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## Abstract

Steepening, harmonics generation, and saturation of a finite amplitude gravity wave are calculated in an approximate fashion by means of a new and elementary approach. The mechanism for these phenomena is nonlinear advection. For simplicity, attention is confined to a single wave or wave packet in a quasi-stationary background. A principal result of the calculation is that harmonics are generated which cause the wave velocity fluctuation to steepen; harmonics of the density fluctuation are neglected. The wave velocity steepens more and more with increasing amplitude until, at a particular amplitude, it breaks and saturates. Energy flows from the primary wave into harmonies down to turbulence. This steepening and saturation resembles the shoaling of an ocean wave. In both cases, steepening of the velocity fluctuation is caused by the advective, v′ · ∇v′ term in the Navier-Stokes equation. Of special interest is that the lapse rate becomes marginally stable at about the same time that the wave velocity begins to break. This conforms to observations that saturation of velocity growth is accompanied by a near adiabatic lapse rate and turbulence. However, here, the adiabatic lapse rate is not utilized to saturate the wave; i.e. unstable lapse rate and saturation are more nearly concurrent than sequential. Turbulence is necessarily produced—by steepened velocity gradient (shear) as well as by unstable lapse rate—but eddy diffusion is not invoked for saturation. The goal is to show that wave-wave interactions (energy cascade) have the possibility of causing saturation at the wave amplitudes observed. Generally speaking, saturation by turbulence is viewed as competitive with wave-wave interactions, with their relative importance depending on wave frequency. In all cases, the superadiabatic lapse rate is a “signature” of saturation. These conclusions pertain to the idealized case considered and can be altered by the presence of mean shears or more than one primary wave. The most serious approximation is neglect of density fluctuation harmonics.

## Abstract

Steepening, harmonics generation, and saturation of a finite amplitude gravity wave are calculated in an approximate fashion by means of a new and elementary approach. The mechanism for these phenomena is nonlinear advection. For simplicity, attention is confined to a single wave or wave packet in a quasi-stationary background. A principal result of the calculation is that harmonics are generated which cause the wave velocity fluctuation to steepen; harmonics of the density fluctuation are neglected. The wave velocity steepens more and more with increasing amplitude until, at a particular amplitude, it breaks and saturates. Energy flows from the primary wave into harmonies down to turbulence. This steepening and saturation resembles the shoaling of an ocean wave. In both cases, steepening of the velocity fluctuation is caused by the advective, v′ · ∇v′ term in the Navier-Stokes equation. Of special interest is that the lapse rate becomes marginally stable at about the same time that the wave velocity begins to break. This conforms to observations that saturation of velocity growth is accompanied by a near adiabatic lapse rate and turbulence. However, here, the adiabatic lapse rate is not utilized to saturate the wave; i.e. unstable lapse rate and saturation are more nearly concurrent than sequential. Turbulence is necessarily produced—by steepened velocity gradient (shear) as well as by unstable lapse rate—but eddy diffusion is not invoked for saturation. The goal is to show that wave-wave interactions (energy cascade) have the possibility of causing saturation at the wave amplitudes observed. Generally speaking, saturation by turbulence is viewed as competitive with wave-wave interactions, with their relative importance depending on wave frequency. In all cases, the superadiabatic lapse rate is a “signature” of saturation. These conclusions pertain to the idealized case considered and can be altered by the presence of mean shears or more than one primary wave. The most serious approximation is neglect of density fluctuation harmonics.

## Abstract

A gap and adjacent bump is predicted to occur in the velocity spectrum of stratified shear flows when the flux Richardson number *R _{f}*, ≲1/2. This prediction is based on a straightforward solution of the spectral energy balance equation with the assumption that the turbulence is locally inertial and nearly stationary. The limitations of the locally inertial assumption are examined critically. The gap occurs at Spectral wavelengths varying from 30 m to more than 1 km, depending on

*R*, the turbulence intensity, and the Brunt-Väisälä frequency. In general, the gap occurs near a wavenumber where the spectral transfer function ε(

_{f}*k*) vanishes. Both the gap and the zero of ε(

*k*) are caused by shear and/or buoyancy production of energy. In addition, an alternative derivation of ε(

*k*) is made from a more general consideration that avoids the locally inertial assumption. Both expressions predict that ε(

*k*) will vanish at a value of

*k*when

*R*≲1/2. A comparison is made with the gaps and adjacent peaks frequently observed in the wind spectra of clear air turbulence (CAT).

