1. 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:

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) 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

5. 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

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

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

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

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

10. Parameterizations and representations of Shear Production
Near the bottom
Bottom stress:

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

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

13. Shear Production from Reynolds’ stresses
Mixing of property S
Mixing of momentum
With ADCP:
and
θ is the angle of ADCP’s transducers -- 20º
Lohrmann et al. (1990, J. Oc. Atmos. Tech., 7, 19)

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

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

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

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

19. Layering Experiment

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

21. Layers in Seno Gala

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

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

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

25. (Monismith’s Lectures)

26. Inertial subrange – transfer of energy by inertial forces
(responsible for dissipation of TKE)
Inertial subrange – transfer of energy by inertial forces
(Monismith’s Lectures)

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

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

29. (Monismith’s Lectures)

30. (Monismith’s Lectures)

31. (Monismith’s Lectures)

32. (Monismith’s Lectures)