Run-up and backwash bore formation from dam-break flow on an inclined plane

Nonlinear shallow water equations are employed to model the inviscid slumping of fluid along an inclined plane and analytical solutions for the motion are derived using the hodograph transformation to reveal the run-up and the inception of a bore on the backwash. Starting from rest, the fluid slumps along the inclined plane, attaining a maximum run-up, before receding and forming a relatively thin and fast moving backwash. This interacts with the less rapidly moving fluid within the interior to form a bore. The evolution of the bore and the velocity and height fields either side of it are also calculated to reveal that it initially grows in magnitude before diminishing and intersecting with the shoreline. This analytical solution reveals features of the solution, such as the onset of the bore and its growth and decline, previously known only through numerical computation and the method presented here may be applied quite widely to the run-up of other initial distributions of fluid.