Stability and the conservation of mass in drainage basin evolution

Abstract

Drainage basin evolution is modeled as the time development of an initial surface subject to conservation of sediment and water and a transport law qs = F(S, q) connecting the sediment flux qs with the local slope S and the discharge of surface water q. Two models are presented. The first is appropriate to a smooth surface on which no discrete channels have formed, and the second is appropriate to a family of V‐shaped valleys, each containing a separate stream of negligible width. For the first model, some solutions are presented that describe the evolution of a long ridge for which the profile is independent of one spatial coordinate. The stability of such surfaces is then discussed. It is shown that, if F/q < ∂F/∂q, disturbances of small amplitude and small lateral scale will grow rapidly and presumably will lead to the formation of closely spaced channels directed down the slope, whereas, if F/q ≥ ∂F/∂q, such channels will tend to disappear. For a surface eroding without change of shape, convex portions are stable, and concave segments are unstable. The second model is less well developed, since conservation principles alone are insufficient to determine its evolution without some additional postulate describing the sideways migration of individual streams. It is shown how, if each stream moves so that the sediment fluxes entering from its two side slopes remain equal, a system of similar parallel valleys is unstable, and neighbors will tend to coalesce on a time scale comparable to that for erosion through a layer as thick as a valley is deep.

Channelization of plumes beneath ice shelves

“…The basic mechanism here is that regions of high plume velocity induce enhanced heat transfer and thus faster melting, which increases the local transverse slope of the ice-shelf and leads to flow focussing. There is a partial analogy with hill-slope erosion (Smith & Bretherton 1972). We seek to determine how a one-dimensional ice-shelf profile is affected by transverse perturbations in the conditions at the grounding line.…”

Mathematical Analysis of a Model of River Channel Formation

“…This positive feedback induces instability, as was shown by SMITH and BRETHERTON (1972), in their pioneering study.…”

“…We complete the previous modeling of the problems by SMITH and BRETHERTON (1972) and FOWLER et al (2007), obtaining a model which involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a superlinear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation.…”

Mathematical Models for Erosion and the Optimal Transportation of Sediment

“…The second group has produced remarkable simulations of evolving channel networks; see [52,53], [18], [48] and [35]. The third group has lead to an increasing understanding of the physical mechanisms that underlie erosion and channel formation; see [40], [42], [30], [36], [28], [27], [29], [43], [22,23,24,25,21], [44,45,46], [39], [50], [9], [15], [7], [41].…”

“…These equations improve the original model in [42] by including a pressure term (in addition to the gravitational and friction terms) that prevents water from accumulating in an unbounded manner in surface concavities; see [28]. They present a representation of the free water surface in a diffusion analogy approximation to the St. Venant equations; see [51].…”