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NASA NTRS1976

Multivariate normality

Sets of experimentally determined or routinely observed data provide information about the past, present and, hopefully, future sets of similarly produced data. An infinite set of statistical models exists which may be used to describe the data sets. The normal distribution is one model. If it serves at all, it serves well. If a data set, or a transformation of the set, representative of a larger population can be described by the normal distribution, then valid statistical inferences can be drawn. There are several tests which may be applied to a data set to determine whether the univariate normal model adequately describes the set. The chi-square test based on Pearson's work in the late nineteenth and early twentieth centuries is often used. Like all tests, it has some weaknesses which are discussed in elementary texts. Extension of the chi-square test to the multivariate normal model is provided. Tables and graphs permit easier application of the test in the higher dimensions. Several examples, using recorded data, illustrate the procedures. Tests of maximum absolute differences, mean sum of squares of residuals, runs and changes of sign are included in these tests. Dimensions one through five with selected sample sizes 11 to 101 are used to illustrate the statistical tests developed.

Crutcher, H. L.↗
NASA NTRS1975

Aerodynamic characteristics of the 10-percent-thick NASA supercritical airfoil 33 designed for a normal-force coefficient of 0.7

A 10-percent-thick supercritical airfoil based on an off-design sonic-pressure plateau criterion was developed and experimental aerodynamic characteristics measured. The airfoil had a design normal-force coefficient of 0.7 and was identified as supercritical airfoil 33. Results show the airfoil to have good drag rise characteristics over a wide range of normal-force coefficients with no measurable shock losses up to the Mach numbers at which drag divergence occurred for normal-force coefficients up to 0.7. Comparisons of experimental and theoretical characteristics were made and composite drag rise characteristics were derived for normal-force coefficients of 0.5 and 0.7 and a Reynolds number of 40 million.

Harris, C. D.↗
NASA NTRS1984

Computational aspects of the nonlinear normal mode initialization of the GLAS 4th order GCM

Using the normal modes of the GLAS 4th Order Model, a Machenhauer nonlinear normal mode initialization (NLNMI) was carried out for the external vertical mode using the GLAS 4th Order shallow water equations model for an equivalent depth corresponding to that associated with the external vertical mode. A simple procedure was devised which was directed at identifying computational modes by following the rate of increase of BAL sub M, the partial (with respect to the zonal wavenumber m) sum of squares of the time change of the normal mode coefficients (for fixed vertical mode index) varying over the latitude index L of symmetric or antisymmetric gravity waves. A working algorithm is presented which speeds up the convergence of the iterative Machenhauer NLNMI. A 24 h integration using the NLNMI state was carried out using both Matsuno and leap-frog time-integration schemes; these runs were then compared to a 24 h integration starting from a non-initialized state. The maximal impact of the nonlinear normal mode initialization was found to occur 6-10 hours after the initial time.

Navon, I. M.↗
NASA NTRS1983

Normal mode study of the earth's rigid body motions

In this paper it is shown that the earth's rigid body (rb) motions can be represented by an analytical set of eigensolutions to the equation of motion for elastic-gravitational free oscillations. Thus each degree of freedom in the rb motion is associated with a rb normal mode. Cases of both nonrotating and rotating earth models are studied, and it is shown that the rb modes do incorporate neatly into the earth's system of normal modes of free oscillation. The excitation formula for the rb modes are also obtained, based on normal mode theory. Physical implications of the results are summarized and the fundamental differences between rb modes and seismic modes are emphasized. In particular, it is ascertained that the Chandler wobble, being one of the rb modes belonging to the rotating earth, can be studied using the established theory of normal modes.

Chao, B. F.↗
NASA NTRS1985

The Vertical Structure of Global Rotational Normal Modes

In a recent study, Lindzen et al. (1984) examined the amplitude and phase evolution of Hough mode projections at 500 mb. The Hough modes represent the simplest normal mode approximation available, and correspond to the neutral eigenfunctions of a shallow water fluid with no mean zonal flow. It was found that when the observed amplitude of a rotational Hough mode was large, it tended to propagate at the phase speed of a normal mode in the presence of mean 500 mb winds, giving a strong indication that normal modes are of relevance to the atmosphere. The vertical structure of the approximately defined rotational normal modes were explored by projecting observed data onto Hough functions at levels other than 500 mb. The stationary and eastward propagating components were filtered out at each level. The evolution of the amplitude in time throughout the troposphere is given for several modes during summer and winter.

