Maximum Storm Tide Response to the Intensity of Typhoons and Bathymetric Changes along the East Coast of Taiwan

A typhoon-induced storm surge is considered one of the most severe coastal disasters in Taiwan. However, the combination of the storm surge and the astronomical tide called the storm tide can actually cause extreme flooding in coastal areas. This study implemented a two-dimensional hydrodynamic model to account for the interaction between tides and storm surges on the coast of Taiwan. The model was validated with observed water levels at Sauo Fish Port, Hualien Port, and Chenggong Fish Port under different historical typhoon events. The model results are in reasonable agreement with the measured data. The validated model was then used to evaluate the effects of the typhoon's intensity, bathymetric change, and the combination of the typhoon’s intensity and bathymetric change on the maximum storm tide and its distribution along the east coast of Taiwan. The results indicated that the maximum storm tide rises to 1.92 m under a typhoon with an intensity of a 100-year return period. The maximum storm tide increased from a baseline of 1.26 m to 2.63 m for a 90% bathymetric rise at Sauo Fish Port under the conditions of Typhoon Jangmi (2008). The combination of the intensity of a typhoon with a 100-year return period and a 90% bathymetric rise will result in a maximum storm tide exceeding 4 m, 2 m, and 3 m at Sauo Fish Port, Hualien Port, and Chenggong Fish Port, respectively. We also found that the distribution of the maximum storm tide on the east coast of Taiwan can expand significantly subject to the bathymetric rise.


Introduction
Storm surges are a result of low-pressure weather systems (cyclones), creating suction over the covered area of the ocean, and the strong winds that are usually present in storms.High sea levels and strong forces, including low pressure, heavy wind, and wave set-up, directly influence humans, property, and habitats by over-topping sea dikes.Storm surges may even cause loss of life; damage; and the loss of habitat, useful land, property, infrastructure, and services [1][2][3].
Storm surge modeling is an important tool for operational flood forecasting systems and flood hazard assessments, particularly for understanding the risks from these moderate to severe storms that have occurred only a few times.In the past, the simplest method of forecasting the maximum storm surge was to use an empirical formula [4,5].Conventional studies on the storm tide or storm surge and the astronomic tide and storm surge were individually simulated and then superposed.However, the interaction caused by the tide and storm surge might be neglected [6].Today, the numerical models of storm surges have been widely applied to predict the surge tides [7][8][9][10][11][12][13]. For example, Xu et al. [14] integrated Monte Carlo and hydrodynamic models to estimate extreme water levels that occurred as a consequence of a storm surge.Murty et al. [15] coupled wave and surge hydrodynamic modeling systems to simulate the storm surge, water level elevation, and wave induced setup with 'Phailin', a very severe cyclonic storm that made landfall in Odisha State on the east coast of India in October 2013.Marujo et al. [16] adopted the advanced circulation model (ADCIRC) to investigate the ability of ADCIRC to model the storm surge that occurred in Viana do Castelo (Portugal) on 14-17 October 1987.Drews and Galarneau [17] examined the storm surge in New York Harbor using the Weather Research and Forecasting (WRF) atmospheric model and the Coupled-Ocean-Atmosphere-Wave-Sediment transport/Regional Ocean Modeling System (COAWST/ROMS).They presented a new technique using a directional analysis to calculate and display maps of a coastline's potential for a storm surge.Zhang and Sheng [18] investigated the extreme sea levels of the 50-year return period associated with storm surges and tides over the Northwest Pacific Ocean using a two-dimensional ocean circulation model.The model forcing included surface wind stress and atmospheric pressure at sea level obtained from the Climate Forecast System Reanalysis fields at 6-hour intervals.Arns et al. [19] assessed the impact of the mean sea level rise on extreme water levels using a numerical tide-surge model that covered the entire North Sea, whereas the bathymetric resolution was highest in the German Bight.Zhang et al. [20] investigated the effect of angular deviations in the forecasted typhoon track on the storm surge height.The simulated results revealed that the uncertainty of the forecasted typhoon track is an important factor to affect the accuracy of storm surge predictions.
To accurately simulate and appropriately describe the physics of storm surges, a hydrodynamic model must treat costal features that can significantly affect the generation and propagation of the storm surge.Therefore, the model domain should incorporate complex costal bathymetry and topography.Shen et al. [21] reported that with the use of unstructured grids, a finite difference model often represented the irregular coastal morphological features better than finite difference models.However, its computational speed is restricted by the stability condition when a very fine resolution grid is used.In this context, a storm surge model that can resolve complex coastal morphological features and maintain computation efficiency is required.
The main objective of this study focuses on the investigation of the maximum storm tide along the east coast of Taiwan with an unstructured grid, two-dimensional hydrodynamic model.The model was validated against the observed water levels at different gauge stations for different historical typhoon events.The validated model was then applied to investigate the maximum storm tide response to a typhoon's intensity and bathymetric change.

