1. Turbulence and mixing in estuaries
Rocky Geyer, WHOI
Acknowlegments:
David Ralston, WHOI
Malcolm Scully, Old Dominion U.

2. wind
Wind-driven
turbulence
velocity
Interfacial, shear-driven turbulence
Boundary-layer
turbulence

3. τB /ρ= Cdub2=u*2 Simplest case: unstratified tidal flow:
Only boundary-layer turbulence
Velocity =
log layer
“eddy viscosity”
stress ub
Bottom stress
τB /ρ= Cdub2=u*2
Turbulent velocity scale
uT ~u* ~ 0.05 ub

4. Mixing Length model for the Eddy Viscosity / Diffusivity
from log layer observations:
define:

5. Now add stratification
“reduced” gravity
Now add stratification
Buoyancy frequency
Velocity
“eddy viscosity” ρ1
enhanced shear
near pycnocline
stratification
damps turbulence
near pycnocline ρ2
stress ub
log layer
near bottom

6. What is the maximum vertical scale for turbulence with stratification?
Bernoulli Function (energy-conserving flow) z
The Ozmidov scale:
maximum size of eddies before
gravity arrests them. 

7. Schematic of turbulence length-scale in a stratified estuary
turbulence suppressed
Ozmidov scaling:
LT=uT/N
distance from bed
u(z)
Boundary layer:
LT ~ kz
So again the conceptual diagram– pretty comfortable with this paradigm, but there is evidence of some non-equilibrium turbulence in the upper bl, but we don’t know what is fuelling it. Now we will look at the wake data from the snohomish---

8. Note that LT  Thorpe overturn scale
Limiting Length-scales in Turbulent Flows
Boundary-Layer Scaling (depth limiting)
Ozmidov Scaling (stratification limiting) h LT z
Now we will get into the “conventional wisdom of the turbulent lenth-scale. In boundary layers, the length-scale comes directly from log-layer theory, and so if we have a log layer, the length-scale varies linearly away from the bed, and it reaches a maximum somewhere in the middle part of the flow that is approximately 1 /(2 pi) of the height. In stratified flows, the Ozmidov scale is limiting, as I mentioned above, and I promised I would talk about. Ok, the ozmidov scale is the largest scale at which there is enough energy in the turbulence to have overturns. At larger scales, there are no overturns. The way I like to think about this is that you put a weight on a bicycle wheel– that represents the stratification- and you start the wheel spinning at a certain speed that is related to the energy level of the turbulence. Lo is the largest radius for that initial velocity that will go all the way over the top– i.e., which has overturns instead of internal waves. And a lot of people, notably Moum, have shown that the ozmidov scale is similar to the thorpe scale, which is the directly measured scale in vertical profiles, shown in this schematic of an overturn.
Note that LT  Thorpe overturn scale
0.2 h
LBL

9. Relative flow
direction

10. ko
spectral density S(k)
Turbulence length-scale LT~ 1/ko

11. Snohomish River Boundary-Layer scaling Ozmidov Length scaling
Scully et al. (2010) Influence of stratification
on estuarine turbulence
Snohomish River
Boundary-Layer scaling
I have to admit, I don’t like non-dimensional plots, because I’m a real world guy, and I like looking at the dimensions of things. (Everyone knows how long a meter is, but how long is the Ozmidov scale in the Merrimack estuary? Well, its 10 to 20 cm, until the stratification gets blown out, and it goes up to 2-3 m). But sometimes we have to plot things non-dimensionally to show that our data matches our theory. So here is a non-dimensional plot of our length-scale data. the points that lie along the diagaonal line agree with ozmidov scaling. If they are above the line, they are bigger, and below the line, smaller. the points that lie along the horizontal line obey boundary layer scaling. This plot basically indicates that the merrimack data, with only a few exceptions, agree with the conventional wisdom. It is interesting that the exceptions have Ri> We consider these to be odd-balls, that are not in equilibrium with their surroundings--i.e., they don’t belong. One way of thinking about this plot in physical space is that the data to the right are toward the bottom, so we are looking sideways at a vertical profile, moving from unstratified to stratified conditions. The transition region from unsratified to stratified is where the shift occurs from boundary-layer scaling to ozmidov scaling. And in these data there is some suggestion that the length-scale is smaller than either assymtote in that transition region.
Ozmidov Length scaling

12. Dissipation: the currency ($ or € ?) of turbulence
ensemble
average of
turbulent
motions
Turbulent dissipation
(conversion of turbulent
motions to heat) =
In a boundary layer, dissipation ~ production ko
“Inertial subrange” method for
estimating dissipation:

