Open boundary value problems in ocean dynamics by finite elements

Open boundary value problems in ocean dynamics by finite elements T O Y O K I T A N A K A , Y O S H I O O N O and. T O S H I K A Z U

ISHISE

Engineering Research Laboratories, Toray Industries, Inc'., Sonoyama 3-chome, Otsu, Shi.qa 520, Japan This paper presents recent results of application of the finite element models to wave overtopping and wave run-up problems in ocean dynamics. Open boundaries are prescribed as natural boundary condition obtained from the continuity equation of the Galerkin finite element formulation. The numerical results are, in general, reasonably good agreements with the histrical field data.

INTRODUCTION Many real problems of coastal engineering, previously intractable, can now be solved by mathematical models, The finite element technique is very suitable for the prediction of the water movement problem, if the boundary conditions are obviously prescribed, In computation of wave propagations, ordinary open boundary value problems were studied by prescribing water elevation. However, run-up behaviour (inundation) caused by tsunami or wave overtopping on a breakwater caused by typhoon surge is fairly difficult to prescribe the boundary conditions, In this paper, flow rate is prescribed along the open boundaries. Because flow rate was empirically or experimentally studied by many coastal engineers. Also, if water elevation cannot be given at open boundaries to the sea, flow rate per unit width (q = V,(h + r/) = r/x/g~ - + q)) must be prescribed there, where q is water elevation from mean sea level, H is depth of mean sea level, g is the acceleration due to gravity, and I/, is normal velocity along the boundaries, Integrating the continuity equation of Galerkin finite element formulation by parts, flow rate is obtained as the natural boundary condition. Two kinds of numerical examples are given in this paper. One is finite elements analysis of typhoon surge including wave overtopping on a breakwater, the other is finite element analysis of tsunami wave run-up behaviour, To evaluate the overtopping flow rate on a breakwater, Goda's graphical technique was adopted, which was obtained by many laboratory investigations, However, using the technique one must derive wave characteristics from sea waves prediction and typhoon surge computation. Numerical prediction of sea waves is derived after Nishimura and Horikawa's numerical significant waves theory which seems to be numerically advanced, compared with S-M-B Method 1 or Wilson's numerical method 2. Then, numerical results of typhoon surge analysis in Ise Bay including overtopping on a breakwater, denote a little difference compared with the results without overtopping. The values of water elevation and velocity in the downstream of a breakwater were a little greater by the effect of overtopping rate.

0309 1708/82/010021 0852.00

©1982 CML Publications

In tsunami wave run-up problems, it is very difficult to give reasonable boundary values on the moving wave front of flooding. At the wave front, flow rate is also given in this paper. Moving boundary effect is simulated by omitting dry elements from the computation and by adding wet elements into the computation. The value of flow rate is determined from conventional formulae, analytically or experimentally obtained by coastal engineers, i.e. flooding flow rate is proportional to h 3/2, where h is water depth in front of a dam or a step water bed 3. The problem is that this formula is derived from steady overflowing on a dam or a step water bed, and that h is an unknown variable. However, validation of the formula was numerically proved by Aida 4. Energy losses due to a sea bed geometry within the original shoreline and also due to obstacles on land are represented by the equivalent friction factor. The computed values of water velocity and water elevation in the sea and on land seem to be reasonable. The results of the computation presented here show that the finite element method provides a powerful approach to the open boundary value problems in ocean dynamics.

GOVERNING EQUATIONS The water pressure distribution is assumed to be given by the hydrostatic pressure in shallow waters. Under this assumption, imcompressive three-dimensional NavierStokes equations and continuity equation are integrated over the depth ofthe sea and reduced to the following twodimensional equations.

~M Rx - - - ' ?,t ~

?(UM) c?x

2(VM) ~- - ~y

- A~.V2M + ?2 Ux/U 2 + V 2 _ ~n

H +r/~P 0

f~(H +q)V+,q(H +~l)?~x -~ p

?x

(1 + k)P"),2 W~x/W~, + W~ = 0 P

Ry=_~N ~-+

~(UN) ~.,x~ +

(1)

9(VN) c~'y

A~V~N+y~Vx/'~+

Adl). Water Resources, 1982, Volume 5, March

21

Open boundary value problems in ocean dynamics." T. Tanaka et al. Wy

Equations (1), (2) and (3) can be written in the discretized form using the Galerkin technique,

~ i n d

press 7

~T ~ x H

y

~W~R~dA=0

(7)

~I4,~.R,.dA = 0

(8)

