Volume 28, Issue 3, March 1985
Index of content:

Variance in the sedimentation speed of a suspension
View Description Hide DescriptionThe variance in the sedimentation speed for a homogeneous suspension of solid spheres in a Stokes fluid is calculated for a particular choice of the distribution function of the spheres. In the infinite particle number limit, the variance is found to be infinite.

Electrostatic ion‐cyclotron instability caused by a nonuniform electric field perpendicular to the external magnetic field
View Description Hide DescriptionA new mechanism that can destabilize kinetic ion‐cyclotron waves in the presence of a nonuniform electric field perpendicular to the uniform ambient magnetic field is given. In the absence of the electric field, the mode energy is positive, while in the presence of a uniform electric field the mode energy could be negative. However, when the electric field is nonuniform, it is possible for a finite region to be a negative wave energy surrounded by regions of positive wave energy. A nonlocal wave packet couples the two regions so that a flow of energy from the region of negative wave energy to the region of positive wave energy will cause the mode to brow. This gives rise to the instability.

Stimulated Raman up‐conversion of electromagnetic waves by a gyrating electron beam
View Description Hide DescriptionA gyrating electron beam supports negative energy modes near the harmonics of electron‐cyclotron frequency. An electromagnetic wave passing through such a beam parametrically up‐converts into high‐frequency electromagnetic modes separated from the pump frequency by the electron‐cyclotron harmonics. The growth rate for this process varies directly as the oscillatory velocity of beam electrons caused by the pump and as square root of the beam density. It has a maximum at values of scattering angle close to 180° and is also implicitly dependent on the beam velocity and the cyclotron frequency of electrons. The effect of a cold electron component is to reduce the growth rate.

On the Smoluchowski paradox in a sedimenting suspension
View Description Hide DescriptionIt is shown, by explicit calculation, that the influence of a plane wall supporting the suspension on the sedimentation velocity is such that the convergence problems of this quantity encountered in an unbounded suspension do not occur—even in the limit of an infinitely distant wall.

A Korteweg–de Vries equation modified by viscosity for waves in a channel of uniform but arbitrary cross section
View Description Hide DescriptionA KdV equation modified by viscosity is derived for weakly nonlinear long waves propagating in a channel of uniform but arbitrary cross section. The case of high Reynolds number is considered, and the method of matched asymptotic expansion is employed. The equation derived here is found to be similar to the corresponding equation for a two‐dimensional layer of liquid derived by previous authors. The only difference is that the dispersive, nonlinear, and viscous terms are multiplied by constants dependent on the cross section geometry of the channel.

Instability of natural convection in a tall vertical annulus
View Description Hide DescriptionAn experimental study has been made of the hydrodynamic stability of viscous fluid flow contained between the differentially heated walls of a tall vertical annulus. Tests were conducted using different glycerol solutions spanning a range of Rayleigh numbers (10^{4}–10^{6}) and moderately high Prandtl numbers (15–150) for a height/gap ratio of 64 and a radius ratio equal to 0.62. Visualization studies show that the unstable flow consists of two separate progressive wave systems, one ascending the hot inner wall and the other descending the cold outer wall. The approximate Rayleigh number, phase speed, and wavelength at the onset of instability for each wave system were determined for several Prandtl numbers. The regular wave patterns observed near the stability boundary persist over only a very limited range of supercritical Rayleigh numbers above which flow dislocations set in. At higher Rayleigh numbers these random dislocations appear in increasing numbers and ultimately render the flow turbulent.

Oscillatory instability in a Bénard problem of two fluids
View Description Hide DescriptionA linear stability analysis for a Bénard problem with two layers is considered. The equations are not self‐adjoint. The system can lose stability to time‐periodic disturbances. For example, it is shown numerically that when the viscosities and coefficients of cubical expansion of the fluids are different, a Hopf bifurcation can occur, resulting in a pair of traveling waves or a standing wave. This may have application in the modeling of convection in the Earth’s mantle.

Collapse and amplification of a vortex filament
View Description Hide DescriptionA single filament with a variable core size parameter is used to model how a vortex tube breaks down in the Euler equations. The first singularity is a self‐similar collapse which brings two antiparallel pieces of filament together at a point. This pairing then quickly encompasses a finite fraction of the initial data and the arclength begins to grow faster than exponential. A local model should exist which would allow one to understand the stage of rapid stretching in terms of simpler processes.

The motion of a spherical particle suspended in a turbulent flow near a plane wall
View Description Hide DescriptionAnalytical solution of the equations of motion of a spherical particle suspended in a turbulent flow near a plane wall has been obtained. The equations include the lift force and wall effects on the drag force. The solution shows that the particle turbulent motion is affected by the wall presence in the following manner: (a) The wall augments the response of the particle to fluid turbulence. The ratio between the particle rms velocity fluctuation near the wall and that of an identical particle in an unbounded flow is always greater than unity. This ratio increases by increasing the particle density and diameter and decreasing the particle distance from the wall. (b) Wall effects in a direction normal to it are more pronounced than those in the parallel direction. This is attributed mainly to the lift force acting in the normal direction. (c) Effects of the drag force on particle intensity are confined close to the wall whereas the lift effects extend to larger distances.

Simultaneous measurement of velocity and temperature by the use of a laser Doppler velocimeter
View Description Hide DescriptionA simple, new method of measuring velocity and temperature simultaneously without imparting any disturbances to the fluid is developed. The key idea is to use a certain liquid that undergoes thermochromism as a test fluid, the temperature of which can be observed directly by its color change or measured quantitatively by a suitable optical method. If a laser Doppler velocimeter is used, the temperature at the measuring point, in addition to the velocity, can be determined simultaneously and instantaneously by detecting the otherwise discarded intensity of the transmitted light of the laser beams. This method was applied to measurements of distributions of velocity and temperature of a plume in a Hele–Shaw cell. Results were in fairly good agreement with similarity laws derived theoretically in this paper.

