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The interaction between small-scale waves and a larger-scale flow can bedescribed by a multi-scale theory that forms the basis for a new class ofparameterizations of subgrid-scale gravity waves (GW) in weather and climatemodels. The development of this theory is reviewed here. It applies to allinteresting regimes of atmospheric stratification, i.e. also to moderatelystrong stratification as occurring in the middle atmosphere, and therebyextends classic assumptions for the derivation of quasi-geostrophic theory. Atstrong wave amplitudes a fully nonlinear theory arises that is complemented bya quasilinear theory for weak GW amplitudes. The latter allows the extension toa spectral description that forms the basis of numerical implementations thatavoid instabilities due to caustics, e.g. from GW reflection. Conservationproperties are discussed, for energy and potential vorticity, as well asconditions under which a GW impact on the larger-scale flow is possible. Thenumerical implementation of the theory for GW parameterizations in atmosphericmodels is described, and the consequences of the approach are discussed, ascompared to classic GW parameterizations. Although more costly than the latter,it exhibits significantly enhanced realism, while being considerably moreefficient than an approach where all relevant GWs are to be resolved. Thereported theory and its implementation might be of interest also for theefficient and conceptually insightful description of other wave-meaninteractions, including those where the formation of caustics presents aspecial challenge.

Multi-Scale Dynamics of the Interaction Between Waves and Mean Flows:  From Nonlinear WKB Theory to Gravity-Wave Parameterizations in Weather and  Climate Models