outline of presentation: richardson number influence on coastal/estuarine mixing derivation of...

Outline of Presentation:

• Richardson number influence on coastal/estuarine mixing• Derivation of stratified “overlap” layer structure • Under-saturated (weakly stratified) sediment suspensions• Critically saturated (Ricr-controlled) sediment suspensions• Hindered settling, over-saturation, and collapse of turbulence

Damping of Turbulenceby Suspended Sediment in the Bottom Boundary Layer

Carl Friedrichs, Virginia Institute of Marine Science

10

30

100 mg/l Time-series of suspended sediment in York River estuary (Friedrichs et al. 2000)

1 meter

2 hours

Gradient RichardsonNumber (Ri) =

density stratification

velocity shear

Shear instabilities occur for Ri < Ricr

“ “ suppressed for Ri > Ricr

Ricr ≈ 1/4

Density & Chlstratified when

Ri > 1/4, mixed when

Ri < 1/4

Hans Paerl et al. (2004)Neuse River Estuary

observations from 2003

OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

g = accel. of gravitys = (s - )/ c = sediment mass conc.s = sediment density

Stratification

Shear=c < 0.3 g/l c > 0.3 g/l

Amazon Shelf (Trowbridge & Kineke, 1994)

Ri ≈ Ricr ≈ O(1/4)

For c > ~ 300 mg/liter

Sedi

men

t gra

dien

t Ric

hard

son

num

ber

Sediment concentration (grams/liter)

0.25

When strong currents are present, mud remains turbulent and in suspension at a concentration that gives Ri ≈ Ricr ≈ 1/4:


density stratification

velocity shear



Ri

-


(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.

Sediment concentration

Hei

ght a

bove

bed

Sediment concentrationH

eigh

t abo

ve b

ed

Ri = Ric Ri < Ric

Ri > Ric

Ri = Ric

Large supply of easily suspended sediment creates negative feedback:


density stratificationvelocity shear



(a) (b)

Are there simple, physically-based relations to predict c and du/dz related to Ri?



Hei

ght a

bove

bed

Ri = Ricr

Ri > Ricr

Ri < Ricr

Consider Three Basic Types of Suspensions

3) Over-saturated -- Settling limited

1) Under-saturated-- Supply limited

2) Critically saturated load






10

30


1 meter

2 hours

(Dyer, 1986)

Dimensionless analysis of bottom boundary layer in the absence of stratification:

h = thickness of bottom boundary layer, n = kinematic viscosity, u* = (tb/r)1/2 = shear velocity

Outer Layer n/(zu*) << 1 ,

but z/h not small

Variables du/dz, z, h, n, u*

~ 1 cm

~ 1 m

n/(zu*) << 1

z/h << 1

h ~ 10 m ~ 10-6 m2/s ~ 1 cm/s

Viscous Layer z/h << 1 ,but n/(zu*) not small

“Overlap” Layer: z/h << 1 & n/(zu*) << 1

(a.k.a. “log-layer”)velocity u(z)


Dimensionless analysis of bottom boundary layer in the absence of stratification:

h = thickness of bottom boundary layer, n = kinematic viscosity, u* = (tb/r)1/2 = shear velocity

Outer Layer n/(zu*) << 1 ,

but z/h not small

Variables du/dz, z, h, n, u*

~ 1 cm

~ 1 m

h ~ 10 m ~ 10-6 m2/s ~ 1 cm/s

Viscous Layer

“Overlap” Layer: z/h << 1 & n/(zu*) << 1

(a.k.a. “log-layer”)

z/h << 1 ,but n/(zu*) not small

i.e.,

Const. = 1/k

k ≈ 0.41

(Dyer, 1986)


Boundary layer - current log layer

(Wright, 1995)

zo = hydraulic roughness

“Overlap” layer

n/(zu*) << 1

z/h << 1

Bottom boundary layer often plotted on log(z) axis:


Dimensionless analysis of overlap layer with (sediment-induced) stratification:

Dimensionless ratio

= “stability parameter”

Additional variableb = Turbulent buoyancy flux

Hei

ght a

bove

the

bed,

z

u(z)

s = (rs – r)/r ≈ 1.6c = sediment mass conc.

w = vertical fluid vel.


Rewrite f(z) as Taylor expansion around z = 0:

= 1 = a≈ 0 ≈ 0

From atmospheric studies, a ≈ 4 - 5

If there is stratification (z > 0) then u(z) increases faster with z than homogeneous case.

Deriving impact of z on structure of overlap (a.k.a. “log” or “wall”) layer






10

30


1 meter

2 hours

Current Speed

Lo

g e

leva

tion

of

he

igh

t a

bo

ve b

ed

as z

well-mixed

stratified

(i)

(iii)

(ii)

is constant in z

well-mixed stratified


as z

-- Case (i): No stratification near the bed (z = 0 at z = z0). Stratification 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 and z decreases 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)

z0

z0

z0

Eq. (1)

-- Case (iii): uniform z with z. Eq (1) integrates to

-- u remains logarithmic, but shear is increased buy a factor of (1+az)

(Friedrichs et al, 2000)


Overlap layer scaling modified by buoyancy flux

Definition of eddy viscosity

Eliminate du/dz and get

-- As stratification increases (larger z), Az decreases

-- If z = const. in z, Az increases like u*z , and the result is still a log-profile.

