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DESIGN CRITERIA OF COASTAL DIKE AROUND TAIWAN COAST

Ho-Shong Hou1 and Tai-Wen Hsu2

1Ministry of Economic Affairs

Taipei, TAIWAN

2Department of Hydraulic and Ocean Engineering National Cheng Kung University

Tainan, TAIWAN

ABSRACT A nearshore hydrodynamic model together with a Reynolds averaged Navier-Stokes (RANS) approach is used to simulate the water level rise and wave setup and run-up mechanics in front of coastal dikes. The sea level changes consist of the maximum astronomic tidal level, storm surge, wind and wave induced-setup and wave run-up and overtopping. The estimated sea water rise is applied to determine the height of coastal dikes. The predicted water level response due to astronomical and meteorological tides and waves is then used as designed criteria of coastal dikes. The advantage of the present designed criteria is its ease, efficiency and accuracy in practical applications when compared with previous empirical methods. Two typical typhoon events with different attacking tracks were demonstrated to validate the predictability of the nearshore numerical models.

INTRODUCTION

Taiwan is an island having a land area of 36,000 km2 and a total length of 1,600 km including offshore islands. In recent years there are growing developmental pressures in the coastal region. This coastal development and the concentration of population near the coastal zone have resulted in increasing demands for the recreational utilization of beaches, and subsequent problems with coastal erosion as homes, hotels, roads, parks, industrial areas and living properties often lie in the path of shoreline recession and the destructive impacts of storm surges and waves. As a result of the growth in coastal development and increasing demand for shore-front properties, countermeasures against beach erosion have become major concerned coastal problems in the government. There are various solutions to coastal defense including hard and soft solutions. The option in reaching to receding shorelines depends on well designed coastal management programs to comprise with natural environment. Beach nourishment combined with proper hard coastal structures such as headland control, coastal structures and geotextile tube is often the best soft response to coastal erosion and may be the effective solution from the view point of sediment transport budget. However, costs are considerable due to the fact that sand added to the beach had rapidly dispersed by storm waves and beach nourishment should be repeated again and again until beach reached its equilibrium state. A soft solution is not always possible for the coastal defense, and a hard solution in the form of structures to protect shoreline may be required in engineering practice. The most commonly used for shore protection are seawall family consisting of wall structures such as bulkheads, revetments, and coastal dikes (sea walls). Such structures are constructed mainly for the purpose of protecting human properties from shoreline erosion and coastal flooding induced by storm surges and waves. Despite their various locations of construction in relation to local beach

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and surf, coastal dikes and sea walls are designed by different shapes with covered layers of armor units for blocking storm surges and dissipating wave energy. In Taiwan, beach erosion has occurred for a long time. Its seriousness has received much attention since the 1960s as the economic development and population increased rapidly in coastal zones. The first coastal dike was built in response to the mitigation of coastal disasters induced by the typhoon event in 1973. Since that time coastal dikes and revetments have been progressively lengthened to about 542m long. Up to now, most of sandy beaches have been surrounded by these hard structures helping to prevent sea water from running over the embankment behind coastal dikes. Some coastal dikes have proven their effectiveness over the years by saving the properties from the erosive action of storm surges and waves during a number of destructive typhoon events. However most of coastal dikes failed to protect shorelines and were needed to be protected against damages to themselves. This is mainly due to the main reason that coastal dikes are rigid structures with vertical or concave faces, as shown in Figure 1, reflecting wave energy upward and backward to the sea. The reflection could lead to great scouring of the front beach and accelerate its disappearance.

Figure 1. Coastal dikes with vertical and concave faces (Kuo, 2001)

