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Directional analysis of sea storms
Ocean Engineering 107 (2015) 45–53
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Directional analysis of sea storms V. Laface a,b, F. Arena a,n, C. Guedes Soares b a b
Natural Ocean Engineering Laboratory, DICEAM, Mediterranea University, Reggio Calabria, Italy Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Tecnico, Universidade de Lisboa, Portugal
art ic l e i nf o
a b s t r a c t
Article history: Received 24 February 2015 Accepted 19 July 2015
The paper deals with the directional analysis of sea storms in the Atlantic and Pacific Oceans and in the Mediterranean Sea. The main focus of the work is to investigate the variability of wave directions during sea storms. The analysis is carried out starting from significant wave height and wave direction time series. At the first stage storms are selected from time series without conditions on wave direction. Subsequently the directional analysis of each storm is performed by considering the wave direction associated to each sea state during the storm. A methodology to classify the “directional storms” pertaining to a certain directional sector is proposed. Finally a technique to determine the main directions of occurrence of the strongest sea storms and the appropriate width of sectors for directional analysis is proposed. Results are useful for different kind of applications such as directional long-term predictions for the design process of angle-dependent structures or for wave energy converter devices. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Significant wave height Wave direction Sea storm
1. Introduction A sea storm is a non-stationary process, with the significant wave height, the wave spectrum and the wave direction not constant in time. It may be defined as a sequence of sea states during which the significant wave height exceeds a fixed constant threshold. In general, a storm starts when the significant wave height goes above this threshold and finishes when it falls below it. Following the definition of Boccotti (2000), (see also Arena and Pavone, 2006, 2009), a sea storm is ‘a sequence of sea states in which the significant wave height exceeds the threshold hcrit and does not fall below it for a continuous time interval greater than 12 h’. The storm threshold hcrit depends upon the considered location, and it may be related to the average significant wave height in the given area. Storms evolve in space and time often changing direction and thus one can choose an Eulerian or a Lagrangian description to model them as discussed by Bernardino et al. (2008). However it is most common to adopt the Eulerian approach, which is the one considered in this paper. Statistical properties of waves during storms were investigated by Borgman (1970, 1973) who determined the cumulative distribution function of the maximum wave height during a storm in an integral form. This result is very important for the long-term
n
Corresponding author. Tel.: þ 39 3355387968; fax: þ 39 09651692260. E-mail address: [email protected] (F. Arena).
http://dx.doi.org/10.1016/j.oceaneng.2015.07.027 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
analysis of extreme waves during storms. For example it is the main concept on which the models of the equivalent storms are based (Arena and Pavone, 2006, 2009; Fedele and Arena, 2010; Arena et al., 2013). They enable to associate to each actual storm an equivalent one defined by means of a parameter representative of storm intensity, which is equal to the maximum significant wave height in the actual storm, and a parameter representative of the storm duration, which is determined by imposing that the maximum expected wave height is the same in the actual and equivalent storms. It is important, for several marine applications, to consider directionality during storms to develop criteria for the prediction of extremes values that take into account the wave direction, and enable to determine the long-term statistics for any directional sector. Recently Jonathan and Ewans (2007) developed an approach to establish appropriate directional criteria and an associated omni-directional criterion. Jonathan et al. (2008) showed that a directional extreme model is generally better than a model that ignores directionality and an omni-directional criterion derived from a directional model are more accurate and should be preferred. In this paper the variability of direction during storm history (which is defined as the significant wave height in time domain during the evolution of the storm) is investigated. Then a criterion to classify “directional storms” pertaining to a certain sector is proposed. Furthermore a methodology for the determination of the center and the width of the directional sector is given. This kind of analysis is useful for several kind of application such as directional long-term predictions, design process of angle-
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V. Laface et al. / Ocean Engineering 107 (2015) 45–53
HIPOCAS(2)
46042 NOAA-NDBC
HIPOCAS(1) RON MAZARA DEL VALLO Fig. 1. (a) Locations of the Mazara Del Vallo buoy from Italian RON network and of the two points from HIPOCAS dataset; (b) location of 46042 buoy from NOAA– NDBC (USA).
dependent devices (Arena et al., 2015), but also to check the correct operation of buoys if data of different kind are compared. The results are of interest to those developing extreme wave criteria for design purposes, as these are often on a storm based approach (Tromans and Vanderschuren, 1995). The methodology proposed has been developed by analyzing wave data in locations that are all in essentially extra-tropical regions where the directional variability in a storm (e.g., Ponce de Leon and Guedes Soares, 2014) is much lower than in a region where tropical storms – hurricane and typhoons – dominate. Different conclusions may be found for such locations and the need of criteria based on different assumptions could be required.
2. Sea storms A sea storm is defined as a sequence of sea states in which the significant wave height Hs exceeds a given threshold. In general, the storm starts when Hs in time domain has an up-crossing related to the given storm threshold and finishes when Hs goes down this threshold. The Boccotti's (2000) definition of sea storm, given in the previous section, admits that a calm period with significant wave height below the threshold hcrit may occur even during the storm evolution. If this period has duration Δt crit smaller than 12 h there is a single storm, otherwise two different storms (note that this value of Δt crit has been proposed for the Mediterranean Sea and the Atlantic and Pacific Oceans; it may change for different locations). The choice of the storm threshold value is done in relation to the characteristics of the considered location. Boccotti (2000) (see also Arena and Puca (2004); Arena and Pavone, 2006, 2009) proposed a
Table 1 Average significant wave height H s , critical threshold hcrit and number of storms with maximum significant wave height H s max Z 2hcrit . Location
H s ðmÞ
hcrit (m)
N1storm (Hs
HIPOCAS(1) HIPOCAS(2) RON MAZARA NDBC 46042
2.40 3.51 0.97 2.21
3.60 5.27 1.45 3.31
137 227 232 34
max 42hcrit)
storm threshold related to the average value of significant wave height H s calculated from time series (note that the average is calculated considering the whole time series of Hs). He assumed a storm threshold equal to 1.5 times H s , but a higher value, for example 2 or 3 times H s may be assumed. It is worth noting that assuming increasing threshold hcrit the number of storms decreases.
2.1. Definition of directional sea storm
To define a “directional storm” pertaining to a certain sector
ϑi 7 Δϑ the variability of wave direction during the storm has to
be investigated. The analysis proposed in this paper (see next section) shows this variability; it is quite high in storm tails (lower sea states), but lower near storm peak (when stronger sea states occur). A directional sea storm may be defined as a sequence of sea states in which the significant wave height exceeds a given threshold hcrit and the wave direction is within a given sector ϑi 7 Δϑ . Because of the strong variability of wave direction in
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
330
14
180 150
6
60
90
90
hcrit
60
Direction
30
0
0 150
120
120
0.5Hsmax
0 0
20
40
60
120
330
0.5Hsmax hcrit
16
14
270
Direction
240
14
210
12
180
10
150
8
360 Hs 0.5Hsmax hcrit Direction
16
300
12 Hs(m)
18
18
360
Hs
HIPOCAS (2)
Direction (°)
22
Hs(m)
100
t(hours)
t(hours)
20
80
330 300 270
240 210
10
180 8
150
120
6
120
6
90
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90
4
60
2
30
0
0 0
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60 2
30
0 0
80
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0 180
t(hours)
t(hours)
7
6
360
360 330
330 6
270
210 180
3
150 120
2
4
Hs(m)
240
Direction (°)
4
300
270
Hs 0.5Hsmax hcrit Direction
5
Hs(m)
5
300
240 210
3
180
150 2
120 90
90
MAZARA DEL VALLO (RON)
30
0
0
0 0
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40
Hs 0.5Hsmax hcrit Direction
1
60
1
50
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60 30 0
50
t(hours)
t(hours)
9
360
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5
180 4
150
3
120
Hs
90
2
hcrit
1 46042-NDBC
0.5Hsmax Direction
0
20
40
60
80
60 30 0 100
t(hours)
Hs(m)
Hs(m)
6
300
270 Direction(°)
7
0
Direction (°)
60
150 Hs
Direction (°)
30
180
270 240 210
5
180 4
150
Direction ( °)
0
210
4
1
30
0
240
2
90
2
270
3
120
4
300
5 Hs(m)
210
Direction (°)
8
240
330
6
270
Hs 0.5Hsmax hcrit Direction
10
360
7
300
12
Hs(m)
8
360
HIPOCAS (1)
Direction (°)
16
47
120
3 Hs 0.5Hsmax hcrit Direction
2
1
90 60 30
0
0
0
10
20
30
40
50
t(hours)
Fig. 2. Some severe storms in the considered locations. Significant wave height Hs and related wave direction during the storms: examples of quite regular (left) and irregular (right) trends of wave direction.
storm tails, it should be appropriate to introduce a directional 0 criterion referring to the wave direction of the sea state above h 0 (where h Z hcrit ).
