Volume 28, Issue 5, May 1985
Index of content:

Spatial localization of bursts of the drift cyclotron loss cone instability
View Description Hide DescriptionSpatial localization of bursts of the drift cyclotron loss cone instability in a mirror machine is caused by asymmetric motion of the background plasma. The dynamic behavior of the plasma determines the appearance of the instability

On the mode equations for the electrostatic ballooning instability in tokamaks
View Description Hide DescriptionSome discrepancies are pointed out for kinetic and fluid dispersion relations of low‐frequency electrostatic modes in tokamaks. It is shown that both electrons and ions contribute to the ballooning term, which is subject to finite ion Larmor radius correction.

Electron‐cyclotron heating of a tokamak reactor at down‐shifted frequencies
View Description Hide DescriptionThe absorption of electron cyclotron waves in a hot, dense tokamak plasma is investigated for the case of the extraordinary mode for outside launching. It is shown that, for electron temperatures T _{ e }≥5 keV, strong absorption occurs for oblique propagation at frequencies significantly below the electron gyrofrequency at the plasma center. A new density dependence of the wave absorption is found which is more favorable for plasma heating than the familiar n ^{−1} _{ e } scaling.

Equilibrium of a liquid in a spherical shell caused by gravity, surface tension, and van der Waals forces
View Description Hide DescriptionA rigorous proof is presented which shows that the classical Young–Laplace equation cannot predict a continuous liquid layer inside a spherical‐shell container. An augmented Young–Laplace equation is in turn derived to describe the equilibrium profiles of a continuous liquid hydrogen layer inside a spherical‐shell inertial fusion target. The augmentation is achieved by adding to the total free energy a term originating from the attractive intermolecular forces of the van der Waals type.

Ultrasonically stimulated low‐frequency oscillation and breakup of immiscible liquid drops: Photographs
View Description Hide DescriptionShape oscillation and fission of drops of an oil in water were studied photographically. Novel theoretical results are also presented concerning the resonance response of drops. The initial radii of the drops observed were typically between 1 and 2 mm. Two‐lobed oscillations of moderate amplitude were driven by the temporally modulated radiation pressure of an ultrasonic wave. The tendencies for drops to elongate and break are discussed qualitatively. The oil is more compressible than water. Radiation stresses repel the oil from pressure nodes of the ultrasonic standing wave. The excitation of a transient deformation that appears to run around the drop’s surface is noted.

Amplitudes and wavelengths of wavy Taylor vortices
View Description Hide DescriptionThe wave height and wavelength of wavy Taylor vortices have been measured in a fluid column of aspect ratio 31 and radius ratio 0.88. The fluid was either water or 10 cS silicone oil. The wave height as well as the wavelength were found to increase with Reynolds number until 1.8R _{ c } for a state with 15 pairs of vortices and four azimuthal waves. Subsequent changes in the wave height and azimuthal wavenumber seem to be primarily caused by changes in the number of vortices which are caused by the increase of the wavelength.

Some bifurcation diagrams for Taylor vortex flows
View Description Hide DescriptionThe numerical continuation and bifurcation methods of Keller [H. B. Keller, in A p p l i c a t i o n s o f B i f u r c a t i o n T h e o r y (Academic, New York, 1977), pp. 359–384] are used to study the variation of some branches of axisymmetric Taylor vortex flow as the wavelength in the axial direction changes. Closed ‘‘loops’’ of solutions and secondary bifurcations are determined. Variations with respect to Reynolds number show the same phenomena. The results presented here show that Taylor vortices with periodic boundary conditions exist in a wider range of wavelengths, λ, than observed in the Burkhalter/Koschmieder experiments [Phys. Fluids 1 7, 1929 (1974)]. They also show that there is possibly a λ subinterval within the neutral curve of Couette flow such that there are no Taylor vortex flows with smallest period in this interval.

A nonlinear study of Kelvin–Helmholtz stability
View Description Hide DescriptionThe Kelvin–Helmholtz stability problem is studied by employing the variational method. With the restriction to a single dominant spatial mode, a set of fully nonlinear evolution equations is derived. The limiting states of these evolution equations are discussed and their stability is analyzed. It is found that at the critical point for the linear stability problem, another stable limiting state with finite amplitude appears. A nonlinear sinusoidal wave state is also found. The nonlinear dispersion relation for this wave is derived. The wavenumber of the dominant spatial mode and its evolution with time can also be determined with the aid of the variational method.

