Mixing

FromStresses

Wind stresses

Bottom stresses

Internal stresses

Non-stress Instabilities

Cooling

DoubleDiffusion

TidalStraining

Shear Production Buoyancy Production

Mixing vs. Stratification

w ind stress bottom stress tida l stra in ing cooling

M ixing from :

To mix the water column, kinetic energy has to be converted to potential energy.

Mixing increases the potential energy of the water column

z

z2

z1

Stratification from:

estuarine circulation tidal straining heating

Potential energy per unit volume: HgV ,

Vol

Potential energy of the water column: HgmV

But )(z

dzzgH

0

The potential energy per unit area of a mixed water column is:

dzzgH

m

0

dzH H

01ˆ

22

32 skg

mmkg

sm

Ψ has units of energy per unit area

The energy difference between a mixed and a stratified water column is:

dzzgH

m

0

)ˆ(

with units of [ Joules/m2 ]

φ is the energy required to mix the water column completely, i.e., the energy required to bring the profile ρ(z) to ρhat

It is called the POTENTIAL ENERGY ANOMALY

z

z2

z1

It is a proxy for stratification

The greater the φ the more stratified the water column

If 0ˆ

no energy is required to mix the water column

We’re really interested in determining whether the water column remains stratified or mixes as a result of the forcings acting on the water column.

For that we need to studydtd

[ Watts per squared meter ]

ncirculatio nalgravitatio from tionstratifica

straining tidal atingcooling/he

tides from mixing windfrom mixing

dt

d

dzz

zK

HyK

yzK

z

zK

HxK

xzw

Hg

dzz

yH''v

Hy'

'vy'

vy

'v

xH''u

Hx'

'ux'

ux

'u

Hg

t

H

F

Hzz

H

h

E

z

F

zz

H

h

D

H

'CCAB

'CCAB

by

sx

yyyy

xxxx

00

0

1

1

1

1

Bx and By are the along-estuary and cross-estuary straining terms

Ax and Ay are the advection terms

Cx and Cy interaction of density and flow deviations in the vertical

C’x and C’y correlation between vertical shear and density variations in the vertical; depth-averaged counterparts of C

E is vertical mixing and D is vertical advection

Hx and Hy are horizontal dispersion; Fs and Fb are surface and bottom density fluxes

De Boer et al (2008, Ocean Modeling, 22, 1)

Burchard and Hofmeister (2008, ECSS, 77, 679)

1-D idealized numerical simulation of tidal straining

0

HE

z

B

dzzz

Kzx

'uHg

t

Burchard and Hofmeister (2008, ECSS, 77, 679)

0 1

Hz dzz

x

H''u

Hx

''u

zw

x

'u

zK

zx'u

H

g

t

stratified entire period

destratified @ end of flood

dzz

zK

HyK

yzK

z

zK

HxK

xzw

Hg

dzz

yH''v

Hy'

'vy'

vy

'v

xH''u

Hx'

'ux'

ux

'u

Hg

t

H

F

Hzz

H

h

E

z

F

zz

H

h

D

H

'CCAB

'CCAB

by

sx

yyyy

xxxx

00

0

1

1

1

1

Bx and By are the along-estuary and cross-estuary straining terms

Ax and Ay are the advection terms

Cx and Cy interaction of density and flow deviations in the vertical

C’x and C’y correlation between vertical shear and density variations in the vertical; depth-averaged counterparts of C

E is vertical mixing and D is vertical advection

Hx and Hy are horizontal dispersion; Fs and Fb are surface and bottom density fluxes

De Boer et al (2008, Ocean Modeling, 22, 1)

Mixing Power From Wind

The power/unit area generated by the wind at a height of 10 m is given by:

31010

10m at wind WCWdt

dDa

But the power/unit area generated by the wind stress on the sea surface is:

*Wdtd

W* is the wind shear velocity at the surface and equals:

1010

210

* WWCWC

W DaDa

2

31010 mWatts

WCWdtd

DaW

00114.0 and ;kg/m 1020 ;kg/m 2.1 with

0.00116 ~ ; of fraction a is

33a

10mat wind

D

W

C

dtd

dtd

23

106103.1

mWatts

Wdtd

aW

Alternatively,

23

1063

10 104.1m

WattsWWk

dtd

aasW

δ is a mixing efficiency coefficient = 0.023ks is a drag factor that equals 6.4x10-5 or ( Cd u / W )

Mixing Power From Wind (cont.)

Mixing Power From Tidal Currents

Can also be expressed in terms of bottom stress.

The power/unit area produced by tidal flow interacting with the bottom is:

average on 3bbbb uCu

But only a fraction of this goes to mixing

flow tidal of amplitude 34

ousinstantane

30

3

uCdtd

uCdtd

bb

bbb

ε is a mixing efficiency [ 0.002, 0037 ]

Cb is a bottom drag coefficient = 0.0025

Tidal Straining

dzzgH

0

)ˆ(

dzztt

gt H

0 ˆ

assuming the along-estuary density gradient is independent of depth, i.e.,xx

ˆ

dzzuux

gt H

0ˆ

Considering advection of mass by ‘u’ only:

xu

txu

t

ˆ

ˆˆ

and ,

01 ˆ

H

udzH

u

We need u(z) from tidal currents to determine the power to stratify/destratify from tidal straining

Tidal Straining (cont.)

