Volume 28, Issue 4, April 1985
Index of content:

Numerical simulation of the thermal turbulent mixing layer
View Description Hide DescriptionInappropriate initial conditions discovered in a recently reported model calculation of the thermal mixing layer are corrected. Good agreement with measurement is restored by a reoptimization of model coefficients.

Modified drag theory of permeability
View Description Hide DescriptionA modified version of the drag theory of permeability for a granular porous medium is presented. The technique employs the Brinkman equation to interpolate between the microscopic (Stokes) equation and the macroscopic (Darcy) equation, but differs from previous treatments in that the medium is built up iteratively beginning with the smallest grains. A discussion of the geometry corresponding to this construction is presented. The result for the permeability takes the form of a nonlinear differential equation which may be integrated numerically for a given choice of porosity and grain size distribution. Numerical results for the case of spherical grains with a log‐normal distribution of sizes are presented, and compared to the predictions of the Kozeny equation for the same geometry. It is found that for narrow distributions the present method gives a somewhat higher permeability than the Kozeny equation, but that for wide distributions it gives a lower permeability. It is shown that in the limits of narrow or wide distribution the differential equation for the permeability may be solved explicitly, and it is suggested that in the latter case the result may be exact.

Oscillations of a compound drop system undergoing rotation
View Description Hide DescriptionA compound liquid drop system is comprised of three immiscible concentric fluids: a core fluid of density ρ^{(i)} surrounded by a shell of density ρ^{(s)} which is embedded in a medium of density ρ^{(} ^{0} ^{)}. In this analysis, the fluids are incompressible and inviscid. The effect of rotation upon the modes of oscillation of a compound drop is investigated. Rotation rate is considered as a small perturbation of the normal modes of the compound drop, thus introducing the effects of a Coriolis force and a centrifugal distortion.

Velocity field measurements of a laminar starting plume
View Description Hide DescriptionThe region of buoyant fluid resulting from the initiation of heating of an infinite fluid is called the starting plume. Here, velocity field measurements of this flow pattern are presented for the first time. The measurements were carried out by processing digitized tracer particle path photographs. Similarity of the velocity field of the starting plume as it rises was found to exist in the limited range of heat injection rates investigated. The vorticity computed from the velocity field was diffuse, there being no evidence of a distinctive core. Additional flow visualization experiments show the spiral structure in the starting plume cap.

Wilton ripples phenomenon with a background current
View Description Hide DescriptionThe Wilton ripples phenomenon with a background current is examined in detail. The case of most interest occurs when the current is near the threshold value such that direct resonance occurs for the Wilton ripples. Interaction between the fundamental mode and its second harmonic occurs on a much faster time scale. Bifurcation diagrams and stability analysis are presented.

Negative viscosity effect in large‐scale flows
View Description Hide DescriptionThis study considers a large‐scale flow maintained by a small‐scale periodic force field. If the field is essentially anisotropic, the system of small‐scale eddies thereby generated turns out to be unstable to long‐wave disturbances, i.e., the effective viscosity of the corresponding large‐scale flow is negative. If the applied field is sufficiently isotropic, the long‐wave instability disappears so that the effective viscosity is positive.

An unsteady momentum integral calculation for the turbulent spot
View Description Hide DescriptionVelocity and wall shear stress measurements were made along the centerline of turbulent spots generated in a laminar boundary layer on a flat plate. A momentum integral calculation method based on two‐dimensional boundary layer theory and the assumption of potential flow above the spot was developed by transforming the unsteady momentum equation into nondimensional similarity coordinates. The centerline wall shear stress, calculated using ensemble‐averaged data for the velocity field, shows fairly good agreement with the measured values.

Joint statistics between temperature and its dissipation in a turbulent jet
View Description Hide DescriptionThe joint probability density function of the temperature θ and its instantaneous dissipation χ is important for the development of turbulent reacting flow models. Measurements are presented, in the nearly self‐preserving region of a turbulent plane jet, of the joint statistics of θ and an approximation χ_{1} to χ. Properties of the three, separately obtained, dissipation components are first compared to justify the choice of χ_{1}. The quantity chosen is an appropriately weighted linear combination of the squared values of the temporal derivative, shown to be an adequate substitution for the derivative in the streamwise direction, and the derivative, in either the lateral or spanwise directions, of temperature. The correlation between θ and χ_{1} is weak, receiving contributions primarily at small frequencies. The assumption of independence between θ and χ_{1}, allowing the joint probability density function of these two quantities to be written as a product of the individual probability density functions, improves with distance from the jet centerline.

