Inertial ranges in three-dimensional quasi-geostrophic turbulence

Abstract

A theory is presented both for spectral energy transfer and for the transfer of spectral components of pseudo-potential enstrophy in a homogeneous quasi-geostrophic turbulent field which is rendered anisotropic by the distortion caused by a random collection of vortices superimposed on the principal motions. The fluid is, thus, subjected to an almost irrotational distortion. The random vortices cause straining effects on turbulent velocity and temperature fluctuations and modify the energy spectrum in the spectral ranges of interest. The strain imposed by the distortion is assumed to be homogeneous. For three-dimensional quasi-geostrophic turbulence that conserves pseudo-potential enstrophy as well as energy, this theory predicts −8/3 and −4 power inertial-range energy spectra.

The predictions favourably corroborate the observed spectrum of energy in the atmosphere in the region of hemispheric wave-numbers 10–16 with a −8/3 slope and at higher wave-numbers with −4 slope on a log-log energy-wave-number diagram. The transfer rates of pseudo-potential enstrophy in the range 10⩽n⩽16 and of energy in the rangen>16 are identically zero, while the transfer of energy in the first range is from higher to lower wave-numbers and that of the pseudo-potential enstrophy in the second range is from lower to higher wave-numbers.

As compared with the earlier two-dimensional turbulence theory of Kraichnan and the quasigeostrophic turbulence theory of Charney, the present theory predicts more realistic shapes of the energy spectra of atmospheric motions at scales shorter than the baroclinic excitation scales.