Volume 25, Issue 12, December 1982
Index of content:

Transition from single to multiple double layers
View Description Hide DescriptionIt is shown that laboratory double layers become multiple double layers when the ratio of Debye length to system length is decreased. This result exhibits characteristics described by boundary layer theory.

Exhaust rate measurements in a divertor with large mirror ratio
View Description Hide DescriptionThe parallel ion fluid velocity in the scrape‐off layer of a poloidal divertor is observed to vary inversely with the mirror ratio in the divertor’s throat for ratios ranging from 1 to 5, in good agreement with models developed for bundle divertors. The density variation on a diverted field line also agrees qualitatively with the models, but the observed electric field does not.

Destabilization of drift waves by a nonuniform radial electric field
View Description Hide DescriptionIt is shown that drift waves can be destabilized in the presence of a nonuniform electrostatic field. This may explain the anomalous diffusion observed in tokamaks.

A two‐grating method for combined beam splitting and frequency shifting in a two‐component laser‐Doppler velocimeter
View Description Hide DescriptionThe use of a rotating radial phase grating to carry out beam splitting and frequency shifting in a laser‐Doppler velocimeter is briefly reviewed. This technique is not new. However, the present design adds a substantial new element by using two overlapping radial gratings to produce a two‐channel system in which channel separation can be accomplished by electronic filtering of the signal from a single detector.

Conical vortices: A class of exact solutions of the Navier–Stokes equations
View Description Hide DescriptionA two‐parameter family of exact axially symmetric solutions of the Navier–Stokes equations for vortices contained within conical boundaries is found. The solutions depend upon the same similarity variable, equivalent to the polar angle φ measured from the symmetry axis, as flows previously discussed by Long and by Serrin, but are distinct from the cases they treated. The conical bounding stream surfaces of the present solution can be located at any angle φ=φ_{0}, where 0<φ_{0}<π. The flows in all of these cases, when solutions exist, are finite everywhere except at the cone vertex which is a source of axial momentum, but not of volume. Solutions are of three types, flow may be (a) towards the vertex on the axis and away from the vertex at the conical boundary, (b) towards the vertex both on the axis and at the cone, or (c) away from the vertex on the axis and towards it at the bounding cone. In the first and second case, strong shear layers form on the cone walls for high Reynolds numbers. In case (c), a region of strong axial shear and strong axial vorticity forms near the axis, even for low Reynolds numbers. The qualitative nature of the possible solutions is deduced, using methods of argument due to Serrin, and examples of flows are numerically computed for cone half‐angles of π/4, π/2 (flows above the plane z=0), and 3π/4. Regions of the parameter space where solutions are proven not to exist are given for the cone half‐angles given above, as well as regions where solutions are proven to exist.

Instability and confined chaos in a nonlinear dispersive wave system
View Description Hide DescriptionCalculations of a discrete nonlinear dispersive wave system show that as the degree of nonlinearity increases, the system experiences in turn, periodic, recurring, chaotic, transitional, and periodic motions. A relationship between the instability of the initial configuration and the long‐time behavior is identified. The calculations further suggest that the corresponding continuous system will exhibit chaotic motions and energy‐sharing among a narrow band of unstable modes, a phenomenon which we call ‘‘confined chaos.’’

Evolution of groups of gravity waves with moderate to high steepness
View Description Hide DescriptionExperimental measurements of evolution of a deep‐water wave group are described. Wave groups are found to change rapidly over a few tens of wavelengths when the initial steepness 0.09≤a _{0} k _{0}≤0.28. The transition creates envelope solitons composed of waves with smaller steepness and lower carrier frequency than the initial state. The carrier frequencies of the envelope solitons can be downshifted as much as 25%. The transition process is irreversible, but does not lead to total randomness.

