Steady-state propagation of gravity currents into a linearly stratified ambient: a generalization of Benjamin's results

The classical results of Benjamin (J. Fluid. Mech. vol. 31, 1968) concerning the propagation of a steady gravity current into a homogeneous ambient fluid, are generalized to the case of a stratified ambient fluid. 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 fluid whose density increases linearly from $\rho_o$ to $\rho_b$. 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