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  • Author or Editor: S. K. Kao x
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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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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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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 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, C. N. Chi, and W. M. Washington

Abstract

An analysis of the three-dimensional, large-scale movement of air particles for the winter months with the NCAR general circulation model indicates that the horizontal movement of particles in the upper troposphere is greatly affected by wave motion in mid- and high latitudes, by the field of horizontal convergence and divergence, and by mean meridional circulation in the tropics. The mean center of mass of particles in both hemispheres generally moves toward respective poles and the mean squire of the meridional component of the particle distances generally decreases with increasing time, indicating the effect of horizontal convergence on particle movement near the subtropics. The vertical movement of the particles is affected by upward motion near the thermal equator and downward motion near the subtropical region in the Northern and Southern Hemispheres. The vertical dispersion is most intense in the tropics and decreases toward the poles. There are two maxima of particle accumulation, one occurring near 15°N, the other near 30°S, and a minimum accumulation of particles appears near the thermal equator, indicating the effects of the divergence field and meridional circulation between the thermal equator and the subtropics.

The mean squares of zonal, meridional and vertical components of the distance for dusty” of particles released at the equator and 45°N appear to consist of two components, a monotonicaly increasing component due essentially to the effect of turbulent diffusion, and a periodic component due primarily to the horizontal velocity convergence and divergence of mean motion.

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S. K. Kao, R. L. Jenne, and J. F. Sagendorf

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 power of the frequency.

The wavenumber spectra of the kinetic energy of the zonal and meridional motion also show a similar distribution at all latitudes and seasons for the respective components of the motion. In the high-wave-number range, the spectra of both the zonal and meridional components of the motion are approximately proportional to the –3 power of the wavenumber, which is characteristic of the wavenumber spectrum for the two-dimensional flow of an incompressible viscous fluid. The fact that the wavenumber and frequency spectra are proportional to different powers of wavenumber and frequency indicates that Taylor's transformation does not apply to the large-scale motion in the atmosphere.

The mean kinetic energy of the zonal motion in the mid-troposphere of the Southern Hemisphere shows a maximum near 40S in winter and 50S in summer, with 75% of the kinetic energy of the zonal motion being associated with the stationary mean zonal motion and 25% with the zonal component of the moving waves.

The mean kinetic energy of the meridional component of the motion shows a maximum at 50S for both the summer and winter seasons. Practically all the kinetic energy of the meridional motion is associated with the moving waves.

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S-K. Kao, C. Y. Tsay, and Larry L. Wendell

Abstract

The wavenumber-frequency spectra of the meridional transport of angular momentum at 100, 200 and 500 mb, at 20, 40, 60 and 80N, show that there exist definite spectral domains of wave interactions between the zonal and meridional velocities at various latitudes. In the middle latitudes near 40N, the spectral band of the meridional transport of angular momentum is oriented from a region of low wavenumbers and low frequencies to a region of high wavenumbers and negative frequencies designated for eastward-moving waves. In low latitudes, however, the spectral band is confined to a narrow band centered near zero frequency.

An analysis of the linear and nonlinear contributions to the meridional transport of angular momentum in various wavenumber-frequency domains indicates that in the mid-troposphere the primary contribution to the nonlinear interactions always involves the interactions of the spectral domain of concern with the mean zonal flow and the stationary planetary waves. It is also found that except in the domain of low-frequency, eastward-moving cyclone waves the following characteristics are in common. 1) the meridional transport of angular momentum is directed toward the north pole; 2) the resultant of the nonlinear interactions due to the longitudinal convergence of the transport provides a poleward flux of angular momentum in the domains of eastward-moving waves, but provides an equatorward transport in the domains of westward-moving waves; 3) the resultant of the nonlinear interactions due to the latitudinal convergence of the transport generally contributes a poleward transport of angular momentum in the domains of westward-moving waves, but contributes an equatorward transport in the domains of eastward-moving waves; 4) the ageostrophic effect always counteracts the nonlinear interactions due to the longitudinal convergence of the transport of angular momentum; and 5) the effects of eddy and molecular stress forces generally work against the ageostrophic effect.

The frequency spectra of the meridional transport of angular momentum indicate that: 1) in the summer most of the transport is accomplished by the moving waves, the eastward-moving waves contributing to most of the poleward transport, and the westward-moving waves to the equatorward transport; 2) in the winter most of the transport is accomplished by the stationary waves, and both the eastward- and westward- moving waves contribute to the poleward transport of angular momentum.

The wavenumber spectra of the transport of angular momentum indicate that in both the summer and winter seasons waves of practically all wavelengths in low and middle latitudes contribute to the poleward transport of angular momentum. In high latitudes, however, only the very long waves contribute to the equatorward transport of angular momentum.

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