Return Periods of Summer-Time Sea Storms in the Central Mediterranean Sea

Return periods of summer-time sea storms in the central Mediterranean Sea

P. Filianoti Department of Mechanics and Materials,

Reggio Calabria University, Italy

Abstract

From an analysis of wave data from the Italian buoy network we obtain the probability of exceedance of the significant wave height in the summer-time.

Then the return period of the summer storms in which the significant wave height exceeds some fixed threshold is obtained by means of Boccotti's equation. Summer waves prove to be much smaller off the Italian south-eastem coast then off the Italian western coast. The return periods obtained in the paper are useful for the design of facilities such as barriers against pollution and mucilage and removable moorings for yachts and sailboats which are used only in summer.

1 Introduction

The significant wave height (H, ) represents a random process of time, gradually variable and strongly not stationary because of a seasonal component (Cunha and Guedes Soares [3]). The analytic form of the exceedance probability P(H, > h) in many locations is well represented by a Weibull in two parameters (Guedes

Soares [4]). Where the wave buoy data are available, the return period of a sea storm in which H, exceeds a fixed threshold is usually obtained by

where At,,, is the sampling interval (Massel [5]).

300 Coastal Engineering V: Computer Modelling of Seas and Coastal Regions

The return period R(H, > h) can be obtained through the closed form solution based on the rule of equivalent triangular storms (Boccotti [l]):

where p(H, = h) is the probability density function relative to the probability of

: exceedance P(H, > h)

and where b(h) is the regression durations-heights of equivalent triangular storms (e.t.s.). To our knowledge there are no studies in literature on the probability P(H, > h) and return periods R(H, > h) in regards of the season. The present paper proposes a study of such nature analysing P(H, > h) and R(H, > h) relative to the summer-time in the Italian seas. P(H, > h) will be estimate from the data of the Italian Sea Wave Measurement Network (SWaN). R(H, > h) will be obtained through eqn (2) which is of general validity. The regression durations- heights which is present in this equation will be obtained for the various locations by examining the hlstory of the storms. To this aim, for each actual storm we will obtain the relative e.t.s. The utility of our study is that many facilities pertaining bathing activities, are put into the sea only in the summer-time. Typically these structures are piers, moorings, and barriers against pollution and mucilage. Designing these structures with the maximum expected wave height relative to all seasons in the lifetime, leads to dimensions really too heavy. The design, as we will see in an example, must be made with the maximum expected wave height during the summer-time in the lifetime and dimensions result much smaller.

2 The probability of H, in the summer-time

Let us define

At(h, summer) P*(H, > h) = total summer time '

At(h, summer) P(H, >h) = total time '

Coastal Engineering V: Computer Modelling of Seas and Coastal Regions 301

where At(h,surnmer) is the time in whch H, is greater than h in the summer time (June, July, August and September). Clearly fiom the definitions we have

P*(H, >h) is shown in fig. 1. The figure shows also the probability of H, relative to the waves of all seasons of the year. We have resorted to the ancillary variables X and Y which are defined by

1 X = lOOln(2Sh) with h inmetres, Y r 100lnln-. (7) P

With this choice if the data points exhbit a linear trend on the plane X - Y then the probability P(H, > h) can be fitted by

that is a Weibull distribution whose parameters u and W depend on the considered location. As we can see the data point shown in fig. 1 exhibit a linear trend starting fiom

Y = 50 (P = 0.19) and the U, W values of table 1 represent this linear trend. The data of H, used for this table and for fig. 1 are those of the Italian SWaN, for the period 1989-1998. These data are taken at a rate of 113 per hour by means of eight directional pitch-roll buoys. The network's efficiency (number of three- hourly observations / number of expected data in whole summer periods) was greater than 94% for all the stations with the exception of Mazara del Vallo (88%).

