Statistical Properties of High-frequency Internal Waves

8/20/2019 Statistical Properties of High-frequency Internal Waves

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Chinese Journal of O ceanologyand Limnology

V o l . 2 0 , N o . l , P . 1 6 - 2 1 , 2 0 0 2

S TA T IS TIC A L P R O P E R T I E S O F H I G H F R E Q U E N C Y I N T E R N A L W A V E S

I N Q I N G D A O O F F S H O R E A R E A O F T H E Y E LL O W S E A ~

W A N G T a o ~ E ~ j~ )

Cali fornia Inst i tu te o f Technology, Pa sade na, CA 91125, USA )

G AO T i a n - f u ~ i ~ )

Inst i tu te o f Acoast ics , Chinese Academ y of Sciences, Bei j ing

100080,

China )

Received June 1, 199 9; revision accepted Ja n. 11, 2001

Abst rac t Dense ly-sampledhermistor chain d ata o btained from a shallow -water acoustics exper-

iment in the Y ellow Sea off the co ast of Qingdao were analyzed to ex amine the statistical properties of the

6 to 520 cpd frequency band internal waves observed. The negative skewness coefficients and the greater-

than-3 kurtosis coefficients indicated n on-Gau ssianity of the internal w av es. The probab ility distributions

were negatively skewed and abnormally high peaks. Nonlinear properties, as exemplified by the asymmet-

ric wav eshapes of the internal wav es in the offshore area a re d escribed quantitatively.

Ke y w ord s: shallow-water internal wa ve, skew ness, kurtosis, non-Gaussianity, nonlinearity

I N T R O D U C T I O N

O bse r va t ions r e ve a l e d t he w ide sp r e a d d i s t r i bu t ion o f i n te r na l w a ve s on the c on t ine n ta l she l f .

M u c h the o r e t i c a l , num e r i c a l , l a bo r a to r y a nd f i eld w or k ha d be e n done on the i r ge ne r a t i on a nd p r op -

a ga t ion . S ign i f ic a n t p r og r e s s ha s be e n a c h i e ve d in she l f i n t e r na l so l it on r e se a r c h . H ow e ve r , s t a t is t i -

c a l p r ope r t ie s o f i n t e r na l w a ve s , e spe c i a l ly i n sha l low w a te r , w e r e s t il l l i tt l e unde r s tood W a ng a nd

G a o , 2 0 0 1 ; X u , 1 9 9 9 ) .

A r e i n t e rna l w a ve s G a n ss i a n? I W E X t e m pe r a tu r e , d i sp l a c e m e n t a nd c u r re n t da t a show e d tha t i n

de e p o c e a n , t he se w a ves w e r e m a in ly G a uss i a n bu t spo r a d i c a l l y non - G a nss i a n . T he de p th a nd loc a -

t i on de pe nd e nc e o f t h i s c onc lu s ion i s no t know n B r i s e oe , 19 77 ) . S t a ti s ti c s o f h igh - f r e qu e nc y in t e r -

na l w a ve s igna ls w i th pe r iods o f le s s t ha n 300 s w e r e a na lyz e d to e xa m ine the c o r r e sponde n c e be -

tw e e n tu r bu le n t m ix ing a nd in t e r na l w a ve s i n t he e qua to r i a l P a c i f i c , w he r e t he ske w ne ss o f t he ho r i -

z on ta l t e m pe r a tu r e g r a d i e n t w a s s ign i f i c a n tly d i ff e r en t f r om z e r o , a nd w a s pos it i ve i n t he de p th a bove

40 m . T he da y to n igh t va r ia b i l i ty o f t he s igna ls w a s i l l u s t ra t e d by the c ha nge in r m s va lue s o f i so -

t h e r m d i s p l ac e m e n t s a n d t h e i r d i s tr i b u ti o n s M o u m e t a l . , 1 9 9 2 ) . A l o n g - te r m p r e d i ct i o n o f i n te n s e

in t e r na l w a ve s w ith a m p l i t ude s l a r ge r t ha n 5 m w a s obse r ve d in t he t r op i c a l r e g ion o f the A t l a n t i c u s -

ing e c ho sounde r m e a su r e m e n t s , show ing tha t the w a ve r e pe t it i on f r e que nc y c ou ld be de sc r ibe d by

t h e P o is s o n l a w f o r e x tr e m e e v e n ts I v a n o v et a l . , 1 9 9 3 ) .

