## 2 Materials and Methods
### Particle-Tracking Model  
The macroalgae migration process is simulated by a particle-tracking model based on the Lagrangian method, and the macroalgae are regarded as discrete particles. The resulting equation, accounting for currents and random dispersion, is written as:  
\[\mathbf{X}_{t\pm\Delta t}=\mathbf{X}_{t}+\mathbf{V}(\mathbf{X}_{t},t)\Delta t+ \Delta\mathbf{X}_{r} \tag{6}\]  
Figure 2: Computational domains with bathymetry and meshes of the (**a**,**d**) Bohai model; (**b**,**e**) Qinhuangdao model; and (**c**,**f**) Jinmeng Bay model. Red lines represent the open boundaries of the three models. where \(\textbf{X}=(x,y)\) is the position of each particle at time \(t\); **V** is the current vector at the position of each particle at time \(t\); \(\Delta\textbf{X}_{r}\) is the horizontal random dispersion for particle motion; and \(\Delta t\) is the calculation time step. As mentioned before, the drifting velocity of floating macroalgae contains the current velocity calculated from the hydrodynamic model and the windage-induced velocity, which is expressed by the 10 m height wind speed multiplied by a windage coefficient. The equation of the drifting velocity of macroalgae is shown as follows:  
\[\textbf{V}=\textbf{v}+(\kappa\textbf{V}_{w}) \tag{7}\]  
where \(v\) is the current vector derived from the hydrodynamic model, \(\kappa\) is the windage coefficient, and \(\textbf{V}_{w}\) is the 10 m height wind vector. The brackets denote that the second term of windage is included for surface-floating macroalgae and excluded for submerged macroalgae. As opposed to the floating green macroalgae in the Yellow Sea, the macroalgae in Jinmeng Bay were spread throughout the water column. In this research, the calculation mesh of the 2 DH model applied a single layer vertically, and therefore a theoretical velocity profile was applied for the vertical redistribution of the current velocity. The velocity profile took into consideration smooth flow and rough flow. Smooth flow included a sub-layer where viscosity controlled the momentum transport and a super-layer where turbulence controlled the same aspect; the sub-layer thickness (\(\delta\)) was more than the bed roughness height (\(z_{0}\)). However, for rough flow the viscous sub-layer was extremely thin and less than the bed roughness height, so only the super-layer velocity profile was taken into consideration. The velocity profile equations for smooth flow (\(\delta>z_{0}\)) are shown as follows:  
\[|\textbf{v}|=\left\{ \begin{array}{l}\frac{u_{*}}{\kappa}\ln\left(\frac{z}{z _{0}}\right),z\geq\delta\\ u_{*}^{2}\frac{z}{v},z<\delta\end{array} \right. \tag{8}\]  
where \(u_{*}\) is the friction velocity; \(\kappa\) is the von Karman constant, and usually taken as 0.4; \(z\) is the vertical coordinate; and \(v\) is the kinematic viscosity. The velocity profile equations for rough flow (\(\delta<z_{0}\)) are shown as follows:  
\[|\textbf{v}|=\left\{ \begin{array}{l}\frac{u_{*}}{\kappa}\ln\left(\frac{z}{z _{0}}\right),z\geq z_{0}\\ 0,z<z_{0}\end{array} \right. \tag{9}\]  
Turbulence-induced dispersion is also included in the particle-tracking calculation by adding a horizontal random dispersion term, \(\Delta\textbf{X}_{r}\), in Equation (6), and the expression of \(\Delta\textbf{X}_{r}\) is shown as follows:  
\[\Delta\textbf{X}_{r}=\textbf{a}R\sqrt{2K_{r}\Delta t} \tag{10}\]  
where \(a\) is a horizontal unit vector with a random direction; \(R\) is a random number following standard normal distribution; and \(K_{r}\) is the dispersion coefficient, which is specified as 2 m\({}^{2}\)/s in the model. The particle-tracking model applies the same computational mesh as the hydrodynamic Jinmeng Bay model (Figure 2c). The current vectors used for the particle position calculation were from the results of the hydrodynamic model.