consist-ing of Hm0 and the mean wave directionθj. The mean wave direction θj at time j is defined as
θj = arctan
{ ∫2π
0 dθ∫0∞df Sj(f, θ) sinθ
∫2π
0 dθ∫0∞df Sj(f, θ) cosθ
}
. (7.11)
The wave roses for 02-04 and 11-01 show that ordinary waves propagate θ ≈ 120.
Note that because wave roses are based on representative values, a peak of the mean of directional spectra S¯and that of the wave rose do not have to be equal. The wave rose for 08-10 shows that extreme waves propagate inθ ≈140.
The relatively small seasonal variation in wave direction suggests that the use of WECs, depending upon wave direction, is possible in the southeast.
At location ×2, in 02-04 and 11-01, joint distributions ofHm0 andTmm10are simi-lar to each other, having simi-largerH¯m0 than seasons 05-07 and 08-10. Means of seasonal directional spectra at ×2in figure 7-13 show more peaks and marked seasonal varia-tions in wave direction than those at×1. A possible cause of several peaks is that the complex environment is exposed to the vast expanse of the ocean but is surrounded by many small islands. The complex environment also results in lower correlation coeffi-cientsσHT/(σHσT)at ×2than at ×1. One of the peaks approximately at S(0.5,¯ 30), is consistently observed in all seasons. In 05-07 and 08-10, peaks at S(0.4,¯ 90) are thought to be caused by typhoons from the south, which is similar to the case in×1.
In 11-01 and 02-04, peaks at S(0.5,¯ 350) are thought to be caused by monsoon from the continent.
Seasonal wave roses at ×2 in figure 7-14 also show marked seasonal variations in wave direction; accordingly, it is suggested that the use of WECs, without depending upon wave direction, is advantageous in the southwest.
Figure 7-9: Joint distributions of (Tmm10, Hm0) and histograms ofHm0 and Tmm10at
×1for 02-04, 05-07, 08-10, and 11-01. Contour lines in joint distributions show wave energy transports for deep water P¯ (kW/m) of indicated on the values vertical axis
Figure 7-10: Seasonal variability of a normalized 10-year mean directional spectra S(f, θ)¯ at ×1 for 02-04, 05-07, 08-10, and 11-01. S(f, θ)¯ are normalized with the maximum values described in each figure.
Figure 7-11: Seasonal variability of wave roses of wave energy transport P at×2for 10 years in 02-04, 05-07, 08-10, and 11-01.
Figure 7-12: Joint distributions of (Tmm10, Hm0) and histograms of Hm0 and Tmm10 at ×2 for 02-04, 05-07, 08-10, and 11-01. Contour lines in joint distributions show wave energy transports for deep water P¯ (kW/m) of indicated on the values vertical
Figure 7-13: Seasonal variability of normalized 10-year mean directional spectra S(f, θ)¯ at ×2 for 02-04, 05-07, 08-10, and 11-01. S(f, θ)¯ are normalized with the maximum values described in each figure.
Figure 7-14: Seasonal variability of wave roses of wave energy transport P at×2for 10 years in 02-04, 05-07, 08-10, and 11-01.
from 1970 to 1984. However, comprehensive analyses of waves around Kyushu were not performed because of the limited number of site measurements. For the future design and deployment of WECs around Kyushu, we calculated waves around Kyushu for 10 years by using SWAN and analyzed statistical wave characteristics at two representative sites south of Kyushu.
The model validation was performed by comparing calculated spectral significant wave height Hm0 and period Tmm10 with the observed zero-crossing significant wave heightH1/3and periodT1/3at two representative locations, Ainoshima and Hososhima for 2009. Comparisons showed that means of calculated spectrum-based statistics could accurately estimate means of measured zero-crossing statistics. The accuracy of estimation might be improved by using highly resolved wind, bathymetry, and ocean current inputs.