R e c e n t s t ud i e s a dd r e s s ing the s t a t i s ti c s o f sha l low - w a te r i n te r na l w a ve s show e d tha t t he c on t i -

ne n ta l s l ope t o the sou th o f t he C e l t ic S e a i s a n a r e a o f i n t e nse a nd c om pl i c a t e d i n t e r na l w a ve a c ti v i -

t y . T o s tudy the p r opa ga t ion o f t he se w a ve s , d i s t r ibu t ions o f t im e in te r va ls be tw e e n the i r oc c u r r e nc e ,

t i da l pha se s a t w h ic h the y w e r e obse r ve d , a nd the i r p ropa ga t ion d i r e c t i ons , w e r e a na ly se d . M or e -

ove r , t he m e a n f l uc tua t ion spe e d a nd ske w ne ss o f t e m pe r a tu r e s igna l s show e d the d i f f e r en t c ha r a c t e r -

i s t ie s o f w a ve pa c ke t s a r ri v ing f r om d i f f er e n t ge ne r a t i on r e g ions H o l t a nd T hor p e , 19 97 ) . T e m pe r a -

tu r e t im e se r i es w e r e m e a su r e d f r om a n a r r a y o f m in i logge rs i n a l i ne a t c ons t a n t de p th a long the s l -

* Project No. 19804013supported by the NSFC.



No. 1 WAN G et al . : STATISTICALPROPERTIES OF INTERNAL WAVES 1 7

o p i n g b o u n d a r y o f L a k e G e n e v a s o u t h e a s t o f O u c h y . D i s t r ib u t i o n s o f t e m p e r a t u r e t i m e d e r i v a ti v e s

w e r e f o u n d t o h a v e a s m a l l n e g a t i v e s k e w n e s s , w h i c h i s c o n s i s t e n t w i t h t h e i n t e r n a l w a v e d y n a m i c s

i n t h e s u rf z o n e ( T h o r p e a n d L e m m i n , 1 9 9 9 ) .

T h i s w o r k i s a i m e d a t i n v e st i g a ti n g t h e s e c o n d - , t h i r d - a n d f o u r t h - o r d e r m o m e n t s a n d p r o b a b i l i -

t y d i s t ri b u t io n s o f i n t e r n a l w a v e s i n s h a l l o w w a t e r , a n d a p p l y i n g t h e d a t a o b t a i n e d t o d e s c r i b e s t a t i s -

t i c a l ly t h e i r n o n l i n e a r p r o p e r t i e s . R e s u l t s o f a s t u d y o n i n t e r n a l w a v e s i n a s t r o n g s e a s o n a l t h e r m o -

c l i n e o f t h e Y e l l o w S e a a r e p r e s e n t e d a n d a n a l y z e d .

D E F I N I T I O N S

C o n s i d e r a s a m p l e fu n c t i o n 7 / ( t ) f r o m a n e r g o d ic r a n d o m p r o c e s s . H e r e , r / ( t ) i s t h e t i m e

h i s to r y o f i n t e r n a l w a v e e l e v a t i o n s . T h e r t h m o m e n t s / z r o f 7?( t ) a r e d e f i ne d b y

,Ur = r1 - ~ ) r P r l ) d 7 ] , r ; 1 , 2 , 3 , 4 , ' ( 1 )

w h e r e P ( 7 1 ) i s t h e p r o b a b i l i t y d e n s i t y f u n c t i o n , a n d ~ is t h e m e a n v a l u e

= f i ~ ' I P ( ' j ) d ' l ( 2 )

