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.

A recent paper by Eckermann (1997, henceforth E97) has launched a major attack on the assumptions and deductions of the Doppler-spread theory (DST) of Hines (1991, 1993), which relates to saturated gravity wave spectra in the middle atmosphere. The challenge extends also to the Doppler-spread parameterization (DSP) of energy and momentum deposition in the middle atmosphere (Hines 1997a,b). I show here that, insofar as the E97 analysis has anything legitimate to say on the issue, it tends to confirm rather than oppose the deductions of the DST and those of the DSP.

The E97 analysis involves deterministic ray tracing, using eikonal theory. It employs an illustrative selection of small-scale test waves (or, more properly, wave packets) propagating through the wind fields of one or a few large-scale waves. In direct contrast, the DST seeks to infer the statistical behavior of waves of every scale (limited only by relevant approximations) propagating through the winds of a broad ensemble of waves of all scales. Despite this intrinsic difference, E97 insists that the latter can be represented by the former—that the DST is legitimately viewed as “a simplified eikonal formulation”—and on that basis E97 draws its general conclusions purportedly adverse to the DST. The alleged equivalence is mythical, many facts are misrepresented, and the E97 analysis suffers major shortcomings of its own that preclude passing adverse judgment on the DST, as will now be described.

It should be noted from the start that, without saying so, E97 tacitly accepts the fundamental position of the DST: the winds induced by other waves are important to an understanding of the propagation of small-scale waves in the middle atmosphere. The only questions at issue are how best to analyze the problem and whether or not the DST as developed to date has been successful in its attempt.

Misrepresentation begins in E97’s introduction, with the case of a single test wave propagating through the wind field of a single large-scale wave. E97 asserts that, in modeling this situation, “Doppler-spread theory assumes that the velocity oscillation of the long wave acts like a time-invariant, vertically varying mean-flow profile that refracts the short wave”. In point of fact, Doppler-spread theory does nothing of the sort. It never even addresses such a situation, nor would it claim an ability to do so. What it does do is make the approximation that, for the statistical purposes it pursues, the total wind field induced by the ensemble of other waves in a broad spectrum may be treated as if it were a time-invariant vertically varying mean flow. This is quite a different assumption. Nevertheless, E97 continues up to the end of its section 7 analyzing the behavior of the test wave in a single-wave environment and drawing conclusions that are alleged to be contrary, here and there, to those that the DST would produce. These early sections have essentially nothing to contribute to an assessment of the DST, despite the adverse claims that they make and despite the interest that they may hold as an academic exercise in their own right. I discuss them no further.

The E97 analysis becomes relevant to the DST, if at all, only on reaching its section 8. There, test waves are taken to propagate through an ensemble of eight large-scale waves. This ensemble gives at least some beginning to the broad spectrum of waves that the DST treats. The most important conclusions are to be found in Fig. 10, where results for five test waves are presented. Only two of the five employ wave frequencies sufficiently less than the buoyancy frequency to permit comparison with the DST as developed to date. These two represent test waves whose initial vertical wavenumbers have magnitudes approximating 0.1 N/σu and N/σu, respectively, where N is the buoyancy frequency and σu is the rms horizontal wind of the ensemble of large-scale waves; the wavenumber N/σu is denoted mC by E97, paralleling a similar symbol in DST. The results reveal that the 0.1 mC test wave suffers very little Doppler spreading (±20% at most), whereas the mC wave suffers spreading by a factor well in excess of 2. This is precisely the sort of behavior that the DST would attribute to these two test waves in a broad ensemble of other waves, although it is contrary to the behavior attributed to middle-atmosphere waves in all competing (mainly linear) theories of gravity wave saturation. Despite the agreement, E97 repeatedly dismisses the DST approximations that first permitted this crucial behavior to be identified and displayed. It would seem to me, rather, that the DST approximations have in fact been confirmed as to their utility, at least to this extent, and I am obliged to wonder why E97 places all its emphasis on “the hole” of what are thought to be their shortcomings while ignoring almost completely “the donut” of their success. Particularly is this so, when I have admitted, indeed, advertised, the shortcomings all along.

