A note on the joint distribution of wave height and period during the growth phase of a storm

Ocean Engng, Vol. 15, No 4, pp. 379-387, 1988.

0029-8t)18/88 $3.00 ~ .00 Pergamon Press plc

Printed in Great Britain.

TECHNICAL

NOTE

A NOTE ON THE JOINT DISTRIBUTION OF WAVE HEIGHT AND PERIOD DURING THE GROWTH PHASE OF A STORM M. A. SROKOSZ Institute of Oceanographic Sciences, Wormley, Godalming, Surrey GU8 5UB, U.K. Abstract--In this note the effect of changes in sea-state, as measured by the significant wave

heigh /4,, on the joint distribution of individual wave height and period are considered. Wave data, obtained from a Waverider buoy during the growth phase of a storm, are used in the analysis. It is found that, by correctly scaling the individual heights and periods, the form of the joint distribution does not depend on H,, but is dependent on the bandwidth of the spectrum. The results obtained also give some indication of the period of individual, high zeroupcrossing waves.

f h~. hz m,, t~ tz H, S(f) Tj, T: T~ Um 5' v

NOMENCLATURE Frequency (Hz) Crest-to-trough wave height Zero-upcrossing wave height nth spectral moment Crest-to-trough wave period Zero-upcrossing wave period Significant wave height Wave spectrum Spectral peak period Mean zero-upcrossing period "Mean" period (inverse of mean frequency) Wind speed measured at 10 m above the surface Spectral bandwidth parameter JONSWAP spectrum peak enhancement factor Spectral bandwidth parameter. 1. INTRODUCTION

IN A RECENT paper, Srokosz and Challenor (1987) examined the joint distribution of individual wave height and period using data continuously recorded at the Scilly Isles from a Waverider buoy. In order to obtain accurate statistics they analysed two long records (approximately 10 hr of data in each case) for which the wave conditions remained roughly stationary with time. As a result of requiring the wave conditions to remain stationary, both records had relatively small values of the significant wave height H,. (3.22 in and 2.38 m). The purpose of this note is to examine whether increasing values of H~ lead to significant changes in the joint distribution of wave height and period (both zero-upcrossing and crest-to-trough height and period are considered here). Furthermore the results obtained also allow the question of what the period associated with individual high waves is to be addressed. In section 2 the data used in this study and the method of analysis are described. 379

3g{)

M A. SROKOSZ

This is followed in section 3 by a presentation of the results obtained and a ciiscussion of their significance. Finally, in section 4. the conclusions of the study are given. 2. DESCRIPTION OF THE DATA AND ANALYSIS In order to study the effect of increasing H, on the height-period distribution, the continuous Scilly Isles Waverider data, recorded by the Institute of Oceanographic Sciences (see Srokosz and Challenor (1987) for more details of the data), were examined to find a period during which H~ increased significantly and the local wind speed also increased; that is, a wave growth phase during a storm. Such a period was identified as occurring on 19 December 1981, from 13.00 hr to 21.00 hr approximateb. During this time H~. increased from 4.3 m to 7.4 m and the wind speed U,) increased from 16.6 msec- ' to 21.9 msec ~. The wave data for this period were recorded continuously in blocks of 2048 data points, at a digitisation rate of 2 Hz. In order to obtain meaningful statistics the data were subdivided into seven sets, each containing four blocks; a total of 28 blocks covered the period of interest. These datasets will be referred to as datasets I-7, each one containing just over 1 hr of wave data. Wind speeds for the relevant period are taken from the U.K. Meteorological Office on the Scilly Isles (hourly mean wind speed and direction). During the period under investigation the wind direction was from approximately 201)°, with little variation from hour to hour. The analysis of the data follows the basic pattern described by Srokosz and Challenor (1987). From each of the datasets the zero-upcrossing heights and periods (h~, t:) and the crest-to-trough heights and periods (h,., t,) were extracted. The resulting data were normalised, binned and contoured to produce the plots of the joint distributions of (h:,. t~) and (h,, t,:) shown in Figs 1 and 2, respectively. At the same time spectra were calculated, together with the spectral moments m() to m4 and the spectral peak period 7),, in order to obtain the wave parameters of interest. The spectral moments m,, are given by

,n,, = j,, S(t) l" df where S(J') is the frequency spectrum and f the frequency (Hz). The parameters of interest are the significant wave height /4, = 4Vmo the wave periods TI = too~m1

and T~ : X/(rn(~/rn2).

