# Lagrangian Modeling of Marine Microplastics Fate and Transport: The State of the Science
## 3 Physical Properties and Processes
### Physical Transport Processes  
In addition to the transformation processes (biofouling, degradation, aggregation), numerical fate and transport models can simulate different physical processes driving microplastics' transport in water such as advection, diffusion, windage, resuspension, beaching, and washing-off. The transport of microplastics is generally governed by a generic advection-diffusion reaction, as follows [98]:  
\[\frac{\partial C}{\partial t}=\
abla.(CU)+\
abla.(K\
abla CU)+\rho(C) \tag{11}\]  
where \(C\) indicates the concentration at time \(t\), \(U\) is the velocity of a surrounding medium, \(K\) represents the dispersion coefficient, and \(\rho(C)\) shows the reaction function.  
Given the aforesaid concept, the main equations used in the reviewed models for addressing the main physical processes controlling microplastics' transport in aquatic systems are provided in Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Mathematical or Empirical Relationships** & **Application** & **Defining Parameters** & **Employed in** & **Ref.** \\ \hline \multirow{6}{*}{7} & \multirow{6}{*}{\(\rho_{p}=\rho_{0}\frac{R_{0}^{2}}{(R_{0}+BT)^{3}}+\rho_{D}\left[1-\frac{R_{0}^{2}}{(R_{0}+ BT)^{3}}\right]\)} & \multirow{6}{*}{Biofouling of spherical particles} & \(\rho_{p}\) (gm\({}^{-3}\)) is the density of a plastic particle, \(R\) (mm) indicates microplastic radius, and \(BT\) and \(\rho_{D}\) represent the thickness (um) and density (gm\({}^{-3}\)) of a biofilm layer, respectively. & \\ \hline \multirow{6}{*}{8} & \multirow{6}{*}{\(\rho_{p}=\rho_{0}\frac{R_{0}^{2}}{(R_{0}+BT)^{2}}+\rho_{D}\left[1-\frac{R_{0}^{2}}{(R_{0}+ BT)^{2}}\right]\)} & \multirow{6}{*}{Biofouling of cylindrical particles} & \(\rho_{p}\) (gm\({}^{-3}\)) is the density of a plastic particle, \(R\) (mm) indicates microplastic radius, and \(BT\) (um) and \(\rho_{D}\) (gm\({}^{-3}\)) represent the thickness and density of a biofilm layer, respectively. & \\ \hline \multirow{6}{*}{9} & \multirow{6}{*}{\(BT=BT_{0}+BR\Delta t\)} & \multirow{6}{*}{Biofouling} & \(BT_{0}\) (um) is the thickness of initial biofouling, \(BR\) is a constant biofilm thickness in stationary biofouling, and \(\Delta t\) indicates the time interval. & \\ \cline{1-1} \cline{3-4}  & & & \(D\) (mm) and \(L\) (mm) are the particle’s diameter and length, respectively. \(Size_{0}\) indicates the particle’s initial diameter and length.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **Mathematical or Empirical Relationships** & **Application** & **Defining Parameters** & **Employed in** & **Ref.** \\ \hline  & & & \(\overset{\widetilde{X}}{X}\) is the 3D position vector with components in the horizontal plane, \(d\) is the total derivative,\(\overset{\widetilde{u}}{X}\) is the 3D velocity vector, and \(\overset{\widetilde{X}}{X}\) is the fluctuation of the position vector. & & \\ \hline  & & & \(V_{adv}\) (ms\({}^{-1}\)) represents the velocity field, which is responsible for advection. \(V_{flow}\) (ms\({}^{-1}\)) is the velocity of flow, and & \\
13 & \(V_{adv}=V_{flow}+V_{windage}\) & Advection velocity & \(V_{windage}=C_{\text{rad}}\) (\(V_{w}-V_{flow}\)). & \(C_{\text{wd}}\) is an empirical wind drag coefficient that is related to particle characteristics and \(V_{w}\) (ms\({}^{-1}\)) represents the wind velocity at 10 above sea level. & \\ \hline  & & & \(K_{h(x,y)}\) is the horizontal component of the diffusion coefficient (m\({}^{2}\)s\({}^{-1}\)) at time \(t\) (which is defined from \(t=0\)), \(a\) (which is equal to the dispersion coefficient for small \(t\)), and & D-WAQ PART [113] \\
14 & \(K_{h(x,y)}=at^{b}\) & component of the diffusion coefficient & \(b\) (\(\in\) [0, 1]) are calibrating coefficients. It should be noted that the upper and lower limits of \(K_{h(x,y)}\) are equal to \(at\) and \(a\) respectively, and \(K_{h(x,y)}\) increases with time. It has been used in Equation (11). & \\ \hline  &