Numerical study of long wave runup on a conical island

VN U J o u rn a l of Sciencc, L arth Scicnccs 24 (2008) 79-86

Numerical study of long wave runup on a conical island

P h u n g D a n g H ieư*

Ccntcrfor Marinc and Occan-Atmosphcrc Intcraction Research

R cccivcd 5 Jn n u ary 2008; rccoivcd in rev ised form 10 July 2008

A b stra ct. A n u m eric aỉ model b ascd on thc 2D shallovv w a tc r e q u a tio n s was d c v c lo p c d u sin g the

F inite V olum c M cth o d . T h e m odcl w as ap p lic d to th e stu d y o f long vvave p ro p a g a tio n an d ru n u p o n a co n ical is ỉa n d . The s im u la tc d r c s u lts b y th c m o d c l w c rc c o m p a r c d w ith p u b lis h e d ex p c rim cn tal d a ta a n d an n ly zc d to u n d c rsta n d m o rc about th e in tcractio n p roccsscs bctwc?en the long vvavcs a n d conical isla n d in tc rm s of w a tc r p ro íile and w a v c ru n u p hcight. 'ITic rc su lts o f the stu d y c o n íirm c d th c cffccts of c d g c vvaves o n thc r u n u p h c ig h t at th c lcc sid e of th c island.

Kcyiuords: C onical island; R u n u p ; rin ite volum e m e th o d ; S haỉlow w a tc r m odcl.

1. Introduction

Sim ulation of tvvo-dim ensional cvolution and ru n u p of long w av es on a sloping beach is a classical problcm of hydrodynam ics. It is u su ally related vvith th e c alc u la tio n of Coastal effects Oi long w av es such as tido and tsunam i. M any research ers h av e contributcd signiíicantly c íío rts to th e d cv elo p m en t of m odels capable of so lving the problcm .

Notable studies can bc mentioncd. Shuto and

Goto (1978) developed a num erical m odcl based on íinite difference m cth o d (FDM) on a staggered schem e |9j. H ibbcrt an d Peregrine (1979) [2] proposed a m odel solving the shallovv vvater eq u atio n in the conservation form using the Lax-W ondroff schem e and allovving for possiblc calculation of w ave brcaking. Hovvever, thcir m odcl had not bcen capable to calculate vvave ru n u p an d obtain

* Tel.: 84-914365198.

E-mail: phungdanghieu@vkttv.cdu.vn

physically realistic solutions. Subsoquontly,

Kobayashi et al. (1987, 1989, 1990, 1992) [3, 4, 5, 6] re íin e d th e m o d e l for p ractical USG, by a d d in g dissipation term s in the íinite- differencc equations, w h at is novv the m ost p o p u la r m ethod for solving the shallovv w atcr oquations. Liu et al. (1995) [7] m odeled the ru n u p of solitary w av e on a d rc u la r island by FDM. Titov an d Synolakis (1995, 1998) [11, 12] proposed m odels to calculate long w av e ru n u p on a sloping bcach and cừcular island using FDM. Wei Gt al. (2006) [13] developed a m odcl based on thc shallovv w atcr eq u atio n s u sin g tho íinite volum e m ethod to sim ulate solitary vvaves ru n u p on a sloping bcach and a cừcular island.

Sim ulated rcsu lts obtaincd by VVei et al.

agreed notably vvith laboratory experimcntal data [13].

M em orablc tsu n am i in Indonesia and Japan caused m illions of dollars in dam ages an d killed th o u sa n d s of peoplc. O n Decem ber 12, 1992, a 7.5-m agnitudc earth q u ak e off

79

80 Phung Dang ỉ ỉieu / VN U Ịounial ọ f Science, r.artli Sciences 24 (200S) 79-86

Flores Island, Indonesia, killed nearly 2500 people and w ash cd aw ay e n tữ c villages (Briggs et al., 1995) Ị1Ị O n JuUy 12, 1993, a 7.8-m agnitude earth q u ak e oíf O kushiri Island, Japan, triggcrod a d cv astatin g tsunam i w ith rGCorded ru n u p as h ig h as 30 m . This tsunam i rcsulted ứì larger p ro p crty dam age than any 1992 tsunam is, and it com pletely in undatcd an village w ith overland flow.

