## 1. Introduction: Analyses, Reanalyses, Fluxes, and Fields

[2] The advent of high quality global scale data sets has finally made it possible to systematically study the space-time statistical properties of the atmosphere as a function of scale. Yet, all empirical analyses suffer from limitations. For example, data from global scale networks are invariably sparse [*Lovejoy et al.*, 1986] and aircraft data are not only limited to 1-D transects, but have nontrivial problems of interpretation due to trajectories which are both fractal and sloping [*Lovejoy et al.*, 2004, 2009c]. Although satellite data are nearly ideal in terms of their coverage from global scales down to kilometric (or less), they measure radiances rather than variables of state and their scale by scale statistical properties have received little attention.

[3] The alternative source of global scale data studied in this paper is the reanalyses. These are hybrid products obtained by using much of the available data from both in situ networks and satellites but which are processed with the help of complex data assimilation algorithms adapted for specific meteorological models. These algorithms use covariance matrices to interpolate the data onto uniform grids. The 4D VAR assimilation scheme relevant here implies that the European Centre for Medium-Range Weather Forecasts (ECMWF) interim products are solutions to the primitive equations but at the relatively low resolution of the grid: there are implicit smoothness and regularity assumptions. Hydrostaticity may also be an issue since the use of (gently sloping) isobars rather than isoheights can potentially lead to scale breaks and the spurious appearance of vertical exponents (see the discussion of isobaric aircraft statistics below). Nevertheless, at least in some cases, such as the hard to measure vertical winds, they provide the only global scale estimates; in any case they complement the “pure” instrumental data sets.

[4] A series of papers involving lidar, drop sonde, aircraft, and satellite data [see *Lovejoy and Schertzer*, 2010, 2011a, and references therein] presents a large body of empirical evidence that the atmosphere is accurately scale invariant from planetary scales continuing down through the mesoscale, continuing possibly to the dissipation scale. As argued by *Schertzer and Lovejoy* [1984, 1985a], such a wide scaling range is only possible if one considers generalized (anisotropic) notions of scaling; here it is sufficient to consider different scalings in different (e.g., orthogonal) directions. This means that while fluctuations Δ*f* in a turbulent quantity *f* (e.g., a component of the wind) are scaling in the horizontal (lag Δ*x*) with exponent then fluctuations in the vertical (lags Δ*z*) are scaling with φ_{h} and φ_{h} are physically different turbulent fluxes. Note that these are fluxes through spherical shells in Fourier space, not fluxes in physical space; this is different from the classical interpretation of the quantities in terms of dissipation; the fluxes are only equal to the dissipation at the small dissipation scales. Study of the fluxes averaged/degraded in resolution show [*Lovejoy et al.*, 2009a; *Lovejoy and Schertzer*, 2010] that up until 5,000–10,000 km their statistics are nearly exactly as predicted by multiplicative cascade models (section 2, equation (1)) and that their exponent functions are nearly the same in the zonal and meridional directions. A model for the velocity field which turns out to be close to the observations takes φ_{h} = ɛ^{1/3}, φ_{v} = φ^{1/5}, *H*_{h} = 1/3 and *H*_{v} = 3/5, where ɛ is the energy and φ is the buoyancy variance flux which thus dominate the dynamics, respectively, along the horizontal and vertical directions [*Schertzer and Lovejoy*, 1985b] (these correspond to the classical Kolmogorov [*Kolmogorov*, 1941] and Bolgiano-Obukhov [*Bolgiano*, 1959; *Obukhov*, 1959] exponents, but in a nonclassical anisotropic framework). As a consequence, *H*_{v} > *H*_{h} for the velocity field and it seems to be a rather general property that corresponds to the fact that structures (defined as *f* isolines) become progressively flatter and flatter at larger and larger scales, yet this stratification occurs without any characteristic size being introduced.

