The interaction between small-scale waves and a larger-scale flow can be described by a multi-scale theory that forms the basis for a new class of parameterizations of subgrid-scale gravity waves (GW) in weather and climate models. The development of this theory is reviewed here. It applies to all interesting regimes of atmospheric stratification, i.e., also to moderately strong stratification as occurring in the middle atmosphere, and thereby extends classic assumption for the derivation of quasi-geostrophic theory. At strong wave amplitudes a fully nonlinear theory arises that is complemented by a quasilinear theory for weak GW amplitude. The latter allows the extension to a spectral description that forms the basis of numerical implementations that avoid instabilities due to caustics, e.g., from GW reflection. Conservation properties are discussed, for energy and potential vorticity, as well as conditions under which a GW impact on the larger-scale flow is possible. The numerical implementation of the theory for GW parameterizations in atmospheric models is described, and the consequences of the approach are discussed, as compared to classic GW parameterizations. Although more costly than the latter, it exhibits significantly enhanced realism, while being considerably more efficient than an approach where all relevant GWs are to be resolved. The reported theory and its implementation might be of interest also for the efficient and conceptually insightful description of other wave-mean interactions, including those where the formation of caustics presents a special challenge.
With horizontal and vertical wavelengths down to at most 1 km and 100 m respectively, mesoscale atmospheric waves such as internal gravity waves (GWs) will not all be simulated explicitly by operational climate models within the foreseeable future. However, without taking their influence into account, climate models miss essential circulation aspects even on the planetary scale.1–3 Hence they must be parameterized, requiring a solid theory for the interaction between the waves and a mean flow. Corresponding studies have led in recent years to the emergence of a new class of GW parameterizations (GWP). The present review is to give an overview on these developments, from theoretical investigations to the implementation of a new GWP into a state-of-the-art climate model. For this purpose we first review the dynamics of large-amplitude, locally monochromatic GWs, then discuss spectra of weak-amplitude GWs, next describe the GW impact in general on the mean flow, touch on conservation properties, and finally give a sketch of the hence resulting numerical developments.
II. LARGE-AMPLITUDE LOCALLY MONOCHROMATIC WAVES
III. A SPECTRUM OF WEAK-AMPLITUDE WAVES
In the weakly nonlinear case n = 1 the wave-action equation is to be supplemented by the effect of wave-wave interactions, of either GWs with GWs or GWs with GMs. To the best of our knowledge, the corresponding scattering integrals have only been worked out fully for Boussinesq dynamics without mean flow,14,15 and first steps for Boussinesq dynamics with non-vanishing mean flows have been taken by Ref. 16. In the atmospheric context they have so far been simply ignored.
IV. WAVE IMPACT ON THE BALANCED MEAN FLOW
V. SUMMARY OF THE RESULTS IN DIMENSIONAL FORM
VI. CONSERVATION PROPERTIES
Neither GW energy is conserved nor is QGPV. The interaction between GWs and synoptic-scale mean flow leads to an exchange between the two so that the corresponding conserved quantity comprises contributions from both components.
B. Potential vorticity
GW amplitudes and wavenumbers are steady,
GW pseudovorticity does not vary horizontally, and
the GWs are not affected by sources or sinks.
Classic GW parameterizations assume steady-state GW fields and they do not take horizontal variations of the GWs fields into account. Hence they rely exclusively on GW sources and GW breaking as processes leading to a GW impact on the resolved flow.
VII. CONSEQUENCES FOR GW PARAMETERIZATIONS
A. Numerical implementation
GW dissipation is simulated by a classic saturation approach based on Ref. 20 and adapted by Refs. 18 and 21 to spectral wave-action dynamics. A static-instability breaking threshold, with a tuning factor close to 1, is determined by the criterion that, within a resolved-flow volume cell, the constructive interference of all ray volumes can lead to a total GW signal with a negative vertical derivative in potential temperature larger than the positive derivative given by the stratification of the resolved flow. Once this threshold is exceeded turbulence is invoked, causing turbulent viscosity and diffusivity that provide a dissipative right-hand side to the spectral wave-action equation (72) so that the GW field is kept at the threshold of static instability.
