Turbulent Kinetic Energy (TKE)

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

4. Multiplying turbulent flow times ui and dropping the primes fluctuating strain rate Turbulent Kinetic Energy (TKE) Equation Total changes of TKE Transport of TKE Shear Production Buoyancy Production Viscous Dissipation 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. Vertical Shears (vertical gradients) Shear production from bottom stress z u bottom

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

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

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

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

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

19. Layering Experiment http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html

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

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

25. (Monismith’s Lectures)

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

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

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) U3

30. (Monismith’s Lectures)

31. (Monismith’s Lectures)

32. (Monismith’s Lectures)