[I+IR¢ dA = 0

(9)

v

. , 4

wate/" F U ~ • ,,-

Figure I.

meansea level

-.,,,,,,

Integrating equation (9) by parts, using Green's theorem the following equation can be introduced.

z

Coordinate system

~f Wf-~l'pt ~zWiM,x -~14~ )

.~vvN~ d.4 = - Q

(10)

k

~q H +qi~Po f,(H +~/)U + g(H + q)= cv + p ~,y (1 + k ) - - 7 ; W , , , w~ + w; = 0 p R,.=~~q + ~M =- + =~N -=0 ~t ~x cv

where (2)

(3)

where

?,_

V2 = ~

72

+ ~--~

fX-

(11)

in which I~,ly are direction cosines of the outward unit normal vector of the boundary. Q is flow rate along the boundary and means as a natural boundary condition. At the open boundaries to land such as wave front of wave run-up and overtopping, the value of the flow rate Q is prescribed. At the open boundary to the sea such as a bay mouth, however, the value of water elevation ,7 is conventionally prescribed in this paper.

( I'-

M=(H+q)U, N=(H+q)V

(4)

in which U,V represent x,y components of vertically averaged velocity;H, distance from mean sea level to the bottom of the seaor land surface (H is negative on land) q, water elevation from mean sea level; P0, atmospheric pressure; f~, the Corioli's parameter, 2co'sin0, where co is angular velocity of the earth and 0 is degree of latitude; A,, horizontal eddy viscosity coefficient; .,2,~,~b,.,2 friction coefficients of sea surface and sea bottom shear stress; 0, the acceleration due to gravity; Wx,W., x,y components of wind velocity; p~,p, densities of atmosphere and water; k, the contribution factor of surface shear stress to the bottom shear stress, The co-ordinate system is shown in Fig. 1. The unknown variables in equations (I), (2) and (3) may be written in terms of linear trial function W(x,y), as

L:-- ~, 14~x,y).Uj~t)

(5)

J: ~

V-- ~ 14~,)'l.V~t) := ~

~1"-~ 14)~x,y)tlj~t) ;= ~ where N is the number of nodes in an element. The Galerkin's method of weighted residuals may be used such that: .~I.R dA =0. i = 1,2 ..... N,

(6)

where W~is weighting function, A the domain area and R the residuals of the equations, N, number of nodes,

22

Q=;W~(H +tl)~U.Ix+ V/y)dS

Adv. Water Resources, 1982, Volume 5, March

T Y P H O O N S U R G E ANALYSIS I N C L U D I N G WAVE O V E R T O P P I N G O N A B R E A K W A T E R As well known, the coastal area of Japan is often attacked by typhoons. Breakwaters have been built or are now under construction in several ports, in order to protect the harbour area by diminishing the energy of rushing water and by reducing the water level elevation inside the basin. The effect of a breakwater in Ise Bay was studied by the authors 5 using the finite element method, but effect of wave overtopping was not included. The phenomena of wave overtopping have close relation with the characteristics of water waves and configuration of a sea bed in front of a breakwater. To evaluate the overtopping flow rate on a breakwater, numerous experiments and investigations were carried out by many coastal engineers 3. The G o d a et al. 65 results of a series of experimental investigations clarified the effects of bottom gradient and wave steepness upon the overtopping rate. Experimental data were compared and supplemented with theoretical calculations by the combination of their random wave breaking model in shallow waters and the weir type overflow model. For example, one of their diagrams for the estimation of overtopping rate for bottom gradient 1/30, equivalent deepwater significant wave steepness (Ho/Lo=O.036), is shown in Fig. 2, in which q is overtopping flow rate per unit width, H equivalent deepwater wave height, L 0 the deepwater wave length, h~ the crown height of a breakwater above the tide level h, the water depth at the toe of a breakwater. These are shown in Fig. 3. If one gives values of Ho, L0, h~ and h, one can obtaine the overtopping flow rate using Fig. 2. The value of H 0 and Lo are determined in this paper using the Nishimura et al. numerical significant wave ~ technique. Details can be referred in their paper. The value