Consider first the identification of the sea storms from the whole time series of significant wave height (whatever the wave direction is). From this whole set, one may define
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
Average direction (°)
Direction at Hs max(°)
48
HIPOCAS (1) 0
2
4
6
8
10
12
14
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
16
0
2
4
6
8
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
HIPOCAS (2) 2
4
6
8
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
10 12 Hs max(m)
14
16
18
MAZARA DEL VALLO (RON) 0
1
2
3
4
5
6
7
8
9
0
2
4
6
8
10
Direction at Hs max(°)
Average direction (°)
1
Fig. 3. Wave direction ϑHs
2
max
3
4
5 6 Hs max(m)
16
0
1
2
3
4
10 12 Hs max(m)
14
16
18
20
5
6
7
8
9
10
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11
Hs max(m)
46042-NDBC 0
14
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
Hs max(m)
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
12
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
20
Average direction (°)
Direction at Hs max(°)
0
10
Hs max(m)
Average direction (°)
Direction at Hs max(°)
Hs max(m)
7
8
9
10
at maximum significant wave height Hs
max
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0 0
11
versus Hs
sea storm as a “ directional storm” pertaining to the sector (i) a ϑi 7 Δϑ , if the wave direction at the maximum significant wave height H smax falls in ϑi 7 Δϑ ;
max
1
2
3
4
5 6 Hs max(m)
7
8
9
(left), average wave direction during the storm versus Hs
max
(right).
as a “ directional storm” pertaining to the sector (ii) a sea storm ϑi 7 Δϑ , if the average wave direction ϑ ðtÞ, calculated for 0 those sea states with H s 4h , falls in ϑi 7 Δϑ .
V. Laface et al. / Ocean Engineering 107 (2015) 45–53 0
Note that in (ii), if h ¼ hcrit the whole storm is considered. Furthermore definition (ii) is related to the wave direction of sea 0 states above the fixed threshold of significant wave height h . In 0 the paper it is applied for thresholds h Z 0:5H s max , because a strong variability of wave direction occurs in storm tails (lower sea states). Fig. 2, shows some examples of sea storms with wave direction during the sea states which characterize the storms. Concerning to the sector ϑi 7 Δϑ it requires to determine both the direction ϑi and the widthΔϑ. It is worth noting that it looks not so relevant for the present analysis to refer to directional sectors of minor relevance from which a few storms with a low intensity (for the given area) occur. For this reason, it is necessary to identify the main sector, which includes the direction of severe storms and possible secondary sectors (if any) from which the significant storms occur. Thus to classify directional storms first the main and secondary directions have to be identified (by means of directionsϑi ) and then the most appropriateΔϑ has to be determined. Increasing Δϑ it will result an increasing number of storms belonging to any considered sector.
49
height in a sea storm. This probability is obtained starting from the crest-to-trough wave height cumulative distribution in a sea state. Let us consider a sea storm as a sequence of Ns sea states, each of them with significant wave height H s ¼ hi and mean period T i (Rice, 1958), duration Dti and dominant wave direction ϑdi . To generalize the Borgman's results to a directional storm associated to a direction ϑi 7 Δϑ, by assuming the stochastic independence of wave heights, the cumulative distribution function of the maximum crest-to-trough wave height is Ns D =T PðH max o H; ϑi 7 ΔϑÞ ¼ ∏ PðH; H s ¼ hi ; ϑi 7 ΔϑÞ ti i i¼1
where PðH; H s ¼ hi ; ϑi 7 ΔϑÞ represents the probability that a wave has height smaller than H occurs in a sea state with significant wave height Hs equal to hi and with wave direction in the sector ϑi 7 Δϑ, which is given by 8 2 < ¼ 1 exp 4 H 1 þ ψ i n hi PðH; Hs ¼ hi ; ϑi 7 ΔϑÞ ¼ : ¼1
if
ϑi Δϑ o ϑdi r ϑi þ Δϑ otherwise
ð1Þ where ϑdi is the dominant wave direction of the i sea state and ψ ni is the narrow bandedness parameter of the spectrum (Boccotti, 2000, 2014) in the sea sate and it is equal to the ratio, in absolute value, between the absolute minimum and the absolute maximum of the covariance function of the sea state. It is equal to 1 for an infinitely narrow spectrum [note that in this condition, Eq. (1) th
2.2. Statistical properties of waves in a directional sea storm
HIPOCAS (2)
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
Direction at Hs max(°)
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
Average direction (°)
Average direction (°)
MAZARA DEL VALLO (RON)
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
46042-NDBC
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
Direction at Hs max(°)
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
HIPOCAS (1)
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
Direction at Hs max(°)
Direction at Hs max(°)
The statistical properties of waves during storms were investigated by Borgman (1970, 1973), who obtained the cumulative distribution function PðH max o HÞ of the extreme individual wave
Average direction (°)
Average direction (°)
Fig. 4. Wave direction ϑHs
max
at maximum significant wave height Hs
max
versus average wave direction during the whole storm.
50
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
40
25 20
Whole storm Storm above 0.5 Hsmax Storm above 0.6Hsmax Storm above 0.7Hsmax Storm above 0.8Hsmax
HIPOCAS (1)
15 10 5
Standard deviation of direction (°)
Standard deviation of direction (°)
30
HIPOCAS (1)
35 30 25 20 15 10 5 0
0 7-8
8-9
9-10
10-11
11-12
12-13
10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-21 22-23
13-14
Hs max(m)
Hs max (m) 50 MAZARA DEL VALLO (RON)
45 40 35 30 25 20
15 10 5
Standard deviation of direction (°)
Standard deviation of direction (°)
50
46042-NDBC
45 40
35 30 25 20 15 10 5 0
0 2-3
3-4
4-5 Hs max(m)
5-6
6-7
6-7
7-8
8-9
9-10
Hs max (m)
Fig. 5. Average value of standard deviation of wave direction calculated (for any storm with Hs max Z 2hcrit) for the whole storm and above a fixed threshold of significant wave height Hs (from 0.5 Hs max to 0.8 Hs max). Each istogram represents storms with maximum significant wave height Hs max in fixed ranges of Hs.
Table 2 Number of directional storms pertaining to directional sector ϑ1 7 Δϑ, classified by means of criteria 2.1 (i) and 2.1 (ii) at HIPOCAS (1).
Table 3 Number of directional storms pertaining to directional sectors ϑ1;2;3 7 Δϑ, classified by means of criteria 2.1 (i) and 2.1 (ii) at HIPOCAS (2).