The problem of wave reflection in front of coastal dike was improved by an inclined mild-slope face from 1:2 to 1:6 with armor units covered on the surface or placed on the toe for dissipating wave energy. Figures 2 and 3 show different shapes of coastal dikes. Notably the greater the length of the coastal dikes, the bigger the volume of sand impounded and the greater impact on adjacent shoreline retreat as discussed by Walton and Sensabaugh (1978). Furthermore, the front face of coastal dikes is very thin concrete covered by armor units in a random placement. Their landscapes and hard structures are not harmonic with coastal environments and degrade people attractive to use the beach. It is noted that coastal dikes are designed for the purpose of preventing property from typhoon events. The height of them should be safe and comfortable with natural environment. A modern concept of designing coastal dikes is thus to provide a harmonious measure of safety, landscape, ecology and attraction to water. To achieve this goal, the design criteria based on the analysis of wave field and storm surge in the front of coastal dikes are desirable. In early time, coastal engineer determined the height of coastal dikes by experience. No criterion can be followed but only some incomplete field data are available to be used in the design. Empirical formulas are frequently employed in the practical applications. However, these formulas are limited because of rare data are recorded in the field to verify their validity at interesting locations. In this paper, we addressed a nearshore hydrodynamic model to account for wave, storm surge, tidal current wind, wave-induced setup and wave run-up to demonstrate design criteria of coastal dikes. The study focuses an estimation of storm surges, wave setup, wave run-up and overtopping on the coastal dikes. The height of coastal dikes is thus determined based on the numerical results of total sea water level rises in front of them.

DESIGNED HEIGHT OF COASTAL DIKES

The designed height of coastal dikes is based on the following total sea levels: (1) the maximum astronomical tide level, (2) the storm surge, (3) the wind and wave setup and (4) the wave run-up. Direct field measurements are probably the most accurate way to monitor the sea water level.

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However, the instruments may be destroyed by severe typhoon events and this method may result in the incompleteness and losses of data. Furthermore, due to the high coast of instruments and high maintenance, the field measurements are relatively difficult. Accordingly, the estimate of sea water level by means of numerical models appears to be a more feasible and efficient tool.

Figure 2. Steep-slope coastal dikes (Kuo, 2001)

Figure 3. Mild-slope coastal dikes (Kuo, 2001)

The principal fluctuation of the sea is the astronomical tide. It is well known that the progressive rise in sea level may bring about coastal erosion and migration of barrier islands. A mean sea level is determined based on field observations by averaging hourly height recording over a number of years. In contrast to the astronomical tide, a meteorological tide is caused by meteorological factors, such as strong wind and rapid atmospheric pressure change. A storm surge is a meteorological tide with abnormal rise of sea water which is mainly induced by tropical low pressure during typhoons.Empirical method is frequently used to calculate the height of a storm surge at a particular location by considering the combined effect of low atmospheric pressure and wind induced setup. However the empirical method is only applicable to the locations where the observed data have been recorded for many years to fit the empirical coefficients in the formula. A numerical model is quite flexible and valuable to forecast the sea surface fluctuations including tidal level, storm surge and wind setup due to typhoons. The POM model developed by Blumberg and Mellor (1983) and FEM model developed by Hsu et al. (1999) were adopted in the present study to calculate sea level changes due to astronomical tide, low atmospheric pressure and wind effect on the sea surfaces. When waves reach the nearshore and eventually break on a beach, wave height and momentum flux will be reduced. Wave breaking produces a compensating force on the water column and this force

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is balanced by an increased mean water surface above the still water level (SWL) which is termed wave setup. It has been shown by Lonquet-Higgins and Stewart (1963, 1964) that the changes in the onshore component of the radiation stresses are balanced by wave setup and setdown before and after wave breaking. In recent years, numerical models describing wave induced setup and nearshore currents have been developed and applied for engineering practice. For the calculation of radiation stresses, a prediction of wave field in the nearshore region is required. Predictions of wave generation and transformation in coastal waters by numerical models become a popular tool, as they may provide the good estimate of the sea conditions at a place for given wind fields. A WWM model developed by Hsu et al. (2005) for wind wave simulations in both large-scale oceanic deep water region and shallow water region is implemented to predict wave field and radiation stresses in the present research. Wave run-up and overtopping have been dealt with by many researchers. Many empirical relations have been proposed and employed in real applications of coastal engineering. However the estimate of wave run-up height for non-breaking and breaking waves before reaching the front of the structure is too complex to be determined through empirical formula. The advanced computer technology enables us to develop powerful numerical models to account for wave run-up and overtopping instead of empirical methods. This paper presents a RANS (Reynold averaged Navier-Stokes) model to simulate the entire processes of wave propagation, breaking, bore formation, run-up and run-down and overtopping. The advantage of the numerical model lies in its ease, efficiency and accuracy of free surface tracking using SPH (smooth particle hydrodynamics) and VOF (volume of fluid) to predict wave height in the front of coastal dikes.

