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Volume 24, Issue 11, November 1981

Further results on the thermal mixing layer downstream of a turbulence grid
View Description Hide DescriptionSimultaneous velocity and temperature measurements in the thermal mixing layer downstream of a partially heated turbulence grid are reported. The temperature data are in good agreement with earlier results. The velocity‐temperature correlations are new and permit a systematic assessment of an earlier similarity theory for the region far downstream of the grid. By selecting the virtual origin appropriately and by reconsidering the role of scalar dissipation in that region, agreement between measurement and prediction with respect to temperature intensity and two mean lateral fluxes, that of temperature and of temperature intensity, is achieved.

Invariants for the one‐point vorticity and strain rate correlation functions
View Description Hide DescriptionAn algorithm to enumerate the number of independent scalars that determine the general tensor formed from n velocity gradients at a point in homogeneous‐isotropic turbulence is elaborated for n = 4. The physical content of the invariants that result as well as their determination from experiment is discussed.

Experiments on the propagation of composite internal wave trains
View Description Hide DescriptionExperiments have been performed in a two‐layer stably stratified flow where the mean velocity surpasses the phase speed of all internal waves of mode one. Results show the existence of two wave trains propagating in the direction of the mean flow (excited by a two‐dimensional wave maker), which are identified by the modulation of the amplitude of the linearly superposed wave trains. Measurements of the composite wave speed and dispersion relation are presented. Wave speeds are shown to be very near the mean. The qualitative behavior of composite wave trains is shown to appear similar to that of unstable Kelvin–Helmholtz waves where spatial growth is dominant, leading to possible misinterpretation.

Structure of shock waves in gas‐particulate fluidized beds
View Description Hide DescriptionThe structure of a two‐phase shock wave in a gas‐particulate fluidized bed is investigated. The present study gives (i) the jump relation across the two‐phase shock, (ii) the critical condition beyond which the stable two‐phase shock breaks up, and (iii) the distribution of the dependent variables through the two‐phase shock and the two‐phase shock thickness.

Heat flux induced wave fronts
View Description Hide DescriptionThe thermalization of a large energy flux W in a medium of density ρ leads to heat waves consisting of compression and expansion fronts, associated with substantial bulk motion. Particles can enter or leave the thermalization front at subsonic, sonic, or supersonic speed, allowing for a wide variety of wave modes. The wave mode depends on power input, thermodynamic response of the medium, and the boundary conditions. The intake Mach number mainly determines the degree of compression achieved across the front. Using macroscopic conservation equations the thermodynamic properties of the wave fronts are discussed in a pressure–density diagram and compared to other thermodynamic processes. The fluid dynamic properties are shown as a function of the absorbed power input W/ρ. Finally, the hydrodynamic properties of the entire two front heat wave are presented for specified boundary conditions in the response plane h–W/ρ, showing the enthalpy h of the heated target material as a function of an external energy flux W/ρ. Examples of the boundary conditions of free and retarded flow and heat transport due to inverse bremsstrahlung are discussed.

Shape of a drop in an electric field
View Description Hide DescriptionThe shape of an axisymmetric dielectric drop in a uniform electric field is computed numerically. The problem is formulated as a nonlinear integro‐differential system of equations. They are discretized and the resulting algebraic system is solved by Newton’s method. The results show that when the dielectric constant ε is larger than a critical value ε_{ c }, the drop develops two obtuse‐angled conical points at its ends for a certain field strength. For ε<ε_{ c }, the drop elongates and retains its original nearly prolate spheroidal shape without developing conical points as the field is increased. The numerical results are in good agreement with the moment and two‐point approximations. The energy, volume, and area of the drop are computed, and the two‐dimensional case is also treated.

Recurrence, dimensionality, and Lagrange stability of solutions of the nonlinear Schrödinger equation
View Description Hide DescriptionThe scope and limitations of elementary methods for establishing qualitative properties of solutions of the nonlinear Schrödinger equation are discussed and clarified. The concept of recurrence is shown to be related to Lagrange and Poisson stability of the system. Lagrange stable motions in any number of spatial dimensions are shown to be recurrent in general.

