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## Abstract

The analysis of an earlier work on the interaction between an internal gravity wave and the wave-induced turbulence is extended here to the case where the wave is generated by vertical wind shear. The initial system is described by a background wind and a temperature distribution such that the Richardson number is less than ¼. A gravity wave is generated by such a dynamically unstable system and grows exponentially in time. The wave modifies the Richardson number and lowers it, particularly in the neighborhood of the critical level. When the generalized Richardson number falls below an assumed critical level, turbulence is assumed to develop and is described by a “1½th order” scheme. A diffusion coefficient can then be calculated which has a mean and a fluctuating part. It is the latter which turns out to be responsible for the positive feedback between the wave and the wave-induced turbulence resulting in the wave growing at a faster rate than the one predicted by linear theory.

## Abstract

The analysis of an earlier work on the interaction between an internal gravity wave and the wave-induced turbulence is extended here to the case where the wave is generated by vertical wind shear. The initial system is described by a background wind and a temperature distribution such that the Richardson number is less than ¼. A gravity wave is generated by such a dynamically unstable system and grows exponentially in time. The wave modifies the Richardson number and lowers it, particularly in the neighborhood of the critical level. When the generalized Richardson number falls below an assumed critical level, turbulence is assumed to develop and is described by a “1½th order” scheme. A diffusion coefficient can then be calculated which has a mean and a fluctuating part. It is the latter which turns out to be responsible for the positive feedback between the wave and the wave-induced turbulence resulting in the wave growing at a faster rate than the one predicted by linear theory.

## Abstract

The simple Kelvin-Helmholtz model for shear zones in the atmosphere is modified, by introducing a solid boundary below to account for the effect of the ground. The new characteristics of neutral and unstable waves that can exist in such configuration are analyzed for various values of wind velocity, depth of the bottom layer, and Brunt-Väisälä frequency. It is shown that the presence of the ground considerably destabilizes waves with long horizontal wavelengths. In particular, long wavelengths are always unstable, so that no neutral stability boundary exists. Furthermore, the solid lower boundary introduces an infinite number of neutral modes, all of which correspond to evanescent waves in the top layer. Finally, the model with the ground is used to calculate the characteristics of the most unstable waves that would be generated for some well-documented observed cases and the calculated values are found to be in reasonable agreement with observations.

## Abstract

The simple Kelvin-Helmholtz model for shear zones in the atmosphere is modified, by introducing a solid boundary below to account for the effect of the ground. The new characteristics of neutral and unstable waves that can exist in such configuration are analyzed for various values of wind velocity, depth of the bottom layer, and Brunt-Väisälä frequency. It is shown that the presence of the ground considerably destabilizes waves with long horizontal wavelengths. In particular, long wavelengths are always unstable, so that no neutral stability boundary exists. Furthermore, the solid lower boundary introduces an infinite number of neutral modes, all of which correspond to evanescent waves in the top layer. Finally, the model with the ground is used to calculate the characteristics of the most unstable waves that would be generated for some well-documented observed cases and the calculated values are found to be in reasonable agreement with observations.

## Abstract

Observations have been made of a stably-stratified nighttime boundary layer perturbed by Kelvin-Helmholtz internal waves with critical levels around 600 m. Significant turbulence intensities were measured although the time-mean gradient Richardson numbers were large and positive. It is shown by constructing energy budgets of wave and turbulent components separately that there is an essential flow of kinetic energy from wave to turbulence and that the mechanics of this exchange process depend upon the nonlinear character of the wave field.

Turbulent energy budgets were followed through a wave cycle and revealed that turbulence production occurred during only one quarter of a wave period, the rest of the time being taken up by redistribution of turbulent kinetic energy (tke) among the three orthogonal components, relaxation under the effects of density stratification and dissipation. The principal path of energy dissipation is through conversion of vertical component tke to density fluctuations, which are in turn dissipated by molecular conductivity. Direct viscous dissipation of tke is negligible in comparison. This behavior is consistent with the quasi-two-dimensional character imposed on the turbulence by the strong stability and is clearly apparent in the behavior of the velocity and temperature spectra.

## Abstract

Observations have been made of a stably-stratified nighttime boundary layer perturbed by Kelvin-Helmholtz internal waves with critical levels around 600 m. Significant turbulence intensities were measured although the time-mean gradient Richardson numbers were large and positive. It is shown by constructing energy budgets of wave and turbulent components separately that there is an essential flow of kinetic energy from wave to turbulence and that the mechanics of this exchange process depend upon the nonlinear character of the wave field.

