# Lagrangian Modeling of Marine Microplastics Fate and Transport: The State of the Science
## 4 Numerical Lagrangian Models
\begin{table}
\begin{tabular}{c c c c c c} \hline
**No.** & **Mathematical or Empirical Relationships** & **Application** & **Defining Parameters** & **Employed in** & **Ref.** \\ \hline
19 & \
\begin{tabular}{c} \(x_{n+1}=x_{n}+U\Delta t_{i}\) \\ \(y_{n+1}=y_{n}+\
abla At_{i}\) \\ \(z_{n+1}=z_{n}+W\Delta t_{i}\) \\ \end{tabular} & \
\begin{tabular}{c} First-order Euler \\ method for \\ advection \\ \end{tabular} &
\begin{tabular}{c} \(\Delta t_{i}\) is an internal time interval \\ and \(U=(U,V,W)\) is the velocity \\ vector. This is a first-order \\ discretization of Equation (11). \\ \end{tabular} & [61] \\ \hline
20 & \begin{tabular}{c} \(x_{n+1}=x_{n}+R\left(2r^{-1}K_{\text{B}}\Delta t_{i}\right)^{1/2}\) \\ \end{tabular} & \begin{tabular}{c} Calculation of \\ horizontal \\ displacement due \\ to diffusion \\ Equation (11) \\ \end{tabular} & \
\begin{tabular}{c} \(R\) indicates a random number \\ with an average of zero and \(r\) is its \\ standard deviation that equals 1, \\ and \(K_{\text{B}}\) is horizontal diffusivity \\ \(\left(\text{m}^{2}\text{s}^{-1}\right)\). \\ \end{tabular} &
\begin{tabular}{c} TrackMPD \\ [61] \\ \end{tabular} \\ \hline
21 & \begin{tabular}{c} \(z_{n+1}=z_{n}+R\left(2r^{-1}K_{\text{B}}\Delta t_{i}\right)^{1/2}\) \\ \end{tabular} & \
\begin{tabular}{c} Calculation of \\ vertical diffusion \\ term Equation (11) \\ \end{tabular} &
\begin{tabular}{c} \(K_{\text{B}}\) is vertical diffusivity \(\left(\text{m}^{2}\text{s}^{-1}\right)\). \\ \end{tabular} & [61] \\ \hline
22 & \begin{tabular}{c} \(U=\frac{U_{f}+U_{w}}{1+\sqrt{\frac{\phi_{w}}{\phi_{w}}}}\frac{\phi_{w}}{\phi_{w}}\) \\ \(1+\sqrt{\frac{\phi_{w}}{\phi_{w}}}\frac{\phi_{w}}{\phi_{w}}\) \\ \end{tabular} & \begin{tabular}{c} Velocity field as a \\ function of current \\ and wave velocities \\ \end{tabular} & \
\begin{tabular}{c} \(U_{f}\) is the current velocity, \(U_{w}\) is the wind velocity, \(\rho_{w}\) and \(\rho_{w}\) \\ indicate the density of air and \\ water, \(S_{above}\) and \(S_{below}\) represent \\ the cross-sectional areas of \\ spherical and cylindrical particles \\ in dry and wet conditions, \\ respectively. Note that Equation (17) and (21) use the same \\ principles and their difference is \\ only in some coefficients. \\ \end{tabular} &
\begin{tabular}{c} TrackMPD \\ \hline
23 & \begin{tabular}{c} \(\frac{S_{above}}{S_{below}}=\frac{2\pi}{(\alpha-\text{sin})}-1\) \\ \end{tabular} & \
\begin{tabular}{c} The ratio of the dry \\ cross-sectional area \\ of the particles to \\ its wet \\ cross-sectional \\ area. \\ \end{tabular} &
\begin{tabular}{c} \(\alpha\) is defined by Equation (24). \\ \end{tabular} & [61] \\ \hline
24 & \begin{tabular}{c} \(\alpha=2\piccos\left(1-\frac{1}{R}\right)\) \\ \end{tabular} & \
\begin{tabular}{c} Parameter for \\ Equation (23) \\ \end{tabular} &
\begin{tabular}{c} \(h\) is the Archimedean force, \(R\) is \\ the radius of the particle. \\ \end{tabular} & [61] \\ \hline
