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- Author or Editor: S. K. Kao x

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## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## Abstract

Analyses of the isotropic and anisotropic diffusion of clusters of fluid particles in the atmosphere are made. Lin's theory of diffusion is subjected to an observational test. The autocorrelation coefficients of the vector acceleration are computed and the value of *B*
_{0} in Lin's one particle theory is found to be 3 and 5 cm^{2} sec^{−3} for two series of experiments. The mean autocorrelation coefficients of the vector relative acceleration are computed and the value of *B* in Lin's isotropic diffusion theory is found to be 68 and 28 cm^{2} sec^{−3} for two series of experiments. The horizontal eddy diffusivities for the isotropic diffusion process are found to range from 0.4×10^{3} to 6.4×10^{3} cm^{2} sec^{−1}. The mean cross-covariance coefficient of the relative acceleration is computed and the value of *B _{ij}* in Lin's anisotropic diffusion theory is found to be 30 cm

^{2}sec

^{−3}. The values of the diffusion tensor for the anisotropic diffusion process are found to range from 0.3×10

^{3}to 1.8×10

^{3}cm

^{2}sec

^{−1}.

## Abstract

Analyses of the isotropic and anisotropic diffusion of clusters of fluid particles in the atmosphere are made. Lin's theory of diffusion is subjected to an observational test. The autocorrelation coefficients of the vector acceleration are computed and the value of *B*
_{0} in Lin's one particle theory is found to be 3 and 5 cm^{2} sec^{−3} for two series of experiments. The mean autocorrelation coefficients of the vector relative acceleration are computed and the value of *B* in Lin's isotropic diffusion theory is found to be 68 and 28 cm^{2} sec^{−3} for two series of experiments. The horizontal eddy diffusivities for the isotropic diffusion process are found to range from 0.4×10^{3} to 6.4×10^{3} cm^{2} sec^{−1}. The mean cross-covariance coefficient of the relative acceleration is computed and the value of *B _{ij}* in Lin's anisotropic diffusion theory is found to be 30 cm

^{2}sec

^{−3}. The values of the diffusion tensor for the anisotropic diffusion process are found to range from 0.3×10

^{3}to 1.8×10

^{3}cm

^{2}sec

^{−1}.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## Abstract

A primitive equation spectral model using spherical harmonics is formulated to study dynamic interactions between the troposphere and stratosphere in association with sudden stratospheric warmings. Using sigma coordinates for five tropospheric layers and log-pressure coordinates for 26 stratospheric and mesospheric layers separate model equations for each system are combined to form single matrix governing equations. The gradual introduction of large scale topography to balanced initial states representative of observed mean winter conditions in the Northern Hemisphere is used for the generation of planetary waves during 40-day time integrations. Results of these integrations indicate that stratospheric warnings can be simulated by this orographic forcing and that mean momentum flux divergence due to zonal mean motion appears to be an essential mechanism of these simulated sudden warmings. It was found that the strength of the polar night jet can be, a determining factor whether a warming becomes “major” or “minor.”

## Abstract

A primitive equation spectral model using spherical harmonics is formulated to study dynamic interactions between the troposphere and stratosphere in association with sudden stratospheric warmings. Using sigma coordinates for five tropospheric layers and log-pressure coordinates for 26 stratospheric and mesospheric layers separate model equations for each system are combined to form single matrix governing equations. The gradual introduction of large scale topography to balanced initial states representative of observed mean winter conditions in the Northern Hemisphere is used for the generation of planetary waves during 40-day time integrations. Results of these integrations indicate that stratospheric warnings can be simulated by this orographic forcing and that mean momentum flux divergence due to zonal mean motion appears to be an essential mechanism of these simulated sudden warmings. It was found that the strength of the polar night jet can be, a determining factor whether a warming becomes “major” or “minor.”

## 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.

## 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.

## 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.

## 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.