The Relationship between Flux Coefficient and Entrainment Ratio in Density Currents

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 (Gamma) over bar, and the bulk Froude number Fr-o in density currents. The main theoretical result is that E = 0.125 (Gamma) over bar Fr-o(2)(C-U(3)/C-L)/cos theta, where theta is the angle of the slope over which the density current flows, C-L is the ratio the turbulent length scale to the depth of the density current, and C-U is the ratio of the turbulent velocity scale to the mean velocity of the density current. In the case of high bulk Froude numbers (Gamma) over bar similar to Fr-o(-2) and (C-U(3)/C-L) = C-epsilon similar to 1, so E similar to 0.1, consistent with observations of a constant entrainment ratio in unstratified jets and weakly stratified plumes. For bulk Froude numbers close to one, (Gamma) over bar is constant and has a value in the range of 0.1-0.3, which means that E similar to Fr-o(2), again in agreement with observations and previous experiments. For bulk Froude numbers less than one, (Gamma) over bar decreases rapidly with bulk Froude number, explaining the sudden decrease in entrainment ratios that has been observed in all field and experimental observations.