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S. K. Kao and H. N. Lee

Abstract

An analysis of the linear and nonlinear interactions of atmospheric motion in the wavenumber-frequency domain indicates that the kinetic energy of the large-scale moving waves is essentially maintained by the nonlinear interactions and the pressure force. In middle latitudes where an eastward mean zonal flow prevails, the supply of kinetic energy to eastward moving waves through the nonlinear interactions is greater than the extraction of kinetic energy through the pressure force, whereas the supply of kinetic energy to westward moving waves through the pressure force is greater than the extraction of kinetic energy through the nonlinear interactions. Near the equator where a weak westward mean zonal Row occurs, the non-linear interactions generally extract kinetic energy from the eastward moving waves, but supply kinetic energy to the westward moving waves; the pressure force, however, supplies kinetic energy to both eastward and westward moving waves.

The primary contribution of the nonlinear interactions to the energy transfer in wavenumber-frequency domain is essentially through the interactions of the slowly moving waves, the stationary long waves and the zonal mean flow. The interactions between the stationary long waves and waves moving in the same (opposite) direction of the mean zonal flow generally extract (supply) kinetic energy from (to) the moving waves, whereas the interactions between the mean zonal flow and waves moving in the same (opposite) direction of the zonal flow generally supply (extract) kinetic energy to (from) the moving waves.

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S-K. Kao and H. D. Woods

Abstract

The wind velocities measured by an aircraft flying parallel and perpendicular to jet streams (Project Jet Stream, 1956–1957) have been analyzed; a smoothing technique has been used to separate the meso-scale turbulence from the mean flow. Eulerian auto-correlation coefficients and energy spectra are computed for the longitudinal and transversal components of the horizontal wind velocities. The distributions of the auto-correlation coefficients and the energy spectra appear to be similar for both the longitudinal and transversal components of the velocities, whereas the corrected meso-scale energy spectrum increases with decreasing wave number and is approximately proportional to k −2 in the range between 10−1 cycles km−1.

An analysis is also made of the distribution of the Richardson number in a cross section perpendicular to the jet stream. A good relationship is found between the areas of turbulence and the regions of small Richardson number.

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S-K. Kao and David C. Powell

Abstract

The characteristics of the large-scale relative particle displacement tensor, the correlation functions, and spectra of the relative particle velocities at 10-, 30-, 50- and 100-mb levels are investigated; pertinent results concerning relative turbulence and diffusion at various levels in both troposphere and stratosphere are discussed and summarized. It is found that a quasi-stationary process exists in the large-scale turbulence diffusion in both the troposphere and stratosphere, the rate of relative particle dispersion being greatest in the tropopause level and generally proportional to the variance of the relative velocity. In general, the auto-correlation functions for the relative zonal velocities in both the troposphere and stratosphere behave like an exponentially decreasing function, whereas those for the relative meridional velocities shows a combination of an exponential function and a cosine function with a damping amplitude. The power spectra of the relative zonal velocities at all levels show the similar characteristics of increasing kinetic energy with decreasing frequency, whereas those of the relative meridional velocities show an energy peak near the frequency of 10−2 cycles hr−1. The high frequency portion of the power spectra of both the zonal and meridional components of the relative velocities at all levels is found to be proportional to the minus third power of the frequency. The principal axis of the large-scale turbulent diffusion in the stratosphere is generally oriented ENE-WSW, whereas in the troposphere it is ESE-WNW.

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S-K. Kao and Larry L. Wendell

Abstract

The wavenumber-frequency spectra of the kinetic energy of the zonal and meridional components of the motion at 100, 200 and 500 mb, at 20, 40, 60 and 8ON, show a definite spectral domain of wave activities in the atmosphere. In middle latitudes, the spectral domain is oriented from a region of low wavenumbers and low frequencies to a region of high wavenumbers and negative frequencies designated for waves moving from west to east. In high latitudes, the domain of wave activities is confined to a region of low wavenumbers and low frequencies. In low latitudes, however, there exist two domains, one similar to that in the middle latitude and the other occurring in a narrow band centered near zero frequency in the medium wavenumber range.

The frequency spectra of the kinetic energy of the zonal motion show similar distributions at all levels and seasons, and are approximately proportional to the minus first power of the frequency in low latitudes but are proportional to the minus second power of the frequency in high latitudes. The wavenumber spectra of the zonal motion a1so show similar distributions at all levels and seasons, and are approximately proportional to the minus third power of the wavenumber in the high wavenumber range. The wavenumber spectra of the meridional motion show an energy peak in the wavenumber range k = 4–10. Again, in the high wavenumber range, the power spectra of the meridional motion are approximately proportional to the minus third power of the wavenumber.

The mean kinetic energy of the zonal motion shows a maximum near 4ON at all levels and seasons, except at 100 mb in the summer where it occurs near 20N. The distribution of the mean kinetic energy of the moving waves indicates a definite shift in the region of wave activities with height; the maximum wave activity occurs near 60N in the troposphere, near 4ON at the tropopause level, and near 6ON in the stratosphere. In winter, the mean kinetic energy of the meridional motion shows a great deal of energy in high latitudes, caused primarily by the winter instability of the polar vortex in the stratosphere.

