Index of content:
Volume 25, Issue 3, March 1982

A generalization of the Rayleigh–Plesset equation of bubble dynamics
View Description Hide DescriptionThe classical Rayleigh‐Plesset equation of spherical bubble dynamics in an incompressible liquid is generalized to include the non‐Newtonian behavior of the liquid and mass exchange processes at the bubble interface.

Effect of long internal waves on the evolution of deep‐water surface gravity waves
View Description Hide DescriptionEquations describing the resonant interaction between long internal waves and short surface waves are derived and represented in wavenumber space. Various evolution equations can arise depending on the slopes of the two wave fields and the ratio of their wavelengths. The existence of the internal waves will affect the stability of a short surface plane wave. A coupled set of transport equations is derived for deterministic internal waves and Gaussian random surface waves. The amplification rate of perturbations can be analyzed for a range of spectral bandwidths.

Nonlinear two‐dimensional sail theory
View Description Hide DescriptionSteady two‐dimensional flow past a sail is considered. The sail is assumed to be supported by two masts. The flow and the shape of the sail are determined as functions of the direction α of the flow at infinity and the Weber number λ. The full nonlinear problem is formulated as an integrodifferential equation for the shape of the sail. This equation is discretized and solved numerically by Newton’s method. Sail profiles, the slack in the sail, and the lift coefficient are presented for various values of α and λ.

Potential flow signature of a turbulent spot
View Description Hide DescriptionHot‐wire velocity measurements in the free stream above turbulent spots generated in a laminar boundary layer on a flat plate show that the regions of largest positive and negative longitudinal potential flow disturbance occur at about 1.5 spot heights above the surface. These regions are located above the forward and rear primary foci previously identified as points of accumulation for particle paths and as extremal points for ensemble mean velocity and temperature perturbations. A three‐dimensional linearized quasisteady potential flow similarity solution for the flow field around a moving, growing bump on the wall, with a shape given by the measured unsteady boundary layer displacement thickness, exhibits the same main features as the measured potential flow field.

Structure of linearized Navier–Stokes equations for compressible flows
View Description Hide DescriptionThe transition from slightly viscous to viscosity‐dominated steady‐state, compressible, small‐disturbance flow is studied in the framework of the linearized Navier–Stokes equations. The classical picture of independent modes for vorticity, pressure, and temperature disturbances is preserved at all disturbance wavelengths down in the molecular scale. Particular transitions do, however, occur within the pressure mode itself, causing the governing equations to display new and qualitatively different features at the smaller disturbance wavelengths.

A general class of self‐similar power‐driven expansions
View Description Hide DescriptionSelf‐similar solutions for the nonadiabatic expansion of a gas in planar, cylindrical, and spherical geometry due to heating of a fairly general form are presented. The solution for an isothermal expansion which has previously been studied in the literature is recovered as a special case. Some additional examples of physical interest which reveal the variety of possible gas profiles resulting from the general solution are discussed.

Current‐induced instabilities in rotating hydromagnetic flows between concentric cylinders
View Description Hide DescriptionThe stability of hydromagnetic flow produced in a thin annular region between concentric cylinders by the interaction of a superimposed radial current and an axial magnetic field is examined both when the bounding cylinders are stationary and when rotating. After deriving the small gap approximation equations for the steady‐state one‐dimensional azimuthal flow and for the corresponding induced electric potential, the equations governing the growth of infinitesimal axisymmetric disturbances are obtained and solved at neutral stability by use of the Galerkin method. Stability results for the nonrotating cylinder case are shown to be expressible in terms of existing results for the stability of hydromagnetic flow produced by an azimuthal pressure gradient. It is found, in our thin film approximation, that the radial current must generally exceed a certain critical value before the one‐dimensional flow breaks down into a three‐dimensional pattern. Rotation of the outer cylinder enhances stability while rotation of the inner cylinder produces a greater tendency toward instability as reflected by lower Dean numbers at the onset of secondary flow.