_{f}## Abstract

A gap and adjacent bump is predicted to occur in the velocity spectrum of stratified shear flows when the flux Richardson number *R _{f}*, ≲1/2. This prediction is based on a straightforward solution of the spectral energy balance equation with the assumption that the turbulence is locally inertial and nearly stationary. The limitations of the locally inertial assumption are examined critically. The gap occurs at Spectral wavelengths varying from 30 m to more than 1 km, depending on

*R*, the turbulence intensity, and the Brunt-Väisälä frequency. In general, the gap occurs near a wavenumber where the spectral transfer function ε(

_{f}*k*) vanishes. Both the gap and the zero of ε(

*k*) are caused by shear and/or buoyancy production of energy. In addition, an alternative derivation of ε(

*k*) is made from a more general consideration that avoids the locally inertial assumption. Both expressions predict that ε(

*k*) will vanish at a value of

*k*when

*R*≲1/2. A comparison is made with the gaps and adjacent peaks frequently observed in the wind spectra of clear air turbulence (CAT).

_{f}## Abstract

Attention is called to a recent calculation of the superadiabatic excess—the ratio of wave amplitude at saturation to wave amplitude at convective instability threshold—caused by a saturated gravity wave. (This excess is also referred to as the degree of supersaturation). The implications of this excess for linear saturation by convective instability are pointed out.

Errors and misprints in the recent calculation are also pointed out and corrected.

## Abstract

Attention is called to a recent calculation of the superadiabatic excess—the ratio of wave amplitude at saturation to wave amplitude at convective instability threshold—caused by a saturated gravity wave. (This excess is also referred to as the degree of supersaturation). The implications of this excess for linear saturation by convective instability are pointed out.

Errors and misprints in the recent calculation are also pointed out and corrected.

## Abstract

The vertical turbulent diffusion coefficient in a stably stratified fluid is derived analytically. This derivation does not require that the flux Richardson number be known or specified. The resulting expression for the diffusion coefficient is compared with the previous stratospheric results of Lilly *et al.* (1974) and its applicability to atmospheric diffusion and clear air turbulence is discussed.

## Abstract

The vertical turbulent diffusion coefficient in a stably stratified fluid is derived analytically. This derivation does not require that the flux Richardson number be known or specified. The resulting expression for the diffusion coefficient is compared with the previous stratospheric results of Lilly *et al.* (1974) and its applicability to atmospheric diffusion and clear air turbulence is discussed.

## Abstract

For stable stratification, it is pointed out that there exists a strong correlation between the intensity of atmospheric turbulence and the energy dissipation rate ε. It is given in terms of the variance of vertical velocity σ_{w}
^{2} and the Brunt-Väisälä frequency ω* _{B}* by ε = 0.4 σ

_{w}

^{2}ω

*. This relation is argued to have a wider range of applicability in the stratosphere than the previous relation ε = β σ*

_{B}_{w}

^{2}, where β is taken to be constant.

## Abstract

For stable stratification, it is pointed out that there exists a strong correlation between the intensity of atmospheric turbulence and the energy dissipation rate ε. It is given in terms of the variance of vertical velocity σ_{w}
^{2} and the Brunt-Väisälä frequency ω* _{B}* by ε = 0.4 σ

_{w}

^{2}ω

*. This relation is argued to have a wider range of applicability in the stratosphere than the previous relation ε = β σ*

_{B}_{w}

^{2}, where β is taken to be constant.

## Abstract

A nonlinear theory of internal gravity waves is extended to slowly varying mean winds *u*
_{0}. This theory determines the wave amplitudes, enhanced diffusion and momentum deposition, for a broad spectrum of waves as well as for a single wave. It is shown that the momentum deposition is not constant with altitude, even above the altitude where a wave saturates (breaks). The deposition is seen to decrease as a critical level is approached. The rate of decrease varies with spectral width. In general, the behavior of saturated waves near a critical level is shown to differ greatly from that of linearly growing waves.

It is then proven that the momentum deposition is always related to the diffusion coefficient in a very simple and general way. Hence, knowledge of one implies knowledge of the other. Both quantities vary with altitude in nearly the same way. A feature of the theoretical momentum deposition is that it sometimes behaves like Rayleigh friction and sometimes not. It is referred to as “generalized Rayleigh friction,” and the different cases are delineated. It is pointed out that the terms “saturation altitude” or “wavebreak altitude” are ambiguous when applied to a broad spectrum because different wavelengths saturate at different altitudes. Hence, there is a range of saturation altitudes. If the spectrum is broad enough, saturation will occur over a wide range of altitudes, thereby causing diffusion and deposition throughout much of the middle atmosphere.

