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Numerical prediction of Normal and Extreme
Waves at Fukushima Offshore Site
To cite this article: Atsushi Yamaguchi and Takeshi Ishihara 2018 J. Phys.: Conf. Ser. 1037 042022

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The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 042022

IOP Publishingdoi:10.1088/1742-6596/1037/4/042022

Numerical prediction of Normal and Extreme Waves at
Fukushima Offshore Site
Atsushi Yamaguchi1 and Takeshi Ishihara2
1

Associate Professor, School of Engineering, The University of Tokyo
Professor, School of Engineering, The University of Tokyo
E-mail: atsushi@bridge.t.u-tokyo.ac.jp
2

Abstract. In this study, wave hindcasting was performed at Fukushima floating offshore
wind turbine demonstration site by using Wave Watch III, and the simulation results werevalidated by using measurement data. Following results were obtained. It was found thatthe use of the computational domain which covers whole the Pacific Ocean improves theprediction accuracy of significant wave height and wave period. By applying correction
factors to the extreme value of the significant wave height results in the accurate
estimation of the average value and the frequency distribution of the significant wave
height. The long-term variations of the annual average significant wave height and period
as well as the extreme value distribution of the annual maxium wave height are estimatedfrom 10 years of hindcasting data. Gumbel fitting can be used for the estimation of theextreme wave height in tropical cyclone prone region. The relationship between extreme
wave height and period is consistent with IEC61400-3. However, when all the event isincluded, the effect of the swell cannot be neglected and a model proposed by Goda should
be used. A model to describe the joint probability distribution of wind speed, wave height
and wave period is proposed. Predicted joint probability distribution based on this modelshows good agreement with measurement.

1. Introduction
Estimation of the extreme wave height with the recurrence period of 50 years and the joint
probability distribution of mean wind speed, significant wave height and peak wave period isneeded for the design of offshore wind turbines. Hindcasting is typically used for the assessment ofwave condition for offshore wind applications [1]. However, several problems are left to be solved.
First of all, along the Pacific coast of Japan, swell plays an important role and computational
domain size needed to simulate the swell component is not clear. For example, it is expected thatthe swell caused by the wind far from the coastline of Japan, may affect the local sea condition,
but there has been no guideline on the computation domain size needed to simulate the swell
component.
Secondly, along the coastline of Japan, the tropical cyclone may play a significant role on theextreme wave height. In case of extreme wind estimation, Ishihara and Yamaguchi [2] showed
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1

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 042022

IOP Publishingdoi:10.1088/1742-6596/1037/4/042022

Gumbel fitting may cause large uncertainty and Monte Carlo simulation of tropical cyclone isneeded to reduce the uncertainty, but the applicability of Gumbel distribution to the extreme
wave distribution is not investigated. Further, in the swell prone region, the applicabilty of therelation between wave height and wave period specified in IEC61400-3 [3] may not be applicableand further analysis is needed.
Lastly, for the evaluation of joint probability distribution of wave height and wave period,
typically, the result of the hindcasting is directly used, but this may result in non-smoothprobability distribution which caused uncertainty in the fatigue load analysis and better method
to estimate the probability distribution is needed.
This study focuses on these three points and proposes methods for the estimation of design wave
condition of offshore wind farm along the coastline of Japan.
2. Wave hindcasting and validation at Fukushima site
In this study, metocean conditions measured in Fukushima Floating Offshore Wind Farm
Demonstration (Fukushima FORWARD) Project[4] were used as validation data. In this project, a
met mast of the height of 60m above sea level is installed on the substation floater together with
Doppler lidar, wave meter is installed in the hull of the substation floater and wave buoy is
installed in the same area. Statistical significant wave height (𝐻1/3) and significant wave period(𝑇1/3) are measured in this project. By cross-validating the wind speed, wave height and wave
period, reliable dataset for one year was created and used in this study.
The third generation wave prediction model Wave Watch III ver 4.18[5]. was used for thehindcasting. Table 1 summarizes the computation domain and The surface wind data which isused as a boundary condition is obtained by mesoscale model WRF embedded in the global
analysis data. Because the output of the wave prediction model is two directional wave spectrum
model, significant wave height and significant wave period have to be calculated for the validation.
In this study, following relationship relationships are assumed to calculate the statisticalsignificant wave height and period from wave spectrum.
𝐻1/3 = 𝐻𝑠 = 𝐻𝑚0 = 4√𝑚0

