NASA Technical Memorandum 104648

Spherical Harmonics Analysis of the

ECMWF Global Wind Fields at the

10-Meter Height Level During 1985:

A Collection of Figures Illustrating Results

Braulio V. Sanchez

Goddard Space Flight Center

Greenbelt, Maryland

Masahiro Nishihama

Hughes STX

Greenbelt, Maryland

National Aeronautics and

Space Administration

Goddard Space Flight Center

Greenbelt, Maryland

1997
I This publication is available from the NASA Center for AeroSpace Information, I 800 Elkridge Landing Road, Linthicum Heights, MD 21090-2934, 301-621-0390. I
NASA TECHNICAL MEMORANDUM 104648

ABSTRACT

Half-daily global wind speeds in the east-west (u) and north-south (v) directions at the 10-meter

height level were obtained from the European Centre for Medium Range Weather Forecasts

(ECMWF) data set of global analyses. The data set covered the period 1985 January to 1995 January.

A spherical harmonic expansion to degree and order 50 was used to perform harmonic analysis of

the east-west (u) and north-south (v) velocity field components.

The resulting wind field is displayed, as well as the residual of the fit, at a particular time. The

contribution of particular coefficients is shown. The time variability of the coefficients up to degree

and order 3 is presented. Corresponding power spectrum plots are given. Time series analyses were

applied also to the power associated with degrees 0-10; the results are included.

..,

III
BRAULIO W. SANCHEZ AND MASAHIRO NISHIHAMA

ACKNOWLEDGMENTS

We express our appreciation to Dr. Nikolaos K. Pavlis of Hughes STX for providing the Fast Fourier

Transform software. We give thanks to Dr. Andrew Y. Au of Hughes STX for facilitating access to

the ECMWF data set. This work was supported by the Space Geodesy Branch at the Goddard Space

Flight Center and the TOPEX/POSEIDON project.

iv
NASA TECHNICALMEMORANDUM104648

TABLE OF CONTENTS

Introduction .................................................................................................................................... 1

Global Wind Fields ......................................................................................................................... 3

Figures 1-I0 ............................................................................................................................. 5

Time Variability of Selected Spherical Harmonics Coefficients .................................................. 17

Figures 11-45 ......................................................................................................................... 19

References .................................................................................................................................... 57

Appendix I .................................................................................................................................... 59

Appendix II ................................................................................................................................... 61

NASA TECI-[NICALMEMORANDUM104648

INTRODUCTION

The general circulation of the ocean is driven by various types of forces, including surface wind

stress. The results presented in this publication were obtained as part of an investigation of the winddriven variability of the ocean circulation. It is possible that these results might be of interest to other

investigators, not only in the limited context of ocean circulation, but also from the larger perspective

of geodynamics, Earth rotation, and space geodesy. With such possibility in mind, it was decided to

document the results. However, the intention is only to present the results obtained in graphical form;

no interpretation is attempted here.

Half-dally global wind speeds in the east-west (u) and north-south (v) directions at the 10-meter

height level were obtained from the European Centre for Medium Range Weather Forecasts

(ECMWF) data set of global analyses. The data set covered the period 1985 January to 1995 January.

The original global grid is 2.5 ° latitude by 2.5 ° longitude, and the data set used in this investigation

has been reprocessed and interpolated onto a 2 ° latitude by 2.5 ° longitude grid, as described by

Schubert et al. (1990).

A spherical harmonic expansion to degree and order 50 was used to perform harmonic analysis of

the east-west (u) and north-south (v) velocity field components. The formal validity of this procedure

was established by Simmonds (1974). Formal developments are given in Appendix I.

The Software used in the analysis implements a technique developed by O. L. Colombo (1981),

based on the Fast Fourier Transform.

NASATECHNICAL MEMORANDUM 104648

GLOBAL WIND FIELDS

Figures 1-10 show the vectors for the wind fields corresponding to various cases. With the exception

of figures 7 and 8, the time epoch is January 1, 0 hour GMT, 1985. This is the time of the first data

entry for 1985.

Figure 1 shows the wind field as obtained from the ECMWF data files, the maximum magnitude is

22 meters/second. This wind field was fitted by means of an expansion in spherical harmonics

including coefficients up to degree and order 10. The resulting field is shown in figure 2; maximum

wind speed is 12 meters/second. The difference between the data and the fit is displayed in figure 3;

the largest discrepancy has a magnitude of 16 meters/second.

Figure 4 exhibits the wind field obtained by fitting the data with spherical harmonics up to degree

and order 50, note the improvement as compared to the degree-10 fit; the maximum wind speed is

now 20 meters/second. The residuals are shown in figure 5; the maximum error has a magnitude of

4.5 meters/second. Note that the largest discrepancies are concentrated around the poles.

