We study the vertical dispersion and distribution of negatively buoyant rigid microplastics within a realistic circulation model of the Mediterranean sea. We first propose an equation describing their idealized dynamics. In that framework, we evaluate the importance of some relevant physical effects (inertia, Coriolis force, small-scale turbulence and variable seawater density), and we bound the relative error of simplifying the dynamics to a constant sinking velocity added to a large-scale velocity field. We then calculate the amount and vertical distribution of microplastic particles on the water column of the open ocean if their release from the sea surface is continuous at rates compatible with observations in the Mediterranean. The vertical distribution is found to be almost uniform with depth for the majority of our parameter range. Transient distributions from flash releases reveal a non-Gaussian character of the dispersion and various diffusion laws, both normal and anomalous. The origin of these behaviors is explored in terms of horizontal and vertical flow organization.

Approximately 8 million tonnes of plastics end up in the oceans
every year

Floating plastics and those that have beached or sedimented on
the seafloor are relatively well studied through field
campaigns

In this paper, we focus
on a certain class of plastic particles, negatively buoyant
rigid microplastics, excluding very small sizes,
and we estimate their vertical distribution
through the water column and their amount in the Mediterranean
Sea. Microplastic particles are among the most important
contributors to marine plastic pollution

We then estimate the amount of microplastic particles in the water column of the open Mediterranean. Our estimates rely on a uniform vertical distribution, which is confirmed by our numerical simulations to be a good approximation for fast-sinking particles. This can be explained by a simple model in which released particles sink with a constant velocity. Detailed consideration of the transient dynamics identifies small non-Gaussian vertical dispersion around this simple sinking behavior, with transitions between anomalous and normal effective diffusion.

The dynamics and the fate of microplastics in the ocean are
largely determined by their material density

Typically, positively buoyant plastic types will remain
floating at the sea surface or close to it and then will not
contribute to the microplastic content in the water column, the
topic we are interested in this paper. However, it has been documented experimentally that biofouling may increase sinking rates of particles up to 81 % and enhances sedimentation

In fact, the fallout from the North
Pacific Garbage Patch almost entirely consists of plastic types
nominally less dense than water

Particles denser than seawater dominantly accumulate at the sea
bottom

There are different classes of microplastic particles denser
than seawater. For example, dense synthetic
microfibers have been found to strongly dominate in sediment
samples far from the coast

We concentrate in the following on dense rigid
microplastic particles. The most abundant particles of this
class are fragments

Whatever their precise equation of motion is, these sinking
particles (directly detected by

While methodological issues make the quantification of
abundance difficult

While the idea of

From a meta-analysis of 39 previous studies,

Thus, the full range of microplastic particle densities in the
ocean, denoted here as

Polymer densities for the most abundant microplastics identified in water bodies

Another relevant property of plastic particles is their
size. By a widely accepted definition, microplastics are particles with a
diameter less than 5 mm without any lower limit

Field data about distributions of size and
quantifiers of shape for negatively buoyant rigid particles in
the water column or deep-sea sediments are not available to
date to the best of our knowledge, except in the Arctic from

For these reasons, we will disregard particles of extremely small size.
To keep our qualitative study sufficiently simple, we will consider all
our modeled particles to have a
radius

In this subsection we indicate the total amount of
dense microplastics entering
the water column in open waters of the Mediterranean. Despite the correlation of plastic source with coastal population density, the rapid fragmentation of small particles along the shoreline

We will take these numbers, 4000 t per year,
37 % of sinking particles
and the proportion of direct release by maritime
activity (6 %), to obtain in Sect.

A standard modeling approach

For the flow velocity

We next show, closely following the treatment of

The simplified MRG equation gives the velocity

In Fig.

The simplified MRG equation, Eq. (

Settling velocities and particle sizes for which
Eq. (

The connection between Eq. (

We can now take the results of

Even if an equation of motion is accurate, the accuracy of its
solution is limited by that of the input data. In particular,
small-scale flow features are absent from oceanic velocity
fields

In this section we analyze the role of a variable seawater
density on the particle settling dynamics. Fluid density is
calculated from the TEOS-10 equations, which is a thermodynamically
consistent description of seawater properties derived from a
Gibbs function, for which absolute salinity is used to describe
salinity of seawater and conservative temperature replaces
potential temperature

We consider particles
of a fixed density

We release

Normalized histogram of the seawater density

The distance, as a function of time, between trajectories obtained with
constant nominal fluid density of

We illustrate the impact of this variable density on particle
trajectories for the summer release in Fig.

