Scaling turbulent atmospheric stratification. III: Space–time stratification of passive scalars from lidar data
Authors: Radkevich A. , Lovejoy S. , Strawbridge K. B. , Schertzer D. , Lilley M. 
Published: March 2008

AbstractIn this third and final part of the series, we concentrate on the temporal behaviour of atmospheric passive scalars. We first recall that—although the full (x, y, z, t) turbulent processes respect an anisotropic scale invariance—that due to advection—the generator will generally not be a diagonal matrix. This implies that the scaling of (1‐D) temporal series will generally involve three exponents in real space: 1/3, 1/2, 3/5, for spectra βτ = 5/3, 2, 11/5, with the first and last corresponding to domination by advection (horizontal and vertical respectively), and the second to pure temporal development (no advection). We survey the literature and find that almost all the empirical βτ values are indeed in the range 5/3 to 2. We then use meteorological analyses to argue that, although pure temporal development is unlikely to be dominant for time‐scales less than the eddy turnover time of the largest structures (about 2 weeks), an intermittent vertical velocity could quite easily explain the occasionally observed βτ ≈ 2 spectra.We then use state‐of‐the‐art vertically pointing lidar data of backscatter ratios from both aerosols and cirrus clouds yielding several (z, t) vertical space–time cross‐sections with resolution of 3.75 m in the vertical, 0.5–30 s in time and spanning 3–4 orders of magnitude in temporal scale. We first test the predictions of the anisotropic, multifractal extension of the Corrsin‐Obukhov law in the vertical and in time, separately finding that the cirrus and aerosol backscatters both followed the theoretical (anisotropic) scalings accurately; three of the six cases show dominance by the horizontal wind, the others by the vertical wind. In order to test the theory in arbitrary directions in this (z, t) space, and in order to get more complete information about the underlying physical scale, we develop and apply a new Anisotropic Scaling Analysis Technique (ASAT) which is based on a nonlinear space–time coordinate transformation. This transforms the original differential scaling into standard self‐similar scaling; there remains only a ‘trivial’ anisotropy. This method is used in real space on 2‐D structure functions. It is applied to both the new (z, t) data as well as the (x, z) data discussed in part II. Using ASAT, we verify the theory to within about 10% over more than three orders of magnitude of space–time scales in arbitrary directions in (x, z) and (z, t) spaces. By considering the high‐ (and low‐) order structure functions, we verify the theory for both weak and strong structures; as predicted, their average anisotropies are apparently the same.Putting together the results for (x, z) and (z, t), and assuming that there is no overall stratification in the horizontal (x, y) plane, we find that the overall (x, y, z, t) space is found to have an effective ‘elliptical dimension’ characterizing the overall space–time stratification equal to Deff, st = 3.21 ± 0.05. Copyright © 2008 Royal Meteorological Society