Figures 6 and 7 exhibit the wind fields from the ECMWF data files at January 1, 0 hour GMT, 1985

and at July 1, 0 hour GMT, 1985; the wind vectors are plotted with equal length in order to show

clearly the wind direction, the magnitudes are indicated by the shading, the lower (higher) wind

speeds corresponding to the darker (brighter) shades. Figure 8 depicts the average wind field for

1985.

Figures 9 and 10 are included to display the effect of some of the coefficients in the spherical harmonic expansion. Figure 9 brings to view the wind field corresponding to the C10 coefficient by

itself. Figure 10 makes apparent the wind field produced by the sum of the C H and S_1 coefficients.

3

NASA TECHNICALMEMORANDUM104648

FIGURES 1-10

NASA TECHNICAL MEMORANDUM 104648

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BRAULIO V. SANCHEZ AND MASAHmO NISHH-IAMA

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NASA TECHNICALMEMORANDUM104648

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BRAULIO V. SANCHEZ AND MASAHIRO NISHH-IAMA

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NASA TECHNICALMEMORANDUM104648

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BRAULIO V. SANCHEZ AND MASAHIRO NISHrI-IAMA

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NASA TECHNICAL MEMORANDUM 104648

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BRAULIO V. SANCHEZ AND MASAHIRO NISFIIHAMA

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NASA TECHNICAL MEMORANDUM 104648

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BRAULIO V, SANCHFAZ AND MASAHIRO NISHIHAMA

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16
NASA TEC_n_ICALMEMORANDUM104648

TIME VARIABILITY OF SELECTED SPHERICAL HARMONIC COEFFICIENTS

Figures 11-26 portray the time variability of the spherical harmonic coefficients up to degree and

order 3. Each figure shows the results for the east-west (u) and north-south (v) velocity components.

There are 730 values over the course of a year for each spherical harmonic coefficient. These time

series were analyzed by means of Fourier expansions in sine and cosine terms, as presented in

Appendix 1I. Plots of power as a function of frequency are given in figures 27-34. In general, most

of the power is concentrated in the low frequency terms, especially the zero frequency component.

The east-west velocity component predominates over the north-south, with the exception of the

coefficients Coo, C3o, and $32.

The time series analysis was applied also to the power associated with degrees 0-10, as given by the

last equation in Appendix I. The results are dispayed in figures 35-45. Each figure presents the

results for the east-west (u) and north-south (v) velocity components. As shown in part (a) of the

figures, most of the power is contained in the zero-frequency term. Part (b) makes evident the power

in the other frequencies by setting the constant part equal to zero.

17

NASA TECHNICALMEMORANDUM104648

FIGURES 11-45

19

U3

W

._J

Figure 11a.

NASA TECHNICAL MEMORANDUM 104648

U10M COEFFICIENT C00 IN 1985

50, 100.

The Coefficient Coo,Year 1985. The East-West Velocity Component (u).

.74

.68

.61

.54

.47

.41

.34

.27

.21

.14

W

=_3.o72

<> .0045

-.063

-.13

-.20

-.26

-.33

-.40

-.47

-.53

-.60

V10M COEFFICIENT COO IN 1985

50. 150, 200, 250.

Days

Figure llb. The Coefficient Coo,Year 1985. The North-South Velocity Component (v).

21
iJJ

.--I

Figure 12a.

0

.0

-.048

-.11

-.18

-.25-

-.31

-.38

-.44

-.51

-.57 4

-.64

-.71

-.77

-.84

-.90

-.97

-.1.0

-.1.1

-.1.2

-.1.2

-.1.3

-.1.4

BRAULIO W. SANCHEZ AND MASAHIRO NISHIHAMA

U10M COEFFICIENT C10 IN 1985

50. 100. 150. 260. 240. 3do.

DAYS

350.365.

The Coefficient C_o, Year 1985. The East-West Velocity Component (u).

V10M COEFFICIENT C10 IN 1985

Figure 12b. The Coefficient C,0 , Year 1985. The North-South Velocity Component (v).

22
NASA TECHNICAL MEMORANDUM 104648

1.1

U10M COEFFICIENT Cll IN 1985

3

<>

.26

.19

.12

.054

-.015

-.084

-.15

-.22

-.29

50 IO0 150 200. 250.

Days

Figure 13a. The Coefficient Cn, Year 1985. The East-West Velocity Component (u).

V10M COEFFICIENT C11 IN 1985

.55

.51

.46

.42

.38

.34

.30

.26

_ .18

_, .14

i i

I'T'Ill "llw' I iil_tll !II

Figure 13b. The Coefficient C n, Year 1985. The North-South Velocity Component (v).

23
X

Figure 14a.

.98

.92

.86

.81

.75

.69

.64

.58

.52

.47

.41

.35

,29

.24

.18

.12

,067

.010

-.047

-.10

-.16

The

BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

U10M COEFFICIENT $11 IN 1985

Coefficient S_1,Year 1985. The East-West Velocity Component (u).