A s s u m e r / ( t ) t a k e s o n d i s c r e t e v a l u e s ~ 1 , r ]2 , , r /N , t h e n

N

/~ , = ~ , , ~ ( ' ] , - ~ ? )~ , r = 1 , 2 , 3 , 4 '

10

2 1 5

2 0

25

30

14 I 6

t i

1 8 20 22 24 26 28

Temperature (~

Fig . 1 M ea n emperature profi le (August 23 14:5 0 -

August 25 16:3 9 , 1992)

t h e r m i s t o r c h a i n w i t h 3 2 c h a n n e l s s p a c e d O . 4 m

a p a r t . T h e s a m p l i n g t i m e i n t e r v a l w a s 6 . 4 s , a n d t h e l e n g t h 4 9 h 4 9 r a i n . I n th e w a t e r o f 3 3 m

d e p t h , a m e a n w e l l - m i x e d s u r f a c e l a y e r e x t e n d e d t o 9 . 5 m , t h e t e m p e r a t u r e t h e n d r o p p e d fr o m

2 6 ~ t o 1 6~ a t a d e p t h o f 1 6 . 5 m , a n d d r o p p e d sl o w l y t o a b o u t 1 5~ a t t h e s e a b e d ( F i g . l ) . A

m a x i m u m B ru nt -V a is ~il ~l f r e q u e n c y o f a b o u t 9 3 0 c p d ( p e r i o d 1 . 5 5 r a i n ) w a s s u p p o r t e d . T h e c h a i n

s p a n n e d t h e w a t e r c o lu m n f r o m 6 . 3 m t o 1 8 . 7 m w h e r e t h e st r o n g s e a s o n a l t h e r m o c l i n e o c c u r re d

3 )

w h e r e

= N , = , ( 4 )

T h e f i r s t m o m e n t i s z e r o f o r a n y r a n d o m p r o -

c e s s; t h e se c o n d m o m e n t is th e v a r i a nc e a 2 ( a ,

s t a n d a r d d e v i a t i o n ) ; t h e t h ir d m o m e n t , w h i c h i s a

m e a s u r e o f t h e a s y m m e t r y o f t h e w a v e s h a p e a b o u t

t h e h o r iz o n t a l a x i s , i s c a l l e d t h e s k e w n e s s ; a n d

t h e f o u r t h m o m e n t , w h i c h g i v e s a n i n d ic a t i o n o f

t h e p e a k e d n e s s o f t h e s t a t i s ti c a l d i s t r i b u t i o n , i s

c a l l e d t h e k u r t o s i s . T h e c o e f f ic i e n t s o f s k e w n e s s

a n d k u r t o s i s a r e

23 ,u3 ,u4

- 3 , 2 , ~ 4 - 2 5 )

~L~2 / /2

O B S E R V A T I O N S A N D D A T A P R O C E S S I N G

W e h a v e c o n d u c t e d s e v e r a l s h a l l o w - w a t e r

a c o u s t i c s e x p e r i m e n t s i n t h e Q i n g d a o o f f sh o r e a r e a

o f t h e Y e l l o w S e a s i n c e 1 9 7 9 . H i g h - q u a l i t y i n t e r -

n a l w a v e d a t a w e r e g a t h e r e d A u g u s t 1 9 9 2 f r o m a



18 CHIN . J. OCEANOL. LIMNOL. 20 1) , 2002 Vol.20

W an g e t a l . , 1999; 20 00 ) . The tempera ture s igna ls of every channel were grouped and averaged over

13 data points. The num ber of data points N of every channel totalled 2155, and the N yquist f requency

was 520 cpd. This averaging removes the fluctuations in thermistor depth due to surface gravity waves and

ship motions, while still keepin g the high-frequency information in the sign al. T he obtained ten-g~erature

time ser ies was then l inearly interpolated , f rom the top channel to the bottom on e, to f ind the d epth t ime

series for a given temperature T, that is the isotherm in the t im e-dep th coordinate sy stem.

