Confluence

Abstract: In this chapter we will present the basic concepts of term rewriting that are needed in this book. More details on term rewriting, its applications, and related subjects can be found in the textbook of Baader and Nipkow [BN98]. Readers versed in German are also referred to the textbooks of Avenhaus [Ave95], Bundgen [Bun98], and Drosten [Dro89]. Moreover, there are several survey articles [HO80, DJ90, Klo92, Pla93] that can also be consulted.

Abstract: 1. Abstract reduction systems 2. First-order term rewriting systems 3. Examples of TRSs and special rewriting formats 4. Orthogonality 5. Properties of rewriting: decidability and modularity 6. Termination 7. Completion of equational specifications 8. Equivalence of reductions 9. Strategies 10. Lambda calculus 11. Higher order rewriting 12. Infinitary rewriting 13. Term graph rewriting 14. Advanced ARS theory 15. Rewriting based languages and systems 16. Mathematical background.

Abstract: River channel confluences form important morphological elements of every river system, being points at which rapid changes in flow, sediment discharge and hydraulic geometry must be accommodated. This article presents results of a quantitative investigation of sediment transport at channel confluences accomplished through both scaled laboratory flume simulation and complementary monitoring of a natural channel confluence. Bed morphology at channel confluences is characterized by three distinct elements: avalanche faces at the mouth of each confluent channel, a deep central scour and a bar within the separation zone formed at the downstream junction corner. These elements are controlled predominantly by the confluence angle and the ratio of discharges between the tributary and mainstream channels. As confluence angle and discharge ratio increase, the sediment contributions from the confluent channels are progressively segregated in their paths through the junction, with sediment being transported around rather than through the centre of the confluence. This segregation of sediment loads is accompanied by the retreat of the main channel avalanche face from the confluence, an increase in the scour depth, a change in the orientation of the scour and an increase in the size of the separation zone bar. Field measurements closely replicate the flume simulation. A model of sediment transport and bed morphology links these features to the fluid dynamics of these sites. An understanding of confluence dynamics is important not only in considerations of channel morphology and design criteria but must form the basis for the interpretation of confluence sediments in the ancient record.