Volume 28, Issue 6, June 1985
Index of content:

The instability of long fingers in Hele–Shaw flows
View Description Hide DescriptionExperiments on steady fingers and their stability in Hele–Shaw cells, are reported. It is shown that the shape of steady fingers scales with a modified capillary number, Ca’, as suggested by McLean and Saffman [J. Fluid. Mech. 1 0 2, 455 (1981) and our previous analysis [J. Fluid Mech. 1 3 9, 291 (1984)]. The behavior at large Ca’ is investigated by using a wide Hele–Shaw cell. It is observed that such fingers are unstable for Ca’>100, in agreement with the prediction by Taylor and Saffman (second symposium on naval hydrodynamics, 1958, p. 277) of instability as Ca’→∞. The mechanism is identified as one of tip‐splitting, which occurs periodically in the weakly supercritical regime, and in a more complex fashion for large Ca’.

Half‐coalescence ideal magnetohydrodynamic instability of the q=1 magnetic island in tokamaks
View Description Hide DescriptionThe ideal magnetohydrodynamic instability of a symmetric q=1 magnetic island is studied by using an initial value method for a pressureless cylindrical tokamak plasma. It is shown that the island is unstable to a half‐coalescence motion which destroys the up–down symmetry of the magnetic configuration. The mode characteristics are qualitatively similar to those predicted by Bussac e t a l. [Phys. Lett. 1 0 5 A, 51 (1984)].

Trapped particle instability induced by axial shear in E×B velocity
View Description Hide DescriptionA new nonflute form of the E×B shear‐driven trapped particle mode is obtained for a simplified square well equilibrium model of a tandem mirror. The mode is more unstable for a larger passing density in contrast to the curvature‐driven trapped particle mode. Under some circumstances this mode can avoid the stabilizing effect of strong anchor curvature by isolating to the center cell and can do so without paying the magnetohydrodynamic (MHD) penalty for bending the field lines, thus requiring a large center cell ion gyroradius for stabilization.

Observation of plasma waves from absolute stimulated Raman scattering
View Description Hide DescriptionTime‐, frequency‐, and wavenumber‐resolved Thomson scattering techniques have been used to identify electron plasma waves driven at the quarter‐critical surface by a CO_{2} laser in a preformed plasma. The clear signature of absolute stimulated Raman scattering was observed, namely ω_{ p }≊ω_{0}/2 and k _{ p }≊k _{0}. In addition, some plasma waves with larger wavenumber were seen, perhaps corresponding to coupling with ion waves produced by stimulated Brillouin scattering. The level of ω_{0}/2 scattered light, calculated from the observed level of plasma waves, should have been several orders of magnitude above the detection threshold, yet no such emission was seen.

Nonideal magnetohydrodynamic instabilities and toroidal magnetic confinement
View Description Hide DescriptionThe marked divergence of experimentally observed plasma instability phenomena from the predictions of ideal magnetohydrodynamics led in the early 1960’s to the formulation of finite‐resistivity stability theory. Beginning in the 1970’s, advanced plasma diagnostics have served to establish a detailed correspondence between the predictions of the finite‐resistivity theory and experimental plasma behavior—particularly in the case of the resistive kink mode and the tokamak plasma. Nonlinear resistive‐kink phenomena have been found to govern the transport of magnetic flux and plasma energy in the reversed‐field pinch. The other predicted finite‐resistivity instability modes have been more difficult to identify directly, and their implications for toroidal confinement are still unresolved.

Long‐wave instability at the interface between two viscous fluids: Thin layer effects
View Description Hide DescriptionThe stability of the interface between two viscous fluids is considered when the depth of the lower fluid is much less than the depth of the upper fluid. A long wavelength perturbation scheme is used to solve the linear stability problem and the equation governing the nonlinear evolution of the interface is deduced. The exact dispersion relation is derived for arbitrary values of wavelength and then simplified for large wavelength values. It is found that the flow is always linearly unstable when the lower fluid is also the more viscous fluid.

Coalescence of stretching vortices
View Description Hide DescriptionThe contour dynamics numerical technique is used to study the coalescence of two equal uniform vortices in the presence of an externally applied stretching strain field. Plane three‐dimensional stretching is found to substantially inhibit vortex coalescence when the plane of the vortex axes is initially perpendicular to the plane of the straining motion. This behavior is interpreted qualitatively in terms of a purely two‐dimensional flow fully equivalent to the stretching configuration.

