Wave and wind data for seakeeping performance assessment

Deift University of Technology

Ship Hydromechanles L&oratory Mekelweg 2 - 2628 CD DELFT

The Netherlands

09 MART 1988

A

,

WAVE AND WIND DATA F OR

SEAKEEPING PERFORMANCE ASSESSMENT

Prepared for International Towing Tank Conference (ITTC) Seakeeping Committee Meeting

Athens, Greece April 1983

Susan L. Bales

Head, Ocean Environment Group Surface Ship Dynamics Branch David W. Taylor Naval Ship R&D Center

INTRODUCTION

This topic remains central in any attempt to predict reliable data for ship seakeeping characteristics in the worldes oceans, and this section

reports on activity and progress since the last ITTC Conference. This

Conference is the first time an attempt is being made to share the task with

the Ocean Engineering Committee. Basically, the Seakeeping Committee is

focusing on those aspects of concern to ship seakeeping performance while the Ocean Engineering Committee is concentrating on matters of importance to ocean engineering applications such as offshore platform performance.

Until recently, the wave and wind environment has played a very minor

role in the design and evaluation of ships and offshore platforms. Generally,

the environment has only been considered in the assessment of the performance of a ship/structure configuration whose architecture has already been fixed. The environment is then introduced to quantify performance in select

conditions for minor hull variations. Even so, the ITTC Seakeeping Committee

ha.s recognized the need to model the environment in some representative fashion for several decades.

In recent years, the potential for designing in both good seakeeping

and good resistance performance has been demonstrated. The implication here

is that ship performance evaluations should reflect more than calm water

resistance and other propulsion system considerations. The need to provide a

good representation of the environment has taken on additional importance. It

is no longer sufficient to evaluate in one or two chosen wave conditions. Now, we must use seaway models that permit the development of both

operationally average performance predictors as well as those cases where

performance is significantly degraded. This permits trade-off studies between

hull form, arrangements, resistance, and propulsion. In some cases, this also

provides the full-scale operator with warnings of conditions which may be hazardous (e.g., bi-directional seas, extreme seas, etc.)

The Committee has not addressed the issue of Sea State numerals since the 14th ITTC when existing confusion and questionable utility for our

profession was emphasized. _{However they remain as popular as ever with}

operators and customers, so current activity on this topic is addressed.

Models of wave directionality continue to be deficient. Ongoing

efforts to parameterize directional characteristics are highlighted.

Other topics briefly discussed are wave and wind persistence, and also wave measurement and generation.

Special topics that may be of greater interest to the Ocean

Engineering Committee are not treated here. Briefly, however, ongoing work on

wave groups, extreme waves, and breaking waves have been reported in [1], [2],

[3], [4], and [5] and in the open literature. Some interest in episodic

waves, particularly for structural applications, has been noted [6] and

Donelan (Centre for Inland Waters - Canada) and Pierson (CUNY - United States) are jointly developing towing tank generation and analysis techniques for

these rare events. _{Generally, [5] provides some guidance for the application}

of. these topics, particularly for ocean engineering applications.

WIND AND WAVE STATISTICS

The primary stimulant for much of the interest in environmental modelling in the United States is due to the U.S. Navy's Spectral Ocean Wave Model (SOWM) Hindcast Wind and Wave Climatology described in the Report of the 16th ITTC. This work has permitted the development of open ocean wind and wave statistics for the North Atlantic, North Pacific, and the Carribbean and

North Seas [7], [8], [9], [10], [11]. Tables 1 and 2 provide interim

statistics associated with the Sea State occurrences in the North Atlantic and Pacific [8].

While the confusion associated with Sea State numeral definitions has sometimes indicated the need to avoid such classifications, most operators and many customers throughout the world use no other means of describing the wave

and wind environment. A recent survey in the United States indicated that many nations utilize the definition proposed by the World Meteorological

Organization (WMO). As most shipboard observations which are currently

published also use WMO codings, it was felt that the WMO definitions would

provide a good standard for Tables 1 and 2.

