On nonlinear cascades in large-scale atmospheric flow

Abstract

Calculations of nonlinear interaction of available potential energy, kinetic energy, enstrophy, and potential enstrophy have been carried out in the domain of wave numbers. The processes of interest are the interactions among waves, the interactions between a given wave and the zonal average, and interactions due to the beta effect for (potential) enstrophy. Results are presented for a three‐month (February through April 1963) period. It is found that available potential energy is cascaded from small to large wave numbers, while kinetic energy is cascaded from intermediate wave numbers to both small and large wave numbers: the greater amounts go to the small wave numbers. Enstrophy and potential enstrophy are cascaded from small to large wave numbers with relatively little accumulation in the middle range (8≤m≤11, where m is the wave number). Verification of the existence of an inertial subrange is inconclusive, although power spectrum analyses of enstrophy and potential enstrophy yield a −1 power law, combined with a −3 slope for the associated kinetic energy spectrum. A value of 2.5×104 m2 sec−1 is obstained for the coefficient of eddy viscosity by means of Leith's formulation, while the value of the cut‐off wavelength for the −3 range is approximately 420 km by means of the formulation by Kraichnan.

A Detailed Normal-Mode Energetics of the General Circulation of the Atmosphere

“…For the barotropic mode, the spectra of K follows the -4 power law, which agrees with Terasaki and Tanaka (2007b) for total energy, and can be explained with the Rossby wave saturation theory according to Tanaka et al, (2004), but the slope for the A spectra is steeper, following approximately a -5 power law. This -5 powerlaw behavior in the spectra of A has been observed in the work of Steinberg (1971), Additionally, Merilees and Warn (1972) argue that any numerical model that has a much finer resolution in the horizontal than the vertical would ultimately have a section of the spectrum where available potential energy would follow the -5 power law.…”

Two-dimensional turbulence properties of the ECMWF reanalyses

“…Since The strong tendency of non-linear 2-dimenthe upscale transfer of energy from synoptic to sional dynamics to transfer energy to large scales planetary scales is a potentially important process in understanding large-scale atmospheric variabil-transfer from atmospheric data (Saltzman and the nature of the non-linear turbulent interactions (Shepherd, 1987b). In the more general baroclinic Fleisher, 1960;Steinberg and Wiin-Nielsen, 1971;Chen and Wiin-Nielsen, 1978;Lambert, 1981). case, the interplay between the development of transients in a mean vertical shear ( baroclinic With the recognition that quasi-geostrophic dynamics share a fundamental property with instability) and quasi-geostrophic turbulence was studied in fairly idealized contexts by Salmon 2-dimensional incompressible dynamics (they both conserve forms of energy and vorticity), it became (1980), Haidvogel and Held (1980), Hoyer and Sadourny (1982), Vallis (1983) and Hua and clear that quasi-geostrophic turbulence was highly relevant to geophysical fluid dynamics in general Haidvogel (1986).…”

The stochastic multiplicative cascade structure of deterministic numerical models of the atmosphere

“…The most important classical spectral exponents are −5/3, −3 for the velocity exponents in regimes dominated by energy and enstrophy fluxes respectively. For example, in analyzing atmospheric models, Steinberg et al (1971), Boer and Shepherd (1983), and Straus and Ditlevsen (1999) calculated spectra and spectral transfers of energy, enstrophy and pseudo-potential enstrophy. The latter paper is particularly pertinent to our discussion since the large atmospheric Published by Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union.…”