ABSTRACT:In this first of a three-part series, we argue that the dynamics of turbulence in a stratified atmosphere should depend on the buoyancy over a wide range of vertical scales and on energy flux over a wide range of horizontal scales; it should be scaling, but anisotropic, not isotropic. We compare the leading statistical theories of atmospheric stratification which are conveniently distinguished by the elliptical dimension D s which quantifies their degree of spatial stratification. This includes the mainstream isotropic 2-D (large scales), isotropic 3-D (small scales) theory but also the more recent linear gravity wave theories (D s = 7/3) and the classical fractionally integrated flux (FIF) 23/9-D unified scaling model. In the latter, the horizontal wind has a k −5/3 spectrum as a function of horizontal wavenumber determined by the energy flux and a k −11/5 energy spectrum as a function of vertical wavenumber determined by the buoyancy force variance flux. In this model, the physically important notion of scale is determined by the turbulent dynamics, it is not given a priori (i.e. the by usual Euclidean distance). The 23/9-D FIF model is the most physically and empirically satisfying, being based on turbulent (spectral) fluxes. The FIF model as originally proposed by Schertzer and Lovejoy is actually a vast family of scaling models broadly compatible with turbulent phenomenology and with the classical turbulent laws of Kolmogorov, Corrsin and Obukov. However, until now it has mostly been developed on the basis of structures localized in space-time. In this paper, we show how to construct extreme FIF models with wave-like structures which are localized in space but unlocalized in space-time, as well as a continuous family of intermediate models which are akin to Lumley-Shur models in which some part of the localized turbulent energy 'leaks' into unlocalized waves.The key point is that the FIF requires two propagators (space-time Green's functions) which can be somewhat different. The first determines the space-time structure of the cascade of fluxes; this must be localized in space-time in order to satisfy the usual turbulence phenomenology. In contrast, the second propagator relates the turbulent fluxes to the observables; although the spatial part of the propagator is localized as before, in space-time it can be unlocalized. (It is still localized in space, now in wave packets.) We display numerical simulations which demonstrate the requisite (anisotropic, multifractal) statistical properties as well as wave-like phenomenologies. In parts II and III we will examine the empirical evidence for the spatial and temporal parts, respectively, of the model using state-of-the-art lidar data of aerosol backscatter ratios (which we use as a surrogate for passive scalar concentration).
ABSTRACT:We critically re-examine existing empirical studies of vertical and horizontal statistics of the horizontal wind and find that the balance of evidence is in favour of the Kolmogorov k x −5/3 scaling in the horizontal, Bolgiano-Obukov scaling k z −11/5 in the vertical corresponding to a D s = 23/9 stratified atmosphere in (x, y, z) space. This interpretation is particularly compelling once one recognizes that the 23/9-D turbulence can lead to long-range biases in aircraft trajectories and hence to spurious statistical exponents in wind, temperature and other statistics reported in the literature. Indeed, we show quantitatively that one is easily able to reinterpret the major aircraft-based campaigns (GASP, MOZAIC) in terms of the model. In part I, we have seen that this model is compatible with 'turbulence waves' which can be close to classical linear gravity waves in spite of their very different nonlinear mechanism. We then use state-of-the-art lidar data of atmospheric aerosols (considered as passive tracers) in order to obtain direct estimates of the effective ('elliptical') dimension of the spatial part: D s = 23/9 = 2.55 ± 0.02. This result essentially rules out the standard 3-D or 2-D isotropic theories or the anisotropic quasi-linear gravity wave theories which have D s = 3, 2, 7/3 respectively.In this paper we focus on the multifractal (intermittency) statistics showing that there is a very small but apparently real variation in the value of D s , ranging for the weak and intense structures so that D s ranges from roughly 2.53 to 2.57. We also show that the passive scalars are well approximated by universal multifractals; we estimate the exponents to be α h = 1.82 ± 0.05, α v = 1.83 ± 0.04, C 1h = 0.037 ± 0.0061 and C 1v = 0.059 ± 0.007 (h for horizontal, v for vertical).
