Abstract
The spectral equations of turbulent kinetic energy and temperature variance have been solved by using Onsager's energy cascade model and by extending Onsager's model to closure of terms that embody the interaction of turbulent and mean flow.
The spectral model yields the following results: In a stably stratified shear flow, the peak wave numbers of the spectra of energy and temperature variance shift toward larger wave numbers as stability increases. In an unstably stratified flow, the peak wave numbers of energy spectra move toward smaller wave numbers as instability increases, whereas the opposite trend is observed for the peak wave numbers of temperature variance spectra. Hence, the peak wave numbers of temperature spectra show a discontinuity at the transition from stable to unstable stratification. At near neutral stratification, both spectra reveal a bimodal structure.
The universal functions of the Monin-Obukhov similarity theory are predicted to behave as Φ m ~ Φ H ~ (- Z/L)-1/3 in an extremely unstable stratification and as Φ m ~ Φ H ~ z/L in an extremely stable stratification. For a stably stratified flow, a constant turbulent Prandtl number is expected.
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Claussen, M. A model of turbulence spectra in the atmospheric surface layer. Boundary-Layer Meteorol 33, 151–172 (1985). https://doi.org/10.1007/BF00123388
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DOI: https://doi.org/10.1007/BF00123388