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Volume 25, Issue 1, January 1982

Motion of a contact line
View Description Hide DescriptionA multiscale expansion approach is applied to the solution of the free‐boundary Stokes’ problem describing flow in the vicinity of a moving contact line. A solution free of singularities is obtained for the case of a liquid advancing into an inviscid medium. The force singularity reported in earlier studies is relaxed as the dynamic contact angle approaches π in the immediate vicinity of the moving contact line. This solution formally breaks down at a nonvanishing viscosity ratio of the receding and advancing fluids, but it still holds approximately for the case of a volatile liquid advancing into gas when Stefan flow in the gas phase is taken into account.

Convective flow with subcritical instability
View Description Hide DescriptionAn asymptotic analysis of subcritical instability in double diffusive convection is presented. Using a modified perturbation method, a Landau equation that determines how the amplitude of the convection evolves in time is derived. From the Landau equation, it is found that in certain cases, stable finite amplitude convection can exist even when the rest state with no flow is locally stable. The perturbation analysis complements and unifies previous work which is primarily qualitative or numerical in character.

A linear analysis of energy budgets in stratified viscous free shear layers
View Description Hide DescriptionThe presence of shear instability can lead to the generation of turbulence in a stably stratified fluid. Using linear hydrodynamic stability theory, a turbulent kinetic energy budget is computed for an unbounded viscous shear layer for which the hyperbolic tangent shear layer is used to define the mean flow. For neutrally stable flows, the turbulent production due to the action of the Reynolds stress on the mean shear flow is balanced by viscous dissipation and the conversion to potential energy through the buoyancy term. The calculations show that maximum production of turbulent energy occurs at the center of the shear layer and that some of this energy is then transported to the edges of the shear layer where viscous dissipation occurs and some of the energy is converted to potential energy near the center of the flow. As the horizontal wavenumber decreases, the viscous dissipation becomes more important than the buoyancy term in removing energy from the flow. As the Reynolds number increases, the production term is concentrated over a narrower region at the center of the shear layer. Streamlines for neutral and stable conditions are presented to show the flow patterns of the disturbances in the x–z plane. Some possible atmospheric and oceanic applications are suggested in the last section.

Investigation of the frozen‐turbulence hypothesis for temperature spectra in a convectively mixed layer
View Description Hide DescriptionTaylor’s frozen turbulence hypothesis is investigated in a laboratory free‐convection mixed layer which can simulate buoyancy‐driven turbulence in an atmospheric mixed layer (of height h) for mean wind speeds of up to several meters per second. For large turbulence intensities the temperature spectra at the height 0.1h are found to be spuriously enhanced at higher wavenumbers based on the frozen turbulence hypothesis, as deduced theoretically by previous investigators. The excess is borrowed from spectral intensities within the energy‐containing range. A theory based on the concept of longitudinal‐temporal isotropy, after proper scaling of the coordinates, is shown to predict a simple shift of the entire spectrum toward higher wavenumbers with increasing turbulence intensity. The failure of the observed spectra to behave this simply is associated with a complicated structure of the correlation coefficient as a function of longitudinal‐temporal lag coordinates when the former is measured relative to a frame moving with the mean flow speed. The temperature spectrum based on the frozen‐turbulence hypothesis appears to be a satisfactory representation of the true spectrum in wavenumber space, in tubulence dominated by thermal convection, if the mean wind speed exceeds 2.7–3.6 times the root‐mean‐square horizontal velocity fluctuation.

Reynolds number dependence of velocity structure functions in turbulent shear flows
View Description Hide DescriptionMoments, up to order eight, of the structure function of the steamwise velocity fluctuation, have been measured in both laboratory and atmospheric turbulent shear flows. The Reynolds number dependence of structure functions evaluated for a separation equal to the Taylor microscale is closely approximated by both lognormal and β models, at least for moments up to order six. Both models predict identical inertial subrange behavior for the sixth‐order structure functions. This prediction is in good agreement with experiment when the value of 0.2 is used for the universal exponent μ. This value has been obtained from correlations of the squared velocity derivative fluctuations, calculated for both laboratory and atmospheric measurements over the inertial subrange. The slope of the correlation is related to that of the sixth‐order structure function, in agreement with a conjecture by Frisch, Sulem, and Nelkin. When plotted against moments of order n+2, moments of even order n exhibit a power‐law behavior whose exponent is in closer agreement with the lognormal than the β model.

Static and dynamic impulses generated by two‐phase detonations
View Description Hide DescriptionThe influence of the mixture density and the reaction zone length on the static and dynamic impulse of the detonations through wheat dust, RDX dust, and decane droplet air mixtures was studied. A numerical solution of the detailed model of the two‐phase detonation initiation and development in the medium, provided the basis for the study. Mixtures with a higher average density were found to produce a significantly greater static impulse, but a reduced dynamic impulse. The reasons for this behavior were discussed.