(1)

𝑚−1
𝑚0

(2)

𝑇1/3 = 𝑇𝑠 = 𝑇𝑚−1,0 =Where 𝑚𝑛 is the nth moment and defined by following equation.
𝑚𝑛 = ∬ 𝑓 𝑛 𝐸(𝑓, 𝜃)𝑑𝑓𝑑𝜃
Table 1 The computation domain of the hindcastingComputation period
Horizontal resolution
Grid number
Bathymetry

Domain 1
Domain 2
Domain 3
Domain 4
From 1 Jan. 2006 to 31 Dec. 2015 (10 years)
0.5 degree
0.2 degree
0.05 degree
0.02 degree
Case 1: 320×240
80×77
80×80
80×80
Case 2: 160×120
ETOPO2ETOPO1

2(3)

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 042022
Surface boundary
condition (wind)Side boundary
condition
Resolution of spectrum

NCEP-FNL(1.0 degree)

IOP Publishingdoi:10.1088/1742-6596/1037/4/042022

WRF(18km)

WRF
(6km)

Close

WRF
(2km)

Nest-down (2 way)

Frequency: 36(0.0345~0.97Hz), Direction 36

Table 1 shows the four level nested grids used for the hindcasting. The effect of the computation
domain was investigated by changing the size of the largest domain (domain 1), The grid intervalof which is 0.5 degree both in latitude and longitude. Two different domain sizes were tested fordomain 1: case 1 covers the entire Pacific Ocean and case 2 covers only north west part of the
Pacific Ocean (Figure 1). Same computational domains are used for the domains 2, 3 and 4. Oneyear simulation for year 2015 was carried out and the simulation results were validated.

Case 2

Case 1Figure 1 Two different computation domain 1: case 1 and case 2
The monthly averaged significant wave height and significant wave period are shown in figure 2
together with the measurement. When only north west Pacific is used for region 1 (case 2), thepredicted significant wave periods are significantly smaller than the measurement while when thewhole Pacific Ocean is covered in domain 1 (case 1), the underestimation of the significant wave
period is improved and the predicted value shows better agreement with the measurement.
20
Measurement
Case 1Case 2(a)4monthly averagesignificant wave period T1/3 (s)monthly averagesignificant wave height H 1/3 (m)

5

3

2

1

(b)
15

10

5

0

0

1

2

3

4

5

6
7
8
Month(2015)

9

10

11

1

12

2

3

4

5

6
7
8
Month(2015)

9

10

11

12

Figure 2 Simulated and measured monthly average significant wave height (a) and period (b)3The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 042022

IOP Publishingdoi:10.1088/1742-6596/1037/4/042022

By using the improved larger domain 1, the hindcasting was carried out for ten years, i.e., fromJanuary 2006 to December 2015. Because the wind speed used as the surface boundary condition
has the averaging time of around three hours[6], the simulated wave height and period also has
the evaluation time of around three hours (180 minutes). Yamaguchi and Ishihara[6] proposed amethod to consider this difference, in which the relationship between the significant wave heightwith the evaluation time of 𝑀 minutes and 20 minutes can be approximated by using following
equation.
𝑀
𝐻1/3
𝑀 − 20 0.3
(4)
=
1
−
0.05
(
)
20
60
𝐻1/3
In this study, this equation is used to estimate the significant wave period with the evaluation time20
180
of 20 minutes (𝐻1/3) from that of 180 minutes (𝐻1/3). Then, a correction factor proposed by Mase𝑀𝑎𝑥
et al.[7] is applied to estimate maximum significant wave period 𝐻1/3
by multiplying 1.19.
𝑀𝑎𝑥
20
𝐻1/3
= 1.19𝐻1/3(5)

These equations are applied when the peak wave height is predicted and within nine hours fromthe time when peak wave height is predicted, the wave height is linearly interpolated to havesmooth change in the wave height. After this correction, predicted wave height show good
agreement with the measurement. Figure 3 shows an example of the comparison of the predictedsignificant wave height before and after correction, predicted significant wave period and wave
direction together with the measurement data.
20
Measurement
Before correction
After correction

8
6
4
2
0

360

(b)

(c)

Measurement
Simulation

15

wave direction (deg)(a)