.66

.61

.56

.51

.45

.40

.35

.30

.25

.20

.15

.094

.042

-.0097

-.061

-.11

-.16

-.22

-.27

-.32

-.37

V10M COEFFICIENT S11 IN 1985

150, 200. 250.

Days

Figure 14b. The Coefficient Sw Year 1985. The North-South Velocity Component (v).

24
NASA TECHNICAL MEMORANDUM 104648

2.7

2.6

2.5

2.4

2.4

2.3

2.2

2.1

2.0

1.9

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

.98

.88

.78

.69

.59

.49

.3g

.29

.20

.098

.0

.0

U10M COEFFICIENT C20 IN 1985

i

50..... 100. 150. 2 O. 25o. 300.' 350365. ''

DAYS

Figure 15a. The Coefficient C20, Year 1985. The East.West Velocity Component (u).

V10M COEFFICIENT C20 IN 1985

.62

.55

.48 4

.41

.33

.26

,19

.12

.052

-.018

-.089

-16

-.23

-.30

-.37 •

-.44

-.51 •

-.58 •

-.65 •

-.72 [

-.80

Figure 15b. The Coefficient C20,Year 1985. The North-South Velocity Component (v).

25
BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

l U10M COEFFICIENT C21 IN 1985

.72

.66 I

.61

.55

.49

.43 t

'Ill _ .20

_, .14

_ .084

Figure 16a. The Coefficient C21 , Year 1985. The East-West Velocity Component (u).

43

.38 I

.34

.29

.25

.21

V10M COEFFICIENT C21 IN 1985

.071

co .026 LU

3 -.o_9

Figure 16b. The Coefficient C_, Year 1985. The North-South Velocity Component (v).

26
NASA TECHNICAL MEMORANDUM 104648

U10M COEFFICIENT $21 IN 1985

.54

.50

.46

.42

.38

.34

.29

.25

.21

_ .17

_, .13

<> .086

.044

.0024 5

-.039

-.081

-.12

-.16

-.21

-.25

-.29

Figure 17a. The Coefficient S21 , Year 1985. The East-West Velocity Component (u).

.39

.34

.30

.26

.21

.17

.13

.082

.038

-.0054

_, -.049

X -.093

-.14

-.18

-.22

-.27

-.31

-.35

-.4o

-.44

-.49

V10M COEFFICIENT $21 IN 1985

Figure 17b. The Coefficient S21,Year 1985. The North-South Velocity Component (v).

27
BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

U10M COEFFICIENT C22 IN 1985

2

150. 250.

Day

Figure 18a. The Coefficient C=, Year 1985. The East-West Velocity Component (u).

V10M COEFFICIENT C22 IN 1985

,21

200. 250.

Doys

=,

X

Figure 18b. The Coefficient C22 , Year 1985. The North-South Velocity Component (v).

28
NASA TECHNICAL MEMORANDUM 104648

.65

.58

.52

.45

.39

.32

.26

.19

.13

_ .061

-.0043

-.ozo

-.13

-.20

-.27

-,33

-.40

-.46

-.53

-.59

-,66

UIOM COEFFICIENT S22 IN 1985

Figure 19a. The Coefficient S22, Year 1985. The East-West Velocity Component (u).

COEFFICIENT S22 IN 1985

5

Figure 19b. The Coefficient $22, Year 1985. The North-South Velocity Component (v).

29
.62

.54

.46

.38

.30

.23

.15

.067

-.012

co -.091

UJ

-.17

_ -.25

-.33

-.41

-.49

-.56

-.84

-.72

-.80

-.88

-.96

Figure 20a. The

BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

U10M COEFFICIENT C30 IN 1985

Coefficient C3o, Year 1985. The East-West Velocity Component (u).

=

<>

.94

.88

.83

.77

.72

.66

.61

.56

.50

.45

.39

.34

.28

.23

.17

.12

.063

.0083

-.046

-.10.

Vl 0M COEFFICIENT C30 IN 1985

Figure 20b. The Coefficient C3o, Year 1985. The North-South Velocity Component (v).

30
NASA TECHNICALMEMORANDUM 104648

U10M COEFFICIENT C31 IN 1985

3

<>

.31

.25

.19

.13

.064

.0022

-.060

-.12

ILll

I

Figure 21a. The Coefficient C3_, Year 1985. The East-West Velocity Component (u).

.31

.27

.23

.19

,14

.100

.057

.014

-.029

-.071

_, -.11

X -.16

-.2o

-.24

-.28

-.33

-.37

-.41

-.46

-.50

-.54

V10M COEFFICIENT C31 IN 1985

Figure 2lb. The Coefficient C3_, Year 1985. The North-South Velocity Component (v).

31
BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

3

X

UIOM COEFFICIENT S31 IN 1985

Figure 22a. The Coefficient $3, Year 1985. The East-West Velocity Component (u).