ϑ1¼ 3051
Δϑ ¼ 101
Δϑ ¼201
Δϑ ¼ 301
ϑ1 ¼ 801
Δϑ ¼101
Δϑ ¼201
Δϑ ¼301
(i) 0 ðiiÞ h ¼ 0:5H s max 0 ðiiÞ h ¼ 0:6H s max 0 ðiiÞ h ¼ 0:7H s max 0 ðiiÞ h ¼ 0:8H s max
52 66 59 58 53
97 99 96 99 100
121 121 122 120 120
(i) 0 ðiiÞ h ¼ 0:5Hs max 0 ðiiÞ h ¼ 0:6Hs max 0 ðiiÞ h ¼ 0:7Hs max 0 ðiiÞ h ¼ 0:8Hs max ϑ2 ¼ 1901 (i) 0 ðiiÞh ¼ 0:5H s max 0 ðiiÞ h ¼ 0:6Hs max 0 ðiiÞ h ¼ 0:7Hs max 0 ðiiÞ h ¼ 0:8Hs max ϑ3 ¼ 3601 (i) 0 ðiiÞ h ¼ 0:5Hs max 0 ðiiÞ h ¼ 0:6Hs max 0 ðiiÞ h ¼ 0:7Hs max 0 ðiiÞ h ¼ 0:8Hs max
80 82 76 80 79 Δϑ ¼101 10 8 8 9 9 Δϑ ¼101 1 0 0 1 2
132 143 139 143 133 Δϑ ¼201 13 12 12 13 15 Δϑ ¼201 1 3 4 7 10
169 168 167 169 167 Δϑ ¼301 15 14 15 14 15 Δϑ ¼301 3 4 7 11 14
gives the Rayleigh distribution, to 0.73 for a mean JONSWAP spectrum (Hasselmann et al., 1973) and to 0.65 for a PiersonMoskowitz (1964) spectrum]. Then, the cumulative distribution function PðH max o H; ϑi 7 ΔϑÞ, following the Borgman's logic, is written in an integral form as
Z D 1 P H; H s ¼ hðtÞ; ϑi 7 ΔϑÞ ln PðH max o H; ϑi 7 ΔϑÞ ¼ exp dt T ½hðtÞ 0 ð2Þ where D is the storm duration. Finally, the maximum expected wave height H max during the sea storm is obtained as the integral over ð0; 1Þ of the probability of exceedance 1 PðH max o H; ϑi 7 ΔϑÞ. 3. Wave data analysis In the paper several kinds of data are considered: buoys data from both NOAA–NDBC (National Oceanic and Atmospheric Administration's – National Data Buoy Center, USA) and Italian buoys network (RON – Rete Ondametrica Nazionale – managed by ISPRA), and wave model data from HIPOCAS project (Guedes Soares et al., 2002; Pilar et al., 2008).
The NOAA manages the NDBC, which consists of many buoys moored along the US coasts, both in the Pacific Ocean and in the Atlantic Ocean. Some buoys were moored in the late 1970s, so that more than 35 years of data are available. The historical wave data give hourly significant wave height, peak and mean period. Only few buoys are directional. In the paper the NOAA directional buoy 46042 (see Fig. 1b), moored off California, is considered. The Italian buoys network (RON) started measurements in 1989, with 8 directional buoys located off the coasts of Italy. Currently, the network consists of 15 buoys, moored in deep water. RON buoys give up to two records per hour; for each record, the data of significant wave height, peak and mean period and
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
dominant direction are given. In the Central Mediterranean Sea, the buoy of Mazara del Vallo is considered (see Fig. 1a). The HIPOCAS project (Hindcast of Dynamic processes of the Ocean and Coastal Areas of Europe) provided a simulation of 44years (1958–2001) wind, waves, sea level data and current climatology. The hindcast wave model used in HIPOCAS is the third generation wave model WAM cycle 4 modified for two-way nesting by Gómez Lahoz and Carretero Albiach (1997), which gives the following output parameters: significant wave height Hs, wave direction, mean period Tm, peak period Tp with a time step of three hours. From HIPOCAS two points in Atlantic Ocean are considered (see Fig. 1a). The paper proposes a directional analysis of sea storms carried out by processing wave data coming from directional buoys of RON and NOAA networks, and by HIPOCAS project. The two locations in North-East Atlantic Ocean from HIPOCAS (Fig. 1a), the one in central Mediterranean Sea (Fig. 1a) from RON network and the one in US coast in Pacific Ocean (Fig. 1b) from NOAA– NDBC are considered. The analysis is performed by processing significant wave height and wave direction time series. At the first stage sea storms are extrapolated from significant wave height time series by means of the definition given in Section (2) which does not take into account the wave direction, but it is only related to the average significant wave height at the considered site (see Table 1). Then the directional analysis is done by considering the sequence of sea states during each sea storm with the sequence of the related wave directions. Fig. 2 shows significant storms for each location, with the directions of the related sea states. Firstly, only the wave direction associated to the peak of the storm Hs max is regarded. Fig. 3 (left) shows the wave direction ϑHs max at the sea state with the maximum significant wave height Hs max (storm peak) versus Hs max (each point represents a storm). From the figure it is clear that at each considered site it is possible to identify one or more main directions from which the strongest storms occurs. The statistical plot in Fig. 3 (left side) enables to identify in a simplified manner the main, the secondary and eventually tertiary directions considering the direction at storms peak. The main direction is the one from which the severest sea storms occur. There are sites like HIPOCAS (1) where there is only one relevant direction, but in some cases it is possible to have more relevant directions like at RON and HIPOCAS (2) sites. In such cases it is needed to define the relevance of each direction and they are classified as main ϑ1 , secondary ϑ2 and tertiary ϑ3 in relation to the severity of the storms coming from the considered direction. In general to identify the main directions different approaches may be adopted. For instance, the energy flux may be calculated for any direction and the relevance of each of them may be established in terms of the contribution of the energy flux to the Table 4 Number of directional storms pertaining to directional sectors ϑ1;2 7 Δϑ, classified by means of criteria 2.1 (i) and 2.1 (ii) at Mazara del Vallo buoy (RON). ϑ1 ¼2801
Δϑ ¼ 101
Δϑ ¼ 201
Δϑ ¼ 301
(i) 0 ðiiÞ h ¼ 0:5H s max 0 ðiiÞ h ¼ 0:6H s max 0 ðiiÞ h ¼ 0:7H s max 0 ðiiÞ h ¼ 0:8H s max ϑ2 ¼1401 (i) 0 ðiiÞ h ¼ 0:5H s max 0 ðiiÞ h ¼ 0:6H s max 0 ðiiÞ h ¼ 0:7H s max 0 ðiiÞ h ¼ 0:8H s max
107 95 105 101 108 Δϑ ¼ 101 28 28 28 28 25
126 116 120 119 129 Δϑ ¼ 201 49 39 41 44 42
137 129 131 129 136 Δϑ ¼ 301 50 41 43 46 49
51
considered direction. Another approach to determine ϑ1 , ϑ2 , ϑ3 is the one proposed by Boccotti (2000) (see also Arena and Fedele, 2002; Arena et al., 2013) by means of cumulative probability distributions functions of significant wave height. For the analysis proposed in the paper the main directions are determined from Fig. 3. In particular, at HIPOCAS (1) the main direction is assumed ϑ1 ¼ 3051, at RON the main and the secondary directions are assumed ϑ1 ¼ 2801, ϑ2 ¼ 1401 respectively. At HIPOCAS (2) three relevant directions are identified ϑ1 ¼ 801,ϑ2 ¼ 1901,ϑ3 ¼ 01. At NOAA 46042 is assumed ϑ1 ¼3001. Subsequently the average wave direction during the whole storm is calculated starting from wave direction of all sea states (Fig. 3 right). The comparison between the average direction and the direction at storm peak is shown in Fig. 4: these two directions are very close to each other for most of the storms, but in some cases differences are observed. Then, a deeper analysis is proposed by considering the strongest storms (Hs max Z2hcrit), by calculating the standard deviation of wave direction and by considering its average values for classes of storm intensity Hs max. The same calculation is done for the whole storm history first and then by considering only sea states 0 above a fixed threshold h (from 0.5Hs max to 0.8Hs max) of significant wave height. Fig. 5 shows the average value of the standard deviation of direction versus any range of H s max . The results show that during the storms a certain variability of wave direction is observed: it is quite relevant if the whole storm history is considered; if we consider only sea states above a fixed thresh0 old h of significant wave height, the general trend (from lowest to 0 highest h ) shows that standard deviation decreases for increasing 0 values of h . The most relevant reduction is observed by comparing the results for the whole storms and for the storms above 0.5Hs max (see Fig. 5). This result is due to the strong variability of wave direction in storm tails (smaller sea states). Starting from the above results the definitions (i) and (ii) in Section 2.1 of directional storm are introduced and applied to classify the identified storms (without any condition of wave direction) as directional storms pertaining to a given directional sector ϑi 7 Δϑ . The direction ϑi is fixed as the wave direction from which the severest storm occurs and for the width Δϑ values of 101, 201, 301 are considered. It is worth noting that for this kind of analysis a too narrow sector (Δϑ less than 101) does not enable to develop criteria on wave direction during sea storms because of the wave direction variability. On the other hand, considering Δϑ greater than 301 the scope of directional analysis became meaningless. Tables 2–5 show the number of storms classified by means of (i) and (ii) (in Section 2.1) as directional storms in the directional sector ϑi 7 Δϑ for each of the main directions ϑi identified, for 0 each considered width Δϑ and for h ¼ 0:5H s max , 0.6Hs max, 0.7Hs , 0.8H . The number of directional storms pertaining to a s max max given directional sector ϑi 7 Δϑ increases for increasing values 0 of the width Δϑ both applying definition (i) and (ii) whatever is h . Furthermore definitions (i) and (ii) tend to give the same result if widths of 301 are considered. For what concerns results obtained 0 applying definition (ii) assuming increasing h it has been seen Table 5 Number of directional storms pertaining to directional sector ϑ1 7 Δϑ, classified by means of criteria 2.1 (i) and 2.1 (ii) at 46042 buoy (NOAA–NDBC). ϑ1 ¼ 3001
Δϑ ¼101
Δϑ ¼201
Δϑ ¼ 301
(i) 0 ðiiÞ h ¼ 0:5Hs max 0 ðiiÞ h ¼ 0:6Hs max 0 ðiiÞ h ¼ 0:7Hs max 0 ðiiÞ h ¼ 0:8Hs max
13 15 13 13 13
22 24 26 26 23
26 27 27 28 28
52
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
Fig. 6. Maximum expected wave height H max for directional storms pertaining to the directional sector ϑ1 7 Δϑ versus maximum expected wave height H max for storm selected without condition about wave direction, for the storms with Hs max Z 2hcrit at HIPOCAS (1).
