Volume 24, Issue 8, August 1981
Index of content:

Görtler instability
View Description Hide DescriptionGörtler instability for boundary‐layer flows over generally curved walls is considered. The full‐linearized disturbance equations are obtained in an orthogonal curvilinear coordinate system. A perturbation procedure to account for second‐order effects is used to determine the effects of the displacement thickness and the variation of the streamline curvature on the neutral stability of the Blasius flow. The streamwise pressure gradient in the mean flow is accounted for by solving the nonsimilar boundary‐layer equations. Growth rates are obtained for the actual mean flow and compared with those for the Blasius flow and the Falkner–Skan flows. The results demonstrate the strong influence of the streamwise pressure gradient and the nonsimilarity of the basic flow on the stability characteristics.

Generalized scalar potentials for linearized three‐dimensional flows with vorticity
View Description Hide DescriptionThe three‐dimensional time‐dependence velocity of incompressible fluids satisfying linearized incompressible Navier–Stokes equations can often be expressed in terms of a single h by v = A∇h+B(l̂×∇h)+Cl×(l×∇h), where l is a constant vector and A,B,C are scalar operators. For B,C≠0,curl v≠0.

Instabilities of a compressible stratified fluid in horizontal sheared motion
View Description Hide DescriptionThe stability characteristics of a Helmholtz velocity profile in a stably stratified, compressible atmosphere in the presence of a lower boundary are studied. A jump in the Brunt–Väisälä frequency is introduced and the level at which this jump occurs is assumed to be different from the shear zone, to simulate sharp temperature discontinuities in the atmosphere. The results are compared with those of Pellacani, Tebaldi, and Tosi and Lindzen and Rosenthal. In the present configuration, new unstable modes with larger growth rates are found. The wavelengths of the most unstable gravity waves for the parameters pertaining to observed cases of clear air turbulence agree quite closely with the experimental values.

Some interesting properties of two‐dimensional turbulence
View Description Hide DescriptionThe effect of superimposed rigid body motions on the structure of two‐dimensional turbulence is examined. It is found that with regard to the fluctuation dynamics of the flow, the rotational behavior of two‐dimensional turbulence is quite different from its three‐dimensional counterpart. The implications that this has on turbulence modeling are discussed briefly.

Spectral transfer and velocity derivative skewness equation for a turbulent velocity field
View Description Hide DescriptionA derivation of the high wavenumber spectral transfer equation is given which indicates the wavenumber range of validity for this equation in terms of Re_{λ}. A similar treatment is given for the commonly used spectral approximation for the velocity derivative skewness and the analogous scalar spectral quantities.

Variational moment solutions to the Grad–Shafranov equation
View Description Hide DescriptionA variational method is developed to find approximate solutions to the Grad–Shafranov equation. The surfaces of the constant poloidal magnetic flux ψ(R, Z) are obtained by solving a few ordinary differential equations, which are moments of the Grad–Shafranov equation, for the Fourier amplitudes of the inverse mapping R(ψ, ϑ) and Z(ψ, ϑ). Analytic properties and solutions of the moment equations are considered. Specific calculations using the Impurity Study Experiment (ISX‐B) and the Engineering Test Facility (ETF)/International Tokamak Reactor (INTOR) geometries are performed numerically, and the results agree well with those calculated using standard two‐dimensional equilibrium codes. The main advantage of the variational moment method is that it significantly reduces the computational time required to determine two‐dimensional equilibria without sacrificing accuracy.

Appendix: Variational moment solutions to the Grad–Shafranov equation [Phys. Fluids 24, 1431 (1981)]
View Description Hide DescriptionAppendix to Variational moment solutions to the Grad–Shafranov equation (AIP)

Thermal heat flux in a plasma for arbitrary collisionality
View Description Hide DescriptionThe thermal heat flux along a uniform magnetic field due to a temperature gradient is calculated using a Monte Carlo solution to the Fokker–Planck equation. This numerical solution, which is computed for a particular electron temperature profile, is valid for arbitrary mean‐free‐path, λ_{mfp}. The calculated heat flux makes a smooth transition between the analytic expressions for the short and long λ_{mfp} limits.

