Comparing the transport-limited and <i>ξ</i>–<i>q</i> models for sediment transport

Abstract

Abstract. Here I present a comparison between two of the most widely used reduced-complexity models for the representation of sediment transport and deposition processes, namely the transport-limited (or TL) model and the under-capacity (or ξ–q) model more recently developed by Davy and Lague (2009). Using both models, I investigate the behavior of a sedimentary continental system of length L fed by a fixed sedimentary flux from a catchment of size A0 in a nearby active orogen through which sediments transit to a fixed base level representing a large river, a lake or an ocean. This comparison shows that the two models share the same steady-state solution, for which I derive a simple 1D analytical expression that reproduces the major features of such sedimentary systems: a steep fan that connects to a shallower alluvial plain. The resulting fan geometry obeys basic observational constraints on fan size and slope with respect to the upstream drainage area, A0. The solution is strongly dependent on the size of the system, L, in comparison to a distance L0, which is determined by the size of A0, and gives rise to two fundamentally different types of sedimentary systems: a constrained system where L<L0 and open systems where L>L0. I derive simple expressions that show the dependence of the system response time on the system characteristics, such as its length, the size of the upstream catchment area, the amplitude of the incoming sedimentary flux and the respective rate parameters (diffusivity or erodibility) for each of the two models. I show that the ξ–q model predicts longer response times. I demonstrate that although the manner in which signals propagates through the sedimentary system differs greatly between the two models, they both predict that perturbations that last longer than the response time of the system can be recorded in the stratigraphy of the sedimentary system and in particular of the fan. Interestingly, the ξ–q model predicts that all perturbations in the incoming sedimentary flux will be transmitted through the system, whereas the TL model predicts that rapid perturbations cannot. I finally discuss why and under which conditions these differences are important and propose observational ways to determine which of the two models is most appropriate to represent natural systems.