Significant wave period 𝑇1/3 (s)Significant wave height 𝐻1/3 (m)

10

10

5

0

15/6/1

15/6/615/6/11

15/6/16DATE

15/6/21

15/6/26

Simulation

180

90

0

15/6/1

15/7/1

Measurement270

15/6/615/6/11

15/6/16

15/6/21

15/6/26

15/7/1

15/6/1

15/6/615/6/11DATE

15/6/16DATE

15/6/21

15/6/26

15/7/1Figure 3 predicted and measured (a) significant wave height, (b) wave period and (c) wave
direction
Figure 4 shows the monthly averaged significant wave height and wave period for all the months
in year 2015. The frequency distribution of the wave height and period are also shown in figure 5.
It is clearly shown that the predicted wave height and period show good agreements withmeasurement for all the month in year 2015.
15

(a)

Obs
Cal

2

1

monthly a ve r ag e s i gn i fi c an t w a ve pe r io d T 1/3 (s)monthly a ve r ag e s i gn i fi c an t w a ve he i gh t H 1/3 (m)

3

0

(b)

Obs
Cal

10

5

0
1

2

3

4

5

6

7

8

9

10

11

12

Annual1Month(2015)

2

3

4

5

6

7

8

Month(2015)

4

9

10

11

1 2 Annual

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 042022

IOP Publishingdoi:10.1088/1742-6596/1037/4/042022

Figure 4 Monthly averaged significant wave height (a) and period (b)30
ObsCal(2015)
Cal(2006-2015)

(a)

25
20
15

10
5

Obs
Cal(2015)
Cal(2006-2015)(b)

25
Probability densityf(%)

Probability density f(%)

30

20
15

10
5

0

0
0

1

2
3
Significant wave height H 1/3 (m)

4

0

5

5
10
Significant wave period T 1/3 (m)15

Figure 5 Frequency distribution of significant wave height (a) and period(b)3. Estimation of the extreme wave height
By using 10 years of the hindcasting, after the correction of the peak value, the one year maximum
significant wave height was extracted and reordered in ascending order, and frequency
distribution of the extreme value are approximated by using Gumbel distribution. Moment
method was used to estimate the parameters of the Gumbel distribution. Figure 6 (a) shows theestimated extreme wave height distribution with the uncertainty, together with the calculated
annual maximum wave heights. The black circles show the extreme wave heights induced bytropical cyclones and the white circles shows those induced by other events. The uncertainty, or
the dispersion of the Gumbel distribution can be calculated based on Gumbel theory as shown inequation (6).
𝜎𝐻2 =

𝜎𝑁2
[1 + 0.885(𝑦 − 𝛾) + 0.6687(𝑦 − 𝛾)2 ]𝑁(6)It is shown that the Gumbel distribution can approximate the distribution and there seems noclear difference between the high wave event induced by tropical cyclones and the other events.
Figure 6 (b) shows the relationship between extreme wave height and wave period for the annual
maximum wave height events. The relationship between wave height and period specified in
IEC61400-3 is also shown in the figure. Most of the data are located within this range, but there
are few exceptions. This is because the extreme wave events are caused by the local wind eventsand the effect of the swell is not significant. However, if all the events are included, the effect of
the swell cannot be ignored and results shows larger variation. Figure 7 shows the relationshipbetween significant wave height and period for all the events. Many data are located outside theregion specified in IEC61400-3. This maybe the source of the uncertainty of the fatigue loadassessment. A model proposed by Goda[8] is also shown in this figure. It is shown that for normal
events, the model proposed by Goda should be used instead of the model specified in IEC61400-3.

5

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 042022
20

15

(a)

Tropical cyclone
Extratropical cyclone
GumbelUncertainty(1σ)

H50 +σH=11.86(m)Significant wave period (s)Annual maximum significant wave height (m)IOP Publishingdoi:10.1088/1742-6596/1037/4/042022

10

5(years)
2

5

10

20

50

3

4

(b)

Tropical cyclone
Extratropical cyclone
IEC

15

10

5

100
0

0
-2

-1

0

1
2
-ln(-ln(F))

0

5

2

4

6

8

10

Significant wave height (m)Figure 6 (a) Frequency distribution of the extreme wave height and (b) relationship between wave
height and period for extreme wave height events

