The fascination of a shallow-water theory for the formation of megaflood-scale dunes and antidunes

Abstract

Exceptional megaflood-scale bedforms on Earth are commonly associated with the catastrophic draining of glacial lakes in the late Pleistocene. The widest studied events have been the Missoula and Altai floods with 300–700 m flow depth, 1–20 m bedform height and 10–300 m wavelength. Nowadays, the Saint-Venant equations have succeeded at simulating the catastrophic glacial-lake drainage process numerically, but we still lack a depth-averaged morphodynamic theory able to predict the growth of dunes and antidunes. The disparity of spatial scales in megafloods prevents the use of non-depth-averaged rotational flow equations, motivating the present shallow-water theory for the formation of megaflood-scale bedforms. We adopt a non-equilibrium sediment transport equation rooted in Einstein's pioneering work. Here we prove that the bed instability triggers to form dunes and antidunes simply by lagging the entrainment term for sediment mass conservation, or the bottom shear stress, with respect to the depth-averaged flow velocity. We formalise this result using a linear stability theory that captures the existence regions of dune and antidune in addition to the roll wave instability. Furthermore, in the spirit of Kennedy (Annu. Rev. Fluid Mech., vol. 1, 1969, pp. 147–168), we derive a closed-form solution of growth rate and wave speed of the bedform. The nondimensional groups controlling the linear instabilities are the Froude number, ℱr, the Shields parameter, Sh, and the grain roughness relative to flow depth, Subsequently, we simulate the drainage of the largest Missoula flood numerically to explain the formation of giant antidunes in the Camas Prairie (Montana, US) during the late stage of the megaflood. Also considered are large fields of gravel dunes in the Kuray-Chuja Lake Basin (Altai Mountains, Siberia). The simulated hydraulic conditions over bedforms in both basins yield values of the nondimensional parameters that lie in the theoretical region of dunes and antidunes according to the proposed theory and in situ measurements in sandy rivers and flume experiments.