Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena

Several simple mathematical models for the turbulent diffusion of a passive scalar field are developed here with an emphasis on the symbiotic interaction between rigorous mathematical theory (including exact solutions), physical intuition, and numerical simulations. The homogenization theory for periodic velocity fields and random velocity fields with short-range correlations is presented and utilized to examine subtle ways in which the flow geometry can influence the large-scale effective scalar diffusivity. Various forms of anomalous diffusion are then illustrated in some exactly solvable random velocity field models with long-range correlations similar to those present in fully developed turbulence. Here both random shear layer models with special geometry but general correlation structure as well as isotropic rapidly decorrelating models are emphasized. Some of the issues studied in detail in these models are superdiffusive and subdiffusive transport, pair dispersion, fractal dimensions of scalar interfaces, spectral scaling regimes, small-scale and large-scale scalar intermittency, and qualitative behavior over finite time intervals. Finally, it is demonstrated how exactly solvable models can be applied to test and design numerical simulation strategies and theoretical closure approximations for turbulent diffusion.