ASTRONOMICAL TIDE The modeling system consists of following two principal components: (1) the quasi-three-dimensional (quasi-3D) hydrodynamic model. Moreover, Princeton Ocean Model (POM), which was developed by Blumberg and Mellor (1983), is adopted and utilized for large scale calculations. With the quasi-3D hydrodynamic model, the variations of astronomical tide as well as meteorological tide can be properly simulated for rapidly varying bathymetry such as east coast of Taiwan. The calculated results are then used for the boundary conditions of inner region. (2) the 2D finite element depth-averaged model (FEM) for near field simulation. Although POM takes most physical components into account, however, its finite differencing scheme refrains it from the simulation of highly irregular coastline. Therefore, this study adopts a finite element depth-averaged model for near field simulation due to its superiority on a geometrically complex domain. The POM uses the sigma coordinate in the vertical direction to take free water surface and smoothly represent the bottom topography, and the curvilinear orthogonal coordinate in the horizontal direction to fit irregular shoreline boundary. The POM contains an embedded second moment turbulence closure sub-model: the Mellor-Yamada turbulence closure model to provide vertical turbulent diffusion coefficient (Mellor and Yamada, 1982). The Smagorinsky formula is used to calculate horizontal diffusion coefficient (Smagorinsky, 1963). Detailed description of the POM model can be referred to Mellor (1998). The governing equations are given by

UD UD 0t x y

∂η ∂ ∂ ∂ω+ + + =

∂ ∂ ∂ ∂σ (1) 2

2 0 0M

x0 0

UD U D UVD U fVD gDt x y x

K U gD gD Dd d DFD x xσ σ

∂ ∂ ∂ ∂ ω ∂η+ + + − +

∂ ∂ ∂ ∂σ ∂

∂ ∂ ∂ ∂ ∂ρ⎡ ⎤= − ρ σ+ σ σ+⎢ ⎥∂σ ∂σ ρ ∂ ρ ∂ ∂σ⎣ ⎦ ∫ ∫ (2)

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2

2 ' ' '0 'My'

0

VD VUD V D V fUD gDt x y y

K V gD D d FD x y D yσ

∂ ∂ ∂ ∂ ω ∂η+ + + + +

∂ ∂ ∂ ∂σ ∂

⎡ ⎤∂ ∂ ∂ ∂ρ σ ∂ ∂ρ⎡ ⎤= − − σ +⎢ ⎥⎢ ⎥∂σ ∂σ ρ ∂ ∂ ∂ ∂σ⎣ ⎦ ⎣ ⎦∫

(3)

HKD UD VD DFt x y D θ

∂θ ∂θ ∂θ ∂θω ∂ ∂θ⎛ ⎞+ + + = +⎜ ⎟∂ ∂ ∂ ∂σ ∂σ ∂σ⎝ ⎠ (4)

HS

KSD SUD SVD S S DFt x y D

∂ ∂ ∂ ∂ ω ∂ ∂⎛ ⎞+ + + = +⎜ ⎟∂ ∂ ∂ ∂σ ∂σ ∂σ⎝ ⎠ (5) where x and y are the horizontal Cartesian coordinates, σ is the terrain-following sigma vertical coordinate, t is the time, U and V are the horizontal velocity components, ω is the transformed vertical velocity, D is total water depth, f is the Coriolis parameter, η is the free surface elevation, ρ is the water density, g is the gravity, xF , yF are the horizontal diffusion terms, MK is the vertical eddy viscosity of turbulent momentum mixing, θ is the potential temperature, S is the salinity, HK is the vertical diffusivity and Fθ , SF are the horizontal heat and salt diffusion terms, respectively. The model equations are based on the assumption of viscous and incompressible flow. Furthermore, to reduce the computational effort, depth-averaged velocities are main variables to be solved. The resulting model equations consist of a continuity equation and momentum equations. These equations are given below:

