1. Virginia Institute of Marine Science
Damping of Turbulence by Suspended Sediment: Ramifications of Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
Richardson number control of saturated suspension
Under-saturated (weakly stratified) sediment suspensions
Critically saturated (Ricr-controlled) sediment suspensions
Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012

2. Ri = For c > ~ 300 mg/liter Ri ≈ Ricr ≈ O(1/4) c > 0.3 g/l
When strong currents are present, mud remains turbulent and in suspension at a concentration that gives Ri ≈ Ricr ≈ 1/4:
Gradient Richardson
Number (Ri) =
density stratification
velocity shear
Shear instabilities occur for Ri < Ricr
“ “ suppressed for Ri > Ricr
Stratification
Shear
c < 0.3 g/l
c > 0.3 g/l Ri = -
0.25
Sediment gradient Richardson number
g = accel. of gravity
s = (s - )/
c = sediment mass conc.
s = sediment density
For c > ~ 300 mg/liter
Sediment concentration (grams/liter)
Ri ≈ Ricr ≈ O(1/4)
Amazon Shelf (Trowbridge & Kineke, 1994)
1/18

3. Large supply of easily suspended sediment creates negative feedback:
Are there simple, physically-based relations to predict c and du/dz related to Ri?
Large supply of easily suspended sediment creates negative feedback:
Gradient Richardson
Number (Ri)
density stratification
Shear instabilities occur for Ri < Ricr
“ “ suppressed for Ri > Ricr =
velocity shear
(a)
(b)
Height above bed
Height above bed
Ri > Ric
Ri = Ric
Ri = Ric
Ri < Ric
Sediment concentration
Sediment concentration
(a) If excess sediment enters bottom boundary layer or bottom stress decreases, Ri beyond Ric, critically damping turbulence. Sediment settles out of boundary layer. Stratification is reduced and Ri returns to Ric.
(b) If excess sediment settles out of boundary layer or bottom stress increases, Ri below Ric and turbulence intensifies. Sediment re-enters base of boundary layer. Stratification is increased in lower boundary layer and Ri returns to Ric.
2/18

4. Ri > Ricr Ri < Ricr Ri = Ricr
Consider Three Basic Types of Suspensions
Height above bed
Ri > Ricr
3) Over-saturated -- Settling limited
Ri < Ricr
Ri = Ricr
1) Under-saturated
-- Supply limited
2) Critically
saturated load
Sediment concentration
3/18

5. Ri > Ricr Ri < Ricr Ri = Ricr
Consider Three Basic Types of Suspensions
Height above bed
Ri > Ricr
3) Over-saturated -- Settling limited
Ri < Ricr
Ri = Ricr
1) Under-saturated
-- Supply limited
2) Critically
saturated load
Sediment concentration
3/18

6. Virginia Institute of Marine Science
Damping of Turbulence by Suspended Sediment: Ramifications of Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
Richardson number control of saturated suspension
Under-saturated (weakly stratified) sediment suspensions
Critically saturated (Ricr-controlled) sediment suspensions
Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012

7. Boundary layer - current log layer
Dimensionless analysis of bottom boundary layer in the absence of stratification:
Variables
du/dz, z, h, n, u*
h = thickness of boundary layer or water depth, n = kinematic viscosity, u* = (tb/r)1/2 = shear velocity
Boundary layer - current log layer
z/h << 1
“Overlap”
layer
n/(zu*)
<< 1
Are thickness of layers to scale? NO. Note thickness of log layer relative to entire flow.
zo = hydraulic roughness
(Wright, 1995)
4/18

8. b = Turbulent buoyancy flux
Dimensionless analysis of overlap layer with (sediment-induced) stratification:
Additional variable
b = Turbulent buoyancy flux
s = (rs – r)/r ≈ 1.6
c = sediment mass conc.
w = vertical fluid vel.
u(z)
Height above the bed, z
Are thickness of layers to scale? NO. Note thickness of log layer relative to entire flow.
Dimensionless ratio
= “stability
parameter”
5/18

9. Deriving impact of z on structure of overlap (a. k. a
Deriving impact of z on structure of overlap (a.k.a. “log” or “wall”) layer
= “stability
parameter”
Rewrite f(z) as Taylor expansion around z = 0:
≈ 0
≈ 0
= 1
= a
From atmospheric studies, a ≈ 4 - 5
If there is stratification (z > 0) then u(z) increases faster with z than homogeneous case.
6/18