Ion‐acoustic solitary waves in relativistic plasmas
View Description Hide DescriptionThis is a sequel to our earlier study on ion‐acoustic waves studied through the augmentation to a modified Korteweg–deVries (K–dV) equation. We have derived a K–dV equation in a plasma, taking account of weakly relativistic effects, and the result shows that the solitary wave does exhibit the relativistic effect in the presence of ion streaming.

Standing envelope solitons in plasmas
View Description Hide DescriptionThe quasistationary slow plasma response to the Langmuir and electromagnetic waves is investigated. Conditions for the wave localization are obtained analytically, and new kinds of soliton profiles are found.

Generation of extraordinary mode radiation by an electrostatic pump
View Description Hide DescriptionIt is shown that an electrostatic wave near the upper‐hybrid resonance frequency can parametrically excite extraordinary (X‐) mode radiation accompanied by a great variety of low‐frequency oscillations. The latter may include the lower‐hybrid, the electron‐acoustic, the ion‐cyclotron, and the shear Alfvén waves. Nonlinear dispersion relations and the growth rates are obtained for each case. Comparison of our investigation to an earlier work and its possible application to space plasmas are pointed out.

Quasistatic heat front and delocalized heat flux
View Description Hide DescriptionNew results concerning the mathematical properties of the Fokker–Planck equation describing the electron distribution function are presented. The validity of the approximations obtained by using a finite number of Legendre polynomials to describe the electron distribution function is discussed. It is shown that, due to the Landau form of the electron‐ion collision operator, it is sufficient to use two or three Legendre polynomials in problems of interest. The theory is applied to the classical albedo problem as a test, and is also applied to determine the distribution and the heat flux in a heat front typical of laser plasma experiments. It is shown that the heat flux can be expressed as a sort of convolution of the Spitzer–Härm heat flux by a delocalization function. The convolution formula leads in a physically relevant way to the saturation and the delocalization of the heat flux.

Induced emission of radiation near 2ω_{ e } by a synchrotron‐maser instability
View Description Hide DescriptionIn the literature, the emission of radiation at 2ω_{ e } , where ω_{ e } denotes the electron plasma frequency, is usually explained as having been produced by the nonlinear interaction of two Langmuir waves via a backscattering process. Since the emission is frequently observed in solar radio bursts, the mechanism has attracted considerable theoretical interest. In the present paper a model is proposed based on a synchrotron‐maser instability excited by a hollow beam of moderately relativistic electrons in a plasma, in which the plasma frequency is much higher than the gyrofrequency. An important conclusion is that, as a result of this instability, unpolarized electromagnetic waves with frequencies near 2ω_{ e } may be amplified.

Resonant absorption in a steep density profile
View Description Hide DescriptionThe electric field pattern is studied in the capacitor model of resonance absorption. Hydrodynamic and kinetic theory are used. The electric field is calculated for L≳10λ_{D}, where L is the density gradient length and λ_{D} the Debye length. The effect of collisional damping is studied. One obtains three different regimes. In the intermediate regime, the electric field amplitude is determined by the thermal convection, while the energy absorption is mainly caused by collisional damping.

Decay instabilities for inhomogeneous plasmas: WKB analysis and absolute instability
View Description Hide DescriptionA general approximate criterion is developed (via WKB analysis) for the absolute instability of two parametrically pumped decay waves with one‐dimensional inhomogeneity. The absolute instability threshold pump power proves to be higher than that required for amplitude amplification R by a factor πk ’ l ^{2}/27R, when k ’ is the phase mismatch dΣk _{ i }/d x at the location of maximum instability and l ^{−} ^{2} is the effective curvature of the necessarily second‐order expansion about that point. Previous results are included as special cases.

Three‐dimensional toroidal equilibria and stability by a variational spectral method
View Description Hide DescriptionThe characteristics of the partial differential equations describing three‐dimensional toroidal magnetohydrodynamic equilibria with nested flux surfaces in inverse flux coordinates are derived and examined. The equilibrium equations are then variationally reduced to a truncated set of ordinary differential equations by decomposing the flux surface geometry into a spectral representation. The magnetic field lines on the flux surfaces are given in terms of a variable stream function to allow optimum choice of the angle coordinates over the flux surfaces and to simplify the treatment in the vicinity of a rational magnetic surface. Analytic properties of the spectral representation and moment equations are considered. Comparative calculations are performed numerically. The results agree well with those calculated using a standard three‐dimensional equilibrium code, but the variational spectral method is substantially faster computationally. The Mercier stability criterion is given.

Electrostatic confinement in a bumpy torus
View Description Hide DescriptionIn a closed‐field‐line device such as a bumpy torus, the combined E×B and ∇B drifts lead to charge separation that is balanced by the ion polarization drift. In this work, we determine self‐consistent potential and density profiles and the condition for electric island formation.

Stochastic particle diffusion in velocity space for a bumpy torus
View Description Hide DescriptionNonadiabatic changes of the magnetic moment μ in the ELMO Bumpy Torus‐Scale (EBT‐S) have been studied both analytically and numerically. Simple forms of Δ μ and gyrophase change were obtained, permitting the changes in these quantities to be studied using an iteration mapping. The mapping results show stochastic behavior for particles having high energy and low initial μ. Otherwise, superadiabatic motion appears. The stochastic diffusion coefficient for the variation of μ was measured numerically by mapping and was also calculated from quasilinear theory. The results are shown to agree well in the stochastic region. For high‐energy particles, the diffusion in μ caused by nonadiabaticity can be comparable to collisional diffusion when stochastic motion occurs for EBT‐S.