CHANGE OF NEARSHORE SIGNIFICANT WAVES IN RESPONSE TO SEA LEVEL RISE. Se-Hyeon Cheon 1 and Kyung-Duck Suh 1

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1 CHANGE OF NEARSHORE SIGNIFICANT WAVES IN RESPONSE TO SEA EVE RISE Se-Hyeon Cheon 1 and Kyung-Duck Suh 1 In thi paper, a method ha been developed for etimating the change of nearhore ignificant wave in repone to long-term ea level rie, by extending the method propoed for regular wave by Townend in The relative change in wavelength, refraction coefficient, hoaling coefficient, and wave height for random wave are preented a function of the relative change in water depth. The change in wavelength and refraction coefficient are calculated by uing the ignificant wave period and principal wave direction in the formula for regular wave. On the other hand, the change in hoaling coefficient and wave height are calculated by uing the formula propoed for tranformation of random wave in the nearhore area including the urf zone. The reult are preented in the form of both formula and graph. In particular, the relative change in ignificant wave height i compared with the reult for regular wave. Keyword: climate change, random wave, ea level rie, wavelength, wave height, wave refraction, wave hoaling INTRODUCTION During the lat everal decade, the international community led by the IPCC (Intergovernmental Panel on Climate Change) ha performed reearche for projecting the emiion of greenhoue gae and the correponding climate change (Marchetti 1977; Schneider and Chen 198; Houghton et al. 1996; Marland et al. 23; Stern 26 among many other). The emiion cenario of the greenhoue gae have been regularly updated by the IPCC, which how different trend depending on the aumption about future technological and economic development. However, all the cenario project the rie of air temperature due to the increae of greenhoue gae emiion and the correponding ea level rie. Accordingly, reearche have been performed for the effect of ea level rie upon variou coatal engineering problem. Coatal tructure are directly influenced by the ea level rie. The effect of water depth increae and wave height change due to ea level rie on the performance and tability of coatal tructure have been invetigated (Klein et al. 1998; Southerland and Wolf 22; Okayau and Sakai 26; Stern 26; Torrean et al. 28; Wigley 29; Reeve 21; Takagi et al. 211; Chini and Stanby 212; Suh et al. 212; Suh et al. 213; ee et al. 213). However, mot of thee tudie ha been performed for a pecific ite uing the ea level rie under a pecific emiion cenario o that it i difficult to ue the reult in different ite ubject to different ea level rie. To reolve thi problem, Townend (1994) propoed a more general dimenionle approach, which can be applied to a wide range of ite and cenario. Expreing the relative change in water depth a, where and are the water depth before and after the ea level rie, he calculated the relative change in wave height, wavelength, hoaling coefficient, and refraction coefficient due to the ea level rie a function of. The approach of Townend (1994), however, i baed on the regular wave theory. In the preent tudy, we extend the Townend approach to irregular wave. The wavelength and refraction coefficient are calculated by the regular wave formula but uing the ignificant wave period and principal wave direction. The hoaling coefficient i calculated by a formula propoed for nonlinear hoaling of irregular wave. The ignificant wave height i calculated by the Goda (1975) approximate formula. METHOD Outline To etimate the effect of ea level rie on wave, a done by Townend (1994), the relative change in water depth due to the ea level rie i ued. Auming a long planar beach with traight and parallel depth-contour, the relative change in wave characteritic (wave height, wavelength, hoaling coefficient, and refraction coefficient) are etimated a function of the relative change in water depth. To extend the Townend approach to irregular wave, equation for nonlinear hoaling coefficient and nearhore ignificant wave height variation including the urf zone are ued. Alo, for convenience of 1 Department of Civil and Environmental Engineering, Seoul National Univerity, 1 Gwanak-ro, Gwanak-gu, Seoul , Republic of Korea 1