3

.63

.56

.49

.42

.35

.28

.21

.14

.072

.0019

-.068

-.14

-.21

-.28

-.36

-.42

-.49

-.56

-.63

-.70

-.77

VIOM COEFFICIENT S31 IN 1985

150.

Doy

Figure 22b. The Coefficient S3x,Year 1985. The North-South Velocity Component (v).

32
.40

.34

.29

.23

.17

.11

.051

-.0073

-.066

-.12

-.18

X -.24

-.30

-.36

-.42

-.48

-.54

-.59

-.65

-.71

-.77

Figure 23a. The

NASA TECHNICALMEMORANDUM104648

U10M COEFFICIENT C32 IN 1985

Coefficient C32,Year 1985. The East-West Velocity Component (u).

3

X

.54

.48

.43

.37

.32

.26

.21

.15

.094

.039

-.016

-.072

-.13

-.18

-.24

-.29

-.35

-.40

-.46

-.52

-.57

V10M COEFFICIENT C32 IN 1985

Figure 23b. The Coefficient C32 , Year 1985. The North-South Velocity Component (v).

33
3

X

Figure 2Aa.

BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

.50

.46

.41

.37

.32

.28

.23

.19

.14

.095

.050

.0049

-.040

-.085

-.13

-.18

-.22

-.27

-.31

-.36

-.40

U10M COEFFICIENT $32 IN 1985

The Coefficient S32 , Year 1985. The East-West Velocity Component (u).

.74

.67

.60

.54

.47

.41

.34

.27

.21

.14

W

_ .074

.0074

-.059

-.13

-.19

-.26

-.32

-.39

-.46

-.52

-.59

V10M COEFFICIENT $32 IN 1985

350365.

Figure 24b. The Coefficient 832 , Year 1985. The North-South Velocity Component (v).

34
NASA TECHNICAL MEMORANDUM 104648

co

LI_I

3

X

.58

.53

.48

.44

.39

.35

.30

.26

.21

.17

.12

.075

.030

-.015

-.061

-.11

-,15

-.20

-.24

-.29

-.33

U10M COEFFICIENT C33 IN 1985

Figure 25a. The Coefficient C33 , Year 1985. The East-West Velocity Component (u).

Figure 25b.

X

,61

.55

.49

.43

.38

,32

.26

.20

.14

.083

.024

-.034

-.093

-.15

-.21

-.27

-.33

-.39

-.44

-.50

-.56

V10M COEFFICIENT C33 IN 1985

The Coefficient C33 , Year 1985. The North-South Velocity Component (v).

35
BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

U10M COEFFICIENT $33 IN 1985

.78

.73

.68

.63

.58

.53

.47

.42

_ .32

_> .22

.17

.11

.06 i

 iii:oI -oV rVr

350365

Figure 26a. The Coefficient $33, Year 1985. The East-West Velocity Component (u).

3

X

.71

.65

.60

.55

.50

.45

.40

.34

.29

.24

.19

.14

.087

.036

-.016

-.068

-.12

-.17

-.22

-.27

-.33

V10M COEFFICIENT $33 IN 1985

Figure 26b. The Coefficient $33,Year 1985. The North-South Velocity Component (v).

36
NASA TECHNICAL MEMORANDUM 104648

U/V 10M POWER SPECTRUM OF C00 OVER 1985

nO

.22

.21

.20

.19

.18

.!7

.16

.14

.13

.12

.11

.10

.089

.078

.067

.056

.044

.033

.022

.011

T '

T ,'

.0

0 2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 27a. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: Nol_h-South Velocity Component (v). The Coefficient Coo.

U/V 1

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.1

rr .97

.88

O .79

.70

.62

.53

.44

.35

.26

.18

.088

.0

0M POWER SPECTRUM OF C10 OVER 1985

0 2' 4 6 8 10121 '''4 161820222 .... '426283032343638404244464850 ....

FREQUENCY

Figure 27b. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient C_o.

37
BRAULIO W. SANCHEZ AND MASAHIRO NISHINAMA

U/V 10M POWER SPECTRUM OF C20 OVER 1985

11.

10.

9.7

9.1

8.6

8.0

7.5

7.0

6.4

rr 5.9

LU

O_ 5.4

o.. 4.8

4.3

3.8

3.2

2.7

2.1

1.6

1.1

.54

'00 -'2--'_ 6 8 10121416182b222_,26283()32343;384()42444;485()

FREQUENCY

Figure 28a. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-Sou_ Velocity Component (v). The Coemcient C_.

U/V 10M POWER SPECTRUM OF C30 OVER 1985

.37

.35

.33

.31

.29

.28

.26

.24

rr .22

.20

n° .18

.17

.15

.13

.11

.092

.074

.055

.037

.018

.0

2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 28b. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient C3o.

