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Volume 24, Issue 9, September 1981

Turbulent solution of the Navier–Stokes equations
View Description Hide DescriptionTo study the nonlinear physics of incompressible turbulent flow, the unaveraged Navier–Stokes equations are solved numerically. Initial three‐dimensional cosine velocity fluctuations and periodic boundary conditions are used. No mean gradients are present. The three components of the mean‐square velocity fluctuations are equal for the initial conditions chosen. The resulting solution shows characteristics of turbulence, such as the nonlinear excitation of small‐scale fluctuations. For the higher Reynolds numbers the initially nonrandom flow develops into an apparently random turbulence. Thus, randomness or turbulence can apparently arise as a consequence of the structure of the Navier–Stokes equations.

A non‐Gaussian model of turbulence (soccer‐ball integrals)
View Description Hide DescriptionThe statistics of the time evolution of a nonlinearly coupled system of first‐order equations representing the Euler equations is studied. The probability distribution of functions is nearly Gaussian, while that of their time derivatives has exponential tails and moments of order 4, 6, and 8 that approach those of the exponential distributions.

Closure for velocity/pressure‐gradient correlations in turbulent shear flow
View Description Hide DescriptionA closure representation is developed for the velocity/pressure‐gradient correlation arising in the Reynolds stress equation through the interaction of the turbulent velocity field with mean shear flow. The development has the advantage of putting the closure on a rational footing. The resulting expression is used in a simple example where nearly steady, homogeneous turbulence is present in a rectilinear mean shear flow. Reynolds stresses and various aspects of the behavior of the velocity/pressure‐gradient correlation seem to be predicted correctly.

A second‐order theory for solitary waves in deep fluids
View Description Hide DescriptionBenjamin and Davis and Acrivos have established the existence of an algebraic solitary wave in deep fluids. This theory is here extended to second order in amplitude. The second‐order correction to the wave amplitude, the second‐order correction to the wave phase speed, and the first‐order correction to the wavelength are all determined for solitary waves propagating in a stratified shear flow. Two special cases are discussed in detail.

Two‐dimensional evolution of packets of short water waves in systems of finite depth
View Description Hide DescriptionThe evolution of packets of water waves described by the long‐wave limit of the Davey–Stewartson equations is considered. The stationary one‐dimensional kink solutions are analyzed with respect to transverse perturbations. They are shown to be stable when surface tension can be neglected. To demonstrate this, the two‐dimensional inverse scattering transform found by Anker and Freeman is applied. The present stability analysis supplements previous investigations in the long‐wavelength limit of water waves where finite depth effects have been ignored.

Measurement of convective cell spectra and the resultant calculated vortex diffusion coefficient
View Description Hide DescriptionThe presence of convective cells in a purely poloidal field Levitated Octupole has been associated with diffusion that scales as D _{ v }∝(T*/n)^{1/2}, independent of B, where T* is an ’’effective temperature,’’ T*∝T. The electric field spectrum of the convective cells can be used to estimate the magnitude of T* and D _{ v }. The results are in reasonable agreement with previous measurements of cross‐field transport, and agree qualitatively with theortical models of vortex diffusion.

Diffusion coefficient for ions in the presence of a coherent lower hybrid wave
View Description Hide DescriptionWhen the amplitude of a coherent lower hybrid wave exceeds a certain stochasticity threshold, the ion motion becomes ergodic. The problem of determining the resulting diffusion coefficient is considered. For large amplitude waves it is found that the diffusion coefficient oscillates with decreasing amplitude about the quasi‐linear value as the wave amplitude is increased. Furthermore, the diffusion coefficient is shown to be subject to narrow resonances at particular values of the wave amplitude.

Theory of the renormalized dielectric for electrostatic drift wave turbulence in tokamaks
View Description Hide DescriptionA theory of collisionless electrostatic drift wave turbulence in a circular cylinder with shear is presented. A renormalized electron drift kinetic equation which is consistent with conservation of energy is derived. For low levels of turbulence, a perturbative solution indicates that the turbulence has a stabilizing effect and that total mode spectrum energy decreases. For strongly turbulent regimes, results quantitatively but not qualitatively different from those of Hirshman and Molvig are found. In particular, additional stabilizing terms lead to a lower saturation amplitude, but the basic picture of turbulent destabilization competing against linear and nonlinear shear damping persists.