3 Storm durations

We have singled out all the sea storms recorded in the summer-time by the eight buoys of the Italian SWaN. By "sea storm" we mean an episode in which H, exceeds the threshold of 1.5< H, > , where < H, > is the mean value of H, in the summer-time. This mean value ranges between 0.45 m for the Jonian Sea and the Adnatic Sea to 0.9 m for the sea off the western coast of Sardinia. The overall number of analysed storms is 1800. Figure 2 shows the history of a storm recorded at Mazara del Vallo and its e.t.s. The duration of e.t.s. is obtained after a few attempts so that the maximum expected wave height of the e.t.s. is equal to the maximum expected wave height of the actual storm. The maximum expected wave height is obtained by Borgman's solution [2].In all the 1800 episodes we have found confirmation of the rule of e.t.s., that is to say: P(H,, >H) of the

302 Coastal Engineering V: Computer Modelling of Seas and Coastal Regions

T Y La Spezia Alghero 250 f 200 1 summer-tme wave - 150 100 100 II Eeasons waves 50 50 - X X

Ponza 250 Y Mazara del Vallo

100

50 0 0 100 200 300

Y Catania

I seasons waves X

Y Monopoli 250 250+ Pescara

150 150

100 100 - VIseasons waves 50 50 - X t X

Figure 1. Italian seas: P*(H, > h) for the summer-time, and P(H, > h) for all seasons of the year. [The ancillary variables X and Y have been defined

by eqn (V1

Coastal Engineering V: Computer Modelling of Seas and Coastal Regions 303

e.t.s. practically coincident with the P(H,, >H) of the actual storm,

P(H,, > H) being the probability that the maximum wave height in the storm exceeds any fixed H (see fig. 2). Our aim is to obtain the regression b(h) relating durations to amplitudes of e.t.s. (that is base b to height a of triangles like that of fig. 2). Thls regression was obtained by Boccotti [l] for all storms, that is for storms of all seasons of the year. He proposed the form

&(a) = K,b,, exp K, - , ( l,) where a,, is the average a of the set of N (= number of years x 10) strongest

storms in the time span under examination, and b,, is the average b of this set. Parameters a,, ,b,, depend on the special location. Parameters K, and K,

depend on the geographic area (Boccotti gives K, and K, for the central Mediterranean Sea, the North Western Atlantic, and the North Eastern Pacific).

Regressions are decreasing, that is K, proves to be negative. Values of K, and K, for the central Mediterranean Sea are

We repeated the examinations of b(a) considering only the summer storms, and we found confirmation of the decreasing monotonic trend. The regression is shown in fig. 3. Each point refers to a single storm: the abscissa represents the

quotient ala,, and the ordinate represents the quotient b/blo . Values of a,, and b,, for each location are given in table 2. Data points of fig. 3 include the

100 heavier storms recorded in each of the 8 locations of the Italian SWaN in ten years. The line represents the function (9) that best-fits the regression. The values of parameters K, and K, are

Equation (9) with values (1 1) of K, and K, and table 2 pennit to calculate the

regression b(a) for each location.

4 Return periods

Using P*(H, > h) of 4 2, b(a) of 3 and eqn (2) we obtain the return periods R(H, > h). Fig. 4 shows the R(H, > h) for the Italian seas. For comparison we

304 Coastal Engineering V: Compu:er Modelling of Seas and Coastal Regions

have represented also the R(H, > h) relative to all seasons of the year.

Table 1. Parameters of P*(H, > h) for the Italian seas.

LOCATION I w [m] I u I

La Spezia 0.580 0.975 Alghero 0.930 1.069

Ponza 1 0.537 1 0.948 1

Mazara del Vallo 1 0.666 1 1.213 1

Catania 1 0.339 1 1.113 1

Crotone 1 0.427 ( 1.195 (

Monopoli 1 0.520 / 1.226 (

Pescara / 0.377 1 0.980 ]

The difference between waves of summer periods and waves of all seasons is much bigger in the south-eastem Italian seas than in the western Italian seas, as fig 4 shows clearly. As an example we can see that the H, which corresponds to a return period of 10 years is

5.8 m all seasons, Catania (Jonian sea) 2.6m summer,

6.3 m all seasons, Crotone (Jonian sea) 3.0m summer,

Figure 2. A sea storm recorded at Mazara del Vallo and its e.t.s. The P(H, >H) of the e.t.s. (points) coincides with the P(H,, >H) of the actual storm (continuous line).

Coastal Engineering V: Computer Modelling of Seas and Coa~talRegions 305

Figure 3 Regression durations-heights (normalised) of the equivalent triangular storms (e.t.s.). Each point represents a single storm.