Significant wave period T 1/3 (s)

20

15

10

Tropic al c y c lones
O t h er ev e nt s
I E C61 4 00- 3
G o d a(wind wave )G o d a( H o/ L o= 0 .0 1 )

5

0
0

2

4

6

8

10

12

14

Significant waveh height H 1/3 (m)Figure 7 The relationship between wave height and period for all the events
Table 2 The model for the parameters of the probability distribution function of significant wave
height and period
Significant wave height
Significant wave periodH  
1/ 3

mean

H

1/ 3,W

H

1/ 3, S2
H1/ 3,W

T 

2
H1/ 3, Sg  9.81m/s 2
0 1

2


,



Standard
deviation

1/ 3

H

1/ 3,W

H

1/ 3, S

1/ 3,W

1/ 3

 0.52m13

 gF  
2.74 U10  
1  1  0.008  2  gU 10     T1/ 3 

0 1

  H1/ 3    H   1/ 3  






 8.86s  T21/ 3, S RT1/ 3,W T1/ 3,W ， RT1/ 3,W  0.14

T

  T1/3 

0 1

 1.34s 1  ln T    2 
1/ 3
T1/ 3
f T1/ 3  
exp     
 2 
 T1/ 32  T1/ 3 T1/ 3
 

1

 
1
T  ln T   T2 ,  T2  ln 1   T  T
2
1/ 3

6

5

g  9.81m/s 2 F  161,000m

 
 
 
1/ 3 

1/ 3

2

 H2  ln 1  
1/ 3

  H2 1/ 3,S

T

1/ 3, S

 1  ln H    2 
1/ 3
H1/ 3
f  H1/ 3  
exp     
 2 
 H1/ 32  H1/ 3 H1/ 3 
1
H1/ 3  ln  H1/ 3   H21/ 3 ,2


1/ 3,W

1/ 3,W

1/ 3,W1

Probability
density H2

 T   T2

  H2 1/ 3,S

0 1

 T1/ 3,S  H2 1/ 3,S

CT1/ 3,W  0.96

T

 RH1/ 3,W  H1/ 3,W ， RH1/ 3,W  0.36

  H1/3

1/ 3,W CT1/ 3,W
1/ 3,WF  161,000m

 1.31m

 H2

T

1/ 3, S H   H2

1/ 3,W

1/ 3

12
 gF  
0.30U102  
1  1  0.004  2  gU 10     H1/ 3

T

1/ 3

1/ 3

1/ 3



2

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 042022

IOP Publishingdoi:10.1088/1742-6596/1037/4/042022

RH1/ 3 T1/ 3  tanh U10    ,   0.047s/m ,   0.278

Correlation

4. Estimation of joint probability distribution of wave height and wave period
For the fatigue load assessment of the offshore wind turbines, the joint probability distribution ofwind speed, wave height and wave period is needed. In this study, the probability distribution ofwave height and wave period is modelled by using log-normal distribution for each wind speed.
Table 2 summarizes the model for the parameters of the probability distribution function ofsignificant wave height and period and figure 8 shows the mean value and the standard deviation of
the significant wave height and period as a function of wind speed. The examples of the
distribution functions are shown in figure 9. The proposed model show good agreement with the
distributions obtained from hindcasting.
10

4Wind wave
Swell
Combined model
CalWind wave
Swell
Combined model
Cal

3

6

σH1/3(m)μH1/3(m)

8

2

4
1

2

0

0
0

5

10

15
U10(m/s)

20

25

30

0

20

10

15
U10(m/s)

20

25

30

15
U10(m/s)

20

25

30

4
Cal
Wind wave
Swell
Combined model

CalWind wave
Swell
Combined model3

σT1/3(s)15

μT1/3(s)

5

10

5

2

1

0

0
0

5

10

15
U10(m/s)

20

25

30

0

5

10

1

1

0.8

0.8

Cumulus Frequency

Cumulus Frequency

Figure 8 The mean value (a) and the standard deviation (b) of the significant wave height fordifferent wind speed; the mean value (c) and the standard deviation (d) of the significant wave
period for different wind speed

