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Outline of Presentation : Tidal sediment transport due to spatial vs. flood/ebb asymmetries

Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Carl Friedrichs Virginia Institute of Marine Science. Outline of Presentation : Tidal sediment transport due to spatial vs. flood/ebb asymmetries

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Outline of Presentation : Tidal sediment transport due to spatial vs. flood/ebb asymmetries

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  1. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Carl Friedrichs Virginia Institute of Marine Science • Outline of Presentation: • Tidal sediment transport due to spatial vs. flood/ebb asymmetries • (1) Minimizing spatial asymmetry → predicts channel convergence rate • (2) Balancing flood and ebb asymmetries → predicts concentration field • Summary of main points Presented at AGU Chapman Conference Reston, VA, 14 November 2012

  2. (1) Spatial asymmetries in bed stress → Net transport toward area of lower stress. -- Equilibrium favors uniform spatial distribution of maximum (tide + river) currents. Tidal advection Higher bottom stress Lower bottom stress Higher sediment concentration Tidal advection Higher bottom stress Lower bottom stress Lower sediment concentration 1/13

  3. (2a) Flood vs. ebb asymmetry → More transport during tidal phase with stronger bed stress. -- Sediment trapping (turbidity max) in region where flood- & ebb-dominance converge. Tidal advection Higher bottom stress Higher bottom stress Higher sediment concentration Tidal advection Lower bottom stress Lower bottom stress Lower sediment concentration 2/13

  4. (2b) Trapping by flood vs. ebb asymmetry → Region of high erodibility at turbidity maximum. -- At equilibrium, advection away high erodibility region cancels trapping by tidal asymmetry. Tidal advection Higher sediment concentration Region of lower concentration & erodibility Region of higher concentration & erodibility (turbidity maximum) Tidal advection Lower sediment concentration Region of lower concentration & erodibility Region of higher concentration & erodibility (turbidity maximum) 3/13

  5. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Carl Friedrichs Virginia Institute of Marine Science • Outline of Presentation: • Tidal sediment transport due to spatial vs. flood/ebb asymmetries • (1) Minimizing spatial asymmetry → predicts channel convergence rate • (2) Balancing flood and ebb asymmetries → predicts concentration field • Summary of main points Presented at AGU Chapman Conference Reston, VA, 14 November 2012

  6. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Main Result 1 (of 2): Uniform distribution of bottom stress → Predicts channel convergence rate (Nichols et al. 1993) Uniform Bed Stress at Equilibrium Predicts: -- Tidal ESTUARINE James River (where URIVER ≈ 0) will have a simple exponential convergence to keep UTIDE ≈ Const. in space at equilibrium. -- Equilibrium convergence concentrates UTIDEas quickly as friction dissipates UTIDE . -- Tidal FRESHWATER James River (where URIVER+ UTIDE≈ Const.) will become less convergent upstream to remain an equilibrium channel. -- Less convergence upstream allows UTIDE to decrease upstream where URIVER is stronger. -- Analytical theory for equilibrium channels predicts observed changes in channel convergence. Tidal ESTUARINE James River Tidal FRESHWATER James River Cross- sectional area (m2) AX-SECT ~ exp(-x/LA) LA ↑ Upstream AX-SECT ~ exp(-x/LA) LA ≈ Const. Cross- sectional area (m2) URIVER+ UTIDE≈ Const. UTIDE≈ Const. Tidal current (cm/s) River current (cm/s) URIVER≈ 0 Tidal current (cm/s) Distance upstream from mouth (km) 4/13

  7. Builds from: Friedrichs (2010). Barotropic tides in channelized estuaries. In: Valle-Levinson (ed.), Contemporary Issues in Estuarine Physics, Cambridge University Press, p. 27-61. b(x,t) = width at high tide Governing equations: (1) Continuity (2) Momentum with Friction factor Solutions for h = const., b(x) ~ w(x) ~ x-sectional area = AX-SECT(x)~ exp(-x/LA) Where LA is the e-folding length-scale over which cross-sectional area decreases. General (linearized) case: look for solutions which are the real part of: T T Where LT is e-folding length-scale over which tidal amplitude and tidal velocity changes. 5/13

  8. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Main Result 1 (of 2): Uniform distribution of bottom stress → Predicts channel convergence rate For UTIDE ≈ const., URIVER << UTIDE , Look for tidal solutions of the form With a = const., U = const., h = const. The result is (Friedrichs, 2010): (Nichols et al. 1993) Tidal ESTUARINE James River Tidal FRESHWATER James River Cross- sectional area (m2) AX-SECT ~ exp(-x/LA) LA ↑ Upstream AX-SECT ~ exp(-x/LA) LA ≈ Const. LA ≈ 20 km Cross- sectional area (m2) 3p g1/2 h3/2 LA= 8 cdUTIDE URIVER+ UTIDE ≈ Const. UTIDE≈ Const. Tidal current (cm/s) River current (cm/s) g = 9.8 m/s2, h = 4 m, UTIDE = 0.6 m/s, cd = 0.0025 →Predicted LA = 20 km URIVER≈ 0 Tidal current (cm/s) Distance upstream from mouth (km) 6/13

  9. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Main Result 1 (of 2): Uniform distribution of bottom stress → Predicts channel convergence rate (b) For UTIDE ≠ const., URIVER ≈ UTIDE , Look for tidal solutions of the form Constraint of URIVER+ UTIDE = Const. Additionally requires LT = - LA. At “transition point” where URIVER= UTIDE , this increases equilibrium LA by a factor of 3 (Nichols et al. 1993) Tidal ESTUARINE James River Tidal FRESHWATER James River T Cross- sectional area (m2) AX-SECT ~ exp(-x/LA) LA ↑ Upstream T AX-SECT ~ exp(-x/LA) LA ≈ Const. LA ≈ 20 km LA ≈ 60 km Cross- sectional area (m2) URIVER+ UTIDE ≈ Const. UTIDE≈ Const. Tidal current (cm/s) River current (cm/s) →Predicted LA = 60 km URIVER≈ 0 Tidal current (cm/s) I.e., x-sect area converges less. Distance upstream from mouth (km) 7/13