13. The Parameter Space of Estuarine Turbulence
Estuaries
Rivers
3 m
lo =1 m
30 cm
10 cm
3 cm
Turbulent Dissipation ε, m2s-3
Continental
Shelf
Lakes
Before I get into the details of our measurements, I want to take a little time to discuss the parameter space in which we are measuring turbulence, and compare it to other oceanic environments. This plot of turbulence dissipation vs. buoyancy frequency (or stratification) has been used by Gibson and Moum, and it can be used to characterize the scales of turbulent motion. Ocean turbulence is down in the lower left corner of the plot, with low dissipation rates and low stratification. The continental shelf is intermediate, and estuaries are at the high end both in terms of energy and frequency. There are several implications of this– one is that the turbulence is often a couple of decades above the viscous limit (unlike oceanic turbulence), which means that there is a well-developed inertial subrange. We will use that characteristic of the turbulence to our advantage. Another is that there is a lot of signal, so we don’t have to use ultra-sensitive devices like shear-probes to measure dissipation rate. In fact this big signal lead to some problems with our micro-conductivity probes when we first used them, and we had to dial down the sensitivity so that we didn’t clip the signal.
Viscous limit
Ocean
lo = ( ε/N3 )1/2
Geyer et al. 2008:
Quantifying vertical
mixing in estuaries
Buoyancy Frequency N, s-1

14. Two different paradigms of estuarine mixing. How important
is the stratified shear layer paradigm in estuarine turbulence?
Stratified boundary layer
Stratified shear layer
u(z)
turbulence
turbulence
u(z)
turbulence
no turbulence
So again the conceptual diagram– pretty comfortable with this paradigm, but there is evidence of some non-equilibrium turbulence in the upper bl, but we don’t know what is fuelling it. Now we will look at the wake data from the snohomish---

15. Shear Instability Thorpe, 1973 gradient Richardson number
necessary condition for stability
Miles, 1961; Howard, 1961
Thorpe, 1973
gradient Richardson number
Richardson, 1920
Smyth et al., 2001

16. Momentum balance of a tilted interfaceus ρ1 hi ρ2 ub
Ri=0.4 u_s=1.2, delta h=2, b=1000 m.
ub=0.6, cd=3e-3, b=1000, L=10 km
0.5-1x10-4 m2s-2 for strong transition zones –
moderate but not intense stress

17. Fraser River salt wedge—early ebb (Geyer and Farmer 1989)
interface
1.2 m/s
meters
weak motion
bottom
Highly stratified estuaries are places where multi-scale interactions are particularly important in controlling the exchange of momentum and salt. This is an echo sounding image from my thesis work in the Fraser river, in which we see several distinct zones of mixing. Each of these zones is associated with a constriction, where the flow goes through a hydraulic transition from subcritical to supercritical flow. The hydraulic transitions result in intensified shear, and the intensified shear leads to

18. Ri<0.25 leading to shear instability 400 200 m 0 1.2 m/s
Connecticut River: Geyer et al. 2010: Shear Instability at high Reynolds number
1.2 m/s
Ri<0.25
leading to
shear
instability m m

19. Day 325--Transect 17 (~ hour 19.1)
river
ocean
Salinity
meters along river
dissipation of TKE
dissipation of salinity variance

20. Salinity contours (black) Salinity variance (dots)M B C
Echo Sounding
at Anchor Station
B: braid
C: core
M: mixing zone M B C #4 #5 #6
Salinity contours (black)
Salinity variance (dots) #4
Salinity
timeseries
~ 60 seconds #5 M B M M B B #6 M M M C C C

21. Staquet, 1995

22. Re~1,000
MIXING
in cores
Re~500,000
MIXING
in braids

23. α ρ1 ρc ρ2
Baroclinicity of the braid accelerates the shear… with plenty of time within the braid…
g’=0.25, alpha=0.3, delta=0.2 m (in braid) = 1.87 Pa
…leading to mixing:

24. New profiler data and acoustic imagery
20 seconds
30 meters

25. Very intense, and very pretty…
100 m
…but is mixing at hydraulic transitions important at the scale of the estuary?

26. Efficiency Rf = B/P = B/(D+B)
Buoyancy flux B = ∫∫∫β g s′w′ dV
fresh
Net
Tidal
Power
“P”
salt
Dissipation D = ∫∫∫ε dV
Energy balance: P = B + D
Efficiency Rf = B/P = B/(D+B)

27. Hudson: ROMS
Merrimack: FVCOM
Massachusetts

28. Merrimack River mixing analysis
In the estuary
U(z)
u’w’
U(z)
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

29. Hudson River mixing analysis
ROMS, Qr = 300 m3/s
Boundary layer
Internal shear
Scully,
unpublished
Boundary layer
Internal shear

30. testing turbulence closure stability functions with Mast data
Canuto et al., 2001
Scully,
unpublished Rf
Kantha and Clayson 1994 Ri

31. Observed buoyancy flux vs. Ri
Modeled buoyancy flux vs. Ri
k-
Mellor-Yamada 2.5 (k-kl)

32. Conclusions and Prospects for the Future
Stratified boundary-layer turbulence is the most important mixing regime in estuaries.
Shear instability is locally important and dramatic but is not the dominant contributor to the total mixing.
Closure models are on the right track. We need more data for testing them.
Estuaries are outstanding natural laboratories for the investigation of stratified mixing processes. We need more measurements of turbulence in these environments!