Weak double layers in ion‐acoustic turbulence
View Description Hide DescriptionThe evolution of weak double layers in ion‐acoustic turbulence in one and two‐dimensional particle simulations is examined. Weak double layers (eφ≲T _{ e }) evolve in simulations when a subthermal electron drift is imposed on a long or nonperiodic system with T _{ e }/T _{ i }≫1. Their growth rate increases with the electron drift, and they decay because of ion trapping. They do not form in weakly magnetized or unmagnetized two‐dimensional (2‐D) systems unless a nonuniformity is introduced in the initial or boundary conditions. When the plasma is strongly magnetized (ω_{ c e }>ω_{ p e }), they emerge from 2‐D ion‐acoustic turbulence as coherent structures localized transversely to the magnetic field.

The whistler mode in a Vlasov plasma
View Description Hide DescriptionIn this study, properties of small‐amplitude parallel and oblique whistler mode waves are investigated for a wide range of plasma parameters by numerically solving the full electromagnetic Vlasov dispersion equation. To investigate the cold‐plasma and electrostatic approximations for the whistler mode, the results are compared with results obtained using these descriptions. For large wavelengths (k c/ω_{ i }≲100), the cold‐plasma description is often accurate, while for short wavelengths (k c/ω_{ i }≳100) and sufficiently oblique propagation, the electrostatic description is often accurate. The study demonstrates that in a Vlasov plasma the whistler mode near resonance has a group velocity more nearly parallel to the magnetic field than that predicted by cold‐plasma theory.

Saturation of drift instabilities by electron dynamics
View Description Hide DescriptionAn analytical explanation is provided for phenomena observed in gyrokinetic particle simulations used to study the nonlinear evolution of the universal instability. The nonlinear E×B advection of nearly resonant electrons is related to the deactivation of the instability mechanism, leading to oscillations of the perturbation amplitude around a nonzero saturated level. An equivalent description is given in terms of nonresonant mode coupling. A self‐consistent evolution equation for the amplitude is obtained for the case where this mechanism predominates, and is successfully compared to the gyrokinetic simulations.

Nonlinear saturation of single‐mode trapped ion instability
View Description Hide DescriptionThe trapped ion instability appears as a single mode in the Columbia Linear Machine and is expected to appear as such in tandem mirrors. Using three‐wave, nonresonant mode coupling of radially nonlocal eigenfunctions, two mode‐coupling equations for the unstable fundamental and stable first‐order radial harmonics are derived and solved. Mode coupling to the damped higher‐order radial harmonics is the principal physical saturation mechanism of the unstable fundamental mode.

Bounce‐averaged Monte Carlo energy and pitch angle scattering operators
View Description Hide DescriptionBeginning with a linearized Coulomb collision operator, the derivations of bounce‐averaged energy and pitch angle scattering terms that have been suitably discretized for Monte Carlo use are described. The derivation assumes that the electrostatic potential is constant along a magnetic field line. The collision operators are applied to electron transport in ELMO Bumpy Torus.

Cavity mode analysis of plasma frequency waves in inhomogeneous cylindrical plasmas
View Description Hide DescriptionA linear, electromagnetic analysis is presented of the cavity mode structure and dispersion relations for waves with frequency near the electron‐plasma frequency in cylindrical plasmas with monotonically decreasing electron density and finite magnetic field. Analytic solutions are obtained which are exact in the limits when either the ratio of electron cyclotron to plasma frequency or the ratio of plasma size to parallel wavelength tends to infinity. Comparison with numerical solutions confirms that the analytic results are highly accurate even for fairly modest values of these ratios. Thermal effects are incorporated, including Landau (damping or) gain, which show the lowest‐order transverse modes to have greatest gain. These modes are highly localized near the cylinder axis so that the plasma itself acts as a cavity, regardless of edge boundary conditions. The theory thus enables an interpretation to be made of maser action in quasicylindrical plasmas such as tokamaks.

