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.