Volume 28, Issue 8, August 1985
Index of content:

Finger breakup in Hele–Shaw cells
View Description Hide DescriptionIt is shown that a boundary integral simulation of fingering in Hele–Shaw cells generates patterns of finger breakup similar to those observed experimentally at low values of reciprocal capillary number τ. It is also shown that, when random noise is introduced in the interface position at a value of τ for which a stable finger would otherwise occur, similar patterns of finger breakup are generated. This supports the hypothesis that the observed behavior is caused by a nonlinear instability driven by experimental or numerical noise at very low values of τ.

Variational formulation for the evolution of a deep‐water wavetrain in two space dimensions
View Description Hide DescriptionThe stability of two‐dimensional perturbations of the two‐dimensional nonlinear Schrödinger equation for deep‐water soliton solutions is investigated. A new variational method is developed that allows calculation of the maximum growth rates close to the cutoff where the conventional variational formulations break down. By a simple numerical evaluation of the variational principle, the growth rate curve is obtained in agreement with previous, purely numerical computations.

Visualization of the structure of a pulsed methane–air diffusion flame
View Description Hide DescriptionExperiments have been carried out in a variable pressure flow facility with the objective of studying the structure of a co‐flowing jet diffusion flame. The flow is visualized using an optical scheme which superimposes the luminous image of the flame on its Schlieren image. This gives a useful picture of the relationship between the bright, yellow‐orange, soot‐laden core flow and the edge of the surrounding hot gas envelope. A loudspeaker is used to force the central fuel jet at several frequencies. In the unforced flow and over most of the driving frequency range in the forced flow, a double structure is observed with two distinct wavelengths: a long wavelength associated with the luminous, buoyancy‐driven core flow and a short wavelength associated with the shear‐driven outer flow. Excitation at the proper frequency causes strong coupling to occur. In this case the core flow pinches off and the flame breaks up into a series of flamelets moving with a single wavelength.

A simultaneous concentration of a vortex and a magnetic field in magnetohydrodynamic flows
View Description Hide DescriptionA three‐dimensional magnetodydrodynamic flow with a combined effect of convection and diffusion is reduced to a two‐dimensional problem by a variable transformation. The original field is composed of a unidirectional vorticity and an aligned magnetic field superimposed on an axisymmetric straining flow in an incompressible, viscous, electrically conducting fluid. By restricting consideration to the axisymmetric vorticity and magnetic fields, an exact solution is presented, which shows time evolution from general initial profiles. In a particular case of constant straining, the solution simultaneously approaches the state of a concentrated cylindrical vortex and a magnetic tube of Gaussian profile sharing a common symmetry axis.

Current enhancement in a conducting channel
View Description Hide DescriptionBy the use of a simple model, it is shown that the transverse motion of a current‐carrying filament in an electrically conducting medium may inductively produce a total current three times as large as that of the filament alone. An explanation of this phenomenon is given, and the implications are explored.

A review of energy confinement and local transport scaling results in neutral‐beam‐heated tokamaks
View Description Hide DescriptionOver the past several years, tokamak neutral beam injection experiments have evolved from the brute force study of the effects of global discharge characteristics (I _{ p }, n̄_{ e }, P _{heat}, etc.) on energy confinement to the appreciation that there are effects more subtle, yet controllable, that may influence confinement dramatically. While this evolution from first to second generation experiments is derived from an empirical understanding of ‘‘low’’ and ‘‘high’’ energy confinement modes and how to achieve them operationally, the underlying physics is still unknown. Several theories with different physical bases appear to describe the global scaling of the low confinement mode discharges quite well. On the other hand, little agreement has been found between theoretical and experimentally deduced values of local transport coefficients. While it is known operationally how to achieve any one of several types of high confinement mode discharges, here too, the underlying physics of the transport associated with these modes is poorly understood.

Stability of rotating liquid drops: II. Homogeneously charged or self‐gravitating drops
View Description Hide DescriptionThe oscillations and instabilities of a uniformly rotating and homogeneously charged or self‐gravitating liquid spheroid with surface tension are investigated. The dispersion relation is obtained by means of an appropriate energy integral better suited to yield the higher modes than the virial method. Criteria are derived for the onset of secular instability that indicate bifurcation toward nonrotational symmetric equilibrium configurations, and of dynamical stability. In the oblate case the validity of the spheroidal approximation can be verified both quantitatively and qualitatively by comparison with two limiting cases: an uncharged or nongravitating rotating drop and a rotating drop without surface tension (astronomical limit). For the prolate spheroids there exists a clear difference between rotating and nonrotating configurations, for the equilibrium as well as for the stability.

