# Turbulent Energy Spectra and Cospectra of Momentum and Heat Fluxes in the Stable Atmospheric Surface Layer

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## Abstract

The turbulent energy spectra and cospectra of momentum and sensible heat fluxes are examined theoretically and experimentally with increasing flux Richardson number (\(\textit{Rf}\)) in the stable atmospheric surface layer. A cospectral budget model, previously used to explain the bulk relation between the turbulent Prandtl number (\(Pr_\mathrm{t}\)) and the gradient Richardson number (\(\textit{Ri}\)) as well as the relation between \(\textit{Rf}\) and \(\textit{Ri}\), is employed to interpret field measurements over a lake and a glacier. The shapes of the vertical velocity and temperature spectra, needed for closing the cospectral budget model, are first examined with increasing \(\textit{Rf}\). In addition, the wavenumber-dependent relaxation time scales for momentum and heat fluxes are inferred from the cospectral budgets and investigated. Using experimental data and proposed extensions to the cospectral budget model, the existence of a ‘\(-1\)’ power-law scaling in the temperature spectra but its absence from the vertical velocity spectra is shown to reduce the magnitude of the maximum flux Richardson number (\(\textit{Rf}_\mathrm{m}\)), which is commonly inferred from the *Rf*–*Ri* relation when \(\textit{Ri}\) becomes very large (idealized with \(\textit{Ri} \rightarrow \infty \)). Moreover, dissimilarity in relaxation time scales between momentum and heat fluxes, also affected by the existence of the ‘\(-1\)’ power-law scaling in the temperature spectra, leads to \(Pr_\mathrm{t} \ne 1\) under near-neutral conditions. It is further shown that the production rate of turbulent kinetic energy decreases more rapidly than that of turbulent potential energy as \(\textit{Rf}\rightarrow \textit{Rf}_\mathrm{m}\), which explains the observed disappearance of the inertial subrange in the vertical velocity spectra at a smaller \(\textit{Rf}\) as compared to its counterpart in the temperature spectra. These results further demonstrate novel linkages between the scale-wise turbulent kinetic energy and potential energy distributions and macroscopic relations such as stability correction functions to the mean flow and the \(Pr_\mathrm{t}\)–*Ri* relation.

## Keywords

Cospectra Energy spectra Flux Richardson number Gradient Richardson number Kolmogorov’s theory Stable atmospheric surface layer Turbulent Prandtl number## Notes

### Acknowledgments

DL acknowledges support from the NOAA (U.S. Department of Commerce) Grant NA08OAR4320752 and the Carbon Mitigation Initiative at Princeton University, sponsored by BP. The statements, findings, and conclusions are those of the authors and do not necessarily reflect the views of the NOAA, the U.S. Department of Commerce or BP. GK acknowledges support from the National Science Foundation (NSF-EAR-1344703, NSF-AGS-1102227), the United States Department of Agriculture (2011-67003-30222), the U.S. Department of Energy (DOE) through the office of Biological and Environmental Research (BER) Terrestrial Ecosystem Science (TES) Program (DE-SC0006967 and DE-SC0011461), and the Binational Agricultural Research and Development (BARD) Fund (IS-4374-11C). EBZ acknowledges support from NSF’s Physical and Dynamic Meteorology Program under AGS-1026636. The experimental data were collected by the Environmental Fluid Mechanics and Hydrology Laboratory of Professor Marc Parlange at L’Ècole Polytechnique Fédérale de Lausanne.

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