2 2 COASTA ENGINEERING 214 application, the graph for the relative change in wave characteritic are expreed a function of the deepwater wave teepne and the water depth relative to deepwater wavelength. Notation The following ymbol are ued in thi paper. D = water depth (m) H = wave height (m) H = deepwater wave height (m) H = ignificant wave height (m) = wavelength (m) = deepwater wavelength (m) C = wave celerity (m/) C = deepwater wave celerity (m/) m = beach lope K = nonlinear hoaling coefficient K i = linear hoaling coefficient K r = refraction coefficient T = wave period () T = ignificant wave period () = principal wave direction ( ) = deepwater principal wave direction ( ) H / = deepwater wave teepne g = gravitational acceleration (m/ 2 ) A prime ( ' ) indicate a value after the ea level rie, while a non-primed value indicate the value before the ea level rie. On the other hand, a lower-cae letter indicate the relative change in the value due to the ea level rie. For example, h H '/ H, where H and H ' are the wave height before and after the ea level rie. Wavelength The wavelength correponding to the ignificant wave period of irregular wave before and after the ea level rie are calculated by the diperion relationhip a g 2 D tanh T (1) and g 2 D' tanh T ' ' repectively. The relative change in wavelength i then given by 4 D / ' e 1 4 D d l tanh 4 D / e 1 l (2) (3) which i an implicit function of l. Refraction coefficient The refraction coefficient correponding to the principal wave direction of random directional wave before and after the ea level rie are given by and co co K r (4) K r ' co co ' (5) repectively. The relative change in refraction coefficient i calculated by the preceding two equation along with the Snell law (i.e. in / C in / C) a Kr ' 1 A kr 2 K 1 ca r 1/4 (6) where A 2 2 tanh (2 D / )in and '/ '/ c C C l.

3 COASTA ENGINEERING Shoaling coefficient Baed on the tudie of Shuto (1974) and Iwagaki et al. (1981), Kweon and Goda (1996) propoed a formula for nonlinear hoaling coefficient a K D H Ki.15 (7) where 2 T and K / (2 ) i C Cg 1.56 theory. The group velocity C g i the linear hoaling coefficient by mall-amplitude wave C g i calculated by 1 4 D / gt 2D 1 tanh 2 inh(4 D / ) 2 (8) The relative change in linear hoaling coefficient due to the ea level rie i given by k i 4 D/ 1 1 inh(4 D/ ) l 4 dd / l 1 inh(4 dd / l ) (9) The relative change in nonlinear hoaling coefficient i given by where k d k k (1) 2.87 ( i ) i.15 K ( D / ) ( H / ) (11) Wave height To calculate the ignificant wave height in nearhore area including the urf zone, the approximate formula of Goda (1975) i ued: H KH : D /.2, min ( H 1D), max H, KH: D / m where max max.92,.32 e m e,.28 (12) 4.2m, and 1.52e. Figure 1. Definition of zone of different wave tranformation

4 4 COASTA ENGINEERING 214 Fig. 1 how the change of ignificant wave height calculated by Eq. 12 for the wave of H = 5 m and T = 13 propagating normally to a planar beach with 1/5 lope. In thi tudy, the nearhore area i divided into hoaling zone, tranition zone, and urf zone a illutrated in Fig. 1. In the tranition zone, the wave are jut about to break and the wave height doe not change with water depth. The contant wave height in the tranition zone i reulted from the approximation of a mooth diagram, in which the wave height lowly change with water depth. However, the difference between the diagram and the calculation by Eq. 12 i jut within everal percent. Baed on Eq. 12 and Fig. 1, the criterion for each zone i given a follow. H H 1 D 1 max max max K : urf zone D, max K : tranition zone : hoaling zone Now, Eq. 12 and 13 are ued for the calculation of the relative change in wave height in each zone: (13) ( d 1) / ( / ( 1( D / )) 1) 1 : urf zone max / ( 1( D / ) / ) : urf/tranition zone h 1 : tranition zone kk / max : tranition/hoaling zone k : hoaling zone (14) Here, the urf/tranition zone indicate the cae where it belong to the urf zone before the ea level rie but belong to the tranition zone after the ea level rie. Similarly, the tranition/hoaling zone indicate the cae where it belong to the tranition zone before the ea level rie but belong to the hoaling zone after the ea level rie. RESUTS AND DISCUSSION Wavelength Figure 2. Relative change in wavelength Fig. 2 how the relative change in wavelength a a function of relative water depth for different relative change in water depth (or different ea level rie). It increae with decreaing water depth and increaing ea level rie. In other word, the wavelength change relatively more in hallower water ubject to greater ea level rie. The relative change in wavelength i le than everal percent in the hoaling zone of D/.3, wherea it rapidly increae with decreaing water depth, becoming greater than 15% in hallow water of D/.1 when d 1.5.

