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

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

Concept of Correlation Function Auto Correlation Function Cross Correlation Function

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

Homogeneous/Stationary I D Correlation Function

Velocity Cross Correlation Function Auto Covariance Function

r 1 r=0 x Time or Space Axis

Concept of Structure Function Integral scale Microscale

Taylor’s Microscale Temporal case Spatial case

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

Concept of Spectrum Temporal Spectrum

Spatial Spectra Terminology

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

3D + Time Spectra

Gradient Spectra

Spatial Spectra Gradient Spectra

Use of the Log-Log plot Linear Plot Log-Log Plot p=-2 100 p=2 p=-2 p=2 20 10 100 p=-2 p=2 1 2 10 20

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

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

From F we create a simulation of and

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

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

The Uncertainty Principle

Developing the Concept of an Eddy Real(u) f

The Eddy

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