A modulated point‐vortex model for geostrophic, β‐plane dynamics
View Description Hide DescriptionA new m o d u l a t e d point‐vortex model is presented for the equivalent barotropic equations of potential vorticity: q _{ t }−ψ_{ y } q _{ x } +ψ_{ x } q _{ y }=0, (q=κ+βy, κ=∇^{2}ψ−γψ). The discrete model conserves ∑_{ m }(q _{ m0} −β_{*} y _{ m })^{2} in analogy with the conserved ∫ ∫κ^{2} d x d y. An analytical study is made for the general two point‐vortex system. For equal vortices, the solution has a monotonic drift ∝(−β). A comparison is made of numerical solutions of the point‐vortex model and corresponding solutions of a finite‐difference model on a periodic domain. For an initially monopolar distribution of ∇^{2}ψ−γ^{2}ψ, an x drift in the direction of sign (−β) and a y drift in the direction of the sign of the extremum of κ is obtained in both cases that are in agreement at short times. The y drift is associated with the development of a spreading Rossby wave wake, as a consequence of the conservation law. For a ‘‘tilted’’ dipolar region of vorticity we observe near‐periodic oscillations. The modulated point‐vortex model is also applicable to drift waves in a plasma.

Point vortex motions with a center of symmetry
View Description Hide DescriptionThe equations of motion for point vortices are well known to preserve certain discrete symmetries of the initial state. The case of a c e n t e r of symmetry is considered in detail here, since this particular instance seems to have been overlooked in the classical literature. This symmetry provides a generalization of the early studies by Gröbli and Greenhill wherein several a x e s of symmetry are present, a case which leads to an effective one‐body problem. The center of symmetry yields an effective two‐body problem which is Hamiltonian and integrable. As an example the ‘‘double alternate ring’’ configurations, circular analogs of the vortex street introduced by Havelock, are considered. A fully nonlinear mode wherein these double rings asymptotically dissolve into freely moving vortex pairs is found analytically. The paper concludes with a discussion of the relevance of such modes to our understanding of the disintegration of vortex streets in two‐dimensional flow.

Small‐amplitude waves produced by a submerged vorticity distribution on the surface of a viscous liquid
View Description Hide DescriptionThe small‐amplitude waves generated by a submerged vorticity distribution on the surface of a viscous fluid are studied. The linearized initial‐value problem is considered, and a closed‐form solution for each monochromatic component is obtained. The results of the theory are illustrated by a numerical synthesis of these components for the case of a vortex filament in water.

Strained spiral vortex model for turbulent fine structure
View Description Hide DescriptionA model for the intermittent fine structure of high Reynolds number turbulence is proposed. The model consists of slender axially strained spiral vortex solutions of the Navier–Stokes equation. The tightening of the spiral turns by the differential rotation of the induced swirling velocity produces a cascade of velocity fluctuations to smaller scale. The Kolmogorov energy spectrum is a result of this model.

Unsteady natural convection about a sphere at small Grashof number
View Description Hide DescriptionUnsteady low‐Grashof‐number natural convection about a sphere is studied when the surface temperature of the sphere is suddenly increased. It is shown that the solutions for the velocity and temperature are, respectively, expressed in terms of three expansions reflecting the existence of three distinct regions in the (r, t) plane, r and t being a nondimensional radial coordinate and nondimensional time, respectively.

Exchange of energy and momentum between gases at different temperatures
View Description Hide DescriptionThe interaction mechanisms between gases at different temperatures are investigated within the thermodynamic theory of mixtures. Conditions a priori on the phenomenological coefficients are derived by having recourse to the Galilean invariance and the entropy principle for mixtures. The results so obtained, which, owing to their thermodynamic origin, must hold for any kinetic derivation, turn out to be satisfied by known kinetic expressions.

Determination of the density perturbation at the wall for the Rayleigh problem
View Description Hide DescriptionThe Rayleigh problem, a fundamental time‐dependent problem of gas kinetics, is studied in the context of the constant collision frequency BGK model. An analytical result, expressed in terms of the stationary solution, is obtained and numerically evaluated. Asymptotic solutions in both the small and large time limit are also presented and compared to results of other theories.