Estừnatcd propcrty d am ag e w as 600 million u s dollars. Rccently, the h ap p en cd at Dccember 26, 2004 S um atra-A ndam an tsunam i-earthquake in the Indian Ocoan vvith 9.3-m agnitude a n d an opicenter off thc wcst coast oí Sum atra, Indonesia had killed m ore than 225,000 pcople in cleven countrics and resulted in m ore th an 1,100,000 people hom cless. Inundation of Coastal areas w as created by vvavcs up to 30 m etcrs in height.

TTiis was tho ninth-deadliest natural disastor in

m odorn history. Indonesia, Sri Lanka, India, Thailand, and M yanm ar w cre hardost hit.

Field survcys of tsu n am i dam agc on both Babi and O kushiri Islands show ed unexpcctcdly Iarge ru n u p heights, especially on tho back or leo sido of the islands, rcspcctively to the incidont tsunam i dừection.

D uring tho Flores Island evcnt, tw o villages locatcd on the Southern sid c of the circular Babi Island, w hose d iam eter is approxim atcly 2 km, w cre vvashcd aw ay by the tsunam i attacking from thc north. Sim ilar p henom cna occurrcd on th e pear-sh ap ed O kushiri Island, w hich is approxim ately 20 km long and 10 km w ide (Liu ct al., 1995) [7Ị.

In this stu d y , the interaction of long vvaves and a conical island is investigated using a num orical m odol bascd on the shallovv w ater cq u atio n an d finite volum e m ethod. The stu d y is to sim ulatc tho processes of vvave p ropagation and ru n u p on thc island in o rd cr to u n d e rsta n d m ore thc ru n u p p h en o m en a on conical islands.

S u p p o rtin g to the sim u lated rcsults by tho m odel, the cxperim ental d ata p ro p o se d by Briggs el al. (1995) Ị1 ] vvere used.

2. N u m erical m odel 2.7. Govcrning cqaation

T h e present stu d y conbiders tvvo- dim cnsional (2D) d c p th -in tc g ra tc d shallovv w ator equations in tho C artesian coordinate system ( x , y ) . T he conservation form of the non-linear shallow w atcr e q u a tio n s is vvritten as [13]:

ỔF dG „

-— = s

dt dx dij 0 )

w h ere

u

is the vcctor of conscrvcd variablos;

F, G is the flux vectors, respcctively, in the

X an d 1/ directions; an d

s

is tho source term . Tho explicit form «f thoso voctors is cxplaincd

as follows:

■

H

■ ^{H u}

u =

^{H u .} F = H u 2 + ị g H 2

H v H u v

G =

H v H u v H v 2 + ị g H :

s =

0

* Õx p

g H Ệ - -_{d\j p}

(2)

vvhere g : gravitational acceleration; p : vvater density; h : still vvater d c p th ; H : total vvater dopth, H = h + T| in w hich ii(.v,Ị/,f) is the displaccm ent of w a ter su ríacc from tho still w atcr level; Tx , T y: bottom shcar stross givon by

Tx = p C f U y Ị i r + V 2 ,

Ty = ọ C f V y j u 2 + v 2 , Cf = S" ⁽³⁾ H^{1 /3}

w h cre n: M anning coc'fficient for tho suríacu roughness.

Phung Dang l licu / VNU lơunial o f Scicncc, i.nrtlĩ Sciciĩces 24 (2008) 79-S6 81

2.2. Numcrical schcmc

The fmito v o lu m c íorm ulation iinposes conservation law s in a control volum e.

Intcgration of Eq. (1) ovcr a cell w ith thc application of the GrceiVs theorem , gives:

í n f r d Q + ỉ r (F” * + G "* ) í í r = í o s d n ' (4) vvhcre Q : ccll d o m a in ; f : b o u n d ary of Q ;

{ n x*n ỵ ) : n o rm al outvvard vector of the boundary.

T aking ti m e in tcg ratio n of Eq. (4) over d u ration At írom t-ị to t2, w c h a v e

J u (x,y, f2 )dQ - u (.Y, \J,í, )dn

h h (5)

+J*J ^(Fỉix + G n v)d r = ỊdtỊ Sdn

h «1

Tho prcscnt m odel usos un iío rm cells vvith dim cnsion A.V, Ai/, thus, tho integrated governing c q u a tio n s (5) \vith a tim o step At can be ap p ro x im ate d vvith a h alí tim e stcp avcragc for tho in teríac c Auxos and sourco torm to bocomo:

At_

Ax

r->kf\j2 "1 . Ạ i^A.tl/2

— J Af Í

5

i./

vvhere i, ị are in d iccs at tho ccll center; k denotes the c u rre n t tim e stcp; the half indices

í + 1 /2 , í - 1 /2 a n d ý + 1 / 2 , / - 1 / 2 indicate tho cell intcríaces; and Ả:+ 1 /2 denotes tho average w ithin a tim o stcp betvvoen k and k + 1. Note that, in Eq. (6) the variables u and source term s arc cell-avcragcd valuos (vve use this m c a n in g from novv on).

