COURSE NOTES AND STUDY GUIDE FOR GEO 432/532 APPLIED GEOMORPHOLOGY

COURSE NOTES AND STUDY GUIDE FOR GEO 432/532 APPLIED GEOMORPHOLOGY STEPHEN T. LANCASTER, ASSOC. PROF., CEOAS 1. General Advice for Exam Preparation I ve included nearly 100 equations in the summary below, not because I expect you to memorize them but, rather, because math is the most straightforward way to represent the concepts, and I expect you to understand those concepts, so I expect you to understand the equations. Many of those equations are variations on a theme, such as the Exner equation. Many others are equations that you have had to use during the term for your assignments or that we have visited repeatedly in class. Regarding what might be considered memorization, things you should know include (but are not necessarily limited to) the basic Exner equation and what it means such that you could, say, make modifications to make it fit some special cases, like some of those in the last few topics covered; the basic form of most bedload sediment transport equations; the most common equations used to calculate bed shear stress; the common implications of the normal flow approximation (i.e., what relationships are valid). Some of the key concepts for the course include (but are not necessarily limited to) sediment and grain size distributions; bankfull channels; normal flow; thresholds for initiation of motion; bed material sediment transport; hydraulic roughness due to skin friction and form drag; morphodynamics and the Exner equation (conservation of mass). Date: Spring 2013. 1

2 LANCASTER 2. Course Summary What follows is a summary of the material covered during the term relative to the outline presented in the syllabus. 1. Introduction: Felix Exner and river morphodynamics (Parker (2004), Ch. 1) Morphodynamics: the class of problems for which the flow over a bed interacts strongly with the shape of the bed, both of which evolve in time. Exner equation describes conservation of bed sediment, (1 λ p ) η t = q t x, (1) which is coupled with the St. Venant shallow water equations for conservation of water mass, H t + UH = 0, (2) x and momentum, UH t + U 2 H x = gh H x gh η x C fu 2, (3) in the downstream direction and a relation between sediment transport rate and flow hydraulics, which Exner wrote as q t = q t (U) (4) Equations (1), (2), (3), and (4) are linked via the sediment transport rate, q t, the flow velocity, U, and the bed elevation, η. 2. Sediment (Parker (2004), Ch. 2) a. Grain-size distributions ψ-scale: base-2 logarithmic scale used for grain size distribution (why?): ψ = φ = log 2 (D), (5) where D s units are mm. Important size ranges for this course are sand (D = 0.0625 2 mm, ψ = 4 1), gravel (D = 2 64 mm, ψ = 1 6), and cobbles (D = 64 256 mm, ψ =6 8). Characterize distributions in terms of N + 1 boundary sizes, D b,i, such that f f,i is the mass fraction that is finer than D b,i and N grain size ranges with sizes, and fractions, in each size range. ψ i = 1 2 (ψ b,i + ψ b,i+1 ), D i = (D b,i D b,i+1 ) 1/2, (6) D x size at which x% is finer, where, e.g., x = 50, 90, etc. f i = f f,i+1 f f,i, (7)

Geometric mean size: Geometric standard deviation: GEO 432/532 COURSE NOTES 3 D g = 2 ψ, ψ = σ g = 2 σ, σ 2 = N ψ i f i. (8) i=1 N ( ψi ψ ) 2 fi. (9) i=1 Submerged specific gravity of sediment, b. Fall velocity R = ρ s ρ 1. (10) Idea is to balance gravitational and drag forces to find terminal velocity: F g = 4 ( ) 3 D 3 πρrg = F D = 1 ( ) 2 D 2 2 πρc D, c D = c D (Re vp), Re vp = v sd 2 ν We can solve for the nondimensional fall velocity, R f = v s / RgD: [ ] 1/2 4 R f = (12) 3c D (Re vp ) We also find that Re vp = R f Re p, where Re p = RgDD/ν, which is the particle Reynolds number. So now the equation gets complicated, partly because the drag coefficient, c D, is a complicated function of Re vp. c. Modes of transport Bed material load: sediment load that exchanges with the bed (and contributes to morphodynamics). Subdivided into bedload and suspended load. Wash load is transported without exchange with the bed (generally silt and clay). Bedload: sliding, rolling, or saltating in ballistic trajectory just above bed, indirect role of turbulence. Suspended load: feels direct dispersive effect of eddies, may be wafted high into water column. 3. Bankfull channel characteristics (Parker (2004), Ch. 3) a. Bankfull concept: Discharge at which flow spills significantly onto floodplain. b. Gravel and sand beds Sand-bed streams have geometric mean (or median) bed sediment size, 0.0625 D sg 2 mm. Gravel-bed streams have D sg > 16 mm, encompasses cobble- and boulder-bed streams. Transitional streams have 2 < D sg 16 mm; few in this category. (11)