Straus, D. M.↗
NASA NTRS1985

Normal mode Rossby waves observed in the upper stratosphere

In recent years, observational evidence has been obtained for westward traveling planetary waves in the middle atmosphere with the aid of global data from satellites. There is no doubt that the fair portion of the observed traveling waves can be understood as the manifestation of the normal mode Rossby waves which are theoretically derived from the tidal theory. Some observational aspects of the structure and behavior of the normal model Rossby waves in the upper stratosphere are reported. The data used are the global stratospheric geopotential thickness and height analyses which are derived mainly from the Stratospheric Sounding Units (SSUs) on board TIROS-N and NOAA satellites. A clear example of the influence of the normal mode Rossby wave on the mean flow is reported. The mechanism considered is interference between the normal mode Rossby wave and the quasi-stationary wave.

Hirooka, T.↗
NASA NTRS1986

A generalized algorithm to design finite field normal basis multipliers

Finite field arithmetic logic is central in the implementation of some error-correcting coders and some cryptographic devices. There is a need for good multiplication algorithms which can be easily realized. Massey and Omura recently developed a new multiplication algorithm for finite fields based on a normal basis representation. Using the normal basis representation, the design of the finite field multiplier is simple and regular. The fundamental design of the Massey-Omura multiplier is based on a design of a product function. In this article, a generalized algorithm to locate a normal basis in a field is first presented. Using this normal basis, an algorithm to construct the product function is then developed. This design does not depend on particular characteristics of the generator polynomial of the field.

Wang, C. C.↗
NASA NTRS1987

An algorithm to design finite field multipliers using a self-dual normal basis

Finite field multiplication is central in the implementation of some error-correcting coders. Massey and Omura have presented a revolutionary design for multiplication in a finite field. In their design, a normal base is utilized to represent the elements of the field. The concept of using a self-dual normal basis to design the Massey-Omura finite field multiplier is presented. Presented first is an algorithm to locate a self-dual normal basis for GF(2 sup m) for odd m. Then a method to construct the product function for designing the Massey-Omura multiplier is developed. It is shown that the construction of the product function base on a self-dual basis is simpler than that based on an arbitrary normal base.

Wang, C. C.↗
NASA NTRS1987

The normal modes of the thermosphere

The linearized momentum, energy, and continuity equations for the thermosphere can be reduced to a form that gives the vertical structure for each horizontal wave mode. The vertical structure equation can be described in terms of the normal modes, or eigenmodes, of the thermosphere. The latter are obtained by using a 27-layer model that includes a realistic temperature profile and the effects of the Lorentz force, viscosity, and heat conduction. The normal modes have one real eigenfrequency for every two complex conjugate eigenfrequency values. The real modes have a dominant rotational wind component and are nonpropagating. The complex modes have comparable divergent and rotational wind components. The complex eigenvalues give vertically propagating modes, primarily associated with the transient response to forcing, and are significantly affected by the dissipation in the upper E region and F region. Results show that the rotational wind component dominates in the steady state when the forcing is due to the two-cell convection pattern at high latitudes and that the normal modes explain the large shears and large winds speeds that are typically observed in the high-latitude E region. The vertical energy flux for the normal modes is also calculated. The results show that the flux is upward above 130 km but downward in the lower E region for the total solution. The downward energy flux is a contribution from the real eigenmode structure.

Larsen, M. F.↗
NASA NTRS1989

A normal-mode approach to Jovian atmospheric dynamic

A nonlinear, quasi-geostrophic, baroclinic model of Jovian atmospheric dynamics is proposed, in which vertical variations of velocity are represented by a truncated sum over a complete set of orthogonal functions obtained by a separation of variables of the linearized quasi-geostrophic potential vorticity equation. A set of equations for the time variation of the mode amplitudes in the nonlinear case is then derived. It is shown that, for a planet with a neutrally stable, fluid interior instead of a solid lower boundary, the barotropic mode represents motions in the interior, and is not affected by the baroclinic modes. One consequence of this is that a normal-mode model with one baroclinic mode is dynamically equivalent to a one-layer model with solid lower topography. It is also shown that, for motions in Jupiter's cloudy lower troposphere, the stratosphere behaves nearly as a rigid lid, so that the normal-mode is applicable to Jupiter. The accuracy of the normal-mode model for Jupiter is tested using the following simple problems: (1) forced, vertically propagating Rossby waves, using two and three baroclinic modes, and (2) baroclinic instability, using two baroclinic modes. It is found that the normal-mode model provides qualitatively correct results, even with only a very limited number of vertical degrees of freedom.