To solve Eq. (6), w e n eed to estim ate thc num erical íluxcs ĩ ^ / ỉ ị , ĩ ị - v ỉ ị an d G ^ l y 2f

at thc cell in teríaces. In this study, vve use the G odunov-typo schem e for this purposo.

According to the G o d u n o v -ty p c scherrte, tho num crical íluxcs at a cell intcríaco could be

+ rc**i/2 1

. *V+1/2,/ 1 /2./ 7 Ay (6)

obtained by solving a local Riem ann problem at tho interíacc.

Since đircct solutions are not available for tw o or threc dim cnsional Ricm ann problcm s, thc present m odel uscs the sccond-order splitting schem e of Strang (1968) [10] to sep arate Eq. (6) into tvvo one-dim cnsionai equations, vvhich are intcgrated scquontially as:

u*}1 = XAI/2YA'X AÍ/2U*/ (7)

vvhere X and

y

den o te the intcgration operators in the X an d y dừections, respectively. Tho cquation in the .V direction is íirst in tcg rated over a half tim c stcp and this is íollovved by integration of a full tim e step in the y direction. These arc expressed as:

Ĩ ] [ k + ì f 2 ) r ỊẢ r » 1 4 p i ♦ 1/ 4 "I

(₈)

*T<S.C

^*-l

^/4

TT^I) = ii*+i/2) _{u ./ - u ./} M - r

*’.1'2 1

W./+1/2 " ^./-1/2

Ạl/ L J (Q\

+*(SJ,)Í.?'2

w here the astcrisk (*) indicatcs partial solutions at the respective tim e increm ents vvithin a tim e stop and Sx , S v arc the source term s in the X dừ ection and

1

/ directions.

lntcgration in thc X dừ ection ovcr the rem ain in g half tim c stop advanccs the solution to the next tim e step:

TT^+l I ĩ(fc+l) ^ í ~ ♦ 3/ 4 xjỉ:+3/4 ~1

u «./ “ Ui./ 9Av|_rí-l/2.i »1/2./J (10) + ^ < S , ) Ĩ Ta + 3 / 4

2 ' -'x ) i.j

T h e partial solutions ư ^ , U jy i/2) and u< ^> , p rovidc the interíace flux torm s ừi equations (8), (9) and (10) through a Ricmann solver in onc-dim ensional problem s. In this study, w e usc the HLL approxim ate Riemann solvcr for the estim ation of num erical íluxes.

For tho w et and dry cell treatm ent, w c use the

82 Phung Dmig llieu / VNU Ịounínl o f Sàence, Enrth Sciences 24 (2008) 79-86

m inim um w et depth, the cell is assum ed to be dry if its w ater d ep th loss th an the m inim um vvct depth (in this stu d y vvc choose m inim um vvct dcpth of 10'5m).

3. S im u latio n resu lts an d d iscu ssio n 3.1. Expcrimeiĩtal condition

A num erical expcrim ent is carried out for the condition sim ilar to the experim ent done by Briggs et al. (1995) [1]. In this expcrim ent, there vvas a conical island Setup in a w ave basin having th e dim cnsion of 30 m vvide and 25 m long. Tho conical island has thc shape of a truncated cone w ith diam etors of 7.2 m at the base and 2.2 m at tho crest. The island is 0.625 m high and has a side slope of 1:4. The suríacc of the island and basin has a sm ooth concrete íinish. Thoro is absorbing m aterials placcd at tho four sidcw alls to reduce w ave roílection. T h e vvater d e p th is h=0.32 m. A solitary vvave w ith tho hcight of A / h = 0.2 was gcncratGd for tho exporimental observation.

Fig. 1 shovvs the sketch of the cxperim ent and w avc gauge location for w ater suríace m casurcm ent. Five tim e-sories data of w atcr suríace elevation vvere collccted for the com parison.