[5] While this horizontal/vertical anisotropy is fundamental in understanding the existence of the wide range horizontal scaling, we do not consider this vertical stratification here. Surprisingly however, it turns out that scaling anisotropy seems to be essential in understanding the horizontal (or more accurately, the isobaric) statistics of the reanalysis fields, although apparently not of the fluxes: i.e., we find *H*_{EW}/*H*_{NS} = *H*_{y} ≈ 0.80 for all fields (with a predicted inversion for the meridional wind; “NS” and “EW” mean “north-south” and “east-west,” respectively) whereas φ_{NS} and φ_{EW} not only have very nearly the same statistics (horizontal isotropy), but also these are very nearly those theoretically predicted for multiplicative cascade processes. While the latter result confirms those made on other reanalyses and products [*Stolle et al.*, 2009], the former is new and helps shed light on the problems encountered by the classical approaches to statistically characterizing reanalyses.

[6] To see this, recall that while the classical approaches [*Boer and Shepherd*, 1983; *Strauss and Ditlevsen*, 1999] suppose a priori the physical nature of the relevant fluxes (e.g., energy, enstrophy flux), our method has the advantage of objectively defining the fluxes using the observed fluctuations. (A disadvantage of such objectively defined fluxes is that the physical meaning of the flux is not obvious: it is a subject of future research.) However, as we review in the next section, it turns out that up until now, understanding the statistical properties of the fields *f* (such as their spectra) has not given any clear result. We argue that that the reason is that the reanalyses are anisotropic in the horizontal with different exponents, i.e., *H*_{EW} ≠ *H*_{NS} (the zonal and meridional directions, respectively) and up until now only isotropic reanalysis spectra (i.e., integrated over all angles) have been considered. Whether, as we suspect, this scaling anisotropy is a spurious artifact of the reanalyses; its discovery will help settle longstanding debates about the statistical nature of the atmosphere (i.e., the classical 2D versus 3D isotropic turbulence model versus the scaling 23/9 D alternative [*Schertzer and Lovejoy*, 1985b]; see *Lovejoy and Schertzer* [2010] for a review).

[7] A final important issue we cover is the nature of the temporal variability and the relation between the spatial and temporal statistics. Since structures in the atmosphere have “lifetimes” which depend on their spatial scales; the two being connected (at least dimensionally) via a velocity, we expect that the spatial scaling of the wind will impose a regime of temporal scaling, at least up to scales corresponding the to the size of the planet; i.e., roughly 5–10 days. Indeed, virtually all meteorological fields have a drastic change in behavior at about this scale; the transition itself has been termed a “dimensional transition” [*Lovejoy and Schertzer*, 2010] and the lower frequency regime where the spectrum is significantly flatter has been called a “spectral plateau” [*Lovejoy and Schertzer*, 1986]. While the fluctuations for high frequencies are dominated by single structures and corresponding lifetimes, the fluctuations for the lower plateau frequencies are consequences of many lifetimes and have much lower statistical interdependences (lower *H*'s, lower spectral exponents) and these are accurately modeled by assuming that the space-time scaling weather cascade model continues to much lower frequencies [*Lovejoy and Schertzer*, 2010]. Interestingly, *Lovejoy and Schertzer* [2011b, 2011c] have given evidence for analogous behavior in the oceans, although with a critical time scale of ∼1 year, so that due to ocean-atmosphere interactions, the behavior in the range of scales ≈ 10 days–1 year is a bit more complex.

[8] This paper is organized as follows. In section 2 we review the basic theory and data analysis techniques. In section 3, we perform systematic cascade analyses on the turbulent fluxes in the two horizontal directions and in time, we examine the statistical relation between space and time (“Stommel” diagrams), their latitudinal dependence, and the “Levy collapse” which is a test of the underlying probability distributions of the fluxes irrespective of the scaling. In section 4 we consider the fields (the observables) taking care to consider their anisotropies. We outline the necessary elements of generalized scale invariance and estimate zonal, meridional and isotropic spectral exponents; we then consider the temporal spectra and the latitudinal dependence of all the exponents. In section 5, we conclude. In Appendix A, we discuss possible biases in our anisotropy analyses due to the use of cylindrical map projections.