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 , a process which forms the so-called biological carbon pump. The downward flux of carbon-rich biogenic particles from the marine surface due to gravitational settling, one of the key process of the biological carbon pump, is responsible (together with the solubility and the physical carbon pumps) for much of the oceans' role in the Earth's carbon cycle . Although most of the organic matter is metabolized and remineralized in surface waters, a significant portion sinks into deeper horizons. It can be sequestered on various timescales spanning a few years to decades in central and intermediate waters, several centuries in deep waters and up to millions of years locked up in bottom sediments . Suitable modeling of the sinking process of particulate matter is thus required to properly assess the amount of carbon sequestered in the ocean and in general to better understand global biogeochemical cycling and its influence on the Earth's climate.

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 . They suggest that the sinking of particles may not be strictly vertical but oblique, meaning that the locations where the particles are formed at the surface may be distant from the location of their deposition in the seafloor sediment. Then presented the concept of statistical funnels which describe and quantify the source region of a sediment trap (subsurface collecting device of sinking particles used to get estimates of vertical fluxes). The validity of this approximation and the influence of different physical processes is however poorly discussed in these analyses.

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 , which highlights the importance of mechanisms beyond passive transport and constant sinking velocity, such as the role of finite size, inertia and history dependence. A major outcome of these studies is that inhomogeneities and particle clustering can arise spontaneously even if the fluid velocity field is incompressible and particles do not interact . Particle clustering and patchiness are indeed observed in the surface and subsurface of the ocean .

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.  we review the main characteristics of marine particles which are relevant for their sinking dynamics. In Sect.  we present the equations of motion describing this process, together with the approximations required to obtain them and the types of particles for which they are valid. In particular, we discuss its validity and the relevance of the different physical processes involved in the range of sizes and densities of marine biogenic particles. In Sect.  we use these equations to study the settling dynamics in a modeled oceanic velocity field produced by a realistic high-resolution regional simulation of the Benguela upwelling system (southwestern Africa). We estimate the relevance of physical processes such as the Coriolis force and the inertia of the particles with respect to the settling velocity. We also observe the spatial distribution of particles falling onto a plane of constant depth above the seabed and we identify clustering of particles that is interpreted with simple geometrical arguments which do not require physical phenomena beyond passive transport and constant terminal velocity. Our main results are finally summarized in the Conclusions section.

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 . However, reasonable estimates of the effective sinking velocities of marine particles can be obtained by taking into account only their size and density . In our Lagrangian setting we thus consider that the two key properties of marine particles controlling their sinking dynamics are their size and density. Here we present the standard classification of marine particles according to the typical range of size and density by compiling different bibliographical sources.

The velocity flow u of the Benguela region was produced by a regional simulation of a hydrostatic free-surface primitive equations model called ROMS (Regional Ocean Modelling System). The configuration used here extends from 12 to 35∘ S and from 4 to 19∘ E (blue rectangle in Fig. 3) and was forced with climatological atmospheric data . The simulation area extends from 12 to 35∘ S and from 4 to 19∘ E (blue rectangle in Fig. ). The velocity field data set consists of 2 years of daily averages of zonal (u), meridional (v), and vertical velocity (w) components, stored in a three-dimensional grid with a horizontal resolution of 1/12∘ and 32 vertical terrain-following levels using a stretched vertical coordinate where the layer thickness increases from the surface/bottom to the ocean interior.

In order to integrate particle trajectories from the velocity in Eq. (), we interpolate linearly u(r,t) from the closest space–time grid points to the actual particle locations. Given the huge disparity between the model resolution and the small particle sizes considered, it is pertinent to parameterize in some way the unresolved scales. This can be done by different approaches, from stochastic Lagrangian modeling , to deterministic kinematic fields . The first approach is adopted by adding a simple white noise to the particle velocity , with different intensity in the vertical and horizontal directions. Thus, we consider this noisy version of the simplified MRG: dr(t)dt=v(t),v=u+vs+τp(β-1)DuDt+2Ω×u+W. W(t)≡2DhWh(t)+2DvWz(t), with (Wh,Wz)=(Wx(t),Wy(t),Wz(t)) a three-dimensional vector Gaussian white noise with zero mean and correlations 〈Wi(t)Wj(t′)〉=δijδ(t-t′), i,j=x,y,z. We consider an horizontal eddy diffusivity, Dh, depending on resolution length scale l according to the Okubo formula : Dh(l)=2.055×104l1.55 (m2 s-1). Thus, when taking l∼ 8 km = 8000 m (corresponding to 1/12∘), we obtain 10 m2 s-1. In the vertical direction we use a constant value of Dv=10-5 m2 s-1 .

