Generalized Scale Invariance
Generalized Scale Invariance
Probability distributions plotted to date from large volumes of high quality atmospheric observations invariably have ‘long tails’: relatively rare but intense events make significant contributions to the mean. Atmospheric fields are intermittent. Gaussian distributions, which are assumed to accompany second moment statistics and power spectra, are not seen. An inherently stochastic approach, that of statistical multifractals, was developed as generalized scale invariance by Schertzer and Lovejoy (1985, 1987, 1991); it incorporates intermittency and anisotropy in a way Kolmogorov theory does not. Generalized scale invariance demands in application to the atmosphere large volumes of high quality data, obtained in simple and representative coordinate systems in a way that is as extensive as possible in both space and time. In theory, these could be obtained for the whole globe by satellites from orbit, but in practice their high velocities and low spatial resolution have to date restricted them to an insufficient range of scales, particularly if averaging over scale height-like depths in the vertical is to be avoided; analysis has been successfully performed on cloud images, however (Lovejoy et al. 2001). Some suitable data were obtained as an accidental by-product of the systematic exploration of the rapid (1–4% per day) ozone loss in the Antarctic and Arctic lower stratospheric vortices during winter and spring by the high-flying ER-2 research aircraft in the late 1980s through to 2000. Data initially at 1Hz and later at 5Hz allowed horizontal resolution of wind speed, temperature, and pressure at approximately 200 m and later at 40 m, with ozone available at 1 Hz, over the long, direct flight tracks necessitated by the distances involved between the airfield and the vortex. Some later flights also had data from other chemical instruments, such as nitrous oxide, N2O, reactive nitrogen, NOy, and chlorine monoxide, ClO, which could sustain at least an analysis for H1, the most robust of the three scaling exponents. Better than four decades of horizontal scale were available for 1Hz and 5Hz data. Since then, a lesser volume of adequate data has been obtained away from the polar regions by the WB57F.
Keywords: Bolgiano scaling, Coriolis parameter, Global Positioning System (GPS) sondes, Richardson instability, fractal instability, nitrous oxide, velocity fluctuations, vertical scaling exponents, vertical scaling of horizontal wind, water observations
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