38
NASA TECHNICALMEMORANDUM104648

U/V 10M POWER SPECTRUM OF Cll OVER 1985

.56

.53

.51

.48

.45

.42

.39

.37

.34 rr

._ .31

n° .28

.25

.23

.20

.17

.14

.11

.084

,056

.028

,0 _ _f-':'-'_ - " _ _ = _ : _ ' ' ' ' ' _ ' ' ' '

0 2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 29a. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient C u.

U/V 10M POWER SPECTRUM OF $11 OVER 1985

.59

.56

.53

.50

.47

.44

.41

.38

cc .35

.32

O .29

.27

.24

.21

.18

.15

.12

.088

.059

.029

.0

0 2 4 6 8 10121416182022242628303234363840424446485n

FREQUENCY

Figure 29b. Power Spectrum.

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient S..

Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

39
BRAULIO V, SANCHF..Z AND MASAHIRO NISHIHAMA

U/V 10M POWER SPECTRUM OF 021 OVER 1985

,018

,017

,016

.015

,014

.013

,012

.012

.011

.0098

.0089

.0080

.0071

.0062

.0053

.0044

.0036

.0018

00089

.0

0 2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 30a. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient C21.

U/V 10M POWER SPECTRUM OF $21 OVER 1985

el"

2

,O26

.025

.024

.022

.021

.020

.018

.017

,016

.014

.013

.012

.011

.0092

.0079

.0066

.0053

.0039

.0026

.0013

.0

ji

02468101214161820222426283032343638404244464850

FREQUENCY

Figure 30b. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient $21.

4O
NASA TECHNICALMEMORANDUM104648

U/V 10M POWER SPECTRUM OF C22 OVER 1985

.34

.33

.31

.29

.27

.26

.24

.22

.21

rr .19

_0 .17

n .15

.14

.12

.10

.086

.069

.051

.034

.017

"00 2 4 6 8 10121_,1618202224262830323436384()4244464850

FREQUENCY

Figure 31a. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient C=.

U/V 10M POWER SPECTRUM OF $22 OVER 1985

.048

.045

.043

.041

.038

.036

.033

.031

rr .029

O_ .026

o...024

.021

.019

.017

.014

.012

.0095

.0072

.0024 t

0 2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 3lb. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient S=.

41
BP._uuo V. Sm_c_x AND M_As,_mo NxsH_

nw

O

o_

U/V 10M POWER SPECTRUM OF C31 OVER 1985

.20

.19

.18

.17

.16

.15

.14

.13

.12

.tl

.10

.091

.081

.071

.061

.051 -

.041

.030 •

.020 .

.010 -

.0 .......... : _ _.-'-_ : _ T : :

0 2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 32a. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient C_x.

U/V 10M POWER SPECTRUM OF $31 OVER 1985

n*

2

0 2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 32b. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient $31.

42
NASA TECHNICALMEMORANDUM104648

U/V 10M POWER SPECTRUM OF C32 OVER 1985

.lOO

.o95

.09o

.085

.080

.075

,07o

.o65

.060 n'-

_: .055

0 .050

.045

.040

.035

.030

.025

.020

.015

,0100

.0050

0 2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

i Ill

Figure 33a. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: Neigh-South Velocity Component (v). The Coefficient C32.

U/V 10M POWER SPECTRUM OF $32 OVER 1985

n--

LLI

(3-

.079

.075

.071

,067

.064

.060

.056

.052

.048

.044

.040

.036

.032

.028

.024

,020

.016

.012

.0079

.0040

.0

0 2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 33b. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: North-South Velocity Component (v). The Coefficient $32.

43
U/V 1

.026

.025

.024

.022

.021

.020

.018

.017

.016

.014

_ .013

.012

.010

.0092

.0079

.0065

.0052

.0039

.0026

.0013

.0

BRAULIO W. SANCHEZ AND MASAHIRO NISHII-IAMA

0M POWER SPECTRUM OF C33 OVER 1985

0 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 34a. Power Spectrum. Frequencies: 0--50. Year 1985. Solid Line: East-West Velocity

Component (u). Dashed Line: Nor_-Sou_ Velocity Component (v). The Coefficient C_.

