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Erik Lindborg

Abstract

The horizontal wavenumber spectra of wind and temperature in the upper troposphere and lower stratosphere display a narrow k −3 range at scales on the order of 1000 km and a broad k −5/3 range at mesoscales on the order of 1 to 500 km. Recently, Tulloch and Smith suggested that a surface quasigeostrophic (SQG) turbulence model can explain the observed spectra. Here, it is first argued that the mesoscale spectra are not likely to be explained by any quasigeostrophic model because the Rossby number corresponding to the mesoscale dynamics is on the order of unity or larger. Then it is argued that the SQG model in particular cannot explain the observations because its mesoscale spectrum displays a k −5/3 dependence only in a very thin layer just below the tropopause. The thickness of this layer can be estimated to be of the order of 10 m, whereas aircraft measurements are typically performed several hundred meters away from the tropopause.

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Erik Lindborg

Abstract

The author shows that the horizontal two-point correlations of vertical vorticity and the associated vorticity wavenumber spectrum can be constructed from previously measured velocity structure functions in the upper troposphere and lower stratosphere. The spectrum has a minimum around k = 10−2 cycles per kilometer (cpkm) corresponding to wavelengths of 100 km. For smaller wavenumbers it displays a k −1 range and for higher wavenumbers, corresponding to mesoscale motions, it grows as k 1/3. The two-point correlation of horizontal divergence of horizontal velocity and the associated horizontal spectrum is also constructed. The horizontal divergence spectrum is of the same order of magnitude as the vorticity spectrum in the mesoscale range and show similar inertial range scaling. It is argued that these results show that the mesoscale motions are not dominated by internal gravity waves. Instead, the author suggests that the dynamic origin of the k 1/3 range is stratified turbulence. However, in contrast to Lilly, the author finds that stratified turbulence is not a phenomenon associated with an upscale energy cascade, but with a downscale energy cascade.

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Qiang Li and Erik Lindborg

Abstract

Recently, it has been discussed whether the mesoscale energy spectra in the upper troposphere and lower stratosphere are generated by weakly or strongly nonlinear dynamics. A necessary condition for weak nonlinearity is that the Rossby number Ro ≡ |ζ z|/f ≪ 1, where ζ z is the vertical vorticity and f is the Coriolis parameter. First, it is shown that Ro can be estimated by integration of the rotational wavenumber energy spectrum E r. Then divergence and rotational energy spectra and their ratio, RE d/E r, are calculated from the Measurement of Ozone and Water Vapor by Airbus In-Service Aircraft (MOZAIC) dataset, and it is shown that at least 1000 flight segments are needed to obtain converged results. It is found that R < 1 in the upper troposphere, ruling out the hypothesis that the spectra are produced by inertia–gravity waves with frequencies larger than f. In the lower stratosphere R is slightly larger than unity. An analysis separating between land and ocean data shows that E d and temperature spectra have somewhat larger magnitude over land compared to ocean in the upper troposphere—a signature of orographically or convectively forced gravity waves. No such effect is seen in the lower stratosphere. At midlatitudes the Rossby number is on the order of unity and at low latitudes it is larger than unity, indicating that strong nonlinearities are prevalent. Also the temperature spectra, when converted into potential energy spectra, have larger magnitude than predicted by the weakly nonlinear wave hypothesis.

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Pierre Augier and Erik Lindborg

Abstract

A new formulation of the spectral energy budget of kinetic and available potential energies of the atmosphere is derived, with spherical harmonics as base functions. Compared to previous formulations, there are three main improvements: (i) the topography is taken into account, (ii) the exact three-dimensional advection terms are considered, and (iii) the vertical flux is separated from the energy transfer between different spherical harmonics. Using this formulation, results from two different high-resolution GCMs are analyzed: the Atmospheric GCM for the Earth Simulator (AFES) T639L24 and the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecast System (IFS) T1279L91. The spectral fluxes show that the AFES, which reproduces quite realistic horizontal spectra with a k −5/3 inertial range at the mesoscales, simulates a strong downscale energy cascade. In contrast, neither the k −5/3 vertically integrated spectra nor the downscale energy cascade are produced by the ECMWF IFS.

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James J. Riley and Erik Lindborg

Abstract

Several existing sets of smaller-scale ocean and atmospheric data appear to display Kolmogorov–Obukov–Corrsin inertial ranges in horizontal spectra for length scales up to at least a few hundred meters. It is argued here that these data are inconsistent with the assumptions for these inertial range theories. Instead, it is hypothesized that the dynamics of stratified turbulence explain these data. If valid, these dynamics may also explain the behavior of strongly stratified flows in similar dynamic ranges of other geophysical flows.

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