Dr =2-2m

h' 0625m

- = 0.2

h

/ i- 0 .3 2 m

- !■ _ ■ Db=7.2m

B * 30rt»

C:t

• G1 L=25m —

[n Fig. 1, the vvave gauge GI is Setup for the m casu rem en t of the incident vvaves; w av e gauges G6 and G9 are for thc vvaves in the shoaling area; and thc vvave gauges G I 6 an d G22 are respcctively, for vvaves on th e right side and lee sidc of the island. T he locations of th e five w a v c gaugcs are given in Tablc 1 in relation vvith tho center of the island.

T ab le 1. L ocation of w a v e g a u g e s

G a u g e n u m . x - x c (m ) \ j - y c (m)

G I 9.00 2.25

G6 3.60 0.00

G9 2.60 0.00

G16 0.00 2.58

G22 -2.60 0.00

(x ( , yc ): c o o rd in a te o f th c cen ter o f th e island

3.2. Numerical simulation and discussion

In thc num erical sim ulation, a com putation d o m ain is sctu p sim ilar to the expcrim ent.

The m csh is regular vvith grid size of 0.1 m in both X an d \J directions. At four sides of th e co m p u tatio n dom ain, radiation b o u n d ary conditions arc u sed in o rdcr to allovv w avcs to go íreely th ro u g h tho sidc bo u n d ary . A solitary vvave is g cn cratcd as the initial condition at a line parallcl w ith the \J direction, and located at the distance of 12.96 m from the center of thc island. The M anning cocffíciont is sct to be constant n= 0.016. Tho initial solitary vvave is created by using the íollovving equation:

T|(.r) = i4sech2

0 ( x~x‘)

⁽¹¹⁾

(12)

Fig. 1. Sketch of the experinìgnt.

whcrc x s is thc ccntcr of the solitary wavc.

The num erical rcsults of w ator suríace elevation at fivo w ave-gauge locations and ru n u p h eig h t on the island arc recordod for

Phung Dang Hicu / VN U Ịoumal ofScicìĩce, Earth Scicìices 24 (2008) 79-86 83

validation of the sim ulation. Fig. 2a show s the tim e proíile of vvater suríaco elev atio n at tho vvave gaugc G l. In this íigure, it is sccn that the incident solitary w av e sim u lated by thc m odel agrees very well w ith th e experim ental data. This gives us a coníidence in com parison of tim e series of vvater suríace elevation at other locations in tho com putation dom ain, as vvell as in com parison of vvavc ru n u p on thc island.

In the Fig. 2b a n d 2c, at th e w av c gaugcs

G6 and G9, it is seen th at thc solitary vvavc is wcll sim ulated on the shoaling rcgion, the w ave com es to the location aítcr about 4 soconds ừ o m thc initial timc. At íirst, thc num crical results an d expcrim ental data agree vcry w cll/ aíter that, there are som e discrepancy appcared. This deílection can bc explaincd d u e to thc reílection from tho side bo u n d arics in thc experim ent donc by Briggs ct al, m u ch largcr than that in thc sim ulation.

0 5 10 15 20

Time (sec)

Time (sec)

Time (sec)

Fig. 2. C o m p a riso n of vvater s u ría c e e lc v a tio n at locations G l, G6, G9: so lỉd th in line: sim u la te d by com nion shallow w a te r c q u a tio n ; so lid th ick linc: s im u la te d by a d d in g B ơussincsq te rm to the shallovv w a tc r cquation.

84 Phung Dang Hieu / VNU Ịơunwỉ o f Science, Earth Scioĩces 24.(2008) 79-86

Time (sec)

Time (sec)

Fig. 3. C o m p ariso n of w a tc r su ría c e elevation at locations G16 a n d G22: so lid th in linc: sim u la te d by coĩìim on shallow vvator cq u atio n ; solid thick linc: sim ulatecỉ by a d d in g B oussinesq tc rm to th e shallovv w a tc r equation.

It can bc coníirm cd írom thc íigure that, the numorical rcsults very soon becom c stablc having non-fluctuation w h en tho w ave goes írecly out of the experim cnt dom ain.

Inversely, the expcrim cntal data havc a long tail of d isturbance and could not be calm aíter 20s (see Fig. 2j at w avo gaugos G6 and G9;

and Fig 3, at vvave gaugos G16 and G22). This Auctuation is d u e to the vvave encrgy dissipation not enough at the sides of tho experim ent basin. Hovvever, the form and height of thc arriving solitary w avc at all locations are vvell m atched bctw ecn experừnental and num erical results. This is very im portant to allovv latcr com parison of w ave ru n u p on tho island.

