Memory effects in wave-induced microplastic transport

Abstract

Microplastics are transported by ocean surface waves in ways that depart significantly from the Stokes drift of fluid parcels, and accurate modeling of this transport requires accounting for forces beyond linear drag. Existing modeling of microplastic transport often neglect the Basset-Boussinesq history force, effectively limiting their use to the smallest particle sizes. Here, we extend the applicability of these models by implementing the history term with a multistep integration scheme, allowing us to capture the transport of larger microplastics in linear surface waves of arbitrary depth. We quantify when the Basset-Boussinesq history force significantly affects microplastic transport by surface gravity waves. We show that memory effects become the leading-order horizontal drag once $S=St/γ^2$ exceeds a critical value $S\approx 0.25$, where $St$ is the Stokes number and $γ$ is the density ratio of the particle and the fluid. The corresponding critical $St$ number is found to be a factor of about three smaller than that given by classical inertial estimates that neglect history effects. These results help provide regime maps that can be used to indicate when history effects can be safely neglected. Our simulations also reveal that history effects significantly increase horizontal transport distances and enhance orbit shearing of particle ensembles.

Figures (13)

• Figure 1: Transport of a microplastic particle by a surface gravity wave, where the particle is located at a wave period endpoint as defined in §\ref{['subsec:param_variation']}
• Figure 2: The Stokes drag and history force contributing to the movement of a small, negatively buoyant particle in a deep water wave, with $S = 0.2$$(St = 0.20611)$, $R = 0.66$ and $\epsilon = 2\upi / 75$. The particle trajectory $(a)$ is shown in the $x$-$z$ plane, with four points in time A-D at different quarters of the central particle orbit, highlighted by the vertical lines in the remaining plots. The forces are split into their horizontal $(b)$ and vertical $(c)$ components. The particle velocity is similarly split into its vertical $(d)$ and horizontal $(e)$ components.
• Figure 3: The Stokes number vs. the signal amplitude $A$$(a)$, and phase angle $\phi$$(b)$ of the horizontal components of the Stokes drag (dashed) and history force (dotted) acting on a particle with $R = 0.66$ and $\epsilon = 2\upi / 75$. The vertical line indicates the position of the Stokes number used in the simulation displayed in Figure \ref{['fig:forces']}
• Figure 4: Contours illustrating the particle radius $a'$ computed with varying Stokes number $St$, varying density ratio $R$, and constant wavelength $\lambda' = 200$m $(a)$, and the Stokes number as a function of the drag force ratio $\chi$, delineating the different regimes according to the significance of the history force. The vertical line in $(a)$ denotes the approximately neutral buoyancy used in the computation of the data points shown in $(b)$, the dot-dashed line corresponds to the bound for the Maxey-Riley equation, and the dashed and dotted lines display numerically computed values of $St$ and $R$ for the respective values of $\chi$. The shaded area in $(a)$ denotes the region applicable to microplastics. In $(b)$, the line corresponds to the inertial approximation \ref{['eq:inertial-corr']}, where $St \approx \widehat{St} \approx S$ for the approximately neutrally buoyant density ratio. The circular markers indicate numerical results, and the filled markers indicate estimations for notable values of $\chi$.
• Figure 5: The average vertical position of the particle $\bar{\bar{z}}$ normalized with the water depth $h$ as a function of the period-averaged horizontal component of the normalized Stokes drift velocity $\bar{u}/\epsilon^2$. The particles are neutrally buoyant ($R = 2 / 3$) with constant wave steepness $\epsilon = \upi / 75$, varying Stokes number, and varying depth. The curves correspond to the analytical solutions for the Stokes drift velocity, and each point corresponds to the numerical solutions averaged over the course of the trajectory of an individual particle. For $h' / \lambda' = 10/3$ (the deep water case), $h = 20\upi / 3$ and $Fr \approx 1$. For $h' / \lambda' = 1/5$ (intermediate depth), $h = 2 \upi / 5$ and $Fr \approx 0.92$. For $h' / \lambda' = 1/10$ (the shallow water case), $h = \upi / 5$ and $Fr \approx 0.75$. The results showed no dependence on $\widehat{S}t$.