A few words are in order with regard to the quality of the hindcast data. Comparisons with visual observations, measurements, and other hindcasts

provide variable correlations. In summary [12],

the hindcasts provide greater occurrences of higher Sea States than visual observations

the hindcasts may show less occurrence of Sea States 0 to 4

the hindcasts may provide somewhat lower Sea State occurrences

for the eastern portion of the North Atlantic

While it is difficult to develop a general conclusion for all conditions and oc.ean regions, clearly the hindcast data set provides comprehensive data to

the naval architectural community. _{And very importantly, it also provides}

directional wave data, heretofore unavailable except for very limited ocean regions.

A Sea State table recommended for small vessel design and developed from the POLARFRONT Weathership [13] provides another interesting comparison

with these hindcasts. _{The POLARFRONT operates at (66° North, 2° East).}

Figures 1 and 2 provide a comparison of the significant wave heights from the

POLARFRONT with those developed using SOWM hindcasts at a location to the

Southwest (62.8° North, 3.9° West). _{While these locations are 250}

nautical miles apart, there is no land mass interference between them and most

likely they experience similar, though not identical, storm systems. Figure 1

shows a comparison of Winter (December through February) heights wile Figure 2

provides the annual cumulative distributions. _{As one woulu expect, the}

hindcast data (location is closer to the major storms of the Northern North

Atlantic) provide somewhat more severe winter conditions, but the annual data

agree well, even though the hindcasts covered fewer years.

A world standard for wave statistics for many years has been that

developed by Hogen and Lumb [14]. Now Hogben and his colleagues, namely

Dacunha (NMI-United Kingdom), are preparing a new wave atlas based on visual

observations. _{The data are processed through the "NMIMET" program which}

smooths the visual observations with gamma distributions and provides a method

for deriving wave period probabilities from wave height statistics without any

usage of the notoriously unreliable visual observations of wave period [15]. This atlas, when published will, without doubt, provide valuable long term wave statistics to the seakeeping community.

A comparison of NMIMET data with visual and SOWM hindcasts [7] for Station India has been developed by Dacunha and Lee (DTNSRDC-United States)

and is provided in Figure 3. _{The correlation between the NMIMET and}

hindcast-derived data is excellent, while the visual data provide somewhat lower percent frequencies of exceedence.

WAVE DIRECTIONALITY

Over the years, a greater appreciation for the structure of the

s&away has gradually crept into the naval architect's thinking. The initial

use of Sea State charts and even the later application of the

Pierson-Moskowitz wave spectrum for fully-developed seas relied solely on the

specification of a significant wave height. The adaption of the two parameter

wave spectrum by the 15th ITTC reflected that wave period was also critical and, in some cases, it has been shown that lower wave heights produce higher

ship responses. This is due to the inclusion of varying distributions of wave

period (or frequency).

Now, with the availability of the directional wave spectra from wave

hindcasts, it is possible to add a third important parameter (or set of

parameters). This, of course, is wave directionality, including spreading.

Ideally, the naval architect should be able to model swell corrupted wind

driven seas, either numerically or in the towing tank. Associated with the

sea and swell components should be some energy spread about a predominant

direction. The value of the hindcasts here is that they can be used to

determine the statistical occurrence of directional conditions which could

S

S

then be incorporated into an analytical multi-parameter model. It should be

pointed out that the Spectral Ocean Wave Model (SOWM) may introduce some

cosine squared bias into the hindcasts. This is presently being investigated

by Neu (VPI-United States).

