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.     

Nonlinear solitary wave in the ionospheric E-region

A nonlinear wave equation for a solitary one-dimensional inhomogeneity is studied. It is similar to the diffusion equation with the diffusion coefficient depending on the phase velocity. The phase velocity depends, in turn, on the electron density. A weak inhomogeneity moves with the velocity close to that of the electric drift. If this velocity exceeds the ion acoustic speed the effective diffusion is negative, and the inhomogeneity grows and contracts. The velocity of the growing inhomogeneity becomes smaller. It absorbs weaker and faster moving inhomogeneities from the back side. In a stationary regime, the ionosphere will be filled with rare but strong inhomogeneities with sharp back sides.     

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.     

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.     

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.     

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_b²/2m³. 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.     

Weak nonlinear theory of the ionospheric response to atmospheric gravity waves in the F-region

The nonlinear ionospheric response to atmospheric gravity waves is studied in an approximate fashion using a new approach. The concept of nonlinear travelling ionospheric disturbances (TIDs) is outlined, and the nonlinear behaviour of atmospheric gravity waves is calculated. A principal result is that harmonics are generated which cause the wave velocity perturbation to deform. The ionospheric response is investigated by solving the continuity equation for ionization in the F-region. The distortion of the TIDs waveform produced by the nonlinear interactions is depicted. The nonlinear TIDs depart seriously from a cosinusoidal wave described by previous linear TID theory. The distorted TIDs appear as ‘sharp peak’ and ‘sawtooth’ waveform shapes. The ‘peaks’ can be upward or downward, and the ‘sawteeth’ forward or backward, depending on the wave parameters. The nonlinearly distorted TIDs show a good agreement with various observed ionospheric irregularities produced by atmospheric gravity waves.     

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 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.     

A quartic dispersion equation for internal gravity waves in the thermosphere

A new quartic dispersion equation in the square of the complex vertical wave number is derived by employing the ‘shallow atmosphere’ approximation and an ion drag approximation. These approximations allow the coefficients of the quartic equation to be given in terms of the corresponding cubic equation, which neglects the Coriolis force and the zonal ion drag component, but modified to take into account these neglected effects. Coupling between the extraordinary viscosity wave mode and the other three wave modes is highlighted and numerical solutions are compared for this quartic equation, an exact eighth order equation and the cubic equation. For the first time the validity of using the ‘shallow atmosphere’ approximation to describe internal gravity wave motions is demonstrated.     

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.     

On retrieving Rossby waves in the middle atmosphere from measurements of thermal outgoing radiation

The first step is made in elaborating special methods to retrieve the planetary-scale waves for the stratosphere and mesosphere from measurements of thermal outgoing radiation. The method is adapted for the nadir sounding of Rossby normal modes of the Lamb wave type in the 15 μm CO₂ band. The main formulae are presented in a dimensionless form. The proposed method consists of EOF filtering to extract a wave-induced signal and of Hermite polynomial expansions to describe the vertical structure of the wave. The accuracy of the retrievals is estimated; it is dependent on the duration of the record as well as on the number of channels. The method is able to provide a higher accuracy than currently available methods.     

A simplified approach to internal gravity wave damping by viscous dissipation

Internal gravity wave perturbation velocity subject to constant dynamic viscosity and constant kinematic viscosity are approximately derived based on an energy conservation principle. When the dynamic viscosity is assumed to be a constant, the velocity at any height relative to the velocity at the saturation height, Z_sat, is found to be solely dependent on the number of scale heights measured from Z_sat. Gravity wave energy dissipation due to constant dynamic viscosity primarily occurs from one scale height below to one scale height above the saturation height. When the kinematic viscosity is assumed to be a constant, the perturbation amplitude either increases or decreases monotonically with height depending on whether the energy dissipation rate due, to viscosity, can offset the effect of the decreasing atmospheric density with increasing height. The derivations are made simple by assuming that the non-dissipative dispersion relation is applicable to the dissipative situation. The condition for the assumption to be approximately valid is also given.     

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.     

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.     

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.     

The 2-day wave in the middle atmosphere as simulated in a general circulation model extending from the surface to 100 km

The 2-day wave observed in the mesosphere and lower thermosphere has been reproduced in a general circulation model of the atmosphere run for fixed January conditions. The wave was confined to the summer hemisphere between 50 and 100 km, and was most strongly evident in the meridional velocity where it caused a reversal in the direction of this wind approximately every 24 h. Similar but smaller fluctuations could be detected in the zonal wind and temperature. The synoptic distributions from the model confirm that the 2-day wave is a zonal wave number 3 phenomenon and that it progresses westwards. These distributions have maximum amplitudes occurring at higher latitudes than observed, probably owing to the mean wind intensity in the model summer hemisphere being slightly underestimated. Quite marked interactions occurred between the high latitude and tropical features of the synoptic meridional velocity distribution as the wave progressed. The wave had a very small phase variation with altitude, and, except for a region near 70 km, exhibited hardly any sign of baroclinic activity. The formulation of the model eliminates atmospheric tides or orography as forcing agents responsible for the excitation of the 2-day wave.     

Interactions between whistler-mode waves and energetic electrons in the coupled system formed by the magnetosphere,ionosphere and atmosphere

The physical mechanism of a cyclotron resonance interaction between trapped energetic electrons and whistler-mode waves in the magnetosphere is discussed. Not only do the electrons have their pitch angles reduced in this interaction, so that they may be precipitated into the upper atmosphere, but also the waves can be amplified. Such a flux of precipitating electrons can, either by direct ionisation or via bremsstrahlung radiation, cause a pimple to be produced on the bottom of the ionosphere. That can significantly modify the amplitude and/or phase of very low frequency radio signals propagating in the Earth-ionosphere waveguide. Various experimental observations that demonstrate the reality of such effects are reviewed. The conditions necessary for a positive feedback situation are discussed, and some evidence for its existence assessed.     

Some effects in the middle atmosphere due to lightning

A brief review is given of some of the electrodynamic responses of the middle atmosphere to lightning. Attention is focused on the precipitation of energetic electrons from the magnetosphere, due to whistler mode electromagnetic waves. The secondary ionisation and bremsstrahlung radiation created, and some of the ways in which such effects can be detected, are also considered. Finally, the possibilities of positive feedback mechanisms operating between the atmosphere and the magnetosphere are investigated.