Description of the Study Area
The computational domain covers the region between the longitude of 114°E to 130°E and the latitude of 19°N to 29°N.This area is composed of the Western Pacific Ocean, the Taiwan Strait, the South China Sea, and the East China Sea, which are located to the east, west, south, and north of Taiwan, respectively (Figure 1a).The bathymetric data were obtained from coastal digital elevation models (DEMs), ETOPO1, and a global relief model.ETOPO1 integrates land topography and the ocean bathymetry of the Earth's surface with 1 arc-minute resolution [22].Figure 1a also shows the bottom elevations for the hydrodynamic model.The bathymetries on the east coast of Taiwan are quite steep.The bottom elevations rapidly vary from several meters at the shoreline to several thousand meters at a distance of 10 km from the coast.A total of 278,630 triangular cells and 142,041 non-overlapping unstructured grids were used in the horizontal plane to fit the complex shoreline of Taiwan and its adjacent small islands.The mesh resolution ranges from 30 km to 300 m.The fine meshes were around the coast of Taiwan and its offshore islands while the coarse meshes were along the boundary of the open ocean (Figure 1b).A time step of 60 s was used in the simulations with no signs of numerical instability for this model grid system.

Hydrodynamic Model
The coastal hydrodynamic model, SELFE [23], was applied to compute the tide and storm surge along the coast of Taiwan.SELFE is the Semi-implicit Eulerian-Lagrangian finite-element model and has been widely used for simulating tsunami propagation [24], assessing water quality and ecosystem dynamics [25], evaluating oil spill dynamics [26], generating inundation maps [27], and inundation modeling of extreme river flow, typhoons, and hurricanes [28,29].
The SELFE used in the present study is a two-dimensional vertically integrated version (SELFE-2D).The government equations in two-dimensional form are given as: ( ) ( ) where ( , , ) τ and sy τ , are the wind stress in the x and y directions, respectively, which can be expressed as: ( ) where s C is the wind drag coefficient; a ρ is the air density; and , x y W W are the wind speed in the , x y components, respectively.The wind drag coefficient, s C , is given by Large and Pond [30]   and Powell et al. [31].
bx τ and by τ are the bottom shear stress in the x and y directions, respectively.The bottom friction force is given by a quadratic drag law: where b C is the bottom drag coefficient and will be determined through the validation of the model.

Global Model for the Prediction of Ocean Tides
For many practical applications of modeling in the coastal environment, accurate predictions of tidal currents or elevations are always indispensable.Due to the necessity of simulating the interaction between astronomical tides and storm surges, the driving forces at the open boundaries of the hydrodynamic model are tidal elevations.In the present study, a global ocean tidal prediction model, TPXO, which was developed by scholars at the Oregon State University, was adopted to extract tidal harmonic constants and then specify the open boundaries of SELFE-2D for simulating the tidal propagation.TPXO uses inverse theory and assimilated measured data from tidal gauges and the TOPEX/Poseidon satellite to achieve the optimum balance between observations and hydrodynamics.The extracted tidal harmonic constants provided complex amplitudes of the earth-relative sea-surface elevation for eight primary tidal constituents (i.e., M2, S2, N2, K2, K1, O1, P1, and Q1).The detailed methodologies used to compute the tides in TPXO can be found in Egbert et al. [32] and Egbert and Erofeeva [33].

