Handout (1.7 MB)

The extension of 23/9D theory from space to space-time predicts Dst=23/9+2/3=29/9. In this paper, we use lidar data to test this prediction in arbitrary directions in (z,t) or (x,z) space, and in order to get more complete information about the underlying physical scale, we developed and applied a new Anisotropic Scaling Analysis Technique (ASAT) which is based on a nonlinear coordinate transformation. This transforms the original differential scaling into standard self-similar scaling; there remains only a trivial anisotropy. This method was used in real space on 2D structure functions as well as in fourier space on spectral densities. It was applied to both (z,t) and (x,z) data. Using the ASAT technique we verified the theory to within about 10% over more than 3 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 elliptical dimension characterizing the overall space-time stratification equal to =3.21±0.05 which is close to the theoretical value Dst =1+1+5/9+2/3=29/9=3.22 corresponding (in conditions with no mean wind) to , , scaling in space and scaling in time.

Finally, we show how these scalings can be used to produce highly realistic multifractal simulations of clouds, including turbulence-wave phenomenologies.

References: Lilley, M., et al. (2004), 23/9 dimensional anisotropic scaling of passive admixtures using lidar aerosol data, Phys. Rev. E, 70, 036307-036301-036307.