Beyond Quasi-Geostrophic Turbulence: Generalized Scale Invariance and (2+Hz)-Dimensional Vorticity Equations

We discuss the claim of Lindborg et al (2009) that the 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 by Lovejoy et al (2009), because this spectrum corresponds to a “well known theory of quasi-geostrophic turbulence developed by Charney (1971)”. Lindborg, et al (2009) also argued that “earlier limitations [of this theory] would have been relaxed in many of the modern models of atmospheric turbulence”. We show that both these statements are irrelevant and that generalized scale invariance (GSI, Schertzer and Lovejoy 1985) is rather indispensable to go beyond the quasi-geostrophic limitations, to go in fact from scale analysis to scaling analysis. This enables us to derive dynamical equations for the vorticity in an embedding space of (fractional) dimension D=2+Hz (0≤ Hz ≤1, 1- Hz measures the scaling stratification of atmospheric turbulence). These equations correspond to an interesting dynamical alternative to quasi-geostrophic approximation and turbulence. References: Charney, J. G. (1971). "Geostrophic Turbulence." J. Atmos. Sci 28: 1087. Lindborg, E., K. K. Tung, G. D. Nastrom, J. Y. N. Cho and K. S. Gage (2009). "Comment on "Reinterpreting aircraft measurements in anisotropic scaling turbulence" by lovejoy et al. (2009)." Atmos. Chem. Phys. Discuss. 9: 22331-22336. Lovejoy, S., A. F. Tuck, D. Schertzer and S. J. Hovde (2009). "Rinterpreting aircraft measurements in anisotropic scaling turbulence." Atmos. Chem. Phys. 9: 5007-5025. Schertzer, D. and S. Lovejoy (1985). "Generalised scale invariance in turbulent phenomena." Physico-Chemical Hydrodynamics Journal 6: 623-635.