## 2 Materials and Methods
### Hydrodynamic Model  
The Lagrangian particle-tracking model used to simulate the migration process of green macroalgae is based on the currents calculated by a hydrodynamic model. The hydrodynamic model was established by applying the integrated numerical model package MIKE21, developed by the Danish Hydraulic Institute (DHI) [42], which is applicable to rivers, lakes, estuaries, bays, and coastal waters. Given that the coverage area of green tides is restricted to the shallow intertidal and subtidal zones, a 2 DH model mesh is adequate to simulate the hydrodynamics of the study area, considering the balance of computational speed and model precision in Qinhuangdao [43; 44]. The hydrodynamic control equations are based on the incompressible Navier-Stokes equations in view of the Boussinesq assumption and hydrostatic assumption. The computational mesh used in this research was an unstructured triangular mesh with the spatial discretization adopting the cell-centered finite volume method. The continuity and momentum equations are  
Figure 1: (**a**) Map of Jinmeng Bay, Qinhuangdao, with bathymetry. Hollow triangles are macroalgae measuring points, and the rectangular shadow area is the seaweed bed, which is the original source of the green tides; (**b**) Beached green macroalgae; (**c**) _Uliva pertusa_; (**d**) _Bryopsis plumose_; (**e**) _Uliva prolifera_. converted into the nonlinear shallow water equations by integrating them over depth, and the 2 DH equations are shown as follows:  
\[\frac{\partial h}{\partial t}+\frac{\partial hu}{\partial x}+\frac{\partial hv}{ \partial y}=hS \tag{1}\]  
\[\frac{\partial hu}{\partial t}+\frac{\partial hu^{2}}{\partial x}+\frac{ \partial hvu}{\partial y}=fvh-gh\frac{\partial\eta}{\partial x}-\frac{h}{ \rho_{0}}\frac{\partial p_{a}}{\partial x}-\frac{gh^{2}}{2\rho_{0}}\frac{ \partial\rho}{\partial x}+\frac{\tau_{sv}}{\rho_{0}}-\frac{\tau_{bv}}{\rho_{0} }-F_{u}+hu_{s}S \tag{2}\]  
\[\frac{\partial hv}{\partial t}+\frac{\partial huv}{\partial x}+\frac{\partial hv ^{2}}{\partial y}=-fuh-gh\frac{\partial\eta}{\partial y}-\frac{h}{\rho_{0}} \frac{\partial p_{a}}{\partial y}-\frac{gh^{2}}{2\rho_{0}}\frac{\partial\rho}{ \partial y}+\frac{\tau_{sy}}{\rho_{0}}-\frac{\tau_{by}}{\rho_{0}}-F_{v}+hv_{ s}S \tag{3}\]  
where \(t\) is the time; \(x\) and \(y\) are the Cartesian coordinates; \(u\) and \(v\) are the depth-averaged velocity components in the \(x\) and \(y\) directions; \(h\) is the sum of the still water depth, \(d\), and the time-varying water level, \(\eta\); \(S\) is the discharge of the point sources; \(f\) is the Coriolis parameter related to the angular velocity of Earth's rotation and the geographic latitude; \(g\) is the gravitational acceleration; \(\rho\) is the water density; \(\rho_{0}\) is the reference water density; \(p_{a}\) is the atmospheric pressure; \(\tau_{sx}\) and \(\tau_{sy}\) are the surface stresses induced by wind; \(\tau_{bx}\) and \(\tau_{by}\) are the bottom stresses; \(F_{u}\) and \(F_{v}\) are the horizontal tress terms due to eddy viscosity; and \(u_{s}\) as well as \(v_{s}\) are the discharge velocities of the point sources. The effect of wind is considered as one of the most important factors affecting the macroalgae drifting, which includes the wind-driven current and the direct frictional drag on the surface-floating macroalgae (defined as windage or leeway) [35, 37, 39, 41]. The wind-driven current is calculated through \(\tau_{s}=\left(\tau_{sx},\tau_{sy}\right)\) in the momentum Equations (2) and (3), which is given by:  
\[\tau_{s}=\rho_{a}c_{f}|\mathbf{u}_{w}|\mathbf{u}_{w} \tag{4}\]  
\[c_{f}=\left\{ \begin{array}{ll}c_{a}&u_{w}<w_{a}\\ c_{a}+\frac{c_{b}-c_{a}}{w_{b}-w_{a}}(u_{w}-w_{a})&w_{a}\leq u_{w}<w_{b}\\ c_{b}&u_{w}\geq w_{b}\end{array} \right.