The spatial distribution of the mean wave energy transportP¯for 10 years revealed that Kyushu has more wave energy in the south than in the north, which was over-looked in previous studies due to the absence of measured data. Mean wave energy transport P¯ is highest in the southwest. The spatial distribution of P¯ for 10 years showed wave energy up to P¯≈ 16in the southeast and P¯ ≈10 in the southeast can be obtained if offshore waves are taken into consideration, which is greater than P¯ estimated by Tabata et al. (1980) and Takahashi et al. (1989).
According to the spatial distribution of seasonal P¯, extreme weather events fre-quently strike this region in 08-10. In 08-10, at ×1which is approximately 10 km off the coast, the monthly P¯ and the ratio of monthly P¯ relative to total yearly P¯ are expected to reach P¯ = 29 and 0.45, respectively. Although the yearly P¯ is expected to be powerful on the southeast coast, a stable high-energy supply cannot be possible, and such extreme weather events may cause severe damage to WECs. Considering less extreme weather events, the site ×2, 10 km off the tip of the peninsula in the south was selected. P at×2is relatively constant and shows the opposite tendency of P at×1. The opposite tendencies betweenP in the southeast and southwest suggest that if WECs can efficiently absorb wave energy including those of extreme waves, wave energy product southeast and southwest of Kyushu can compensate each other
to supply high stable energy.
Wave characteristics at×1and×2were well described by using the joint distribu-tion and histogram ofHm0 and Tmm10, the mean of the seasonal directional spectrum S(f, θ), and the wave rose which consists of(P, θ). Statistical wave characteristics in the southeast and southwest represented by×1and×2also showed different tenden-cies in terms of seasonal wave direction variation.
It is suggested that WECs in the southeast should be able to efficiently absorb ordinary waves concentrated approximately at S(0.45,¯ 90), and should be designed and deployed to withstand extreme waves at approximately S(0.45,¯ 150). The rela-tively small seasonal variation in wave direction at×1suggests that the use of WECs, depending upon wave direction, is possible in the southeast, whereas marked seasonal variations in wave direction at ×2 suggest that the use of WECs without depending upon the wave direction is advantageous southwest of Kyushu.
It is hoped that the analyses presented in this study will contribute to the devel-opment of wave energy generation.
Chapter 8 Future Plan
8.1 Future Plan: unsteady wave motions
To construct symmetric and asymmetric traveling waves, we started by considering the velocity potential of superposed sin waves propagating in only two directions. We would like to formulate unsteady wave motions such that the waves do not propagate in only two directions but arbitrary directions. Because such unsteady wave motions cannot be expressed by either sin or cos function, we will expand the solutions using full Fourier series. To write simply we define three-dimensional cartesian coordinate x = (x⊥, x3) = (x, y, z) where x⊥ is its horizontal coordinate x⊥ = (x1, x2) = (x, y) and x3 =z, then the dynamic condition and kinematic condition become
ρ=ϕt+1
2ϕxnϕxn+gx3 = 0 on x3 =η(x⊥, t), (8.1) Dρ
Dt =ϕtt+ 2ϕxnϕxnt+ϕxnϕxmϕxnxm +gϕx3 = 0 on x3 =η(x⊥, t). (8.2) We look the dimensional velocity potential ϕ for a solution in a form:
ϕ(x, t) =
∑N j=−N
∑N k=−N
ajk(t)ψjk(x), (8.3)
where
ψjk(x) =rjk(x3) exp[i(jκ1x1+kκ2x2)], (κ1, κ2) =
(2π λ1,2π
λ2
)
.
To write simply we define
b= (a)t, (8.4)
whereb anda are matrices with elementbjk and ajk, respectively. Substituting (8.3) into the dynamic condition (8.1) and the kinematic condition (8.2), we have
P(x⊥, η(x⊥, t);a(t),b(t)) =bjkψjk+ 1
2ajkaˆjˆk(ψjk)xn(ψˆjˆk)xn+gη(x⊥, t) = 0, (8.5) Q(x⊥, η(x⊥, t);a(t),b(t)) = (bjk)tψjk+ 2ajkbˆjˆk(ψjk)xn(ψˆjˆk)xn +gajk(ψjk)x3
+ajkaˆjˆkaˆˆjˆˆk(ψjk)xn(ψˆjkˆ)xm(ψˆˆjˆˆk)xnxm = 0, (8.6) wherej, k,ˆj,k,ˆ ˆˆj,ˆˆk are integers showing wavenumbers, and the summation sign Σfor these integers is omitted in above equations.
