1. Turbulent properties:
- vary chaotically in time around a mean value
exhibit a wide, continuous range of scale variations
cascade energy from large to small spatial scales
“Big whorls have little whorls
Which feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity.” (Richardson, ~1920)

2. - Use these properties of turbulent flows in the Navier Stokes equations
The only terms that have products of fluctuations are the advection terms
All other terms remain the same, e.g.,

3. are the Reynolds stresses
arise from advective (non-linear or inertial) terms

4. Turbulent Kinetic Energy (TKE)
An equation to describe TKE is obtained by multiplying the momentum equation for turbulent flow times the flow itself (scalar product)
Total flow = Mean plus turbulent parts =
Same for a scalar:

5. Turbulent Kinetic Energy (TKE) Equation
Multiplying turbulent flow times ui and dropping the primes
Total changes of TKE
Transport of TKE
Shear
Production
Buoyancy
Production
Viscous
Dissipation
fluctuating strain rate
Transport of TKE. Has a flux divergence form and represents spatial transport of TKE. The first two terms are transport of turbulence by turbulence itself: pressure fluctuations (waves) and turbulent transport by eddies; the third term is viscous transport

6. interaction of Reynolds stresses with mean shear;
represents gain of TKE
represents gain or loss of TKE, depending on covariance
of density and w fluctuations
represents loss of TKE

7. In many ocean applications, the TKE balance is approximated as:

8. - distance from boundary
The largest scales of turbulent motion (energy containing scales) are set by geometry:
- depth of channel
- distance from boundary
The rate of energy transfer to smaller scales can be estimated from scaling:
u velocity of the eddies containing energy
l is the length scale of those eddies
u2 kinetic energy of eddies
l / u turnover time
u2 / (l / u ) rate of energy transfer = u3 / l ~
At any intermediate scale l,
But at the smallest scales LK,
Kolmogorov length scale
Typically,
so that

9. Shear production from bottom stressz u
Vertical Shears (vertical gradients)
bottom

10. Shear production from wind stressz W u
Vertical Shears (vertical gradients)

11. Shear production from internal stressesz
Vertical Shears (vertical gradients) u1 u2
Flux of momentum from regions of fast flow to regions of slow flow

12. Parameterizations and representations of Shear Production
Near the bottom
Law of the wall
Bottom stress:

13. Florida Intracoastal Waterway
Data from Ponce de Leon Inlet
Florida Intracoastal Waterway
Florida

14. Law of the wall may be widely applicable
(Monismith’s Lectures)
Law of the wall may be widely applicable

15. (Monismith’s Lectures)
Ralph
Obtained from velocity profiles and best fitting them to the values of z0 and u*

16. Shear Production from Reynolds’ stresses
Mixing of property S
Mixing of momentum
Munk & Anderson (1948, J. Mar. Res., 7, 276)
Pacanowski & Philander (1981, J. Phys. Oceanogr., 11, 1443)

17. With ADCP:
and
θ is the angle of ADCP’s transducers -- 20º
Lohrmann et al. (1990, J. Oc. Atmos. Tech., 7, 19)

18. Souza et al. (2004, Geophys. Res. Lett., 31, L20309)
(2002)

19. Souza et al. (2004, Geophys. Res. Lett., 31, L20309)
Day of the year (2002)

20. Souza et al. (2004, Geophys. Res. Lett., 31, L20309)

21. Buoyancy Production from
Cooling and Double Diffusion
S1, T1
S2, T2
S2 > S1
T2 > T1

22. Layering Experiment

23. Data from the Arctic
From Kelley et al. (2002, The Diffusive Regime of Double-Diffusive Convection)

24. Layers in Seno Gala

25. Dissipation from strain in the flow (m2/s3)
(Jennifer MacKinnon’s webpage)

26. Production of TKE Dissipation of TKE From:
Rippeth et al. (2003, JPO, 1889)
Dissipation of TKE

27. Example of Spectrum – Electromagnetic Spectrum

28. (Monismith’s Lectures)

29. Other ways to determine dissipation (indirectly)
Kolmogorov’s K-5/3 law
S (m3 s-2)
Wave number K (m-1)

30. P Kolmogorov’s K-5/3 law (Monismith’s Lectures) equilibrium range
inertial
dissipating range

31. Kolmogorov’s K-5/3 law -- one of the most important results of turbulence theory
(Monismith’s Lectures)

32. Stratification kills turbulence
In stratified flow, buoyancy tends to:
i) inhibit range of scales in the subinertial range
ii) “kill” the turbulence

33. (Monismith’s Lectures)

34. (Monismith’s Lectures)

35. (Monismith’s Lectures)

37. At intermediate scales --Inertial subrange –
(responsible for dissipation of TKE)
At intermediate scales --Inertial subrange –
transfer of energy by inertial forces
(Monismith’s Lectures)

38. Other ways to determine dissipation (indirectly)
(Monismith’s Lectures)