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NASA NTRS1975

Variational calculations for resonance oscillations of inhomogeneous plasmas

The electrostatic resonance properties of an inhomogeneous plasma column are treated by the Rayleigh-Ritz method. In contrast to Parker, Nickel & Gould (1964), who carried out an exact computation, the present treatment uses a description of the RF equation of motion and pressure term that allows one to express the system of equations in Euler-Lagrange form. The Rayleigh-Ritz procedure is then applied to the corresponding Lagrangian, to obtain approximate resonance frequences and eigenfunctions. An appropriate set of trial coordinate functions is defined, which leads to frequency and eigenfunction estimates in excellent agreement with the work of Parker et al. (1964).

Peng, Y.-K. M.↗
NASA NTRS1977

Extraction of the large eddy structure of a turbulent boundary layer

Much attention is currently being focused on turbulence modeling by separation of large-scale and small-scale turbulence. One major difficulty with this method is the problem of separating and defining the large-scale structure. In this paper, a method is presented which does this separation by analyzing time-averaged, two-point double velocity correlation measurements. The basis of the method is a maximization of the inner product of a candidate velocity field with the measured turbulent velocity field. An eigenvalue problem results, whose solution provides a set of orthogonal eigenfunctions associated with specific modes of turbulent motion. The large-scale eddy, containing most of the turbulent energy, is associated with the first (largest) mode. The turbulent velocity field is then represented as a sum of the eigenfunctions. A short description of the equations is given, the methods of solution are discussed, and some results for a two-dimensional fully developed turbulent boundary layer are presented.

Lemmerman, L. A.↗
NASA NTRS1977

Theoretical intensity parameters for the vibration-rotation bands of ClO

Intensity parameters for the vibration-rotation bands of ClO were calculated by evaluating matrix elements of the vibrational eigenfunctions using the theoretical dipole moment curve. These vibrational eigenfunctions are based on realistic Klein-Dunham potentials deduced from experimental values (Coxon et al, 1976) for certain vibrational energy levels. The data can be applied to the interpretation of remote or in situ experiments to deduce stratospheric ClO concentrations from vibration-rotation absorption spectra.

Langhoff, S. R.↗
NASA NTRS1983

Elasticity solutions for a class of composite laminate problems with stress singularities

A study on the fundamental mechanics of fiber-reinforced composite laminates with stress singularities is presented. Based on the theory of anisotropic elasticity and Lekhnitskii's complex-variable stress potentials, a system of coupled governing partial differential equations are established. An eigenfunction expansion method is introduced to determine the orders of stress singularities in composite laminates with various geometric configurations and material systems. Complete elasticity solutions are obtained for this class of singular composite laminate mechanics problems. Homogeneous solutions in eigenfunction series and particular solutions in polynomials are presented for several cases of interest. Three examples are given to illustrate the method of approach and the basic nature of the singular laminate elasticity solutions. The first problem is the well-known laminate free-edge stress problem, which has a rather weak stress singularity. The second problem is the important composite delamination problem, which has a strong crack-tip stress singularity. The third problem is the commonly encountered bonded composite joints, which has a complex solution structure with moderate orders of stress singularities.

Wang, S. S.↗
NASA NTRS1985

Numerical Descriptions of Cosmic-Ray Transport

The behavior of energetic particles in the solar system is described by a well known Fokker-Planck equation. Although analytic methods yield insight into the nature of its solutions, especially in the diffusion regime, calculations that go beyond diffusion are very complicated. The reliability of these calculations is of concern, because numerical methods are notorious for their errors and artifacts. The well known Milne problem of classical transport theory was analyzed with the aid of three different numerical methods. These are: (1) The method of eigenfunctions in which the distribution function is approximated by a sum of eigenfunctions of the scattering operator, (2) Numerical solutions of a finite difference aquation; and (3) Direct simulation of the scattering and streaming of individual particles with the aid of Monte Carlo methods.

Earl, J. A.↗
NASA NTRS1985

Parameter identification in distributed systems

This paper describes a method for the identification of the parameters entering into the equations of motion of distributed systems. Because the motion of distributed systems is described in terms of partial differential equations, these parameters are in general continuous functions of the spatial variables. For vibrating systems, these parameters ordinarily represent the mass, stiffness and damping distributions. In this paper, these distributions are expanded in terms of finite series of known functions of the spatial variables multiplied by undetermined coefficients. It is assumed that the nature of the equations of motion is known and that a limited number of eigenvalues and eigenfunctions is identified in advance. Use is then made of the least squares method, in conjunction with the eigenfunctions' orthogonality, to compute the undetermined coefficients, thus identifying the system distributed parameters. A method for the identification of the eigensolution is also presented. The procedure for the identification of the eigensolution and of the system parameters is demonstrated via a numerical example.

