Gravity Wave Saturation Research Articles

Comments on “Influence of Wave Propagation on the Doppler Spreading of Atmospheric Gravity Waves”

A recent paper by Eckermann claims to have produced, via eikonal analysis, conclusions that undermine the Doppler-spread theory (DST) of middle-atmosphere gravity wave saturation and consequently undermine a certain parameterization based upon it. Here it is pointed out that his analysis in fact supports the underlying thesis of the DST, supports its quantitative estimates once a suitable and realistic criterion for the onset of instability is adopted, suffers from inadequacies of analysis even on its own terms, and suffers from a basic failure of the eikonal method in application to the broad wave spectrum of the middle atmosphere, such that it is rendered inadequate as an adverse test of alternative approximate methods like those employed by the DST to date.

Reply to Hines' comments on “Testing theories of atmospheric gravity wave saturation and dissipation”

Abstract Hines' (1998, henceforth H98) comments on the paper “Testing theories of atmospheric gravity wave saturation and dissipation” (Gardner, 1996, henceforth G96) have certainly sharpened the discussion on this important issue. He has provided additional insights about several mechanisms which are believed by some to dominate wave saturation and dissipation in the middle atmosphere. However, I disagree with several of his assertions which I shall now address in the order in which they appear in H98.

Comments on the paper by C. S. Gardner, “Testing theories of atmospheric gravity wave saturation and dissipation”

Gardner (1996, henceforth G96), examines a number of theories that seek to account for the vertical-wavenumber (m) spectrum of horizontal wind perturbations (u′) induced by gravity waves in the middle atmosphere. He notes that all the theories predict much the same spectra, and so he seeks to “identify the experimental data required to test the fundamental physics upon which these theories are based …” While I applaud Gardner's goal and will at a later stage contribute to it, I must first comment adversely on some aspects of his preliminary summaries and of his observational inferences. My remarks are limited almost exclusively to his discussion of the linear instability theory of Dewan and Good (1986, henceforth DG86) and Smith et al. (1987, henceforth SFV87), called by him the LIT, the diffusive filtering theory (DFT) of Gardner (1994, henceforth G94), and the Doppler spread theory (DST) initiated by Hines (1991b). In all of the theories under discussion, the u′ power spectral density (PSD) is taken in G96 to exhibit an increase with m at m values less than some characteristic value m∗ and a decrease, more-or-less proportional to m−3, in a ‘tail’ region at greater m values up to some upper limit mM, whereafter turbulence is said to set in. The transition at m∗ may be thought of, for convenience of discussion, as being abrupt, though each theory in fact either assumes or derives a smooth transition. The various theories differ from one another in the physical processes that are assumed to establish the tail portion of the spectrum, the tail's range in m, and the intensity of the PSD of the tail. That intensity has been said observationally to be invariant under changes of circumstance and height, a characteristic that is widely attributed to a ‘saturation’ of the spectrum or of the corresponding waves. The tail portion of the spectrum is therefore said by many to be saturated, though the theory of G94 purports to disagree with this concept and in fact provides possible alternatives to the m−3 form (as, indeed, does DG86). The LIT, DFT and DST will be discussed below in sequence.

Testing theories of atmospheric gravity wave saturation and dissipation

Numerous mechanisms have been proposed to account for the dissipation and saturation of gravity waves in the atmosphere. We review the leading wave dissipation paradigms and identify the experimental data required to test definitively the fundamental physics upon which these theories are based. We also examine the separability of the joint vertical wave number ( m) and temporal frequency (ω) spectrum and the separability of the unambiguous two-dimensional horizontal wave number spectrum. Definitive tests of the Linear Instability (Dewan and Good, 1986), Saturated-Cascade (Dewan, 1994), and Diffusive Filtering Theories (Gardner, 1994) and separability of the ( m, ω) spectrum are within the observational capabilities of modern remote sensing instruments. The Diffusive Damping (Weinstock, 1990) and Doppler Spreading Theories (Hines, 1991) are untestable in their present forms. Separability of the two-dimensional horizontal wave number spectrum is difficult to test using current technology, although analysis of airglow images may provide some insight.

