For several different airglow emissions, the deviations from H&T's approximate relation (due to chemical time constants) were found to be small. So, in this aspect of the theory, dynamics and chemistry are decoupled!
After first attempts to compare observations with model results by Hines and Tarasick (1987) on modulus and phase of Krassovsky's ratio (e.g., Ammosov (1992)), Reisin (1994) and Reisin and Scheer (1996) have shown that their observations of the O2 and the OH emissions (and reports by other researchers, too) are consistent with the prediction of the H&T theory about phase and vertical wavelength. The phase shifts observed by Reisin and Scheer for the tidal period range have later been confirmed by other authors (Taylor et al. 2001). The mean vertical wavelength Reisin and Scheer derived for tides (about 28 km) could in principle have been checked against an independent determination via the phase difference of the temperature oscillations at the two airglow layers, if the layer separation were known precisely. And indeed, a study based on 12-hour horizontal wind oscillations observed optically in New Zealand (Hernandez et al. 1995), in OH and the atomic oxygen green line emission, supported by emission altitudes measured from the UARS satellite, had previously obtained vertical wavelengths between 20 and 40 km. This is consistent with Reisin and Scheer's results obtained from the Hines and Tarasick theory.
However, in spite of the formal equivalence (Walterscheid et al. 1994) between the Hines-Tarasick and competing theories, the sign of the phase shift in the other models depended (sometimes delicately) on wave period, wave type (gravity or tide), airglow layer characteristics, atomic oxygen profile, or other factors (see Reisin and Scheer 1996, for more references about airglow/wave models).
Hines (1997) and Hines and Tarasick (1997) attributed some of the diverging results of the other theories to erroneous procedures or numerical problems avoided by their own approach.
A few general remarks may be in order, here:
In hydrodynamics, two different descriptions can be applied, either with a spatially fixed reference frame ( Eulerian approach), or a frame that moves with the medium ( Lagrangian approach). Formally, both approaches are equivalent. However, under certain conditions the Lagrangian approach leads to a much simpler description, while the Eulerian description can become very complex (for a very insightful discussion of the relation between the frame of reference and the linearity, or nonlinearity, of waves, see Hines (2001, 2002)).
In order to take advantage of the greater inherent simplicity of the Lagrangian approach (in this context), the H&T theory uses a hybrid (semi-Lagrangian) scheme, although the observational conditions, i.e. optical remote sensing of a fixed column volume, makes a purely Eulerian approach seem the natural choice. However, this better match of the Euler view is partly an illusion, because of the material that moves, vertically and horizontally, into and out of a fixed volume, complicating things. The philosophy behind H&T's technique is illustrated most clearly in Hines and Tarasick (1997).
The more recent model by Swenson and Gardner (1998; abbreviated below as "S&G"), for the OH and Na emissions, also predicts a dependence of phase shift on the sense of vertical propagation. This paper also supplied an interpretation more accessible to intuition than the H&T treatment, by considering the deviation, with respect to the center of the emission layer, of the altitude of maximum intensity variation (and the opposite one, for temperature). This is why the intensity variation sees an upward-propagating wave first, while the temperature oscillation sees a downward-propagating wave first (in terms of phase propagation, whereas vertical energy propagation of atmospheric waves, including tides and planetary waves, is opposite to phase propagation, in general).
The comparison with the scant observational material available in the early phase of modeling must have been not very encouraging for modelers: The evidence for observed gravity waves in the atmosphere (some published results, and more unpublished ones) seemed to suggest -or, were summarized in that sense- that either zero phase shift, or antiphase, between the airglow intensity and temperature oscillations prevailed, whereas practically all the models predicted some non-zero phase shift.
Although zero phase shift in the presence of ducting, or wave evanescence, can be viewed as being consistent with the H&T theory (although this was carefully excluded in formulating the original theory), Hines and Tarasick (1994) gave this case a special treatment, concluding that real values of eta (and therefore, in a purely mathematical argument, phase shifts of zero or 180°) were consistent with theory, thereby establishing formal consistency with the perceived observational situation.
And indeed, ducting conditions which produce reflection and therefore standing waves in the vertical direction are expected to lead to waves that should be easy to observe because of their high amplitudes (for being trapped in a duct, or wave guide), and the absence of phase cancellation caused by the averaging effect of the emission layer (since observed intensities are column-integrated).
Schubert et al. (1999) reported the observation of cases (for the atomic oxygen green-line emission) with phase shift "near minus 180°" [sic!], taken as evidence of reflection. All the 13 values were between +11° (the most positive) and -155° (most negative value). When the published data are plotted in the complex eta plain, it becomes clear that the data are closest to the real axis only near zero phase shift.
Reisin (1994), and Reisin and Scheer (2001) provided evidence (from a large number of short-period gravity waves contained in their several years of data) that phase shifts of the sign typically seen for tides are most frequent, but that the opposite sign and intermediate values are also very often observed, so that small phase shifts consistent with zero are a natural part of the statistical distribution. However, phase shifts of 180° are definitely exceptions in their data set, and may even be statistical outliers, contradicting the conclusions of Schubert et al. (1999).
Correctly dealing with phases is not as trivial as it might seem. So, any statement about empirically derived phase values must be seen as part of the whole complex quantity (i.e., including amplitude, or in the present context, modulus of eta). Therefore, the widespread habit of presenting and plotting amplitude and phase results separately is dangerous. Especially, for small amplitude, the combined amplitude and phase uncertainty can make phase completely undefined, if the uncertainty distribution in the complex plain imcludes the zero point.