In order to quantify some of the characteristics associated with wave directionality, a set of new parameters has been developed by Cummins

(DTNSRDC) and applied to the wave hindcasts [16], [17]. _{The derivation of the}

three parameters m2, p2. and q are provided in the Appendix. In short,

m2 is a measure of the angular spread or width of the

directionality and, in a sense, corresponds to a variance; for a unidirectional swell, it has a value of 0; Figure 4 shows m2

as a function of half spreading angle for the cosine

squared distribution (m2 = 0.25 corresponds to + 90 degree

spreading as recommended for usage by the 16th ITTC)

p2 is a measure of the "centralness" of the spreading and it corresponds to a moment of inertia of the wave spreading; large values of p2 (e.g., p2 > 0.1) suggests a non-central

distribution of wave directionality such as is present when two or more systems propagate into an area

q is a measure of skewness; small values reflect slightly skewed central distributions while large values, particularly in

conjunction with large values of p2. suggest wave systems from multiple sources

Distributions of these three parameters have been developed for the three grid

points identified in Figure 5 for about a 10 year hindcast data set. Figure 6

shows cumulative distributions of m2 for three significant wave height

classes 0-2, 2-5, and 5-10 m. Overall patterns at the three locations are

similar, suggesting a stable interrelationship between angular spread and wave height. While differences are observed (e.g., the medians vary) which are statistically significant, their significance is not otherwise evident.

Figure 7 provides cumulative distributions of p2. The variation

between locations, while statistically significant, is small, so only one

curve is drawn. _{The median occurs at p2 of 0.05, and p2 exceeds 0.1 about}

25 percent of the time. This implies that two or more systems are evident

about half the time, and that about one quarter of the time the effect is large. From these data and histograms of m2 versus p2 (not shown), it is concluded that about one-third of the hindcast directional distributions cannot be fit with a cosine squared distribution.

Figure 8 shows the cumulative distribution of _q for grid point 149

for various wave heights. More skew is shown for lower wave heights at this

location but otherwise the pattern is similar to the other two locations.

In conclusion, it is suggested that about half of the hindcast directional spectra for these three locations can be approximated with a cosine squared distribution with half spreading angles near 90 degrees.

During 1983 and 1984 this investigation will be continued to include locations

in the North Pacific. Based on the results available to date, it is too soon

to suggest any improvement to the spreading function recommended by the 16th I T.T C.

PERSISTENCE

Distributions of the duration or persistence of environmental parameters have long been required in the application of long-term

statistics. Fatigue analysis, while not of direct concern to the Seakeeping

Committee, is of importance to structural design. As the hindcasts are

essentially time histories of wave spectra at six hour time increments for locations throughout the ocean, they can be used to develop this important

type of data. Figures 9 and 10 are samples developed for a location near

Station India in the North Atlantic [18]. Similar data sets are available for

larger ocean areas [7], [10].

S

SPECTRAL FAMILIES

The development of a stratified sample of hindcast wave spectra is

being conducted by Cummins. Such a sample provides a statistically unbiased

sample of directional wave spectra representative of all sea seventies and

ocean areas. The application of this sample to ship seakeeping prediction is

under investigation and will be reported at a later time.

The 16th ITTC recommends application of a two-parameter wave spectral

formulation. This is similar to that developed by Bretschneider and

recommended by the ISSC for open ocean conditions. At this time, there is no

evidence that a more representative spectral model is available. However, the

formulation developed by Huang and his colleagues [19] at NASA's Wallops

Flight Center may be of future interest. The Wallops spectrum incorporates a

variable bandwidth dependent on the significant slope of the wave field and is of a similar form to the JONSWAP spectrum.

WAVE MEASUREMENT

The need for improvements in wave measuring techniques for seakeeping applications is focused on measuring wave directionality for

full-scale seakeeping trials to better correlate measured

full-scale ship responses to those derived in towing tanks or by numerical prediction

validation of directional wave models such as the SOWM used to develop the Hindcast Wind and Wave Climatology

While many measuring techniques are suggested by the oceanographic community (e.g., hull mounted radars, discus buoys, cloverleaf buoys,

overflying/rernote sensors, etc.), very few systems are easily deployable by a

moving ship. For this reason, the U.S. Navy has contributed to the support of

the development of a small, relatively inexpensive roll-pitch buoy by the

ENDECO Corporation. Other such buoys may also be available in Europe. While

the buoy is still undergoing evaluation, initial results indicate that it provides comparable point spectral data to that derived by other buoys [20].

In May 1982, the Royal Netherlands Navy hosted a cooperative

Dutch/U.S. Navy full-scale trial aboard the Hr. Ms. TYDEMAN. The disposable

buoy developed by Prof. Gerritsma and his colleagues at Delft University [21], the DTNSRDC/ENDECO buoy, and a sophisticated Datawell directional buoy

prototype were tested aside-by-side" at locations noted in Figure 11. Generally, the point wave spectra from the three substantially different systems agreed well, see Figures 12, 13, and 14, and the significant wave

heights were also in agreement, see Figure 15. As the directional data from

this trial becomes available, they too will be reported as well as compared to the forecasts provided by the SOWM.

A workshop on wave measurement requirements and techniques was held in April 1981 and the results provide a comprehensive overview of those topics [22]. While great emphasis was placed on remote rather than in-situ devices, many useful summaries were provided.

WAVE GENERATION

Since the 16th ITTC, little real progress in wave generation can be

r8ported from the United States. More importantly, certain deficiencies are

noted. The most obvious one is the need for shallow water tank facilities for

testing vehicles at speed, in a directional wave system. Such a facility is

operational at the Norwegian Institute of Technology.

Efforts to generate directional waves in DTNSRDC's MASK facility have been undertaken and involve the variation of phase angles of adjacent wave makers. No final results or conclusions as to this potential capability are yet available.

COASTAL AND FETCH LIMITED WAVES

The 16th ITTC recommends the application of the JONSWAP spectrum for

fetch limited conditions. Also recommended by the ISSC [5] but in a slightly

An on-going field program by the U.S. Navy indicates the difficulty

of applying any spectral form to shallow water. The program provides for the

long-term collection of wave data at both an inshore and an offshore station

near a major navigation channel. Transformation of waves from deep to shallow

water involves complex physics which may require inclusion of topographic,

current and other local effects [23]. Wave directionality is also a

complicating factor. In general, it has been observed that the JONSWAP

Spectrum does not adequately represent specific wave spectra derived from

measurements of this site (nor does any other known formulation). It is

speculated that generalized spectral formulations are less applicable to site specific shallow water seakeeping investigations than those for the open ocean.

While publications on this subject may not yet be available, it is reported that the Max-Planck Institute in Hamburg is developing a hindcast wind and wave climatology for portions of the North Sea.

Additionally, the U.S. Army Corps of Engineers has published an atlas of wave hindcasts for 1956-1975 at 13 stations adjacent to the U.S. Atlantic

coast (deep water) [24] and for some near shore locations [25]. The deep

water data [24] provide somewhat higher wave heights than developed by SOWM hindcasts [7] due to a refinement in the wind field.

REFERENCES

Proceedings of the International Symposium on Hydrodynamics in Ocean Engineering, Norwegian Hydrodynamic Laboratories, Trondheim, Norway, 24 - 28 August 1981.

Proceedings of the Conference on Directional Wave Spectra, University of California, Berkeley, California, 14 - 16 September 1981.

Proceedings of Ocean Structural Dynamics Symposium 182, Oregon State University, Corvallis, Oregon, 8 - 10 September 1982.

Conference Record of Oceans '82, Washington, D.C., 20 - 22 September 1982.

Report of Environmental Committee (1.1) of the 8th ISSC, Gdansk, 1982.

Buckley, W.H. and A.B. Stavovy, "Progress in the Development of Structural Load Criteria for Extreme Waves, SSC/SNAME Extreme Loads Response Symposium, Washington, D.C., 19 - 20 October 1981.

Bales, S.L., W.T. Lee, and J.M. Voelker, "Standardized Wave and Wind Environments for NATO Operational Waters," Report

DTNSRDC/SPD-0919.-Ol, July 1981.

Bales, S.L., "Designing Ships to the Natural Environment," Naval Engineers Journal, Vol. 95, No. 2, March 1983.

Report identifying worst season environmental data for 16 global "hot spots," to be published approximately in September 1983.

Report similar to reference 7 except for the North Pacific, to be published approximately in September 1983.

