Frequency-wavenumber Space Research Articles

Harmonic Wave Propagation in a Periodically Layered, Infinite Elastic Body: Antiplane Strain

The propagation of horizontally polarized shear waves through a periodically layered elastic medium is analyzed. The dispersion equation is obtained by using Floquet’s theory and is shown to define a surface in frequency-wave number space. The important features of the surface are the passing and stopping bands, where harmonic waves are propagated or attenuated, respectively. Other features of the spectrum, such as uncoupling at the ends of the Brillouin zones, conical points, and asymptotic values at short wavelengths, are also examined.

A Wavenumber-Frequency Analysis of Large-Scale Tropospheric Motions in the Extratropical Northern Hemisphere

Abstract The geopotential height and temperature at 500 and 300 mb at various northern latitudes are analysed in wavenumber-frequency space, for a summer and a winter season. The power spectra at individual wave-numbers are flattish or peaked at low frequencies, but above a cutoff frequency (which increases with wavenumber from 0.03 to 0.3 cycles per day) they fall off sharply, with a −3 to −4 power law for the geopotential height at mid-latitudes. Temperature spectra are less steep than height spectra, and summer spectra less steep than winter spectra. The quadrature spectra show higher eastward phase speeds in the winter than in the summer. Westward-moving ultralong waves at mid-latitudes are visible in the quadrature spectra of height, but not of temperature. By means of quadrature spectral analysis, separate wavenumber spectra are computed for travelling and stationary waves. The stationary-wave spectra of the temperature fall almost monotonically with wave-number, and at mid-latitudes tend to a −3 po...

Space-time scales of internal waves: A progress report

We present a revised model for the distribution of internal wave energy in wave number frequency space. The model is empirical, guided by the following measurements: moored spectra and moored coherences for horizontal and vertical separations (MS, MHC, MVC as functions of frequency), towed spectra and towed vertical and time-lagged coherences (TS, TVC, TLC as functions of horizontal wave number), and dropped spectra and dropped horizontal and lagged coherences (DS, DHC, DLC as functions of vertical wave number). Measurements are available for all but TLC and DHC. There is some indication of universality, suggesting perhaps a saturation limit.

Interaction Subrange Spectra of Turbulent Wind Over an Air-Water Interface

Abstract The nonlinear interaction of wave-induced air motion and the turbulent wind is examined in spectral space with the aid of a model developed through two averaging processes. The averaging periods are determined by the time scales of turbulent and of wave-induced air motion. It is shown that an interaction subrange spectrum characterized by a −1 slope may exist in the turbulent wind over the air-water interface. A method which involves the calculation of energy spectra in frequency and in wavenumber-frequency space is described. It permits an assessment of nonlinear interactions in these spectral domains. We are concerned particularly with interactions with the wave-induced motion and with the contribution which this may make to the total turbulent kinetic energy.

Short-period seismic noise structure at the Norwegian Seismic Array

abstract Power spectral analysis in frequency-wavenumber space and coherence studies in lag space have shown that the noise recorded by the short period Øyer subarray at NORSAR is critically dependent on the weather situation in the North Atlantic Ocean. In addition to the low-frequency noise from the west, there is observed 2-sec microseisms from the Baltic Sea. Because of the nonisotropic noise, the coherence is usually strongly azimuthal dependent, being represented in lag space by ellipses. Large time variations of the coherence are demonstrated.

The Kinetic Energy of Large-Scale Atmospheric Motion in Wavenumber-Frequency Space: 11. Mid-Troposphere of the Southern Hemisphere

Abstract The wavenumber-frequency spectra of the kinetic energy of the zonal and meridional components of the motion in the mid-troposphere of the Southern Hemisphere show a definite spectral domain of wave activities. This spectral domain is generally oriented from a region of low wavenumbers and low frequencies to a region of high wavenumber and negative frequencies designated for waves moving from west to east. The wavenumber-frequency spectra of the large-scale motion indicate that wave activities in the summer have the same intensity as in the winter in the Southern Hemisphere, whereas in the Northern Hemisphere the wave intensity in summer is about 50% of that in winter. The frequency spectra of the kinetic energy of the zonal and meridional components of the motion show similar distributions at all latitudes and seasons for the respective components of the motion. In the high-frequency range, the frequency spectra of both the zonal and meridional motion are approximately proportional to the –1 powe...