Relative effect on horizontal and vertical particle positions after 10 and 20 d of integration, averaged over 78 803 particles released over the whole Mediterranean at 1 m depth, of replacing the actual seawater density by a nominal value

A summary of the average relative differences on horizontal and
vertical particle positions between using the
location-dependent seawater density and a nominal constant
value

In brief, we
see that the effect of location-dependent density may be a
relevant effect to evaluate microplastic transport.
At least, the traditional value of seawater density may be biased, which may be reflected in the particle trajectories.
We recall,
however, that we used parameters for the particle properties
for which they are slowly falling. The impact of variable
density on particles that sink faster will be smaller. Also, the
effects reported in Table

We will first estimate the total mass of negatively buoyant
rigid microplastics in the water column of the open
Mediterranean Sea by assuming a uniform vertical distribution;
then we will justify this assumption by running numerical
simulations according to the conclusion of
Sect.

For estimating the total mass, we take the quantities of
Sect.

The next step is to estimate the time during which these microplastic
particles remain in the water column before reaching the sea
bottom. We take the mean depth for the Mediterranean to be

We emphasize the many uncertainties affecting this result
(Sect.

The result for the total mass is independent of the horizontal
distribution of particle release, which is quite inhomogeneous

The above estimates are rather rough
as a result of the uncertainties mentioned.
The assumption of a uniform distribution in the vertical
direction has not yet been justified either,
but we will show it to be appropriate by means of
our
simulations of particle release starting at 1 m depth over
the whole Mediterranean. Instead of performing a continuous
release of new particles at each time step, and computing
statistics over this growing number of sinking particles, we
approximate this by the statistics of all positions at all time
steps of a set of particles deployed in a single release event.
This assumes a time-independent fluid flow, but this approximation
is appropriate, since the dispersion of an ensemble of particles
released in a single event follows rather well-defined
statistical laws (see Sect.

Figure

A uniform distribution of plastics in

The continuous line is the area that the Mediterranean
has at each depth

We now analyze in detail the
transient evolution of particle
clouds initialized by flash
releases at a fixed depth.
Numerically we proceed by
releasing

Figure

The probability density function, estimated from a histogram of bin size

Since a non-Gaussian distribution is usually linked to
anomalous dispersion

According to Fig.

Variance of depth reached by the particles as a function of time.
Straight lines represent power laws for reference, with exponents

We start our analysis with the fastest-sinking particles (v153;
Fig.

We explain this last crossover as resulting from a different
mean sinking velocity in diverse regions of the Mediterranean,
associated with up- and down-welling. This can be modeled in an
effective way by writing the vertical position of particle

To evaluate Eq. (

The different regimes are not as clear in the v68 case as for
v153; see Fig.

For v6 (Fig.

While this is one possible explanation for the earlier timing
of the initial transition from anomalous to normal diffusion for
higher settling velocity, one cannot exclude that a
depth-dependent organization of the flow is more in play; note
that

Variance of depth reached by the particles as a function of their mean depth.

Note in Fig.

We have discussed the different types of plastics occurring in the water column, pointing out gaps in our knowledge about the sources, transport pathways and properties of such particles. It would be highly beneficial to have distributions of size, polymer type and quantifiers of shape recorded separately for the dynamically different classes of microplastics.

We have focused our attention on rigid microplastic particles with negative buoyancy. We have argued that the simplified MRG equation approximates the dynamics of such particles sufficiently well for qualitative estimations.

We have then analyzed the importance of different effects in
this equation and concluded that the Coriolis and the inertial
terms are negligible. When a velocity field of large-scale
nature is input to the equation (such that small-scale
turbulence is not resolved), or when the variability in
seawater density is neglected, moderate but possibly
non-negligible errors emerge

When the velocity field of the Mediterranean Sea is approximated by realistic simulation, this equation of motion results in a nearly uniform steady distribution along the water column, except perhaps at extremely low settling velocities. The corresponding total amount of plastic present in the water column is relatively small, close to 1 % of the floating plastic mass, but it may be an important contribution to the microplastic pollution in deep layers of the ocean and is subject to several uncertainties.