The development of a solitary wave in a channel of variable elliptical cross section
View Description Hide DescriptionIn this paper the disintegration of a solitary wave in an elliptical channel is studied when it propagates from one uniform section of the channel into another through a transition section. It is found that for an elliptical channel with constant width, fission of solitons is also possible when the solitary wave moves into a uniform section of larger depth.

Channel cooling by turbulent convective mixing
View Description Hide DescriptionResults from a series of experiments are described which show that hot, reduced‐density channels in the atmosphere usually cool by a process of turbulent convective mixing. Five different types of channels were created: (a) by the interaction of a pulsed CO_{2} laser with aerosols in the atmosphere, (b) by electric discharges in the atmosphere, (c) by laser‐guided electric discharges in the atmosphere, and (d) and (e) by the absorption of CO_{2} laser radiation in nitrogen doped with sulfur hexafluoride. For channels in which the energy deposition was almost cylindrically symmetric and axially uniform, (e), the rate of cooling, after reaching pressure equilibrium, was within an order of magnitude of thermal conduction. But for channels in which the energy deposition was asymmetric and/or axially nonuniform, the rate of cooling was typically one thousand times faster than thermal conduction (for channels whose radius at pressure equilibrium was ∼1 cm). These channels were seen to be turbulent and to cool by mixing cold surrounding air into the hot channel. Such turbulence has been explained by Picone and Boris [Phys. Fluids 2 6, 365 (1983)] in terms of a residual vorticity that is caused by the noncylindrical energy deposition. A simple empirical formula is deduced relating the rate of cooling (growth of channel envelope) to the radius of the channel at pressure equilibrium and to the ambient sound speed, which indicates that the effect of vorticity/turbulence saturates for variations in the energy deposition of greater than about 2 to 1.

Gas flow in vertical slots with large horizontal temperature differences
View Description Hide DescriptionPerfect gas exact solutions to the steady Navier–Stokes equations are given for laminar convective motion in open and closed vertical slots with large temperature differences using Sutherland law transport properties. The solutions are valid a few slot widths away from the ends in the asymptotic region where the opposite hot and cold wall boundary layers are fully merged. It is found that the static pressure (in the closed slot) and temperature and velocity distributions (in all cases) are very sensitive to property variations, even though the heat flux may not be. We observe the net horizontal and vertical heat fluxes to be the same as those obtained from the Boussinesq equations. Comparisons with constant property solutions and the well‐known Boussinesq limiting solution for small temperature differences are given for examples using air.

Finite‐amplitude ion‐acoustic solitons in weakly inhomogeneous plasmas
View Description Hide DescriptionThe properties of an ion‐acoustic soliton in a weakly inhomogeneous plasma are studied. Unlike previous analyses, the soliton amplitude is not required to be small. The ion generation rate is assumed to be either proportional to the electron density or to be uniform. Assuming that the local soliton width is small compared with the scale length of the plasma inhomogeneity, a perturbation theory is developed which gives the local speed and amplitude of the soliton. For a small amplitude soliton, simple expressions for the local soliton speed, peak soliton potential, and peak soliton ion density are derived. It is shown that both the soliton velocity relative to the ion drift velocity and the peak soliton potential do not vary greatly as the soliton moves from the plasma center toward the sheath. The peak ion density, however, varies over a wider range. These results are shown to agree with those from direct numerical integration of the basic equations.

Asymptotic theory of ion conic distributions
View Description Hide DescriptionThe formation of ion conics is discussed in terms of a model which permits an asymptotic determination of the ion distribution function. This model is applicable to a class of observed ion conic events where the ion energization is attributable to lower hybrid wave activity. A two‐stage process is considered wherein the upward moving component of the ambient ion population is first subjected to a region of weak turbulence and is then allowed to drift adiabatically up the geomagnetic field. Identification of the ratio of ion thermal speed to mean wave speed as a small parameter leads to a uniformly valid solution of the quasilinear diffusion equation. The resulting analytic form for the ion conic is then available for further analytic work. Implications and limitations of the model will be discussed.

Statistical theory of cubic Langmuir turbulence
View Description Hide DescriptionThe cubic direct interaction approximation is applied to a truncated (in Fourier space) version of the cubically nonlinear Schrödinger equation model of Langmuir physics. The results are compared (in the three‐mode case) to those for an ensemble of numerical solutions of the dynamical equations with 10 000 different sets of Gaussianly distributed initial conditions. In the undriven, undamped case, the statistical theory (but not the ensemble) evolves to a state of thermal equilibrium. In the driven, damped case, the statistical theory appears to evolve to a state close to that corresponding to one of the limit cycles of the dynamical equations.