Physical mechanism of wave‐particle resonances in an inhomogeneous magnetic field. I. Linear theory
View Description Hide DescriptionA physical description is given of the wave‐particle resonance which can occur when particles undergoing a ∇ B drift are in phase with a wave propagating across the magnetic field (i.e., ω ∼ k⋅V_{∇ B }, where V_{∇ B } is the ∇ B drift velocity). Specifically, the physics of the energy exchange mechanism is discussed and a general equation for the wave damping/growth rate is derived based upon physical arguments. The theory is applied to the lower‐hybrid‐drift instability and collisionless trapped electron instability.

Variational theory of electrical conductivity and kinetic tearing modes
View Description Hide DescriptionA variational theory for Ohm’s law, valid for arbitrary collision frequency, is presented for a slab geometry using a pitch angle scattering collision operator. Shear and temperature gradient effects are included but toroidicity is neglected. The variational conductivity is used to examine the effects of shear and electron temperature gradients on tearing mode instabilities. It is shown that shear has a strong stabilizing effect on the so‐called ’’collisional’’ and ’’semicollisional’’ electron temperature gradient tearing mode instabilities. In addition, the stabilizing effect of shear on the classical resistive and inertial tearing modes is discussed.

Omnigenous equilibria
View Description Hide DescriptionThe conditions are obtained for confining a plasma of arbitrary asymmetry, plasma beta, and collisionality, in an equilibrium configuration free of neoclassical transport. Such omnigeneous scalar pressure equilibria are shown to exist if, and only if, the divergence of the parallel current density vanishes, while for an anisotropic pressure other mild restrictions are needed. Explicit constraints are derived for axisymmetric tori and long‐thin, low‐beta conventional and tandem mirrors.

Rayleigh–Taylor instabilities of an accelerating thin plasma slab
View Description Hide DescriptionRayleigh–Taylor instabilities of an accelerating plasma sheet are studied by application of normal mode analysis to the one‐fluid magnetohydrodynamic equations. The equilibrium unperturbed states are determined by specifying the resistivity law of the current carrying plasma, although resistivity effects are neglected in the developing instabilities. The dispersion curves for the current filamentation and interchange type instabilities are given.

Tilting instability of a cylindrical spheromak
View Description Hide DescriptionThe stability of a low‐beta spheromak with a perfectly conducting cylindrical boundary of length L and radius R is analyzed in terms of force‐free fields with J = λB(λ = const). The axisymmetric equilibrium is found unstable to tilting when the elongation L/R is larger than about 1.67. Numerical solutions of the time‐dependent ideal magnetohydrodynamic equations confirm this result.

Plasma frequency radiation in tokamaks
View Description Hide DescriptionRadiation from relativistic runaway electrons is considered as a source for plasma frequency radiation in tokamaks. Two specific mechanisms, Cerenkov emission and radiation produced by nonlinear coupling of plasma and acoustic waves, are studied. In many cases the Cerenkov emission provides a reasonable spectral fit. It can also be used to measure the runaway current, and to estimate the runaway cutoff velocity. The nonlinear emission is found to be negligible unless the acoustic waves are enhanced by about two orders of magnitude above the thermal level (or the plasma waves correspondingly enhanced above their superthermal level). Some observations, though, indicate the need for significant nonlinear (or other) emission in addition to the Cerenkov emission. For some typical Alcator data, the Cerenkov model gives a runaway current of 2.5% of the Ohmic current, and a cutoff energy of 1.5 MeV.