Nonparallel stability of heated two‐dimensional boundary layers
View Description Hide DescriptionThe method of multiple scales is used to determine three partial differential equations describing the modulation of the amplitude and complex wavenumbers of three‐dimensonal (3‐D) waves propagating in two‐dimensional (2‐D) heated liquid layers. These equations are solved numerically along the characteristics subject to the condition that the ratio of the complex group velocities in the streamwise and transverse directions be real. A new criterion for the most dangerous frequency is proposed. For an n factor of 9, F=25×10^{−} ^{6} is found to be the most dangerous frequency for the Blasius flow. Three‐dimensional waves yield lower n factors than 2‐D waves, irrespective of the heating distribution. For a power‐law heating distribution of the form T=T _{ e }+A x ^{ N }, one cannot make a general statement on the effect of N on the stability. Numerical results are presented that show the n factor to increase with an increase or a decrease in N.

Design and testing of axisymmetric nozzles for ion‐molecule reaction studies between 20 °K and 160 °K
View Description Hide DescriptionAxisymmetric contoured Laval nozzles have been calculated in order to obtain uniform supersonic flow. For nitrogen and oxygen gases the calculation uses only similarity techniques, but for helium, complete calculations of the isentropic core and the boundary layer were performed for two sets of boundary conditions. Experimental testing has shown that the nitrogen or oxygen flows exhibit a good uniformity but that the helium flow was uniform only for one set of boundary conditions. The purpose of this work is to provide flow reactors for studies of ion‐molecule reactions in the real temperature range of interstellar clouds (T<80 °K).

A kinetic theory analysis of evaporation and condensation of a diatomic gas
View Description Hide DescriptionThe evaporation and condensation problem of diatomic gas was studied based on the BGK–Morse kinetic model equation. The results showed that (1) the macroscopic jump coefficients were insensitive to the variation in the relaxation time of internal energy, (2) the pressure jump caused by evaporation (condensation) or by heat transfer was almost the same as the value of a monatomic gas, and (3) the temperature jumps of a diatomic gas were smaller than those of a monatomic gas in accordance with the quantities of specific heats. Approximate expressions of macroscopic jump coefficients of a diatomic gas for arbitrary values of the internal degrees of freedom and those of accommodation coefficients of mass, translational energy, and internal energy are obtained using the jump coefficients of a monatomic gas.

Flow of a rarefied gas past a circular cylinder
View Description Hide DescriptionThe flow of a rarefied gas past a circular cylinder at low Mach numbers is studied on the basis of the linearized Bhatnager–Gross–Krook model equation. The flow field is divided into two regions: the kinetic region adjacent to the cylinder and the near continuum region outside it. The Oseen–Stokes equation is treated in the near continuum region. In the kinetic region, the simultaneous integral equations for the flow velocity, density, and temperature are derived from the model equation. These equations are solved numerically by matching with the Oseen–Stokes solution. The distributions of the flow velocity are obtained for a wide range of the Knudsen numbers. The drag on the cylinder is also calculated and is favorably compared with the previous formulas valid for small and large Knudsen numbers.

Numerical study of the initial condition dependence of Burgers’ turbulence
View Description Hide DescriptionThe initial condition dependence of the statistical properties of Burgers’ turbulence is investigated by numerical simulations. As a result, the energy spectrum of the form k ^{ a } in the lowest wavenumber range is shown not to change in time as long as the value of a is in the range [0,2]. It is also shown that the appearance of the k ^{−} ^{2} spectrum and the time dependence of Taylor’s microscale are the universal properties of Burgers’ turbulence, and are independent of the initial conditions. On the other hand, the energy decay law and the time development of the skewness of the velocity derivative are dependent on the initial conditions. Especially, the energy decay law is shown to take the form, E(t)∝t ^{−p }, where p has a value in the range [0.63–1.14] depending on the large‐scale structure of the turbulence.

A passive scalar field convected by turbulence
View Description Hide DescriptionClassical statistical mechanics is applied to the study of a passive scalar field convected by isotropic turbulence. A complete set of independent real parameters and dynamic equations are worked out to describe the dynamic state of the passive scalar field. The corresponding Liouville equation is solved by a perturbation method based upon a Langevin–Fokker–Planck model. The closure problem is treated by a variational approach reported in earlier papers. Two integral equations are obtained for two unknown functions: the scalar variance spectrum F(k) and the effective damping coefficient Ω(k). The appearance of the energy spectrum of the velocity field in the two integral equations represents the coupling of the scalar field with the velocity field. As an application of the theory, the two integral equations are solved to derive the inertial‐convective‐range spectrum, obtaining F(k)=0.61 χε^{−} ^{1} ^{/} ^{3} k ^{−} ^{5} ^{/} ^{3}. Here χ is the dissipation rate of the scalar variance and ε is the dissipation rate of the energy of the velocity field. This theoretical value of the scalar Kolmogorov constant, 0.61, is in good agreement with experiments.

