outline of presentation: tidal sediment transport due to spatial vs. flood/ebb asymmetries (1)...

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

Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels

Carl FriedrichsVirginia Institute of Marine Science

Presented at AGU Chapman ConferenceReston, VA, 14 November 2012

Higher sediment concentration

Lower sediment concentration

Tidal advection

Tidal advectionHigher bottom stress Lower bottom stress

(1) Spatial asymmetries in bed stress → Net transport toward area of lower stress. -- Equilibrium favors uniform spatial distribution of maximum (tide + river) currents.

Higher bottom stress Lower bottom stress

1/13



Higher bottom stress Higher bottom stress

Lower bottom stress Lower bottom stress

(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

Tidal advection

2/13

Region of lowerconcentration& erodibility

(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

Tidal advection

Region of higherconcentration & erodibility

(turbidity maximum)

Region of lowerconcentration& erodibility

Region of higherconcentration & erodibility

(turbidity maximum)



3/13

Distance upstream from mouth (km)

Tidal current (cm/s)

Cross-sectional area (m2)

(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 UTIDE as 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.

URIVER + UTIDE ≈ Const.

URIVER ≈ 0

UTIDE ≈ Const.

AX-SECT ~ exp(-x/LA)

LA ≈ Const.


LA ↑ Upstream

Tidal FRESHWATER James River Tidal ESTUARINE James River


Main Result 1 (of 2):

Uniform distribution of bottom stress → Predicts channel convergence rate

River current (cm/s)



4/13

b(x,t) = width at high tide

Governing equations: (1) Continuity (2) Momentum with Friction factor

General (linearized) case: look for solutions which are the real part of:

T

Where LT is e-folding length-scale over which tidal amplitude and tidal velocity changes.

T

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.

Builds from: Friedrichs (2010). Barotropic tides in channelized estuaries. In: Valle-Levinson (ed.), Contemporary Issues in Estuarine Physics, Cambridge University Press, p. 27-61.

5/13

g = 9.8 m/s2, h = 4 m, UTIDE = 0.6 m/s, cd = 0.0025

→ Predicted LA = 20 km




(a) For UTIDE ≈ const., URIVER << UTIDE , Look for tidal solutions of the form

With a = const., U = const., h = const. The result is (Friedrichs, 2010):








URIVER ≈ 0

UTIDE ≈ Const.


LA ≈ Const.


LA ↑ Upstream


LA ≈

20 km

LA = 3p g1/2 h3/2

8 cd UTIDE


6/13




(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








URIVER ≈ 0

UTIDE ≈ Const.


LA ≈ Const.


LA ↑ Upstream


LA ≈

20 km

T

T


LA ≈ 60 km

→ Predicted LA = 60 km

I.e., x-sect area converges less.7/13

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.

River Tamar, UK

Observed Conc. from Uncles et al. (1989)(normalized by (tidal amplitude)2)

Equilibrium Concentration Predicted by Analytical Model

50 ppm/m2


Main Result 2 (of 2): Flood and ebb asymmetries → Predicts concentration field

8/13

Governing equations: (1) Continuity (2) Momentum with quadratic friction

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, UKh = 2.4 m, Lw = 4.7 km, a/h = 0.6

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.

tb = r cd |u| u

w(x) = wo exp(-x/Lw)

wo

w(x)

(3) Sediment Transport

C = sediment conc.K = along-channel diffusionE = (a/Tc)u2 = erosionD = C/Tc = depositionTc = settling time-scale = 45 min a = bed erodibility

9/13

Perturbation solution approach:

h = a {h0 + eh1 + O(e2) }u = U {u0 + eu1 + O(e2) }

C = c {C0 + eC1 + O(e2) }

Distance from mouth (km)

Tidal phase (deg)

High water

(m)

Maxflood(m/s)

hM4 rel phase(deg)

uM4 rel phase(deg)

uM4 /uM2

hM4 /hM2

Observations and analytical solution:

River Tamar, UK

10/13



50 ppm/m2

Distance from mouth (km)

Dispersion away fromtidal turbidity max Flood dominance

from tidal asymmetryEbb dominancefrom river flow

SedimentConc.

Analytic solution predicts equilibrium along-channel variation in erodibility, where am scales total suspendable sediment in system

(At lowest order, C ≈ a u2 )

11/13





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 UTIDE as 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.


URIVER ≈ 0

UTIDE ≈ Const.


LA ≈ Const.


LA ↑ Upstream








12/13

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.

River Tamar, UK



50 ppm/m2


Main Result 2 (of 2): Flood and ebb asymmetries → Predicts concentration field

13/13

outline of presentation: tidal sediment transport due to spatial vs. flood/ebb asymmetries (1)...

Documents

tidal sediment transport

tidal asymmetry

equilibrium convergence

equilibrium channels

tidal phase

uriver utide

spatial asymmetry

flood ebb