Implications of Monin–Obukhov Similarity Theory for Scalar Quantities

Abstract

Monin-Obukhov similarity theory of surface-layer turbulence has been extended to include all scalar quantities. The tenets of this theory, as it is presently practiced, are followed to their logical conclusions, which produce some novel results. First, the similarity theory is applied to the variance of a scalar quantity and its correlation with another scalar quantity. For this similarity to apply to all scalar quantities, it is shown that the dimensionless functions of stability are all equal, that the correlation coefficient of any two scalars is +1 or −1, and that the correlation of two scalars has the same sign as the product of their fluxes. These results also apply to structure parameters and cross-structure parameters as well as to the dissipation rates of the scalar's variances and correlations. It is shown that the fluctuations of all the scalar quantities are proportional to one another. On the basis of Monin–Obukhov similarity, the Obukhov–Corrsin constants (i.e., Kolmogorov constants) of scalar spectra and cospectra are shown to be equal, and are expressed as a function of stability. The dimensionless fluxes of scalar variance and correlation are shown to be the same for all scalars. Likewise the dimensionless functions for gradient production of scalar variance and covariance are the same for all scalars. All the above demonstrate that internal consistency of the similarity theory of the horizontally homogeneous surface layer. The above results, however, show in what manner the similarity theory becomes an overidealization when it is extended to all scalar quantities. In particular, the internal dynamics of the surface itself is not considered in the similarity theory; yet it is reasonable that such internal dynamics causes violations of the similarity theory.