Fractal analysis of gravity wave spectra in the middle atmosphere

Wind fluctuations in the middle atmosphere behave like colored noise processes. They have a continuum of scales without dominant features and a power spectrum density (PSD) that often decays with frequency ƒ as ƒ^−β. Spectral index β is generally obtained through least-square fit to PSD estimated by Fourier methods. Graphs of colored noise have fractal plane-filling properties depending on β. An efficient method for finding β using the fractal dimension (D), based on analysis of 1/ƒ noise in galactic X-ray luminosities by McHardy I. and Czerny B., (1987, Nature325, 696), is described. An empirical relation is found between D and β and its validity is confirmed in limiting cases. Then D is obtained from power-law dependence of a length metric L(μ) on scale μ. The method is applied to middle-atmospheric velocity data from the Poker Flat radar in Alaska. Variations of D follow those in β, from an earlier analysis by Bemraet al., (1986, Handbook for MAP20, 216), but show an offset of 0.1–0.2 even after corrections for outliers, gaps, and additive noise. Usefulness of this method for screening data as an aid to spectral analysis is examined.     

Observational evidence of a saturated gravity wave spectrum in the mesosphere

Five vertical profiles of scalar horizontal winds have been measured at high resolution (25m) in the range from 80–95 km during the last salvo of the MAC/SINE campaign in the summer 1987 at Andenes, Northern Norway (69.3°N). Our purpose in this study is to examine the consistency of the motion spectrum with the saturated spectrum of gravity waves proposed by Smith S. A., Fritts D. C. and Van Zandt T.E., (1987, J. atmos. Sci. 44, 1404). An analysis of vertical wavenumber spectra of the five horizontal wind profiles is presented and it is found that (a) the average slope of the five vertical wavenumber spectra is −3.0 ± 0.2 for wavelengths in the range from 6.4 km to 100 m. The slope is considerably steeper than the vertical wavenumber spectra of the horizontal velocity discussed in the literature, (b) the average vertical wavenumber spectrum shows that there is excellent agreement between the observed spectrum and the saturated spectrum in both slope and amplitude, suggesting that saturation processes do indeed act to control spectral amplitudes at large wavenumbers, and (c) a dominant vertical wavelength of 6.4 km is found in the mesosphere. Taken together, our observations provide further support for the saturated spectrum theory.     

The nonlinear mechanism of gravity wave generation by meteorological motions in the atmosphere

Using asymptotic expansions of the hydrodynamic equations in the Rossby number and the method of multiple time scales, we derive approximate expressions for the inhomogeneous “forcing” terms which describe the continuous generation of inertio-gravity waves by quasi-geostrophic motions. As a result of numerical modelling applied to the evolution of tropospheric meso- and macro-scale wave sources, the values of these forcing terms are estimated. A three-dimensional numerical simulation of wave propagation from a mesometeorological tropospheric eddy into the upper atmosphere was done to estimate the gravity wave response to the sources described. The results of the calculations show that the most part of the wave energy propagates quasi-horizontally carried by two-dimensional inertio-gravity waves. At the same time, a part of the energy is transported into the upper atmosphere by internal-gravity waves which can create regions of wave disturbance in the upper atmosphere at considerable distances from the source site. The amplitudes of these waves increase with increasing intensity and decreasing time scales of the wave sources and can reach the values observed in the upper atmosphere.     

Climatology of gravity waves in the middle atmosphere

An attempt is made at the statistical analysis of small-scale disturbances in the stratosphere and mesosphere with the aid of meteorological rocket observations at many stations from 77°N to 8°S for several years.By applying a high-pass filter to daily rocket data in the height range 20–65 km, wind and temperature fluctuations with characteristic vertical scales close to or less than 10 km are obtained, which are considered to be due to internal gravity waves. Results are expressed in terms of parameters which tend to emphasize smallscale vertical fluctuations and which should provide qualitative measures of gravity wave activity.It is found that the gravity wave activity shows a notable annual cycle in higher latitudes with the maximum in wintertime, while it shows a semiannual cycle in lower latitudes with the maxima around equinoxes. It is also found from the standard deviation around the monthly mean that the temporal variability of gravity waves is very large.     