Parametric Cyclone Model
The meteorological boundary conditions for storm surge calculations consist of the wind fields and atmospheric pressure of the typhoon.In practice, a common way to generate the wind fields and air pressure fields of typhoons is to reconstruct them using the analytical parametric cyclone model.Over the years, many parametric cyclone models have been developed to provide meteorological information for storm surge modeling [34][35][36][37][38]. Jakobsen and Madsen [39] investigated and compared parametric cyclone models based on the cyclone position, central pressure, maximum wind speed, and radius to maximum wind speed.They found that the analytical models provide very similar air pressure and tangential wind speed distributions for a cyclone.Therefore, a parametric cyclone model presented by Holland [35] was employed: ( ) where A P is the air pressure; n P is the ambient pressure; c P is the central air pressure of the typhoon; max R is the radius to maximum wind speed; r is the radial distance from typhoon center; W is the wind speed; a ρ is the air density; f is the Coriolis factor; , x y W W are the wind speeds in the , x y components, respectively; θ is the azimuthal angle with respect to the typhoon's eye; and B is a parameter that characterizes the scale of the typhoon.The formula of ( ) To estimate the minimum central pressure with different return periods, two approaches for frequency analysis, the Gumbel distribution [41] and the Weibull distribution [42], were adopted for analyzing the extreme values.The formulas can be represented as: where x is the sample variables; ( ) f x is the probability density function; ( ) F x is the cumulative probability function; σ is the scale parameter; ξ is the local parameter; α is the shape parameter ( 0 > α ); and β is the scale parameter of the distribution ( 0 > β ).The formula for calculating the difference between the ambient and typhoon central pressure, c P Δ , that corresponds to each return period can be expressed as: ( ) where T is the return period.

Indicators of Mode Performance
Three criteria are adopted to evaluate the model performance for water level simulation.The criteria are the mean absolute error (MAE), the root mean square error (RMSE), and percent bias (PBIAS) [43].The optimal value of PBIAS is 0 with low-magnitude values, indicating accurate simulations of the model.Positive values of PBIAS indicate that the model overestimates the water level, while negative values indicate that the model underestimates the water level [44].The equations for these three criteria are shown as follows: where η is the simulated water level, and m i η is the measured water level.

Model Validation
To confirm the model's capability and ability to simulate storm tides along the east coast of Taiwan, ten historical typhoon events, including Typhoon Tim (1994), Typhoon Bilis ( 2000  The information of the track and central pressure of the ten typhoon events used to construct the wind and air pressure fields can be obtained from the Regional Specialization Meteorological Center (RSMC) Tokyo-Typhoon Center best track database.The hourly water level data measured at three tide gauge stations was provided by the Central Weather Bureau (CWB).Figures 3-5 present the model-data comparisons of the water levels for different typhoon events at Sauo Fish Port (Figure 3), Hualien Port (Figure 4), and Chenggong Fish Port (Figure 5).The water level hydrographs indicate that the simulations have good agreement with the observations for both astronomical tides and storm tides, except at Hualien Port during Typhoon Jangmi (2008) (Figure 4c).The model simulations underestimated the water level around the maximum storm tide, and these underestimations may be due to the inaccuracy of the wind and air pressure fields provided by the parametric cyclone model.The parametric cyclone model is sometimes difficult to use to represent the structure of the typhoon well at a certain location around landfall.Feng et al. found a similar problem regarding the inaccuracy of the wind and air pressure forcing.The statistical errors for the differences between the simulated and observed water levels at the three tidal gauge stations can be found in Table 1.The maximum MAE, RMSE, and PBIAS are 0.14 m, 0.17 m, and -11.46%, respectively, at Hualien Port. Figure 6 shows the comparison of the simulated and measured maximum storm tides for historical typhoon events at three tide gauge stations.The maximum storm tides for simulation and measurement are 1.49m and 1.41 m, respectively, and occurred at Chenggong Fish Port during Typhoon Bilis (2000) (Figure 6c).The maximum difference between the simulations and measurements of the maximum storm tide is 0.09 m (Figure 6c) and also occurred at Chenggong Fish Port for Typhoon Kaemi (2006).The phase error between the simulated and measured maximum storm tide is 1 h (i.e., lag 1 h, Figure 6b) at Hualien Port during Typhoon Krosa (2007) and Typhoon Fungwong (2008).Overall, the simulated results quantitatively agree with the observations of astronomical tides and storm surges.Throughout the model validation procedure, the bottom drag coefficient was set to 0.005.