A numerical study of vortex merging in mixing layers
View Description Hide DescriptionNumerical solutions are presented for forced spatially developing axisymmetric and two‐dimensional mixing layers. The numerical scheme employs quadratic upwind differencing for convection and a Leith type of temporal differencing in order to solve the incompressible Navier–Stokes and continuity equations. The applied forcing function is derived from linear inviscid stability theory. The resulting large‐scale vortex dynamics is visualized by means of streakline and isovorticity contour plots. It is seen that the vortex merging behavior in both types of mixing layers is determined by the subharmonics present in the forcing function. Manipulation of the vortex dynamics in a predictable fashion is possible through alterations in the frequency content of this applied forcing. Reynolds number is shown to be of only minor importance.

Nonlinear dispersive periodic waves in the presence of instability and damping
View Description Hide DescriptionPeriodic solutions of a nonlinear dispersive equation involving both growth and damping mechanisms are investigated numerically and theoretically. General features of equilibrium periodic states for a strongly dispersive case are found to be well explained by means of a perturbation analysis of the cnoidal wave solutions.

Random walk models for particle displacements in inhomogeneous unsteady turbulent flows
View Description Hide DescriptionFirst a small time analysis is developed for the first and second moments of the velocity (W) and displacement (Z) in one direction of particles marked at a given point in an inhomogeneous unsteady turbulent flow, in terms of the local energy dissipation rate, and the local derivatives of the second and third moments of the vertical component of the velocity field, ∂∼(u ^{2} _{3})/∂z and ∼(∂u ^{3} _{3})/∂z. Then the appropriate form of a Langevin equation in inhomogeneous turbulence is suggested, namely, d W=(−W/T _{L}+a _{1})d t+a ^{1/2} _{2} dω_{ t } where a _{1}, a _{2}, and T _{L} are functions of the particle position and time, and dω_{ t } is a random Gaussian velocity increment with ∼(dω_{ t })=0 and ∼((dω_{ t }))^{2}=d t. For simplicity, only one component of the particle motion, W(t), is considered. The functions a _{1} and a _{2} are determined by relating the random walk model to the Eulerian conservation equations for the mass of the contaminant and volume of the flow (i.e., the continuity equation), using the Fokker–Planck equation and the Eulerian equations for the moments of a vertical velocity. The coefficients a _{1} and a _{2} reduce to the same form as that obtained by the statistical analysis, namely a _{1}=∂∼(u ^{2} _{3})/∂z, (≊d W̄/d t, when t → 0) and a _{2}=2∼(u ^{2} _{3})/T _{L} +d∼(u ^{2} _{3})/d t (≊2∼(W ’ ^{2})/T _{L}+d∼(W ’ ^{2})/d t, when t → 0).
It is shown that the random walk model has the correct behavior as t/T _{L} → 0. The theory is shown to agree reasonably well with the measurements of mean height and mean vertical displacement of particles released in a convective boundary layer [Q. J. R. Meteorol. Soc. 1 0 2, 427 (1976)]. Yaglom’s [Isv. Atmos. Oceanic Phys. 8, 333 (1972)] surface similarity result is recovered as a special case. For t ≫ T _{L}, and in a zero‐skewness steady turbulence, the random walk model reduces to the familiar K‐diffusion equation. Some examples are presented to show that mean and mean square particle displacements from the random walk model are virtually identical to those obtained from the analytical solution of the corresponding Eulerian moment equations. Careful analysis is still required when concentration distributions in turbulent flows near a boundary are evaluated using random walk models.

Turbulent normal velocity fluctuations close to a wall
View Description Hide DescriptionMeasurements of the normal velocity fluctuations in the immediate vicinity of a wall are made using a technique which does not interfere with the flow. It is found that (∼(v ^{+} ^{2}))^{1} ^{/} ^{2} =0.005y ^{+} ^{2} for y ^{+}→0. Values of the normal intensity obtained by extrapolating measurements made with thermal probes are shown to be in considerable error.

Motion of a strong shock wave in an exponential medium
View Description Hide DescriptionIn this paper, corrections to the Chester, Chisnell, and Whitham characteristic exponent have been computed for the motion of a strong shock wave in a medium with exponentially varying density and ray‐tube area. The similarity solution of Hayes is used to find the various interaction terms at different points of the flow.