Open boundary r'alue problems in ocean dynamics: T. Tanaka et al. 2 [--H, iz,.0.036 I I I I l l IF I:,,,],.] I I I ~l I ,,q.2_, ~]°"l- i i I...---'t'-----~ILJ/)~ , , * a ~ z i * i I I/K_.~'] ~ b:, 2^_,1---- 1- - - ~ ."~.~/ ~. . . . -I.',,.i FI--L0_.~ . I..t III '/I/~ 0_,~l ~ ' -5 ~ _ / "%".x~/~o~--~.,..---4,~.r'~t~Jj,,~,J~ : j ~ / , - ; e ./ s .~- / . a --4 / %~ l ~ . -G :L : , ¢ ,~,-,/ I / iV ," .X,~:'7 ,...--.~",lNl'd&2s~LI ~ /l//..:r~'*-7" I 1,7,-, "~ I ~ I/ ;A.,/ ',,-.~.~-r" I ~NI,J'NI ,, ,~q,~/l~ I/!A'Z-U//)i,.~ I ~'~ •".-d /I /I,'1[ / X ~ ~,~,llllll/./,5,¢~,Z I I I I I,;~ ,o ILZ I z / : F . ' / I ,-~[~] ~ E ' t ' ~ L l r 2 z ~ . ' . . ~ r i , I~ t t " ; . ; r I//,'U/I ~*~E~,7'-iT~i~ZI I i l l I:1, 10-.,I- ,1t.;1~. l ! ~ . . . I [l till I I"r'.gl~l:,Y..I...I I . . H I..I,.,l~o-, .

--0.5

0

.

.

0.5

.

.

.

.

.

1.0

.

.

.

1.5

.

.

2

:

3

4 56

,

,

II1010-~

r ,1

J

, ,1,1

SlO'~

/,/Ho

i

, 1 4

510-=2 ¢(=V='

of h w a s n u m e r i c a l l y p r e d i c t e d u s i n g finite e l e m e n t s . N u m e r i c a l r e s u l t s of t y p h o o n s u r g e a n a l y s i s in lse Bay. i n c l u d i n g o v e r t o p p i n g r a t e o n a b r e a k w a t e r , a r e s h o w n in Figs. 4 8. C o m p u t a t i o n a l c o n d i t i o n is the s a m e as t h a t in our earlier paper 5 except overtopping. Figure 4 shows velocities distribution near a b r e a k w a t e r l o c a t e d in the n o r t h - e a s t b a y h e a d ( N a g o y a P o r t ) o f l s e Bay. Effect o f a b r e a k w a t e r u p o n the v e l o c i t i e s d i s t r i b u t i o n is s h o w n in this F i g u r e .

5 10 -~

,e¢)

Figure 2. Expected overtopping rate for a vertical wall. I f h / H o and h J H o are given, overtopping rate q is obtained from the Figure, (after Goda et al. 1970)

N F4T E R

EL EVRT [ 0 N

METERS

q. O00 L

Lo

3. 2 0 0

_1

2.400

1.600 0.800 ~ -0.800

Figure 3. A experiments

breakwater

used

in

the

overtopping

,,'~

~

,,.n

, ~

~

I

~

~

~

re

~

ru

re

re

ru

o0

to

o

--

r~

~

,=

(..n

~

HOURS

'"

-/",

'~

(a)

.

,

Figure 5. Computed water elevation time histories at the point, 900 metres south-west Jrom a breakwater. -- , Neglected overtopping: x , considered oeertopping

;- '; ; ;' "" [ "l /:.'-...

ii

(b)

)

ll.O

llq,~l£=l,

;/;:

I

>

I~ I~II[C l .

,tl,l J::":-."":;,":-:'" (c)

ill t,'

"

(d)

Figure 4. Velocities distribution in the vicinity of a breakwater in Nagoya Port, which are computed by including wave overtopping on the breakwater on condition that the 1959 Typhoon in lse Bav hit the present basin geometry in the same condition. (a) A t 19:00, Sept. 26, 1977, (b) at 20:00, Sept. 26, 1977; (c) at 21:00, Sept. 26, 1977; (d) at 22:00, Sept. 26, 1977

Adv. Water Resources, 1982, Volume 5, March

23

Open boundary value problems in ocean dynamics." T. Tanaka et al.

14R T E R

E L_F V A 1" 113 N

METEAS

ooo

~

q. 3. 200 2.q00

/ /

/

\ .. ~ . .

1.600 0.e00 -0.800

~ ~, ~ ~ ~ r~ <, &, ~ o7 o~ ~ oo ~o o - ~o ~

= ~ ~

.ouRs

Figure 6. Computed water elecation-time histories at the point, 450 metres north-east ]i'om a breakwater. Ne#lected ot'ertoppin,q: x , considered ot,ertopping

VE k 3 C [ T ,T >

'~\ k9

• 2c,

:~

/.2

~

3

~/sec:

o.~

~, \\

,23

P R @F ~ L E 5 .

