The overdeepening theory in river morphodynamics: Two decades of shifting interpretations

Theoretical analysis of mathematical equations is complementary to numerical solution as it offers additional insights into the fundamental behaviour of the corresponding physical system. An example is the theory of overdeepening in river morphodynamics. This theory explains why outer-bend pools can be locally deeper than the equilibrium outer-bend water depth in infinitely long river bends. Its essence lies in the superimposition of a steady alternate bar pattern on a streamwise uniform river bed topography. The same steady alternate bar pattern arises in a straight channel in response to a local permanent geometrical disturbance. Basically, the overdeepening theory describes river bed topography by a double-periodic function with an amplitude decaying exponentially along the river. This function yields two problems of interpretation. First, certain parameter combinations produce a growing rather than a decaying amplitude. Which phenomena are related to this situation? Originally, the growing amplitudes were explained as an indication of incipient braiding, but this interpretation was soon discarded because growing bed deformations in a linear theory may still reach a finite amplitude by nonlinear effects. The second problem of interpretation was whether the double-periodic function holds upstream or downstream of the local disturbance, or both. Originally it was assumed to hold only downstream, because of the very short adaptation length of the upstream flow accelerating towards a local constriction. Later upstream influence was revealed to occur as well, albeit with initially differing opinions on the corresponding parameter ranges. We review the history of the shifting interpretations of the overdeepening theory. In this way we seek to demonstrate that mathematical analyses of physical systems do not produce straightforward answers or predictions, but inherently require interpretations based on experimental evidence and a thorough understanding of the underlying physical processes.