Effect of overtaking disturbances on the motion of a shock wave due to an intense explosion
View Description Hide DescriptionThe Chester, Chisnell, and Whitham approximation has been applied to the Sedov problem of an explosion in a medium of uniform density. It is noted that the error in the propagation parameter K is 70% or more. The reason for this discrepancy has been explained.

Highly supersonic ion pulses in a collisionless magnetized plasma
View Description Hide DescriptionThe initial transient response of a collisionless plasma to a high positive voltage step is investigated. Four different pulses are observed. An electron plasma wave pulse is followed by an ion burst. The latter is overtaken and absorbed by a highly supersonic ion pulse. Thereafter, an ion rarefaction pulse with roughly the ion acoustic velocity propagates into the other direction. The supersonic pulse is tentatively explained as the result of a transient Buneman instability. Together with the rarefaction pulse, it forms the first cycle of an instability which is customarily considered a current‐ driven ion acoustic one, but is, in fact, a potential relaxation instability.

Curvature‐induced interchange mode in an axisymmetric plasma
View Description Hide DescriptionA theoretical and experimental description of the curvature‐induced electrostatic interchange mode in a simple mirror confined low‐β plasma is given. The frequency, growth rate , and nonflute‐like effects have been measured and compared with theory. Effects due to radial electric field, finite ion Larmor radius, line‐tying, wall radius, and parallel electron response are discussed.

Formation and dynamics of a rotating proton ring in a magnetic mirror
View Description Hide DescriptionExperimental results are presented on the formation and dynamics of a rotating energetic proton ring in a magnetic mirror field. An annular 430 keV proton beam from a magnetically insulated diode is injected through a cusp‐like magnetic field to form a rotating proton beam in a 2 m long solenoidal magnetic field. With 15–400 mT of neutral gas in the 40 cm diam experimental chamber up to 83% of the 350 J total beam energy is in rotation, the axial velocity dispersion is small, and the beam is sharply defined radially with inner and outer radii of 7 and 13 cm, respectively. The beam is 90%–100% axially current neutralized by currents induced in the beam‐generated plasma. Azimuthal plasma currents are observed in air, but not in hydrogen. In hydrogen a ring containing 5×10^{15} protons is formed with sufficiently small axial velocity dispersion so that the protons are confined axially by their own 3% diamagnetic well as they propagate about 1.3 m in an 8 kG solenoidal field. In air, up to 15% of the beam axial energy is inductively coupled to the induced plasma currents, and ≳50% reflections are achieved from a 1.23 mirror ratio magnetic mirror located 2 m from the ion injector. About 1% inductive energy coupling to a wall resistor array was observed in both air and hydrogen. A peak ring diamagnetism of 875 G was observed 50 cm from the injector at a point where the applied field was 11 kG (near the peak of an upstream mirror); this ring contained about 1×10^{16} protons.

The Tormac V experiment
View Description Hide DescriptionTormac (T o r o i d a l M a g n e t i c C u s p) is a plasma confinement concept combining the favorable magnetohydrodynamic stability properties of a cusp geometry with the good particle confinement inherent to closed field geometry. A conceptual Tormac plasma has two regions; an interior region in which a toroidal bias or stuffing field is embedded, and an exterior or surface region confined by mirror trapping along open field lines. The combination of these two regions is expected to lead to a configuration having confinement substantially superior to that of a mirror, and to allow the plasma to be stable to high β. The Tormac V experiment is an attempt to establish such a configuration and to investigate the characteristic behavior of the Tormac plasma. The Tormac concept, the Tormac V experimental setup, and the results are described.

Evaluation of multipole moments over the current density in a tokamak with magnetic probes
View Description Hide DescriptionZakharov and Shafranov’s method for deriving plasma displacement and averaged shape in a tokamak from magnetic probes positioned along a contour around the plasma cross section is extended in several ways. Formulas are derived which allow vertical as well as horizontal displacements with any ratio of minor contour width to major radius, asymmetric as well as symmetric multipole moments, and explicit relations for rectangular as well as circular contours. The relations in the literature for a circular contour are also re‐examined and corrected equations are given. It is shown that a rectangular contour can provide the same information.

Wall losses for a single‐species plasma near thermal equilibrium
View Description Hide DescriptionThe effect of a radially bounding wall on a magnetically confined single‐species plasma near thermal equilibrium is considered. Solutions to the like‐particle collisional transport equation are obtained; the boundary conditions at the wall follow from simple physical considerations. Integral constraints on the plasma evolution imply that only a fraction of the plasma can ever be lost to the wall. Analytic estimates and numerical solutions give the scaled wall flux in terms of the unperturbed equilibrium density at the radius of the wall.

