Volume 31, Issue 3, March 1988
Index of content:

Effect of a strong‐current ion ring on spheromak stability
View Description Hide DescriptionThe stability of a spheromak with an energetic ion ring, carrying a current comparable to the plasma current, to the tilt mode is considered. For small departures from sphericity a perturbative approach is applied to an appropriate energy principle in order to calculate the lowest nontrivial kinetic contribution of the ion ring. An analytic stability criterion is obtained. It is seen that the prolate configuration becomes more stable while the oblate one is less stable than in the absence of the ring. The prolomak becomes stable when the ring kinetic energy exceeds the magnetic energy within the separatrix.

Electron diamagnetism and toroidal coupling of tearing modes
View Description Hide DescriptionUsing a simple model for the layer of the tearing mode, it is demonstrated that toroidally coupled tearing modes with two rational surfaces are most unstable when the ω*’s of the electrons at the rational surfaces are equal. The onset of instability may then occur because of the tuning of ω* rather than the passage of Δ’‐like quantities through zero. This mechanism for the onset of instability is sharp since the resonance is narrow. The effect of toroidal rotation is also discussed.

Nonlinear unstable viscous fingers in Hele–Shaw flows. II. Numerical simulation
View Description Hide DescriptionThe nonlinear stages of two‐dimensional immiscible displacement processes in Hele–Shaw flows are investigated by means of large scale numerical simulations based on a purely Lagrangian vortex method. The vortex sheet at the interface between the two fluid phases is discretized into circular arcs with a continuous distribution of circulation, which renders our numerical technique highly accurate. A complicated unsteady growth mechanism is observed for the emerging viscous fingers, involving a combination of spreading, shielding, and tip splitting. As the surface tension is further reduced, smaller length scales arise and the fingertip exhibits a new splitting pattern in which three new lobes emerge instead of two. Monitoring the velocity as well as the radius of curvature at the fingertip demonstrates that the instability of the finger evolves in an oscillatory fashion. The two‐lobe and the three‐lobe splitting can thus be explained as different manifestations of the same instability mode. Comparison with experiment shows good qualitative but only fair quantitative agreement. By imposing a constraint on the curvature at the fingertip, experimental results, which show fingers of width considerably smaller than half the cell width and exhibit ‘‘dendritic’’ instability modes, are reproduced.

Bubble competition in Rayleigh–Taylor instability
View Description Hide DescriptionThe penetration of a front of light fluid into a heavy fluid in a Rayleigh–Taylor unstable flow is studied by using a model that simulates the competition among the bubbles formed in the interface when the density ratio of the two fluids is very large. Several different initial conditions have been considered, and it is found that the front moves with constant acceleration. The values obtained for the acceleration of the front are in very good agreement with experimental results obtained by Read [Physica D 1 2, 45 (1984)].

The dynamics of bubble growth for Rayleigh–Taylor unstable interfaces
View Description Hide DescriptionA statistical model is analyzed for the growth of bubbles in a Rayleigh–Taylor unstable interface. The model is compared to solutions of the full Euler equations for compressible two phase flow, using numerical solutions based on the method of front tracking. The front tracking method has the distinguishing feature of being a predominantly Eulerian method in which sharp interfaces are preserved with zero numerical diffusion. Various regimes in the statistical model exhibiting qualitatively distinct behavior are explored.

Fluid flow due to a stretching cylinder
View Description Hide DescriptionThe fluid flow outside of a stretching cylinder is studied. The problem is governed by a third‐order nonlinear ordinary differential equation that leads to exact similarity solutions of the Navier–Stokes equations. Because of algebraic decay, an exponential transform is used to facilitate numerical integration. Asymptotic solutions for large Reynolds numbers compare well with numerical results. The heat transfer is determined.

Chaotic advection in pulsed source–sink systems
View Description Hide DescriptionThe onset of chaos in passive advection of particles by flow caused by a pulsed source–sink system is documented. This type of model is of interest in various applications. It is of fundamental interest as the first example of a flow without circulation about any contour at any instant displaying chaotic particle paths. Standard chaos diagnostics such as Poincaré sections and Lyapunov exponents are studied as are more conventional flow visualization measures such as streaklines. Numerical stirring experiments for various collections of particles are performed and the properties of a certain one‐dimensional map induced by the two‐dimensional flow are examined.