Providing that we have values a(t) and b(t), we can numerically solve (8.5) at arbitrary given position x⊥ for η(x⊥, t) using Newton’s method. We respectively have(2N−1)2 time-dependent variablesajk andbjk whereN is the truncated number defined in (8.3). We have(2N−1)2equations as (8.4) and another(2N−1)2equations by using Galerkin’s method:
(bµν)trµν−Nµν(a,b) =
∫ λ1
0
dx1
∫ λ2
0
dx2 Q(x⊥, η(x⊥, t), t) exp [i(µκ1x1+νκ2x2)] = 0.
(8.7) To update variablesaj,kandbj,kwe solve (8.4) and (8.7) using an explicit Runge-Kutta method.
set initial valuesa(t), b(t)
find!(x!,t)fromP
(
x!,!(x!,t),t;a(t),b(t))
=0 using Neton's methodfind
a(t+!t) b(t+!t)
!
"
# $
%& from
b(t)=a
t(t)
!" #$0 = b
1,1(t)!N
1,1
(
a(t),b(t))
b1,2(t)!N
1,2
(
a(t),b(t))
!
!
"
##
##
$
%
&
&
&
&
' ( ))
* ) ))
using an explicity Runge- Kutta method update time
Figure 8-1: Flowchart of the computational method for unsteady wave motions.
Appendix A
Ingredients for Computing Direct Numerical Solutions
A.1 Initial solution
We used the 3rd order perturbation solution as the initial solution guess for com-puting direct numerical solutions:
(Aj,k, C, G,Γ) =
∑3 m=1
(
A(m)j,k , C(m), G(m),Γ(m)). (A.1)
From (6.5),A(1)1,1andA(1)1,−1 are approximately given as Although the following solution is obtained by the perturbation method introduced in chapter 5 for asymmetric waves, it can be used for symmetric waves, setting A(1)1,1 = A(1)1,−1: ε = ˆε. The 2nd and 3rd order Fourier modes A(2)j,k, A(3)j,k of perturbation solutions (velocity potential) Ψare
A(2)2,2 = A(1)21,1
(
2−3r21,1,1+r1,1,2
)
2(4− rr2,2,11,1,1) , A(2)2,−2 = A(1)21,−1(2−3r1,1,12 +r1,1,2)
2(4−rr2,2,11,1,1) ,
A(2)2,0 = A(1)1,−1A(1)1,1(−2 + 4p2−3r21,1,1+r1,1,2)
4− rr2,0,11,1,1 ,
(A.2)
A(3)3,1 =A(1)1,1
A(2)2,0r1,1,1(−10r1,1,1r2,0,1+r2,0,2+ 12p2)
18r1,1,1−2r3,1,1 − B2,0(2)(r21,1,1−r1,1,2) 18r1,1,1−2r3,1,1
+A(1)1,−1