Fig. 7. Maximum expected wave height H max for directional storms pertaining to the directional sectors ϑ1 7 Δϑ (top) and ϑ2 7 Δϑ (bottom) versus maximum expected wave height H max for storm selected without condition about wave direction, for the storms with Hs max Z 2hcrit at Mazara Del Vallo (RON).
that there are no significant variations of the number of directional storms if Δϑ is assumed equal to 301, especially for the main and the secondary sectors. For this reason it could be more appropriate 0 to utilize definition (ii) with h ¼ 0:5H s max in order to take into account a larger number of sea states. It is worth noting that the main purpose of the analysis proposed here is related to the introduction of a physically based approach which enables us to associate each storm to a given directional sector. Consider the storms with maximum significant wave height Hs max Z2hcrit (see Table 1), by applying definition (i) or (ii) with a width Δϑ of 301 and h' ¼ 0:5H s max it is possible to classify more than 80% of the storms as directional storms at both HIPOCAS (1) , HIPOCAS (2) and at 46042-NDBC, and more than 70% at Mazara del Vallo RON. Finally to test the validity of the criterion the maximum expected wave heights H max calculated for the whole storm (storm selected without any condition on wave direction) and for directional storm are calculated and compared
(see Figs. 6 and 7). Results show that the maximum expected wave height of directional storm is less or equal to the one calculated without condition on wave direction (whole storm). This could leads for example to a reduction of design wave height for angledependent devices. It is worth noting that for increasing width Δϑ the maximum expected wave height of the whole storm and of the directional storm tend to converge.
4. Conclusions The paper has investigated the variability of wave direction during sea storms, showing how it is large for lower sea states (storm tails) and it is reduced by considering the wave direction associated to sea states above increasing thresholds of significant wave height (storm peak). A definition of directional sea storm pertaining to a given directional sector is introduced. The
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
comparison between direction at Hs max (maximum significant wave height during the storm) and average wave direction during storm has revealed that for most of the storms these two directions are similar. It has been found that the number of directional storms pertaining to a given directional sector increases for increasing values of the width of the directional sector. Finally it has been proved that an appropriate amplitude for the directional sectors may be of 601 because it enables to classify as directional storms most of the storms. Acknowledgments The data for this paper are available at NOAA and at idromare.it websites. The work has been partially supported by project LARGE MULTIPURPOSE PLATFORMS FOR EXPLOITING RENEWABLE ENERGY IN OPEN SEAS (PLENOSE) funded by EU IRSES Marie Curie action (project PIRSES-GA-2013-612581). The PhD programme of Valentina Laface is supported by European Commission, European Social Fund (FSE) and Regione Calabria via the scholarship International mobility for young graduates and researchers – POR Calabria FSE 20011/2013 Intervento D.5 – Operational Target M2 Encouraging individual highlevel education Programmes for young Graduates and Researchers in Eminent National and International Research Centers. References Arena, F., Pavone, D., 2006. Return period of nonlinear high wave crests. J. Geophys. Res. 111, C08004. Arena, F., Pavone, D., 2009. A generalized approach for the long-term modelling of extreme sea waves. Ocean Model. 26, 217–225. Arena, F., and Fedele, F., 2002. Intensity and duration of sea storms off the Californian coast. In: Proceedings of Solutions to Coastal Disasters'02 of ASCE (the American Society of Civil Engineers), San Diego, 24–27 February, pp. 126– 141. Arena, F., Puca, S., 2004. The reconstruction of significant wave height time series by using a neural network approach. ASME J. Offshore Mech. Arct. Eng. 126, 213–219.
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Arena, F., Malara, G., Barbaro, G., Romolo, A., Ghiretti, S., 2013. Long-term modelling of wave run-up and overtopping during sea storms. J. Coast. Res. 29, 419–429. Arena, F., Laface, V., Malara, G., Romolo, A., Viviano, A., Fiamma, V., Sannino, G., Carillo, A., 2015. Wave climate analysis for the design of wave energy harvesters in the Mediterranean Sea. Renew. Energy 77, 125–141. Bernardino, M., Boukhanovsky, A., and Guedes Soares, C., 2008. Alternative Approaches to Storm Statistics in The Ocean. In: Proceedings of the 27th International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2008); Estoril, Portugal. New York, USA: ASME; OMAE2008-58053. Boccotti, P., 2000. Wave Mechanics for Ocean Engineering. Elsevier Science, New York. Boccotti, P., 2014. Wave Mechanics and Wave Loads on Marine Structures. Elsevier Science, Butterworth-Heinemann, Oxford. Borgman, L. E., 1970. Maximum wave height probabilities for a random number of random intensity storms. In: Proceedings of the 12th Conference of Coastal Engineering. Washington, DC, ASCE, 53–64. Borgman, L.E., 1973. Probabilities for highest wave in hurricane. J. Waterw. Harb. Coast. Eng. Div. Am. Soc. Civ. Eng. 99, 185–207. Fedele, F., Arena, F., 2010. The equivalent power storm model for long-term predictions of extreme wave events. J. Phys. Oceanogr. 40, 1106–1117. Gómez Lahoz, M., Carretero Albiach, J.C., 1997. A two-way nesting procedure for the WAM model: application to the Spanish coast. J. Offshore Mech. Arct. Eng. 119, 20–24. Guedes Soares, C., Weisse, R., Carretero, J.C., and E. Alvarez, 2002. A 40 years hindcast of wind, sea level and waves in European waters. In: Proceedings of the 21st International Conference on Offshore Mechanics and Arctic Engineering (OMAE'02), ASME Paper OMAE2002-SR28604. Hasselmann, K., Barnett, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, E., Ewing, J.A., Gienapp, H., Hasselmann, D.E., Krusemann, P., Meerburg, A., Müller, P., Olbers, D.J., Richter, K., Sell, W., Walden, H., 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z A12, 1–95. Jonathan, P., Ewans, K., 2007. The effect of directionality on extreme waves design criteria. Ocean Eng. 34, 1977–1994. Jonathan, P., Ewans, K., Forristall, G., 2008. Statistical estimation of extreme ocean environments: the requirements for modelling directionality and other covariate effects. Ocean Eng. 35, 1211–1225. Pierson, W.J., Moskowitz, L.A., 1964. A proposed spectral form for fully developed waves based on the similarity theory of S. A. Kitaigorodskii. J. Geophys. Res. 69, 5181–5190. Pilar, P., Guedes Soares, C., Carretero, J.C., 2008. Hindcast for the North East Atlantic European coast. Coast. Eng. 55, 861–871. Ponce de Leon, S., Guedes Soares, C., 2014. Extreme wave parameters under North Atlantic extratropical hurricanes. Ocean Model. 81, 78–88. Rice, 1958. Distribution of the duration of fades in ration transmission. Bell Syst. Tech. J. 37, 581–635. Tromans, P. S., and Vanderschuren, L., 1995. Response based design conditions in the North Sea: Application of a new method. OTC 7683, 27th OTC, Houston.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Directional analysis of sea storms V. Laface a,b, F. Arena a,n, C. Guedes Soares b a b
Natural Ocean Engineering Laboratory, DICEAM, Mediterranea University, Reggio Calabria, Italy Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Tecnico, Universidade de Lisboa, Portugal
art ic l e i nf o
a b s t r a c t
Article history: Received 24 February 2015 Accepted 19 July 2015
The paper deals with the directional analysis of sea storms in the Atlantic and Pacific Oceans and in the Mediterranean Sea. The main focus of the work is to investigate the variability of wave directions during sea storms. The analysis is carried out starting from significant wave height and wave direction time series. At the first stage storms are selected from time series without conditions on wave direction. Subsequently the directional analysis of each storm is performed by considering the wave direction associated to each sea state during the storm. A methodology to classify the “directional storms” pertaining to a certain directional sector is proposed. Finally a technique to determine the main directions of occurrence of the strongest sea storms and the appropriate width of sectors for directional analysis is proposed. Results are useful for different kind of applications such as directional long-term predictions for the design process of angle-dependent structures or for wave energy converter devices. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Significant wave height Wave direction Sea storm
1. Introduction A sea storm is a non-stationary process, with the significant wave height, the wave spectrum and the wave direction not constant in time. It may be defined as a sequence of sea states during which the significant wave height exceeds a fixed constant threshold. In general, a storm starts when the significant wave height goes above this threshold and finishes when it falls below it. Following the definition of Boccotti (2000), (see also Arena and Pavone, 2006, 2009), a sea storm is ‘a sequence of sea states in which the significant wave height exceeds the threshold hcrit and does not fall below it for a continuous time interval greater than 12 h’. The storm threshold hcrit depends upon the considered location, and it may be related to the average significant wave height in the given area. Storms evolve in space and time often changing direction and thus one can choose an Eulerian or a Lagrangian description to model them as discussed by Bernardino et al. (2008). However it is most common to adopt the Eulerian approach, which is the one considered in this paper. Statistical properties of waves during storms were investigated by Borgman (1970, 1973) who determined the cumulative distribution function of the maximum wave height during a storm in an integral form. This result is very important for the long-term
n
Corresponding author. Tel.: þ 39 3355387968; fax: þ 39 09651692260. E-mail address: [email protected] (F. Arena).