"U.S. Navy Hindcast Spectral Ocean Wave Model Climatology Atlas: North Atlantic," Publication NAVAIR 50-1C-538, Naval Oceanography

Command Detachment, Asheville, N.C., to be published in June 1983. 10

To be published by Ship Structures Committee, approximately at end of 1983.

Kjeldsen, S.P., "Design Waves," Report NHL1-8l008, 30 January 1981.

Hogben, N. and F.E. Lumb, "Ocean Wave Statistics," Her Majesty's Stationary Office, London, 1967.

Andrews, K.S., N.M.C. Dacunha, and N. Hogben, "Wave Climate Synthesis," Report NMI R-l49, January 1983.

Cummins, W.E. and S.L. Bales, "Extreme Value and Rare Occurrence Statistics for Northern Hemispheric Shipping Lanes," Proceedings, Fifth SNAME Ship Technology and Research (STAR) Symposium, Coronado, California, June 1980.

Cummins, W.E., S.L. Bales, and D.M. Gentile, "Hindcasting Waves for Engineering Applications," presented at International Symposium on Hydrodynamics in Ocean Engineering, Trondheim, August 1981 [1].

Bales, S.L., W.E. Cummins, and E.N. Comstock, "Potential Impact of Twenty Year Hindcast Wind and Wave Climatology," Marine Technology, Vol. 19, No. 2, April 1982.

Huang, N.E., S.R. Long, C-C. Tung, V. Yuen, and L.E. Bliven, "A Unified Two-Parameter Wave Spectral Model for a General Sea State,"

Journal of Fluid Mechanics, Vol. 112, pp. 203-224, 1981.

Foley, E.W., R.J. Bachman, and S.L. Bales, "Open Ocean Wave Buoy Comparisons in the North Atlantic," Proceedings of the MTS/NDBC 1983

Symposium on Buoy Technology, New Orleans, Louisiana, 27 - 29 April

1983.

Buitenhek, M. and J. Ooms, "An Updated Design of a Disposable Wave

Buoy," Technische Hogeschool Delft, Report No. 463, May 1978.

"Measuring Ocean Waves," Proceedings of a Symposium and Workshop on Wave Measurement Technology, National Research Council, Washington, D.C., April 1981.

Lai, R.J., and S.L. Bales, "Effects of Shoaling Zone on Wave Transformation and Ship Motion," presented at First International Conference on Meteorology and Air/Sea Interaction of the Coastal Zone, The Hague, 10 - 14 May 1982.

Corson, W.D., D.T. Resio, R.M. Brooks, B.A. Ebersole, R.E. Jensen, D.S. Ragsdale, and B.A. Terry, "Atlantic Coast Hindcast, Deepwater, Significant Wave Information," Waterways Experiment Station, WIS Report 2, January 1981.

Corson, W.D., D.T. Resio, R.M. Brooks, B.A. Ebersole, R.E. Jensen, D.S. Ragsdale, and B.A. Terry, "Atlantic Coast Hindcast, Phase II, Wave Information," Waterways Experiment Station, WIS Report 6, March 1982.

.

.

.

TABLE 1

- ANNUAL SEA STATE OCCURRENCES IN THE OPEN OCEAN

NORTH ATLANTIC

Sea _State

Number

Significant Wave

Height (m)

Sustained Wind Speed (Knots)* Percentage Probability

of Sea State

Modal Wave Period (Sec) Range**

Most Probable*** Range Mean Range Mean

0-1

0-0.1

0.05

0-6

3 0

=

-2 0.1 -0.5 0.3 7 - 10 8.5 7.2 3.3 - 12.8 7.5 3 0.5 - 1.25 0.88 11 - 16 13.5 22.4 5.0 - 14.8 4 1.25 - 2.5 1.88 17 - 21 19 28.7 6.1 - 15.2 8.8 5 2.5 - 4 3.25 22 - 27 24.5 15.5 8.3 - 15.5 9.7 6 4 - 6 5 28 - 47 37.5 18.7 9.8 - 16.2 12.4 7 6 - 9 7.5 48 - 55 51.5 6.1 11.8 - 18.5 15.0 8 9 - 14 11.5 56 - 63 59.5 1.2 14.2 - 18.6 16.4 >8 >14 >14 >63 >63 <0.05 18.0 - 23.7 20.0

*Ambient wind sustained

at 19.5 m above surface to generate fully-developed

seas. To convert to another

altitude, H2. apply V2

= V1(H2/19.5)7

**Minimum is 5 percentile and

maximum is 95 percentile for periods given

wave height range.

