Competition between kinematic and dynamic waves in floods on steep slopes

We present a theoretical stability analysis of the flow after the sudden release of a fixed mass of fluid on an inclined plane formally restricted to relatively long time scales, for which the kinematic regime is valid. Shallow-water equations for steep slopes with bed stress are employed to study the threshold for the onset of roll waves. An asymptotic solution for long-wave perturbations of small amplitude is found on background flows with a Froude number value of 2. Small disturbances are stable under this condition, with a linear decay rate independent of the wavelength and with a wavelength that increases linearly with time. For larger values or the Froude number it is shown that the basic flow moves at a different scale than the perturbations, and hence the wavelength of the unstable modes is characterized as a function of the plane-parallel Froude number Fr(p) and a measure of the local slope of the free-surface height phi by means of a multiple-scale analysis in space and time. The linear stability results obtained in the presence of small non-uniformities in the flow, phi > 0, introduce substantial differences with respect to the plane-parallel flow with phi = 0. In particular, we find that instabilities do not occur at Froude numbers Fr(cr) much larger than the critical value 2 of the parallel case for some wavelength ranges. These results differ from that previously reported by Lighthill & Whitham (Proc. R. Soc. A, vol. 229, 1955, pp. 281-345), because of the fundamental role that the non-parallel, time-dependent characteristics of the kinematic-wave play in the behaviour of small disturbances, which was neglected in their stability analyses. The present work concludes with supporting numerical simulations of the evolution of small disturbances, within the framework of the frictional shallow-water equations, that are superimposed on a base state which is essentially a kinematic wave, complementing the asymptotic theory relevant near the onset. The numerical simulations corroborate the cutoff in wavelength for the spectrum that stabilizes the tall of the darn-break flood.