Gravity wave activity in the lower atmosphere: Seasonal and latitudinal variations

A climatology of gravity wave activity in the lower atmosphere based on high‐resolution radiosonde measurements provided by the Australian Bureau of Meteorology is presented. These data are ideal for investigating gravity wave activity and its variation with position and time. Observations from 18 meteorological stations within Australia and Antarctica, covering a latitude range of 12°S – 68°S and a longitude range of 78°E–159°E, are discussed. Vertical wavenumber power spectra of normalized temperature fluctuations are calculated within both the troposphere and the lower stratosphere and are compared with the predictions of current gravity wave saturation theories. Estimates of important model parameters such as the total gravity wave energy per unit mass are also presented. The vertical wavenumber power spectra are found to remain approximately invariant with time and geographic location with only one significant exception. Spectral amplitudes observed within the lower stratosphere are found to be consistent with theoretical expectations but the amplitudes observed within the troposphere are consistently larger than expected, often by as much as a factor of about 3. Seasonal variations of stratospheric wave energy per unit mass are identified with maxima occurring during the low‐latitude wet season and during the midlatitude winter. These variations do not exceed a factor of about 2. Similar variations are not found in the troposphere where temperature fluctuations are likely to be contaminated by convection and inversions. The largest values of wave energy density are typically found near the tropopause.

A New Theory of the Saturated Gravity Wave Spectrum for the Middle Atmosphere

Abstract A new theory of the saturated vertical wavenumber spectrum of gravity waves for the middle atmosphere is developed. The theory is based on a hypothesis that the saturated spectrum is a result of balance among various spectral sources and sinks of gravity waves in the middle atmosphere. The spectral source included in this paper is the amplification of upward propagating gravity waves due to the density effect. Two spectral sinks are the off-resonant wave-wave interactions and scale-dependent radiative damping. It is found that the algebraic approximation of the saturated vertical wavenumber spectrum of horizontal winds, F(μ), can be written as F(μ) ∼ μ−3(θ + 2.5/1 + μ2)−1,where μ is the dimensionless vertical wavenumber, and θ is a parameter measuring the ratio of the dynamic effect to the radiative effect on the dissipation. The theory predicts the widely cited −3 power law for both small and large vertical wavenumbers. However, when θ < 0.1 a significant range of wavenumbers exists for which F(...

Vertical structure of atmospheric gravity waves revealed by the wavelet analysis

It is well known that the vertical wavenumber (m) spectra of temperature and horizontal wind fluctuations have a steep slope (almost proportional to m−3) in the middle atmosphere. These spectra are considered to be due to saturated gravity waves. However, since power spectral analysis assumes sinusoidal waves with constant amplitudes in the height region for which the spectra are calculated, the information on the height position, where the disturbance having each wavenumber is dominant, is lost. In this paper the dominant height position is examined using a recently developed wavelet method. An analysis is made of temperature observation data by radiosondes and wind data by a mesosphere, stratosphere, and troposphere (MST) radar in the lower stratosphere. It is shown that the height where the variance is largest increases with the wavenumber. This variation with wavenumber is explained well by a monochromatic inertia‐gravity wave whose vertical wavenumber changes due to the vertical shear of the large‐scale background wind during its vertical propagation. Thus we consider theoretically the vertical wavenumber spectrum of a single saturated gravity wave which propagates upward in a large‐scale background wind with a vertical shear. The slope of the theoretical spectrum is steep (proportional to m−4 or m−3 depending on m for a linear shear) and similar to the observations. The result suggests that the vertical shear of large‐scale wind structure plays an important role in determining the shape of universal m spectra in the middle atmosphere.

Saturated gravity wave spectra measured with balloons in Mendoza (Argentina)

Spectral results from a sounding of temperature and wind velocity between 11.9 and 23.1 km performed with an open type stratospheric balloon near the Andes Mountains, are reported. The use of this kind of balloons and high performance instruments, allowed to attain better spatial resolution (around 2.5 meters during the descent) than previous results and other sounding techniques. This high resolution permitted to study the complete interval of altitude data in three consecutive subintervals (below, around and above the tropopause). Each of these subintervals shows different saturation characteristics. Some differences have been found with the general pattern of power spectral densities reported by different theories. A large amplitude wave of around 1.8 km and possibly originated in the mountain relief, is clearly observed in the temperature, vertical and shear wind velocity components.

