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NASA NTRS1972

Interpretation of dispersion relations for bounded systems.

Treatment of the problem of constructing normal modes for an arbitrarily bounded system from roots of the linear dispersion relation D(omega, k) = 0 for the corresponding infinite or periodically bounded system. For a system described by continuous macroscopic variables, and of general cylindrical form, each transverse eigenmode gives rise to a set of axial normal modes constructed from a pair of dominant roots of D = 0 satisfying the boundary conditions which are characterized by complex reflection coefficients for the dominant waves. The implications of the results for the interpretation of experiments on plasma waves and instabilities on finite cylinders are discussed, with particular reference to the effects of end-plate damping and axial current on Q-machines.

Rognlien, T. D.↗

NASA NTRS1973

Quantum theory of an optical maser. VI - Transient behavior.

The transient behavior of a laser is discussed using the quantum theory as did Scully and Lamb. The formal solution of the density-matrix equation is expressed in terms of exponentially decaying eigenmodes. Some of the lower decay constants are obtained numerically. The equations for the moments of the density matrix are then derived and solved by a truncation method. The equations of motion are integrated numerically for the case where the average number of photons in a laser cavity has the realistically large value 1.3 x 100,000. An alternative Fokker-Planck-equation approach is discussed.

Wang, Y. K.↗

NASA NTRS1974

Energetic electrons in Jupiter's magnetosphere

A theoretical model for the energetic electron fluxes in the Jovian magnetosphere is developed. Electrons are transported inward from the solar wind or Jovian magnetospheric tail by radial diffusion. The radial diffusion is driven by fluctuating ionospheric dynamo electric fields associated with a neutral-wind tidal eigenmode at ionospheric altitudes. The tidal mode is excited by the electromagnetic coupling of the solar wind to the polar ionosphere. Two injection models are considered: (1) electron penetration through the dayside magnetopause - low-energy model; and (2) injection of electrons from an assumed magnetospheric tail - high-energy model. Both thermal solar-wind electrons and energetic solar-flare electrons are considered.

Coroniti, F. V.↗

NASA NTRS1974

Some analytical results for radiative transfer in thick atmospheres

Singular eigenmode expansions are a convenient analytical tool with which to study problems of monochromatic radiative transfer in thick or semi-infinite atmospheres. Some closed-form solutions are presented for anisotropic scattering, with the neglect of polarization effects. A basic ingredient for applications to the semi-infinite medium is Chandrasekhar's H-function, which is best defined through the Wiener-Hopf factorization. The solutions of the Milne and albedo problems are discussed in order that the method of matched asymptotic approximations may be used to describe the solution for a thick atmosphere adjacent to a diffusely reflecting ground. Expressions for the emerging distributions are quoted. A possible extension of the general scheme to problems involving polarization is indicated.

Kuscer, I.↗

NASA NTRS1976

Whistler wave propagation in a large magnetoplasma

A large collisionless quiescent plasma source is developed for investigating the phase and amplitude distribution of antenna-launched whistler waves in a specified parameter regime relating wave frequency to electron cyclotron frequency. Wave dispersion is studied both by interferometer techniques with monochromatic waves and by propagation of short phase-coherent wave bursts. The wave damping mechanism is examined by propagating perfectly ducted whistler waves. The dispersion of single frequency waves and wave packets is demonstrated. Trough ducting for wave frequency to electron cyclotron frequency ratio greater than 1/2 is verified, and new eigenmodes in nonuniform plasmas at ratio values less than 1/2 are observed. It is shown that geometric effects due to ray divergence and wave refraction dominate over collisional damping.

Stenzel, R. L.↗

NASA NTRS1976

The eigenvalue spectrum of the Orr-Sommerfeld problem

A numerical investigation of the temporal eigenvalue spectrum of the ORR-Sommerfeld equation is presented. Two flow profiles are studied, the plane Poiseuille flow profile and the Blasius boundary layer (parallel): flow profile. In both cases a portion of the complex c-plane bounded by 0 less than or equal to CR sub r 1 and -1 less than or equal to ci sub i 0 is searched and the eigenvalues within it are identified. The spectra for the plane Poiseuille flow at alpha = 1.0 and R = 100, 1000, 6000, and 10000 are determined and compared with existing results where possible. The spectrum for the Blasius boundary layer flow at alpha = 0.308 and R = 998 was found to be infinite and discrete. Other spectra for the Blasius boundary layer at various Reynolds numbers seem to confirm this result. The eigenmodes belonging to these spectra were located and discussed.

