Global ocean surface mass balance model

We introduce a simple box model to quantify buoyant macroplastics (>0.5 cm) at the surface of the global ocean. We consider synthetic polymer production data from 1950 to 2015 and compute the fraction of positively buoyant polymers (65.5% of total) used by different market sectors associated with product lifetime distributions7. Starting in 1950 and for every year, we compute the plastic population age distribution of material mass that is reaching end of lifetime and is therefore discarded. We define the global mass of plastic D, produced in year y 0 and discarded in year y as follow:

$$D(y,{y}_{0})=\mathop{\sum }\limits_{\sigma }^{\begin{array}{c}market\,\\ sectors\end{array}}\,Production({y}_{0})\,\ast \,Market\,Share(\sigma )\,\ast \,LifeTime(y-{y}_{0},\sigma )$$ (1)

With Production(y 0 ), global plastic production for year y 0 , MarketShare(σ), percentage of market share in global plastic production for sector σ and, Lifetime(y-y 0 , σ), probability density function of product lifetime for sector σ and plastic age y-y 0 as proposed by Geyer et al.7 with market sectors including “Packaging”, “Transportation”, “Building and Construction”, “Electrical/Electronic”, “Consumer & Institutional Products”, “Industrial Machinery” and “Other”. More information on the market share and probability density functions for lifetime of plastic objects is provided in Supplementary Table 1.

Some of the discarded plastic waste may enter the global ocean in coastal areas6. Once at sea, buoyant plastic may strand back on the shoreline or sink from fouling-induced loss of buoyancy. At deeper depths, debris may experience rapid defouling followed by resurfacing as floating debris16. In shallower depth, however, debris has a higher chance to reach the seabed. In this framework, we define the coastal waters as the area with bathymetry between high tide line and the euphotic depth (typically shallower than 200 m water depth). Our model considers that when in the coastal environment a fraction of floating plastic mass is captured by the landmass, undergoing repeated episodes of stranding and release at the shoreline or settling and resurfacing from the seabed. Some of the floating plastic remaining at the surface may be transported to offshore waters. As time passes, the mass of stranded, settled and floating plastic degrades into microplastics, thus leaving the model domain. Our model primarily focuses on buoyant macroplastics and considers mass loss from degradation into microplastics as a permanent sink.

We divide the global marine environment in three surface domains (Fig. 1): the shoreline (S), the coastal surface waters (C) and the offshore surface waters (O). Our model includes 6 mass compartments: S M , C M , O M for macroplastics and S m , C m , O m for secondary microplastics. The model conserves mass. For any year between 1950 to present, the accumulated mass of plastic that has been introduced in the global ocean is equal to the sum of these 6 mass compartments.

Figure 1 Predicting quantities of positively buoyant macroplastics (>0.5 cm) in the ocean environment using a global ocean emission-transport-degradation model. Every year, a fraction i of discarded plastic material is emitted into the coastal surface layer (C M ). Material present in coastal waters can strand or settle around shorelines (S M ) with probability s and material from S M can leak back into C M with release probability r. Material from C M can escape the continental shelf and enter the ocean surface layer O M with transport probability t. Finally, fractions d S , d C and d O of macroplastics present in the marine environment enter a permanent sink by degradation into microplastics (<0.5 cm) from the shoreline (S m ), the coastal surface layer (C m ) and the offshore surface layer (O m ). The processes are repeated annually from 1950 to 2015. Full size image

The model is initiated with input into coastal environments starting in 1950. Then from year y-1 to year y, we compute the net mass input of plastic produced in year y 0 , into the surface waters of the global coastal environment:

$$\Delta (y,{y}_{0})=i\,\ast \,D(y,{y}_{0})+\,r\,\ast \,(1-{d}_{S})\,\ast \,{S}_{M}(y-1,\,{y}_{0})+(1-{d}_{C})\,\ast \,{C}_{M}(y-1,{y}_{0})$$ (2)

The first term constitutes the direct inputs from discarded plastic leaking into the environment. Model parameter i is the annual mass fraction of discarded plastic reaching the coastal ocean. The two other terms represent respectively material released from the shoreline S M and existing floating material in C M left from previous years that has not yet degraded into microplastics. Model parameters d C and d S are regarded as the annual mass fraction of respectively floating and stranded or settled macroplastics degrading into microplastics, while r is the annual mass fraction of stranded or settled macroplastics that is released back into surface waters of coastal environments. The resulting mass of plastic produced in year y 0 that is present in coastal surface waters during year y is then computed as:

$${C}_{M}(y,{y}_{0})=(1-s)\,\ast \,(1-t)\,\ast \,\Delta (y,{y}_{0})$$ (3)

Where s is the annual mass fraction of floating plastic that strands and settles around shorelines and t the annual mass fraction of remaining floating plastic that is transported offshore. The total mass of plastic produced in year y 0 and stranded or settled around the global shoreline during year y is the sum of newly stranded or settled material and previously accumulated plastic that was not degraded into microplastics and not released back into coastal waters:

$${S}_{M}(y,{y}_{0})=(1-r)\,\ast \,(1-{d}_{S})\,\ast \,{S}_{M}(y-1,{y}_{0})+s\,\ast \,\,\varDelta (y,{y}_{0})$$ (4)