Modeling Investigation and Discussion
Two typhoon events, Typhoon Bilis (2000) for Chenggong Fish Port and Typhoon Jangmi (2008) for Sauo Fish Port and Hualien Port, induced maximum storm tides and were selected as baseline typhoon events to investigate the influence of a typhoon's intensity and bathymetric change on the maximum storm tide and storm tide distribution.

Maximum Storm Tide Response to Typhoon's Intensity
To determinate a valid intensity of a typhoon, i.e., the minimum central pressure, the typhoon events that passed Taiwan within a distance of 50 km from 1994 to 2014 were adopted, and the annual minimum central pressure was then selected for an extreme value analysis.Based on the frequency analysis methods, the parameters, σ , ξ , α , and β are estimated to be 14.911, 73.357, 4.434, and 88.15, respectively, for the Gumbel distribution and the Weibull distribution in Eqs. ( 9)-( 12).
Figure 7 illustrates the relationship between the cumulative probability and pressure difference between the ambient pressure (1013.25 mb) and the central pressure of a typhoon, , using the Weibull distribution and Gumbel distribution.Figure 7 shows that the correlation coefficient (R=0.99)from the Weibull distribution is higher than that from the Gumbel distribution (R=0.97);thus, the frequency analysis method using Weibull distribution is selected as the suitable approach for the present study.Table 2 indicates that the values of c P Δ under different return periods depended on the frequency analysis.Table 2 shows that the The simulated maximum storm tides under baseline, 20-year, 50-year, and 100-year return periods are shown in Figure 8.Note that the simulated maximum storm tides at the gauge stations at Sauo Fish Port and Hualien Port adopt the track path of Typhoon Jangmi (2008), while the simulated maximum storm tide at Chenggong Fish Port applies the track path of Typhoo Bilis (2000).Compared to baseline and the 20-year return period, the simulated maximum storm tide significantly increased.However, the increment of the maximum storm tide with the changes in central pressures with the 20-year, 50-year, and 100-year return periods is small.The small increment may be the reason that the central air pressure of typhoons for the 20-year, 50-year, and 100-year return periods is similar (Table 2).Compared to baseline and the central pressure of the 100-year return period, the maximum increment of the maximum storm tide is only 0.57 m at Hualien Port (Figure 8).
Figure 9 shows the water level hydrographs for Sauo Fish Port, Hualien Port, and Chenggong Fish Port under baseline and c P Δ specified with different return periods.It indicates that the water level increases as increasing the return period.The figure also shows the phase shift of maximum storm tide under baseline and c P Δ specified with different return periods at the Sauo Fish Port (Figure 9a) and Hualien Port (Figure 9b) during Typhoon Jangmi (2008), while the phase keeps the same at maximum storm tide at the Chenggong Fish Port during Typhoon Bilis (2000).It is the reason that the track of Typhoon Jangmi (2008) passes through the path between Sauo Fish Port and Hualien Port (see Figures 2a and 2b) and the storm tide is significantly influenced by wind stress at these two stations.As the c P Δ value increases (i.e.c P decreases), the radius of maximum wind speed ( max R ) decreases, resulting in stronger wind stress [46,47] to alter the phase of maximum storm tide.The track of Typhoon Bilis (2000) passes through the Chenggong Fish Port (see Figure 2c).The storm tide is mainly affected by air pressure therefore the phase of maximum storm tide keeps the same for baseline and different return periods (Figure 9c).In general, air pressure affects the storm surge by the way of inverse barometer [48].The effect of the air pressure is only evident near the typhoon center where the minimum air pressure occurs.Feng et al. [45] used the Advanced Circulation model (ADCIRC) to investigate the sensitivity of Tianjin storm surge to the four paths using the validated storm surge simulation system and found that the hurricane path had significant influence on storm surge height.In the current study, the path of Typhoon Bilis (2000) closely approached to Chenggong Fish Port resulting in highest maximum storm tide under the conditions of baseline and different return periods (see Figure 8).

