Scaling Relationships for Sand Wave Development in Unidirectional Flow

Sand wave development on an initially flat bottom under steady unidirectional flow is examined on the basis of laboratory measurements. A common scaling relationship P0(t)/Pe= (t/T)α is proposed for the length and height of waves [P0(t) and Pe denote the average length or height of sand waves after time t and in an equilibrium state that is attained after time T]. A similar relationship σz(δx, t)/σz0= (t/T)αf[(δx/δx0)/(t/T)α/H] is proposed to describe the scaling behavior of the root-mean-square deviations σz(δx, t) of bottom elevation Z(x, t). Here, σz0 is σz(δx, t) when δx≥δx0 and t≥T, and δx0 is δx when the function σz(δx, T) becomes saturated, i.e., during the transition from the regime where σz∝δxH to the regime where σz=σz0= const. The scaling exponents α and H equaled 0.28 and 1.0, respectively, for the laboratory experiments.