Turbulent energy budgets were followed through a wave cycle and revealed that turbulence production occurred during only one quarter of a wave period, the rest of the time being taken up by redistribution of turbulent kinetic energy (tke) among the three orthogonal components, relaxation under the effects of density stratification and dissipation. The principal path of energy dissipation is through conversion of vertical component tke to density fluctuations, which are in turn dissipated by molecular conductivity. Direct viscous dissipation of tke is negligible in comparison. This behavior is consistent with the quasi-two-dimensional character imposed on the turbulence by the strong stability and is clearly apparent in the behavior of the velocity and temperature spectra.

## Abstract

We describe aircraft turbulence-atmospheric gravity wave events which occurred during a 2-day period over the Continental Divide. The waves are observed by two microbarograph networks an each side of the divide and last for several hours at a time. We show them to be unstable modes of the jet stream, corresponding to propagating internal gravity waves. We also show that the position of aircraft-reported turbulence coincides with the critical levels of the waves.

## Abstract

We describe aircraft turbulence-atmospheric gravity wave events which occurred during a 2-day period over the Continental Divide. The waves are observed by two microbarograph networks an each side of the divide and last for several hours at a time. We show them to be unstable modes of the jet stream, corresponding to propagating internal gravity waves. We also show that the position of aircraft-reported turbulence coincides with the critical levels of the waves.

## Abstract

During the 1978 PHOENIX experiment at the Boulder Atmospheric Observatory in Colorado, the presence of atmospheric gravity waves was detected by various independent remote sensing instruments. Fluctuations in the zenith atmospheric radiation were measured at 22.235 and 55.45 GHz in the water vapor and oxygen absorption bands and compared with corresponding fluctuations of surface pressure and the height of FM-CW radar echo returns. These fluctuations are explained, qualitatively and quantitatively, in terms of an internal gravity wave generated by wind shear above the boundary layer. The analysis shows that the oscillations at 22.235 GHz are essentially due to fluctuations of water vapor in the antenna beam while those at 55.45 GHz are due to temperature variations.

## Abstract

During the 1978 PHOENIX experiment at the Boulder Atmospheric Observatory in Colorado, the presence of atmospheric gravity waves was detected by various independent remote sensing instruments. Fluctuations in the zenith atmospheric radiation were measured at 22.235 and 55.45 GHz in the water vapor and oxygen absorption bands and compared with corresponding fluctuations of surface pressure and the height of FM-CW radar echo returns. These fluctuations are explained, qualitatively and quantitatively, in terms of an internal gravity wave generated by wind shear above the boundary layer. The analysis shows that the oscillations at 22.235 GHz are essentially due to fluctuations of water vapor in the antenna beam while those at 55.45 GHz are due to temperature variations.

## Abstract

We present the results of an analytical and numerical calculation of the interaction between an internal gravity wave and a wave-induced turbulence. The initial atmospheric state, assumed horizontally homogeneous, is statically and dynamically stable with the background Richardson number Ri_{0} approaching ¼ over some height regions. An initial non-singular neutral gravity wave propagates through such a system and modifies the Richardson number. The new Richardson number Ri may become smaller than ¼ and turbulence may develop. Using a “1½th order” scheme for the turbulence, we calculate the mean and the fluctuating part of the eddy diffusion coefficient. We show that the fluctuating part of the diffusion coefficient, because of its amplitude and phase, may overcome the damping effect of its mean part and force the original wave to grow in time. As the wave grows, it may further lower the Richardson number, increase the intensity of the turbulence, and further strengthen its interaction with it. At least in its initial stages, wave-induced turbulence appears to be an effective mechanism for transfer of energy from the background state into the wave. By showing that the early stages of the wave-induced turbulence interaction can lead to energy being transferred into the wave, we strengthen the case for gravity waves as important elements in the generation of turbulence in the atmosphere. The values we obtain for the eddy diffusion coefficients suggest that the process is quite capable of producing the empirically observed mixing rates at substantial heights above the ground. While the present calculations cannot describe the long-time limit of the wave-turbulence system, one may suggest that the often observed atmospheric conditions in which turbulence and waves appear to co-exist for several hours may result from a sort of equilibrium between the roles of the mean and the fluctuating parts of the eddy diffusion coefficient in taking away from and feeding energy into the wave.

