## 2 Materials and Methods  
### Fetswcm  
The tide and storm surge were simulated using the FETSWCM (finite element tide-surge-wave coupled model) model [11; 12]. The FETSWCM model is a two-dimensional hydrodynamic model encompassed with a specific typhoon wind model for the Taiwan Strait [13]. FETSWCM has demonstrated satisfactory performance in storm surge forecasting along the Fujian Coast, which is on the western bank of the Taiwan Strait. Using a right-handed Cartesian coordinate system, combined GWCE (generalized wave continuity equation) [14] and VIMEs (vertically integrated momentum equations), the governing equations of FETSWCM in two-dimensional form are given as:  
\[\frac{\partial^{2}\zeta}{\partial t^{2}}+\tau_{0}\cdot\frac{\partial\zeta}{ \partial t}+\frac{\partial}{\partial x}\bigg{(}\frac{\partial(U\cdot H)}{ \partial t}+\tau_{0}\cdot U\cdot H\bigg{)}+\frac{\partial}{\partial y} \bigg{(}\frac{\partial(V\cdot H)}{\partial t}+\tau_{  
\(t=\) time;  
\(\mathbf{x}\), \(\mathbf{y}=\) longitude (east) and latitude (north), respectively;  
\(U\), \(V=\) the east and north components, respectively, of depth averaged velocity;  
\(H=\) the total water depth, where \(H=h+\zeta\);  
\(h=\) the depth of the undisturbed water;  
\(f=\) the Coriolis parameter;  
\(\mathbf{g}=\) acceleration due to gravity;  
\(P_{s}=\) sea surface atmospheric pressure;  
\(\rho_{\text{a}}=\) the density of air;  
\(\rho_{0}=\) the density of sea water;  
\(W=\) the wind velocity vector at the height of 10 m above sea surface;  
\(\tau_{sx}\), \(\tau_{sy}=\) the east and north components, respectively, of wind stress friction;  
\[\tau_{s}=\rho_{\text{a}}\text{C}_{s}\cdot\left|\overset{\rightarrow}{\text{W }}\right|\cdot\overset{\rightarrow}{\text{W}} \tag{4}\]  
where C\({}_{s}\) is the wind stress coefficient, which varies with wind velocity as follows:  
\[\text{C}_{s}=\left\{\begin{array}{ccc}1.052\times 10^{-3}&&\left|\overset{ \rightarrow}{\text{W}}\right|\leq 6\ m/s\\ \left(0.638+0.069\left|\overset{\rightarrow}{\text{W}}\right|\right)\times 10^{-3 }&&6\ m/s\leq\left|\overset{\rightarrow}{\text{W}}\right|\leq 30\ m/s\\ 2.708\times 10^{-3}&&\left|\overset{\rightarrow}{\text{W}}\right|\geq 30\ m/s \end{array}\right. \tag{5}\]  
\(\tau_{bx}\), \(\tau_{by}=\) the east and north components, respectively, of bottom friction.  
\[\tau_{bx}=\rho_{0}\cdot\text{C}_{d}\cdot U\cdot\sqrt{(U^{2}+V^{2})} \tag{6}\]  
\[\tau_{by}=\rho_{0}\cdot\text{C}_{d}\cdot V\cdot\sqrt{(U^{2}+V^{2})} \tag{7}\]  
where \(C_{d}\) is the bottom friction coefficient, which was taken as 0.0015 in this study.  
\(\tau_{wx}\), \(\tau_{wy}=\) the east and north components, respectively of wave force (gradient of the wave radiation stress) which are calculated through SWAN model (Section 2.2) as follows:  
\[\tau_{wx}=-\frac{\partial S_{xx}}{\partial x}-\frac{\partial S_{xy}}{\partial y} \tag{8}\]  
\[\tau_{wx}=-\frac{\partial S_{yx}}{\partial x}-\frac{\partial S_{yy}}{\partial y} \tag{9}\]  
where \(S_{xx}\), \(S_{xy}\), \(S_{yx}\), \(S_{yy}=\) the radiation stress tensor (m/s).  
In GWCE, the spatially differentiated momentum equation in its conservative form is combined with the temporally differentiated continuity equation, and the continuity equation is multiplied by the numerical parameter \(\tau_{0}\), which is a constant that controls the balance of the wave equation and the primitive continuity equation. Extensive analysis of the GWCE has demonstrated that the scheme is stable [15].