10. (e.g., halocline being mixed away from below)
Eq. (1)
= “stability
parameter”
(i)
well-mixed
-- Case (i): No stratification near the bed (z = 0 at z = z0). Stratification effects and z increase with increased z.
-- Eq. (1) gives u increasing faster and faster with z relative to classic well-mixed log-layer.
(e.g., halocline being mixed away from below)
-- Case (ii): Stratified near the bed (z > 0 at z = z0). Stratification effects and z decrease with increased z.
-- Eq. (1) gives u initially increasing faster than u, but then matching du/dz from neutral log-layer.
(e.g., fluid mud being entrained by wind-driven flow)
stratified
 as z z0
(ii)
well-mixed
stratified
Log elevation of height above bed
 as z z0
-- Case (iii): uniform z with z. Eq (1) integrates to
(iii)
stratified
well-mixed
-- u remains logarithmic, but shear is increased buy a factor of (1+az)
 is constant in z z0
(Friedrichs et al, 2000)
Current Speed
7/18

11. Then A <,>,= 1 determines shape of u profile
Stability parameter, z, can be related to shape of concentration profile, c(z):
(Friedrichs et al, 2000)
z = const. in z if
(i)
well-mixed
Fit a general power-law to c(z) of the form
stratified
A < 1
Then
 as z
If A < 1, c decreases more slowly than z-1
z increases with z,
stability increases upward,
u is more concave-down than log(z)
If A > 1, c increases more quickly than z-1
z decreases with z, stability
becomes less pronounced upward,
u is more concave-up than log(z)
If A = 1, c ~ z-1
z is constant with elevation
stability is uniform in z,
u follows log(z) profile z0
(ii)
well-mixed
stratified
A > 1
Log elevation of height above bed
 as z z0
(iii)
stratified
well-mixed
A = 1
 is constant in z z0
If suspended sediment concentration, C ~ z-A
Then A <,>,= 1 determines shape of u profile
Current Speed
(7/18)

12. If suspended sediment concentration, C ~ z-A
A < 1 predicts u more concave-down than log(z)
A > 1 predicts u more concave-up than log(z)
A = 1 predicts u will follow log(z)
Eckernförde Bay,
Baltic Coast,
Germany
Testing this relationship using observations from bottom
boundary layers:
STATAFORM mid-shelf site,
Northern California, USA
(Friedrichs & Wright, 1997;
Friedrichs et al, 2000)
Inner shelf,
Louisiana
USA
8/18

13. If suspended sediment concentration, C ~ z-A
A < 1 predicts u more convex-up than log(z)
A > 1 predicts u more concave-up than log(z)
A = 1 predicts u will follow log(z)
STATAFORM mid-shelf site,
Northern California, USA,
1995, 1996
Inner shelf, Louisiana, USA,
1993
A ≈ 1.0
A ≈ 3.1
A ≈ 0.73
A ≈ 0.35
A ≈ 0.11
-- Smallest values of A < 1 are associated with concave-downward velocities on log-plot.
-- Largest value of A > 1 is associated with concave-upward velocities on log-plot.
-- Intermediate values of A ≈ 1 are associated with straightest velocity profiles on log-plot.
9/18

14. Normalized log of sensor height above bed
Observations showing effect of concentration exponent A on shape of velocity profile
Normalized log of sensor height above bed
Normalized burst-averaged current speed
Observations also show: A < 1, concave-down velocity
A > 1, concave-up velocity
A ~ 1, straight velocity profile
(Friedrichs et al, 2000)
10/18

15. Virginia Institute of Marine Science
Damping of Turbulence by Suspended Sediment: Ramifications of Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
Richardson number control of saturated suspension
Under-saturated (weakly stratified) sediment suspensions
Critically saturated (Ricr-controlled) sediment suspensions
Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012

16. Relate stability parameter, z, to Richardson number:
Definition of gradient Richardson number associated with suspended sediment:
Original definition and application of z:
Relation for eddy viscosity:
Definition of eddy diffusivity:
Assume momentum and mass are mixed similarly:
Combine all these and you get:
So a constant z with height also leads to a constant Ri with height.
Also, if z increases (or decreases) with height Ri correspondingly increases (or decreases).
11/18