Simulation of gravity wave effects under solstice conditions using a 3-D circulation model of the middle atmosphere

Vertically propagating gravity waves can transport momentum and energy from the troposphere up to the mesosphere and thus modify the circulation of the middle atmosphere. The effects of regional gravity wave sources, together with temporal changes of gravity wave activity, are studied under solstice conditions in a 3-D circulation model using a simplified parameterization scheme for the gravity momentum deposition. In this way we can reproduce the reversal of the mean zonal wind with height and very low temperatures at the summer mesopause region. Using a stochastic forcing by taking the gravity wave parameters at random, characteristic oscillations are found with periods in the planetary scale range (2, 4 and 5 days) and in the tidal range (1 day, 16 h and 12 h).     

Generation of gravity waves by inhomogeneous heating of the atmosphere

Local variation of atmospheric heating which might occur in inhomogeneities of various constituents such as ozone or molecular oxygen may generate gravity waves. These perturbations are induced by the terminator crossing constituent inhomogeneities of short lifetime. The quasi-point heating model developed here shows that the largest amplitude must appear vertically above the source, where the perturbation frequency is close to the Vaisala-Brunt frequency. Numerical calculations not band limited in the frequency suggest several characteristics of the perturbation.     

A numerical model for gravity wave dissipation in the thermosphere

Two simplified models of internal gravity wave dissipation due to viscosity, thermal conduction and ion-drag, in a multilayered, isothermal thermosphere are developed. Each of these models uses the WKB approximation, ray theory and the time-averaged equation of energy conservation, but whereas one of the models (A) employs all of the gravity wave equations appropriate to a dissipative atmosphere, the other (B) does not. Results derived from these models for one particular wave are compared to each other and also to some previously published results of Klostermeyer, which employed a full-wave, model. A breakdown of the WKB approximation in the lower, non-isothermal thermosphere leads to models A and B underestimating the total dissipation there. In the middle thermosphere model A estimates the dissipation reasonably well, while model B grossly overestimates the dissipation. In the upper thermosphere model A underestimates the total upward energy flux, probably as a result of the neglect of coupling into the dissipative waves at these levels, while no energy remains in model B. Results from model A show that when dissipation due to viscosity and thermal conduction are included correctly and simultaneously, the dissipation due to viscosity can exceed that due to thermal conduction by a factor of three. It is argued that ray theory may either overestimate or underestimate the energy flux reaching the upper boundary of a thermospheric model depending on both its height and the particular thermospheric model used.     

Gravity wave saturation and eddy diffusion in the middle atmosphere

A discussion is given of gravity wave saturation and its relation to eddy diffusion in the middle atmosphere. Attention is focused on the saturation process and some of its observable manifestations. It does not serve as a review of all related work. Although a theoretical point of view is taken, the emphasis is on which wave parameters need be measured to predict quantitatively the influence of gravity waves on eddy transport. The following considerations are stressed: the variation of spectra with observation time T; that eddy diffusivities are determined by velocity spectra; the anisotropic nature of diffusivity; a unified approach to saturation; an attempt to make eddy diffusivity more precise; the relationship between eddy diffusivity and wave dissipation.The subjects of ‘wave drag’ (momentum flux deposition) and heat flux need only be treated briefly, because they are related to eddy diffusivity in simple ways. Consideration is also given to two different theoretical mechanisms of wave saturation—wave induced convective instability and strong nonlinear wave interactions. The saturation theory is then used to predict a globally averaged height profile of vertical diffusivity in the middle atmosphere. This calculation shows that gravity waves are a major contributor to eddy diffusion from heights of 40–110 km, and that they are significant down to 20 km. A more detailed calculation of wave induced eddy diffusion, including latitudinal and seasonal variations, can be made if wave velocity spectra become available. The paper closes with recommendations for future research.     