4 LANCASTER c. Characteristic dimensionless parameters: Dimensionless bankfull discharge: ˆQ = Q bf gds50 D 2 s50 (13) Dimensionless bankfull depth: Good relationship with ˆQ Ĥ = H bf D s50 (14) Dimensionless bankfull width: Good relationship with ˆQ ˆB = B bf D s50 (15) Bankfull Froude number: describes transition between subcritical (Fr < 1) and supercritical (Fr > 1) flow Fr bf = Q bf B bf H bf ghbf = U bf ghbf (16) Bankfull Shields number (estimated): Typically 0.05 for gravel-bed and 1.9 for sand-bed streams τ bf50 = H bfs RD s50 (17) Bankfull Chezy resistance coefficient: note that it s actually an inverse roughness, i.e., larger values mean lower resistance (also note how similar it is to Froude number) Cz bf = Q bf B bf H bf ghbf S = U bf ghbf S Particle Reynolds number (surrogate for grain size): (18) Re p50 = RgDs50 D s50 ν (19) 4. Conservation of bed sediment (more Exner-type equations) (Parker (2004), Ch. 4) Note list of variables and parameters and explanation of coordinate system a. Bedload (uniform grain size): nearly identical to equation (1) (1 λ p ) η t = q b x (20) b. Bedload + suspended load (1 λ p ) η t = q b x + v s (c b E) (21) c. Bedload + suspended load + subsidence or uplift ( ) η (1 λ p ) t + σ = q b x + v s (c b E) (22)

d. Sediment in suspension t GEO 432/532 COURSE NOTES 5 ( H 0 ) c dz + q s x = v s (E c b ) (23) e. Total load + subsidence or uplift (1 λ p ) ( ) η t + σ = q t x (24) f. Active layer concept: Active, exchange, or surface layer approximation: g. Mixtures Sediment grains in active layer (thickness L a )have constant, finite probability of entrainment. Sediment grains below active layer (in substrate) have zero probability of entrainment. Size fractions: F i in active layer have no vertical structure f i in substrate do not vary in time interfacial exchange f li at substrate-active layer boundary p bi = q bi /q bt in transport, q bt = N q bi (25) i=1 Conservation of mixture with active layer: [ (1 λ p ) f li t (η L a) + ] t (F il a ) = q bt p bi x Sum over i just gives equation (1), and for evolution of active layer grain size distribution, multiply (1) by f li and subtract from (26): [ F i (1 λ p ) L a t + (F i f li ) L ] a = q bt p bi q bt + f li (27) t x x where the exchange fractions depend on whether there s erosion or deposition: η f i z=η La, f li = t < 0 η (28) αf i + (1 α) p bi, t > 0, 0 α 1 5. River hydraulics (Parker (2004), Ch. 5) (26)

6 LANCASTER a. Boundary resistance and steady, uniform flow Hydraulic radius and wide-channel approximation: R h = HB B + 2H = H 1 + 2 H H, B H (29) B Shields number: scales as the ratio of drag (impelling) and friction (resisting) forces τ = τ b (30) ρrgd Average velocity: Shear velocity: Dimensionless resistance coefficient: Chezy resistance coefficient: U = u = C f = Q BH τb ρ (31) (32) τ b ρu 2 (33) Cz = U u = C 1/2 f (34) Resistance for hydraulically rough flow, Keulegan: Cz = C 1/2 f = 1 ( κ ln 11 H ) k s (35) Manning-Strickler: Cz = C 1/2 f = α r ( H k s ) 1/6 α r = 8.1, k s n k D s50 for flat bed, n k = 2. b. Normal flow: equilibrium state defined by a perfect balance between downstream gravitational impelling force and resisting bed force; flow is constant in time and in the downstream direction, i.e., uniform and steady. Conservation of mass and momentum reduce to depth-slope product rule for normal flow: τ b = ρghs, u = ghs (37) Combine momentum conservation and resistance forms: ρc f U 2 ghs = ρghs U = (38) C f With mass conservation: H = (36) ( ) Cf qw 2 1/3 (39) gs

