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Colin O. Hines

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 the time. The present paper derives an analytic form for the cutoff function, illustrates it in application, and provides quantitative support for a value possibly as low as 0.5 in the stratosphere and a value possibly as high as 1.0 in the mesosphere. In addition, it slightly recasts the heuristic approach to the Doppler-spread analysis, and it admits to certain difficulties, associated with the largest-m waves, whose circumvention appears to require a far more detailed analysis of wave-wave interaction through the advective nonlinearity.

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Colin O. Hines

Abstract

The theory of mountain waves is usually discussed for the case of a steady background wind. Here, the consequences of a superimposed diurnal (or other periodic) background wind variation are considered in outline. They are found to be sufficiently complicated as to warrant avoidance in detailed case studies. They include, however, the production of freely propagating waves and may be of interest on that account for other purposes.

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Colin O. Hines

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Colin O. Hines

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Colin O. Hines

Abstract

The standard current criterion for the generation of turbulence by atmospheric gravity waves and for the associated limitation on wave growth is based upon the standard criterion for static instability of the unperturbed atmosphere, namely, that the vertical gradient of potential temperature be negative. This criterion fails to recognize that a slantwise static instability may be available in the presence of gravity waves and, if so, could be of importance. New criteria, involving estimated growth times, are developed here and reveal slantwise instability to be a likely mechanism of turbulence production. It is also found that the most favored axes for the development of the slantwise instability are quasi-horizontal, which suggests that the resultant turbulent motions may well be quasi-horizontal, at least at the energy-input portion of the turbulence spectrum.

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Colin O. Hines

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 the tail are said to result from this process.

The arguments in favor of this view are questioned in the present paper and found wanting (though the claim of instability remains unchallenged and is even reinforced). The waves of the tail are then recognized as being subject to a strong wave–wave interaction arising from the Eulerian advective nonlinearity—from the Doppler shifts that can be imposed upon them by the larger-scale winds of the wave system—a fact recognized in the corresponding oceanographic literature for about a decade now.

In a companion paper, a rudimentary analytic approximation to the advective nonlinearity is introduced, and its consequences are shown to yield a spectral form and intensity quite similar to those obtained observationally. The linear instabilities (and some formulas) of the present paper are then invoked to establish the length, rather than the form and intensity, of the tail, at least below the turbopause.

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Colin O. Hines

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 being taken to be horizontal, horizontally stratified and constant in time, but otherwise that field is taken to have the statistical characteristics that would be expected of a wave-induced spectrum, including a propensity for growth with height.

It is found that waves with relatively large vertical wavenumber m exceeding a characteristic value mc (comparable to or greater than N 0/2σ r , where N 0 is buoyancy frequency and σ r is rms wind speed) are substantially Doppler spread in vertical wavenumber, most particularly into a large-m tail. At sufficiently large m, the waves are taken to be obliterated by dissipative processes. The net effect is to leave a tail that will be universal and can be identified readily with the observed tail (subject to further approximation in the treatment of the wave obliteration).

If the tail extends to a sufficiently large m—a well defined value m Minst—the spectrum as a whole renders itself unstable. The length of the tail is then taken to be limited by the instability, any m values observed beyond m Minst being attributed to turbulence.

The implications of these conclusions are built into spectral models of the “modified Desaubies” form for future application to middle-atmosphere modeling.

The theory is backed by appeal to observations and is brought briefly into contact with related views in oceanographic studies.

The body of the paper ignores the Coriolis force and the contribution of vertical motions to the advective nonlinearity, but their effects are touched on in two Appendices. It also assumes, for the most part, azimuthal isotropy in the propagation of the waves, but the consequences of abandoning this assumption are outlined in a third Appendix.

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Colin O. Hines

Abstract

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.

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Colin O. Hines

Abstract

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.

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Colin O. Hines

Abstract

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