(h )u (h )v 0t x y

∂η ∂ +η ∂ +η+ + =

∂ ∂ ∂ (6)

bx wxu u u 1 pu v g fvt x y x x

∂ ∂ ∂ ∂η ∂+ + = − + − τ + τ −

∂ ∂ ∂ ∂ ρ ∂ (7)

by wyv v v 1 pu v g fut x y y y

∂ ∂ ∂ ∂η ∂+ + = − − − τ + τ −

∂ ∂ ∂ ∂ ρ ∂ (8) in which (u, v) the depth-averaged velocity components, η the tidallevel deviation, h the water depth, p the pressure, g the gravitational acceleration, f the Coriolis parameter ( 2 sin )= ω ϕ with ω and ϕ respectively being earth’s rotation speed and latitude, t the time, B B

x y( , )τ τ the bottom friction components, S S

x y( , )τ τ the surface stress components. More specifically, the bottom shear stress components can be expressed as

2 2Bx 2

gu u vC (h )

+τ =

+η (9)

2 2By 2

gv u vC (h )

+τ =

+ η (10) where C is Chezy coefficient give by 2 1 3 2

zC h n−= and n is Manning’s coefficient chosen as 0.025 in this study. Moreover, surface shear stress components read as

2 2a x x yS

x

W W W(h )

γρ +τ =

ρ + η (11) 2 2

a y x ySy

W W W(h )

γρ +τ =

ρ + η (12) where γ is the surface stress coefficient chosen as 32.6 10−γ ≈ × , aρ is the air density 3 3

a( 1.22 10 gr cm )−ρ ≈ × , ρ is the salt-water density 3 3( 1.033 10 gr cm )−ρ ≈ × and x y(W , W ) are the wind velocity components.

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The governing equations (6)-(8) are then solved by select lumping finite element method (Kawahara et al., 1980; Hsu et al., 1999). This algorithm allows for the extremely flexible spatial discretizations over the entire computational domain and has demonstrated excellent stability characteristics (Kawahara et al., 1982). Specifically, the flexibility of element size is characterized by the larger elements used in the open-ocean regions where less resolutions are needed and the smaller elements applied in the near shore and estuary areas where finer resolutions are required to resolve hydrodynamic details. Since the details of the numerical scheme can be found in the literature, we shall not repeat them here.

WWM MODEL

In WWM, the evolution of the wave spectrum is described by the spectral action balance equation, which is expressed by Cartesian coordinates as follows (e.g., Hasselmann et al., 1973).

x y totalN (C N) (C N) (C N) (C N) St x y σ θ

∂ ∂ ∂ ∂ ∂+ + + + =

∂ ∂ ∂ ∂σ ∂θ (13) where N N(t, x, y, , )= σ θ is the wave action density spectrum; t is the time; xC and yC are the wave propagation velocities in x and y space, respectively; Cσ and Cθ are the wave propagation velocities in σ and θ space, respectively; σ is the relative frequency; θ is the wave direction;

totaltotalS S ( , )= σ θ σ% is the source term; and totalS% is the spectral density of wave energy. The first term on the left-hand side of equation (13) represents the time rate of change of the action density. The second and third term represent propagation of the action density in the geographical space. The fourth term represents shifting of the relative frequency due to variations in the depth and the current. The fifth term represents the refraction induced by variations in the depth and the current. The term at the right-hand side is the source term in terms of the energy density, representing the combined effects of wave generation, dissipation, and nonlinear wave-wave interactions. For applications on large-scale oceanic regions, the action balance equation, equation (13), needs to be reformulated in terms of the spherical coordinates. The longitude-latitude formulation of the action balance equation is given by

N̂t

∂+

∂1 ˆ(cos ) ( cos N)− ∂

φ φ φ∂φ

ˆ( N)∂+ λ∂λ

& ˆ( N)∂+ σ∂σ

& totalˆ( N) S∂

+ θ =∂θ

&

(14) where N̂ N̂( , , , , t)= φ λ σ θ is the wave action density spectrum for spherical coordinates; φ is the latitude; λ is the longitude; φ& is the time rate of change of φ ; λ& is the time rate of change of λ ; σ& is the time rate of change of the relative frequency; and θ& is the time rate of change of the propagation direction. The wave action density spectrum N̂ is related to the normal spectral density N with respect to a local Cartesian coordinates through N̂d d d dσ θ φ λ Nd d dxdy= σ θ , or

