Wavelength selection and symmetry breaking in orbital wave ripples

Abstract

Sand ripples formed by waves have a uniform wavelength while at equilibrium and develop defects while adjusting to changes in the flow. These patterns arise from the interaction of the flow with the bed topography, but the specific mechanisms have not been fully explained. We use numerical flow models and laboratory wave tank experiments to explore the origins of these patterns. The wavelength of “orbital” wave ripples (λ) is directly proportional to the oscillating flow's orbital diameter (d), with many experimental and field studies finding λ/d ≈ 0.65. We demonstrate a coupling that selects this ratio: the maximum length of the flow separation zone downstream of a ripple crest equals λ when λ/d ≈ 0.65. We show that this condition maximizes the growth rate of ripples. Ripples adjusting to changed flow conditions develop defects that break the bed's symmetry. When d is shortened sufficiently, two new incipient crests appear in every trough, but only one grows into a full-sized crest. Experiments have shown that the same side (right or left) wins in every trough. We find that this occurs because incipient secondary crests slow the flow and encourage the growth of crests on the next flank. Experiments have also shown that when d is lengthened, ripple crests become increasingly sinuous and eventually break up. We find that this occurs because crests migrate preferentially toward the nearest adjacent crest, amplifying any initial sinuosity. Our results reveal the mechanisms that form common wave ripple patterns and highlight interactions among unsteady flows, sediment transport, and bed topography.