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

Abstract

The turbulent energy spectra and cospectra of momentum and sensible heat fluxes are examined theoretically and experimentally with increasing flux Richardson number (Rf) inthestableatmosphericsurfacelayer.Acospectralbudgetmodel,previouslyusedtoexplain the bulk relation between the turbulent Prandtl number (Prt) and the gradient Richardson number (Ri) as well as the relation between Rf and Ri, is employed to interpret field measure- ments 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 Rf .I n addition, the wavenumber-dependent relaxation time scales for momentum and heat fluxes are inferred from the cospectral budgets and investigated. Using experimental data and pro- posedextensionstothecospectralbudgetmodel,theexistenceofa'−1'power-lawscalingin the temperature spectra but its absence from the vertical velocity spectra is shown to reduce the magnitude of the maximum flux Richardson number (Rfm), which is commonly inferred from the Rf-Ri relation when Ri becomes very large (idealized with Ri →∞ ). 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 Prt � 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 Rf → Rfm ,w hich explains the observed disappearance of the inertial subrange in the vertical velocity spectra at a smaller 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 Prt-Ri relation.