Achterberg, Richard K.↗
NASA NTRS1992

Far-infrared luminosity functions of normal galaxies

A volume-limited sample is constructed from the Zwicky catalog and IRAS data base to examine the FIR luminosity functions of normal galaxies, and to investigate possible relationships between FIR emission and galaxy morphology. Quantitative and unbiased treatment is provided by 'survival analysis' statistical methods. It is found that the FIR distributions of normal galaxies are better fit by lognormal than Schechter functions. The total FIR emissivity (8 to 115 microns) of normal galaxies is approximately equal to half their emission in the B plus V optical bands. Normal galaxy FIR emission is uncorrelated with the basic S0-Sm Hubble sequence of spiral galaxy morphology, but appears to be affected by de Vaucouleurs' (1959) revised morphological classifications based on inner rings and S-shaped arms. Spirals with bars and inner rings are systematically fainter than unbarred spirals. It is suggested that bars and rings reduce the amount or spatially confine the dust in spiral disks, resulting in lower efficiency conversion of optical and UV photons into the IR.

Isobe, Takashi↗
NASA NTRS1992

Quaternion normalization in spacecraft attitude determination

Methods are presented to normalize the attitude quaternion in two extended Kalman filters (EKF), namely, the multiplicative EKF (MEKF) and the additive EKF (AEKF). It is concluded that all the normalization methods work well and yield comparable results. In the AEKF, normalization is not essential, since the data chosen for the test do not have a rapidly varying attitude. In the MEKF, normalization is necessary to avoid divergence of the attitude estimate. All of the methods of the methods behave similarly when the spacecraft experiences low angular rates.

Deutschmann, Julie↗
NASA NTRS1993

The barotropic normal modes in certain shear flows and the traveling waves in the atmosphere

It is shown analytically and numerically that in certain shear flows the linearized nondivergent barotropic vorticity equation has a limited number of neutral normal modes. The latitudinal structures of these shear flows can be expressed as polynomials of the sine of latitude. The first few such shear flows resemble the gross features of the zonal winds in the atmosphere of the earth at different times and altitudes. The spatial structures of the neutral normal modes in these shear flows are spherical harmonics, and, as a consequence, these modes are also the exact solutions of the fully nonlinear equation because the nonlinear interaction term vanishes identically. The spatial structures of the observed 5-, 4-, 2-, and 16-day free traveling waves in the atmosphere are often identified with the spherical harmonics with indices of (m, n) = ( 1, 2), (2, 3), (3, 3), and ( 1, 4), which are known previously as the neutral normal modes of the nondivergent barotropic vorticity equation in a motionless background state. Our results could explain why these free traveling waves can survive the shearing effects of zonal flows that are far different from rest because these spherical harmonics are also normal modes in certain shear flows that resemble the observations of the atmosphere.

Chen, Ping↗
NASA NTRS1993

Normalized vertical ice mass flux profiles from vertically pointing 8-mm-wavelength Doppler radar

During the FIRE 2 (First International Satellite Cloud Climatology Project Regional Experiment) project, NOAA's Wave Propagation Laboratory (WPL) operated its 8-mm wavelength Doppler radar extensively in the vertically pointing mode. This allowed for the calculation of a number of important cirrus cloud parameters, including cloud boundary statistics, cloud particle characteristic sizes and concentrations, and ice mass content (imc). The flux of imc, or, alternatively, ice mass flux (imf), is also an important parameter of a cirrus cloud system. Ice mass flux is important in the vertical redistribution of water substance and thus, in part, determines the cloud evolution. It is important for the development of cloud parameterizations to be able to define the essential physical characteristics of large populations of clouds in the simplest possible way. One method would be to normalize profiles of observed cloud properties, such as those mentioned above, in ways similar to those used in the convective boundary layer. The height then scales from 0.0 at cloud base to 1.0 at cloud top, and the measured cloud parameter scales by its maximum value so that all normalized profiles have 1.0 as their maximum value. The goal is that there will be a 'universal' shape to profiles of the normalized data. This idea was applied to estimates of imf calculated from data obtained by the WPL cloud radar during FIRE II. Other quantities such as median particle diameter, concentration, and ice mass content can also be estimated with this radar, and we expect to also examine normalized profiles of these quantities in time for the 1993 FIRE II meeting.

Orr, Brad W.↗
NASA NTRS1995

Earthquake recurrence on the southern San Andreas modulated by fault-normal stress

Earthquake recurrence data from the Pallett Creek and Wrightwood paleoseismic sites on the San Andreas fault appear to show temporal variations in repeat interval. We investigate the interaction between strike-slip faults and auxiliary reverse and normal faults as a physical mechanism capable of producing such variations. Under the assumption that fault strength is a function of fault-normal stress (e.g. Byerlee's Law), failure of an auxiliary fault modifies the strength of the strike-slip fault, thereby modulating the recurrence interval for earthquakes. In our finite element model, auxiliary faults are driven by stress accumulation near restraining and releasing bends of a strike-slip fault. Earthquakes occur when fault strength is exceeded and are incorporated as a stress drop which is dependent on fault-normal stress. The model is driven by a velocity boundary condition over many earthquake cycles. Resulting synthetic strike-slip earthquake recurrence data display temporal variations similar to observed paleoseismic data within time windows surrounding auxiliary fault failures. Our simple model supports the idea that interaction between a strike-slip fault and auxiliary reverse or normal faults can modulate the recurrence interval of events on the strike-slip fault, possibly producing short term variations in earthquake recurrence interval.