5 COASTA ENGINEERING Refraction coefficient Fig. 3 how the relative change in refraction coefficient a a function of relative water depth and deepwater principal wave direction for different relative change in water depth (or different ea level rie). It increae with deepwater principal wave direction and ea level rie, becoming greater than 1.2 in water depth of D/.2 ~.3 when 8 and d 1.5. The increae of refraction coefficient due to the ea level rie of d 1.5 i le than 5% when either the wave incident angle i maller than 5 or the relative water depth i maller than.5. For a mall ea level rie of d 1.1, the increae i le than 5% in all water depth regardle of the incident wave angle. The maximum relative change in refraction coefficient occur in water depth of D/.1~.2 for the incident wave angle up to about 6, and it move to deeper water of D/.2 ~.3 a the wave angle further increae. Figure 3. Relative change in refraction coefficient: (a) d = 1.1 Figure 3. Relative change in refraction coefficient: (b) d = 1.3

6 6 COASTA ENGINEERING 214 Figure 3. Relative change in refraction coefficient: (c) d = 1.5 Shoaling coefficient The relative change in hoaling coefficient can be calculated by Eq. 1. It can alo be calculated uing Fig. 4, from which k i and can be read off graphically. The relative change in linear hoaling coefficient, k i, i read off from the right ordinate for given d and D/. On the other hand, i read off from the oblique line in Fig. 4 for given D/ and H /. Note that the value of in Fig. 4 include the effect of K which i alo a function of D/ and H /. Fig. 4 indicate that the relative change in hoaling coefficient increae with decreaing relative water depth and increaing deepwater wave teepne. However, it i meaningful only outide the urf zone where wave do not break. Figure 4. Relative change in hoaling coefficient.

7 COASTA ENGINEERING Wave height Fig. 5 i the diagram for calculating the relative change in wave height due to the ea level rie in nearhore area with different bottom lope. In thee figure, the haded area indicate the tranition zone, while it left and right ide indicate the urf zone and hoaling zone, repectively. If the water depth increae due to the ea level rie, the boundarie of the tranition zone are hifted to the right. The amount of hift i indicated by the cale bar of d at the upper and lower edge of the tranition zone. To calculate the relative change in wave height uing thee figure, which zone the wave belong to hould be determined for given m, d, D/ and H /. If the wave belong to the urf zone, the.62 value of / ( D / ) i read off from the figure. The relative change in wave height i then calculated by the firt equation in Eq. 14, which can be rewritten a h d exp(2m 4.2 m) 1 If the wave belong to the hoaling zone, the relative change in wave height i the ame a the relative change in hoaling coefficient, which can be calculated by Eq. 1 with and k i read off from the figure. If the wave belong to the urf/tranition zone, tranition zone, or tranition/hoaling zone, h i calculated by Eq (15) Fig. 1. Relative change in wave height: (a) m = 1/1 To compare the relative change in wave height between the preent tudy and Townend (1994) method, two cae of d 1.1 and d 1.5 are examined on a beach with 1/5 lope. In real ituation, the relative change in water depth will be cloe to d 1.5 in an area cloe to the horeline, while it will be cloe to d 1.1 in deeper water. The percent difference between the two method i defined by h ht % Difference 1% (16) h T where h i the relative change in wave height calculated by the preent method, and h T i the relative change by the Townend method. The percent difference are hown in Fig. 6 for d 1.1 and d 1.5.

8 8 COASTA ENGINEERING 214 In both cae, the maximum difference occur along the boundary between the urf zone and tranition zone, and the Townend method alway calculate the greater wave height change. Inide the urf zone, the difference increae with the wave teepne for the ame relative water depth. For the ame wave teepne, the difference decreae with decreaing relative water depth in the outer urf zone but it turn to the increae in the inner urf zone. Figure 2. Relative change in wave height: (b) m = 1/2 Figure 3. Relative change in wave height: (c) m = 1/3 When d 1.1, the maximum difference of about 6% occur both near the horeline and along the boundary between the urf zone and tranition zone. When d 1.5, the maximum difference of about

9 COASTA ENGINEERING % occur along the boundary between the urf zone and tranition zone and a difference of about 25% occur near the horeline. In the hoaling zone, the difference rapidly decreae with increaing relative water depth and decreaing wave teepne o that a ignificant difference i only oberved near the boundary with the tranition zone for mall wave teepne. When d 1.1 and d 1.5, the maximum difference become about 5% and 2%, repectively, for very mall wave teepne. Figure 4. Relative change in wave height: (d) m = 1/5 Figure 5. Relative change in wave height: (e) m = 1/1 The previou reult how that the maximum percent difference between the two method i about 35% at the offhore boundary of the urf zone, when d 1.5. A mentioned earlier, however, thi