Note that only those microplastic particles are considered here
that have not yet sedimented on the bottom, and the plastic
amount sedimented on the seafloor is large

As for the vertical distribution profile, its approximate uniformity may be linked to the weak vertical dispersion of particles that is found in our simulations, started with a flash release over the whole surface of the Mediterranean sea. The shape of the emerging transient vertical distribution exhibits deviations from a Gaussian, which are related to anomalous diffusive laws that dominate the vertical dispersion process in some phases.

The different diffusive laws are related to the properties of
the decay in the Lagrangian velocity autocorrelation defined
along the trajectories of the sinking particles. An important
example is the transition from initial superdiffusion to a
longer phase of normal diffusion, occurring around

We quantitatively assess the impact of deviations from a spherical shape through a correction to the settling velocity

Most generally, the settling velocity vector

To characterize the correction in the settling velocity for a given nonspherical particle (with a given density

Note that it is always possible to define an

The shape of rigid microplastic particles is not usually described in the literature, but we can see photographs of some examples in, for example,

Under this assumption, the particle size will correspond to the longest edge,

We can substitute either of these choices of

After substituting all these expressions in Eq. (

We plot

We have left the question of which orientation is relevant open so far. In small-scale isotropic turbulence, which is certainly present in the ocean, nonspherical particles have a preferential alignment with certain characteristics of the flow but undergo rotation

Even though the real advection of the particles will become more complicated as a result of the ever-changing orientation and may thus be beyond the scope of the MRG equation (cf. the discussion in the main text about the settling velocity of irregular particles), we have found that changing orientation introduces minor variations in the value of the settling velocity. Together with the absence of order-of-magnitude corrections that may arise from a nonspherical shape (but comparing shapes under the assumption of the same particle density), this gives quantitative support for the applicability of a spherical shape in Eq. (

Finally, we briefly comment on the more general Eqs. (

We present here the detailed numerical analysis of the relevance of a finite
time of response (Stokes time,

We incorporate the first two effects to a single equation,

The effect of unresolved scales will also be estimated by
keeping the original NEMO velocity field

Solid line indicates the average horizontal distance

Same as Fig.

The statistical properties are chosen to be
similar to the ones expected for oceanic motions below the
scales resolved by the numerical model

Average horizontal and vertical pairwise particle distances

Average horizontal distance

Same as Fig.

In order to compare the different equations of motion, we
release a large number

In Fig.

To evaluate the different effects more quantitatively, we
summarize in Table

We can also observe the time evolution of the

We investigate here if the finite and spatially varying depth
of the basin (the bathymetry) could affect the conclusions in
Sect.

Variance of depth reached by all sinking particles (in black) and an
adaptively restricted subset of them (in yellow) as a function of time. See text for details.
Straight lines represent power laws for reference, with exponents

We start with effect (i) by comparing, in
Fig.

The time evolution of

Variance of depth reached by all sinking particles (in black)
and those initialized in the Sea of Sardinia (in blue; longitudes in

According to Fig.

Based on these analyses, we believe that the findings of
Sect.

The velocity field from the NEMO simulation used in our study can be downloaded from

All authors designed research. RdlF performed the simulations. RdlF and GD analyzed data. RdlF, GD, EHG and CL prepared the first draft, and all authors reviewed and edited the paper.

The authors declare that they have no conflict of interest.

We thank two anonymous referees for useful comments.

This research has been supported by the H2020 European Research Council (TOPIOS grant no. 715386) and the Spanish State Research Agency through the María de Maeztu Program for Units of Excellence in R&D (grant no. MDM-2017-0711). Rebeca de la Fuente was also supported by MINECO, Spain, through the FPI program (fellowship no. BES-2016-078416). Gábor Drótos was also supported by CAIB and the European Social Fund through its postdoctoral program (grant no. PD/020/2018) and by NKFIH, Hungary (grant no. NKFI-124256). The publication fee was supported by the CSIC Open Access Publication Support Initiative through its Unit of Information Resources for Research (URICI).

This paper was edited by Piers Chapman and reviewed by two anonymous referees.