A 1-d Ising model for ripple formation
  A 1-d Ising model is shown to reproduce qualitatively the dynamics of ripple formation. Saltation effect is imposed using a Kawasaki dynamics and a pair interaction over some distance l. Within this model, the ripple state turns out to be metastable in agreement with cellular automata simulations as well as recent under water experiments. A dynamical phase diagram is obtained. A mean-field solution of the problem is given in terms of the ripple size. A mapping is then performed onto a 2-dimensional l x infinity static problem. 
Nicolas Vandewalle and Serge Galam