The three‐dimensional boundary layer in the entry region of curved pipes with finite curvature ratio
View Description Hide DescriptionThe major flow development in the region within a distance O((a R)^{1/2}) from the entrance of a curved pipe occurs near the pipe wall, where a is the radius of the pipe cross section, assumed circular, and R is the radius of curvature of the central axis of the pipe. A three‐dimensional boundary‐layer solution is obtained for elucidating the physics of this developing flow; in particular, the effect of nonzero curvature ratio α=a/R on the geometric similarity of the flow. The numerical results show that the series solution in terms of α is valid only when α≤0.1 and s≤0.1 (a R)^{1/2}, where s is the distance from the inlet along the pipe axis. The crossover of the axial wall shear is purely a geometric property and its location is a strong function of α. It is also demonstrated that (a R)^{1/2} is the proper length scale by showing that the solution of the first region, s∼O(a), is included in that of the second, s∼O(a R)^{1/2}.

Bifurcation in axisymmetric Czochralski natural convection
View Description Hide DescriptionNumerical simulations using a finite volume method with primitive variables formulation are presented for a natural convection flow in the Czochralski melt. In the limit of very small Prandtl numbers it is shown that unsteadiness appears in the form of regular oscillations for sufficiently high values of the Rayleigh number. Such regular oscillations are preceded by a multicell motion structure in the melt, with flow separation at the wall. The critical value of the Rayleigh number for the onset of the oscillations is determined by carrying out a series of time dependent calculations.

Nonlinear particle diffusion in a time‐dependent host medium
View Description Hide DescriptionThe dynamics of a gas in a time‐dependent host medium is studied by means of a generalized Boltzmann equation. Removal and regeneration events, as well as linear external sources, are taken into account. The corresponding continuity equation is solved, and the time evolution of the system is investigated, with particular attention paid to its asymptotic regime.

Direct numerical simulations of the turbulent mixing of a passive scalar
View Description Hide DescriptionThe evolution of scalar fields, of different initial integral length scales, in statistically stationary, homogeneous, isotropic turbulence is studied. The initial scalar fields conform, approximately, to ‘‘double‐delta function’’ probability density functions (pdf ’s). The initial scalar‐to‐velocity integral length‐scale ratio is found to influence the rate of the subsequent evolution of the scalar fields, in accord with experimental observations of Warhaft and Lumley [J. Fluid Mech. 8 8, 659 (1978)]. On the other hand, the pdf of the scalar is found to evolve in a similar fashion for all the scalar fields studied; and, as expected, it tends to a Gaussian. The pdf of the logarithm of the scalar‐dissipation rate reaches an approximately Gaussian self‐similar state. The scalar‐dissipation spectrum function also becomes self‐similar. The evolution of the conditional scalar‐dissipation rate is also studied. The consequences of these results for closure models for the scalar pdf equation are discussed.

Couette flow of a binary gas mixture
View Description Hide DescriptionThe linearized binary model described by Hamel [Phys. Fluids 8, 418 (1964)] is used to obtain a set of kinetic equations and boundary conditions for the Couette flow problem. The derived set of two coupled integrodifferential equations is solved by iteration implementing standard discretization techniques. Highly accurate numerical results are presented for the mass velocity distribution and the total shear stress of the binary gas system.

Conditions for the validity of unmagnetized‐plasma theory in describing weakly magnetized plasmas
View Description Hide DescriptionAn investigation is made of the conditions under which the expression for the dielectric tensor from magnetized‐plasma theory may be approximated by that from unmagnetized‐plasma theory. Surprisingly, the conditions usually quoted are shown to be inaccurate chiefly because they make no reference to the form of the plasma distribution function. Corrected conditions depend strongly on the distribution function and their application is thus highly problem‐dependent. For completely arbitrary distributions these conditions are generally stronger than the commonly quoted ones for weakly damped waves. By contrast, the conditions applicable to axisymmetric distributions can be weaker than some of those commonly quoted. A geometric interpretation of the revised conditions for axisymmetric distributions is given in terms of the resonance ellipses of cyclotron maser theory and is illustrated with reference to dispersion and instability of Langmuir and Bernstein waves in weakly magnetized plasmas. These results apply to all weakly magnetized plasmas that can be described by the linearized Vlasov equations and imply that some investigations which have used the commonly quoted conditions to justify use of unmagnetized‐plasma theory may be in error on this point.

Kinetic theory of electron drift vortex modes
View Description Hide DescriptionMagnetic electron drift vortex modes are examined using a kinetic theory. The dispersion is found to be much different than that obtained from the fluid theory. For frequencies much smaller than the electron plasma frequency, the waves are heavily Landau damped. The modes are weakly damped at frequencies just below the plasma frequency and hence have an electromagnetic rather than magnetic character.