The i sotherms conta ined mot ions in three dis t inc t f requency bands , the synopt ic band inc luding

synopt ic weather pa t te rn and iner t ia l wav es , the tida l band inc luding diurna l and s emidiurna l t ides ,

and the high- f requen cy band con ta ining inte rna l wave s . The low-f requency component was of inte res t

in a previous pape r by W ang e t a l . 20 00 ) ; the present investiga t ion was concent ra ted on the high-

f requenc y wave com pon ent . A h ighpass But te rwor th f i l te r was f ir s t used to remove the mear i depth

of every i sotherm af te r wh ich, > 6 cp d mot ions were then removed f rom the t ida l and synopt ic

band s ; and the poss ible e f fec ts of surface t ide on tempe ra ture var ia tions were removed as well Ap el

e t a l . , 199 7; Pr ingle , 19 99 ) . In coas ta l r egions away f rom f jords and es tuar ies , var ia t ions in sea

water densi ty a re de te rmined m ainly by temp era ture , so the e f fec t of sa l ini ty can b e neglec ted

Zh ao , 19 92 ) . Therefore , the resul t ing isotherms w ere regarded as e leva t ions of the inte rna l waves

Sa gg io a nd I m be r ge r , 1998 ; Sm a l l e t a l . , 19 99 ) . I t shou ld be po in t ed out tha t the ve r ti c a l a x i s o f

Fig . 1 , extending downw ards , r epresents the depth from the s t il l water sur fac e . To be consis tent

wi th t radi tiona l def ini t ions , the u pward inte rna l wave displacemen ts f rom the i r mea n leve ls were tak-

en as posi t ive and vice versa . Ver t ica l displacements of the high pass- f i l te red 2 5 .0 - 17 .5 ~C iso-

therms w i th wave he ights of up to 5 m are shown in Fig . 2 .

4 ̂

2

~2

0 5 113 15 20 25 30 35 40 45

T i me h )

F i g . 2 V e r t i c a l d i s p l a c e m e n t s o f h i g h p a s s - f i l t e r e d is o t h e r m s a t 0 . 5 ~ C i n t e r v a l s w i t h t h e l o w e s t i s o t h e r m a t 1 7 . 5 ~ C .

T h e c u r v e s w e r e s e q u e n t i a l l y o f f se t b y 2 . 5 m

STATISTICS

The s tandard devia t ion, and skewness and kur tos is coef f ic ients of each i sotherm are given in



No. I WA NG et al. : STATISTICAL ROPER TIES OF INTERNA LWAVES 19

Ta b le 1 . The m e a n de p th o f t he i so the r m be f o re t he h ighpa s s f i l te r ing i s show n in t he t a b l e a s w e l l .

The in t e r na l w a ve e le va t ion m a y be r e ga r de d a s a c om bina t ion o f a s t a t i c o r t im e - inva r i a n t c om -

pone n t a nd a dyna m ic o r f l uc tua t ing c om pon e n t . The s t a t i c c om pone n t i s de sc r ibe d by the m e a n va l -

ue 7 /, a nd the dy na m ic c om pone n t by the s t a nda r d de v ia t i on 6 . The inc r e a se o f ~ w a s 3 . 2 1 m w he n

T d e c r e as e d f ro m 2 5 . 0 ~ t o 1 7 . 5 ~ T h e m e a n g r a d ie n t w a s - 0 . 4 3 m / ~ T h e s t a n d a rd d e v i a -

t i o ns o f t h e 2 5 . 0 ~ - 1 7 . 5 ~ i s ot h er m s r a n g e d f r om 0 . 4 3 m t o 0 . 5 8 m , i n d i c at i n g t h a t t h e w a v e

e ne r gy d id no t va r y g r e a t ly w i th de p th .