ABSTRACT:In this third and final part of the series, we concentrate on the temporal behaviour of atmospheric passive scalars. We first recall that -although the full (x, y, z, t) turbulent processes respect an anisotropic scale invariance -that due to advection -the generator will generally not be a diagonal matrix. This implies that the scaling of (1-D) temporal series will generally involve three exponents in real space: 1/3, 1/2, 3/5, for spectra β τ = 5/3, 2, 11/5, with the first and last corresponding to domination by advection (horizontal and vertical respectively), and the second to pure temporal development (no advection). We survey the literature and find that almost all the empirical β τ values are indeed in the range 5/3 to 2. We then use meteorological analyses to argue that, although pure temporal development is unlikely to be dominant for time-scales less than the eddy turnover time of the largest structures (about 2 weeks), an intermittent vertical velocity could quite easily explain the occasionally observed β τ ≈ 2 spectra.We then use state-of-the-art vertically pointing lidar data of backscatter ratios from both aerosols and cirrus clouds yielding several (z, t) vertical space-time cross-sections with resolution of 3.75 m in the vertical, 0.5-30 s in time and spanning 3-4 orders of magnitude in temporal scale. We first test the predictions of the anisotropic, multifractal extension of the Corrsin-Obukhov law in the vertical and in time, separately finding that the cirrus and aerosol backscatters both followed the theoretical (anisotropic) scalings accurately; three of the six cases show dominance by the horizontal wind, the others by the vertical wind. In order to test the theory in arbitrary directions in this (z, t) space, and in order to get more complete information about the underlying physical scale, we develop and apply a new Anisotropic Scaling Analysis Technique (ASAT) which is based on a nonlinear space-time coordinate transformation. This transforms the original differential scaling into standard self-similar scaling; there remains only a 'trivial' anisotropy. This method is used in real space on 2-D structure functions. It is applied to both the new (z, t) data as well as the (x, z) data discussed in part II. Using ASAT, we verify the theory to within about 10% over more than three orders of magnitude of space-time scales in arbitrary directions in (x, z) and (z, t) spaces. By considering the high-(and low-) order structure functions, we verify the theory for both weak and strong structures; as predicted, their average anisotropies are apparently the same.Putting together the results for (x, z) and (z, t), and assuming that there is no overall stratification in the horizontal (x, y) plane, we find that the overall (x, y, z, t) space is found to have an effective 'elliptical dimension' characterizing the overall space-time stratification equal to D eff,st = 3.21 ± 0.05.
Radiative heating rates computed with cloud properties derived from passive and active sensors are investigated. Zonal monthly radiative heating rate anomalies computed using both active and passive sensors show that larger variability in longwave cooling exists near the tropical tropopause and near the top of the boundary layer between ~50°N to ~50°S. Aerosol variability contributes to increases in shortwave heating rate variability. When zonal monthly mean cloud effects on the radiative heating rate computed with both active and passive sensors and those computed with passive sensor only are compared, the latter shows cooling and heating peaks corresponding to cloud top and base height ranges used for separating cloud types. The difference of these two sets of cloud radiative effect on heating rates in the middle to upper troposphere is larger than the radiative heating rate uncertainty estimated based on the difference of two active sensor radiative heating rate profile data products. In addition, radiative heating rate contribution to generation of eddy available potential energy is also investigated. Although radiation contribution to generation of eddy available potential energy averaged over a year and the entire globe is small, radiation increases the eddy available potential energy in the northern hemisphere during summer. Two key elements that longwave radiation contribute to the generation of eddy potential energy are (1) longitudinal temperature gradient in the atmosphere associated with land and ocean surface temperatures contrasts and absorption of longwave radiation emitted by the surface and (2) cooling near the cloud top of stratocumulus clouds.
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