What is the allegedly new shortcoming? If E97’s Fig. 10 is taken at face value, then it implies substantially less Doppler spreading than the DST would imply for the one relevant mC wave. But this is precisely the faulty consequence that I have already attributed to the DST as developed to date: the DST produces m−1 spectra instead of m−3 spectra at large m, which means that it is spreading the spectra to greater m values than it should. And I have attributed that failing to a breakdown of the DST approximations, which is precisely what E97 concludes the cause to be. So, even here, E97 is qualitatively confirming rather than opposing what I have already written about the DST!

(E97 concludes that the breakdown results from the time-invariance assumption and contrasts that conclusion with its own claim that I had attributed the breakdown to the neglect of vertical wind components. I made no such restricted attribution, but rather chose for extended comment the vertical-wind breakdown as a source of error that could be readily demonstrated. Moreover, since E97 employs large-scale waves that are of relatively low frequency, it minimizes the role that might be played by their vertical wind components and so can scarcely arbitrate between the two sources of error for a realistic middle-atmosphere spectrum.)

The disparity of spreadings implied by the two analyses, that of E97 and that of the DST, becomes quantitative and becomes relevant to the middle-atmosphere DSP, when consideration is given to the possible spreading of the test wavenumber mC to the dissipative wavenumber mM. For the purpose, E97 chose mM = 200 mC, adapted from section 4 of Hines (1991). It then found that the mC wave was never spread as far as mM, and so would be free of dissipation, whereas the DST would take the mC wave to be spread as far as mM over a small but significant portion of space–time and so to be subject to some dissipation.

It should first be noted that the presence or absence of dissipation in specifically mC waves is of little concern to the DST, dissipation being but a slight modifier of the most appropriate specification of mC. The spectral tail would be formed as before, and it would begin in the neighborhood of mC as before. Only minor details would change if dissipation were to set in only at shorter scales, but details are subject to correction in any event because of the previously recognized shortcomings of the DST approximations.

The factor 200 employed in specifying mM was the most detrimental available from Hines (1991) for adaptation by E97. It came initially from Smith et al. (1987) but was never favored by me. Instead, I have mainly employed a factor of 2.4 or 11.5 for illustrative purposes, values that were based on a requirement for appreciable instability (Hines 1991). Indeed, the DSP incorporates precisely that range and attaches no credence whatever to the 200 value (Hines 1997a). Had the 2.4 factor been employed, E97 could not possibly have reached the conclusion that “there is again no tendency for short waves at [|m| > mC] to be removed via refraction to small vertical scales [|m| > mM].” On the contrary, the consequences of the DST approximations would have been fully confirmed for application to the mC waves, at least to the extent that they could have been confirmed in the absence (from E97) of a full probability distribution of spreading.

The claim relating to “short waves at [|m| > mC]” was actually never justified in application to the lower-frequency waves that the DST treats: no such waves were examined by E97. Had they been, had even a wave at 1.2 mC been examined, substantially increased spreading would have been found, certainly to 2.4 mC if not also to 11.5 mC. The distinction between mC and 1.2 mC, as a transition point that marks the onset of appreciable dissipation, is of little concern either to the DST or to the DSP. And in any event, the DSP makes explicit provision for that transition to occur anywhere in a range from 0.75 N/σh to 1.15 N/σh (where σh is closely analogous to the σu of E97 but defined for more general circumstances). The DSP therefore escapes quite unscathed from E97’s attack.

In summary to this point, one can say that, to whatever extent E97 may be relevant, its results confirm the spreading theory of the DST, confirm the tendency to a greatly enhanced spreading as the test wavenumber increases to and through a characteristic value determined by the rms horizontal wind induced by a number of other waves, and confirm that the spreading becomes sufficient to impose significant dissipation on waves at |m| > mC if a rational criterion is chosen for the onset of instability. Neither the DST nor the DSP could ask for much more support than that from an eikonal analysis.

The relevance of an eikonal analysis will in fact be questioned in due course. Before that is done, however, for the sake of argument I shall for a moment accept the E97 analyis on its own terms and list a number of its shortcomings in its role as a putative test of the DST.