## Abstract

We present the results of an analytical and numerical calculation of the interaction between an internal gravity wave and a wave-induced turbulence. The initial atmospheric state, assumed horizontally homogeneous, is statically and dynamically stable with the background Richardson number Ri_{0} approaching ¼ over some height regions. An initial non-singular neutral gravity wave propagates through such a system and modifies the Richardson number. The new Richardson number Ri may become smaller than ¼ and turbulence may develop. Using a “1½th order” scheme for the turbulence, we calculate the mean and the fluctuating part of the eddy diffusion coefficient. We show that the fluctuating part of the diffusion coefficient, because of its amplitude and phase, may overcome the damping effect of its mean part and force the original wave to grow in time. As the wave grows, it may further lower the Richardson number, increase the intensity of the turbulence, and further strengthen its interaction with it. At least in its initial stages, wave-induced turbulence appears to be an effective mechanism for transfer of energy from the background state into the wave. By showing that the early stages of the wave-induced turbulence interaction can lead to energy being transferred into the wave, we strengthen the case for gravity waves as important elements in the generation of turbulence in the atmosphere. The values we obtain for the eddy diffusion coefficients suggest that the process is quite capable of producing the empirically observed mixing rates at substantial heights above the ground. While the present calculations cannot describe the long-time limit of the wave-turbulence system, one may suggest that the often observed atmospheric conditions in which turbulence and waves appear to co-exist for several hours may result from a sort of equilibrium between the roles of the mean and the fluctuating parts of the eddy diffusion coefficient in taking away from and feeding energy into the wave.

## Abstract

We present a climatological study of gravity waves and other coherent disturbances at the Boulder Atmospheric Observatory, during the period mid-March-mid-April 1984. The data were collected by a network of microbarographs, and by sensors on the 300 m tower. The total observational period was divided into 522 time segments of 5120 s each. Coherent and incoherent motions were identified on the basis of a cross-correlation coefficient, calculated from the microbarograph network for each time segment and frequency band analyzed, using the assumption that the atmospheric state can be described by an equivalent plane wave. Five passbands were considered in the period range 1–20 min.

The analysis indicates that the atmospheric state at these passbands displays highly coherent structure, most of the time. During the interval from 0800 to 1800 LST, coherent motions with cross-correlation coefficient larger than 0.5 are present about 25% of the time for periods between 1 and 5 min and more than 80% of the time for periods between 10 and 20 min. In the remaining hours of the day, the percentages rise to more than 40% and 95% of the time, respectively.

A relationship is illustrated between the turbulent kinetic energy measured on the tower and the amplitude of the rms pressure field at the ground for disturbances having up to 5 min periods. For longer periods, such a relationship appears to be absent, indicating that the longer the scales, the deeper the atmospheric zone important to the dynamics of the pressure fluctuations.

## Abstract

We present a climatological study of gravity waves and other coherent disturbances at the Boulder Atmospheric Observatory, during the period mid-March-mid-April 1984. The data were collected by a network of microbarographs, and by sensors on the 300 m tower. The total observational period was divided into 522 time segments of 5120 s each. Coherent and incoherent motions were identified on the basis of a cross-correlation coefficient, calculated from the microbarograph network for each time segment and frequency band analyzed, using the assumption that the atmospheric state can be described by an equivalent plane wave. Five passbands were considered in the period range 1–20 min.

The analysis indicates that the atmospheric state at these passbands displays highly coherent structure, most of the time. During the interval from 0800 to 1800 LST, coherent motions with cross-correlation coefficient larger than 0.5 are present about 25% of the time for periods between 1 and 5 min and more than 80% of the time for periods between 10 and 20 min. In the remaining hours of the day, the percentages rise to more than 40% and 95% of the time, respectively.

A relationship is illustrated between the turbulent kinetic energy measured on the tower and the amplitude of the rms pressure field at the ground for disturbances having up to 5 min periods. For longer periods, such a relationship appears to be absent, indicating that the longer the scales, the deeper the atmospheric zone important to the dynamics of the pressure fluctuations.

## Abstract

The characteristics of internal gravity waves generated by tropospheric jet streams are analyzed and discussed. By solving numerically the equations of motion in the linear, inviscid and Boussinesq limit, it is shown that a modal structure exists. Some of these modes have the ability to propagate vertically away from the jet and are likely to he responsible for some of the observed wave activities in the ionosphere as well as at the ground. For selected values of the minimum Richardson number of the flow, growth rates and horizontal phase velocities are given as functions of the horizontal wavenumber, for jet streams of varying width. Finally, a brief study of the stability of the so-called low-level jet, whose spectrum of generated waves undoubtedly will contribute to the dynamics of the nocturnal boundary layer, is undertaken.