GEO 432/532 COURSE NOTES 7 For Shields stress: τ = τ ( ) b ρrgd = Cf qw 2 1/3 S 2/3 g RD Also yields, with Manning-Strickler roughness, g U = α H 2/3 S 1/2 = H2/3 S 1/2 n k 1/6 s (40) (41) c. St. Venant shallow water equations: equations (2) and (3) d. Gradually varied flow and backwater curves Steady, gradually varied flow, St. Venant equations reduce to the backwater equation: dh dx = S S f 1 Fr 2, S = η x, S f = C f Fr 2 (42) Critical flow depth: Normal flow depth: H c = ( ) q 2 1/3 w (43) g H n = ( Cf q 2 w gs ) 1/3 or H n = (k 1/3 s q 2 w α 2 rgs ) 3/10 (44) If H n > H c, (Fr) n < 1: normal flow is subcritical, bed slope is mild. If H n < H c, (Fr) n > 1: normal flow is supercritical, bed slope is steep. For mild slope, three cases: M1: H 1 > H n > H c, flow downstream deepening into standing water, shallows and approaches normal flow upstream M2: H n > H 1 > H c, flow downstream approaches critical depth and waterfall, flow upstream deepens and approaches normal flow M3: H n > H c > H 1, flow downstream goes through hydraulic jump, shallows upstream Also family of steep curves. 6. Initiation of motion (Parker (2004), Ch. 6) a. Angle of repose: downslope gravitational force equals net Coulomb resistive force tan θ r = µ c (45) b. Critical Shields number, τ c : Shields number at threshold of motion from modified Shields diagram In limit of fully rough turbulent flow (large Re p ), τ c 0.03

8 LANCASTER c. Law of the wall for turbulent flow: u = 1 ( ) z u κ ln + B k s ( ) u k s ν where u k s /ν is the roughness Reynolds number: k s 8.5, > 8.62 or u k s > 100 (hydraulically rough) δ v ν B = 5.5 + 1 ( ) κ ln u k s k s, < 0.26 or u k s < 3 (hydraulically smooth) ν δ v ν read off the graph, otherwise (transitional flow) Law of the wall for rough turbulent flow and balance of friction and drag forces produces estimate of critical Shields number near 0.03. d. Significant streamwise slope: τ c = τ c0 cos α ( 1 tan α ) µ c e. Sediment mixtures: coarser grains are harder to move, but not as much harder as their size alone would dictate. Hiding function: τsci τscg = ( Di D sg (46) (47) (48) ) γ, 0.65 γ 0.90 (49) γ = 0: grain indepence, i.e., each size fraction behaves as if part of uniform bed γ = 1: equal threshold (or mobility) condition, i.e., hiding so effective that all grains move at the same critical boundary stress f. Condition for significant suspension: u sus = v s, or τ sus = v2 s RgD = R f 2 (Re p ) (50) g. Really a critical shear stress, or threshold of motion? Use of threshold may underpredict transport at low rates. 7. Bedload transport, mainly gravel on plane beds (Parker (2004), Ch. 7) a. Dimensionless Einstein bedload number: qb q b = (51) RgDD Seek transport as a function of Shields stress, q b = q b (τ ) or q b = q b (τ τ c ) b. Meyer-Peter and Müller bedload transport relation is typical of many (use corrected form of Wong, 2003): q b = 3.97 (τ τ c ) 3/2, τ c = 0.0495 (52) The main problem: data used for MP&M correspond to extreme flood flows in gravelbed rivers. Its threshold for initiation of motion is similar to τbf50 for most gravel-bed