  1. In the DST (as distinct from the DSP), the essential role of mC is to mark the onset not of severe dissipation, but of the severely diminished amplitudes that characterize the tail region of observed spectra. The diminution, represented by a certain transfer function and illustrated by Fig. 2 of Hines (1991), necessarily accompanies spreading. Had it been sought by E97 (in the form of probability distributions), it would have been found and would have given further credence to the DST. But this crucial test was never made, and the DST was precluded from passing it.

  2. In keeping with its assumed broad spectrum, the DST employs a Gaussian distribution of wave-induced winds, not one that stops abruptly at 4σu as in E97’s eight-wave model. If the DST is to be tested by an eight-wave model, it should be reworked with the corresponding distribution of winds to determine, among other things, whether N/σu is still an appropriate choice for the onset of strong spreading and diminished spectral densities. (There would not even be such thing as mM by which to judge mC, because E97’s restricted spectrum of background waves precludes the occurrence of instability as inferred from standard criteria.)

  3. The DST assumes random phasing of the background waves. In E97, the test wave packets were inserted at a point of space–time where each of the eight large-scale waves, individually, had zero wind speed. This automatically biased the analysis against the availability of substantial winds for Doppler-spreading purposes. The problem would be particularly acute in cases (arising by random selection) when four of the eight waves started “in phase” and the other four in antiphase to them. Since all had equal amplitudes, the zero-wind condition would then tend to persist unnaturally.

  4. All large-scale waves were taken by E97 to propagate into a single azimuth, either “against” the test wave or “with” it (thereby accounting, in a predictable fashion, for the differences between Figs. 10a and 10b, respectively). They hardly constitute a random superposition in a broad spectrum.

  5. E97 ignored the standard growth of wave amplitude with height. Its depiction of test waves propagating from ground level to 100 km without dissipation (in its Fig. 11) is therefore highly misleading at best. The σu produced by the large-scale waves would have grown with height, if permitted to, and the test waves would have been Doppler spread to a correspondingly reduced mM and obliterated very shortly, regardless of the numerical factor (2.4, 11.5, or even 200) employed in the specification of mM. The only question at issue would then be whether or not obliteration occurred near the height at which the DST would claim. The answer is clear, coming as it does from the imagined calculation proposed above for a test wave at 1.2 mC. That calculation would correspond to a wave packet with initial |m| equal to the initial mC, after it had propagated upward little more than 2 km.

  6. The eikonal development, by its very nature, focuses attention on a single point of space at a single time:the centroid of the wave packet it seeks to follow. But a wave packet has finite extent, and its energy reaches into spatial regions that may well be prohibited to its centroid. Conditions at the centroid are therefore inadequate by themselves to determine with assurance such matters as the potential for dissipation, but they are all that eikonal theory provides. In a different context, neglect of the spatial extension of wave packets would permit eikonal theory to deny the reality of, for example, knife-edge diffraction.

Some of the foregoing shortcomings of E97 can be overcome by further computation, others not. But I hesitate to suggest that eikonal methods be pursued further, at least for an intended application either to the DST or to the irregular winds of the middle atmosphere. The reason is simple. Those methods are inapplicable except for establishing some “feeling” for the consequences of the Eulerian advective nonlinearity: the scale separation that they demand is quite unavailable. For example, in the standard “modified Desaubies” spectrum of middle-atmosphere waves, characterized by a wavenumber, m∗ (which corresponds crudely to E97’s mC), fully half of the wind variance derives from |m| > m∗. Only a fifth of the wind variance derives from |m| < m∗/3, a minimum separation from m∗, one might think, for ray-tracing methods to be invoked at m∗, given that the background waves also have horizontal wavenumbers and frequencies that overlap those of the relevant test waves. With scale separation unavailable to them, the approximations inherent in an eikonal development are simply invalid and inelligible as a means of testing the validity of any other set of approximations, such as those that the DST adopts.

I turn now to the discussion section of E97.

