Amplitude equations

The integration of the 3D hydrodynamic equations is too expensive in computation time. It is possible to reproduce the dynamics of ripples with simplified local equations of evolution, such as:

The linear dynamics of such equation is similar to that of the complete model while the computation of non-linear dynamics is much faster. These simulations allow to observe the long time evolution of ripples, such as the phenomenon of 3D coarsening. You can see below the evolution of an a small number of ripples (the color indicating the height of the surface). One can observe the coarsening, reducing the number of ripples from 6 to only 1 in the final state.