Stochastic Properties of Ocean Waves

Chapter 3

Stochastic Properties of Ocean Waves 3.1

Introduction

In Section (2.5) the statistical variability of the water surface elevation was modelled in terms of the Fourier series (2.45). In this chapter the statistical distribution of the Fourier amplitudes, ai will be investigated.

3.2 3.2.1

Probability Distribution of Wave Heights The Rayleigh Distribution

The basic theory to be adopted for the distribution of wave heights was originally developed by Rice (1954). It is assumed that the spectrum of the water surface elevation is narrow banded. That is, the energy is confined to a relatively narrow range of frequencies. For such a spectrum it can be shown that the probability density function for wave heights follows a Rayleigh distribution p{H) = ^ e - ^

(3.1)

where a^ = / F ( / ) d / is the variance of the record. Equation (3.1) must satisfy the requirement that J p{H)dH = 1. Longuet-Higgins (1952) derived relationships between a number of characteristic wave heights based on (3.1). The mean and root-mean-square (rms) wave heights are, respectively /•OO

H=

Jo

Hp{H)dH = V2^ 25

(3.2)

26

CHAPTER 3. STOCHASTIC H^^, = m=

PROPERTIES

H'p{H)dH

OF OCEAN

= 8a'

Jo

WAVES (3.3)

Adopting (3.3), the Rayleigh distribution (3.1) can be written in the alternate form ,„,

2H

- ^

(3.4)

rms

The cumulative form of (3.4) (i.e. probability that H is greater than a value H)\s P{H >H)=

/•OO

JH

p{H)dH

u±

(3.5)

The average height of all waves greater than H, written as H{H) is given by (3.6)

3.2.2

Representative Measures of Wave Height

Rather than wishing to know the average height of all waves greater than a particular value, it is more common to consider the average height of the highest 1/N waves. This can be calculated from the Rayleigh distribution by determining the wave height which has a probability of exceedence given by 1/A^. A full analysis is given by Goda (1985). Typical results for various values of N appear in Table 3.1.

N

HI/N/O-

HI/N/H

Hl/N/Hrms

100

6.67

2.66

2.36

50

6.24

2.49

2.21

20

5.62

2.24

1.99

10

5.09

2.03

1.80

5

4.50

1.80

1.59

3

4.00

1.60

1.42

2

3.55

1.42

1.26

1

2.51

1.00

0.87

Comments

Highest 1/lOth wave Significant wave Mean wave

Table 3.1: Representative wave heights calculated from the Rayleigh distribution [after Goda (1985)]

3.2. PROBABILITY

DISTRIB UTION OF WAVE HEIGHTS

27

The significant wave height, Hs is defined in Table 3.1 as //1/3, the average of the highest 1/3 of the waves. The term significant wave height is historical as this value appeared to correlate well with visual estimates of wave height from experienced observers. Table 3.1 indicates that Hg = 4<7. From (2.49), the variance, cr^ can also be defined in terms of the spectrum as a'^ = J F{f)df. Therefore, the significant wave height can be calculated from the spectrum, its value is 4 times the square root of the area under the spectral curve. The probability density function for wave height, as defined by the Rayleigh distribution (3.4) is shown in Figure 3.1. Also shown is the significant wave height, Hs. From Table 3.1, Hs = 1.42//rms- The probability of a wave with height greater than Hs is represented by the shaded area in Figure 3.1 and is defined by the cumulative distribution (3.5). The probability, P{H > Hs) takes a value of 0.135, that is, there is a 13.5% chance of a wave with height greater than the significant wave height. As stated previously, the significant wave height equates to the wave height which would be visually estimated by an experienced observer. On average, 13.5% of waves, or roughly 1 in 7, would be higher than this value. This agrees well, with the common sailors tale, that every 7th wave is large.

X

0.4 h

Figure 3.1: The probability density function, p{H) for wave height as defined by the Rayleigh distribution (3.4). Also shown is the significant wave height, Hs = 1.427^,^,.

28

3.3

CHAPTER 3. STOCHASTIC

PROPERTIES

OF OCEAN

WAVES

Global Distribution of Wave Properties

In Chapter 4 the detailed processes responsible for the generation and evolution of waves and methods for the prediction of waves will be investigated. Independent of these predictive capabilities, long term measurements of waves provides the opportunity to investigate regional variability of the wave climate. As an example, long term measurements at a point using an in situ instrument, such as a wave buoy, would provide information on the wave climate at that point. Such information is invaluable for engineering applications and long term measurement programs have been established at many sites around the world. By their nature, such measurements are point specific and cannot hope to give full global coverage. This type of coverage can, however, be obtained from Earth orbiting satellites. The details of such systems are discussed in Chapter 9. Young (1994b), Young and Holland (1996a) and Young and Holland (1996b) have presented global statistics of significant wave height obtained from the Radar altimeter which was flown on the GEOSAT mission. This analysis extended more limited satellite climatologies presented by Mognard et al. (1983), Challenor et al. (1990) and Carter et al. (1991). The data set was, however, based on only 3 years data and hence limited in its application. Young and Holland (1998) extended the data set to approximately 9 years by combining data from 3 altimeter missions: GEOSAT, ERSl and TOPEX. The data were processed to yield mean monthly values of Hg and exceedence probabilities for Hs (i.e. probability that Hg will be above a given value). Data were obtained for the full globe on a 2° x 2° grid. Mean monthly values of //g, obtained in this manner are shown in Figures 3.2 - 3.5 for the full year. The 9 year data set also provides a means of determining the probability of obtaining wave heights of a particular value at any location. Figures 3.6 - 3.8 represent contour charts of the values of Hs which could be expected to be exceeded 10% • • • 90% of the time, respectively.