U/V 10M POWER SPECTRUM OF $33 OVER 1985

.13

12

12

.11

.10

.097

.090

.084

.077

n- .071

.064

O .058

.052

.045

.039

.032

.026

.019 ,

.0t3 '

.0064 _

2 4 6 8 101214161820222426283032343638404244464850

FREQUENCY

Figure 34b. Power Spectrum. Frequencies: 0-50. Year 1985. Solid Line: East-West Velocity

Component (n). Dashed Line: North-South Velocity Component (v). The Coefficient S33°

44
NASA TECHNICALMEMORANDUM104648

DEGREE 0 POWER SPECTRUM FOR 1985

.058

.055

.052

.049

.046

.044

.041

.038

.035

.032

.029

.026

.023

.020

.0t7

.015

.012

.0087

.0058

.0029

.0

0

)b

10 20 3O 40 50

FREQUENCY

Figure 35a. Degree 0, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

DEGREE 0

.012

.011

.010

.0098

.0092

.0086

.0081

.0075

.0069

rr .0063

.._.0058

O .0052

.0046

.0040

.0035

.0029

.0023

POWER SPECTRUM FOR 1985

(P = 0AT F = 0)

•0012"0017 II,,i .00058

,0 -

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 35b. Degree 0, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: tt-41), Power Set to Zero at Zero

Frequency.

45
uJ

©

n

BRAULIO V. SANCHEZ AND MASAHIRO NISHII-IAMA

DEGREE 1

3.2

3.0 I

2.9

2.7

2.6

2.4

2.2.

2.1

1.9

1.8

1.6

1.4

1.3

1.1

.96

.80

.64

,48

.32

.16

.0 _ --

0 10

POWER SPECTRUM FOR 1985

i i

20 30 40 50

FREQUENCY

Figure 36a. Degree 1, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

rr

Q.

DEGREE 1

,17

.16

.15

.14

.13

.13

.12

.11 •

.10 •

.092

.084

.076

.067

.059

.050

.042

.034

.025

.017

.0084

.0

POWER SPECTRUM FOR 1985

(P = 0 AT F = 0)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 36b. Degree 1, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0--40, Power Set to Zero at Zero

Frequency.

46
NASA TECHNICALMEMORANDUM104648

DEGREE 2

37.

35.

33,

32.

30,

28.

26.

24,

22,

20.

Ill

19.

©

o_ 17.

15.

13.

11.

9.3

7.4.

5.6

3.7

1.9

0

POWER SPECTRUM FOR 1985

: _ i i i

10 20 30 40 50

FREQUENCY

Figure 37a. Degree 2, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

n2

DEGREE 2

.58 •

.55

.521

.49"

.46 •

.43 .

.41 1

.38

,35 l

.32 *

.29

,26-

.23.

.20"

.17.

.14 -

.12-

.087_

.058.

.029-

.£

0

POWER SPECTRUM FOR 1985

(P- o AT F = 0)

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 37b. Degree 2, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-40, Power Set to Zero at Zero

Frequency.

47
rr

©

[3_

B RAULIO V, SANCHEZ AND MASAHIRO NISHIHAMA

DEGREE 3

1.3

1.2

1.2

1.1

1.0

.98

.91

.85

.78

.72

.65

.59

.52

.46

.39

.33

26

.20

.13

.065

.0 _:-_ -- _

0

POWER SPECTRUM FOR 1985

i i i p i

10 20 30 40 50

FREQUENCY

Figure 38a. Degree 3, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

DEGREE 3

o'sTl

°'STI/

°'ql/

°'61i/

°'511/

o,411I

-°,ill I

.°,,tl 1

.o,otlI

o_

_..0083 J,l I

.0074.

.0046,'0055"0065. /,,,',

.0037.

.0028 i

.0018. " "L_

.00092,

.0

POWER SPECTRUM FOR 1985

(P =o AT F = 0)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 38b. Degree 3, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0--40, Power Set to Zero at Zero

Frequency.

48
LU

©

[3.

NASA TECHNICALMEMORANDUM104648

DEGREE 4

25.

23.

22.

21.

20.

18.

17.

16.

15.

14.

12.

1I.

9.8

8.6

7.4

6.1

4.9

3.7.

2.5.

1.2,

.C

POWER SPECTRUM FOR 1985

10 20 30 40 50

FREQUENCY

Figure 39a. Degree 4, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

rr"

2

DEGREE 4

.27 \

,25 \

.24

.23

.21

.20

.19

.17

.16

.15

.13

.12

.11

.094

.080

.067

.053

.040

.027

.013

.0

POWER SPECTRUM FOR 1985

(P = o AT F = 0)

A

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 39b. Degree 4, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-40, Power Set to Zero at Zero

Frequency. 49
BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

DEGREE 5

15.

14.

13.

13,

12.

11.

10.

9.7

9.0

8,2

7.5

6.7

6.0

5.2

4.5

3.7

3.0

2.2

1.5

.75

.0

0

POWER SPECTRUM FOR 1985

10 20 30 4O

FREQUENCY

i

5O

Figure 40a. Degree 5, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 11-50.

DEGREE 5 POWER SPECTRUM FOR 1985

(P = 0AT F = 0)

.92

.87

.82

.78

.73"

.68'

,63

.58

m .53

._ .48

n° .44

.39

.34

.29

.24

.19

.o,;y

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 40b. Degree 5, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-40, Power Set to Zero at Zero

Frequency.

50
NASA TECHNICALMEMORANDUM104648

DEGREE 6

23.