0.6

0.4
Cal(4<U10<5m/s)
Cal(12<U10<13m/s)
Log-normal dist.(4<U10<5m/s)
Log-normal dist.(12<U10<13m/s)

0.2

0.6

0.4

0.2

0

0
0

1

2

3

4

5

0

H1/3(m)5
T1/3(s)10

15

Figure 9 Examples of the frequency distribution of the significant wave height (a) and significant
wave period (b)7The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 042022

IOP Publishingdoi:10.1088/1742-6596/1037/4/042022

In addition, the correlation between wave height and wave period is also modelled as a function ofmean wind speed. This correlation is needed when Monte-Carlo simulation is performed togenerate synthetic set of wind speed, wave height and wave period. Figure 10 shows the modelledcorrelation and the correlations obtained from hindcasting. Proposed model shows good
agreement with the hindcasting.
1

相関係数RH1/3-T1/3

0.8

0.6

0.4

0.2

Cal
tanh(αU10+β)

0
0

5

10

15

20

25

U10(m/s)

Figure 10 The correlation between significant wave height and period
Finally, based on these models, by using Monte-Carlo simulation, synthetic sets of mean windspeed, significant wave height and wave period were created. Figure 11 shows the joint
probability distribution obtained by (a) measurement and (b) Monte-Carlo simulation. The
estimated probability distribution by using Monte-Carlo simulation shows similar result with themeasurement but smoother results with more data, which contribute to reduce the uncertainty of
the fatigue load assessment.

(a) Measurement(b) Monte-Carlo simulation
Figure 11 Measured and predicted joint probability distribution of wind speed, wave height andperiod8

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 042022

IOP Publishingdoi:10.1088/1742-6596/1037/4/042022

5. Conclusion
In this study, wave hindcasting was performed at Fukushima floating offshore wind turbine
demonstration site by using Wave Watch III, and the simulation results were validated by using
measurement data. Following results were obtained.
1. It was found that the use of the computational domain which covers whole the Pacific Ocean
improves the prediction accuracy of significant wave height and wave period. By applying
correction factors to the extreme value of the significant wave height results in the accurate
estimation of the average value and the frequency distribution of the significant wave height.
2. The long-term variations of the annual average significant wave height and period as well as
the extreme value distribution of the annual maxium wave height are estimated from 10 yearsof hindcasting data. Gumbel fitting can be used for the estimation of the extreme wave heightin tropical cyclone prone region. The relationship between extreme wave height and period is
consistent with IEC61400-3. However, when all the event is included, the effect of the swellcannot be neglected and a model proposed by Goda should be used.
3. A model to describe the joint probability distribution of wind speed, wave height and wave
period is proposed. Predicted joint probability distribution based on this model shows good
agreement with measurement.
Acknowledgement
This study was carried out as a part of Fukushima FORWARD project, funded by ministry of
economy, trade and industry, Japan.
6. References[1] J.F.Manwell, C.N.Elkinton, A.L.Rogers and J.G.McGowan, Review of design conditions
applicable to offshore wind energy systems in the United States, Renewable and Sustainable
Energy Reviews, 11(2), 210-234, 2007
[2] Takeshi ISHIHARA, Atsushi YAMAGUCHI. Prediction of the extreme wind speed in themixed climate region by using Monte Carlo simulation and Measure-Correlate-Predict
method, Wind Energy, 18(1): 171-186, 2015.
[3] IEC61400-3, Wind Turbine - Part 3: Design requirements for offshore wind turbines, 2009.
[4] A. Yamaguchi and T. Ishihara, Current status of research activity on floating offshore windturbine in Japan, OFFSHORE2013, 2013.
[5] Tolman H. L.: User manual and system documentation of WAVEWATCH III version 4.18.
NOAA/NWS/NCEP/ MMAB Technical Note 316.,2014.
[6] A. Yamaguchi and T. Ishihara, Effect of sampling time and disjunct sampling of significant
wave height for the design of offshore wind farm,Wind Energy Symposium, Tokyo, 2012 (inJapanese).
[7] H. Mase, N. Mori, T. Yasuda, J. Sakunaka and T. Utsunomiya, Assessment of Coastal Design
Wave utilizing North Pacific Wave Analysis Data, Jour. of JSCE, Ser. B2 (Coastal Eng.),
Vol.65, pp.146–150, 2009 (in Japanese)[8] Y.Goda: Random Seas and Design of Maritime Structures 3rd Edition, Advanced Series on
Ocean Engineering,Vol.33, 2009.

9