There, E97 states, “Thus, while Doppler-spread theory is conceived to be a comprehensive model of broadband nonlinear advective processes, we view its current mathematical formulations as a simplified version of the‘eikonal’ (ray tracing) spectral models of oceanic gravity waves.” This is a view, unsupported in any way, that I simply cannot share because of the restrictions that apply to eikonal legitimacy, imposed by scale separation and wave-packet extension in physical space. Indeed, I view eikonal theory as a simplified and inherently incomplete version of what the DST at least seeks to come to grips with.

E97 next notes my deference to the analysis of Allen and Joseph (1989, henceforth AJ), recently adapted to an atmospheric context by Chunchuzov (1996), as the proper means of deriving the asymptotic form m−3 under the effects of Doppler spreading. E97 then remarks,“However, the AJ model differs from eikonal models. It does not include refraction [by the winds of other waves is intended], but instead. . . .” The fact that AJ differs from the eikonal models preferred by E97 is thereby presented as a shortcoming of the AJ analysis, rather than, as it is, a shortcoming of eikonal models. The AJ analysis provides a meticulously rigorous determination of the asymptotic effects (at large m) of the advective nonlinearity of the Eulerian fluid-dynamic equations, for circumstances in which there is no relevant nonlinearity in the Lagrangian system. It does not include refraction of one wave by the winds of another wave simply because, in the Lagrangian system, there is no nonlinearity and no such refraction to be included. But, having provided a relevant spectrum in the Lagrangian system, AJ transform that spectrum into the Eulerian system in a fully rigorous fashion that must incorporate all wave–wave interactions that might have been attributed to the waves by anyone who insisted on an Eulerian description: refraction, reflection, diffraction, scattering, whatever. If AJ’s Eulerian results could not be replicated by an Eulerian eikonal analysis applied to AJ’s Lagrangian spectrum, it would be the eikonal analysis that would be in error. Unfortunately, the mindset evidenced in E97 is that eikonal analysis is the only valid analysis and that any other must conform to its conclusions to be judged acceptable, despite its many shortcomings. This is a failing of E97, not of AJ.

Next, E97 notes that the large-m tail of AJ’s Eulerian wind spectrum derives from what I have called “wavulence,” whose frequencies and wavenumbers are not related by a well-defined dispersion relation but rather have diffused off the 3D gravity wave dispersion surface that exists in 4D frequency-wavenumber space. E97 then asserts, “Conversely, no violations of the dispersion relation occur for waves in an eikonal formulation within the Eulerian frame: the system always remains entirely wavelike . . . even as waves are refracted to larger |m| values to produce tail spectra.” Here, E97 simply misses the point and speaks at cross-purposes. The dispersion relation referred to by AJ is a relation between wavenumber and wave frequency as observed in a single Eulerian reference frame, derived with nonlinear interactions ignored (as is appropriate to a true dispersion relation and, indeed, is virtually demanded of it). E97’s test waves succeed in matching this relation only by moving its Eulerian frame of reference backward and forward in response to the changing wave-induced winds in which the test wavepacket finds itself and, indeed, by adopting different reference frames simultaneously for the test waves and the large-scale waves, and different sequences of reference frames for successive realizations of its model. Neither AJ nor the DST permits itself this luxury, nor the luxury of dealing separately with individual cases rather than statistically with ensemble results, nor are such luxuries normally available to observers. If E97 were to confine itself to frequencies as observed in a single reference frame, and to the linear-theory dispersion relation linking those frequencies to wavenumbers, it too would find that the dispersion relation had been violated. The quoted comment is simply inappropriate: there is no distinction between AJ and E97 in this respect, except to the extent that AJ would infer a rigorously correct diffusion off the dispersion surface whereas E97’s diffusion would suffer from the inadequacies of the eikonal method (just as the DST’s diffusion, or Doppler spreading, suffers from the inadequacies of the DST approximations).