Enhanced lower‐hybrid heating caused by frequency modulation
View Description Hide DescriptionThe effect of frequency modulation during stochastic ion heating induced by lower‐hybrid waves is examined. The modulation occurs either in the ion‐cyclotron frequency because of the variation of the magnetic field in toroidal devices, or it can be externally imposed on the frequency of the lower‐hybrid waves. It has already been observed numerically [Phys. Fluids 2 7, 184 (1984)] that a small variation in the ion‐cyclotron frequency can induce velocity diffusion for wave amplitudes well below the stochasticity threshold in a uniform magnetic field. Here a detailed study reveals that to the lowest order in the small parameters, the modulational effects can be incorporated in a two‐dimensional Hamiltonian. This allows the derivation of the new stochasticity thresholds. It is found that a small amount of modulation, Δω/ω≲1%, produces an order of magnitude reduction in the stochasticity threshold relative to the constant frequency case. The stochastic regime in velocity space also grows in size, resulting in a considerable increase of number of heated particles in the case of devices with modest aspect ratio. Both ion‐cyclotron and wave‐frequency modulation lead to similar results. The modulation of the wave frequency offers the ability to control and optimize the modulation parameters and is proposed as a method to enhance radio frequency (rf) heating.

Nonlinear circularly polarized electromagnetic waves in a warm magnetoplasma
View Description Hide DescriptionThis paper considers the nonlinear interaction of a background warm plasma with intense, circularly polarized radiation along an external magnetic field. It is shown that the couplings of the adiabatic particle as well as the hydrodynamic plasma motions with the radiation field can lead to localized wave‐envelope packets. The resulting self‐consistent equilibria consist of finite‐amplitude electron density perturbations which are nonlinearly driven by electromagnetic fields.

Kinetic theory of resistive ballooning modes
View Description Hide DescriptionA linear and nonlinear kinetic theory of resistive ballooning modes that includes diamagnetic drifts and finite Larmor radius effects is presented. The linear stability of resistive ballooning modes is examined analytically and numerically. A renormalized resistive ballooning equation is derived, and the saturation level of the instabilities is analytically calculated. Finally, a calculation of the electron thermal conductivity for the large ω_{*} regime is presented.

Confinement properties of positive ambipolar potential equilibria in a bumpy torus
View Description Hide DescriptionThe confinement properties of particles and energy in the ELMO Bumpy Torus in the collisionless‐ion regime, characterized by positive ambipolar potential in the core region, have been investigated using a one‐dimensional transport code. It is found that both particle and energy confinement can be significantly improved over the conventional negative‐potential‐regime values. A centrally peaked radial profile of microwave heat deposition is found to be an essential condition for positive potential equilibria.

The effect of alpha particles on the stability of a bumpy torus reactor
View Description Hide DescriptionThe macroscopic stability of an ignited ELMO Bumpy Torus (EBT) reactor is investigated by studying the effects of the alpha particles generated by the deuterium–tritium (D–T) fusion reaction on the background interchange mode, the interacting interchange mode, and the high‐frequency compressional Alfvén and coupled modes. A fluid description is used for the background plasma while a kinetic treatment is utilized for the hot‐electron species and the alpha particles. It is shown that the alphas tend to mildly destabilize the interacting interchange while stabilizing the background interchange because of their sizable Larmor radii. The destabilization is most pronounced when the beta of the alpha particles is highest, i.e., at birth, and recovery of stabilization takes place as these particles slow down toward thermalization. It is also shown that the alphas completely stabilize the high‐frequency modes, so that it can safely be concluded that fusion alphas present no detrimental effects on the stability of an EBT reactor that possesses an appropriate hot‐electron ring for macroscopic stability.

The effect of finite β on stellarator transport
View Description Hide DescriptionA theory of the modification of stellarator transport caused by the presence of finite plasma pressure is developed and applied to a range of stellarator configurations. For typical planar‐axis stellarators, transport can change by a factor of about 4–5 in the progression from zero pressure to the equilibrium β limit of the device. For transport‐optimized configurations, the factor can be over an order of magnitude. Thus, a stellarator with transport‐optimized vacuum fields can have poor confinement at the desired operating β. Without an external compensating field, increasing β tends to degrade confinement, unless the initial field structure is very carefully chosen. The theory permits correct determination of this vacuum structure, in terms of the desired structure of the field at a prescribed operating β. With a compensating external field, the deleterious effect of finite β on transport can be partially eliminated.