Atmospheric type modes in laser fusion targets
View Description Hide DescriptionThe fluid stability of laser‐fusion targets is considered as a stability problem of a pseudo‐planetary atmosphere. Three atmospheric type modes are studied; acoustic, gravity, and Lamb modes. The changing character (i.e., growing, oscillatory, or growing‐oscillatory) of each is investigated as a function of the density‐gradient scale length H of the fluid. A growing class of modes is found which is distinct from the gravity (i.e., Taylor) mode if a gradient in entropy exists in the fluid. These modes are shown to be overstable Lamb modes. Also, the gravity mode is only stable for a distinct band of values of H. These values, at which the density and Lamb modes change character, are derived from the dispersion relation for the modes. Finally, the consequences for laser targets are discussed.

Microwave heating of the ELMO Bumpy Torus relativistic electron ring
View Description Hide DescriptionA model for microwave heating of electron rings in the ELMO Bumpy Torus configuration is analyzed using a relativistically correct quasi‐linear formulation. The spatial locations of heating by the different electron‐cyclotron harmonics are calculated. The steady‐state ring energy and the microwave power required to sustain the rings are determined by balancing the line‐averaged heating rate against classical collisional and radiative energy loss processes. Although ring formation is generally attributed to the second harmonic electron‐cyclotron resonance, the calculations show that fundamental heating also plays a critical role in ring start‐up and steady state. The model predicts ring power requirements for EBT which are consistent with previous estimates.

Perturbation analysis of a simple model of magnetic island structures
View Description Hide DescriptionSingular perturbation theory is used to construct a class of stationary solutions to the resistive plasma equilibrium equations in a plane slab geometry. These are nonlinearly saturated magnetic island structures which bifurcate from the basic Ohmic state when Δ′(k) is positive and small in some sense. The relationship between the nondimensional wavenumber k and the size of the islands is obtained by a self‐consistent, matched asymptotic expansion. The assumed resisitivity profile is found to determine the relationship uniquely. It is found that either two bifurcating branches exist or none at all depending upon the profile chosen. The structure of the plasma current due to the islands is evaluated in terms of two functions which are independent of the resistivity profile. It is found that for sufficiently large Δ′(k) and suitable resistivity, a self‐consistent, steady magnetic island state cannot be found. A possible connection between this phenomenon and disruptive instabilities observed in experimental plasmas is discussed very briefly. The proposed model complements the work done on magnetic islands by various groups using powerful numerical codes. In contrast to such codes, the current distributions due to a wide class of resistivity profiles have been calculated using the model. The solutions constructed shed some light on how nonlinear saturation in the island states can occur.

Some nonlinear optical phenomena
View Description Hide DescriptionSome nonlinear optical phenomena are investigated, especially stimulated scattering, from the point of view of the kinetic theory of radiation (i.e., photon transport theory). Kinetic theory provides a perspective, different from Maxwell’s wave theory, from which an examination of these complex matters may proceed with some advantages: (i) considerable mathematical simplication in some instances, (ii) clear and natural separation of microscopic versus macroscopic nonlinearities, (iii) kinetic theory couples the radition field nonlinearly to a formally exact description of the matter field, and (iv) it is believed that the mathematical model provided by kinetic theory is perhaps better suited for numerical studies of the effect of diverse nonlinear optical phenomena upon laser‐fusion implosion dynamics than Maxwell’s wave theory. Although the main emphasis is upon stimulated scattering, the incorporation of other nonlinearities into the kinetic model is discussed briefly.

Hamiltonian formulation of guiding center motion
View Description Hide DescriptionA Hamiltonian theory of guiding center motion which uses rectangular coordinates in physical space and noncanonical coordinates in phase space is presented. The averaging methods preserve two important features of Hamiltonian systems, viz., conservation of energy (for time‐independent fields) and Liouville’s theorem. These features are sacrificed by the traditional averaging methods. The methods also relieve much of the burden of higher order perturbation calculations, and the drift equations for fully electromagnetic fields are extended to one higher order than they have been known in the past. The first correction to the relativistic magnetic moment is also calculated. Many applications are anticipated, both to single particle motion and to kinetic theory.

Observations of the onset of flow instabilities in constricted tubes
View Description Hide DescriptionA general stability boundary is determined which establishes the minimum Reynolds number R _{ i } and percent area ratio for the onset of flow instabilities. Both nonaxisymmetrix and axisymmetric disturbances were observed. The existance of a finite extent turbulence region is also described.