Arrested coarsening of granular roll waves

We study a system in which granular matter, flowing down an inclined chute with periodic boundary conditions, organizes itself in a train of roll waves of varying size. Since large waves travel faster than small ones, the waves merge, and their number gradually diminishes. This coarsening process, however, does not generally proceed to the ultimate one-wave state: Numerical simulations of the dynamical equations (being the granular analogue of the shallow water equations) reveal that the process is arrested at some intermediate stage. This is confirmed by a theoretical analysis, in which we show that the roll waves cannot grow beyond a certain limiting size (which is fully determined by the system parameters), meaning that on long chutes the material necessarily remains distributed over more waves. We determine the average lifetime τN of the successive N-wave states, from the initial state with typically N = 50 waves (depending on the length of the periodic domain) down to the final state consisting of only a handful of waves (N = Narr). At the latter value of N, the lifetime τN goes to infinity. At this point the roll waves all have become equal in size and are traveling with the same speed. Our theoretical predictions for the successive lifetimes τN and the value for Narr show good agreement with the numerical observations.