21.

20.

19.

18.

17.

16.

15.

14.

rr 12.

UJ

11.

o

n lO.

9.0

7.9

6.8

5.6

4.5

3.4

2.3

1.1

.0

0

POWER SPECTRUM FOR 1985

lo 20 3o 4o

FREQUENCY

Figure 41a. Degree 6, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

DEGREE 6

.21"

.20"

.19'

.18+

.17

.16

.15

POWER SPECTRUM FOR 1985

(P--- o AT F = 0)

/i

.13

.12

.11 rr"

_O .10

a.. .093

.083.

.073-

,062.

.052

.041

.031

.021

.010

.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 4lb. Degree 6, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-40, Power Set to Zero at Zero

Frequency.

51
BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

DEGREE 7 POWER SPECTRUM FOR 1985

rr

LU

(3-

11.

10.

9.8

9.3

8.7

8.2

7.6

7.1

6.5

6.0

5.5

4,9

4.4

3.8

3.3

2.7

2.2

1.6

1.1

.55

.0 i i 4

10 20 30 40 50

FREQUENCY

Figure 42a. Degree 7, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

DEGREE 7

.10

.095

.090

.085

.080

.075'

.070 I

,065

POWER SPECTRUM FOR 1985

(P = o AT F = 0)

.060

.055.

.050, .045,

.040,

.035,

.030. !i

.025_ :"

.020, ,

.oo5o°1°°15.o ::i,""''"',,,'"""il '",'>

0 2 4 6 8 10'12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 42b. Degree 7, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-40, Power Set to Zero at Zero

Frequency.

52
NASA TECHNICALMEMORANDUM104648

DEGREE 8 POWER SPECTRUM FOR 1985

13.

12.

11.

11.

10.

9.4

8.8

8.2

7.6

n.- 6.9

ILl

O_ 6.3

o.. 5.7

5.0

4.4

3.8

3.1

2.5

1.9

1.3

.63

.0

10 20 30 40

FREQUENCY

5O

Figure 43a. Degree 8, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

DEGREE 8

.064

.061

.058

.054

.051

.048

.045

.042

.038

.035.

_ .032,

_ .029.

.026.

.022

.019

.016

.013

.0096

.0064

.0032

.0

POWER SPECTRUM FOR 1985

(P=0ATF=0)

] I

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 43b. Degree 8, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-40, Power Set to Zero at Zero

Frequency. 53
BRAULIO V. SANCHEZ AND MASAHIRO NISHIHAMA

DEGREE 9

5.4.

5.1.

4.8.

4.6.

4.3

4.0

3.8

3.5

3.2

rr 3.0

LU

O_ 2.7

a_ 2.4

2.2

1.9

1.6

1.3

1.1

.81

.54

.27

.0 _

0

POWER SPECTRUM FOR 1985

10 20 30 4O 50

FREQUENCY

Figure 44a. Degree 9, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

DEGREE 9 POWER SPECTRUM FOR 1985

.10

.100

.094

.089

.084

.079

.073

.068

.063

rr ,058.

._ .052

°n .047

.042

.037

,031

.026

.021

(P- 0 AT F = 0)

.0052,olo.0161 ,,,i, .0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

FREQUENCY

Figure 44b. Degree 9, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 11-411,Power Set to Zero at Zero

Frequency.

54
NASA TECHNICALMEMORANDUM104648

DEGREE 10

4.1.

3.9,

3.7

3.5.

3.3,

3.1

2.9

2,7

2.5

rr 2.3

LLI

2,1

O

1.9

1.7

1.4

1.2

1.0

.83

.62

.41

.21

.0

POWER SPECTRUM FOR 1985

4 I I I

0 10 20 30 40 50

FREQUENCY

Figure 45a. Degree]0, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-50.

DEGREE 10

.0101

.00971

.00921

.0086t

.00814

.0076:

.0071"

.0066'

.0061.

.0056.

_.0051,

_ .0046.

.0041.

.0036.

.0031.

.0025.

.0020,

.0015

,0010

.00051

.0

POWER SPECTRUM FOR 1985

(p = o AT F = 0)

q_

rl

f_

i I

bt

tl

fl

ii

'i

, ii :]

i ' It r • q I

, , ..... , ',,/

, 1

n

II

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

FREQUENCY

Figure 45b. Degree 10, Power Spectrum. Year 1985. Solid Line: East-West Velocity Component (u).

Dashed Line: North-South Velocity Component (v). Frequencies: 0-40, Power Set to Zero at Zero

Frequency.

55

NASA TECHNICALMEMORANDUM104648

References

Colombo, O. L., "Numerical Methods for Harmonic Analysis on the Sphere," Dept. of Geodetic

Science and Surveying, Report 310, The Ohio State University, Columbus, Ohio, 1981.