Cascade properties of shear Alfvén wave turbulence
View Description Hide DescriptionNonlinear three‐wave interactions of linear normal modes are investigated for two‐dimensional incompressible magnetohydrodynamics and the weakly three‐dimensional Strauss equations in the case where a strong uniform background field B _{0} is present. In both systems the only resonant interaction affecting Alfvén waves is caused by the shear of the background field plus the zero frequency (ω=k ⋅ B _{0}=0) components of the perturbation. It is shown that the Alfvén waves are cascaded in wavenumber space by a mechanism equivalent to the resonant absorption at the Alfvén resonance. For large wavenumbers perpendicular to B _{0}, the cascade is described by Hamilton’s ray equations, d k/d t=−∂ω/∂r, where ω includes the effects of the zero frequency perturbations.

Forced magnetic reconnection
View Description Hide DescriptionBy studying a simple model problem, the time evolution of magnetic field islands which are induced by perturbing the boundary surrounding an incompressible plasma with a resonant surface inside is examined. The reconnection and island formation process for sufficiently small boundary perturbations occurs on the tearing mode time scale defined by Furth, Killeen, and Rosenbluth [Phys. Fluids 6, 459 (1963)]. For larger perturbations the time scale is that defined by Rutherford [Phys. Fluids 1 6, 1903 (1973)]. The resulting asymptotic equilibrium is such that surface currents in the resonant region vanish. A detailed analytical picture of this reconnection process is presented.

Convectively unstable two‐plasmon decay modes in a weakly inhomogeneous plasma
View Description Hide DescriptionThe behavior of the convective two‐plasmon decay instability is studied using a kinetic theory formulation which is applicable to high‐temperature plasmas. Theoretical values for the Langmuir wave amplification are obtained in several limits. The temperature‐dependent effects of Landau damping and kinetic theory corrections to the coupling coefficients are studied. Landau damping significantly reduces the spectrum of unstable modes for T>1 keV, while kinetic corrections become important for T>10 keV. The nonresonant scattering of Langmuir waves by thermal electrons is calculated and is found to be comparable to the resonant convective process for T>20 keV.

Theory of semicollisional drift‐interchange modes in cylindrical plasmas
View Description Hide DescriptionResistive interchange instabilities in cylindrical plasmas are studied, including the effects of electron diamagnetic drift, perpendicular resistivity, and plasma compression. The analyses are pertinent to the semicollisional regime where the effective ion gyroradius is larger than the resistive layer width. Both analytical and numerical results show that the modes can be completely stabilized by the perpendicular plasma transport. Ion sound effects, meanwhile, are found to be negligible in the semicollisional regime.

The electron‐acoustic mode
View Description Hide DescriptionThis paper examines electrostatic modes in an unmagnetized, homogeneous, Vlasov plasma with three Maxwellian components: ions, hot electrons, and cool electrons. In such a plasma, the electron‐acoustic mode with frequencies between the ion and electron plasma frequencies may propagate with light damping. The conditions that allow propagation of this mode, which is distinct from the well‐known ion‐acoustic and Langmuir waves, are given in detail; approximate necessary conditions are 10≲T _{ h }/T _{ c } and 0<n _{ c }<0.8n _{ e }, where the subscripts c, h, and e refer to the cool and hot electron components and the total electron population, respectively.

Finite orbit analysis for long wavelength modes in a plasma with a hot component
View Description Hide DescriptionThe z‐pinch model is used to calculate finite Larmor radius effects of a plasma with a hot component plasma annulus. The equations are analyzed for layer modes and the finite Larmor radius stabilization condition is calculated. Stability requires k ^{2} ρ^{2} _{ h } Rβ_{ h }/Δ≳1, where k is the wavenumber in the z direction, ρ_{ h } the hot species Larmor radius, β_{ h } the hot particle beta, and Δ the thickness of the pressure profile. In addition a new instability is found, caused by the interaction of the precessional modes associated with inner and outer edges of the hot particle pressure profile.

Relativistic Hartree condition for magnetrons: Theory and comparison with experiments
View Description Hide DescriptionThe Hartree voltage U _{H} is derived for a cylindrical magnetron in the relativistic limit with and without an axial current. This voltage corresponds to the breakdown voltage of the magnetron in the presence of a rotating perturbation field. In general, U _{H} is less than the cutoff voltage U _{ I } for magnetic insulation in the absence of a perturbation field. Comparisons with a number of different experiments at zero axial current indicate that the maximum microwave power occurs for voltages slightly larger than U _{H}, but well below U _{ I }. With axial current, two different Hartree voltages are derived, U _{HA} and U _{HB}<U _{HA}, based on different assumptions. Recent experiments at Physics International show little microwave power for voltages less than U _{HB} and high power for voltages at or slightly larger than U _{HA}. The absence of large microwave power outputs at large axial currents may be caused by the finite axial length of the magnetron.