3.3. GLOBAL DISTRIBUTION OF WAVE PROPERTIES

29

Hs(m) December

Figure 3.2: Contours of mean monthly values of Hs for the months of December, January and February (Northern Hemisphere Winter).

30

CHAPTER 3. STOCHASTIC PROPERTIES OF OCEAN WAVES

Hs(m) March

300 Hs(m) May

Figure 3.3: Contours of mean monthly values of Hs for the months of March, April and May (Northern Hemisphere Spring).

31

3.3. GLOBAL DISTRIBUTION OF WAVE PROPERTIES

Hs(m) June

Hs(m) July

100

200

250

300

350

250

300

350

Hs (m) August

100

150

200

Figure 3.4: Contours of mean monthly values of H^ for the months of June, July and August (Northern Hemisphere Summer).

32

CHAPTER 3. STOCHASTIC PROPERTIES OF OCEAN WAVES

Hs(m) Septembei

350 Hs(m) October

150

200

250

300

350

250

300

350

Hs(m) November

150

200

Figure 3.5: Contours of mean monthly values of Hs for the months of September October and November (Northern Hemisphere Autumn).

3.3. GLOBAL DISTRIBUTION OF WAVE PROPERTIES

33

Hs(m) 10%

50

100

150

200

250

300

350

Hs(m) 30%

350

Figure 3.6: Contours of Hs which could be expected to be exceeded 10%, 20% and 30% of the time.

34

CHAPTER 3, STOCHASTIC PROPERTIES OF OCEAN WAVES

Hs(m) 40%

50

100

150

200

250

300

350

^0

300

350

Hs(m) 50%

50

100

150

200

Hs(m)

Figure 3.7: Contours of Hs which could be expected to be exceeded 40%, 50% and 60% of the time.

3.2. GLOBAL DISTRIBUTION OF WAVE PROPERTIES

35

Hs(m) 7 0 %

150

200

250

300

350

Figure 3.8: Contours of Hs which could be expected to be exceeded 70%, 80% and 90% of the time.

36

CHAPTER 3. STOCHASTIC

PROPERTIES

OF OCEAN

WAVES

Examination of Figures 3.2 - 3.8 reveals many fascinating features of the global wave climate. The most striking feature of the figures is the north-south seasonal variability. Maximum waves occur at the higher latitudes in each hemisphere during their respective Winters. This is clear in Figure 3.9, which shows the yearly variation in Hs at points with the same absolute values of latitude from both hemispheres. The peak values for both hemispheres are approximately the same at 6m. The Southern Hemisphere, however, has much less seasonal variability with Summer wave heights falling to only 4m whereas in the Northern Hemisphere Summer wave heights are only 2.3m. This clearly reflects sailors observations that the Southern Ocean is rough all year round.

J J Month Figure 3.9: The annual variation in values of mean monthly Hs from points in the Southern Ocean (-50^,66°, dashed Une) and the North Atlantic (50°, 326°, solid line). These points represent the most extreme wave climates for each of the hemispheres. In contrast to the extreme conditions at high latitudes, the equatorial regions are, not surprisingly, calm all year round. There is little seasonal variability in this region. As one moves either north or south from the equator both the extreme values and the seasonal variability increase. Young and Holland (1996a) also presented global variations in wind speed obtained from the GEOSAT altimeter. Although the distribution of wind speed generally parallels that for wave height, there are notable exceptions. In particular, a clear band of higher wind speeds exists in both hemispheres at a latitude

3.3. GLOBAL DISTRIBUTION

OF WAVE

PROPERTIES

37

of approximately 20°. This band is associates with the trade winds. A similar band of higher wave heights does not exist for wave height in any of Figures 3.2 to 3.8. This occurs since waves at any particular point are a combination of locally generated wind seas and swell propagating from distant points. In the trade wind belts, swell propagating from high latitudes is sufficiently large to mask any local elevation in wave height caused by the stronger local trade winds. The variation in wave height with latitude is clearly evident in Figure 3.10. This figure shows the value of Hs which could be expected to be exceeded 10% of the time along a line drawn north-south through the Pacific Ocean at longitude 209°E. The variation is approximately symmetric about the equator with the maximum waves occurring at latitude 50° in both hemispheres and the minimum at the equator.

