On the propagation of internal bores

According to classical hydraulic theory, the energy losses within an external bore must occur within the expanding layer. However, the application of this theory to describe the propagation of internal bores leads to contradiction with accepted gravity-current behaviour in the limit as the depth of the expanding layer ahead of the bore becomes small. In seeking an improved expression for the propagation of internal bores, we have rederived the steady front condition for a bore in a two-layer Boussinesq fluid in a channel under the assumption that the energy loss occurs within the contracting layer. The resulting front condition is in good agreement with available laboratory data and numerical simulations, and has the appropriate behaviour in both the linear long-wave and gravity-current limits. Analysis of an idealized internal bore assuming localized turbulent stresses suggests that the energy within the expanding layer should, in fact, increase. Numerical simulations with a two-dimensional non-hydrostatic model also reveal a slight increase of energy within the expanding layer and suggest that the structure of internal bores is fundamentally different from classical external bores, having the opposite circulation and little turbulence in the vicinity of the leading edge. However, if there is strong shear near the interface between layers, the structure and propagation of internal jumps may become similar to their counterparts in classical hydraulic theory. The modified jump conditions for internal bores produce some significant alterations in the traditional Froude-number dependence of Boussinesq shallow-water flow over an obstacle owing to the altered behaviour of the upstream-propagating internal bore.