E97 continues, “Given that the Doppler-spread formulation . . . is viewed here as being a simplified eikonal formulation, then the degree to which Doppler-spread theory corresponds to the AJ model is unclear at present.” This “given” is one that I refuse to concede, and clarity will come only if it is abandoned. What should become clear, with its abandonment, is that the DST seeks by approximate means to determine the effects of the Eulerian advective nonlinearity at all m in a broad spectrum, while the AJ analysis, and that of Chunchuzov, seeks and successfully determines by rigorous means those very effects in the asymptotic behavior at large m. It is as simple as that, and any attempt to compare DST with AJ by way of eikonals does nothing but obscure the comparison.

E97 goes on, “While Doppler-spread theory now defers to the AJ model in many areas, the original formulation of Hines (1991, 1993) is still used to predict the cutoff wavenumber mC.” In point of fact, I defer to AJ now, as always, for its production of the m−3 form at large m, and I am unaware that I have had occasion to defer to it beyond that (though I would, where appropriate). My continued use of my own mC—which, incidentally, is consistent with an order-of-magnitude estimate made by AJ for the “knee” of the Eulerian spectrum—is based largely on calculations such as those of Fig. 7 of Hines (1993), which illustrates the use of the factors 2.4 and 11.5 in the specification of mM. And, contrary to what E97 next states on the basis of its own choice of mM, the E97 analysis supports the use of the DST and its mC if those factors are employed, as already noted.

Because of its rigor, I have had no hesitation in deferring to the AJ analysis in setting a goal for the DST at large |m| (in the absence of dissipation, which AJ does not include). That goal is unlikely to be reached except via an empirical marriage of DST relations at midrange with the relations of AJ (or Chunchuzov) at large |m|, a marriage currently under negotiation. Or, if the AJ analysis is extended in due course to provide explicitly the behavior at midrange—specifically, by deriving the full form of the Doppler-spread spectrum imposed on a delta-function test wave (analogous to but superseding the transfer function of the DST)—then the basic approximations incorporated in the DST to date can be abandoned. For the moment, they are the best we have. And they appear to be perfectly adequate for their purposes, even on the basis of E97’s computations, given a perfectly rational choice for mM.

Note added in proof: Dr. Chunchuzov and I, separately but in repeated close consultation, have concluded that the work of AJ, while rigorous, contains a serious error. When corrected, it produces an m−1 asymptotic behavior. This conclusion leads to a variety of consequences, including an inherent requirement for the inclusion of dissipation. We will be reporting the corrections and their implications elsewhere, jointly and seperately. The need for correction of AJ does nothing to undermine my statements made above, relating the eikonal approach of E97 to the rigorous approach of AJ.


Allen, K. R., and R. I. Joseph, 1989: A canonical statistical theory of oceanic internal waves. J. Fluid Mech., 204, 185–228.
Chunchuzov, I. P., 1996: The spectrum of high-frequency internal waves in the atmospheric waveguide. J. Atmos. Sci., 53, 1798–1814.
Eckermann, S. D., 1997: Influence of wave propagation on the Doppler spreading of atmospheric gravity waves. J. Atmos. Sci., 54, 2554–2573.
Hines, C. O., 1991: The saturation of gravity waves in the middle atmosphere. Part II: Development of Doppler-spread theory. J. Atmos. Sci., 48, 1360–1379.
——, 1993: The saturation of gravity waves in the middle atmosphere. Part IV: Cutoff of the incident wave spectrum. J. Atmos. Sci., 50, 3045–3060.
——, 1997a: Doppler-spread parameterization of gravity-wave momentum deposition in the middle atmosphere. Part 1: Basic formulation. J. Atmos. Solar-Terr. Phys., 59, 371–386.
——, 1997b: Doppler-spread parameterization of gravity-wave momentum deposition in the middle atmosphere. Part 2: Broad spectra and quasi monochromatic waves, and implementation. J. Atmos. Solar-Terr. Phys., 59, 387–400.
Smith, S. A., D. C. Fritts, and T. E. VanZandt, 1987: Evidence for a saturated spectrum of atmospheric gravity waves. J. Atmos. Sci., 44, 1404–1410.


Corresponding author address: C. O. Hines, 15 Henry Street, Toronto, ON M5T 1W9, Canada.