Scaling Properties of Temperature Spectra and Heat-Flux Cospectra in the Surface Friction Layer Beneath an Unstable Outer Layer

Abstract

Temperature variance and temperature power spectra in the unstable surface layer have always presented a problem to the standard Monin-Obukhov similarity model. Recently that problem has intensified with the demonstration by Smedman et al. (2007, Q J Roy Meteorol Soc 133: 37–51) that temperature spectra and heat-flux cospectra can have two distinct peaks in slightly unstable conditions, and by McNaughton et al. (2007, Nonlinear Process Geophys 14: 257–271) who showed that the wavenumber of the peak of temperature spectra in a convective boundary layer (CBL), closely above the surface friction layer (SFL), can be sensitive to the CBL depth, zi. Neither the two-peak form at slight instability nor the dependence of peak position on zi at large instability is compatible with the Monin-Obukhov model. Here we examine the properties of temperature spectra and heat-flux cospectra from between these extremes, i.e. from within the unstable SFL, in two experiments. The analysis is based on McNaughton’s model of the turbulence structure in the SFL. According to this model, heat is transported through most of the SFL by sheet plumes, created by the action of impinging outer eddies. The smallest and most effective of these outer eddies have sizes that scale on SFL depth, zs. The zs-scale eddies and plumes are organised within the overall convection pattern in the CBL, and in turn they organise the motion of smaller eddies within the SFL, whose sizes scale on height, z. The main experimental results are: (1) the peak amplitudes of the temperature spectra in the SFL are collapsed with a scaling factor \({(z_{\rm s}z)^{1/3}\varepsilon_{\rm o}^{2/3}}\) divided by the square of the surface temperature flux, where \({\varepsilon_{\rm o}}\) is the dissipation rate of turbulent energy in the outer CBL (above the SFL); (2) the peak wavenumbers of the temperature spectra are collapsed with the mixed length scale (zizs)1/2; (3) the peak wavenumbers of the heat-flux cospectra are collapsed with the doubly-mixed length scale (zizs)1/4z1/2; (4) for z/zs < 0.03, the peak in the cospectrum is replaced by another peak at a wavenumber about a magnitude larger. This peak’s position scales on z; (5) all these findings are consistent with the observations of Smedman et al.