2N̂ NR cos= φ (15) where R is the radius of the earth. The expressions of φ& , λ& , σ& , and θ& are given, respectively, by

g(C cosφ = θ+&U 1/ north)R−

(16a)

g(C sinλ = θ+&U 1/ east)(R cos )−φ

(16b)

1gC sin tan R (−θ = θ φ +&

k×k 2)k−

(16c)

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tσ = ∂σ ∂&

(16d) where gC is the group velocity, θ is the wave direction measured clockwise relative to true north, U is the current velocity vector, k is the wave number vector and k = |k| is the wavenumber. In equations (16a) and (16b), “north” and “east” represent Latitude and Longitude of the earth, respectively. Equation (14) is the basic transport equation which will be used in the wave prediction model. The boundary conditions in WWM, both in the geographic space and the spectral space, are fully absorbing boundaries. The wave energy is set to leave the computational domain or cross a coastal line. For coastal regions the incoming wave energy is only provided along the deepwater boundary. The spectral densities are assumed to be zero along the lateral boundaries. To avoid the propagation of numerical errors into the computational domain, the lateral boundaries are placed sufficiently far away from the area of interest.

WAVE SETUP MODEL

The wave-induced setup and nearshore currents are calculated based on the computed wave field. The 2D depth-averaged governing equations for a mean flow are written as

(UD) (VD) 0t x y

∂ζ ∂ ∂+ + =

∂ ∂ ∂ (17)

x x xU U UU V F M R g 0t x y x

∂ ∂ ∂ ∂ζ+ + + − + + =

∂ ∂ ∂ ∂ (18)

y y yV V VU V F M R g 0t x y y

∂ ∂ ∂ ∂ζ+ + + − + + =

∂ ∂ ∂ ∂ (19) where D is the total water depth, i.e. D = ζ+h , (U , V) are x and y component of the mean current, ζ is the mean water surface elevation, xF and yF are bottom friction terms in the x and y direction and are given by (Nishimura, 1982)

FCD

WW

UW

Vxf b b= + +⎧⎨⎩

⎫⎬⎭

( cos ) cos sinω

αω

α α2

22

(20)

FCD

WW

VW

Uyf b b= + +⎧⎨⎩

⎫⎬⎭

( sin ) cos sinω

αω

α α2

22

(21)

2 2 2

b b

2 2 2b b

W { U V 2(U cos Vsin )

U V 2(U cos Vsin ) }

= + +ω + α+ α ω

+ + +ω − α+ α ω　　　 　2 (22)

where fC is the friction coefficient, α is the breaking wave angle, ω π= H T kh/ sinh , H is the local wave height, and T is the wave period.

xM and yM are lateral mixing terms written as

xU UM ( ) ( )

x x y y∂ ∂ ∂ ∂

ε + ε∂ ∂ ∂ ∂

＝ (23)

yV VM ( ) ( )

x x y y∂ ∂ ∂ ∂

ε + ε∂ ∂ ∂ ∂

＝ (24)

in which ε is the momentum change coefficient and is assumed in the form proposed by Longuet-Higgins (1970).

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N gDε = l (25) where N is a constant and value smaller than 0.016, l is a length characterized by D tan= βl .

xR and yR are radiation stresses, and are given by xyxx

x

SS1R ( )D x y

∂∂= +ρ ∂ ∂ (26)

yx yyy

S S1R ( )D x y

∂ ∂= +ρ ∂ ∂ (27)

where ρ is the density of water, xxS , xyS , Syx and yyS denote the radiation stress components.