http://dx.doi.org/10.1016/j.oceaneng.2015.07.027 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
analysis of extreme waves during storms. For example it is the main concept on which the models of the equivalent storms are based (Arena and Pavone, 2006, 2009; Fedele and Arena, 2010; Arena et al., 2013). They enable to associate to each actual storm an equivalent one defined by means of a parameter representative of storm intensity, which is equal to the maximum significant wave height in the actual storm, and a parameter representative of the storm duration, which is determined by imposing that the maximum expected wave height is the same in the actual and equivalent storms. It is important, for several marine applications, to consider directionality during storms to develop criteria for the prediction of extremes values that take into account the wave direction, and enable to determine the long-term statistics for any directional sector. Recently Jonathan and Ewans (2007) developed an approach to establish appropriate directional criteria and an associated omni-directional criterion. Jonathan et al. (2008) showed that a directional extreme model is generally better than a model that ignores directionality and an omni-directional criterion derived from a directional model are more accurate and should be preferred. In this paper the variability of direction during storm history (which is defined as the significant wave height in time domain during the evolution of the storm) is investigated. Then a criterion to classify “directional storms” pertaining to a certain sector is proposed. Furthermore a methodology for the determination of the center and the width of the directional sector is given. This kind of analysis is useful for several kind of application such as directional long-term predictions, design process of angle-
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V. Laface et al. / Ocean Engineering 107 (2015) 45–53
HIPOCAS(2)
46042 NOAA-NDBC
HIPOCAS(1) RON MAZARA DEL VALLO Fig. 1. (a) Locations of the Mazara Del Vallo buoy from Italian RON network and of the two points from HIPOCAS dataset; (b) location of 46042 buoy from NOAA– NDBC (USA).
dependent devices (Arena et al., 2015), but also to check the correct operation of buoys if data of different kind are compared. The results are of interest to those developing extreme wave criteria for design purposes, as these are often on a storm based approach (Tromans and Vanderschuren, 1995). The methodology proposed has been developed by analyzing wave data in locations that are all in essentially extra-tropical regions where the directional variability in a storm (e.g., Ponce de Leon and Guedes Soares, 2014) is much lower than in a region where tropical storms – hurricane and typhoons – dominate. Different conclusions may be found for such locations and the need of criteria based on different assumptions could be required.
2. Sea storms A sea storm is defined as a sequence of sea states in which the significant wave height Hs exceeds a given threshold. In general, the storm starts when Hs in time domain has an up-crossing related to the given storm threshold and finishes when Hs goes down this threshold. The Boccotti's (2000) definition of sea storm, given in the previous section, admits that a calm period with significant wave height below the threshold hcrit may occur even during the storm evolution. If this period has duration Δt crit smaller than 12 h there is a single storm, otherwise two different storms (note that this value of Δt crit has been proposed for the Mediterranean Sea and the Atlantic and Pacific Oceans; it may change for different locations). The choice of the storm threshold value is done in relation to the characteristics of the considered location. Boccotti (2000) (see also Arena and Puca (2004); Arena and Pavone, 2006, 2009) proposed a
Table 1 Average significant wave height H s , critical threshold hcrit and number of storms with maximum significant wave height H s max Z 2hcrit . Location
H s ðmÞ
hcrit (m)
N1storm (Hs
HIPOCAS(1) HIPOCAS(2) RON MAZARA NDBC 46042
2.40 3.51 0.97 2.21
3.60 5.27 1.45 3.31
137 227 232 34
max 42hcrit)
storm threshold related to the average value of significant wave height H s calculated from time series (note that the average is calculated considering the whole time series of Hs). He assumed a storm threshold equal to 1.5 times H s , but a higher value, for example 2 or 3 times H s may be assumed. It is worth noting that assuming increasing threshold hcrit the number of storms decreases.
2.1. Definition of directional sea storm
To define a “directional storm” pertaining to a certain sector
ϑi 7 Δϑ the variability of wave direction during the storm has to
be investigated. The analysis proposed in this paper (see next section) shows this variability; it is quite high in storm tails (lower sea states), but lower near storm peak (when stronger sea states occur). A directional sea storm may be defined as a sequence of sea states in which the significant wave height exceeds a given threshold hcrit and the wave direction is within a given sector ϑi 7 Δϑ . Because of the strong variability of wave direction in
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
330
14
180 150
6
60
90
90
hcrit
60
Direction
30
0
0 150
120
120
0.5Hsmax
0 0
20
40
60
120
330
0.5Hsmax hcrit
16
14
270
Direction
240
14
210
12
180
10
150
8
360 Hs 0.5Hsmax hcrit Direction
16
300
12 Hs(m)
18
18
360
Hs
HIPOCAS (2)
Direction (°)
22
Hs(m)
100
t(hours)
t(hours)
20
80
330 300 270
240 210
10
180 8
150
120
6
120
6
90
4
90
4
60
2
30
0
0 0
20
40
60
60 2
30
0 0
80
30
60
90
120
150
0 180
t(hours)
t(hours)
7
6
360
360 330
330 6
270
210 180
3
150 120
2
4
Hs(m)
240
Direction (°)
4
300
270
Hs 0.5Hsmax hcrit Direction
5
Hs(m)
5
300
240 210
3
180
150 2
120 90
90
MAZARA DEL VALLO (RON)
30
0
0
0 0
10
20
30
40
Hs 0.5Hsmax hcrit Direction
1
60
1
50
60
70
0
80
10
20
30
40
60 30 0
50
t(hours)
t(hours)
9
360
9
330
8
360 330
8
300
7
240
6
210
5
180 4
150
3
120
Hs
90
2
hcrit
1 46042-NDBC
0.5Hsmax Direction
0
20
40
60
80
60 30 0 100
t(hours)
Hs(m)
Hs(m)
6
300
270 Direction(°)
7
0
Direction (°)
60
150 Hs
Direction (°)
30
180
270 240 210
5
180 4
150
Direction ( °)
0
210
4
1
30
0
240
2
90
2
270
3
120
4
300
5 Hs(m)
210
Direction (°)
8
240
330
6
270
Hs 0.5Hsmax hcrit Direction
10
360
7
300
12
Hs(m)
8
360
HIPOCAS (1)
Direction (°)
16
47
120
3 Hs 0.5Hsmax hcrit Direction
2
1
90 60 30
0
0
0
10
20
30
40
50
t(hours)
Fig. 2. Some severe storms in the considered locations. Significant wave height Hs and related wave direction during the storms: examples of quite regular (left) and irregular (right) trends of wave direction.
storm tails, it should be appropriate to introduce a directional 0 criterion referring to the wave direction of the sea state above h 0 (where h Z hcrit ).