Radar studies of gravity waves and tides in the middle atmosphere: a review

This review deals with recent radar studies of gravity waves and tides in the middle atmosphere, roughly over regions of 10–30 and 60–90 km. The techniques are briefly discussed and their limitations are pointed out. In the troposphere-stratosphere region, buoyancy oscillations, gravity-wave critical-layer interactions, and gravity waves excited by cumulus convection have been observed. Pronounced short-period (10–20 min) waves have frequently been detected in the mesosphere, and in some cases these have been identified as evanescent and trapped gravity wave modes. Diurnal and semidiurnal tides have been observed in the stratosphere and mesosphere at low and mid latitudes, but the corresponding tidal modes are not unambiguously resolved. The need for obtaining more comprehensive data bases with the existing radar systems is emphasized for further tidal and wave studies in the middle atmosphere.     

Pseudosaturation of gravity waves in the middle atmosphere: an interpretation of certain lidar observations

Irregular winds of the middle atmosphere, commonly attributed to gravity waves, often exhibit a vertical-wavenumber (m) spectral form approximating to Km⁻³ at sufficiently large m, with K a constant found to be relatively unvarying with time, location and even height. This behavior is widely believed to result from some saturation process, but the physical mechanism remains a matter for debate. There now exist three theories:

• 1.(a) linear instability,
• 2.(b) nonlinear wave-induced diffusion, and
• 3.(c) nonlinear waveinduced Doppler spreading. Each has produced the Km⁻³ form (but only as an approximation in the case of Doppler spreading) and values of K within a factor of three of one another and of observed values.

New data have revealed circumstances in which an approximation to the form Km⁻³ is again found, suggestive of saturation, but with values of K that increase by a factor of 5 or 10 on moving from the stratopause to the mesopause region. This height variation is incompatible with theories (a) and (b) if the m⁻³ form is taken to be induced by the corresponding saturation process (rather than by source spectra), but is shown here to be compatible with the Doppler-spread theory. Because of the continued growth of K with height, which must ultimately cease, the observations and corresponding theory are taken to represent pseudosaturation rather than fully developed saturation per se.     

Daily tidal,planetary wave,and gravity wave amplitudes over the Canadian prairies

Fifteen months of daily mean winds, semi-diurnal tides, and gravity waves observed from the Canadian Prairies MF radar network were examined to find correlations in their amplitude fluctuations. In all cases, the correlations of the wave/winds between the three sites were greater than 95% significant. Generally, while the correlations between the various wave/winds types were much weaker, there was evidence of gravity wave/tidal and tidal/mean wind interactions.     

A spherical model of wave generation in the atmosphere by the solar terminator

With the help of equations written in a spherical coordinate system moving with the terminator and which describe acoustic-gravitational wave generation, the general solution for pressure field perturbation is obtained. A spherical model of the solar terminator is presented. Types of nonstationary waves generated by the terminator when it crosses atmosphere inhomogeneities have been studied.     

Applying weakly nonlinear internal wave theories to solitary waves in the atmosphere

The theory of weakly nonlinear internal solitary waves is summarized in a simple way, with the view to applying it to solitary waves in the atmosphere. Two observations [Smith and Morton (1984) Australian Meteor. Mag. 33, 185; and Lin and Goff (1988) J. atmos. Sci. 45, 194] are considered in more detail. It is shown that for mesoscale solitary waves compressibility is important, and that the so-called ‘Morning Glory’ waves may possibly be described by shallow-fluid as well as deep-fluid solitary wave theory.     

