Quasi-geostrophic turbulence and generalized scale invariance, a theoretical reply 1Université Paris-Est, Ecole des Ponts ParisTech, LEESU, Marne-la-Vallée, France
05 Jan 2012
2McGill U., Physics dept, Montreal, Canada
Received: 23 November 2010 – Published in Atmos. Chem. Phys. Discuss.: 31 January 2011 Abstract. Lindborg et al. (2010) claim that the apparent spectrum power law E(k) ≈
k−3 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 D=2+Hz (0 ≤ Hz ≤ 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 + Hz)-D turbulence (with 0 < Hz < 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.
Revised: 21 December 2011 – Accepted: 21 December 2011 – Published: 05 January 2012
Citation: Schertzer, D., Tchiguirinskaia, I., Lovejoy, S., and Tuck, A. F.: Quasi-geostrophic turbulence and generalized scale invariance, a theoretical reply, Atmos. Chem. Phys., 12, 327-336, doi:10.5194/acp-12-327-2012, 2012.