Consider first the identification of the sea storms from the whole time series of significant wave height (whatever the wave direction is). From this whole set, one may define
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
Average direction (°)
Direction at Hs max(°)
48
HIPOCAS (1) 0
2
4
6
8
10
12
14
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
16
0
2
4
6
8
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
HIPOCAS (2) 2
4
6
8
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
10 12 Hs max(m)
14
16
18
MAZARA DEL VALLO (RON) 0
1
2
3
4
5
6
7
8
9
0
2
4
6
8
10
Direction at Hs max(°)
Average direction (°)
1
Fig. 3. Wave direction ϑHs
2
max
3
4
5 6 Hs max(m)
16
0
1
2
3
4
10 12 Hs max(m)
14
16
18
20
5
6
7
8
9
10
10
11
Hs max(m)
46042-NDBC 0
14
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
Hs max(m)
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
12
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
20
Average direction (°)
Direction at Hs max(°)
0
10
Hs max(m)
Average direction (°)
Direction at Hs max(°)
Hs max(m)
7
8
9
10
at maximum significant wave height Hs
max
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0 0
11
versus Hs
sea storm as a “ directional storm” pertaining to the sector (i) a ϑi 7 Δϑ , if the wave direction at the maximum significant wave height H smax falls in ϑi 7 Δϑ ;
max
1
2
3
4
5 6 Hs max(m)
7
8
9
(left), average wave direction during the storm versus Hs
max
(right).
as a “ directional storm” pertaining to the sector (ii) a sea storm ϑi 7 Δϑ , if the average wave direction ϑ ðtÞ, calculated for 0 those sea states with H s 4h , falls in ϑi 7 Δϑ .
V. Laface et al. / Ocean Engineering 107 (2015) 45–53 0
Note that in (ii), if h ¼ hcrit the whole storm is considered. Furthermore definition (ii) is related to the wave direction of sea 0 states above the fixed threshold of significant wave height h . In 0 the paper it is applied for thresholds h Z 0:5H s max , because a strong variability of wave direction occurs in storm tails (lower sea states). Fig. 2, shows some examples of sea storms with wave direction during the sea states which characterize the storms. Concerning to the sector ϑi 7 Δϑ it requires to determine both the direction ϑi and the widthΔϑ. It is worth noting that it looks not so relevant for the present analysis to refer to directional sectors of minor relevance from which a few storms with a low intensity (for the given area) occur. For this reason, it is necessary to identify the main sector, which includes the direction of severe storms and possible secondary sectors (if any) from which the significant storms occur. Thus to classify directional storms first the main and secondary directions have to be identified (by means of directionsϑi ) and then the most appropriateΔϑ has to be determined. Increasing Δϑ it will result an increasing number of storms belonging to any considered sector.
49
height in a sea storm. This probability is obtained starting from the crest-to-trough wave height cumulative distribution in a sea state. Let us consider a sea storm as a sequence of Ns sea states, each of them with significant wave height H s ¼ hi and mean period T i (Rice, 1958), duration Dti and dominant wave direction ϑdi . To generalize the Borgman's results to a directional storm associated to a direction ϑi 7 Δϑ, by assuming the stochastic independence of wave heights, the cumulative distribution function of the maximum crest-to-trough wave height is Ns D =T PðH max o H; ϑi 7 ΔϑÞ ¼ ∏ PðH; H s ¼ hi ; ϑi 7 ΔϑÞ ti i i¼1
where PðH; H s ¼ hi ; ϑi 7 ΔϑÞ represents the probability that a wave has height smaller than H occurs in a sea state with significant wave height Hs equal to hi and with wave direction in the sector ϑi 7 Δϑ, which is given by 8 2 < ¼ 1 exp 4 H 1 þ ψ i n hi PðH; Hs ¼ hi ; ϑi 7 ΔϑÞ ¼ : ¼1
if
ϑi Δϑ o ϑdi r ϑi þ Δϑ otherwise
ð1Þ where ϑdi is the dominant wave direction of the i sea state and ψ ni is the narrow bandedness parameter of the spectrum (Boccotti, 2000, 2014) in the sea sate and it is equal to the ratio, in absolute value, between the absolute minimum and the absolute maximum of the covariance function of the sea state. It is equal to 1 for an infinitely narrow spectrum [note that in this condition, Eq. (1) th
2.2. Statistical properties of waves in a directional sea storm
HIPOCAS (2)
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
Direction at Hs max(°)
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
Average direction (°)
Average direction (°)
MAZARA DEL VALLO (RON)
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
46042-NDBC
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
Direction at Hs max(°)
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
HIPOCAS (1)
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
Direction at Hs max(°)
Direction at Hs max(°)
The statistical properties of waves during storms were investigated by Borgman (1970, 1973), who obtained the cumulative distribution function PðH max o HÞ of the extreme individual wave
Average direction (°)
Average direction (°)
Fig. 4. Wave direction ϑHs
max
at maximum significant wave height Hs
max
versus average wave direction during the whole storm.
50
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
40
25 20
Whole storm Storm above 0.5 Hsmax Storm above 0.6Hsmax Storm above 0.7Hsmax Storm above 0.8Hsmax
HIPOCAS (1)
15 10 5
Standard deviation of direction (°)
Standard deviation of direction (°)
30
HIPOCAS (1)
35 30 25 20 15 10 5 0
0 7-8
8-9
9-10
10-11
11-12
12-13
10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-21 22-23
13-14
Hs max(m)
Hs max (m) 50 MAZARA DEL VALLO (RON)
45 40 35 30 25 20
15 10 5
Standard deviation of direction (°)
Standard deviation of direction (°)
50
46042-NDBC
45 40
35 30 25 20 15 10 5 0
0 2-3
3-4
4-5 Hs max(m)
5-6
6-7
6-7
7-8
8-9
9-10
Hs max (m)
Fig. 5. Average value of standard deviation of wave direction calculated (for any storm with Hs max Z 2hcrit) for the whole storm and above a fixed threshold of significant wave height Hs (from 0.5 Hs max to 0.8 Hs max). Each istogram represents storms with maximum significant wave height Hs max in fixed ranges of Hs.
Table 2 Number of directional storms pertaining to directional sector ϑ1 7 Δϑ, classified by means of criteria 2.1 (i) and 2.1 (ii) at HIPOCAS (1).
Table 3 Number of directional storms pertaining to directional sectors ϑ1;2;3 7 Δϑ, classified by means of criteria 2.1 (i) and 2.1 (ii) at HIPOCAS (2).
