Turbulence and mixing in estuaries
Rocky Geyer, WHOI
Acknowlegments:
David Ralston, WHOI
Malcolm Scully, Old Dominion U.
wind
velocity
Wind-driven
turbulence
Interfacial, sheardriven turbulence
Boundary-layer
turbulence
Simplest case: unstratified tidal flow:
Only boundary-layer turbulence
Velocity =
log layer
“eddy viscosity”
stress
ub
Bottom stress
Turbulent velocity scale
uT ~u* ~ 0.05 ub
τB /ρ= Cdub2=u*2
Mixing Length model for the Eddy Viscosity / Diffusivity
K m  u* z (1  z / h)
from log layer observations:
 uT 
define:
uT  u*
  z (1  z / h)
 max  0.1 h at z  h / 2
z
 (z )
uT

g' 
“reduced” gravity

Now add stratification
Buoyancy frequency

N2  
g
g 
 z
Velocity
enhanced shear
near pycnocline
ρ1
“eddy viscosity”
stratification
damps turbulence
near pycnocline
ρ2
stress
ub
log layer
near bottom
What is the maximum vertical scale for turbulence with stratification?
Bernoulli Function (energy-conserving flow)
B
z
z   o
i
1 2
u  gz  p
2
h1  z
1
 i ui 2   i gz  g   ( z )dz
2
h1
1 
g
(z ) 2
2 z
 ui  Nz

take uT  u1
uT
o 
N

The Ozmidov scale:
maximum size of eddies before
gravity arrests them.
Schematic of turbulence length-scale in a stratified estuary
distance from bed
turbulence suppressed
Ozmidov scaling:
LT=uT/N
u(z)
Boundary layer:
LT ~ kz
Limiting Length-scales in Turbulent Flows
Boundary-Layer Scaling (depth limiting)
1/ 2
 z
LT  LBL  z 1  
 h
Ozmidov Scaling (stratification limiting)
  
LT  LO   3 
N 
1/ 2
h
LT
z
Note that LT  Thorpe overturn scale
0
LBL
0.2 h
Relative flow
direction
spectral density S(k)
Turbulence length-scale LT~ 1/ko
ko
Scully et al. (2010) Influence of stratification
on estuarine turbulence
Snohomish River
Boundary-Layer scaling
Dissipation: the currency ($ or € ?) of turbulence
Turbulent dissipation
(conversion of turbulent
motions to heat)
ui
 
x j
2
=
In a boundary layer, dissipation ~ production
  u*2
u
z
“Inertial subrange” method for
estimating dissipation:
S (k )  ao 2 / 3 k 5 / 3
ko
ensemble
average of
turbulent
motions
The Parameter Space of Estuarine Turbulence
Turbulent Dissipation ε, m2s-3
Estuaries
Continental
Shelf
lo = ( ε/N3 )1/2
Geyer et al. 2008:
Quantifying vertical
mixing in estuaries
Buoyancy Frequency N, s-1
Two different paradigms of estuarine mixing. How important
is the stratified shear layer paradigm in estuarine turbulence?
Stratified boundary layer
Stratified shear layer
turbulence
turbulence
u(z)
u(z)
turbulence
no turbulence
Shear Instability
Thorpe, 1973
Ri 
N2
gradient Richardson number
Richardson, 1920
 u 



z


Ri  0.25 necessary condition for stability
2
Miles, 1961; Howard, 1961
Smyth et al., 2001
Momentum balance of a tilted interface
us
ρ1

hi
ub
ρ2
 2 u ' w' g ' hi

2
z
 x
hi
1
u ' w'max  g 
8
x
0.5-1x10-4 m2s-2 for strong transition zones –
moderate but not intense stress
Fraser River salt wedge—early ebb (Geyer and Farmer 1989)
meters
interface
1.2 m/s
weak motion
bottom
Connecticut River: Geyer et al. 2010: Shear Instability at high Reynolds number
1.2 m/s
Ri<0.25
leading
to
shear
instability
200
180 m
400
160
200 m
0
Day 325--Transect 17 (~ hour 19.1)
river
ocean
Salinity
meters along river
dissipation of TKE
dissipation of salinity variance
M
Echo Sounding
at Anchor Station
M
M
M
B
C
B
C
B
C
M
M
B: braid
C: core
M: mixing zone
M
Salinity contours (black)
Salinity variance (dots)
M
C
B
M
C
B
M
M
#4
C
M
B
#5
#6
#4
Salinity
timeseries
~ 60 seconds
M
M
B
C
M
M
B
C
M
M
C
B
#5
#6
Staquet, 1995
Re~1,000
MIXING
in cores
Re~500,000
MIXING
in braids
α
ρ1
ρc
ρ2
Baroclinicity of the braid accelerates the shear… with plenty of time within the braid…
 u  g 

t z

…leading to mixing:
Tacc
1 1

 Ri o 1 ~ 0.7
Tadv 8
u ' w'max 
 max 

8

8
g  ~ 1  2 Pa
g u ~ 5 10 3 m 2 s 3
New profiler data and acoustic imagery
20 seconds
30 meters
Very intense, and very pretty…
100 m
…but is mixing at hydraulic transitions important at the scale of the estuary?
Buoyancy flux B =
∫∫∫β g s′w′ dV
fresh
salt
Dissipation D =
∫∫∫ε dV
Energy balance: P = B + D
Efficiency Rf = B/P = B/(D+B)
Net
Tidal
Power
“P”
Hudson: ROMS
Merrimack: FVCOM
Massachusetts
Merrimack River mixing analysis
U(z)
u’w’
U(z)
In the estuary
u’w’
Ralston et al., 2010
Turbulent mixing in a
strongly forced salt
wedge estuary.
volume-integrated buoyancy flux
Boundary layer
Internal shear
Boundary layer
Hudson River mixing analysis
ROMS, Qr = 300 m3/s
Boundary layer
Internal shear
Scully,
unpublished
Boundary layer
Internal shear
testing turbulence closure stability functions with Mast data
Rf
Canuto et al., 2001
Kantha and Clayson 1994
Ri
Scully,
unpublished
Observed buoyancy flux vs. Ri
  B dz dt
ebb depth
Modeled buoyancy flux vs. Ri
k-
Mellor-Yamada 2.5 (k-kl)
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 of
stratified mixing processes. We need more measurements of turbulence
in these environments!