An experimental shock wave study of aerosol droplet evaporation in the transition regime
View Description Hide DescriptionThe evaporation rate of submicron aerosol droplets suspended in argon was studied behind incident shock waves at initial droplet Knudsen numbers 0.1≤Kn_{ G }≤0.9. In this case the Knudsen number is formulated with the mean free path of the surrounding carrier gas and the droplet radius. The reduction in the size of the particles in the post‐shock relaxation zone was measured by laser light scattering, simultaneously recorded under four fixed angles. Mie theory was applied to determine the time‐dependent droplet size during the evaporation process. By comparison with an analytical description of the transfer process of a single droplet with the surrounding gas, a physical interpretation of the measurements could be established. The experimental results are consistent with analytical first‐order correction terms of the mass flux in the transition regime.

Analysis of thermal, sound, and shear waves according to a relativistic kinetic model
View Description Hide DescriptionIn this paper a kinetic model describing a relativistic gas is considered. The collision frequency is assumed to depend upon the particle speed in a way suggested in a previous paper by the authors. The new model has more satisfactory properties than one previously studied. Several results of both analytic and numerical character concerning the sound and heat waves with frequencies comparable to the intermolecular collision frequency are given.

Kinetic and fluid approaches to low‐frequency magnetohydrodynamics: A comparison
View Description Hide DescriptionKinetic and fluid approaches to the problem of low‐frequency magnetohydrodynamic flute stability of plasmas where the equilibrium electron gradient‐B drift is larger than the curvature drift are compared. It is shown that the two approaches are practically identical for frequencies below the curvature drift frequency. The identical nature of the approaches has to do with the fact that the large gradient‐B drift has the primary effect of maintaining the electron temperature constant along its path; since this fact is independent of the collision frequency, the same equation of state applies to both approaches.

Finite‐Larmor‐radius‐corrected collision operator: A new approach to solving the kinetic equation
View Description Hide DescriptionThe kinetic equation for a magnetized plasma is usually treated by transforming the kinetic operator ∂/∂t+v ⋅ ∇_{ x }+(e/m)(E+v×B/c) ⋅ ∇_{ v } to guiding‐center variables, while evaluating the collision operator at the particle position. This introduces apparent nonlocality into a local equation. A method is presented for transforming the kinetic equation into a local equation, i.e., an equation in which the kinetic operator and the collision operator are evaluated at the same point (the guiding‐center position). The collision operator is transformed thereby into a new operator, called the FLR‐corrected collision operator, which is a power series in the displacement between a particle and its guiding center. This transformation to a local equation, plus the use of the guiding‐center distribution function, make the method significantly more efficient than standard techniques. The method is illustrated by a calculation of transport caused by ion–neutral collisions. The technique is sufficiently general that it could be applied in a variety of contexts, for example, neoclassical transport, or the effect of collisions on a wave.

Matrix elements of the linearized Fokker–Planck operator
View Description Hide DescriptionGenerating functions for the matrix elements of the linearized Fokker–Planck operator in the Burnett function basis are considered. They are shown to be governed by recursion relations, which allow speedy numerical evaluation of the matrix elements.

On the relation between ‘‘mixing length’’ and ‘‘direct interaction approximation’’ theories of turbulence
View Description Hide DescriptionA systematic procedure is outlined for deriving a spectral equation from the renormalized theory of turbulence based on the so‐called direct interaction approximation. This spectral equation should be valid for a large class of plasma and fluid systems.

Drift‐Alfvén vortices
View Description Hide DescriptionThe solitary vortex solution is shown to exist for the set of nonlinear equations describing low‐frequency fluid motion in a magnetized plasma.

Finite beta stabilization of the kinetic ion mixing mode
View Description Hide DescriptionThe full finite‐β local dispersion relation of the ion mixing mode is derived and analyzed, using kinetic theory. The physical mechanism of finite‐β stabilization of a single (i.e., fixed wavenumber, k _{ z } r _{ n } and b _{ i } constant) ion mixing mode is discussed. The distortion of the resonant region, in velocity space (by ∇B drifts) and the coupling to the shear Alfvén wave are shown to be the important stabilization mechanisms. It is found that high values of β (β>β_{ c }, where β_{ c } =0.1→1.0, for relevant parameters) are necessary for stabilization.