An analytical model of a two‐phase liquid metal magnetohydrodynamic generator
View Description Hide DescriptionAn analytical model of a two‐phase liquid‐metal magnetohydrodynamic generator is developed. The model is based on the averaging equations governing the two‐phase flow in the channel of a liquid‐metal magnetohydrodynamic generator. The inhomogeneity of the distribution of the gas phase over the channel cross section is taken into account by introducing the correlation coefficients β_{1}, β_{2}, and β_{3}: 〈αu〉 = β_{1}〈α〉〈u〉, 〈σu〉 = β_{2}〈σ〉〈u〉, 〈ρu〉 = β_{3}〈ρ〉〈u〉. Expressions for these coefficients in terms of the correlation coefficient β_{1} are obtained. The calculated characteristics of the generator were compared with experimental data obtained at the Argonne National Laboratory. The comparison shows that the calculated and measured characteristics are in fairly good (±15%) agreement.

Effects of an inhomogeneous impurity distribution in a field‐reversed theta pinch
View Description Hide DescriptionA 140‐kJ theta‐pinch device originally constructed for spectroscopic investigations of highly ionized atoms is described. Atoms of solid elements are introduced into the initial plasma by laser‐driven ablation from solid targets. The stability of the field‐reversed plasma configuration is investigated with and without impurities. An inhomogeneous impurity distribution leads to the ejection of the plasma as a whole. This phenomenon is interpreted as the result of asymmetric field line reconnection at the ends.

Study of driven lower‐hybrid waves in the Alcator A tokamak using CO_{2} laser scattering
View Description Hide DescriptionMeasurements are presented of the properties of lower‐hybrid waves excited at the edge of plasmas in the Alcator A tokamak by a two‐waveguide array operating at microwave powers up to 70 kW. The waves are studied directly in the plasma by measuring the small angle scattering of CO_{2} laser radiation. A broad N _{∥} spectrum is observed in the range 2⩽N _{∥}≲8, with more intensity at lower N _{∥} (here, N _{∥} is the parallel index of refraction). The waves are distributed throughout the minor cross section and do not appear to be localized in resonance cones. The wave amplitude is linear in net microwave power into the plasma. A power‐independent frequency width of several mega‐Hertz and a downshift in the frequency of the waves of the order of 1 MHz are observed. Finally, the wave amplitude is observed to be independent of the relative phase of the exciting, two‐waveguide array. Most of these results are explained by taking into account the scattering of the lower‐hybrid waves from the large‐amplitude, low‐frequency density fluctuations previously observed near the plasma edge.

Tearing mode stability of elongated field‐reversed plasmas without toroidal field
View Description Hide DescriptionThe marginal stability of long thin field‐reversed plasmas to axisymmetric tearing modes is examined. It is found that stability improves greatly as flux is excluded from the plasma. Such naturally occurring states may account for the lack of tearing mode activity in many experiments. Consideration is also given as to whether, in three dimensions, the tearing mode is an instability or a condition for the breakdown of equilibrium. An example is given that demonstrates that it is an instability. Idealized analytic results and more realistic numerical results are presented.

Electromagnetic kinetic toroidal eigenmodes for general magnetohydrodynamic equilibria
View Description Hide DescriptionA comprehensive analysis of low‐frequency, high‐toroidal‐mode‐number linear eigenmodes for tokamaks is presented. The most significant new features of this stability study are that the calculation is interfaced with a general numerical magnetohydrodynamic equilibrium, and that it is fully electromagnetic. The ballooning formalism is employed and all important kinetic effects, including those of trapped particles, are retained. In particular, the familiar trapped‐electron drift‐wave frequency regime is considered and results are presented for three sequences of artificial equilibria; one of increasing β(≡plasma pressure/magnetic pressure) values; one of varying equilibrium shape, from inverse‐D to circular to normal‐D; and one of increasing vertical ellipticity. The analysis is then applied to a realistic (self‐consistent) high‐β equilibrium generated with data obtained from the ISX‐B tokamak experiment. Here it is found that for the usual trapped‐electron branch, kinetic microinstabilities appear to be present over a wide range of toroidal mode numbers.

Stochasticity and the random phase approximation for three electron drift waves
View Description Hide DescriptionThe interaction of three nonlinearly coupled drift waves is investigated for the occurrence of stochastization of the phases and the applicability of the random phase approximation. The drift wave nonlinearities include the E×B and polarization drift couplings for waves that are linearly unstable for appropriate values of the perpendicular wavenumber. The conservation properties and sample numerical solutions for the exact three wave interaction are given along with the conservation properties and solutions of the corresponding random phase approximation equations.