GEO 432/532 COURSE NOTES 9 streams, so it predicts little or no transport of larger grain sizes, which do in fact move at bankfull, so MP&M relation, and those like it, are not the best relations for gravel-bed streams. c. Similarity of many relations motivates formulation of K (or W ): ( ) q b u 2 3/2 = K or K = Rgq b RgDD RgD u 3 (53) d. Sheet flow: τ < τsheet : Bedload localized to rolling, sliding, and saltating grains that exchange only with immediate bed surface. τ > τsheet : Bedload layer devolves into sliding layer of grains up to several grains thick. Estimates: τsheet 0.5 1.5, decreases with increasing Froude number. e. Mechanistic derivations: based on ξ bl : Volume of sediment in transport per unit area; u bl : Mean velocity of bedload particles; E bl : Volume rate of entrainment of bed particles into bedload movement (not suspension) per bed area; L sbl : Average step length of bedload particle, from entrainment to deposition, typically over many saltations. From continuity: q b = ξ bl u bl (cf. Bagnold), q b = E bl L sbl (cf. Einstein) (54) f. Calculations with bedload transport relations: Specify 1) sediment submerged gravity, R; 2) representative, or characteristic, surface grain size, e.g., D sg or D s50, for D in the relation; 3) value for shear velocity, u (and thus τ b ) from flow field with techniques of Ch. 5, e.g., ( ) u 2 k 1/3 3/10 ( ) s qw 2 = g 7/10 S 7/10 τ ks 1/3 3/10 qw 2 S 7/10 = (55) αrg 2 RD Compute α 2 r 1) τ = u2 RgD 2) q b = q b (τ ) 3) q b = RgDDq b

10 LANCASTER g. Alternative dimensionless bedload transport: W W is just K from above: W = q b τ = Rgq b 3/2 u 3 Most bedload transport relations can be expressed in terms of W h. Surface-based bedload transport formulation for mixtures: For active layer with thickness, L a = n a D s90, n a 1 2, divide bed material into N grain size ranges with characteristic size D i ; Let F i be fraction of material in surface (active) layer in i th size range; Volume of bedload transport per unit width in i th size range is q bi, total bedload is N q bt = q bi ; (57) i=1 Fraction of bedload in i th size range is p bi, where Define dimensionless terms for i th size range: (56) p bi = q bi q bt (58) τi = τ b = u2 ρrgd i RgD i (59) qbi q bi = RgDi D i F i (60) W i i. Parker bedload relation for gravel mixtures: Must renormalize F i s so that sand is removed. = Rgq bi u 3 F i (61) Based on surface geometric mean size, D sg (8), surface arithmetic standard deviation of ψ, σ s (9), (not geometric standard deviation, 2 σs ) computed from renormalized F i s, calculate transport in terms of a mobility function, G: W i = (0.00218)G(φ i ) (62) ( ) 0.0951 Di φ i = ωφ sg0, φ sg0 = τ sg, τ D sg τssrg sg = u2, τssrg = 0.0386 (63) RgD sg ( 5474 1 0.853 ) 4.5, φ > 1.59 φ G(φ) = exp [ 14.2 (φ 1) 9.28 (φ 1) 2] (64), 1 φ 1.59 φ 14.2, φ < 1 σ s ω = 1 + σ 0 (φ sg0 ) [ω 0(φ sg0 ) 1] (65)

GEO 432/532 COURSE NOTES 11 ω 0 and σ 0 are set functions of φ sg0, computed numerically. The Parker transport relation is implemented as Acronym1 in the Excel workbook RTe-bookAcronym1.xls. (Note: ACRONYM isn t actually an acronym for anything; it s just a result of Gary s sense of humor.) There are three ways to run the model: Acronym1: User inputs specific gravity of sediment, R + 1, shear velocity, u, and the grain size distribution of the bed (surface) material. The code computes the magnitude and size distribution of the bedload transport. Acronym1 R: User inputs R + 1, flow discharge Q, bed slope S, channel width B, parameter n k relating roughness height k s to surface size D s90, and the grain size distribution of the bed (surface) material. The code computes shear velocity with Manning-Strickler resistance and assumption of normal flow and, then, magnitude and size distribution of the bedload transport. Acronym1 D: Uses the same formulation as Acronym1 R but allows specification of flow-duration curve so that average annual gravel transport rate and grain size distribution are computed. j. Wilcock and Crowe bedload relation for sand-gravel mixtures: Sand is not excluded; method based on D sg and fraction of sand in surface (active) layer, F s : Wi = G(φ i ) (66) 0.002φ, φ < 1.35 ( G(φ) = 14 1 0.894 ) 4.5, φ > 1.35 φ 0.5 (67) φ i = τ sg ( ) b Di, τ τssrg ssrg = 0.021 + 0.015 exp ( 20F s ), b = D sg 0.67 1 + exp (1.5 D i /D sg ) Note the strong dependence on sand fraction, F s, in the reference Shields stress, τ ssrg. Also, the exponent, b, on the hiding ratio, D i /D sg, is not constant but, rather, depends on the value of the ratio, and b 0.1 0.4, for small to large grain sizes. Wilcock & Crowe isn t set up to simply calculate magnitude and size distribution of transport, but the workbook, RTe-bookAgDegNormGravMixPW.xls, can be hacked to provide these outputs, as described in Exercise 2 for this class. k. Relative dimensionless shear stress, φ i : It s a bit tricky to sort out the meaning of the hiding ratio, D i /D sg, in the Parker and Wilcock & Crowe relations. First, note that b γ. So, what does the value of b signify relative to the extremes of equal mobility (γ = 1) and grain independence (γ = 0)? Let s break it down. For Wilcock & Crowe, φ i = τ sg τ ssrg ( Di D sg (68) ) b (69)

