We study the problem of sinking particles in a realistic oceanic flow, with major energetic structures in the mesoscale, focussing on the range of particle sizes and densities appropriate for marine biogenic particles. Our aim is to evaluate the relevance of theoretical results of finite size particle dynamics in their applications in the oceanographic context. By using a simplified equation of motion of small particles in a mesoscale simulation of the oceanic velocity field, we estimate the influence of physical processes such as the Coriolis force and the inertia of the particles, and we conclude that they represent negligible corrections to the most important terms, which are passive motion with the velocity of the flow, and a constant added vertical velocity due to gravity. Even if within this approximation three-dimensional clustering of particles can not occur, two-dimensional cuts or projections of the evolving three-dimensional density can display inhomogeneities similar to the ones observed in sinking ocean particles.

The sinking of small particles suspended in fluids is a topic of both
fundamental importance and of practical implications in diverse fields ranging
from rain nucleation to industrial processes

In the oceans, photosynthesis by phytoplankton in surface waters uses
sunlight, inorganic nutrients and carbon dioxide to produce organic matter
which is then exported downward and isolated from the atmosphere

Size and classification of marine particles (adapted from

This is a challenging task that involves the downward transport of particles
of many different sizes and densities by turbulent ocean flows which contain
an enormous range of interacting scales. In the oceanographic community,
numerous studies approached this problem by considering biogenic particles
transported in oceanic flow as passive particles with an added constant
velocity in the vertical to account for the sinking dynamics

In the physical community, the framework to model sinking particles is based
on the Maxey–Riley–Gatignol equation for a small spherical particle moving
in an ambient flow

Here we consider the theory of small but finite-sized particles driven by
geophysical flows, which is, as mentioned above, conveniently based on the
Maxey–Riley–Gatignol equation. In Sect.

In theory, the sinking velocities of biogenic particles depend on various
intrinsic factors (such as their sizes, shapes, densities, and porosities)
which can be modified along their fall by complex bio-physical processes
(e.g., aggregation, ballasting, trimming by remineralization) as well as by
the three-dimensional flow field

Because of the diversity of the shapes, the size of a particle refers to the
diameter of a sphere of equivalent volume (equivalent spherical diameter;

Originally, the size classification of particles was based on the minimal
pore size of the nets used for their collection, which is about

In the following, our focus is thus on particulate matter larger than
1.0

Thereafter zooplankton consume live phytoplankton and inert particles and
produce fecal pellets and dead bodies. Most fecal materials have enough size
to sink rapidly by their own

Finally, there are the so-called organic aggregates which occur in the size
range of 1

Simplified categorization of marine biogenic particles, and their associated sizes.

Krill fecal pellets: length between 400

10

Macrozooplankton:

Mesozooplankton:

Microzooplankton:

Microphytoplankton:(size

Nanophytoplankton:(

Picophytoplankton:(

Macroscopic (marine snow):

Microscopic:1

Submicron:

The density of marine particles depends on their composition, which can be
divided into a mineral and an organic fraction

Considering all these estimates together, the density of marine particles
ranges approximately between

To describe the sedimentation of biogenic particles, we need to study the
motion of single particles driven by fluid flow. A milestone to analyze the
dynamics of a small spherical rigid particle of radius

The full MRG is very complicated to manage. A further simplification is
usually performed based on the single assumption of very small particles
(what this exactly means will be discussed later on). With this, the
Faxén corrections and, as commented on below, also the history term
(since

Equation (

We now discuss the validity of the MRG equation Eq. (

Another condition to be satisfied for the validity of the MRG equation is
that the so-called Reynolds particle number,

Summarizing, both the MRG and its approximation Eq. (

Sinking velocity versus particle radius for different

We are interested in applying Eq. (

Two apparent forces arise in the equation, the Coriolis force

The ratio between the particle response time and the Kolmogorov timescale is
the Stokes number

It is worth recalling that

A further discussion of Eq. (

The velocity flow

Map of region of study. Color corresponds to bathymetry. The blue
rectangle is the region used for simulations of the ROMS model. The orange
rectangle
is the region for the clustering numerical experiment of
Sect.

In order to integrate particle trajectories from the velocity in
Eq. (

In order to obtain quantitative assessment of the relative effects of the
different physical terms in Eq. (

For the numerical experiments we will consider a set of six values of

Table

Mean and standard deviation of the set of depths attained,
according to Eqs. (

We now perform a more stringent test going beyond the analyses of mean
displacements by considering differences between individual particle
trajectories. To assess the impact of the Coriolis and of the inertial
effects, we compare the positions

Figure

The horizontal and vertical differences

In all cases, the differences (both in vertical and horizontal) between the
simple dynamics (Eq.

Root mean square difference per particle, as a function of time,
between horizontal particle positions computed with
Eq. (

Root mean square difference per particle between final positions (at
times

Root mean square difference per particle between final positions (at
times

In summary, for the range of sizes and densities of the marine
particles considered here, the sinking dynamics is essentially
given by the velocity

Compressibility of the particle-velocity field, i.e.,

We now reproduce numerically a typical situation in which clustering of marine
particles is observed. We release particles uniformly in an horizontal layer
close to the surface, we let them sink within the oceanic
flow and we finally observe the distribution of the locations where
they touch another horizontal deeper layer. The domain chosen is the
rectangle 12 to 35

Results of the clustering numerical experiments of
Sect.

We explain the observed particle clustering by considering the field
displayed in Fig.

A simple way to confirm that this clustering arises from the
two-dimensionality of the measurement is to estimate the changes in the
horizontal density of evolving particle layers as if they were produced just
by the horizontal part of the velocity field. This is only correct if an
initially horizontal particle layer remains always
horizontal during the sinking process, which is not true. But, given the huge
differences in the values of the horizontal and vertical velocities in the
ocean, we expect this approximation to capture the essential physics and
provide a qualitative explanation of the observed clustering. We expect the
approximation to become better for increasing

Figure

We have studied the problem of sinking particles in a realistic oceanic flow,
focussing on the range of sizes and densities appropriate for marine biogenic
particles. Starting from a modeling approach in terms of the MRG
Eq. (

Corrections arising from the Coriolis force turn out to be about 100 times
larger than the ones coming from inertial effects, in agreement with the
results in

If the fluid flow field

Data results are available
in

The authors declare that they have no conflict of interest.

We acknowledge support from the LAOP project, CTM2015-66407-P (AEI/FEDER, EU), from the Office of Naval Research, grant no. N00014-16-1-2492, and from a Juan de la Cierva Incorporación fellowship (IJCI-2014-22343) granted to Vincent Rossi. Edited by: Vicente Perez-Munuzuri Reviewed by: two anonymous referees