10 1 COASTA ENGINEERING 214 value of d i unreaonably large in the outer urf zone where the water depth i relatively large. Therefore, the difference in thi area i almot meaningle for d 1.5. However, the percent difference of 25% in the inner urf zone for d 1.5 i poible becaue the water depth i relatively mall there. On the other hand, the percent difference hown in Fig. 6(a) for d 1.1 i poible in the outer urf zone where the water depth i relatively large. In ummary, the Townend method would overetimate the relative change in wave height by everal ten of percent in the inner urf zone and only a few percent in the outer urf zone. Fig. 6. Percent difference of relative change in wave height between preent method and Townend method (Townend, 1994): (a) d = 1.1

11 COASTA ENGINEERING Fig. 7. Percent difference of relative change in wave height between preent method and Townend method (Townend, 1994): (b) d = 1.5 CONCUSION In thi tudy, the Townend (1994) method for regular wave wa extended to irregular wave to etimate the change of nearhore ignificant wave due to long-term ea level rie. The change in wavelength and refraction coefficient were calculated by the regular wave formula with the ignificant wave period and principal wave direction of irregular wave. The hoaling coefficient wa calculated by a formula propoed for nonlinear hoaling of irregular wave, and the ignificant wave height wa calculated by the Goda (1975) approximate formula. It wa found that the Townend method baed on regular wave theory would overetimate the relative change in wave height by everal ten of percent in the inner urf zone and only a few percent in the outer urf zone. ACKNOWEDGMENTS Thi reearch wa upported by Baic Science Reearch Program through the National Reearch Foundation of Korea (NRF) funded by the Minitry of Science, ICT and Future Planning(NRF- 214R1A2A2A17921). The reearch wa conducted at the Engineering Reearch Intitute of Seoul National Univerity. REFERENCES Chini, N., Stanby, P.K., 212. Extreme value of coatal wave overtopping accounting for climate change and ea level rie. Coatal Engineering, 65, Goda, Y., Irregular wave deformation in the urf zone. Coatal Engineering in Japan, 18, Houghton, J.T., Meira Filho,.G., Callander, B.A., Harri, N., Kattenberg, A., Makell, K. (Ed.), Climate change 1995: The cience of climate change. Cambridge Univerity Pre, Cambridge, UK. Iwagaki, Y., Shiota, K., Doi, H., Shoaling and refraction of finite amplitude wave. In: Proceeding of 28th Japanee Conference on Coatal Engineering, (in Japanee). Klein, R.J.T., Smit, M.J., Gooen, H., Hulbergen, C.H., Reilience and vulnerability: Coatal dynamic or Dutch dike?, The Geographical Journal, 164, Kweon, H.M., Goda, Y., A parametric model for random wave deformation by breaking on arbitrary beach profile. In: Proceeding of 25th International Conference on Coatal Engineering, ASCE, ee, C.-E., Kim, S.-W., Park, D.-H., Suh, K.-D., 213. Rik aement of wave run-up and armor tability of inclined coatal tructure ubject to long-term ea level rie. Ocean Engineering, 71, Marchetti, C., On geoengineering and the CO2 problem. Climatic Change, 1, Marland, G., Boden, T.A., Andre, R.J., 23. Global, regional, and national CO2 emiion. In: Trend: A compendium of data on global change, Carbon Dioxide Information Analyi Center, Oak Ridge National aboratory, U.S. Department of Energy, Oak Ridge, Tenneee, USA. Okayau, A., Sakai, K., 26. Effect of ea level rie on liding ditance of a caion breakwater: Optimization with probabilitic deign method. In: Proceeding of 3th International Conference on Coatal Engineering, World Scientific, Singapore, Reeve, D., 21. On the impact of climate change for port deign. In: Proceeding of 26th International Conference for Seaport & Maritime Tranport. Schneider, S.H., Chen, R.S., 198. Carbon dioxide warming and coatline flooding: Phyical factor and climatic impact. Annual Review of Energy, 5, Shuto, N., Nonlinear long wave in a channel of variable ection. Coatal Engineering in Japan, 17, Stern, N., 26. Stern review on the economic of climate change. Cambridge Univerity Pre, Cambridge, UK. Suh, K.-D., Kim, S.-W., Mori, N., Mae, H., 212. Effect of climate change on performance-baed deign of caion breakwater. Journal of Waterway, Port, Coatal and Ocean Engineering, 138, Suh, K.-D., Kim, S.-W., Kim, S., Cheon, S., 213. Effect of climate change on tability of caion breakwater in different water depth. Ocean Engineering, 71, Sutherland, J., Wolf, J., 22. Coatal defence vulnerability 275. Report SR 59, HR Wallingford,

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