The Saturation of Gravity Waves in the Middle Atmosphere. Part IV: Cutoff of the Incident Wave Spectrum

Abstract A Doppler-spread theory for the saturation of middle atmosphere gravity waves was presented in an earlier member of this sequence of papers. It employed a model in which a broad spectrum of waves subject to linear theory is incident from below. The spectral distribution (in vertical wavenumber m) is deformed, as it propagates upward, in response to the growing importance of the Eulerian advective nonlinearity imposed on each wave by the total wave-induced wind. The deformation is such as to statistically spread the spectrum towards larger m, with the largest-m waves being progressively obliterated in quasi-critical-layer interactions. The model invoked a cutoff of the incident spectrum at a vertical wavenumber specified as lying in the range 0.5–1.0 times the local buoyancy frequency divided by the rms wind speed, with the choice 0.5 being adopted tentatively. A qualitative argument for the chosen cutoff wavenumber was presented but was not supported by any more certain quantitative analysis at t...

Gravity waves from O 2 nightglow during the AIDA '89 campaign III: effects of gravity wave saturation

The O 2 atmospheric (0–1) night airglow emitted within the gravity wave saturation region at ∼90–100 km can serve as a means of studying the wave activity. In this analysis, the atmospheric motions were described by a mean spectral model and an algorithm was developed to infer the wave kinetic energy density and momentum flux from variations in O 2 (0–1) airglow emission rate and rotational temperature. The method was applied to eight nights of data collected by MORTI, a mesopause oxygen rotational temperature imager, during the AIDA campaign of 1989 in Puerto Rico (18°N, 67°W). The observed r.m.s. fractional fluctuations of airglow emission rate and rotational temperature were of the order of ∼0.07–0.15 and ∼0.02–0.04, respectively, and the characteristic vertical wavelengths were estimated at ∼10 2 -20 km. The inferred r.m.s. horizontal velocities and velocity variances were found to be ∼12–25 m/s and ∼150–600 m 2/s 2, with the majority of the horizontal velocity and its variance associated with low-frequency, large-scale wave motions. The estimated momentum fluxes, mainly contributed by high-frequency, small-scale waves, were ∼2–10 m 2/s 2. These results are in good agreement with those obtained from other measurements using different observational methods at low and mid-latitudes.

Saturated gravity wave spectra derived from foil chaff experiments performed during campaign ‘DYANA’ at 44°N (Biscarrosse, Southwest France)

Two series of measurements of horizontal wind velocity, one performed in winter 1990 at 44°N latitude, and the other in summer 1988 at 69°N latitude, were subjected to a spectral analysis. The two series covered a height interval of 11–15 km centred at about the same height (92.5km in winter, 91.5km in summer). In both cases the slopes of the spectra were close to −3. A dominant wavelength (if it existed) must have been larger than the height interval covered.

Saturated Gravity Wave Spectrum in the Polar Summer Lower Thermosphere Observed by Foil Chaff during Campaign “Sodium 88”

Abstract Ten vertical profiles of horizontal winds have been measured at high resolution (25 m) in the range from 85 to 103 km during the sodium campaign in summer 1988 at Andenes, northern Norway (69°N). The purposes of this study are to examine the hypothesis that the vertical wavenumber spectrum is controlled by saturation processes and to identify the mechanisms that contribute to saturation of the atmospheric motion spectrum within the observed height range. An analysis of vertical wavenumber spectra and vertical wind shears of the ten horizontal wind measurements is presented. Our results indicate a rather good agreement between the observed and predicted velocity spectra in both amplitude and slope for vertical wavelengths up to about 12.8 km. (This wavelength is given by the height interval covered by the experiment.) Very large wind shears, as high as 40–90 m s−1 km−1, are observed to be present at various heights, and at certain heights the shears even exceed 100 m s−1 km−1. Such strong wind she...

Numerical simulations of the saturated gravity wave spectra in the atmosphere

A two dimensional numerical model is used to compute the saturation of small scale gravity waves in the region near the critical level. The vertical wave number spectrum of horizontal velocity fluctuations in the unstable region (USR) where shear instability develops is found to be governed by wave-shear interaction and follows a theoretical saturation spectrum ~ ω b 2/2 m 3. Wave-shear interaction is also found to be responsible for the observed fact that the variance of vertical velocity fluctuations is significantly lower than the level predicted by linear gravity wave theory. On the other hand, the corresponding spectrum in the stable region (SR) following a much shallower spectrum ~m −2 is found to result from the combined effects of wave-wave interactions and eddy diffusion. The key step in our simulation is the separate parameterization of horizontal and vertical eddy diffusion coefficients instead of a constant molecular viscosity coefficient.

