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All sourcesNASA NTRSDOE OSTI
NASA NTRS1984

Empirical orthogonal functions and normal modes

An attempt to provide physical insight into the empirical orthogonal function (EOF) representation of data fields by the study of fields generated by linear stochastic models is presented in this paper. In a large class of these models, the EOFs at individual Fourier frequencies coincide with the orthogonal mechanical modes of the system - provided they exist. The precise mathematical criteria for this coincidence are derived and a physical interpretation is provided. A scheme possibly useful in forecasting is formally constructed for representing any stochastic field by a linear Hermitian model forced by noise.

North, G. R.↗
NASA NTRS1984

A simple derivation of the formula to calculate synthetic long-period seismograms in a heterogeneous earth by normal mode summation

A simple modification of Gilbert's formula to account for slight lateral heterogeneity of the earth leads to a convenient formula to calculate synthetic long period seismograms. Partial derivatives are easily calculated, thus the formula is suitable for direct inversion of seismograms for lateral heterogeneity of the earth. Previously announced in STAR as N83-29893

Tanimoto, T.↗
NASA NTRS1985

Numerical solution of the vertical structure equation in the normal mode method

In the present model of multilayered stability stratification, aimed at obtaining the analytic eigensolutions of the vertical structure equation, each layer is characterized by its own static stability value. By requiring continuity of pressure and vertical velocity across each interface level, and by imposing suitable upper and lower boundary conditions, matching eigensolutions are obtained in terms of the Bessel functions. Attention is given to an explicit example of a double-layered stratified atmosphere which demonstrates the mathematical manipulations involved; the resultant vertical structure functions are used to check the accuracy of the numerical solutions by the finite difference and finite element methods.

Sasaki, Y. K.↗
NASA NTRS1986

Vertical normal modes of a mesoscale model using a scaled height coordinate

Vertical modes were derived for a version of the Colorado State Regional Atmospheric Mesoscale Modeling System. The impacts of three options for dealing with the upper boundary of the model were studied. The standard model formulation holds pressure constant at a fixed altitude near the model top, and produces a fastest mode with a speed of about 90 m/sec. An alternative formulation, which allows for an external mode, could require recomputation of vertical modes for every surface elevation on the horizontal grid unless the modes are derived in a particular way. These results have bearing on the feasibility of applying vertical mode initialization to models with scaled height coordinates.

Lipton, A. 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 NTRS1992

Influence of mass moment of inertia on normal modes of preloaded solar array mast

Earth-orbiting spacecraft often contain solar arrays or antennas supported by a preloaded mast. Because of weight and cost considerations, the structures supporting the spacecraft appendages are extremely light and flexible; therefore, it is vital to investigate the influence of all physical and structural parameters that may influence the dynamic behavior of the overall structure. The study primarily focuses on the mast for the space station solar arrays, but the formulations and the techniques developed in this study apply to any large and flexible mast in zero gravity. Furthermore, to determine the influence on the circular frequencies, the mass moment of inertia of the mast was incorporated into the governing equation of motion for bending. A finite element technique (MSC/NASTRAN) was used to verify the formulation. Results indicate that when the mast is relatively flexible and long, the mass moment inertia influences the circular frequencies.

Armand, Sasan C.↗
NASA NTRS1992

On the normal modes of Laplace's tidal equations for zonal wavenumber zero

The characteristic differences between two different rotational modes of Laplace's tidal equations for wavenumber m = 0, called the K- and the S-modes, are compared in their energy ratio and structures. It is shown that the K-mode representation captures most of the observed zonal energy with a few terms, whereas the S-mode representation requires many terms. For small vertical scale components, the K-mode series converges faster than the S-mode series. Attention is also given to the differences between the energy spectra projected upon the K- and S-modes and the merits of each set as expansion functions for the zonal atmospheric motions.

