What? River response to base level rise. The morphodynamic system. Why? Channel-forming discharge. Flow. u = What s in a name. Flow Sediment transport

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1 River response to base level rise and other boundary conditions Dr. Maarten Kleinhans Summer course climate change and fluvial systems Course materials of Prof. Gary Parker Flow Sediment transport What? Mass conservation and equilibrium profile Effects of changing boundary conditions Play with the models of Gary Parker /~parker/morphodynamics_e-book.htm and have fun! 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS Gary Parker November, 24 Why? We need basic (trained) intuition of the effects of conservation of mass Input = output - storage flow The morphodynamic system Introduction River flood waves Hydraulic roughness Bedforms Equilibrium river profile is helpful concept to test effects of changing boundary conditions Upstream: discharge, sediment feed Downstream: base level Along the river: initial conditions of slope, sediment composition, entrenchment Sediment tr. morphology Sediment transport Mixture effects Channel patterns Downstream fining Bars, bends, islands Overbank sedimentation Hydraulic geometry example connections Flow from old laws: 2/ 3 1/ 2 ( R S ) Flow u = What s in a name n Manning law u = C RS Chezy law 8gRS u = f Darcy-Weisbach law Here: Manning-Strickler for friction H C = α r kc 1 /6 Channel-forming discharge common frequency years definitions based on: sediment transport frequency channel dimensions Herein: assume simple channel dimensions! because channel margins (levees, banks) formed at discharge which just floods the banks ~ bankful discharge equilibrium assumed! 1

Sediment moved over time S=u 3 or u 5 S S avg Q Qchannel forming Bankfull (channel forming) flow discharge: the flux Q = Au A Wh Q = Whu H depth bankfull Q flow discharge m 3 /s u flow velocity Area

2 Sediment moved over time S=u 3 or u 5 S S avg Q Qchannel forming Bankfull (channel forming) flow discharge: the flux Q = Au A Wh Q = Whu H depth bankfull Q flow discharge m 3 /s u flow velocity Area (bankfull) Width wetted Perimeter Q X probability Intermittency of bankfull discharge Approach to full discharge regime Intermittency of bankfull discharge Typically.3-.1 for small flashy to large rivers Compute sediment transport for I Backwater curve Subcritical flow, decrease to S= (basin) water surface elevation (base level) is raised at t = by e.g. installation of a dam Q low flow flood t sediment supply remains constant at q sa antecedent equilibrium bed profile established with load q sa before raising base level About Froude: subcritical and supercritical flow slow fast downstream control no downstr. control Fr < 1 Fr > 1 bed shear stress (and sediment mobility) τ = ρgr sin τ* = ( S ) τ ρgrs [( ρ s ρ ) gd ] Shear stress Flow shear stress on the bed (Newton) 5 Shields number: Sediment-entraining force vs. sediment-detraining force 2

3 Sediment transport Meyer-Peter and Mueller type: q nt ( τ τ ) τ > τ t = α t c, c α=8, n=1.5 Einstein parameter: q t = q [( ρ ρ) / ρ ] gd D s t s About sediment transport: Shields mobility number 1 1,1 silt,1 sand Shields criterion Suspended transport Bed load transport RIJN ALLIER GRENSMAAS gravel Grain size/diameter (mm) About channel pattern: Seven equations needed (1) 1. Boundary conditions: Q, H (downstream for Fr<1) Allier Meuse Rhine Volga Stream power Braiding, often Fr~1 (Van den Berg, 1995) Grain size Meandering, often Fr<<1 2. Water continuity: Q=uwh (= mass conservation) 3. Chezy (or Darcy-Weisbach or Manning) u=c(hs) 1/2 4. Slope S 1. either as fixed boundary condition (mountains, large disequilibrium rivers) or from sediment transport and input! 2. Sediment continuity: 5. Roughness predictor C 6. Sediment transport predictor 7. Width of the channel W S qb (1 λp) = - t x transport gradient λ=poros =bed level q b =transp t=time x=location Channel change General rate of change: qb ( 1 λp ) = - 1. Exner: h/ t~ q b / x t x 2. So after a sudden change the gradient (and thus q b / x) is large 3. Therefore morph change fast 4. But then gradient decreases and morph change less fast 5. y(t) = y equil +/- ae -b t 6. Exponential decrease or increase with representative T: Note: Seven equations needed (3) 1. Slope of equilibrium channel: 1. More water, less sediment input: smaller slope 2. Less water, more sediment input: larger slope 3. Sea-level rise/fall: specify as boundary condition H 4. Climate change: specify as boundary condition Q (q b ) 5. Tectonics: specify as raising/falling bed level Slope reacts slowly parameter ~63 % of change accomplished at T time 2. Equilibrium slope diffusive character: 1. Bump ->local flow acceleration ->increase sediment transport ->bump removed! 2. BUT: bedforms and bars! Other extra mechanisms involved 3

water depth h grain flow thickness h g discharge sediment feeder and slope i Laboratory Seven equations needed (3) water depth h 1 sediment bed Note: 3. Roughness predictor: 1. Grain size 2. Bedforms!