Observational evidence of local turbulence production due to gravity wave saturation in the summer mesosphere

Five foil chaff and two falling sphere rockets flown during the MAC/SINE Campaign on 15 July 1987 at Andenes, Northern Norway (69°17′N). From these rocket measurements, turbulent energy dissipation rates, vertical wind shears and Richardson numbers as functions of height were derived in the range from 82 to 92km. Turbulent energy dissipation rates generally range from 1.4 × 10 −5 to 2.0 × 10 −2W/kg and are consistent with other experiments performed at the same latitude. Strong wind shears of the order of 50–90 m/s/km are observed at various heights. Good correspondence between turbulence intensity peaks, regions of strong wind shear and low Richardson number is found. Vertical wavenumber spectra of the five scalar winds measured by the foil chaff rockets indicate that there is an excellent agreement with the saturation hypothesis, suggesting that the turbulence intensity peaks measured in this salvo are linked directly to the saturation of gravity wave motions via dynamical instabilities.

Rocketsonde Observations of the Middle Atmosphere Dynamics at Uchinoura(31.DEG.N,131.DEG.E) during the DYANA Campaign. Part II: Characteristics of Gravity Waves.

We describe in this paper the height variations of the gravity wave characteristics in the middle atmosphere, using the profiles of wind velocity and temperature collected at 20-90km altitude by means of a series of rocketsonde experiments and the MU (middle and upper atmosphere) radar during the DYANA campaign. The dominant vertical scales of the wind velocity fluctuations generally increased with altitude from 2-3km in the lower stratosphere to 10-15km in the mesosphere. The vertical wavenumber spectra of the wind velocity fluctuations, constructed in five altitude regions, showed height variations in the spectral shape, which was fairly consistent with the model spectrum based upon the saturation of gravity waves. The wind velocity variance due to gravity waves generally increased in the stratosphere, diminished just above the stratopause, greatly increased again in the lower mesosphere, and finally became nearly constant in the upper mesosphere above about 65km. The gravity waves with relatively small vertical scales were significantly weakened just above the stratopause, suggesting the important effect of the vertical structure of the static stability on the propagation characteristics of gravity waves.

Spectral analysis of temperature and Brunt‐Väisälä frequency fluctuations observed by radiosondes

We observed profiles of the temperature, T, and Brunt‐Väisälä frequency squared, N2, from 0 to 30 km in altitude using radiosondes with 150 m height resolution launched from the MU observatory, Japan (34°51′N, 136°06′), from September 27, 1986, to February 24, 1989. We analyzed vertical wavenumber spectra of the normalized temperature T/T and N2 fluctuations in the 2.0–8.5 km (troposphere) and 18.5–25.0 km (lower stratosphere) altitude ranges and compared them with model spectra based on saturated gravity wave theory. In the winter stratosphere the slope of the mean T/T spectra in the wavenumber range from 7.0×10−4 to 2.0×10−3 (cycles per meter) was very close to the −3 predicted by the model, and the spectral amplitudes were 1.3–1.9 times larger than the predicted values, which is within the possible range of variability of the model. On the other hand, in the summer stratosphere the spectral slope ranged from −2.2 to −2.4, which is more gradual than the model, and the spectral amplitudes were only 0.4 to 0.5 of the predictions. The spectral shape in the troposphere did not show a significant difference between summer and winter. The spectral amplitudes, however, exceeded the model values by factors of about 3.2 and 1.9 in winter and summer, respectively. The overall shape of the profile for fluctuations with vertical scales from 150 to 900 m was generally similar to the shape of the background value of N4, consistent with the saturated gravity wave model, but the details of the altitude variations were rather complicated. That is, just below the tropopause usually exceeded the model by a factor of 2 to 4, and it became significantly smaller than the model in the summer stratosphere and in a region above 25 km in winter.