The large-scale meridional transport of sensible heat in wavenumber frequency space

The longitude-time co-spectra of the meridional transport of sensible theat across 20°, 40°, 60°, and 80° N at 100, 200, 500, and 850 mb for the summer and winter seasons of 1964 have been computed and analyzed. It is found that there exists a preferred spectral band in the wave-number frequency spectra at various latitudes, which indicates the wave-number frequency domain of interaction between the meridional velocity and temperature. In the middle latitudes, the spectral band is oriented in a domain extending from the region of low wave-numbers and frequencies to that of high wave-numbers and negative frequencies. In the high latitudes, the heat transport is confined to a domain of low wave-numbers and frequencies. In the lower latitudes, however, the transport is confined to a very narrow band centered near the zero frequency. The total meridional transport of sensible heat at all four levels is mostly directed toward the North Pole, except at 20° N there is a weak equatorward transport of sensible heat at the 100, 200, and 850 mb levels. In the troposphere, the maximum heat transport is found to occur near 40° N, whereas in the stratosphere it occurs near 60° N. It is also found that most of the poleward transport of sensible heat in both the troposphere and stratosphere is accomplished by the stationary and eastward moving, long and medium waves, and that the equatorward transport is mostly accomplished by the westward moving, low frequency waves. Most of the convergence and divergence of the meridional transport in the troposphere is the contribution of the transient waves, whereas in the stratosphere it is mainly the result of the stationary waves. DOI: 10.1111/j.2153-3490.1970.tb01519.x

A study of large-scale atmospheric turbulent kinetic energy in wave-number frequency space

A study of large-scale atmospheric turbulent kinetic energy in wave-number frequency space

A study of large-scale atmospheric turbulent kinetic energy in wave-number frequency space

Large-scale atmospheric turbulence is examined through the application of a wave-number frequency Fourier analysis of the velocity distributions on latitude circles. The primary advantage of the wave-number frequency energy spectra is that it permits analysis of the transient eddies in terms of their length scale and in terms of the speed and direction of their motion. Thus, the relative importance of retrogressing waves may be considered in the analysis. The kinetic energy spectra of the meridional component of velocity is found to contain a distinct band of energy for each wave number which shifts to higher negative frequencies as wave number increases from 4 through 10, at 40° N during winter season 1964. This indicates that the motion of these waves is predominantly from west to east. During the summer season, waves moving in the opposite direction carry a relatively large portion of the transient eddy kinetic energy. The terms of the component form of the kinetic energy equations in wave-number frequency space are evaluated with horizontal nondivergent velocities. The form of the equations used involves nonlinear interaction terms, which represent an inertial transfer process; ageostrophic terms, which represent the work done by the pressure and Coriolis forces; and the eddy frictional terms. Of the nonlinear interaction terms, the one which provides the largest positive or negative contribution to the spectral energy is the inertial term involving the longitudinal derivative. The largest interaction combination within each of these terms involves the spectral components of the kinetic energy production terms. This condition causes a positive contribution to the energy of waves moving from west to east and a negative contribution to waves moving from east to west. The contributions to the spectral energy through the interaction terms involving latitudinal derivatives are smaller with less consistency in sign. The interaction combinations representing the kinetic energy production in these terms have the same magnitude as several other interaction combinations. The contributions from the ageostrophic terms serve to balance the contributions from the interaction terms. The magnitude of these balancing contributions is from 2 to 5 times the magnitude of the spectral energy. DOI: 10.1111/j.2153-3490.1969.tb00484.x