Palmer, Randy↗
NASA NTRS1995

Delayed detonation models for normal and subluminous type Ia sueprnovae: Absolute brightness, light curves, and molecule formation

We compute optical and infrared light curves of the pulsating class of delayed detonation models for Type Ia supernovae (SN Ia's) using an elaborate treatment of the Local Thermodynamic Equilbrium (LTE) radiation transport, equation of state and ionization balance, expansion opacity including the cooling by CO, Co(+), and SiO, and a Monte Carlo gamma-ray deposition scheme. The models have an amount of Ni-56 in the range from approximately or equal to 0.1 solar mass up to 0.7 solar mass depending on the density at which the transition from a deflagration to a detonation occurs. Models with a large nickel production give light curves comparable to those of typical Type Ia supernovae. Subluminous supernovae can be explained by models with a low nickel production. Multiband light curves are presented in comparison with the normally bright event SN 1992bc and the subluminous events Sn 1991bg and SN 1992bo to establish the principle that the delayed detonation paradigm in Chandrasekhar mass models may give a common explosion mechanism accounting for both normal and subluminous SN Ia's. Secondary IR-maxima are formed in the models of normal SN Ia's as a photospheric effect if the photospheric radius continues to increase well after maximum light. Secondary maxima appear later and stronger in models with moderate expansion velocities and with radioactive material closer to the surface. Model light curves for subluminous SN Ia's tend to show only one 'late' IR-maximum. In some delayed detonation models shell-like envelopes form, which consist of unburned carbon and oxygen. The formation of molecules in these envelopes is addressed. If the model retains a C/O-envelope and is subluminous, strong vibration bands of CO may appear, typically several weeks past maximum light. CO should be very weak or absent in normal Sn Ia's.

Hoflich, P.↗
NASA NTRS1997

A Near-Wall Reynolds-Stress Closure Without Wall Normals

Turbulent wall-bounded complex flows are commonly encountered in engineering practice and are of considerable interest in a variety of industrial applications. The presence of a wall significantly affects turbulence characteristics. In addition to the wall effects, turbulent wall-bounded flows become more complicated by the presence of additional body forces (e.g. centrifugal force and Coriolis force) and complex geometry. Most near-wall Reynolds stress models are developed from a high-Reynolds-number model which assumes turbulence is homogenous (or quasi-homogenous). Near-wall modifications are proposed to include wall effects in near-wall regions. In this process, wall normals are introduced. Good predictions could be obtained by Reynolds stress models with wall normals. However, ambiguity arises when the models are applied in flows with multiple walls. Many models have been proposed to model turbulent flows. Among them, Reynolds stress models, in which turbulent stresses are obtained by solving the Reynolds stress transport equations, have been proved to be the most successful ones. To apply the Reynolds stress models to wall-bounded flows, near-wall corrections accounting for the wall effects are needed, and the resulting models are called near-wall Reynolds stress models. In most of the existing near-wall models, the near-wall corrections invoke wall normals. These wall-dependent near-wall models are difficult to implement for turbulent flows with complex geometry and may give inaccurate predictions due to the ambiguity of wall normals at corners connecting multiple walls. The objective of this study is to develop a more general and flexible near-wall Reynolds stress model without using any wall-dependent variable for wall-bounded turbulent flows. With the aid of near-wall asymptotic analysis and results of direct numerical simulation, a new near-wall Reynolds stress model (NNWRS) is formulated based on Speziale et al.'s high-Reynolds-stress model with wall-independent near-wall corrections. Moreover, only one damping function is used for flows with a wide range of Reynolds numbers to ensure that the near-wall modifications diminish away from the walls.

Yuan, S. P.↗
NASA NTRS1996

Dynamic Visual Acuity While Walking in Normals and Labyrinthine-Deficient Patients

We describe a new, objective, easily administered test of dynamic visual acuity (DVA) while walking. Ten normal subjects and five patients with histories of severe bilateral vestibular dysfunctions participated in this study. Subjects viewed a visual display of numerals of different font sizes presented on a laptop computer while they stood still and while they walked on a motorized treadmill. Treadmill speed was adapted for 4 of 5 patients. Subjects were asked to identify the numerals as they appeared on the computer screen. Test results were reasonably repeatable in normals. The percent correct responses at each font size dropped slightly while walking in normals and dropped significantly more in patients. Patients performed significantly worse than normals while standing still and while walking. This task may be useful for evaluating post-flight astronauts and vestibularly impaired patients.

Hillman, Edward J.↗