***Based on periods associated

with central frequencies included in Hindcast

TABLE 2 - ANNUAL SEA

STATE OCCURRENCES IN

THE OPEN OCEAN

NORTH PACIFIC

Sea _State

Number

Significant Wave

Height Im)

Sustained Wind Speed (Knots)* Percentage Probability _{of Sea State} Modal Wave Period (Sec) Range**

Most Probable*** Range Mean Range Mean

0-1

0-0.1

0.05

0-6

3 0

-2 0.1 - 0.5 0.3 7 - 10 8.5 4.1 3.0 - 15.0 7.5 3 0.5 - 1.25 0.88 11 - 16 13.5 16.9 5.2 15.5 7.5 4 1.25 - 2.5 1.88 17 - 21 19 27.8 5.9 - 15.5 8.8 5 2.5 - 4 3.25 22 - 27 24.5 23.5 7.2 - 16.5 9.7 6 4 - 6 5 28 - 47 37.5 16.3 9.3 - 16.5 13.8 7 6 - 9 7.5 48 - 55 51.5 9,1 10.0 - 17.2 13.8 8 9 - 14 11.5 56 - 63 59.5 2.2 13.0 - 18.4 18.0 >8 >14 >14 >63 >63 0.1 20.0 20.0

*Am blent wind

sustained at 19.5

m above surface to

generate fully-developed

seas. To convert to another

altitude, H2, apply V2

= V1(H2/19.5)117

**MinimtJrn is 5

percentile and maximum is

95 percentile for

periods given

wave height range.

***Based

on periods associated with

central frequencies

included in Hindcast

100 80 V 0 WINTER (Dec - Feb) 2 I 4 6 8

SIGNIFICANT WAVE HEIGHT, M

Figure 1 - POLARFRONT/SOWM Winter Wave Height Comparisons 8 60 40 2 0 ANNUAL

.

S

2 L

L

J

4 6 8

SIGNIFICANT WAVE HEIGHT, M

Figure 2 - P0LARFR0NT/S0TN Annual Wave

100

80

60

40

20

TEN YEAR HINDCASTS GRID POINT 127 1959 /6 9

VISUAL OBS FOR 5° x 5 AREA AROUND OWS INDIA

'NNIMET' data for

50 50 area around India 0 1 J 4 ₈ 12 16

SIGNIFICANT WAVE HEIGHT, M

Figure 3

600

Figure 5 - Location of Three Grid Points (279, 149, 127)

+. DEG.

Figure 4 - Spreading Parameter m2, and Parameter _{p2 Versus Cosine}

Squared Spreading Angle,

_±

300 _W00E 0.5 0.4 0.3 0.2 0.1 0.0 0.5 0.4 0.3 0.2 0.1 'V p..,

-/

/-/

/

/

- ---p2

0 20 40 60 80 100 120 140 160 180 08 0.8

Figure 6 - Cumulative Frequency

Distribution of Spreading Parameter, for Various Significant Wave Height Ranges for Three

1.0 0.8 0.6 0.4 0.2 0.0 D GP 127 o GP 149 GP 279 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 7 - Cumulative Frequency Distributions of Parameter p2 for Three Grid Points, 1959 to 1969

