1. A unification of channel incision laws in steep, coarse catchments based on high Shields numbers D. E. J. Hobley, H. D. Sinclair, P. A. Cowie 1. The transport-limited hypothesis for channels is valid (e.g., Whipple & Tucker, 2002), i.e., sediment transport capacity, Q c, is a power function of bed shear stress,  : and 2a. Dimensionless transport capacity, Q*, is a power function of the Shields number,  *: where and 2b. The power, p, incorporates the incision threshold. We use parameters reflecting a modified version of the Meyer-Peter-Muller expression (Sinha & Parker, 1996): But note this is only valid only close to a reference stress, i.e., close to  c. We work forward with this formulation, treating  Q as a constant. 3. Behaviour under perturbation Suppression of  at high d  Amplification of  at low d , very powerful at low  * Erosion only permitted under this equation while  * < 1 Change in shear stress downstream 4. Behaviour under steady state (i.e., graded) conditions Whipple & Tucker, 2002  t =  d + (  c /  0 – 1) Then under graded conditions where  0 is constant & incrementally greater than  c  d –  t → +0 i.e., we predict no slope-area scaling break between detachment- and transport-limited systems should be obvious under steady state conditions, for any values of  c and  * (c.f., Fig. 4.2; Whipple and Tucker, 2002). However, transport-limited reaches will naturally occur downstream of detachment- limited reaches under these conditions. However, in the general case, from the assumptions we may express the intrinsic concavities of transport- and detachment limited systems as: 2. Key assumptions 1. Rationale Valley floor glacial sediments, Basgo valley, Lahakh Thick incising glacial sediments, Leh valley, Lahakh Typical river channel, Basgo valley, Lahakh Incised debris flow sediments, Phyang valley, Lahakh Landslide loaded channel, Young River, NZ http://www.geonet.org.nz/images/news/2008/ landslide_view.jpg Incision into coarse, very poorly sorted loose material is a very common channel situation in upland environments. Examples may be: -in glacial environments, where rivers cut into valley drapes of diamicton, -where landslides and debris flows feed rivers directly, -where jointed rock is exposed and plucking may be important. However, this situation is poorly described by existing analytical modelling approaches, primarily since erosion must proceed by moving blocks of fixed sizes (i.e., a transport-limited problem), but where the threshold of incision is very high. In such settings, mobilising sediment may be of comparable difficulty to transporting it away. We gain new insight into the dynamics of transport-limited systems by analytically modelling such situations, explicitly considering the threshold of motion of the sediments,  c, and the Shields Number,  *, in the calculations. Plucking from riverside outcrop, Ladakh (FoV 50cm) A1. Glossary of terms 5. A natural definition of a detachment-limited river Form of Fig. 3.1 indicates that for appropriate settings, erosional response will be greatest in intermediate intensity shear stress events (c.f., Hartshorn et al., 2002), an entirely novel result for a transport-limited system. This form is extremely reminiscent of experimental erosional trends obtained by Sklar & Dietrich (2001), and of erosional functions modelled by these authors (2004) and Turowski et al. (2007) (see Fig. 3.2). Such nonlinearities are known to have significant consequences for channel evolution through time (e.g., Gasparini et al., 2006); our equation will have similarly significant consequences which we intend to model subsequently. In Fig. 3.1a we see a transition from expected smoothly increasing erosional signals where  * is small (black and red curves) through to a highly nonlinear “humped” curve as  * approaches 1 (purple and cyan). This complex behaviour is induced by the comparable difficulties of detaching clasts versus transporting them away, represented by the Shields number. This situation may also be representative of channels where erosion proceeds by plucking. Many studies of steep mountain rivers hypothesised to be responding in a detachment-limited or hybrid manner have reported elevated  * values compared to gravel-bed transport-limited rivers, which typically exhibit  * = 0.06-0.12 (e.g., Figs. 5.2, 5.3). We suggest that these values can provide a quantitative assessment of the erosional style occurring in the channel. Fig. 5.3. Data from channel in Ladakh, hypothesised to show nonlinear transport-limited behaviour. Fig. 5.2. Shields numbers for rivers in Italian Appenines, interpreted as eroding in a purely detachment-limited style (from Whittaker et al., 2007). NB: - equation is only valid for small values of d  /  c. This is reasonable in settings where sediments are very coarse and partially mobile, as surface