4 water depth h grain flow thickness h g discharge sediment feeder and slope i Laboratory Seven equations needed (3) water depth h 1 sediment bed Note: 3. Roughness predictor: 1. Grain size 2. Bedforms! (no bedforms in large grains, large bedforms in small grains) 3. Bars (braid bars, meanders, etc.) Very uncertain 4. Sediment transport predictor (bedload, suspended load) 5. Width of the channel 1. NO PHYSICAL PREDICTOR AVAILABLE Uncertain Depends on channel pattern 2. Bank erosion and sediment uptake 3. Bank stability: soil type, antecedent deposits, vegetation Note: a river may react in various ways to changing Q,Qs, and how is not well known 1. Morphological change 2. River pattern change 3. Meander/bar wavelength change 4. Sediment composition (e.g. coarse top-layer or fine deposit) 2-4 are all ignored in the computer exercises. River models in practice Upstream specification Q and q b Downstream specification H (or h) Along river specification of D grain size and W width Along river specification of initial S Empirical roughness predictor is calibrated (check H) Empirical sediment transport predictor is calibrated (check rate of bedlevel change) So no bank erosion; assume fixed banks (Dutch canals ) Examples: Sobek, Wendy Long profiles of rivers Often concave! But straight slope expected? h (m) Long Profile of the Amazon River Quasi-equilibrium long profiles quasi implies not equilibrium where sediment output equals input over each reach. That would nearly always give a straight slope. Causes of concavity: Subsidence Sea level rise -> downstr. slope decreases Delta progradation -> downstr. slope decreases Downstream sorting of sediment -> fining Abrasion of sediment -> fining Effect of tributaries: increase of discharge! Antecedent relief: drop from mountains to the plain x (km) 4

Effect concavity on width The Kosi River flows into a zone of rapid subsidence. Subsidence forces a streamwise decline in the sediment load in a similar way to sea level rise.

5 Effect concavity on width The Kosi River flows into a zone of rapid subsidence. Subsidence forces a streamwise decline in the sediment load in a similar way to sea level rise. Note how the river width decreases noticeably in the downstream direction. Kosi River and Fan, India (and adjacent countries). Image from NASA; Response to base level rise Backwater curve and sea level rise Together generate accomodation space Ultimate water surface Initial water surface Response to change in sediment supply Increase in load (but Q unchanged): aggradation Decrease: degradation final equilibrium bed profile in balance with load q t > q ta transient aggradational profile Ultimate bed Initial bed transient bed profile (prograding delta) sediment supply increases from qta to q t at t = antecedent equilibrium bed profile established with load q ta Elevation in m Examples aggradation/degradation Bed evolution yr 5 yr 12 1 yr 15 yr 1 2 yr 25 yr 8 Ultimate Distance in m aggradation Elevation in m degradation Bed evolution yr 2 yr 4 yr 6 yr 8 yr 1 yr Ultimate Distance in m Elevation in m Delta progradation Bed evolution (+ Water Surface at End of Run) Distance in m bed yr bed 2 yr bed 4 yr bed 6 yr bed 8 yr bed 1 yr bed 12 yr ws 12 yr 5

Example delta progradation Response to sudden faulting Back to equilibrium Time scale depends on transport rate and fault height Missouri River prograding into Lake Sakakawea, North Dakota.

Response to downstream base level RTe-book1DRiverwFPRisingBaseLevelNormal.xls Optional 3. Gilbert-type delta building RTe-bookAgDegBW.xls 4. Response to faulting RTe-bookAgDegNormalFault.

6 Example delta progradation Response to sudden faulting Back to equilibrium Time scale depends on transport rate and fault height Missouri River prograding into Lake Sakakawea, North Dakota. Image from NASA website: Computer exercises 1. Response to upstream Q and qs RTe-bookAgDegNormal.xls 2. Response to downstream base level RTe-book1DRiverwFPRisingBaseLevelNormal.xls Optional 3. Gilbert-type delta building RTe-bookAgDegBW.xls 4. Response to faulting RTe-bookAgDegNormalFault.xls Computer exercises - sample Calculation of River Bed Elevation Variation with Normal Flow Assumption Calculation of ambient river conditions (before imposed change) Assumed parameters Q 7 m^3/s Flood discharge (Qf) (Inter) If.3 Intermittency The colored boxes: (B) B 25 Channel Width indicate the parameters m you must specify. (D) D 3 mm Grain Size The rest are computed for you. (lamp) λ p.35 Bed Porosity (kc) k c 75 Roughness Height mm If bedforms are absent, set k c = k s, where k s = n k D and n k is an order-one factor (e.g. 3). (S) S.8 Ambient Bed Slope Otherwise set k c = an appropriate value including the effects of bedforms. Computed parameters at ambient conditions H m Flow depth (at flood) τ * Shields number (at flood) q* Einstein number (at flood) q t.4859 m^2/s Volume sediment transport rate per unit width (at flood) G t 3.5E+5 tons/a Ambient annual sediment transport rate in tons per annum (averaged over entire year) Calculation of ultimate conditions imposed by a modified rate of sediment input G tf 7.E+5 tons/a Imposed annual sediment transport rate fed in from upstream (which must all be carried during floods) q tf m^2/s Upstream imposed volume sediment transport rate per unit width (at flood) τult Ultimate equilibrium Shields number (at flood) S ult.1427 Ultimate slope to which the bed must aggrade Click the button to perform a calculation H ult m Ultimate flow depth (at flood) Calculation of time evolution toward this ultimate state L 1 m length of reach Ntoprint 2 Number of time steps to printout qt,g m^2/s sediment feed rate (during floods) at ghost node Nprint 5 Number of printouts x 1.67E+2 m spatial step M 6 Intervals t.1 year time step.5 Here 1 = full upwind,.5 = central difference αu Duration of calculation 1 years Computer exercises - sample 6

Modelling of flow and sediment transport in rivers and freshwater deltas Peggy Zinke

1 Modelling of flow and sediment transport in rivers and freshwater deltas Peggy Zinke with contributions from Norwegian and international project partners 2 Outline 1. Introduction 2. Basic ideas of flow