In order to obtain quantitative assessment of the relative effects of the different physical terms in Eq. (), we will compare trajectories obtained from the following expressions which only consider some of the terms of the full Eq. (): v(0)=u+vs+W,v(co)=u+vs+τp(β-1)2Ω×u+W,v(in)=u+vs+τp(β-1)DuDt+W. Besides the random noise term, the first Eq. () only considers the settling velocity, Eq. () resolves the settling velocity plus the Coriolis effect, and Eq. () considers the settling plus the inertial term.

For the numerical experiments we will consider a set of six values of vs ranging from 5 to 200 m day-1, with different integration times to have in all the cases a sinking to about 1000–1100 m depth. The stochastic Eq. () with Eqs. ()–() is written in spherical coordinates and numerically integrated using a second-order Heun method with time step of 4 h . We use R=6371 km for the Earth's radius, g= 9.81 m s-2, and the angular velocity Ω is a vector pointing in the direction of the Earth's axis and modulus |Ω|=7.2722×10-5 s-1. We take vs and τp constant in each experiment because, although water density may increase with depth, this variation is at most of 10 kg m-3 in the range of depths we are considering here and then the impact on vs is below 0.1 %. We use as initial starting date 17 September 2008. The numerical experiments consist in launching N=6000 particles from initial conditions randomly chosen in a square of size 1/6∘ centered at 10.0∘ E 29.12∘S and -100.0 m depth (red rectangle in Fig. ), and in letting them evolve for a given time tf (stated in Table ) following Eq. () with Eqs. ()–(). We use in each case identical initial conditions and the same sequence of random numbers for the noise terms. In this way we guarantee that any difference in particle trajectories arise from the inclusion or not of the inertial and Coriolis terms. We obtain the time-dependent positions of all the particles for each approximation to the dynamics: ri(t), ri(0)(t), ri(co)(t), and ri(in)(t), i=1,…,N, following, respectively, Eqs. ()–() and the corresponding final positions at t=tf.

Table  gives the mean and the standard deviation of the depths attained by the set of particles in each numerical experiment as obtained from Eqs. () and (). We find that the use of the different approximations ()–() gives virtually the same results. The only differences larger than 1 cm in mean or standard deviation are the ones for the smallest unperturbed settling velocity considered, vs= 5 m day-1, and are also reported in Table . The measured differences are negligible as compared with the traveled distance or even with the model grid size. Indeed, small changes in the ROMS model configuration or in the velocity interpolation procedure would have an impact larger than this. The mean displacements in the horizontal obtained with the different approximations (not shown) also differ by less than 0.1%. We thus conclude that the simplest approximation Eq. () which only considers passive transport and an added constant sinking velocity already provides a good description of the sinking process for the type of marine particles and the range of space scales and timescales considered here. Note that the depth attained by the particles is always slightly shallower than z=-1100 m, which is the depth that would be reached in a still fluid. It is still debated under which conditions fluid flows enhance or reduce the settling velocity .

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 ri(co)(t) and ri(in)(t) with the simpler dynamics Eq. () which gives ri(0)(t) for each time t. To do so we compute the root mean square difference in position per particle and time, which we separate into vertical and horizontal components: rh(k)(t)=1N∑i=1Nxi(0)(t)-xi(k)(t)2,rv(k)(t)=1N∑i=1Nzi(0)(t)-zi(k)(t)2, with xi=(xi,yi), the horizontal position vectors, and the superindex (k) takes the values (co) or (in).