Taking, Hz );425.015.1(ˆ )( 2 uu

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.5 1 1.5

u (m/s)

No

rma

lize

d D

ep

th

u

1.15

0.72

(Bowden and Fairbairn, 1952,Proc. Roy. Soc. London, A214:371:392.)

)425.015.0(ˆ)425.0115.1(ˆˆ 22 uuuu

Tidal Straining (cont.)

dzzuux

gt H

0ˆ

)425.015.0(ˆ)425.0115.1(ˆˆ 22 uuuu

031.0

0

1

22 425.015.0ˆ

dux

gHdtd S

ux

Hgdtd S ˆ 031.0 2

The water column will stratify at ebb as is positive, and vice versau

dzzuux

gt H

0ˆ

Taking again:

Gravitational Circulation

0 ˆ because ˆ uuuu

323

89148

xA

gHu

zand using

20/3

0

1

32252

89148

d

xAHg

dtd

z

E

252

3201

xA

Hgdtd

z

E

will tend to stratify the water column

Heating/Cooling

p

C

cgHQ

dtd

2

In addition to buoyancy from heating, it may come from precipitation (rain)

rP P

gHdtd

2

Δρ is the density contrast between fresh water and sea water

Pr is the precipitation rate (m/s)

α is the thermal expansion coefficient of seawater ~ 1.6x10-4 °C-1

cp is the specific heat of seawater ~ 4x103 J/(kg °C)

Q is the cooling/heating rate (Watts/m2)

In estuaries, however, the main input of freshwater buoyancy is from river discharge.

There is no simple way of dealing with feshwater input as specified by the discharge rate R because R is not distributed uniformly over a prescribed area (as is the case forwind, bottom stress, rain, heat).

The alternative way of representing the riverine influence on stratification is by assuming that increased R enhances Δρ / Δx.

This may be parameterized withdtd E

Caution! Increased R does not necessarily mean increased gradients

dtd

dtd

dtd

dtd

dtd

dtd

dtd PCESbW

Assuming that each stratifying/destratifying mechanism can be superimposed separately:

Example: Let’s compare the stratifying tendencies of rain as compared to a low heating rate of 10 W/m2

p

C

cgHQ

dtd

2

α = 1.6x10-4 °C-1

cp = 4x103 J/(kg °C)H = 10 m

255 mW 1021096.1 dtd C

25 mW 1022

rP P

gHdtd

If the contrast between rain water and sea water is 20 kg/m3, then

mm/d 1.7m/s 102mW 1022 825

gHPr

A precipitation rate of 1.7 mm per day is comparable to a heating rate of 10 W/m2

Where can this happen?

dtd

dtd

dtd

dtd

dtd

dtd

dtd PCESbW

Competition between buoyancy from Heating and mixing from Bottom Stress

dtd

dtd

dtd Cb

p

b cHQg

uCdtd

234 3

0

If stratification remains unaltered (or if buoyancy = mixing),

pb c

QHguC

dtd

234

and ,0 30

For a prescribed Q, the only variables are H and u0

303

8

u

HQg

Cc bp

If Q increases, u0 needs to increase to keep H/u03 constant

If u0 does not increase then stratification ensues

H/u03 is then indicative of regions where mixed waters meet stratified

Simpson-Hunter parameter

H/u03 ~ 1.6x104/ Q

Line where mixed waters are separated from stratified waters.

LOG10 (H / U3)

z

Bowman and Esaias, 1981, JGR, 86(C5), 4260.

Loder and Greenberg, 1996, Cont. Shelf Res., 6(3), 397-414.

M2 -----------------M2-N2 ------------M2-N2-S2--------M2+N2+S2------

Loder and Greenberg, 1996, Cont. Shelf Res., 6(3), 397-414.

Restrictions of the approach?

Dominant Stratifying Power from Heating

Dominant Destratifying Power from bottom stresses

Another example:

Assume a system with Δρ / Δx of 10 kg/m3 over 50 km = 2x10-4, H = 10 m, Az =0.005 m2/s

24252

mW 104.2320

1

xA

Hgdtd

z

E

In order to balance that stratifying power, we need a wind power of:

4310

6 104.2104.1 Wdtd

aW

m/s 2.510 W

or a current power of: 430

3 104.2101.2 udtd B

m/s 49.00 u

Another example:

Assume a system with Δρ / Δx of 1 kg/m3 over 3 km, H = 20 m, Av =0.001 m2/s

2252

mW 1.0320

1

xA

Hgdtd

z

E

In order to balance such stratifying power, we need a wind power of:

105.0104.1 310

6 Wdtd

aW

m/s 7.3910 W

or a current power of: 105.0101.2 30

3 udtd B

m/s 7.30 u

From Heating/Cooling

From Density Gradient (grav circ)

Examples of successful applications of this approach:

Simpson et al. (1990), Estuaries, 13(2), 125-132.

Lund-Hansen et al. (1996), Estuar. Coast. Shelf Sci., 42, 45-54.