Al tho ugh s tandard de via t ions of d i f fe rent i so the rm s va r ied l i t t le , the skew ness coef f ic ients va r -

i e d s ign i f ic a n t ly f r om - 0 . 0 6 to - 0 . 5 0 . I t c a n be exp la ine d tha t t he w a ve e ne r gy w a s no t un i f o r m ly

d i s t r ibu t e d a long t im e a nd c on c e n t r a t e d on the l a r ge - a m p l i t ude w a ve s be tw e e n 3 h a nd 8 h a nd those

be tw e e n 28 h a n d 33 h . The i r ve r t ic a l l y a sym m e t r i c w a ve fo r m s de t e r m ine d m a in ly t he ske w ne ss va l -

ue o f t he c o r r e spond ing i so the r m F i g . 2 ) . The ne ga t ive ske w ne ss va lue s sugge s t e d t ha t t he c re s t

w i th pos i t i ve d i sp l a c e m e n t ) he igh t s o f t he i n t e rna l w a ve s w e re le s s tha n the i r t r ough w i th ne ga t ive

d i sp l a c e m e n t ) de p th s , t ha t i s t o s a y , t he t r oughs w e r e sha r p a nd de e p w he r e a s t he c r e s t s w e r e r a the r

low a nd b r oa d .

F u r the r m or e , t he non - z e r o ske w ne ss va lue s i nd i c a t e d t ha t t he i n t e rna l w a ve s i n t he Q ingda o of f -

sho r e a r e a w e r e non - G a uss i a n , s i nc e a G a uss i a n no r m a l ) r a ndom p r oc e s s ha s ~3 = 0 . The l ine a r

the o r y o f r a ndom se a i n t e r na l w a ve s w a s ba se d on the supe r pos i ti on o f in f in i te s im a l a m p l i t ude w a ve

c om pone n t s w i th un i f o rm ly a nd r a ndom ly d i s t r ibu t e d pha se a ng le s . H ow e ve r , i f t he p r oc e s s is non -

l i ne a r , som e o f t he pha se a ng le s a r e no t i nde p e nde n t , bu t ha ve a f ixe d r e l a t ionsh ip t o e a c h o the r .

The in t e r na l w a ve c om pone n t s be c om e c oup le d to e a c h o the r a nd non l ine a r c om pone n t s a r e p r e se n t a s

ind i c a t e d by the ske w e d w a ve sha pe s .

T a b l e S t a t i s ti c s o f i n t e r n a l w a v e s in Qingdao o f f sh o r e a r e a o f t h e Y e l l o w Sea

T oC ) ~ m) a m ) 3,3 X4

25.0 11 .84 0 .46 - 0 .36 4 .12

24 .5 12 .11 0 .45 - 0 .37 4 .33

24 .0 12 .33 0 .44 - 0 .25 4 .27

2 3 .5 1 2 .5 1 0.4 3 - 0 .2 5 4 .4 4

23 .0 12 .68 0 .44 - 0 .30 4 .55

22 .5 12 .82 0 .44 - 0 .34 4 .66

22 .0 12 .95 0 .45 - 0 .39 4 .83

21 .5 13 .07 0 .45 - 0 .44 5 .0 l

21 .0 13 .19 0 .46 - 0 .46 5 .05

20 .5 13 .30 0 .46 - 0 .48 5 .0 l

20 .0 13 .43 0 .47 - 0 .50 4 .92

19 .5 13 .56 0 .48 - 0 .50 4 .85

19 .0 13 .72 0 .50 - 0 .46 4 .71

18 .5 13 .91 0 .52 - 0 .38 4 .78

18 .0 14 .24 0 .58 - 0 .25 5 .29

17 .5 15 .05 0 .54 - 0 .06 3 .24

The ku r to s i s c oe f f i c ie n t s o f t he i n t e r na l w a ves r a nge d fr om 3 . 2 4 to 5 . 2 9 , g r e a t e r t ha n the no r -

r e al va lue 3 , a nd w a s a no the r i nd i c a t i on o f t he non - G a uss i a n i ty o f t he w a v e s . I t is e xpe c t e d t ha t t he

s t a ti s t ic a l d i s t r i bu t ions o f t he i n t e r na l w a ve s ha ve h igh e r t ha n no r m a l pe a ks .