Effect of stratification (via z) on eddy viscosity (Az)

Connect stability parameter, z, to shape of concentration profile, c(z):

Definition of z: Rouse balance (Reynolds flux

= settling):

Combine to eliminate <c’w’> :z = const. in z if

(Assuming ws is const. in z)


If suspended sediment concentration, C ~ z-A

Then A <,>,= 1 determines shape of u profile (Friedrichs et al, 2000)

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 elevationstability is uniform in z,u follows log(z) profile

z = const. in z if

Fit a general power-law to c(z) of the form

Then

Current Speed

Lo

g e

leva

tion

of

he

igh

t a

bo

ve b

ed

as z

well-mixed

stratified

(i)

(iii)

(ii)

is constant in z



z0

z0

z0

as z

A < 1

A = 1

A > 1


STRATAFORM mid-shelf site, Northern California, USA

Inner shelf, Louisiana

USA

Eckernförde Bay,Baltic Coast,

Germany


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)

Testing this relationship using observations from bottom boundary layers:

(Friedrichs & Wright, 1997;Friedrichs et al, 2000)


STATAFORM mid-shelf site, Northern California, USA,

1995, 1996

Inner shelf, Louisiana, USA,1993

-- 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 velocities on log-plot.

A ≈ 0.11

A ≈ 3.1

A ≈ 0.35

A ≈ 0.73

A ≈ 1.0


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)

(Friedrichs et al, 2000)


Eckernförde Bay, Baltic Coast, Germany, spring 1993

-- Salinity stratification that increases upwards cannot be directly represented by c ~ z-A. Friedrichs et al. (2000) argued that this case is dynamically analogous to A ≈ -1.

(Friedrichs & Wright, 1997)


Observations showing effect of concentration exponent A on shape of velocity profile

Normalized burst-averaged current speed

Nor

mal

ized

log

of s

enso

r hei

ght a

bove

bed

Observations also show: A < 1, concave-down velocityA > 1, concave-up velocity

A ~ 1, straight velocity profile




10

30


1 meter

2 hours



Relate stability parameter, z, to Richardson number:

Definition of gradient Richardson number associated with suspended sediment:

Original definition and application of z:

Assume momentum and mass are mixed similarly:

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).

Relation found for eddy viscosity:

Definition of eddy diffusivity:

Combine all these and you get:


(Friedrichs et al, 2000) Current Speed

Lo

g e

leva

tion

of

he

igh

t a

bo

ve b

ed

and Ri as z

well-mixed

stratified

(i)

(iii)

(ii)

and Ri are constant in z



z0

z0

z0

A < 1

A = 1

A > 1

and Ri as z


then A <,>,= 1 determines shape of u profileand also the vertical trend in z and Ri

z and Ri const. in z if

thenDefine

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 elevationstability is uniform in z,u follows log(z) profile



Hei

ght a

bove

bed

Ri = Ricr

Ri > Ricr

Ri < Ricr

Now focus on the case where Ri = Ricr (so Ri is constant in z over “log” layer)





Eliminate Kz and integrate in z to get

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:

But we already know when Ri = const.

So and when Ri = Ricr


when Ri = Ricr . This also means that when Ri = Ricr :


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)


(a) Eel shelf, 60 m depth, winter 1995-96(Wright, Friedrichs, et al. 1999)

Velocity shear du/dz (1/sec)

10 - 40 cm40 - 70 cm

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410-2

10-1

100

101

Ricr = 1/4Se

dim

ent g

radi

ent R

icha

rdso

n nu

mbe

r

(b) Waiapu shelf, NZ, 40 m depth, winter 2004(Ma, Friedrichs, et al. in 2008)

Ricr = 1/4

18 - 40 cm


Application of Ricr log-layer equations fo Eel shelf, 60 m depth, winter 1995-96

(Souza & Friedrichs, 2005)




10

30


1 meter

2 hours




Hei

ght a

bove

bed

Ri = Ricr

Ri > Ricr

Ri < Ricr

Now also consider over-saturated cases:





More Settling

Starting at around 5 - 8 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 (vertical sediment gradient) to form.

A lutecline with hindered settling can cause turbulent collapse. The sediment can’t leave the water column, so dC/dz keeps increasing, creating positive feedback. Ri increases further above Ricr, and more sediment to settles. Then there is more hindered settling and a stronger lutecline, increases Ri further.

1 10 100.01

.1

1

ws (

mm

/s)


(Mehta & McAnally, 2008)

(Winterwerp, 2011)

-- 1-DV k-e model based on components of Delft 3D-- Sediment in density formulation-- Flocculation model-- Hindered settling model

Ri ≤ Ricr

during flood

Ri ≤ Ricr

during flood

Ri >> Ricr

around slack

Ri >> Ricr

around slack

Ri ≈ Ricr

at peak ebb

Ri ≈ Ricr

at peak ebb

Observed

Modeled


(Ozedemir, Hsu & Balachandar, in press)

U ~ 60 cm/sC ~ 10 g/liter

“large eddy simulation” model

Fixed sediment supply

ws = 0.45 mm/s

ws = 0.75 mm/s


(Ozedemir, Hsu & Balachandar, in press)

U ~ 60 cm/sC ~ 10 g/liter“large eddy

simulation” model

ws = 0.45 mm/s ws = 0.75 mm/s

Profiles of flux Richardson number at time of max free stream U

3.4 cm

2.9 cm

Non

-dim

ensi

onal

ele

vati

on

Non

-dim

ensi

onal

ele

vati

on

The Richardson number is of “order” critical (relatively close to 0.25) near top of suspended layer




10

30


1 meter

2 hours



outline of presentation: richardson number influence on coastal/estuarine mixing derivation of...

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