WAVE RUN-UP AND OVERTOPPING MODEL

Wave run-up and overtopping models based on the solution of the full Navier-Stokes have become increasingly popular. This approach can provide full details of the flow in the surf zone, ranging from the breaking process to bore formation, run-up and backwash. The large eddy simulation (LES) lies the extreme of Reynolds averaged Navier-Stokes (RANS) and attempts to capture the large scale motion, which is thought to contain most of the energy and momentum. The LES mass and momentum conservation equations for the particle scale flow are derived through the filtering operation of the corresponding Navier-Stokes equation. By using a flattop spatial filter, the following Lagrangian particle scale equations are obtained

1 DDtρ+∇

ρu 0=

(28)

D 1Dt

= − ∇ρ

P 2g+ + υ∇ u 1+ ∇ρτ (29)

where ‘u’ denotes velocity components and τ = SPS stress tensor with elements given by i jij i j(u u u u )τ = ρ − (30)

where “ ” represents time average quantities. An eddy viscosity assumption is used to model the sub-particle tensor as

ijijt ij

22 S k3

τ= υ − δ

ρ (31) where tυ = turbulence eddy viscosity and S is SPS strain tensor with elements ijS and k is the SPS turbulent kinetic energy. In subsequent computations, it is convenient to incorporate k from τ into the pressure term when solving the momentum equation (29).

MODEL APPLICATIONS In order to demonstrate the predictability of all numerical models, sea water changes caused by two historical typhoons are simulated and the results are compared to the observed data. The two typhoon events are Typhoon Mindulle (2004) and Haitang (2005) with different tracks as shown in Figure 4.

u

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(a) Typhoon Mindulle (2004)

(b) Typhoon Haitang (2005)

Figure 4. The attacking tracks of (a) Mindulle and (b) Haitang typhoon events

Figure 5. The domain for outer layer of near field simulation.

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In particular, Typhoon Mindulle belongs to track 4 and Typhoon Haitang is of track 1 according to the classification of Central Weather Bureau of Taiwan. Typical results of tidal level for model prediction and field data at three measured stations (see Fig. 5) are respectively shown in Figures 6(a)-(c) for Typhoons Mindulle and Haitang. It is seen that the overall agreement including the magnitude of amplitude and phase between numerical results and field data are very good, indicating that the coupling systems of POM and finite element depth-averaged model can properly predict the water level due to storm surge. The coupling modeling system including POM and finite-element depth-averaged model is configured in Taiwan coastal region to investigate the storm surge responses due to Typhoon. Clearly, the success and quality of the results obtained with mathematical models depend equally on the techniques’ efficiency and numerical methods used. Figures 7 and 8 present the variations of wave height, wave setup, tidal level and wind speeds during Typhoons Mindulle and Haitang. It is worthwhile to note that the sea level changes are associated with wind speed. A greater wind speed could produce a larger sea level rise during typhoon events.

(a)

(b)

(c) Figure 6. Simulation of tidal currents by POM compared with measurement

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Figure 7. Sea level rises including (a) wave height; (b) wave setup; (c) tidal level and (d) wind speeds during Typhoon Mindulle

Figure 8. Sea level rises including (a) wave height; (b) wave setup; (c) tidal level and (d) wind speeds during Typhoon Haitang

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(a)

(b)

(c)

(d) Figure 9. Computed surface profiles of wave during shoaling, run-up and run-down and overtopping The particle configurations are shown in Figures 9(a)-(d). Note that only the upper part of the flow over the slope is plotted in the figure. It is seen that the entire processes of wave shoaling, run-up and run-down and overtopping are well reproduced by the VOF model with very good agreement in the profile. It is shown in Figure 9(d) that a strong backwash occurs near the original shoreline where the free surface deforms significantly. There is an indication of a second wave breaking during the run-down process. Meanwhile, a reflected wave is created and the remaining thin water layer over the slope retreats at a slower rate than the main flow.