9.3 m all seasons, Alghero (West Sardinia) 7.6m summer,

6.4 m all seasons, La Spezia (Ligurian sea) 6.0m summer.

That is the quotient

H. (10 years) for the waves of summer - time H, (10 years) for the waves of all seasons

is about 0.46 for the Jonian Sea and ranges from 0.82 for the sea off West Sardinia to 0.94 for the Ligurian Sea.

5 Conclusions

The design of maritime structures is made with the lughest waves that we may expect in the lifetime. By waves we mean those of whichever season, because the

fixed structures (i.e. detached breakwaters, offshore structures, etc.) typically stay in the sea all the seasons of the year. For some structures pertaining summer facilities, which stay in the sea only in the summer period, we must take the maximum wave height in the summer seasons. For example if the structure has a lifetime of 50 years, it must be considered the maximum expected wave in a

period equal to 50 successive summers. To this aim the results obtained in 2-3- 4 can be used. These results are valid for Italian seas; the logic used is of general validity.

306 Coastal Engineering V: Computer Modelling of Seas and Ccastal Regions

Table 2 Parameters a,, and b,, for eqn (7) of b(a) .

LOCATION a,, [m] 40 [hours]

La Spezia 2.2 73

l Alghero 1 3.2 1 98 1

Mazara del Vallo 2.0 92 ( 1 1 1 Catania 1.2 67

Crotone 1.4 59

l Monopoli 11.7 168 1

l ~escara 1 1.6 1 57 1

As an example let us suppose to design a summer single point mooring off Catania (we mean a single point mooring that is put in at the beginning of summer and is removed at the end of the season). Following the guidelines on the design of maritime structures (Puertos del Estado [6]) we fix lifetime L equal to

25 years and encounter probability B equal to 0.10, from whlch we obtain the following return period

to which corresponds an H, of 3.5 m (see fig. 4) that will be the H, of the design sea state for our summer single point mooring. If we consider waves of all seasons instead of waves of summer periods, to the same return period of 237 years corresponds an H, of 8.4 m (see again fig. 4). Conclusion, with the traditional approach (waves of all seasons) the design H, is 8.4 m, with the most appropriate approach (waves of summer periods) the design H, is 3.5 m . The difference as we can see is really big. Some smaller differences would be in the other Italian seas and particularly in the Ligurian Sea. This result should be borne in mind in designing maritime structures used prevalently in summer. When the difference between the design wave of all seasons and the design wave of summer-time is very big like at Catania, it can be worth, at least for some kind of structures like piers and single point moorings, to resort to structures which can be installed at the beginning of the season and removed at the end of summer.

Coasral Engineering V: Compurer Modelling of Seas and Coastal Regions 307

Alghero 1000 summer-ume storm summcr-ume storms

0.1 0.1 0 4 8 12

R Ponza R [yeafiIh Mazara del Vallo

R [years] Catania R [years] Crotone

100 l0y10 , , > 10

1 all seasons storms I %asons:stotms 1 0.1 h [m1 12 0.1 0 4 8 12

Monopoli R Pescara 1000

0 1 0 4 8 12

Figure 4 Italian seas: R(H, >h) of summer-time sea storms, and R(H, > h) of

ail seasons sea storms.

308 Coastal Engineering V: Computer Modelling of Seas and Coastal Reg~ons

References

[l] Boccotti, P., Wave Mechanics for Ocean Engineering. Elsevier

Oceanography Series, pp. 1-495,2000. [2] Borgman, L.E., Maximum wave height probabilities for a random number of random intensity storms. Proc. 121h Conf Coastal. Eng., ASCE, pp. 53-64, 1970.

[3] Cunha C. & Guedes Soares C., On the choice of data transformation for modelling time series of significant wave height. Ocean Eng., 26, pp. 489- 506,1999. [4] Guedes Soares C., Assessment of the uncertainty in visual observations of wave height. Ocean Eng., 6, pp. 37-56, 1986.

[5] Massel S.R., Ocean surface waves: their physics and prediction. World Scientific, 1996. [6] Puertos del Estado, Maritime Works Recommendations-ROM, Madrid, 0.2- 90, 1990.