The interaction between the parameterized GWs and the resolved large-scale flow takes two directions. The contribution of the large-scale winds ⟨u⟩ to the group velocity cg and to the wavenumber velocity influences the development of the wave-action density. In the other direction, GWs influence the resolved flow in its thermodynamics via the divergence of the GW entropy flux (77). They also change its momentum via the divergence of GW momentum flux (78)–(82), and via the elastic term, i.e., the last term in (76). That term can be obtained from the GW entropy-flux convergence as well. The wavenumber integrals in the fluxes are presently estimated to first-order accuracy, by evaluating the integrand at the center of the ray volume and multiplying it by its wavenumber volume. So far implementations have been into finite-volume dynamical model cores for resolved-flow dynamics. In this context the fluxes are then projected onto the finite-volume cell faces and finally used there.
It would seem attractive to avoid the projection of the fluxes from the Lagrangian GW model onto the resolved-flow finite volume cells, and also the interpolation of the resolved-flow winds to the location of each ray volume, by a straightforward finite-volume implementation of the spectral wave-action equation (71) in flux form. This alternative has been studied by Ref. 18. It turns out that the Lagrangian approach is computationally much more efficient. Two factors contribute to this: Firstly, in a finite-volume approach, one must span a six-dimensional phase space, with often a substantial fraction of cells not contributing essentially to the GW fluxes. Secondly, GW refraction and reflection are only captured well in the finite-volume approach if the resolution in wavenumber space is excessively high.
B. Comparison with classic GWP
As follows from the non-acceleration result discussed in Subsection VI C the classic approach using the single-column and steady-state approximations, the first neglecting GW horizontal propagation and horizontal GW fluxes, and the second neglecting GW transience, relies exclusively on GW dissipation as a process allowing a GW impact on the resolved flow. The relevance of GW transience has been investigated by Refs. 18, 19, and 21. As shown by Ref. 18, the resolved-flow impact induced by the breaking of a single GW packet is often described quite incorrectly if GW transience is not taken into account. The consequence of this for GW parameterization in a global climate model has been studied by Refs. 19 and 21. The result is that the statistics of simulated GW fluxes, exhibiting in measurements distributions with long tails of strong fluxes, are affected significantly by a steady-state assumption, to the point that the distributions are distorted significantly away from the observational findings.
The consequences of the single-column approximation have been investigated in the mean time as well (Voelker et al. and Kim et al., both to be submitted elsewhere). As with regard to the neglect of wave transience, they appear to be of leading order as well. The horizontal distribution of GW fluxes in the middle atmosphere (above 15 km altitude) is very different between the single-column and the general parameterization approach so that also the large-scale winds differ significantly. In the tropics this leads to a significantly changed variability, with the quasi-biennial oscillation (e.g., Ref. 23) differing in its period conspicuously between simulations using the two approaches. Hence it appears that the general approach is considerably more realistic.
VIII. FINAL DISCUSSION
The multi-scale theory outlined above has led to the development of an extended approach, in weather and climate models, for the parameterization of subgrid-scale GWs. As is clear by now it is considerably more realistic than classic GW parameterizations, where the GW impact has been simplified by focusing on GW pseudomomentum, where GW transience is neglected, and where the effects of horizontal GW propagation and of horizontal GW fluxes are neglected. This comes at a computational prize: Simulations with the new approach are slower, by about an order of magnitude at global-model resolutions tested so far, than those using a classic GW parameterization. Yet, they are more realistic, and if one wanted to resolve the GWs that turn out to matter, simulations would be and are (e.g., Refs. 24 and 25) by many orders of magnitude slower than even this (Ref. 21). Hence the approach discussed here seems to be a reasonable compromise between the realism of GW permitting simulations, certainly of their own value as a data source for many studies, and the efficiency of climate simulations using classic GW parameterizations. Moreover, in view of the increasing complexity of ever higher resolved weather and climate models26 there is an ever-increasing need for a hierarchy of models that help us gain conceptual insight into the complex atmosphere.27 In this hierarchy the more general approach for GW parameterizations should have its place.