25 & \
\begin{tabular}{c} \(\left(\frac{h}{R}\right)^{2}\cdot\left(3-\frac{h}{R}\right)=4\frac{\rho}{\rho _{w}}\) \\ \end{tabular} & Archimedean force &
\begin{tabular}{c} \(\rho\) (gm\({}^{-3}\)) and \(\rho_{w}\) (gm\({}^{-3}\)) are the \\ density of the particle and \\ water, respectively. \\ \end{tabular} & [61] \\ \hline
26 & \
\begin{tabular}{c} \(R_{j}=R_{jmax}\left(\frac{\tau}{\tau_{\text{min}}}-1\right)\) \\ \end{tabular} & Resuspension flux &
\begin{tabular}{c} \(R_{j}\) is the maximum \\ resuspension constant for \\ microplastics. and \(\tau_{\text{cr}}\) are shear \\ stress and critical shear \\ stress, respectively. \\ \end{tabular} & [116] \\ \hline
27 & \begin{tabular}{c} \(\tau=\rho_{w}(\frac{\rho_{w}}{\phi_{w}})^{2}\) \\ \end{tabular} & Actual shear stress & \begin{tabular}{c} \(\rho_{w}\) (gm\({}^{-3}\)) is the density of \\ seawater, \(g\) (ms\({}^{-2}\)) is \\ gravitational acceleration, \(v_{w}\) is \\ the mean velocity of the seawater, \\ and _Chezy_ is the Chezy coefficient \\ (m\({}^{0.5}\text{s}^{-1}\)). \\ \end{tabular} & \begin{tabular}{c} where \(v_{p}\) is the vertical \\ displacement velocity of particles, \\ \(v_{i}\) is particle velocity at the \\ computational cells, \(\alpha_{i}\) is vertex, \\ and \(v_{i}\) is particles settling/rising \\ velocity (obtained from \\ Equation (5)). \\ \end{tabular} & \
\begin{tabular}{c} \(\rho_{w}\) (gm\({}^{-3}\)) is the density of \\ seawater, \(g\) (ms\({}^{-2}\)) is \\ gravitational acceleration, \(v_{w}\) is \\ the mean velocity of the seawater, \\ and _Chezy_ is the Chezy coefficient \\ (m\({}^{0.5}\text{s}^{-1}\)). \\ \end{tabular} &
\begin{tabular}{c} CaMPSim-3D \\ \end{tabular} \\ \hline
28 & \begin{tabular}{c} \(v_{p}=\frac{\tau^{5}}{\tau_{\text{min}}-1}\frac{1}{R}\tau_{i}+v_{p}\) \\ \(\frac{1}{\tau_{i}-1}\frac{1}{R}\tau_{i}\) \\ \end{tabular} & \begin{tabular}{c} Vertical \\ displacement \\ velocity of particles \\ \end{tabular} & \
\begin{tabular}{c} where \(v_{p}\) is the vertical \\ displacement velocity of particles, \\ \(v_{i}\) is particle velocity at the \\ computational cells, \(\alpha_{i}\) is vertex, \\ and \(v_{i}\) is particles settling/rising \\ velocity (obtained from \\ Equation (5)). \\ \end{tabular} &
\begin{tabular}{c} where \(K\) is the diffusion \\ coefficient (obtained from \\ Equation (14)), \(\Delta t\) is the time step, \\ \(R\) is a random number, and \(K_{p}\) is \\ the diffusion coefficient. \\ \end{tabular} & [98] \\ \hline
29 & \begin{tabular}{c} \(Diff=\frac{3R}{\Delta t}\Delta t+R\left\{\sqrt{2K_{p}\left[X_{p}(t)+\frac{1}{2} \frac{4R}{\Delta t}\Delta t\right]}\Delta t\right\}\) \\ \end{tabular} & \
\begin{tabular}{c} Calculation of \\ diffusion term \\ \end{tabular} &
\begin{tabular}{c} where \(K\) is the diffusion \\ coefficient (obtained from \\ Equation (14)), \(\Delta t\) is the time step, \\ \(R\) is a random number, and \(K_{p}\) is \\ the diffusion coefficient. \\ \end{tabular} & [98] \\ \hline \end{tabular}
\end{table}