Logarithmic scaling of higher-order temperature moments in the atmospheric surface layer

A generalized logarithmic law for high-order moments of passive scalars is proposed for turbulent boundary layers. This law is analogous to the generalized log law that has been proposed for high-order moments of the turbulent longitudinal velocity and is derived by combining the random sweeping decorrelation hypothesis with a spectral model informed by the attached eddy hypothesis. The proposed theory predicts that the high-order moments of passive scalar fluctuations within the inertial sublayer will vary logarithmically with wall-normal distance ($z$). The proposed theory is evaluated using high frequency time-series measurements of temperature and streamwise velocity fluctuations obtained in the first meter of the atmospheric surface layer (ASL) under near-neutral thermal stratification. The logarithmic dependence with $z$ within the inertial sublayer is observed in both the air temperature and velocity moments, with good agreement to the predictions from the proposed theory. Surprisingly, the proposed theory appears to be as, if not more, valid for transported passive scalars than for the longitudinal velocity.