12 LANCASTER Note that, in the first ratio, both numerator and denominator are relative to the geometric mean grain size, D sg. Because we can always multiply by one, we can write φ i = τ sg ( ) γ ( ) 1 γ b D sg Di Di. (70) τssrg D i D sg D sg Then, from (49) and the definition of the Shields number, φ i = τ ( ) si 1 γ b Di (71) τssri D sg Okay, bear with me. If we suppose that a default value of γ is, say, γ = 0.6, then larger values of b effectively get us closer to equal mobility. If smaller grains have smaller γ, then they are more likely to be transported at lower bed shear stresses than are the particles with D sg. If larger grains have larger γ, then they are more likely to be transported at a bed shear stress more similar to that at which particles with D sg are transported. Note that we still haven t specified a value for γ, but we don t actually need to; we can still understand b in these terms: smaller b, less hiding of smaller grains and less protrusion of larger grains; larger b, more hiding of smaller grains and more protrusion of larger grains; if smaller grains have a smaller value of b than larger grains, that means smaller grains are effectively less hidden, and larger grains effectively more exposed. l. Annual bedload yield: Divide annual hydrograph into M bins, Specify characteristic discharge, Q k, for each bin, and Fraction of time, p Qk, the discharge occurs. Sum over discharges and, for mixtures, grain size fractions. 8. Bedforms (Parker (2004), Ch. 8) a. Ripples: b. Dunes: Small-scale bedforms Migrate downstream Asymmetric with gentle stoss and steep lee faces Require reasonably well-defined viscous sublayer, so slow flow, and ripples don t interact with water surface Criterion: Above threshold for motion, but grain diameter less than thickness of viscous sublayer, or D δ v = 11.6ν/u, or τ > τc and ( ) 2 11.6 τ < (72) Re p Above criterion says, then, D 0.8 mm; in fact, usually find D 0.6 mm.

Common in sand-bed rivers GEO 432/532 COURSE NOTES 13 Wavelengths up to hundreds of meters, heights up to 5 m Asymmetric with gentle stoss and steep lee faces Characteristic of subcritical flow Migrate downstream Interact weakly with water surface, which is out of phase with dunes: flow accelerates over dune crests, where water surface is slightly lower. c. Antidunes: Can occur in sand-bed and gravel-bed rivers High (but not necessarily supercritical) Froude number Most commonly migrate upstream Nearly symmetric Interact strongly with water surface, which is in phase with antidunes: flow typically accelerates down face and into trough, where water surface is lower. Train of symmetrical surface (standing) waves indicative of antidunes on the bed d. Potential flow formulation, Froude number, and the dune-antidune transition Subcritical flow: water surface perturbation is approximately out of phase with bed perturbation Supercritical flow: water surface is approximately in phase with bed perturbation. Define Fr 0 in terms of the average flow depth, H 0, Fr 0 = q 2 w gh0 H 0 (73) and wavenumber of the bedforms, k = 2πH 0 /λ, which is like frequency. According to potential flow theory, the border between subcritical and supercritical flow is function of both Fr 0 and k, i.e., critical flow is defined by Fr 2 0 = tanh k (74) k For long-wavelength bedforms, k 0, and critical flow is at Fr 0 1, which is the shallow-water limit. For typical values of k, the critical flow transition is at Fr 0 < 1, e.g., for λ = 5H 0, k = 1.26, the threshold Fr 0 = 0.82. Another threshold from potential flow theory defines a regime in which water surface and flow depth are out of phase: Fr 0 2 = 1 k tanh k So, we have three regions defined by Froude number and wavenumber (plus two planebed regimes): (75)

