A quantitative evaluation of Playfair's law and its use in testing long‐term stream erosion models

Abstract

Playfair's law (J. Playfair, illustrations of the Huttonian Theory of the Earth, 1802) requires any two tributaries in a river network to lower at the same rate near their junction. Although this law holds exactly at the junction, it is unclear how well it holds in the vicinity of the junction. This issue has practical importance because Playfair's law has been used to estimate parameters for detachment-limited models of erosion. If the incision rate of a stream is modelled asˇA m S n , whereˇis an erodibility parameter, A is the area drained by the stream, and S is the local gradient of the channel, then the ratio of the parameters m/n can be estimated from junctions by assuming that Playfair's law holds over the distance used to determine S for each tributary. In this paper, Playfair's law and associated m/n estimates are evaluated for simulated basins with constant and temporally varying uplift rates (or baselevel lowering rates). The results demonstrate that estimates of m/n may be biased for basins with upward-concave stream profiles because the local slope must be approximated with an average upstream slope. In addition, when uplift rate varies temporally, knickpoints are shown to travel through the basins with constant vertical velocity. Because incision rates vary within the basin, Playfair's law only holds exactly at the junctions. These effects are more important when slopes are measured over longer distances. Finally, measurement techniques are presented which address these potential biases.

How concave are river channels?

“…Therefore, any change in incision rate on the main stem channel will be transmitted to the upstream tributaries. Using simple geometric relationships, Niemann et al (2001) showed that a knickpoint should migrate upstream with a horizontal celerity (Ce h , in length per time) of…”

River profile response to normal fault growth and linkage: an example from the Hellenic forearc of south-central Crete, Greece

“…Herein, we utilize the variable τ to indicate to readers that the time component associated with τ determinations in our study is relative, not absolute, as it is based on empirically derived constants that may not represent actual values. The calculation of τ is valuable in areas of spatially variable uplift rate such as south-central Crete because the elevation of knickpoints from the same relative base-level fall event will not lie at the same elevation, as is expected if uplift rate is uniform (Niemann et al, 2001). It is also worth noting that if n = 1 and K is uniform, then χ effectively represents the relative response time of a river network and thus knickpoints from a common base-level fall event will cluster around the same χ distance.…”

Multi-phased uplift of the southern margin of the Central Anatolian plateau, Turkey: A record of tectonic and upper mantle processes

“…The downstream portion of the channel, which is adjusted to the new uplift rate, is typically characterized by a higher k s (e.g., Snyder et al, 2000). In areas of relatively uniform climate and lithology and no dependence of θ on uplift rate, the vertical component of knickpoint migration should be constant (Niemann et al, 2001). Observing knickpoints at a similar elevation within a drainage network is, there-fore, one line of evidence to support a past base level drop or increase in uplift rate Wobus et al, 2006).…”