From Fig. 2 and Fig. 3, it is also seen that, the w ave hcight at the lee side (gaugc G22, Fig. 3b) of th e island is still very high in com parison w ith tho height at thc íront side (gaugc G6, G9, Fig. 2b, 2c) of the island, and m uch biggcr th an that at the right side (gauge G16, Fig. 3a) of the island. T hese results give

us a confidcncc in co níirm ing thai tho vvave height at lce sid e of an circular island can be large also. In Fig. 2 and Fig. 3, tvvo sots of num crical results are plottcd. O ne is sim ulated by the com m on non-linear shallow vvatcr equation (NSW ), and thc othcr is simulatGd by a d d in g the Boussinosq dispersion term [8] into thc NSW. From the íiguros, it is co n íirm cd that tho m odcl using the Boussinesq app ro x im atio n can givc sim ulated rcsults m u ch b etter than th c com m on NSW bascd m odel. T hus, for thc practical p u rp o se of sim ulation non-lincar long w av c problem , tho Boussinesq ap proxim ation torm s should bc considorcd.

Fig. 4 show s th e sn ap sh o t of vvatcr suríacc displaccment on tho computation domain. From the íigure, w e can sec that, aítor tho solitary w av c comcs to the island, thc w av c roíraction ap p cars d u e to the variation of w ater depth.

B chlnd the island, the cdgo vvavcs comc from tw o sides of the islan d d u e to w avcs bcn d in g a ro u n d the island an d m atching togcther at

Phuììg Dang iìicu / VNU Ịounuìỉ ọ f Science, Larth S c ì c ỉ ì c c s 24 (2008) 79-86 85

tho leeside of the island. Then, they íorm an aroa of very high w ave ru sh in g up to thc lcc side coast of the island. This m echanism can bc cxplainod for tho unexpectedly largc ru n u p heights on thứ leosidtí of tho Babi and O kushiri Islands d u c to the tsunam i.

Fig. 5 is tho com parison of w avc ru n u p aro u n d the island, betvveen num crical sim ulation and experim ent. The horixontal axis in tho íiguro indicatcs tho anglc betvveen the line drasving from thc conter of the island to the point of ru n u p m casu rem cn t and tho \J

diroction. Thi} anglo of 0 degree m cans th at the m easuring point is at tho right side of tho island and on Ihc line through tho centcr of the island and norm al to the incident w ave diroction (i.e. parallol to tho \J direction). It is shovvn from tho íigurc that, thc ru n u p is highcst at the ío resid e of thc island, tho

m axim um sim ulatcd ru n u p hcight is somevvhat less than oxperim ontal data. At the lecside of the island, thorc is an area vvith ru n u p h ig h cr than both sides of the island.

The num erical results of ru n u p hcight in this area are also sm aller th an cxperim ental data.

These m ight be d u c to the fact that tho com putational m esh not fine en ough to capture highly non-linear interactions of edge w aves at the Ieeside. In overall, tho num crical m odcl can sim ulatc woll tho ru n u p height at m any locations aro u n d tho island. Especially, thc tendcncy of tho ru n u p variation and ru n u p location are well sim ulatcd by the prescnt num erical m odcl. This m eans that, the model dovelopcd in this study has potential íeatures to apply to the study of practical problem s rolatcd w ith long w aves, such as in u n d atio n of tsu n am i on Coastal areas.

Fig. 4. S n ap sh o ts o f the w a tc r su ríac c d isp lace m e n t d u c to the so lita ry w ave.

Angle (deg)

Fig. 5. R unup of w a te r a ro u n d th e islan d d u c to th e solitary w av e (270 deg.: at ío rc sid c in thc n o rm al directio n of w a v e p ro p a g atio n ; 90 dcg.: at thc leeside of the island; 0 dcg.: at th c rig h t sid c o f th c island;

an d 180 dcg.: at thc left sid c of th e island).

