Simplified Models For Turbulent Diffusion: Theory, Numerical Modelling, And Physical Phenomena

Physics Reports 314 (1999) 237}574 Simpli"ed models for turbulent di!usion: Theory, numerical modelling, and physical phenomena Andrew J. Majda*, Peter R. Kramer New York University, Courant Institute, 251 Mercer Street, New York, NY 10012, USA Received August 1998; editor: I. Procaccia Contents 1. Introduction 2. Enhanced di!usion with periodic or shortrange correlated velocity "elds 2.1. Homogenization theory for spatiotemporal periodic #ows 2.2. E!ective di!usivity in various periodic #ow geometries 2.3. Tracer transport in periodic #ows at "nite times 2.4. Random #ow "elds with short-range correlations 3. Anomalous di!usion and renormalization for simple shear models 3.1. Connection between anomalous di!usion and Lagrangian correlations 3.2. Tracer transport in steady, random shear #ow with transverse sweep 3.3. Tracer transport in shear #ow with random spatio-temporal #uctuations and transverse sweep 3.4. Large-scale e!ective equations for mean statistics and departures from standard eddy di!usivity theory 3.5. Pair-distance function and fractal dimension of scalar interfaces 4. Passive scalar statistics for turbulent di!usion in rapidly decorrelating velocity "eld models 4.1. De"nition of the rapid decorrelation in time (RDT) model and governing equations 240 243 245 262 285 293 304 308 316 342 366 389 413 4.2. Evolution of the passive scalar correlation function through an inertial range of scales 4.3. Scaling regimes in spectrum of #uctuations of driven passive scalar "eld 4.4. Higher-order small-scale statistics of passive scalar "eld 5. Elementary models for scalar intermittency 5.1. Empirical observations 5.2. An exactly solvable model displaying scalar intermittency 5.3. An example with qualitative "nite-time corrections to the homogenized limit 5.4. Other theoretical work concerning