−A(2)2,2(−10r1,1,1r2,2,1+r2,2,2+ 24p2−12) 2(rr3,1,1
1,1,1 −9) − B2,2(2)(r21,1,1 −r1,1,2) 18r1,1,1−2r3,1,1
+A(1)1,−1A(1)21,1
(
r21,1,1(33r1,1,2−48p2+ 12)−3r1,1,1r1,1,3+ 8p4−2
)
8(rr3,1,1
1,1,1 −9) ,
A(3)3,3 =A(1)1,1
−A(2)2,2(−10r1,1,1r2,2,1+r2,2,2+ 12) 2(rr3,3,1
1,1,1 −9) − B2,2(2)(r21,1,1−r1,1,2) 18r1,1,1−2r3,3,1
+A(1)31,1 ((11r1,1,2−12)r1,1,12 −r1,1,3r1,1,1+ 2) 8(rr3,3,1
1,1,1 −9) ,
A(3)1,−3 =A(1)1,1
B2,(2)−2(r21,1,1−r1,1,2)
2 (r1,1,1−r1,3,1) − A(2)2,−2(−2r1,1,1r2,2,1+r2,2,2+ 8p2−4) 2(rr1,3,1
1,1,1 −1)
− A(1)1,−1B0,2(2)(r21,1,1 −r1,1,2) 2 (r1,1,1−r1,3,1) +A(1)1,1A(1)21,−1r1,1,1
(
r1,1,12 (−3r1,1,2+ 16p2−4) +r1,1,1r1,1,3+ 8p4−16p2+ 6)
8 (r1,1,1−r1,3,1) ,
A(3)3,−3 =A(1)1,−1
−A(2)2,−2(−10r1,1,1r2,2,1+r2,2,2+ 12) 2(rr3,3,1
1,1,1 −9) − B2,(2)−2(r21,1,1−r1,1,2
)
18r1,1,1−2r3,3,1
+A(1)31,−1((11r1,1,2−12)r21,1,1−r1,1,3r1,1,1+ 2) 8(rr3,3,1
1,1,1 −9) ,
A(3)3,−1 =A(1)1,−1
A(2)2,0r1,1,1(−10r1,1,1r2,0,1 +r2,0,2+ 12p2)
18r1,1,1−2r3,1,1 −B2,0(2)(r1,1,12 −r1,1,2) 18r1,1,1 −2r3,1,1
+A(1)1,1
−A(2)2,−2(−10r1,1,1r2,2,1+r2,2,2+ 24p2−12) 2(rr3,1,1
1,1,1 −9) − B2,(2)−2(r1,1,12 −r1,1,2) 18r1,1,1−2r3,1,1 +A(1)21,−1(r21,1,1(33r1,1,2−48p2+ 12)−3r1,1,1r1,1,3+ 8p4−2)
8(rr3,1,1
1,1,1 −9)
,
(A.3) where
rj,k,n=
αnj,k n is even αnj,ktanh (αj,kD) n is odd
,
andαj.k =√
j2p2+k2q2, andBj,k(1), Bj,k(2) are the 1st order and 2nd order Fourier mode of perturbation solutions (wave height) H:
B1,1(1) =A(1)1,1r1,1,1, B1,(1)−1 =A(1)1,−1r1,1,1,
(A.4)
B0,2(2) = 1
2A(1)1,−1A(1)1,1r1,1,1
(
r1,1,12 −2p2+ 1), B2,2(2) = 1
4A(1)21,1 r1,1,1(3r1,1,12 −1)+ 2A(2)2,2r1,1,1, B2,(2)−2 = 1
4A(1)21,−1r1,1,1(3r21,1,1−1)+ 2A(2)2,−2r1,1,1, B2,0(2) = 1
2A(1)1,−1A(1)1,1r1,1,1(3r1,1,12 −2p2 + 1)+ 2A(2)2,0r1,1,1.
(A.5)
Bj,k(1) is already substituted into (A.2) and (A.3). From (6.5) and (A.4),A(1)1,1 andA(1)1,−1 are approximately given as
A(1)1,1 = rB1,1(1)
r1,1,1 ≈ rˆεD 2r1,1,1, A(1)1,−1 = B1,(1)−1
r1,1,1 ≈ εD 2r1,1,1.