Centrifugal separation of a multispecies pure ion plasma
View Description Hide DescriptionConsider an unneutralized column of ions (a pure ion plasma) confined by an axial magnetic field. Because of space charge, there is a large radial electric field and a consequent rotation of the plasma column. For a multispecies ion plasma, the rotation tends to produce centrifugal separation of the plasma into its component species. Self‐consistent thermal equilibrium states which exhibit various degrees of separation are discussed.

Excitation of multiple ion‐acoustic shocks
View Description Hide DescriptionMultiple electrostatic ion‐acoustic shock‐like density perturbations are observed experimentally to be launched from a large plate in a quiescent collisionless plasma. The formation of the fastest shock is due to ions expelled from the edge of the ’’transient sheath’’ surrounding the plate. The second shock then evolves as a result of the surrounding ions filling in the region of density depression created by the first shock. Reflected ions and turbulent ion‐acoustic noise at the shock front and amplitude modulation in the trailing dispersing oscillations are only observed for the leading shock.

Parametric instabilities in an electron beam plasma system
View Description Hide DescriptionThe excitation of low‐frequency parametric instabilities by a finite wavelength pump in a system consisting of a warm electron plasma traversed by a warm electron beam is investigated in a fluid dissipationless model. The dispersion relation for the three‐dimensional problem in a magnetized plasma with arbitrary directions for the waves is derived, and the one‐dimensional case is analyzed numerically. For the one‐dimensional back‐scattering decay process, it is found that when the plasma‐electron Debye length (λ_{D} ^{ p }) is larger than the beam‐electron Debye length (λ_{D} ^{ b }), two low‐frequency electrostatic instability branches with different growth rates may exist simultaneously. When λ_{D} ^{ p }≃λ_{D} ^{ b }, the large growth rate instability found in the analysis depends strongly on the amplitude of the pump field. For the case λ_{D} ^{ p }<λ_{D} ^{ b }, only one low‐frequency instability branch is generally excited.

Variational principle for low‐frequency stability of collisionless plasmas
View Description Hide DescriptionAn analysis of the stability of an arbitrary β collisionless plasma to modes with wavelengths greater than the ion gyroradius is presented. The stability of such a plasma to perturbations that grow on the hydrodynamic time scale is determined by the Kruskal–Oberman energy principle. However, a configuration which is predicted to be stable on the basis of this kinetic energy principle may still be unstable to modes that grow with a frequency comparable to the diamagnetic or curvature drift frequency. A new variational principle that gives sufficient conditions for instability of these low‐ frequency modes is derived. The new principle indicates that two types of instabilities are possible; the first corresponds to the low‐frequency electrostatic, trapped particles mode, and the second is the low‐frequency limit of magnetohydrodynamic (interchange and ballooning) modes. The kinetic modifications to the interchange (Mercier) criterion are evaluated and the effect of the kinetic terms on ballooning modes is estimated.

Theory and simulation of stimulated Brillouin scatter excited by nonabsorbed light in laser fusion systems
View Description Hide DescriptionThe noise spectrum from which stimulated Brillouin scatter grows has two sources in laser fusion plasmas; a broadband source due to ion‐acoustic fluctuations, and a line source, usually much larger, which is the nonabsorbed light returning from the plasma critical surface. We give a theoretical description of stimulated Brillouin backscatter when the fluctuation source may be neglected and the scatter grows exclusively from the nonabsorbed light. Gradients of background density, velocity, and temperature are allowed. Theoretical predictions are compared to numerical simulations of scatter for parameters of recent experiments. It is found that stimulated Brillouin scatter can be greatly enhanced by the presence of a critical surface and that it can become an important part of the total energy balance.

Integral‐equation formulation for drift eigenmodes in cylindrically symmetric systems
View Description Hide DescriptionA method for solving the integral eigenmode equation for drift waves in cylindrical (or slab) geometry is presented. A leading‐order kinematic effect that has been noted in the past, but incorrectly ignored in recent integral‐equation calculations, is incorporated. The present method also allows electrons to be treated with a physical mass ratio (unlike earlier work that is restricted to artificially small m _{ i }/m _{ e } owing to resolution limitations). Results for the universal mode and for the ion‐temperature‐gradient driven mode are presented. The kinematic effect qualitatively changes the spectrum of the ion mode, and a new ’’second region of instability’’ for k _{⊥} ρ_{ i }≳1 is found.

