Investigating inertial particle transport for application in trackingplastic litter in the ocean

Birgit Sutzl1,*, Pavel Berloff1, Erik van Sebille21Imperial College London, 2Utrecht University

*b.sutzl@imperial.ac.uk

Motivation

Around 250,000 metric tonnesof plastic debris are floating onthe ocean surface and studieshave shown that they accumu-late in the ocean gyres, howeverthe reason for this clustering isstill up to debate [5]. Math-ematical modelling of oceanictransport of small sized objects(particles) can help us to anal-yse patterns of movement andinvestigate their origins.

Figure 1: Schematic figure ofplastic accumulation in the fiveocean gyres, image from [5].

Surface particle transport in the ocean

The ocean currents determine how floating particles movearound on the ocean surface, which is described by the ve-locities of the ocean flow field. Thus, for the modelling ofparticle transport in the ocean we need to model the oceanflow and the reaction of the particle to it, which results inparticle movement. We use a quasi-geostrophic ocean model,which gives a simplified model of a general ocean circulationwith rich features of turbulent mesoscale flows. The model rep-resents a wind-driven double-gyre circulation with a subtropicalocean gyre (anticyclonic circulation) in the South and a sub-polar ocean gyre (cyclonic circulation) in the North. [1]

Figure 2: Particle trajectories of nine particlesfor 4000 days in the double-gyre flow field.

Typically the move-ment of floating sur-face particles is mod-elled as passive tracerparticles. A particlemoves with the oceanflow and adapts in-stantaneously to ve-locity changes:

dXdt = u(X(t), t),

where X is the particleposition and u the flowvelocity of the ocean.

PARCELS Particle tracking software

Parcels [3] is a Lagrangian particle tracking code for the sim-ulation of particle pathways in a given velocity field. In thestandard setup it models the movement of passive fluid parti-cles. This project included the implementation of inertial par-ticle transport into Parcels.

The simplified Maxey-Riley model for inertial particle transport

This project presents the modelling of floating plastic debris as inertial particles, and how these particlesdiffer from particles without inertial properties in a numerical study. Inertial particles are finite-size and finite-massobjects, which interact with ambient fluid and experience flow drag. The dynamics describing inertial particle movement hasbeen rigorously derived by Maxey & Riley [4]. A simplified model of these equations was used for implementation:

dXdt = V(t)dVdt = 3R

2DuDt (X(t), t) − 1

τ(V(t) − u(X(t), t))

with R = mf(mp + 1

2mf ) and τ = 2a2

9Rν .

X and V are particle position and velocity, u and Du/Dt the flow field’s velocity and acceleration, R the mass ratio of the particleand fluid, and τ is the relaxation time, which is the typical timescale at which the particle adapts its velocity to the flow velocitysurrounding the particle. The parameters are: particle diameter a, fluid and particle mass mf and mp, kinematic viscosity ν.

Simulation results

(a) Passive particles. (b) Plastic-like particles, τ = 1 hour.

(c) Light particles, τ = 1 hour. (d) Heavy particles, τ = 1 hour.

Figure 3: Particle location histograms showing the frequency of particles’ position over whole simulation time.

Passive particles are spread fairly evenly across the domain, no clustering occurs. Light particles and plastic-like particles areattracted to the western boundary of the ocean basin, and particularly to the regions around the western origin of the jet, wherethe eddying flow is most energetic. This roughly corresponds to accumulation inside the gyres, somewhat off the gyres. Heavyparticles particularly show a tendency to move away from the centre of circulation and towards the edges of the domain.

Parameters and simulation setup

The simulations are run for 4000 days (approx. 11 years) onthe double gyre ocean velocity field with 900 particles, whichwere initially spread across the full domain. Time step ofintegration is ∆t = 30 s. Simulations were carried out withthe following combination of parameters:

Mass ratio R 4/9 2/3 8/9 0.7Particle type heavy neutral light plastic

Particle density 1.6 g/cm3 1 g/cm3 0.6 g/cm3 0.9 g/cm3

Table 1: Values for mass ratio R and corresponding particle characteristics.

Relaxation time τ 10 min. 1 hour 12 hours 1 dayParticle diameter 3-5 cm 8-12 cm 29-42 cm 42-59 cm

Table 2: Values for relaxation time τ and corresponding particlecharacteristics.

Conclusions

I Simulations showed that inertial particle transport can signifi-cantly differ from passive particle transport.

I Key parameters that influence the inertial particle movementare the particle density and particle size.

I Accumulation for lighter particles is consistent with observa-tional data, which show higher plastic particle clustering insideocean gyres and in particular higher density of particles off thegyre centres.

I Solutions suggest that particle inertia could play a role in in de-termining the denser plastic accumulation zones and thus thatthe model can present an improvement for particle transportsimulations.

I Further research is required to determine the parameter rangesfor which application of the inertial particle model is meaning-ful, particularly considering plastic.

I Studying the theoretical background of equations’ dissipativedynamics in double gyre ocean flow could provide further in-sight.

References

1. Berloff, P. Dynamically consistent parameterization of mesoscale eddies. Part I:Simple model. Ocean Modelling 87, 1–19 (2015).

2. Cartwright, J. H. et al. Nonlinear Dynamics and Chaos: Advances and Perspectives(eds Thiel, M. et al.) 51–87 (Springer Berlin Heidelberg, 2010).

3. Lange, M. et al. Parcels v0.9: prototyping a Lagrangian Ocean Analysis frameworkfor the petascale age. Geoscientific Model Development Discussions (2017).

4. Maxey, M. R. et al. Equation of motion for a small rigid sphere in a nonuniformflow. The Physics of Fluids 26, 883–889 (1983).

5. van Sebille, E. The oceans’ accumulating plastic garbage. Physics today 68, 60–61(2015).