1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 08 SKEWNESS PARAMETER. Il

Figure 8 - Cumulative Frequency Distributions of Skewness Parameter, _qj,

for Various Significant Wave Height Ranges for Grid Point 149, 1959 to 1969

55 + 48 41 34 28 22 17 11 7 4 0 TOTALS 24+ 20 16 14 12

I-ilO

(3 3 2 0 6 12 18 24 30 36 42 48 _{54 60} 66

TOTALS DURATION HOURS

50.0 72.7 10,1 4.3 73 1 0 7 3 3 1 01 .02 04

Figure 9 - _{nnual Persistence of Significant Wave Height for}

Grid Point 128, 1959 to 1969 GRID POINT 128 TOTAL SAMPLES 13,333 0 72 79+ TOTALS 35 24 42 19 16 12 10 8 10 7 1 1 193 233 90 67 14 6 4 1 2 475 202 110 02 45 20 7 6 4 3 1 1 641 403 162 52 10 8 1 1 1 ₆₃₀ 366 165 72 31 18 6 7 1 666 371 105 99 46 12 8 4 4 2 1 732 365 91 35 13 6 1 511 419 97 37 9 9 3 ₅₇₄ 351 107 20 21 13 2 1 ₅₂₃ 317 131 36 16 5 4 1 510 327 07 39 17 8 2 1 1 491 7110 15 48 22 16 I 1 1 ₄₆₄ 708 102 45 18 12 7 1 I 1 1 316 145 67 37 23 18 6 3 I I ₂₀₃ 61 42 19 13 1 9 5 4 I 1 1 1 1 165 lOt 14 5 2 5 1 7 130 4204 1667 742 319 176 74 51 75 22 11 2 2 3 7352 7 2 9 28 9 3 2 1 41 75 26 17 0 1 127 181 99 33 16 7 1 2 ₁ 340 377 163 68 21 12 10 1 1 2 661 615 243 115 59 33 18 12 2 4 4 2 1110 d29' 133 60 20 20 5 8 4 1 1 ₁₇₉₀ 1 702 312 154 50 54 25 16 13 5 6 2 3 3 1385 506 254 107 44 26 8 12 5 5 3 1 1 10 1062 366 135 35 24 10 5 3 1 543

: :

::

_i

::3 _6815: 32

2:14

7 3 10

GRID POINT 126 _{TOTAL SAMPLES 13,333}

6 12 18 24 30 36 42 48 54 60 66 72 78 TOTALS

DURATION HOURS

562 23.1 10.0 4,9 2.4 1.3 .8 .5 .3 .2 .1 .05 04 .1

Figure 10 - Annual Persistence of Wind Speed for Grid Point 128, 1959 to 1969 (Dashed Line Indicates Duration of Constant

Wind _{Over Infinite Fetch Required for Fully}

Developed Seas) 100 75 rn 1) 11 502 25 100 75 m 50 2 25 0

Figure 11 - TYDEMAN Transit Route

1 3 MAY It)...I 7 MAY (j)...1LI MAY®lB MAY

a-15 MAY MAY

MAY>..-2Q MAY

DELI-i ( (. ) 0.3 ENDECO----(t) 2.32 U

-

1/1 WAVCC---( )

32

-.

0.6 _{. 6} '(4 w '

0.4

0./4_

-J ft I 'I LU I

'.'

LU 1

0.2

('

_'1.21.6

_°:

y

WAVE FREQUENCY (rad/sec)

Figure 12 Spectral Comparisons of Three _{Figure 14 - Spectral Comparisons of Three}

Buoys Using the Delft Analysis for Run 25 _{Buoys Using the Deift Analysis for Run 31}

0 0.4 0.8 .2 1.6 WAVE FREQUENCY (rdd/tcc) 2.0 3 'a DELFT ENDECO(6) 1,31.71 WAVEC --(c) 1J 0.8 1.2 1.6

WAVE FREQUENCY (rad/sec)

o ENDECO o WAVEC

A DELFT

g

AD

0 0

Figure 13 _{Spectral Comparisons of Three}

I

Buoys Using the Deift Analysis for Run 30 13 i1 15 16 17 18 19 20

MAY 1982

2.0

Figure 15 - Significant Wave Height Comparisons Using the Deift Analysis Procedure