By use or rocket data, the theoretical momentum deposition is estimated to be 40 m a day^{−1} near the stratopause (about 45 km) and more than 100 m s^{−1} day^{−1} in regions of the mesosphere. It is very small below 40 km altitude because the waves are linear there. The (vertical) wave diffusion coefficient is about 10 m^{2} s^{−1} just below the stratopause, more than 200 m^{2} s^{−1} in regions of the mesosphere, and very small below 40 km. In addition, the horizontal diffusion is shown to be much larger than the vertical, as was pointed out in a previous paper. It is also found that the diffusion and momentum deposition vary greatly with the spectral width, being much smaller for a very broad spectrum than for a narrow spectrum. The spectral width has a crucial affect on the height profiles of diffusion and deposition. The theory is compared to previous theories and to some observations.

## Abstract

A nonlinear theory of internal gravity waves is extended to slowly varying mean winds *u*
_{0}. This theory determines the wave amplitudes, enhanced diffusion and momentum deposition, for a broad spectrum of waves as well as for a single wave. It is shown that the momentum deposition is not constant with altitude, even above the altitude where a wave saturates (breaks). The deposition is seen to decrease as a critical level is approached. The rate of decrease varies with spectral width. In general, the behavior of saturated waves near a critical level is shown to differ greatly from that of linearly growing waves.

It is then proven that the momentum deposition is always related to the diffusion coefficient in a very simple and general way. Hence, knowledge of one implies knowledge of the other. Both quantities vary with altitude in nearly the same way. A feature of the theoretical momentum deposition is that it sometimes behaves like Rayleigh friction and sometimes not. It is referred to as “generalized Rayleigh friction,” and the different cases are delineated. It is pointed out that the terms “saturation altitude” or “wavebreak altitude” are ambiguous when applied to a broad spectrum because different wavelengths saturate at different altitudes. Hence, there is a range of saturation altitudes. If the spectrum is broad enough, saturation will occur over a wide range of altitudes, thereby causing diffusion and deposition throughout much of the middle atmosphere.

By use or rocket data, the theoretical momentum deposition is estimated to be 40 m a day^{−1} near the stratopause (about 45 km) and more than 100 m s^{−1} day^{−1} in regions of the mesosphere. It is very small below 40 km altitude because the waves are linear there. The (vertical) wave diffusion coefficient is about 10 m^{2} s^{−1} just below the stratopause, more than 200 m^{2} s^{−1} in regions of the mesosphere, and very small below 40 km. In addition, the horizontal diffusion is shown to be much larger than the vertical, as was pointed out in a previous paper. It is also found that the diffusion and momentum deposition vary greatly with the spectral width, being much smaller for a very broad spectrum than for a narrow spectrum. The spectral width has a crucial affect on the height profiles of diffusion and deposition. The theory is compared to previous theories and to some observations.

## Abstract

For a highly idealized condition, the spectrum of saturated and unsaturated gravity waves at each height is calculated directly from the wave equation. A principal feature of this wave equation is the inclusion of wave dissipation, although in an approximate form. In the absence of wave absorption, reflection, radiation, wind shears, resonant wave–wave interactions and other sources and sinks, this dissipation at each height is determined solely by the “turbulent” or chaotic state caused by off-resonant wave–wave interactions and instability of the (broad) wave spectrum at that height. The dissipation is approximately accounted for by a diffusion term. The appropriate diffusivity is self-consistent with the continuum of spectral waves that cause the chaotic state and is argued to be scale dependent.

An inverse calculation is also made of what the observed spectra imply for wave dissipation—again assuming that many wave dissipations can be approximately described by a scale-dependent diffusion process. The relationship of middle atmospheric spectra to (averaged) gravity wave sources in the troposphere is investigated.

## Abstract

For a highly idealized condition, the spectrum of saturated and unsaturated gravity waves at each height is calculated directly from the wave equation. A principal feature of this wave equation is the inclusion of wave dissipation, although in an approximate form. In the absence of wave absorption, reflection, radiation, wind shears, resonant wave–wave interactions and other sources and sinks, this dissipation at each height is determined solely by the “turbulent” or chaotic state caused by off-resonant wave–wave interactions and instability of the (broad) wave spectrum at that height. The dissipation is approximately accounted for by a diffusion term. The appropriate diffusivity is self-consistent with the continuum of spectral waves that cause the chaotic state and is argued to be scale dependent.

An inverse calculation is also made of what the observed spectra imply for wave dissipation—again assuming that many wave dissipations can be approximately described by a scale-dependent diffusion process. The relationship of middle atmospheric spectra to (averaged) gravity wave sources in the troposphere is investigated.