Maximum Storm Tide Response to Bathymetric Change
To investigate the influence of bathymetric changes on storm surges, three model experiments with bathymetry rising 50%, 70% and 90% in the entire computational domain were examined using the validated hydrodynamic model.Two baseline typhoons, Typhoon Bilis (2000) and Typhoon Jangmi (2008), were used in all of the scenario simulations.Figure 10 shows the water level hydrographs for Sauo Fish Port (Figure 10a), Hualien Port (Figure 10b), and Chenggong Fish Port (Figure 10c) using different bathymetric rise conditions.Figure 10 indicates that the maximum storm tide increases with rising bathymetry, particularly for the 90% bathymetric rise.It also can be observed that both positive and negative storm surges can be enlarged once the bathymetries become shallow.Weaver and Slinn [49] developed a one-dimensional analytical solution for waves and water level.They found that if the amplitudes of the bathymetric fluctuation are less than 60% of the original depth, the variation in the storm surge is within 10% ± of the original storm surge.Our model experiments for the storm tide response to bathymetric change are similar to the findings of Weaver and Slinn [49].The meteorological forcing terms that affect storm surges include the wind stress and horizontal atmospheric pressure gradient.If we only consider the effects of the atmospheric pressure gradient on the water level with one-dimension, Eq. ( 2) can be simplified as: Eq. ( 17) reveals that the contribution of A P Δ to the water level in the storm surge is irrelevant to the water depth.Dean and Dalrymple [50] explained that the water level in a storm surge is related to wind stress ( s τ ), bottom stress ( b τ ), and total water depth ( h η + ) using a simple steady one-dimensional momentum equation: As shown in Eq. ( 18), the water surface slope is proportional to the difference between wind stress and bottom stress and inversely proportion to the total water depth.If the difference between wind stress and bottom stress remains constant, the decreased water depth can result in an increased water level gradient.Flather and Williams [51] also documented that the water depth could influence the tidal component by changing its wavelength and modifying the propagation and dissipation of the tidal energy.The decreased water depth also affected the generation, propagation, and dissipation of the storm surge component, perhaps by slightly increasing the surge magnitude.
According to the Eq. ( 18) and the report by Flather and Williams [51], we found that the tidal amplitude slightly increased at Soui Fish Port (Figure 10a), Hualien Port (Figure 10b) and Chenggong Fish Port (Figure 10c) due to the rising bathymetry (i.e., the decreasing water depth).Another interesting phenomenon is that the phase of the tidal wave lags behind the baseline condition when the bathymetric rise reaches 90%.Tidal waves are regarded as shallow water waves, and the tidal wave speed is independent of the wavelength [52].The formula for the tidal wave speed, c , can be represented as: The tidal wave speed, c , decreases as a result of a decrease in water depth, resulting in the phase lag shown in Figure 10.

The Combined Effect of a Typhoon's Intensity and Bathymetric Change
To comprehend the influence of the combination of a typhoon's intensity and bathymetric change on storm surges, the combined effect of c P Δ specified with a 100-year return period and a 90% bathymetric rise using the tracks of baseline typhoons were examined.Figure 11 illustrates the comparison of the simulated water level under baseline conditions and different scenarios.Figure 11 shows that that the maximum storm tides exceed 4 m, 2 m, and 3 m at Sauo Fish Port, Hualien Port, and Chenggong Fish Port, respectively, as a result of the combination of c P Δ specified with a 100-year return period and a 90% bathymetric rise.The maximum storm tide induced by a 90% bathymetric rise is higher than that induced by c P Δ with a 100-year return period at Sauo Fish Port (Figure 11a) and Chenggong Fish Port (Figure 11c).Compared to the maximum storm tide induced by a 90% bathymetric rise and by c P Δ with a 100-year return period, they are similar to Hualien Port (Figure 11b).
Figure 12 and Figure 13 show the distributions of the maximum storm tide in the horizontal plane under baseline conditions and different simulation scenarios, respectively, for Typhoon Bilis (2000) and Typhoon Jangmi (2008).We found that the combination of c P Δ specified with a 100-year return period and a 90% bathymetric rise produced the largest extent of storm tide (Figure 12d and Figure 13d).Compared to the baseline (Figure 12a and Figure 13a) and to c P Δ specified with a 100-year return period (Figure 12b and Figure 13b), the storm tides induced by a 90% bathymetric rise exhibit a longer (north-south) distribution along the shoreline (Figure 12c and Figure 13c) due to the shallower coast.