U 'Ii In a >- I-0.8 0.6 I / I I DEL _()1131.68m 1 48m

ENDEC0()i,3

WAVEC--(ç)173.4.70m--(I\ LU _Is 0.4 Ii I-I, 4j LU cL. LI) _0,2 LU '4 V '. /4 - -0 4) 5/ 5-Lii :1: F-5-, (I) 0 o 0 o 0 1.0 1 .0

4

Appendix: _{Parameters for Angular Distribution [17]}

In [161, a set of parameters for treating the directional spread of

wave energy was introduced. _{These parameters are reviewed and}

extended here, as knowledge of the definitions and properties is

essential to the understanding of this paper. The circular or

cyclic geometry limits the usefulness of the usual statistical moments, so these are replaced by more or less analogous parameters

based upon a simple analogy from classical mechanics. The spectral

weights associated with the twelve 30 degree directional sectors

are considered to be on a unit circle, see Figure A-i. _{The centroid}

of this

weighted

_{circle, with coordinates p,, p or}

c' 8c

is taken to define the mean wave direction. _{The radius of gyration}

about the diameter through this centroid, designated by m, is a measure of the angular spread about this direction, and is an analog

of the standard deviation. _{The parameter m2, which is a}

nor-malized moment of inertia about this axis, is preferred for use in here. _{A second normalized moment of inertia, about the axis}

perpen-dicular to the reference diameter at the _{centroid, is also used,}

and is designated by p2 . These three parameters satisfy the

identity

P2 + = 1

(A-i)

as this sum is the normalized polar moment of inertia of the weighted circle about its center, which of course, has the value

one. Specifically, =

_{(h/E)E1(e1) sin Oj}

(A-2)

cos ej (A-3) (A-4) ec =

arctan(P,/)

_(A-5)

m2 = ('/E)E1 sin2 (Gj - e)

_(A-6)

2

,2

2

4

S

S

Figure A-i-Definition of Angular Distribution Parameters on the Unit Circle.

To this set is added a "skewness parameter,"

q =

('/E)Ej sin 2(Gj

- 8)

(A-8)

In the above, the angles are taken as compass direction, the X axis

is toward the east, and Y toward the north.

The dimensionless parameters

p0, e'

m2, q ore0, m2 2 q,

together with the spectral weight, E, are sufficient to fix the first five coefficients of a Fourier series for the directional distribution, and conversely.

If

c equals zero, the parameters are undefined. This degenerate

case has not been observed in any hindcasts. _{However, small values}

of

p0,

_{while rare, are sometimes found, and the determination of}

the parameters is then poorly conditioned. _{Hindoast data is far}

from precise, and slight shifts in the _{E can cause large shifts}

in

_e0,

for example, in the near degenerate case. _{This should be}

remembered when analyzing statistical combinations for which + p2 is near 1.

Many of the parameter sets are consistent with an assumption of a

single system, with a distribution proportional to cos2 over a

range e ₊

D'

-

This would have the form

S(e)= E/0

cos2[((e_ec)/o)(,T/2)]

for

a

-The parameters m2 and p2 for this distribution are shown in

Figure 11 of the main text. Matching a pair of parameters for a

cosine squared distribution is no proof that an actual distribution is of this type, but if q is also very small, there is a strong implication that the distribution does not depart greatly from this form. Such distributions are described as "central," in the sense that they are primarily from a single weather source, with little

contamination from secondary sources.

The five parameters E, _ec, m2, p2, and q do not fix the

directional distribution, any more than a finite set of moments

fixes a statistical distribution. However, if certain assumptions

are made as to the character of the directional distribution, a set of parameter values limits the possibilities in a useful manner.

It may be noted that two unidirectional wave systems can be

superimposed to yield any set of e(,, m2, p2 values which is

consistent with equation (A-7). i m ₊ p2 is <i, either or

both of the unidirectional distributions can be replaced by a

symmetric distribution of cosine squared form. The addition of a

value q, together with the assumption that one of the wave systems

is unidirectional, is sufficient to fix the angle for the