Chaotic (strange) and periodic behavior in instability saturation by the oscillating two‐stream instability
View Description Hide DescriptionThe nonlinear Schrödinger equation with linear growth and damping is truncated to three waves. The resulting system of nonlinear ordinary differential equations describes the excitation of linearly damped waves by the oscillating two‐stream instability driven by a linearly unstable pump wave. This system represents a simple model for the nonlinear saturation of a linearly unstable wave. The model is examined analytically and numerically as a function of the dimensionless parameters of the system. It is found that the model can exhibit a wealth of characteristic dynamical behavior including stationary equilibria, Hopf bifurcations to periodic orbits, period doubling bifurcations, chaotic solutions characteristic of a strange attractor, tangent bifurcations from chaotic to periodic solutions, transient chaos, and hysteresis. Many of these features are shown to be explainable on the basis of one‐dimensional maps. In the case of chaotic solutions, evidence for the presence of a strange attractor is provided by demonstrating Cantor set‐like structure (i.e., scale invariance) in the surface of section.

Finite Larmor radius model for axisymmetric compact toroids
View Description Hide DescriptionStability equations for axisymmetric compact toroidal configurations without a toroidal magnetic field are derived in a variational form. The application of a particular ordering procedure gives equations which include the coupling of geometry to finite Larmor radius as well as including resonant ion effects. The equations are readily obtained by applying the ordering assumptions to a variational dispersion functional form of the Vlasov‐fluid equations. Application of the dispersion functional to find necessary and sufficient conditions for stability are made for the case of small, but finite Larmor radius and for the case of zero Larmor radius. The mathematical result that the zero Larmor radius case is more optimistic is physically understood in the context of an interesting stabilizing mechanism involving parallel kinetic effects.

Plasma equilibrium with rational magnetic surfaces
View Description Hide DescriptionThe self‐consistent classical plasma equilibrium with diffusion is studied in a toroidal geometry having a sheared magnetic field. Near each rational surface it is found that the pressure gradient is zero unless the Fourier component of 1/B ^{2}, which resonates with that surface, vanishes. Despite the resonances, the overall plasma confinement is, in practice, only slightly modified by the rational surfaces.

Resistive ballooning modes
View Description Hide DescriptionResistive ballooning modes are analyzed in the neighborhood of the magnetic axis. There are two mode branches, one connecting to the usual tearing mode and the other to the magnetohydrodynamic ballooning mode. Unstable modes satisfy the generalized tearing dispersion relation. Resistive ballooning modes can be unstable for a wide range of beta, but may be stabilized by interchange effects and finite sound speed. The analytic results are in qualitative agreement with low mode number numerical solutions.

Kinetic ballooning‐interchange modes in tandem mirrors
View Description Hide DescriptionA hybrid fluid‐gyrokinetic derivation of the kinetic modifications to the ballooning‐interchange eigenmode equation is presented which removes the large and small wave frequency restrictions of Tang and Catto. The derivation employs a long‐thin ordering and an eikonal ansatz, but otherwise allows arbitrary asymmetry, beta, pressure anisotropy, and unperturbed parallel and perpendicular electrostatic field.

Finite Larmor radius stabilization of ballooning modes in tokamaks
View Description Hide DescriptionA ballooning mode equation that includes full finite Larmor radius effects has been derived from the Vlasov equation for circular tokamak equilibrium. A numerical solution of this equation shows that finite Larmor radius effects are stabilizing.

Currents generated by lower hybrid waves
View Description Hide DescriptionElectron currents can be driven in a linear plasma by the absorption of lower‐hybrid waves excited primarily in one direction. Current‐drive has been demonstrated both for collisional and resonant‐electron absorption. The magnitude of the excited current is compared with the predictions from an electron kinetic equation with a Lorentz collision operator in the regime k _{∥} v _{ t e }/ω≪1.

Derivation of the mode conversion‐tunneling equation from the Vlasov equation
View Description Hide DescriptionInstead of using the uniform warm plasma dispersion relation with an inverse Fourier transform to form the mode conversion‐tunneling equation, the equation is developed directly from the Vlasov equation and Maxwell’s equations. The plasma is assumed uniform parallel to B_{0}. The expansion parameters are the Larmor orbit and the scale length, keeping terms to order ρ^{2} _{L} and L ^{−1} = (1/ω_{ c })(dω_{ c }/d x). For the special case with ω≃2ω_{ c } in a single species plasma, the asymptotic form of the mode conversion‐tunneling equation is unchanged, but localized third and first derivative terms appear even as absorption vanishes.

