1. Mean and Fluctuating Quantities
Ocean Surface
3D Turbulence
Current Meter
Mean Flow
Fluctuating Flow
One Dimensional Measurement
time

2. Three Types of Averages Ensemble
Time
Space
Ergodic Hypothesis: Replace ensemble average by either a space or time average

3. Concept of Correlation Function
Auto Correlation Function
Cross Correlation Function

4. Correlation Function R
Concept of Spatial Homogeneity and Temporal Stationarity
Time or Space
Correlation Function R

5. Homogeneous/Stationary I D Correlation Function

6. Velocity Cross Correlation Function
Auto Covariance Function

7. r 1
r=0 x
Time or Space Axis

8. Concept of Structure Function
Integral scale
Microscale

9. Taylor’s Microscale
Temporal case
Spatial case

10. How to Calculate Correlation Functions from Data
x, t
L,T

11. Concept of Spectrum
Temporal Spectrum

12. Spatial Spectra
Terminology

13. x r Normalized Correlation Function and Spectra
Integral Scale = Area under curve r x
You can show that:

14. 3D + Time Spectra

15. Gradient Spectra

17. Spatial Spectra
Gradient Spectra

18. Use of the Log-Log plot Linear Plot Log-Log Plot p=-2 100 p=2 p=-2 p=220 10
100
p=-2
p=2

19. Spectra Area under curve Interpret k as eddy of size Gradient Spectra

20. Consider Model Correlation Function
is the area under w
Consider Model Correlation Function

21. From F we create a simulation of and

24. Calculation of Spectra
Spectra = Decomposition of Variance into contributions by sines/cosines L

28. Calculation of Spectra of u’Dx
1. Choose Sampling (digitizing) Dx
2. Calculate f the DFT (Discrete Fourier Transform) of sections of u’
3. Estimate Spectra by
where & 1 2 3 n

29. The Uncertainty Principle

30. Developing the Concept of an Eddy
Real(u) f

31. The Eddy

33. Q

34. kQ

35. kQ