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All sourcesNASA NTRSDOE OSTI
NASA NTRS1982

Determination of normal modes from measured complex modes

A technique is presented for computing a set of normal modes from a set of measured complex modes. The number of elements in the modal vectors, which is equal to the number of measurements, can be larger than the number of modes under consideration. It is also shown that the practice of normal mode approximation to complex modes can lead to very large errors when the modes are too complex. A numerical example and a simulated experiment illustrate the concepts discussed and support the theory presented.

Ibrahim, S. R.↗
NASA NTRS1985

Computation of Normal Modes from Identified Complex Modes

A technique is presented to compute a set of normal modes from a set of measured (damped) complex modes. The number of elements in the modal vectors, which is equal to the number of measurements, can be larger than the number of modes under consideration. It is also shown in this paper that the practice of normal mode approximation to complex modes can lead to considerably large errors when the modes are too complex. A numerical example and a simulated experiment are presented to illustrate the concepts discussed and to support the theory presented.

Ibrahim, S. R.↗
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

Global normal-mode Rossby waves observed in stratospheric ozone data

Westward-propagating Rossby normal-mode planetary waves are documented in stratospheric ozone data using Solar Backscatter Ultraviolet (SBUV) satellite measurements. These modes are evidenced by enhanced spectral power and near-global coherence for westward-traveling zonal wave 1 oscillations with periods of 5-10 days. The ozone waves have maxima in high latitudes of the middle stratosphere (due to transport) and over midlatitudes in the upper stratosphere (due to photochemistry). These modes are nearly continuous throughout the eight years of SBUV observations, with maximum global coherence during the equinoxes. The upper-stratospheric waves are symmetric (in phase) between hemispheres, even for modes previously identified as antisymmetric in geopotential height. This behavior is due to differing wave vertical structure in each hemisphere; the planetary temperature waves are nearly in phase in the upper stratosphere, even though the height waves are out of phase. The observed ozone waves are furthermore compared to calculations based on linear wave transport and photochemistry, incorporating derived wind and temperature fields. Good agreement is found, showing that normal modes provide an idealized context to study the linear wave behavior of trace constituents in the real atmosphere.

Randel, William J.↗
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 NTRS1985

Resistive wave dissipation on magnetic inhomogeneities Normal modes and phase mixing

Numerical solutions of the linearized, resistive MHD equations indicate the presence of two characteristic forms of wave decay on magnetic inhomogeneities. Solutions for relatively long-wavelength disturbances and large values of the resistivity have the behavior of decaying normal modes. In the opposite limits, phase mixing becomes important, with the eventual build-up of large spatial gradients in which the resistive terms dominate. This composite behavior is shown to be conceptually consistent with analytic results, which predict that the complete solution consists of the sum of collective and noncollective contributions. The numerical simulations go beyond analytic theory by defining where each contribution prevails and computing wave decay when phase mixing is effective. The dispersion relation in the normal-mode regime is also determined analytically, with some approximations, and is in good agreement with the numerical solution. The decay time of the normal modes varies with Lundquist number at S1/6, while the phase-mixed decay time scales as S1/3.

Steinolfson, R. S.↗
NASA NTRS1986

Energetics analysis of the observed and simulated general circulation using three-dimensional normal mode expansions

The energetics characteristics of the observed and simulated general circulation are analyzed using three-dimensional normal mode expansions. The data sets involved are the Goddard Laboratory for Atmospheres (GLA) analysis and simulation data and the Geophysical Fluid Dynamics Laboratory (GFDL) analysis data. The spectral energy properties of the Rossby and gravity modes and energy transformations are presented. Significant influences of model characteristics and the assimilation techniques are observed in the barotropic energy spectrum, particularly for the gravity mode. Energy transformations of the zonal mean field in the GLA analysis and simulation are similar, but distinctly different from that in the GFDL analysis. However, overall, the energy generation in the baroclinic mode is largely balanced by the sink in the barotropic mode. The present study may demonstrate utilities of the three-dimensional normal mode energetics in the analysis of the general circulation.

Tanaka, Hiroshi↗
NASA NTRS1985

Recent Developments in Nonlinear Normal Mode Initialization

The importance of a balanced initial condition upon GLAS GCM forecasts and assimilation cycles was assessed. An effort to combine previous work on normal mode initialization at GLA is underway to develop an initialization process for the production version of the GLAS 4th order GCM. The major aspects of this work fall into two parts: vectorization of the linear projector code and the insertion of the mode projector and Machenhauer iteration algorithm into the full GLAS GCM. Memory and paging constraints place restrictions on the number of horizontal modes stored for initialization purposes, and on the manner in which they are stored. Only the first five vertical structures of the gravity modes are used. Differing phase and normalization conventions provided many elusive coding errors. A Machenhauer nonlinear normal mode initialization technique is used. This method entails the insertion of a modified version of the mode projector into the full GCM, and the modification of the GCM to allow for iterative calls to the projector.

