Run-Up of a Vortex Hydrodynamic Bore onto the Shore

The run-up of a vortex hydrodynamic bore onto an inclined beach is the subject of this study. To theoretically analyze this problem, we use the Benny equations, which, within the shallow water model, allow us to take into account the distribution of horizontal fluid velocity along the depth of the fluid layer. We show that the presence of a shear flow behind a bore significantly modifies the different regimes of bore motion toward the shoreline depending on its strength. The subsequent collapse of the bore near the shoreline with the release of a high-speed run-up jet onto the dry shore is also significantly modulated by the degree of vorticity of the fluid flow. The maximum flooding length and run-up height increase significantly with increasing vorticity of the fluid flow. We use a theoretical model based on the characteristic Whitham rule for a bore, supplemented by the laws of conservation of mass and the momentum of a liquid crossing a shock wave. It is assumed that the wave run-up that appears after the "collapse" of the bore is determined by gravity. As a result, the maximum value of the wave run-up, its speed, the influence of flow vorticity, and its structure as a whole are estimated. The acceptable agreement of the simulation results with experimental data can serve as a justification for the applicability of our model to the calculation of the bore run-up onto a sloping beach.