Pattern formation in a vibrated granular layer: the pattern selection issue

We present a numerical study of a surface instability occurring in a bidimensional vibrated granular layer. The driving mechanism for the formation of stationary waves is closely followed. Two regimes of wavelength selection are identified: a dispersive regime where the wavelength decreases with increasing frequency and a saturation regime where the value of the wavelength is a constant depending on the number of grains in the vertical direction. For the dispersive regime an empirical relation is proposed, based on dimensional arguments involving transport properties in the layer. A comparison is made with existing experimental results in two and three dimensions. For the saturation regime, a connection is established between the pattern formation and an intrinsic instability occurring spontaneously in dissipative gases. The observed dependence on the layer height is linked to a detailed dissipation mechanism for the collisions between grains.