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

Abstract

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 $\roh_c$ propagates, with speed $U$, at the bottom of a long horizontal channel of height $H$, into the unperturbed ambient whose density increases linearly from $\roh_o$ (at the top) to $\roh_b$ (at the bottom). The reduced gravity is $g^{\prime} \,{=}\, (\roh_c/\roh_o -1)g$ and the governing parameters are $a \,{=}\, h/H$ and $S \,{=}\, (\roh_b-\roh_o)/(\roh_c-\roh_o)$, 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: $\hbox{\it Fr} \,{=}\, U/(g^{\prime}h)^{1/2}$, head loss (dissipation), and criticality of $U$ with respect to the fastest internal wave mode. The classical results of Benjamin are fully recovered for $S \rightarrow 0$. For small $S$ and fixed $a$, the values of $\hbox{\it Fr}$ 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.