## Abstract

The characteristics of internal gravity waves generated by tropospheric jet streams are analyzed and discussed. By solving numerically the equations of motion in the linear, inviscid and Boussinesq limit, it is shown that a modal structure exists. Some of these modes have the ability to propagate vertically away from the jet and are likely to he responsible for some of the observed wave activities in the ionosphere as well as at the ground. For selected values of the minimum Richardson number of the flow, growth rates and horizontal phase velocities are given as functions of the horizontal wavenumber, for jet streams of varying width. Finally, a brief study of the stability of the so-called low-level jet, whose spectrum of generated waves undoubtedly will contribute to the dynamics of the nocturnal boundary layer, is undertaken.

## Abstract

This paper summarizes the results of a detailed study from the Cooperative Convective Precipitation Experiment (CCOPE) of the vertical structure of mesoscale gravity waves that disturbed a sizable part of the troposphere and that played a significant role in the generation of a mesoscale convective complex. These bimodal waves displayed periods of 148 (50) min, wavelengths of 135 (60) km, and phase speeds of 15.2 (19.8) m s^{−1}. A comparison is made between wave-induced pressure perturbation fields derived from triple-Doppler wind fields within regions of essentially nonconvective precipitation, pressure perturbation fields obtained by bandpass filtering of surface mesonetwork data, and the vertical structure of the pressure eigenfunctions as predicted from a linear stability analysis. It is believed that this represents the first such application of the Doppler radar pressure retrieval technique to the study of gravity waves. In addition, an analysis of the potential for shear instability was performed on all of the special CCOPE soundings taken on this day to determine the representativeness of the chosen soundings for the theoretical analysis and the likelihood that a wave maintenance mechanism endured throughout the 33-h wave event.

The analysis of the potential for shear instability and the eigenfunctions both indicate that the bimodal waves were able to efficiently extract energy from the mean flow near several closely spaced critical levels in the 4.0– 6.5-km layer to maintain their coherence for many wave cycles. This result serves as the explanation for the observed ability of the waves to organize precipitation into long convective bands whose axes were along and just ahead of the wave crests. The eigenvalue analysis predicts unstable modes that are hydrostatic, nondispersive, ducted gravity waves characterized by half of a vertical wavelength contained between the ground and the lowest critical level (at *z* = 4 km). Eigenfunctions of pressure and other variables all display negligible tilt below 2.3–3.3 km, above which a sudden reversal in phase occurs.

The vertical structure of the Doppler-derived fields associated with one of these gravity waves agrees in terms of the following respects with the eigenfunction predictions and/or the surface mesoanalyses: (a) the vertical wavelength, horizontal structure, and amplitude of the perturbation horizontal wind and pressure fields, and (b) the in-phase covariance between the pressure and horizontal wind fields at levels below 2.5 km. On the other hand, the theory predicted a much more abrupt vertical transition in phase in the pressure fields and weaker amplitudes aloft than were evident in the Doppler analyses. In addition, the size of the multiple-Doppler analysis domain was too small to capture an entire horizontal wavelength of the 135-km-scale gravity wave, which made direct comparisons difficult. Furthermore, the linear theory predicts much smaller amplitudes and somewhat longer horizontal wavelengths for the vertical motions characterizing both wave modes than those seen in the Doppler winds, which likely also contain nonwave effects. These discrepancies are largely due to the combined effects of weak convection, turbulence, and data sampling problems. Despite these drawbacks, the findings from this and other recent studies using Doppler radars and ground-based radiometers suggest that remote sensing of mesoscale gravity waves that occupy a significant fraction of the troposphere should be exploited further.

## Abstract

This paper summarizes the results of a detailed study from the Cooperative Convective Precipitation Experiment (CCOPE) of the vertical structure of mesoscale gravity waves that disturbed a sizable part of the troposphere and that played a significant role in the generation of a mesoscale convective complex. These bimodal waves displayed periods of 148 (50) min, wavelengths of 135 (60) km, and phase speeds of 15.2 (19.8) m s^{−1}. A comparison is made between wave-induced pressure perturbation fields derived from triple-Doppler wind fields within regions of essentially nonconvective precipitation, pressure perturbation fields obtained by bandpass filtering of surface mesonetwork data, and the vertical structure of the pressure eigenfunctions as predicted from a linear stability analysis. It is believed that this represents the first such application of the Doppler radar pressure retrieval technique to the study of gravity waves. In addition, an analysis of the potential for shear instability was performed on all of the special CCOPE soundings taken on this day to determine the representativeness of the chosen soundings for the theoretical analysis and the likelihood that a wave maintenance mechanism endured throughout the 33-h wave event.

