The Relationship between Flux Coefficient and Entrainment Ratio in Density Currents

Abstract

The authors explore the theoretical and empirical relationship between the nonlocal quantities of the entrainment ratio E, the appropriately depth- and time-averaged flux coefficient Γ, and the bulk Froude number Fro in density currents. The main theoretical result is that E = 0.125 Γ Fro2(CU3/CL)/cos?, where ? is the angle of the slope over which the density current flows, CL is the ratio the turbulent length scale to the depth of the density current, and CU is the ratio of the turbulent velocity scale to the mean velocity of the density current. In the case of high bulk Froude numbers Γ ? Fro?2 and (CU3/CL) = C? ? 1, so E ? 0.1, consistent with observations of a constant entrainment ratio in unstratified jets and weakly stratified plumes. For bulk Froude numbers close to one, Γ is constant and has a value in the range of 0.1?0.3, which means that E ? Fro2, again in agreement with observations and previous experiments. For bulk Froude numbers less than one, Γ decreases rapidly with bulk Froude number, explaining the sudden decrease in entrainment ratios that has been observed in all field and experimental observations.