Conclusions
The two-dimensional hydrodynamic model, SELFE-2D, was implemented to simulate the storm tide along the coast of Taiwan.The model was validated with measured astronomical tides and storm tides at three tidal gauge stations for ten historical typhoon events.We found that the hydrodynamic model satisfactorily reproduced the tidal levels at Sauo Fish Port, Hualien Port, and Chenggong Fish Port, which are affected by astronomical tides and storm surges.The model is also capable of mimicking the maximum storm tide.
The frequency analysis was adopted to determine the pressure difference between the ambient pressure (1013.25mb)and the center pressure of a typhoon, , for different return periods.The validated hydrodynamic model was then applied to investigate the maximum storm tide along the east coast of Taiwan in response to changes in the typhoon's intensity and bathymetry.The simulated results reveal that the maximum increment of the maximum storm tide reaches 0.57 m compared to the baseline condition, and c P reached 888.85 mb for a typhoon with a 100-year return period at Hualien Port.The maximum storm tide increases as a result of rising bathymetry.The phase of the tidal wave lags behind the baseline condition when the bathymetric rise reaches 90%.The maximum storm tides exceed 4 m, 2 m, and 3 m at Sauo Fish Port, Hualien Port, and Chenggong Fish Port, respectively, as a result of the combination of c P Δ from a typhoon with a 100-year return period and 90% bathymetric rise.The simulated results also indicate that the storm tides induced by the condition of 90% bathymetric rise produced a longer (north-south) distribution along the east coast of Taiwan.

Figure 1 .
Figure 1.(a) Bathymetry and (b) unstructured grids of the computational domain.
), Typhoon Nockten (2004), Typhoon Kaemi (2006), Typhoon Sepat (2007), Typhoon Krosa (2007), Typhoon Fungwong (2008), Typhoon Jangmi (2008), Typhoon Sinlaku (2008) and Typhoon Fanapi (2010), were used for validating the two-dimensional hydrodynamic model.The tracks of these ten Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 28 February 2017 doi:10.20944/preprints201702.0107.v1typhoons are shown in Figure 2.To make the comparison of water levels between the simulation and observations, three tide gauge stations, Sauo Fish Port, Hualien Port, and Chenggong Fish Port, which are located on the east coast of Taiwan, were selected for the model validation.Figure2also illustrates the locations of the tidal gauge stations and their corresponding four typhoon events for the model validation.The observed water levels of the four typhoons at each tidal gauge station were used to validate the hydrodynamic model.

Figure 2 .
Figure 2. Tracks of typhoons for model validation and their corresponding tide gauge stations at (a) Sauo Fish Port, (b) Hualien Port, and (c) Chenggong Fish Port.

Figure 6 .
Figure 6.Model-data comparison of maximum storm tides for different typhoon events at (a) Sauo Fish Port, (b) Hualien Port, and (c) Chenggong Fish Port.
values sequentially increase by 2 mb only for the 100-year, 150-year, and 200-year return periods.Therefore, the c P Δ values for the 20-year, 50-year, and 100-year return periods were used in following simulations, and their corresponding central pressure ( c P ) values are 900.35mb, 893.35   mb, and 888.85 mb, respectively (see Table2, .These three central pressure values combined with the fixed track paths, Typhoon Bilis (2000) and Typhoon Jangmi (2008), were used for different scenario simulations.

Figure 7 .
Figure 7.The relationship between the cumulative probability and pressure difference ( c P Δ ) using different frequency analysis models (i.e., the Gumbel distribution and Weibull distribution).

Figure 8 .Figure 9 .
Figure 8.The comparison of the simulated maximum storm tide under baseline, 20-year, 50-year, and 100-year return periods at different tide gauge stations.

PreprintsFigure 11 .
Figure 11.Comparison of the simulated water level under baseline,

Table 1 .
Statistical error between simulated and measured water levels.

Table 2 .
Computed c P Δ in different return periods with the Weibull distribution.