turbulence and mixing in estuaries rocky geyer, whoi acknowlegments: david ralston, whoi malcolm...
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Turbulence and mixing in estuaries
Rocky Geyer, WHOI Acknowlegments:
David Ralston, WHOI Malcolm Scully, Old Dominion U.
Wind-driventurbulence
Interfacial, shear-driven turbulence
Boundary-layerturbulence
wind
velocity
Velocity =log layer
stress
Bottom stress
τB /ρ= Cdub2=u*
2Turbulent velocity scale
uT ~u* ~ 0.05 ub
ub
“eddy viscosity”
Simplest case: unstratified tidal flow:Only boundary-layer turbulence
Mixing Length model for the Eddy Viscosity / Diffusivity
2/1.0
)/1(
)/1(
max
*
*
hzath
hzz
uu
u
hzzuK
T
T
m
Tu
from log layer observations:
define:
)(z
z
Velocity
stress
ub
“eddy viscosity”
Now add stratification
log layernear bottom
stratification damps turbulencenear pycnocline
enhanced shearnear pycnocline
ρ1
ρ2
z
gN
gg
2
'“reduced” gravity
Buoyancy frequency
What is the maximum vertical scale for turbulence with stratification?
z
pgzuB 2
2
1
Bernoulli Function (energy-conserving flow)
N
u
uutake
zNu
zz
g
dzzgzgu
To
T
i
zh
h
iii
1
2
2
)(2
1
)(2
1 1
1
The Ozmidov scale:maximum size of eddies beforegravity arrests them.
oz
i
Boundary layer: LT ~ kz
turbulence suppressed
Ozmidov scaling:LT=uT/N
Schematic of turbulence length-scale in a stratified estuary
dist
ance
fro
m b
ed
u(z)
h
0
z
LBL
Limiting Length-scales in Turbulent Flows
Boundary-Layer Scaling (depth limiting)
LT
Note that LT Thorpe overturn scale
Ozmidov Scaling (stratification limiting)
0.2 h
2/1
1
h
zzLL BLT
2/1
3
N
LL OT
Relative flow direction
kosp
ectr
al d
ensi
ty S
(k)
Turbulence length-scale LT~ 1/ko
Ozm
idov
Len
gth
scal
ing
Boundary-Layer scaling
Snohomish River
Scully et al. (2010) Influence of stratification on estuarine turbulence
Turbulent dissipation (conversion of turbulent motions to heat)
2
j
i
x
u
In a boundary layer, dissipation ~ production
z
uu
2*
ensembleaverage of turbulent motions
=
Dissipation: the currency ($ or € ?) of turbulence
ko
3/53/2)( kakS o
“Inertial subrange” method for estimating dissipation:
Viscous limit
10 cml o
=1
m
Tur
bule
nt D
issi
patio
n ε,
m2 s
-3
Buoyancy Frequency N, s-1
ContinentalShelf
Ocean
Riv
ers
Lakes
3 cm30
cm3 m
Estuaries
The Parameter Space of Estuarine Turbulence
lo = ( ε/N3 )1/2
Geyer et al. 2008: Quantifying vertical mixing in estuaries
Stratified boundary layer
u(z)u(z)
Stratified shear layer
no turbulence
Two different paradigms of estuarine mixing. How importantis the stratified shear layer paradigm in estuarine turbulence?
turbulence
turbulence turbulence
Shear Instability
25.0
Ri 2
2
Ri
zu
N
necessary condition for stabilityMiles, 1961; Howard, 1961
gradient Richardson numberRichardson, 1920
Thorpe, 1973
Smyth et al., 2001
ub
hi
us ρ1
ρ2
x
hgwu
x
hg
z
wu
i
i
8
1''
'''
max
2
2
Momentum balance of a tilted interface
0.5-1x10-4 m2s-2 for strong transition zones –moderate but not intense stress
meters
1.2 m/s
weak motion
bottom
interface
Fraser River salt wedge—early ebb (Geyer and Farmer 1989)
1.2 m/s
400 200 m 0
200 180 m 160
Ri<0.25leadingto shearinstability
Connecticut River: Geyer et al. 2010: Shear Instability at high Reynolds number
Salinity
dissipation of TKE
dissipation of salinity variance
Day 325--Transect 17 (~ hour 19.1)
riverocean
meters along river
MB
MB
M
BC C C
MMM
M
B
M
B
M
BC C C
#4
#5
#6MMM
M BM BM BC C C
#4#5#6
MMM
Echo Soundingat Anchor Station
Salinity contours (black)Salinity variance (dots)
B: braidC: coreM: mixing zone
Salinitytimeseries~ 60 seconds
Staquet, 1995
Re~1,000
Re~500,000
MIXINGin cores
MIXINGin braids
Baroclinicity of the braid accelerates the shear… with plenty of time within the braid…
α
ρc
ρ2
ρ1
323max
max
11
105~8
21~8
''
7.0~Ri8
1
smug
Pagwu
T
Tg
z
u
t oadv
acc
…leading to mixing:
20 seconds
30 meters
New profiler data and acoustic imagery
Very intense, and very pretty…
…but is mixing at hydraulic transitions important at the scale of the estuary?
100 m
fresh
salt
NetTidal Power “P”
Buoyancy flux B = ∫∫∫β g s′w′ dV
Dissipation D = ∫∫∫ε dV
Energy balance: P = B + D
Efficiency Rf = B/P = B/(D+B)
Hudson: ROMS
Merrimack: FVCOM
Massachusetts
Boundary layer
Internal shear
U(z)U(z) u’w’u’w’
In the estuaryMerrimack River mixing analysis
volume-integrated buoyancy flux
Boundary layer
Ralston et al., 2010Turbulent mixing in astrongly forced salt wedge estuary.
Hudson River mixing analysisROMS, Qr = 300 m3/s
Scully, unpublished
Boundary layer
Boundary layer
Internal shear
Internal shear
Ri
Rf
testing turbulence closure stability functions with Mast data
Kantha and Clayson 1994
Canuto et al., 2001 Scully, unpublished
k- Mellor-Yamada 2.5 (k-kl)
ebb depth
dtdzB
Observed buoyancy flux vs. Ri
Modeled buoyancy flux vs. Ri
Conclusions and Prospects for the Future
1. Stratified boundary-layer turbulence is the most important mixing regime in estuaries.
2. Shear instability is locally important and dramatic but is not the dominant contributor to the total mixing.
3. Closure models are on the right track. We need more data for testing them.
4. Estuaries are outstanding natural laboratories for the investigation ofstratified mixing processes. We need more measurements of turbulencein these environments!