The resolution of turbulent pressures at the wall of a boundary layer

This paper is a discussion of the resolution by a piezo-electric transducer of the local pressure fluctuations at the wall of a turbulent boundary layer. The first part applies to transducers whose sensing area is large in comparison with the boundary layer thickness, and the second part to transducers whose sensing area is either of the order of, or smaller than, the boundary layer thickness. For large transducers, the main result, derived for a sensing surface of arbitrary shape and with an arbitrary spatial sensitivity distribution, is that under circumstances which are perhaps unrealistically stringent, the attenuation of a given frequency component will increase as the cube of the transducer face linear dimension. In general, it should be expected to increase only as the square of the dimension. In the second part, the probable dependence of the resolution on frequency and on transducer size is indicated by using both recent experimental evidence and physical reasoning bearing on the nature of the pressure cross-spectral density. Both approaches suggest that in a co-ordinate system translated downstream at an appropriate velocity, as the streamwise wave number increases, the spectral density in wave number-frequency space is associated with proportionately increasing spanwise wave numbers and almost proportionately increasing frequencies. Experimental evidence suggests departure from this finding if the fixed axis longitudinal wave length is very large (of the order of the boundary layer thickness) in which case both the characteristic time in convected axes and the characteristic lateral coherence length appear to reach asymptotic finite values. For shorter waves, the strong relationship between the two space scales and the convective time scale necessarily implies that the resolution of the frequency spectral density by a finite transducer continues to deterioriate as the frequency increases. It is concluded that the attenuation predictions previously published by the author are not likely to be seriously in error.

Spectrum of hydromagnetic waves in the exosphere

A disturbance in the exosphere generates waves in three partially separable modes. The modes are described by considering the vorticity about a line of force, the two-dimensional divergences of velocity in the plane perpendicular to the line of force, and the component of velocity along the line of force. The propagation of vorticity is one-dimensional and there is no geometrical attenuation. It is suggested that this mode is associated with the sudden commencement of a magnetic storm. The propagation of the longitudinal component velocity is almost one-dimensional, but it is heavily damped at high frequencies. In a gravitational field, the medium is no longer uniform, and at low frequencies the modes are coupled in a complicated way. The influence of gravity is characterized by the frequency of oscillation of a small parcel of fluid when displaced from its zero-order position in the gravitational field. For parallel magnetic and gravitational fields the vorticity mode is still separable, and gravity leads to anisotropic dispersion in the other modes. The propagation at low frequencies is no longer cylindrically symmetrical if the lines of force are at an arbitrary angle to the direction of gravity. Detailed investigations of the influence of the gravitational field on hydromagnetic wave propagation is carried out for the case where the sound velocity is small compared to the Alfven wave velocity. The principal aim of the study is an attempt to provide a background for the separation of source effects from those of the medium as an aid to the spectral study of magnetic records. The relative proportion of the three modes generated by a specific source is estimated. Consideration of the regions of allowed solutions in frequency-wave number space show that the vorticity mode will have a narrow spectrum. It is suggested that high-frequency micropulsations are associated with vorticity mode propagation. Examples of vorticity generating sources include the turbulent region in the vicinity of a finite amplitude wave, the outgassing of a rocket motor, and a wake of a satellite in the lower exosphere. Energy may be trapped within the exosphere in various free vorticity mode oscillations. The free periods of these oscillations are comparable to the periods observed in the low frequency micropulsations. The limiting conditions for the propagation of the various modes in the exosphere are considered. Because of the long-range nature of the Coulomb forces, the charged particle may travel distances short compared with its mean free path and still exchange a substantial proportion of its momentum. For representative models of the exosphere, disturbances at several earth radii must have a length scale large compared with several kilometers for the propagation of the longitudinal component of velocity. The limiting conditions on the vorticity and transverse modes involve the ratio of the length scale of the disturbance to the Larmor radius. These modes are propagated if the disturbances have length scales of some hundreds of meters in the outer regions of the exosphere.