Temperature effects on harmonic generation in laser‐irradiated plasmas
View Description Hide DescriptionThe generation of second and third harmonics at the oblique incidence of a p‐polarized wave on an inhomogeneous plasma (resonance absorption) is studied. The relevant wave equations are solved numerically. First, the dependence of the emitted harmonics on the temperature, the density scale length, and the angle of incidence is investigated for a linear density profile. The strong oscillations of the harmonic emission in a certain temperature range are attributed to the position of the point where the condition for phase matching is fulfilled. In the second part a modified density profile with a caviton is considered, and it is shown that a large enhancement of harmonic emission is possible because of the formation of electrostatic resonant structures. This effect will become important under experimental nonstationary conditions when the density profile is deformed by the ponderomotive force.

Collisional shear Alfvén waves in sheared magnetic fields
View Description Hide DescriptionThe structure of the shear Alfvén waves is investigated in a collisional plasma with shear. The fourth‐order equation obtained by combining Ampere’s law and the quasi‐neutrality condition is solved by the method of matched asymptotic expansions for k _{⊥} ρ_{ s }≫≪1 to obtain the dispersion relation. A hierarchy of damped, localized modes are found which can have either even or odd parity. The solutions basically have the structure of kinetic Alfvén modes trapped between the two Alfvén cutoffs on either side of the rational surface. The mode damping arises from Ohmic dissipation by electrons near the rational surface. The relation between these modes, microtearing modes, and the Alfvén continuum is discussed.

Destabilitization of low mode number Alfvén modes in a tokamak by energetic or alpha particles
View Description Hide DescriptionWith the inclusion of finite Larmor radius effects in the shear Alfvén eigenmode equation, the continuous Alfvén spectrum, which has been extensively discussed in ideal magnetohydrodnamics, is removed. Neutrally stable, discrete radial eignmodes appear in the absence of sources of free energy dand dissipation. Alpha (or energetic) particle toroidal drifts destabilize these modes, provided the particles are faster than the Alfvén speed. Although the electron Landau resonance contributes to damping, a stability study of the parametric variation of the energy and the density scale length of the energetic particles shows that modes with low radial mode numbers remain unstable in most cases. Since the alpha particles are concentrated in the center of the plasma, this drift‐type instability suggests anomalous helium ash diffusion. Indeed, it is shown that stochasticity of alpha orbits due to the overlapping of radially neighboring Alfvén resonances is induced at low amplitudes, e _{ i }φ/T _{ i }≳0.05, implying a diffusion coefficient D _{ r } ^{α}≳4.4×10^{3} cm^{2}/sec.

Interpretation of the fine structure of electrostatic waves excited in space
View Description Hide DescriptionPreliminary results are presented from an active wave experiment performed with a double‐dipole probe aboard the F4 rocket launched during the PORCUPINE campaign. They show that electrostatic cyclotron waves can be excited linearly around the half‐harmonics of the electron gyrofrequency. The most striking fact is that, under certain plasma conditions, the amplitude of waves excited at these peculiar frequencies can be even stronger than those observed at the characteristic resonances of the magnetoplasma. The fine structure of the signals received is also observed and interpreted. Theoretical numerical calculations of the potential created by a dipole antenna in a Maxwellian plasma are presented; good agreement is obtained with the complicated pattern of the various experimental results.

Proton beam‐target interaction at pellet fusion power densities
View Description Hide DescriptionThe interaction of proton beams at pellet fusion power densities (∼100 TW/cm^{2}) with initially solid targets is calculated. Radiation losses and conduction are shown to have important effects at these power densities.

Stability of colliding ion beams
View Description Hide DescriptionConditions are determined for the stability of two identical colliding ion beams in the presence of neutralizing electrons, but no background ions. Such a situation is envisioned for the Counterstreaming Ion Torus. The ion beams are taken to be Maxwellian in their frames of reference. The approximation of decoupled electrostatic and electromagnetic modes is made. The stability of the electrostatic modes depends on the relation between the ion‐electron temperature ratio and the relative beam velocities. The stability of the electromagnetic mode depends on the relation between the ion plasma β and the relative beam velocities.