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S-K. Kao and A. A. al-Gain

Abstract

The characteristics of the relative particle displacement tensor, the correlation functions, and spectra of the relative particle velocities at 200-, 500- and 850-mb levels are investigated. It is found that similarity and stability exist for the autocorrelation functions as well as the power spectra at various levels, which indicate that a quasi-stationary process exists in the large-scale relative diffusion in the atmosphere. The high frequency portion of the power spectra of both the zonal and meridional components of the relative velocities is found to be more or less proportional to k −8; the relative meridional velocity shows a maximum at the low frequency end. The low frequency portion of the cross spectra of the zonal and meridional relative velocities at 850 mb shows an opposite transfer to that at 200 mb. The mean square of the relative zonal displacement at a level is found to be about twice that of the relative meridional displacement at that level. It is also found that anisotropy exists in the field of the large-scale turbulent dispersion and that the major axis of the dispersion is generally oriented in the ESE to WNW direction.

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H. N. Lee and S. K. Kao

Abstract

A dynamic turbulent boundary-layer model in the neutral atmosphere is constructed, using a dynamic turbulent equation of the eddy viscosity coefficient for momentum derived from the relationship among the turbulent dissipation rate, the turbulent kinetic energy and the eddy viscosity coefficient, with aid of the turbulent second-order closure scheme. A finite-element technique was used for the numerical integration. In preliminary results, the behavior of the neutral planetary boundary layer agrees well with the available data and with the existing elaborate turbulent models, using a finite-difference scheme. The proposed dynamic formulation of the eddy viscosity coefficient for momentum is particularly attractive and can provide a viable alternative approach to study atmospheric turbulence, diffusion and air pollution.

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S-K. Kao and William R. Hill

Abstract

An analysis of the Eulerian and Lagrangian velocities at the 200-mb level in the Southern Hemisphere is made. It is found that: 1) the zonal component of the eddy diffusivity in the mid-atmosphere in the Southern Hemisphere is about 50% greater than that in the Northern, whereas the meridional component of the eddy diffusivity in the Southern Hemisphere is about 50% smaller than that in the Northern; 2) the coefficient for the Eulerian-Lagrangian time-scale transformation in the Southern Hemisphere is about 0.6 which is of the same order of magnitude as that in the Northern; 3) the autocorrelation functions and energy spectra of the Eulerian and Lagrangian velocities in the Southern Hemisphere are similar to those in the Northern; and 4) the peak of the energy spectrum of the meridional component of the Lagrangian velocity in the Southern Hemisphere occurs near the frequency 1.8 × 10−2 cycle hr−1, about the same as that in the Northern.

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S. K. Kao and C. N. Chi

Abstract

An analysis of the forces and motion at 500 mb, between 30 and 60°N, in wavenumber-frequency domain, indicates that there exist definite cycles in the generation, transport and dissipation of the kinetic and available potential energies associated with long- and synoptic-scale waves. The growth and decay of the kinetic energy of long- and synoptic-scale waves are primarily controlled by the transport of kinetic energy to and from the waves through the nonlinear wave interactions, while the contribution to the kinetic energy through energy conversion tends to balance the effects of the Reynolds and frictional stresses. The evolution of the available potential energy associated with the long and synoptic waves is essentially the consequence of the transfer of thermal energy to and from the wave through the interaction between the velocity and temperature waves, while the transfer of thermal energy through the interactions between the velocity waves and the gradient of the zonal mean temperature tends to balance the effects of diabatic heating or cooling and energy conversion. The growth and decay of the kinetic energy of the zonal flow are primarily the result of the interaction between the velocity waves and the gradient of the mean zonal velocity, while the energy conversion from available potential to kinetic energy tends to balance the effects of the Reynolds and frictional stresses. The evolution of available potential energy associated with the zonal flow is essentially controlled by the interaction between the velocity waves and the gradient of the zonal mean temperature, while the effect of diabatic heating tends to balance the effect of energy conversion between the kinetic and available potential energies.

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S-K. Kao and R. J. Kuczek

Abstract

The wavenumber-frequency spectra of the kinetic energy of the zonal and meridional components of the motion at 200- and 500-mb levels in the tropics show that there exists a band of wave activities which is oriented from a region of low wavenumber, and frequencies to a region of high wavenumbers and low frequencies. This orientation is distinctly different from what is found at higher latitudes where the band extends from a region of low wavenumbers and frequencies to a region of high wavenumbers and negative frequencies.

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W. S. Kau, H. N. Lee, and S. K. Kao

Abstract

Statistical models for surface-wind predictions at a mountain and a valley station near Anderson Creek, California, have been constructed. It is found that the surface wind speed depends primarily on the slope wind, cross-isobaric angle, surface thermal stability and geostrophic wind. The correlations between the calculated and observed surface wind speeds are found to be high for all time periods of the day and night.

Because the variability of wind direction, which is greatly affected by topography, geostrophic wind and turbulent motion, is generally larger than that of the surface wind speed, statistical models for wind direction are more complicated than those for the wind speed. It is found that wind direction depends primarily on the geostrophic wind direction, aspect angle of the topography, up-canyon direction and cross-isobaric angle in the boundary layer.

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