Journal of Statistical Physics, Vol. 93, Nos. 3/4, 1998 Motions of individual particles within the stripe and square patterns formed in oscillated granular media are studied using numerical simulations. Our event- driven molecular dynamics simulations yield standing wave patterns in good accord with those observed in experiments at the same frequency and accelera- tion amplitude. The patterns are subharmonic and so return to their initial macroscopic state after two external cycles. However, simulations reveal that individual particles do not return to their initial position. In addition to diffusive motion, an organized flow of particles within the patterns is found; associated with each peak and each valley of the pattern is a pair of counterrotating con- vection rolls. The diffusion is anisotropic: transport perpendicular to stripes is enhanced over that parallel to stripes. This enhancement is computed as a func- tion of the layer depth, acceleration amplitude, frequency, and coefficient of restitution of the particles, and is attributed to the effect of the advective motion. Velocity distributions, granular temperature, and the dependence of the diffu- sion coefficient parallel to the stripes on the average granular temperature are studied. KEY WORDS: Granular media; pattern; convection; enhanced diffusion; granular temperature; numerical simulation. Just as vertically oscillated fluids exhibit an instability to patterns of sub- harmonic standing waves, (1–3) so too do vertically oscillated granular media. (4–7) Patterns observed in experiments on granular media include 0022-4715/98/1100-0449$15.00/0 © 1998 Plenum Publishing Corporation 449 1 Center for Nonlinear Dynamics and Department of Physics, University of Texas, Austin, Texas 78712. 2 Permanent address: Department of Physics and Physical Oceanography, Memorial Univer- sity of Newfoundland, St. John's, Newfoundland, Canada. I. INTRODUCTION Convection and Diffusion in Patterns in Oscillated Granular Media C. Bizon, 1 M. D. Shattuck, 1 John R. de Bruyn, 1, 2 J. B. Swift, 1 W. D. McCormick, 1 and Harry L. Swinney 1 Received February 2, 1998