Finally, the total mass of plastic produced in year y 0 and floating in offshore waters during year y is the sum of previously accumulated plastic that has not degraded into microplastics and new debris leaked from the coastal waters.

$${O}_{M}(y,{y}_{0})=\,\,(1-{d}_{O})\,\ast \,{O}_{M}(y-1,{y}_{0})+t\,\ast \,(1-s)\ast \,\varDelta (y,{y}_{0})$$ (5)

With d O the degradation rate for macroplastics in offshore surface waters. For each year, the three mass sink terms are populated with input from degradation into microplastics from coastal, shoreline and offshore environments.

$${C}_{m}(y,{y}_{0})=(1+{d}_{C})\,\ast \,{C}_{M}(y-1,{y}_{0})$$ (6)

$${S}_{m}(y,{y}_{0})=(1+{d}_{S})\,\ast \,{S}_{M}(y-1,{y}_{0})$$ (7)

$${O}_{m}(y,{y}_{0})=(1+{d}_{O})\,\ast \,{O}_{M}(y-1,{y}_{0})$$ (8)

In this study, we assumed the degradation rates d C , d S and d O to be equal. The degradation term is called thereafter d.

$$d={d}_{C}={d}_{S}={d}_{O}$$ (9)

We note that in nature, these values may be different particularly for the shoreline where the degradation rate could be greater than in surface waters. These values will also likely differ between polymer composition and dimension of objects. We acknowledge that global degradation into secondary microplastics is far more complex than described by our model. With current available data, however, we are limited to propose a whole-ocean average degradation rate for the total macroplastic mass. Specifications of degradation rates by environments and polymer types will require more experimental research.

Another major assumption here is that model parameters do not show any interannual variability and that the dynamics of degradation, stranding, release and recirculation into the coastal environment is independent from the age and characteristics of plastic objects. A crucial parameter is the fraction i of new plastic waste generated on land that reaches the ocean. This parameter has a substantial influence when constraining the values of s (stranding on shoreline), r (release from shoreline) and t (offshore transport). We constrain parameter i by using estimates of global input from land into the ocean for 2010 with 4.8 to 12.7 million metric tons of input6. For a global plastic waste generation of 274 million metric tons in 20107, this translates to a fraction of annual discarded plastic reaching the ocean ranging from i = 1.7–4.6%. We therefore used this reported range to define the confidence interval for the results presented here.

Age distribution of ocean plastic

To study the persistency of macroplastics, the age distribution of plastic in the different compartments of our model is compared to the age distribution of plastic debris collected in a large oceanic gyre. In 2015, a multivessel expedition collected marine plastics debris floating in the Great Pacific Garbage Patch located in the North Pacific subtropical gyre15. The expedition landed 664 kg of positively buoyant macroplastics (debris larger than 0.5 cm) back to shore. Of the 83,144 collected pieces (>0.5 cm), 427 had a recognizable inscription for which 11 languages and 50 dates of production could be identified. Here, we consider the distribution of production dates found on these samples to be representative of plastic age distribution in oceanic gyres. By exploring all possible combinations of our five model parameters ranging from 0% to 100% at every 1%, we observe that only the degradation rate d significantly impacts the relative age distribution of positively buoyant macroplastics in offshore surface waters (Fig. 2). This is because degradation into microplastics is the only permanent sink considered by our model i.e. it is the only natural mechanism that removes material from our model domain. Therefore, we use the observed plastic age distribution to constrain the parameter d. Note that when fixing parameters s, r and t to 0%, we reproduce a global ocean emission-degradation model for macroplastics introduced in a previous study23. We compare the modelled decadal distribution with our samples for a degradation rate d ranging from 0% to 100%. Only one plastic object with production date in the 2010s was identified in the samples which were collected in 2015. We explain this by the minimum time taken for objects to reach the area that we estimate from dispersal model trajectories to be between 5 to 10 years to be representative of sources. Therefore, accounting for a minimum delay of 5 years, we compared observed and modelled decadal distributions of production dates from the 1950s to the 2000s. We computed the sum of squared residuals between observed and modelled age distribution by decades and varied model parameter d for minimization. Best fit was found for d = 3% of mass of positively buoyant macroplastics annually degraded into microplastics (Supplementary Fig. 1). A small degradation rate is in good agreement with field experiments, estimating a mass loss ranging from 0.65% to 1.9% of total mass, depending on polymers, for samples immersed at sea for a period of 12 months26.