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CONCLUSIONS Coastal dikes are often used for the purpose of preventing beach erosion and other damage induced by storm surges and wave action. As noted above, coastal dikes have been constructed at a number of places during the last 35 years as the main measure against beach erosion in Taiwan. To design a coastal dike, the engineering prior must clearly define the principal function of the structure, and the sea level rise at the location where the structure is to be built. Sea level rise including the maximum astronomical tidal level, storm surge, wind and wave induced setup and run-up are key factors to be considered for determining the height of the structure. Failures of coastal dikes are mainly due to a narrow sandy beach quickly disappeared as a result of wave reflection in the front of the structure. Ti is of great importance that the sea level and wave conditions should be prior accurately estimated before deciding the shape and height of the structure. In early time, many empirical formulas were used to calculate sea level changes and wave induced setup in engineering practice. The empirical methods are usually based on some field data at different locations. They are not applicable to the locations where the bathymetry and coastal lines are complicate or the recorded data are rare. In recent years the advanced computer technology permits us to carry out the numerical simulations of sea water level changes and wave conditions. The hydrodynamics of numerical model has the advantage of ease, efficiency and accuracy in practical applications. In this paper, a nearshore hydrodynamics model together with a RANS approach is used to simulate the sea water level and wave induced setup and runup in front of coastal dikes. The estimated sea water rise is applied to determine the height of coastal dikes. The predicted water level response due to astronomical and meteorological tides and waves is then used as designed criteria of coastal dikes. The advantage of the present designed criteria is its ease efficiency and accuracy in practical applications when compared with empirical methods. Two typical typhoon events with different attacking tracks were demonstrated to validate the predictability of the nearshore numerical models.

REFERENCES

Blumberg, A.F. and Mellor, G.L., “Diagnostic and prognostic numerical circulation studies of the South Atlantic Bight,” Journal of Geophysical Research, Vol. 88, C8, pp. 4579-4592 (1983). Hasselmann, K., Barnett, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, K., Ewing, J.A., Gienapp, H., Hasselmann, D.E., Kruseman, P., Meerburg, A., Muller, P., Olbers, D.J., Richter, K., Sell, W., Walden, H., “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP),” Dtsch. Hydrogr. Z., Suppl. 12, A8 (1973). Hsu T.W., Ou, S.H. and Liau, J.M., “Hindcasting nearshore wind waves using a FEM Code for SWAN,” Coastal Engineering, Vol. 52, No. 2, pp. 177-195 (2005). Hsu, T.W., Liau, J.M. and Lee, Z.S., “Storm surge prediction by FEM on the north-east coast of Taiwan,” Journal of the Chinese Institute of Civil and Hydraulic Engineering, Vol. 11, No. 4, pp. 849-857 (1999). (in Chinese) Kawahara, M., Hirano, H., Tsubota, K. and Inagaki, K., “Selective lumping finite element method for shallow water flow,” International Journal for Numerical Methods in Engineering, Vol. 2, pp. 89-112 (1982).

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Kawahara, M., Nakazawa, S., Ohmori, S. and Tagaki, T., “Two-step explicit finite element method for storm surge propagation analysis,” International Journal for Numerical Methods in Engineering, Vol. 15, pp. 1129-1148 (1980). Kuo, C.T. and Chien, C.H., “Performance assessment of coastal defence work in Taiwan,” Research Report, Department of Hydraulic and Ocean Engineering, NCKU, 173p (2001). (in Chinese) Longuet-Higgins, M.S., “Longshore currents generated by obliquely incident sea waves,” Journal of Geophysical Research, 75, pp. 6778-6801 (1970). Longuet-Higgins, M.S. and Stewart, R.W., “Radiation stress in water wave: a physical discussion with applications,” Deep-Sea Research, Vol. 11, No. 4, pp. 529-562 (1964). Longuet-Higgins, M.S. and Stewart, R.W., “A note on wave set-up,” J. Mar. Res., Vol. 21, No. 1, pp. 1-14 (1963). Mellor, G., “Users guide for a three-dimensional, primitive equation, numerical ocean model,” Princeton, Princeton University, 1998. Mellor, G.L. and Yamada, T., “Development of a turbulence closure model for geophysical fluid problems,” Rev. Geophys. Space Phys., Vol. 20, pp. 851-875 (1982). Nishimura, H., “Numerical simulation of nearshore circulations,” Proceeding of the 29th Japanese Conference on Coastal Engineering, Sendai, JSCE, pp. 333-337 (1982). (in Japanese) Smagorinsky, J., “General circulation experiments with the primitive equations. I. The basic experiment,” Monthly Weather Rev., Vol. 91, pp. 99-164 (1963). Walton, T.L. and Sensabaugh, W., “Seawall design on sandy beaches,” University of Florida Sea Grant Report (1978).