Antar, B. N.↗

NASA NTRS1977

The sensitivity of nonradial p mode eigenfrequencies to solar envelope structure

Eigenfrequencies are calculated for nonradial p mode oscillations in the sun. These frequencies are shown to depend on the adiabat of the solar convective envelope. The frequency dependence is greatest for the p2 and p3 modes described by spherical harmonics with l = 500 to 1000. The solar envelope model includes a realistic representation of the chromosphere out to the base of the corona. For the chromospheric model of Vernazza, Avrett, and Loeser (1976) chromospheric modes with a distinct locus of eigenfrequencies exist near an eigenfrequency of 0.026 Hz. The actual frequencies of these eigenmodes depend on the thickness of the chromosphere. Because of the difference in the dependence of eigenfrequency on l for the chromospheric modes as compared with the p modes, the two types of mode can be distinguished observationally. If detected, the chromospheric modes would provide a good way of determining the thickness of the chromosphere.

Ulrich, R. K.↗

NASA NTRS1978

Wave production in an ultrarelativistic electron-positron plasma

In this paper we calculate the eigenmodes associated with an ultrarelativistic electron-positron beam traversing a low-energy electron-positron plasma under physical conditions that may exist along open magnetic field lines above pulsar polar caps. We assume that both beam and plasma are cold and charge neutral, and that magnetic field strength and particle density decrease as the cube of (1/R). In the superstrong magnetic fields near the stellar surface where cyclotron frequencies exceed plasma frequencies, electrostatic waves and associated transverse low-frequency waves are driven unstable by the two-stream type particle distribution. At larger distances above the stellar surface the slow beam cyclotron wave is driven unstable. At even greater radial distance (approximately when particle energy density exceeds magnetic energy density) both the Ordinary and Alfven modes become unstable. All of these instabilities can lead to generation of radiation. If the particle distribution as generated at the stellar surface, assumed to consist of two streams, is modified only by plasma effects, then electrostatic and associated transverse waves may only be unstable near the stellar surface. The beam cyclotron mode which is driven by a two-stream or an inverted particle distribution may be stable.

Hardee, P. E.↗

NASA NTRS1977

Small disturbances in a compressor with strong swirl

A detailed time resolved study of the flow field upstream and downstream of a high work transonic compressor rotor shows that the flow field is dominated by the downstream evolution of the viscous flow shed from the rotor blades under the influence of the strong mean swirl. The dominant periodicity in the flow changes from blade passing to 1.4 times blade passing within one chord from the blade row. A possible explanation is that the wakes evolve to a shear eigenmode of the swirling flow, as suggested by perturbation theory. Another possibility is a 'propagating stall' of 16 cells, but the rotor operated near its design point. Treatments of 'turbulence' in turbomachines should account for such phenomena, which originate in the strong mean swirl.

Kerrebrock, J. L.↗

NASA NTRS1979

The effect of a radial rotational velocity gradient on p-mode eigenfrequencies

The splitting of p-mode eigenfrequencies due to a depth-dependent solar rotation rate is considered. It is shown that to a first approximation this splitting can be treated as resulting from the translation of the wave pattern across the field of view with a velocity equal the actual velocity at a depth called the effective depth. The effective depths for a model solar envelope with l/H = 3 are presented as a function of position on the (kh, omega)-plane. The combination of omega near 0.025/s and kh near 0 yields the deepest probe of the solar envelope. For this same combination of kh and omega it is possible to resolve individual groups of eigenmodes in frequency but not in spatial wavenumber. Achievement of this resolution observationally may permit a determination of the lifetime of these particular modes.

Ulrich, R. K.↗

NASA NTRS1982

Circumferentially segmented duct lines optimized for axisymmetric and standing wave sources

Optimum and off-optimum properties of circumferentially segmented duct liners are compared with those of uniform liners to identify any potential benefits of circumferentially segmented liners. High- and low-order spinning-mode sources are considered in the study. The solution for the segmented liner is obtained by a multimodal expansion of the segmented-liner eigenmodes in terms of a series of hardwall duct models. The coefficients in the hard-wall series are obtained by using Galerkin's method. Results show that for some frequencies and duct lengths, circumferentially segmented liners scatter energy equally between a higher and lower order circumferential wave number. Studies for higher order spinning-mode sources show that an optimized segmented liner with a hard-wall/soft-wal admittance variation representing an optimum configuration gives better performance than an optimized uniform liner. Overall, the greatest benefit of the segmented liner over the uniform liner occurs under off-optimum conditions. The optimized segmented liner gives more effective broadband performance than the optimized uniform liner.

Watson, W. R.↗

NASA NTRS1981

Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the bifurcation point. The plate is discretized by using displacement finite element (or finite difference) models. The special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region. The vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors, and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the bifurcation point and the associated eigenmode of the panel. In the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new (updated) sets of basis vectors and reduced equations are generated. In the third stage the reduced equations are used to trace the post-buckling paths. The effectiveness of the proposed technique for predicting the bifurcation and post-buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements.

