# An Empirical Scaling Analysis of Heat and Momentum Cospectra Above the Surface Friction Layer in a Convective Boundary Layer

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## Abstract

We report empirical scaling results of heat and momentum cospectra at heights above the surface friction layer but well within the bottom tenth of the convective boundary layer (CBL), also known as the local free-convection layer. We perform scaling analysis using two field datasets, based on McNaughton et al. (Nonlinear Process Geophys 14:257–271, 2007), and discuss the results using their structural model of CBL turbulence as an untested hypothesis. Using their scaling parameter set, the scaled heat cospectra from the two experiments collapse in small, mid, and large wavenumber ranges as in the reported temperature spectra. The empirical results show: (1) at small wavenumbers, the heat cospectra scale with a length scale \( \lambda \) and an amplitude scale \( H_{0} (z/\lambda )^{1/2} \), where \( \lambda \) is the peak wavelength of the streamwise velocity spectrum, \( z \) is the height above the surface, and \( H_{0} \) is the kinematic heat flux at the surface, (2) at mid wavenumbers, the heat-cospectral peak positions and heights scale with a doubly-mixed length scale \( \lambda^{1/4} z^{3/4} \) and an amplitude scale \( H_{0} (z/\lambda )^{1/12} \), and (3) at large wavenumbers, the heat cospectra scale with a length scale \( z \) and an amplitude scale \( H_{0} \). However, the momentum cospectra resist a similar scaling analysis and display erratic peaks at small wavenumbers. The wavelet analysis of heat and momentum flux time series shows this is related to poor sampling of the momentum flux carried at small wavenumbers.

## Keywords

Cospectral analysis Local free convection Mixed length scale Spectral scaling analysis Surface friction layer## Notes

### Acknowledgements

S.C. and T.V.P. gratefully acknowledge the support of the Ministry of Earth Science (MoES). The authors would also like to thank the four anonymous reviewers for their helpful comments. The wavelet MatLab code used in this paper can be found at http://www.glaciology.net/wavelet-coherence. The SLTEST and CAIPEEX datasets are also available to the interested researchers by contacting K.G.M. and S.C. at keith@mcnaughty.com and subharthi1987@gmail.com.

## Supplementary material

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