## Abstract

A theoretical investigation is made of turbulence in the buoyancy subrange of stably stratified sheer flows. This theory is based on a new calculation of the buoyancy flux spectrum *B* (*k*). In the Lumley-Shur theory, it was assumed that *B* (*k*) had a universal form which could be determined by a simple dimensional consideration. That assumption is shown to be incorrect. One new result of the present calculation is that *B* (*k*) has a fairly sharp transition at a wavenumber *k _{B}* = 0.8

^{½}ω

_{B}/

*v*, where ω

_{m}_{B}is the Brunt-Väisälä frequency and

*v*the root-mean-square velocity in the equilibrium range. Physically, this transition is “interpreted” as an emission of incoherent gravity waves fed by the kinetic energy of vertically fluctuating air particles. When

_{m}*k*<

*k*, the emitted gravity waves are undamped and absorb a good part of an air particle's energy. When, however,

_{B}*k*>

*k*, the gravity waves are strongly damped and consequently contain very little energy. The transition of the energy spectrum

_{B}*E*(

*k*) at

*k*is found usually to be gradual. It is also found that the power law governing

_{B}*E*(

*k*) in the buoyancy subrange is not universal. It depends on the flux Richardson number R

_{f}and upon the rms velocity, and is quite variable. Such variability seems to conform with several observations of

*E*(

*k*) made in the stratosphere and upper troposphere. Present theory is compared with a previous theory and with observations in the atmosphere.

## Abstract

A theoretical investigation is made of turbulence in the buoyancy subrange of stably stratified sheer flows. This theory is based on a new calculation of the buoyancy flux spectrum *B* (*k*). In the Lumley-Shur theory, it was assumed that *B* (*k*) had a universal form which could be determined by a simple dimensional consideration. That assumption is shown to be incorrect. One new result of the present calculation is that *B* (*k*) has a fairly sharp transition at a wavenumber *k _{B}* = 0.8

^{½}ω

_{B}/

*v*, where ω

_{m}_{B}is the Brunt-Väisälä frequency and

*v*the root-mean-square velocity in the equilibrium range. Physically, this transition is “interpreted” as an emission of incoherent gravity waves fed by the kinetic energy of vertically fluctuating air particles. When

_{m}*k*<

*k*, the emitted gravity waves are undamped and absorb a good part of an air particle's energy. When, however,

_{B}*k*>

*k*, the gravity waves are strongly damped and consequently contain very little energy. The transition of the energy spectrum

_{B}*E*(

*k*) at

*k*is found usually to be gradual. It is also found that the power law governing

_{B}*E*(

*k*) in the buoyancy subrange is not universal. It depends on the flux Richardson number R

_{f}and upon the rms velocity, and is quite variable. Such variability seems to conform with several observations of

*E*(

*k*) made in the stratosphere and upper troposphere. Present theory is compared with a previous theory and with observations in the atmosphere.

## Abstract

It is shown that the Lumley-Phillips buoyancy subrange theory predicts the temperature spectrum to be proportional to *k*
^{−3} at small scalar wavenumbers *k*—-consistent with oceanic experiments (although those experiments measure vertical wavenumber *k*
_{3} rather than *k*). A previous derivation by Phillips does not apply to such small *k*. The predicted temperature spectrum is also shown proportional to the velocity spectrum in the buoyancy subrange. Although its predictions are consistent with experiments, the *a priori* validity of buoyancy subrange theory still remains in question.

## Abstract

It is shown that the Lumley-Phillips buoyancy subrange theory predicts the temperature spectrum to be proportional to *k*
^{−3} at small scalar wavenumbers *k*—-consistent with oceanic experiments (although those experiments measure vertical wavenumber *k*
_{3} rather than *k*). A previous derivation by Phillips does not apply to such small *k*. The predicted temperature spectrum is also shown proportional to the velocity spectrum in the buoyancy subrange. Although its predictions are consistent with experiments, the *a priori* validity of buoyancy subrange theory still remains in question.

## Abstract

Vertically propagating, compressible, internal gravity waves are shown to have a vertical Stokes drift which is proportional to the vertical wave energy flux. In regions of the atmosphere dominated by upward propagating waves, such as the summer mesosphere, this Stokes drift will be upward. For the Lagrangian mean parcel motion to be small, a downward mean Eulerian velocity must exist to largely oppose the upward Stokes drift. These. results may explain the downward mean Eulerian velocity observed at Poker Flat, Alaska in the summer mesosphere.

## Abstract

Vertically propagating, compressible, internal gravity waves are shown to have a vertical Stokes drift which is proportional to the vertical wave energy flux. In regions of the atmosphere dominated by upward propagating waves, such as the summer mesosphere, this Stokes drift will be upward. For the Lagrangian mean parcel motion to be small, a downward mean Eulerian velocity must exist to largely oppose the upward Stokes drift. These. results may explain the downward mean Eulerian velocity observed at Poker Flat, Alaska in the summer mesosphere.