The non-linear interaction of a gravity wave with the vortical modes

The wave-wave interaction theory has been used successfully in describing one class of weakly non-linear wave phenomena. The application of this theory to the atmosphere shows the possibilities of energy and momentum transfer among three interacting gravity waves, as well as from the gravity wave to the other modes of motion. It has been found that the non-resonant interaction of a gravity wave with two vortical modes can proceed at a reasonably rapid rate. With the gravity wave viewed as the primary wave and the two vortical modes as the secondary waves, the interaction equation can be linearized and solved. The resulting analytic formula gives the growth rate of the interaction. In the absence of the Earth's rotation, the growth is limited to a threshold effect. The theory shows that whenever the horizontal air parcel velocity of a gravity wave exceeds a factor of √2 times the horizontal trace velocity of the wave, energy and momentum transfer from the gravity wave to the vortical modes can proceed. The rotation of the Earth will blur this threshold effect by making the interaction more likely to occur. Thus, through this mechanism, a gravity wave can transfer its energy and momentum to the horizontal velocity field in the vortical mode. In this sense, the small scale vortical motions would serve as the sink of both energy and momentum of a propagating gravity wave. When scales of vortical modes reach sufficiently small values, dissipation through viscosity becomes important. At this scale and smaller, the vortical modes are damped out quickly and its energy spectrum must exhibit a sharp decay.     

Dynamic coupling between the lower and upper atmosphere by tides and gravity waves

Results of a General Circulation Model simulation of the dynamics of the middle atmosphere are shown focusing our attention to the tidal wave mean flow interaction and propagation of migrating diurnal and semidiurnal tides in the model. It is shown that migrating tidal waves are well simulated and the amplitude growth with height is effectively suppressed by the convective adjustment in the model. It is also shown that the dissipating solar diurnal tide plays an important role in inducing mean zonal winds in the low latitude region of the lower thermosphere. The behavior of non-migrating diurnal tides is also analyzed to show that non-migrating diurnal tides have significant amplitudes in the lower thermosphere. It is suggested that the non-migrating diurnal tide, which propagates against background mean zonal winds, has the possibility to propagate into the middle to high latitude region due to the Doppler effect.     

Dominant vertical scales of gravity waves in the middle atmosphere observed with the MU radar and rocketsondes

We have simultaneously observed wind motions in the altitude range of 5–90 km by means of the MU radar, rocketsondes and radiosondes. Dominant vertical scales of wind fluctuations due to gravity waves were 2–5 km in the lower stratosphere, about 5–15 km in the upper stratosphere and longer than 15 km in the mesosphere. The increase in the vertical scale with altitude is interpreted in terms of the saturation of upward propagating gravity waves. In the stratosphere, the observed vertical wavenumber spectra showed smaller amplitudes and more gradual slopes than the model values. Furthermore, the wind velocity variance in the stratosphere increases exponentially with an e-folding height of about 9 km, implying that the gravity waves were not fully saturated. On the other hand, the spectra in the upper stratosphere and mesosphere agreed fairly well with the model spectra. The variance in the mesosphere seems to cease increase of the wave amplitudes and agrees reasonably well with the model value.     

An assessment of the effect of gravity waves on the width of radar Doppler spectra

Radars can be used to obtain turbulence parameters by measuring the width of the Doppler spectrum of returned power. Other effects can contribute to the Doppler-spectrum width, including short-period gravity waves, making turbulence parameter measurements more difficult. In this paper, an experimental study of the effect of gravity waves on the width of the Doppler spectrum is presented. The data were obtained using the Poker Flat 50 MHz VHP radar. A parameter Y_i is used to describe the discrepancy between the width of spectra sampled over a long period of time and that taken over a shorter time. If Y_i is not significantly different from 1.0, no contamination is present. The experimental data considered were of a time resolution that allowed the period and amplitude of the contaminating gravity wave to be determined. A theoretical expression for gravity wave contamination proposed by Hocking [(1988) J. geophys. Res.93, 2475–2491] was tested and found to agree with measurements. It was also found that non-unity values of Y_i occurred in some cases. This suggests that, at commonly used sampling times, short-period gravity waves can contaminate spectral width estimates.     

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.