Schubert, S., C. K. Park, W. Higgins, S. Moorthi, and M. Suarez, "An Atlas of ECMWF Analysis"

NASA Tech. Memo., TM-100747, 1990.

Simmonds, I., "Spectral Representation of Horizontal Wind in Numerical Models of the Atmosphere," Journal of Applied Meteorology, Vol. 13, pp. 221-226, March 1974.

57

NASATECHNICAL MEMORANDUM 104648

APPENDIX I

HARMONIC ANALYSIS OF FUNCTIONS DEFINED ON THE GLOBE

where,

Then,

Let frO, 2_)represent a function defined globally at a set of discrete points, that is,

frO, _,) = f(O i, _j)

i=0, 1, ... ,n-1

j=0,1 ..... m-1

_.j " longitude _+1 - _'j = 2rdm

0 i " colatitude 0i+ 1 -0 i = rdn

Assume that frO, )Q can be expanded in a series of spherical harmonics,

f(0, _,) : Z n _m _ (Cnm cos(n_) + Snm sin(mk) t Pnm(0)

Pnm(0) " fully normalized Legendre functions of degree n,

order m, i.e.

_ Prim (0) t COS(m_), sin(m_.) t Pk, (0) _ cos(IX), sin(l_.) t sin(0) dO d_.

= (471:)8ak 8ml

Cnm : (1/4_) f _ f(0, _.) Pnm(0) cos(re_X) sin0 dO d_.

Sam = (1/4re) _ _ frO, 3.) Pnm(0) sin(rn_) sin0 dO d_.

The power associated with the terms of degree n is given by,

W n -- Zm (Cnm)2 -4- (Sam)2

59

NASA TECHNICAL MEMORANDUM 104648

APPENDIX II

POWER SPECTRUM OF FUNCTIONS DEFINED ON THE GLOBE

Let F denote a function such as Cnm, S m, or W, as defined in Appendix I. If certain basic

conditions are satisfied, then F can be expanded in a Fourier series:

F = a0 + a_ cos(t) + b_ sin (t) + a2 cos (2t) + b 2 sin (2t) +... + a365cos(365t) + b365 sin(365t).

for 0 _<t _ 2x

If the data covers a time span of one year, then the zero frequency term corresponds to the average

over the year, frequency 1 corresponds to a periodicity of one year, frequency 2 corresponds to a

periodicity of 6 months, and so on. The power corresponding to each frequency is given by:

Pi = (ai)2 + (bi)2

61




i

REPORT DOCUMENTATION PAGE Form Approved

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1. AGENCY USE ONLY (Leave b/ank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

July 1997 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Spherical Harmonics Analysis of the ECMWF Global Winds Fields at

the 10-Meter Height Level During 1985: A Collection of Figures

Illustratin_ Results Code 926

6. AUTHOR(S)

Braulio V. Sanchez and Masahiro Nishihama

7. PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)

Goddard Space Flight Center

Greenbelt, Maryland

9. SPONSORING/ MONITORINGAGENCYNAME(S)ANDADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

8. PEFORMING ORGANIZATION

REPORT NUMBER

97B00064

10. SPONSORING I MONITORING

AGENCY REPORT NUMBER

TM-104648

11. SUPPLEMENTARY NOTES

Braulio V. Sanchez, emaih g5bvs@charney.gsfc.nasa.gov

Masahiro Nishihama: Hughes STX, 7701 Greenbelt Road, Suite 400, Greenbelt, Maryland

12a. DISTRIBUTION I AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category: 46

This report is available from the NASA Center for AeroSpace Information,

800 Elkrid_e Landin_ Road, Linthicum Heights, MD 21090;301-621-0390

13. ABSTRACT (Maximum 200 words)

12b. DISTRIBUTION CODE

Half-daily global wind speeds in the east-west (u) and north-south (v) directions at the 10-meter height level

were obtained from the European Centre for Medium Range Weather Forecasts (ECMWF) data set of global

analyses. The data set covered the period 1985 January to 1995 January.

A spherical harmonic expansion to degree and order 50 was used to perform harmonic analysis of the eastwest (u) and north-south (v) velocity field components.

The resulting wind field is displayed, as well as the residual of the fit, at a particular time. The contribution of

particular coefficients is shown. The time variability of the coefficients up to degree and order 3 is presented.

Corresponding power spectrum plots are given. Time series analyses were applied also to the power associated with degrees 0-10; the results are included.

14. SUBJECT TERMS

Spherical Harmonics, Wind Fields, ECMWF

17. SECURITYCLASSIRCATION18. SECURITYCLASSlRCATION

OFREPORT OFTHISPAGE

Unclassified Unclassified

NSN7540-01-280-5500

19. SECURITY CLASSIRCATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

62

16. PRICE CODE

20. LIMITATION OF ABSTRAC'i

UL

Standard Form 298 (Rev. 2-89)