Figure  shows the influence of the Coriolis term in the horizontal component for each sinking velocity as a function of time. We observe an exponential growth in a wide range of times, which reveals the chaotic behavior of each of the compared trajectories. The value of the exponent 0.08 days-1 is in agreement with the order of magnitude of the Lyapunov exponent calculated using the same ROMS velocity model and region . Similar exponential growth with the same growth rate were observed for the inertial terms and the vertical components (not shown), although the absolute magnitude of these mean root square differences was much smaller.

The horizontal and vertical differences rh,v(co) at the final integration time tf (i.e., the time at which the particles reach an approximate depth of 1000 m for each value of vs) are displayed in Fig. , both as a function of vs and of tf. Similarly, the values of rh,v(co) are presented in Fig. . The behavior can be understood as resulting from two factors: on the one hand smaller vs requires larger tf to reach the final depth, and larger integration time tf allows for accumulation of larger differences between trajectories. On the other hand the Coriolis and inertial terms in Eqs. ()–() are proportional to τp(β-1)=vs/g so that their magnitude decreases for smaller vs. The combination of these two competing effects shapes the curves in Figs.  and , which for the vertical-difference case turn out to be non-monotonic in vs or tf.

In all cases, the differences (both in vertical and horizontal) between the simple dynamics (Eq. ) and the corrected ones in Eqs. () and () are negligible when compared with typical particle displacements, or even with model grid sizes. For example, we imposed in our simulations a vertical displacement close to 1000 m, whereas the mean root square difference with respect to simple sinking is below 1 m for the Coriolis case (Fig. ) and below 1 cm for the inertial case (Fig. ). In the horizontal direction, displacements during those times are of the order of hundreds of km, whereas the corrections introduced by the Coriolis and inertial terms are in the worst cases of the order of a few kilometers or of tens of meters, respectively. In particular, the most important impact (horizontal differences of tens of kilometers) is attributed to the Coriolis term for particles sinking at 5 m day-1 (Fig. ). It is worth noting that although the small value of Rossby numbers ≃0.01 for mesoscale processes might indicate a strong influence of the Coriolis force in Eq. (), its influence on particle dynamics becomes negligible because it is multiplied by τp or, equivalently, the Stokes number, which is significantly small for biogenic particles. Nevertheless, the Rossby number coincides with the ratio of inertial term to Coriolis term in Eq. () and its value ≃0.01 explains the difference of 2 orders of magnitude among the corrections arising from the inertial force and from Coriolis. The trajectories of the full dynamics ruled by Eq. () are nearly identical to the ones under the approximation which keeps only the sinking term and Coriolis, so that the corresponding comparison to ri(0) gives a figure essentially identical to Fig.  (not shown).

In summary, for the range of sizes and densities of the marine particles considered here, the sinking dynamics is essentially given by the velocity v=u+vs, which has been the one used in some oceanographic studies . Note however that a new question arises: what is then the reason for the observed clustering of falling particles ? The argument of the non-inertial dynamics of the particles does not serve since ∇⋅v=∇⋅u=0. A possible response is explored in the next section.

Compressibility of the particle-velocity field, i.e., ∇⋅v≠0, which can arise from inertial effects even when the corresponding fluid-velocity field is incompressible, ∇⋅u=0, has been identified as one of the mechanisms leading to preferential clustering of particles in flows . This is so because ρ(t), the particle density at time t at the location r=r(r0,t) of a particle that started at r0 at time zero, satisfies ρ(t)=ρ(0)δ-1, where δ is a dilation factor equal to the determinant of the Jacobian |∂r∂r0|, which satisfies 1δDδDt=∇⋅v or, using δ(0)=1: δ(tf)=e∫0tfdt∇⋅v. Thus, particles will accumulate (i.e., higher ρ(tf)) in final deep locations receiving particles whose trajectories have predominantly travelled through regions with ∇⋅v<0. We have seen however that to a good approximation ∇⋅v≈∇⋅u=0 since inertial effects can be neglected for the type of marine particles we consider here, and then the three-dimensional particle-velocity field is incompressible.