Also of interest are the spectral width parameters v and ~, given by v 2 = (m2mo/rr~,)

- 1

and e: = I - ( m ~ / r n ( , m 4 ) .

Technical note

3~1

These, with the appropriate wind speed, are listed in Table 1 for the seven datasets together with the parameters for the two datasets, A and B, studied by Srokosz and Challenor (1987). Finally, from each dataset the maximum zero-upcrossing wave height and associated period were extracted and these are listed, in normalised form, in Table 2 together with the corresponding results for datasets A and B of Srokosz and Challenor (1987). The normalisation used throughout this paper is that the wave heights are normalised by X/m., while the periods are normalised by T,, as calculated for each individual dataset. It should be noted that, due to the shorter length of the records used in the analysis here, the resulting contour plots of the joint height-period distributions are less smooth

(a) hz 6.0

4.0

4.0

2.0

20

0.0

. . . . 0.0

i

. . . .

0.5 NU = 0.366

i

. . . .

1.0

i

'

'tz

0.0

1.5

. . . . 0.0

i

. . . .

0.5 NU = 0.385

E P S I L O N = 0.770

I

'

'

1.0

1.5

E P S I L O N = 0.802

(d)

(c) 6.0-

6.0

4.0

4.0

2.0

2.0

0.0

. . . . . .t . . . 0.5 NU = 0.363

i

. . . .

1.0

E P S I L O N = 0.793

I ' '

15

C

....

0.0 0.0

i .... 0.5

N U = 0.386

i .... 1.0

i'' 1.5

E P S I L O N = 0.826

FIG. I. (hz, t~) distributions for datasets 1-7, (a-g) respectively; with Longuet-Higgins' (1983) distribution (h), for average value of v. Height normalised by X/m, and period by T,. Outermost contour for 0.01, subsequent contours at 0.05, 0.07, 0.1, 0.2, 0.3, 0.5, 0.7, 1.0, 1.2.

3S2

M.A.

SROKO,,J

than those displayed by Srokosz and Challenor (1987). An alternative approach to thai taken here would be to combine data from different times having the same value of H~ but, as discussed by Srokosz and Challenor (1987), this leads to difficulties in interpretation since the data come from different sea-states (H, on its <~wn is an inadequate discriminator between sea-states). The task of finding data from sufiicientlv similar sea-states is difficult to say the least. As a result the primary inadequacy of the method employed here, that is short record lengths, cannot easily be avoided. (e)

(f)

hz 6.0

60 7

4.0

40 i

2.0

2.0-

............... 0.5 1.0

0.0 0.0

tz

0.0 O0

1.5

05

1.0

15

NU = 0.394 EPSILON = 0838

NU = 0391 EPSILON = 0.805

(r

(g) 6.0

6,0

4.0

40

,

,

~

2.0

0.0

. . . .

0.0

i . . . .

0.5

I

1.0

0.0

i''

....

0.0

1.5

0.5

1.0

1.5

NU = 0.388 EPSILON = 0.810

NU = 0.433 EPSlLON = 0.839

Flc~. l. contd.

Technical note

3~3

3. RESULTS AND DISCUSSION By examining the results in Table 1 it can be seen that during the period under consideration, H, increased from 4.3 m to 7.4 m, as UH, increased from 16.6 msec ~ to 21.9 msec-~. Simultaneously the period of the waves, as measured by Tp, T1, or T~, increased and the spectral bandwidth, as measured by v or ~, also increased. However, on examining the results given in Figs 1 and 2 for the normalised (h~, tz) and (h,, t,) distributions there appears to be little change in the form of these despite the changes evident in the wave parameters listed in Table 1.

