Linear stability of the 1D Saint-Venant equations and drag parameterizations

Abstract

The stability of the homogeneous and steady flow based on the one-dimensional Saint-Venant equations for free surface and shallow water flows of constant slope is derived and displayed through graphs. With a suitable choice of units, the small and large drag limits, respectively, correspond to the small and large spatio-temporal scales of a linear system only controlled by the Froude number and two other dimensionless numbers associated with the bottom drag parameterization. Between the small drag limit, with the two families of marginal and non-dispersive shallow water waves, and the large drag limit, with the marginal and non-dispersive waves of the kinematic wave approximation, dispersive roll waves are detailed. These waves are damped or amplified, depending on the value of the three control parameters. The spatial generalized dispersion relations are also derived indicating that the roll-wave instability is of the convective type for all drag parameterizations.