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 rho(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 rho(o) (at the top) to rho(b) (at the bottom). The reduced gravity is g' = (rho(c)/rho(o) - 1)g and the governing parameters are a = h/H and S = (rho(b) - rho(o))/(rho(c) - rho(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.