The Saturation of Gravity Waves in the Middle Atmosphere. Part II: Development Of Doppler-Spread Theory

Abstract The irregular winds of the middle atmosphere are commonly attributed to an upwardly propagating system of atmospheric gravity waves. Their one-dimensional (in vertical wavenumber m) power spectrum has been reported to exhibit a nearly universal behavior in its “tail” region of large m: both the form (∼m−3) and the intensity of the tail are approximately invariant with meteorological conditions, time, place and height. This universality is often described as resulting from “saturation” of the system, with the physical cause of saturation being left for separate identification and analysis. Here the cause is attributed to nonlinear interaction between the waves of the full spectrum, most specifically to the advective nonlinearity of the Eulerian fluid-dynamic equations. This nonlinearity has the effect of Doppler shifting the local intrinsic frequency of any given wave in the wind field imposed by all waves. Only an approximation to its effects is sought here, the wind field of the full spectrum be...

The Saturation of Gravity Waves in the Middle Atmosphere. Part III: Formation of the Turbopause and of Turbulent Layers beneath It

A Doppler-spread theory for the “saturation” of middle-atmosphere gravity–wave spectra (in vertical wave-number m) is presented in a companion paper. It includes a formula for the large-m limit that would be imposed by the onset of instability. At sufficiently great heights, however, this limit may not be attained because of dissipation imposed by molecular viscosity and conductivity. Here, the transition between the two regimes is taken to mark the turbopause—the level at which turbulence ceases—and relevant relations are obtained. These are shown to be consistent not only with observations at turbopause levels, but also, after extrapolation downward through five orders of magnitude in atmospheric density (with the use of the Doppler-spread theory), with similar observations in the middle stratosphere. In a second application, the concepts behind the Doppler-spread theory are applied to circumstances that would be found on individual occasions (rather than in statistical ensembles, which the basic theory treats). Horizontally stratified layers of intensified turbulence are then found to be expected, perhaps in the stratosphere and more probably in the mesosphere, as is observed. The layers are often observed by medium-frequency, partial-reflection radar techniques, including in particular spaced-receiver “drift” techniques that are often taken to represent the ambient winds. Such an interpretation is confirmed to be appropriate to the present model.

The Saturation of Gravity Waves in the Middle Atmosphere. Part I: Critique of Linear-Instability Theory

Abstract The irregular winds of the middle atmosphere are commonly attributed to an upwardly propagating system of atmospheric gravity waves. Their one-dimensional (in vertical wavenumber m) power spectrum has been reported to exhibit a nearly universal behavior in its “tail” region of large m: both the form (∼m−3) and the intensity of the tail are approximately invariant with meteorological conditions, time, place and height. This universality is often described as resulting from “saturation” of the system, with the physical cause of saturation being left for separate identification and analysis. Of current theories as to physical cause, the most fully developed and widely employed assumes that saturation results from linear instability: that the waves of the tail grow in amplitude with height (in response to the decrease of atmospheric density) until the system as a whole, or each portion of its tail, is rendered unstable and prevented from growing further. Initially the form and then the intensity of t...

Further study of a saturated gravity wave spectrum in the mesosphere

Mesospheric wind profiles with a height resolution of 25 m derived from five foil chaff experiments performed during the last salvo of the MAC/SINE (Middle Atmosphere Cooperation/Summer in Northern Europe) campaign in the summer of 1987 at Andøya Rocket Range, in Northern Norway (69°17′N, 16°01′E), allow us to study the character of the motion spectrum in the mesosphere. The wind data have shown that (1) the average slope of the five vertical wave number spectra was −3.0 ± 0.2 for wavelengths in the range from 6.4 km to 100 m; (2) a dominant vertical wavelength of more than 6.4 km was found; and (3) there was good agreement between the observed spectrum and the saturated spectrum presented by Dewan and Good (1986) and Smith et al. (1987). In this paper, we further show that (1) very large wind shears, as high as 50–90 m s−1 km−1, are observed to be present at various heights (such large shears yield Richardson numbers smaller than 0.25, suggesting that the saturated gravity wave spectrum is associated with the shear instabilities); and (2) when the Richardson number profile is compared with the altitude profile of the mesosphere‐stratosphere‐troposphere (MST) radar (SOUSY) echo power, the low Richardson numbers are observed in the region of most intense backscatter. It is suggested from our measurements that the enhanced echo power is linked to wave saturation processes.