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∘ S and 4 to 19∘ E (orange rectangle in Fig. ). We divide the domain horizontally into squares of side 1/25∘, then initialize 1000 particles at random positions in each of them on 20 August 2008 at depth z=-100 m (i.e., the bottom of the euphotic layer, starting point of our biogenic particles), and then integrate each trajectory until it reaches -1000 m depth. We use Eq. () for the velocity, with vs=50 m day-1. In order to avoid any small fluctuating compressibility arising from the noise term, we put W=0, but we have checked that the result in the presence of noise is virtually indistinguishable (not shown). At the bottom layer (z=-1000 m) we count how many particles arrive to each of the 1/25∘ boxes and display the result in Fig. a. Despite ∇⋅v=0 we see clear preferential clustering of particles in some regions related to eddies and filaments. We note that our horizontal boxes have a latitude-dependent area so that distributing particles at random in them produces a latitude-dependent initial density which could lead to some final inhomogeneities. We have checked however that for the range of displacements of the particles, this effect is everywhere smaller than 5% and thus can not be responsible for the large clustering observed in Fig. a. Nevertheless, this effect will be taken into account later.

We explain the observed particle clustering by considering the field displayed in Fig. a as a projection in two dimensions of a density field (the cloud of sinking particles) which evolves in three dimensions. Even if the three-dimensional divergence is zero, and then an homogeneous three-dimensional density will remain homogeneous, a two-dimensional cut or projection can be strongly inhomogeneous. This mechanism has been proposed to explain clustering and inhomogeneities in the ocean surface , but we show here that it is also relevant for the crossing of a horizontal layer by a set of falling particles.

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 vs, because of the shorter sinking time during which vertical deformations could develop. Thus we compute the two-dimensional version of the dilation field, δh(x,tf), at each horizontal location x in the deep layer at z=-1000 m: δh(x,tf)=e∫0tfdt∇h⋅v with the horizontal divergence ∇h⋅v≡∂vx∂x+∂vy∂y=∂u∂x+∂v∂y=-∂w∂z, where in the second equality we have used Eq. () from which ∇h⋅v=∇h⋅u and the third one is a consequence of ∇⋅u=0. In order to get the values of δh on a uniform grid on the -1000m depth layer at the arrival date tf of the particles in the previous simulation, we integrate backwards in time trajectories from grid points separated 1/50∘ at z=-1000 m until they reach -100 m. The starting date (tf) of the backwards integration was 7 September 2008, i.e., 18 days after the release date used in the previous clustering experiment. This value correspond to the average duration time of trajectories in that experiment. Then δh was computed integrating in time the values of ∇h⋅v along every trajectory using Eq. ().

Figure b displays the quantity δ(x,tf)-1cos⁡(θf)/cos⁡(θ0), which gives the ratio between densities in the upper and lower layers, corrected with the angular factors controlling the area of the horizontal boxes so that this can be compared with the ratio between the particle numbers displayed in Fig. a. θf is the latitude of point x, and θ0 is the latitude of the corresponding trajectory in the upper z=-100 m layer. As stated before, the latitudinal corrections by the cosine terms are always smaller than a 5%. Although there is no perfect quantitative agreement, there is clear correspondence between the main clustered structures in panels (a) and (b) of Fig. , confirming that they originate from the horizontal dynamics in an incompressible three-dimensional velocity field. We have checked in specific cases that locations with larger differences between Fig. a and b correspond to places with large dispersion in the arrival times to the bottom layer, indicating deviations from the horizontality assumption.

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. (), our conclusion is that the simplest approximation given by Eq. () in which particles move passively with the fluid flow with an added constant settling velocity in the vertical direction is an accurate framework to describe the sinking process in the types of flows and particles considered. A re-assessment of these assumptions may be required if more complex processes (such as aggregation/disaggregation) are included and when super-high resolution (submesoscale and below) mimicking the real ocean becomes available.

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 or in , but both of them are negligible when compared to the effects of passive transport by the fluid velocity plus the added gravity term, except for very slowly sinking particles at high latitudes.

If the fluid flow field u(r,t) has vanishing divergence, then the same is true for the particle velocity field defined by the approximation in Eq. (). Then, no three-dimensional clustering can occur within this approximation. Nevertheless, we have shown that two-dimensional cuts or projections of evolving three-dimensional particle clouds display horizontal clustering.