Baruh, H.↗
NASA NTRS1985

Sum rules and other properties involving resonance projection operators

A sum rule is derived for the auxiliary eigenvalues of an equation whose eigenspectrum pertains to projection operators which describe electron scattering from multielectron atoms and ions. The sum rule's right-hand side depends on an integral involving the target system eigenfunctions. The sum rule is checked for several approximations of the two-electron target. It is shown that target functions which have a unit eigenvalue in their auxiliary eigenspectrum do not give rise to well-defined projection operators except through a limiting process. For Hylleraas target approximations, the auxiliary equations are shown to contain an infinite spectrum. However, using a Rayleigh-Ritz variational principle, it is shown that a comparatively simple aproximation can exhaust the sum rule to better than five significant figures. The auxiliary Hylleraas equation is greatly simplified by conversion to a square root equation containing the same eigenfunction spectrum and from which the required eigenvalues are trivially recovered by squaring.

Berk, A.↗
NASA NTRS1985

Application of satellite data to tropic/subtropic moisture coupling

The objective is to develop analysis tools for use of satellite data to interpret synoptic-scale systems in data-void regions. Interim goals are to: (1) quantify the synoptic information content of satellite data; and (2) utilize these data in the diagnosis of moisture bursts in the eastern tropical Pacific Ocean. Researchers developed and implemented a statistical procedure for using TIROS N microwave data to infer infrared channel data for overcast conditions; they used the same procedure for deducing full TIROS N channel radiance profiles from NOAA 5 VTPR channel data over regions where the TIROS N data are missing. An empirical orthogonal function analysis of twice-daily channel radiance fields over the tropical eastern Pacific was completed. The vertically oriented eigenfunctions were interpreted in terms of typical meteorological events. The horizontal distribution of the eigenfunction amplitudes relates these meteorological signals to moisture bursts. A pair of moisture burst climatologies is complete: one of four years using infrared imagery (including the highly anomalous 1982 to 83 cold season); the other implementing 850 to 200 mb wind analyses in conjunction with GOES imagery. A number of different evaluations of the synoptic evolution of moisture fields (enhanced infrared imagery, moisture channel data, FGGE humidity analysis, and in situ station and sounding observations) are compared. All have limitations; all can be utilized together; all together are still less than adequate in the tropical Pacific.

Mcguirk, J. P.↗
NASA NTRS1987

An analysis of the vertical structure equation for arbitrary thermal profiles

The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations.

Cohn, Stephen E.↗
NASA NTRS1986

Methods for the solution of radiative transfer equation

To obtain an exact solution of the radiative-transfer equation in media where both absorption and scattering are significant, the usual approach is to use a numerical method. Three methods are known in the literature: invariant imbedding, eigenvalue-eigenfunction, and matrix doubling. This paper examines the practical application of these methods to the problem of emission from an inhomogeneous (Rayleigh) layer, the effects of layer parameters on the stability. It is found that invariant imbedding is most suitable for computing emission from an inhomogeneous layer with a temperature profile but tends to be unstable as the optical thickness of the layer increases beyond 0.5. On the other hand, the matrix-doubling method is stable for arbitrary optical thickness but is not suitable for handling multilayers. The eigenvalue-eigenfunction method is more stable than the invariant imbedding as optical thickness increases up to 2.0. It also permits temperature profile in the layer, but the computation is much more complicated. It is less stable than the matrix-doubling method when optical thickness is larger than 2.0. In general, the choice of a method is dependent on the nature of the problem.

Chen, M. F.↗
NASA NTRS1987

A linear thermal stability analysis for the vertical structure of alpha model accretion disks

The linear equations that describe the thermal stability of the convective and radiative vertical structure of an alpha model accretion disk are derived. For a specific model chosen to be representative of a dwarf nova disk, it is found that the eigenfunction for the temperature shows large excursions in the zone of partial ionization, but that the eigenfunctions for the unstable fundamental mode perturbations of the vertical distance and flux are nearly constant over the vertical extent of the disk. The growth of any instability is thus nearly homologous and the stability criteria derived for vertically averaged structures represent an adequate approximation.

Saio, Hideyuki↗
NASA NTRS1987

Solar oscillations - A method for deriving nonlinear effects

The frequencies of solar oscillations are so closely spaced that nonlinear interactions among modes are probable. The rate of interaction is proportional to an integral involving the eigenfunctions of the interacting modes, which usually are known only numerically. An approximation in which the eigenfunctions are strictly sinusoidal functions of a suitably defined radial variable, and the numerical details of stellar structure are banished to a coefficient in the integrand, are here explored. The physical assumptions are the same as in the asymptotic approximation of p- or g-modes. This method should allow a general investigation as to likely nonlinear interactions and which modes may participate in such interactions. The coupling of two p-modes by possible large-scale internal magnetic fields is introduced as an anisotropic pressure response to a displacement. Pairs of modes differing in frequency by less than the fraction magnetic/thermal pressure are strongly coupled, and energy appears to oscillate slowly between the two associated spherical harmonics. Potentially, upper limits may be derived for internal magnetic fields.