(a)

(b) 6.0-

6.0-

hc 4.0-

4.0-

2.0-

2.0-

o.o

. . . .

0.0

i . . . . 0.5

i . . . . 1.0

i'' 1.5

o.o

to

0.0

. . . . . . I. . . 0.5

i . . . . 1.0

i'' 1.5

NU = 0.385 EPSILON = 0.802

NU = 0,366 EPSILON = 0.770

(d)

(c)

6.0-

6.0-

4.0-

4.0--

2.0-

2.0-

0.0 0.0

I 0.5

I 1.0

I 1.5

NLJ = 0.363 EPSILON = 0.793

0.0

v

. . . . 0.0

i . . . . 0.5

i . . . . 10

i'' 1.5

NU = 0.386 EPSILON : 0.826

FIG. 2. (h,, 6.) distributions for datasets 1-7, (a-g) respectively: with Cavani6 et al.'s (1976) distribution (It). for average values of v and ~. Normalisation and contouring as for Fig. 1.

M.A.

384

S~oKosz

Comparison of the results with those of Srokosz and Challenor (1987) indicates that for the data considered here the spectrum is a narrower band and this leads to some slight differences between their (hz, t:) distributions and the ones presented here. Primarily the peak (or mode) of their distributions is at (h:, t:) = (1,0.5), whereas here (Fig. 1) there is the suggestion of a second peak at (h:, q) ~ (3,1). In contrast, the results for the (h,., t,.) distributions show little difference between the two studies. In Fig. lh Longuet-Higgins' (1983) theoretical joint height-period distribution is plotted, based on the avarege value of v for the seven datasets; his distribution is dependent only on that parameter under the normalisation used here. It can be seen that, while the overall shape of the distribution is similar to that of the data. the position of the mode is not the same.

(e)

60-[

S

60-

0 q

4.0

)

4.0-

J

J

20-

0.o 0.0

0.5 NU = 0.391

''

' i 15

1.0

O0

tc

.... 0.0

EPSILON = 0.805

i .... 0.5

~ ~ - t 10

-~ ~i,5

NU = 0,394 EPSILON = 0 838

(n)

(g) 6.0-

4.0-

4.0

"'\'\,il )

2.0-

0.0

I

0.0

I

"

I'

0,5

'

•

'

I

•|

1.0

,

,

,

,

]

,

0.0

i

1.5

[ .... 0.0

NU = 0.433 EPSILON = 0.839

; .... 0.5

i

1.0

'

'

~

'd

'

15

NU = 0.388 EPSILON = 0810

F~G. 2, ¢ontd,

'

385

Technical note

TABLE 1 Dataset

H~.(m)

Te(sec )

Tl(sec)

Tz(sec)

v

e

U.,(msec ')

1 2 3 4 5 6 7

4.3 4.9 6.0 6.8 7.1 7.3 7.4

8.26 9.01 10.00 10.64 11.11 11.36 11.63

7.50 7.81 8.35 8.89 8.99 9.05 9.35

6.81 7.30 7.84 8.29 8.33 8.38 8.61

0.366 0. 385 0.363 0. 386 0.391 0. 394 0.433

0.770 0. 802 0.793 0. 826 0.8(15 0. 838 (I.839

16.6 17.7 18.3 20.2 21.1 21.9 2t .9

A B

3.2 2.4

11.90 9.80

8.31 7.47

7.47 6.38

0.486 0.572

0.868 0.886

---

Wave parameters for the datasets 1-7 analysed in this note, together with results for datasets A and B from Srokosz and Challcnor (1987).