17. If A < 1, c decreases more slowly than z-1
z and Ri const. in z if
well-mixed
Define
stratified
then
A < 1
If A < 1, c decreases more slowly than z-1
z and Ri increase with z,
stability increases upward,
u is more concave-down than log(z)
If A > 1, c decreases more quickly than z-1
z and Ri decrease with z, stability
becomes less pronounced upward,
u is more concave-up than log(z)
If A = 1, c ~ z-1
z and Ri are constant with elevation
stability is uniform in z,
u follows log(z) profile
 and Ri as z z0
(ii)
well-mixed
stratified
A > 1
Log elevation of height above bed
 and Ri as z z0
(iii)
stratified
well-mixed
A = 1
If suspended sediment concentration, C ~ z-A
then A <,>,= 1 determines shape of u profile
and also the vertical trend in z and Ri
 and Ri are constant in z z0
(Friedrichs et al, 2000)
Current Speed
(7/18)

18. Ri > Ricr Ri < Ricr Ri = Ricr
Now focus on the case where Ri = Ricr (so Ri is constant in z over “log” layer)
Height above bed
Ri > Ricr
3) Over-saturated -- Settling limited
Ri < Ricr
Ri = Ricr
1) Under-saturated
-- Supply limited
2) Critically
saturated load
Sediment concentration
(3/18)

19. Earlier relation for eddy viscosity:
Connection between structure of sediment settling velocity to structure of “log-layer” when Ri = Ricr in z (and therefore z is constant in z too).
Rouse Balance:
Earlier relation for eddy viscosity:
Eliminate Kz and integrate in z to get
But we already know
when Ri = const.
when Ri = Ricr
and So
12/18

20. when Ri = Ricr . This also means that when Ri = Ricr :
13/18

21. STATAFORM mid-shelf site, Northern California, USA Mid-shelf site off
Waiapu River, New Zealand
(Wright, Friedrichs et al., 1999;
Maa, Friedrichs, et al., 2010)
14/18

22. Sediment gradient Richardson numbercm cm
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4 10 -2 -1 1
Ricr = 1/4
(a) Eel shelf, 60 m depth, winter
(Wright, Friedrichs, et al. 1999)
Sediment gradient Richardson number
(b) Waiapu shelf, NZ, 40 m depth, winter 2004
(Ma, Friedrichs, et al. in 2008)
Ricr = 1/4 cm
Velocity shear du/dz (1/sec)
15/18

23. Application of Ricr log-layer equations fo Eel shelf, 60 m depth, winter 1995-96
(Souza & Friedrichs, 2005)
16/18

24. Virginia Institute of Marine Science
Damping of Turbulence by Suspended Sediment: Ramifications of Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
Richardson number control of saturated suspension
Under-saturated (weakly stratified) sediment suspensions
Critically saturated (Ricr-controlled) sediment suspensions
Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012

25. Ri > Ricr Ri < Ricr Ri = Ricr
Now also consider over-saturated cases:
Height above bed
Ri > Ricr
3) Over-saturated -- Settling limited
Ri < Ricr
Ri = Ricr
1) Under-saturated
-- Supply limited
2) Critically
saturated load
Sediment concentration
(3/18)

26. More Settling
(Mehta & McAnally, 2008)
Starting at around grams/liter, the return flow of water around settling flocs creates so much drag on neighboring flocs that ws starts to decrease with additional increases in concentration.
At ~ 10 g/l, ws decreases so much with increased C that the rate of settling flux decreases with further increases in C. This is “hindered settling” and can cause a strong lutecline to form.
Hindered settling below a lutecline defines “fluid mud”. Fluid mud has concentrations from about 10 g/l to 250 g/l. The upper limit on fluid mud depends on shear. It is when “gelling” occurs such that the mud can support a vertical load without flowing sideways.
17/18

27. -- 1-DV k-e model based on components of Delft 3D
(Winterwerp, 2011)
-- 1-DV k-e model based on components of Delft 3D
-- Sediment in density formulation
-- Flocculation model
-- Hindered settling model
18/18

28. Virginia Institute of Marine Science
Damping of Turbulence by Suspended Sediment: Ramifications of Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
Richardson number control of saturated suspension
Under-saturated (weakly stratified) sediment suspensions
Critically saturated (Ricr-controlled) sediment suspensions
Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012