Nonlinear gyrokinetic equations for low‐frequency electromagnetic waves in general plasma equilibria
View Description Hide DescriptionA nonlinear gyrokinetic formalism for low‐frequency (less than the cyclotron frequency) microscopic electromagnetic perturbations in general magnetic field configurations is developed. The nonlinear equations thus derived are valid in the strong‐turbulence regime and contain effects due to finite Larmor radius, plasma inhomogeneities, and magnetic field geometries. The specific case of axisymmetric tokamaks is then considered and a model nonlinear equation is derived for electrostatic drift waves. Also, applying the formalism to the shear Alfvén wave heating scheme, it is found that nonlinear ion Landau damping of kinetic shear‐Alfvén waves is modified, both qualitatively and quantitatively, by the diamagnetic drift effects. In particular, wave energy is found to cascade in wavenumber instead of frequency.

The high‐β universal drift mode
View Description Hide DescriptionA linear kinetic analysis of the universal drift mode, valid for arbitrary values of β( = 8πp/B ^{2}), is presented. The three high‐β effects on the universal, ∇B drifting particles, coupling to the drift Alfvén wave and compressional magnetic fluctuations are examined in detail. For small values of β, the drift wave behavior is dominated by coupling to the Alfvén wave, while for β∼O(1) the wave behavior is determined by ∇B effects. For most plasma and wave parameters, the mode is stable for β_{ i }⩾7%. The stabilization mechanism is the Alfvén induced frequency downshift which reduces the parallel phase velocity of the wave until the (slightly) ∇B broadened ion Landau resonance damps the wave. For η_{ e }( = ∂ ln T _{ c }/∂ ln N)≳0, a peculiar case is found where the mode remains unstable for β_{ i }⩽0(50%). This is caused by the ∇B frequency upshift coupled with inverse transit time damping.

Axisymmetric magnetohydrodynamic equilibria in local polar coordinates
View Description Hide DescriptionThe Grad–Shafranov equation for an ideal magnetohydrodynamic axisymmetric toroidal configuration is solved analytically in a local polar coordinate system using a novel method which produces solutions valid up to the second order in the inverse aspect ratio expansion.

Establishment of magnetic coordinates for a given magnetic field
View Description Hide DescriptionA method is given for expressing the magnetic field strength in magnetic coordinates for an arbitrary toroidal, scalar pressure, equilibrium with magnetic surfaces. The field strength in magnetic coordinates is central to the study of equilibrium, stability, and transport in asymmetric plasmas. While doing these calculations, it is assumed that the plasma equilibrium is known.

Reduction of transport in a tokamak reactor by externally driven turbulence
View Description Hide DescriptionThe possibility of curtailing trapped particle modes and reducing ripple diffusion in a tokamak reactor by external turbulence is investigated. A ’’classical’’ analysis of the problem shows that the power level required is low for ripple detrapping, but it is very high for suppression of trapped particle modes. If, however, strong turbulence effects including clump formation are considered, periodically pulsed turbulence can be used to eliminate trapped particles and significantly reduce the cross‐field diffusion at an average power level which may be acceptable in a reactor.

Mode coupling and anomalous dissipation in magnetohydrodynamic turbulence
View Description Hide DescriptionAn energy‐conserving model of magnetohydrodynamic turbulence is described that predicts both mode coupling and turbulent dissipation. The dissipation takes the form of anomalous resistivity and viscosity due to turbulent magnetic fields. The model predicts a dual cascade of energy to large wavenumbers and magnetic flux to small wavenumbers. Turbulent rearrangement of equilibrium magnetic shear generates resonant fluctuations via a mixing length process. The effect of the turbulence on the disruptive instability in tokamaks is discussed.

Effects of electrostatic trapping on neoclassical impurity transport in a collision‐dominated plasma
View Description Hide DescriptionIt has been shown by Hazeltine and Ware that the existence of a reasonable high‐Z impurity species can produce electrostatic trapping and drift comparable to the magnetic effects, assuming that the main plasma components are in the plateau regime. The calculation is extended to the collisional regime, modeling the edge region of the tokamak column and significant modification to the neoclassical impurity transport coefficients is found. A fluid approach is developed and the transport coefficients are found to be nonlinear in density and temperature gradients.