There is growing evidence from observations that mixing occurs in hydraulic jumps, or flow transitions, downstream of sills within channels connecting deep ocean basins or within submarine canyons on the flanks of mid-ocean ridges. Models with continuous profiles of velocity and density upstream and downstream of a transition, but conforming to continuity conditions, are devised to represent the mixing that occurs in a hydraulic jump in a stratified shear flow of finite depth moving over a horizontal boundary in a deep fluid. These are used to assess the conditions in which transitions may occur and to provide an estimate of the loss in the flux of energy carried by the flow. An increase in the thickness of the stratified flow layer is necessary as water passes through a transition. The rate of loss of energy flux per unit channel width in a transition is typically of order 6ρh(gβh)3/2, where h is a measure of the thickness of the flow before transition, g the acceleration due to gravity and β = Δρ/ρ (≪1), where Δρ is half the difference in density between that in the flow approaching the transition and that in the overlying fluid, and ρ is the mean density. The mean rate of loss of energy in a transition in the flow of Antarctic Bottom Water over just one of the 6 – 8 sills in the Romanche Fracture Zone is estimated to be of order 60 MW (6 × 107 W).