20 CH IN. J . O CEANO L. LIMNO L. , 20 (1 ) , 2002 Vol . 20

P R O B A B I L I T Y D I ST R I B U T IO N S

T h e p r o b a b i l i ty d i s t r ib u t i o n s o f t h e i s o t h e r m d i s p l a c e m e n t s , n o r m a l i z e d w i t h r e s p e c t t o z , a r e

c o m p a r e d w i t h t h e c o r r e s p o n d i n g G a u s s i a n d is t r i b u t io n s in F i g . 3 .

0 4

0 2

0

2 4 5 ~

C

C

0

6

(

]

2 4 O ~

2

C

4 6 f

0 2

O

C

C

C

0 .6

o

0 4

22 C

- 4 -2 0 2 4

C

4 2 0 2

C

4 2 0 2

q / a

1 7 5 ~

4 4 2 0 2

Fig. 3 Com parisonbetw een internal w av e (histogram ) and no rm al (solid fine) distributions

I t i s s e e n t h at t h e s e i n t e r n a l w a v e d i s p l a c e m e n t d i s t r ib u t i o n s w e r e a l l n e g a t i v e l y s k e w e d . T h e

d i s t r ib u t i o n o f t h e 1 7 . 5 ~ i s o t h e r m is s l ig h t l y d e v i a t e d f ro m t h e n o r m a l , a n d t h e d e v i a t i o n s b e c o m e

s i g n if i ca n t f o r t h e 2 0 . 0 a n d 1 9 . 5 i s o t h e r m s . M o r e o v e r , p e a k s o f t h e w a v e d i s p l a ce m e n t d i st r i-

b u h o n s a r e h i g h e r t h a n t h o s e o f t h e n o r m a l . T h e 1 7 . 5 ~ i s o th e r m h a s t h e l o w e s t p e a k o f d is t ri b u -

t i o n , w h i l e t h e 1 8 . 0 i s o th e r m h a s t h e h i g h e st p e a k a n d a l o n g t a il i n t h e s i d e t l / a < 0 i n c o m p e n -

s a t i o n . T h e r e s u l t s w e r e c o n s i s t e n t w i t h th o s e f r o m a n a l y s i s o f t h e i n t e r n a l w a v e s t a t i s t i c s .

D I S C U S S I O N S

T h e s t a t i s ti c s a n d p r o b a b i l it y d i s t r i b u ti o n s o f i n t e r n a l w a v e s i n t h e Q i n g d a o o f f s h o r e a r e a o f t h e

Y e l l o w S e a a r e p r es e n t ed a n d a n a l y z e d . O u r f i n di n g t h at t h e s k e w n e s s c o e f f i ci e n t s w e r e n o n - z e r o a n d

t h a t t h e k u r t o s i s c o e f f i c i e n t s w e r e gr e at er t h a n 3 , i n d i c a t e d s k e w e d w a v e s h a p e s a n d n o n - G a u s s i a n

d i s t r i b u t i o n s o f t h e i n t e r n a l w a v e e l e v a t i o n s .

S t a t is t ic s h a d b e e n w i d e l y u s e d t o d e s c r ib e n o n b r e a k i n g sh o a l i n g s e a su r f a c e w a v e s ( L i u e t a l . ,

1 9 9 8 ) . I t i s s u g g e s t e d th a t t h e p r e s e n t s t u d y o n i n t er n a l gr a v it y w a v e s c o u l d b e e x t e n d e d t o c o n s t r u c t

n o n l i n e a r m o d e l s f o r s h a l l o w - w a t e r i n t e r n a l w a v e s .

A C K N O W L E D G E M E N T S

T h e a u t h o r s a r e gr a te fu l t o P r o f e s s o r T h e o d o r e Y . W u ( C a l i f o r n i a I n s t it u t e o f T e c h n o l o g y ) a n d

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