Cyclotron wave generation and phase velocity control on an intense relativistic electron beam
View Description Hide DescriptionThe axisymmetric, negative energy branch of the Doppler‐shifted electron cyclotron eigenmode has been excited on an unneutralized, 15 kA, 2.25 MV, relativistic electron beam propagating along a guide magnetic field interior to a vacuum conducting wave guide. Large amplitude waves have been generated by means of a traveling wave interaction with helical slow wave structures. The growth process has been identified as the resonant coupling of the negative energy cyclotron mode with the positive energy helix modes. The large amplitude wave produced was propagated on the beam and then identified to be the desired symmetric, negative energy cyclotron wave by direct measurement of its azimuthal symmetry, wavelength, and phase velocity direction. The on‐axis axial electric field component of the wave was inferred to be in excess of 10 MV/m. Finally, as is necessary for collective ion acceleration by these waves, the cyclotron wave phase velocity has been controlled, in the range 0.20–0.06 times the speed of light, by spatially changing the guide magnetic field strength.

Nonlinear saturation of the drift cyclotron loss‐cone instability. II. Comparison with PR‐6 data
View Description Hide DescriptionThe single‐mode time‐asymptotic nonlinear saturation of the drift‐cyclotron loss‐cone instability is considered. In this model, there is a uniform magnetic field as well as mirror losses. The linear stability boundary, and the saturation levels of unstable oscillations and floating potential near this boundary, are determined. The theory is then compared with observations on the PR‐6 mirror experiment. Good agreement is obtained using a simple energy model for the slow time variation of the equilibrium distribution function.

Relativistic plasma half‐space with an external magnetic field
View Description Hide DescriptionThe boundary value problem involving a magnetized plasma half‐space confined by a perfectly reflecting interface is solved using the relativistic Vlasov–Maxwell system. The external field is assumed to be constant and along the normal of the boundary, and only the linear responses to obliquely incident vacuum waves are considered. When the incidence is normal, only a pair of purely transversal modes, similar to the ordinary and extraordinary modes of the infinite systems, can be excited. When the incidence is oblique, two additional hybrid p and s modes may exist, which are decoupled into the longitudinal and transversal parts only when there is no external field. The p mode vanishes if the field at the boundary, B(0), is in the plane of incidence, and the s mode is zero if B(0) is in the plane which is perpendicular to both the interface and the plane of incidence. Reflection, transmission, and absorption coefficients are calculated for the case of normal incidence.

Field‐reversed experiments (FRX) on compact toroids
View Description Hide DescriptionEquilibrium, stability, and confinement properties of compact toroids produced in field‐reversed theta‐pinch experiments (FRX) are reported. Two experimental facilities, FRX‐A and FRX‐B, have been used to study highly elongated compact toroid plasmas confined in a purely poloidal field geometry. Spatial scans and fill pressure scaling of the equilibrium plasma parameters are presented. Plasma conditions range from T _{ e }∼150 eV, T _{ i }∼800 eV, n _{ m }∼1×10^{15} cm^{−3} to T _{ e }∼100 eV, T _{ i }∼150 eV, n _{ m }∼4×10^{15} cm^{−3}. Typical confined plasma dimensions are: major radius R∼4 cm, minor radius a∼2 cm, and total length 35–50 cm. The plasma configuration remains in a stable equilibrium for up to 50 μsec followed by the destructive n = 2 rotational instability. The stable period prior to the onset of the rotational mode is up to one hundred times greater than characteristic Alfvén transit times of the plasma. This stable period increases and the mode growth rate decreases with increased a/ρ_{ i } (where ρ_{ i } is the ion gyroradius). Agreement of experimental and theoretical mode frequencies for the instability is observed. Preferential particle loss has been proposed as a likely cause of rotation. The particle inventory at the onset of the instability is consistent with this hypothesis. The particle loss rate is also consistent with the predicted anomalous transport near the separatrix. Contributions to rotational instability from diffusion, end‐shorting, equipartition, and compression are also discussed.

Anomalous heat conduction at the ends of a theta pinch
View Description Hide DescriptionIn a small theta pinch, a density pulse propagating to the end is found to form after the compression phase. Doppler‐shift measurements of the He i i, λ = 4686 Å line indicate that axial particle flow is present only during the implosion, thermal conduction being the dominant end‐loss mechanism at later times. Comparison of the actual heat flux with that expected from classical theory, the presence of an enhanced forbidden component as well as plasma satellites of the He i, λ = 4922 Å line outside the coil region, and a flat‐top electron distribution function obtained by Thomson scattering, lead to the conclusion that thermal conduction is anomalous. The concept of a replacement wave is successfully applied to account for the observed phenomena.