Transient attractors: towards a theory of the graded stream for alluvial and bedrock channels
We construct a theory in which we interpret stable, self-similar, intermediate asymptotic solutions to PDEs representing conservation of mass over stream long profiles as graded streams. The theory applies to alluvial channels for which the transport of sediment is modeled by qs?q?wS?, where qw is the discharge of water and S the slope, and to bedrock channels for which the transport of sediment is modeled by (?qs/?x)?q?wS?, where x is the distance downstream. The parameters ? and ? of these similarity solutions, z(x, t)=??F(x/??), where ?=?0+?t and t is time, may be derived using dimensional analysis and are representable in terms of conditions of sediment removal at the lower boundary of the profile and (?, ?). Conditions on the physical realizability of profiles lead to constraints on admissible values of (?, ?, ?, ?, ?). All alluvial self-similar profiles and an important subset of bedrock self-similar profiles are stable and act as transient attractors when boundary conditions are unchanging. Their forms are independent of the details of their initial conditions. The remaining bedrock channels are unstable because of the spontaneous emergence of shocks that migrate upstream as breaks in slope. Two regimes of profile behavior exist: for ? 0, corresponding to low energy environments, profiles have decelerating relative loss rates and infinite life. Changes in profile elevations over time may be decomposed into upper boundary and lower boundary effects controlled, respectively, by ?? and ?? and depending ultimately on ? and ?. We examine explicit profiles for geomorphically important sets of boundary conditions: desert mountain/pediment (?+?=0); hanging valley (?=0); fixed Davisian base-level (?=0); and steady state (?=1), for alluvial and bedrock channels and various tectonic conditions. We investigate numerically the effects of changes from hanging valley to fixed base level boundary conditions, showing that profiles associated with the hanging valley attractor retreat upstream as profiles associated with the fixed based level attractor replace them from below. We demonstrate numerically the extension of the theory to streams with tributaries. The theory provides the natural definition of grade for our models, and indicates that it is a real and deep property of the associated profiles. The theory also resolves many contentious issues concerning the concept of grade. We conjecture that defining grade in terms of stable, self-similar, intermediate asymptotic solutions to conservation equations generalizes over reasonable extensions to the transport conditions of our models.
graded streams intermediate asymptotics similarity solutions fluvial geomorphology longitudinal profiles