Nonparallel spatial stability of shallow water flow down an inclined plane of arbitrary slope
Roll waves are known to occur in the frictional flow of a thin layer of water down an inclined solid surface. For a layer of constant depth, the formation of these waves on a solid plane with small slope angle have been explained as a hydrodynamic instability occurring above a critical Froude number by analyzing the temporal stability of the constant velocity flow. Here we analyze the linear, spatial stability of the shallow-water flow down an inclined plane of arbitrary slope including the first order effects of the gradients of both the velocity and the water depth. For constant water depth (parallel case) we reproduce previous results of the temporal stability analysis. Then we apply the nonparallel stability formulation to the kinematic wave approximation of the shallow-water flow down an inclined plane of arbitrary slope and characterize, analytically, the frequency and the wavelength of the most unstable waves as a function of the Froude number, slope angle, velocity and velocity gradient. We find important qualitative differences with respect to the parallel (constant depth) case. These stability results are used to discuss the numerical solution to the nonlinear shallow-water flow equations for the dam-break problem on an inclined surface of arbitrary slope. A good agreement between the waves resulting from the numerical simulations and the predictions of the stability analysis is found.