Topographic expressions of bars in channels with variable width

A three dimensional quasi-analytical model is introduced to determine the flow field and the altimetric response of movable-bed channels subject to periodic width variations. The basic assumptions underlying the analysis are those of small amplitude of width variations and wide channel, so that non linear effects and side wall effects are neglected. The aim of the work is to determine the conditions under which the channel is planimetrically stable or unstable, i.e. it tends to damp (or enhance) a given initial (infinitesimal) perturbation of the channel width due to bank erosion. A simple bank erosion model is adopted whereby the rate of bank retreat is related to the excess shear stress at the banks. Theoretical results suggest that the equilibrium bottom profile is mainly constituted by two components. The first component represents a purely longitudinal bottom deformation, which induces deposition at the widest section and scour at the constraint, where the cross sectionally averaged velocity attains its maximum value. The second component is mainly originated by three dimensional effects and induces a transverse deformation of the bed in the form of a central bar. Its relative position with respect to the former component changes with the length of width variations: under suitable conditions the flow divergence induced by the central bar leads to a maximum velocity at the banks in wide sections, which implies that width variations tend to amplify.