It is known from observations that the horizontal kinetic energyspectrum follows a k-5/3 power law in the mesoscale phenomena inthe upper troposphere. Nastrom et al. (1984) found the k-5/3power law in the scales less than about 500 km from the observation inthe upper troposphere. From the aircraft observation, Cho etal. (1999) showed that this spectral slope can be applied to theatmospheric motion with the hoizontal scales less than 100 km in thetroposphere except the boundary layer. Lilly (1984) attemped tounderstand the energy spectrum in the mesoscales from inverse energycascades in stratified turbulence. However the results from numericalsimulations of stratified turbulence indicate that inverse energycascades does not occur unless the Coriolis parameter f is extremelylarge (Herring and Metais; 1989, Metais et al., 1994). On the otherhand, Vallis et al. (1997) reproduced the above spectral slope in themesoscales with the regional model including water vapor and Koshykand Hamilton (1999) did with GFDL SKYHI GCM. These models include manyphysical processes so that it is difficult to understand their resultsas energy cascades in stratified turbulence.
We conduct numerical experiments with a three dimensionalnonhydrostatic model assuming the Boussinesq approximationin in order toconsider upscale and/or downscale energy cascades in stratifiedturbulence. The results obtained in these experiments are asfollows. When the dynamical forcing function has a peak at thehorizontal scale of 20 km, inverse energy cascading due to interactionsbetween horizontal vortices is too weak to explain the spectral slopein the mesoscales, even if the amplitude of the forcing function andthe stratification are strong. When the forcing is given at the domainsize (400--800 km), the k-5/3 spectral slope are formed as theresult of downscale energy cascades only if the forcing amplitude issufficiently large. However, in this case the spectral amplitude isabout two order larger than that obtained by the observationalstudies. For the weak forcing, the spectral slope tends to be steeperand this result is not improved by the smaller eddy viscosity.
We also examine the Smagorinsky-Lilly parameterization and the 1.5order TKE parameterization, which are well known as subgrid-scaleturbulent parameterization. In both parameterizations, the kineticenergy at the grid scale is not sufficiently removed because of tooweak eddy viscosity at this scale, while the energy spectrum in thescales larger than 20--30 km is close to that of the observations. Wewill also discuss details of energy cascade processes and bettersubgrid-scale parameterization. |
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