Electromagnetic effects on parametric instabilities of Langmuir waves
View Description Hide DescriptionElectromagnetic effects on parametric instabilities of Langmuir waves in unmagnetized plasmas are investigated. A fully electromagnetic treatment of these instabilities removes discontinuities of frequencies that are found to be present in the wave vector space of the electrostatic dispersion equation. Furthermore, it was found that a pair of novel parametric instabilities of Langmuir pump waves emerge owing to the electromagnetic effects. Both of them excite electromagnetic plasma waves near the plasma frequency. One of them is the hybrid modulational instability, which is a four‐wave up‐conversion process. As the wave vector of the pump wave increases the hybrid parametric decay instability becomes dominant. This is a three‐wave down‐conversion instability, which has been investigated previously [Space Sci. Rev. 2 6, 3 (1980); Phys. Rev. A 2 7, 552 (1983); Astrophys. J. 3 0 8, 954 (1986)].

Linear spectrum of magnetostatic and thermally conducting planar plasmas
View Description Hide DescriptionThe linear spectrum of 1‐D magnetostatic and thermally conducting equilibrium plasmas is analyzed. It is found that the influence of thermal conduction is fundamentally different on the various parts of the linear spectrum of ideal magnetohydrodynamics and that it is by far the most profound for the slow magnetoacoustic part. The ideal Alfvén continuous spectrum is unaffected by thermal conduction. However, the ideal slow continuous spectrum is replaced by the isothermal slow continuous spectrum. This new continuous spectrum owes its existence to thermal conduction but is independent of κ and involves a different range of continuum frequencies. In addition to these two continuous parts, the spectrum consists of discrete slow and fast magnetoacoustic modes and thermal modes. The point eigenvalues of the fast magnetoacoustic modes are slightly distorted in proportion to κ. However, the point eigenvalues of the slow magnetoacoustic modes lie on well‐defined curves in the complex plane that are independent of κ and controlled by the ideal slow and isothermal slow continua. The discrete slow magnetoacoustic spectrum hangs, as it were, on the ideal slow and isothermal slow continua and is determined by the nonuniformity of the equilibrium. The thermal modes are the result of the inclusion of the nonideal effect of thermal conduction in the energy equation.

Nonlinear resonance of two‐dimensional ion layers
View Description Hide DescriptionA nonlinear theory of wave resonances in a two‐dimensional ion layer confined under the surface of liquid helium is presented. The ion layer is modeled as a two‐dimensional cold plasma fluid. In addition to the usual nonlinearities present in the continuity equation and the equation of motion, the theory considers a nonlinear dependence of the mass of a plasma particle on its velocity, as suggested by indirect experimental evidence. Secular perturbation theory is used to find the plasma response when the damped, nonlinear system is driven externally. For typical experimental parameters, the mass nonlinearity is found to be the dominant nonlinear effect, giving rise to a backbending of the resonance curve.

Structure of subcritical perpendicular shock waves
View Description Hide DescriptionThe structure of quasineutral one‐dimensional stationary perpendicular shock waves is investigated within a two fluid model. Dissipation is considered to occur by resistivity. The influence of the entropy increase on the shock structure and especially the evolution of Sagdeev’s potential for shocks of finite strength is discussed quantitatively. The global structure of shocks with small resistivity is treated analytically.

Tearing modes in toroidal geometry
View Description Hide DescriptionThe separation of the cylindrical tearing mode stability problem into a resistive resonant layer calculation and an external marginal ideal magnetohydrodynamic (MHD) calculation (Δ’ calculation) is generalized to axisymmetric toroidal geometry. The general structure of this separation is analyzed and the marginal ideal MHD information (the toroidal generalization of Δ’) required to discuss stability is isolated. This can then, in principle, be combined with relevant resonant layer calculations to determine tearing mode growth rates in realistic situations. Two examples are given: the first is an analytic treatment of toroidally coupled (m=1, n=1) and (m=2, n=1) tearing modes in a large aspect ratio torus; the second, a numerical treatment of the toroidal coupling of three tearing modes through finite pressure effects in a large aspect ratio torus. In addition, the use of a coupling integral approach for determining the stability of coupled tearing modes is discussed. Finally, the possibility of using initial value resistive MHD codes in realistic toroidal geometry to determine the necessary information from the ideal MHD marginal solution is discussed.

Power requirements for current drive
View Description Hide DescriptionGeneral formulas for the efficiency of current drive in toroidal plasmas are derived using entropy arguments. The highest possible efficiency for current drive in which a high‐energy electron tail is formed is shown to be p=E _{ r } j, with p and j the power and current densities and E _{ r }≊0.09n _{1} _{4} V/m with n _{1} _{4} the electron density in units of 10^{1} ^{4}/cm.^{3} The electric field required to maintain the current in a runaway discharge is also shown to equal E _{ r }. If the plasma current is carried by near‐Maxwellian electrons, waves that have a low phase velocity, compared to the energy of the electrons with which they interact, can drive a current with Ohmic efficiency, p=ηj ^{2}. Such waves were first discussed in the context of current drive by Fisch [Rev. Mod. Phys. 5 9, 175 (1987)].