,\,o~,,~

~'-k

~

Figure 7. Computed celocity-time histories at the point, 900 metres south-west]?om a breakwate,'. ,, Neglected overtoppiml, ---*, considered overtopping

peninsula. Tsunami waves deform due to changes of bay shape as well as depth. The relation between the predominant tsunami period and the natural period of the bay has a very important role in the tsunami height variation inside the bay. And that, wave inundation or wave run-up onto land causesa tremendous number of damages in the vicinity of coastal area. Therefore prediction of a tsunami propagation is very important for coastal inhabitants. For further discussion the reader should refer to Horikawa's book 8. Tsunami w a v e p r o p a g a t i o n h a s a l r e a d y b e e n a n a l y s e d numerically 49-~2 using the finite difference method. Finite element approach has recently been made by Kawahara ~3 and others. As the numerical examples, the 1946 Nankai tsunami and the 1960 Chilean tsunami that hit the coast of Susaki, Kochi prefecture, have been adopted in this study. The present paper is especially concerned with the application of merits of the finite element techniques to a tsunami wave run-up problem including moving boundaries. Using the conventional Galerkin procedures, the finite element method is applied to space functions of velocity and water elevation. The interpolation function is a linear polynomial based on the three nodes triangular finite elements. To discretize time, a two step Lax Wendroffmethod is used after Kawahara x3. The time step At is chosen to be 10 sec satisfying the stability condition. As the boundary conditions shown in Fig. 9, the reflection condition is used along the coastal lines. That is, the normal velocity to the coastline or flow rate Q is assumed to be zero.

Q=0 VELSCITx

P~gFILES. O.S

t~'SEC;

or

U'lx+ V l r = O on Sl

x-

~,-~"

"<--~'~:.

:

Fiqure 8. Computed celocity time histories at the point 450 metres north-east li'om a breakwater. ,, Neglected overtopping, - ---< considered ocertopping

(12)

The enforced wave elevation is conventionally prescribed along the open boundary to the sea, such as the vicinity of bay mouth S_~. The wave of the incident tsunami is assumed to be sinusoidal. q = G + q,,- sin(2n/T,,-t} on S,

The comparison of overtopping with non-overtopping is shown in Figs. 5-8. The effect of overtopping appears more significant in the downstream of a breakwater than in the upstream. However, numerical results of typhoon surge analysis in lse Bay including overtopping on a breakwater, have made little difference compared with the results without overtopping. This is due to the fact that overtopping flow rate is much smaller than the water volume near a breakwater. These trials seem to be significant for increasing numerical accuracy of typhoon surge computation.

open

S 1 /l z

f

boundary/" _ S3t,' Q=Q /

~

~ ~ \S 3 bay

S i~

/

~,,,._ j i'~ S

T S U N A M I ANALYSIS BY FINITE E L E M E N T S Tsunamis are long waves generated crustal movement from submarine earthquakes, by the eruption of submarine volcanos, or by large scale landslides on the coast, A tsunami propagates from its origin towards coastal area, with the speed of the long wave celerity. Diffraction occurs in the vicinity of an island as well as that of a

24

Adv. Water Resources, 1982, Volume 5, March

~ ,

the Figure 9.

s

~ b r ~ ! " " e

Q=O a

S2 Boundary conditions

~

(13)

Open boundary value problems in ocean dynamics: T. Tanaka et el.

Step i. Estimation of run-up

reflection condition /~~~~

n : water surface estimated run-up (h>0) ~o: water surface not estimated~/ run-up (h<0) W : heighth=H+n_wOf a barrier

0 -

~h ~ |j / , / , / ,

" [~--I ~_,I~ I.-~'-H"

shoreline Step 2. Prescription of Q on the boundary_ Q = Cob gCTff.B

B : width of a step bed or a step structure. ~ - -

--

/fundamental tide level in computation. Step 3. Re-estimation of run-up

~

/

~

~

water thichness (H+~)>0 J

g f f J f

new s h o r e l i n e ~///

//

extended

f

computational

region

~

land

new shoreline

Fiyure 10.

