Probability Density Functions of Decaying Passive Scalars in Periodic Domains : An Application of Sinai-Yakhot Theory

Employing the formalism introduced by Sinai and Yakhot [PRL, 63(18), p. 1962, 1989], we study the probability density functions (pdf's) of decaying passive scalars in periodic domains under the influence of smooth large scale velocity fields. The particular regime we focus on is one where the normalized scalar pdf's attain a self-similar profile in finite time, i.e., the so called strange or statistical eigenmode regime. In accordance with the work of Sinai and Yakhot, the central regions of the pdf's are power laws. But the details of the pdf profiles are dependent on the physical parameters in the problem. Interestingly, for small Peclet numbers the pdf's {\it resemble} stretched or pure exponential functions, whereas in the limit of large Peclet numbers, there emerges a universal Gaussian form for the pdf. Numerical simulations are used to verify these predictions.