ϑ1¼ 3051
Δϑ ¼ 101
Δϑ ¼201
Δϑ ¼ 301
ϑ1 ¼ 801
Δϑ ¼101
Δϑ ¼201
Δϑ ¼301
(i) 0 ðiiÞ h ¼ 0:5H s max 0 ðiiÞ h ¼ 0:6H s max 0 ðiiÞ h ¼ 0:7H s max 0 ðiiÞ h ¼ 0:8H s max
52 66 59 58 53
97 99 96 99 100
121 121 122 120 120
(i) 0 ðiiÞ h ¼ 0:5Hs max 0 ðiiÞ h ¼ 0:6Hs max 0 ðiiÞ h ¼ 0:7Hs max 0 ðiiÞ h ¼ 0:8Hs max ϑ2 ¼ 1901 (i) 0 ðiiÞh ¼ 0:5H s max 0 ðiiÞ h ¼ 0:6Hs max 0 ðiiÞ h ¼ 0:7Hs max 0 ðiiÞ h ¼ 0:8Hs max ϑ3 ¼ 3601 (i) 0 ðiiÞ h ¼ 0:5Hs max 0 ðiiÞ h ¼ 0:6Hs max 0 ðiiÞ h ¼ 0:7Hs max 0 ðiiÞ h ¼ 0:8Hs max
80 82 76 80 79 Δϑ ¼101 10 8 8 9 9 Δϑ ¼101 1 0 0 1 2
132 143 139 143 133 Δϑ ¼201 13 12 12 13 15 Δϑ ¼201 1 3 4 7 10
169 168 167 169 167 Δϑ ¼301 15 14 15 14 15 Δϑ ¼301 3 4 7 11 14
gives the Rayleigh distribution, to 0.73 for a mean JONSWAP spectrum (Hasselmann et al., 1973) and to 0.65 for a PiersonMoskowitz (1964) spectrum]. Then, the cumulative distribution function PðH max o H; ϑi 7 ΔϑÞ, following the Borgman's logic, is written in an integral form as
Z D 1 P H; H s ¼ hðtÞ; ϑi 7 ΔϑÞ ln PðH max o H; ϑi 7 ΔϑÞ ¼ exp dt T ½hðtÞ 0 ð2Þ where D is the storm duration. Finally, the maximum expected wave height H max during the sea storm is obtained as the integral over ð0; 1Þ of the probability of exceedance 1 PðH max o H; ϑi 7 ΔϑÞ. 3. Wave data analysis In the paper several kinds of data are considered: buoys data from both NOAA–NDBC (National Oceanic and Atmospheric Administration's – National Data Buoy Center, USA) and Italian buoys network (RON – Rete Ondametrica Nazionale – managed by ISPRA), and wave model data from HIPOCAS project (Guedes Soares et al., 2002; Pilar et al., 2008).
The NOAA manages the NDBC, which consists of many buoys moored along the US coasts, both in the Pacific Ocean and in the Atlantic Ocean. Some buoys were moored in the late 1970s, so that more than 35 years of data are available. The historical wave data give hourly significant wave height, peak and mean period. Only few buoys are directional. In the paper the NOAA directional buoy 46042 (see Fig. 1b), moored off California, is considered. The Italian buoys network (RON) started measurements in 1989, with 8 directional buoys located off the coasts of Italy. Currently, the network consists of 15 buoys, moored in deep water. RON buoys give up to two records per hour; for each record, the data of significant wave height, peak and mean period and
V. Laface et al. / Ocean Engineering 107 (2015) 45–53
dominant direction are given. In the Central Mediterranean Sea, the buoy of Mazara del Vallo is considered (see Fig. 1a). The HIPOCAS project (Hindcast of Dynamic processes of the Ocean and Coastal Areas of Europe) provided a simulation of 44years (1958–2001) wind, waves, sea level data and current climatology. The hindcast wave model used in HIPOCAS is the third generation wave model WAM cycle 4 modified for two-way nesting by Gómez Lahoz and Carretero Albiach (1997), which gives the following output parameters: significant wave height Hs, wave direction, mean period Tm, peak period Tp with a time step of three hours. From HIPOCAS two points in Atlantic Ocean are considered (see Fig. 1a). The paper proposes a directional analysis of sea storms carried out by processing wave data coming from directional buoys of RON and NOAA networks, and by HIPOCAS project. The two locations in North-East Atlantic Ocean from HIPOCAS (Fig. 1a), the one in central Mediterranean Sea (Fig. 1a) from RON network and the one in US coast in Pacific Ocean (Fig. 1b) from NOAA– NDBC are considered. The analysis is performed by processing significant wave height and wave direction time series. At the first stage sea storms are extrapolated from significant wave height time series by means of the definition given in Section (2) which does not take into account the wave direction, but it is only related to the average significant wave height at the considered site (see Table 1). Then the directional analysis is done by considering the sequence of sea states during each sea storm with the sequence of the related wave directions. Fig. 2 shows significant storms for each location, with the directions of the related sea states. Firstly, only the wave direction associated to the peak of the storm Hs max is regarded. Fig. 3 (left) shows the wave direction ϑHs max at the sea state with the maximum significant wave height Hs max (storm peak) versus Hs max (each point represents a storm). From the figure it is clear that at each considered site it is possible to identify one or more main directions from which the strongest storms occurs. The statistical plot in Fig. 3 (left side) enables to identify in a simplified manner the main, the secondary and eventually tertiary directions considering the direction at storms peak. The main direction is the one from which the severest sea storms occur. There are sites like HIPOCAS (1) where there is only one relevant direction, but in some cases it is possible to have more relevant directions like at RON and HIPOCAS (2) sites. In such cases it is needed to define the relevance of each direction and they are classified as main ϑ1 , secondary ϑ2 and tertiary ϑ3 in relation to the severity of the storms coming from the considered direction. In general to identify the main directions different approaches may be adopted. For instance, the energy flux may be calculated for any direction and the relevance of each of them may be established in terms of the contribution of the energy flux to the Table 4 Number of directional storms pertaining to directional sectors ϑ1;2 7 Δϑ, classified by means of criteria 2.1 (i) and 2.1 (ii) at Mazara del Vallo buoy (RON). ϑ1 ¼2801
Δϑ ¼ 101
Δϑ ¼ 201
Δϑ ¼ 301
(i) 0 ðiiÞ h ¼ 0:5H s max 0 ðiiÞ h ¼ 0:6H s max 0 ðiiÞ h ¼ 0:7H s max 0 ðiiÞ h ¼ 0:8H s max ϑ2 ¼1401 (i) 0 ðiiÞ h ¼ 0:5H s max 0 ðiiÞ h ¼ 0:6H s max 0 ðiiÞ h ¼ 0:7H s max 0 ðiiÞ h ¼ 0:8H s max
107 95 105 101 108 Δϑ ¼ 101 28 28 28 28 25
126 116 120 119 129 Δϑ ¼ 201 49 39 41 44 42
137 129 131 129 136 Δϑ ¼ 301 50 41 43 46 49
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considered direction. Another approach to determine ϑ1 , ϑ2 , ϑ3 is the one proposed by Boccotti (2000) (see also Arena and Fedele, 2002; Arena et al., 2013) by means of cumulative probability distributions functions of significant wave height. For the analysis proposed in the paper the main directions are determined from Fig. 3. In particular, at HIPOCAS (1) the main direction is assumed ϑ1 ¼ 3051, at RON the main and the secondary directions are assumed ϑ1 ¼ 2801, ϑ2 ¼ 1401 respectively. At HIPOCAS (2) three relevant directions are identified ϑ1 ¼ 801,ϑ2 ¼ 1901,ϑ3 ¼ 01. At NOAA 46042 is assumed ϑ1 ¼3001. Subsequently the average wave direction during the whole storm is calculated starting from wave direction of all sea states (Fig. 3 right). The comparison between the average direction and the direction at storm peak is shown in Fig. 4: these two directions are very close to each other for most of the storms, but in some cases differences are observed. Then, a deeper analysis is proposed by considering the strongest storms (Hs max Z2hcrit), by calculating the standard deviation of wave direction and by considering its average values for classes of storm intensity Hs max. The same calculation is done for the whole storm history first and then by considering only sea states 0 above a fixed threshold h (from 0.5Hs max to 0.8Hs max) of significant wave height. Fig. 5 shows the average value of the standard deviation of direction versus any range of H s max . The results show that during the storms a certain variability of wave direction is observed: it is quite relevant if the whole storm history is considered; if we consider only sea states above a fixed thresh0 old h of significant wave height, the general trend (from lowest to 0 highest h ) shows that standard deviation decreases for increasing 0 values of h . The most relevant reduction is observed by comparing the results for the whole storms and for the storms above 0.5Hs max (see Fig. 5). This result is due to the strong variability of wave direction in storm tails (smaller sea states). Starting from the above results the definitions (i) and (ii) in Section 2.1 of directional storm are introduced and applied to classify the identified storms (without any condition of wave direction) as directional storms pertaining to a given directional sector ϑi 7 Δϑ . The direction ϑi is fixed as the wave direction from which the severest storm occurs and for the width Δϑ values of 101, 201, 301 are considered. It is worth noting that for this kind of analysis a too narrow sector (Δϑ less than 101) does not enable to develop criteria on wave direction during sea storms because of the wave direction variability. On the other hand, considering Δϑ greater than 301 the scope of directional analysis became meaningless. Tables 2–5 show the number of storms classified by means of (i) and (ii) (in Section 2.1) as directional storms in the directional sector ϑi 7 Δϑ for each of the main directions ϑi identified, for 0 each considered width Δϑ and for h ¼ 0:5H s max , 0.6Hs max, 0.7Hs , 0.8H . The number of directional storms pertaining to a s max max given directional sector ϑi 7 Δϑ increases for increasing values 0 of the width Δϑ both applying definition (i) and (ii) whatever is h . Furthermore definitions (i) and (ii) tend to give the same result if widths of 301 are considered. For what concerns results obtained 0 applying definition (ii) assuming increasing h it has been seen Table 5 Number of directional storms pertaining to directional sector ϑ1 7 Δϑ, classified by means of criteria 2.1 (i) and 2.1 (ii) at 46042 buoy (NOAA–NDBC). ϑ1 ¼ 3001
Δϑ ¼101
Δϑ ¼201
Δϑ ¼ 301
(i) 0 ðiiÞ h ¼ 0:5Hs max 0 ðiiÞ h ¼ 0:6Hs max 0 ðiiÞ h ¼ 0:7Hs max 0 ðiiÞ h ¼ 0:8Hs max
13 15 13 13 13
22 24 26 26 23
26 27 27 28 28
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Fig. 6. Maximum expected wave height H max for directional storms pertaining to the directional sector ϑ1 7 Δϑ versus maximum expected wave height H max for storm selected without condition about wave direction, for the storms with Hs max Z 2hcrit at HIPOCAS (1).
