The elliptical dimension of space-time atmospheric stratification of passive admixtures using lidar data

State-of-the-art airborne lidar data of passive scalars have shown that the spatial stratification of the atmosphere is scaling: the vertical extent (Δz) of structures is typically ≈ΔxHz where Δx is the horizontal extent and Hz is a stratification exponent. Assuming horizontal isotropy, the volumes of the structures therefore vary as ΔxΔxΔxHz=ΔxDs where the “elliptical dimension” Ds characterizes the rate at which the volumes of typical non-intermittent structures vary with scale. Work on vertical cross-sections has shown that 2+Hz=2.55±0.02 (close to the theoretical prediction 23/9). In this paper we extend these (x, z) analyses to (z, t). In the absence of overall advection, the lifetime Δt of a structure of size Δx varies as ΔxHt with Ht=2/3 so that the overall space-time dimension is Dst=29/9=3.22…. However, horizontal and vertical advection lead to new exponents: we argue that the temporal stratification exponent Ht≈1 or ≈0.7 depending on the relative importance of horizontal versus vertical advection velocities. We empirically test these space-time predictions using vertical-time (z, t) cross-sections using passive scalar surrogates (aerosol backscatter ratios from lidar) at 3 m resolution in the vertical, 0.5–30 s in time and spanning 3–4 orders of magnitude in scale as well as new analyses of vertical (x, z) cross-sections (spanning over 3 orders of magnitude in both x, z directions). In order to test the theory for density fluctuations at arbitrary displacements in (Δz, Δt) and (Δx, Δz) spaces, we developed and applied a new Anisotropic Scaling Analysis Technique (ASAT) based on nonlinear coordinate transformations. Applying this and other analyses to data spanning more than 3 orders of magnitude of space-time scales we determined the anisotropic scaling of space-time finding the empirical value Dst=3.13±0.16. The analyses also show that both cirrus clouds and aerosols had very similar space-time scaling properties. We point out that this model is compatible with (nonlinear) “turbulence” waves, hence potentially explaining the observed atmospheric structures.