Reconnection conditions for a coaxial plasma gun
View Description Hide DescriptionA fluid model for the flow conditions necessary to form a compact torus from the plasma ejected by a coaxial plasma gun is developed. This is done by finding the conditions for which the steady‐flow equations break down. Results are found for two cases; variable external flux and variable outer‐wall radius.

Relaxation toward states of minimum energy in a compact torus
View Description Hide DescriptionA finite‐difference, resistive, magnetohydrodynamic code is used to follow the long‐time evolution of decaying nonequilibrium states inside a rigid, perfectly conducting cylindrical boundary. The energy‐to‐magnetic helicity ratio decays toward a minimum, in accord with a conjecture of Taylor. The magnetic Reynolds number is considerably higher than the mechanical Reynolds number for the regime considered. The energy, which is mostly magnetic, tends to decay in bursts associated with current filamentation and magnetic reconnection.

Tilt and shift mode stability in spheromaks with line tying
View Description Hide DescriptionMagnetohydrodynamic stability of force‐free spheromak plasmas to n = 1 free boundary tilt and radial shift modes is studied. The exterior region of open field lines is treated either as a conducting plasma or as a vacuum. In the former case the effect of line tying of the open field lines is present and is found to improve stability greatly. The equilibria which have optimum stability properties to the tilt and shift modes have a length‐to‐diameter ratio of 0.6, and can be stable with conducting walls at a relatively large distance.

Long time plasma equilibrium
View Description Hide DescriptionPlasma equilibrium with dissipation is governed by more equations than the one used for stability problems in ideal magnetohydrodynamics. Consequently, as transport processes became more important in fusion experiments, it is necessary to determine whether an equilibrium will eventually be reached after a long time at the end of the transport processes. A study is made of a cylindrical plasma with an ohmic heating current as an approximation to a large aspect‐ratio tokamak. For comparison, both single‐fluid theory and two‐fluid theory are used. It is found that no equilibrium exists in the single‐fluid theory with a plasma temperature dropping to zero at a finite radius, while such an equilibrium does exist in two‐fluid theory provided that the electron temperature is higher than twice the ion temperature. Both the low‐pressure and high‐pressure cases are studied in the tokamak parameter range. Analytical results for the plasma behavior near its edge are also obtained.

Electrostatic modification of variational principles for anisotropic plasmas
View Description Hide DescriptionThe kinetic energy principle is modified by the presence of an equilibrium electrostatic potential and the inclusion of the quasineutrality condition. By identifying terms that are fluid‐like (local) and kinetic (nonlocal), one can express the change in potential energy, ΔW, in a form which is both variational and minimizing and also displays, clearly, the electrostatic effects on the various free energy sources. The kinetic energy principle determines the stability of the plasma to perturbations that grow on the time scale associated with the streaming of particles along field lines. Recently, a new variational principle has been proposed which determines the stability of the plasma to perturbations that grow on the time scale associated with the drifting of particles across field lines. The modifications of this principle due to electrostatic effects will also be considered, and cases in which necessary and sufficient conditions for instability can be found will be discussed.

Banana drift transport in tokamaks with ripple
View Description Hide DescriptionRipple transport in tokamaks is discussed for the ’’banana drift’’ collisionality regime, which lies below the ripple plateau regime treated earlier. The physical mechanisms that dominate banana drift transport are found to differ from those considered in previous work on this regime, and consequently the resulting transport coefficients can differ by several orders of magnitude.

Thermal equilibrium and stability of an Ohmically heated plasma
View Description Hide DescriptionSteady‐state temperature profiles in a magnetically confined tokamak type plasma are derived by balancing Ohmic power input, thermal diffusion, and radiation. A comprehensive study is presented of the impact that different electron conductivities, K _{⊥} boundary conditions, and radiation may have on thermal stability. In particular, their influence on the bifurcation in the position of the minor radius as a function of the temperature at the center T _{ m } is considered. Points of bifurcation correspond to marginally stable equilibria. However, it is also possible to have both branches without marginally stable equilibrium. Critical values of parameters that cause the onset of a new branch or instability are determined. Growth rates are not calculated, but domains of stabilty of all possible equilibria are determined. It is shown that Ohmically heated plasma in the classical scaling is thermally unstable to convective boundary conditions or if the plasma is cold at the edge. It is shown that a finite temperature at the edge will generate a stable, lower branch, but the upper branch will be unstable as long as p<0.5, where K _{⊥}(T)∼T ^{ p }.