14 LANCASTER Fr 0 tanh k/k: Lower-regime plane bed, where subcritical flow is too low to create ripples or dunes; gravel-bed streams usually fall in this regime. Upper end of this regime is τ 2.72τ c. So, for coarse material (Re p 1), dunes will not form unless τ > 0.0816, which is uncommon. Fr 0 < tanh k/k: Lower-regime, subcritical flow; water surface elevation and flow depth are in phase with one another, and both are out of phase with the bed elevation, and we have downstream-migrating dunes. tanh k/k Fr0 < 1/k tanh k: Upper-regime plane bed, near critical flow, where the competition between the effects of bedload, favoring dunes, and suspended load, favoring antidunes, cancel each other out. tanh k/k < Fr0 < 1/k tanh k: Upper-regime, supercritical flow; water surface elevation, flow depth, and bed elevation are all in phase, and we have upstreammigrating antidunes, although for very low k, cyclic steps may occur in this regime. 1/k tanh k < Fr0 : Upper-regime, supercritical flow; water surface and bed elevation are in phase, and both are out of phase with the flow depth, and we have downstream-migrating antidunes. Rare, but observed. 9. Form drag and sand transport (Parker (2004), Ch. 9) a. Skin friction and form drag: Skin friction: generated by the viscous shear stress acting tangentially to a body. Form drag: generated by the normal stress (mostly pressure) acting on a body. Total shear stress is τ ij = pδ ij + τ v,ij, τ v,ij = ρν ( ui + u ) j x j x i and the drag force on a body is given by integrating the shear stress times the outward normal vector over the surface: D i = τ ji n j ds (77) Drag force is the total of skin friction and form drag, or D i = D si + D fi. Skin friction is from the τ v,ji s, and form drag is from the pressure: ( ui D si = τ v,ji n j ds = ρν + u ) j n j ds (78) x j x i D fi = pn i ds (79) b. Einstein decomposition: subtract out form drag to perform bedload calculations. Given the two basic forms we have for bed shear stress, (76) τ b = ρc f U 2 = ρghs, (80)

GEO 432/532 COURSE NOTES 15 For flow with same mean velocity, U, and bed slope, S, with and without bedforms, the friction coefficient and depth due to skin friction alone must be smaller than with bedforms, or C f = C fs + C ff and H = H s + H f, so that τ b = τ bs + τ bf = ρ (C fs + C ff ) U 2 = ρg (H s + H f ) S (81) A decomposition like this is conceptually suspect, but this is generally the way it s done. Note that, while empirical formulations (e.g., shown below) exist for dunes in sand-bed rivers, methods for incorporating other sorts of forms, e.g., logs and boulders, are not well established, and finding the form drag due to a set of several or many such form drag elements is not generally a simple matter of adding together the contributions from each element. Perhaps the best treatment of this subject to date is that of Kean and Smith (2006a,b). c. Skin friction: calculate from grain size distribution and depth, with Keulegan (35) or Manning-Strickler (36). d. Form drag of dunes: Wright and Parker: skin friction as a function of total stress τs = 0.05 + 0.7 ( ( τ Fr 0.7) ( ) ) 0.7 0.8 0.8 HS U = 0.05 + 0.7, (82) gh which can be rewritten as [ H = Γ RD s50 S Use with skin friction predictor, C 1/2 fs = RD s50 ( ) ] g 0.7 20/13 ( ) τ 5/4, Γ = s 0.05 (83) U 0.7 U ghs S = 8.32 α strat ( Hs k s ) 1/6, k s = 3D 90, (84) where α strat is a correction for flow stratification, set to 1 without other information. This can be rewritten (without stratification) as U = 8.32 ( ) 1/6 Hs gh s S (85) Computation of depth-discharge relation proceeds as follows: Compute k s from D s90. Assume a value (or series of values) of H s. Compute velocity U from the skin friction predictor (85). Compute Γ and H from (83). Compute the water discharge per unit width, q w = UH. Plot H vs. q w. k s