Noor, A. K.↗

NASA NTRS1982

A nonlocal theory of an electrostatic sinusoidal density drift instability

The stability of space plasmas in which the macroscopic variables vary in the direction x perpendicular to the magnetic field is usually studied by means of the local approximation, in which the fluctuating potential is assumed to be independent of x. To remove this approximation, a nonlocal problem is studied in which the background density, and hence the fluctuating potentials, are periodic in x. From the linear Vlasov/Poisson equations, a set of coupled, linear, homogeneous, algebraic equations relating the Fourier amplitudes of the eigensolution is derived. The equations are solved numerically and a hierarchy of exact eigenmodes characterized by different growth rates and spatial structures is found. At points where the density gradient is locally zero, the mode amplitude is generally several orders of magnitude lower than the peak amplitude. The dependence of these solutions on the parameters which define the background density perturbation is studied and a correspondence between these nonlocal results and those obtained with the local approximation is demonstrated.

Thomsen, M. F.↗

NASA NTRS1981

A diurnal resonance in the ocean tide and in the earth's load response due to the resonant free 'core nutation'

The effects of the oceans, which are subject to a resonance due to a free rotational eigenmode of an elliptical, rotating earth with a fluid outer core having an eigenfrequency of (1 + 1/460) cycle/day, on the body tide and nutational response of the earth to the diurnal luni-tidal force are computed. The response of an elastic, rotating, elliptical, oceanless earth with a fluid outer core to a given load distribution on its surface is first considered, and the tidal sea level height for equilibrium and nonequilibrium oceans is examined. Computations of the effects of equilibrium and nonequilibrium oceans on the nutational and deformational responses of the earth are then presented which show small but significant perturbations to the retrograde 18.6-year and prograde six-month nutations, and more important effects on the earth body tide, which is also resonant at the free core notation eigenfrequency.

Wahr, J. M.↗

NASA NTRS1982

Sunspot time series - Spectrum from square law modulation of the Hale cycle

The spectrum of the sunspot index can be qualitatively accounted for by a very simple time series model which the Hale 22 year cycle dominates; the series is amplitude modulated at the 90 year Gleissberg period with an index of 25%. The square of the modulated Hale cycle reproduces all of the spectral lines of the Zurich sunspot index spectrum, provided that a small offset is included in the Hale carrier. Without this, evidence of the 22 year periodicity disappears. The squaring explains the dominance of the 11 year cycle which is an artifact of the modulation as is the 45 year line. The offset suggests a small relict magnetic field in the sun's core. Lastly Gilliland (1981) finds a radial eigenmode in the sun with period of 76 + or - 8 years suggestive of a physical basis for the Gleissberg modulation.

Sonett, C. P.↗

NASA NTRS1983

Hall current effect on tearing mode instability

From a linear 2-D eigenmode analysis, it is found that the Hall current effect on collisional tearing mode instability becomes important for the thin magnetic reversal layer whose width is comparable to the ion inertia length; Hall currents produce a three-dimensional field structure and increase the reconnection (growth) rate. Since the magnetaic reversal layer widths both in the magnetopause and in the magnetotail are reported to become as thin as the ion inertial length (several hundred km) when the reconnection process is supposed to occur, the Hall current effect may explain the appearance of the dawn-dusk component of the magnetic field in the magnetotail reconnection region.

Terasawa, T.↗

NASA NTRS1983

A linear analysis of the transition curve for the baroclinic annulus

A primitive equation linear model is shown to be capable of quantitatively predicting the neutral curve for a baroclinic annulus. Flows unstable to small nonaxisymmetric perturbations are assumed to lie on the nonaxisymmetric side of a transition curve, while the stable flows are on the axisymmetric side. The stability in the presence of perturbations is tested with linearized hydrostatic primitive equations. Attention is given to the fastest growing eigenmodes, demonstrating that the linear and nonlinear wave have similar structures at a particular point in the nonaxisymmetric regime. All the waves are similar to the Eddy wave except at the extreme to the transition curve, where the waves maintain little structure with height. The eddy kinetic energy is baroclinic with a potential energy suppresed by the large static stability of the basic state. Eddy dissipation is noted to be significant near all boundaries.

Miller, T. L.↗

NASA NTRS1984

Supermode analysis of phase-locked arrays of semiconductor lasers

The optical characteristics of phase-locked semiconductor laser arrays are formulated in terms of the array supermodes, which are the eigenmodes of the composite-array waveguide, by using coupled-mode theory. These supermodes are employed to calculate the near fields, the far fields, and the difference in the longitudinal-mode oscillation wavelengths of the array. It is shown that the broadening in the far-field beam divergence, as well as the broadening of each of the longitudinal modes that were observed in phase-locked arrays, may arise from the excitation of an increasing number of supermodes at increasing pumping levels.

Kapon, E.↗