armouring allows the threshold to track imposed shear stresses. Assume an instantaneous perturbation, i.e., neglect uplift and variation in width, porosity. Then by substitution: Turowski et al., 2007 Sklar & Dietrich, 2001 University of Edinburgh, UK. Email: dan.hobley@ed.ac.uk Fig. 3.1. Graphs showing output from equation (3.1) and assuming constant d  /dx, for (a) constant  c = 50 Pa, and (b) constant  * = 0.99. A2. Mathematical approach Under the assumptions above and solving the equations analytically as outlined in box A1 we show: Hence (A2.1) (A2.2 a,b) (A2.3) Then (2.1b) is solvable by logarithmic differentiation and the chain rule as: (2.1 a,b) (2.2) (2.3 a,b) (2.4 a,b) (3.1) (4.1) (4.2) References  * << 1: Normal transport- limited conditions.  * → 1: Non-linear transport- limited response; detachment limited response becoming important. Hybrid behaviour.  * > 1: Fully detachment- limited conditions.  * = 1 ** …but is k* = 1 in (2.3b)? Fig. 5.1. Fig. 4.1. Schematic of equilibrium channel long profiles produced by (3.1); normal width scaling and Hack’s Law (h=2) assumed. Dark blue curve exhibits normal concavity (  =0.5); curves become more linear with increasing U. Increasing U U = 0 Equation (3.1) can produce stable steady state profiles, provided climate storminess does not change through time. However, only with no uplift will channel long profile exhibit “normal” slope-area scaling. Increasing uplift gives a more linear profile. Fig. 4.2. Schematic diagram of previously assumed scaling relations between equilibrium detachment- and transport-limited systems. We argue the illustrated scaling break will never be visible. Under our assumptions we suggest that transport- and detachment- limited processes should be considered as active simultaneously in a channel. Relative value of  * determines whether the river “sees” its bed as primarily loose clasts or solid material it must detach (Fig. 5.1). Fig. 3.2. Graphs showing nonlinear dynamics suggested by other authors for detachment- limited systems. The upper results are experimental; the lower are theoretical. A: upstream drainage area D char : Characteristic grain size of eroded sediment S: channel slope Q c : sediment flux capacity Q*: dimensionless Einstein sediment transport rate U: uplift rate of channel bed w: channel width x: downstream distance z: channel bed elevation  : erosion rate  : shear stress on bed  c : critical shear stress for sediment mobilisation  *: dimensionless “Shields-type” number (c.f. 1.3b) d  : increment of shear stress away from threshold “0” subscript denotes a reference value of the variable g: acceleration due to gravity K t, K t ’, k*: constants of proportionality (k* is dimensionless)  : shear stress exponent p : sediment porosity  s : sediment density  w : water density  : drainage area exponent : slope exponent  d : intrinsic concavity of detachment limited system  t : intrinsic concavity of transport limited system (=(  -1)/ )) F: constant reflecting erosive efficiency (ratio of d  /dx to U) k h : constant in Hack’s law k  : constant reflecting size of incision threshold at steady state Gasparini, N. M., Bras, R. L., and Whipple, K. X., 2006, Numerical modeling of non-steady-state river profile evolution using a sediment-flux-dependent incision model, in Willett, S. D., Hovius, N., Brandon, M. T., and Fisher, D., eds., Tectonics, Climate, and Landscape Evolution: Geological Society of America Special Paper, p. 127-141. Hartshorn, K., Hovius, N., Dade, W. B., and Slingerland, R. L., 2002, Climate-Driven Bedrock Incision in an Active Mountain Belt: Science, v. 297, p. 2036-2038. Sinha, S. K., and Parker, G., 1996, Causes of concavity in longitudinal profiles of rivers: Water Resources Research, v. 32, no. 5, p. 1417-1428. Sklar, L. S., and Dietrich, W. E., 2001, Sediment and rock strength controls on river incision into bedrock: Geology, v. 29, no. 12, p. 1087-1090. -, 2004, A mechanistic model for river incision into bedrock by saltating bed load: Water Resources Research, v. 40, no. 6, p. 22. Turowski, J. M., Lague, D., and Hovius, N., 2007, Cover effect in bedrock abrasion: A new derivation and its implications for the modeling of bedrock channel morphology: Journal of Geophysical Research, v. 112, no. F4, p. 16. Whipple, K. X., and Tucker, G. E., 2002, Implications of sediment-flux-dependent river incision models for landscape evolution: Journal of Geophysical Research, v. 107, no. B2, p. 18. Whittaker, A. C., Cowie, P. A., Attal, M., Tucker, G. E., and Roberts, G. P., 2007, Contrasting transient and steady-state rivers crossing active normal faults: new field observations from the Central Apennines, Italy: Basin Research, v. 19, no. 4, p. 529-556.  * = 0.12  * = 0.6  * = 0.9  * = 0.95  * = 0.98  * = 0.99  c = 50 Pa  c = 100 Pa  c = 150 Pa  c = 200 Pa  c = 250 Pa  * = 0.99) (a)(b)