Wentzel, Donat G.↗
NASA NTRS1988

Sampling inhomogeneous turbulent fields

The reconstruction of an inhomogeneous random process from a finite number of discrete samples can be performed in terms of the Karhunen-Loeve (KL) expansion for that process. The n(th) eigenfunction has n - 1 zero crossings which are the sampling points for the inhomogeneous process. The rapid variation of the KL eigenfunctions makes it unnecessary to have a high density of sampling (or grid points) near the wall. However, this result should not be construed to indicate that with spectral simulations significantly fewer grid points are required with the KL expansion as compared to other orthogonal expansions. Moin and Moser (1989) have shown that the advantage of the KL expansion over Chebychev expansion rapidly diminishes when high percentage (say 90 percent) energy recovery is demanded.

Adrian, R. J.↗
NASA NTRS1989

An analysis of the vertical structure equation for arbitrary thermal profiles

The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations.

Cohn, Stephen E.↗
NASA NTRS1989

The normal modes of a resonant cavity containing discrete inhomogeneities - The influence of fibril magnetic fields on the solar acoustic oscillations

Motivated by considerations of the interaction between fibril magnetic fields and solar p-modes, the acoustic spectrum of a cylindrical cavity filled with ideal gas in which a number of magnetic flux tubes are embedded is studied. A formalism, based on the T-matrix approach to acoustic scattering, is developed which can be used to determine the eigenfrequencies and eigenfunctions for any arbitrary distribution of flux tubes. For weak scatterers, the frequency shifts and velocity eigenfunctions are calculated using perturbation theory for the cases of a single flux tube and a random distribution of up to 100 flux tubes. The results of this 'exact' approach are used to give a critical appraisal of the predictions of theories based on some form of averaging, such as the one discussed recently by Bogdan and Zweibel (1987).

Bogdan, Thomas J.↗
NASA NTRS1991

The inviscid compressible Goertler problem

The growth rate is studied of Goertler vortices in a compressible flow in the inviscid limit of large Goertler number. Numerical solutions are obtained for 0(1) wavenumbers. The further limits of large Mach number and large wavenumber with 0(1) Mach number are considered. It is shown that two different types of disturbance modes can appear in this problem. The first is a wall layer mode, so named as it has its eigenfunctions trapped in a thin layer away from the wall and termed a trapped layer mode for large wavenumbers and an adjustment layer mode for large Mach numbers, since then this mode has its eigenfunctions concentrated in the temperature adjustment layer. The near crossing of the modes which occurs in each of the limits mentioned is investigated.

Dando, Andrew↗
NASA NTRS1992

Diskoseismology: Probing accretion disks. II - G-modes, gravitational radiation reaction, and viscosity

A scalar potential is used to derive a single partial differential equation governing the oscillation of a disk. The eigenfunctions and eigenfrequencies of a variety of disk models are found to fall into two main classes which are analogous to the p-modes and g-modes in the sun. Specifically, the eigenfunctions and eigenfrequencies of isothermal disks are computed, and the way in which these results can be generalized to other disk models is indicated. The (assumed) relatively small rates of growth or damping of the modes due to various mechanisms, in particular gravitational radiation reaction and parameterized models of viscosity are also computed. It is found that for certain parameters the p-modes are unstable to gravitational radiation reaction (CFS instability), while both the p-modes and g-modes are unstable to viscosity unless highly anisotropic viscosity models are considered.

Nowak, Michael A.↗
NASA NTRS1992

Diskoseismology - Signatures of black hole accretion disks

General relativity requires the existence of a spectrum of oscillations which are trapped near the inner edge of accretion disks around black holes. We have developed a general formalism for analyzing the normal modes of such acoustic perturbations of arbitrary thin disk models, approximating the dominant relativistic effects via a modified Newtonian potential (these modes do not exist in Newtonian gravity). The eigenfunctions and eigenfrequencies of a variety of disk models are found to fall in to two main classes, which are analogous to the p-modes and g-modes in the sun. In this work, we compute the eigenfunctions and eigenfrequencies of isothermal disks. The (relatively small) rates of growth or damping of these oscillations due to gravitational radiation and parameterized models of viscosity are also computed.

Nowak, Michael↗