Bloom, S. C.↗
NASA NTRS1979

A theory of stationary long waves. III - Quasi-normal modes in a singular waveguide

The present paper deals with the fundamental issue of whether one can treat waves as normal modes when critical surface, where the phase speed of the wave matches the zonal wind speed, are present. In particular the question of whether a Rossby critical level (such as the zero-wind line for stationary waves) is absorbing or reflecting is raised and subsequently addressed. It is found that the critical level is never totally absorbing; Rossby waves are partially reflected even if the critical layer is dominated by dissipative processes. The relevance of nonlinearity in planetary-scale Rossby wave critical layers is also discussed and it is found to be the dominant mechanism. With the relative magnitudes of nonlinearity versus viscosity relevant to the earth's atmosphere it is found that the steady-state critical level should be almost perfectly reflecting to incident Rossby waves. Consequently, normal-mode solutions can be found; the quantization condition for these waves is also derived.

Tung, K. K.↗
NASA NTRS1982

Utilization of normal mode initial conditions for detecting errors in the dynamics part of primitive equation global models

When a global atmospheric basic state has constant angular velocity and its temperature varies with altitude only, there exist normal mode solutions to the linearized global primitive equations. The use of these normal modes, which have known behavior in time, is superior to the use of the Rossby-Haurwitz wave as initial conditions for detecting errors in the dynamics part of primitive equation global models. With these initial conditions, integration through only one time step is sufficient to detect many formulation and coding errors. Other tests are still required for detecting problems of nonlinear instability and conservation of integral properties, however.

Chao, W. C.↗
NASA NTRS1981

Rossby normal modes in nonuniform background configurations. I Simple fields. II - Equinox and solstice conditions

An investigation is conducted regarding the influence of mean field variations on the realization of planetary normal modes, taking into account the mode response and structure in the presence of simple background nonuniformities. It is found that mean field variations have the combined effect of depressing, shifting, and broadening the characteristic response of Rossby normal modes. While nonuniformities in both the mean wind and temperature fields contribute to the reduction in peak response, the former are primarily responsible for translation and spectral broadening. An investigation is conducted to determine which modes may be realized in actual atmospheric configurations and which may be identified. For both the equinox and solstice configurations, response peaks corresponding to all of the first four modes of wavenumbers 1, 2, and 3 are readily visible above the noise.

Salby, M. L.↗
NASA NTRS1989

MHD surface waves in high- and low-beta plasmas. I - Normal-mode solutions

Since the first paper by Barston (1964) on electrostatic oscillations in inhomogeneous cold plasmas, it has been commonly accepted that all finite layers with a continuous profile in pressure, density and magnetic field cannot support normal surface waves but instead the waves always decay through phase mixing (also called resonant absorption). The problem is reanalyzed by studying a compressible current sheet of a general structure with rotation of the magnetic field included. All inhomogeneous layers considered in the high-beta plasma limit do not support normal modes. However, in the limit of a low-beta plasma there are some cases when normal-mode solutions are recovered. The latter means that the process of resonant absorption is not common for all inhomogeneous layers.

Musielak, Z. E.↗
NASA NTRS1989

A design criterion for avoiding resonance in lumped mass normal mode systems

A simple design criterion that determines whether a normal mode multiple degree of freedom damped linear lumped parameter system can or cannot resonate is presented. The relations are derived based on criteria for resonance in the single degree of freedom case, and on the definiteness of certain combinations of coefficient matrices. An example follows that both numerically verifies the derivation and illustrates the simplicity of implementing the result as a design criterion.

Ross, A. D. S.↗
NASA NTRS1986

Global energetics analysis using 3-dimensional normal mode decomposition

The Goddard Laboratory for Atmospheric Sciences (GLAS) analysis of the FGGE observations for a 25-day period in January 1979 is examined using the normal mode energetics scheme. The results from the energetics analysis are compared with the data of Tanaka (1985). Kinetic energy spectra of the barotropic mode in the meridional-mode and frequency domains, and the total diabatic process are studied. It is observed that there are significant differences between GLAS and GFDL analyses results in the barotropic energy of gravity modes and the normal energetics scheme is applicable as a diagnostic tool.

Tanaka, Hiroshi↗
NASA NTRS1997

Application of Normal Mode Expansion to AE Waves in Finite Plates

Breckenridge et al. (1975), Hsu (1985) and Pao (1978) adapted approaches from seismology to calculate the response at the surface of an infinite half-space and an infinite plate. These approaches have found use in calibrating acoustic emission (AE) transducers. However, it is difficult to extend this theoretical approach to AE testing of practical structures. Weaver and Pao (1982) considered a normal mode solution to the Lamb equations. Hutchinson (1983) pointed out the potential relevance of Mindlin's plate theory (1951) to AE. Pao (1982) reviewed Medick s (1961) classical plate theory for a point source, but rejected it as useful for AE and no one seems to have investigated its relevance to AE any further. Herein, a normal mode solution to the classical plate bending equation was investigated for its applicability to AE. The same source-time function chosen by Weaver and Pao is considered. However, arbitrary source and receiver positions are chosen relative to the boundaries of the plate. This is another advantage of the plate theory treatment in addition to its simplicity. The source does not have to be at the center of the plate as in the axisymmetric treatment. The plate is allowed to remain finite and reflections are predicted. The importance of this theory to AE is that it can handle finite plates, realistic boundary conditions, and can be extended to composite materials.

Gorman, M. R.↗