Formation of sand bars under surface waves

A quantitative theory is described for the formation mechanism of sand bars under surface water waves. By assuming that the slopes of waves and bars are comparably gentle and sediment motion is dominated by the bedload, an approximate evolution equation for bar height is derived. The wave field and the boundary layer structure above the wavy bed are worked out to the accuracy needed for solving this evolution equation, It is shown that the evolution of sand bars is a process of forced diffusion. This is unlike that for sand ripples which is governed by an instability. The forcing is directly caused by the non-uniformity of the wave envelope, hence of the wave-induced bottom shear stress associated with wave reflection, while the effective diffusivity is the consequence of gravity and modified by the local bed stress. During the slow formation, bars and waves affect each other through the Bragg scattering mechanism, which consists of two concurrent processes: energy transfer between waves propagating in opposite directions and change of their wavelengths. Both effects are found to be controlled locally by the position of bar crests relative to wave nodes. Comparison with available laboratory experiments is discussed and theoretical examples are studied to help understand the coupled evolution of bars and waves in the field.