The analysis of the potential for shear instability and the eigenfunctions both indicate that the bimodal waves were able to efficiently extract energy from the mean flow near several closely spaced critical levels in the 4.0– 6.5-km layer to maintain their coherence for many wave cycles. This result serves as the explanation for the observed ability of the waves to organize precipitation into long convective bands whose axes were along and just ahead of the wave crests. The eigenvalue analysis predicts unstable modes that are hydrostatic, nondispersive, ducted gravity waves characterized by half of a vertical wavelength contained between the ground and the lowest critical level (at *z* = 4 km). Eigenfunctions of pressure and other variables all display negligible tilt below 2.3–3.3 km, above which a sudden reversal in phase occurs.

The vertical structure of the Doppler-derived fields associated with one of these gravity waves agrees in terms of the following respects with the eigenfunction predictions and/or the surface mesoanalyses: (a) the vertical wavelength, horizontal structure, and amplitude of the perturbation horizontal wind and pressure fields, and (b) the in-phase covariance between the pressure and horizontal wind fields at levels below 2.5 km. On the other hand, the theory predicted a much more abrupt vertical transition in phase in the pressure fields and weaker amplitudes aloft than were evident in the Doppler analyses. In addition, the size of the multiple-Doppler analysis domain was too small to capture an entire horizontal wavelength of the 135-km-scale gravity wave, which made direct comparisons difficult. Furthermore, the linear theory predicts much smaller amplitudes and somewhat longer horizontal wavelengths for the vertical motions characterizing both wave modes than those seen in the Doppler winds, which likely also contain nonwave effects. These discrepancies are largely due to the combined effects of weak convection, turbulence, and data sampling problems. Despite these drawbacks, the findings from this and other recent studies using Doppler radars and ground-based radiometers suggest that remote sensing of mesoscale gravity waves that occupy a significant fraction of the troposphere should be exploited further.

## Abstract

In order to gain insight into the complex dynamics of a convective system interacting with a gravity wave train, we have carried out an experiment in northeast Colorado during July and August, 1983, utilizing data from several program areas in NOAA. Pressure data from the PROFS mesonetwork of microbarograph stations were combined with velocity profiles from the Wave Propagation Laboratory UHF wind profiler (ST) radar at Stapleton Airport in Denver and convective cell location data from the NWS Limon weather radar. Several events were clearly visible in the microbarograph data, from which four (called Events A, B, C and D) in late July were selected for further study. These events differed from each other in fundamental ways.

In each event the waves represent oscillations of a substantial depth of the troposphere and seem to appear and disappear together with the convective cells. In Events A and B the waves have a critical level and are probably unstable modes generated by wind shear in the jet stream, from which they extract energy. We suggest that the convective cells cause the selection of some modes over others in a system that is initially dynamically unstable. In Event A the wave appears to be locked together with the convective cells, which move at the same velocity as the phase velocity of the wave. The wave and the cells seem to grow and evolve synergetically. In Event B the wave and convective cells commence at about the same time, but the cell velocities are quite different from the wave phase velocity. The cell velocities vary substantially over the time of the event and appear to be controlled by the local winds.

In the Events C and D, the waves move faster than the maximum wind in the jet and at least twice as fast as the convective cells. It is suggested that these are nonsingular neutral modes whose excitation depends on a number of mechanisms, such as vertical convective motions and acceleration in the jet flow.

## Abstract

In order to gain insight into the complex dynamics of a convective system interacting with a gravity wave train, we have carried out an experiment in northeast Colorado during July and August, 1983, utilizing data from several program areas in NOAA. Pressure data from the PROFS mesonetwork of microbarograph stations were combined with velocity profiles from the Wave Propagation Laboratory UHF wind profiler (ST) radar at Stapleton Airport in Denver and convective cell location data from the NWS Limon weather radar. Several events were clearly visible in the microbarograph data, from which four (called Events A, B, C and D) in late July were selected for further study. These events differed from each other in fundamental ways.

In each event the waves represent oscillations of a substantial depth of the troposphere and seem to appear and disappear together with the convective cells. In Events A and B the waves have a critical level and are probably unstable modes generated by wind shear in the jet stream, from which they extract energy. We suggest that the convective cells cause the selection of some modes over others in a system that is initially dynamically unstable. In Event A the wave appears to be locked together with the convective cells, which move at the same velocity as the phase velocity of the wave. The wave and the cells seem to grow and evolve synergetically. In Event B the wave and convective cells commence at about the same time, but the cell velocities are quite different from the wave phase velocity. The cell velocities vary substantially over the time of the event and appear to be controlled by the local winds.

In the Events C and D, the waves move faster than the maximum wind in the jet and at least twice as fast as the convective cells. It is suggested that these are nonsingular neutral modes whose excitation depends on a number of mechanisms, such as vertical convective motions and acceleration in the jet flow.