TABLE 2

Dataset

h:, .....

t~

t.T~/T:

GTt/T n

T~/Tz

Tn/Tz

T~/TI,

1 2 3 4 5 6 7

6.20 6.14 6.38 6.12 6.66 5.98 6.26

0.84 1.03 0.97 1.07 1.01 0.89 1.19

0.88 1.10 1.03 1.15 1.09 0.96 1.30

0.73 0.90 0.81 0.90 0.82 0.71 0.95

1.05 1.07 1.06 1.07 1.08 1.08 1.(/9

1.21 1.23 1.28 1.28 1.33 1.36 1.35

0.87 0.87 0.84 (/.84 0.81 0.80 0.80

A B

7.16 7.45

(/.99 1.12

1.10 1.30

0.69 (I.85

1.11 1.17

1.59 1.53

0.70 (/.76

mean(9) mean(7)

---

1.01 1.00

1.10 1.(17

0.82 0.83

1.(t9 1.07

1.35 1.29

(I.8t 0.83

Results for the highest zero-upcrossing wave in a record: h: normatised by

\/m.

and t~ by' T~. Mean(9) is the mean of all

9 records, while mean(7) is that of datascts 1-7.

Similarly in Fig. 2h the theoretical joint height-period distribution of Cavani6 et al.'s (1976) is plotted for comparison with the (he, t,.) distributions obtained from the data. Average values of v and ~ for the seven datasets are used to specify the distribution. This distribution shows some similarities to the (hc, t,,) distributions obtained from the data but does not appear to provide a very accurate representation of them. Srokosz and Challenor (1987) give a comprehensive discussion of the theoretical distributions, so they will not be discussed further here. In addition to studying the joint distribution of wave height and period it is possible to use the data to consider the question of the period to be associated with individual high zero-upcrossing waves. This is a question of some interest to engineers designing offshore structures, who require knowledge of both the height and the period of the extreme waves that the structure is likely to encounter. The problem of predicting extreme wave height has been extensively studied, whereas the question of the period

386

M. A. SkOKOSZ '1 .,x~l.I 3

y

7;,;1;

/,/i

/c~.

1

1.41

{t77

1.09

043

3.3 7

1.29 1.21

0.,~3 (l.SS

1.07 I.O(~

{!.3~ (~35

Results |or the JONSWAP spcclrum [or three values ol the peak tanhancement factor y. Note that the results are independent of H, and lhat the other JONSWAP parameters take slandard values (sec ('artet. 19821.

associated with extreme waves has received less attention, despite its importance in the design of offshore structures. From Table 2 it can be seen that the normatised period, t~, of the highest wave in each dataset is close to unity and in fact its mean value is unity. This implies that the period of individual high zero-upcrossing waves will be approximately T~. This is consistent with the theoretical prediction of Longuet-Higgins (1983), that high waves should have period Tj. By considering the joint (h> t~) distributions (in Fig. 1 and those given by Srokosz and Challenor, 1987) it can be seen that there will be some spread about this value. The results from Table 2 give a range of 0.84-l.19 T~. In terms of Tz, a more commonly used period parameter, the period of the high waves lies in the range 0.88-1.30 ~ , with a mean of 1.1 T:. These results differ from the findings of Bell (1972) and Su and Bergin (1983) who give values of 1.27 T= and 1.35 ~ , respectively, for the period associated with individual high zero-upcrossing waves. Bell (19721 obtained his result from the analysis of 23 15-min wave records taken in the North Sea, while Su and Bergin (19831 analysed nearly 600 20-min wave records taken in the Gulf of Mexico. In contrast, the results presented here are based on three long records (one subdivided into seven sections); whether this leads to the differences in the results is not clear. Another possible explanation for the differences may be that the data analysed here is oceanic data recorded at the Scilly Isles, whereas both the North Sea and The Gulf of Mexico are essentially enclosed seas; again the question of whether this would affect the results remains open. As the datasets 1-7 studied here represent a growing sea it seems reasonable to compare the results obtained with those that can be calculated using a J O N S W A P spectrum. The J O N S W A P results are based on the work of Carter (1982) and are given in Table 3. It can be seen, by comparing the values in Table 3 with those in Tables 1 and 2, that the results obtained here are consistent with those obtained from the J O N S W A P spectrum. In particular, the ratios Tp/T,, T~/T~, and Tt/Tz show the same trend with v, the bandwidth parameter. (Note that m 4 does not exist for the J O N S W A P spectrum, so ~ is undefined.) 4. CONCLUSIONS The results presented here, when considered in conjunction with those of Srokosz and Challenor (1987), lead to the following conclusions (a) the joint distributions of (h,, t,) and (h,, t,.) show no significant change With increasing values of H,., if they are scaled by ~/mo for the height and T~ for the period,

