Transcript pptx

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
(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
(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
(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
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
Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels
Main Result 1 (of 2):
(Nichols et al. 1993)
Uniform distribution of bottom stress
→ Predicts channel convergence rate
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
Tidal current
(cm/s)
Crosssectional
area (m2)
AX-SECT ~ exp(-x/LA)
LA ↑ Upstream
-- Tidal FRESHWATER James River (where URIVER +
UTIDE ≈ Const.) will become less convergent
upstream to remain an equilibrium channel.
Cross-- Less convergence upstream allows UTIDE to
decrease upstream where URIVER is stronger. sectional
area (m2)
-- Analytical theory for equilibrium channels
predicts observed changes in channel
River current
convergence.
(cm/s)
Tidal ESTUARINE James River
AX-SECT ~ exp(-x/LA)
LA ≈ Const.
URIVER + UTIDE ≈ Const.
UTIDE ≈ Const.
Tidal
current
(cm/s)
URIVER ≈ 0
Distance upstream from mouth (km)
4/13
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
Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels
Main Result 1 (of 2):
(Nichols et al. 1993)
Uniform distribution of bottom stress
→ Predicts channel convergence rate
(a) For UTIDE ≈ const., URIVER << UTIDE ,
Look for tidal solutions of the form
Tidal FRESHWATER James River
Tidal ESTUARINE James River
Crosssectional
area (m2)
AX-SECT ~ exp(-x/LA)
LA ↑ Upstream
With a = const., U = const., h = const.
The result is (Friedrichs, 2010):
3p g1/2 h3/2
LA =
8 cd UTIDE
g = 9.8 m/s2, h = 4 m,
UTIDE = 0.6 m/s, cd = 0.0025
→ Predicted LA = 20 km
Crosssectional
area (m2)
River current
(cm/s)
Tidal current
(cm/s)
LA ≈
20 km
URIVER + UTIDE
≈ Const.
AX-SECT ~ exp(-x/LA)
LA ≈ Const.
UTIDE ≈ Const.
Tidal
current
(cm/s)
URIVER ≈ 0
Distance upstream from mouth (km)
6/13
Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels
Main Result 1 (of 2):
(Nichols et al. 1993)
Uniform distribution of bottom stress
→ Predicts channel convergence rate
(b) For UTIDE ≠ const., URIVER ≈ UTIDE ,
Look for tidal solutions of the form
Tidal FRESHWATER James River
T
LA ↑ Upstream
Constraint of URIVER + UTIDE = Const.
Additionally requires LT = - LA .
Crosssectional
area (m2)
At “transition point” where
URIVER = UTIDE , this increases
→ Predicted LA = 60 km
Crosssectional
area (m2)
AX-SECT ~ exp(-x/LA)
T
equilibrium LA by a factor of 3
Tidal ESTUARINE James River
River current
(cm/s)
Tidal current
(cm/s)
I.e., x-sect area converges less.
LA ≈ 60 km
URIVER + UTIDE
≈ Const.
LA ≈
20 km
AX-SECT ~ exp(-x/LA)
LA ≈ Const.
UTIDE ≈ Const.
Tidal
current
(cm/s)
URIVER ≈ 0
Distance upstream from mouth (km)
7/13
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
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
From: Friedrichs et al. (1998). Hydrodynamics and equilibrium sediment dynamics of shallow, funnelshaped 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) = wo exp(-x/Lw)
9/13
Observations and analytical solution:
Perturbation solution approach:
h = a {h0 + eh1 + O(e2) }
u = U {u0 + eu1 + O(e2) }
Tidal
phase
(deg)
C = c {C0 + eC1 + O(e2) }
High
water
(m)
Max
flood
(m/s)
River Tamar, UK
hM4 rel
phase
(deg)
uM4 rel
phase
(deg)
uM4 /uM2
hM4 /hM2
Distance from mouth (km)
10/13
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 )
Flood dominance
from tidal asymmetry
Dispersion away from
tidal turbidity max
Ebb dominance
from river flow
Sediment
Conc.
Distance from mouth (km)
11/13
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
Hydrodynamics, Morphology and Sediment Transport in Equilibrium Tidal Freshwater Channels
Main Result 1 (of 2):
(Nichols et al. 1993)
Uniform distribution of bottom stress
→ Predicts channel convergence rate
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
Tidal current
(cm/s)
Crosssectional
area (m2)
AX-SECT ~ exp(-x/LA)
LA ↑ Upstream
-- Tidal FRESHWATER James River (where URIVER +
UTIDE ≈ Const.) will become less convergent
upstream to remain an equilibrium channel.
Cross-- Less convergence upstream allows UTIDE to
decrease upstream where URIVER is stronger. sectional
area (m2)
-- Analytical theory for equilibrium channels
predicts observed changes in channel
River current
convergence.
(cm/s)
Tidal ESTUARINE James River
AX-SECT ~ exp(-x/LA)
LA ≈ Const.
URIVER + UTIDE ≈ Const.
UTIDE ≈ Const.
Tidal
current
(cm/s)
URIVER ≈ 0
Distance upstream from mouth (km)
12/13
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