12.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 V. Evolution of Depositional Alluvial River Profiles A.Recap:Essentials from Flow Mechanics and Sediment Transport 1. Conservation of Momentum, Steady Uniform Flow ghSb!"=dxdzS !=2uCfb!"=21*!=fCuu!"bu =*Note: Unsteady: tu!!xuu!!is usually small; exceptions: dam break floods, etc. Non-Uniform: ; xhg!!"-- important in flow around bends, over point bars, across abrupt changes in channel slope or width 2. Velocity Profiles ( )ozzkuzu ln*=hzouzhkuu37.*1ln==!!"#$$%&'=()3. Sediment Transport Shield’s Stress ( )gDsb!!""#=*steady-uniform flow: ( )( )DhSs!!!"#=*Critical Shear Stress for Initiation of Motion Shield’s diagram (explicit particle Reynold’s number) For Gravel: ( )epcrRf=*!06.003.0*!"cr#112.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 Non-Dimensional Sediment Transport wQqss=qs*=qs!s"!( )!( )gD D=qsRep#Bedload Transport (gravel) ( )23***8crsq!!"=Total Load Sand Transport qs*=0.05Cf!*2.5B. Exner Equation (Erosion Equation): Conservation of Mass (Sediment) Derivation of conservation of mass – the erosion equation SKETCH: Control reach, width Δy, length Δx, qs_in at x, qs_out at x + Δx If more sediment comes in than out, bed elevation goes up – deposition If more sediment goes out than in, bed elevation goes down – erosion per unit time Δt, sediment volume in = qs_inΔtΔy per unit time Δt, sediment volume out = qs_outΔtΔy Change in sediment volume per unit time ytqytqVoutsinss!!"!!=!__212.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 Change in bed volume per unit time poutsinspsbedytqytqVzyxV!!"##"##="#=###=#11__For change in bed elevation, divide through by ΔxΔyΔt: ( ) ( )!"#$%&''((=!!"#$$%&'((=''xqxqqtzspoutsinsp))1111__( )xqtzsp!!""=!!#11Note from basics of sediment transport that Accordingly, we can expand the erosion equation using the chain rule: ( )bsfq!=( )xqtzbbsp!!!!""=!!##$11Thus we can see immediately that erosion will occur in regions of increasing shear stress (i.e., not necessarily in regions of high shear stress) and deposition will occur in regions of decreasing shear stress (not necessarily low shear stress). C. Channel Width Closure Problem: all relations above derived in terms of flow depth, shear stress, discharge per unit width, but channel width changes downstream with: changing Qw, changing slope (S), changing D50, changing vegetation, etc.: To solve problem of alluvial river profile evolution we must specify how channel width evolves downstream. 1. Hydraulic Geometry (Leopold et al, 1950s) Empirical: 5.0Qw!2. Equilibrium (Threshold) Straight, Gravel-bed Channels (Parker, 1978) Concept: channel will widen until banks are just stable, just below the threshold for motion (erosion = widening) SKETCH 312.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 Critical or Equilibrium Condition: ; ( )c**1!"!+=4.02.0 !="Summary Condition at Bankfull flow: Mobile bed, stable banks, generally low transport stage Thus, for given Qw, D50, S, Cf, Width ( )c**1!"!+=(w) increases downstream such that h reduces to establish 3. Sandy, Meandering Channels Theory not well developed, but some evidence indicates near constant shield’s stress: ( )c**1!"!+=; 4.12.1 !="Otherwise, generally the assumption that c**!!>>is often reasonably accurate. D. Relations for Alluvial Plain Slope (Paola, 1992; with corrections) DEFINITION SKETCH 412.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 1. Conservation of Mass (Water) hwuQ =!huVwhuqVQwvw===; wVw!"Note qv denotes water discharge per unit valley width, as opposed to q which we have used previously for water discharge per unit channel width. Parallel drainage: Vw = constant; no lateral inflows of water or sediment, no loss of water to infiltration or evaporation. 0=!!xQ2. Conservation of Mass (Sediment) wqQsS=!swssvwsqVwqqVQ===where qsv is sediment transport per unit valley width [m2/s] xqtzsvp!!""=!!#113. Conservation of Momentum 512.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 ghSb!"=dxdzS !=2uCfb!"=*21uuCf=!4. Sediment Transport (Bedload = gravel) ( )( )( )23***8crsssDgDqq!!"""#=#=Dimensional sediment flux per unit channel width: ( )( )( )238!!"#$$%&'''=gDDgDqscrss(())(((( )( )( )23238crssgq!!""""##=5. Channel width closure: Braided, gravel-bedded channel -- ( )cr!"!+= 1!"#$%&+='(()))1crMeandering, sand-bedded channel – !!!="cr6. Relation for sediment flux using channel closure rule(s) ( )( )23238!""""gcqsws#=231!"#$%&+=''wc1=wcBraided, gravel-bedded channels Meandering, sand-bedded channels Write this relation in terms of slope and water discharge per unit valley width, qv 2123!!!=612.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 Using two relations for conservation of momentum: ( )21223uCxzghf!!"##$=Using conservation of mass for water: xzqCgxzuhCgvff!!"=!!"=#$$%232323Substitute into relation for sediment flux (per unit channel width): ( )( )xzCqcqsfvws!!""=###$8xzKqfs!!"=( )( )!!!"#=sfvwfCqcK8; (fluvial transport coefficient) No dependence on grainsize? Why? -- Channel width closure accounts for grainsize, ie. channel width adjusts according to grainsize. Note qs is sediment transport per unit channel width. 7. Conservation of Mass (sediment) (from above) xqtzsvp!!""=!!#11recall qsv is sediment transport per unit valley width xzKqqfssv!!"==##!"#$%&''''(=''xzKxtzfp)*11Assume: constant Vw, Cf, q (and note βis not constant, but cancels out) (diffusion equation) Effective Diffusivity: 221 xzKtzpf!!"=!!#$712.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 ( )( )psfvwpffCqcKD!"""!#$$=$=18122xzDtzf!!=!!Sediment inflow (delivered from upstream erosional source area) is Qso. Steady-state condition: xqtzsv!!==!!0Thus sediment flux per unit valley width qsv is constant at steady state: ( )wsoxsvsvVQqxq ===0Whether steady-state or not, slope at inflow must be sufficient to carry all sediment: 00 ==!!"==xfwsoxsvxzKVQq#Thus inlet slope must be (under all conditions): ( )( ) ( )QQCcCqcwQwKQKVQxzsofwsfvwssofsofwsox!!"#$$%&''=''='='=((=))))))**880where the term in brackets collects physical constants. Inlet slope is linearly dependent on the ratio Qso/Q. Note that valley width and sediment porosity do not influence the inlet slope. Steady state condition: