged particles in external random longitudinal electric fields
View Description Hide DescriptionCharged particle diffusion in a given one‐dimensional Gaussian random longitudinal electric field is investigated by three different methods; quasilinear theory, resonance broadening theory, and direct numerical integration of trajectories. Analytical predictions of both theories are expressed using a random walk approach. An analytic frequency and wavenumber spectrum of constant shape but variable amplitude, related to experimental observations, is used for comparison between the three methods. They all agree at low amplitude. At larger amplitude, resonance broadening theory shows the right tendency for the maximum diffusion coefficient to increase less rapidly than the mean square field. However, the exact value diverges from quasilinear and resonance broadening predictions as soon as the product of the quasilinear correlation time multiplied by mean bounce frequency increases above 0.4.

Wave heating models for ion‐cyclotron heating in ELMO Bumpy Torus‐Scale
View Description Hide DescriptionWave heating of the ELMO Bumpy Torus‐Scale (EBT‐S) experiment in the ion‐cyclotron range of frequencies will be strongly influenced by the geometry of the plasma. In particular, the short finite length of the mirror sections means that the electron bounce frequency is of comparable magnitude to the ion‐cyclotron frequency. Consequently, the bouncing motion of trapped particles impacts the electron absorption of wave energy. Furthermore, the varying magnetic field strength along magnetic field lines influences the ion‐cyclotron absorption of waves because the ion‐cyclotron resonance conditions are satisfied only at discrete points along the field lines. Expressions are given for trapped and passing electron absorption as well as ion‐cyclotron absorption. A numerical example is also discussed.

Analytic solution of the two‐dimensional Fokker–Planck equation governing stochastic ion heating by a lower‐hybrid wave
View Description Hide DescriptionThe two‐dimensional Fokker–Planck equation describing the ion motion in a coherent lower‐ hybrid wave above the stochasticity threshold is analytically solved. An expression is given for the steady‐state power dissipation.