Tsunami wave run-up alyorithm

where r/$ is land subsidence height, qe and T~ are amplitude and period of the incident tsunami respectively. The amplitude and period are unknown and cannot be determined at the beginning. After several numerical trials, the amplitudes and the periods on the boundary $2 are determined to be 2.55 m and 35 min at the 1946 tsunami, and 2.50 m and 45 min at the 1950 tsunami, respectively. These values are quite reasonable considering the limited historical field reports, and will be

indispensable for the simulation in the present or future coastal geometry. Other open boundaries $3 shown in Fig. 9 are especially important in this paper. These are non-linear and moving boundaries because of a moving wave front. Along this open boundary, flow rate Q is also prescribed in this paper. The steady state flow rate of running up on a step structure is proportional to the three-half power of the water thickness at the toe of a step structure. The relation

Adv. Water Resources, 1982, Volume 5, March

25

Open boundary value problems in ocean dynamics: T. Tanaka et al.

element descritization in the computational region are shown in Fig. 11. The numerical results obtained in the way described above, are shown in Figs. 12 and 13. Figure 12 shows the velocity distribution of the 1946 tsunami including run-up onto land. In this Fig. the solid line inside the

_ ~

0 ,

,

,

'

Figure 11. Finite element discretization (1333 elements, 858 nodes) and distribution of the assumed friction factor for

is empirically or experimentally introduced by many coastal engineers3, which is written as follows, Q = C o B hx/gh

(14)

where h=H+q-

W

on $3

s

~

.

"

~,,~,

"" " "

" -

~ ,

-

"

Figure 12. Computed water velocities at 85 minutes after initiation of the incidence of ttle 1946 Nankai Tsunami model waves from the southern end ofdiscretization system. The solid line inside the computational region shows the original shoreline between the sea and land. t t ~ , 4.0 m/sec

(15)

in which COis empirical constant 0.5, B the width of a step structure, g the acceleration due to gravity, and W height of a barrier, shown in Fig. 10. In this equation q is the unknown variable so that the boundary condition is nonlinear. We estimate the value of r/from the computation using the reflection boundary condition of $1 instead of $3. The value seems to be overestimated, but the effect might be slightly diminished by adopting elementaveraged value in the finite element procedure. The algorithm of wave run-up is shown in Fig. 10. Moving boundary effect at the wave front, is simulated by omitting dry elements from the computation or by adding wet elements into the computation. If the sign of water thickness (H +q) on an element is positive, the element assumes to be wet. Energy losses due to a sea bed geometry within the original shorelines and also due to obstacles on land are represented by the equivalent friction factor after Aida 4. The value of friction factor of the bayin tsunami analysis is approximately 1.5 times as large as that in astronomical tidal analysis, which is presented by Kajiura ~4, therefore~ is determined to be 0.0035 in the bay. On the other hand, the value of bottom friction coefficient to land is distributed from 0.01 to 0. l according to the surface geometry of land, which is also after Aida '~. The distributed values of friction coefficient and finite

26 Adv. Water Resources, 1982, Volume 5, March

original

h \" x, k _ ~ ~.~ ~

_ ~ _ ~ ~ " r",-/,*~' - . . I S" ~,

~

. ~'ii,~

~ I ~"""

i

!

~

':j.~.?!!

/

original

~, ,, . . shoreline .~/ , #'' ~ i . .'\ . ~". . . ~. ~ . . . . ~ . ', ' ' - ~. ' ' " " -. . . . " . / " ~ ." . , , •

1'

b

,

Figure 13. Computed water velocities at 65 minutes after the initiation of the incidence of the 1960 Chilean Tsunami model wavesJrom the southern end ofdiscretization system. I I-'*, 4 m/sec

Open boundary value problems in ocean dynamics: T. ~maka et el. Tablel. The ef]ect c?[the incident tsunami period upon the computed runup he~qhts {m)

Observed* point [see Fig. 151

Incident tsunami period

Observed* height Im)

45 min

50 rain

55 min

3.25 2.83 2.04 2.75 3.05

3.37 3.08 2.05 2.76 2.99

2.66 3.14 2.02 2.63 2.88

2.68 3.15 1.98 2.53 2.75

c1 c3 c4 c6 CI0

tsunami in 1946 and Chilean tsunami in 1960 were practical examples in computations. Some assumptions were presented in this paper. These must be verified comparing with the field data or experimental results. The results of the computation presented here show that the finite element method provides a powerful approach to

I

CI[3.25

* 1960 Chilean tsunami.

N

!~

\\

observed(meters)

(~.37) comp~t~(,,~te~sl

,

c6 2 -75 h_/-v ~ (2.761 'r-c,~2.o, 2 . 7 5 7C7 \(2.05}

NIl5.10 observed(meters) (5.50) .......a(...... )

(....)