Stellarator equilibria with weak helical curvature
View Description Hide DescriptionGeneral low‐β stellarator equilibria having weak helical curvature are calculated analytically. Important properties of these equilibria are also calculated; separatrix location, stability toward transverse shifts, and flute interchange stability.

Ideal magnetohydrodynamic stability of axisymmetric mirrors
View Description Hide DescriptionThe governing partial differential equation for general mode‐number pressure‐driven ballooning modes in a long‐thin, axisymmetric plasma is derived within the context of ideal magnetohydrodynamics. It is shown that the equation reduces in special limits to the Hain–Lüst equation, the high‐m diffuse p(ψ) ballooning equation, and the low‐m sharp‐boundary equation. A low‐β analytic solution of the full partial differential equation is presented for quasiflute modes in an idealized tandem mirror model to elucidate the relationship of the various limiting cases.

Effect of ion collisionality on ion‐acoustic waves
View Description Hide DescriptionThe frequency and damping ratio of ion‐acoustic waves in plasmas with moderately collisional ions is obtained by numerical solution of the linearized Fokker–Planck equation for electron‐ion temperature ratios, Z T _{ c }/T _{ i }=4, 8, 16, and 32.

Energy and momentum deposition in plasmas due to the lower hybrid wave by a finite source
View Description Hide DescriptionHeating and current generation due to the lower hybrid wave are studied using particle simulation. In contrast with previous work, where only a single mode is treated, the main interest of this work is focused on the physical problems of a propagation cone consisting of many Fourier‐expanded modes. It is found that the trajectory of the propagation cone is well described up to the lower hybrid resonance layer using both the cold plasma approximation and the WKB method. An ion cross‐field drift due to the ponderomotive force is observed. A main discovery of this work is that the modes in the upper portion of the spectrum of the antenna play a key role in the creation of the ion high‐energy tail. This process cannot be explained by the linear theory and is called the cascade process judging from the time variation of the damping of each mode. The particle model is significantly improved using the elongated grid and the quadratic spatial interpolation. Applications of this model to simulations of other problems are expected to be fruitful in research on plasma physics and nuclear fusion.

Role of the relativistic mass variation in electron cyclotron resonance wave absorption for oblique propagation
View Description Hide DescriptionThe role of the relativistic mass variation on wave absorption in the electron cyclotron range of frequencies is investigated. It is first shown that the validity of the nonrelastivistic linear dispersion relation for a Maxwellian plasma is restricted by the conditions N ^{2} _{∥}≫T _{ e }/m c ^{2} and N ^{2} _{∥}≫‖1−ω^{2} _{ c }/ω^{2}‖. A numerical investigation of wave damping in a plasma slab located in an inhomogeneous tokamak‐like magnetic field shows that for most angles of practical interest the latter condition is easily violated and, therefore, the nonrelativistic dispersion relation yields inaccurate results. The problem of the validity of the nonrelativistic quasilinear equation for oblique propagation is also discussed. Using a quasilinear model equation, it is shown that the inclusion of the relativistic mass variation in the diffusion coefficient results in a basic change of the wave–particle selective interaction compared to the nonrelativistic approximation for any value of T _{ e } or N _{∥}.

Electrostatic drift modes with stochastic particle diffusion
View Description Hide DescriptionThe effect of stochastic particle diffusion on the collisionless universal mode in slab geometry is considered. Cross‐field particle diffusion is assumed to be caused by a model static magnetic field turbulence. The resultant eigenmode equation is an integral equation for the perturbed electrostatic potential, which is solved numerically. As stochastic particle diffusion is gradually turned on, its effect on the usual drift branch is found to be stabilizing. At the same time, however, there appears a new, stochasticity‐induced mode which, under certain circumstances, may become unstable.