Technical note

387

(b) the f o r m of the (hz, tz) d i s t r i b u t i o n a p p e a r s to d e p e n d on the b a n d w i d t h of the s p e c t r u m , as m e a s u r e d by v and e. This is not so o b v i o u s in the case of the (h,., t,.) distribution, (c) for individual high z e r o - u p c r o s s i n g waves the d i s t r i b u t i o n of p e r i o d n a r r o w s down a r o u n d the value TI. This m a y be t r a n s l a t e d into a range of values in t e r m s of T~ via the ratio TI/T~, which is d e p e n d e n t on the s p e c t r u m . The results o b t a i n e d here differ s o m e w h a t f r o m those of Bell (1972) a n d Su and Bergin (1983), but are consistent with the t h e o r e t i c a l p r e d i c t i o n s of L o n g u e t - H i g g i n s (1983) and the p a r a m e t e r values o b t a i n e d from the J O N S W A P s p e c t r u m . O n the basis of the results p r e s e n t e d h e r e , t a k e n t o g e t h e r with those of S r o k o s z and C h a l l e n o r (1987) and L o n g u e t - H i g g i n s (1983), it s e e m s r e a s o n a b l e to assume that the p e r i o d of individual high z e r o - u p c r o s s i n g waves m a y be a p p r o x i m a t e d by T~, with s o m e s p r e a d a b o u t this value. T h e differences b e t w e e n the results given here and those of Bell (1972) a n d Su and Bergin (1983) d e s e r v e f u r t h e r investigation. N e i t h e r of those studies give values for T~, so direct c o m p a r i s o n with their results is difficult as it is d e p e n d e n t on the ratio T~/T~. It is possible that the difference b e t w e e n the results might be e x p l a i n e d by noting that d a t a s e t s 1-7 are for the growth phase of a s t o r m , w h e r e a s Bell (1972) and Su and Bergin (1983) c o m b i n e d a t a from m a n y different sea-states, including d e c a y i n g ones. F u r t h e r w o r k is n e c e s s a r y to e x a m i n e this aspect of the p r o b l e m . Acknowledgements--The author is grateful to David Carter and Peter Challenor for a number of useful discussions during the course of this work. The study was partly financed by the U.K. Department of Energy.

REFERENCES BELL, A. O. 1972. North Sea wave spectra. North Sea Environmental Study Group. CARTER, D. J. T. 1982. Estimation of wave spectra from wave height and period. I.O.S. Report No. 135. 19p. CAVANH~,A., ARHAN, M. and EZRATY, R. 1976. A statistical relationship between individual heights and periods of storm waves. In Proc. Conf. on Behaviour of Offshore Structures. Trondheim, Norway, pp. 354-360. Norwegian Institute of Technology. LONGUET-HIGGINS, M. S. 1983. On the joint distribution of wave periods and amplitudes in a random wavefield. Proc. R. Soc. Lond. A389, 241-258. SROKOSZ, M. A. and CHALLENOR,P. G. 1987. Joint distributions of wave height and period; a critical comparison. Ocean Engng 14, 295-311. Su. M. Y. and BERGIN, M. T. 1983. Storm wave characteristics in the Gulf of Mexico. Proc. Syrup. Buoy Technol., New Orleans, U.S.A. pp. t47-152, Marine Technological Society.