  10. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Carl Friedrichs Virginia Institute of Marine Science • Outline of Presentation: • Tidal sediment transport due to spatial vs. flood/ebb asymmetries • (1) Minimizing spatial asymmetry → predicts channel convergence rate • (2) Balancing flood and ebb asymmetries → predicts concentration field • Summary of main points Presented at AGU Chapman Conference Reston, VA, 14 November 2012

  11. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Main Result 2 (of 2): Flood and ebb asymmetries → Predicts concentration field Assuming Tidal Freshwater Conditions, then Tidal Asymmetries Predict: -- Upstream transport in lower river by flood dominance due to tidal nonlinearities. -- Downstream transport in upper river by ebb dominance due to river flow. -- Turbidity maximum forms at point where asymmetries are equal and opposite. -- Enhanced erodibility at turbidity maximum disperses sediment away from turbidity maximum, allowing equilibrium. -- Analytic solution predicts location and intensity of turbidity maximum as well as its dependence on river flow. Observed Conc. from Uncles et al. (1989) (normalized by (tidal amplitude)2) Equilibrium Concentration Predicted by Analytical Model River Tamar, UK 50 ppm/m2 8/13

  12. From: Friedrichs et al. (1998). Hydrodynamics and equilibrium sediment dynamics of shallow, funnel-shaped tidal estuaries. In: Dronkers & Scheffers (eds.), Physics of Estuaries and Coastal Seas, Balkema Press, p. 315-328. Governing equations: (1) Continuity (2) Momentum with quadratic friction tb = r cd |u| u C = sediment conc. K = along-channel diffusion E = (a/Tc)u2= erosion D = C/Tc= deposition Tc = settling time-scale = 45 min a = bed erodibility (3) Sediment Transport Keep “Order” e = a/h non-linear tidal fluctuations in h, u ∂u/∂x, and u2 . Assume <h>= const., and b(x) = w(x) ~ exp(-x/Lw) Field example: River Tamar, UK h = 2.4 m, Lw = 4.7 km, a/h = 0.6 wo w(x) w(x) = woexp(-x/Lw) 9/13

  13. Observations and analytical solution: Perturbation solution approach: h = a{h0 + eh1 + O(e2) } u = U{u0+ eu1 + O(e2) } C = c{C0+ eC1+ O(e2) } Tidal phase (deg) High water (m) Max flood (m/s) hM4rel phase (deg) River Tamar, UK uM4rel phase (deg) uM4 /uM2 hM4 /hM2 Distance from mouth (km) 10/13

  14. Analytic solution predicts equilibrium along-channel variation in erodibility, where am scales total suspendable sediment in system Observed Conc. from Uncles et al. (1989) (normalized by (tidal amplitude)2) Equilibrium Concentration Predicted by Analytical Model 50 ppm/m2 (At lowest order, C ≈ a u2 ) Dispersion away from tidal turbidity max Flood dominance from tidal asymmetry Ebb dominance from river flow Sediment Conc. Distance from mouth (km) 11/13

  15. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Carl Friedrichs Virginia Institute of Marine Science • Outline of Presentation: • Tidal sediment transport due to spatial vs. flood/ebb asymmetries • (1) Minimizing spatial asymmetry → predicts channel convergence rate • (2) Balancing flood and ebb asymmetries → predicts concentration field • Summary of main points Presented at AGU Chapman Conference Reston, VA, 14 November 2012

  16. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Main Result 1 (of 2): Uniform distribution of bottom stress → Predicts channel convergence rate (Nichols et al. 1993) Uniform Bed Stress at Equilibrium Predicts: -- Tidal ESTUARINE James River (where URIVER ≈ 0) will have a simple exponential convergence to keep UTIDE ≈ Const. in space at equilibrium. -- Equilibrium convergence concentrates UTIDEas quickly as friction dissipates UTIDE . -- Tidal FRESHWATER James River (where URIVER+ UTIDE≈ Const.) will become less convergent upstream to remain an equilibrium channel. -- Less convergence upstream allows UTIDE to decrease upstream where URIVER is stronger. -- Analytical theory for equilibrium channels predicts observed changes in channel convergence. Tidal ESTUARINE James River Tidal FRESHWATER James River Cross- sectional area (m2) AX-SECT ~ exp(-x/LA) LA ↑ Upstream AX-SECT ~ exp(-x/LA) LA ≈ Const. Cross- sectional area (m2) URIVER+ UTIDE≈ Const. UTIDE≈ Const. Tidal current (cm/s) River current (cm/s) URIVER≈ 0 Tidal current (cm/s) Distance upstream from mouth (km) 12/13

  17. Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels Main Result 2 (of 2): Flood and ebb asymmetries → Predicts concentration field Assuming Tidal Freshwater Conditions, then Tidal Asymmetries Predict: -- Upstream transport in lower river by flood dominance due to tidal nonlinearities. -- Downstream transport in upper river by ebb dominance due to river flow. -- Turbidity maximum forms at point where asymmetries are equal and opposite. -- Enhanced erodibility at turbidity maximum disperses sediment away from turbidity maximum, allowing equilibrium. -- Analytic solution predicts location and intensity of turbidity maximum as well as its dependence on river flow. Observed Conc. from Uncles et al. (1989) (normalized by (tidal amplitude)2) Equilibrium Concentration Predicted by Analytical Model River Tamar, UK 50 ppm/m2 13/13

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