Simpli"ed models for turbulent di!usion: Theory, numerical modelling, and physical phenomena

Several simple mathematical models for the turbulent di!usion of a passive scalar "eld 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 "elds and random velocity "elds with short-range correlations is presented and utilized to examine subtle ways in which the #ow geometry can in#uence the large-scale e!ective scalar di!usivity. Various forms of anomalous di!usion are then illustrated in some exactly solvable random velocity "eld 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 superdi!usive and subdi!usive transport, pair dispersion, fractal dimensions of scalar interfaces, spectral scaling regimes, small-scale and large-scale scalar intermittency, and qualitative behavior over "nite 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 di!usion. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 47.27.Qb; 05.40.#j; 47.27.!i; 05.60.#w; 47.27.Eq; 02.70.Lq A.J. Majda, P.R. Kramer / Physics Reports 314 (1999) 237}574 239