0 Latitude Figure 3.10: The variation in Hs with latitude which could be expected to be exceeded 10% of the time along a north-south line drawn through the Pacific Ocean at longitude 209° E. There are many regional features of interest apparent in these global distributions of wave height. For the purposes of illustration, interest here is confined to examples of extreme events. As mentioned earlier, the Southern Ocean clearly has the most extreme, year round, wave conditions. Figure 3.11 shows the variation in mean monthly values of Hs along latitude —50°. Values for each of the four seasons are shown. Although the Southern Ocean is largely free of significant land masses, the wave height is not constant. As indicated above, the peak waves occur in Winter (July) in the area to the south of the Indian Ocean (between South

38

CHAPTER 3. STOCHASTIC

PROPERTIES

OF OCEAN

WAVES

Africa and Australia). Winds in the Southern Ocean are generally from the west. Despite the fact that as one moves east from this point the fetch could be expected to increase, Hs actually decreases. The occurrence of the maxima and this gradual decrease in Hs appears to be associated with the distribution of wind speed rather than fetch. The results of Young and Holland (1996a) show that the wind speed follows a similar trend. This indicates that the progression of low pressure systems which track across the Southern Ocean tend, on average, to be most intense in the region between South Africa and Australia and gradually weaken as they propagate further east. In Spring and Autumn this intense wind and wave region weakens but moves further to the east.

I

I

Australia N.Z.

50

100

150 200 Longitude

I

I

S. America

250

300

350

Figure 3.11: Variation in //^ as a function of longitude across the Southern Ocean at latitude —50°. Each of the four seasons is shown. Also shown at the top of the figure are the approximate locations and extent of the Southern Hemisphere land masses which lie to the north of this area. It is clear from Figure 3.11 that H3 is significantly smaller to the east of South America than to the west. This is a consistent feature for many continents. Examination of the global distributions shows that the west coast wave climates of Africa, Australia and South America are all significantly higher that their respective east coast wave climates. This is largely due to the fact that conditions on the east coasts are fetch limited under the prevailing westerly winds (see Chapter 5). Young and Holland (1996a), however, show that the winds on the east coasts of these continents are also generally weaker than on their respective west coasts.

3.4. LIMITATIONS

OF GLOBAL

STATISTICS

39

As stated earlier, the wave climate in equatorial and sub-equatorial regions is generally benign. One notable exception to this is the region of the Arabian Sea. In the period from June to September a very localized maximum develops close to the African Coast (see Figure 3.4). In this region mean monthly values of Hg reach 5m. Such values are typically only found at high latitudes. Figure 3.12 shows the variation in Hs with latitude along a line drawn at longitude 62° E, through the Arabian Sea. With the exception of the month of July, Hs consistently decreases with decreasing latitude. The localized maximum in the Arabian Sea is clearly apparent in July. This characteristic is caused by the strong south-west winds which develop in this region as a result of the Asian Summer Monsoon. These strong winds are confined to a jet relatively close to the coast and typically have mean monthly wind speeds of approximately 12m/s (Anderson and Prell, 1992).

1

1

1

1

1

1

1

7 October

6

.7 "

I4 w

X

July

: ^ " \

5

/

3 2r -

^ April

NX''^ v>

\

January

^ • • • • • .

• -s

i

-60

•••./*

^ - N ^

'^

1 1

1

-40

J

.

^'

H

1

-20 Latitude

20

40

Figure 3.12: Variation in Hs with latitude along a north-south line drawn at longitude 62°E through the Arabian Sea. Values are shown for each season. Note the localized maximum which develops in the Arabian Sea during July.

3.4

Limitations of Global Statistics

Results such as those presented above are obtained purely from observation without any requirement to understand the physical processes responsible for the generation or evolution of ocean waves. Nevertheless, they provide a valuable data source. Extensive climatologies, such as those presented by Young and Holland (1998)

40

CHAPTER 3. STOCHASTIC

PROPERTIES

OF OCEAN

WAVES

are valuable resources for a host of applications. The use of this material should, however, be approached cautiously as the applicability of such data is limited by the manner in which it was obtained and processed. Its use should reflect these limitations. The data set presented in Figures 3.2 to 3.8 is based on a total of 9 years of satellite observations. In addition, the satellite tracks are such that individual points are typically imaged, of order, once every 10 days. Ground tracks are typically separated by hundreds of kilometers. As a result, the data set will be typically too short to ensure that rare but extreme events are measured. In addition, intense but geographically confined events such as hurricanes may not be sampled by the satellites. Such events will have little influence on mean monthly values of Hsj such as presented in Figures 3.2 - 3.5. These same events will, however, be critical in determining exceedence probabilities which are typically used for engineering design. As an example, the data set may be unsuitable for determining the extreme probability distributions required for oflFshore structure design (i.e. return periods of 1 in 50 or 1 in 100 years). As a result, the exceedence probabilities presented in Figures 3.6 - 3.8 are shown at the most extreme level of only 10%. Although the number of individual satellite observations used to construct these climatologies is enormous, in order to obtain stable statistical values, it has been averaged on a 2° X 2° grid. Such a grid is relatively course and hence many localized features may not be resolved. This might include local variations in the wave climate close to shore or variations caused by currents. As the period for which satellite data exists gradually increases, these limitations will decrease and the applications of such data will expand.