Direct interaction approximation for Vlasov turbulence from the Kadomtsev weak coupling approximation
View Description Hide DescriptionThe weak coupling approximation of Kadomtsev is applied to Vlasov turbulence and shown to lead straightforwardly to the equations of the direct interaction approximation. This application of the weak coupling approximation was apparently incorrectly carried out in Kadomtsev’s book. The method has the advantage of simplicity and makes contact with other more conventional approximations.

Reduced DIA equations for the weak warm beam instability in the strong mode‐coupling limit
View Description Hide DescriptionThe direct interaction approximate (DIA) is considered for a one‐component, one‐dimensional turbulent plasma excited by wave–particle interactions in the limit of a weak warm beam instability. The situation where strong mode coupling effects take place is explicitly studied. In this limit the DIA equations for the nonlinear evolution of the instability are reduced to a dimensionless set of equations for the spectral harmonics of the correlation functions in which all time dependence has been scaled out. The distinctive polarization terms and mean field coupling renormalization effects of the DIA are seen to be of the same order as the usual quasi‐Gaussian or ‘‘test particle’’ renormalization effects. From these equations it is shown that the true growth rate γ_{ k } is given by γ_{ k } =A _{[k]}γ^{ql} _{ k }, where γ^{ql} _{ k } is the quasilinear growth rate; the renormalizing function A _{[k]} is shown to be larger than unity with A _{[k]} =O(1), corresponding to a net increase of energy transfer between the waves and the particles as compared with the quasilinear predictions. The function A _{[k]} may be a slowly varying function of k, but is independent of beam parameters. The basic physical effects are the promotion of mode‐coupling effects to the same order as the quasilinear terms by the intermediation of resonant particles in the mode‐coupling matrix elements, plus the consideration of the non‐Gaussian character of the self‐consistent electric field fluctuations. Arguments are presented that the reduced DIA equations are exact in this limit.

Generalized gyrokinetics
View Description Hide DescriptionA nonlinear gyrokinetic formalism is developed which admits mean velocities comparable to thermal speeds in arbitrary magnetic field geometry. The theory is fully electromagnetic and does not employ an eikonal ansatz. The moments of the gyrokinetic distribution function, as well as the gyrokinetic equation it satisfies, are derived to first order in the guiding center expansion parameter. This large mean velocity formulation is important whenever the E×B drift over the thermal speed cannot be ordered small. For example, upon summing over all species, the moment description yields a set of generalized magnetohydrodynamic equations which are valid to one order higher in the expansion parameter.

Resistive evolution of general plasma configurations
View Description Hide DescriptionThe resistive evolution through equilibrium states of general plasma configurations with closed magnetic field lines is described. Cases where the magnetic field forms magnetic surfaces and where the magnetic field is ergodic are treated. In the former case a simple equation for the rate of change of rotational transform at fixed values of toroidal flux is obtained, as is already known. In the latter case the evolution of the equilibrium is naturally described in terms of the magnetic helicity, using the formalism of relaxed states introduced by J. B. Taylor [Phys. Rev. Lett. 1 3 3, 1139 (1974)]. The equation for rate of change of magnetic helicity is shown to be a general law of resistive evolution, implying the former equation for rotational transform in the case of magnetic surfaces. In principle, the resistive evolution model provides a complete description of global long‐time‐scale plasma behavior in the limit where the plasma density vanishes. In this limit, the magnetohydrodynamic description of a plasma is not practical because of the vanishing of the inertial time scale.

Magnetohydrodynamic ballooning instabilities excited by energetic trapped particles
View Description Hide DescriptionA new branch of magnetohydrodynamic ballooning modes is shown to be destabilized by energetic trapped particles. Both the real frequencies and growth rates of the instabilities are comparable to the trapped‐particle precession frequencies. The theoretical results are also shown to be consistent with the high‐frequency (∼100 kHz) oscillations observed during the high‐power beam‐injection experiments in the tokamak experiment PDX.

Thermal structures on resonance cones in a weakly collisional plasma
View Description Hide DescriptionResonance cones in the range of the lower‐hybrid frequency are investigated experimentally and theoretically. Two competing models, which alternatively relate the cone structure to the curvature of the dispersion branch or to the onset of damping are considered. In an extended model the superposition of both effects is studied by numerical integration of the potential. The cone structures related to the second effect turn out to be sensitive to collisional damping.