Open issues remain, both on the theoretical and on the applied side. One aspect that seems to deserve consideration is the interaction between GWs and the geostrophic vortical mode that is the other constituting component of mesoscale atmospheric dynamics (e.g., Refs. 9 and 12). Present approaches for the description of this interaction, using tools of wave-turbulence theory,15 do not take the presence of a leading-order mean flow into account. In the ocean context this is possible, but in the atmosphere mean winds enter to leading order. This also holds for another aspect of relevance, the so-far neglect of GW–GW interactions. It is not clear how relevant this process will be in the end, and to the best of our knowledge this has not been investigated yet. Here as well available theories14 suffer from the neglect of a mean flow. It would be interesting to obtain a corresponding extension of the theory, e.g., following the route indicated by Ref. 16. Such work would close a conceptual gap that we still seem to have in the weakly nonlinear regime between the quasilinear regime (allowing a spectral approach) and the large-amplitude regime (where only locally monochromatic GW fields are possible). Finally, the handling of GW breaking by the saturation approach is very crude and a theory encompassing GW-turbulence interaction still needs to be derived from the basic equations. Similar considerations hold for the emission of GWs by various processes, be it flow over orography, convection, or emission by jets and fronts, to just name the most often discussed candidates. In all of these instances closed mathematical descriptions are still to be derived from multi-scale approaches, that hopefully would be able to replace present schemes. Finally, we think that in the numerical implementation of the general approach the last word has not been spoken. It would be helpful if experts in numerical mathematics gave it a closer look, especially with regard to accuracy and efficiency.
Finally we also want to point out that it might be of interest to consider and extend the tools and techniques outlined here for other challenging problems of wave-mean-flow-interaction theory. The quasi-biennial oscillation of the equatorial zonal-mean zonal winds,23 e.g., is not only due to GWs but also to larger-scale tropical Kelvin and Rossby–Gravity waves. Moreover, Kelvin waves also interact with GWs in the tropics.28 The interaction between mesoscale GWs, larger-scale tropical waves and the zonal-mean flow could be an interesting problem to be studied using these methods. Another field of application could be caustics in general, e.g., in nonlinear acoustics,29 quantum mechanics,30 general relativity,31 or plasma physics,32 where the spectral approach outlined here could help the development of closed and comparatively simple treatments.
U.A. thanks the German Research Foundation (DFG) for partial support through the research unit “Multiscale Dynamics of Gravity Waves” (MS-GWaves, Grant Nos. AC 71/8-2, AC 71/9-2, and AC 71/12-2) and CRC 301 “TPChange” (Project No. 428312742, Projects B06 “Impact of small-scale dynamics on UTLS transport and mixing” and B07 “Impact of cirrus clouds on tropopause structure”). Y.-H.K. and U.A. thank the German Federal Ministry of Education and Research (BMBF) for partial support through the program Role of the Middle Atmosphere in Climate (ROMIC II: QUBICC) and through Grant No. 01LG1905B. U.A. and G.S.V. thank the German Research Foundation (DFG) for partial support through the CRC 181 “Energy transfers in Atmosphere an Ocean” (Project No. 274762653, Projects W01 “Gravity-wave parameterization for the atmosphere” and S02 “Improved Parameterizations and Numerics in Climate Models.”) U.A. is furthermore grateful for support by Eric and Wendy Schmidt through the Schmidt Futures VESRI “DataWave” project.
Conflict of Interest
The authors have no conflicts to disclose.
U. Achatz: Conceptualization (equal); Funding acquisition (equal); Writing – original draft (equal); Writing – review & editing (equal). Y.-H. Kim: Writing – review & editing (equal). G. S. Voelker: Writing – review & editing (equal).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.