86 Phung D ang Hicu / VNU Ịountal ọ f Sôence, Larth Scieììces 24 (2008) 79-86

4. Conclusions

A 2D num erical m odel based on thc shallow w ater cq u atio n has been successíully developed for tho sim ulation of long vvave propagation, d eío rm atio n and ru n u p on the conical island. The num erical results sim ulated by NSW m odel and by Boussinesq m odel revealed that by a d d in g Boussinesq tcrm s to the NSW m odel, sim ulated results oí long w ave pro p ag atio n and d eíorm ation can bc signiíicantly im proved. Thercíore, it is vvorth to m cntion that Boussinesq approxim ation should be considered in a practical p roblcm related w ith long vvaves.

Tho m odcl also has potcntial íeatures to apply to the study of practical problem s related to long vvaves, such as inundation of tsunam i on Coastal arcas.

Simulated rosults in this study also

contìrm that tho arca bchind an island can be attacked by big w aves Corning from the opposite sidc of the island d u e to non-linear intcraction of edge vvaves rcsulted from rcíraction processes.

Acknowledgments

This p ap er w as com pleted w ithin thc fram ew ork of Fundam cntal Research Project 304006 íu n d c d by V ietnam M inistry of Science and Technology.

Reíerences

[1] M.J. B riggs ct al, I-ab o rato ry cx p c rim cn ts of tsu n am i r u n u p o n a circ u lar island, Purc Applied Geophỵs. 144 (1995) 569.

[2] s . H ibbert, D.11. P oregrine, S u rf and r u n u p o n a bcach: a u n iío rm bore, Ịoum aỉ o f riu id Mechanics 95(1979) 323.

[3| N . K obayashi, A.K. O tta, I. Roy, VVave rcílection a n d ru n u p o n ro u g h slo p es, /. W atenvnỵ, Port, Coastal a)ĩd Ocertì ¹ Lngỉncering 113 (1987) 282.

[4] N . K obnyashi, G .s. DeSilva, K.D. VVattson, W av e tra n sío rm a tỉo n a n d sw a sh oscỉllations on g c n tlc a n d s tc e p slopes, lounuil o f Geophysics Research 94(1989) 951.

[5J N . K obayashi, D.T. Cox, A. W urjanto, Irrc g u la r w ave rcflcction and ru n u p on rough im pcrm cablc slopes, ]oun\nl o f Watcrway, Port, Coastal ntid Oceaìì Engineering 116 (1990) 708.

[6] N . K ob ay ash i, A. VVurịanto, Irrcg u lar w a v c s c tu p a n d r u n u p on beaches, Ịúuntal Waterw(ìỊ/, Pưrt, Coastal and Occan Eìỉghiecrbíg 118 (1992) 368.

[7] P.L-F Liu ct al, R u n u p o f so lita ry w avo on a circ u lar islan d , Ịountaỉ o f Ị'luid Mechỉĩĩĩics 302 (1995) 259.

|8] P.A. M a d sc n , O.R. S orenscn, 11.A. Schaffor, S urf zo n e d y n a m ic s s im u la te d by B oussinesq ty p e m o d el, P a rt 1: M odc) d e sc rip tio n and cross-shorc m o tio n of rc g u la r vvaves, Coastal Eìígineerbĩg 32

(1997) 255.

[9] N. S h u to , c . G oto, N um orical sim u latio n t>f tsu n am i ru n u p , Coastal Engincering Ịouriíaỉ-Ịapaĩi 21 (1978) 13.

[10] G. S tran g , O n th c co n stru c tio n an d c o m p a riso n of d iffc rcn c c schem es, SI A M (Soc. Int. A p p l.

M ath.) Ịounuỉl o f Numcrícnỉ Aĩiaỉỵsis 5 (1968) 506.

[11] v .v . T ito v , C.E. Synolakis, M o d o lin g o í b rc ak in g and n o n -b re a k in g long-vvavc cvolution a n d ru n u p u s in g VTCS-2, louninỉ o f Waterwaì/, ỉyort, Coastal ntid Occmi r.ĩiỊỊÌncerÌHg 121 (1995) 308.

[12] v .v . T ỉtov, C.E. S ynolakis, N u m ericaỉ m o d elin g of tid al vvave ru n u p , Ịoum aỉ o f Watenoa\f, Port, Coastal (ì)ĩd Occtììì Eìiginecriìíg 124 (1998) 157.

[13] Y. W ci/ x .z . M ao, K.p. C h cu n g , VVcll-balanccd

íinỉtc-volumc model for long-wavc runup.

Ịountnỉ o f Wntcrwn\f, Port, Coastal rnd Occaìì Eìỉgbìccrhĩg 132 (2(X)6) 114.