Self-contained two-layer shallow water theory of strong internal bores

Abstract

We show that interfacial gravity waves comprising strong hydraulic jumps (bores) can be described by a two-layer hydrostatic shallow-water (SW) approximation without invoking additional front conditions. The theory is based on a new SW momentum equation which is derived in locally conservative form containing a free parameter $\alpha.$ This parameter, which defines the relative contribution of each layer to the pressure at the interface, affects only hydraulic jumps but not continuous waves. The Rankine-Hugoniot jump conditions for the momentum and mass conservation equations are found to be mathematically equivalent to the classical front conditions, which were previously thought to be outside the scope of SW approximation. Dimensional arguments suggest that $\alpha$ depends on the density ratio. For nearly equal densities, both layers are expected to affect interfacial pressure with approximately equal weight coefficients, which corresponds to $\alpha\approx0.$ The front propagation velocity for $\alpha=0$ agrees well with experimental and numerical results in a wide range of bore strengths. A remarkably better agreement with high-accuracy numerical results is achieved by $\alpha=\sqrt{5}-2,$ which yields the largest height that a stable gravity current can have.