On gravity currents in a linearly stratified ambient: a generalization of Benjamin's steady-state propagation results

This paper presents a generalization of the classical results of T. B. Benjamin ( J. Fluid. Mech. vol. 31, 1968, p. 209) concerning the propagation of a steady gravity current into a homogeneous ambient, to the case of a stratified ambient. The current of thickness h and density c propagates, with speed U , at the bottom of a long horizontal channel of height H , into the unperturbed ambient whose density increases linearly from ₒ (at the top) to b (at the bottom). The reduced gravity is g^ \,=\, (c/ₒ -1)g and the governing parameters are a \,=\, h/H and S \,=\, (b-ₒ)/(c-ₒ) , with $0<a<1, 0 < S < 1 $ ; here g is the acceleration due to gravity. For a Boussinesq high-Reynolds two-dimensional configuration, a flow-field solution of Long's model, combined with flow-force balance over the width of the channel, are used for obtaining the desired results, in particular: Fr \,=\, U/(g^h)1/2 , head loss (dissipation), and criticality of U with respect to the fastest internal wave mode. The classical results of Benjamin are fully recovered for S → 0 . For small S and fixed a , the values of Fr and head loss are shown to decrease with S like (1-2S/3)1/2 and $(1-2S/3)$ , respectively, and the propagation is supercritical. For larger S several solutions are possible (for a given geometry a ), mostly in the subcritical regime. Considerations for the physical acceptability of the multiple results are presented, and the connection with observations from lock-release experiments are discussed. The conclusion is that the present results provide a reliable and versatile generalization of the classical unstratified problem.