Chapter 6: Vertical stratification and anisotropic scaling 1 Models of vertical stratification: local isotropy, trivial and scaling anisotropy




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10/17/12 Chapter 6

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Chapter 6: Vertical stratification and anisotropic scaling

6.1 Models of vertical stratification: local isotropy, trivial and scaling anisotropy

6.1.1 The isotropy assumption, historical overview


We have argued that atmospheric scaling holds over a wide range in the horizontal, this is clearly not possible if the turbulence is isotropic since it would imply the existence of roundish structures hundreds or even thousands of kilometres thick. Such wide range scaling is only possible because of the stratification. Following a brief historical overview, in this chapter we discuss the nature of the stratification.

Initially motivated by mathematical convenience, starting with (Taylor, 1935), the paradigm of isotropic and scaling turbulence was developed initially for laboratory applications, but following (Kolmogorov, 1941), three dimensional isotropic turbulence was progressively applied to the atmosphere. However, there are several features (including gravity, the Coriolis force and stratification) that bring into question the simultaneous relevance of both isotropy and scaling. In particular since the atmosphere is strongly stratified, a model with a single wide range of scaling which is both isotropic and scaling is not possible so that theorists had to immediately choose between the two symmetries: isotropy or scale invariance. Following the development of models of two dimensional isotropic turbulence ((Fjortoft, 1953) and (Kraichnan, 1967) but especially quasi geostrophic turbulence (Charney, 1971) which can be seen as quasi-2D turbulence (see section 2.6), the mainstream choice was to first make the convenient assumption of isotropy and to drop wide range scale invariance; this could be called the “isotropy primary” paradigm. In ch. 2 we saw how - starting at the end of the 1970’s, this has lead to a series of increasingly complex isotropic 2D/isotropic 3D models of atmospheric dynamics and we noted that justifications for these approaches have focused almost exclusively on the horizontal statistics of the horizontal wind in both numerical models and analyses and from aircraft campaigns, especially the highly cited GASP (Nastrom and Gage, 1983), (Gage and Nastrom, 1986; Nastrom and Gage, 1985) and MOZAIC (Cho and Lindborg, 2001) experiments. Since understanding the anisotropy clearly requires comparisons between horizontal and vertical statistics/structures it is not surprising that this focus has had deleterious consequences.

Over the same thirty year period that 2D/3D isotropic models were being elaborated, evidence slowly accumulated in favour of the opposite theoretical choice: to drop the isotropy assumption but to retain wide range scaling. This change of paradigm from isotropy primary to scaling primary was explicitly proposed by (Schertzer and Lovejoy, 1985), and (Schertzer and Lovejoy, 1987) who considered strongly anisotropic scaling so that vertical sections of structures become increasingly stratified at larger and larger scales albeit in a power law manner. Similarly, although not turbulent, anisotropic wave spectra were proposed in the ocean by (Garrett and Munk, 1972) and an anisotropic buoyancy driven wave spectrum was proposed in the atmosphere by ((Van Zandt, 1982). Related anisotropic wave approaches may be found in (Dewan and Good, 1986; Fritts et al., 1988), (Tsuda et al., 1989), (Gardner et al., 1993), (Hostetler and Gardner, 1994) and (Dewan, 1997). These authors used anisotropic scaling models not so much for theoretical reasons, but rather because the data could not be explained without it. In addition, many experiments found non-standard vertical scaling exponents thus implicitly supporting this position, see the review (and many additional references) in (Lilley et al., 2008). Below we shall see that state-of-the-art lidar vertical sections of passive scalars or satellite vertical radar sections of clouds (section 6.5) give direct evidence for the corresponding scaling (power law) stratification of structures. These analyses show that the standard bearer for isotropic models - 3D isotropic Kolmogorov turbulence - apparently doesn’t exist in the atmosphere at any scale - at least down to 5 m in scale - or at any altitude level within the troposphere (Lovejoy et al., 2007), section 6.1.5. In ch. 1 and 4 we used large quantities of high quality satellite data to directly demonstrate the wide range horizontal scaling of the atmospheric forcing (long and short wave radiances) and showed that reanalyses and atmospheric models display nearly perfect scaling cascade structures over the entire available horizontal ranges. This shows also that the source/sink free “inertial ranges” used in the classical models are at best academic idealizations and at worst unphysical: atmospheric dynamics has multiple energy sources that do not prevent it from being scaling.

At a theoretical level, quasi-geostrophic and similar systems of equations purporting to approximate the synoptic scale structure of the atmosphere are justified by “scale” analysis of various terms in the dynamical equations in which “typical” large scale values of atmospheric variables (usually their fluctuations) are assumed and terms eliminated if they are deemed too small. However, in section 2.3.2 we showed that by replacing such scale analyses – valid at most over narrow ranges of scales – by scaling analysis, focusing on horizontal and vertical exponents valid over wide ranges, that the equations were symmetric with respect to anisotropic scale changes. However, just as the classical isotropic scaling analysis of the Navier-Stokes equations is insufficient to determine the Kolmogorov exponent 1/3 – for this we also need dimensional analysis on the scale by scale conserved energy flux – so here, the value of the new vertical exponents depend on a new turbulent flux that is the subject of this section.
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