Fig. 7. Maximum expected wave height H max for directional storms pertaining to the directional sectors ϑ1 7 Δϑ (top) and ϑ2 7 Δϑ (bottom) versus maximum expected wave height H max for storm selected without condition about wave direction, for the storms with Hs max Z 2hcrit at Mazara Del Vallo (RON).
that there are no significant variations of the number of directional storms if Δϑ is assumed equal to 301, especially for the main and the secondary sectors. For this reason it could be more appropriate 0 to utilize definition (ii) with h ¼ 0:5H s max in order to take into account a larger number of sea states. It is worth noting that the main purpose of the analysis proposed here is related to the introduction of a physically based approach which enables us to associate each storm to a given directional sector. Consider the storms with maximum significant wave height Hs max Z2hcrit (see Table 1), by applying definition (i) or (ii) with a width Δϑ of 301 and h' ¼ 0:5H s max it is possible to classify more than 80% of the storms as directional storms at both HIPOCAS (1) , HIPOCAS (2) and at 46042-NDBC, and more than 70% at Mazara del Vallo RON. Finally to test the validity of the criterion the maximum expected wave heights H max calculated for the whole storm (storm selected without any condition on wave direction) and for directional storm are calculated and compared
(see Figs. 6 and 7). Results show that the maximum expected wave height of directional storm is less or equal to the one calculated without condition on wave direction (whole storm). This could leads for example to a reduction of design wave height for angledependent devices. It is worth noting that for increasing width Δϑ the maximum expected wave height of the whole storm and of the directional storm tend to converge.
4. Conclusions The paper has investigated the variability of wave direction during sea storms, showing how it is large for lower sea states (storm tails) and it is reduced by considering the wave direction associated to sea states above increasing thresholds of significant wave height (storm peak). A definition of directional sea storm pertaining to a given directional sector is introduced. The
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comparison between direction at Hs max (maximum significant wave height during the storm) and average wave direction during storm has revealed that for most of the storms these two directions are similar. It has been found that the number of directional storms pertaining to a given directional sector increases for increasing values of the width of the directional sector. Finally it has been proved that an appropriate amplitude for the directional sectors may be of 601 because it enables to classify as directional storms most of the storms. Acknowledgments The data for this paper are available at NOAA and at idromare.it websites. The work has been partially supported by project LARGE MULTIPURPOSE PLATFORMS FOR EXPLOITING RENEWABLE ENERGY IN OPEN SEAS (PLENOSE) funded by EU IRSES Marie Curie action (project PIRSES-GA-2013-612581). The PhD programme of Valentina Laface is supported by European Commission, European Social Fund (FSE) and Regione Calabria via the scholarship International mobility for young graduates and researchers – POR Calabria FSE 20011/2013 Intervento D.5 – Operational Target M2 Encouraging individual highlevel education Programmes for young Graduates and Researchers in Eminent National and International Research Centers. References Arena, F., Pavone, D., 2006. Return period of nonlinear high wave crests. J. Geophys. Res. 111, C08004. Arena, F., Pavone, D., 2009. A generalized approach for the long-term modelling of extreme sea waves. Ocean Model. 26, 217–225. Arena, F., and Fedele, F., 2002. Intensity and duration of sea storms off the Californian coast. In: Proceedings of Solutions to Coastal Disasters'02 of ASCE (the American Society of Civil Engineers), San Diego, 24–27 February, pp. 126– 141. Arena, F., Puca, S., 2004. The reconstruction of significant wave height time series by using a neural network approach. ASME J. Offshore Mech. Arct. Eng. 126, 213–219.
53
Arena, F., Malara, G., Barbaro, G., Romolo, A., Ghiretti, S., 2013. Long-term modelling of wave run-up and overtopping during sea storms. J. Coast. Res. 29, 419–429. Arena, F., Laface, V., Malara, G., Romolo, A., Viviano, A., Fiamma, V., Sannino, G., Carillo, A., 2015. Wave climate analysis for the design of wave energy harvesters in the Mediterranean Sea. Renew. Energy 77, 125–141. Bernardino, M., Boukhanovsky, A., and Guedes Soares, C., 2008. Alternative Approaches to Storm Statistics in The Ocean. In: Proceedings of the 27th International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2008); Estoril, Portugal. New York, USA: ASME; OMAE2008-58053. Boccotti, P., 2000. Wave Mechanics for Ocean Engineering. Elsevier Science, New York. Boccotti, P., 2014. Wave Mechanics and Wave Loads on Marine Structures. Elsevier Science, Butterworth-Heinemann, Oxford. Borgman, L. E., 1970. Maximum wave height probabilities for a random number of random intensity storms. In: Proceedings of the 12th Conference of Coastal Engineering. Washington, DC, ASCE, 53–64. Borgman, L.E., 1973. Probabilities for highest wave in hurricane. J. Waterw. Harb. Coast. Eng. Div. Am. Soc. Civ. Eng. 99, 185–207. Fedele, F., Arena, F., 2010. The equivalent power storm model for long-term predictions of extreme wave events. J. Phys. Oceanogr. 40, 1106–1117. Gómez Lahoz, M., Carretero Albiach, J.C., 1997. A two-way nesting procedure for the WAM model: application to the Spanish coast. J. Offshore Mech. Arct. Eng. 119, 20–24. Guedes Soares, C., Weisse, R., Carretero, J.C., and E. Alvarez, 2002. A 40 years hindcast of wind, sea level and waves in European waters. In: Proceedings of the 21st International Conference on Offshore Mechanics and Arctic Engineering (OMAE'02), ASME Paper OMAE2002-SR28604. Hasselmann, K., Barnett, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, E., Ewing, J.A., Gienapp, H., Hasselmann, D.E., Krusemann, P., Meerburg, A., Müller, P., Olbers, D.J., Richter, K., Sell, W., Walden, H., 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z A12, 1–95. Jonathan, P., Ewans, K., 2007. The effect of directionality on extreme waves design criteria. Ocean Eng. 34, 1977–1994. Jonathan, P., Ewans, K., Forristall, G., 2008. Statistical estimation of extreme ocean environments: the requirements for modelling directionality and other covariate effects. Ocean Eng. 35, 1211–1225. Pierson, W.J., Moskowitz, L.A., 1964. A proposed spectral form for fully developed waves based on the similarity theory of S. A. Kitaigorodskii. J. Geophys. Res. 69, 5181–5190. Pilar, P., Guedes Soares, C., Carretero, J.C., 2008. Hindcast for the North East Atlantic European coast. Coast. Eng. 55, 861–871. Ponce de Leon, S., Guedes Soares, C., 2014. Extreme wave parameters under North Atlantic extratropical hurricanes. Ocean Model. 81, 78–88. Rice, 1958. Distribution of the duration of fades in ration transmission. Bell Syst. Tech. J. 37, 581–635. Tromans, P. S., and Vanderschuren, L., 1995. Response based design conditions in the North Sea: Application of a new method. OTC 7683, 27th OTC, Houston.