Internal bores, or internal hydraulic jumps, arise in many atmospheric and oceanographic phenomena. The classic single-layer hydraulic jump model accurately predicts the bore height and propagation velocity when the difference between the densities of the expanding and contracting layers is large (i.e., water and air), but fails in the Boussinesq limit. A two-layer model, which conserves mass separately in each layer and momentum globally is more accurate in the Boussinesq limit, but it requires a fourth jump condition for closure--- typically an assumption about the loss of energy across a bore. It is widely believed that bounds on the bore speed can be found by restricting the energy loss entirely to one of the two layers, but under some circumstances, both bounds over-predict the propagation speed. A front velocity slower than both bounds implies that, somehow, the expanding layer is gaining energy. We directly examine the flux of energy within internal bores using two- and three-dimensional direct numerical simulations and find that although there is a global loss of energy across a bore, a transfer of energy from the contracting to the expanding layer causes a net energy gain in the expanding layer. For Boussinesq bores, the energy transfer is largely the result of turbulent mixing at the interface. For non-Boussinesq flows, undular waves generated at the bore's front may dominate over the effects of turbulent mixing, and the expanding layer loses energy across the bore. Based on these results, we propose improved two-layer models that provides accurate propagation velocities as function of the geometrical parameters, the Reynolds number, and the Schmidt number.

We also propose a new model for internal bores, as well as gravity currents (a related flow) on the basis of mass and momentum conservation alone, without the need for considering energy. This is accomplished by combining the streamwise and vertical momentum equations to obtain a vorticity relation that no longer involves pressure. Hence, this vorticity equation, in combination with the conservation of mass, is sufficient for evaluating the bore velocity. The energy loss across the bore can then be predicted by the streamwise energy equation and compared to the assumptions underlying earlier models. The flux of vorticity across an internal bore predicted by this new model is seen to be in close agreement with direct numerical simulation results. Any discrepancies with experimentally measured bore velocities are shown to be due to the effects of downstream mixing.