Similarity Relations for $$C_{T}^2 $$ C T 2 in the Unstable Atmospheric Surface Layer: Dependence on Regression Approach, Observation Height and Stability Range

Abstract

A great variety of similarity functions for the structure parameter of temperature ( $$C_{T}^2$$ ) have been proposed in the literature. They differ in the way they were derived from the data and in the characteristics of the dataset used for their derivation (surface type, observation level, stability range). In this study, we use one single dataset (CASES-99 experiment) and investigate the impact on the similarity functions of applying various regression approaches, and measuring at different heights and within different stability ranges. We limit ourselves to similarity functions under unstable conditions, and evaluate only the most common shape that describes the relation with two coefficients ( $$f\left( z/L\right) =c_{1} \left( 1-c_{2} {z}/{L}\right) ^{-2/3}$$ , where $$z$$ is the height, and $$L$$ is the Obukhov length and a measure of the stability, and $$c_{1}$$ and $$c_{2}$$ are the regression coefficients). The results show that applying various regression approaches has an impact on the regression coefficients $$c_{1}$$ and $$c_{2}$$ . Thus studies should always specify the regression approach when presenting similarity relations. We suggest use of an orthogonal distance regression method such that uncertainties in $$-z/L$$ are also taken into account, to apply this to the logarithmic transformation of both dimensionless groups, and to use a weighted dataset such that unreliable data points have a smaller influence on the fit. Dividing the dataset into eight height ( $$z$$ ) and eight stability ( $$-{1/L}$$ classes) classes, we show that the observation height and the stability range has an impact on the coefficients too. This implies that variations in $$c_{1}$$ and $$c_{2}$$ found in the literature may result from variations in the height and stability ranges among the datasets. Furthermore, application of the coefficients on a dataset obtained at a different height or within a different stability range has to be made with care. Finally, the variation in the coefficients between the classes indicates that the Monin–Obukhov similarity function for $$C_{T}^2$$ is not sufficiently described by the two-coefficient function used here.