Paula Agudelo

Turbulence, Intermittency and Chaos in High-Resolution Data,

Collected At The Amazon Forest.

The data used consists of the wind velocity components along the three orthogonal directions and the temperature, all obtained using fast response sonic instruments.

60m tower built in the Rebio Jaru reserve in (10º04'S 61º56'W), Brazilian state of Rondonia.

(The Large Scale Biosphere-Atmosphere Experiment in the Amazon)

Data were collected as part of a LBA project

frequency of 60 Hz. (60 Samples/second) (9min)

DATA

Data at 21m and 66m

SERIES

6th

7thMarch/1999

12pm

3pm

6pm

9pm12am U V

W T

Histograms

Examples

Examples

Profiles

Fourier Vs WaveletsFourier transform

Decompose a time series in sines and cosines

of different frequencies.

Wavelet transform Decompose a time series in different functions

Since sines and cosines are infinite functions,It only gives information of frequency

The wavelet function goes to zero, giving information of frequency and localization in time

7 March, 12pm at 66m

Kolmogorov law of -5/3 (n=2)

33

nrn

n

n

i Ku

2

2

i

j

j

i

x

u

x

u

8 March, 12pm at 66m

Kolmogorov law of 5/3

)2ln(2

)(

2)(

dyiWTKE

m

m

2

122)(4)(

)2ln(2)(

iWTiWT

dyKSD mmmE

)(

)()(

m

mEmE KE

KSDKCV

nm

nm

n

dy

iWTxrx

22

)(~)()(

22

4

)(iWT

iWTRFF

m

m

m

Removing intermittencyWT:Wavelet Coefficients (Result of the transform) Sum over all WT = Series Variance

Km=Wave Number

Spectral density function

Standard deviation

Coefficient of energy variation

Structure Function

Flatness Factor (Similar to Kurtosis)

Results

Results

Chaotic BehaviorPhase Space reconstruction

how to go from scalar observations to multivariate phase space

to apply the embedding theorem

to say that what time lag (time delay) to use and what dimension to use are the central

issues of this reconstruction.

Average Mutual Information

)()(

),(log),( 2

, iBiA

iiAB

baiiABAB bPaP

baPbaPI

ii

Embedding dimension dE.

Global False Nearest Neighbors

7 March, 12pm at 21m

7 March, 12pm at 66m

Mutual Information

False Nearest Neigbors, 12pm 09/07 21m

0

20

40

60

80

100

1 2 3 4 5 6 7

Dimension

Per

cen

tag

e o

f F

NN

False Nearest Neigbors, 12pm 09/07 66m

0

20

40

60

80

100

1 2 3 4 5 6 7

Dimension

Per

cen

tag

e o

f F

NN

False Nearest Neigbors, 9pm 09/08 21m

0

20

40

60

80

100

1 2 3 4 5 6 7

Dimension

Per

cen

tag

e o

f F

NN

Embedding dimension

False Nearest Neigbors, 9pm 09/08 21m

0

20

40

60

80

100

1 2 3 4 5 6 7

Dimension

Perc

enta

ge o

f FN

N

Lorenz Attractor

U Component12am

U Component12pm

T Component5pm

T Component5pm