Care must also be taken to avoid (or correct for) any systematic (spurious) correlations (or anticorrelations) between band intensity and temperature oscillation, if band intensity is not determined independently of temperature, but extrapolated from the measured parts of the molecular band. Otherwise, such a correlation might shift the experimental data parallel to the real axis (see Reisin (1994), Reisin and Scheer (2001)).
Reisin and Scheer (2001) also concluded from their observed phase shifts, that a considerable fraction of the gravity waves with (observed) periods between 1000 and 10000 seconds propagate their energy downwards, namely 39% at 87 km (OH emission), and 30% at 95 km (O2 emission). Independent observations available at that time could only be used to show qualitatively that this evidence of a large proportion of downward-propagating waves made sense at all, but not to confirm it quantitatively. Only recently, meteor radar observations (using vertical profiles of zonal wind, that is, a completely different technique) by Oleynikov et al. (2005) has given practically the same result, that is, 35% downward propagation, as an average over the height range 80 - 100 km. There are some minor differences in the range of wave parameters observed: Oleynikov et al. report observed periods greater than 30 min, and the cutoff at short vertical wavelengths is smaller than with the airglow data (where it is due to the thickness of the airglow layers). In spite of this, the good quantitative agreement with the airglow results lends additional support to the validity of Hines' and Tarasick's vertical wavelength relation.
So it looks as if the observations for gravity waves with periods shorter than the tidal range are also consistent with the H&T and S&G theories, and there is no evidence of a major influence of the horizontal scale or period of the waves.
Needless to say, the relationship between vertical wavelength and Krassovsky's eta converts zenith airglow observations into a powerful diagnostic tool for mesopause region dynamics, and Reisin and Scheer have made much use of it (Scheer 1995, Scheer and Reisin 1998, 2001, 2002, 2007, Smith et al. 2006).
The H&T theory also sheds light on the frequently observable similarity between the shapes (without apparent phase shifts) of the nocturnal variations of O2 intensity and OH temperature, as noted and discussed by Scheer (1995). This similarity is especially striking when the variations are not simply sinusoidal, and while the other parameters have different shapes, and/or phase shifts. A few more examples are shown in Scheer and Reisin (1998). This feature has called the attention of other investigators (and manifests itself in published data), but no attempt at explaining it has otherwise been made, in the past. Only the recent paper by Smith et al. (2010) does not only show a particularly impressive example, but also points out the relation to the S&G findings.
From the new perspective of the S&G theory, it becomes more obvious that the sense of vertical propagation of the waves plays no role in explaining this similarity: According to the S&G model, the level of maximum sensitivity to temperature variations is always above the altitude for maximum intensity variations (a fact that has also been confirmed by satellite observations, Marsh et al. 2006). Then, the surprising similarity (at least, as far as the lack of phase shift between O2 intensity and OH temperature is concerned) stems simply from a coincidence between the heights of maximum intensity sensitivity for the (higher) O2 layer, and of maximum temperature sensitivity for the (lower) OH layer. This means, the lack of phase shift holds for all periods and vertical wavelengths, and for upward- and downward-propagating waves. It depends only on the suitable separation between the airglow layers.
Therefore, this similarity phenomenon is a rather straightforward confirmation of the validity of the S&G and H&T theories, and the usual assumptions about the nominal (relative) emission altitudes of the OH and O2 airglow layers. This is why the phenomenon is (or should be) also observable when the intensity fluctuations of other O2 bands (infrared, blue or violet), or the OI green line, are involved (and compared with OH temperature).
Unfortunately, the importance of the H&T and S&G results for the interpretation of airglow observations has occasionally not been appreciated in the literature. For example, in a study of tidal oscillations with OH airglow, Won et al. (2001) concluded (correctly!) that "...the temperature modulations lead the airglow brightness modulation..." in phase. Although they did not say so, this is of course consistent with the growing body of observations, the upward energy (and downward phase) propagation of tides, and the H&T and S&G models. However, these authors stated that their findings were "contrary to predicted values for tides", and did not cite the H&T or S&G theories. Since, according to these theories, the phase shift depends on vertical wavelength only, there should be nothing special in tides as opposed to smaller-scale gravity waves that would warrant a different behaviour, in this respect (and the complication of the polarisation relations by the introduction of Coriolis terms was shown to be inconsequential; Reisin, 1994).
The growing body of observational evidence, partly thanks to the wider use of airglow instruments that simultaneously observe OH and O2 emissions (like, e.g. SATI and other new instruments), is now leading to wider confirming use of the H&T or S&G theories. E.g., López-González et al. (2005), Chung et al. (2006), Taori and Taylor (2006), Guharay et al. (2008), and Guharay et al. (2009) have successfully applied the H&T theory.
A step forward in the development of the theory of airglow-dynamics relations has been the incorporation of energy dissipation by Liu and Swenson (2003). A main conclusion is that wave damping contributes to the phase shift between the intensity and temperature oscillations, while conserving the fundamental qualitative relationship between the sign of the phase shift and the sense of vertical wave propagation. This is also the first theoretical treatment of two different airglow emissions together, and has repercussions on the detailed understanding of the similarity phenomenon described above.
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