ACP Atmospheric Chemistry and Physics ACP Atmos. Chem. Phys. 1680-7324 Copernicus Publications Göttingen, Germany 10.5194/acp-12-327-2012 Quasi-geostrophic turbulence and generalized scale invariance, a theoretical reply Schertzer D. 1 Tchiguirinskaia I. 1 Lovejoy S. 2 Tuck A. F. 1 Université Paris-Est, Ecole des Ponts ParisTech, LEESU, Marne-la-Vallée, France McGill U., Physics dept, Montreal, Canada 05 01 2012 12 1 327 336 Copyright: © 2012 D. Schertzer et al. 2012 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://acp.copernicus.org/articles/12/327/2012/acp-12-327-2012.html The full text article is available as a PDF file from https://acp.copernicus.org/articles/12/327/2012/acp-12-327-2012.pdf

Lindborg et al. (2010) claim that the apparent spectrum power law <i>E(k)</i> ≈ k<sup>&minus;3</sup> on scales ≥600 km obtained with the help of commercial jetliner trajectory deviations (GASP and Mozaic databases) could not be brought into question (Lovejoy et al., 2009a), because this spectrum corresponds to "a well known theory of quasi-geostrophic turbulence developed by Charney (1971)". Lindborg et al. (2010) also claim that "limitations [of this theory] have been relaxed in many of the modern models of atmospheric turbulence". We show that both claims are irrelevant and that generalized scale invariance (GSI) is indispensable to go beyond the quasi-geostrophic limitations, to go in fact from scale analysis to scaling analysis in order to derive better analytical models. In this direction, we derive vorticity equations in a space of (fractal) dimension <i>D</i>=2+<i>H<sub>z</sub></i> (0 ≤ <i>H<sub>z</sub></i> ≤ 1), which corresponds to a first step in the derivation of a dynamical alternative to the quasi-geostrophic approximation and turbulence. The corresponding precise definition of fractional dimensional turbulence already demonstrates that the classical 2-D and 3-D turbulence are not the main options to understand atmospheric dynamics. Although (2 + <i>H<sub>z</sub></i>)-D turbulence (with 0 < <i>H<sub>z</sub></i> < 1) has more common features with 3-D turbulence than with 2-D turbulence, it has nevertheless very distinctive features: its scaling anisotropy is in agreement with the layered pancake structure, which is typical of rotating and stratified turbulence but not of the classical 3-D turbulence.

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