Semiempirical theories of turbulent diffusion in an atmospheric surface layer
Authors: Tsarenko V. M. , Yaglom A. M. 
Published: July 2002

The equation for the mean concentration 〈c〉=C of a passive admixture in a turbulent flow is known not to be closed, since it includes, besides the unknown C and the mean velocity vector U=(U1, U2, U3), the turbulent fluxes of the admixture which are new unknowns. (These fluxes are given as Si=〈−u′ic′〉, i=1, 2, 3, where c′ and ui are the turbulent fluctuations of concentration and velocity components, and the angular brackets symbolize averaging.) Therefore, the equation for C cannot be solved without using some closure hypothesis.
The simplest closure hypothesis is based on Boussinesq’s assumption of the proportionality of turbulent flux S=(S1, S2, S3) and the gradient of admixture concentration. According to Schmidt (1925) and Richardson (1926) this assumption can be written Si=Ki(∂C/∂xi), i=1, 2, 3. (1) Of course, (1) can be interpreted as the definition of new unknowns Ki instead of the turbulent fluxes Si. Therefore, without any supplementary hypotheses these equations are useless. Moreover, it is easy to see that Eqs. (1) contradict some known experimental facts which can be explained only if (1) are replaced by more general equations, Si=𝒥j=13 Kij (∂C)/(∂xij), i=1, 2, 3, (2) which include nine unknown eddy diffusivities. In the meteorological literature the off-diagonal components of the eddy diffusivity tensor Kij are usually neglected and the diagonal components are determined with the aid of crude, purely speculative hypotheses.
A preferable approach that fits well with the modern development of semiempirical turbulence theory is based on the application of second-order closures for the equations of turbulence mechanics. Such closures supplement the exact equations for the mean fields by the equation for the second moments of turbulent fluctuations, where the new unknowns (third-order moments, correlations with pressure fluctuations) are approximated by simple functions of mean fields and second moments. It is easy to show that the so-called algebraic models of second-order closure, based on local equilibrium approximation (i.e., neglect of time variations and turbulent diffusion of the second moments), lead automatically to an equation of the form (2) and permit one to determine all the eddy diffusivities Kij.
A number of different algebraic models of second-order closure can be found in the available literature and many of them require further modifications to become consistent with the experimental data for wall turbulent flows. However, the analysis of the algebraic models shows that different models consistent with the experimental data often lead to close values of turbulent diffusivities Kij. A survey of the results obtained from the evaluation of eddy diffusivities Kij for a nonstratified and a thermally stratified atmospheric surface layer with the aid of second-order closures will be reported. Some results of the study of turbulent diffusion by more general differential models of second-order closures will be briefly outlined. These results permit one to estimate approximately the accuracy of the assumptions (2).