.

~

2

C$ X 2 . 0 0

{2.931 12.05)

3

3 0~

~ ,~.~) 3.4Q 9N14 TM

,~:~L7 ~ "

f3

l elk 3.25

,.... )

:

1].691

N8~ 3,40 (21.59) N2,~ 5.?0 5.68

~15

/

~

/ ,\

N7 X 4,70

~ (4"32)7~

2.83

C ~[3.0B)

N6~'13.40

~ "

. .o ......v V

<

15.451

with the computed maximum heights o/ 1960 Chilean tsunami. H eights are measured relative to the mean sea level at the time of the arrival o( the tsunami Figure 14. Comparison of the observed inundation heights with the computed maximum heights of 1946 Nankai Tsunami. Heights are measured relative to the tide level at the time of the arrival of the tsunami

computational region shows the original shoreline between the sea and land. Figure 13 shows the velocity distribution of the 1960Chilean tsunami including run-up onto land. It may be considered from the two Figs. that the effect of the incident predominant tsunami period and shoreline geometry appears sensitively in the velocities distribution. The effect of the incident tsunami period upon the computed run-up maximum height is shown in Table 1. As is evident from Table 1, the computed heights agree best with the observed ones in case that the incident tsunami period is 45 min. The comparisons of the numerical results with the historical field data are presented in Figs. 14 and 15. From these Figures, measured inundation heights and their distribution are fairly well reproduced by the present numericalmodeI. Therefore we will be able to predict and study the dynamic behaviour of tsunami waves in the present or future coastline of Susaki. One of the examples is shown in Fig. 16. CONCLUSIONS A numerical model was developed in which open boundary value problems such as wave overtopping or wave run-up were solved using the Galerkin finite element approximations. Typhoon in Ise Bay in 1969, Nankai

,velocity

,, .~ ~.o ,,/~c~. _..f'-,J~.,'CJ,t ' ~ . . . S / - , ~t,, , j~, ~ ~ ' ~ t' ~-;/ ,,',', ~ ' ~",,"~~ ' 3 - % } /x,,, j-J-7~ (~,~ "" " , '-, ~ ~ - - , a "'/~'~ ~e~'s'~'''~' ~ ~ ~'.___,_ ~ ~, .~,~,~,~.~,:,~ ' ~ ' ~ ~ a . ~" ~ ~, ..-.-...~r~,:... . . . . " ~ ' ~ ' . : ~ ~ , " i,~-originalshoreline f ; " ~'," 'k'" ' ~ , f 2 - " "~-5 . , ' _ ', ~ " ,~ V---~.

)

Figure 16. Predicted water velocittes at 65 minutes after the initiation of the incidence q] the 1960 Chilean tsunami model waves from the .southern end o/ discretization system. The solid line inside the computational region shows the original shoreline between the sea and land. An imaginary peninsula is located near the bay centre, hatched in this Fig., as compared with Fig. 13

Adv. Water Resources, 1982, Volume 5, March

27

Open boundary value problems in ocean dynamics." T. Tanaka et al.

the ocean dynamics. We believe that the techniques used

4

in t h i s s t u d y a r e a l s o a p p l i c a b l e t o t h e p r o b l e m s o f r i v e r flood.

5

ACKNOWLEDGEMENTS

6

The authors wish to express their sincere appreciation t o P r o f e s s o r K. K a j i u r a a n d D o c t o r I. A i d a , t h e I n s t i t u t e of

7

Earthquake Research, University of Tokyo, for their s u g g e s t i o n to t h i s s t u d y . T h e w o r k w a s d o n e u n d e r t h e c o n t r a c t of P o r t s a n d H a r b o u r s D e p a r t m e n t , K o c h i P r e f e c t u r e , J a p a n . T h a n k s a r e d u e to e n g i n e e r s o f t h e

Department for providing the authors with valuable materials and useful suggestions.

8 9 10 11

REFERENCES 1 2 3

28

Sverdrup, H.U. andMunk. W.H Wind, sea and swell; Theory of relations for forecasting, US Nat')' Hydrographic Office, Pub., No. 601, 1947 Wilson. B. W~ Numerical prediction ofocean waves in the North Atlantice for December, 1959, Deut. Hydrogr. Z. Jahrgang 1965, 18, t3~, 114 Yoshikawa, H. [Ed.~ Hydraulic Formulae. Japan Society of Civil Engineering, 1971

Adv. Water Resources, 1982, Volume 5, M a r c h

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