In this first of a three-part series, we argue that the dynamics of turbulence in a stratified atmosphere should depend on the buoyancy over a wide range of vertical scales and on energy flux over a wide range of horizontal scales; it should be scaling, but anisotropic, not isotropic. We compare the leading statistical theories of atmospheric stratification which are conveniently distinguished by the elliptical dimension D(s) which quantifies their degree of spatial stratification. This includes the mainstream isotropic 2-D (large scales), isotropic 3-D (small scales) theory but also the more recent linear gravity wave theories (D(s) = 7/3) and the classical fractionally integrated flux (FIF) 23/9-D unified scaling model. In the latter, the horizontal wind has a k(-5/3) spectrum as a function of horizontal wavenumber determined by the energy flux and a k(-11/5) energy spectrum as a function of vertical wavenumber determined by the buoyancy force variance flux. In this model, the physically important notion of scale is determined by the turbulent dynamics, it is not given a priori (i.e. the by usual Euclidean distance). The 23/9-D FIF model is the most physically and empirically satisfying, being based on turbulent (spectral) fluxes. The FIF model as originally proposed by Schertzer and Lovejoy is actually a vast family of scaling models broadly compatible with turbulent phenomenology and with the classical turbulent laws of Kolmogorov, Corrsin and Obukov. However, until now it has mostly been developed on the basis of structures localized in space - time. In this paper, we show how to construct extreme FIF models with wave-like structures which are localized in space but unlocalized in space - time, as well as a continuous family of intermediate models which are akin to Lumley - Shur models in which some part of the localized turbulent energy 'leaks' into unlocalized waves. The key point is that the FIF requires two propagators (space - time Green's functions) which can be somewhat different. The first determines the space - time structure of the cascade of fluxes; this must be localized in space - time in order to satisfy the usual turbulence phenomenology. In contrast, the second propagator relates the turbulent fluxes to the observables; although the spatial part of the propagator is localized as before, in space - time it can be unlocalized. (It is still localized in space, now in wave packets.) We display numerical simulations which demonstrate the requisite (anisotropic, multifractal) statistical properties as well as wave-like phenomenologies. In parts II and III we will examine the empirical evidence for the spatial and temporal parts, respectively, of the model using state-of-the-art lidar data of aerosol backscatter ratios (which we use as a surrogate for passive scalar concentration). Copyright (c) 2008 Royal Meteorological Society.