(A.6)
G(m) and Γ(m) are given as
G(0) = 1 r1,1,1, G(1) = 0, G(2) =
A(1)1,−1
(
A(2)2,0
(−14r2,0,2−p2
r1,1,1 + 12r2,0,1
)
+B2,0(2)
(
r1,1,2
4r1,1,12 − 14)+B0,2(2)
(
1
4 − 4rr1,1,22
1,1,1
))
A(1)1,1 +
A(1)1,1
(
A(2)2,0
(−14r2,0,2−p2
r1,1,1 + 12r2,0,1
)
+B2,0(2)
(
r1,1,2
4r1,1,12 − 14)+B0,2(2)
(
1
4 −4rr1,1,22
1,1,1
))
A(1)1,−1 +A(1)21,−1
(
r1,1,1
( 9
16r1,1,2−2p2− 1 4
)
+p4−p2+38 r1,1,1 − 3
16r1,1,3
)
+A(1)21,1
(
r1,1,1
( 9
16r1,1,2−2p2− 1 4
)
+ p4 −p2+ 38 r1,1,1 − 3
16r1,1,3
)
+A(2)2,−2
(1
2r2,2,1+ −14r2,2,2−1 r1,1,1
)
+A(2)2,2
(1
2r2,2,1+ −14r2,2,2−1 r1,1,1
)
+B2,−2(2)
( r1,1,2 4r21,1,1 −1
4
)
+B2,2(2)
( r1,1,2 4r21,1,1 −1
4
)
,
G(3) = A(1)1,1
(
B2,0(3)−B0,2(3))
r1,1,2
4r1,1,12 +14(B0,2(3)−B2,0(3))
A(1)1,−1
+ A(1)1,−1
(
B(3)2,0−B0,2(3)
)
r1,1,2
4r21,1,1 + 14(B0,2(3)−B2,0(3))
A(1)1,1 +
(
B2,(3)−2+B2,2(3)
)
r1,1,2
4r1,1,12 +1 4
(−B2,(3)−2−B2,2(3)),
(A.7)
Γ(0) = 0, Γ(1) = 0,
Γ(2) = A(1)1,−1(A(2)2,0(−14r1,1,1r2,0,1+18r2,0,2+ p22)+B0,2(2)(8rr1,1,2
1,1,1 −18r1,1,1
)
+B2,0(2)(18r1,1,1 −8rr1,1,21,1,1)) A(1)1,1
+A(1)1,1(A(2)2,0(14r1,1,1r2,0,1− 18r2,0,2− p22)+B2,0(2)(8rr1,1,2
1,1,1 − 18r1,1,1)+B0,2(2)(18r1,1,1− 8rr1,1,21,1,1)) A(1)1,−1
+A(1)21,−1
(
r1,1,12
(
− 3
32r1,1,2+p2−5 8
)
+ 1
32r1,1,3r1,1,1+ 1 16
(−8p4 + 8p2−1))
+A(1)21,1
(
r21,1,1
( 3
32r1,1,2−p2+ 5 8
)
− 1
32r1,1,3r1,1,1+ 1 16
(
8p4−8p2+ 1)) +A(2)2,2
(
−1
4r1,1,1r2,2,1+ 1
8r2,2,2+ 1 2
)
+A(2)2,−2
(1
4r1,1,1r2,2,1−1
8r2,2,2− 1 2
)
+B2,−2(2)
( r1,1,2 8r1,1,1 −1
8r1,1,1
)
+B2,2(2)
(1
8r1,1,1− r1,1,2 8r1,1,1
)
,
Γ(3) = A(1)1,−1
1
8
(
B2,0(3)−B0,2(3))r1,1,1+
(
B(3)0,2−B2,0(3)
)
r1,1,2
8r1,1,1
A(1)1,1
+ A(1)1,1
1
8
(
B0,2(3)−B2,0(3))r1,1,1+
(
B2,0(3)−B0,2(3)
)
r1,1,2
8r1,1,1
A(1)1,−1 +1
8
(
B2,2(3)−B2,(3)−2)r1,1,1+
(
B2,(3)−2−B(3)2,2)r1,1,2
8r1,1,1 .
(A.8) All Γ(m) become zero in the symmetric wave case, A(1)1,1 =A(1)1,−1.