Figure 2 Comparison between observed and predicted plastic age distribution. Observations correspond to the relative age distribution of macroplastics collected from the North Pacific subtropical gyre in 201515. This distribution is derived from production date labels identified on debris (N = 50). Model predicted age distributions are given for a range of the degradation variable d (3%, 10%, 50% and 90%) and parameters s, r and t set to 0%, reproducing a simple global emission-degradation model. Whiskers extend to all possible values by computing all possible combinations of s, r, and t varying from 0% to 100%, showing that the plastic age distribution is mostly sensitive to the degradation rate parameter d. We compute the least square sum for decadal distribution of observed and modelled plastic from 1950s to 2000s; minimum value is found for d = 3% (Supplementary Fig. 1). Pearson p-test values for d = 3%, 10%, 50%, 90% are respectively p = 0.009, 0.0133, 0.0361 and 0.0310. Underrepresentation of objects produced between 2010 and 2015 (the year when the samples were collected) is explained by the minimum time required for plastic debris to travel to oceanic gyres (~5 years). Full size image

Model sensitivity analysis

We study model convergence by varying parameters s, r and t from 0% to 100%. The model is considered convergent when the confidence interval of model predicted mass floating on the global ocean surface (e.g. C M + O M ) includes values on the order of hundreds of thousand metric tons of material (i.e. <106 metric tons). The model is generally converging for large values of stranding probability (s) and low values of offshore transport (t). Converging values for coastal release (r) are inversely proportional to the value of stranding probability (s). This was to be expected, given that these parameters are intrinsically connected as their difference measures the capture efficiency of the continental mass. Note that parameter s must be higher than r to reproduce accumulation on the world’s beaches. To constrain the model parameters s and t, we investigate trajectories of Lagrangian particles from a global dispersal model reproducing 20-year of surface circulation25,15. Particles are released from significant point sources (Fig. 3) near the coast based on population27 and waste management data6. A proportional number of particles is attributed to each country based on the estimated amount of mismanaged waste the country’s coastal population generates over a year. The particle release locations are derived from coastal population density and the timing of release is randomly distributed throughout the year. Particles are advected using different model forcing components, including sea surface currents, stokes drift and variable influences of wind.

Figure 3 Lagrangian dispersal model source locations and global ocean surface model domains. Amplitude and location of model particle sources are derived from predicted inputs of plastic from land into the ocean6 and population changes from 1993 to 201227. The separation between coastal and offshore surface waters in our model is shown with areas of respectively light and dark blue color. Coastal surface waters represent the continental shelf with bottom depths shallower than the photic zone (i.e. depths <200 m). Full size image

For stranding probability (s), we follow particles from their day of release until they spend two consecutive days near the shoreline. The model parameter (s) is defined as the fraction of model particles that have spent at least two consecutive days near the shoreline after one year since their initial release over the total number of particles present in the model. A particle is considered near the shoreline when it is located at a distance smaller than the hydrodynamic model cell size from a land cell (1/16°, several kms depending on latitude). With no wind influence and after one year, 96% of the 2,510,918 particles investigated have spent at least two consecutive days near the shoreline. This value increases when adding wind forcing, for a windage coefficient of 2% (of 10 m height wind speed value), 98% of particles have transited around the coast for at least two days. This increase in beaching probability for high windage debris is in good agreement with observations reporting mainly low-windage debris accumulating in oceanic gyres24. Here, we considered that if a particle spends more than two consecutive days in contact to the shoreline it is likely stranded as it would have gone through at least one full tidal cycle. We used these results to define the stranding probability in our model with s = 96–98%. For offshore transport probability (t), we locate the fraction of particles that stay on the continental shelf one year after release. The continental shelf is defined in our model by a water depth shallower than 200 m. We used gridded bathymetry data from the General Bathymetric Chart of the Ocean28 to determine ocean water depth. After one year of release, between 32% and 34% of modelled non-beaching particles have escaped the continental shelf. Values decrease with windage coefficient, except for the first month which we attribute to a rapid presorting of model particles depending on emissions location. We used these results to estimate the annual offshore transport probability with t = 32–34%. Using midpoint values for s and t (i.e. 97% and 33%, respectively), a coastal release parameter r = 1% results in convergence (Fig. 4) and explains the discrepancies between emissions estimate and observed mass on the global ocean surface layer. An overview of the model parameters, with description and selected values is given in Table 1.

Figure 4 Sensitivity analysis and model convergence. (a) Stranding probability (s) determined from Lagrangian particle trajectories starting in coastal environments. The value of s is defined as the percentage of particles present in the model that have spent two consecutive days in close proximity (<1/16°) to the shoreline. Intervals correspond to different forcing scenarios with influence of wind ranging from 0% (thick dark line) to 2% (thin dark line) of 10 m height wind speed. Probability of stranding increases with windage coefficient. (b) Offshore transport (t) estimated from the same Lagrangian trajectories. The parameter t is defined as the percentage of model particles that are located outside the continental shelf (>200 m water depth) after one year of release. Intervals also correspond to different wind forcing scenarios ranging from 0% (thick dark line) to 2% (thin dark line). Generally, the probability of transport to offshore waters decreases with windage coefficient. (c) Sensitivity analysis and model convergence. Parameters s, t, and r vary from 0% to 100%. The model is considered convergent (blue colored area) when the confidence range for mass estimate on the global ocean surface layer (O M + C M ) overlaps with values below 106 metric tons, assuming a degradation rate d = 3% and emission rates i = 1.7–4.6%. Our model is converging with estimated value s = 97%, t = 33%, and r = 1% (black diamond). Full size image