16 LANCASTER e. Prediction of bedload transport with dunes: Substitute for τ with τ s in bedload transport relation, where for normal flow (note typo in Parker (2004)), τs = H ss (86) RD Proceed from depth-discharge relation, e.g., given discharge, slope, and grain size, find H s. There are other predictors, and we can generalize to gradually varied (i.e., not normal) flow. 10. Aggradation and degradation in alluvial streams a. Channel degradation and bank erosion (Parker (2004), Ch. 15) A 1D Exner equation can incorporate the effect of bank erosion on channel degradation. For bed lowering by an amount η, the contribution from the bed to the volume (per distance downstream) of sediment removal is (1 λ p ) B b η. Contribution from each sidewall is (1 λ p ) B s η = (1 λ p ) (η t η) η/s s. Volume rate of supply from both sidewalls per stream length is η I s = 2 (1 λ p ) B s t = 2 (1 λ p) η t η η (87) S s t The Exner equation for sediment concentration for a degrading channel with sidewall erosion and fraction, f bb, contributing to bed material load reduces to (1 λ p ) η t = 1 q t η t η (88) 1 + 2f x bb B b S s where the second term in the denominator of the first ratio on the right-hand side is the extra due to bank erosion. Note that, for a given flux divergence (the partial derivative on the right-hand side), sidewall erosion slows the rate of channel bed degradation. b. Aggradation and channel shift (Parker (2004), Ch. 15) Channel shift: Gradual shift is migration. Sudden shift is avulsion. Sinuosity, Ω: ratio between average down-channel distance x and average down-valley distance x v between the same points. For meandering rivers, typically 1.5 Ω 3. Assuming bed material load is transported within a channel of constant width B and sinuosity Ω, but is deposited across the floodplain width, B f, reach-averaged long-term Exner equation for aggradation is (1 λ p ) η t = Ω B q t B f x As long as Ω < B f /B, which is usually a pretty safe bet, then for a given flux divergence, floodplain deposition slows the rate of channel bed aggradation. Note that we re still only considering bed material load, so floodplain deposition here is really only from lateral accretion of point bars, etc., by a laterally migrating (meandering) channel. (89)

c. Longitudinal river profiles (Parker (2004), Ch. 25) Profile shapes: GEO 432/532 COURSE NOTES 17 A river profile is upward concave if slope decreases in the downstream direction, i.e., S x = η 2 x < 0 (90) 2 Otherwise, i.e., slope increases downstream, the profile is upward convex. Transient long profiles: If sediment inflow is greater than sediment outflow, i.e., if ( Q t / x) < 0, profile is upward-concave during aggradation toward a new equilibrium. If sediment inflow is less than sediment outflow, i.e., if ( Q t / x) > 0, profile is upward-convex during degradation toward a new equilibrium. Many rivers have upward-concave quasi-equilibrium long profiles ( quasi in the sense that sediment inflow and outflow are not equal). Reasons: Subsidence: filling hole left by subsidence makes ( Q t / x) < 0 over long term, so profile is upward-concave over long term. Sea level rise: Sea level rise of 120 m in last 12,000 yrs has same effect as subsidence. Delta progradation: forces long-term aggradation. Downstream sorting of sediment: Finer sediment is easier to transport, so same volume can be transported at a lower slope. Sediment is typically sorted such that median grain size decreases downstream. Finer sediment downstream lower slope downstream means upward-concave profile. Abrasion of sediment: Gravel is broken into finer particles, including sand, silt, and smaller gravel, all of which can be transported in the same volume at lower slopes, so effect is similar to downstream sorting. Effect of tributaries: Tributaries typically add proportionally more water than sediment supply (this is partly a way of saying that tributaries tend not to all provide high sediment concentrations at the same time), so sediment concentration decreases downstream and requires lower slopes for transport, and the tendency is for upward-concave profiles. For river with constant sea level rise: Reach length, L; floodplain width B f B; sand-bed grain size D; Transport Q tbf only during fraction I f of year, so annual yield is Q btf I f. Deposition over width of floodplain; Λ units of wash load (or floodplain material load) deposited per unit of bed material load deposited; no mass balance for wash load; assumed adequate supply.