Tanaka, H. L.↗
NASA NTRS1993

An analysis of nearfield normal mode amplitude anomalies of the Landers earthquake

The 1992 Landers earthquake (M(sub w) = 7.3) occurred in the middle of the TERRAscope network. Long-period Rayleigh waves recorded at the TERRAscope stations (delta less than or = 3 deg) after traveling around the Earth show large amplitude anomalies, one order of magnitude larger than spherical Earth predictions up to a period of about 600 s. The ground motions over the epicentral region at and after the arrival of R4-5 are in phase at all stations. These observations are inconsistent with the nearly vertical strike slip mechanism of the Landers earthquake. Synthetic seismograms for a rotating, elliptic, and laterally heterogeneous Earth model calculated by the variational method agree well with the observed waveforms. Calculations for various 3D Earth models demonstrate that the amplitudes are very sensitive to the large scale aspherical structure in the crust and the mantle. The anomalies for modes shorter than 300 s period can be explained by lateral heterogeneity shallower than the upper mantle. Rotation of the Earth and lower mantle heterogeneity are required to explain mode amplitudes at longer periods. Current whole mantle seismic tomographic models can fully explain the observed amplitudes longer than 300 s. To assess the effect of the high order lateral heterogeneity in the mantle more precise estimate of the crustal correction is required.

Watada, Shingo↗
NASA NTRS2001

Atmospheric Excitation of Planetary Normal Modes

The objectives of this study were to: (1) understand the phenomenon of continuous free oscillations of the Earth and (2) examine the idea of using this phenomenon for planetary seismology. We first describe the results on (1) and present our evaluations of the idea (2) in the final section. In 1997, after almost forty years since the initial attempt by Benioff et al, continuous free oscillations of the Earth were discovered. Spheroidal fundamental modes between 2 and 7 millihertz are excited continuously with acceleration amplitudes of about 0.3-0.5 nanogals. The signal is now commonly found in virtually all data recorded by STS-1 type broadband seismometers at quiet sites. Seasonal variation in amplitude and the existence of two coupled modes between the atmosphere and the solid Earth support that these oscillations are excited by the atmosphere. Stochastic excitation due to atmospheric turbulence is a favored mechanism, providing a good match between theory and data. The atmosphere has ample energy to support this theory because excitation of these modes require only 500-10000 W whereas the atmosphere contains about 117 W of kinetic energy. An application of this phenomenon includes planetary seismology, because other planets may be oscillating due to atmospheric excitation. The interior structure of planets could be learned by determining the eigenfrequencies in the continuous free oscillations. It is especially attractive to pursue this idea for tectonically quiet planets, since quakes may be too infrequent to be recorded by seismic instruments.

Tanimoto, Toshiro↗
NASA NTRS1985

A review of time-domain and frequency-domain component mode synthesis method

Hurty (1965) has conducted a dynamic analysis of structural systems using component modes. The component mode synthesis (CMS) procedure considered by him represents a form of substructure coupling analysis which is often utilized in structural dynamics. Time-domain CMS methods employing real modes are discussed, taking into account real component modes, normal modes, redundant constraint modes, rigid-body modes, attachment modes, inertia-relief modes, statically-complete interface mode sets, dynamic component mode supersets, component modal models, the coupling of components, and the classification of methods. Attention is also given to the experimental determination of component mode synthesis parameters, time-domain CMS methods for damped systems, and frequency-domain CMS methods for damped systems.

Craig, R. R., Jr.↗
NASA NTRS1988

Dynamical criterion for a marginally unstable, quasi-linear behavior in a two-layer model

A two-layer quasi-geostrophic flow forced by meridional variations in heating can be in regimes ranging from radiative equilibrium to forced geostrophic turbulence. Between these extremes is a regime where the time-mean (zonal) flow is marginally unstable. Using scaling arguments, it is concluded that such a marginally unstable state should occur when a certain parameter, measuring the strength of wave-wave interactions relative to the beta effect and advection by the thermal wind, is small. Numerical simulations support this proposal. A transition from the marginally unstable regime to a more nonlinear regime is then examined through numerical simulations with different radiative forcings. It is found that transition is not caused by secondary instability of waves in the marginally unstable regime. Instead, the time-mean flow can support a number of marginally unstable normal modes. These normal modes interact with each other, and if they are of sufficient amplitude, the flow enters a more nonlinear regime.

Ebisuzaki, W.↗