18 LANCASTER Constant rate of sea level rise, ξ d ; constant sinuosity, Ω; normal flow. Exner (mass balance) equation: (1 λ p ) η t = ΩI f(1 + Λ) which differs from (89) only by the intermittency factor, I f, and the addition of the wash load ratio, Λ, in the numerator on the right-hand side. Relation for sediment transport in sand-bed stream as a function of channel-forming Shields number, τ form : B f Q t x (91) Q tbf = α EHτform Q RC 1/2 bf S (92) f A linear function of slope! Which means the Exner equation turns into a linear diffusion equation: η t = κ 2 η d x, κ 2 d = I fω(1 + Λ) α EH τform Q bf (1 λ p ) B f RC 1/2 f Decomposing (91) into constant sea level rise and deviations from the steady-state solution, we eliminate the deviations part by supposing that there is a steady-state solution to be found, and we get (the now rather simple ordinary differential equation): (93) which integrates to dq tbf dx = (1 λ p) B f I f Ω(1 + Λ) ξ d (94) Q tbf = Q tbf,feed (1 λ p) B f I f Ω(1 + Λ) ξ dx (95) From this solution, some interesting things can be shown: Slope decreases downstream, S = (1 βˆx)s u. Channel width decreases downstream, B bf Q tbf. Channel depth increases downstream, H bf Q 1 tbf. 11. Meandering and alluvial channel planform morphology (Lancaster and Bras, 2002) (see http://www.geo.oregonstate.edu/~lancasts/publications.html) a. Meandering, or lateral bend migration, is a fundamental process, whatever the planform (e.g.,, combined with mid-channel bar formation and chute cut-off formation, makes braided river planform). b. Appears to require sufficient bedload to create point bars (Shepherd and Schumm, 1974). c. Point bars produce topographic steering, which transfers streamwise momentum laterally across the channel to the opposite bank.

GEO 432/532 COURSE NOTES 19 d. Transverse slope from point bar to pool, S T, can be explained in terms of helical flow, induced by centrifugal force, and is proportional to channel curvature, C and average flow depth, H: where, for gravel-bed channels (Ikeda, 1989), ( ) τ 0.228 K = 0.361 Cf S T = KHC (96) τ c (97) or, for sand-bed channels (Lancaster and Bras, 2002), K = τ sf τ [ ( sf 0.228 11.0τ ) τ τc κ ln sf H τ D s50 ] 0.361 (98) e. Lateral transfer of downstream momentum means the high-velocity flow moves across the channel and impinges on the outer bank and creates a large lateral gradient of downstream velocity and, thus, a large shear stress on the bank. With distance downstream, the gradient and shear stress are dissipated by development of the rough turbulent boundary layer. f. General approach to modeling lateral migration: Changes in near-bank velocity proportional to channel curvature Effect on near-bank velocity at a point along the channel centerline, s, is proportional to a weighted average of local and upstream curvature: u b(s) = F 1 (U, b, C f ) C(s) + F 2 (U, H, b, C f ) s 0 C(s )e 2C f H (s s ) ds (99) Note that the model that simulates the 2D planform is essentially a 1D morphodynamic model. (Model of Lancaster and Bras (2002) is substantially different in its formulation of the shear stress on the bank but similar in integrating upstream effects of curvature.) Bank erosion modeled as proportional to u b (s); deposition on point bar assumed to keep pace with outer bank erosion. Different models generally similar in that Small bends migrate downstream Medium bends migrate outward Large bends migrate upstream Process model selects a characteristic scale for the meander wavelength (twice the distance from crossover to crossover)

20 LANCASTER References Ikeda, S. (1989), Sediment transport and sorting at bends, in River Meandering, vol. Water Resources Monograph 12, edited by S. Ikeda and G. Parker, pp. 103 126, American Geophysical Union, Washington. Kean, J. W., and J. D. Smith (2006a), Form drag in rivers due to small-scale natural topographic features: 1. regular sequences, Journal of Geophysical Research, 111 (F4), doi: 10.1029/2006JF000467. Kean, J. W., and J. D. Smith (2006b), Form drag in rivers due to small-scale natural topographic features: 2. irregular sequences, Journal of Geophysical Research, 111 (F4), doi:10.1029/2006jf000490. Lancaster, S. T., and R. L. Bras (2002), A simple model of river meandering and its comparison to natural channels, Hydrological Processes, 16 (1), 1 26. Parker, G. (2004), 1D Sediment Transport Morphodynamics with Applications to Rivers and Turbidity Currents, University of Illinois, http://hydrolab.illinois.edu/people/ parkerg//morphodynamics_e-book.htm, visited April, 2012. Shepherd, R. G., and S. A. Schumm (1974), Experimental study of river incision, Geological Society of America Bulletin, 85 (2), 257 268. Note that the list of references does not include many of those that are referenced in Parker (2004).